Commun. Math. Phys. 283, 1–24 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0556-8
Communications in
Mathematical Physics
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds Fuquan Fang1, , Yuguang Zhang1, , Zhenlei Zhang2 1 Department of Mathematics, Capital Normal University, Beijing, P.R. China.
E-mail:
[email protected]
2 Nankai Institute of Mathematics, Weijin Road 94, Tianjin 300071, P.R. China.
E-mail:
[email protected] Received: 5 April 2007 / Accepted: 21 April 2008 Published online: 22 July 2008 – © Springer-Verlag 2008
Abstract: We consider the maximum solution g(t), t ∈ [0, +∞), to the normalized Ricci flow. Among other things, we prove that, if (M, ω) is a smooth compact symplectic 4-manifold such that b2+ (M) > 1 and let g(t), t ∈ [0, ∞), be a solution to (1.3) on M whose Ricci curvature satisfies that |Ric(g(t))| ≤ 3 and additionally χ (M) = 3τ (M) > 0, then there exists an m ∈ N, and a sequence of points {x j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence, dG H
(M, g(tk + t), x1,k , . . . , xm,k ) −→ (
m
N j , g∞ , x1,∞ , . . . , xm,∞ ),
j=1
t ∈ [0, ∞), in the m-pointed Gromov-Hausdorff sense for any sequence tk −→ ∞, where (N j , g∞ ), j = 1, . . . , m, are complete complex hyperbolic orbifolds of complex dimension 2 with at most finitely many isolated orbifold points. Moreover, m the convergence is C ∞ in the non-singular part of m N and Vol (M) = j g 0 1 j=1 Volg∞ (N j ), where χ (M) (resp. τ (M)) is the Euler characteristic (resp. signature) of M. 1. Introduction Let (M, g) be a compact Riemannian manifold. The Perelman λ-functional λ M (g) =
inf ∞
f ∈C (M)
e− f dvolg = 1},
{F(g, f ) :
(1.1)
M
The first author was supported by NSFC Grant No.10671097 and the Capital Normal University.
Current address: Department of Mathematical Sciences, Korea Advanced Institute of Science and Tech-
nology, Daejeon, Republic of Korea. E-mail:
[email protected]
2
F. Fang, Y. Zhang, Z. Zhang
where F(g, f ) = M (Rg + |∇ f |2 )e− f dvolg and Rg is the scalar curvature of g. Note that λ M (g) is the lowest eigenvalue of the operator −4 + Rg . By [Pe1] the gradient flow of the Perelman λ-functional is Hamilton’s Ricci-flow evolution equation ∂ g(t) = −2Ric(g(t)). ∂t
(1.2)
The normalized Ricci flow equation on an n-manifold M reads 2R ∂ g(t) = −2Ric(g(t)) + g(t), ∂t n
(1.3)
Rdv
where Ric (resp. R) denotes the Ricci tensor (resp. the average scalar curvature M dv ). M Note that (1.2) and (1.3) differ only by a change of scale in space and time, and the volume 2 Vol(g(t)) is constant in t along the flow (1.3). If dim M = n, λ M (g) = λ M (g)Volg (M) n is invariant up to rescaling the metric. Perelman [Pe1] has proved that λ M (g(t)) is nondecreasing along the Ricci flow g(t) whenever λ M (g(t)) ≤ 0. This leads to the Perelman invariant λ M by taking the supremum of λ M (g) in the set M of all Riemannian metrics on M. By [AIL] the Perelman invariant λ M is equal to the Yamabe invariant whenever λ M ≤ 0, after the earlier estimations (cf. [An5, Pe2, Le4, FZ and Kot]). In particular, if (M, g) is a smooth compact oriented 4-manifold with a Spinc -structure c which is a monopole class (i.e., the associated Seiberg-Witten equation possesses an irreducible solution) so
that c12 (c)[M] > 0, by [FZ] λ M ≤ − 32π 2 c12 (c)[M]. Moreover, g is a Kähler-Einstein metric of negative scalar curvature if and only if λ M (g) = − 32π 2 c12 (c)[M]. However, there are plenty of 4-manifolds where the Perelman invariant λ M = − 32π 2 c12 (c)[M] but do not admit any Kähler Einstein metric. It is natural to study 4-manifolds with these extremal properties. For such a 4-manifold M, to seek for an “optimal” Riemannian metric on M with respect to the Perelman functional λ M : M → R, we want to consider a maximal solution g(t) which is a solution of the Ricci flow (1.3). We call a longtime solution g(t), t ∈ [0, +∞), to the Ricci flow (1.3) a maximum solution if lim λ M (g(t)) = λ M . For a compact 3-manifold, by Perelman [Pe2] all solutions of the t→∞ Ricci flow (1.2) with surgery exist for longtime and are maximum solutions, provided λ M ≤ 0. In the paper [FZZ] obstructions are found for the longtime solutions with bounded curvature to (1.3). In this paper we are going to study the maximum solutions of (1.3) with bounded Ricci curvatures instead. To avoid technique terminology we only state our results for symplectic 4-manifolds by using the celebrated work of Taubes [Ta]: if (M, ω) is a compact symplectic manifold with b2+ (M) > 1 (the dimension of self-dual harmonic 2-forms of M), the spinc -structure induced by ω is a monopole class. Moreover, in this situation c12 (c)[M] = 2χ (M) + 3τ (M), where χ (M) (resp. τ (M)) is the Euler characteristic (resp. signature) of M. Theorem 1.1. Let (M, ω) be a smooth compact symplectic 4-manifold satisfying that b2+ (M) > 1 and 2χ (M) + 3τ (M) > 0. If g(t), t ∈ [0, ∞), is a solution to (1.3) such that |Ric(g(t))| ≤ 3, and lim λ M (g(t)) = − 32π 2 (2χ (M) + 3τ (M)), t→∞
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
3
then there exists an m ∈ N, and sequences of points {x j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence, dG H
(M, g(tk + t), x1,k , . . . , xm,k ) −→ (
m
N j , g∞ , x1,∞ , . . . , xm,∞ ),
j=1
t ∈ [0, ∞), in the m-pointed Gromov-Hausdorff sense for any sequence tk −→ ∞, where (N j , g∞ ), j = 1, . . . , m, are complete Kähler-Einstein orbifolds of complex dimension 2 with at most finitely many isolated orbifold points. The scalar curvature (resp. volume) of g∞ is 1
−Volg0 (M)− 2
m 32π 2 (2χ (M) + 3τ (M)) (resp. Volg0 (M) = Volg∞ (N j )). j=1
Moreover, the convergence is C ∞ in the non-singular part of
m 1
Nj.
We first remark that, if the diameters diamg(tk ) (M) possess a uniform upper bound, then m = 1, and N1 is a compact Kähler-Einstein orbifold. Secondly, if the Ricci curvature bound in the above theorem is replaced by a uniform bound of sectional curvature, then every (N j , g∞ ), j = 1, . . . , m are complete Kähler-Einstein manifolds. By the same arguments as in [An5,An6], mj=1 N j can weakly embed in M, mj=1 N j ⊂⊂ M, i.e. for any compact subset K ⊂ mj=1 N j , there is a smooth embedding FK : K −→ M. Furthermore, there exists a sufficiently large compact subset K ⊂ mj=1 N j such that M\K admits an F-structure of positive rank. This type of geometric decomposition seems very useful to understand the diffeomorphism type of 4-manifolds. Theorem 1.2. Let (M, ω) be a smooth compact symplectic 4-manifold such that b2+ (M) > 1 and let g(t), t ∈ [0, ∞), be a solution to (1.3) such that |R(g(t))| ≤ 12. If in addition χ (M) = 3τ (M) > 0, then lim λ M (g(t)) = − 32π 2 (2χ (M) + 3τ (M)). t→∞
Moreover, if |Ric(g(t))| ≤ 3, the Kähler-Einstein metric g∞ in Theorem 1.1 is complex hyperbolic. To conclude the section we point out that the main result in Theorem 1.1 (resp. Theorem 1.2) holds if the manifold is not symplectic but a compact oriented 4-manifold with a monopole class c1 (i.e. with a spinc -structure with non-vanishing Seiberg-Witten invariant) so that c12 = 2χ (M) + 3τ (M) > 0. 2. Preliminaries 2.1. Monopole class. Let (M, g) be a compact oriented Riemannian 4-manifold with a Spinc structure c. Let b2+ (M) denote the dimension of the space of self-dual harmonic 2-forms in M. Let Sc± denote the Spinc -bundles associated to c, and let L be the determinant line bundle of c. There is a well-defined Dirac operator D A : (Sc+ ) −→ (Sc− ).
4
F. Fang, Y. Zhang, Z. Zhang
Let c : ∧∗ T ∗ M −→ End(Sc+ ⊕ Sc− ) denote the Clifford multiplication on the Spinc bundles, and, for any φ ∈ (Sc± ), let 1 q(φ) = φ ⊗ φ − |φ|2 id. 2 The Seiberg-Witten equations read D A φ = 0, c(FA+ ) = q(φ),
(2.1)
where A is an Hermitian connection on L, and FA+ is the self-dual part of the curvature of A. A solution of (2.1) is called reducible if φ ≡ 0; otherwise, it is called irreducible. If (φ, A) is a resolution of (2.1), one calculates easily that 1 |FA+ | = √ |φ|2 , 2 2
(2.2)
0 = −2|φ|2 + 4|∇ A φ|2 + Rg |φ|2 + |φ|4 ,
(2.3)
The Bochner formula reads
where Rg is the scalar curvature of g. The Seiberg-Witten invariant can be defined by counting the irreducible solutions of the Seiberg-Witten equations (cf. [Le2]). Definition 2.2 ([K]). Let M be a smooth compact oriented 4-manifold. An element α ∈ H 2 (M, Z)/torsion is called a monopole class of M if and only if there exists a Spinc structure c on M with first Chern class c1 ≡ α(mod torsion), so that the Seiberg-Witten equations have a solution for every Riemannian metric g on M. By the celebrated work of Taubes [Ta], if (M, ω) is a compact symplectic 4-manifold with b2+ (M) > 1, the canonical class of (M, ω) is a monopole class. 2.3. Kato’s inequality. Let (M, g) be a Riemannian Spinc -manifold of dimension n; the following Kato inequality is useful: Proposition 2.4 (Proposition 2.2 in [BD]). Let φ be a harmonic Spinc -spinor on (M, g), i.e. D A φ = 0, where D A is the Dirac operator and A is an Hermitian connection on the determinant line bundle. Then n−1 A 2 |∇ φ| ≤ |∇ A φ|2 |∇|φ||2 ≤ (2.4) n at all points where φ is non-zero. Moreover, |∇|φ||2 = |∇ A φ|2 occurs only if ∇ A φ ≡ 0. Note that the arguments in the proof of Proposition 2.2 in [BD] can be used to prove this proposition without any change, where the same conclusion was derived for Spinspinor φ. For any > 0, let |φ|2 = |φ|2 + 2 . If φ is harmonic, by the above proposition, |∇|φ| |2 ≤
|φ| n−1 A 2 |∇ φ| ≤ |∇ A φ|2 |∇|φ||2 ≤ |φ| n
(2.5)
at points where φ( p) = 0. Since { p ∈ M : φ( p) = 0} is dense in M for harmonic φ, we conclude that (2.5) holds everywhere in M.
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
5
2.5. Chern-Gauss-Bonnet formula and Hirzebruch signature formula. Let (M, g) be a compact closed oriented Riemannian 4-manifold, χ (M) and τ (M) are the Euler number and the signature of M respectively. The Chern-Gauss-Bonnet formula and the Hirzebruch signature theorem say that Rg2 1 1 + |Wg |2 − |Ricº|2 )dvg , and ( 8π 2 M 24 2 1 τ (M) = (|Wg+ |2 − |Wg− |2 )dvg , 12π 2 M
χ (M) =
(2.6) (2.7)
R
where Ricº = Ric(g) − 4g g is the Einstein tensor, Wg+ and Wg− are the self-dual and anti-self-dual Weyl tensors respectively (cf. [B]). If g is a Kähler-Einstein metric, then Rg2 = 24|Wg+ |2 ,
(2.8)
(cf. [B]) which will be used in the proof of Theorem 1.1. By Chern-Gauss-Bonnet formula, one has an L 2 -bound of the curvature operator Rm(g) by the bounds of Ricci curvature, i.e. if |Ric(g)| < C, then |Rm(g)|2 dvg ≤ 8π 2 χ (M) + C1 V ol g (M), (2.9) M
where C and C1 are constants independent of (M, g). Let (N , g) be a complete Ricci-flat Einstein 4-manifold. Assume that |Rm(g)|2 dvg < ∞, and Volg (Bg (x, r)) ≥ Cr 4 ,
(2.10)
N
for all r > 0, a point x ∈ N , and a positive constant C. By Theorem 2.11 of [N], (N , g) is ALE. (i.e, Asymptotically Locally Euclidean space) of order 4. It is well-known that N is asymptotic to the cone on the spherical space form S 3 / , where ⊂ S O(4) is a finite group. The Chern-Gauss-Bonnet formula implies that 1 1 χ (N ) = (2.11) |Rm(g)|2 dvg + 2 8π N || (cf. [N] and [An1]).
2.6. Curvature estimates for 4-manifolds. Now let’s recall a result of [CT], which is important to the proof of Theorem 1.1. Let (M, g) be a complete Riemannian 4-manifold. A subset U ⊂ M such that for all p ∈ U , sup Ric(g) ≥ −3, is called -collapsed if for all p ∈ U ,
Bg ( p,1)
V ol g (Bg ( p, 1)) ≤ . By Theorem 0.1 in [CG], there is a constant ε4 such that if U is -collapsed with sectional curvature |K g | ≤ 1 and ≤ ε4 , then U carries an F-structure of positive rank.
6
F. Fang, Y. Zhang, Z. Zhang
Theorem 2.7 (Remark 5.11 and Theorem 1.26 in [CT]). There exist constants δ > 0, c > 0 such that: if (M, g) is a complete Riemannian 4-manifold with |Ric(g)| ≤ 3 and |Rm(g)|2 dvg ≤ C, M
and if E ⊂ M is a bounded open subset such that T1 (E) = {x ∈ M : dist(x, E) ≤ 1} is ε4 -collapsed with |Rm(g)|2 dvg ≤ δ ( f or all T1 (E)), Bg (x,1)
then
|Rm(g)|2 dvg ≤ cV ol g (A0,1 (E)), E
where A0,1 (E) = T1 (E)\E. 3. The Limiting Behavior of Ricci Flow In this section we study the limiting behavior of Ricci-flow with bounded Ricci curvatures on 4-manifolds. We will assume in this section that M is a smooth closed oriented 4-manifold with λ M < 0, and g(t), t ∈ [0, +∞), is a longtime solution of the normalized Ricci flow (1.3) with bounded Ricci-curvature. By normalization we may assume that |Ric(g(t))| ≤ 3. By (2.9) there is a constant C independent of t such that |Rm(g(t))|2 dvg(t) ≤ C. M
˘ Let us denote by V the volume Volg(0) (M) = Volg(t) (M), and R(g(t)) = min x∈M
˘ R(g(t))(x) the minimum of the scalar curvature of g(t). It is easy to see that R(g(t)) ≤ 1 −2 λM V < 0. ˘ Lemma 3.1. (3.1.1) lim λ M (g(t)) = lim R(g(t)) = lim R(g(t)) = R∞, t→∞ t→∞ t→∞ (3.1.2) lim M |R(g(t)) − R(g(t))|dvg(t) = 0, t→∞ (3.1.3) lim M |Ricº(g(t))|2 dvg(t) = 0. t→∞
Proof. By Perelman [Pe1] λ M (g(t)) is a non-decreasing function on t, therefore the limit lim λ M (g(t)) exists since λ M < 0. Now let us denote by R ∞ the limit lim λ M (g(t)).
t→∞
t→∞
1
Note that R ∞ ≤ λ M V − 2 < 0. To prove (3.1.1), we first prove that both lim R(g(t)) t→∞
˘ exist and take values R ∞ . By the same arguments as in the proof of and lim R(g(t)) t→∞ Proposition 2.6 and Lemma 2.7 of [FZZ] we get that ˘ lim R(g(t)) − R(g(t)) = 0.
t→∞
˘ (cf. [KL] (92.3)). Therefore lim R(g(t)) = Observe that R(g(t)) ≥ λ M (g(t)) ≥ R(g(t)) ˘ R ∞ = lim R(g(t)). This proves (3.1.1). t−→∞
t→∞
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
7
Note that ˘ |R(g(t)) − R(g(t))|dvg(t) ≤ (R(g(t)) − R(g(t)))dv (R(g(t)) g(t) + M
M
M
˘ − R(g(t)))dv g(t) ˘ = 2(R(g(t)) − R(g(t)))V. Equation (3.1.2) follows from (3.1.1). By Lemma 3.1 in [FZZ], ∞ |Ricº(g(t))|2 dvg(t) dt < ∞, 0
M
and, by Lemma 1 in [Y], we have d 2 |Ricº(g(t))| dvg(t) ≤ −2 |∇ Ricº(g(t))|2 dvg(t) dt M M +4 |Rm||Ricº(g(t))|2 dvg(t) < D, M
where D is a constant independent of t. By the same argument as in the proof of Proposition 2.6 in [FZZ] (3.1.3) follows. The following is the main result of this section, which is an analogy of Theorem 10.5 in [CT], where the same conclusion was derived for closed oriented Einstein 4-manifolds with the same negative Einstein constant. The key point in our case is to use Lemma 3.1 to get non-collapsing balls and to prove the limiting metric is an Einstein metric (cf. Lemma 3.3 and Lemma 3.4 below). Proposition 3.2. Let M be a smooth closed oriented 4-manifold with λ M < 0. If g(t), t ∈ [0, ∞) is a solution to (1.3) such that |Ric(g(t))| ≤ 3, and {tk } is a sequence of times which tends to infinity such that diamgk (M) −→ ∞, when k −→ ∞, where gk = g(tk ), then there exists an m ∈ N, and sequences of points {x j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence, dG H
(M, gk , x1,k , . . . , xm,k ) −→ (
m
N j , g∞ , x1,∞ , . . . , xm,∞ )
j=1
in the m-pointed Gromov-Hausdorff sense for k → ∞, where (N j , g∞ ) j = 1, . . . , m are complete Einstein 4-orbifolds with at most finitely many isolated orbifold points {qi }. The scalar curvature (resp. volume) of g∞ is R ∞ = lim λ M (g(t)), t−→∞
(r esp. V = Volg0 (M) =
m
Volg∞ (N j )).
j=1
Furthermore, in the regular part of N j , {gk } converges to g∞ in both L 2, p (resp. C 1,α ) sense for all p < ∞ (resp. α < 1).
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F. Fang, Y. Zhang, Z. Zhang
We divide the proof of Proposition 3.2 into several useful lemmas. A key result in the paper [CT] shows that, for any compact oriented Einstein 4-manifold (X, g) with Einstein constant −3, there exists a constant C depending only on the Euler number of X , and a point x ∈ X such that Volg (Bg (x, 1)) ≥ CVolg (X ) (cf. Theorem 0.14 [CT]). Cheeger-Tian remarked that the same result continues to hold for 4-manifolds which are sufficiently negatively Ricci pinched. The following lemma is an analogy of the result for the metric gk in Proposition 3.2. Lemma 3.3. There exists a constant v > 0, and a sequence {xk } ⊂ M such that Volgk (Bgk (xk , 1)) ≥ v. Proof. Let ε4 > 0 be the critical constant of Cheeger-Tian (cf. §1 [CT]), i.e., if X is a Riemannian 4-manifold which is ε4 -collapsed with locally bounded curvature, then X carries an F-structure of positive rank. We may assume that, for all x ∈ M and gk , Volgk (Bgk (x, 1)) < ε4 . By a standard covering argument, for any k, there exist finitely many points q1 , . . . , ql such that E = M\ li=1 Bgk (qi , 1) satisfies the hypothesis of Theorem 2.7. Moreover, l ≤ Cδ −1 , where C and δ are the constants in Theorem 2.7. Therefore, by Theorem 2.7 we conclude that, there is a constant C1 independent of k such that l |R(gk )|2 dvk ≤ 6 |Rm(gk )|2 dvk ≤ C1 Volgk (Bgk (qi , 1)). (3.1) E
E
i=1
On the other hand, by Lemma (3.1.2) k→∞ 2 2 | (R(gk ) − R(gk ) )dvk | ≤ 24 |R(gk ) − R(gk )|dvk −→ 0. E
E
Therefore 1 2 R ∞ Volgk (E) − R(gk )2 dvk ≤ R(gk )2 Volgk (E) − R(gk )2 dvk 2 E E = (R(gk )2 − R(gk )2 )dvk
(3.2)
E
1 2 ≤ R∞ V 4 1
for sufficiently large k since R ∞ ≤ λ M V − 2 < 0. By inserting (3.1) we get that 1 2 1 2 1 2 1 2 R ∞ (V − V ol gk (Bgk (qi , 1)) − R ∞ V ≤ R ∞ Volgk (E) − R ∞ V 2 4 2 4 l
i=1
≤ C1
l
Volgk (Bgk (qi , 1)),
i=1
and V ≤ C2
l
Volgk (Bgk (qi , 1)),
i=1
where C2 is a constant independent of k. Therefore, there is at least a ball among the l balls whose volume is at least CV2 l . The desired result follows.
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
9
Assuming that diamgk (M) → ∞ for k → ∞, by using the technique developed in [An3], the analogue of Theorem 3.3 in [An2] holds (cf. Theorem 2.3 in [An4]), i.e. there exist a sequence of points {xk } ⊂ M such that, by passing to a subsequence, dG H
{(M, gk , xk )} −→ (N∞ , g∞ , x∞ ), where N∞ is a 4-orbifold with only isolated orbifold points {qi }, g∞ is a complete C 0 orbifold metric, and g∞ is a C 1,α ∩ L 2, p Riemannian metric on the regular part of N∞ , for all p < ∞ and α < 1. Furthermore, {gk } converges to g∞ in the L 2, p (resp. C 1,α ) sense on the regular partof N∞ , i.e. for any r 1 and k, there is a smooth embedding Fk,r : Bg∞ (x∞ , r )\ i Bg∞ (qi , r −1 ) ⊂ N∞ → M such that, by passing to ∗ g converge to g in both L 2, p and C 1,α senses. a subsequence, Fk,r k ∞ Lemma 3.4. g∞ is an Einstein orbifold metric with scalar curvature R ∞ . Proof. We first prove that g∞ is an Einstein metric with scalar curvature R ∞ on the ∗ g converge to g 2, p (resp. C 1,α ) sense on regular part of N∞ . Since Fk,r k ∞ in the L −1 Bg∞ ( p∞ , r )\ i Bg∞ (qi , r ), for any r , by Lemma 3.1, we obtain that 2 0≤ |Ricº(g∞ )| dv∞ ≤ lim |Ricº(gk )|2 dvk = 0,
0≤
Bg∞ ( p∞ ,r )\
Bg∞ ( p∞ ,r )\
i
i
Bg∞ (qi ,r −1 )
Bg∞ (qi ,r −1 )
k−→∞ M
|R(g∞ )− R ∞ |dv∞ ≤ lim
k−→∞ M
|R(gk )− R(gk )|dvk = 0.
Therefore g∞ is a C 1,α Riemannian metric on Bg∞ ( p∞ , r )\ i Bg∞ (qi , r −1 ) which satisfies the Einstein equation in the weak sense. By elliptic regularity theory, g∞ is a smooth Einstein metric with scalar curvature R ∞ . Since g∞ is a C 0 -orbifold metric, i.e. for any orbifold point qi ∈ N∞ , there is a neighborhood Ui ∼ g∞ is a C 0 -Riemannian metric on = B(0, r )/ of qi such that B(0, r ) ⊂ R4 , where ⊂ S O(4) is a finite subgroup acting freely on S 3 , and g∞ | B(0,r )\{0} is the pull-back that g∞ is a smooth Einstein metric on metric of g∞ . Note g∞ )|2 dv < C < ∞. By the arguments as in B(0, r )\{0} satisfying that B(0,r ) |Rm( g∞ [An1] and [Ti], g∞ is a C ∞ Einstein metric on B(0, r ) (cf. the proof of Theorem C in [An1], and Sect. 4 in [Ti]). Hence g∞ is an Einstein orbifold metric. By the discussion before Lemma 3.4 we may choose sequences of points {x j,k } ⊂ k→∞
M, j = 1, . . . , , such that distgk (xi,k , x j,k ) −→ ∞ for any i = j, and dG H
{(M, gk , x1,k , . . . , x,k )} −→ (
N j , g∞ , x1,∞ , . . . , x,∞ ),
(3.3)
j=1
where (N j , g∞ , x j,∞ ), j = 1, . . . , are complete Einstein 4-orbifolds with only isolated singular points and scalar curvatures R ∞ . Furthermore, {gk } converges to g∞ in both L 2, p (resp. C 1,α ) sense on the regular parts of N j , j = 1, . . . , . Note that V ≥
i=1
Volg∞ (N j ).
(3.4)
10
F. Fang, Y. Zhang, Z. Zhang
Lemma 3.5. The number of orbifold points of
N j is less than a constant depending
j=1
only on the Euler characteristic χ (M).
Proof. For each orbifold point q ∈ N j , there exists a sequence {qk } ⊂ M, and two constants r r1 > 0 such that: (3.5.1) q ∈ Bg∞ (x j,∞ , r ); (3.5.2) Bg∞ (q, r1 )\Bg∞ (q, σ ) lies in the regular part of Bg∞ (x j,∞ , r ) for any σ < r1 ; C 1,α
(3.5.3) (Bgk (qk , r1 )\Bgk (qk , σ ), gk ) −→ (Bg∞ (q, r1 )\Bg∞ (q, σ ), g∞ ). By the definition of harmonic radius (cf. [An3]), the harmonic radii of all points in Bgk (qk , r1 )\Bgk (qk , σ ) have a uniform lower bound, saying µ > 0, a constant depending on σ but independent of k. Clearly, there is a positive constant v0 (e.g., 21 Volg∞ (Bg∞ (x j,∞ , r ))) such that Volgk (Bgk (x j,k , r )) ≥ v0 . Note that the Sobolev constants C S,k of Bgk (x j,k , r ) are bounded from below by a constant depending only on v0 , r (cf. [An2] and [Cr]). Therefore, by [An2] again we get that Volgk (Bgk (qk , s)) ≥ Cs 4 for any s 1, where C is independent of k. Let us denote by rh,k the infimum of the harmonic radii of gk in the ball Bgk (qk , r1 ). k→∞
Note that rh,k −→ 0 since q is an orbifold point (cf. [An3]). Therefore, there is a point q¯k ∈ Bgk (qk , σ ) so that rh (q¯k ) = rh,k for sufficiently large k. −2 Consider the normalized balls (Bgk (qk , r1 ), rh,k gk ), which have harmonic radii at least 1. By passing to a subsequence if necessary, C 1,α
−2 (Bgk (qk , r1 ), rh,k gk , q¯k ) −→ (W, g¯ ∞ , q), ¯
where (W, g¯ ∞ ) is a complete Ricci-flat 4-manifold satisfying that ¯ r )) ≥ Cr 4 Volg¯∞ (Bg¯∞ (q,
(3.5)
for any r > 0. It is obvious that |Rm(g¯ ∞ )|2 dvg¯∞ ≤ lim inf |Rm(gk )|2 dvk ≤ C. k−→∞
W
M
Therefore (W, g¯ ∞ ) is an Asymptotically Locally Euclidean space (cf. Theorem 2.11 in [N] or [An1]), which is asymptotic to a cone of S 3 / , where ⊂ S O(4) is a finite group acting freely on S 3 . By the Chern-Gauss-Bonnet formula 1 1 . (3.6) |Rm(g¯ ∞ )|2 dvg¯∞ + χ (W ) = 8π 2 W || By [An1] W is isometric to R4 , provided || = 1. Since the harmonic radius of g¯ ∞ at q¯ is 1, hence g¯ ∞ cannot be the Euclidean metric. Hence || ≥ 2. It is easy to verify that χ (W ) ≥ 1. By (3.6) we get that |Rm(g¯ ∞ )|2 dvg¯∞ ≥ 4π 2 . W
This proves that every orbifold point contributes to lim inf k−→∞
M
|Rm(gk )|2 dvk at least
4π 2 . By the rescaling invariance of the integral we conclude that the number of orbifold points β ≤ 4πC 2 .
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
11
The following lemma is an analogue of a result in Cheeger-Tian [CT]. Lemma 3.6. < χ (M)+β +1, where β := #{number of orbifold points in Lemma 3.5}. Proof. Suppose not, i.e, ≥ χ (M) + β + 1, by definition there are at least χ (M) + 1 components of 1 N j which are smooth complete non-compact Einstein 4-manifolds of finite volume, for simplicity saying N1 , . . . , Ns , where s ≥ χ (M) + 1. By Theorem 4.5 in [CT], for each 1 ≤ j ≤ s, |Rm(g∞ )|2 dvg∞ ≥ 8π 2 . Nj
L 2, p
Since (M, gk , xk, j ) −→ (N j , g∞ , x∞, j ), by the Chern-Gauss-Bonnet formula and (3.1.3) in Lemma 3.1 we get that 8π 2 χ (M) = lim |Rm(gk )|2 dvgk ≥ |Rm(g∞ )|2 dvg∞ ≥ 8π 2 (χ (M) + 1). k−→∞ M
Nj
A contradiction. Let m denote the maximal value of all possible choices of the base point sequences in (3.3), which has a upper bound by Lemma 3.6. Lemma 3.7. Let Mk,r = M\ mj=1 Bgk (x j,k , r ). For sufficiently large r , there is a constant C independent of r such that lim Volgk (Mk,r ) ≤ C
m
k→∞
m
j=1
r Volg∞ (N j \Bg∞ (x j,∞ , )), 2
Volg∞ (N j ) = V.
(3.7)
(3.8)
j=1
Proof. We may choose r 1 such that, for any y ∈ mj=1 (N j \Bg∞ (x j,∞ , r − 1)), Volg∞ (Bg∞ (y, 1)) ≤ 21 ε4 , where ε4 > 0 is the critical constant of Cheeger-Tian (cf. proof of Lemma 3.3 or §1 [CT]). Now we claim that there is a constant k0 1 such that, for any k > k0 and any x ∈ Mk,r , Volgk (Bgk (x, 1)) ≤ ε4 . If it is false, without loss of generality we may assume a sequence of points {yk } ⊂ Mk,r such that Volgk (Bgk (yk , 1)) > ε4 .
(3.9)
Observe that the distance distgk (yk , xj,k ) → ∞ as k → ∞ for all 1 ≤ j ≤ m. Otherwise, −1 assuming distgk (yk , xj,k ) < ρ for some j and ρ > 0, we get that F j,k,ρ (yk ) → y∞ ∈ Bg∞ (x j,∞ , ρ)\Bg∞ (x j,∞ , r − 1), and so Volgk (Bgk (yk , 1)) → Volg∞ (Bg∞ (y∞ , 1)) ≤
1 ε4 2
(3.10)
12
F. Fang, Y. Zhang, Z. Zhang
∗ when k → ∞, since F j,k,ρ gk C 1,α -converges to g j,∞ , where
F j,k,ρ : Bg∞ (x j,∞ , ρ)\
Bg∞ (qi , ρ −1 ) ⊂ N∞ → M
(3.11)
i ∗ is a smooth embedding so that F j,k,ρ gk converges to g∞ in the C 1,α -sense (cf. the discussion before Lemma 3.4). A contradiction to (3.9). dG H
Note that (M, gk , yk ) −→ (N∞ , g∞ , y∞ ) where N∞ is a complete 4-orbifold different from each of N j , 1 ≤ j ≤ m. This violates the choice of maximality of m. Hence we have proved the claim. By a standard covering argument, for any k, there exist finitely many points I z 1,k , . . . , z I,k such that E k,r = Mk,r \ i=1 Bzi,k (1) satisfies the hypothesis of Theorem 2.7, where I is independent of k. By Theorem 2.7, there is a constant C independent of k such that
|R(gk )| dvk ≤ 6
|Rm(gk )|2 dvk ≤ C(
2
E k,r
E k,r
I
Volgk (Bgk (z i,k , 1))
i=1
+Volgk (A0,1 (Mk,r ))). By Lemma 3.1, for k 1, we have |R(gk ) − R(gk )|dvk < |R(gk ) − R(gk )|dvk −→ 0. E k,r
(3.12)
M
By (3.2) we get 1 2 R Volgk (E k,r ) − 2 ∞
R(gk )2 dvk ≤ E k,r
(R(gk )2 − R(gk )2 )dvk E k,r
≤ 24
|R(gk ) − R(gk )|dvk . E k,r
Since Volgk (Ek,r ) ≥ Volgk (Mk,r ) −
I
Volgk (Bgk (zi,k , 1)), by the above together we get
i=1
immediately that Volgk (Mk,r ) ≤ C(
I i=1
Volgk (Bgk (zi,k , 1)) + Volgk (A0,1 (Mk,r )))
|R(gk ) − R(gk )|dvk .
+24
(3.13)
E k,r
If distgk (zi,k , xj,k ) → ∞ for all 1 ≤ j ≤ m, by the same argument as above we get that Volgk (Bgk (zi,k , 1)) → 0 when k → ∞. Otherwise, there exists a subsequence ks → ∞ and an index j such that distgks (zi,ks , xj,ks ) < ρ
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
13
for some constant ρ. In both cases, we obtain lim sup Volgk (Bgk (zi,ks , 1)) ≤ k→∞
m j=1
r Volg∞ (Nj \Bg∞ (xj,∞ , )) 2
for r ρ. Therefore, by (3.12) and (3.13) we conclude immediately (3.7). By (3.7) it follows that lim Volgk (Mk,r ) → 0. Hence (3.8) follows. k,r→∞
By now Proposition 3.2 follows by the above lemmas. 4. Smooth Convergence on the Regular Part The main result of this section is the following: Proposition 4.1. Let M be a closed 4-manifold satisfying that λ¯ M < 0 and let g(t), t ∈ [0, ∞), be a solution to the normalized Ricci flow equation (1.3) on M with uniformly dG H
bounded Ricci curvature. If (M, g(tk ), pk ) −→ (N∞ , g∞ , p∞ ), where tk → ∞ and C 1,α
N∞ is a 4-dimensional orbifold, and g(tk ) −→ g∞ on the regular part R of N∞ (the compliment of the orbifold points), then, by passing to a subsequence, for all t ∈ [0, ∞), dG H
(M, g(tk + t), pk ) −→ (N∞ , g∞ (t), p∞ ), where g∞ (t) is a family of smooth metrics on R solving the normalized Ricci flow equation on R with g∞ (0) = g∞ . Moreover, the convergence is smooth on R × [0, ∞). In [Se] the convergence of Kähler-Ricci flow on compact Kähler manifolds with bounded Ricci curvature was studied. It seems that the arguments in [Se] could be applied to prove Proposition 4.1, but the authors can not follow completely her line. Therefore, we give a quite different approach, where we first give a curvature estimate of the Ricci flow similar to Perelman’s pseudolocality theorem. Using this curvature estimation we prove the limit Ricci flow exists on R × [0, ∞). Finally, we prove that R is exactly the regular part of every subsequence limit of (M, g(tk + t), pk ), for all t ∈ [0, ∞). We point out that our approach works only in dimension 4. We now give a curvature estimate for the Ricci flow which is an analogy of Perelman’s pseudolocality theorem (cf.[Pe1],Thm. 10.1). The difference is that here we use the hypothesis of local almost Euclidean volume growth, instead of the almost Euclidean isoperimetric estimate. The proof is much easier than that of Perelman’s pseudolocality theorem. Theorem 4.2. There exist universal constants δ0 , 0 > 0 with the following property. Let g(t), t ∈ [0, ( P r0 )2 ], be a solution to the Ricci flow equation (1.2) on a closed n-manifold M and x0 ∈ M be a point. If the scalar curvature R(x, t) ≥ −r0−2
whenever dist g(t) (x0 , x) ≤ r0 ,
and the volume Volg(t) (Bg(t) (x, r )) ≥ (1 − δ0 )Vol(B(r )) for all Bg(t) (x, r ) ⊂ Bg(t) (x0 , r0 ), where B(r ) denotes a ball of radius r in the n-Euclidean space and Vol(B(r )) denotes its Euclidean volume, then the Riemannian curvature tensor satisfies |Rm|g(t) (x, t) ≤ t −1 ,
whenever dist g(t) (x0 , x) < 0 r0 , and 0 < t ≤ ( 0 r0 )2 .
In particular, |Rm|g(t) (x0 , t) ≤ t −1 for all time t ∈ (0, ( 0 r0 )2 ].
14
F. Fang, Y. Zhang, Z. Zhang
Proof. We use Claim 1 and Claim 2 of Theorem 10.1 in [Pe1] and adopt a contradiction argument. For any given small constants , δ > 0, set 0 = , δ0 = δ, then there is a solution to the Ricci flow equation (1.2), say (M, g(t)), not satisfying the conclusion of the theorem. After a rescaling, we may assume that r0 = 1. Denote by M¯ the nonempty set of pairs (x, t) such that |Rm|g(t) (x, t) > t −1 , then as in Claim 1 and Claim 2 of Theorem 10.1 in [Pe1], we can choose another space time point (x, ¯ t¯) ∈ M¯ with 1 0 < t¯ ≤ 2 , dist g(t¯) (x0 , x) ¯ < 10 , such that |Rm|g(t) (x, t) ≤ 4Q whenever t¯ −
1 −1 1 Q ≤ t ≤ t¯, dist g(t¯) (x, (100n )−1 Q −1/2 , ¯ x) ≤ 2n 10
where Q = |Rm|g(t¯) (x, ¯ t¯). It is remarkable that from the proof of Claim 2 of Theorem 10.1 in [Pe1], each such a space time point (x, t) satisfies dist g(t) (x, x0 ) < dist g(t¯) (x0 , x) ¯ + (100n )−1 Q −1/2 <
1 1 + (100n)−1 < . 10 2
Now choosing sequences of positive numbers k → 0 and δk → 0, we obtain a sequence of solutions (Mk , gk (t)), t ∈ [0, k2 ] and a sequence of points x0,k , x¯k ∈ Mk and times t¯k , with each satisfying the assumptions of the theorem and the properties described above. In particular, we have that Q k = |Rm k |gk (t¯k ) (x¯k , t¯k ) → ∞. Consider the sequence of pointed Ricci flow solutions (Bgk (t¯k ) (x¯k ,
1 1 −1/2 ¯ (100n k )−1 Q k ), Q k gk (Q −1 , 0]. k t + tk ), x¯ k ), t ∈ [− 10 2n
Using Hamilton’s compactness theorem for solutions to the Ricci flow, we can extract a subsequence which converges to a complete Ricci flow solution (M∞ , g∞ (t), x¯∞ ), 1 t ∈ (− 2n , 0], with |Rm ∞ |g∞ (0) (x¯∞ , 0) = 1. By assumption, the balls Bgk (t¯k ) (x¯k ,
1 1 −1/2 (100n k )−1 Q k ) ⊂ Bgk (t) (x0,k , ) 10 2
1 1 for any t ∈ [t¯ − 2n Q −1 , t¯], so the scalar curvature Rk (x, t) ≥ − 1 for t ∈ [t¯ − 2n Q −1 , t¯] −1/2 1 (100n k )−1 Q k ) and Volgk (t) (Bgk (t) (x, r )) ≥ (1−δk )Vol(B(r )) and x ∈ Bgk (t¯k ) (x¯k , 10 −1/2 1 1 (100n k )−1 Q k ), t ∈ [t¯ − 2n Q −1 , t¯]. for any metric ball Bgk (t) (x, r ) ⊂ Bgk (t¯k ) (x¯k , 10 Passing to the limit, we see that g∞ (t) has scalar curvature R∞ ≥ 0 everywhere and local volume Volg∞ (t) (Bg∞ (t) (z, r )) ≥ Vol(B(r )) for any balls Bg∞ (t) (z, r ) at time 1 t ∈ (− 2n , 0]. Then the local variation formula of the volume implies that R∞ ≡ 0 1 on M∞ × (− 2n , 0], see [STW] for details. By the evolution of the scalar curvature ∂ 1 R = R + 2|Ric∞ |2 , we get that Ric∞ ≡ 0 over M∞ × (− 2n , 0]. Then the ∞ ∞ ∂t Bishop-Gromov volume comparison theorem implies that g∞ (t) are flat solutions to the Ricci flow, which contradicts the fact that |Rm ∞ |(x¯∞ , 0) = 1. This ends the proof of the theorem.
The next lemma provides a comparison of the curvature of the normalized and unnormalized Ricci flow. By assumption, there is C¯ < ∞ such that |Ric| ≤ C¯ everywhere along the flow (M, g(t)). Note that by Lemma 3.1, there is some time T < ∞ such that 2R ∞ ≤ R(g(t)) ≤ 21 R ∞ < 0 whenever t > T . Fix any such time t¯ > T and let h(t) and ˜ t˜) be the solutions to the normalized and unnormalized Ricci flow with initial metric h(
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
15
˜ h(0) = h(0) = g(t¯) respectively, where t˜ = t˜(t) is the corresponding rescaled time for t. Denote by Rm t¯, Rict¯, Rt¯ and Rm t¯, Rict¯, R˜ t¯ the corresponding Riemannian curvature, Ricci curvature and scalar curvature of them, where |Rict¯| ≤ C¯ since h(t) = g(t¯ + t). Then we have ˜ t˜) exists for all time t˜ ∈ [0, ∞). Furthermore, there exist Lemma 4.3. The solution h( ¯ such that constants C and τ depending on λ¯ M and C, Rm t¯|(x, t˜), whenever t ≤ τ. t ≤ t˜ ≤ Ct, | Rm t¯|(x, t˜) ≤ |Rm t¯|(x, t) ≤ C| ˜ t˜) Proof. The solution h(t) has average scalar curvature R(t¯ + t) ≤ 21 R ∞ < 0, so h( ˜ ˜ d ˜ t˜)) = − R, also has average scalar curvature R < 0. From the evolution ln Vol(h( d t˜
˜ t˜)) increases strictly in t˜, so to normalize it, we need to compress the the volume Vol(h( ˜ t˜) exists space and time. Thus t˜ ≥ t and | Rm t¯|(x, t˜) ≤ |Rm t¯|(x, t) for all (x, t). So h( for all time. The last assertion means that the scaling factor from normalized Ricci flow to the unnormalized one is less than C on the time interval [0, τ ]. Consider the evolution of ˜ t˜): the average scalar curvature R( d ˜ R= d t˜
M (2| Rict¯|
2
− R˜ t2¯ )dvk
V olh˜ ( t˜) (M)
2 + R˜ ≤ ,
˜ ≤ ¯ since | ¯ | R˜ t¯| ≤ |Rt¯| ≤ C, ¯ | R| for some constant = (C), Rict¯| ≤ |Rict¯| ≤ C, ˜ 1 ¯ Note that the initial value R(0) = R(g(t¯)) ≤ R ∞ , so there is some constant |R| ≤ C. 2 ˜ t˜) ≤ 1 R for t˜ ∈ [0, τ˜ ]. Thus the scaling factor from normalized τ˜ = τ˜ () such that R( 4
∞
R(h(t)) ˜ t˜) , is less than 8 on the time interval R( = τ8˜ and C = 8.
Ricci flow to the unnormalized one, which equals t˜ ∈ [0, τ˜ ]. Now the result follows, by setting τ
The following lemma gives the estimation of the local volume along the Ricci flow. As in [Se], the proof uses Theorem A 1.5 of [CC]. By assumption, we have a solution (M, g(t)) to the normalized Ricci flow (1.3) and a sequence of times tk → ∞ and points dG H
C 1,α
pk such that (M, g(tk ), pk ) −→ (N∞ , g∞ , p∞ ) with g(tk ) −→ g∞ on the regular part R of the orbifold N∞ . For the space M or N∞ , let R ,ρ be the set of points x such that dG H (B(x, r ), B(r )) < r for any r ≤ u, where u ≥ ρ is some constant depending on x. Here and after, B(r ) denotes a ball of radius r in 4-Euclidean space and B(x, r ) the metric ball of radius r with center x in a metric space. A weak version is WR ,ρ , the set of points x such that there is u ≥ ρ with dG H (B(x, u), B(u)) < u. Lemma 4.4. For each q ∈ R, choose a sequence qk ∈ M that converges to q. Then for any > 0, there exist k0 , η, ρ > 0 such that
Vol(Bg(tk +t) (qk , r )) ≥ (1 − )Vol(B(r )), ∀r < ρ, k0 < k,
whenever Bg(tk +t) (qk , r ) ⊂ Bg(tk ) (qk , ρ) and t ∈ [−η, η].
16
F. Fang, Y. Zhang, Z. Zhang
Proof. By the boundedness of the Ricci tensor, there is a universal constant ¯ > 1 such that Bg(t) ( p, −1 r ) ⊂ Bg(s) ( p, r ) ⊂ Bg(t) ( p, r ) for all = (C) t, s ∈ [tk − 1, tk + 1], p ∈ M and r > 0. By Theorem A.1.5 of [CC], for fixed > 0, there are δ = δ( , n), ρ = ρ( , n) > 0 such that x ∈ WRδ,r implies Vol(Bg(t) (x, r )) ≥ (1 − )Vol(B(r )) for each r ≤ ρ and x ∈ M. So by definition, it suffices to show qk ∈ Rδ,ρ with respect to each metric g(t), t ∈ [tk − η, tk + η], whe never qk ∈ Bg(tk ) (qk , ρ), for some constant η > 0. The constant ρ may be modified by a smaller one if necessary. Using Theorem A.1.5 of [CC] again, for fixed δ as above, there is δ1 = δ1 (δ, n) > 0 such that qk ∈ WR (2 +1)ρ implies qk ∈ Rδ,ρ for any qk ∈ Bg(t) (qk , 2 ρ). So it δ1 ,
1−δ
reduces to show qk ∈ WR
2
+1)ρ δ1 , (1−δ
with respect to each time t ∈ [tk −η, tk +η] for some
η > 0 small enough. In fact, as shown in [Se], dG H (Bg(tk ) (qk , ρ1 ), B(ρ1 )) < 21 δ1 ρ1 for some small number ρ1 and all k large enough. By the boundedness of the Ricci tensor again, there is a constant η ≤ 1 such that for each time t ∈ [−η, η], we have dG H (Bg(tk +t) (qk , ρ1 ), Bg(tk ) (qk , ρ1 )) < 21 δ1 ρ1 for all k. Thus dG H (Bg(tk +t) (qk , ρ1 ), 1 . B(ρ1 )) < δ1 ρ1 for each t ∈ [−η, η]. Now the result follows by setting ρ = (1−δ)ρ 2 +1 Note that in the proof, the constant δ1 = δ1 ( , n), so the constant η depends only on ¯ By assumption, there is a compact exhaustion {K i }∞ of R and a sequence , n and C. i=1 of smooth embeddings Fi : K i → M such that Fi ( p∞ ) = pi and Fi∗ g(ti ) converges to g∞ in the local C 1,α sense. Following the lines described in [Se], we can prove Lemma 4.5. Denote by K i,k = Fk (K i ), then for any > 0 and i, there are k0 , η, ρ > 0 such that
Vol(Bg(tk +t) (qk , r )) ≥ (1 − )Vol(B(r )), ∀qk ∈ K i,k , k0 < k, t ∈ [−η, η] and r < ρ. Now we are ready to prove Proposition 4.1. Proof of Proposition 4.1. Assume that p∞ ∈ K i for each i. Set = δ0 in the previous lemma, where δ0 is just the constant in Theorem 4.2, then for one fixed K i , there exist k0 , η, ρ > 0 such that Vol(Bg(tk +t) (q, r )) ≥ (1−δ0 )Vol(B(r )) whenever q ∈ K i,k , k0 < k, t ∈ [−η, η] and r < ρ. Modifying ρ and η by smaller constants, we assume ( 0 ρ)2 ≤ 2η < τ , where τ and 0 are constants in Lemma 4.3 and Theorem 4.2 respectively. Let h k (t˜) be the corresponding solutions to the unnormalized Ricci flow equation with initial value h k (0) = g(tk − η), then Vol(Bh k (t˜) (q, r )) ≥ (1 − δ0 )Vol(B(r )) whenever q ∈ K i,k , r < ρ, k0 < k and t˜ satisfying t (t˜) ∈ [0, 2η], since the inequality Vol(B(q, r )) ≥ (1 − δ0 )Vol(B(r )) is scale invariant and Bh k (t˜) ⊂ Bg(tk +t (t˜)) (q, r ) for k large enough such that tk ≥ T +η for T chosen as above. Denote by Rm k the Riemannian curvature tensor of h k , then by Theorem 4.2 and Lemma 4.3, we have |Rm|(q, tk + t) ≤ C| Rm k |(q, t˜) ≤ C(t˜)−1 ≤ C(t − tk + η)−1 , for all q ∈ K i,k . Hence |Rm|(q, t) is uniformly bounded on K i,k × [tk − η2 , tk + η2 ]. By Hamilton’s compactness theorem of the Ricci flow solution, {(K i,k , g(tk + t), pk )}∞ k=1 converge along a subsequence to a solution to the normalized Ricci flow (K i,∞ , gi,∞ (t), pi,∞ ), t ∈ (− η2 , η2 ), in the local C ∞ sense. When we consider the time t = 0, then using a diagonalization argument, a subsequence of {(K i,k , g(tk ), pk )}i,k
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
17
will converge in the local C ∞ sense to a smooth Riemannian manifold (K ∞ , g∞ , p∞ ), which is just (R, g∞ ), by the uniqueness of the limit space. For fixed i, there is a family of metrics gi,∞ (t), t ∈ (− η2 , η2 ), on K i . As shown in [Se], we translate the time by η4 , say considering the sequence {(K i,k , g(tk + η4 + ∞ Cloc
t), pk )}k , and repeat the above argument, then obtain that {(K i,k , g(tk + t), pk )}k −→ (K i,∞ , gi,∞ (t), pi,∞ ) along another subsequence, on the time interval t ∈ (− η2 , η4 + η2 ). The essential point is that the estimate dG H (Bg(tk ) (qk , ρ1 ), B(ρ1 )) < 21 δ1 ρ1 in the proof of Lemma 4.4 holds for some constant ρ1 , simultaneously the time tk is replaced by tk + η4 , but the constant η in Lemma 4.5 is fixed in this procedure. Iterating this process infinite times we obtain the convergence on K i for all t ∈ [0, ∞). Then do the same thing for each K i , i = 1, 2, . . ., and after a diagonalization argument, we get that a ∞ Cloc
subsequence of {(K i,k , g(tk + t), pk )}k , say (K i,ki , g(tki + t), pki ) −→ (R, g∞ (t), p∞ ) for all t ∈ [0, ∞), with g∞ (0) = g∞ . ¯ t, We finally show that the completion of R with respect to the metric g∞ (t), say R is just N∞ , for each time t ∈ [0, ∞). Denote by S = N∞ \R the set of singular points ¯ t = R ∪ S for fixed time t. Assume of (N∞ , g∞ (0)), then it suffices to show that R Q S = {ql }l=1 , where Q ≤ β for β = β(M) by Lemma 3.5, and let ε > 0 be any small constant such that Bg∞ (0) (qi , ε) ∩ Bg∞ (0) (q j , ε) = ∅ whenever i = j. Denote by K ε = R\ pl Bg∞ (0) ( pl , ε), then using |Ric∞ | ≤ C¯ on R × [0, ∞) and by the ¯
evolution of the distance function, we obtain dG H ((R\K ε , g∞ (t)), S) ≤ e2Ct ε and ¯ t = R ∪ S, by letting ε → 0. consequently R We pose one remark at the end of this section: Remark. Perelman’s pseudolocality theorem still holds for the normalized Ricci flow in our situation, so if the solution does not spatially local collapse along a subsequence of space time points ( pk , tk ) with tk → ∞, then passing a subsequence, (M, g(tk + t), pk ) will converge to a family of orbifolds (N∞ , g∞ (t), p∞ ) with finite isolated singularities for all t ≥ 0. The convergence is smooth on the regular part. If λ M (g(t)) ≤ 0 for all time, then by the same argument as in [FZZ], the limits (N∞ , g∞ (t)) are Einstein; while if not, then the limits g∞ (t) will have nonnegative scalar curvature on the regular part and positive totally L 1 scalar curvature. We point out that in the second case, (M, g(t)) does locally non-collapsing for each time t, by Perelman’s local non-collapsing theorem. Furthermore, it seems that under this situation, the limit may satisfy a shrinking Ricci soliton equation and has smooth metric in the orbifold sense. We don’t have a detailed argument for this now and it forms part of a future study on normalized Ricci flow which exists for all time. 5. Proofs of Theorems 1.1 and 1.2 The main result of this section is the following Theorem 5.1. Let (M, c) be a smooth oriented closed 4-manifold with a Spinc -structure c. Assume that the first Chern class c1 (c) of c is a monopole class of M satisfying that c12 (c)[M] ≥ 2χ (M) + 3τ (M) > 0. Let g(t), t ∈ [0, ∞), be a solution to (1.3) so that |Ric(g(t))| ≤ 3, and lim λ M (g(t)) = − 32π 2 c12 (c)[M]. t→∞
(5.1)
(5.2)
18
F. Fang, Y. Zhang, Z. Zhang
Then there exists an m ∈ N, and sequences of points {x j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence, dG H
(M, g(tk + t), x1,k , . . . , xm,k ) −→ (
m
N j , g∞ , x1,∞ , . . . , , xm,∞ ),
j=1
t ∈ [0, ∞), in the m-pointed Gromov-Hausdorff sense for any tk → ∞, where (N j , g∞ ) j = 1, . . . , m are complete Kähler-Einstein orbifolds of complex dimension 2 with at most finitely many isolated orbifold points {qi }. The scalar curvature (resp. volume) of g∞ is 1
−Volg0 (M)− 2
32π 2 c12 (c)[M] (r esp.
V = Volg0 (M) =
m
Volg∞ (N j )).
j=1
Furthermore, in the regular part of N j , {g(tk + t)} converges to g∞ in C ∞ -sense. Comparing with Proposition 3.2, Theorem 5.1 shows that the Einstein orbifolds are actually Kähler Einstein orbifolds under the additional assumptions. The key point in the proof is that the sequence of the self-dual parts of the curvatures of the connections on the determinant line bundles given by the irreducible solutions in the Seiberg-Witten equations converges to a non-trivial parallel self-dual 2-form on every component N j , which is a candidate of the Kähler form. ˘ Let (M, c) and g(t) be the same as in Thoerem 5.1, and let V , m, tk , x j,k , R(g(t)), gk , g∞ , N j and F j,k,r be the same as in Sect. 3. Assume that, for each k, (φk , Ak ) is an irreducible solution to the Seiberg-Witten equations (2.1). Let | · |k denote the norm with respect to the metric gk = g(tk ). The following lemma shows that the L 2 -norms of the self-dual parts FA+k tends to zero. Lemma 5.2.
lim
k−→∞ M
|∇ k FA+k |2k dvk = 0,
where ∇ k is the connection on 2 T ∗ (M) induced by Levi-civita connection. Proof. The Bochner formula implies that 1 R(gk ) 1 |φk |2k + |φk |4k . 0 = − k |φk |2k + |∇ Ak φk |2k + 2 4 4 By taking integration we get that R(gk ) 1 |φk |2k )dvk = − (|∇ Ak φk |2k + |φk |4k dvk . 4 4 M M
(5.3)
Since λ M (gk ) is the lowest eigenvalue of the operator −4k + R(gk ), for any 1 > 0, by definition λ M (gk ) |φk |2k, dvk ≤ (4|∇|φk |k, |2 + R(gk )|φk |2k, )dvk , (5.4) M
M
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
19
where | · |2k, = | · |2k + 2 . By Kato’s inequality (cf. (2.5)) and letting → 0, λ M (gk ) |φk |2k dvk ≤ (4|∇ Ak φk |2k + R(gk )|φk |2k )dvk = − |φk |4k dvk ≤ 0. M
M
M
As λ M (gk ) ≤ 0, by Schwarz inequality, 1 1 1 λ M (gk )( |φk |4k, dvk ) 2 = λ M (gk )Volgk (M) 2 ( |φk |4k, dvk ) 2 M M ≤ λ M (gk ) |φk |2k, dvk . M
Therefore
λ M (gk )( M
1 |φk |4k, dvk ) 2
≤ M
(4|∇|φk |k, |2 + R(gk )|φ|2k, )dvk .
Thus 1 Ak 2 2 4 4 (|∇ φk |k − |∇|φk |k, | )dvk ≤ − |φk |k dvk − λ M (gk )( |φk |4k, dvk ) 2 . M
M
M
(5.5) From (2.5), |∇|φk |k, ≤ Hence, by letting −→ 0, we have 1 1 |∇ Ak φk |2k dvk ≤ −(( |φk |4k dvk ) 2 + λ M (gk ))( |φk |4k dvk ) 2 . |2
3 2 Ak 4 |∇ φk |k .
M
M
(5.6)
M
+ denotes the self-dual part of the harmonic form representing the first Chern class If c1,k c1 (c) of c, by the Seiberg-Witten equation we get that + 2 |φk |4k dvk = 8 |FA+k |2 dvk ≥ 32π 2 [c1,k ] [M] ≥ 32π 2 c12 (c)[M]. (5.7) M
M
Note that, by the standard estimates for Seiberg-Witten equations, ˘ k ) ≥ |φk |2k − R(g
and, by Theorem 1.1 in [FZ], 32π 2 c12 (c)[M] + λ M (gk ) is non-positive. Hence 1 Ak 2 2 2 |∇ φk |k dvk ≤ −( 32π c1 (c)[M] + λ M (gk ))( |φk |4k dvk ) 2 M M 1 2 ˘ k )V 2 ( 32π 2 c (c)[M] + λ M (gk )) −→ 0, ≤ R(g (5.8) 1 when k −→ ∞, by (5.2) and Lemma 3.1. By the second one of the Seiberg-Witten equations again (cf. [Le2]), |∇ k FA+k |2k ≤
1 |φk |2k |∇ Ak φk |2k , 2
(5.9)
where ∇ Ak is the connection on (Sc) induced by the Levi-Civita connection. Hence 1 ˘ |∇ k FA+k |2k dvk ≤ | R(g(t ))| |∇ Ak φk |2k dvk −→ 0, k 2 M M when k −→ ∞.
20
F. Fang, Y. Zhang, Z. Zhang
Regard FA+k as self-dual 2-forms of gk on U j,r = Bg∞ (x j,∞ , r )\ i Bg∞ (qi, j , r −1 ), ∗ where gk = F j,k,r +1 gk , and qi, j are the orbifold points of N j . Since |FA+k |2k =
1 1 ˘ 2 |φk |4k ≤ R(g k ) ≤ C, 8 8
(5.10)
where C is a constant independent of k, FA+k ∈ L 1,2 (gk ), and FA+k L 1,2 (g ) ≤ C , k
where C
is a constant independent of k. Note that · L 1,2 (g∞ ) ≤ 2 · L 1,2 (g ) for k 1 k
C 1,α since gk −→
g∞ on U j,r . Thus, by passing to a subsequence, a self-dual 2-form with respect to g∞ .
L 1,2 FA+k −→
j ∈ L 1,2 (g∞ ),
Lemma 5.3. For any j, j is a smooth self-dual 2-form on U j,r \∂U j,r such that ∇ ∞ j ≡ 0, and | j |∞ ≡ cont. = 0, where ∇ ∞ is the connection induced by the √ Levi-Civita connection of g∞ . Hence, g∞ is a Kähler metric with Kähler form 2 | jj | on U j,r . Proof. By Lemma 5.2 ∞ 2 0≤ |∇ j |∞ dv∞ = lim
k−→∞ U j,r
U j,r
|∇
∞
FA+k |2∞ dv∞ ≤ 2
lim
k−→∞ M
|∇ k FA+k |2k dvk = 0.
It is easy to see that j is a weak solution of the elliptic equation ∇ ∞ j = 0 on U j,r . By elliptic equation theory, j is a smooth self-dual 2-form on U j,r \∂U j,r , ∇ ∞ j ≡ 0, and | j |∞ ≡ cont. Now we claim that, for any j and r 1, U j,r | j |2∞ dv∞ = 0. If not, there exist js , s = 1, . . . , m 0 , m 0 ≤ m, such that U j ,r | js |2∞ dv∞ ≡ 0. By Lemma 3.1, s
˘ k ) = λ M V − 21 , which is the scalar curvature of g∞ , R ∞ = lim R(gk ) = lim R(g k−→∞
k−→∞
i.e. R ∞ = R(g∞ ). Note that, by (5.10) and Lemma 3.7, | j |2∞ dv∞ = lim
k−→∞ U j,r
U j,r
|FA+k |2k dvk
1 ˘ k )2 Volg (Uj,r ) lim R(g k 8 k−→∞ 1 2 = R ∞ Volg∞ (Uj,r ), 8
1 ˘ k )2 Volgk (M\ lim R(g |FA+k |2k dvk | ≤ Fk,j,r (Uj,r )) 8 k−→∞ ≤
lim |
k−→∞
M
|FA+k |2k dvk −
m j=1 U j,r
j
≤
1 2 C R∞ 8
m
Volg∞ (Nj \Uj, 2r ),
j=1
and, by Lemma 3.1, 2 2 lim | (R(gk ) − R ∞ )dvk | ≤ 24 lim (|R(gk ) − R(gk )| + |R ∞ k−→∞
M
k−→∞ M
−R(gk )|)dvk = 0,
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
21
where C is a constant independent of k. Hence, we obtain 2
R∞
Volg∞ (Uj,r ) ≥
j= j1 ,..., jm 0
m j=1 U j,r
8| j |2∞ dv∞ = lim
k−→∞
≥ lim
k−→∞ M
m
2
8|FA+k |2k dvk − C R ∞
j=1 U j,r
m
8|FA+k |2k dvk
Volg∞ (Nj \Uj, 2r )
j=1 2
≥ 32π 2 c12 (c)[M] − C R ∞
m
Volg∞ (Nj \Uj, 2r ).
j=1
The last inequality is obtained by (5.7). Thus, by (5.1), 2
R∞
2
j= j1 ,..., jm 0
Volg∞ (Uj,r ) ≥ 32π 2 (2χ (M) + 3τ (M)) − CR∞
m
Volg∞ (Nj \Uj, 2r ).
j=1
By the Chern-Gauss-Bonnet formula and the Hirzebruch signature theorem, 1 1 1 2χ (M)+3τ (M) ≥ 2 ( R(gk )2 + 2|W + (gk )|2k )dvk − 2 |Ricº(gk )|2 dvk . 4π Uk,r 24 8π M L 2, p
By Lemma 3.1, and the fact that gk −→ g∞ on U j,r , we obtain that 2 R∞
m Volg∞ (Uj,r ) ≥ 8
j= j1 ,..., jm 0
2
R ( ∞ + 2|W + (g∞ )|2∞ )dv∞ U j,r 24
j=1 2
−C R ∞
m
Volg∞ (Nj \Uj, 2r ).
j=1
Note that, on any U j,r , j = j1 , . . . , jm 0 , ∇ ∞ j ≡ 0, | j |∞ ≡ cont. = 0, and j √ is a self-dual 2-form. Thus g∞ is a Kähler metric with Kähler form 2 | jj | on U j,r , 2
j = j1 , . . . , jm 0 . It is well known that R ∞ = 24|W + (g∞ )|2∞ for Kähler metrics (cf. [B]). Thus 2
R∞
j= j1 ,..., jm 0
2
Volg∞ (Uj,r ) ≥ R ∞
j= j1 ,..., jm 0
+
js = j1 ,..., jm 0 2
≥ R∞ +
1 3
2
Volg∞ (Uj,r ) − CR∞
m
Volg∞ (Nj \Uj, 2r )
j=1
2
R 8 ( ∞ + 2|W + (g∞ )|2∞ )dv∞ U j,r 24
j= j1 ,..., jm 0
js = j1 ,..., jm 0
2
Volg∞ (Uj,r ) − CR∞
m j=1
2
R ∞ Volg∞ (Ujs ,r ).
Volg∞ (Nj \Uj, 2r )
22
F. Fang, Y. Zhang, Z. Zhang
Note that, for r 1, 2
1 3C R ∞
m
Volg∞ (Nj \Uj, 2r ) ≥
j=1
2
js = j1 ,..., jm 0
R ∞ Volg∞ (Ujs ,r ).
A contradiction. Thus, for all j, U j,r | j |2∞ dv∞ = 0, and ∇ ∞ j ≡ 0, | j |∞ ≡ cont. = 0. Thus we obtain the conclusion. Proof of Theorem 5.1. First, assume that diam g(tk ) (M) −→ ∞, when k −→ ∞. By Proposition 3.2 and Proposition 4.1, there exists a m ∈ N, and a sequence of points {x j,k ∈ M}, k ∈ N, j = 1, . . . , m, satisfying that, by passing to a subsequence, (M, g(tk + t), x1,k , . . . , xm,k ), t ∈ [0, ∞), converges to {(N1 , g∞ , x1,∞ ), . . . , (Nm , g∞ , xm,∞ )} in the m-pointed Gromov-Hausdorff sense, when k −→ ∞, where (N j , g∞ ) j = 1, . . . , m are complete Einstein 4-orbifolds with finite isolated orbifold points {qi }. The scalar curvature of g∞ is R ∞ = lim λ M (g(t)), and t−→∞
V = Volg0 (M) =
m
Volg∞ (N j ).
j=1
By Lemma 5.2, g∞ is a Kähler-Einstein metric in the non-singular part of
m
Nj.
j=1
Then by the same arguments as in Sect. 4 of [Ti], g∞ is actually a Kähler-Einstein m N j , {g(tk + t)}, t ∈ [0, ∞), orbifold metric. Furthermore, in the non-singular part of j=1
C ∞ -converges to g∞ by Proposition 4.1. If diam gk (M) < C for a constant C independent of k, we can also obtain the conclusion by the similar, but much easier, arguments as above. Theorem 5.4. Let (M, c) be a smooth compact closed oriented 4-manifold with a Spinc structure c. Assume that the first Chern class c1 (c) of c is a monopole class of M satisfying c12 (c)[M] = 2χ (M) + 3τ (M) > 0, and χ (M) = 3τ (M). If M admits a solution g(t), t ∈ [0, ∞) to (1.3) with |R(g(t))| ≤ 12, then lim λ M (g(t)) = − 32π 2 c12 (c)[M]. t−→∞
Furthermore, if |Ric(g(t))| ≤ 3, the Kähler-Einstein metric g∞ in Theorem 5.1 is a complex hyperbolic metric. Proof. Let V = V ol g(t) (M). By the Chern-Gauss-Bonnet formula and the Hirzebruch signature theorem, 1 1 2χ (M) − 3τ (M) ≥ ( R(g(t))2 + 2|W − (g(t))|2 4π 2 M 24 1 − |Ricº(g(t))|2 )dvg(t) , (5.11) 2 where W − is the anti-self-dual Weyl tensor. Note that 2 R(g(t))2 dvg(t) ≥ R(g(t))2 V −→ R ∞ V = lim λ M (g(t))2 , M
t−→∞
(5.12)
Maximum Solutions of Normalized Ricci Flow on 4-Manifolds
23
when t −→ ∞, by Schwarz inequality and Lemma 3.1. By (5.11), (5.12), Lemma 3.1 and Theorem 1.1 in [FZ], 1 1 |W − (g(t))|2 dvg(t) + lim λ M (g(t))2 2χ (M) − 3τ (M) ≥ lim inf 2 t−→∞ 2π 96π 2 t−→∞ M 1 1 ≥ lim inf |W − (g(t))|2 dvg(t) + c12 (c)[M] t−→∞ 2π 2 M 3 1 1 = lim inf |W − (g(t))|2 dvg(t) + (2χ (M) + 3τ (M)). t−→∞ 2π 2 M 3 Since χ (M) = 3τ (M), we obtain
lim λ M (g(t)) = − 32π 2 c12 (c)[M],
t−→∞
and lim inf t−→∞
1 2π 2
|W − (g(t))|2 dvg(t) = 0. M
Now, assume that |Ric(g(t))| ≤ 3. Let tk , N j , gk , and g∞ be the same as above. For any j and compact subset U of the regular part of N j , − 2 0≤ |W (g∞ )|∞ dv∞ ≤ lim inf |W − (g(tk ))|2k dvk = 0, U
k−→∞
M
L 2, p
since g(tk ) −→ g∞ on U . Hence g∞ is a Kähler-Einstein metric with W − (g∞ ) ≡ 0. This implies that g∞ is a complex hyperbolic metric (cf. [Le1]). The desired result follows. Proof of Theorem 1.1 and Theorem 1.2. By the work of Taubes [Ta], if (M, ω) is a compact symplectic manifold with b2+ (M) > 1, the spinc -structure induced by ω is a monopole class. Moreover, since in this situation c12 (c)[M] = 2χ (M) + 3τ (M), Theorem 1.1 (resp. Theorem 1.2) is an obvious consequence of Theorem 5.1 (resp. Theorem 5.4). References [An1] [An2] [An3] [An4] [An5] [An6] [AIL] [B] [BD]
Anderson, M.T.: Ricci curvature bounds and einstein metrics on compact manifolds. J. Amer. Math. Soc. 2, 455–490 (1989) Anderson, M.T.: The l 2 structure of moduli spaces of einstein metrics on 4-manifolds. G.A.F.A. 2(1), 29–89 (1992) Anderson, M.T.: Convergence and rigidity of manifolds under ricci curvature bounds. Invent. Math. 102, 429–445 (1990) Anderson, M.T.: Degeneration of metrics with bounded curvature and applications to critical metrics of Riemannian functionals. Proc. Symp. Pure Math. 54, 53–79 (1993) Anderson, M.T.: Canonical metrics on 3-manifolds and 4-manifolds. Asian J. Math. 10, 127–163 (2006) Anderson, M.T.: Extrema of curvature functionals on the space of metrics on 3-manifolds. Calc. Var. and PDE 5, 199–269 (1997) Akutagawa, K., Ishida, M., LeBrun, C.: Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds. Arch. Math. 88(1), 71–76 (2006) Besse, A.L.: Einstein manifolds. Ergebnisse der Math. Berlin-New York:Springer-Verlag, 1987 Bär, C., Dahl, M.: Small eigenvalues of the conformal Laplacian. Geom. Funct. Anal. 13, 483–508 (2003)
24
[CC]
F. Fang, Y. Zhang, Z. Zhang
Cheeger, J., Colding, T.H.: On the structure of space with Ricci curvature bounded below I. J. Diff. Geom. 45, 406–480 (1997) [CG] Cheeger, J., Gromov, M.: Collapsing Riemannian Manifolds while keeping their curvature bounded I. J. Diff. Geom. 23, 309–364 (1986) [Cr] Croke, C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ecole Norm. Sup. 13(4), 419–435 (1980) [CT] Cheeger, J., Tian, G.: Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Amer. Math. Soci. 19, 487–525 (2006) [FZ] Fang, F., Zhang, Y.G.: Perelman’s λ-functional and the Seiberg-Witten equations. Front. Math. China, 2(2), 191–210 (2007) [FZZ] Fang, F., Zhang, Y.G., Zhang, Z.L.: Non-singular solutions to the normalized Ricci flow equation. Math. Ann. 340(3), 647–674 (2008) [H1] Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982) [H2] Hamilton, R.: A compactness property for solutions of the Ricci flow. Amer. J. Math. 117, 545– 574 (1995) [K] Kronheimer, P.B.: Minimal genus in S 1 × M 3 . Invent. Math. 135(1), 45–61 (1999) [KL] Kleiner, B., Lott, J.: Notes on Perelman’s papers. http://arxiv.org/list/math.DG/0605667, 2006 [Kot] Kotschick, D.: Monopole classes and Perelman’s invariant of four-manifolds. http://arxiv.org/list/ math.DG/0608504, 2006 [Le1] LeBrun, C.: Einstein metrics and Mostow rigidity. Math. Res. Lett. 2, 1–8 (1995) [Le2] LeBrun, C.: Four-Dimensional Einstein Manifolds and Beyond. In: Surveys in Differential Geometry, Vol. VI: Essays on Einstein Manifolds, Boston:International Press, 1999, pp. 247–285 [Le3] LeBrun, C.: Ricci curvature, minimal volumes, and Seiberg-Witten theory. Invent. Math. 145, 279– 316 (2001) [Le4] LeBrun, C.: Kodaira dimension and the Yamabe probblem. Comm. Anal.Geom. 7, 133–156 (1999) [N] Nakajima, H.: Self-duality of ale Ricci-flat 4-manifolds and positive mass theorem. Adv. Stud. Pure Math. 18-I, 385–395 (1990) [Pe1] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http://arxiv.org/ list/math:/0211159, 2002 [Pe2] Perelman, G.: Ricci flow with surgery on three-manifolds. http://arxiv.org/list/math:DG/0303109v1, 2003 [Se] Sesum, N.: Convergence of a Kähler-Ricci flow. http://arxiv.org/list/math.DG/0402238v1, 2004 [STW] Sesum, N., Tian, G., Wang, X.D.: Notes on Perelman’s paper on the entropy formula for the Ricci flow and its geometric applications. Preprint [Ta] Taubes, C.H.: More constraints on symplectic forms from Seiberg-Witten invariants. Math. Res. Lett. 2, 9–13 (1995) [Ti] Tian, G.: On Calabi’s conjecture for complex surface with positive first Chern class. Invent. Math. 101, 101–172 (1990) [W] Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1, 809–822 (1994) [Y] Ye, R.: Ricci flow, Einstein metrics and space forms. Trans. Amer. Math. Soc. 338(2), 871–896 (1993) Communicated by P. Sarnak
Commun. Math. Phys. 283, 25–92 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0555-9
Communications in
Mathematical Physics
Cardy Condition for Open-Closed Field Algebras Liang Kong1,2,3 1 Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig, Germany 2 Institut Des Hautes Études Scientifiques, Le Bois-Marie, 35, Route De Chartres,
F-91440 Bures-sur-Yvette, France
3 Max-Planck-Institute for Mathematics, Vivatsgasse 7, D-23111 Bonn, Germany.
E-mail:
[email protected] Received: 10 April 2007 / Accepted: 1 April 2008 Published online: 24 July 2008 – © Springer-Verlag 2008
Abstract: Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that CV , the category of V -modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain C-extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation S : τ → − τ1 on the space of intertwining operators of V . We then derive a graphical representation of S in the modular tensor category CV . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy CV |CV ⊗V -algebra. In the end, we give a categorical construction of the Cardy CV |CV ⊗V -algebra in the Cardy case. Contents 0. 1.
2.
Introduction . . . . . . . . . . . . . . . Partial Dioperads . . . . . . . . . . . . . 1.1 Partial dioperads . . . . . . . . . . . 1.2 Conformal full field algebras . . . . 1.3 Open-string vertex operator algebras Swiss-Cheese Partial Dioperad . . . . . 2.1 2-colored (partial) dioperads . . . . 2.2 Swiss-cheese partial dioperads . . . 2.3 Open-closed field algebras over V . 2.4 Ishibashi states . . . . . . . . . . .
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26
L. Kong
3.
Cardy Condition . . . . . . . . . . . . . . . 3.1 The first version . . . . . . . . . . . . . 3.2 The second version . . . . . . . . . . . 4. Modular Tensor Categories . . . . . . . . . 4.1 Preliminaries . . . . . . . . . . . . . . 4.2 Graphical representation of S : τ → − τ1 5. Categorical Formulations and Constructions 5.1 Modular invariant CV L ⊗V R -algebras . . 5.2 Cardy CV |CV ⊗V -algebras . . . . . . . . 5.3 Constructions . . . . . . . . . . . . . . A. The Proof of Lemma 4.30 . . . . . . . . . .
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51 51 56 64 65 69 76 76 80 84 88
0. Introduction This work is a continuation of the works [HKo1-HKo3,Ko1,Ko2] and a part of an openstring extension of a program on the closed conformal field theory via the theory of vertex operator algebra. This program was initiated by I. Frenkel and largely developed by Huang [H1]-[H11]. Zhu’s work [Z] is also very influential in this development. Segal defined the (closed) conformal field theory [Se1] as a projective monoidal functor from the category of finite ordered sets with morphisms being the conformal equivalent classes of Riemann surfaces with parametrized boundaries to the category of locally convex complete topological vector spaces. This definition is very difficult to work with directly. Taking advantage of the theory of vertex operator algebra, Huang suggested to first construct all necessary structures on a dense subspace of the relevant complete topological vector space [H3], then complete it properly later [H5,H6]. This idea guided us in all our previous works [HKo1-HKo3,Ko1,Ko2], in particular in our formulation of the notion of algebra over the Swiss-cheese partial operad which catches only some genus-zero information of the whole structure on the dense subspace. This seemingly temporary structure on the dense subspace does not necessarily follow from that on the complete topological vector space, thus has its own independent values and is worthwhile to be formulated properly and extended to a theory of all genus. We will call such a theory on the dense subspace a (closed) partial conformal field theory. More precisely, a partial conformal field theory is a projective monoidal functor F : RS → GV between two partial categories RS, GV in which the compositions of morphisms are not always well-defined. RS is the category of finite ordered sets, and MorRS (S1 , S2 ) for any pair of such sets S1 , S2 is the set of the conformal equivalent classes of closed Riemann surfaces with |S1 | positively oriented punctures and local coordinates and |S2 | negatively oriented punctures and local coordinates [H3], and the compositions of morphisms in RS are given by the sewing operations on oppositely oriented punctures ([H3]), where the sewing operations are only partially defined. GV is the category of graded vector spaces with finite dimensional homogeneous spaces and a weak topology induced from the restricted dual spaces. For any pair of A, B ∈ Ob(GV), MorGV (A, B) is the set of continuous linear maps from A to B := n∈G B B(n) , where g
the abelian group G B gives the grading on B. For any pair of morphisms A − → B f
and B − → C in GV, using the projector Pn : B → B(n) , we define f ◦ g(u) := n∈G B f (Pn g(u)), ∀u ∈ A, and f ◦ g is well-defined only when the sum is absolutely convergent for all u ∈ A. We remark that by replacing the surfaces with parametrized boundaries in Segal’s definition by the surfaces with oriented punctures and local coor-
Cardy Condition for Open-Closed Field Algebras
27
dinates in RS we have enlarged the morphism sets. Thus a Segal’s functor may not be extendable to a functor on RS. The above definition can be easily extended to include open strings by adding Riemann surfaces with (unparametrized!) boundaries and both oriented interior punctures and punctures on the boundaries to RS. We will call this open-string extended theory an open-closed partial conformal field theory. We denote the graded vector spaces associated to interior punctures (closed strings at infinity) and boundary punctures (open strings at infinity) by Vcl and Vop respectively. The sets of all genus-zero closed surfaces (spheres) with an arbitrary number of positively and negatively oriented punctures form a structure of sphere partial dioperad K (see the definition in Sect. 1.1). It includes as a substructure the sphere partial operad [H3], which only includes spheres with a single negatively oriented puncture. The sphere partial dioperad allows all sewing operations as long as the surfaces after sewing are still genus-zero. Hence to construct genus-zero partial conformal field theory amounts to construct projective K-algebras or algebras over a certain extension of K. In [HKo2], we introduced the notion of a conformal full field algebra over V L ⊗ V R equipped with a nondegenerate invariant bilinear form, where V L and V R are two vertex operator algebras of central charge c L and c R respectively and satisfy certain finiteness and reductivity conditions. Theorem 2.7 in [Ko1] can be reformulated as follows: a conformal full field algebra Vcl over V L ⊗ V R equipped with a nondegenerate invariant bilinear form ˜ cL ⊗K ˜ c R , which is a partial dioperad extension canonically gives on Vcl an algebra over K of K. The sets of all genus-zero surfaces with one (unparametrized!) boundary component (disks) and an arbitrary number of oriented boundary punctures form the so-called disk partial dioperad denoted by D (see the definition in Sect. 1.1). F restricted on D induces a structure of algebra over a certain extension of D on Vop . In [HKo1], Huang and I introduced the notion of an open-string vertex operator algebra. We will show in Sect. 1.3 that an open-string vertex operator algebra of central charge c equipped with a ˜ c , which is a nondegenerate invariant bilinear form canonically gives an algebra over D partial dioperad extension of D. The sets of all genus-zero surfaces with only one (unparametrized!) boundary component and arbitrary number of oriented interior punctures and boundary punctures form the so-called Swiss-Cheese partial dioperad S (see the definition in Sect. 2.2). A typical elements in S is depicted in Fig. 1, where boundary punctures are drawn as an infinitely long strip (or an open string) and interior punctures are drawn as an infinitely long tube (or a closed string). Let V be a vertex operator algebra of central charge c satisfying the conditions in Theorem 0.1. We will show in Sect. 2.3 that an open-closed field algebra over V , which contains an open-string vertex operator algebra Vop and a conformal full field algebra Vcl , satisfying a V -invariant boundary condition [Ko2] and equipped with nondegenerate invariant bilinear forms on both Vop and Vcl , canonically gives an algebra over S˜ c , which is an extension of S. Note that all surfaces of any genus with an arbitrary number of boundary components, interior punctures and boundary punctures can be obtained by applying sewing operations to elements in S. Therefore, except for some compatibility conditions coming from different decompositions of the same surfaces with higher genus, an algebra over S˜ c if extendable uniquely determine the entire theory of all genus. In particular, the famous Ishibashi states [I] can be obtained in an open-closed field algebra equipped with nondegenerate invariant bilinear forms. An Ishibashi state is a coherent state ψ
28
L. Kong
Fig. 1. A typical element in the Swiss-Cheese partial dioperad S
in Vcl such that (L L (n) − L R (−n))ψ = 0, ∀n ∈ Z. In physics, Ishibashi states are obtained by solving the above equation. In Sect. 2.4, we show how to obtain Ishibashi states constructively and geometrically from vacuum-like states in Vop . Other surfaces which are not included in S only provide additional compatibility conditions. In 2-d topological field theories, only three additional compatibility conditions are needed to ensure the consistency of a theory of all genus [La,Mo1,Se2,MSeg,AN, LP]. The first compatibility condition says that both Vop and Vcl are finite dimensional. This guarantees the convergence of all higher genus correlation functions. The second condition is the modular invariance condition for 1-point correlation functions on torus. It is due to two different decompositions of the same torus as depicted in Fig. 2. This condition is automatically satisfied in 2-d topological field theories, but nontrivial in conformal field theories. The third condition is the famous Cardy condition which is again due to two different decompositions of a single surface as shown in Fig. 3.
Fig. 2. Modular invariance condition for 1-pt correlation functions on torus
= Fig. 3. Cardy condition
Cardy Condition for Open-Closed Field Algebras
29
Now we turn to the compatibility conditions in conformal field theory. In this case, both Vcl and Vop in any nontrivial theory are infinite dimensional. We need to require the convergence of all correlation functions of all genus. This is a highly nontrivial condition and not easy to check for examples. So far the only known convergence results are in genus-zero [H7] and genus-one theories [Z,DLM,Mi1,Mi2,H9]. We recall a theorem by Huang. Theorem 0.1 ([H10,H11]). If (V, Y, 1, ω) is a simple vertex operator algebra V satisfying the following conditions: 1. V is C2 -cofinite; 2. Vn = 0 for n < 0, V(0) = C1, V is isomorphic to V as V -module; 3. all N-gradable weak V -modules are completely reducible, then the direct sum of all inequivalent irreducible V -modules has a natural structure of intertwining operator algebra [H4], and the category CV of V -modules has a structure of vertex tensor category [HL1]-[HL4,H2] and modular tensor category [RT,T]. For the intertwining operator algebra given in Theorem 0.1, Huang also proved in [H7,H9] that the products of intertwining operators and their q-traces have a certain nice convergence and analytic extension properties. These properties are sufficient for the construction of genus-zero and genus-one correlation functions in closed partial conformal field theory [HKo2,HKo3]. Since the modular tensor category CV supports an action of mapping class groups of all genus, it is reasonable to believe that the above conditions on V are also sufficient to guarantee the convergence of correlation functions of all genus. Assumption 0.2. In this work, we fix a vertex operator algebra V with central charge c, which is assumed to satisfy the conditions in Theorem 0.1 without further announcement. Besides the convergence condition and the axioms of projective K-algebra, Sonoda [So] argued on a physical level of rigor that it is sufficient to check the modular invariance condition in Fig. 2 in order to have a consistent partial conformal field theory of all genus. This modular invariance condition was studied in [HKo3]. In the framework of conformal full field algebra over V L ⊗ V R , it was formulated algebraically as a modular invariance property of an intertwining operator of V L ⊗ V R . For an open-closed theory, besides the convergence condition, the axioms of algebra over S˜ c and the modular invariance condition, Lewellen [Le] argued on a physical level of rigor that the only remaining compatibility condition one needs is the Cardy condition. The Cardy condition in an open-closed conformal field theory is more complicated than that in topological theory and has never been fully written down by physicists. In Sect. 3.1, we derive the Cardy condition from the axioms of open-closed partial conformal field theory by writing out two sides of the Cardy condition in Figure 3 explicitly in terms of the ingredients of an open-closed field algebra (see Definition 3.4). Using results in [H9,H10], we can show that the Cardy condition can be reformulated as an invariance condition of the modular transformation S : τ → − τ1 on intertwining operators. There are still more compatibility conditions which were not discussed in [So,Le]. One also need to prove a certain algebraic version of uniformization theorems (see [H1,H3] for the genus-zero case). Such results for genus larger than 0 are still not available. But it seems that no additional assumption on V is needed. They should follow automatically from the properties of the Virasoro algebra and intertwining operators. This uniformization problem and convergence problems are not pursued further in this work.
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L. Kong
In order to take advantage of some powerful tools, such as the graphic calculus in tensor category, in the study of the modular invariance condition and the Cardy condition, we would like to obtain categorical formulations of these conditions. It requires us to know the action of the modular transformation S in CV . Although the action of S L(2, Z) in a modular tensor category is explicitly known [MSei3,V,Ly,Ki,BK2], its relation to the modular transformation of the q-trace of the products of intertwining operators of V is not completely clear. This relation was first suggested by I. Frenkel and studied by Moore and Seiberg in [MSei3] but only on a physical level of rigor. Using Huang’s results on the modular tensor category [H10,H11], we derive a graphical representation of S in CV . This result enables us to give categorical formulations of the modular invariance condition and the Cardy condition. We incorporate them with the categorical formulation of open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy CV |CV ⊗V -algebra. As we discussed in previous paragraphs, it is reasonable to believe that open-closed partial conformal field theories of all genus satisfying the V -invariant boundary condition [Ko2] are classified by Cardy CV |CV ⊗V -algebras. However, to construct the high-genus theories explicitly is still a hard open problem which is not pursued in this work. In the end, we give an explicit construction of Cardy the CV |CV ⊗V -algebra in the so-called Cardy case in the physics literature (see for example [FFRS]). Note that this work is somewhat complementary to the works of Fjelstad, Fuchs, Runkel, Schweigert [FS,FRS1-FRS4,FjFRS1,FjFRS2]. We will leave a detailed study of the relationship between these two approaches to [KR]. The layout of this work is as follows: in Sect. 1, we introduce the notion of a sphere partial dioperad and disk partial dioperad and study algebras over them; in Sect. 2, we introduce the notions of the Swiss-cheese partial dioperad S and its C-extension S˜ c , and show that an open-closed conformal field algebra over V equipped with nondegenerate invariant bilinear forms canonically gives an algebra over S˜ c , and we also construct Ishibashi states in such algebras; in Sect. 3, we give two formulations of the Cardy condition; in Sect. 4, we derive a graphic representation of the modular transformation S; in Sect. 5, we give the categorical formulations of the nondegenerate invariant bilinear forms, the modular invariance condition and the Cardy condition. Then we introduce the notion of the Cardy CV |CV ⊗V -algebra and give a construction. Convention of notations: N, Z, Z+ , R, R+ , C denote the set of natural numbers, integers, positive integers, real numbers, positive real numbers, complex numbers, respecˆ C ˆ and H ˆ be the tively. Let H = {z ∈ C|Imz > 0} and H = {z ∈ C|Imz < 0}. Let R, one point compactification of real line, complex plane and up-half plane (including the R-boundary) respectively. Let R+ and C× be the multiplication groups of positive real and nonzero complex numbers respectively. The ground field is always chosen to be C. Throughout this work, we choose a branch cut for logarithm as follows: log z = log |z| + iArg z,
0 ≤ Arg z < 2π.
(0.1)
We define power functions of two different types of complex variables as follows: z s := es log z ,
z¯ s := es log z ,
∀s ∈ R.
(0.2)
1. Partial Dioperads In Sect. 1.1, we recall the definition of (partial) dioperad and algebra over it, and intro˜ cL ⊗ K ˜ cR , duce sphere partial dioperad K, disk partial dioperad D and their extensions K
Cardy Condition for Open-Closed Field Algebras
31
˜ cL ⊗ K ˜ c as examples. In Sect. 1.2, we discuss an algebra over K ˜ c R from a conformal D L R full field algebra over V ⊗ V . In Sect. 1.3, we discuss an algebra over D˜ c from an open-string vertex operator algebra. 1.1. Partial dioperads. Let us first recall the definition of dioperad given by Gan [G]. Let Sn be the automorphism group of the set {1, . . . , n} for n ∈ Z+ . Let m = m 1 +· · ·+m n be an ordered partition and σ ∈ Sn . The block permutation σ(m 1 ,...,m n ) ∈ Sm is the permutation which permutes n intervals of lengths m 1 , . . . , m n in the same way as σ permutes 1, . . . , n. Let σi ∈ Sm i , i = 1, . . . , k. We view the element (σ1 , . . . , σk ) ∈ Sm 1 ×· · ·×Sm k naturally as an element in Sm by the canonical embedding Sm 1 × · · · × Sm k → Sm . For ˆ j) = j any σ ∈ Sn and 1 ≤ i ≤ n, we define a map iˆ : {1, . . . , n −1} → {1, . . . , n} by i( ˆ j) = j + 1 if j ≥ i and an element i(σ ˆ ) ∈ Sn−1 by if j < i and i( −1 (i)( j). ˆ )( j) := iˆ−1 ◦ σ ◦ σ i(σ
Definition 1.1. A dioperad consists of a family of sets {P(m, n)}m,n∈N with an action of Sm × Sn on P(m, n) for each pair of m, n ∈ Z+ , a distinguished element IP ∈ P(1, 1) and substitution maps γ(i 1 ,...,i n )
P(m, n) × P(k1 , l1 ) × · · · × P(kn , ln ) −−−−−→ P(m − n + k1 · · · + kn , l1 + · · · + ln ) (P, P1 , . . . , Pn ) → γ(i1 ,...,in ) (P; P1 , . . . , Pn ) (1.1) for m, n, l1 , . . . , ln ∈ N, k1 , . . . , kn ∈ Z+ and 1 ≤ i j ≤ k j , j = 1, . . . , n, satisfying the following axioms: 1. Unit properties: For P ∈ P(m, n), (a) left unit property: γ(i) (IP ; P) = P for 1 ≤ i ≤ m, (b) right unit property: γ(1,...,1) (P; IP , . . . , IP ) = P. 2. Associativity: for P ∈ P(m, n), Q i ∈ P(ki , li ), i = 1, . . . , n, R j ∈ P(s j , t j ), j = 1, . . . , l = l1 + · · · + ln , we have γ(q1 ,...,ql ) γ( p1 ,..., pn ) (P; Q 1 , . . . , Q n ); R1 , . . . , Rl = γ( p1 ,..., pn ) (P; P1 , . . . , Pn ) (1.2) where Pi = γ(ql1 +···+li−1 +1 ,...,ql1 +···+li ) (Q i ; Rl1 +···+li−1 +1 , . . . , Rl1 +···+li ) for i = 1, . . . , n. 3. Permutation property: For P ∈ P(m, n), Q i ∈ P(ki , li ), i = 1, . . . , n, (σ, τ ) ∈ Sm × Sn , (σi , τi ) ∈ Ski × Sli , i = 1, . . . , n, γ(i1 ,...,in ) ((σ, τ )(P); Q 1 , . . . , Q n ) = ((σ, τ(k1 −1,...,kn −1) ), τ(l1 ,...,ln ) )γ(iτ (1) ,...,iτ (n) ) (P; Q τ (1) , . . . , Q τ (n) ), γ(i1 ,...,in ) (P; (σ1 , τ1 )(Q 1 ), . . . , (σn , τn )(Q n )) = ((id, iˆ1 (σ1 ), . . . , iˆn (σn )), (τ1 , . . . , τn ))γ −1 (P; Q 1 , . . . , Q n ). −1 (σ1 (i 1 ),...,σn (i n ))
We denote such a dioperad as (P, γP , IP ) or simply P. Remark 1.2. We define compositions i ◦ j as follows: P i ◦ j Q := γ(1,...,1, j,1,...1) (P; IP , . . . , IP , Q, IP , . . . , IP ). It is easy to see that γ(i1 ,...,in ) can be reobtained from i ◦ j . In [G], the definition of dioperad is given in terms of i ◦ j instead of γ(i1 ,...,in ) .
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Definition 1.3. A partial dioperad has a similar definition as that of dioperad except the map γ(i1 ,...,in ) or i ◦ j is only partially defined and the same associativity holds whenever both sides of (1.2) exist. A (partial) nonassociative dioperad consists of the same data as those of a (partial) dioperad satisfying all the axioms of a (partial) dioperad except the associativity. The notion of homomorphism and isomorphism of a (partial pseudo-) dioperad are naturally defined. Remark 1.4. In the case of a partial dioperad, the definition using γ(i1 ,...,in ) or i ◦ j may have subtle differences in the domains on which γ(i1 ,...,in ) or i ◦ j is defined (see Appendix C in [H2] for more details). These differences have no effect on those algebras over partial dioperads considered in this work. So we will simply ignore these differences. Remark 1.5. Notice that a (partial nonassociative) dioperad {P(m, n)}m,n∈N naturally contains a (partial nonassociative) operad {P(1, n)}n∈N as a substructure. Definition 1.6. A subset G of P(1, 1) is called a rescaling group of P if the following conditions are satisfied: 1. For any g, g1 , . . . , gn ∈ G, Q ∈ P(m, n), γ(i) (g; Q) and γ(1,...,1) (Q; g1 , . . . , gn ) are always well-defined for 1 ≤ i ≤ m. 2. IP ∈ G and G together with the identity element IP and multiplication map γ(1) : G × G → G is a group. Definition 1.7. A G-rescalable partial dioperad is a partial dioperad P such that for any P ∈ P(m, exist gi ∈ G, i = 1, . . . , n such n), Q i ∈ P(ki , li ), i = 1, . . . , n there that γ(i1 ,...,in ) P; γ(i1 ) (g1 ; Q 1 ), . . . , γ(in ) (gn ; Q n ) is well-defined. The first example of partial dioperad in which we are interested in this work comes from K = {K(n − , n + )}n − ,n + ∈N [Ko2], a natural extension of the sphere partial operad K [H3]. More precisely, K(n − , n + ) is the set of the conformal equivalent classes of a sphere with n − (n + ) ordered negatively (positively) oriented punctures and local coordinate map around each puncture. In particular, K(0, 0) is an one-element set consisting of the conformal equivalent class of a sphere with no additional structure. We simply ˆ We use denote this element as C. (−1)
Q = ( (z −1 ; a0
(−n − )
, A(−1) ), . . . , (z −n − ; a0
, A(−n − ) ) |
(z 1 ; a0(1) , A(1) ) . . . (z n + ; a0(n + ) , A(n + ) ) )K ,
(1.3)
ˆ a (i) ∈ C× , A(i) ∈ C∞ for i = −n − , . . . , −1, 1, . . . , n + , to denote a where z i ∈ C, 0 ˆ with positively (negatively) oriented punctures at z i ∈ C ˆ for i = 1, . . . , n + sphere C (i = −1, . . . , −n − ), and with local coordinate map f i around each puncture z i given by: ∞ (i) (i) x ddx A j x j+1 ddx j=1 (a0 ) x if z i ∈ C, (1.4) f i (w) = e =e
∞
(i) j+1 d j=1 A j x dx
(i) d (a0 )x d x x
x=w−z i x= −1 w
if z i = ∞.
(1.5)
Cardy Condition for Open-Closed Field Algebras
33
We introduce a useful notation Q¯ defined as follows: (−n ) (−1) Q¯ = ( (¯z −1 ; a0 , A(−1) ), . . . , (¯z −n − ; a0 − , A(−n − ) )| (1)
(n + )
(¯z 1 ; a0 , A(1) ) . . . (¯z n + ; a0
, A(n + ) ) )K ,
(1.6)
where the “overline” represents complex conjugations. We denote the set of all such Q as TK (n − , n + ). Let TK := {TK (n − , n + )}n − ,n + ∈N . There is an action of S L(2, C) on TK (n − , n + ) as Mobius transformations. It is clear that K(n − , n + ) = TK (n − , n + )/S L(2, C).
(1.7)
We denote the quotient map TK → K as πK . The identity IK ∈ K(1, 1) is given by (1.8) IK = πK ( (∞, 1, 0)|(0, 1, 0) ), ∞ where 0 = (0, 0, . . . ) ∈ n=1 C. The composition i ◦ j is provided by the sewing operation i ∞− j [H3]. In particular, for n 1 , m 2 ≥ 1, P ∈ K(m 1 , n 1 ) and Q ∈ K(m 2 , n 2 ), Pi ∞− j Q is the sphere with punctures obtained by sewing the i th positively oriented puncture of P with the j th negatively oriented puncture of Q. The Sn − × Sn + -action on K(n − , n + ) (or T (n − , n + )) is the natural one. Moreover, the set {( (∞, 1, 0)|(0, a, 0) )|a ∈ C× }
(1.9)
together with multiplication 1 ∞1 is a group which can be canonically identified with group C× . It is clear that K is a C× -rescalable partial dioperad. We call it sphere partial dioperad. ˜ cL ⊗ K ˜ c R for c, c L , c R ∈ C, are trivial line The C-extensions of K, such as Kc and K × bundles over K with natural C -rescalable partial dioperad structures (see Sect. 6.8 in ˜ cL ⊗ K ˜ c R as ψK . [H2]). Moreover, we denote the canonical section K → K The next example of partial dioperad is D = {D(n − , n + )}n − ,n + ∈N , which is an extension of the partial operad of a disk with strips ϒ introduced [HKo1]. More precisely, D(n − , n + ) is the set of conformal equivalent classes of disks with ordered puncture on their boundaries and local coordinate map around each puncture. In particular, D(0, 0) is an one-element set consisting of the conformal equivalent class of a disk with no ˆ We use additional structure. We simply denote this element as H. (−n − )
Q = ( (r−n − ; b0
, B (−n − ) ), . . . , (r−1 ; b0(−1) , B (−1) )| (1)
(n + )
(r1 ; b0 , B (1) ) . . . (rn + ; b0
, B (n + ) ) )D ,
(1.10)
(i)
ˆ a ∈ R+ , B (i) ∈ R∞ for i = −n − , . . . , −1, 1, . . . , n + , to denote a where ri ∈ R, 0 ˆ with positively (negatively) oriented punctures at ri ∈ R ˆ for i = 1, . . . , n + disk H (i = −1, . . . , −n − ), and with local coordinate map gi around each punctures ri given by: ∞ (i) B j x j+1 ddx (i) x ddx j=1 gi (w) = e (b0 ) x if ri ∈ R, (1.11) =e
∞
(i) j+1 d j=1 B j x dx
(i) d (b0 )x d x x
x=w−ri x= −1 w
if ri = ∞.
(1.12)
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L. Kong
We denote the set of all such Q as TD (n − , n + ). Let TD := {TD (n − , n + )}n − ,n + ∈N . The automorphism group of H, S L(2, R), naturally acts on TD (n − , n + ). It is clear that D(n − , n + ) = TK (n − , n + )/S L(2, R).
(1.13)
We denote the quotient map as πD . The identity ID ∈ D(1, 1) is given by ID = πD ( (∞, 1, 0)|(0, 1, 0) ).
(1.14)
The composition i ◦ j is provided by the sewing operation i ∞− j [HKo1]. In particular, for n 1 , m 2 ≥ 1, P ∈ D(m − , m + ) and Q ∈ D(n − , n + ), Pi ∞− j Q is the disk with strips obtained by sewing the i th positively oriented puncture of P with the j th negatively oriented puncture of Q. The Sn − × Sn + -action on D(n − , n + ) (or T (n − , n + )) is the natural one. The set {( (∞, 1, 0)|(0, a, 0) )|a ∈ R+ }
(1.15)
together with multiplication 1 ∞1 is a group which can be canonically identified with group R+ . It is clear that D is a R+ -rescalable partial dioperad. We call it the disk partial dioperad. D can be naturally embedded to K as a sub-dioperad. The C-extension Dc of D for c ∈ C is just the restriction of the line bundle Kc on D. Dc is also a R+ -rescalable partial dioperad and a partial sub-dioperad of Kc . We denote the canonical section on D → Dc as ψD . Now we discuss an example of a partial nonassociative dioperad which is important for us. Let U = ⊕n∈J U(n) be a graded vector space and J an index set. We denote the projection U → U(n) as Pn . Now we consider a family of spaces of multi-linear maps EU = {EU (m, n)}m,n∈N , where EU (m, n) := HomC (U ⊗m , U ⊗n ).
(1.16)
( j)
For f ∈ EU (m, n), g j ∈ EU (k j , l j ) and u p j ∈ U , 1 ≤ p j ≤ l j , j = 1, . . . , n, we say that (1)
(n)
(i1 ,...,in ) ( f ; g1 , . . . , gn )(u 1 ⊗ · · · ⊗ u ln ) (1) (1) (n) (n) f Ps1 g1 (u 1 ⊗ · · · ⊗ u l1 ) ⊗ · · · ⊗ Psn gn (u 1 ⊗ · · · ⊗ u ln ) := s1 ,...,sn ∈J
is well-defined if the multiple sum converges absolutely. This gives rise to a partially defined substitution map, for 1 ≤ i j ≤ k j , j = 1, . . . , n, i1 ,...,in : EU (m, n) ⊗ EU (k1 , l1 ) ⊗ · · · ⊗ EU (kn , ln ) → EU (m + k − n, l), where k = k1 + · · · + kn and l = l1 + · · · + ln . In general, the compositions of three substitution maps are not associative. The permutation groups actions on EU are the usual one. Let = { i1 ,...,in }. It is clear that (EU , , idU ) is a partial nonassociative dioperad. We often denote it simply by EU . Definition 1.8. Let (P, γP , IP ) be a partial dioperad. A P-algebra (U, ν) consists of a graded vector space U and a morphism of partial nonassociative dioperad ν : P → EU .
Cardy Condition for Open-Closed Field Algebras
35
When U = ⊕n∈J U(n) is a completely reducible module for a group G, J is the set of equivalent classes of irreducible G-modules and U(n) is a direct sum of irreducible G. G-modules of equivalent class n ∈ J , we denote EU by EU Definition 1.9. Let (P, γP , IP ) be a G-rescalable partial dioperad. A G-rescalable G is so that ν| P-algebra (U, ν) is a P-algebra and the morphism ν : P → EU G : G → End U coincides with the given G-module structure on U .
1.2. Conformal full field algebras. Let (V L , YV L , 1 L , ω L ) and (V R , YV R , 1 R , ω R ) be two vertex operator algebras with central charge c L and c R respectively, satisfying the conditions in Theorem 0.1. Let (Vcl , m cl , ιcl ) be a conformal full field algebra over V L ⊗ V R . A bilinear form (·, ·)cl on Vcl is invariant [Ko1] if, for any u, w1 , w2 ∈ Vcl , ¯ 1 )cl (w2 , Y f (u; x, x)w = (Y f (e−x L
L (1)
x −2L
L (0)
⊗ e−x¯ L
R (1)
x¯ −2L
R (0)
u; eπi x −1 , e−πi x¯ −1 )w2 , w1 )cl , (1.17)
or equivalently, (Y f (u; eπi x, e−πi x)w ¯ 2 , w1 )cl = (w2 , Y f (e x L
L (1)
x −2L
L (0)
⊗ e x¯ L
R (1)
x¯ −2L
R (0)
u; x −1 , x¯ −1 )w1 )cl .
(1.18)
We showed in [Ko1] that an invariant bilinear form on Vcl is automatically symmetric. Namely, for u 1 , u 2 ∈ Vcl , we have (u 1 , u 2 )cl = (u 2 , u 1 )cl .
(1.19)
Vcl has a countable basis. We choose it to be {ei }i∈N . Assume that (·, ·)cl is also nondegenerate, we also have the dual basis {ei }i∈N . Then we define a linear map cl : C → Vcl ⊗ Vcl as follows: cl : 1 →
ei ⊗ ei .
(1.20)
i∈N
The correlation functions maps m (n) cl , n ∈ N of Vcl are canonically determined by Y and the identity 1cl := ιcl (1 L ⊗ 1 R ) [Ko2]. For Q ∈ TK (n − , n + ) given in (1.3), we define, for λ ∈ C, νcl (λψK (πK (Q))) (u 1 ⊗ · · · ⊗ u n + ) in the following three cases:
(1.21)
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L. Kong
1. If z k = ∞ for k = −n − , . . . , −1, 1, . . . , n + , (1.21) is given by −L R (0) L (−1) R L (−1) (n +n ) (−1) (−1) λ ei1 , 1cl , m cl − + (e−L + (A )−L + (A ) (a0 )−L (0) a0 i 1 ,...,i n − ∈N
. . . , e−L + (A L
(−n − ) )−L R (A(−n − ) ) +
e−L + (A L
. . . , e−L + (A L
(−n − ) −L L (0) (−n − ) ) a0
(a0
(1) )−L R (A(1) ) +
(n + ) )−L R (A(n + ) ) +
(1)
(a0 )−L
L (0)
(1)
a0
(n + ) −L L (0) (n + ) ) a0
(a0
z −1 , z¯ −1 , . . . , z −n − , z¯ −n − , z 1 , z¯ 1 , . . . , z n + , z¯ n + )
−L R (0)
−L R (0)
−L R (0)
ein− ,
u1,
u n+ ;
e ⊗ · · · ⊗ ei n − , (1.22) ∞ R j=1 A j L + ( j) for i1
cl
∞ where L +L (A) = A L L ( j) and L +R (A) = ∞ j=1 j + A = (A1 , A2 , . . . ) ∈ j=1 C; 2. If ∃ k ∈ {−n − , . . . , −1} such that z k = ∞ (recall (1.5)), (1.21) is given by the formula obtained from (1.22) by exchanging 1cl in (1.22) with e−L + (A L
(k) )−L R (A(k) ) +
(k)
(a0 )−L
L (0)
(k)
a0
−L R (0)
ei−k ;
(1.23)
3. If ∃ k ∈ {1, . . . , n + } such that z k = ∞ (recall (1.5)), (1.21) is given by the formula obtained from (1.22) by exchanging 1cl in (1.22) with e−L + (A L
(k) )−L R (A(k) ) +
(k)
(a0 )−L
L (0)
(k)
a0
−L R (0)
uk .
(1.24)
The following result is proved in [Ko1]. Proposition 1.10. The map νcl is S L(2, C)-invariant. ˜ cL ⊗ K ˜ c R → EC× , which is still denoted as νcl . Some Hence νcl induces a map K Vcl interesting special cases are listed below: ˆ = νcl (ψK (C)) νcl (ψK (πK (( (∞, 1, 0)| )K )) = νcl (ψK (πK (( |(∞, 1, 0), (0, 1, 0) )K )))(u ⊗ v) = νcl (ψK (πK (( (∞, 1, 0), (0, 1, 0)| )K ))) = νcl (ψK (πK (( (∞; 1, 0)|(z; 1, 0), (0; 1, 0) )K )))(u ⊗ v) =
(1cl , 1cl )cl idC , 1cl , (u, v)cl , cl , Y(u; z, z¯ )v,
ˆ is the single element in K(0, 0). where C ˜ cL ⊗ K ˜ c R -algebra (U, ν) is called smooth if Definition 1.11. A K 1. U = ⊕m,n∈R U(m,n) is a completely reducible C× -module, where z · u = z m z¯ n u, ∀z ∈ C× , u ∈ U(m,n) . 2. dim U(m,n) < ∞, ∀m, n ∈ R and dim U(m,n) = 0 for m or n sufficiently small. 3. ν is linear on fiber and smooth on the base space K. Theorem 2.7 in [Ko1] can be restated as the following theorem. ˜ cL ⊗ K ˜ c R -algebra. Theorem 1.12. (Vcl , νcl ) is a smooth K
Cardy Condition for Open-Closed Field Algebras
37
1.3. Open-string vertex operator algebras. Let (Vop , Yop , 1op , ωop ) be an open-string vertex operator algebra. For r > 0 and v1 , v2 ∈ Vop , we define Yop (v1 , −r )v2 by Yop (v1 , −r )v2 := e−r L(−1) Yop (v2 , r )v1 .
(1.25)
Remark 1.13. Taking the analogy between the open-string vertex operator algebra and associative algebra, Yop (·, −r )· corresponds to the opposite product [HKo1]. An invariant bilinear form on an open-string vertex operator algebra Vop is a bilinear form (·, ·)op on Vop satisfying the following properties: (v3 , Yop (v1 , r )v2 )op = (Yop (e−r L(1) r −2L(0) v1 , −r −1 )v3 , v2 )op , (Yop (v1 , r )v3 , v2 )op = (v3 , Yop (e
−r L(1) −2L(0)
r
v1 , −r
−1
)v2 )op
(1.26) (1.27)
for r > 0 and v1 , v2 , v3 ∈ Vop . Lemma 1.14. (v1 , v2 )op = (v2 , v1 )op
(1.28)
Proof. The proof is exactly the same as that of Proposition 2.3 in [Ko1].
We further assume that (·, ·)op is nondegenerate. Let { f i }i∈R be a basis of Vop and { f i }i∈R its dual basis. We define linear map op : C → Vcl ⊗ Vcl as follows: fi ⊗ f i . (1.29) op : 1 → i∈R
The open-string vertex operator algebra (Vop , Yop , 1op , ωop ) naturally gives a bound(n) ary field algebra (Vop , m op , dop , Dop ) in which the correlation-function maps m op , n ∈ N are completely determined by Yop and 1op [Ko2]. For any Q ∈ TD (n − , n + ) given in (1.10). Let α be a bijective map α
{−n − , . . . , −1, 1, . . . , n + } − → {1, . . . , n − + n + }
(1.30)
so that s1 , . . . , sn − +n + , defined by si := rα −1 (i) , satisfy ∞ ≥ s1 > · · · > sn − +n + ≥ 0. Then we define, for λ ∈ C, νop (λψD (πD (Q)))(v1 ⊗ · · · ⊗ vn + )
(1.31)
as follows: 1. If rk = ∞, ∀k = −n − , . . . , −1, 1, . . . , n + , (1.31) is given by (n +n ) 1op , m op− + (w1 , . . . , wn − +n + ; s1 , . . . , sn − +n + ) λ
op
i 1 ,...,i n − ∈R
f i1 ⊗ · · · ⊗ f in− , (1.32)
(B ( p) )
( p)
(B (q) )
(q)
(b0 )−L(0) f i− p and wα(q) = e−L + (b0 )−L(0) vq for where wα( p) = e−L + p = −1, · · · − n − and q = 1, . . . , n + ; 2. If ∃ k ∈ {−n − , . . . , −1, 1, . . . , n + } such that rk = ∞, (1.31) is given by the formula obtained from (1.32) by exchanging the 1op with wα(k) .
38
L. Kong
Proposition 1.15. νop is S L(2, R) invariant. Proof. The proof is the same as that of Proposition 1.10.
˜ c → ER+ , which is still denoted as νop . Some interesting Hence νop induces a map D Vop special cases are listed explicitly below: ˆ = (1op , 1op )op idC , νop (ψD (H)) νop (ψD (πD (( (∞, 1, 0)| )))) = 1op , νop (ψD (πD (( |(∞, 1, 0), (0, 1, 0) ))))(u ⊗ v) = (u, v)op , νop (ψD (πD (( (∞, 1, 0), (0, 1, 0)| )))) = op , νop (ψD (πD (( (∞; 1, 0)|(r ; 1, 0), (0; 1, 0) ))))(u ⊗ v) = Yop (u, r )v, ˆ is the single element in D(0, 0) and r > 0. where H ˜ c -algebra (U, ν) is called smooth if Definition 1.16. A D 1. U = ⊕n∈R U(n) is a completely reducible R+ -module, where r · u = r n u, ∀r ∈ R+ , u ∈ U(m,n) . 2. dim U(n) < ∞, ∀n ∈ R and dim U(n) = 0 for n sufficiently small. 3. ν is linear on fiber and smooth on the base space D. ˜ c -algebra. Theorem 1.17. (Vop , νop ) is a smooth D Proof. The proof is same as that of Theorem 1.12.
2. Swiss-Cheese Partial Dioperad In Sect. 3.1, we introduce the notion of a 2-colored partial dioperad and algebra over it. In Sect. 3.2, we study a special example of a 2-colored partial dioperad called a Swiss-cheese partial dioperad S and its C-extension S˜ c . In Sect. 3.3, we show that an open-closed field algebra over V equipped with nondegenerate invariant bilinear forms canonically gives an algebra over S˜ c . In Sect. 3.4, we define boundary states in such algebra and show that some of the boundary states are Ishibashi states.
2.1. 2-colored (partial) dioperads. Definition 2.1. A right module over a dioperad (Q, γQ , IQ ), or a right Q-module, is a family of sets {P(m, n)}m,n∈N with an Sm × Sn -action on each set P(m, n) and substitution maps: γ(i 1 ,...,i n )
P(m, n) × Q(k1 , l1 ) × · · · × Q(kn , ln ) −−−−−→ P(m + k1 · · · + kn − n, l1 + · · · + ln ), (P, Q 1 , . . . , Q n ) → γ(i1 ,...,in ) (P; Q 1 , . . . , Q n ) (2.1) for m, n, l1 , . . . , ln ∈ N, k1 , . . . , kn ∈ Z+ and 1 ≤ i j ≤ k j , j = 1, . . . , n, satisfying the right unit property, the associativity and the permutation axioms of dioperad but with the right action of P on itself in the definition of dioperad replaced by that of Q.
Cardy Condition for Open-Closed Field Algebras
39
Homomorphism and isomorphism between two right Q-modules can be naturally defined. The right module over a partial dioperad can also be defined in the usual way. Definition 2.2. Let Q be a G-rescalable partial dioperad. A right Q-module is called (i) (i) (i) G-rescalable if for any P ∈ P(m − , m + ), Q i ∈ Q(n − , n + ) and 1 ≤ ji ≤ n − , i = 1, . . . , m + , there exist g j ∈ G, j = 1, . . . , m + such that γ( j1 ,..., jm + ) P; γ( j1 ) (g1 ; Q 1 ), . . . , γ( jm + ) (gm + ; Q m + ) (2.2) is well-defined. Definition 2.3. A 2-colored dioperad consists of a dioperad (Q, γQ , IQ ) and a family B , n B |n I , n I ) equipped with a S × S × S × S -action for n B , n I ∈ N, of sets P(n − B I + − + ± ± n− n +B n− n +I a distinguished element IP ∈ P(1, 1|0, 0), and maps (1)
(1) (1)
(1)
(m + )
P(m − , m + |n − , n + ) × P(k− , k+ |l− , l+ ) × · · · × P(k− γ(iB ,...,i m
(m + ) (m + ) (m + ) |l− , l+ )
, k+
+)
−−−−−−→ P(m − − m + + k− , k+ |n − + l− , n + + l+ ), 1
(1)
(1)
(n )
(n)
P(m − , m + |n − , n + ) × Q( p− , p+ ) × · · · × Q( p− + , p+ ) γ(Ij
1 ,..., jn + )
−−−−−−→ P(m − , m + |n − − n + + p− , p+ ), (1)
(m )
(1)
(m )
(2.3) (1)
(n )
where k± = k± + · · · + k± + , l± = l± + · · · + l± + and p± = p± + · · · + p± + , for (r ) (s) 1 ≤ ir ≤ k− , r = 1, . . . , m + , 1 ≤ js ≤ p− , s = 1, . . . , n + , satisfying the following axioms: 1. The family of sets P := {P(m − , m + )}m − ,m + ∈N , where P(m − , m + ) := ∪n − ,n + ∈N P(m − , m + |n − , n + ), equipped with the natural Sm − × Sm + -action on P(m − , m + ), together with identity element IP and the family of maps γ B := {γ(iB1 ,...,im ) } gives an dioperad. +
2. The family of maps γ I := {γ(Ij1 ,..., jn ) } gives each P(m − , m + ) a right Q-module + structure for m − , m + ∈ N.
We denote such a 2-colored dioperad as (P|Q, (γ B , γ I )). The 2-colored partial dioperad can be naturally defined. If the associativities of γQ , γ B , γ I do not hold, then it is called a 2-colored (partial) nonassociative dioperad. Remark 2.4. If we restrict to {P(1, m|0, n)}m,n∈N and {Q(1, n)}n∈N , they simply gives a structure of 2-colored (partial) operad [V,Kont,Ko2]. Now we discuss an important example of the 2-colored partial nonassociative dioperad. Let J1 , J2 be two index sets, Ui = ⊕n∈Ji (Ui )(n) , i = 1, 2 be two graded vector spaces. Consider two families of sets, EU2 (m, n) = HomC (U2⊗m , U2⊗n ), ⊗m −
EU1 |U2 (m − , m + |n − , n + ) = HomC (U1⊗m + ⊗ U2⊗n + , U1
⊗n −
⊗ U2
),
40
L. Kong
for m, n, m ± , n ± ∈ N. We denote both of the projection operators U1 → (U1 )(n) , (i) U2 → (U2 )(n) as Pn for n ∈ J1 or J2 . For f ∈ EU1 |U2 (k− , k+ |l− , l+ ), gi ∈ EU1 |U2 (m − , ( j) ( j) (i) (i) (i) m + |n − , n + ), i = 1, . . . , k+ , and h j ∈ EU2 ( p− , p+ ), j = 1, . . . , l+ , we say that (1)
(k )
(iB1 ,...,ik ) ( f ; g1 , . . . , gk+ )(u 1 ⊗ · · · ⊗ v (k++ ) ⊗ v1 ⊗ · · · ⊗ vl+ ) + n+
(1) (1) (1) (1) := f Ps1 g1 (u 1 ⊗ · · · ⊗ u (1) ⊗ v1 ⊗ · · · ⊗ v (1) )⊗ m+
s1 ,...,sk ∈J1
··· ⊗
(k ) Psk+ gk+ (u 1 +
(k ) ⊗ · · · ⊗ u (k+ + ) m+
n+
(k ) ⊗ v1 +
(k ) ⊗ · · · ⊗ v (k++ ) ) ⊗ v1 n+
⊗ · · · ⊗ vl+
(I j1 ,..., jl ) ( f ; h 1 , . . . , h l+ )(u 1 ⊗ · · · ⊗ u k+ ⊗ w1(1) ⊗ · · · ⊗ w (l(l++) ) ) + p+
(1) (1) := f u 1 ⊗ · · · ⊗ u k+ ⊗ Pt1 h 1 (w1 ⊗ · · · ⊗ w (1) ) p+
t1 ,...,tl ∈J2
⊗··· ⊗
Ptl+ h l+ (w1(l+ )
⊗ · · · ⊗ w (l(l++) ) ) p+
(i) (i) for u (i) j ∈ U1 , v j , w j ∈ U2 , are well-defined if each multiple sum is absolutely convergent. These give rise to partially defined substitution maps: (1)
(1)
(1)
(1)
(k )
(k )
(k )
(k )
EU1 |U2 (k− , k+ |l− , l+ )⊗ EU1 |U2 (m − , m + |n − , n + )⊗. . .⊗ EU1 |U2 (m − + , m + + |n − + , n + + ) (iB
1 ,...,i k+ )
−−−−−−→ EU1 |U2 (k− − k+ + m − , m + |l− + n − , n + ), (1)
(1)
(l )
(l )
EU1 |U2 (k− , k+ |l− , l+ ) ⊗ EU2 ( p− , p+ ) ⊗ · · · ⊗ EU2 ( p−+ , p+ − ) (I j
1 +···+ jl+ )
(1)
(l )
(1)
(l )
−−−−−−→ EU1 |U2 (k− , k+ |l− − l+ + p− + · · · + p−+ , p+ + · · · + p+ + ), (1)
(k )
(1)
(k )
where m ± = m ± + · · · + m ± + and n ± = n ± + · · · + n ± + . In general, (iB1 ,...,ik (I j1 +···+ jl
+
+)
and
) do not satisfy the associativity. Let
EU1 |U2 = {EU1 |U2 (m − , m + |n − , n + )}m ± ,n ± ∈N , EU2 = {EU2 (n)}n∈N , B := { (iB1 ,...,in ) } and I := { (I j1 ,..., jn ) }. It is obvious that (EU1 |U2 |EU2 , ( B , I )) is a 2-colored partial nonassociative operad. Let U1 be a completely reducible G 1 -module and U2 a, completely reducible G 2 -module. Namely, U1 = ⊕n 1 ∈J1 (U1 )(n 1 ) , U2 = ⊕n 2 ∈J2 (U2 )(n 2 ) where Ji is the set of equivalent classes of irreducible G i -modules and (Ui )(n i ) is a direct sum of irreducible G |G G i -modules of equivalent class n i for i = 1, 2. In this case, we denote EU1 |U2 by EU11|U22 . Definition 2.5. A homomorphism between two 2-colored (partial, nonassociative) dioperads (Pi |Qi , (γiB , γiI )), i = 1, 2 consists of two (partial, nonassociative) dioperad homomorphisms: νP1 |Q1 : P1 → P2 ,
and
νQ1 : Q1 → Q2
such that νP1 |Q1 : P1 → P2 , where P2 has a right Q1 -module structure induced by a dioperad homomorphism νQ1 , is also a right Q1 -module homomorphism.
Cardy Condition for Open-Closed Field Algebras
41
Definition 2.6. An algebra over a 2-colored partial dioperad (P|Q, (γ B , γ I )), or a P|Q-algebra consists of two graded vector spaces U1 , U2 and a 2-colored partial nonassociative dioperad homomorphism (νP |Q , νQ ) : (P|Q, (γ B , γ I )) → (EU1 |U2 |EU2 , ( B , I )). We denote this algebra as (U1 |U2 , νP |Q , νQ ). Definition 2.7. If a 2-colored partial dioperad (P|Q, γ ) is so that P is a G 1 -rescalable partial operad and a G 2 -rescalable right Q-module, then it is called G 1 |G 2 -rescalable. Definition 2.8. A G 1 |G 2 -rescalable P|Q-algebra (U1 |U2 , νP |Q , νQ ) is a P|Q-algebra G |G G2 so that νP |Q : P → EU11|U22 and νQ : Q → EU are partial nonassociative dioper2 ad homomorphisms such that νP |Q : G 1 → End U1 coincides with the G 1 -module structure on U1 and νQ : G 2 → End U2 coincides with the G 2 -module structure on U2 . 2.2. Swiss-cheese partial dioperads. A disk with strips and tubes of type (m − , m + ; n − , n + ) [Ko2] is a disk consisting of m + (m − ) ordered positively (negatively) oriented punctures on the boundary of the disk, and n + (n − ) ordered positively (negatively) oriented punctures in the interior of the disk, and local coordinate map around each puncture. Two disks are conformal equivalent if there exists a biholomorphic map between them preserving order, orientation and local coordinates. We denote the moduli space of disks with strips and tubes of type (m − , m + ; n − , n + ) as S(m − , m + |n − , n + ). We use the following notation: −m ( (r−m − , b0 − , B (−m − ) ), . . . , (r−1 , b0−1 , B (−1) ) |
(r1 , b1 , B (1) ), . . . , (rm , bm + , B (m + ) ) ) 0
+
−n ( (z −n − , a0 − ,
A
0 (−n − )
), . . . , (z −1 , a0−1 , A(−1) ) |
(z 1 , a01 , A(1) ), . . . , (z n + , a0n + , A(n + ) ) )
S
, (2.4)
ˆ b(i) ∈ R× , B (i) ∈ R and z j ∈ H, a ( j) ∈ C× , A( j) ∈ C for all where ri ∈ R, 0 0 k l i = −m − , . . . , −1, 1, . . . , m + , j = −n − , . . . , −1, 1, . . . , n + and k, l ∈ Z+ , to represent a disk with strips at ri with local coordinate map f i and tubes at z j with local coordinate map g j given as follows: f i (w) = e =e g j (w) = e
k
(i)
Bk x k+1 ddx
(i) k+1 d k Bk x dx
k
( j)
Ak x k+1 ddx
(i)
d
(i)
d dx
(b0 )x d x x|x=w−ri (b0 )x ( j)
x|x= −1 w
if ri ∈ R ,
(2.5)
if ri = ∞,
(2.6)
d
(a0 )x d x x|x=w−z j .
(2.7)
We denote the set of all such disks given in (2.4) as TS (m − , m + |n − , n + ). The automorphisms of the upper half plane, which is S L(2, R), change the disk (2.4) to a different but conformal equivalent disk. It is clear that we have S(m − , m + |n − , n + ) = TS (m − , m + |n − , n + )/S L(2, R). Let S = {S(m − , m + |n − , n + )}m − ,m + ,n − ,n + ∈N . The permutation groups (Sm − × Sm + ) × (Sn − × Sn + )
(2.8)
42
L. Kong
act naturally on S(m − , m + |n − , n + ). There are so-called boundary sewing operations [HKo1,Ko2] on S, denoted as i ∞−B j , which sews the i th positively oriented boundary puncture of the first disk with the j th negatively oriented boundary puncture of the second disk. Boundary sewing operations naturally induce the following maps: (1)
(1) (1)
(1)
(m + )
S(m − , m + |n − , n + ) × S(k− , k+ |l− , l+ ) × · · · × S(k− γ(iB ,...,i m
(m + ) (m + ) (m + ) |l− , l+ )
, k+
+)
−−−−−−→ S(m − − m + + k− , k+ |n − + l− , n + + l+ ), 1
(1)
(m )
(1)
(m )
(r )
where k± = k± + · · · + k± + , l± = l± + · · · + l± + for 1 ≤ ir ≤ k− , r = 1, . . . , m + . It is easy to see that boundary sewing operations or γ(iB1 ,...,im ) , together with permutation + group actions on the order of boundary punctures, provide S with the structure of a partial dioperad. There are also so-called interior sewing operations [HKo1,Ko2] on S, denoted as I th positively oriented interior puncture of a disk with the j th negai ∞− j , which sew the i tively oriented puncture of a sphere. The interior sewing operations define a right action of K on S: (1) (n + ) S(m − , m + |n − , n + ) × K( p− , p+(1) ) × · · · × K( p− , p+(n + ) ) γ(Ij
1 ,..., jn + )
−−−−−−→ S(m − , m + |n − − n + + p− , p+ ),
(2.9)
(1) (n + ) (s) + · · · + p± , for 1 ≤ js ≤ p− , s = 1, . . . , n + . Such action gives S a where p± = p± right K-module structure. Let γ B = {γ(iB1 ,...,in ) } and γ I = {γ(iI 1 ,...,in ) }. The following proposition is clear.
Proposition 2.9. (S|K, (γ B , γ I )) dioperad.
is
a
R+ |C× -rescalable
2-colored
partial
We call (S|K, (γ B , γ I )) Swiss-cheese partial dioperad. When it is restricted on S = {S(1, m|0, n)}m,n∈N , it is nothing but the so-called Swiss-cheese partial operad [HKo1,Ko2]. In [HKo1,Ko2], we show that the Swiss-cheese partial operad S can be naturally embedded into the sphere partial operad K via the so-called doubling map, denoted as δ : S → K . Such doubling map δ obviously can be extended to a doubling map S → K, still denoted as δ. In particular, the general element (2.4) maps under δ to (−1)
( (z −1 , a0
(−n − )
, A(−1) ), . . . , (z −n − , a0 (−1)
(¯z −1 , a0
(−n − )
, A(−1) ), . . . (¯z −n − , a0 (−1)
(r−1 , b0 (1)
, A(−n − ) ),
(n + )
(z 1 , a0 , A(1) ), . . . , (z n + , a0
, A(−n − ) ), (−m − )
, B (−1) ), . . . , (r−m − , b0
, B (−m − ) ) |,
, A(n + ) ),
(1)
(n + )
(¯z 1 , a0 , A(1) ), . . . , (¯z n + , a0 (1) (r1 , b0 ,
The following proposition is clear.
, A(n + ) ), (m + )
B (1) ), . . . , (rm + , b0
, B (m + ) ) )K .
(2.10)
Cardy Condition for Open-Closed Field Algebras (i)
43
(i)
(i)
(i)
Proposition 2.10. Let Pi ∈ S(m − , m + |n − , n + ), i = 1, 2 and Q ∈ K(m − , m + ). If I Q exists for 1 ≤ i ≤ m (1) , 1 ≤ j ≤ m (2) and 1 ≤ k ≤ n (1) , P1 i ∞−B j P2 and P1 k ∞−l + + − 1 ≤ l ≤ m − , then we have δ(P1 i ∞−B j P2 ) = δ(P1 )
(1) 2n + +i
∞−(2n(2) + j) δ(P2 ), −
I δ(P1 k ∞−l Q) = (δ(P1 ) k ∞−l Q )
(1) n + +m + −1+k
¯ ∞−l Q.
(2.11)
By the above Proposition, we can identify S as its image under δ in K with boundary sewing operations replaced by ordinary sewing operations in K and interior sewing operations replaced by double-sewing operations in K as given in (2.11). The C-extension S˜ c (m − , m + |n − , n + ) of S(m − , m + |n − , n + ) is defined to be the pull˜ c (2n − + m − , 2n + + m + ). We denote the canonical section on S˜ c , which back bundle of K ˜ c , as ψS . The boundary (interior) sewing operations can be natis induced from that on K c B (∞ I ) and the corresponding substitution urally extended to S˜ . We denote them as ∞ ˜c⊗K ˜ c¯ on S˜ c defined by γ˜ I . maps as γ˜ B (γ˜ I ). There is a natural right action of K The following proposition is also clear. ˜ c ⊗K ˜ c¯ , (γ˜ B , γ˜ I )) is a R+ |C× -rescalable 2-colored partial dioProposition 2.11. (S˜ c |K perad. ˜c ⊗K ˜ c¯ , (γ˜ B , γ˜ I )) a Swiss-cheese partial dioperad We will call the structure (S˜ c |K with central charge c. Definition 2.12. An algebra over S˜ c viewed as a partial dioperad, (U, ν), is called smooth if 1. U = ⊕n∈R U(n) is a completely reducible R+ -module, where r · u = r n u, ∀r ∈ R+ , u ∈ U(m,n) . 2. dim U(n) < ∞, ∀n ∈ R and dim U(n) = 0 for n sufficiently small. 3. ν is linear on fiber and smooth on the base space S. ˜c ⊗K ˜ c¯ -algebra (U1 |U2 , ν Definition 2.13. An S˜ c |K ˜ c ⊗K ˜ c¯ , νK ˜ c ⊗K ˜ c¯ ) is smooth if both S˜ c |K (U1 , ν ˜ c ˜ c ˜ c¯ ) and (U2 , ν ˜ c ˜ c¯ ) as algebras over partial dioperads are smooth. S |K ⊗K
K ⊗K
2.3. Open-closed field algebras over V . Let (Vop , Yop , ιop ) be an open-string vertex operator algebra over V and (Vcl , Y, ιcl ) a conformal full field algebra over V ⊗ V . Let ( (Vcl , Y, ιcl ), (Vop , Yop , ιop ), Ycl−op )
(2.12)
be an open-closed field algebra over V [Ko2]. We denote the formal vertex operators assof ciated with Yop and Y as Yop and Y f respectively. Let ω L = ιcl (ω ⊗ 1), ω R = ιcl (1 ⊗ ω) and ωop = ιop (ω). We have Y f (ω L ; x, x) ¯ = Y f (ω L , x) = L(n) ⊗ 1x −n−2 , n∈Z
¯ = Y (ω , x) ¯ = Y (ω ; x, x) f
R
f Yop (ωop , x)
f
=
n∈Z
R
1 ⊗ L(n)x¯ −n−2 ,
n∈Z
L(n)x
−n−2
.
(2.13)
44
L. Kong
We also set L L (n) = L(n) ⊗ 1 and L R (n) = 1 ⊗ L(n) for n ∈ Z. For u ∈ V , we showed in [Ko2] that Ycl−op (ιcl (u ⊗ 1); z, z¯ ) and Ycl−op (ιcl (1 ⊗ u); z, z¯ ) are holomorphic and antiholomorphic respectively. So we also denote them simply by Ycl−op (ιcl (u ⊗ 1), z) and Ycl−op (ιcl (1 ⊗ u), z¯ ) respectively. By the V -invariant boundary condition [Ko2], we have Ycl−op (ω L , r ) = Yop (ωop , r ) = Ycl−op (ω R , r ).
(2.14)
We assume that both Vop and Vcl are equipped with nondegenerate invariant bilinear forms (·, ·)op and (·, ·)cl respectively. Lemma 2.14. For any u ∈ Vcl and v1 , v2 ∈ Vop and z ∈ H, we have (v2 , Ycl−op (u; z, z¯ )v1 )op = (Ycl−op (e−z L(1) z −2L(0) ⊗ e−¯z L(1) z¯ −2L(0) u; −z −1 , −¯z −1 )v2 , v1 )op .
(2.15)
Proof. Using (1.26), for fixed z ∈ H, we have, (v2 , Ycl−op (u; z, z¯ )v1 )op = (v2 , Ycl−op (u; z, z¯ )Yop (1, r )v1 )op = (v2 , Yop (Ycl−op (u; z − r, z¯ − r )1, r )v1 )op = (e−r
−1 L(−1)
Yop (v2 , r −1 )e−r L(1) r −2L(0) Ycl−op (u; z − r, z¯ − r )1, v1 )op (2.16)
for |z| > r > |z − r | > 0. Notice that e−r v2 = 1op , it is easy to see that
−1 L(−1)
∈ Aut(V op ), ∀r ∈ C. By taking
e−r L(1) r −2L(0) Ycl−op (u; z − r, z¯ − r )1
(2.17)
is a well-defined element in V op for |z| > r > |z − r | > 0. Because of the chirality splitting property of Ycl−op (see (1.72), (1.73) in [Ko2]), it is easy to show that (2.17) equals Ycl−op (e−z L(1) z −2L(0) ⊗ e−¯z L(1) z¯ −2L(0) u, r −1 − z −1 , r −1 − z¯ −1 )1
(2.18)
for r > |z − r | > 0. By the commutativity I of the analytic open-closed field algebra proved in [Ko2], we know that for fixed z ∈ H, e−r
−1 L(−1)
Yop (v2 , r −1 )Ycl−op (e−z L(1) z −2L(0)
⊗e−¯z L(1) z¯ −2L(0) u, r −1 − z −1 , r −1 − z¯ −1 )1 and e−r
−1 L(−1)
Ycl−op (e−z L(1) z −2L(0)
⊗e−¯z L(1) z¯ −2L(0) u, r −1 − z −1 , r −1 − z¯ −1 )Yop (v2 , r −1 )1 = e−r
−1 L(−1)
Ycl−op (e−z L(1) z −2L(0)
⊗e−¯z L(1) z¯ −2L(0) u, r −1 − z −1 , r −1 − z¯ −1 )er
−1 L(−1)
v2
(2.19)
Cardy Condition for Open-Closed Field Algebras
45
converge in different domains for r , but are an analytic continuation of each other along a path in r ∈ R+ . Moreover, using the L(−1) property of the intertwining operator and chirality splitting property of Ycl−op again, the right hand side of (2.19) equals Ycl−op (e−z L(1) z −2L(0) ⊗ e−¯z L(1) z¯ −2L(0) u, −z −1 , −¯z −1 )v2 for |r −1 − z −1 | > |r −1 |. Therefore, both sides of (2.15) as constant functions of r are analytic continuation of each other. Hence (2.15) must hold identically for all z ∈ H. For u 1 , . . . , u l ∈ Vcl , v1 , . . . , vn ∈ Vcl , r1 , . . . , rn ∈ R, r1 > · · · > rn and z 1 , . . . , zl ∈ H, we define (l;n)
m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) (l;n)
:= e−rn L(−1) m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 − rn , z¯ 1 − rn , . . . , zl − rn , z¯l − rn ; r1 − rn , . . . , rn−1 − rn , 0).
(2.20)
(l;n)
We simply extend the definition of m cl−op to a domain where some of ri can be negative. Note that such a definition is compatible with L(−1)-properties of m cl−op . Lemma 2.15. For u 1 , . . . , u l ∈ Vcl , v, v1 , . . . , vn ∈ Vcl , r1 , . . . , rn ∈ R, r1 > · · · > rn = 0 and z 1 , . . . , zl ∈ H, (l;n)
(v, m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ))op (l;n)
= (vn , m cl−op (F1 u 1 , . . . , Fl u l ; v, G 1 v1 , . . . , G n−1 vn−1 ; −1 ))op , −z 1−1 , −¯z 1−1 , . . . , −zl−1 , −¯zl−1 ; 0, −r1−1 , . . . , −rn−1
(2.21)
where −2L(0)
Fi = e−zi L(1) z i Gj =
−2L(0)
⊗ e−¯zi L(1) z¯ i
−2L(0) e−r j L(1)r j ,
, i = 1, . . . , l,
j = 1, . . . , n − 1.
(2.22)
Proof. By Lemma 2.14, (2.21) is clearly true for l = 0, 1; n = 0, 1. By (1.26) and (1.27), (2.21) is true for l = 0, n = 2. We then prove the lemma by induction. Assume that (2.21) is true for l = k ≥ 0, n = m ≥ 2 or l = k ≥ 1, n = m ≥ 1. Let l = k and n = m + 1. It is harmless to assume that 0 < rn−1 , |z i | < rn−2 , i = l1 + 1, . . . , l for some l1 ≤ l. Using the induction hypothesis, we obtain (l;n)
(v, m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ))op (l ;n−1)
(l−l ;2)
1 1 = (v, m cl−op (u 1 , . . . , u l1 ; v1 , . . . , vn−2 , m cl−op (u l1 +1 , . . . , u l ; vn−1 , vn ;
zl1 +1 , z¯l1 +1 , . . . , zl , z¯l ; rn−1 , 0); z 1 , z¯ 1 , . . . , zl1 , z¯l1 ; r1 , . . . , rn−2 , 0))op (l ;n−1)
1 = (m cl−op (F1 u 1 , . . . , Fl1 u l1 ; v, G 1 v1 , . . . , G n−1 vn−1 ;
−1 , −¯zl−1 ; 0, −r1−1 , . . . , −rn−1 ), −z 1−1 , −¯z 1−1 , . . . , −zl−1 1 1 (l−l ;2)
1 m cl−op (u l1 +1 , . . . , u l ; vn−1 , vn ; zl1 +1 , z¯l1 +1 , . . . , zl , z¯l ; rn−1 , 0))op
(l−l ;2)
(l ;n−1)
1 1 = (m cl−op (Fl1 +1 u l1 +1 , . . . , Fl u l ; m cl−op (F1 u 1 , . . . , Fl1 u l1 ;
46
L. Kong −1 v, G 1 v1 , . . . , G n−1 vn−1 ; −z 1−1 , −¯z 1−1 , . . . , −zl−1 , −¯zl−1 ; 0, −r1−1 , . . . , −rn−2 ), 1 1 −1 G n−1 vn−1 ; −zl−1 , −¯zl−1 , . . . , −zl−1 , −¯zl−1 ; 0, −rn−1 ), vn )op 1 +1 1 +1 −1 −1 −rn−1 L(−1) (l−l1 ;2) −rn−2 L(−1) (l−l1 ;n−1) = (e m cl−op (Fl1 +1 u l1 +1 , . . . , Fl u l ; Ps e m cl−op (F1 u 1 , s∈R −1 −1 . . . , Fl1 u l1 ; v, G 1 v1 , . . . , G n−1 vn−1 ; −z 1−1 + rn−2 , −¯z 1−1 + rn−2 , −1 −1 −1 −1 −1 −1 . . . , −zl−1 + rn−2 , −¯zl−1 + rn−2 ; rn−2 , −r1−1 + rn−2 , . . . , −rn−3 + rn−2 , 0), 1 1 −1 −1 G n−1 vn−1 ; −zl−1 + rn−1 , −¯zl−1 + rn−1 , 1 +1 1 +1 −1 −1 −1 . . . , −zl−1 + rn−1 , −¯zl−1 + rn−1 ; rn−1 , 0), vn )op . (2.23) −1
Note that the position of Ps and e−rn−2 L(−1) can not be exchanged in general. Because if we exchange their position, the sum may not converge and then the associativity law does not hold. We want to use analytic continuation to move it to a domain such that we can freely apply the associativity law. By our assumption on V , both sides of (2.23) are restrictions of the analytic function of the zl1 +1 , ζl1 +1 , . . . , zl , ζl , rn−1 on ζl1 +1 = z¯l1 +1 , . . . , ζl = z¯l . Let z˜l1 +1 , . . . , z˜l , r˜n−1 satisfy the following conditions: −1 −1 −1 −1 −1 −1 −1 −1 | − z˜ −1 p + rn−2 |, r˜n−1 − rn−2 > | − z i + rn−2 |, rn−2 , −r j + rn−2 ,
(2.24)
for all i = 1, . . . , l, j = 1, . . . , n − 3 and p = l1 + 1, . . . , l. Note that such a condition define a nonempty open subset on Hl × R+ . Choose a path γ1 in the complement of the diagonal in Hl from initial point (zl1 +1 , . . . , zl ) to (˜zl1 +1 , . . . , z˜l ) and a path γ2 in R+ from rn−1 to r˜n−1 . We also denote the complex conjugate of path γ1 as γ¯1 , which is a path l in H . Combine γ1 , γ¯1 with γ2 , we obtain a path γ from (zl1 +1 , z¯l1 +1 , . . . , zl , z¯l , rn−1 ) to (˜zl1 +1 , z¯˜l1 +1 , . . . , z˜l , z¯˜l , r˜n−1 ). Analytically continuing the right hand side of (2.23) (l;n) along the path γ , we obtain, by the properties of m cl−op [Ko2], −1
(l−l ;2)
(l−l ;n−1)
1 1 (Fl1 +1 u l1 +1 , . . . , Fl u l ; Ps m cl−op (e−˜rn−1 L(−1) m cl−op
(F1 u 1 , . . . , Fl1 u l1 ;
−1 −1 , −¯z 1−1 + rn−2 , v, G 1 v1 , . . . , G n−1 vn−1 ; −z 1−1 + rn−2 −1 −1 −1 −1 −1 −1 . . . , −zl−1 + rn−2 , −¯zl−1 + rn−2 ; rn−2 , −r1−1 + rn−2 , . . . , −rn−3 + rn−2 , 0), 1 1 −1 −1 G˜ n−1 vn−1 ; −˜zl−1 + rn−1 , −z¯˜l−1 + rn−1 , 1 +1 1 +1
−1 −1 −1 −1 . . . , −˜zl−1 + rn−1 , −z¯˜l−1 + rn−1 ; r˜n−1 − rn−2 , 0), vn )op ,
(2.25)
−2L(0)
where G˜ n−1 = e−˜rn−1 L(1)r˜n−1 . Using the associativity of open-closed field algebra and L(−1)-properties of m cl−op , we see that (2.25) further equals (l;n+1)
(m cl−op (F1 u 1 , . . . , Fl u l ; v, G 1 v1 , . . . , G˜ n−1 vn−1 ; −z 1−1 , −¯z 1−1 , . . . , −1 , −¯zl−1 , −˜zl−1 , −z¯˜l−1 , . . . , −zl−1 , −¯zl−1 ; 0, −r1−1 , . . . , −˜rn−1 ), vn )op . −zl−1 1 1 1 +1 1 +1 (2.26)
By analytically continuing (2.26) along the path −γ , which is γ reversed, we obtain the right hand side of (2.21). Hence (2.23) and the right hand side of (2.21) are analytic continuation of each other along path (−γ ) ◦ (γ ) which is a constant path. Hence (2.21) holds for l = k, n = m + 1.
Cardy Condition for Open-Closed Field Algebras
47
Now let l = k + 1, n = m. The proof is similar to the case l = k, n = m + 1. We only point out the difference. Using the smoothness of m cl−op , it is enough to prove the case when |z i | = |z j | for i, j = 1, . . . , l and i = j. Without losing generality, we assume that |z 1 | > · · · > |zl | > 0. Let n 1 ≤ n be the smallest so that 0 < r j < |zl | for j ≥ n 1 . Then we have (v, m (l;n) cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , z l , z¯ l ; r1 , . . . , rn ))op (l−1;n 1 ) 1 +1) = (v, m cl−op (u 1 , . . . , u l−1 ; v1 , . . . , vn 1 −1 , m (1;n−n (u l ; vn 1 , . . . , vn ; cl−op
zl , z¯l ; rn 1 , . . . , rn ); z 1 , z¯ 1 , . . . , zl−1 , z¯l−1 ; r1 , . . . , rn 1 −1 ))op .
(2.27)
We can then apply (2.21) as in (2.23) for the case l ≤ k, n ≤ m, which is true by our induction hypothesis. The rest of the proof is entirely the same as that of the case l = k, n = m + 1. We define a map, for z, ζ ∈ C and z = ζ , ιcl−op (z, ζ ) : Vcl → V op as ιcl−op (z, ζ )(u) = Ycl−op (u; z, ζ )1op . We denote its adjoint as ι∗cl−op (z, ζ ). Namely, ι∗cl−op (z, ζ ) : Vop → Vcl is given by (ι∗cl−op (z, ζ )(w), u)cl = (w, ιcl−op (z, ζ )(u))op
(2.28)
for any u ∈ Vcl and w ∈ Vop . Let Q be an element in TS (n − , n + |m − , m + ) of form (2.4). Let α be the map (1.30) so that s1 , . . . , sn − +n + , defined as si := rα −1 (i) , satisfy ∞ ≥ s1 > · · · > sn − +n + ≥ 0. Then we define νcl−op (λψS (Q))(u 1 ⊗ · · · ⊗ u m + ⊗ v1 ⊗ · · · ⊗ vn + )
(2.29)
as follows: 1. If s1 = ∞, (2.29) is given by (m − +m + ;n − +n + ) 1op , m cl−op λ (u −1 , . . . , u −m − , u 1 , . . . , u m + ; i 1 ,...,i m − ; j1 ,..., jn −
w1 , . . . , wn − +n + ; z −1 , z¯ −1 , . . . , z −m − , z¯ −m − , z 1 , z¯ 1 , . . . , z m + , z¯ m + ; s1 , . . . , sn − +n + ) op · ei1 ⊗ · · · ⊗ ein− ⊗ f j1 ⊗ · · · ⊗ f jm − , (2.30) where u p = e−L + (A L
u q = e−L + (A L
wα(k) = wα(l) =
( p) )−L R (A( p) ) + (q) )−L R (A(q) ) +
( p)
(a0 )−L (q)
(a0 )−L
L (0)
L (0)
( p)
−L R (0)
a0
(q)
a0
−L R (0)
ei− p ,
uq ,
−L + (B (k) )
(k) e (b0 )−L(0) f j−k , (l) (l) e−L + (B ) (b0 )−L(0) vl ,
(2.31)
for p = −1, . . . , −n − , q = 1, . . . , n + , k = −1, . . . , −m − and l = 1, . . . , m + . 2. When rk = ∞ for some k = −m − , . . . , −1, 1, . . . , m + . Equation (2.29) is given by the formula obtained from (2.30) by exchanging 1op with wα(k) .
48
L. Kong
Lemma 2.16. νcl−op is S L(2, R)-invariant. Proof. The S L(2, R) is generated by the following three transformations 1. w → aw, ∀a ∈ R+ ; 2. w → w − b, ∀b ∈ R; 3. w → −1 w . That νcl−op is invariant under the first two transformations simply follows from the L(0)- and L(−1)-properties of m cl−op . That νcl−op is invariant under the third transformation is proved in Lemma 2.15. ×
R + |C Hence νcl−op induces a map S˜ c → EVop |Vcl , which is still denoted as νcl−op . We list a few interesting cases:
νcl−op ψS ( (∞, 1, 0)| ) ( |(z, 1, 0)) S = ιcl−op (z, z¯ ),
νcl−op ψS ( |(∞, 1, 0)) ((z, 1, 0)| ) = ι∗ (z, z¯ ), (2.32) S
and for b ∈ R+ , B ∈ R∞ , a ∈ C× , A ∈ C∞ and v ∈ Vop , we have
νcl−op ψS ( |(∞, b, B)) ((z, a, A)| ) S = a −L
L (0)
a¯ −L
R (0)
e−
∞
j=1 (−1)
j [A
jL
L (− j)+A
jL
R (− j)]
ι∗ (z, z¯ )(e−L + (B) b−L(0) v). (2.33)
˜c ⊗ Theorem 2.17. (Vop |Vcl , νcl−op , νcl ) is an R+ |C× -rescalable smooth S˜ c |K Kc¯ -algebra. Proof. The smoothness is automatic. We showed in [Ko2] that (Vop |Vcl , νcl−op , νcl ) is ˜ c | K˜ c ⊗ K c¯ -algebra. The rest of the proof is similar to an R+ |C× -rescalable smooth S that of Theorem 1.12 in [Ko1]. We omit the detail here. 2.4. Ishibashi states. As we mentioned in the introduction, an open-closed field algebra over V equipped with nondegenerate invariant bilinear forms for both open theory and closed theory contains all the data needed to grow to an open-closed partial field theory of all genus. Without adding more compatibility conditions, itself is already an interesting object to study. We show in this subsection that the famous “Ishibashi states” [I] can be studied in the framework of such an algebra. Throughout this subsection, we fix an open-closed algebra over V given in (2.12) and equipped with nondegenerate invariant bilinear forms (·, ·)cl and (·, ·)op . For u ∈ Vop and z 0 ∈ H, we define the boundary state Bz 0 (u) ∈ Vcl associated with u and z 0 by Bz 0 (u) = e L(−1) (¯z 0 − z 0 ) L(0) ⊗ e L(−1) z¯ 0 − z 0
L(0) ∗ ιcl−op (z 0 , z¯ 0 )(u).
(2.34)
Proposition 2.18. If u ∈ Vop is a vacuum-like vector [LL], i.e. L(−1)u = 0, then, for z 0 ∈ H, Bz 0 (u) is an Ishibashi state, i.e. (L L (n) − L R (−n))Bz 0 (u) = 0,
∀n ∈ Z.
(2.35)
Proof. For v ∈ Vcl , the following two functions of z : (u, Ycl−op (ω L , z + z 0 )Y(v; z 0 , z¯ 0 )1op )op , (u, Ycl−op (ω R ; z + z 0 )Y(v; z 0 , z¯ 0 )1op )op
(2.36)
Cardy Condition for Open-Closed Field Algebras
49
can be extended to a holomorphic function and an antiholomorphic function in {z|z+z 0 ∈ H, z = 0} respectively by our assumption on V . We denote the extended functions by g1 (ω L , z) and g2 (ω R , z¯ ) respectively. The following two limits:
4 z lim 1− g1 (ω L , z), z+z 0 →r z¯ 0 − z 0
4 z 1− g2 (ω R , z¯ ) (2.37) lim z+z 0 →r z¯ 0 − z 0 exist for all r ∈ R. Using (2.15), it is easy to see that the above two limits also exist ˆ if and only if u is vacuum-like. Hence, by V -invariant (or conformal for r = ∞ ∈ R invariant) boundary condition, we have
4
4 z z lim 1− 1− g1 (ω L , z) = lim g2 (ω R , z) (2.38) z+z 0 →r z+z 0 →r z¯ 0 − z 0 z¯ 0 − z 0 ˆ when L(−1)u = 0. for all r ∈ R On the other hand, for |z + z 0 | > |z 0 |, we have (u, Ycl−op (ω L , z + z 0 )Ycl−op (v; z 0 , z¯ 0 )1op )op = (u, Ycl−op (Y(ω L , z)v; z 0 , z¯ 0 )op 1op )op = (u, ιcl−op (z 0 , z¯ 0 )(Y(ω L , z)v))op = (ι∗cl−op (z 0 , z¯ 0 )(u), Y(ω L , z)v)cl = (Bz 0 (u), e L(1) (¯z 0 − z 0 )−L(0) ⊗ e L(1) z¯ 0 − z 0
−L(0)
Y(ω L , z)v)cl .
(2.39)
Note that one should check the convergence property of each step in (2.39). In particular, in the last step, the convergence and equality follow from the convergence of early −L(0) −L(−1) e ∈ Aut Vcl . For steps and the fact that (¯z 0 − z 0 )−L(0) e−L(−1) ⊗ z¯ 0 − z 0 0 < |z| < |Imz 0 |, it is easy to show that −L(0)
e L(1) (¯z 0 − z 0 )−L(0) ⊗ e L(1) z¯ 0 − z 0 Y(ω L , z)v
−2L L (0) z (1− z )L L (1) 1− = Y(e z¯0 −z0 · z¯ 0 − z 0 z 1 L ·(¯z 0 − z 0 )−L (0) ω L , z )v1 z¯ 0 − z 0 1 − z¯ 0 −z 0 −4
z z 1 = 1− (¯z 0 − z 0 )−2 Y(ω L , z )v1 , z¯ 0 − z 0 z¯ 0 − z 0 1 − z¯ 0 −z 0 where v1 = e L(1) (¯z 0 − z 0 )−L(0) ⊗ e L(1) z¯ 0 − z 0 and |z + z 0 | > |z 0 |, we obtain
−L(0)
(2.40)
v. Hence, for all 0 < |z| < |Imz 0 |
(u, Ycl−op (ω L , z + z 0 )Ycl−op (v; z 0 , z¯ 0 )1op )op −4
z = (Bz 0 (u), 1 − (¯z 0 − z 0 )−2 Y(ω L , f (z))v1 )cl , z¯ 0 − z 0
(2.41)
50
L. Kong
where f is the composition of the following maps: w → w + z 0 → −
w (w + z 0 ) − z 0 1 = w (w + z 0 ) − z¯ 0 z¯ 0 − z 0 1 − z¯ 0 −z 0
which maps the domain H − z 0 to the unit disk. Since g1 (ω L , z) is analytic and free of singularities for z + z 0 ∈ H\{z 0 }, the right hand side of (2.41) can also be extended to an analytic function in z ∈ H − z 0 \{0}. If we view f (z) as a new variable ξ , then the right hand side of (2.41) can be extended to an analytic function on {ξ |1 > |ξ | > 0}, which has a Laurent series expansion. By the uniqueness of Laurent expansion, the right hand side of (2.41) gives exactly such a Laurent expansion and thus is absolutely convergent ˆ By the propin {ξ |1 > |ξ | > 0}. Moreover, lim z+z 0 →r g1 (ω L , z) exists for all r ∈ R. erties of Laurent series, the right hand side of (2.41) must converge absolutely for all ˆ f (z) ∈ {ξ ||ξ | = 1} to the function given by lim z+z 0 →r g1 (ω L , z), r ∈ R. Similarly, for all 0 < |¯z | < |Imz 0 | and |z + z 0 | > |z 0 |, we have (u, Ycl−op (ω R , z + z 0 )Ycl−op (v; z 0 , z¯ 0 )1op )op −L(0)
= (Bz 0 (u), e L(1) (¯z 0 − z 0 )−L(0) ⊗ e L(1) z¯ 0 − z 0 Y(ω R , z¯ )v)cl
−2L R (0) z¯ z R (1− z −¯ )L R (1) z 0 0 1− = (Bz 0 (u), Y(e (z 0 − z¯ 0 )−L (0) ω R , g(¯z ))v1 )cl z 0 − z¯ 0
−4 z¯ = 1− (z 0 − z¯ 0 )−2 (Bz 0 (u), Y(ω R , g(¯z ))v1 )cl , (2.42) z 0 − z¯ 0 where g is the composition of the following maps: w → w + z¯ 0 → −
w (w + z¯ 0 ) − z 0 (w + z¯ 0 ) − z¯ 0 1 → − = w (w + z¯ 0 ) − z¯ 0 (w + z¯ 0 ) − z 0 z 0 − z¯ 0 1 − z 0 −¯ z0
which maps the domain −H − z¯ 0 to the unit disk. Moreover, the right hand side of (2.42), as a Laurent series of g(¯z ), is absolutely convergent for all g(¯z ) ∈ {ξ ||ξ | = 1} to ˆ lim z+z 0 →r g2 (ω R , z), r ∈ R. Also notice that g(r − z¯ 0 ) =
1 ∈ {eiθ |0 ≤ θ < 2π } f (r − z 0 )
(2.43)
ˆ Using (2.38) and by replacing z in (2.41) by r − z 0 and z¯ in (2.42) by for all r ∈ R. r − z¯ 0 , we obtain the following identity: (Bz 0 (u), Y(ω L , eiθ )v1 )cl = (Bz 0 (u), Y(ω R , e−iθ )v1 )cl e−4iθ ,
(2.44)
where eiθ = f (r − z 0 ), for all 0 ≤ θ < 2π . Notice that the existence of both sides of (2.44) follows directly from (2.38), which further follows from the condition of u being vacuum-like. Then we obtain (Bz 0 (u), L L (n)v1 )cl eiθ(−n−2) = (Bz 0 (u), L R (−n)v1 )cl eiθ(−n−2) n∈Z
n∈Z
for all 0 ≤ θ < 2π . Notice that v1 can be arbitrary. Therefore, we must have (2.35) when L(−1)u = 0.
Cardy Condition for Open-Closed Field Algebras
51
In physics, boundary states are usually obtained by solving Eq. (2.35). The solutions of such equation was first obtained by Ishibashi [I]. They are called Ishibashi states. The definition of boundary states we give in (2.34) is more general. Boundary conditions are also called “D-branes” in string theory. If u is not a vacuum-like vector, the boundary state (2.34) associated with u is also very interesting in physics (see for example [FFFS1]). Such boundary states are associated to the geometry on D-branes. In the end of Sect. 5.2, we will give a more natural (or algebraic) definition of D-brane. 3. Cardy Condition In this section, we derive the Cardy condition from the axioms of open-closed partial conformal field theory by writing out the algebraic realizations of the both sides of Fig. 3 explicitly. Then we reformulate the Cardy condition in the framework of the intertwining operator algebra. Throughout this section, we fix an open-closed field algebra over V given in (2.12) equipped with nondegenerate invariant bilinear forms (·, ·)op and (·, ·)cl . 3.1. The first version. In the Swiss-cheese dioperad, we exclude an interior sewing operation between two disks with strips and tubes and a self-sewing operation between two oppositely oriented boundary punctures on a single disk. The surface obtained after these two types of sewing operations can be the same cylinder or annulus. The axioms of open-closed partial conformal field theory require that the algebraic realization of these two sewing operations must coincide. This gives a nontrivial condition called Cardy condition (recall Fig. 3). Although the Cardy condition only involves genus-zero surfaces, its algebraic realization is genus-one in nature. This fact is manifest if we consider the doubling map δ. A double of a cylinder is actually a torus. Hence the Cardy condition is a condition on the equivalence of two algebraic realizations of two different decompositions of a torus. This is nothing but a condition associated to modularity. That an annulus can be obtained by two different sewing operations is also shown in Fig. 4. In particular, the surface (A) in Fig. 4 shows how an annulus is obtained by sewing two oppositely oriented boundary punctures on the same disk with strips and tubes in S(1, 3|0, 0), and surface (C) in Fig. 4, viewed as a propagator of the close string, can be obtained by sewing an element in S(0, 1|1, 0) with an element in S(0, 1|0, 1) along the interior punctures. We only show in Figure 4 a simple case in which there are only two boundary punctures and no interior puncture. In general, the number of boundary punctures and interior punctures can be arbitrary. However, all general cases can be reduced to this simple case by applying associativities. Notice that the two boundary punctures in this simple case can not be reduced further by the associativities. We only focus on this case in this work. The conformal map f between the surface (A) and (B) and g between the surface (C) and (B) in Fig. 4 are given by 1 log w, 2πi −τ log w. g(w) = 2πi f (w) =
(3.1) 1
It is also useful to know their inverses f −1 (w) = e2πiw , g −1 (w) = e2πi(− τ )w . For any 1 z ∈ C, we set qz := e2πi z and pz := e2πi(− τ )z . The radius of the outer circle of the
52
L. Kong
τ
s2
f
s1 −1
qs2
0
qτ
qs1
1 0
1/2
g
(A)
1
(B)
ps2 0
1
ps1
(C) Fig. 4. Cardy condition: two different sewings give same annulus
surface (C) in Fig. 4 is | ps1 | = 1 and that of the inner circle is 1
| ps2 | = eπi(− τ ) = q
1/2 . − τ1
(3.2)
As we have argued and will show more explicitly later, the Cardy condition is deeply related to modularity. In [H9], Huang introduced the so-called geometrically modified intertwining operators, which is very convenient for the study of modularity. He was motivated by the fact that it is much easier to study modularity in the global coordinates. Namely, one should choose the local coordinates at s1 , s2 in surface (B) in Fig. 4 as simple as possible. More precisely, we choose the local coordinates at s1 , s2 as πi
w → e 2 (w − s1 ), πi
w → e− 2 (w − s2 )
(3.3)
respectively. Correspondingly, the local coordinates at punctures qs1 , qs2 are
πi 1 1 log 1 + x , f qs1 (w) = e 2 2πi qs1 x=w−qs1
1 1 − πi 2 log 1 + x f qs2 (w) = e 2πi qs2 x=w−qs 2
respectively.
(3.4)
Cardy Condition for Open-Closed Field Algebras
53
Notice that both local coordinates f qs1 (w) and f qs2 (w) are real analytic. Hence
(i) ∃B j
(i)
∈ R, b0 ∈ R+ , i = 1, 2 such that f qs1 (w) = e x , x=w−qs1 ∞ (2) B j x j+1 ddx (2) x ddx j=1 f qs2 (w) = e (b0 ) x . ∞
j=1
(1)
B j x j+1 ddx
d (b0(1) )x d x
(3.5)
x=w−qs2
Then the algebraic realization of the surface (A) gives a map Vop ⊗ Vop → C. We assume 1 > |qs1 | > |qs2 | > |qτ | > 0. By the axiom of open-closed conformal field theory, this map must be given by (recall (1.25)) v1 ⊗ v2 → TrVop Yop (Tqs1 v1 , qs1 )Yop (Tqs2 v2 , qs2 )qτL(0) ,
(3.6)
where Tqs1 = e− Tqs2 = e− by
∞
j=1
∞
j=1
(1)
B j L( j) (2)
B j L( j)
(1)
(b0 )−L(0) , (2)
(b0 )−L(0) .
(3.7)
We need to rewrite Tqs1 and Tqs2 . Let A j , j ∈ Z+ , be the complex numbers defined log(1 + y) = e
∞
j=1
d A j y j+1 dy
y.
It is clear that A j ∈ R. Hence we also obtain:
d 1 1 log 1 + x = z −x d x e j∈Z+ 2πi z
A j x j+1 ddx
d
(2πi)−x d x x.
(3.8)
Lemma 3.1. Tqs1 = (qs1 ) L(0) e− Tqs2 = (qs2 ) L(0) e−
∞
A j L( j)
∞
(2πi) L(0) e− 2
A j L( j)
(2πi) L(0) e−2πi L(0) e
j=1 j=1
∞
πi
L(0)
, πi 2
∞
L(0)
L(0)
.
(3.9)
L(0)
= e j=1 D j L( j) d0 for any C j , Proof. By results in [BHL], if e j=1 C j L( j) c0 D j ∈ C, c0 , d0 ∈ C, we must have C j = D j and c0 = d0 . Therefore, by moving ∞ the factor (qs1 ) L(0) to the right side of e− j=1 A j L( j) in (3.9) and similarly moving the factor (qs2 ) L(0) in (3.9), we see that it is enough to show (1)
πi
(b0 ) L(0) = (qs1 ) L(0) (2πi) L(0) e− 2 (1) (b0 ) L(0)
L(0)
,
= (qs2 ) L(0) (2πi) L(0) e−2πi L(0) e
πi 2
L(0)
.
(3.10)
Using our conventions (0.1)(0.2), it is easy to check that the above identities hold.
54
L. Kong
Let W be a V -module. Huang introduced the following operator in [H9]: U(x) := x L(0) e−
∞
A j L( j)
j=1
(2πi) L(0) ∈ (EndW ){x}.
Thus (3.6) can be rewritten as follows: v1 ⊗ v2 → TrVop Yop (U(qs1 )v1 , qs1 )Yop (U(qs2 )e−2πi L(0) v2 , qs2 )qτL(0) , πi
(3.11)
πi
where v1 = e− 2 L(0) v1 and v2 = e 2 L(0) v2 . Now we consider the algebraic realization of the surface (C) in Fig. 4 obtained from an interior sewing operation between an element in S(0, 1|1, 0) and an element in S(0, 1|0, 1). Lemma 3.2. ∀r ∈ (0, 1) ⊂ R+ , the surface C in Fig. 4 is conformally equivalent to a surface Q 1 1 ∞1I Q 2 , where ˆ and local coordinates: 1. Q 1 ∈ S(0, 1|0, 1) with punctures at z 1 ∈ H, ∞ ∈ R p w−z s1 1 , r w − z¯ 1
πi w − z1 −τ log : w → e 2 ; 2πi w − z¯ 1
f z1 : w → (1) f∞
(3.12) (3.13)
ˆ with local coordinates 2. Q 2 ∈ S(0, 1|1, 0) with punctures at z 2 ∈ H, ∞ ∈ R
−r w − z 2 , (3.14) f z2 : w → ps2 w − z¯ 2
πi −τ w − z¯ 2 (2) : w → e− 2 . (3.15) f∞ log 2πi w − z2 Proof. Let us first define two disks D1 and D2 . D1 is the unit disk, i.e. D1 := {z ∈ C||z| ≤ 1}, which has punctures at ps1 , 0 and local coordinates: g ps1 : w → e
πi 2
−τ log w − s1 , 2πi
g0 : w → r −1 w.
(3.16) (3.17)
ˆ D2 is the disk {z ∈ C||z| ≥ | ps2 |} which has punctures at ps2 , ∞ and local coordinates: g ps2 : w → e g∞ : w →
− πi 2
−r . w
−τ log w − s2 , 2πi
(3.18) (3.19)
It is not hard to see that the surface C in Fig. 1 can be obtained by sewing the puncture 0 ∈ D1 with the puncture ∞ ∈ D2 according to the usual definition of the interior sewing operation. Then it is enough to show that D1 and D2 are conformally equivalent
Cardy Condition for Open-Closed Field Algebras
55
to P and Q respectively. We define two maps h 1 : Q 1 → D1 and h 2 : Q 2 → D2 as follows: w − z1 h 1 : w → ps1 , w − z¯ 1 w − z¯ 2 h 2 : w → ps2 . w − z2 It is clear that h 1 and h 2 are both biholomorphic. We can check directly that h 1 and h 2 map punctures to punctures and preserve local coordinates as well. Using (2.33), we obtain the algebraic realization of the annulus C in Fig. 1 as follows: v1 ⊗ v2 → ((T1L ⊗ T1R )∗ ι∗cl−op (z 1 , z¯ 1 )(T2 v1 ), (T3L ⊗ T3R )∗ ι∗cl−op (z 2 , z¯ 2 )(T4 v2 ))cl , (3.20) where T1L ,R , T2 , T3L ,R , T4 are conformal transformations determined by local coordi(2) (1) (2) nates f z 1 , f ∞ , f z 2 , f ∞ , and (TiL ⊗ Ti R )∗ is the adjoint of TiL ⊗ Ti R with respect to πi
the bilinear form (·, ·)cl for i = 1, 3, and v1 = e− 2
L(0)
v1 and v2 = e
πi 2
L(0)
v2 .
Lemma 3.3. Recall the convention (0.2), we have −L(0) p −L(0) L(0) L(1) ps1 s1 T1L = (z 1 − z¯ 1 ) L(0) e L(1) , T1R = (z 1 − z¯ 1 ) e , r r
1 L(0) T2 = e z¯ 1 L(1) (z 1 − z¯ 1 )−L(0) U(1) − , τ L(0) p L(0) L(0) −L(1) ps2 s2 , T3R = (¯z 2 − z 2 ) e , T3L = (¯z 2 − z 2 ) L(0) e−L(1) r r L(0)
1 T4 = e z 2 L(1) (¯z 2 − z 2 )−L(0) U(1) − . τ Proof. From (3.12) and (3.14), we obtain f z 1 : w → (z 1 − z¯ 1 )
p x
d dx
r p −x
d dx
−x ddx −x 2 ddx
e
d
f z 2 : w → (¯z 2 − z 2 )−x d x e x
2 d dx
s1
s2
r
x , x=w−z 1 x .
(3.21) (3.22) (3.23) (3.24)
(3.25)
x=w−z 2
Then (3.21) and (3.23) is obvious. Notice that the expression (3.21) is independent of our choice of branch cut as long as we keep the convention (0.2). From (3.13) and (3.15), we obtain
−x d ∞ dx πi d 1 A j x j+1 ddx (1) −¯z 1 x 2 ddx x ddx −x ddx x dx j=1 2 − f ∞ (w) = e (z 1 − z¯ 1 ) e (2πi) e x , τ x= −1 w
(2) f∞ (w) = e−z 2 x
2 d dx
d
(¯z 2 − z 2 )x d x e
∞
j=1
A j x j+1 ddx
−x d dx d 1 (2πi)−x d x − τ
(3.26) πi d e− 2 x d x x . x= −1 w
(3.27)
56
L. Kong (1)
(2)
Recall that f ∞ , f ∞ are both real analytic. Similar to the proof of Lemma 3.1, to show (3.22) and (3.24) it is enough to show that (1) L(0) ) (b∞ (2) L(0) (b∞ ) (1)
1 L(0) − πi L(0) = (z 1 − z¯ 1 ) (2πi) e 2 , − τ
1 L(0) πi L(0) = (¯z 2 − z 2 )−L(0) (2πi) L(0) − e2 τ −L(0)
L(0)
(3.28) (3.29)
(2)
for some b∞ , b∞ ∈ R+ . Using our convention (0.1) and (0.2), it is a direct check that (3.28) and (3.29) holds. c − 24
Combining (3.11), (3.20) and additional natural factors qτ
,q
c − 24
− τ1
(see [Z,H9]),
which is due to the determinant line bundle on torus [Se1,Kr], we obtain the following formulation of the Cardy condition: Definition 3.4. The open-closed field algebra over V given in (2.12) and equipped with nondegenerate bilinear forms (·, ·)op and (·, ·)cl is said to satisfy Cardy condition if the left hand sides of the following formula, ∀z 1 , z 2 ∈ H, v1 , v2 ∈ Vop ,
TrVop Yop (U(qs1 )v1 , qs1 )Yop (U(qs2 )e−2πi L(0) v2 , qs2 )qτL(0)−c/24
−c/24 L R ∗ ∗ L R ∗ ∗ = (T1 ⊗ T1 ) ιcl−op (z 1 , z¯ 1 )(T2 v1 ), q 1 (T3 ⊗ T3 ) ιcl−op (z 2 , z¯ 2 )(T4 v2 ) −τ
cl
(3.30) converge absolutely when 1 > |qs1 | > |qs2 | > |qτ | > 0, and the right hand side of (3.30) converge absolutely for all s1 , s2 ∈ H satisfying Re s1 = 0, Re s2 = 21 . Moreover, Eq. (3.30) holds when 1 > |qs1 | > |qs2 | > |qτ | > 0. Remark 3.5. The dependence of z 1 , z 2 , r of the right hand side of (3.30) is superficial as required by the independence of z 1 , z 2 , r of the left hand side of (3.30). We will see it more explicitly later. Using the definition of boundary states (2.34), (3.30) can also be written as follows: TrVop Yop (U(qs1 )v1 , qs1 )Yop (U(qs2 )e−2πi L(0) v2 , qs2 )qτL(0)−c/24
ps2 L(0) −c/24 ps2 L(0) ⊗ q 1 Bz 2 (T4 v2 ) . (3.31) = Bz 1 (T2 v1 ), −τ − ps1 − ps1 cl
3.2. The second version. In this subsection, we rewrite the Cardy condition (3.30) in the framework of the intertwining operator algebra. Since V satisfies the conditions in Theorem 0.1, it has only finite number of inequivalent irreducible modules. Let I be the set of equivalence classes of irreducible V -modules. We denote the equivalence class of the adjoint module V as e, i.e. e ∈ I. Let Wa be a chosen representative of a ∈ I.
Cardy Condition for Open-Closed Field Algebras
57
For any V -module (W, YW ), we denote the graded dual space of W as W , i.e. = ⊕n∈C (W(n) )∗ . There is a contragredient module structure on W [FHL] given by , which is defined as follows: a vertex operator YW W
YW (u, x)w , w := w , YW (e−x L(1) x −2L(0) u, −x −1 )w
(3.32)
) (or simply W ) is the only module structure on for u ∈ V, w ∈ W, w ∈ W . (W , YW . We denote the equivalent class of W we use in this work. So we can set YW := YW Wa as a . It is harmless to set Wa = Wa . Moreover, W is canonically identified with W . Hence a = a for a ∈ I. By assumption on V , V ∼ = V , i.e. e = e. From [FHL], there is a nondegenerate invariant bilinear form (·, ·) on V such that (1, 1) = 1. This bilinear form specifies a unique isomorphism from V to V . In the rest of this work, we identify V with V using this isomorphism without mentioning it explicitly. For any triple V -modules W1 , W2 , W3 , we have isomorphisms W3 W3 → VW , r : VW 1 W2 2 W1
∀r ∈ Z
given as follows: r (Y)(w2 , z)w1 = e z L(−1) Y(w1 , e(2r +1)πi z)w2 , for Y ∈
W3 VW 1 W2
(3.33)
and wi ∈ Wi , i = 1, 2. The following identity r ◦ −r −1 = −r −1 ◦ r = id
(3.34)
is proved in [HL3]. W3 and r ∈ Z, a so-called r -contragredient operator Ar (Y) was introFor Y ∈ VW 1 W2 duced in [HL3]. Here, we use two slightly different operators A˜ r (Y) and Aˆ r (Y) introduced in [Ko1] and defined as follows: A˜ r (Y)(w1 , e(2r +1)πi x)w3 , w2 = w3 , Y(e x L(1) x −2L(0) w1 , x −1 )w2 , Aˆ r (Y)(w1 , x)w3 , w2 = w3 , Y(e−x L(1) x −2L(0) w1 , e(2r +1)πi x −1 )w2 ,
(3.35)
and w1 ∈ W1 , w2 ∈ W2 , w3 ∈ W3 . In particular, when W1 = V and = Y , ∀r ∈ Z. If Y ∈ V W3 , W2 = W3 = W , we have A˜ r (YW ) = Aˆ r (YW ) = YW W W1 W2
for Y ∈
W3 VW 1 W2
W then A˜ r (Y), Aˆ r (Y) ∈ VW 2W for r ∈ Z and 1
Let Y
3
A˜ r ◦ Aˆ r (Y) = Aˆ r ◦ A˜ r (Y) = Y. a3 ∈ Va1 a2 . We define σ123 := r ◦ A˜ r . It is easy to see wa 3 , Y(wa1 , x)wa2 = e−x
−1 L(−1)
(3.36) that
σ123 (Y)(wa 3 , x −1 )e−x L(1) x −2L(0) wa1 , wa2
(3.37)
for wa1 ∈ Wa1 , wa2 ∈ Wa2 , wa 3 ∈ Wa 3 . It is also clear that σ123 is independent of r ∈ Z. 3 = id a . We also denote σ −1 as σ It is proved in [Ko1] that σ123 132 . Clearly, we have 123 Va 3a 1 2
σ132 = Aˆ r ◦ −r −1 and
σ132 (Y)(w1 , x)w3 , w2 for wa1 ∈ Wa1 , wa2 ∈
= w3 , e−x
−1 L(−1)
Wa2 , wa 3
Wa 3 .
∈
Y(w2 , x −1 )e−x L(1) x −2L(0) w1
(3.38)
58
L. Kong
For any V -module W , we define a V -module map θW : W → W by θW : w → e−2πi L(0) w.
(3.39)
For Wa , we have θWa = e−2πi h a idWa , where h a ∈ C is the lowest conformal weight of Wa . respectively. Let We denote the graded dual space of Vcl and Vop by Vcl and Vop ϕcl : Vcl → Vcl and ϕop : Vop → Vop be the isomorphisms induced from (·, ·)cl and (·, ·)op respectively. Namely, we have (u 1 , u 2 )cl = ϕcl (u 1 ), u 2 , (v1 , v2 )op = ϕop (v1 ), v2
(3.40)
for u 1 , u 2 ∈ Vcl and v1 , v2 ∈ Vop . Vcl as a conformal full field algebra over V ⊗ V can be expanded as follows: Ncl Wr L (i) ⊗ Wr R (i) , Vcl = ⊕i=1
(3.41)
where r L , r R : {1, . . . , Ncl } → I. For a ∈ I, we choose a basis {ea;α }α∈N of Wa and a } dual basis {ea;α α∈N of Wa . Then {er L (i),α ⊗ er R (i),β }i=1,...,Ncl ,α,β∈N
(3.42)
{ϕcl−1 (er L (i);α ⊗ er R (i),β )}i=1,...,Ncl ,α,β∈N
(3.43)
is a basis of Vcl and
is its dual basis with respect to the nondegenerate bilinear form (·, ·)cl . Let T : CV ⊗V → CV be the tensor bifunctor. We showed in [Ko2] that there is a morphism ιcl−op : T (Vcl ) → Vop in CV (see (3.81),(3.82) in [Ko2] for the definition). We define a morphism ιcl−op : T (Vcl ) → Vop as a composition of maps as follows: −1 T (ϕcl )
ιcl−op
ιcl−op : T (Vcl ) −−−−→ T (Vcl ) −−−→ Vop .
(3.44)
W3 By the universal property of the tensor product [HL1-HL4], VW and HomV (W1 1 W2 W2 , W3 ) for any three V -modules W1 , W2 , W3 are canonically isomorphic. Given a morphism m ∈ HomV (W1 W2 , W3 ), we denote the corresponding intertwining operator as Ym . Conversely, given an intertwining operator Y, we denote its corresponding morphism as m Y . Therefore, we have two intertwining operators Yιcl−op and Yιcl−op corresponding to morphisms ιcl−op and ιcl−op respectively.
Lemma 3.6. For z ∈ H, we have ιcl−op (z, z¯ )(er L (i),α ⊗ er R (i),β ) = e z¯ L(−1) Yιcl−op (er L (i),α , z − z¯ )er R (i),α .
(3.45)
Cardy Condition for Open-Closed Field Algebras
59
Proof. It is proved in [Ko2] that m Ycl−op = m Y f ◦ (ιcl−op idVop ).
(3.46)
op
Using (3.46), when z ∈ H, ζ ∈ H and |ζ | > |z − ζ | > 0, we have ιcl−op (z, ζ )(er L (i),α ⊗ er R (i),β ) = Ycl−op (er L (i),α ⊗ er R (i),β ; z, ζ )1op f
= Yop (Yιcl−op (er L (i),α , z − ζ )er R (i),β , ζ )1 = eζ L(−1) Yιcl−op (er L (i),α , z − ζ )er R (i),β . (3.47) By the convergence property of the iterate of two intertwining operators, the right hand side of (3.47) is a power series of ζ absolutely convergent for |ζ | > |z − ζ | > 0 . By the property of power series, the right hand side of (3.47) must converge absolutely for all z ∈ H, ζ ∈ H. Because analytic extension in a simply connected domain is unique, we obtain that the equality (3.47) holds for all z ∈ H, ζ ∈ H. When ζ = z¯ , we obtain (3.45). Now we consider both sides of the Cardy condition (3.30) for an open-closed field algebra over V . On the left hand side of (3.30), we have qs2 < 0. Using (1.25) and (3.33), we obtain, ∀v3 ∈ Vop , Yop (U(qs2 )v2 , qs2 )v3 = e−|qs2 |L(−1) Yop (v3 , |qs2 |) U(qs2 )v2 = −1 (Yop )(U(qs2 )v2 , eπi |qs2 |)v3 . f
(3.48)
Hence we can rewrite the left hand side of (3.30) as follows: f f TrVop Yop (U(qs1 )v1 , qs1 )−1 (Yop )(U(qs2 )e−2πi L(0) v2 , eπi |qs2 |)qτL(0)−c/24 (3.49) for qs1 > |qs2 | > |qτ | > 0. We have the following result for the right hand side of (3.30). Proposition 3.7. For s1 , s2 ∈ H, Re s1 = 0, Re s2 = 0, −
c
((T1L ⊗ T1R )∗ ι∗cl−op (z 1 , z¯ 1 )(T2 v1 ), q 124 (T3L ⊗ T3R )∗ ι∗cl−op (z 2 , z¯ 2 )(T4 v2 ))cl −τ L(0)
N cl 1 = TrWr R (i) e−2πi L(0) Y1 U(q− 1 s1 ) − v1 , q− 1 s1 τ τ τ i=1
1 L(0) L(0)− c Y2 U(q− 1 s2 ) − v2 , q− 1 s2 q 1 24 , −τ τ τ τ where Y1 and Y2 are intertwining operators of types
Wr R (i) Vop Wr (i)
and
L
tively and are given by Y1 = σ123 (Yιcl−op ) ◦ (ϕop ⊗ idW
r L (i)
),
Y2 = 0 (σ132 (Yιcl−op )) ◦ (ϕop ⊗ idWr R (i) ).
Wr (i) L Vop Wr R (i)
(3.50)
respec-
(3.51) (3.52)
60
L. Kong
Proof. Let z 3 := z 1 − z¯ 1 and z 4 := z 2 − z¯ 2 . By (2.28) and (3.45), the left hand side of (3.50) equals ((T1L ⊗ T1R )∗ ι∗cl−op (z 1 , z¯ 1 )(T2 v1 ), (T3L ⊗ T3R )∗ ι∗cl−op (z 2 , z¯ 2 )(T4 v2 ))cl =
Ncl
(T3L ⊗ T3R )∗ ι∗cl−op (z 2 , z¯ 2 )(T4 v2 ), er L (i),α ⊗ er R (i),β
i=1 α,β
=
ϕcl−1 (er L (i),α ⊗ er R (i),β ), (T1L ⊗ T1R )∗ ι∗cl−op (z 1 , z¯ 1 )(T2 v1 )
Ncl
T4 v2 , ιcl−op (z 2 , z¯ 2 )(T3L er L (i),α ⊗ T3R er R (i),β )
i=1 α,β
=
ιcl−op (z 1 , z¯ 1 )(ϕcl−1 (T1L er L (i),α ⊗ T1R er R (i),β )), T2 v1
Ncl
=
cl
op
op
T4 v2 , e z¯ 2 L(−1) Yιcl−op (T3L er L (i),α , z 2 − z¯ 2 )T3R er R (i),β
e z¯ 1 L(−1) Yιcl−op (T1L er L (i),α , z 1 − z¯ 1 )T1R er R (i),β , T2 v1
Ncl
cl
i=1 α,β
op
op
ϕop (T4 v2 ), e z¯ 2 L(−1) Yιcl−op (T3L er L (i),α , z 2 − z¯ 2 )T3R er R (i),β
i=1 α,β
e z¯ 1 L(−1) Yιcl−op (T1L er L (i),α , z 1 − z¯ 1 )T1R er R (i),β , ϕop (T2 v1 ) =
Ncl
−1
e−z 4
L(−1)
σ123 (Yιcl−op )(ϕop (e−¯z 2 L(1) T4 v2 ), z 4−1 )
i=1 α,β −1
z 4−2L(0) e−z 4
L(1)
T3L er L (i),α , T3R er R (i),β −1
T1R er R (i),β , e−z 3 −2L(0) −z 3−1 L(1)
z3 =
e
Ncl
L(−1)
σ123 (Yιcl−op )(ϕop (e−¯z 1 L(1) T2 v1 ), z 3−1 )
T1L er L (i),α −2L(0) −z 4−1 L(1)
σ123 (Yιcl−op )(ϕop (e−¯z 2 L(1) T4 v2 ), z 4−1 )z 4
e
T3L er L (i),α ,
i=1 α,β −1
ez4
L(1)
−1
T3R er R (i),β er R (i),β , (T1R )∗ e−z 3
L(−1)
·
−2L(0) −z 3−1 L(1)
σ123 (Yιcl−op )(ϕop (e−¯z 1 L(1) T2 v1 ), z 3−1 )z 3 =
Ncl
−1
er L (i),α , (T3L )∗ e z 4
L(−1) −2L(0) ˜ z4 A0
e
T1L er L (i),α −1
◦ σ123 (Yιcl−op )(e−z 4
L(1) 2L(0) z4
·
i=1 α,β −1
ϕop (e−¯z 2 L(1) T4 v2 ), eπi z 4 )e z 4
L(1)
−1
T3R er R (i),β er R (i),β , (T1R )∗ e−z 3 −1
σ123 (Yιcl−op )(ϕop (e−¯z 1 L(1) T2 v1 ), z 3−1 )z 3−2L(0) e−z 3
L(1)
T1L er L (i),α .
L(−1)
· (3.53)
Cardy Condition for Open-Closed Field Algebras
61
We define two intertwining operators as follows: (0)
Y1
r L (i)
(0)
Y2
= σ123 (Yιcl−op ) ◦ (ϕop ⊗ idW
),
= A˜ 0 (σ123 (Yιcl−op )) ◦ (ϕop ⊗ idW
r R (i)
).
(3.54)
Using (3.21),(3.22),(3.23) and (3.24), we further obtain that the left hand side of (3.53) equals Ncl p L(0) s2 −2L(0) er L (i),α , (¯z 2 − z 2 ) L(0) z 4 · r i=1 α,β
1 L(0) (0) 2L(0) −L(0) Y2 (z 4 (¯z 2 − z 2 ) U(1) − v2 , eπi z 4 ) τ L(0) p −L(0) L(0) L(0) ps2 s1 z¯ 2 − z 2 er R (i),β er R (i),β , (z 1 − z¯ 1 ) r r L(0)
p −L(0) 1 s1 Y2(0) (z 3−L(0) U(1) − v1 , z 3−1 )z 3−L(0) er L (i),α τ r
=
Ncl
TrW
r L (i)
i=1
Y2(0) (U( ps2 )(−1/τ ) L(0) v2 , E 1 )E 2 Y1(0) (U( ps1 )(−1/τ ) L(0) v1 , ps1 ), (3.55)
where E 1 = ps2 (¯z 2 − z 2 )z 4−2 eπi z 4 and −2L(0)
(¯z 2 − z 2 ) L(0) z 4 E 2 = psL(0) 2
(¯z 2 − z 2 )
L(0)
ps2 L(0) ps1 −L(0) z 1 − z¯ 1
L(0) −L(0) −L(0) z3 ps1 .
For E 1 , since z 4−1 is obtained by operations on the intertwining operator where z 4 is treated formally, z 4−1 really means |z 4 |−1 e−i E 1 = ps2 e
3πi 2
πi 2
. Therefore, we have
e−πi eπi e
πi 2
= e2πi ps2 .
(3.56)
For E 2 , keep in mind (0.1) and (0.2), we have ps 2L(0) L(0) L(0) E 2 = 2 (¯z 2 − z 2 ) L(0) (z 2 − z¯ 2 )−2L(0) (¯z 2 − z 2 ) (z 1 − z¯ 1 ) (z 1 − z¯ 1 )−L(0) ps1 = q L(0) e 1 −τ
=q
L(0) − τ1
3πi 2
3πi πi πi L(0) −2 πi 2 L(0) − 2 L(0) − 2 L(0) − 2 L(0)
e
e
e
e
e−2πi L(0) .
Therefore, we obtain that the left hand side of (3.53) further equals L(0)
Ncl 1 (0) TrW 20 (Y2 ) U( ps2 ) − v2 , ps2 r L (i) τ i=1
1 L(0) L(0) −2πi L(0) (0) q 1 e Y1 v1 , ps1 , U( ps1 ) − −τ τ
(3.57)
(3.58)
62
L. Kong
where we have used the fact that Y(·, e2πi x)· = 20 (Y)(·, x)· for any intertwining operator Y. By using the property of trace, it is easy to see that (3.58) multiplying q
nothing but the right hand side of (3.50).
c − 24
− τ1
is
Remark 3.8. It is easy to check that the absolute convergence of the left hand side of (3.53) by our assumption easily implies the absolute convergence of each step in (3.53). Notice that the absolute convergence of the right side of (3.50) is automatic because V is assumed to satisfy the conditions in Theorem 0.1. Hence, by tracing back the steps in above the proof, we see that the absolute convergence of the left hand side of (3.50) is also automatic. Now we recall some results in [H9,H10]. We follow the notations in [HKo3]. We denote the unique analytic extension of c L(0)− 24
a ;(1)
TrWa1 Yaa1 1 ;i (U(e2πi z )wa , e2πi z )qτ in the universal covering space of 1 > |qτ | > 0 as
c L(0)− 24
a ;(1)
E(TrWa1 Yaa1 1 ;i (U(e2πi z )wa , e2πi z )qτ
).
By [Mi2,H9], above formula is independent of z. Consider the map: for wa ∈ Wa , a ;(1)
c L(0)− 24
a ;(1)
1 (Yaa1 1 ;i ) : wa → E(TrWa1 Yaa1 1 ;i (U(e2πi z )wa , e2πi z )qτ
).
(3.59)
a ;(1)
We denote the right hand side of (3.59) as 1 (Yaa1 1 ;i )(wa ; z, τ ). Notice that we choose to add z in the notation even though it is independent of z. We define an action of S L(2, Z) on the map (3.59) as follows:
ab a ;(1) (1 (Yaa1 1 ;i )) (wa ; z, τ ) cd L(0)
c 1 L(0)− 24 a1 ;(1) 2πi z 2πi z = E TrWa1 Yaa1 ;i U(e ) wa , e qτ , (3.60) cτ + d where
τ
=
aτ +b cτ +d
and
z
=
z cτ +d ,
for
ab cd
∈ S L(2, Z) and wa ∈ Wa . The following
theorem is proved in [Mi2,H7]. ij
Theorem 3.9. There exists a unique Aa2 a3 ∈ C for a2 , a3 ∈ I such that, for wa ∈ Wa ,
a ;(1) TrWa1 Yaa1 1 ;i
E =
a3 ∈I
where τ =
aτ +b cτ +d
U(e
2πi z
1 ) cτ + d
L(0) wa , e
2πi z
c L(0)− 24
a2 ;(2) Aa1 a2 E(TrWa1 Yaa (U(e2πi z )wa , e2πi z )qτ 2; j ij
and z =
c L(0)− 24 qτ
z cτ +d .
),
(3.61)
Cardy Condition for Open-Closed Field Algebras
63
0 1 on (3.59) induces, for each a ∈ I, an −1 0 a1 automorphism on ⊕a1 ∈I Vaa 1 , denoted as S(a). Namely, we have
In particular, the action of S =
a ;(1)
a ;(1)
S(1 (Yaa1 1 ;i )) = 1 (S(a)(Yaa1 1 ;i )).
(3.62)
a1 Combining all such S(a), we obtain an automorphism on ⊕a,a1 ∈I Vaa 1 . We still denote it a3 a as S, i.e. S = ⊕a∈I S(a). Then S can be further extended to a map on ⊕a,a3 ∈I Vaa 3 ⊗Va1 a2 given as follows: a ;(1)
a ;(1)
a;(2)
a;(2)
S(Yaa3 3 ;i ⊗ Ya1 a2 ; j ) := S(Yaa3 3 ;i ) ⊗ Ya1 a2 ; j .
(3.63)
There is a fusing isomorphism map [H4]: ∼ =
a4 → ⊕b∈I Vba ⊗ Vab1 a2 F : ⊕a∈I Vaa14a ⊗ Vaa2 a3 − 3
(3.64)
for a1 , a2 , a3 , a4 ∈ I. Using the isomorphism F, we obtain a natural action of S on ⊕b,a3 Vaa13b ⊗ Vab2 a3 . It is shown in [H7] that the following 2-points genus-one correlation function, for a1 a a1 , a2 , a3 , a4 ∈ I, i = 1, . . . , Naa 1 , j = 1, . . . , Na2 a3 and wak ∈ Wak , k = 2, 3, a ;(1)
a;(2)
c L(0)− 24
TrWa1 Yaa1 1 ;i (U(qz 2 )Ya2 a3 ; j (wa2 , z 1 − z 2 )wa3 , qz 2 )qτ
(3.65)
is absolutely convergent when 1 > |e2πi z 2 | > |qτ | > 0 and 1 > |e2πi(z 1 −z 2 ) | > 0 and single-valued in the chosen branch. It can be extended uniquely to a single-valued analytic function on the universal covering space of M12 = {(z 1 , z 2 , τ ) ∈ C3 |z 1 = z 2 + pτ + q, ∀ p, q ∈ Z , τ ∈ H}. This universal covering space is denoted by M˜ 12 . We denote this single-valued analytic function on M˜ 12 as a ;(1)
a;(2)
c L(0)− 24
E(TrWa1 Yaa1 1 ;i (U(qz 2 )Ya2 a3 ; j (wa2 , z 1 − z 2 )wa3 , qz 2 )qτ
).
We denote the space spanned by such functions on M˜ 12 by G1;2 . a ;(1) a;(2) a1 a For Yaa1 1 ;i ∈ Vaa 1 and Ya2 a3 ; j ∈ Va2 a3 , we now define the following linear map: a ;(1)
a;(2)
2 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ) :
⊕b2 ,b3 ∈I Wb2 ⊗ Wb3 → G1;2
as follows: the map restricted on Wb2 ⊗ Wb3 is defined by 0 for b2 = a2 or b3 = a3 , and by c L(0)− 24
a1 ;(1) E(TrWa1 Yaa (U(qz 2 )Yaa;(2) (wa2 , z 1 − z 2 )wa3 , qz 2 )qτ 1 ;i 2 a3 ; j
),
(3.66)
for all wak ∈ Wak , k = 2, 3. The following identity was proved in [H9]:
1 L(0) 1 1 L(0) 1 1 a1 ;(1) a;(2) − − z1, − z2 ; − wa2 ⊗ − wa3 2 (Yaa1 ;i ⊗ Ya2 a3 ; j ) τ τ τ τ τ a ;(1) a;(2) = 2 (S(Yaa1 1 ;i ⊗ Ya2 a3 ; j ))(wa2 ⊗ wa3 ) (z 1 , z 2 , τ ). (3.67)
64
L. Kong
One can also produce 2-point genus-one correlation functions from a product of two intertwining operators. It is proved in [H7] that ∀wak ∈ Wak , k = 1, 2, a ;(1)
a ;(2)
c L(0)− 24
TrWa4 Ya14a3 ;i (U(qz 1 )wa1 , qz 1 )Ya23a4 ; j (U(qz 2 )wa2 , qz 2 )qτ
,
(3.68)
is absolutely convergent when 1 > |qz 1 | > |qz 2 | > |qτ | > 0. Equation (3.68) has a unique extension to the universal covering space M˜ 12 , denoted as a ;(1)
a ;(2)
c L(0)− 24
E(TrWa4 Ya14a3 ;i (U(qz 1 )wa1 , qz 1 )Ya23a4 ; j (U(qz 2 )wa2 , qz 2 )qτ
).
(3.69)
Such functions on M˜ 12 also span G1;2 . We define a map a ;(1)
a ;(2)
2 (Ya14a3 ;i ⊗ Ya23a4 ; j ) : ⊕b1 ,b2 ∈I Wb1 ⊗ Wb2 → G1;2
(3.70)
as follows: the map restricted on Wb1 ⊗ Wb2 is defined by 0 for b1 = a1 , b2 = a2 , and by (3.69) for wa1 ∈ Wa1 , wa2 ∈ Wa2 . It was proved by Huang in [H7] that the fusing isomorphism (3.64) gives the following associativity: c L(0)− 24
E(TrWa4 Yaa14a;(1) (U(qz 1 )wa1 , qz 1 )Yaa23a;(2) (U(qz 2 )wa2 , qz 2 )qτ 3 ;i 4; j a4 ;(1) a3 ;(2) a4 ;(3) F(Ya1 a3 ;i ⊗ Ya2 a4 ; j , Ya5 a4 ;k ⊗ Yaa15a;(4) ) = 2 ;l
)
a5 ∈I k,l
a ;(3)
a ;(4)
c L(0)− 24
E(TrWa4 Ya54a4 ;k (U(qz 2 )Ya15a2 ;l (wa1 , z 1 − z 2 )wa2 , qz 2 )qτ
), (3.71)
where F(Yaa14a;(1) ⊗ Yaa23a;(2) , Yaa54a;(3) ⊗ Yaa15a;(4) ) is the matrix representation of F in the 3 ;i 4; j 4 ;k 2 ;l
⊗ Yaa23a;(2) }i, j , {Yaa54a;(3) ⊗ Yaa15a;(4) }k,l . Therefore, ∀wak ∈ Wak , k = 1, 2, we basis {Yaa14a;(1) 3 ;i 4; j 4 ;k 2 ;l obtain
1 L(0) 1 1 L(0) 1 1 a3 ;(1) a;(2) − − z1, − z2 ; − 2 (Ya1 a;i ⊗ Ya2 a3 ; j ) wa1 ⊗ − wa2 τ τ τ τ τ a ;(1) a;(2) = 2 (S(Ya13a;i ⊗ Ya2 a3 ; j ))(wa1 ⊗ wa2 ) (z 1 , z 2 , τ ). (3.72)
Combining (3.49), (3.50), (3.51), (3.52) and (3.72) and Remark 3.8, we obtain a simpler version of the Cardy condition. Theorem 3.10. The Cardy condition can be rewritten as follows: θWr R (i) ◦ σ123 (Yιcl−op ) ◦ (ϕop ⊗ idW ) ⊗ 0 (σ132 (Yιcl−op )) ◦ (ϕop ⊗ idWr R (i) ) r L (i) f f −1 Yop ⊗ −1 (Yop ) ◦ (θVop ⊗ idVop ) . (3.73) =S 4. Modular Tensor Categories This section is independent of the rest of this work. The tensor product theory of modules over a vertex operator algebra has been developed by Huang and Lepowsky [HL1][HL4,H2]. In particular, the notion of vertex tensor category is introduced in [HL1]. Huang later proved in [H11] that CV is a modular tensor category for V satisfying conditions in Theorem 0.1. In Sect. 4.1, we review some basic ingredients of the modular tensor category CV . In Sect. 4.2, we show how to find in CV a graphical representation of the modular transformation S : τ → − τ1 discussed in Sect. 3.2.
Cardy Condition for Open-Closed Field Algebras
65
4.1. Preliminaries. We recall some ingredients of vertex tensor category CV and those of modular tensor category structure on CV constructed in [H11]. There is an associativity isomorphism A, A : W1 (W2 W3 ) → (W1 W2 ) W3 , for each triple of V -modules W1 , W2 , W3 . The relation between the fusing isomorphism F (recall (3.64)) and the associativity isomorphism A in CV is described by the following commutative diagram: ⊕a5 ∈I Vaa14a5 ⊗ Vaa25a3
∼ =
/ HomV (Wa1 (Wa2 Wa3 ), Wa4 ) (A−1 )∗
F
⊕a6 ∈I Vaa64a3 ⊗ Vaa16a2
(4.1)
∼ =
/ HomV ((Wa1
Wa2 ) Wa3 , Wa4 )
where the two horizontal maps are canonical isomorphisms induced from the universal property of . We recall the braiding structure on CV . For each pair of V -modules W1 , W2 , there is also a natural isomorphism, for z > 0, R+P(z) : W1 P(z) W2 → W2 P(z) W1 , defined by P(z)
R+
(w1 P(z) w2 ) = e L(−1) T γ+ (w2 P(−z) w1 ),
(4.2)
where γ+ is a path from −z to z inside the lower half plane as shown in the following graph
(4.3) P(z)
The inverse of R+
P(z)
is denoted by R− , which is characterized by
P(z) R− (w2 P(z) w1 ) = e L(−1) T γ− (w1 P(−z) w2 ),
(4.4)
where γ− is a path in the upper half plane as shown in the following graph
(4.5) P(1)
We denote R± simply as R± . The natural isomorphisms R± give CV two different braiding structures. We choose R+ as the default braiding structure on CV . Sometimes, we will denote it by (CV , R+ ) to emphasize our choice of braiding isomorphisms. Notice that our choice of R± follows that in [Ko2], which is different from that in [H8,H11,Ko1]. For each V -module W , (3.39) defines an automorphism θW : W → W
66
L. Kong
called a twist. A V -module W is said to have a trivial twist if θW = idW . The twist θ and braiding R+ satisfy the following three balancing axioms θW1 W2 = R+ ◦ R+ ◦ (θW1 θW2 ), θV = idV , θW = (θW )∗ ,
(4.6) (4.7) (4.8)
for any pair of V -modules W1 , W2 . a } be a basis of V a for all a ∈ I such that it coincides with the vertex operator Let {Yea ea a = Y . We choose a basis YWa , which defines the V -module structure on Wa , i.e. Yea Wa a } of V a as follows: {Yae ae a a = −1 (Yea ). Yae
(4.9)
e } of V e as We also choose a basis {Yaa aa e e a a ˆ Yaa = Yaa = A 0 (Yae ) = σ132 (Yea ).
(4.10)
Notice that these choices are made for all a ∈ I. In particular, we have
a Yaa e = −1 (Yea ),
Yae a = Yae a = Aˆ 0 (Yaa e ).
The following relation was proved in [Ko1]. e −2πi h a e Yae a = e2πi h a 0 (Yaa −1 (Yaa ) = e ).
(4.11)
W3 For any V -modules W1 , W2 , W3 and Y ∈ VW , we denote by m Y the morphism 1 W2 in HomV (W1 W2 , W3 ) associated to Y under the identification of two spaces induced by the universal property of . Now we recall the construction of duality maps [H11]. We will follow the convention in [Ko1]. Since CV is semisimple, we only need to discuss irreducible modules. For a ∈ I, the right duality maps ea : Wa Wa → V and i a : V → Wa Wa for a ∈ I are given by
ea = m Y e , m Y e ◦ i a = dim a idV , aa
aa
where dim a = 0 for a ∈ I (proved by Huang in [H10]). The left duality maps ea : W a Wa → V and i a : V → Wa W a are given by ea = m Y e , m Y e ◦ i a = dim a idV . aa aa In a ribbon category, there is a powerful tool called graphic calculus. One can express various morphisms in terms of graphs. In particular, the right duality maps i a and ea are denoted by the following graphs:
Cardy Condition for Open-Closed Field Algebras
67
the left duality maps are denoted by
and the twist and its inverse, for any object W , are denoted by
The identity (4.11) proved in [Ko2] is nothing but the following identity:
(4.12) This formula (4.12) is implicitly used in many graphic calculations in this work. a
3 a ;(1) Na a
A basis {Ya13a2 ;i }i=11 2 of Vaa13a2 for a1 , a2 , a3 ∈ I induces a basis {eaa13a2 ;i } of Hom (Wa1 Wa2 , Wa3 ). One can also denote eaa13a2 ;i as the following graph:
(4.13) Note that we will always use a to represent Wa and a to represent Wa in graphs for simplicity. By the universal property of P(z) , the map 0 : Vaa13a2 → Vaa23a1 induces a linear map 0 : HomV (Wa1 Wa2 , Wa3 ) → HomV (Wa2 Wa1 , Wa3 ) given as follows:
(4.14) Let us choose a basis { f
a3 a 1 a 2 N a1 a2 a3 ; j j=1
}
of HomV (Wa3 , Wa1 Wa2 ), denoted by
(4.15) such that
(4.16)
68
L. Kong
The following identity is proved in [Ko1]
(4.17) We prove a similar identity below. Lemma 4.1.
(4.18) Proof. Using the first balancing axiom (4.6), we have
(4.19) Then the lemma follows from the following relations:
(4.20)
(4.21) Similar to 0 , A˜ 0 , σ123 and σ132 can also be described graphically as proved in [Ko1]. We recall these results below.
Cardy Condition for Open-Closed Field Algebras
69
Proposition 4.2. a2
a3 A˜0 :
i
i
a1
a2
a1
a3
(4.22)
(4.23)
(4.24) 4.2. Graphical representation of S : τ → − τ1 . In [HKo3], we defined an action of α, β a1 ;(1) ⊗ Yaa;(2) ). More precisely, on 2 (Yaa 1; p 2 a3 ;q a ;(1)
a;(2)
(4.25)
a ;(1)
a;(2)
(4.26)
α(2 (Yaa1 1 ; p ⊗ Ya2 a3 ;q )) : ⊕a2 ,a3 ∈I Wa2 ⊗ Wa3 → G1;2 , β(2 (Yaa1 1 ; p ⊗ Ya2 a3 ;q )) : ⊕a2 ,a3 ∈I Wa2 ⊗ Wa3 → G1;2 are defined by a ;(1)
a;(2)
(α(2 (Yaa1 1 ; p ⊗ Ya2 a3 ;q )))(w2 ⊗ w3 ) a ;(1)
a;(2) ))(w2 2 a2 ;q
= (2 (Yaa1 1 ; p ⊗ Ya
⊗ w3 ; z 1 , z 2 − 1; τ ),
a1 ;(1) (β((Yaa ⊗ Yaa;(2) )))(w2 ⊗ w3 ) 1; p 2 a3 ;q a1 ;(1) ⊗ Yaa;(2) ))(w2 ⊗ w3 ; z 1 , z 2 + τ ; τ ) = (2 (Yaa 1; p 2 a3 ;q
(4.27)
if w2 ⊗ w3 ∈ Wa2 ⊗ Wa3 , and by 0 if otherwise. a1 a We also showed in [HKo3] that α induces an automorphism on ⊕a∈I Vaa 1 ⊗ Va2 a3 given as follows: a ;(1) a;(2) a ;(1) a;(2) a ;(3) b;(4) Yaa1 1 ;i ⊗ Ya2 a3 ; j → e−2πi h a3 F −1 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ; Ya21b;k ⊗ Ya3 a1 ;l ) b,c∈A k,l, p,q a ;(3)
b;(4)
a ;(5)
c;(6)
F(Ya21b;k ⊗ 2−1 (Ya3 a1 ;l ); Yca11 ; p ⊗ Ya2 a3 ;q ) a ;(5)
c;(6)
Yca11 ; p ⊗ Ya2 a3 ;q .
(4.28)
a1 a We still denote this automorphism and its natural extension on ⊕a,a1 ∈I Vaa 1 ⊗ Va2 a3 by α. The following lemma follows immediately from (4.28).
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Lemma 4.3. α can also be expressed graphically as follows:
(4.29) For β, we prefer to use maps A˜ 0 and Aˆ 0 defined in (3.35) instead of the map A0 used in [HKo3]. We obtain the following lemma, which is proved in the Appendix. a1 a Lemma 4.4. β also induces an automorphism on ⊕a,a1 ∈I Vaa 1 ⊗ Va2 a3 given by
a ;(1)
a;(2)
Yaa1 1 ;i ⊗ Ya2 a3 ; j →
b∈I k,l c∈I p,q
a ;(1)
a ;(3)
F −1 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ; Ya21b;k ⊗ Ya3 a1 ;l ) a;(2)
b;(4)
a ;(3) b;(4) b ;(5) c;(6) F( A˜ 0 (Ya21b;k ) ⊗ A˜ 0 (Ya3 a1 ;l ), Ycb ; p ⊗ Ya2 a3 ;q )
b ;(5) c;(6) Aˆ 0 (Ycb ; p ) ⊗ 20 (Ya2 a3 ;q ).
(4.30)
We still denote this automorphism by β. Lemma 4.5. β can be expressed graphically as follows:
(4.31) Proof. Using (4.30), we can see that β is the composition of following maps
a1 a ⊕a,a1 ∈I Vaa 1 ⊗ Va2 a3
F −1
/ ⊕b,a ∈I V a1 ⊗ Vab a 1 a2 b 3 1
b ⊗ Vc o ⊕b,c∈I Vcb a2 a3
A˜ 0 ⊗ A˜ 0
/ ⊕b,a ∈I V b ⊗ V a1 (4.32) 1 a3 b a2 a1
Aˆ 0 ⊗20
F
b ⊗ Vc . ⊕b,c∈I Vcb a2 a3
Cardy Condition for Open-Closed Field Algebras
71
By the commutative diagram (4.1), (4.32) can be rewritten graphically as follows:
(4.33) a1 a We have introduced S, α, β all as isomorphisms on ⊕a,a1 ∈I Vaa 1 ⊗Va2 a3 . They satisfy the following well-known equation [MSei1,MSei3,H10,HKo3]:
Sα = β S.
(4.34)
We proved in [HKo3] that S is determined by the identity (4.34) up to a constant See . We will solve Eq. (4.34) for S graphically below. Proposition 4.6.
(4.35) Proof. Since we know that Eq. (4.34) determines S up to an overall constant See , we only need to check that (4.35) gives a solution to (4.34).
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Combining (4.31) with (4.35), we obtain that
(4.36) The diagram in the right hand side of (4.36) can be deformed as follows:
(4.37) By (4.18), we have
(4.38) On the other hand, combining (4.29) with (4.35), we obtain
(4.39) Notice the diagram on the right hand side of (4.39) is exactly a deformation of the diagram on the right hand side of (4.38). Hence we obtain that the map defined as (4.35) gives a solution to the equation Sα = β S. To determine S completely, we need to determine See . This can be done by using other identities satisfied by S. Proposition 4.7. a ;(1) a ;(1) S 2 (a)(Yaa1 1 ;i ) = A˜ 0 (Yaa1 1 ;i )
(4.40)
Cardy Condition for Open-Closed Field Algebras
73 a ;(1)
a1 Proof. By the definition of S-cation on Yaa1 1 ;i ∈ Vaa 1 , we have
1 (S
2
a1 ;(1) (a)(Yaa ))(wa ; z, τ ) 1 ;i
=
a1 ;(1) 1 (Yaa ) 1 ;i
τ
L(0)
−1 τ
L(0) wa ; −z, τ
(4.41)
for wa ∈ Wa . Keep in mind our convention on the branch cut for a logarithm. We have
τ L(0)
−1 τ
L(0)
wa = eπi L(0) wa .
Hence we obtain a ;(1)
a ;(1)
1 (S 2 (a)(Yaa1 1 ;i ))(wa ; z, τ ) = 1 (Yaa1 1 ;i )(eπi L(0) wa ; −z, τ ).
(4.42)
By (A.47), we also have a ;(1)
1 (Yaa1 1 ;i )(eπi L(0) wa ; −z, τ ) c L(0)− 24 a ;(1) = E TrWa1 Yaa1 1 ;i (U(e−2πi z )eπi L(0) wa , e−2πi z )qτ c L(0)− 24 a1 ;(1) = E Tr(Wa1 ) A˜ 0 (Yaa )(eπi L(0) U(e2πi z )wa , eπi e2πi z )qτ 1 ;i c L(0)− 24 a ;(1) = E Tr(Wa1 ) A˜ 0 (Yaa1 1 ;i )(U(e2πi z )wa , e2πi z )qτ a1 ;(1) ))(wa ; z, τ ). = 1 ( A˜ 0 (Yaa 1 ;i
(4.43)
Therefore, combining (4.42) and (4.43), we obtain (4.40).
The following lemma is proved in [BK2]. Lemma 4.8. Let D 2 = a∈I dim2 a. Then D = 0 and we have
(4.44) Proposition 4.9. (See )2 =
1 . D2
(4.45)
Proof. By (4.35), we have
(4.46)
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Apply (4.12) to the graph in the right hand side of (4.46), we obtain
(4.47) By (4.44), the right-hand side of (4.47) equals
(4.48) By (4.40) and (4.22), we obtain (4.45).
So far, we have determined See up to a sign. Now we consider the relation between S and another generator of the modular group T : τ → τ + 1. We define a T -action on a1 1 (Yaa ) as follows: 1 ;i a1 a1 T (1 (Yaa ))(wa ; z, τ ) := 1 (Yaa )(wa ; z, τ + 1). 1 ;i 1 ;i
(4.49)
a1 It is clear that this action induces an action of T on Vaa 1 for all a, a1 ∈ I, given by c
2πi(h a − 24 ) a1 = e , T |Vaa
(4.50)
1
where h a is the lowest conformal weight of Wa . Lemma 4.10. S and T satisfy the following relation: (T −1 S)3 = S 2 = T −1 S 2 T.
(4.51)
Proof. Let wa ∈ Wa . We have a1 ;(1) (T −1 S)3 ((1 (Yaa ;i )))(wa ; z, τ ) ⎛1 ⎞ L(0)
L(0) −1 −1 a1 ;(1) ⎝ L(0) ) τ wa , −z, τ ⎠ . = 1 (Yaa −1 1 ;i τ − 1 − 1 τ −1
(4.52)
Keeping in mind our choice of branch cut, then it is easy to show that τ
L(0)
−1 −1 τ −1
−1
L(0)
−1 τ −1
L(0)
wa = eπi L(0) wa .
By (4.42), we obtain the first equality of (4.51). The proof of the second equality (4.51) is similar.
Cardy Condition for Open-Closed Field Algebras
Proposition 4.11. Let p± =
See =
a∈I
75
e±2πi h a dim2 a. Then we have
1 2πic/8 1 −2πic/8 e = e . p− p+
(4.53)
Proof. In the proof of Theorem 3.1.16 in [BK2], Bakalov and Kirillov proved an identity, which, in our own notation, can be written as follows: c 2c 1 1 p+ e−2πi 24 ST −1 S = e 2 e2πi 24 T ST. (See )2 D 2 Se D
By (4.51) and the fact p− p+ = D 2 which is proved in [BK2], we simply obtain that See =
1 2πic/8 e . p−
Using (4.45) and p+ p− = D 2 , we also obtain the second equality.
We thus define D := p− e−2πic/8 = p+ e2πic/8 .
(4.54) D 2 . Then the action of mod-
Notice that this notation is compatible with the definition of ular transformation S(a) on ⊕a1 ∈I HomV (Wa Wa1 , Wa1 ) can be expressed graphically as follows:
(4.55) Proposition 4.12.
(4.56) Proof. Composing the map (4.55) with (4.56), we obtain a map given as follows:
(4.57) Apply (4.44) to the graph in (4.57), it is easy to see that the above map is the identity map. Remark 4.13. Bakalov and Kirillov obtained the same formula (4.55) in [BK2] by directly working with the modular tensor category and solving equations obtained in the so-called Lego-Teichmüller game [BK1]. In our approach, we see the direct link between the modular transformations of q-traces of the product (or iterate) of intertwining operators and their graphic representations in a modular tensor category.
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Proposition 4.14.
(4.58)
(4.59) Proof. We only prove (4.58). The proof of (4.59) is analogous to that of (4.58). It is enough to show that the pairing between the image of (4.35) and that of (4.58) still gives δi j . This can be proved as follows:
(4.60) By (4.44), the right hand side of (4.60) equals
5. Categorical Formulations and Constructions In this section, we give a categorical formulation of modular invariant conformal full field algebra over V L ⊗ V L , open-string vertex operator algebra over V equipped with nondegenerate invariant bilinear forms and Cardy condition. Then we introduce a notion called Cardy CV |CV ⊗V -algebra. In the end, we give a categorical construction of such an algebra in the Cardy case [FFFS2]. 5.1. Modular invariant CV L ⊗V R -algebras. We first recall the notion of coalgebra and Frobenius algebra ([FS]) in a tensor category. Definition 5.1. A coalgebra A in a tensor category C is an object with a coproduct ∈ Mor(A, A ⊗ A) and a counit ∈ Mor(A, 1C ) such that ( ⊗ id A ) ◦ = (id A ⊗ ) ◦ , ( ⊗ id A ) ◦ = id A = (id A ⊗ ) ◦ , which can also be expressed in term of the following graphic equations:
(5.1)
Cardy Condition for Open-Closed Field Algebras
77
Definition 5.2. Frobenius algebra in C is an object that is both an algebra and a coalgebra and for which the product and coproduct are related by (id A ⊗ m) ◦ ( ⊗ id A ) = ◦ m = (m ⊗ id A ) ◦ (id A ⊗ ),
(5.2)
or as the following graphic equations:
(5.3) A Frobenius algebra is called symmetric if the following condition is satisfied:
(5.4) Let V L and V R be vertex operator algebras satisfying the conditions in Theorem 0.1. Then the vertex operator algebra V L ⊗ V R also satisfies the conditions in Theorem 0.1 [HKo1]. Thus CV L ⊗V R also has a structure of modular tensor category. In particular, we choose the braiding structure on CV L ⊗V R to be R+− which is defined in [Ko2]. The twist θ+− for each V L ⊗ V R -module is defined by θ+− = e−2πi L
L (0)
⊗ e2πi L
R (0)
.
(5.5)
Duality maps are naturally induced from those of CV L and CV R . The following theorem is proved in [Ko1]. Theorem 5.3. The category of conformal full field algebras over V L ⊗V R equipped with nondegenerate invariant bilinear forms is isomorphic to the category of commutative Frobenius algebra in CV L ⊗V R with a trivial twist. Remark 5.4. In a ribbon category, it was proved in [FFRS] that a commutative Frobenius algebra with a trivial twist is equivalent to a commutative symmetric Frobenius algebra. Let I L and I R denote the set of equivalent class of irreducible V L -modules and R V -modules respectively. We use a and ai for i ∈ N to denote elements in I L and we use e to denote the equivalent class of V L . We use a¯ and a¯ i for i ∈ N to denote elements in I R , and e¯ to denote the equivalent class of V R . For each a ∈ I L (a¯ ∈ I R ), we choose a representative Wa (Wa¯ ). We denote the vector space of intertwining operators of type Wa W¯ 3 3 and W aW as Vaa13a2 and V¯ aa¯¯13a¯ 2 respectively, the fusion rule as Naa13a2 and Na¯a¯13a¯ 2 Wa 1 Wa 2 a¯ 1 a¯ 2 respectively. A conformal full field algebra over V L ⊗ V R , denoted as Acl , is a direct sum of irreducible modules of V L ⊗ V R , i.e. N Acl = ⊕α=1 Wr (α) ⊗ Wr¯ (α) ,
(5.6)
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c } where r : {1, . . . , N } → I L and r¯ : {1, . . . , N } → I R , for some N ∈ Z+ . Let {eab;i ¯ and {e¯c¯ } be basis for V c and V¯ c¯ , and { f ab } and { f¯a¯ b } be the dual basis respectively. ¯ j a¯ b;
a¯ b¯
ab
c; ¯ j
c;i
Then the vertex operator Y can also be expanded as follows: γ r¯ (γ ); j r (α)r (β) r¯ (α)¯r (β) r (γ );i Y= dαβ ( fr (γ );i , f¯r¯ (γ ); j ) er (α)r (β) ⊗ e¯r¯ (α)¯r (β) ,
(5.7)
α,β,γ i, j
γ
where dαβ defines a bilinear map r (γ ) r¯ (γ ) (Vr (α)r (β) )∗ ⊗ (V¯ r¯ (α)¯r (β) )∗ → C
for all α, β, γ = 1, . . . , N for some N ∈ N. diag Since the trace function picks out γ = β terms, we define Yα by β r¯ (β); j r (α)r (β) r¯ (α)¯r (β) r (β);i Ydiag := dαβ ( fr (β);i , f¯r¯ (β); j ) er (α)r (β) ⊗ e¯r¯ (α)¯r (β) . α β
(5.8)
i, j
diag Let Ydiag := α Yα . Of course, it is obvious to see that such defined Ydiag is independent of the choice of basis. We denote the representation of the modular transformation b b¯ L R ¯ respectively. S : τ → −1 ¯ I R Va¯ b¯ , by S (a) and S (a) τ on ⊕b∈I L Vab and ⊕b∈ In [HKo3], we defined the notion of modular invariant conformal full field algebra over V L ⊗ V R (see [HKo3] for the precise definition). It basically means that the n-point genus-one correlation functions built out of q-q-traces ¯ are invariant under the action of modular group S L(2, Z) for all n ∈ N. Moreover, we proved the following results in [HKo3]. Proposition 5.5. Acl , a conformal full field algebra over V L ⊗ V R , is modular invariant if it satisfies c L − c R = 0 mod 24 and = Ydiag S L (r (α)) ⊗ (S R (¯r (α)))−1 Ydiag α α
(5.9)
for all α = 1, . . . , N . We denote the morphism in HomC V L ⊗V R (Wr (α) ⊗ Wr¯ (α) ) (Wr (β) ⊗ Wr¯ (β) ), Wr (β) ⊗ Wr¯ (β) diag
which corresponds to Yα gorical condition:
by m Ydiag . Then (5.9) is equivalent to the following cateα
S L (r (α)) ⊗ (S R (¯r (α)))−1 m Ydiag = m Ydiag α
α
(5.10)
for all α = 1, . . . , N . Acl a⊗a;i ¯ Now we choose a basis {ba⊗ a;i ¯ } of HomC V L ⊗V R (Wa ⊗ Wa¯ , Acl ) and a basis {b Acl } of HomC V L ⊗V R (Acl , Wa ⊗ Wa¯ ) as follows:
(5.11)
Cardy Condition for Open-Closed Field Algebras
79
satisfying the following conditions: for all a ∈ I L , a¯ ∈ I R ,
(5.12) Then the condition (5.10) can be further expressed graphically as follows:
(5.13) for all a ∈ I L , a¯ ∈ I R , where D L and D R are the D defined by (4.54) in CV L and CV R respectively. Definition 5.6. Let V L and V R be so that c L − c R = 0 mod 24. A modular invariant CV L ⊗V R -algebra is an associative algebra (A, µ A , ι A ) satisfying the condition (5.13). Some properties of the modular invariant algebra follow immediately from the above definition, such as the following famous condition [MSei3]: a∈I L ,a∈ ¯ IR
(S L (e))ab Nbb¯ (S R (e)) ¯ −1 = Na a¯ , b¯ a¯
(5.14)
where Na a¯ is the multiplicity of Wa ⊗ Wa¯ in Acl for a ∈ I L , a¯ ∈ I R . We leave a more systematic study of modular invariant CV L ⊗V R -algebras to elsewhere. Theorem 5.7. Let V L and V R be so that c L − c R = 0 mod 24. The following two notions are equivalent: 1. modular invariant conformal full field algebra over V L ⊗ V R equipped with a nondegenerate invariant bilinear form, 2. modular invariant commutative Frobenius algebra with a trivial twist. Proof. The theorem follows from Theorem 5.3 and the equivalence between (5.9) and (5.13) immediately.
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Remark 5.8. In the case V L ∼ =C∼ = V R , a modular invariant commutative Frobenius algebra with a trivial twist in CV L ⊗V R is simply a commutative Frobenius algebra over C, which is equivalent to a 2-dimensional topological field theory (see for example [BK2]). In this case, the modular invariance condition holds automatically.
5.2. Cardy CV |CV ⊗V -algebras. For an open-string vertex operator algebra Vop over V equipped with a nondegenerate invariant bilinear form (·, ·)op , there is an isomorphism ϕop : Vop → Vop induced from (·, ·)op (recall (3.40)). f In this case, Vop is a V -module and Yop is an intertwining operator. By comparing (1.26) with (3.37), and (1.27) with (3.38), we see that the conditions (1.26) and (1.27) can be rewritten as −1 Yop = ϕop ◦ σ123 (Yop ) ◦ (ϕop ⊗ idVop )
(5.15)
−1 ◦ σ132 (Yop ) ◦ (idop ⊗ ϕop ). = ϕop
(5.16)
f
f
f
Remark 5.9. The representation theory of the open-string vertex operator algebra can f a right V -module structure and be developed. In that context, σ123 (Yop ) gives Vop op Eq. (5.15) is equivalent to the statement that ϕop is an isomorphism between two right f a left V -module structure and Eq. (5.16) Vop -modules. Similarly, σ132 (Yop ) gives Vop op is equivalent to the statement that ϕop is an isomorphism between two left Vop -modules. But we do not need it in this work. Theorem 5.10. The category of open-string vertex operator algebras over V equipped with a nondegenerate invariant bilinear form is isomorphic to the category of symmetric Frobenius algebras in CV . Proof. We have already shown in [HKo1] that an open-string vertex operator algebra over V is equivalent to an associative algebra in CV . Let Vop be an open-string vertex operator algebra over V . Giving a nondegenerate invariant bilinear form (recall (1.26) and (1.27)) on Vop is equivalent to giving an satisfying the conditions (5.15) and (5.16). If we define isomorphism ϕop : Vop → Vop
(5.17)
(5.18)
Cardy Condition for Open-Closed Field Algebras
81
then (5.15) and (5.16) can be rewritten as
(5.19)
(5.20) Using the map ϕop and its inverse, we can obtain a natural coalgebra structure on F defined as follows:
(5.21) Similar to the proof of [Ko1, thm.4.15], (5.19) and (5.20) imply that such defined Vop and Vop give Vop a Frobenius algebra structure. Moreover, we also showed in [Ko1] that (5.19) implies the equality between ϕop and the left hand side of (5.4). Similarly, using (5.20), we can show that the right hand side of (5.4) also equals ϕop . Thus Vop has a structure of symmetric Frobenius algebra. We thus obtain a functor from the first category to the second category. Conversely, given a symmetric Frobenius algebra in CV , in [HKo1], we showed that it gives an open-string vertex operator algebra over V . It is shown in [FRS2] that either side of (5.4) is an isomorphism. Take ϕop to be either side of (5.4). Then (5.19) and (5.20) follow automatically from the definition (5.17) and (5.18). They are nothing but the invariance properties (recall (5.15)(5.16)) of the bilinear form associated with ϕop . Thus we obtain a functor from the second category to the first category. It is routine to check that these two functors are inverse to each other. Now we consider an open-closed field algebra over V given in (2.12) equipped with nondegenerate invariant bilinear forms (·, ·)op and (·, ·)cl . We assume that Vcl and Vop have the following decompositions: N
Ncl op Vcl = ⊕i=1 Wr L (i) ⊗ Wr R (i) , Vop = ⊕i=1 Wr (i) ,
where r L , r R : {1, . . . , Ncl } → I and r : {1, . . . , Nop } → I. We denote the embedding V
(i)
Vcl : Wr L (i) Wr R (i) → b(i)op : Wr (i) → Vop , the projection bVop : Vop → Wr (i) and b(i)
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L. Kong
T (Vcl ) by the following graphs:
(5.22) We denote the map ιcl−op : T (Vcl ) → Vop [Ko2] by the following graph:
(5.23) Now we can express the Cardy condition (3.73) in graphs. The left hand side of (3.73) can be expressed by:
(5.24) By the universal property of tensor product, for ai ∈ I, i = 1, . . . , 6, we have a canonical isomorphism: ∼ =
⊕a∈I Vaa14a ⊗ Vaa2 a3 − → HomV (Wa1 (Wa2 Wa3 ), Wa4 ) Y1 ⊗ Y2 → m Y1 ◦ (idWa1 m Y2 ).
(5.25)
Under this canonical isomorphism, the Cardy condition (3.73) can be viewed as a condition on two morphisms in HomV (Vop (Vop Wr R (i) ), Wr R (i) ). In particular, the left hand side of (3.73) viewed as a morphism in HomV (Vop (Vop Wr R (i) ), Wr R (i) ) can be expressed as follows:
(5.26)
Cardy Condition for Open-Closed Field Algebras
83
We define a morphism ι∗cl−op : Vop → T (Vcl ) by
(5.27) Then using the morphism
ι∗cl−op ,
we can rewrite the graph in (5.26) as follows:
(5.28) Using (4.55), (5.26) and (5.28), we obtain a graphic version of the Cardy condition (3.73) as follows:
(5.29) Using the Frobenius properties of Vop , one can show that (5.29) is equivalent to the following condition:
(5.30) The asymmetry between chiral and antichiral parts in (5.29)(5.30) is superficial. Using (4.44), one can show that (5.30) is further equivalent to the following condition:
(5.31) We recall a definition in [Ko2].
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Definition 5.11. An open-closed CV |CV ⊗V -algebra, denoted as (Aop |Acl , ιcl−op ), consists of a commutative symmetric associative algebra Acl in CV ⊗V , an associative algebra Aop in CV and an associative algebra morphism ιcl−op : T (Vcl ) → Vop , satisfying the following condition:
(5.32) The following theorem is proved in [Ko2]. Theorem 5.12. The category of open-closed field algebras over V is isomorphic to the category of open-closed CV |CV ⊗V -algebras. Definition 5.13. A Cardy CV |CV ⊗V -algebra is an open-closed CV |CV ⊗V -algebra (Aop |Acl , ιcl−op ) such that Acl is modular invariant commutative symmetric Frobenius algebra in CV ⊗V and Aop a symmetric Frobenius algebra in CV and the Cardy condition (5.30) or (5.31) hold. Remark 5.14. Notice that, in the case V = C, the Cardy CV |CV ⊗V -algebra (using (5.31)) exactly coincides with the usual algebraic formulation of 2-dimensional open-closed topological field theory [La,Mo1,Mo2,Se2,MSeg,AN,LP]. As we discussed in the introduction, we believe that open-closed partial conformal field theories of all genus satisfying the V -invariant boundary condition [Ko2] are classified by Cardy CV |CV ⊗V algebras. The following result is clear. Theorem 5.15. The category of open-closed field algebras over V equipped with nondegenerate invariant bilinear forms and satisfying the modular invariance condition and the Cardy condition is isomorphic to the category of Cardy CV |CV ⊗V -algebras. Definition 5.16. A V -invariant D-brane associated to a closed algebra Acl in CV ⊗V is a pair (Aop , ιcl−op ) such that the triple (Aop |Acl , ιcl−op ) gives a Cardy CV |CV ⊗V algebra. D-branes usually form a category as we will see in an example in the next subsection. 5.3. Constructions. In this section, we give a categorical construction of Cardy CV |CV ⊗V -algebra. This construction is called the Cardy case in the physics literature [FFFS2]. Let us first recall the diagonal construction of the close algebra Vcl [FFRS,HKo2, Ko1,HKo3]. We will follow the categorical construction given in [Ko1]. Let Vcl be the object in CV ⊗V given as follows: Vcl = ⊕a∈I Wa ⊗ Wa .
(5.33)
The decomposition of Vcl as a direct sum gives a natural embedding V ⊗ V → Vcl . We denote this embedding as ιcl . We define a morphism µcl ∈ HomV ⊗V (Vcl Vcl , Vcl ) by
Cardy Condition for Open-Closed Field Algebras
µcl =
a
Na13a2
a1 ,a2 ,a3 ∈A i, j=1
a a2 ; j
where eaa13a2 ;i and f a31 pairing given by
85
a a
a
f aa31;ia2 , f a 1; j2 eaa13a2 ;i ⊗ ea3 a ; j , 3
(5.34)
1 2
are basis vectors given in (4.13) and (4.15) and ·, · is a bilinear
(5.35) Notice that Vcl has the same decomposition as Vcl in (5.33). They are isomorphic as V ⊗ V -modules. There is, however, no canonical isomorphism. Now we choose a particular isomorphism ϕcl : Vcl → Vcl given by ϕcl = ⊕a∈I
D e−2πi h a idWa ⊗Wa . dim a
(5.36)
The isomorphism ϕcl induces a nondegenerate invariant bilinear form on Vcl viewed as V ⊗ V -module. The following theorem is a categorical version of Theorem 5.1 in [HKo3]. We give a categorical proof here. Theorem 5.17. (Vcl , µcl , ιcl ) together with the isomorphism ϕcl gives a modular invariant commutative Frobenius algebra in CV ⊗V with a trivial twist. Proof. It was proved in [Ko1] that (Vcl , µcl , ιcl ) together with the isomorphism ϕcl gives a commutative Frobenius algebra with a trivial twist. It remains to show the modular invariance. First, the bilinear pairing ·, · given in (5.35) can be naturally extended to a bilinear form, still denoted as ·, ·, on ⊕a,a1 ∈I (Wa1 , Wa Wa1 ) as follows: aa
1 1 1 f aaa , f bbb := δab δa1 b1 f aaa , f a ; j1 . 1 ;i 1; j 1 ;i
(5.37)
1
Then it is easy to see that to prove the modular invariance of Vcl is equivalent to prove that the bilinear form ·, · on ⊕a,a1 ∈I (Wa1 , Wa Wa1 ) defined above is invariant under 1 1 the action of (S −1 )∗ ⊗ S ∗ . Clearly, when b = a , (S −1 (a))∗ f aaa , (S(b))∗ f bbb = 0. 1 ;i 1; j When b = a , we have (using (4.58), (4.59) and (4.44))
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Now we define Vop . Let X be a V -module. Let e X : X X → V and i X : V → X X be the duality maps defined in [Ko1]. Vop := X X has a natural structure of symmetric Frobenius algebra [FS] with ιop := i X , µop := id X e X id X , op := e X and op := id X i X id X . Now we define a map ιcl−op : T (Vcl ) → Vop by
(5.38)
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87
Lemma 5.18.
(5.39) Proof. By (5.27) and (5.38), we have
(5.40) It is easy to see that the last figure in (5.40) can be deformed to that on the right hand side of (5.39). Theorem 5.19. (Vcl , Vop , ιcl−op ) is a Cardy CV |CV ⊗V -algebra. Proof. Recall that T (Vcl ) together with the multiplication morphism µT (Vcl ) = T (µcl )◦ ϕ2 and the morphism ιT (Vcl ) = T (ιcl ) ◦ ϕ0 is an associative algebra. We first prove that ιcl−op is an algebra morphism. It is clear that ιcl−op ◦ ιT (Vcl ) = ιop . It remains to show the following identity ιcl−op ◦ µT (Vcl ) = µop ◦ (ιcl−op ιcl−op ).
(5.41)
By the definition of ϕ2 , that of µcl and (5.38), we obtain
(5.42) It is easy to see that the right hand side of (5.42) equals
(5.43)
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Using (4.17) to sum up the indices c1 and i, we obtain that (5.43) further equals
(5.44) Using (4.17) again, we obtain
(5.45) the right hand side of which is nothing but µop ◦ (ιcl−op ιcl−op ). The commutativity (5.32) follows from the following identity:
(5.46) In summary, we have proved that the triple (Vcl , Vop , ιcl−op ) is an open-closed CV |CV ⊗V -algebra. It remains to show that the Cardy condition (5.30) holds. We use (5.38), (5.39) and the definition of µop and op to express both sides of (5.30) graphically. Then it is easy to see that both sides are the deformations of each other. Theorem 5.19 reflects the general fact that consistent open theories (or D-branes) for a given closed theory are not unique. Instead they form a category. There are many good questions one can ask about Cardy CV |CV ⊗V -algebra, for example its relation to the works of Fuchs, Runkel, Schweigert and Fjelstad [FS,FRS1-FRS4,FjFRS1,FjFRS2]. We leave such topics to [KR] and future publications. A. The Proof of Lemma 4.30 Lemma A.1. For w1 ∈ Wa1 , wa ∈ Wa2 and wa 3 ∈ (Wa3 ) and Yaa13a2 ∈ Vaa13a2 , we have wa 3 , Y(U(x)wa1 , x)wa2 = A˜ r (Y)(U(e(2r +1)πi x −1 )e−(2r +1)πi L(0) wa1 , e(2r +1)πi x −1 )wa 3 , wa2 .
(A.47)
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89
Proof. Using the definition of Aˆ r , we see that the left hand side of (A.47) equals A˜ r (Y)(e−x L(1) x −2L(0) U(x)wa1 , e(2r +1)πi x −1 )wa 3 , wa2 .
(A.48)
In [H8], the following formula e x L(1) x −2L(0) e(2r +1)πi L(0) U(x)e−(2r +1)πi L(0) = U(x −1 )
(A.49)
is proved. Applying (A.49) to (A.48), we obtain (A.47) immediately. Now we are ready to give a proof of Lemma 4.4. Proof. We have, for wa2 ∈ Wa2 , wa3 ∈ Wa3 , a1 ;(1) ⊗ Yaa;(2) ))(wa2 ⊗ wa3 ))(z 1 , z 2 + τ, τ ) ((2 (Yaa 1 ;i 2 a3 ; j a ;(1) = E TrWa1 Yaa1 1 ;i (U(e2πi(z 2 +τ ) )· L(0)−
c
24 (wa2 , z 1 − (z 2 + τ ))wa3 , e2πi(z 2 +τ ) )qτ ·Yaa;(2) 2 a3 ; j a ;(1) a;(2) a ;(3) b;(4) F −1 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ; Ya21b;k ⊗ Ya3 a1 ;l ) =
b∈I k,l
a ;(3) E TrWa1 Ya21b;k (U(e2πi z 1 )wa2 , e2πi z 1 ) c L(0)− 24
·Ya3 a1 ;l (U(e2πi(z 2 +τ ) )wa3 , e2πi(z 2 +τ ) )qτ b;(4)
.
(A.50) b;(4)
Using the L(0)-conjugation formula, we can move qτ from the right side of Ya3 a1 ;l to the left side of Yab;(4) . Then using the following property of trace: 3 a1 ;l TrWa1 (AB) = TrWb (B A),
(A.51)
for all A : Wb → Wa1 , B : Wa1 → Wb whenever the multiple sums in either side of (A.51) converge absolutely, we obtain that the left hand side of (A.50) equals a1 ;(1) ;(3) F −1 (Yaa ⊗ Yaa;(2) ; Yaa21b;k ⊗ Yab;(4) ) 1 ;i 2 a3 ; j 3 a1 ;l b∈I k,l
c L(0)− 24 b;(4) a ;(3) . E TrWb Ya3 a1 ;l (U(e2πi z 2 )wa3 , e2πi z 2 )Ya21b;k (U(e2πi z 1 )wa2 , e2πi z 1 )qτ (A.52)
Now apply (A.47) to (A.52). We then obtain that (A.52) equals a ;(1) a;(2) a ;(3) b;(4) F −1 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ; Ya21b;k ⊗ Ya3 a1 ;l ) b∈I k,l
a ;(3) E Tr(Wb ) A˜ r (Ya21b;k )(U(e(2r +1)πi e−2πi z 1 )e−(2r +1)πi L(0) wa2 , e(2r +1)πi e−2πi z 1 )
c L(0)− 24 b;(4) . A˜ r (Ya3 a1 ;l )(U(e(2r +1)πi e−2πi z 2 )e−(2r +1)πi L(0) wa3 , e(2r +1)πi e−2πi z 2 )qτ
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Now apply the associativity again and be careful about the branch cut as in [H8], then the left hand side of (A.50) further equals to a ;(1) a;(2) a ;(3) b;(4) F −1 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ; Ya21b;k ⊗ Ya3 a1 ;l ) b∈I k,l c∈I p,q
a ;(3) b;(4) b ;(5) c;(6) ×F( A˜ r (Ya21b;k ) ⊗ A˜ r (Ya3 a1 ;l ), Ycb ; p ⊗ Ya2 a3 ;q ) c L(0)− 24 b ;(5) ×E Tr(Wb ) qτ Ycb ; p (U(e(2r +1)πi e−2πi z 2 )
c;(6) ·Ya2 a3 ;q (e−(2r +1)πi L(0) wa2 , eπi (z 1 − z 2 ))e−(2r +1)πi L(0) wa3 , e(2r +1)πi e−2πi z 2 ) . a ;(1) a;(2) a ;(3) b;(4) = F −1 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ; Ya21b;k ⊗ Ya3 a1 ;l ) b∈I k,l c∈I p,q
a ;(3) b;(4) b ;(5) c;(6) ×F( A˜ r (Ya21b;k ) ⊗ A˜ r (Ya3 a1 ;l ), Ycb ; p ⊗ Ya2 a3 ;q ) c L(0)− 24 b ;(5) ×E TrWb qτ Aˆ r (Ycb ; p )(e(2r +1)πi L(0)
−(2r +1)πi L(0) πi −(2r +1)πi L(0) 2πi z 2 ×Yac;(6) (e w , e (z − z ))e w , e ) . a 1 2 a 2 3 a ;q 2 3
(A.53)
Choosing r = 0 and using Y(·, e2πi x)· = 20 (Y)(·, x)·, we obtain a ;(1)
a;(2)
((2 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ))(wa2 ⊗ wa3 ))(z 1 , z 2 + τ, τ ) a ;(1) a;(2) a ;(3) b;(4) F −1 (Yaa1 1 ;i ⊗ Ya2 a3 ; j ; Ya21b;k ⊗ Ya3 a1 ;l ) = b∈I k,l c∈I p,q
;(3) b ;(5) c;(6) ×F( A˜ 0 (Yaa21b;k ) ⊗ A˜ 0 (Yab;(4) ), Ycb ; p ⊗ Ya a ;q ) 3 a1 ;l 2 3 c L(0)− 24 b ;(5) c;(6) 2 ˆ ×E TrWb A0 (Ycb ; p )(0 (Ya2 a3 ;q )(wa2 , z 1 − z 2 )wa3 , e2πi z 2 )qτ .
(A.54) By the linear independency proved in [H10] of the last factor in each term of the above sum, it is clear that β induces a map given by (4.30). Acknowledgement. The results in Sect. 2.4 and 3.1 are included in author’s thesis. I want to thank my advisor Yi-Zhi Huang for introducing me to this interesting field and for his constant support and many important suggestions for improvement. I thank C. Schweigert for telling me the meaning of boundary states from a physical point of view. I also want to thank I. Frenkel, J. Fuchs, A. Kirillov, Jr. and C. Schweigert for some inspiring conversations on the subject of Sect. 4.2.
References [AN] [BHL] [BK1] [BK2] [C1]
Alexeevski, A., Natanzon, S.M.: Noncommutative two-dimensional topological field theories and hurwitz numbers for real algebraic curves. Sel. Math. 12(3–4), 377–397 (2006) Barron, K., Huang, Y.-Z., Lepowsky, J.: Factorization of formal exponential and uniformization. J. Alg. 228, 551–579 (2000) Bakalov, B., Kirillov, Jr., A.: On the lego-teichmüller game. Transform. Groups 5, 207 (2000) Bakalov, B., Kirillov, Jr., A.: Lectures on Tensor Categories and Modular Functors. University Lecture Series, Vol. 21, Providence, RI: Amer. Math. Soc., 2001 Cardy, J.L.: Conformal invariance and surface critical behavior. Nucl. Phys. B 240, 514–532 (1984)
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Cardy, J.L.: Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories. Nucl. Phys. B275, 200–218 (1986) [C3] Cardy, J.L.: Operator content of two-dimensional conformal invariant theories. Nucl. Phys. B270, 186–204 (1986) [C4] Cardy, J.L.: Boundary conditions, fusion rules and the verlinde formula. Nucl. Phys. B324, 581–596 (1989) [DLM] Dong, C.-Y., Li, H.-S., Mason, G.: Modular-invariance of trace functions in orbifold theory and generalized moonshine. Commun. Math. Phys. 214, 1–56 (2000) [FFFS1] Felder, G., Fröhlich, J., Fuchs, J., Schweigert, C.: The geometry of wzw branes. J. Geom. Phys. 34, 162–190 (2000) [FFFS2] Felder, G., Fröhlich, J., Fuchs, J., Schweigert, C.: Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology. Compositio Math. 131, 189–237 (2002) [FFRS] Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Correspondences of ribbon categories. Adv. Math. 199(1), 192–329 (2006) [FjFRS1] Fjelstad, J., Fuchs, J., Runkel, I., Schweigert, C.: Tft construction of rcft correlators v: proof of modular invariance and factorisation. Theory and Appl. of Categ 16, 392–433 (2006) [FjFRS2] Fjelstad, J., Fuchs, J., Runkel, I., Schweigert, C.: Uniqueness of open/closed rational CFT with given algebra of open states. http://arXiv.org/listhep-th/0612306, 2006 [FHL] Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 104, 1993 [FRS1] Fuchs, J., Runkel, I., Schweigert, C.: Conformal correlation functions, frobenius algebras and triangulations. Nucl. Phys. B624, 452–468 (2002) [FRS2] Fuchs, J., Runkel, I., Schweigert, C.: Tft construction of rcft correlators I: partition functions. Nucl. Phys. B678, 511 (2004) [FRS3] Fuchs, J., Runkel, I., Schweigert, C.: Tft construction of rcft correlators III: simple currents. Nucl. Phys. B694, 277 (2004) [FRS4] Fuchs, J., Runkel, I., Schweigert, C.: Tft construction of rcft correlators IV: structure constants and correlation functions. Nucl. Phys. B 715(3), 539–638 (2005) [FS] Fuchs, J., Schweigert, C.: Category theory for conformal boundary conditions. Fields Institute Commun. 39, 25 (2003) [G] Gan, W.L.: Koszul duality for dioperads. Math. Res. Lett. 10(1), 109–124 (2003) [H1] Huang, Y.-Z.: Geometric interpretation of vertex operator algebras. Proc. Natl. Acad. Sci. USA 88, 9964–9968 (1991) [H2] Huang, Y.-Z.: A theory of tensor products for module categories for a vertex operator algebra, iv. J. Pure Appl. Alg. 100, 173–216 (1995) [H3] Huang, Y.-Z.: Two-dimensional conformal geometry and vertex operator algebras. Progress in Mathematics, Vol. 148, Boston: Birkhäuser, 1997 [H4] Huang, Y.-Z.: Generalized rationality and a “jacobi identity” for intertwining operator algebras. Selecta Math. (N. S.) 6, 225–267 (2000) [H5] Huang, Y.-Z.: A functional-analytic theory of vertex (operator) algebras. I. Commun. Math. Phys. 204(1), 61–84 (1999) [H6] Huang, Y.-Z.: A functional-analytic theory of vertex (operator) algebras. II. Commun. Math. Phys. 242(3), 425–444 (2003) [H7] Huang, Y.-Z.: Differential equations and intertwining operators. Commun. Contemp. Math. 7, 375–400 (2005) [H8] Huang, Y.-Z.: Riemann surfaces with boundaries and the theory of vertex operator algebras. Fields Institute Commun. 39, 109 (2003) [H9] Huang, Y.-Z.: Differential equations, duality and modular invariance. Commun. Contemp. Math. 7, 649–706 (2005) [H10] Huang, Y.-Z.: Vertex operator algebras and the verlinde conjecture. Commun. Contemp. Math. 10(1), 103–154 (2008) [H11] Huang, Y.-Z.: Rigidity and modularity of vertex tensor categories. math.QA/0502533, 2005 [HKo1] Huang, Y.-Z., Kong, L.: Open-string vertex algebra, category and operad. Commun. Math. Phys. 250, 433–471 (2004) [HKo2] Huang, Y.-Z., Kong, L.: Full field algebras. Commun. Math. Phys. 272, 345–396 (2007) [HKo3] Huang, Y.-Z., Kong, L.: Modular invariance for conformal full field algebras, math.QA/0609570, 2006 [HL1] Huang, Y.-Z., Lepowsky, J.: Tensor products of modules for a vertex operator algebra and vertex tensor categories, In: Lie Theory and Geometry, in honor of Bertram Kostant, R. Brylinski, J.-L. Brylinski, V. Guillemin, V. Kac (Eds.) Boston: Birkhäuser, 1994, pp. 349–383
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[Z]
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Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra, I. Selecta Math. (N.S.) 1, 699–756 (1995) Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra, II. Selecta Math. (N.S.) 1, 757–786 (1995) Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra, III. J. Pure Appl. Alg. 100, 141–171 (1995) Ishibashi, N.: The boundary and crosscap states in conformal field theories. Mod. Phys. Lett. A4, 251 (1989) Kong, L.: Full field algebras, operads and tensor categories. Adv. Math. 213, 271–340 (2007) Kong, L.: Open-closed field algebras. Commun. Math. Phys. 280, 207–261 (2008) Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35–72 (1999) Kirillov,, Jr. A.: On an inner product in modular tensor categories. J. Amer. Math. Soc 9(4), 1135–1169 (1996) Kriz, I.: On spin and modularity of conformal field theory. Ann. Sci. École Norm. Sup. (4) 36(1), 57–112 (2003) Kong, L., Runkel, I.: Cardy algebras and sewing constraints, I, II, in preparation Lazaroiu, C.I.: On the structure of open-closed topological field theory in two dimensions. Nucl. Phys. B603, 497–530 (2001) Lewellen, D.C.: Sewing constraints for conformal field theories on surfaces with boundaries. Nucl. Phys. B372, 654 (1992) Lyubashenko, V.: Modular transformations for tensor categories. J. Pure Appl. Alg. 98(3), 297–327 (1995) Lepowsky, J., Li, H.-S.: Introduction to vertex operator algebras and their representations. Progress in Mathematics, 227, Boston, MA: Birkhäuser Boston, Inc. 2004 Lauda, A., Pfeiffer, H.: Open-closed strings: two-dimensional extended tqfts and frobenius algebras. Topology Appl 155(7), 623–666 (2008) Miyamoto, M.: Modular invariance of vertex operator algebras satisfying c2 -cofiniteness. Duke Math. J. 122(1), 51–91 (2004) Miyamoto, M.: Intertwining operators and modular invariance. math.QA/0010180, 2000 Moore, G.: Some comments on branes, g-flux, and k-theory. Internat. J. Mod. Phys. A16, 936–944 (2001) Moore, G.: D-branes, RR-Fields and K-Theory, I, II, III, VI. Lecture notes for the ITP miniprogram: The duality workshop: a Math/Physics collaboration, June, 2001; http://online.itp.ucsb. edu/online/mp01/moore1 Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989) Moore, G., Seiberg, N.: Naturality in conformal field theory. Nucl. Phys. B313, 16–40 (1989) Moore, G., Seiberg, N.: Lecture on RCFT. In: Physics Geometry and Topology. Edited by H.C. Lee, New York: Plenum Press, 1990 Moore, G., Segal, G.: D-branes and K-theory in 2D topological field theory. hep-th/0609042, 2006 Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials of quantum groups. Invent. Math. 103(3), 547–597 (1991) Segal, G.: The definition of conformal field theory, Preprint, 1988; also In: Topology, geometry and quantum field theory. ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge: Cambridge University Press, 2004, 421–577 Segal, G.: Topological structures in string theory. R. Soc. Lond. Philos. Trans. A359, 1389–1398 (2001) Sonoda, H.: Sewing conformal field theories, I, II. Nucl. Phys. B311, 401–416, 417–432 (1988) Turaev.: Quantum invariant of knots and 3-manifolds, de Gruyter Studies in Mathematics, Vol. 18, Berlin: Walter de Gruyter, 1994 Voronov, A.A.: The Swiss-cheese operad. In: Homotopy invariant algebraic structures, in honor of J. Michael Boardman, Proc. of the AMS Special Session on Homotopy Theory, Baltimore, 1998, ed. J.-P. Meyer, J. Morava, W. S. Wilson, Contemporary Math., Vol. 239, pp. 365–373 Amer. Math. Soc., Providence, RI, 1999 Zhu, Y.-C.: Modular invariance of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)
Communicated by N.A. Nekrasov
Commun. Math. Phys. 283, 93–125 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0456-y
Communications in
Mathematical Physics
Sharp Threshold of Global Existence and Instability of Standing Wave for a Davey-Stewartson System Zaihui Gan1,2 , Jian Zhang1 1 College of Mathematics and Software Science, Sichuan Normal University,
Chengdu 610068, P. R. China. E-mail:
[email protected]
2 The Institute of Mathematical Sciences, The Chinese University of Hong Kong,
Shatin, N.T., Hong Kong Received: 20 July 2007 / Accepted: 10 September 2007 Published online: 7 March 2008 – © Springer-Verlag 2008
Abstract: This paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave φ(t, x) = eiωt u(x) for the system: iφt + φ + a|φ| p−1 φ + bE 1 (|φ|2 )φ = 0, t ≥ 0, x ∈ R N ,
(DS)
+2 where a > 0, b > 0, 1 < p < (NN−2) + and N ∈ {2, 3}. Firstly, by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and +2 blowup of the solution to the Cauchy problem for (DS) provided 1 + N4 ≤ p < (NN−2) +. Secondly, by using the scaling argument, we show how small the initial data are for the global solutions to exist. Finally, we prove the strong instability of the standing waves with finite time blow up for any ω > 0 by combining the former results.
1. Introduction In this paper we study the generalized Davey-Stewartson system: iφt + φ + a|φ| p−1 φ + bE 1 (|φ|2 )φ = 0, t ≥ 0, x ∈ R N ,
(1.1)
where φ = φ(t, x) is a complex-valued function of (t, x) ∈ R+ × R N , N ∈ {2, 3}, +2 is the usual Laplacian operator in R N , a and b are positive constants, 1 < p < (NN−2) +
+2 + = N − 2 when (we use the convention: (NN−2) + = ∞ when N = 2, and (N − 2) N = 3), E 1 is the singular integral operator with symbol σ1 (ξ ) = ξ12 /|ξ |2 , ξ ∈ R N , This work is supported by NSFC (No. 10726034, 10771151), Sichuan Youth Science and Technology Foundation (07ZQ026-009) and The Institute of Mathematical Sciences at The Chinese University of Hong Kong.
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E 1 (ψ) = F −1 ξ12 /|ξ |2 Fψ, F −1 and F are the Fourier inverse transform and Fourier transform on R N , respectively (see [5,6,14,15]). The Davey-Stewartson system (1.1) has its origin in fluid mechanics, where for p = 3 and N = 2, it appears as mathematical models for the evolution of shallow-water waves having one predominant direction of travel (see[4–6]). Moreover, (1.1) is the N -dimensional extension of the Davey-Stewartson systems in the elliptic-elliptic case, iφt + λφx x + µφ yy + a|φ|p−1 φ = b1 φψx , (1.2) νψx x + ψ yy = −b2 |φ|2 x , where λ, µ, ν > 0, b1 , b2 are positive constants and a ∈ R. In this case, (1.2) describes the time evolution of two-dimensional surface of water waves having one propagation preponderantly in the x-direction (see [4–6]). In addition, according to the signs of µ and ν, system (1.2) may be classified as: − elliptic − elliptic : −elliptic − hyperbolic : −hyperbolic − elliptic : −hyperbolic − hyperbolic :
µ > 0, µ > 0, µ < 0, µ < 0,
ν > 0, ν < 0, ν > 0, ν < 0,
(1.3) (1.4) (1.5) (1.6)
although the last case does not seem to occur in the context of water waves [see also [4]]. A large amount of work [14,15,17,20–23,29,30] has been devoted to the study of the Davey-Stewartson systems. The Cauchy problem for the Davey-Stewartson systems in all physical relevant cases ((1.3)-(1.5)) has been studied in [14] by using functional analytical methods, in which Ghidaglia and Saut proved the solvability in the Sobolev spaces H 1 = H 1 (R2 ). In the case (1.4), Tsutsumi in [29] obtained the L p (R2 )-decay estimates of solutions of (1.1) (2 < p < ∞). In [23], Ozawa obtained the exact blowup solutions of Cauchy problem for (1.2). Ohta in [20–22] discussed the existence and nonexistence of stable standing waves for (1.1) under some conditions. Moreover, Guo and Wang in [15] studied blow-up in a finite time and global existence of the solutions to the Cauchy problem for the generalized Davey-Stewartson system with the case (1.3); Wang and Guo in [30] investigated the initial value problem and scattering of solutions to the generalized Davey-Stewartson systems; Li, Guo and Jian in [17] exploited the global existence and blowup of solutions to degenerate Davey-Stewartson equations. From the view-point of physics, the following problems are very important. Under what conditions will the water waves become unstable to collapse (blowup)? And under what conditions will the water waves be stable for all time (global existence)? Especially the sharp thresholds for blowup and global existence are pursued strongly (see Zhang [32–34] and their references). In the present paper, we investigate the sharp threshold of blowup and global existence of the Cauchy problem to the generalized Davey-Stewartson system (1.1) and the +2 instability of the standing waves for (1.1) provided 1 + N4 ≤ p < (NN−2) + . To our knowledge, although there are some results [9,33,34] concerning the sharp condition for global existence of the solutions to the Cauchy problem for nonlinear Schrödinger equations, on the study of sharp threshold of blowup and global existence of the solutions to the Cauchy problem of the generalized Davey-Stewartson system (1.1), there is little work in the literature. For 1+ N4 ≤ p ≤ 3, in [7,8,27], the authors obtained the sharp threshold of global existence to the Cauchy problem for (1.1) by using the potential well argument
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
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[24] and concavity method [16]. However, the methods in [7,8,27] can not be used to +2 solve the above problems for (1.1) for 3 < p < (NN−2) + . Therefore, it is interesting to +2 introduce some other methods to solve the related problems for 3 < p < (NN−2) + . In the present paper, motivated by the study of nonlinear Schrödinger equations [1,28,31,34], we construct a type of cross-constrained variational problem and establish its property, then apply it to (1.1). By defining the corresponding cross-invariant manifolds under the flow generated by the Cauchy problem of the Davey-Stewartson system (1.1), we establish the sharp threshold for global existence and blowup of the solutions to the Cauchy +2 problem for (1.1) with 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. In addition, by using the scaling arguments, we show how small the initial data are for the global solution of the Cauchy problem for (1.1) to exist. Finally, applying the sharp threshold and the property of the cross constrained variational problem, we can also prove the strong instability of +2 standing waves for (1.1) with finite time blowup for 1+ N4 ≤ p < (NN−2) + and N ∈ {2, 3}. +2 To our knowledge, these results are new for system (1.1) with 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. As for the standing waves for (1.1), Cipolatti [4] treated the standing waves with the existence of ground state by means of P. L. Lion’s concentration-compactness method [18,19]. By standing waves we mean special periodic solutions of the form
φ(t, x) = eiωt ϕ(x), where ω ∈ R and ϕ is a ground state of the problem: −ψ + ωψ − a|ψ| p−1 ψ − bE 1 (|ψ|2 )ψ = 0, x ∈ R N , ψ ∈ H 1 (R N ), ψ ≡ 0.
(1.7)
(1.8)
The so-called ground states are standing waves which minimize the action among all nontrivial solutions of the form (1.7). Both for physical and mathematical reasons, in recent years, many authors paid much attention to the stability and instability of the standing waves for the nonlinear Schrödinger equation. Berestycki and Cazenave [1] investigated the instability of ground states; Cazenave and Lions [3] as well as Grillakis, Shatah and Strauss [13] obtained the existence of stable standing waves; Shatah and Strauss [26] established the instability of nonlinear bound states. Our idea is initiated by the works of Berestycki and Cazenave [1] as well as Weinstein [31]. In [1] and [31], the related variational problems have to be solved, the Schwarz symmetrization and complicated variational computation have to be conducted. But in the present paper, using the variational argument proposed in Zhang [34], we can refrain from solving the attaching variational problems, and directly establish the sharp threshold for global existence and blowup of solutions to the Cauchy problem of system (1.1) +2 for 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. Moreover, initiated by the work of Soffer and Weinstein [28], we also discuss the instability of the standing waves for (1.1) with finite +2 time blow up for 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. The major difficulties in the analysis of the Davey-Stewartson system (1.1) are the nonlinearity including the singular integral operator E 1 . Due to the singular integral operator E 1 in system (1.1) which is very different from the harmonic potential −|x|2 in Zhang [34], although the idea in the present paper is motivated mainly by Zhang [34], there are still many difficulties which need to be solved in order to obtain our results. The key to the idea is to construct a type of cross-constrained minimization problem, and establish so-called cross-invariant manifolds, which require us to search for proper
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functionals. By the definitions of I (u), S(u) and Q(u) in Sect. 3, it is evident that they are different from those in [34]. In Zhang [34], every functional only includes a high+2 order nonlinearity, but in the present paper, since 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}, every functional has two high-order nonlinearities with different degrees, and one of the nonlinearities is the singular integral operator E 1 . For these reasons, compared to the argument in [34], our discussion will be more complicated and skillful. In addition, for the time being, as we have mentioned, the results in the present paper, especially those results concerning the sharp threshold for global existence and blowup are new +2 for system (1.1) under the conditions 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. It should be pointed out that system (1.1) has its origin in Fluid Mechanics for a ∈ R and b > 0. But to our knowledge, in the present paper, we only discuss the case a > 0 and b > 0. Furthermore, it is easily verified that our results in this paper can still cover the cases a = 0 and b > 0 as well as a > 0 and b = 0. For the time being, as far as the case a < 0 is concerned, whether the result concerning the sharp threshold for global existence holds or not remains open. At the end of this section, we give a brief outline of the rest of the paper. In Sect. 2 we collect some elementary facts which are useful for our analysis later. In Sect. 3 we establish the cross-invariant manifolds. The sharp threshold for global existence and blowup is treated in Sect. 4. Finally, we investigate the strong instability of the standing waves. 2. Preliminaries In this section, we will recall some known facts and give some elementary results which will be used and play important roles later. Firstly, we endow (1.1) with the initial data φ(0, x) = φ0 , x ∈ R N .
(2.1)
For system (1.1), Guo and Wang [15] as well as Cazenave [2] established the local well-posedness of the Cauchy problem in energy class H 1 (R N ). Now we state the following result about the local existence of weak solutions to the Cauchy problem (1.1)-(2.1) in energy space H 1 (R N ) (see also [5],Theorem 2.1). Proposition 2.1. Let N ∈ {2, 3} and 1 ≤ p <
N +2 (N −2)+ .
Then the following holds:
(1) For any φ0 ∈ H 1 (R N ), there exists a unique solution φ of the Cauchy problem (1.1)-(2.1) on a maximal time interval [0, T ) such that φ ∈ C([0, T ), H 1 (R N )) and either T = ∞ or else T < ∞ and lim φ H 1 (R N ) = ∞.
t→T
(2) We have conservation of charge and energy, that is |φ|2 d x = |φ0 |2 d x,
(2.2)
E(φ(t)) = E(φ0 )
(2.3)
for all t ∈ [0, T ), where a 1 1 E(φ) = |∇φ|2 d x − |φ| p+1 d x − b |φ|2 E 1 (|φ|2 )d x. (2.4) 2 p+1 4
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Here and hereafter, for simplicity, we denote R N ·d x by ·d x. For more specific results concerning the Cauchy problem (1.1)-(2.1), we refer the reader to [14]. In addition, by a direct calculation (see also Ohta [20–22]) we have Proposition 2.2. Let φ0 ∈ H 1 (R N ), |x|φ0 ∈ L 2 (R N ) and φ(t) be a solution of the Cauchy problem (1.1)-(2.1) on [0, T ). Put J (t) := |x|2 |φ|2 d x. (2.5) Then one has J (t) = −4I m
¯ x, xφ∇ φd
(2.6)
and by (2.4), we have p−1 J (t) = 8 |∇φ|2 d x − 4N a |φ| p+1 d x − 2N b |φ|2 E 1 (|φ|2 )d x p+1 4N ( p − 1) − 16 a |φ| p+1 d x = 16E(φ0 ) − p+1 (2.7) −(2N − 4)b |φ|2 E 1 (|φ|2 )d x. Thus the following result is true. +2 Corollary 2.1. Let φ0 ∈ H 1 (R N ) and N ∈ {2, 3}. For 1 + N4 ≤ p < (NN−2) + , when E(φ0 ) < 0, the solution φ of the Cauchy problem (1.1)-(2.1) blows up in finite time.
Proof. We prove this corollary by contradiction. Suppose that the maximal existence time T of the solution φ to the Cauchy problem (1.1)-(2.1) is infinity. Since N ∈ {2, 3}, +2 and 1 + N4 ≤ p < (NN−2) + , by (2.7) we have J (t) ≤ 16E(φ0 ), 0 ≤ t < ∞.
(2.8)
Through a classical analysis, the following identity is true: t J (t) = J (0) + J (0)t + (t − s)J (s)ds, 0 ≤ t < ∞.
(2.9)
0
From (2.8) it follows that J (t) ≤ 8E(φ0 )t 2 + J (0)t + J (0), 0 ≤ t < ∞.
(2.10)
Noting that J (t) is a nonnegative function, and J (0) = |x|2 |φ0 |2 d x, J (0) = −4I m xφ0 ∇ φ¯0 d x,
(2.11)
by E(φ0 ) < 0 and (2.10) we get that there exists T ∗ < ∞ such that lim∗ J (t) = 0. t→T
Namely,
lim
t→T ∗
|x|2 |φ|2 d x = 0,
which together with (2.2) leads us to the contradiction.
(2.12)
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Remark 2.1. For system (1.1) without the singular integral operator E 1 , by Glassey [12] and (2.2), the following conclusion holds: Let φ0 ∈ H 1 (R N ). Then for 1 ≤ p < 1 + N4 , the Cauchy problem (1.1)-(2.1) has +2 a unique bounded global solution. For 1 + N4 ≤ p < (NN−2) + , when φ0 H 1 (R N ) is sufficiently small, the Cauchy problem (1.1)-(2.1) has a unique bounded global solution. Therefore, for system (1.1) without the singular integral operator E 1 , from Corollary 2.1 we see that p = 1 + N4 is the critical nonlinearity index for blowup and global existence of the Cauchy problem (1.1)-(2.1). Thus, in that case, we call p = 1 + N4 a critical +2 case, 1 ≤ p < 1 + N4 a subcritical case, 1 + N4 < p < (NN−2) + a supercritical case. But for system (1.1) with the singular integral operator E 1 , when 1 ≤ p < 1 + N4 , the Cauchy problem (1.1)-(2.1) has a unique bounded global solution only for N = 1 using the method in Glassey [12], and whether the Cauchy problem (1.1)-(2.1) has a unique +2 bounded solution or not for N ∈ {2, 3} remains open. For 1 + N4 ≤ p < (NN−2) + , it is evident that the Cauchy problem (1.1)-(2.1) has global solutions and blowup solutions. Therefore, it is natural to search for the sharp threshold for global existence and blowup +2 of the solutions to the Cauchy problem (1.1)-(2.1) for 1 + N4 ≤ p < (NN−2) + , which is interesting and remains open before we solve it in this paper. Remark 2.2. From (2.7), when N = 2, for the critical case p = 3 one has J (t) = 8E(φ0 )t 2 + J (0)t + J (0), which is a quadratic function about time t. This coincides with the case of system (1.1) without the singular integral operator E 1 . Now we give some known facts in Cipolatti [4,5]. Lemma 2.1. (Cipolatti [4]). Let E 1 be the singular integral operator defined in Fourier variables by F{E 1 (ψ)}(ξ ) = σ1 (ξ )F{ψ}(ξ ), where σ1 (ξ ) = ξ12 /|ξ |2 , ξ ∈ R N and F denotes the Fourier transform in R N : N 2 F{ψ}(ξ ) = (1/2π ) e−iξ x ψ(x)d x. For 1 < p < ∞, E 1 satisfies the following properties: i) E 1 ∈ L(L p , L p ), where L(L p , L p ) denotes the space of bounded linear operators from L p to L p . ii) If ψ ∈ H s , then E 1 (ψ) ∈ H s , s ∈ R. iii) If ψ ∈ W m, p , then E 1 (ψ) ∈ W m, p and ∂k E 1 (ψ) = E 1 (∂k ψ), iv)
k = 1, . . . , N .
E 1 preserves the following operations: -translation: E 1 (ψ(· + y))(x) = E 1 (ψ)(x + y), y ∈ R N , -dilatation: E 1 (ψ(λ·))(x) = E 1 (ψ)(λx), λ > 0, -conjugation: E 1 (ψ) = E 1 (ψ), where ψ is the complex conjugate of ψ.
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Thus Lemma 2.1 yields directly the following properties. Remark 2.3. (Cipolatti [4] and Ohta[21]). Let B1 be the quadratic functional on L 2 defined by B1 (ψ) =
σ1 (ξ )|F(ψ)(ξ )|2 dξ.
It follows from the Parseval identity
f · gd x =
F[ f ]F[g]dξ, dξ = (2π )−N d x
(2.13)
that B1 (ψ) =
E 1 (ψ)ψd x,
and in particular we have B1 (ψ) ≤
|ψ|2 d x,
(2.14)
B1 ∈ C ∞ (L 2 , R), with B1 = 2E 1 . Therefore, from the definition of E 1 , the Parseval identity (2.13) and (2.14), we have
|ψ|2 E 1 (|ψ|2 )d x ≤
|ψ|4 d x
(2.15)
and
|ψ|2 E 1 (|ψ|2 )d x =
=
|ψ|2 F −1 σ1 (ξ )F(|ψ|2 )d x σ1 (ξ )|F(|ψ|2 )|2 dξ > 0.
Finally, we state an elementary lemma. Lemma 2.2. (Cipolatti [5]). For all φ ∈ S(R N , R) (the Schwartz space of rapidly decreasing functions), the following identities hold: (i) (ii) (iii)
φx · ∇φd x = −
N 2
|φ|2 d x. N |φ| p−1 φx · ∇φd x = − |φ| p+1 d x. p+1 N 2 E 1 (|φ| )φx · ∇φd x = − |φ|2 E 1 (|φ|2 )d x. 4
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3. The Cross-Constrained Variational Problem and the Cross-Invariant Manifolds In this section, we first define some functionals and manifolds, then consider the constrained minimization problems. Since system (1.1) includes the singular integral operator E 1 , our arguments are more difficult than the related discussions on the system (1.1) without the singular integral operator E 1 . Throughout this paper, we limit our arguments to the case N ∈ {2, 3}. +2 For u ∈ H 1 (R N ), ω > 0 and 1 < p < (NN−2) + , we define the following functionals: ω a b 1 2 2 p+1 I (u) : = |∇u| d x + |u| d x − |u| d x − |u|2 E 1 (|u|2 )d x, 2 2 p+1 4 (3.1) 2 2 p+1 2 2 S(u) : = |∇u| d x + ω |u| d x − a |u| d x − b |u| E 1 (|u| )d x, (3.2) N ( p − 1) N a |u| p+1 d x − b |u|2 E 1 (|u|2 )d x. Q(u) : = |∇u|2 d x − (3.3) 2( p + 1) 4 From Remark 2.3 it follows that 2 2 |u| E 1 (|u| )d x ≤ |u|4 d x.
(3.4)
By (3.4) and Sobolev’s embedding theorem, the above functionals are well-defined. In addition, we define a manifold N := u ∈ H 1 (R N ) \ {0}, S(u) = 0 . We first define the constrained variational problem: dN := inf I (u).
(3.5)
N
Thus, we have the following results. Lemma 3.1. Let 1 < p <
N +2 (N −2)+
and N ∈ {2, 3}, then dN > 0.
Proof. From (3.1),(3.2) and (3.5), on N we get p−1 I (u) = a |u| p+1 d x + 2( p + 1) p−3 a |u| p+1 d x + = 4( p + 1) and
b |u|2 E 1 (|u|2 )d x, 4 1 (|∇u|2 + ω|u|2 )d x, 4
(|∇u|2 + ω|u|2 )d x = a
Since 1 < p < I (u) ≥
N +2 (N −2)+ ,
1 min 2
p−1 1 , p+1 2
(3.6)
|u| p+1 d x + b
|u|2 E 1 (|u|2 )d x.
a |u| p+1 d x + b |u|2 E 1 (|u|2 )d x .
(3.7)
(3.8)
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By the Sobolev embedding theorem and (3.4), one gets a |u| p+1 d x + b |u|2 E 1 (|u|2 )d x p+1 ≤ a |u| d x + b |u|4 d x |∇u| d x + ω
2 |u| d x 2
δ
|∇u| d x + ω 2
2
|u|2 d x
2
+C2 ≤C
p+1
|∇u|2 d x + ω
≤ C1
|u| d x 2
2
,
(3.9)
where δ = max{ p + 1, 4} ≥ 4 or δ = min{ p + 1, 4} > 2 since p > 1. Here and hereafter, C > 0 denotes various positive constants. From (3.7) and (3.9), it follows that p+1 a |u| d x + b |u|2 E 1 (|u|2 )d x
δ 2 , ≤ C a |u| p+1 d x + b |u|2 E 1 (|u|2 )d x thus
δ −1 2 p+1 2 2 ≥ C > 0. a |u| d x + b |u| E 1 (|u| )d x Since
δ 2
− 1 > 0, (3.10) implies that p+1 a |u| d x + b |u|2 E 1 (|u|2 )d x ≥ C > 0.
Therefore (3.8), (3.11) and 1 < p <
N +2 (N −2)+
(3.10)
(3.11)
imply that I (u) ≥ C > 0, that is,
dN ≥ C > 0.
(3.12)
Lemma 3.2. There exists u ∈ H 1 (R N ) \ {0} such that S(u) = 0 and Q(u) = 0. Proof. From Ohta [21,22], it follows that there exists u ∈ H 1 (R N ) \ {0} such that u is a solution of the following Euler-Lagrangian equation: − u + ωu − a|u| p−1 u − bE 1 (|u|2 )u = 0.
(3.13)
Thus S(u) = 0, which is obtained from multiplying (3.13) by u and then integrating with respect to x on R N . Moreover, by (3.13) and Lemma 2.2, we have the Pohozaev identity, 2 1 N −2 a |u| p+1 d x − b |u|2 E 1 (|u|2 )d x = 0, |∇u|2 d x +ω |u|2 d x − N p+1 2 (3.14)
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which is obtained from multiplying (3.13) by x · ∇u and then integrating with respect to x on R N . Noting that S(u) = 0, we get Q(u) = 0. Now we define a cross-manifold in H 1 (R N ) as follows: M := u ∈ H 1 (R N ), S(u) < 0, Q(u) = 0 .
(3.15)
Then the following result is true. Lemma 3.3. Let N ∈ {2, 3}. ThenM is not empty provided 1 +
4 N
≤p<
N +2 (N −2)+ .
Proof. We divide the proof into two steps: i) p = 3 and ii) p = 3. Step 1. We first treat the case i) p = 3. In this case, from Lemma 3.2, it follows that there exists u ∈ H 1 (R N ) \ {0} such that both S(u) = 0 and Q(u) = 0. Let vλ = λu(λx) for λ > 0, then vλ ∈ H 1 (R N ) \ {0}. By (3.2) and (3.3), we get S(vλ ) = λ4−N |∇u|2 d x + λ2−N ω |u|2 d x 4−N 4 4−N −λ a |u| d x − λ b |u|2 E 1 (|u|2 )d x
= λ4−N |∇u|2 d x − a |u|4 d x − b |u|2 E 1 (|u|2 )d x (3.16) +λ2−N ω |u|2 d x, Q(vλ ) = λ4−N
|∇u|2 d x −
N a 4
|u|4 d x −
N b 4
|u|2 E 1 (|u|2 )d x .
(3.17)
Thus S(u) = 0 implies that there exists λ∗ > 1 such that S(vλ∗ ) < 0. On the other hand, from λ∗ > 1 and Q(u) = 0, we still have Q(vλ∗ ) = 0. So vλ∗ ∈ M. This proves M is not empty for i) p = 3. +2 Step 2. Next we treat case ii) p = 3. In this case, by 1 + N4 ≤ p < (NN−2) + , we divide this case into the following three cases: (a) 3 < p < ∞, 7 ≤ p < 3, (b) 3 (c) 3 < p < 5,
when N = 2; when N = 3; when N = 3.
From Lemma 3.2, it follows that there exists u ∈ H 1 (R N ) \ {0} such that both S(u) = 0 and Q(u) = 0. For λ > 1, let v = λu, then v ∈ H 1 (R N ) \ {0}. Thus (3.2) and (3.3) imply that S(v) = |∇v|2 d x + ω |v|2 d x − a |v| p+1 d x − b |v|2 E 1 (|v|2 )d x 2 2 2 =λ |∇u| d x + λ ω |u|2 d x −λ p+1 a |u| p+1 d x − λ4 b |u|2 E 1 (|u|2 )d x, (3.18)
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
Q(v) =
|∇v|2 d x −
= λ2 −λ4
N ( p − 1) a 2( p + 1)
|∇u|2 d x − λ p+1 N b 4
|v| p+1 d x −
N ( p − 1) a 2( p + 1)
N b 4
103
|v|2 E 1 (|v|2 )d x
|u| p+1 d x
|u|2 E 1 (|u|2 )d x.
(3.19)
N ( p−1) +2 From 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}, it follows that 2( p+1) < 1. Thus by S(u) = 0 and Q(u) = 0, we can always choose λ > 1 suitably such that S(v) < 0, Q(v) < 0,
N ( p − 1) a |v| p+1 d x > 0, 2( p + 1) N |∇v|2 d x − b |v|2 E 1 (|v|2 )d x > 0, 4 2 |∇v| d x − a |v| p+1 d x < 0, |∇v|2 d x −
(3.20) (3.21) (3.22)
and
|∇v| d x − b 2
|v|2 E 1 (|v|2 )d x < 0.
(3.23)
Step 2.1. We first prove case (a) 3 < p < ∞ and N = 2. 2
(a) Let vβ = β p−1 v(βx). Thus we have S(vβ ) = β
4 p−1
|∇v| d x − a
6−2 p p−1
|v| d x − β
|v| 10−2 p p−1
p+1
dx
|v|2 E 1 (|v|2 )d x,
4 p−1 Q(vβ ) = β p−1 a |v| p+1 d x |∇v|2 d x − p+1 1 10−2 p − β p−1 b |v|2 E 1 (|v|2 )d x. 2 +β
ω
2
2
b
(3.24)
(3.25)
p 10−2 p 6−2 p 6−2 p 4 > 10−2 By 3 < p < ∞, it follows that p−1 p−1 , p−1 > p−1 and p−1 < 0. ∗ Therefore Q(v) < 0, (3.20) and (3.25) imply that there exists β > 1 such that Q(vβ ∗ ) = 0. On the other hand, from β ∗ > 1, S(v) < 0, (3.22) and (3.24), it still has S(vβ ∗ ) < 0. So vβ ∗ ∈ M. Step 2.2. Next we prove case (b) 73 ≤ p < 3 and N = 3. In this case, one has 13 ≤ p − 2 < 1.
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(b) Let vµ = µv(µx). Thus one has
S(vµ ) = µ |∇v|2 d x − b |v|2 E 1 (|v|2 )d x +µ−1 ω |v|2 d x − µ p−2 a |v| p+1 d x,
3 Q(vµ ) = µ |∇v|2 d x − b |v|2 E 1 (|v|2 )d x 4 3( p − 1) p−2 µ a |v| p+1 d x. − 2( p + 1)
(3.26)
(3.27)
Therefore by Q(v) < 0, from (3.21) and (3.27) it follows that there exists µ∗ > 1 such that Q(vµ∗ ) = 0. On the other hand, from µ∗ > 1, S(v) < 0, (3.23) and (3.26), it still follows that S(vµ∗ ) < 0. So vµ∗ ∈ M. Step 2.3. Finally, we prove case (c) 3 < p < 5 and N = 3. In this case, we have 5 − p > 11 − 3 p and 7 − 3 p < 0. 2
(c) Let vξ = ξ p−1 v(ξ x). Thus we have
5− p S(vξ ) = ξ p−1 |∇v|2 d x − a |v| p+1 d x 7−3 p 11−3 p 2 p−1 p−1 +ξ ω |v| d x − ξ b |v|2 E 1 (|v|2 )d x,
5− p 3( p − 1) 2 p+1 p−1 a |v| d x Q(vξ ) = ξ |∇v| d x − 2( p + 1) 3 11−3 p − ξ p−1 b |v|2 E 1 (|v|2 )d x. 4
(3.28)
(3.29)
Therefore, from (3.20), (3.29) and Q(v) < 0, it follows that there exists ξ ∗ > 1 such that Q(vξ ∗ ) = 0. On the other hand, from ξ ∗ > 1, S(v) < 0, (3.22) and (3.28), it still follows that S(vξ ∗ ) < 0. So vξ ∗ ∈ M. From the above arguments of Step 1 and Step 2, we obtain that M is not empty +2 provided 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. So far, we have completed the proof of Lemma 3.3. Now let us consider the cross-constrained minimization problem: dM := inf I (u),
(3.30)
M
then we have: Lemma 3.4. Let N ∈ {2, 3}. Then dM > 0 provided 1 +
4 N
≤p<
N +2 (N −2)+ .
Proof. Let u ∈ M. From S(u) < 0, it follows that u = 0. By Q(u) = 0, we obtain N ( p − 1) − 4 I (u) = a |u| p+1 d x 4( p + 1)
1 N ω − b |u|2 E 1 (|u|2 )d x + + (3.31) |u|2 d x. 8 4 2
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
105
+2 Since 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}, (3.31) and u ≡ 0 imply that dM ≥ 0. In the following, we prove dM > 0 by dividing the proof into two cases:
1)
The critical case. p = 1 +
4 N;
2)
The supercritical case. 1 +
4 N
N +2 (N −2)+ .
We first consider case 1) The critical case: p = 1 + N4 . Since we have showed dM ≥ 0, in this case, we argue by contradiction. If dM = 0, then from (3.30) there would exist a sequence {u n , n ∈ Z+ } ⊂ M such that Q(u n ) = 0, S(u n ) < 0 and I (u n ) → 0 as n → ∞. Since p = 1 + N4 , (3.31) implies that 2 2 |u n |2 d x → 0 as n → ∞. (3.32) |u n | E 1 (|u n | )d x → 0, From the Gagliardo-Nirenberg inequality
N ( p−1)
|v|
p+1
dx ≤ C
4
|∇v| d x 2
|v| d x 2
p+1 − N ( p−1) 2
4
, v ∈ H 1 (R N ), (3.33)
for p = 1 +
4 N
and u n we have
|u n | p+1 d x ≤ C
|∇u n |2 d x ·
2 |u n |2 d x
N
.
(3.34)
According to S(u n ) < 0, |u n |2 E 1 (|u n |2 )d x ≤ |u n |4 d x (2.15) and (3.34), we get for p = 1 + N4 , 2 2 p+1 |∇u n | d x + ω |u n | d x < a |u n | d x + b |u n |2 E 1 (|u n |2 )d x
|∇u n | d x ·
≤C
2 |u n | d x
2
2
N
N
4−N 2 2 2 · . |∇u n | d x |u n | d x 2
+C
(3.35)
For C in (3.35), by (3.32), we have when n sufficiently large, |∇u n |2 d x + ω |u n |2 d x
≥C
|∇u n | d x · 2
+C
2 |u n | d x 2
N
N
4−N 2 2 2 · . |∇u n | d x |u n | d x 2
(3.36)
It is obvious that (3.35) and (3.36) are contradictory. Since we have showed dM ≥ 0, thus we get dM > 0 for p = 1 + N4 .
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Next we treat the supercritical case 2) 1 + Sobolev embedding inequality
4 N
|u|
p+1
dx ≤ C
N +2 (N −2)+ .
p+1
|∇u| d x + ω
|u| d x
2
In this case, we use the
2
2
.
(3.37)
From S(u) < 0, (2.15) and (3.37), it follows that |∇u|2 d x + ω |u|2 d x < a |u| p+1 d x + b |u|2 E 1 (|u|2 )d x ≤C
|∇u| d x + ω
≤C
p+1
|u| d x
2
2
|∇u| d x + ω
|u| d x 2
|u|4 d x
+b
p+1
2
2
2
|∇u| d x + ω
+C
2
2
|u| d x 2
⎧ 2 W 2 ⎪ ⎪ C |∇u| d x + ω |u| d x , ⎨ when |∇u|2d x + ω |u|2 d x ≥ 1, ≤ m 2 2 ⎪ ⎪ ⎩ C |∇u| d x + ω |u| d x2 , when |∇u| d x + ω |u|2 d x < 1, where W = max{( p + 1)/2, 2} ≥ 2 and m = min{( p + 1)/2, 2}. Since 1 + N +2 2 5 (N −2)+ and N ∈ {2, 3}, we get ( p + 1)/2 ≥ 1 + N ≥ 3 > 1. Thus we get 2 |∇u| d x + ω |u|2 d x ≥ C > 0.
4 N
< p<
(3.38)
In order to prove dM > 0, we divide the following proof into two cases: 2-i) N = 3, 73 < p < 5; 2-ii) N = 2, 3 < p < ∞. First we consider 2-i) N = 3, 73 < p < 5. In this case, by (3.31) we get 1 ω 3p − 7 p+1 2 2 a |u| d x + b |u| E 1 (|u| )d x + I (u) = |u|2 d x. 4( p + 1) 8 2 1 Let D = min{ 4(3 p−7 p+1) , 8 } > 0. Then by
1 D=
7 3
(3.39)
< p < 5, we get
when 3 8, 3 p−7 when 4( p+1) ,
≤ p < 5, 7 3
< p < 3.
Thus by (3.39), we obtain
ω I (u) ≥ D a |u| p+1 d x + b |u|2 E 1 (|u|2 )d x + |u|2 d x. 2 From S(u) < 0 it follows that |∇u|2 d x + ω |u|2 d x ≤ a |u| p+1 d x + b |u|2 E 1 (|u|2 )d x,
(3.40)
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
107
which together with (3.40) implies that
ω 2 2 I (u) ≥ D |u|2 d x. |∇u| d x + ω |u| d x + 2
(3.41)
By (3.38) and (3.41), we get I (u) ≥ C > 0.
(3.42)
Thus when N = 3, (3.30) implies that dM > 0 for 73 < p < 5. Next we consider 2-ii) N = 2, 3 < p < ∞. In this case, by (3.31) and (3.32), 2p − 6 ω a |u| p+1 d x + I (u) = (3.43) |u|2 d x. 4( p + 1) 2 In the following, we discuss by contradiction. If dM = 0, then from (3.43) there would exist a sequence {u n , n ∈ Z + } ⊂ M such that Q(u n ) = 0, S(u n ) < 0 and I (u n ) → 0 as n → ∞. Since 3 < p < ∞, (3.43) implies that |u n | p+1 d x → 0, |u n |2 d x → 0 as n → ∞. (3.44) From the Gagliardo-Nirenberg inequality for N = 2 and 3 < p < ∞,
( p−1)
|v| p+1 d x ≤ C
2
|∇v|2 d x
|v|2 d x , v ∈ H 1 (R2 ),
(3.45)
thus u n satisfies
( p−1)
|u n | p+1 d x ≤ C
|∇u n |2 d x
2
|u n |2 d x .
(3.46)
According to S(u n ) < 0, |u n |2 E 1 (|u n |2 )d x ≤ |u n |4 d x (2.15) and (3.46), using the Sobolev embedding inequality, we get |∇u n |2 d x + ω |u n |2 d x < a |u n | p+1 d x + b |u n |2 E 1 (|u n |2 )d x
( p−1)
≤C
2
|∇u n |2 d x
|∇u n |2 d x ·
+C
|u n |2 d x
|u n |2 d x.
(3.47)
For C in (3.47), from (3.44), it follows that when n sufficiently large, 2 |∇u n | d x + ω |u n |2 d x
( p−1)
≥C
|∇u n | d x
|u n | d x 2
+C
2
2
|∇u n |2 d x ·
|u n |2 d x.
(3.48)
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It is obvious that (3.47) and (3.48) are contradictory. Since we have showed dM ≥ 0, thus we get dM > 0 for N = 2 and 3 < p < ∞. +2 So far, we have proved that dM > 0 for the supercritical case 1 + N4 < p < (NN−2) +. 4 Therefore from the above arguments of 1) and 2), we obtain dM > 0 for 1 + N ≤ p < N +2 (N −2)+ . This completes the proof of Lemma 3.4. Now we define d := min{dN , dM }.
(3.49)
Remark 3.1. For the time being, to our knowledge, for a > 0, b > 0, ω > 0, 1 + N4 ≤ +2 p < (NN−2) + and N ∈ {2, 3}, the question remains open: Under what conditions will it have d = dN , and under what conditions will it have d = dM ? But for some special cases, it is known that d = dN (that is, dM ≥ dN ). For example, +2 Case 1 (Zhang [35]). a > 0, b = 0, ω > 0, 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. Case 2 (Shu and Zhang [27]). a > 0, b > 0, ω = 1, p = 3 and N ≥ 3. Thus by Lemma 3.1 and Lemma 3.4, the following result is true. Proposition 3.1. Let N ∈ {2, 3}. Then d > 0 provided 1 + We further define K := φ ∈ H 1 (R N ),
I (φ) < d, S(φ) < 0,
4 N
≤p<
N +2 (N −2)+ .
Q(φ) < 0 .
(3.50)
Thus we have Lemma 3.5. Let N ∈ {2, 3}. Then K is not empty provided 1 +
4 N
≤p<
N +2 (N −2)+ .
Proof. From Lemma 3.2, there exists u ∈ H 1 (R N ) \ {0} such that both S(u) = 0 and Q(u) = 0. Put u λ (x) = λu(x), then by (3.1), (3.2) and (3.3), one has
S(u λ ) = λ2 |∇u|2 d x + ω |u|2 d x (3.51) −λ p+1 a |u| p+1 d x − λ4 b |u|2 E 1 (|u|2 )d x, N ( p − 1) p+1 λ a |u| p+1 d x Q(u λ ) = λ2 |∇u|2 d x − 2( p + 1) N 4 − λ b |u|2 E 1 (|u|2 )d x, (3.52) 4
1 I (u λ ) = λ2 |∇u|2 d x + ω |u|2 d x 2 1 a λ p+1 |u| p+1 d x − λ4 b |u|2 E 1 (|u|2 )d x. (3.53) − p+1 4 +2 Since d > 0 and 1 + N4 ≤ p < (NN−2) + , for λ > 1 large enough, in view of (3.51), (3.52) and (3.53) as well as S(u) = 0 and Q(u) = 0, one always has that S(u λ ) < 0, Q(u λ ) < 0 and I (u λ ) < d. Thus u λ ∈ K. This completes the proof of Lemma 3.5.
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
109
Furthermore, we have +2 Proposition 3.2. Let 1 + N4 < p < (NN−2) + and N ∈ {2, 3}. Then K is an invariant manifold of the Cauchy problem (1.1)-(2.1). That is, if φ0 (x) ∈ K, then the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) satisfies φ(t, ·) ∈ K for any t ∈ [0, T ).
Proof. Let φ0 ∈ K. According to Proposition 2.1, there exists a unique φ(t, ·) ∈ C([0, T ); H 1 (R N )) with T ≤ ∞ such that φ(t, x) is a solution of the Cauchy problem (1.1)-(2.1). From (2.2), (2.3), (2.4) and (3.1), it follows that I (φ(t, ·)) = I (φ0 ),
t ∈ [0, T ).
Thus I (φ0 ) < d implies that I (φ(t, ·)) < d for any t ∈ [0, T ).
(3.54)
We first show S(φ(t, ·)) < 0 for t ∈ [0, T ) by contradiction. If otherwise, from the continuity, there were a t ∗ ∈ (0, T ) such that S(φ(t ∗ , ·)) = 0. By (3.54), it has φ(t ∗ , ·) ≡ 0. From (3.5) and (3.49), it follows that I (φ(t ∗ , ·)) ≥ dN ≥ d. This is contradictory to I (φ(t ∗ , ·)) < d for t ∈ [0, T ). Therefore S(φ(t, ·)) < 0 for all t ∈ [0, T ). Next we prove Q(φ(t, ·)) < 0 for t ∈ [0, T ) by contradiction. If otherwise, from the continuity, there were a t¯ ∈ (0, T ) such that Q(φ(t¯, ·)) = 0. Since we have showed S(φ(t¯, ·)) < 0, it follows that φ(t¯, ·) ∈ M. Thus (3.30) and (3.49) imply that I (φ(t¯, ·)) ≥ dM ≥ d. This is contradictory to I (φ(t¯, ·)) < d for t ∈ [0, T ). Therefore Q(φ(t, ·)) < 0 for all t ∈ [0, T ). From the above argument, we proved that φ(t, ·) ∈ K for any t ∈ [0, T ). Thus the proof of Proposition 3.2 is completed. By the same argument as Proposition 3.2, we get Proposition 3.3. Let N ∈ {2, 3}. Define K+ := φ ∈ H 1 (R N ), I (φ) < d, S(φ) < 0, Q(φ) > 0 , R− := φ ∈ H 1 (R N ), I (φ) < d, S(φ) < 0 , R+ := φ ∈ H 1 (R N ), I (φ) < d, S(φ) > 0 . +2 If 1 + N4 ≤ p < (NN−2) + , then K+ , R− and R+ are all invariant manifolds of the Cauchy problem (1.1)-(2.1).
Remark 3.2. For these manifolds defined in Proposition 3.3, we call R− and R+ invariant manifolds of the Cauchy problem (1.1)-(2.1). In the course of nature, we call K and K+ cross-invariant manifolds of the Cauchy problem (1.1)-(2.1). By the definitions of K, K+ and R+ , as well as (3.5), (3.30) and (3.49), the following result holds. Proposition 3.4. Let N ∈ {2, 3} and 1 +
4 N
≤p<
N +2 (N −2)+ .
Then
φ ∈ H 1 (R N ) \ {0}, I (φ) < d = R+ ∪ K+ ∪ K.
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Z. Gan, J. Zhang
4. Sharp Threshold for Global Existence and Blowup In this section, we discuss the sharp sufficient condition for blowup and global existence. Firstly, we give a sufficient condition for blowup of the solutions to the Cauchy problem (1.1)-(2.1). +2 Theorem 4.1. Let N ∈ {2, 3} and 1 + N4 ≤ p < (NN−2) + . If φ0 (x) ∈ K and satisfies 2 N |x|φ0 ∈ L (R ), then the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) blows up in finite time.
Proof. According to Ginibre and Velo [10,11], from |x|φ0 (x) ∈ L 2 (R N ), one has |x|φ ∈ L 2 (R N ). By φ0 ∈ K, Proposition 3.2 implies that φ(t, .) ∈ K for t ∈ [0, T ). Thus we have I (φ) < d, S(φ) < 0 and Q(φ) < 0. Now we put J (t) = |x|2 |φ|2 d x, (4.1) Equations(2.7) and (3.3) imply that J (t) = 8Q(φ(t, .)), t ∈ [0, T ).
(4.2)
Fix t ∈ [0, T ) and denote φ(t, .) = φ. Thus φ satisfies that I (φ) < d, Q(φ) < 0 and S(φ) < 0. In the following, we continue to prove Theorem 4.1 through two steps: Step 1. p = 3; Step 2. p = 3. We first consider Step 1. p = 3. N
Let φλ = λ 2 φ(λx). Then S(φλ ) = λ2 |∇φ|2 d x + ω |φ|2 d x N 4 N −aλ |φ| d x − λ b |φ|2 E 1 (|φ|2 )d x, N Q(φλ ) = λ2 |∇φ|2 d x − λ N a |φ|4 d x 4 N − λ N b |φ|2 E 1 (|φ|2 )d x. 4
(4.3)
(4.4)
Since S(φ) < 0, it yields that there exists 0 < λ∗ < 1 such that S(φλ∗ ) = 0, and when λ ∈ (λ∗ , 1], S(φλ ) < 0. For λ ∈ [λ∗ , 1] and Q(φ) < 0, Q(φλ ) has the following three possibilities: 1-i) Q(φλ ) < 0 for λ ∈ [λ∗ , 1]; 1-ii) Q(φλ∗ ) = 0; 1-iii) There exist µ ∈ (λ∗ , 1) such that Q(φµ ) = 0. For the case 1-i) and 1-ii), we both have S(φλ∗ ) = 0 and Q(φλ∗ ) ≤ 0. It follows from (3.5), (3.30) and (3.49) that I (φλ∗ ) ≥ dN ≥ d. Moreover, we have 1 I (φ) − I (φλ∗ ) = (1 − λ∗ 2 ) |∇φ|2 d x 2 1 − (1 − λ∗ N ) a |φ|4 d x + b |φ|2 E 1 (|φ|2 ))d x , (4.5) 4
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
Q(φ) − Q(φλ∗ ) = (1 − λ∗ 2 )
111
|∇φ|2 d x N ∗N 4 2 2 − (1 − λ ) a |φ| d x + b |φ| E 1 (|φ| ))d x . 4
(4.6)
According to 0 < λ∗ < 1 and N ∈ {2, 3}, (4.5) and (4.6) imply that 1 1 1 Q(φ) − Q(φλ∗ ) ≥ Q(φ). (4.7) 2 2 2 For the case 1-iii), we have Q(φµ ) = 0 and S(φµ ) < 0. Thus φµ ∈ M. From (3.30) and (3.49), it follows that I (φµ ) ≥ dM ≥ d. In addition, I (φ) − I (φλ∗ ) ≥
1 1 1 Q(φ) − Q(φµ ) ≥ Q(φ). 2 2 2 Since I (φλ∗ ) ≥ d and I (φµ ) ≥ d, in view of (4.7) and (4.8), we get ⎧ ⎨ Q(φ) ≤ 2(I (φ) − I (φµ )) ≤ 2[I (φ) − d], I (φ) − I (φµ ) ≥
⎩ Q(φ) ≤ 2(I (φ) − I (φ ∗ )) ≤ 2[I (φ) − d]. λ
(4.8)
(4.9)
From (2.2), (2.3), (2.4) and (3.1), it has I (φ) = I (φ0 ).
(4.10)
Thus by φ0 ∈ K and (4.2), we have J (t) = 8Q(φ) < 16[I (φ0 ) − d] < 0. Obviously J (t) can not verify the above inequality for all time (see also Glassey [12]). Therefore, from Proposition 2.1, it must be the case that T < ∞ , which implies lim φ(t, .) H 1 (R N ) = ∞. t∈T
That is, the solution φ of the Cauchy problem (1.1)-(2.1) blows up in finite time for p = 3 and N ∈ {2, 3}. +2 Now we consider Step 2. p = 3. In this case, by N ∈ {2, 3} and 1+ N4 ≤ p < (NN−2) +, we divide the proof into two cases: +2 2-1) 3 < p < (NN−2) + and N ∈ {2, 3}; 2-2) 73 ≤ p < 3 and N = 3. Step 2-1) We first consider the case 2-1) 3 < p < In this case, let φβ = β S(φβ ) = β
N p+1
N +2− p(N −2) p+1
φ(βx). Then
N +2 (N −2)+
and N ∈ {2, 3}.
|∇φ| d x − a
|φ| p+1 d x (1− p)N (3− p)N +β p+1 ω |φ|2 d x − β p+1 b |φ|2 E 1 (|φ|2 )d x, N +2− p(N −2) p+1 Q(φβ ) = β |∇φ|2 d x N ( p − 1) N (3− p)N − a |φ| p+1 d x − β p+1 b |φ|2 E 1 (|φ|2 )d x. 4( p + 1) 4 2
(4.11)
(4.12)
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Since 3 < p <
N +2 (N −2)+
and N ∈ {2, 3}, we have
N + 2 − p(N − 2) (3 − p)N > , p+1 p+1
N + 2 − p(N − 2) > 0, p+1
(3 − p)N < 0. p+1
Thus Q(φ) < 0 implies that there exists β ∗ > 1 such that Q(φβ ∗ ) = 0, and when β ∈ [1, β ∗ ), Q(φβ ) < 0. For β ∈ [1, β ∗ ], because S(φ) < 0, S(φβ ) has the following two possibilities: 2-1-i) S(φβ ) < 0 for β ∈ [1, β ∗ ]. 2-1-ii) There exists 1 < µ ≤ β ∗ such that S(φµ ) = 0. For the case 2-1-i), we have Q(φβ ∗ ) = 0 and S(φβ ∗ ) < 0, that is, φβ ∗ ∈ M. From (3.30) and (3.49), it follows that I (φβ ∗ ) ≥ dM ≥ d. Moreover, we have
N +2− p(N −2) 1 p+1 1 − β∗ |∇φ|2 d x 2
(1− p)N ω 1 − β ∗ p+1 + |φ|2 d x 2
(3− p)N 1 1 − β ∗ p+1 b |φ|2 E 1 (|φ|2 )d x, − 4
N +2− p(N −2) p+1 Q(φ) − Q(φβ ∗ ) = 1 − β ∗ |∇φ|2 d x
(3− p)N N b |φ|2 E 1 (|φ|2 )d x. 1 − β ∗ p+1 − 4 I (φ) − I (φβ ∗ ) =
Thus from β ∗ > 1, N ∈ {2, 3} and 3 < p < I (φ) − I (φβ ∗ ) ≥
N +2 (N −2)+ ,
(4.13)
(4.14)
it follows that
1 1 1 Q(φ) − Q(φβ ∗ ) = Q(φ). 2 2 2
(4.15)
For the case 2-1-ii), we have S(φµ ) = 0 and Q(φµ ) ≤ 0. Thus (3.5) and (3.49) imply that I (φµ ) ≥ dN ≥ d. Referring to (4.13) and (4.14), we have I (φ) − I (φµ ) ≥
1 1 1 Q(φ) − Q(φµ ) ≥ Q(φ). 2 2 2
(4.16)
For the case 2-1-i) and the case 2-1-ii), since I (φβ ∗ ) ≥ d, I (φµ ) ≥ d, from (4.15) and (4.16), it follows that
Q(φ) ≤ 2[I (φ) − I (φβ ∗ )] ≤ 2[I (φ) − d], Q(φ) ≤ 2[I (φ) − I (φµ )] ≤ 2[I (φ) − d].
(4.17)
From (2.2), (2.3), (2.4) and (3.1), one has I (φ) = I (φ0 ). Thus by φ0 ∈ K and (4.2), we have J (t) = 8Q(φ) < 16[I (φ) − d] = 16[I (φ0 ) − d] < 0.
(4.18)
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
Step 2-2) Now we deal with the case 2-2), 3 4
7 3
113
≤ p < 3 and N = 3. In this case, let
φη = η φ(ηx). Then 3 p−9 1 S(φη ) = η 2 |∇φ|2 d x − η 4 a |φ| p+1 d x − 23 2 +η ω |φ| d x − b |φ|2 E 1 (|φ|2 )d x, (4.19) 1 3( p − 1) 3 p−9 3 η 4 a |φ| p+1 d x − b |φ|2 E 1 (|φ|2 )d x. Q(φη ) = η 2 |∇φ|2 d x − 4( p + 1) 4 (4.20) From
7 3
≤ p < 3, it follows that 3p − 9 < 0, 4
3p − 9 1 3 ≥− >− . 4 2 2
Thus Q(φ) < 0 implies that there exists η∗ > 1 such that Q(φη∗ ) = 0, and when η ∈ [1, η∗ ), Q(φη ) < 0. For η ∈ [1, η∗ ], by S(φ) < 0, we get S(φη ) has the following two possibilities: 2-2-a) S(φη ) < 0 for η ∈ [1, η∗ ]. 2-2-b) There exists 1 < λ ≤ η∗ such that S(φλ ) = 0. For the case 2-2-a), we have Q(φη∗ ) = 0 and S(φη∗ ) < 0, that is, φη∗ ∈ M. From (3.30) and (3.49), it follows that I (φη∗ ) ≥ dM ≥ d. In addition, one has 1 3 1 ω 1 − η∗ 2 1 − η∗ − 2 |∇φ|2 d x + |φ|2 d x I (φ) − I (φη∗ ) = 2 2
(3 p−9 1 ∗ 4 a |φ| p+1 d x, 1−η − (4.21) 4 3 p−9 1 3 Q(φ) − Q(φη∗ ) = 1 − η∗ 2 1 − η∗ 4 a |φ| p+1 d x. (4.22) |∇φ|2 d x − 4 Thus from η∗ > 1 and
7 3
≤ p < 3, it follows that
I (φ) − I (φη∗ ) ≥
1 1 1 Q(φ) − Q(φη∗ ) = Q(φ). 2 2 2
(4.23)
For the case 2-2-b), we have S(φλ ) = 0 and Q(φλ ) ≤ 0. Thus (3.5) and (3.49) imply that I (φλ ) ≥ dN ≥ d. Referring to (4.21) and (4.22), we have I (φ) − I (φλ ) ≥
1 1 1 Q(φ) − Q(φλ ) ≥ Q(φ). 2 2 2
(4.24)
For the case 2-2-a) and the case 2-2-b), since I (φη∗ ) ≥ d, I (φλ ) ≥ d, from (4.23) and (4.24), it follows that Q(φ) ≤ 2[I (φ) − I (φη∗ )] ≤ 2[I (φ) − d], (4.25) Q(φ) ≤ 2[I (φ) − I (φλ )] ≤ 2[I (φ) − d]. From (2.2), (2.3), (2.4) and (3.1), one has I (φ) = I (φ0 ). Thus by φ0 ∈ K and (4.2), we have J (t) = 8Q(φ) < 16[I (φ) − d] = 16[I (φ0 ) − d] < 0.
(4.26)
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Thus, from the arguments of Step 2-1 and Step 2-2, in view of (4.18) and (4.26), obviously J (t) can not verify the above inequality for all time (see also Glassey [12]). Therefore, from Proposition 2.1, it must be the case that T < ∞ , which implies lim φ(t, .) H 1 (R N ) = ∞. t∈T
That is, the solution φ of the Cauchy problem (1.1)-(2.1) blows up in finite time for p = 3 and N ∈ {2, 3}. From the arguments of Step 1 and Step 2, the proof of Theorem 4.1 is completed. Remark 4.1. In Theorem 4.1, we developed a new argument to obtain the blowup property of the solutions to the Cauchy problem (1.1)-(2.1). Since d > 0, we see that Theorem 4.1 is quite different from Corollary 2.1. If d ≤ 21 |φ0 |2 d x, then by (2.2), (2.3), (2.4), (3.1) and I (φ) < d, we can obtain E(φ0 ) < 0. Thus in this case, from Corollary 2.1, it can follow the result of Theorem 4.1. On the other hand, if d > 21 |φ0 |2 d x, then from I (φ) < d, it may conclude three cases: (1) E(φ0 ) > 0; (2) E(φ0 ) = 0; (3) E(φ0 ) < 0. So Theorem 4.1 includes the result that when initial energy is nonnegative, the solution φ of the Cauchy problem (1.1)-(2.1) can also blow up in finite time. This kind of results in Theorem 4.1 generalize the result in Corollary 2.1. In the following, we give a sufficient condition for global existence of the solutions to the Cauchy problem (1.1)-(2.1). +2 Theorem 4.2. Let N ∈ {2, 3} and 1 + N4 ≤ p < (NN−2) + . If φ0 ∈ K+ ∪ R+ , then the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) exists globally in t ∈ [0, ∞).
Proof. We prove this theorem by two steps. Step 1. We prove the case φ0 ∈ K+ . Step 2. We prove the case φ0 ∈ R+ . Firstly, we consider Step 1. φ0 ∈ K+ . From φ0 ∈ K + , Proposition 3.3 implies that the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) satisfies that φ(t, ·) ∈ K+ for t ∈ [0, T ). For fixed t ∈ [0, T ), we denote φ(t, ·) = φ. So we have I (φ) < d, Q(φ) > 0 and S(φ) < 0. Thus φ ≡ 0. According to (3.1) and (3.3), I (φ) < d and Q(φ) > 0 yield that
1 2 ω − |∇φ|2 d x + |φ|2 d x 2 N ( p − 1) 2
1 1 − b |φ|2 E 1 (|φ|2 )d x < d. (4.27) + 2( p − 1) 4 In the following, we divide the proof into six cases: 1) N = 2 and p = 3; 2) N = 3 and p = 3; 3) N = 2 and 3 < p < ∞; 4) N = 3 and p = 73 ; 5) N = 3 and 73 < p < 3; 6) N = 3 and 3 < p < 5. Step 1-1. We first consider the case 1), N = 2 and p = 3. By (4.27), we have ω (4.28) |φ|2 d x < d. 2
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1 In the following, we prove that |∇φ|2 d x is bounded. Put φλ = λ 2 φ(λx). It follows from (3.3) that 1 1 Q(φλ ) = λ |∇φ|2 d x − a |φ|4 d x − b |φ|2 E 1 (|φ|2 )d x. (4.29) 2 2 By Q(φ) > 0, it follows that when λ → 0, Q(φλ ) < 0 and λ → 1, Q(φλ ) > 0. Thus by continuity, there exists a 0 < λ∗ < 1 such that Q(φλ∗ ) = 0. According to (3.1) and (3.3), we have ω ω ∗ −1 2 ∗ ∗ I (φλ ) = (4.30) |φ|2 d x. |φλ | d x = λ 2 2 From (4.28), it follows that I (φλ∗ ) < λ∗ −1 d.
(4.31)
Also, by (3.2) and (3.3), it follows from Q(φλ∗ ) = 0 and S(φ) < 0 that 2 ∗ ∗ S(φλ ) = ω |φλ | d x − |∇φλ∗ |2 d x = ωλ∗ −1 |φ|2 d x − λ∗ |∇φ|2 d x,
(4.31*)
which has two possibilities: 1-1-i) S(φλ∗ ) < 0. 1-1-ii)
S(φλ∗ ) ≥ 0.
We first treat 1-1-i), S(φλ∗ ) < 0. In this case, noting that Q(φλ∗ ) = 0, we get φλ∗ ∈ M. Thus by (3.30) and (3.49), one has I (φλ∗ ) ≥ dM ≥ d > I (φ),
(4.32)
I (φ) − I (φλ∗ ) < 0.
(4.33)
which implies that
That is,
1 1 ∗ − λ 2 2
|∇φ| d x + 2
By (4.28), it has |∇φ|2 d x <
1 1 ∗ −1 ω |φ|2 d x < 0. − λ 2 2
λ∗ −1 − 1 ω 1 − λ∗
|φ|2 d x < 2
λ∗ −1 − 1 d. 1 − λ∗
(4.34)
(4.35)
∗−1
Since 0 < λ∗ < 1, one has λ1−λ−1 ∗ > 0. Now we consider 1-1-ii), S(φλ∗ ) ≥ 0. In this case, from (4.31) it follows that I (φλ∗ ) −
1 S(φλ∗ ) < λ∗ −1 d. 4
(4.36)
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Thus by (4.30) and (4.31)∗ , one gets 1 1 |∇φλ∗ |2 d x + ω |φλ∗ |2 d x < λ∗ −1 d. 4 4 Namely, λ
∗
|∇φ| d x + λ
which implies that
2
∗ −1
ω
(4.37)
|φ|2 d x < 4λ∗ −1 d,
|∇φ|2 d x < 4λ∗ −2 d.
(4.38)
By (4.35) and (4.38), we obtain that |∇φ|2 d x is bounded for any t ∈ [0, T ), which together with (4.28) yields that φ is bounded in H 1 (R N ). So Proposition 2.1 implies that the solution φ of the Cauchy problem (1.1)-(2.1) globally exists on t ∈ [0, ∞) for φ0 ∈ K+ , p = 3 and N = 2. Step 1-2. Secondly, we treat the case 2), N = 3 and p = 3. By (4.27), we get 1 ω (4.39) |∇φ|2 d x + |φ|2 d x < d, 6 2 which implies that the solution φ of the Cauchy problem (1.1)-(2.1) is bounded in H 1 (R N ) for any t ∈ [0, T ). So Proposition 2.1 implies that the solution φ of the Cauchy problem (1.1)-(2.1) globally exists on t ∈ [0, ∞) for φ0 ∈ K+ , p = 3 and N = 3. Step 1-3. Thirdly, we deal with the case 3), N = 2 and 3 < p < ∞. In this case, it follows from Q(φ) > 0 that p−1 1 2 p+1 a |φ| d x + b |φ|2 E 1 (|φ|2 )d x. |∇φ| d x > p+1 2 Thus (3.1) implies that
p−1 1 a |φ| p+1 d x + b |φ|2 E 1 (|φ|2 )d x 2( p + 1) 4 ω 1 1 2 p+1 + |φ| d x − a |φ| d x − b |φ|2 E 1 (|φ|2 )d x 2 p+1 4
1 p−1 ω − a |φ| p+1 d x + = |φ|2 d x 2( p + 1) p+1 2 p−3 ω a |φ| p+1 d x + = |φ|2 d x, 2( p + 1) 2
I (φ) ≥
which together with I (φ) < d yields that p−3 ω a |φ| p+1 d x + |φ|2 d x < d. 2( p + 1) 2 Since 3 < p < ∞, we obtain ω 2
|φ|2 d x < d.
(4.40)
Global Existence Sharp Threshold and Instability in Davey-Stewartson System
In the following, we prove that Put φλ = λ 4
117
|∇φ|2 d x is bounded.
2 p+1
Q(φλ ) = λ p+1
φ(λx). It follows from (3.3) that p−1 1 6−2 p a |φ| p+1 d x − λ p+1 b |φ|2 E 1 (|φ|2 )d x. |∇φ|2 d x − p+1 2 (4.41)
By Q(φ) > 0, one has when λ → 0, Q(φλ ) < 0, and when λ → 1, Q(φλ ) > 0. Thus by continuity, there exists 0 < λ∗ < 1 such that Q(φλ∗ ) = 0. According to (3.1) and (3.2), one has 4 1 ω 2−2 p |φ|2 d x I (φλ∗ ) = λ∗ p+1 |∇φ|2 d x + λ∗ p+1 2 2 1 1 6−2 p a |φ| p+1 d x − λ∗ p+1 b |φ|2 E 1 (|φ|2 )d x, (4.42) − p+1 4 2−2 p 4 ∗ p+1 2 ∗ p+1 ∗ S(φλ ) = λ |∇φ| d x + ωλ |φ|2 d x 6−2 p (4.43) −a |φ| p+1 d x − λ∗ p+1 b |φ|2 E 1 (|φ|2 )d x. From S(φ) < 0, it follows that S(φλ∗ ) has two possibilities: 1-3-a) S(φλ∗ ) < 0; 1-3-b) S(φλ∗ ) ≥ 0. We first consider 1-3-a), S(φλ∗ ) < 0. In this case, noting that Q(φλ∗ ) = 0, one has φλ∗ ∈ M. Thus by (3.30) and (3.49), one has I (φλ∗ ) ≥ dM ≥ d > I (φ).
(4.44)
I (φ) − I (φλ∗ ) < 0.
(4.45)
Equation (4.44) implies that
That is,
4 1 1 ∗ p+1 − λ 2 2
|∇φ|2 d x
p 1 1 ∗ 2−2 − λ p+1 ω |φ|2 d x 2 2
p 1 1 ∗ 6−2 λ p+1 − b |φ|2 E 1 (|φ|2 )d x + 4 4 < 0.
+
6−2 p
Since 3 < p < ∞ and 0 < λ∗ < 1, one has λ∗ p+1 So (4.40) and (4.46) imply that
p 1 1 ∗ 2−2 1 2 p+1 |∇φ| d x < λ / − − 2 2 2 < 2[(λ∗
2−2 p p+1
4
> 1, λ∗ p+1 < 1 and λ∗ 4 1 ∗ p+1 λ 2 4
− 1)/(1 − λ∗ p+1 )]d.
(4.46)
2−2 p p+1
> 1.
ω
|φ|2 d x (4.47)
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Secondly, we deal with 1-3-b), S(φλ∗ ) ≥ 0. In this case, let φβ = β p+1 φλ∗ (βx). By (3.2), we get 2−2 p 4 S(φβ ) = β p+1 |∇φλ∗ |2 d x + β p+1 ω |φλ∗ |2 d x 6−2 p −a |φλ∗ | p+1 d x − β p+1 b |φλ∗ |2 E 1 (|φλ∗ |2 )d x. (4.48) From S(φλ∗ ) ≥ 0, it has S(φ1 ) = S(φλ∗ ) ≥ 0 for β = 1 and S(φβ ) < 0 for β close to zero. Therefore, there exists β ∗ ∈ (0, 1] such that S(φβ ∗ ) = 0. Thus by (3.5) and (3.49), it has I (φβ ∗ ) ≥ dN ≥ d > I (φ).
(4.49)
Since I (φβ ∗ ) =
2−2 p 4 1 ∗ ∗ p+1 ω (β λ ) |∇φ|2 d x + (β ∗ λ∗ ) p+1 |φ|2 d x 2 2 6−2 p 1 1 a |φ| p+1 d x − (β ∗ λ∗ ) p+1 b |φ|2 E 1 (|φ|2 )d x, − p+1 4
which together with (4.49) implies that 4 1 1 − (β ∗ λ∗ ) p+1 2
2−2 p ω 1 − (β ∗ λ∗ ) p+1 |∇φ|2 d x + |φ|2 d x 2
6−2 p 1 (β ∗ λ∗ ) p+1 − 1 b |φ|2 E 1 (|φ|2 )d x + 4 < 0.
(4.50)
From 3 < p < ∞ and β ∗ λ∗ ∈ (0, 1) since β ∗ ∈ (0, 1] and λ∗ ∈ (0, 1), (4.40) and (4.50) yield that 2−2 p 4 (4.51) |∇φ|2 d x < 2[((β ∗ λ∗ ) p+1 − 1)/(1 − (β ∗ λ∗ ) p+1 )]d. By (4.40), (4.47) and (4.51), we get that φ is bounded in H 1 (R N ). So Proposition 2.1 implies that the solution φ of the Cauchy problem (1.1)-(2.1) globally exists on t ∈ [0, ∞) for φ0 ∈ K+ , N = 2 and 3 < p < ∞. Step 1-4. We then consider the case 4), N = 3 and p = 73 . In this case, by (4.27) we have ω 1 (4.52) |φ|2 d x + b |φ|2 E 1 (|φ|2 )d x < d. 2 8 9
We put φµ = µ 10 φ(µx). Then we get 4
Q(φµ ) = µ 5
3 |∇φ|2 d x − a 5
10 3 3 |φ| 3 d x − µ 5 b 4
|φ|2 E 1 (|φ|2 )d x. (4.53)
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119
Thus Q(φ) > 0 implies that there exists a µ∗ ∈ (0, 1) such that Q(φµ∗ ) = 0. By (3.1), (3.3) and Q(φµ∗ ) = 0, we have ω 1 |φµ∗ |2 d x + b |φµ∗ |2 E 1 (|φµ∗ |2 )d x 2 8 ω ∗ − 65 1 ∗ 35 2 = µ |φ| d x + µ b |φ|2 E 1 (|φ|2 )d x. 2 8
I (φµ∗ ) =
(4.54)
It follows from (4.53) and µ∗ ∈ (0, 1) that 6
I (φµ∗ ) < µ∗ − 5 d.
(4.55)
Now we consider S(φµ∗ ), which has two possibilities: 1-4-a) S(φµ∗ ) < 0; 1-4-b) S(φµ∗ ) ≥ 0. At first, we consider 1-4-a), S(φµ∗ ) < 0. In this case, noting that Q(φµ∗ ) = 0, we have φµ∗ ∈ M, which together with (3.30) and (3.49) implies that I (φµ∗ ) ≥ dM ≥ d > I (φ).
(4.56)
I (φ) − I (φµ∗ ) < 0,
(4.57)
So one has
namely,
1 1 ∗ 45 1 2 ∗ − 65 − µ 1−µ ω |φ|2 d x |∇φ| d x + 2 2 2 3 1 − 1 − µ∗ 5 b |φ|2 E 1 (|φ|2 )d x < 0, 4
which implies that
∗ − 65 ∗ 45 ω |φ|2 d x −1 / 1−µ |∇φ| d x < µ 3 4 1 + 1 − µ∗ 5 / 1 − µ∗ 5 b |φ|2 E 1 (|φ|2 )d x. 2 2
(4.58)
By (4.52) and µ∗ ∈ (0, 1), we get |∇φ|2 d x < C.
(4.59)
Now we deal with 1-4-b), S(φµ∗ ) ≥ 0. In this case, from (4.55), it follows that I (φµ∗ ) −
6 3 S(φµ∗ ) < µ∗ − 5 d. 10
(4.60)
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It follows that 1 ∗ 45 µ 5
1 ∗ − 65 ω |φ|2 d x |∇φ| d x + µ 5 1 ∗ 35 + µ b |φ|2 E 1 (|φ|2 )d x 20 2
6
< µ∗ − 5 d. Thus
|∇φ|2 d x < C.
(4.61)
Therefore, (4.52), (4.59) and (4.61) show that φ is bounded in H 1 (R N ) for any t ∈ [0, T ). Thus by Proposition 2.1, we get that the solution φ of the Cauchy problem (1.1)-(2.1) exists globally in t ∈ [0, ∞) for φ0 ∈ K+ , N = 3 and p = 73 . Step 1-5. We further treat the case 5), N = 3 and 73 < p < 3. In this case, by (4.27), we get
2 1 ω 2 − |∇φ| d x + |φ|2 d x 2 3( p − 1) 2
1 1 − b |φ|2 E 1 (|φ|2 )d x + 2( p − 1) 4 < d. (4.62) 2 1 > 0 and 2( p−1) − 41 > 0. Therefore, (4.62) implies Since 73 < p < 3, it has 21 − 3( p−1) that the solution φ of the Cauchy problem (1.1)-(2.1) is bounded in H 1 (R N ) for any t ∈ [0, T ). Thus Proposition 2.1 implies that the solution φ of the Cauchy problem (1.1)-(2.1) exists globally in t ∈ [0, ∞) for φ0 ∈ K+ , N = 3 and 73 < p < 3. Step 1-6. At last, we investigate the case 6), N = 3 and 3 < p < 5. In this case, from (3.3) and Q(φ) > 0, it follows that 1 1 p−1 b |φ|2 E 1 (|φ|2 )d x > − a |φ| p+1 d x, (4.63) |∇φ|2 d x + 4 3 2( p + 1)
which together with (3.1) and I (φ) < d implies that 1 ω p−3 2 2 a |φ| p+1 d x < d. |∇φ| d x + |φ| d x + 6 2 2( p + 1) Since 3 < p < 5, (4.64) yields that 1 ω |∇φ|2 d x + |φ|2 d x < d. 6 2
(4.64)
(4.65)
Equation (4.65)gives that φ is bounded in H 1 (R N ) for any t ∈ [0, T ). Thus Proposition 2.1 implies that the solution φ of the Cauchy problem (1.1)-(2.1) exists globally in t ∈ [0, ∞) for φ0 ∈ K+ , N = 3 and 3 < p < 5. +2 Thus from Step 1-1-Step 1-6, for φ0 ∈ K+ , N ∈ {2, 3} and 1 + N4 ≤ p < (NN−2) + , we have proved that the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) exists globally in t ∈ [0, ∞).
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121
Now we deal with Step 2. φ0 ∈ R+ . By φ0 ∈ R+ , Proposition 3.3 implies that the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) satisfies that φ(t, ·) ∈ R+ for t ∈ [0, T ). Then we have I (φ) < d and S(φ) > 0. By S(φ) > 0 and (3.2), one has
− a |φ| p+1 d x − b |φ|2 E 1 (|φ|2 )d x > − |∇φ|2 d x + ω |φ|2 d x . (4.66) +2 From I (φ) < d and 1 + N4 ≤ p < (NN−2) + , one has the following two results: (1) For N ∈ {2, 3} and 3 ≤ p < ∞, it follows from I (φ) < d that ω 1 |∇φ|2 d x + |φ|2 d x 2 2 a 1 p+1 − |φ| d x − b |φ|2 E 1 (|φ|2 )d x 4 4 ≤ I (φ) < d.
(2) For N = 3 and 73 ≤ p < 3, it follows from I (φ) < d that ω 1 |∇φ|2 d x + |φ|2 d x 2 2 a 1 p+1 b |φ|2 E 1 (|φ|2 )d x − |φ| d x − p+1 p+1 < I (φ) < d. Therefore (4.66), (4.67) and (4.68) imply that ω 1 2 |∇φ| d x + |φ|2 d x < d, 4 4 or
1 1 − 2 p+1
|∇φ| d x + ω
(4.68)
(4.69)
2
(4.67)
|φ| d x 2
< d.
(4.70)
+2 1 N In view of 1 + N4 ≤ p < (NN−2) + , (4.69) and (4.70) imply that φ is bounded in H (R ) for any t ∈ [0, T ). Thus by Proposition 2.1, we obtain that the solution φ of the Cauchy +2 problem (1.1)-(2.1) exists globally in t ∈ [0, ∞) for φ0 ∈ R+ , 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. Through the arguments of Step 1 and Step 2, we complete the proof of Theorem 4.2.
By Theorem 4.1 and 4.2, using Proposition 3.4, we can get a necessary and sufficient condition for blowup of the solution to the Cauchy problem (1.1)-(2.1) for 1 + N4 ≤ p < N +2 (N −2)+ and N ∈ {2, 3}. +2 Theorem 4.3. Let 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. If φ0 satisfies I (φ0 ) < d, then the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) blows up in finite time if and only if φ0 ∈ K.
In addition, by Theorem 4.2 and using the scaling argument, we can also get another sufficient condition for the global existence of the solution to the Cauchy problem (1.1)(2.1).
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Corollary 4.1. (Small Data Criterion). Let 1 + N4 ≤ p < ω > 0. If φ0 ∈ H 1 (R N ) and satisfies |∇φ0 |2 d x + ω |φ0 |2 d x < 2d,
N +2 (N −2)+ ,
N ∈ {2, 3} and
(4.71)
then the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) exists globally in t ∈ [0, ∞). Proof. According to (3.1) and (4.71), we can get I (φ0 ) < d. In addition, we assert that S(φ0 ) > 0. If otherwise, from (3.2), there were a 0 < λ ≤ 1 such that S(λφ0 ) = 0. Thus by (3.5) and (3.49), I (λφ0 ) ≥ dN ≥ d. On the other hand,
(4.72)
|∇(λφ0 )| d x + ω |λφ0 |2 d x
= λ2 |∇φ0 |2 d x + ω |φ0 |2 d x 2
< 2λ2 d ≤ 2d,
(4.73)
which implies that I (λφ0 ) < d.
(4.74)
Equations (4.72) and (4.74) are contradictory. So S(φ0 ) > 0 holds. From the above argument, we get φ0 ∈ R+ , thus Theorem 4.2 yields the result of Corollary 4.1. Remark 4.2. Although the condition for global existence in Corollary 4.1 is not sharp, Corollary 4.1 gives an answer to the question: How small are the initial data, when the solution of the Cauchy problem (1.1)-(2.1) exists globally? 5. Instability of the Standing Waves Using the methods in Berestycki and Cazenave [1] as well as Rabinowitz [25], one can easily obtain that the constrained variational problem (3.5) is attained. Let u be a solution of (3.5), that is we have dN = min I (u).
(5.1)
u∈N
Then u ∈ H 1 (R N ) \ {0} is a solution of (3.13). Thus φ(t, x) = eiωt u(x)
(5.2)
is a standing wave solution of (1.1). From (5.1), it follows that u(x) is a ground state solution of (3.13). Now we hope to study the instability of the standing wave (5.2). In general, it depends upon the frequency ω and the solvability of the following variational problem: dQ =
inf
{u∈H 1 (R N )\{0},Q(u)=0}
I (u).
(5.3)
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123
In the present paper, by Proposition 3.1 and Theorem 4.1, we can refrain from solving the problem (5.3), and show the instability of the standing wave (5.2), which commonly depends upon the frequency ω (see Cipolatti [5], Ohta [21,22]). In order to obtain the instability result, we first consider two lemmas which are key to our analysis later. Lemma 5.1. Let φ ∈ H 1 (R N ) \ {0}. Then there exists a unique µ > 0 such that S(µφ) = 0 and I (µφ) > I (λφ) for any λ > 0 and λ = µ. Proof. Let λ > 0. Then we have
2 2 2 S(λφ) = λ |∇φ| d x + ω |φ| d x p+1 p+1 4 −λ a |φ| d x − λ b |φ|2 E 1 (|φ|2 )d x,
1 I (λφ) = λ2 |∇φ|2 d x + ω |φ|2 d x 2 1 1 λ p+1 a |φ| p+1 d x − λ4 b |φ|2 E 1 (|φ|2 )d x. − p+1 4
(5.4)
(5.5)
Noting that (5.4) and (5.5), we obtain d I (λφ) = λ−1 S(λφ). dλ
(5.6)
Thus by (5.4) and (5.6), the result of Lemma 5.1 is true.
Lemma 5.2. Let u be a minimizer of (5.1). Then Q(u) = 0. Proof. Since u is a minimizer of (5.1), thus u is also a solution of (3.13) or (1.8). So multiplying (3.13) by x · ∇u, we get N −2 2 1 a |u| p+1 d x − b |u|2 E 1 (|u|2 )d x = 0, |∇u|2 d x +ω |u|2 d x − N p+1 2 (5.7) which is called Pohozaev identity. Note that S(u) = 0, (5.7) implies that Q(u) = 0.
Using Lemma 5.1 and Lemma 5.2, on the standing wave (5.2), we get the following instability theorem. +2 Theorem 5.1. For 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}, Let ω be such that dM ≥ dN . Then for the minimizer u of (3.5) and any ε > 0, there exists φ0 ∈ H 1 (R N ) with φ0 − u H 1 (R N ) < ε such that the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) blows up in finite time.
Proof. Since dM ≥ dN , one has d = dN by (3.49). Because u is the minimizer of (3.5), it follows from Lemma 3.2 that there exists u ∈ H 1 (R N ) \ {0} such that both S(u) = 0 and Q(u) = 0. Thus by (3.2) and (3.3), for any λ > 1 we have S(λu) < 0,
Q(λu) < 0,
λ > 1.
(5.8)
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Z. Gan, J. Zhang
On the other hand, from Lemma 5.1, S(u) = 0 implies that I (λu) < I (u) for any λ > 1. Note that I (u) = dN = d. Thus for any λ > 1, we have λu ∈ K. Now we take λ > 1, and λ is sufficiently close to 1 such that λu − u H 1 (R N ) = (λ − 1) u H 1 (R N ) < ε.
(5.9)
Then take φ0 = λu(x). From Theorem 4.1, it follows that the solution φ(t, x) of the Cauchy problem (1.1)-(2.1) blows up in finite time. Remark 5.1. Under the condition dM ≥ dN , Theorem 5.1 gives the strong instability of the ground state standing wave (5.2) for system (1.1) with finite time blow up when +2 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. For the related results on the instability of the standing wave (5.2) for system (1.1), there has been a lot of work (see Cipolatti [5] and Ohta [20–22]). Cipolatti [5] proved that if p ≥ 3 and N ∈ {2, 3}, then the standing wave (5.2) is unstable for any ω ∈ (0, ∞) and that if N = 2 and p = 3, then the standing wave (5.2) is strongly unstable for any ω ∈ (0, ∞). After that, the author [21] proved that if p ≥ 1 + N4 and N ∈ {2, 3}, then (5.2) is unstable for any ω ∈ (0, ∞), and if N = 3 and 1 < p < 73 , then there exists a positive constant ω0 = ω0 (a, p) such that (5.2) is unstable for any ω ∈ (ω0 , ∞). In addition, the author [20] showed that if N = 3 and 73 < p < 5, then (5.2) is strongly unstable for any ω ∈ (0, ∞). On the other hand, when N = 2 and p ≤ 3, the author [20] proved the existence of stable standing waves of (1.1). Further, under the condition N = 2 and p > 3 or N = 3 and p = 73 , the author [22] obtained that for any ω ∈ (0, ∞), (5.2) is strongly unstable in the sense of Definition 1.1 of [22]. It should be pointed out that in the present paper, we introduce the cross-invariant manifold to discuss the instability of standing waves for system (1.1) with finite time blow up. These kind of results partially answer the open problem proposed in [21, Remark 8]. In Theorem 5.1, by limiting frequency ω to satisfy dM ≥ dN , we get the strong +2 instability of (5.2) for 1 + N4 ≤ p < (NN−2) + and N ∈ {2, 3}. Of course, the condition dM ≥ dN is still vague. Further we need to determine for which ω, dM ≥ dN is true. Moreover, if dM < dN , is the standing wave (5.2) orbital stable? These problems remain open. Acknowledgements. This work is partially done when the first author visited the IMS of the Chinese University of Hong Kong. She would like to express her thanks for the hospitality of the IMS. Especially, she would like to express her deep gratitude to Professor Zhouping Xin for his valuable suggestions and constant encouragements.
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7. Gan, Z.H., Zhang, J.: Sharp conditions of global existence for the generalized Davey-Stewartson system in three dimensional space. Acta Mathematica Scientia, Series A 26(1), 87–92 (2006) 8. Gan, Z.H., Zhang, J.: Sharp conditions of global existence for the generalized Davey-Stewartson system in two space dimensions. Math. Appl. 17(3), 360–365 (2004) 9. Gan, Z.H., Zhang, J.: Sharp conditions of global existence and collapse for coupled nonlinear Schrödinger equations. J. Part. Diff. Eqs. 17, 207–220 (2004) 10. Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. J. Funct. Anal. 32, 1–71 (1979) 11. Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation. Ann.Inst. H. Poincare. ´ Anal. Non Lineaire ´ 2, 309–327 (1985) 12. Glassey, R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 18(9), 1794–1797 (1977) 13. Grillakis, M., Shatah, J., Strauss, W.A.: Stability theory of solitary waves in presence of symmetry I. J. Funct. Anal. 74, 160–197 (1987) 14. Ghidaglia, J.M., Saut, J.C.: On the initial value problem for the Davey-Stewartson systems. Nonlinearity 3, 475–506 (1990) 15. Guo, B.L., Wang, B.X.: The Cauchy problem for Davey-Stewartson systems. Commun Pure Appl. Math. 52(12), 1477–1490 (1999) 16. Levine, H.A.: Instability and non-existence of global solutions to nonlinear wave equations of the form Pu tt = −Au + F(u). Trans. Amer. Math. Soc. 192, 1–21 (1974) 17. Li, Y.S., Guo, B.L., Jiang, M.R.: Global existence and blowup of solutions to a degenerate Davey-Stewartson equations. J. Math. Phys. 41(5), 2943–2956 (2000) 18. Lions, P.L.: The concentration-compactness principle in the calculus of variations, the locally compact case, Part I. Ann. Inst. H. Poincaré. Analyse Non linéaire 1, 109–145 (1984) 19. Lions, P.L.: The concentration-compactness principle in the calculus of variations, the locally compact case, Part I. Ann. Inst. H. Poincaré. Analyse Non linéaire 1, 223–283 (1984) 20. Ohta, M.: Stability of standing waves for the generalized Davey-Stewartson system, J. Dynam. Differ. Eqs. 6(2), 325–334 (1994) 21. Ohta, M.: Instability of standing waves for the generalized Davey-Stewartson system. Ann. Inst. Henri Poincare´ 62, 69–80 (1995) 22. Ohta, M.: Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in R 2 . Ann. Inst. Henri Poincare´ 63, 111–117 (1995) 23. Ozawa, T.: Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems. Proc. Roy. Soc. London Ser. A 436(1897), 345–349 (1992) 24. Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22(3-4), 273–303 (1975) 25. Rabinowitz, P. H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992) 26. Shatah, J., Strauss, W.A.: Instability of nonlinear bound states. Commun. Math. Phys. 100, 173–190 (1985) 27. Shu, J., Zhang, J.: conditons of global existence for the generalized Davey-Stewartson system. IMA J. Appl. Math. 72(1), 36–42 (2007) 28. Soffer, A., Weinstein, M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 36, 9–74 (1999) 29. Tsutsumi, M.: Decay of weak solutions to the Davey-Stewartson systems. J. Math. Anal. Appl. 182(3), 680–704 (1994) 30. Wang, B.X., Guo, B.L.: On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems. Sci. in China 44(8), 994–1002 (2001) 31. Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolations estimates. Commun. Math. Phys. 87, 567–576 (1983) 32. Zhang, J.: On the finite-time behaviour for nonlinear Schrödinger equations. Commun. Math. Phys. 162, 249–260 (1994) 33. Zhang, J.: Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations. Nonlinear Anal. 48, 191–207 (2002) 34. Zhang, J.: Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential. Commun. in PDE 30, 1429–1443 (2005) 35. Zhang, J.: Cross-constrained variational problem and nonlinear Schrödinger equation. In: Proceedings of the Smalefest 2000, F. Cucker, J. M. Rojas (eds.), Foundations of Computational Math. Series 457–469, River Edge, NJ: World Scientific, 2002 Communicated by P. Constantin
Commun. Math. Phys. 283, 127–167 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0564-8
Communications in
Mathematical Physics
The Dirac System on the Anti-de Sitter Universe Alain Bachelot Université de Bordeaux, Institut de Mathématiques, UMR CNRS 5251, F-33405 Talence Cedex, France. E-mail:
[email protected] Received: 20 July 2007 / Accepted: 20 March 2008 Published online: 18 July 2008 – © Springer-Verlag 2008
Abstract: We investigate the global solutions of the Dirac equation on the Antide-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass M of the field and the cosmological constant > 0: it appears a critical value, /12, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When M 2 ≥ /12 there exists a unique unitary dynamics. On the contrary, for the light fermions satisfying M 2 < /12, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy. I. Introduction There has been much recent interest in the field theory in the covering space of the Anti-de-Sitter space-time C Ad S, that appears as the ground state of the gauged supergravity group [15]. This lorentzian manifold is the maximally symmetric solution of the Einstein equations with cosmological constant − < 0 included. Its topology is Rt × R3X , but its causality is non-trivial because it is non-globally hyperbolic: the Cauchy data on {t = 0} × R3 determines the evolution of the fields only in a region D, bounded by a null hypersurface, a Cauchy horizon. More precisely D is defined called 3 π by | t |< 2 − arctan . Thus we can think that to specify the physics 3 | X | apart from D, we have to impose some asymptotic constraint at infinity as | X |→ ∞. Since the conformal boundary of C Ad S is timelike, this condition can be considered as a boundary condition. It is exactly the case for the massless, conformally coupled scalar fields that are conformally invariant in C Ad S, and these fields have been studied in this spirit by Avis, Isham and Storey in [1]. For the massive fields the situation is different because the gravitational potential relative to any origin increases at large spatial
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distances from the origin. It causes confinement of massive particles and prevents them from escaping to infinity. In fact, the situation is rather subtle and depends on the ratio between the mass from the field and the cosmological constant. This phenomenon has been discovered by Breitenlohner and Freedman [7,8], who have showed the existence of two critical values, the B-F bounds, for the scalar fields; the first one assures the positivity of the energy, and the second one assures the uniqueness of the dynamics. In this paper, we establish a similar result for the Dirac fields.1 The square of the mass of the spinors is compared with a unique B-F bound that is equal to /12. We shall see that the physics of the heavy fermions (M 2 ≥ /12) is uniquely determined, but there exists a lot of possible dynamics for the light fermions (M 2 < /12), involving the asymptotic forms, at the C Ad S infinity, of classical boundary conditions, local or non-local: MIT-bag, Chiral, APS conditions, etc. From the mathematical point of view, the solutions of the initial value problem are given in D by the Leray-Hadamard theorem for the hyperbolic equations ∂t = H(X, ∂ X ), and on the whole space-time, we solve the Cauchy problem by a spectral approach, i.e. we look for the solutions formally given by (t) = eitH (0). Therefore we have to construct self-adjoint extensions of the Dirac hamiltonian H(X, ∂ X ). This method was used by A. Ishibashi and R.M. Wald [21,22], for the integer spin fields. The paper is organized as follows. In Part II, we briefly describe the Anti-de-Sitter manifold, mainly the different systems of coordinates and the properties of the null and time-like geodesics. The explicit forms of the Dirac equation on C Ad S are described in Sect. III, and we state the main result, Theorem III.4. We perform the spinoidal spherical harmonics decomposition in the following part. The asymptotic conditions and the selfadjoint extensions are discussed in the final section. In a short appendix, we present a new proof of the B-F bounds for the Klein-Gordon equation. We end this introduction with some bibliographic information. Above all, we have to mention the works treating the scalar fields on C Ad S, [1,7,8,22]. We refer to [15,19,29] for a presentation of the Anti-de-Sitter universe. There are many mathematical works on the one-half spin field on curved space-time, in particular [4,17,18,25–28]. The gravitational potential plays the role of a variable mass that tends to the infinity at the space infinity; the rather similar Dirac equation on Minkowski space with increasing potential has been considered in [23,34,38]. The literature on the boundary value problems for the Dirac system is huge ; among important contributions, we can cite [5,6,9,10,16,20]. There are few papers concerning the deep problem of the global existence of fields on the non-globally hyperbolic lorentzian manifolds, in particular [2,11,13,21,32,37] .
II. The Anti-de-Sitter Space Time Given > 0, the anti-de-Sitter space Ad S is defined as the quadric (X 1 )2 + (X 2 )2 + (X 3 )2 − U 2 − V 2 = −
3
embedded in the flat 5-dimensional space R5 with the metric ds 2 = dU 2 + d V 2 − (d X 1 )2 − (d X 2 )2 − (d X 3 )2 . 1 The author thanks the anonymous referee for his valuable comments on the B-F bounds.
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Ad S is the maximally symmetric solution of Einstein’s equations with an attractive cosmological constant − < 0. To describe Ad S it is convenient to set U = R cos T , V = R sin T , 3 3 then we can see that Ad S = ST1 × R3(X 1 ,X 2 ,X 3 ) , 2 2 R dT + d R 2 − (d X 1 )2 − (d X 2 )2 − (d X 3 )2 , 3 3 R = (X 1 )2 + (X 2 )2 + (X 3 )2 + .
ds 2Ad S =
For constant T , the slice {T } × R3 is exactly the 3-dimensional hyperbolic space H3 that is the upper sheet of the hyperboloid (X 1 )2 + (X 2 )2 + (X 3 )2 − W 2 = − 3 in the Minkowski space R4(X 1 ,X 2 ,X 3 ,W ) with the metric (d X 1 )2 + (d X 2 )2 + (d X 3 )2 − dW 2 . It is useful to use the spherical coordinates r=
(X 1 )2 + (X 2 )2 + (X 3 )2 ∈ [0, ∞[, and i f 0 < r, ω =
1 1 2 3 (X , X , X ) ∈ S 2 , r
for which the hyperbolic metric becomes 2 −1 2 2 dsH3 = 1 + r dr + r 2 dω2 , 3 where dω2 is the euclidean metric on the unit two-sphere S 2 , dω2 = dθ 2 + sin2 θ dϕ 2 , 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π. We shall use the nice picture of the hyperbolic space, the so called Poincaré ball. We introduce 1 j 1 ≤ j ≤ 3, x = X j, 3 1 + 1 + r2 3
=
r 3 1+ 1+
2 3r
∈ [0, 1[,
then H3 can be seen as the unit ball B = {x := (x 1 , x 2 , x 2 ) ∈ R3 ; 2 = (x 1 )2 + (x 2 )2 + (x 3 )2 < 1} endowed with the metric 2 dsH 3 =
4 3 d2 + 2 dω2 , 0 ≤ < 1, ω ∈ S 2 . 1 − 2
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We note that the time coordinate T is periodic, and this property implies the existence of closed timelike curves. To avoid this unpleasant fact, we replace T ∈ S 1 by t ∈ R, i.e. we change the topology, and we consider in this paper the Universal Covering Space of the anti-de-Sitter space-time, that is the lorentzian manifold C Ad S := (M, g) defined by M = Rt × R3(X 1 ,X 2 ,X 3 ) = Rt × B(x 1 ,x 2 ,x 3 ) , with the metric, −1 gµν d x µ d x ν = 1 + r 2 dt 2 − 1 + r 2 dr 2 − r 2 dω2 , 0 ≤ r < ∞, ω ∈ S 2 , 3 3 2
4 3 1 + 2 2 2 2 2 d , 0 ≤ < 1, ω ∈ S 2 . dt − + dω = 1 − 2 1 − 2 It will be useful to introduce a third radial coordinate, r = 2 arctan . x = arctan 3
(II.1)
Then the Anti-de-Sitter manifold can be described by: (t, x, θ, ϕ) ∈ R × [0,
π [×[0, π ] × [0, 2π [, 2
gµν d x µ d x ν = 1 + tan2 x g˜ µν d x µ d x ν , where g˜ is given by g˜ µν d x µ d x ν = dt 2 −
3 2 d x + sin2 xdθ 2 + sin2 x sin2 θ dϕ 2 .
Therefore, if the 3-sphere S 3 is parametrized by (x, θ, ϕ) ∈ [0, π ] × [0, π ] × [0, 2π [, and S+3 is the upper hemisphere [0, π2 [x ×[0, π ]θ × [0, 2π [ϕ , C Ad S can be considered = Rt × S+3 of the Einstein cylinder as conformally equivalent to the submanifold M (E, g), ˜ E := Rt × S 3 ,
π
(II.2)
2 = Rt × x = and the crucial point is that the boundary ∂ M 2 × Sθ,ϕ is time-like. Nevertheless, we should note that, unlike the black-hole horizon of the Schwarzschild metric (that is a characteristic submanifold of the Kruskal space-time), the time-like infinity of C Ad S, like the cosmological horizon of the De Sitter universe, (or a rainbow) is seen in the same way by any observer: since C Ad S is frame-homogeneous (i.e. any Lorentz frame on C Ad S can be carried to any other by the differential map of an isometry of C Ad S), no point is privileged. Finally we recall that the null geodesics of Ad S are straight lines in R5(X 1 ,X 2 ,X 3 ,U,V )
and the timelike geodesics are ellipses, intersection of Ad S with the 2-planes of R5 passing through the origin 0. As a consequence, C Ad S is geodesically complete, and time oriented by the Killing vector field ∂t , but its causality is not at all trivial: (1)
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given a point P on the slice t = 0, the future-pointing null geodesics starting from P form a curving cone of which the boundary approaches but does not reach the slice t =
π 2
2π
3 .
3 ,
hence C Ad S is not globally hyperbolic; (2) the future-pointing timelike geodesics on C Ad S starting from P, all meet a conjugate point Q at t = π 3 , P and Q project on antipodal points of Ad S. Therefore the time-like geodesics on C Ad S can be parametrized by (t, x(t))t∈R , where the function t → x(t) is t-periodic, with period These unusual properties yield important consequences for the propagation of
the fields: (1) suggests that we could have to add some condition at the “infinity” S 2 = ∂B to solve an initial value problem, at least for the massless fermions; nevertheless, since the massive particles propagate along the time-like geodesics, (2) seems to imply that such a condition is not necessary for the massive fields. In fact, the situation is rather subtle and depends on the ratio between the square of the mass of the fermion, and the cosmological constant. We shall see that no asymptotic constraint at infinity is necessary for the heavy spinors, but there are many possible physical constraints for the light masses. In all the cases, the spectrum of the hamiltonian of the massive fields is discrete. III. The Dirac Equation on C Ad S We consider the Dirac equation with mass M ∈ R on a 3+1 dimensional lorentzian manifold (M, g): µ
iγ(g) ∇µ ψ − Mψ = 0.
(III.1) µ
The notations are the following. ∇µ are the covariant derivatives, γ(g) , 0 ≤ µ ≤ 3, are the Dirac matrices, unique up to a unitary transform, satisfying: µ
µ
0∗ 0 ν ν = γ(g) , γ(g) = −γ(g) , 1 ≤ j ≤ 3, γ(g) γ(g) + γ(g) γ(g) = 2g µν 1. γ(g) j∗
j
(III.2)
Here A∗ denotes the conjugate transpose of any complex matrix A. We make the following choices for the Dirac matrices on the Minkowski space time R1+3 : γ µ are the 4 × 4 matrices of the Pauli-Dirac representation given for µ = 0, 1, 2, 3 by: γ = 0
where
I =
I 0 0 −I
, γ = j
0 σj −σ j 0
,
10 1 0 01 0 −i 1 2 3 , σ = , σ = , σ = . 01 0 −1 10 i 0
We also introduce another Dirac matrix that plays an important role in the boundary problems: 0I , (III.3) γ 5 := −iγ 0 γ 1 γ 2 γ 3 = I 0 that satisfies γ 5 γ µ + γ µ γ 5 = 0, 0 ≤ µ ≤ 3.
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We know that when the metric is spherically symmetric,
1 gµν d x µ d x ν = F(r )dt 2 − dr 2 − r 2 dθ 2 + sin2 θ dϕ 2 , F(r ) µ
then, if we choose the local orthonormal Lorentz frame {ea , a = 0, 1, 2, 3} defined by 1 ea µ = g µµ 2 , i f µ = a, ea µ = 0 i f µ = a, the Dirac equation has the following form in (t, r, θ, ϕ) coordinates (see e.g. [26–28]): 1 1 ∂ ∂ 1 F
i 1 ∂ i F− 2 γ 0 + i F 2 γ 1 + + + γ2 + ∂t ∂r r 4F r ∂θ 2 tan θ ∂ i γ3 − M ψ = 0. + r sin θ ∂ϕ For the Anti-de-Sitter manifold we have
F(r ) = 1 + r 2 , 3
and it is convenient to make a first change of spinor ; we use the radial coordinate (II.1), and we put 1 4
(t, x, θ, ϕ) := r 1 + r 2 ψ(t, r, θ, ϕ). 3
(III.4)
Then we obtain the Dirac equation on the Anti-de-Sitter universe with the coordinates t ∈ R, x ∈ [0, π2 [, θ ∈ [0, π ], ϕ ∈ [0, 2π [: ∂ 1 1 1 0 3 ∂ 3 ∂ ∂
+ γ 0γ 1 + γ 0γ 2 + + γ γ
∂t ∂x sin x ∂θ 2 tan θ sin θ ∂ϕ i 3 0 + M γ = 0. (III.5) cos x Since the part of this differential operator involving ∂t , ∂x is with constant coefficients, the form of this equation is convenient to make a separation of variables by using the generalized spin spherical harmonics. But this decomposition has an inconvenience: since the one-half spin harmonics are not smooth functions on S 2 , the functional framework involves spaces that are different from the usual Sobolev spaces on S 2 as we shall see in the following part. It will also be useful to write the Dirac equation with the coordinates (t, , θ, ϕ) ∈ R × [0, 1[×[0, π ] × [0, 2π [. We put
(t, , θ, ϕ) := (t, x, θ, ϕ), : and the Dirac equation becomes: 3 ∂ 1 + 2
+ γ0 ∂t : 2 1 2 ∂ 1 1 2i M 3 1 ∂ 3 ∂ + γ + + γ + × γ
= 0. ∂ ∂θ 2 tan θ sin θ ∂ϕ 1 − 2 :
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In the part involving γ 1 , γ 2 , γ 3 , we recognize the usual Dirac operator in spherical coordinates on R3 with the euclidean metric. This is a nice way to get the Dirac operator on C Ad S in cartesian coordinates. Adapting an approach of [33], we introduce 1
a := I − γ 1γ 2 − γ 2γ 3 − γ 3γ 1 , 2 ϕ
S(θ, ϕ) := e 2 γ
1γ 2
θ
e2γ
3γ 1
a.
(III.6)
We easily check that aa ∗ = I, SS ∗ = I, γ 1 a = aγ 2 , γ 2 a = aγ 3 , γ 3 a = aγ 1 . We put γ 1 (, θ ) := γ 1 , γ 2 (, θ ) := :
:
1 2 1 γ , γ 3 (, θ ) := γ 3, : sin θ
⎧ 1 ⎪ γ˜ (, θ, ϕ) := cos ϕ sin θ γ 1 + sin ϕ sin θ γ 2 + cos θ γ 3 , ⎪ ⎪ ⎪ ⎪ 1
⎨ 2 cos ϕ cos θ γ 1 + sin ϕ cos θ γ 2 − sin θ γ 3 , γ˜ (, θ, ϕ) := ⎪ ⎪ ⎪ 1
⎪ 3 ⎪ − sin ϕγ 1 + cos ϕγ 2 . ⎩ γ˜ (, θ, ϕ) := sin θ Tedious calculations give: 1 ≤ j ≤ 3, S(θ, ϕ)γ j (, θ ) = γ˜ j (, θ, ϕ)S(θ, ϕ). :
The cartesian coordinates x := (x 1 , x 2 , x 3 ) on B being x 1 = cos ϕ sin θ, x 2 = sin ϕ sin θ, x 3 = cos θ, ˜ on B by the relations we define the spinors , ˜ (x 1 , x 2 , x 3 ) = (, θ, ϕ) :=
1 S(θ, ϕ) (, θ, ϕ), :
and the Dirac operators ⎧ ∂ ∂ ∂ ⎪ D : = γ1 1 + γ2 2 + γ3 3, ⎪ ⎪ ⎪ ∂ x ∂ x ∂ x ⎪ ⎪ ⎨ ∂ ∂ ∂ ˜ : = γ˜ 1 + γ˜ 2 + γ˜ 3 , D ∂ ∂θ ∂ϕ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ∂ 1 ⎪ 1 2 ⎩D + γ3 . :=γ +γ + : : ∂ : : ∂θ 2 tan θ ∂ϕ We omit the direct calculus that gives the links between these operators:
(III.7)
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Lemma III.1.
1 ˜ ˜ (, θ, ϕ) = S(θ, ϕ) D (, θ, ϕ). (D) (x 1 , x 2 , x 3 ) = D : : We denote S the operator that relates the spinors in cartesian and spherical coordinates: S : → S = , (x 1 , x 2 , x 3 ) :=
1 S(θ, ϕ) (x, θ, ϕ). tan x2
(III.8)
Then, if (t, .) is a solution of (III.5), the Dirac equation satisfied by (t, .) := S (t, .) for t ∈ R, x = (x 1 , x 2 , x 3 ) ∈ B has the form: 1 + 2 3 0∂ 3 2i M 1 ∂ 2 ∂ 3 ∂ γ + +γ +γ + γ = 0. (III.9) 1 2 3 2 ∂t 2 ∂x ∂x ∂x 1 − Since the charge of the spinor is the formally conserved L 2 norm, it is natural to introduce the Hilbert space: 4 2 L2 := L 2 B, dx , (III.10) 1 + 2 and given 0 ∈ L2 we want to solve the initial problem, i.e. to find a unique ∈ C 0 (Rt ; L2 )
(III.11)
(t = 0, .) = 0 (.),
(III.12)
∀t ∈ R, (t) L2 = 0 L2 .
(III.13)
solution of (III.9) satisfying:
and the conservation law:
Moreover, since
∂ ∂t
is a Killing vector field on C Ad S, it is natural to assume that t ∈ R −→ (0 → (t)) ,
(III.14)
is a group acting on L2 . Therefore we look for strongly continuous unitary groups U (t) on L2 that solve (III.5). According to the Stone theorem, the problem consists in finding self-adjoint realizations on L2 of the differential operator 3 1 + 2 2i M 0 1 ∂ 2 ∂ 3 ∂ γ γ +γ +γ + H M := i , (III.15) 1 2 3 2 2 ∂x ∂x ∂x 1− with domain
D(H M ) = ∈ L2 ; H M ∈ L2 ,
(III.16)
by adding suitable constraints at the CAdS infinity = 1. The answer crucially depends on the mass of the spinor. First we discuss the massless case. When M = 0, the Dirac system is conformal invariant, and it is equivalent to solving the Cauchy problem in the half of the Einstein
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135
cylinder, Rt × S+3 . Therefore we can extend the initial data from the hemisphere S+3 to the whole sphere S 3 , and solve the Cauchy problem on the Einstein cylinder Rt × S 3 . This is tantamount to solving Eq. (III.9) on Rt ×R3x , instead of Rt ×Bx . This approach was used by S.J. Avis, C.J. Isham, D. Storey [1] for the scalar field, and later, by Y. Choquet-Bruhat for the Yang-Mills-Higgs equations [11]. By this way, we impose no boundary condition at the C Ad S infinity, or, in other words, a “perfectly transparent” boundary condition, and we easily obtain global solutions on C Ad S. We have to remark that since there exists a lot of ways to extend the initial data, such a solution is not uniquely determined by the Cauchy data on S+3 . Moreover the effect of this “perfectly transparent” condition is to recirculate the energy: the conserved charge is the L 2 -norm on S 3 while the L 2 -norm on S+3 is changing in time, and so (III.13) is not satisfied. In order to assure the conservation (III.13), we can take another route, and impose some “reflecting” boundary conditions on {x = π2 } × S 2 . In [1], several conditions are discussed for the scalar massless field. For the Dirac equation, we note that when M = 0, Eq. (III.9) has smooth coefficients up to the boundary | x |= 1. Therefore, in the massless case, we deal with a classical mixed hyperbolic problem, and different boundary conditions for the Dirac system with regular potential are well known (see e.g. [5,6,9,10,16,20]). We recall an important local boundary condition for the Dirac spinors defined on some open domain of the space-time, the so called generalized MIT-bag condition: n µ γ µ (t, x 1 , x 2 , x 3 ) = ieiαγ (t, x 1 , x 2 , x 3 ), (t, x 1 , x 2 , x 3 ) ∈ ∂, 5
where n µ is the outgoing normal quadrivector at ∂ and α ∈ R is the chiral angle. When α = 0 this is the MIT-bag condition for the hadrons and when α = π this is the Chiral condition. Another fundamental boundary condition is the non-local APS condition introduced by M.F. Atiyah, V. K. Patodi, and I. M. Singer (see e.g. [6]) and defined by 1]0,∞[ (D∂ ) = 0 on ∂, where D∂ is the Dirac operator on ∂. More recently, O. Hijazi, S. Montiel, A. Roldan [20] have introduced the mAPS condition: 1]0,∞[ (D∂ ) I d − n µ γ µ = 0 on ∂. For = Rt × B, these boundary conditions become B(t, ω) = 0, (t, ω) ∈ R × S 2 ,
(III.17)
MIT − bag condition : BMIT = γ˜ 1 + i I d,
(III.18)
Chiral condition : BCHI = γ˜ 1 − i I d,
(III.19)
APS condition : BAPS = 1]0,∞[ D S 2 ,
(III.20)
mAPS condition : BmAPS = 1]0,∞[ D S 2 γ˜ 1 + I d ,
(III.21)
where
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A. Bachelot
where D S 2 is the intrinsic Dirac operator on the two-sphere: 0 2 ∂ 3 ∂ ˜ + γ ˜
. γ ˜ D = iγ 2 S ∂θ ∂ϕ We conclude that there exists many unitary dynamics for the massless spin- 21 field on C Ad S, that we can easily construct by solving (III.9) with M = 0, (III.12), (III.17), by invoking the classical theorems on the mixed hyperbolic problems. In consequence, our work is mainly concerned with the massive field, and in the sequel, we consider only this case. When M = 0 the situation is very different because the potential blows up as → 1. The analogous situation of the infinite mass at the infinity of the Minkowski space has been investigated in [23,34]. In our case, the key result is the asymptotic behaviour, near the boundary, of the spinors of D(H M ). We note that it is sufficient to consider only the case of the positive mass, because the chiral transform −→ γ 5 changes M into −M since we have γ 5 H M γ 5 = H−M . We remark that the MIT-bag and the Chiral conditions are exchanged by the chiral transform, and the APS condition is chiral invariant. Theorem III.2. Let be in D (H M )with M ∈ R∗ . Then 4
1 , ∈ C 0 ]0, 1[ ; H 2 (Sω2
1 0
When M 2 >
12 ,
(ω) 2H 1 (S 2 ) d ≤ H M 2L2 . ω
we have (ω) L 2 (Sω2 ) = O
When M 2 =
12 ,
we have
1 − , → 1.
(III.22)
(III.23)
(III.24)
(III.25) ( − 1) ln (1 − ) , → 1. 1 4 When 0 < M 2 < 12 , we put m := M 3 , and there exists − ∈ H 2 (S 2 ) , 4 4 + ∈ L 2 (S 2 ) , and ψ ∈ C 0 [0, 1] ; L 2 (Sω2 ) satisfying (ω) L 2 (Sω2 ) = O
(ω) = (1 − )−m − (ω) + (1 − )m + (ω) + ψ(ω),
(III.26)
γ˜ 1 − + i− = 0, γ˜ 1 + − i+ = 0,
(III.27)
ψ(ω) L 2 (Sω2 ) = o
1 − , → 1.
(III.28)
1 4 1 4 Conversely, for any − ∈ H 2 +m (S 2 ) , + ∈ H 2 −m (S 2 ) satisfying (III.27), there exists ∈ D(H M ) satisfying (III.26) and (III.28).
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137
Remark III.3. We shall see that (III.24) can be improved and when M 2 > 12 the el 1 d 2 liptic estimate below (III.32) implies that 0 (ω) L 2 (S 2 ) (1−)2 < ∞. When ω 4 M 2 ≥ 12 , then ∈ C 0 (]0, 1] ; L 2 (Sω2 ) , but the trace of on ∂B does not exist for . Moreover we see with (III.23) that when M = 0, H M = 0 implies = M 2 < 12 4
1 for 0.The situation is different when M = 0: we have ∈ C 0 ]0, 1] ; H − 2 (Sω2
∈ D(H0 ), and this result is optimal: there exists ∈ L2 , = 0, with H0 = 0 and 4 4 1 (ω) ∈ H − 2 (Sω2 ) \ ∪s>− 1 H s (Sω2 ) . 2
, the elements of the domain of H M satisfy the homogeneous We note that when M 2≥ 12 Dirichlet Condition on ∂B. We shall see that H M is self-adjoint. On the contrary, when
, the trace of on ∂B is not defined, the leading term (1 − )−m − satis0 < M < 12 m fies the MIT-bag Condition and the next term (1 − ) + satisfies the Chiral Condition 12
(and the converse for −
< M < 0). We introduce natural generalizations of the
classic boundary conditions in terms of asymptotic behaviours near S 2 :
B(ω) L 2 (Sω2 ) = o 1− ,
(III.29)
and we consider the operators HB , B = B M I T , BC H I , B A P S , Bm A P S , defined as the differential operator H M endowed with the domain
D (HB ) := ∈ D(H M ); B(ω) L 2 (Sω2 ) = o 1− . We remark that (III.26), (III.27) and (III.28) imply: D HB M I T := { ∈ D(H M ); + = 0 i f M > 0, − = 0 i f M < 0} , D HBC H I := { ∈ D(H M ); − = 0 i f M > 0, + = 0 i f M < 0} , D HB A P S = D HBm A P S = ∈ D(H M ); 1]0,∞[ D S 2 + = 1]0,∞[ D S 2 − = 0 . We now construct a large family of asymptotic conditions, generalizing the previous one, by imposing a linear relation between − and + . If we denote ± 1 , ψ 2 , ψ 3 , ψ 4 ), the constraints of polarization (III.27) allow to express ψ 3,4 =t (ψ± ± ± ± ± 1,2 by using ψ± :
3 (ω) ψ± 4 (ω) ψ±
ω.σ = ± iω.σ
1 (ω) ψ± 2 (ω) , ω.σ := ψ±
3
ωjσ j. 1
We consider two densely defined self-adjoint operators (A± , D(A± )) on L 2 (S 2 ) × L 2 (S 2 ), satisfying 1
1
D(A+ ) = L 2 (S 2 ) × L 2 (S 2 ), D(A− ) ⊃ H 2 (S 2 ) × H 2 (S 2 ),
(III.30)
1 1 A± C ∞ (S 2 ) × C ∞ (S 2 ) ⊂ H 2 ±m (S 2 ) × H 2 ±m (S 2 ).
(III.31)
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A. Bachelot 1
As an example, we can choose A− any hermitian matrix of H 2 (S 2 ; C2×2 ), and A+ any 1 hermitian matrix of H 2 +m ∩ L ∞ (S 2 ; C2×2 ). We define the operators (HA+ , D(HA+ )), (HA− , D(HA− )), where 1 " 1 ! ψ∓ ± ψ± = A . D (HA± ) := ∈ D(H M ); 2 2 ψ∓ ψ± For A− = A+ = 0, we obviously have HA∓ = HB M I T , HA± = HBC H I if ± M > 0. Furthermore, the chiral transform → γ 5 leads to the exchanges M → −M, HB M I T → HBC H I , HBC H I → HB M I T , HB A P S → HB A P S , HBA± → HBω.σ A±ω.σ . We now state the main theorem of this paper. Theorem III.4 (Main result). Given M ∈ R∗ , we consider the massive Dirac hamilto nian H M defined by (III.15), (III.16). When M 2 ≥ 12 , H M is essentially self-adjoint on ∞ 4 4 1 2 C0 (B) , and if M > 12 , then D (H M ) = H0 (B) , and for all ∈ D (H M ), we have the following elliptic estimate: H M L2 ≥ | M | − (III.32) ∇x L2 . 12 12 , HA+ , HA− , HB A P S , HBm A P S are self-adjoint on L2 , and HB A P S When M 2 < 12 = HBm A P S . The resolvent of any self-adjoint realization of H M , M ∈ R∗ , is compact on L2 , and so, the spectrum of these operators is discrete. We see that 12 is an important critical value. It plays exactly the same role that the bounds that Breitenlohner and Freedman have discovered for the scalar massive fields [7,8]. We recall that these authors have considered the Klein-Gordon equation 1 1 | g |− 2 ∂µ | g | 2 g µν ∂ν u − α 3 u = 0, for which α = 2 corresponds to the massless case. By a sharp analysis of the modes, they have established, among other results, that: (i) the natural energy is positive when α ≤ 9/4 , in particular for the light tachyons associated with 2 < α < 9/4; (ii) the dynamics is unique when α ≤ 5/4 ; (iii) there exists a lot of unitary dynamics when 5/4 < α < 9/4. For the sake of completeness we give in the one-page Appendix, a new and very simple proof of these results, based on a Hardy estimate and on the Kato-Rellich theorem. For the spin- 21 field with real mass, the most important conserved quantity is the L 2 -norm that is always positive, hence one bound will suffice to distinguish the different cases: it is 12 . We have to emphasize that this value was already presented in the discussion of the massive O Sp(1, 4) scalar multiplet in [7,8]. This multiplet consists√of a Dirac spinor with mass M, and two Klein-Gordon fields for which α = 2 ± M 3/ − 3M 2 /. We can easily check that α ≤ 9/4 for any M ∈ R, and α ≤ 5/4 iff M 2 ≥ /12. Therefore our own result is coherent with this particular model of Anti-de Sitter supersymmetry: the constraints for the uniqueness of the dynamics are simultaneously satisfied for the spin field and the scalar fields. The case α > 9/4 describes the heavy tachyons in C Ad S, and corresponds to the case of an imaginary mass for the Dirac field. This regime seems to be unphysical since the energy of a scalar tachyon is not positive, and the L 2 -norm of a spin- 21 field with an imaginary mass is not conserved. Of the mathematical point of view, it is doubtful that the global Cauchy problem with these parameters is well posed, and of the physical
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139
point of view, we could suspect that the Ad S background is not stable with respect to the fluctuations of such fields. We do not address this situation in this paper. We now turn over to the Cauchy problem. Theorem III.5. Given 0 ∈ L2 , there exist solutions of (III.9), (III.11), (III.12), and all the solutions are equal for
3 π − 2 arctan . (III.33) (t, x) ∈ R × B, | t |< 2 When M 2 ≥ 12 , the Cauchy problem (III.9), (III.11), (III.12) has a unique solution. This solution satisfies (III.13).
We achieve this part with a result of equipartition of the energy. We know, [3], that the solutions ∈ C 0 (Rt ; L 2 (R3 ; C4 )) of the massive Dirac equation on the Minkowski space-time, satisfy ∗ γ 0 γ 5 (t, x)dx = 0. lim |t|→∞ R3
Since the spectrum of the possible hamiltonians for the massive fermions on C Ad S is discrete, we cannot expect such an asymptotic behaviour. Nevertheless, we establish the existence of a similar limit, in the weaker sense of Cesaro:
Theorem III.6. Let ∈ C 0 (Rt ; L2 ) be a solution of (III.9), given by (t)=e where H is a self-adjoint realization of H M , M ∈ R∗ . Then we have: 1 T lim ∗ γ 0 γ 5 (t, x)dxdt = 0. T →∞ T 0 B
it
3H
(0),
(III.34)
The proofs of these results are presented in Parts V and VI. They are made much easier by the use of the spherical coordinates. Operator S, defined by (III.8), that relates the spinors within the two systems of coordinates, is an isometry from π 4 L2 := L 2 [0, [x ×[0, π ]θ × [0, 2π [ϕ , sin θ d xdθ dϕ , (III.35) 2 onto L2 , and satisfies the intertwining relation H M S = SHm , where Hm is the differential operator ∂ i 1 1 0 3 ∂ 0 1 ∂ 0 2 + γ γ + + γ γ Hm := iγ γ ∂ x sin x ∂θ 2 tan θ sin θ ∂ϕ 3 m 0 γ , m=M . − cos x
(III.36)
(III.37)
The problem essentially consists in finding self-adjoint realizations of Hm in L2 . The difficulty comes from the blow-up of the gravitational interaction on the boundary. We see that H0 is just the Dirac operator on the 3-sphere S 3 ↔ [0, π ]x × [0, π ]θ × [0, 2π [ϕ , restricted to the upper hemisphere S+3 ↔ [0, π2 [x ×[0, π ]θ × [0, 2π [ϕ . The key result, Theorem V.1, deals with the asymptotic behaviour of ∈ D(Hm ) at the equatorial 2-sphere S 2 = ∂ S+3 , as x → π2 . The tool is a careful analysis based on the diagonalization of D S 2 by the spinoidal spherical harmonics.
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A. Bachelot
IV. The Spinoidal Spherical Harmonics We start by introducing several tools based on the spinor representation of the rotation 2 2 group (see[14,26,35]). It is well known thatthere exists two Hilbert bases of L (S ), given by: T 1l (θ, ϕ) 2 ,n
, Tl
− 21 ,n
(l,n)∈I
! I := (l, n); l ∈ N + ! = (l, n); l ∈ N +
(θ, ϕ)
, (l,n)∈I
" 1 1 , n ∈ Z + , l− | n |∈ N 2 2 " 1 , n = −l, −l + 1, . . . , l , 2
(IV.1)
T±l 1 ,n (θ, ϕ) = e−inϕ P±l 1 ,n (cos θ ), 2
where P l
± 21 ,n
2
can be expressed in terms of generalized Jacobi functions:
P±l 1 ,n (X ) = Al±,n (1 − X )
±1−2n 4
2
(1 + X )
∓1−2n 4
d l−n l∓ 21 l± 21 (1 − X ) , (1 + X ) d X l−n
and the constant Al±,n
# $ $ l ∓ 1 !(l + n)! 2l + 1 (−1) i 2 % = 4π l ± 21 !(l − n)! 2l l ∓ 21 ! l∓ 12 n∓ 21
is chosen to normalize the basis functions (in comparison with the notations adopted in √ l are multiplied by (2l + 1)/4π ): the book [14], the functions Pm,n
2π 0
π 0
T±l 1 ,n (θ, ϕ)T l
± 12 ,n
2
(θ, ϕ) sin θ dθ dϕ = δl,l δn,n .
Therefore we can expand any function f ∈ L 2 (S 2 ) on both these bases bases f (θ, ϕ) = (l,n)∈I
u l±,n ( f )T±l 1 ,n (θ, ϕ), u l±,n ( f ) ∈ C, 2
and by the Plancherel formula: f 2L 2 =
| u l+,n ( f ) |2 = (l,n)∈I
(l,n)∈I
| u l−,n ( f ) |2 .
More generally, for s ∈ R, we introduce the Hilbert spaces W±s defined as the closure of the space ⎧ ⎫ ⎨ ⎬ W± u l±,n T±l 1 ,n ; u l±,n ∈ C (IV.2) f := ⎩ ⎭ 2 f inite
The Dirac System on the Anti-de Sitter Universe
for the norm f 2W s := ±
(l,n)∈I
141
1 2s l l+ | u ±,n ( f ) |2 . 2
We note that the basis functions are not smooth on S 2 since T l
± 21 ,n
(θ, 2π ) = −T l
± 21 ,n
(θ, 0) # 0. Hence is not a classical Sobolev space on We state some properties of these spaces. Firstly it is easy to prove that for
s ≥ 0 =⇒ W±s = f ∈ L 2 S 2 ; f W±s < ∞ , W±s
S2.
and the topological dual of W±s can be isometrically identified with W±−s :
s ∈ R, W±s = W±−s . Secondly we show that W±s contains the test functions on ]0, π [θ ×]0, 2π [ϕ . To see that, we recall the differential equations satisfied by the basis functions:
∂ 1 + ∂θ 2 tan θ
T±l 1 ,n 2
n 1 l T 1 −i l + =± T∓l 1 ,n , sin θ ± 2 ,n 2 2
(IV.3)
∂ l T 1 = −inT±l 1 ,n . ∂ϕ ± 2 ,n 2
(IV.4)
If f ∈ C0∞ (]0, π [θ ×]0, 2π [ϕ ) then (∂θ + 21 cot θ ∓ sini θ ∂ϕ ) f ∈ C0∞ (]0, π [θ ×]0, 2π [ϕ ) and for any integer N , the differential equation (IV.3) assures that 1 2N l ∂ 1 i ∂ 2N N l l+ + ∓ u ±,n ( f ) = (−1) u ±,n f ∈ l 2 (I ). 2 ∂θ 2 tan θ sin θ ∂ϕ We conclude that any test function belongs to W±s for any real s, and the series I u l±,n T l 1 ∈ W±s converges in the sense of the distributions on ]0, π [θ ×]0, 2π [ϕ , in parti± 2 ,n
cular for all s < 0. We deduce that (∂θ + 21 cot θ ∓ sini θ ∂ϕ ), acting in the sense of the distributions, is an isometry from W±s onto W∓s−1 . But we have to be careful since the set of the test functions is not dense in general in W±s , s > 0: we cannot identify W −s with a subspace of distributions, and there can exist f ∈ W±−s \ {0} which is null in the 1
sense of the distributions on ]0, π [θ ×]0, 2π [ϕ . For instance, since (sin θ )− 2 ∈ L 2 (S 2 ), we have √ 1 1 l T±l 1 ,n ∈ W±−1 , f ± W −1 = 2π, l+ u ∓,n √ f ± := ± 2 2 sin θ (l,n)∈I
but its restriction on the test functions is the null distribution because 1 i ∂ 1 ∂ + ∓ f ± |C ∞ (]0,π [θ ×]0,2π [ϕ ) = i √ 0 ∂θ 2 tan θ sin θ ∂ϕ sin θ = 0 in D (]0, π [×]0, 2π [).
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A. Bachelot
Finally we investigate the links between W+s and W−s . We know that P 1l ,n = P−l 1 ,−n , 2
2
and Pl
± 21 ,n
1
= (−1)n∓ 2 P±l 1 ,n , 2
hence 1
u l±,n ( f ) = (−1)n∓ 2 u l∓,−n ( f ), and we have s ∈ R,
f ∈ W±s ⇐⇒ f ∈ W∓s , f W±s = f W∓s .
We warn that in general W+s = W−s . Indeed, given f ± ∈ W±1 , we have by (IV.3): ∂ 1 i ∂ 1 + ∓ f± = u l±,n ( f ± )T∓l 1 ,n ∈ L 2 (S 2 ). −i l + ∂θ 2 tan θ sin θ ∂ϕ 2 2 (l,n)∈I
We deduce that f ∈ W+1 ∩ W−1 =⇒ Then if we consider T 1
we see that
1 2 sin θ ∂ϕ T 1 , 1
1 2 1 1 2,2
(θ, ϕ) =
1 ∂ f ∈ L 2 (S 2 ). sin θ ∂ϕ
3 −i ϕ θ e 2 cos ∈ W+1 , 4π 2
∈ / L 2 (S 2 ), and we conclude that
2 2
W+1 = W−1 . Therefore it is convenient to introduce the isometry J on L 2 (S 2 ) defined by l J T+ 1 ,n = T−l 1 ,n . 2
Then we have
2
J ∗ T−l 1 ,n = T+l 1 ,n , 2
2
and J is an isometry from W+s onto W−s . We now return to the Dirac field. In: the same way, we can expand any spinor defined on S 2 , (θ, ϕ) ∈ L 2 (S 2 ; C4 ): ⎞ ⎛ l u 1,n T l 1 (θ, ϕ) − 2 ,n ⎟ ⎜ l ⎜ u 2,n T l 1 (θ, ϕ) ⎟ ⎟ ⎜ + 2 ,n
(θ, ϕ) = ⎟ , u lj,n ∈ C. ⎜ l ⎜ u 3,n T l 1 (θ, ϕ) ⎟ − 2 ,n (l,n)∈I ⎝ ⎠ u l4,n T l 1 (θ, ϕ) + 2 ,n
The Dirac System on the Anti-de Sitter Universe
143
The main interest of this expansion is the following: if we consider the angular part of the hamiltonian Hm , ∂ 1 i ∂ 0 2 + + γ 0γ 3 , D := iγ γ ∂θ 2 tan θ sin θ ∂ϕ an elementary but tedious computation shows that: ⎛ l ⎞ u 4,n T l 1 (θ, ϕ) − 2 ,n ⎜ u l T l (θ, ϕ) ⎟ ⎟ 3,n 1 ⎜ ⎜ ⎟ + 21 ,n l+ D (θ, ϕ) = ⎜ l ⎟. 2 ⎜ u 2,n T−l 1 ,n (θ, ϕ) ⎟ (l,n)∈I ⎝ ⎠ 2 u l1,n T l 1 (θ, ϕ)
(IV.5)
+ 2 ,n
Hence it is natural to introduce the Hilbert spaces W s := W−s × W+s × W−s × W+s endowed with the norm: 4
W s := 2
j=1 (l,n)∈I
1 2s l 2 l+ | u j,n | . 2
(IV.6)
W s is also the closure for this norm, of the subspace − + + W f := W − f × Wf × Wf × Wf .
As a differential operator, D acts from W s to W s−1 and D endowed
with the domain W 1 is self-adjoint on W 0 . We see that the spectrum of (D, W 1 ) is ± l + 21 , l ∈ N , , its positive subspace L 2+ S 2 ; C4 is spanned by the eigenvectors T l 1 , 0, 0, T l 1 − 2 ,n + 2 ,n 0, T l 1 , T l 1 , 0 , (l, n) ∈ I , and the negative subspace L 2− S 2 ; C4 is spanned + 2 ,n − 2 ,n l l l l by the eigenvectors T 1 , 0, 0, −T 1 , 0, T 1 , −T 1 , 0 , (l, n) ∈ I . We can − 2 ,n
+ 2 ,n
+ 2 ,n
characterize these spaces by using the operator J :
L 2± S 2 ; C4
− 2 ,n
⎧⎛ ⎫ ⎞ ψ ⎪ ⎨ ⎬
⎪ ⎜ χ ⎟ = ⎝ . , ψ, χ ∈ L 2 S 2 ⎠ ⎪ ⎪ ⎩ ±J∗χ ⎭ ±J ψ
We easily obtain the orthogonal projectors K± on L 2± S 2 ; C4 : ⎛
1 1⎜ 0 K± = ⎝ 0 2 ±J ∗
0 0 1 ±J ∗ ±J 1 0 0
⎞ ±J 0 ⎟ . 0 ⎠ 1
(IV.7)
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A. Bachelot
K± can be extended into bounded operators on W s , s ∈ R. These operators are used to define the global boundary conditions of M.F. Atiyah, V. K. Patodi, and I. M. Singer (see e.g. [6]): K± = 0,
(IV.8)
and the boundary condition introduced by O. Hijazi, S. Montiel, A. Roldan [20],
K+ I d + γ 1 = 0. W s is also invariant by the operator 5
Bα := γ 1 + ieiαγ , α ∈ R, involved in the local MIT-bag boundary condition: B0 = 0, and the chiral condition: Bπ = 0. If we consider the operator := γ 0 γ 2 D as a positive, unbounded, selfadjoint operator on W 0 with domain W 1 , then for 0 ≤ s ≤ 1, W s is the domain of s , that is to say, these spaces are spaces of interpolation (see e.g. [24]): W s = W 1, W 0 , 0 ≤ s ≤ 1. 1−s
The link between this space and the usual Sobolev spaces on S 2 is given by the following: Proposition IV.1. For any s ∈ R, the linear map
(θ, ϕ) −→ (x 1 , x 2 , x 3 ) = S(θ, ϕ) (θ, ϕ), (x 1 , x 2 , x 3 ) ∈ S 2 , 4 defined from W f to L 2 S 2 , where S is given by (III.6) and x j , θ , ϕ are related to 4 (III.7), can be extended into a bounded isomorphism from W s onto H s (S 2 ) . Proof of Proposition IV.1. A tedious but elementary calculation shows that: S11 S12 , S(θ, ϕ) = S21 S22 where
ϕ ⎞ ⎛
ϕ θ θ θ θ −i iϕ i −i ϕ 1 ⎝ (1 + i) e 2 cos 2 + e 2 sin 2 (1 + i) e 2 cos 2 − e 2 sin 2 ⎠ S11 = S22 = , ϕ ϕ ϕ ϕ 2 (1 − i) −e−i 2 cos θ + ei 2 sin θ (1 − i) ei 2 cos θ + e−i 2 sin θ 2 2 2 2 (IV.9) S12 = S21 =
00 . 00
The Dirac System on the Anti-de Sitter Universe
145
Following [36], p.337, formula (3) with n = 0, we have √ √ θ l+ 12 l Pm,0 l + 1P 1 1 (cos θ ) = l − m + 1 cos (cos θ ) m− 2 ,− 2 2 √ θ l Pm−1,0 (cos θ ), + l + m sin 2 then since l l Pm,n = (−1)m+n Pn,m ,
we get for l ∈ N, m ∈ Z, −l ≤ m ≤ l + 1: 3 θ l+ 21 −i ϕ2 m−1 l − m + 1 x + 1 l T 1 Ym (θ, ϕ) cos e 1 (θ, ϕ) = (−1) − 2 ,m− 2 2 l +1 2 l + m x1 − i x2 l Ym−1 (θ, ϕ), +(−1)m−1 l +1 2
e
i ϕ2
1 2 θ l+ 12 m−1 l − m + 1 x + i x T 1 Yml (θ, ϕ) sin 1 (θ, ϕ) = (−1) − 2 ,m− 2 2 l +1 2 3 m−1 l + m 1 − x l Ym−1 (θ, ϕ). +(−1) l +1 2
In the same way, with [36], p.337, formula (4) with n = 0, we have √ √ θ l+ 12 l Pm,0 l + 1P 1 1 (cos θ ) = − l − m + 1 sin (cos θ ) m− 2 , 2 2 √ θ l Pm−1,0 (cos θ ), + l + m cos 2 then we get for l ∈ N, m ∈ Z, −l + 1 ≤ m ≤ l: 1 2 θ l+ 21 i ϕ2 m+1 l − m + 1 x + i x T1 Yml (θ, ϕ) e cos 1 (θ, ϕ) = (−1) 2 l +1 2 2 ,m− 2 3 m l +m x +1 l Ym−1 (θ, ϕ), +(−1) l +1 2
e
−i ϕ2
3 θ l+ 21 m+1 l − m + 1 1 − x T1 Yml (θ, ϕ) sin 1 (θ, ϕ) = (−1) 2 l +1 2 2 ,m− 2 1 2 m l + m x − ix l Ym−1 +(−1) (θ, ϕ). l +1 2
Since f → x j f is bounded on H s (S 2 ) and f belongs to H s (S 2 ) iff ∞
l
αl,m Yml ,
f = l=0 m=−l
l 2s | αl,m |2 < ∞, l,m
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we conclude that the linear map → S(θ, ϕ) = is bounded from W f endowed 4 with the norm of W s to H s (S 2 ) , hence it can be extended into a continuous linear 4 map S : → from W s to H s (S 2 ) . Then S∗ is a bounded linear map from −s 2 4 4 H (S ) to W −s for any s ∈ R. Since S∗ = S ∗ (θ, ϕ) for ∈ C0∞ (S 2 ) , and S ∗ (θ, ϕ) = S −1 (θ, ϕ), we conclude that SS∗ = I d H s , S∗ S = I dW s . V. Asymptotic Behaviour at the Boundary In this part we investigate the properties of the spinors that belong to the natural domain of the hamiltonian, especially the asymptotic behaviours near the boundary. We begin with its form in spherical coordinates, Hm given by (III.37), and D (Hm ) := ∈ L2 ; Hm ∈ L2 . (V.1) Theorem V.1. For any ∈ D (Hm ) we have
π 1
∈ C 0 [0, [x ; W 2 , 2 (x, .)
1
W2
2
0
When
1 2
√ x , x → 0,
(V.3)
dx ≤ Hm 2L2 . sin x
(V.4)
=O
and when 0 < m we have π (x, .) 2W 1
(V.2)
< m, we have (x, .) L 2 (S 2 ) = O
When m = 21 , we have: (x, .) L 2 (S 2 ) = O
x−
π π −x , x → . 2 2
π π π ln −x , x → . 2 2 2 1
1
(V.5)
(V.6)
When 0 < m < 21 , there exists ψ− ∈ W−2 , χ− ∈ W+2 , ψ+ , χ+ ∈ L 2 (S 2 ), and φ ∈ C 0 [0, π2 ]x ; L 2 (S 2 ; C4 ) satisfying ⎛ ⎞ ψ− (θ, ϕ) −m
π ⎜ χ− (θ, ϕ) ⎟ −x
(x, θ, ϕ) = ⎝ −iψ (θ, ϕ) ⎠ − 2 iχ− (θ, ϕ) ⎛ ⎞ ψ+ (θ, ϕ) m
π ⎜ χ (θ, ϕ) ⎟ −x ⎝ + + φ(x, θ, ϕ), (V.7) + iψ+ (θ, ϕ) ⎠ 2 −iχ+ (θ, ϕ)
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147
φ(x, .) L 2 (S 2 ) = o 1
1
+m
π π −x , x → . 2 2 1
+m
Conversely, for any ψ− ∈ W−2 , χ− ∈ W+2 , ψ+ ∈ W−2
∈ D(Hm ) satisfying (V.7) and (V.8). When m = 0, then
π 1
∈ C 0 [0, ]x ; W − 2 . 2
−m
(V.8) 1
, χ+ ∈ W+2
−m
there exists
(V.9)
Remark V.2. Equation (V.4) shows that when m > 0, Hm = 0 implies = 0. On the contrary, when m = 0, the left member of (V.4) can be infinite even if H0 = 0. 1 Furthermore the space W − 2 is optimal for the traces on x = π2 : there exists ∈ D(H0 ) / ∪s>− 1 W s . As an example, we consider a sequence Cl,n (l,n)∈I ⊂ C such that ( π2 ) ∈ 2 such that 2 2 1 −1 1 s Cl,n < ∞, −1 < s ⇒ Cl,n = ∞, l+ l+ 2 2 (l,n)∈I
(l,n)∈I
we can take for instance Cl,n =
√ 1 , l log(l+1)
and we put ⎛
(x, θ, ϕ) =
Cl,n tan
x l+ 1
(l,n)∈I
2
(θ, ϕ)
⎞
1 ⎜ − 2 l,n ⎟ ⎜ −i T 1 (θ, ϕ) ⎟ ,n ⎜ ⎟. 2
⎝
2
Tl
0 0
⎠
Then we easily check that
π 2
∈ L2 , H0 = 0, 0 < s ⇒ 1 π
( , .) ∈ W − 2 \ ∪s>− 1 W s . 2 2
0
(x, .) 2W s d x = ∞,
Remark V.3. For 0 < m < 21 , the leading terms of satisfy the MIT-bag or the Chiral boundary condition since: ⎛ ⎞ ψ+ m
π ⎜ χ ⎟ − x ⎝ + ⎠ + B0 φ(x), B0 (x) = 2i iψ+ 2 −iχ+ ⎛ ⎞ ψ− −m
π ⎜ χ− ⎟ Bπ (x) = −2i −x ⎝ −iψ ⎠ + Bπ φ(x). − 2 iχ−
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A. Bachelot
Proof of Theorem V.1. We expand any spinor (x, θ, ϕ) in the previous way: ⎞ ⎛ l u 1,n (x)T l 1 (θ, ϕ) − 2 ,n ⎟ ⎜ l ⎜ u 2,n (x)T l 1 (θ, ϕ) ⎟ ⎟ ⎜ + 2 ,n
(x, θ, ϕ) = ⎟, ⎜ l ⎜ u 3,n (x)T l 1 (θ, ϕ) ⎟ − 2 ,n (l,n)∈I ⎝ ⎠ u l4,n (x)T l 1 (θ, ϕ) + 2 ,n
and we have: 4
2L2 =
j=1 (l,n)∈I
Furthermore, for ∈ D(Hm ), (IV.5) gives: ⎛ ⎜ ⎜ ⎜ Hm (x, θ, ϕ) = ⎜ ⎜ (l,n)∈I ⎝
u lj,n 2L 2 (0, π ) . 2
l (x)T l f 1,n
(θ, ϕ)
⎞
− 12 ,n ⎟ l f 2,n (x)T l 1 (θ, ϕ) ⎟ ⎟ + 2 ,n ⎟, l (x)T l f 3,n 1 (θ, ϕ) ⎟ − 2 ,n ⎠ l (x)T l (θ, ϕ) f 4,n + 12 ,n
with
⎧
l+ 21 ⎪ m l l l ⎪ ⎪ i u 3,n + sin x u l4,n − cos ⎪ x u 1,n = f 1,n , ⎪
⎪ ⎪
⎪ l+ 21 ⎪ m l l ⎨ −i u l + u l − cos 4,n x u 2,n = f 2,n ,
sin x 3,n , l+ 12 ⎪ l
⎪ m l l l ⎪ u + cos x u 3,n = f 3,n , ⎪ ⎪ i u 1,n + sin
x 2,n ⎪ ⎪ 1
⎪
l+ ⎪ 2 ⎩ −i u l m l l l 2,n + sin x u 1,n + cos x u 4,n = f 4,n ,
and 4
Hm L2 = 2
j=1 (l,n)∈I
l f j,n 2L 2 (0, π ) . 2
For 1 ≤ h, k ≤ 4, we put l,± l l l l u l,± hk,n = u h,n ± iu k,n , f hk,n = f h,n ± i f k,n .
We have
l + 1 im l,∓ l,∓ 2 l,± u l,± u 12,n = u − i f 34,n , 12,n ∓ sin x cos x 34,n
u l,± 34,n
∓
l + 21 l,± im l,∓ l,∓ u u =− − i f 12,n . sin x 34,n cos x 12,n
(V.10)
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Given w+l ∈ L 2 (0, π2 ), any solution v+l of l + 21 l d l π v+ − v = w+l , 0 < x < , dx sin x + 2 1 (]0, π ]) ⊂ C 0 (]0, π ]) and v l can be written: belongs to Hloc + 2 2
v+l (x)
=
x l+ 1 π2 2 tan − 2 2 x
π v+l ( )
tan tan
x l+ 21 2y
w+l (y)dy.
(V.11)
2
On the one hand, by integrating we get: π | v+l ( ) |2 ≤ C(l + 1)( v+l 2L 2 + w+l 2L 2 ). 2 On the other hand, we easily show that for 0 < x ≤
(V.12)
π 2
x 2l
y −2l−1 1 x −2l 1 − tan , tan tan dy ≤ 2 2l 2 2
π 2
x
therefore since tan(x/2) ≤ x on [0, π2 ], we obtain that:
x l+ 1 2
x 2l l 2 l π l 2 tan 2l v+ (x) − v+ , ≤| x | w+ L 2 (x, π2 ) 1 − tan 2 2 2 and we conclude that π v+l ( ) = 0 =⇒ l | v+l (x) |2 ≤| x | w+l 2L 2 . 2
(V.13)
l of Now the solutions v−
l + 21 l π d l l v− + v = w− ∈ L 2 (0, ), dx sin x − 2
(V.14)
have the form
x −l− 1 x 2 l v− (x) = C tan + 2 0
tan tan
y l+ 21 2x
l w− (y)dy.
(V.15)
2
Then, when v− ∈ L 2 (0, π2 ) and l ≥ 0, we have C = 0. Since for 0 ≤ x ≤ x 2l+1 y 1 x 2l+2 tan tan dy ≤ , 2 l +1 2 0
π 2
we have
we obtain that the L 2 solutions of (V.14) satisfy: l l (x) |2 ≤| x | w− 2L 2 . (l + 1) | v−
For any χ ∈ C0∞ ([0, π2 [), we apply the previous estimates to l l = χ u l,± v± 12(34),n , w± = +(−)
im l,∓ χ u l,∓ − iχ f 34(12),n − χ u l,± 12(34),n . cos x 34(12),n
(V.16)
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A. Bachelot
From (V.13) and (V.16), we deduce 4
|
l
2 χ (x)u l,± hk,n (x) | ≤
C(χ ) | x |
hk=12,34
l u lj,n 2L 2 + f j,n 2L 2 ,
(V.17)
j=1
where C(χ ) > 0 depends only on χ . We get (V.2) and (V.3) that are consequences of (V.17). When m = 0, we can take l,∓ l l v± = u l,± 12(34),n , w± = −i f 34(12),n ,
and we get from (V.12) and (V.16) that 4
(l + 1)−1
2 | u l,± hk,n (x) | ≤ C | x |
l u lj,n 2L 2 + f j,n 2L 2 .
hk=12,34
(V.18)
j=1
This estimate yields (V.9). Now we have
l + 21 l,± m l,± l,∓ u l,± u 13,n = ± f 13,n u +i , 13,n ∓ cos x sin x 24,n
u l,± 24,n
±
l + 21 l,± m l,± l,∓ u 24,n = ∓ f 24,n u −i . cos x sin x 13,n
Given m ≥ 0, w+l ∈ L 2 (0, π2 ), any solution v+l of d l m l π v+ + v+ = w+l , 0 < x < , dx cos x 2 1 ([0, π [) ⊂ C 0 ([0, π [) and when belongs to Hloc 2 2
v+l (0) = 0, v+l can be written: v+l (x)
= 0
x
tan
π
4 tan π4
− −
m
x 2 y 2
w+l (y)dy.
(V.19)
Therefore the Cauchy-Schwarz estimate yields 1 < m =⇒| v+l (x) |≤ C w+l L 2 2 1 m = =⇒| v+l (x) |≤ C w+l L 2 2 0≤m<
π − x, 2
π
π − x ln −x , 2 2
m
π 1 =⇒| v+l (x) |≤ C w+l L 2 −x . 2 2
(V.20)
(V.21)
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151
We make this last estimate precise for 0 ≤ m < 21 : m π2 π
y −m l l −m π −x − tan w+ (y)dy v+ (x) − 2 2 4 2 0
π m+2 π ≤ C w+l L 2 + w+l L 2 (x, π ) −x −x , 2 2 2
(V.22)
in particular we have 0 < m =⇒ limπ v+l (x) = 0.
(V.23)
x→ 2
l of On the other hand the solutions v−
d l m l π l v − v = w− , 0<x< , d x − cos x − 2 have the form l v− (x)
π π 2 x −m − = Cl tan − 4 2 x
tan
π
4 tan π4
− −
y 2 x 2
m l w− (y)dy,
(V.24)
thus, −m
l ≤ wl L 2 (x, π ) π − x, v (x) − Cl tan π − x − − 2 4 2 2
(V.25)
and l v− ∈ L 2 (0,
0≤m<
1 π ), ≤ m =⇒ Cl = 0, 2 2
1 l (0) + =⇒ Cl = v− 2
(V.26)
π y m l tan w− (y)dy. − 4 2
π 2
0
We pick χ ∈ C0∞ (]0, π2 ] such that χ ( π2 ) = 1, and we apply the previous estimates to l,∓(±)
l,∓(±)
l,±(∓)
l l v± = χ u 13(24),n , w± = w13(24),n := ∓(±)χ f 13(24),n +(−)i
l + 21 l,∓(±) l,∓(±) χu −χ u 13(24),n . sin x 24(13),n
From (V.23) we deduce that when m > 0: lim u l1,n (x) − iu l3,n (x) = limπ u l2,n (x) + iu l4,n (x) = 0,
x→ π2
x→ 2
hence
limπ u l1,n (x)u l2,n (x) + (u l3,n (x)u l4,n (x) = 0.
x→ 2
(V.27)
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A. Bachelot
Now multiplying (V.10) by u lj,n and taking the real part we get: l + 1 4 d l l 2 u 1,n u 2,n + (u l3,n u l4,n + | u lj,n |2 dx sin x 1
l l l l l l l = f 1,n u 4,n + f 2,n u 3,n + f 3,n u 2,n + f 4,n u l1,n , and thanks to (V.17) and (V.27) we obtain π 1 2 4 2 l+2 | u lj,n (x) |2 d x ≤ sin x 0 j=1
4 l f j,n 2L 2 , j=1
that proves (V.4). We also see that: 4
l,∓(±) w13(24),n
L 2 ≤ C(χ )
l f j,n L 2 .
(V.28)
j=1
Therefore when m ≥ 21 , (V.5) and (V.6) follow from (V.20), (V.21), (V.25) and (V.26). On the other hand, when 0 < m < 21 , (V.22), (V.25) and (V.28) assure there exists l,∓(± ϕ13(24),n ∈ C 0 ([0, π2 ]) such that: l,−(+)
u 13(24),n (x) =
π 2
−x
m
π 2
=
π 2
−x
−m
π 2
limx
x→ 2
π 4
−
y −m l,−(+) w13(24),n (y)dy 2
π − x, 2
tan
0
l,+(−) +ϕ13(24),n (x)
2 tan
0
l,−(+) + ϕ13(24),n (x)
u l,+(−) 13(24),n (x)
π 2
m y l,+(−) tan − w13(24),n (y)dy 4 2 − x2
−x π 4
π
π − x, 2 2 l,∓(±) ϕ13(24),n (x) = 0.
(l,n)∈I
We deduce that there exists ψ± , χ± ∈ L 2 (S 2 ) such that can be expressed according to (V.7), (V.8). It remains to prove the regularity of ψ− and χ− . We consider
cos x m
1 + iγ 1 . (x, θ, ϕ) := 1 + sin x Equations (V.7), (V.8) assure that ⎛
⎞ ψ− π ⎜ χ ⎟ (x, .) −→ ⎝ − ⎠ in W 0 as x → . −iψ− 2 iχ−
(V.29)
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153
We calculate
cos x m
∂ 1 1 0 Hm − (x, .) = D . 1 + iγ γ ∂x 1 + sin x sin x Since ∈ L 2 [0, π2 [x ; W 1 by (V.4), we deduce that
π ∈ L 2 [1, [x ; W 1 , 2
π ∂ ∈ L 2 [1, [x ; W 0 . ∂x 2 The theorem of the intermediate derivative ([24], p. 23) shows that π 0 1 0 ∈ C [1, ]x ; W , W 1 . 2 2 1 1 1 Recalling that W 1 , W 0 1 = W 2 , we conclude by (V.29) that ψ− ∈ W−2 , χ− ∈ W+2 . 2
1
1
∓m
∓m
Finally we consider ψ± ∈ W−2 , χ± ∈ W+2 , and we want to construct
∈ D(Hm ) satisfying (V.7) and (V.8). We choose f ∈ C0∞ ([0, 1[) such that f (0) = 1, and we put
(x) =
π 2
−x
−m
⎛
⎞ ⎛ ⎞ ψ− ψ+ m
π ⎜ χ− ⎟ ⎜ χ ⎟ − x ⎝ + ⎠ + φ(x), ⎝ −iψ ⎠ + iψ+ − 2 iχ− −iχ+
where ⎛ φ(x) =
π 2
−x
−m (l,n)∈I
π − [f l 2
u l−,n (ψ− )T l
− 21 ,n ⎜ l l ⎜ ⎜ u +,n (χ− )T+ 21 ,n x − 1] ⎜ ⎜ −iu l−,n (ψ− )T l 1 − 2 ,n ⎝ iu l+,n (χ− )T l 1 + 2 ,n
⎛
+
π 2
−x
m (l,n)∈I
π − [f l 2
u l−,n (ψ+ )T l
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
− 21 ,n ⎜ l ⎜ u +,n (χ+ )T l 1 ⎜ + 2 ,n x − 1] ⎜ l ⎜ iu −,n (ψ+ )T l 1 − 2 ,n ⎝ −iu l+,n (χ+ )T l 1 + 2 ,n
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠
We use the fact that 1∓2m π 2 π 1±2m l ( π2 −x ) 2
− x − 1 ≤ −x | f (t) | 1∓2m dt l 1±2m , f l 2 2 0
154
A. Bachelot
to get φ(x, .) 2L 2 (S 2 ) ≤ 2
π 2
−x
π l ( 2 −x )
(l,n)∈I
1−2m
| f (t) |
2 1−2m
dt
0
l 1+2m | u l−,n (ψ− ) |2 + | u l+,n (χ− ) |2 1+2m π
π l ( 2 −x ) 2
1+2m −x | f (t) | dt +2 2 0 (l,n)∈I
l 1−2m | u l−,n (ψ+ ) |2 + | u l+,n (χ+ ) |2 . The dominated convergence theorem assures that φ satisfies (V.8), and so ∈ L2 . To achieve the proof, we have to show that Hm ∈ L2 . We calculate: ⎛ π −m−1 2 −x 1− −x Hm (x) = m 2 cos x
π
(l,n)∈I
π f l 2 ⎛
−
π 2
−x
−m (l,n)∈I
π −x l f l 2
π 2
−x
−m
1 sin x
(l,n)∈I
− 21 ,n ⎜ l ⎜ u +,n (χ− )T l 1 ⎜ + 2 ,n ⎜ ⎜ −iu l−,n (ψ− )T l 1 − 2 ,n ⎝ iu l+,n (χ− )T l 1 + 2 ,n
π −x l+ f l 2
π 2
−x
m (l,n)∈I
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
iu l+,n (χ− )T l
⎞
− 21 ,n l l ⎟ ⎜ ⎜ 1 ⎜−iu −,n (ψ− )T+ 21 ,n⎟ ⎟ ⎟ ⎜ 2 ⎜ u l+,n (χ− )T−l 1 ,n ⎟ ⎠ ⎝ 2 u l−,n (ψ− )T l 1 + 2 ,n
u l−,n (ψ+ )T l
− 21 ,n
⎞
⎜ ⎟ ⎟ ⎜ u l+,n (χ+ )T l 1
π ⎜ + 2 ,n ⎟ −x ⎜ l f l ⎟ ⎜iu −,n (ψ+ )T l 1 ⎟ 2 − 2 ,n ⎠ (l,n)∈I ⎝ −iu l+,n (χ+ )T l 1 + 2 ,n ⎞ ⎛ l l u −,n (ψ+ )T 1 − 2 ,n ⎟ ⎜ l ⎟
π ⎜ u +,n (χ+ )T l 1 ⎟ ⎜ + ,n 2 f l −x l⎜ l ⎟ l ⎜ iu −,n (ψ+ )T 1 ⎟ 2 − 2 ,n ⎠ ⎝ −iu l+,n (χ+ )T l 1
π m−1 −x −m −x 1− 2 2 cos x
−
⎞
u l−,n (ψ− )T l
⎛
π
⎞
− 21 ,n ⎜ ⎟ ⎜ u l+,n (χ− )T l 1 ⎟ ⎜ + 2 ,n ⎟ −x ⎜ ⎟ ⎜−iu l−,n (ψ− )T l 1 ⎟ − 2 ,n⎠ ⎝ iu l+,n (χ− )T l 1 + 2 ,n
⎛
+
u l−,n (ψ− )T l
+ 2 ,n
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155
⎛ +
π 2
−x
m
1 sin x
(l,n)∈I
⎜ 1 ⎜
π ⎜ −x l+ f l ⎜ 2 2 ⎜ ⎝
−iu l+,n (χ+ )T l
− 21 ,n l iu −,n (ψ+ )T l 1 + 2 ,n u l+,n (χ+ )T l 1 − 2 ,n u l−,n (ψ+ )T l 1 + 2 ,n
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠
In this sum, the leading terms have the form ±
(x, θ, ϕ) =
π 2
−x
±m (l,n)∈I
1
π −x l+ g ± (θ, ϕ), h l 2 2 l,n
where h ∈ C0∞ ([0, 1[) and
(l,n)∈I
1 1∓2m ± 2 ± l+ gl,n L 2 (S 2 ) < ∞, gl,n (ω)gl± ,n (ω)dω = δl,l δn,n . 2 S2
Taking account of the support of h, we evaluate
±
2L 2 (]0,π [×S 2 )
π
±2m 2 1 2 π ± 2 l+ −x = gl,n L 2 (S 2 ) π 1 2 2 2−l (l,n)∈I 2 h l 1 − x d x 2 1 1 1∓2m ± 2 ≤ l+ t ±2m | h(t) |2 dt gl,n L 2 (S 2 ) < ∞. 2 0 (l,n)∈I
Proof of Theorem III.2. Since the map S given by (III.8) satisfies (III.36), (III.22) and (III.23) follow from Proposition IV.1 and (V.2), and (V.4). Moreover, since π2 − x ∼ 1 2 (1 − ), (V.5) and (V.6) imply (III.24) and (III.25). Now, if we put ⎛
⎞ ψ± (θ, ϕ) ⎜ χ (θ, ϕ) ⎟ ± (ω) = S(θ, ϕ) ⎝ ± , ±iψ± (θ, ϕ) ⎠ ∓iχ± (θ, ϕ) (III.26) and (III.27) are consequences respectively of (V.7), and: ⎛ ⎞ ψ± (θ, ϕ)
⎜ χ (θ, ϕ) ⎟ = 0. γ˜ 1 ∓ i I d ± (ω) = S(θ, ϕ) γ 1 ∓ i I d ⎝ ± ±iψ± (θ, ϕ) ⎠ ∓iχ± (θ, ϕ)
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A. Bachelot
Finally, for any − ∈
4 1 4 1 H 2 +m (S 2 ) , + ∈ H 2 −m (S 2 ) we define ± (θ, ϕ)
1
:= S ∗ (θ, ϕ)(ω) ∈ W 2 ∓m . When ± satisfy (III.27), then ± have the form ⎛ ⎞ ψ± (θ, ϕ) 1 1 +m +m ⎜ χ (θ, ϕ) ⎟
± (θ, ϕ) = ⎝ ± , ψ− ∈ W−2 , χ− ∈ W+2 , ⎠ ±iψ± (θ, ϕ) ∓iχ± (θ, ϕ) 1
ψ+ ∈ W−2
−m
1
, χ+ ∈ W+2
−m
,
and there exists ∈ D(Hm ) satisfying (V.7) and (V.8). We conclude that := S , belongs to D(H M ) and satisfies (III.26) and (III.28). At last, Remark III.3 directly follows from (V.9), Proposition IV.1 and Remark V.2. We end this part by an important result of compactness: Proposition V.4. Let K be the set K := ∈ D(Hm ), 2L2 + 2L2 ≤ 1 .
(V.30)
Then, when m > 0, K is a compact of L2 . Proof of Proposition V.4. We consider a sequence ( ν )ν∈N in K . We write ⎛
l u l,ν 1,n T
− 21 ,n
⎞
⎜ ⎟ ⎜ u l,ν T l ⎟ ⎜ ⎟ 1 2,n + ,n ν ⎜ ⎟, 2
= ⎜ u l,ν T l ⎟ 1 ⎜ 3,n (l,n)∈I ⎝ − 2 ,n ⎟ ⎠ l u l,ν 4,n T 1 + 2 ,n
⎛
l,ν l f 1,n T
⎞
− 21 ,n
⎜ ⎟ ⎜ f l,ν T l ⎟ ⎜ ⎟ 1 2,n + ,n ν ⎜ ⎟, 2 Hm = ⎜ f l,ν T l ⎟ 1 ⎜ 3,n (l,n)∈I ⎝ − 2 ,n ⎟ ⎠ l,ν l f 4,n T 1 + 2 ,n
and we have: 4 j=1 (l,n)∈I
l,ν 2 2 u l,ν j,n L 2 (0, π ) + f j,n L 2 (0, π ) ≤ 1. 2
2
The Banach-Alaoglu theorem assures that there exists ∈ K and a sub-sequence denoted ( ν )ν∈N again, such that ⎛ l ⎞ u 1,n T l 1 ⎜ l −l 2 ,n ⎟ ⎜ u 2,n T 1 ⎟ ⎜ + 2 ,n ⎟
ν = ⎜ l ⎟, ⎜ u 3,n T l 1 ⎟ − 2 ,n ⎠ (l,n)∈I ⎝ u l4,n T l 1 + 2 ,n ⎛ l l ⎞ f 1,n T 1 − 2 ,n ⎜ l l ⎟ ⎜ f 2,n T 1 ⎟ ⎜ ⎟ + ,n Hm ν Hm = ⎜ l l 2 ⎟ in L2 − ∗, ν → ∞. ⎜ f 3,n T 1 ⎟ − 2 ,n ⎠ (l,n)∈I ⎝ l Tl f 4,n 1 + 2 ,n
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l,ν π l l 2 Since for any (l, n) ∈ I , j = 1, . . . 4, u l,ν j,n u j,n , f j,n f j,n , in L (0, 2 ) − ∗ as ν → ∞, we deduce from (V.11), (V.15), (V.19) and (V.24), that π l,ν l ∀x ∈ [0, ], u l,ν j,n (x) → u j,n (x), sup supπ | u j,n (x) |< ∞. 2 ν x∈[0, ] 2
Therefore l π u l,ν j,n − u j,n L 2 (0, ] → 0, ν → ∞.
(V.31)
2
Moreove, since m > 0, (V.4) implies: sup ν
For l ∈ N +
1 2
(l,n)∈I
1 2 l+ 2
4 l 2 u l,ν j,n − u j,n L 2 (0, π ) < ∞.
(V.32)
2
j=1
we put 4
ε
l,ν
l l 2 u l,ν j,n − u j,n L 2 (0, π ) .
:=
2
j=1 n=−l
Equations (V.31) and (V.32) show that 1 ∀l ∈ N + , εl,ν → 0, ν → ∞, A := sup 2 ν
1 2 l,ν l+ ε < ∞. 2 1
l∈N+ 2
−2 / Since εl,ν ≤ A l + 21 , the dominated convergence theorem implies that l εl,ν → 0, as ν → ∞, that is to say, ν strongly tends to in L2 . VI. Self-Adjoint Extensions When 0 < m < 21 , we define the linear map
⎞ ψ− 1 ⎜χ ⎟ : ∈ D(Hm ) −→ ( ) = ⎝ − ⎠ ∈ W 2 , ψ+ χ+ ⎛
where ψ± and χ± are given by (V.7), and we put ( ) = 0 when Theorem V.1 assures that
1 2
≤ m. We note that
1 1 1 1 1 +m +m −m −m ∀m ∈]0, [, W−2 × W+2 × W−2 × W+2 ⊂ (D(Hm )) . 2
We introduce the matrix
⎛
0 ⎜0 0 5 Q := −γ γ = ⎝ 1 0 The basic tool is a nice Green formula:
0 0 0 1
−1 0 0 0
⎞ 0 −1 ⎟ . 0 ⎠ 0
(VI.1)
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˜ ∈ D(Hm ) we have Lemma VI.1. Given 0 < m, for any , ˜ >L2 = 2 < ( ), Q( ) ˜ >L2 − < , Hm ˜ >W 0 . < Hm ,
(VI.2)
Proof of Lemma VI.1. Equation (V.4) assures that any ∈ D(H m ) belongs to L 2 [0, π2 ]x ; W 1 , hence for any ε > 0, ∈ H 1 ]ε, π2 − ε[x ; W 0 . Since (D, W 1 ) is selfadjoint on W 0 , we evaluate ˜ >L2 = lim < iγ 0 γ 1 ( ˜ >L2 − < , Hm < Hm , ε→0
˜
(
π ˜ − ε) >W 0 − < iγ 0 γ 1 (ε), (ε) >W 0 , 2
π − ε), 2
and taking account of (V.3), (V.5), (V.6), (V.7) and (V.8) we get (VI.2).
We now investigate the self-adjoint extensions (H, D(H)) of Hm , with C0∞ (]0, π2 [x ×]0, π [θ ×]0, 2π [ϕ ; C4 ) ⊂ D(H). The adjoint H∗ is just Hm with domain D(H∗ ) ⊂ D(Hm ), and we have: ˜ ∈ D(H∗ ), < ( ), Q( ) ˜ >W 0 = 0. ∀ ∈ D(H), ∀ When m ≥
1 2
we immediately obtain a first result of self-adjointness of Hm on L2 :
Proposition VI.2. When 21 ≤ m, Hm is essentially self-adjoint on C0∞ ]0, π2 [x×]0, π [θ×] 4 0, 2π [ϕ . Proof of Proposition VI.2. Let H be the operator defined by the differential operator Hm endowed with the domain D(H) = C0∞ (]0, π2 [x ×]0, π [θ ×]0, 2π [ϕ ; C4 ). On the one hand, H is obviously symmetric, and on the other hand, its adjoint H∗ is just Hm with domain D(Hm ). Let any ± be in D(Hm ) such that H∗ ± ± i ± = 0, satisfies ∓2i 2L2 =< Hm ± , ± >L2 − < ± , Hm ± >L2 , and we conclude by (VI.2) that ± = 0.
When 0 < m < 21 , the situation is much more interesting: there exists a lot of self-adjoint realizations of Hm . First, we introduce the operators H M I T and HC H I respectively associated with the MIT-bag and the Chiral boundary conditions. They are defined as Hm endowed with the domains D (H M I T ) " ! π π −x , x → , := ∈ D(Hm ); γ 1 (x, .) + i (x, .) W 0 = o 2 2 (VI.3) D (HC H I ) " ! π π −x , x → . := ∈ D(Hm ); γ 1 (x, .) − i (x, .) W 0 = o 2 2 (VI.4)
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In fact these asymptotic conditions are reduced to linear constraints on the asymptotic profiles ± : we check by (V.7) that γ 1 (x, θ, ϕ) ± i (x, θ, ϕ) = ±2i
π 2
−x
±m
⎛
⎞ ψ± (θ, ϕ)
⎜ χ± (θ, ϕ) ⎟ 1 ⎝ ±iψ (θ, ϕ) ⎠ + γ ± i ϕ(x, θ, ϕ). ± ∓iχ± (θ, ϕ)
Thus (V.8) implies that D (H M I T ) = { ∈ D(Hm ); ψ+ = χ+ = 0} , D (HC H I ) = { ∈ D(Hm ); ψ− = χ− = 0} . We now construct a large family of self-adjoint extensions that are non-local generalizations of the MIT-bag and Chiral conditions. We consider densely defined self-adjoint operators (A± , D(A± )) on L 2 (S 2 ) × L 2 (S 2 ), satisfying 1
1
W−2 × W+2 ⊂ D(A− ), D(A+ ) = L 2 (S 2 ) × L 2 (S 2 ),
(VI.5)
1 1 ±m ±m A± C0∞ (]0, π [×]0, 2π [; C2 ⊂ W−2 × W+2 .
(VI.6)
We introduce the operators H A± defined as Hm endowed with the domain " ! ψ± ψ∓ = A± . D (H A± ) := ∈ D(Hm ); χ∓ χ± In particular, we have H A− = 0 = H M I T and H A+ =0 = HC H I . Proposition VI.3. When 0 < m < 21 , H A+ and H A− are self-adjoint on L2 . ˜ ∈ D(H A± ): Proof of Proposition VI.3. Since A± are self-adjoint, we have for , 1 0 ψ± ψ˜ ∓ ψ˜ ± ˜ >W 0 = , − A± = 0. (VI.7) < ( ), Q( ) χ± χ˜ ∓ χ˜ ± L 2 (S 2 ;C2 ) ˜ ∈ D(H∗ ± ), we have (VI.7) for any Therefore H A± are symmetric. Moreover, given A ∞
∈ D(H A± ) again. For all ψ± , χ± ∈ C0 (]0, π [×]0, 2π [), (VI.1) and (VI.6) assure there exists ∈ D(H A± ) such that ( ) = A+ (ψ+ , χ+ ), ψ+ , χ+ or ψ− , χ− , A− (ψ− , χ− ) . Therefore
0
ψ± χ±
1 ψ˜ ∓ ψ˜ ± , − A± = 0. χ˜ ∓ χ˜ ± L 2 (S 2 ;C2 )
˜ ∈ D(H A± ). We conclude that
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Finally we consider the operators H A P S , Hm A P S associated with the APS and mAPS boundary conditions: " ! π π −x , x → , D (H A P S ) := ∈ D(Hm ); K+ (x, .) W 0 = o 2 2 D (Hm A P S ) " !
π π −x , x → , := ∈ D(Hm ); K+ I d + γ 1 (x, .) W 0 = o 2 2 where K+ is defined by (IV.7). Proposition VI.4. When 0 < m < 21 , we have D (H A P S ) = D (Hm A P S ) = { ∈ D(Hm ); K+ + = K+ − = 0} , and H A P S = Hm A P S is self-adjoint on L2 . Proof of Proposition VI.4. By (V.7), we have ⎞ 1 ⎜ −iJ ∗ ⎟ −x K+ (x) = ⎝ −i ⎠ (ψ− + iJ χ− ) 2 J∗ ⎛ ⎞ 1
π m ∗ ⎜ iJ ⎟ + −x ⎝ (ψ − iJ χ+ ) + K+ ϕ(x), i ⎠ + 2 J∗
π
K+ I d + γ 1
−m
⎛
⎞ 1 ∗ ⎜ −iJ ⎟
(x) = (1 − i) −x ⎝ −i ⎠ (ψ− + iJ χ− ) 2 J∗ ⎛ ⎞ 1
π m ∗ ⎜ iJ ⎟ +(1 + i) −x ⎝ (ψ − iJ χ+ ) i ⎠ + 2 ∗ J
1 +K+ I d + γ ϕ(x),
π
−m
⎛
thus we deduce from (V.8) that
π π 1 K+ (x, .) W 0 = o − x ⇔ K+ I d + γ (x, .) W 0 = o −x 2 2 ⇔ ψ± = ±iJ χ± ⇔ K+ + = K+ − = 0. (VI.8)
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˜ ∈ D (H A P S ), we have < , Q >W 0 = 0, i.e. This equality assures that for , ˜ ∈ D H∗ H A P S is symmetric. Moreover, for any ∈ D (H A P S ), A P S , we have < χ+ , χ˜ − − iJ ∗ ψ˜ − > L 2 (S 2 ) − < χ− , χ˜ + + iJ ∗ ψ˜ + > L 2 (S 2 ) = 0.
(VI.9)
Since C0∞ (]0, π [×]0, 2π [) ⊂ W+1 , for any χ± ∈ C0∞ (]0, π [×]0, 2π [), J χ± belongs to W−1 and by (VI.1) there exists ∈ D(Hm ) such that ⎞ ⎛ −iJ χ− ⎜ χ− ⎟ . ( ) = ⎝ iJ χ+ ⎠ χ+ But such a satisfies (VI.8), that means that is in the domain of H A P S . Since χ± are arbitrary, (VI.9) implies χ˜ ± ± iJ ∗ ψ˜ ± = 0, ˜ ∈ D (H A P S ). that is equivalent to (VI.8). We conclude that
The remainder of the article is devoted to the demonstrations of the theorems of Part 3. As we have explained above, it is sufficient to consider only the case M > 0, since the chiral transform changes the sign of the mass. Proof of Theorem III.4. We denote by H the operator H M endowed with the domain 4 D(H) := C0∞ (B) . Since H M = SHm S−1 , Proposition VI.2 assures that H M is . essentially self-adjoint on S C0∞ 0, π2 x ×]0, π [θ ×]0, 2π [ϕ ; C4 when M ≥ 12 Proposition IV.1 and the Sobolev Imbedding Theorem imply that this set is included in ∞ 4 C0 (B) . Since H is symmetric, we deduce that it is essentially self-adjoint. To determine its domain and establish the elliptic estimate, we prove an inequality of Hardy type. Given a real valued function f ∈ C01 (]0, 1[), an integration by part gives: 1 1 2 1 1 2 2 f () f () d f () d = − f () d + 2 2 2 2 (1 − ) 2 0 1− 1− 0 0 1 1 2 2 1 1 2 2 ≤ f () d + f ()d, 2 0 (1 − 2 )2 2 0 4 1 ∈ L2 when ∈ H01 (B) , and we have the hence by density we get that 1− following Hardy estimate: 1 1 2 | (x) | dx ≤ | ∇x (x) |2 dx. (VI.10) ∀ ∈ H0 (B), (1− | x |2 )2 B B 4 Thus we see that H01 (B) ⊂ D(H M ) and the graph norm of H M is bounded by the H01 4 norm. Conversely, for ∈ C0∞ (B) , we use the Fourier transform of , the Parseval formula and the anticommutations relations (III.2) to remark that ∂i ∗ γ i γ j ∂ j + ∂ j ∗ γ j γ i ∂i dx = 0, B 1≤i< j≤3
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then we calculate 2 3 2i M 12M 2 j 2 dx = | ∇ | + | |2 γ ∂ j + x 1 − 2 (1 − 2 )2 B B 3 4i M x j ∗ γ j dx. + 2 2 (1 − ) Therefore the Hardy inequality (VI.10) shows that when M > 12 , the elliptic estimate (III.32) holds: 2 12 2 H M L2 ≥ 1 − M | ∇x |2 dx, B 4 and the H01 -norm on C0∞ (B) is bounded by the graph norm of H M . Since H is essentially self-adjoint, we have H∗ = H. On the one hand D(H∗ ) = D(H M ). On the 4 other hand D(H) is the closure of C0∞ (B) for the graph norm. We conclude that 4 D(H M ) = H01 (B) when M > 12 and the first part of the theorem is proved. ∗ A± S , and A± satisfy (III.30) and (III.31), then A± = S11 Now when 0 < M < 12 11 where S11 is defined by (IV.9), satisfy (VI.5) and (VI.6). We deduce fom Proposition VI.3 that HA± = SH A± S−1 is self-adjoint. On the other hand, we have HB A P S = SH A P S S−1= SHm A P S S−1 = HBm A P S that is self-adjoint by Proposition VI.4. Finally ∈ D(H M ), 2L2 + H M 2L2 ≤ 1 is equal to SK where K defined by (V.30) is compact by Proposition V.4. We conclude that the resolvent of any self-adjoint realization of H M is compact.
Proof of Theorem III.5. Theorem III.4 provides a lot of solutions of the initial value it
H
problem: if H is a self-adjoint realization of H M , (t) = e 3 0 is a solution of (III.9), (III.11), (III.12) and (III.13). 2 Since the maximal globally hyperbolic domain in E including {t = 0}×[0, π2 [x ×Sθ,ϕ is given by 0 ≤| t |< 3 π2 − x , the maximal globally hyperbolic domain in M including {t = 0} × R3 is defined by the same relation, that is in (t, x) coordinates: 0 ≤| t |< 3 π2 − 2 arctan . We show that all the solutions are equal in this domain. Given satisfying (III.9), (III.11), we introduce for all ε > 0, 1 t+ε ε (t) = (s)ds. ε t
It is clear that ε ∈ C 1 (Rt ; L2 ), ε → in C 0 (Rt ; L2 ) as ε → 0. Moreover we can see that ε is a solution of (III.9), thus H M ε ∈ C 0 (Rt ; L2 ) and 3 ∂ 3 1 + 2 ∂ ∗ 0 j | ε |2 + ε γ γ ε = 0. ∂t 2 ∂x j j=1
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! " where 0 We integrate this equality on (t, x), 0 ≤ t ≤ T, ≤ tan π4 − T 12 < T < π2 3 , and applying the Green formula we get | ε (T, x) |2 dx = | ε (0, x) |2 dx ≤tan( π4 −T
−
12 )
B
3 π 0≤t= ( 2 −2 arctan )≤T
| (t, x) |2 −
xj ∗ 0 j γ γ (t, x)dσ.
The last integral is non-negative since | x j γ j |≤ | |, and taking the limit as ε → 0, we obtain 2 | (T, x) | dx ≤ | (0, x) |2 dx. ≤tan( π4 −T
12 )
B
We conclude that = 0 for 0 ≤| t |< 3 π2 − 2 arctan if 0 = 0. Finally when M ≥ 12 , we use the fact that H M is self-adjoint to write d −it H −it H 3 M (t) 3 M e H M ε (t) + ∂t ε (t) = 0, =e −i ε dt 3
and we deduce that ε (t) = e that (t) = e
it
3 HM
0 .
it
3 HM
ε (0). Taking the limit in ε again, we conclude
Proof of Theorem III.6. Since the spectrum of H is discrete, and 0 is not an eigenvalue when M > 0, there exists an orthonormal basis of eigenvectors, (k )k∈N , with H M k = λk
3 k ,
λk ∈ R∗ . Now the crucial point is that k∗ γ 0 γ 5 k (x)dx = 0.
(VI.11)
B γ 0γ 5
= −γ 0 γ 5 H M , and we write 1 < k , γ 0 γ 5 k >L2 = < H M k , γ 0 γ 5 k >L2 λk 1 = − < k , γ 0 γ 5 H M k >L2 λk = − < k , γ 0 γ 5 k >L2 .
To see that, we note that H M
We can expand on this basis: ck eiλk t k (x), ck ∈ C,
(t, x) = k∈N
| ck |2 < ∞, k∈N
and taking advantage of (VI.11) we evaluate i(λ p −λq )T − 1 1 T ∗ 0 5 ∗e γ γ (t, x)dxdt = c p cq ∗ γ 0 γ 5 p (x)dx. T 0 B i(λ p − λq )T B q λ p =λq
The dominated convergence theorem assures that this sum tends to 0 as T → ∞.
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VII. Appendix. Breitenlohner-Freedman Bounds for the Scalar Waves We consider the Klein-Gordon equation on the Anti-de Sitter space-time
1 1 | g |− 2 ∂µ | g | 2 g µν ∂ν u − α u = 0, 3 where α ∈ R is a coefficient linked to the mass ; the equation with α = 2 is conformally invariant and corresponds to the massless case. Using the radial coordinate x given by
(II.1), we introduce f (t, x, ω) := r u(t h := −∂x2 +
3 , r, ω)
that is solution of ∂t2 f + h f = 0 with
2−α 1 S2 . − cos2 x sin2 x ω
(VII.1)
First we investigate the positivity of the potential energy E( f ) :=
π 2
S2
0
| ∂ x f |2 +
2−α 1 | f |2 + 2 | ∇ Sω2 f |2 d xdω. cos2 x sin x
To estimate the second term, we employ a Hardy inequality. Given φ ∈ C01 ([0, 21 [; R) an integration by part gives 0
π 2
π π 2 1 1 φ(x)
1 2 2 φ φ φ 2 (x)d x (x)d x = − 2 (x) sin xd x ≤ 2 cos x cos x 2 0 cos2 x 0 π 2 2φ 2 (x) sin2 xd x, + 0
hence
π 2
0
1 φ 2 (x)d x ≤ 4 cos2 x
π 2
φ 2 (x)d x.
0
We deduce that for all f ∈ C0∞ (]0, π2 [x ×Sω2 ), < h f, f > L 2 ≥ min(9 − 4α, 1)
π 2
+ 0
S2
π 2
0
S2
| ∂x f |2 d xdω
1 1 9 | ∇ Sω2 f |2 d xdω ≥ min( − α, ) f 2L 2 , 2 4 4 sin x
and we conclude that the operator h endowed with the domain D(h) = C0∞ (]0, π2 [x ×Sω2 ), is (strictly) positive when α is (strictly) smaller than the upper bound of BreitenlohnerFreedman: 9 9 (r espectively α < ). (VII.2) 4 4 cos x We note that for α = 9/4 and f (x, ω) = 1+sin x , we have E( f ) = 0, hence ∗ f ∈ K er (h ) = {0}. α≤
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To study the self-adjointness, we expand f (x, .) on the basis of the spherical harmonics Ylm l,m by writing ∞ m=l
π 2
π 2 L 2 ]0, [x ×Sω2 = Ll2 , Ll2 := L 2 ]0, [x ⊗ Ylm , 2 2 l=0 m=−l 3∞ therefore h is unitarily equivalent to l=0 hl where: m=l 2 d2 2 − α l(l + 1) π + C0∞ (]0, [) ⊗ Ylm . hl := − 2 + , D(hl ) = 2 dx cos2 x 2 sin x m=−l l(l+1) l(l+1) 2−α 2−α Since cos is a real valued function, bounded on ]0, π2 [, the 2 x + sin2 x − ( π −x)2 − x2 2 symmetric form of the Kato-Rellich theorem (see [31], Theorem X.13) assures that hl is essentially self-adjoint iff
kl := −
m2 =l d2 2−α l(l + 1) π + + , D(k ) = C0∞ (]0, [) ⊗ Ylm , l d x 2 ( π2 − x)2 x2 2 m = −l
is essentially self-adjoint. By Theorem X.10 of [31], kl is in the limit point case at zero when l ≥ 1, and in the limit point case at π2 if 2 − α ≥ 43 , i.e. α is smaller than the lower bound of Breitenlhoner-Freedman 5 (VII.3) α≤ , 4 and if α > 45 , kl is in the limit circle case at π2 . Then the Weyl’s limit point-limit circle criterion (see e.g. [30], Theorems 6.3 and 6.5), assures that kl is essentially self-adjoint when l ≥ 1, α ≤ 45 , and there exists an infinity of self-adjoint extensions associated with boundary conditions at π2 when l ≥ 1, α > 54 . The case l = 0 is particular. For α < 49 , the 1 1 + 94 −α − 94 −α π π π
−2
2 2 solutions of −u +(2−α)( 2 −x) u = 0 are u = c( 2 −x) +c ( 2 −x) , therefore k0 is always in the limit circle case at π2 and there exists a lot of self-adjoint
extensions. By the Kato-Rellich theorem ([31], Theorem X.12), the same results are true for hl . Since the spherically symmetric fields play a peculiar role, we introduce their orthogonal space " 2 ! ∞ π π 2 2 2 2 f (x, ω)g(x)d xdω = 0 = L∗ := f ∈ L (]0, [×S ); ∀g ∈ L (]0, [), Ll2 , 2 2 l=1
C0∞ (]0, π2 [×S 2 ) ∩ L∗2 , and
and h∗ denotes h endowed with the domain D(h∗ ) = considered as a densely defined operator on L∗2 . Since this operator is strictly positive when α < 49 , it is essentially selfadjoint iff its range is dense ([31], Theorem X.26). We easily ⊥
⊥
∞ (Ran(h )) Ll , and we conclude that h is essentially prove that (Ran(h∗ )) L∗ = ⊕l=1 l ∗ self-adjoint when α ≤ 45 . Finally we have proved the following: 2
2
Theorem VII.1. When α ≤ 49 (resp. α < 49 ), h is a positive (resp. strictly positive) symmetric operator on L 2 (]0, π2 [×S 2 ). When 45 < α < 49 there exists an infinity of self-adjoint extensions of h∗ on L∗2 , associated with boundary conditions on { π2 } × S 2 . When α ≤ 45 , h∗ is essentially self-adjoint.
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32. Segev, I.: Dynamics in stationary, non-globally hyperbolic spacetimes. Class. Quantum Grav. 21, 2651–2668 (1994) 33. Shishkin, G.V., Villalba, V.M.: Dirac Equation in external vector fields: separation of variables. J. Math. Phys. 30, 2132–2143 (1989) 34. Schmidt, K.M., Yamada, O.: Spherically symmetric Dirac operators with variable mass and potential infinite at infinity. Publ. Res. Inst. Math. Sci. Kyoto Univ. 34, 211–227 (1998) 35. Vilenkin, N.J.: Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs, Volume 22, Providence, RI: Amer. Math. Soc., 1968 36. Vilenkin, N.J., Klimyk, A.U.: Representation of Lie Groups and Special Functions, Vol. 1. Mathematics and Its Applications (Soviet Series), Dordrecht: Kluwer Academic Publishers, 1991 37. Wald, R.M.: Dynamics in nonglobally hyperbolic, static space-times. J. Math. Phys. 21(12), 2802–2805 (1980) 38. Yamada, O.: On the spectrum of Dirac operators with the unbounded potential at infinity. Hokkaido Math. J. 26, 439–449 (1997) Communicated by G.W. Gibbons
Commun. Math. Phys. 283, 169–226 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0563-9
Communications in
Mathematical Physics
Quasi-Conformal Actions, Quaternionic Discrete Series and Twistors: SU(2, 1) and G 2(2) M. Günaydin1,2 , A. Neitzke2 , O. Pavlyk3 , B. Pioline4,5 1 Physics Department, Pennsylvania State University, University Park, PA 16802, USA.
E-mail:
[email protected]
2 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA.
E-mail:
[email protected]
3 Wolfram Research Inc., 100 Trade Center Dr., Champaign, IL 61820, USA.
E-mail:
[email protected]
4 Laboratoire de Physique Théorique et Hautes Energies , Université Pierre et
Marie Curie - Paris 6, 4 place Jussieu, F-75252 Paris cedex 05, France. E-mail:
[email protected]
5 Laboratoire de Physique Théorique de l’Ecole Normale Supérieure ,
24 rue Lhomond, F-75231 Paris cedex 05, France Received: 22 July 2007 / Accepted: 14 March 2008 Published online: 22 July 2008 – © Springer-Verlag 2008
Abstract: Quasi-conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane, to Hermitian symmetric tube domains based on arbitrary Jordan algebras, and further to arbitrary Freudenthal triple systems. In the mathematics literature, quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionicKähler spaces. These two constructions are essentially identical, as we show explicitly for the two rank 2 cases SU (2, 1) and G 2(2) . We obtain explicit results for certain principal series, quaternionic discrete series and minimal representations of these groups, including formulas for the lowest K -types in various polarizations. We expect our results to have applications to topological strings, black hole micro-state counting and to the theory of automorphic forms. 1. Introduction and Summary Despite much recent progress, classifying the unitary representations of a real reductive group G remains a challenging task, which has only been addressed in a few, typically low-rank cases, including for example S L(2, R) [1–3], S L(3, R) [4,5], SU (2, 1) [6], SU (2, 2) [7] and G 2(2) [8]. Even in well-charted cases, the available mathematical description is often not directly useful to physicists, who are in general more interested in explicit differential operator realizations than abstract classifications. The goal of this paper is to give an explicit description of aspects of principal and discrete series representations (and continuations of the discrete series) which arise when G is a quaternionic real form of a semi-simple group. For the sake of explicitness, we restrict to the rank 2 case – so G = SU (2, 1) or G = G 2(2) . Quaternionic real forms arise as symmetries of supergravity theories with 8 supercharges in three dimensions [9–13], and therefore Unité mixte de recherche du CNRS UMR 7589. Unité mixte de recherche du CNRS UMR 8549.
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as spectrum-generating symmetries for black holes in four dimensional supergravity; a detailed discussion of this relation, which indeed motivated our interest in the first place, can be found in [14–16] and references therein. The simplest discrete series representations1 arise when K = U (1)× M, so that G/K is a Hermitian symmetric domain. The simplest example is G/K = S L(2, R)/U (1): G acts holomorphically on the upper half-plane by fractional linear transformations τ → (aτ + b)/(cτ + d), which preserve the Kähler potential K = − log[(τ − τ¯ )] up to Kähler transformations. This construction can be extended to all Hermitian symmetric tube domains using the language of Jordan algebras. Indeed, consider the “upper half plane” τ ∈ J + i J + , where J is a Euclidean Jordan algebra J of degree n and J + is the domain of positivity of J , equipped with the Kähler potential K = − log N (τ − τ¯ ), where N is the norm form of J 2 . The corresponding metric is invariant under a non-compact group G = Conf(J ), the “conformal group” associated to J , acting holomorphically by generalized fractional linear transformations on τ [24,25]. The ), resulting space is the Hermitian symmetric tube domain G/K = Conf(J )/Str(J ) = U (1) × Str 0 (J ) is the compact real form of the structure group3 of where Str(J J , which coincides with the maximal compact subgroup of G. The action of G on sections of holomorphic vector bundles over G/K leads to the holomorphic discrete series representations. The next simplest case, of interest in this paper, arises when G is in its quaternionic real form, such that K = SU (2) × M, and its Lie algebra decomposes as g = su(2)⊕M⊕(2, V ), where V is a pseudo-real representation of M. The symmetric space G/K , of real dimension 4d, is now a quaternionic-Kähler space, and does not generally admit a G-invariant complex structure. The twistor space Z = G/U (1) × M, a bundle over G/K with fiber CP1 = SU (2)/U (1), does however carry a G-invariant complex structure. In [26] this complex structure was exploited to construct a family of representations πk of G, labeled by k ∈ Z: namely, πk is the sheaf cohomology H 1 (Z, O(−k)) of a certain line bundle O(−k) over Z.4 πk is a representation of G, with Gelfand-Kirillov (functional) dimension 2d + 1, equal to the complex dimension of Z. For k ≥ 2d + 1, πk are discrete series unitary representations of G, called “quaternionic discrete series.” The πk for k < 2d + 1 are also of interest. In [26] special attention was paid to quaternionic groups of type G 2 , D4 , F4 , E 6 , E 7 , E 8 , such that d = 3 f + 4 with f = − 23 , 0, 1, 2, 4, 8. For these groups, it was shown that πk is irreducible and unitarizable even for k ≥ d + 1 (although it no longer belongs to the quaternionic discrete series for k < (2d +1)). Moreover, for selected smaller values of k, namely k = 3 f +2, k = 2 f +2 and k = f +2, πk was shown to be reducible but to admit a unitarizable submodule πk , of smaller Gelfand-Kirillov dimension, 2d = 6 f + 8, 5 f + 6, 3 f + 5 = d + 1, respectively. 1 Discrete series representations arise as discrete summands in the spectral decomposition of L 2 (G) under the left action of G. A basic result of Harish-Chandra [17,18] states that G admits discrete series representations if and only if its maximal compact subgroup K has the same rank as G itself. 2 The generalized upper half planes associated to Jordan algebras are sometimes called Köcher half-spaces since Köcher pioneered their study [19]. In particular, he introduced the linear fractional groups of Jordan algebras [20], which were interpreted as conformal groups of generalized spacetimes defined by Jordan algebras and extended to Jordan superalgebras in the physics literature [21–23]. The terminology stems from the well-known action of the conformal group S O(4, 2) on Minkowski spacetime, which arises when J is the Jordan algebra of 2 × 2 Hermitian matrices [21]. Choosing J = R leads instead to our original example G/K = S L(2, R)/U (1). 3 The structure group Str(J ) leaves the norm form N of the Jordan algebra J invariant up to an overall scaling. It decomposes as an Abelian factor R times the reduced structure group Str 0 (J ) which leaves the norm invariant. 4 For a definition of sheaf cohomology see e.g. [27].
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The smallest of these representations is the “minimal” or “ladder” representation of G; the latter name refers to the structure of its K -type decomposition;5 a review of minimal representations can be found in [28]. Independently of these mathematical developments, it was shown in [25] that the conformal realization of the group Conf(J ) attached to a Jordan algebra J of degree 3, preserving a generalized light-cone N3 (τ − τ¯ ) = 0, could be extended to an action of a larger non-compact group, preserving a “quartic light-cone” ¯ + (α − α¯ + , ) ¯ 2 = 0, N4 ≡ I4 ( − )
(1.1)
where = (ξ I , ξ˜ I ) = (ξ 0 , ξ , ξ˜ , ξ˜0 ) is an element of the Freudenthal triple system ¯ a symplectic pairing invariant under the F = C ⊕ JC ⊕ JC ⊕ C associated to J , , linear action of Conf(J ) on , I4 a quartic polynomial invariant under this same action6 and α an additional complex variable of homogeneity degree two. This larger group was called the “quasi-conformal group” QConf(J ) attached to the Jordan algebra J , or more appropriately to the Freudenthal triple system F; its geometric action on (, α) was called the “quasi-conformal realization”. When J is Euclidean, QConf(J ) is a noncompact group in its quaternionic real form; other real forms can be similarly obtained from Jordan algebras of indefinite signature [25]. Moreover, it was observed in [29] for G = E 8(8) , and generalized to other simple groups in [30–32], that this quasi-conformal action on 2d + 1 variables could be reduced to a representation on functions of d + 1 variables, obtained by first adding one more variable (symplectizing) and then quantizing the resulting 2d + 2-dimensional symplectic space. This smaller representation was identified as the minimal representation of G = QConf(J ). Let us now comment briefly on the physics. Euclidean Jordan algebras J of degree three made their appearance in supergravity a long time ago [9,10,33]. MaxwellEinstein supergravity theories with N = 2 supersymmetry in D = 5 and a symmetric moduli space G/H such that G is a symmetry group of the action are in one-to-one correspondence with Euclidean Jordan algebras J of degree three. Their symmetry group in D = 5 is simply the reduced structure group Str 0 (J ) of J . Upon reduction to D = 4 and D = 3, the symmetry groups are extended to the conformal Conf(J ) and quasi-conformal groups QConf(J ), respectively. The corresponding moduli spaces are given by the quotient of the respective symmetry group by its maximal compact subgroup, and are special real, special Kähler and quaternionic-Kähler manifolds, respectively [9,10,13]. An explicit description of the quasi-conformal action QConf(J ) of D = 3 MaxwellEinstein supergravity theories with symmetric target spaces was obtained in [31]. Finally, it was observed in [15] that the minimal unitary representation of QConf(J ) is closely related to the vector space to which the topological string partition function naturally belongs. 5 We remind the reader of a few definitions. Suppose ρ is a representation of a real Lie group G on a vector space V . Let VK denote the space of K -finite vectors in V (i.e. those which generate a finite-dimensional subspace of V under the action of the maximal compact subgroup K ⊂ G). Then any representation of K which occurs in VK is called a K -type of ρ. Parameterizing the K -types by their highest weights µ, a “lowest K -type” is one with the minimal value of µ + 2ρ K 2 , where ρ K is half the sum of positive roots of K . A discrete series representation always has a unique lowest K -type. If the lowest K -type is the trivial representation, then it is also called the spherical vector and the representation ρ is deemed spherical. The Gelfand-Kirillov dimension measures the growth of the multiplicities of the K -types; morally it counts the number of variables xi needed to realize V as a space of functions. Finally, if all K -types in ρ occur with multiplicity 1 and lie along a ray in the weight space of K , then ρ is called a ladder representation. 2 6 I is expressible in terms of N via 8I () = ξ 0 ξ˜ − ξ ξ˜ 0 − 4ξ ξ˜ + 4ξ N3 (ξ ) + 4ξ˜0 N3 (ξ˜ ), 4 3 4 0
where ξ is related to ξ by the (quadratic) adjoint map, defined by (ξ ) = N (ξ ) ξ .
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The minimal representation has also appeared in another physical context: in [34], a connection between the harmonic superspace (HSS) formulation of N = 2, d = 4 supersymmetric quaternionic Kähler sigma models that couple to N = 2 supergravity and the minimal unitary representations of their isometry groups was established. In particular, for N = 2 sigma models with quaternionic symmetric target spaces of the form QConf(J )/C onf(J ) × SU (2) there exists a one-to-one mapping between the quartic Killing potentials that generate the isometry group QConf(J ) under Poisson brackets in the HSS formulation, and the generators of the minimal unitary representation of QConf(J ). It would be important to understand physically how the minimal representation may arise by quantizing the sigma-model in harmonic superspace. The main goal of the present work is to explain the relation between the twistorial construction of the quaternionic discrete series in [26] and the quasi-conformal actions discovered in [25], and moreover to elucidate the sense in which the minimal representation is obtained by “quantizing the quasi-conformal action”. While our results are unlikely to cause any surprise to the informed mathematician, we hope that our exegesis of [26] will be useful to physicists, e.g. in subsequent applications to supergravity and black holes, and possibly to mathematicians too, e.g. in obtaining explicit formulae for automorphic forms along the lines of [35]. As the main body of this paper is fairly technical, we summarize its content below, including some open problems and possible applications: (i) The key observation is that the variables (, α) of the quasi-conformal realization have a natural interpretation as complex coordinates on the twistor space Z = G/U (1) × M over the quaternionic-Kähler space G/SU (2) × M, adapted to the action of the Heisenberg algebra in the nilpotent radical of the Heisenberg parabolic subgroup of G. The logarithm of the “quartic norm” N4 provides a Kähler potential (2.56) for the G-invariant Einstein-Kähler metric on Z.7 Using the Harish-Chandra decomposition, we also construct the complex coordinates adapted to another Heisenberg algebra related by a Cayley-type transform, whose center is a compact generator rather than a nilpotent one. These coordinates are the analogue of the Poincaré disk coordinates for S L(2, R)/U (1), and it would be interesting to give a Jordan-type description of the corresponding Kähler potential, given in (2.39) for G = SU (2, 1). (ii) It is also possible to view (, α) as real coordinates on G/P, where P is a parabolic subgroup of G with Heisenberg radical. This G/P arises as a piece of the boundary of Z or of G/K . (iii) The quasi-conformal action admits a continuous deformation by a parameter k∈C, corresponding to the action of G on sections of a line bundle over G/P induced from a character of P. This action provides a degenerate principal series representation, which is manifestly unitary for k ∈ 2+iR (SU (2, 1)) or k ∈ 3+iR (G 2(2) ). We tabulate the formulas for the infinitesimal action of g in this representation, and determine the spherical vector. (iv) When k is an integer, one can also consider sections on G/P which can be (in an appropriate sense) extended holomorphically into Z. Thus, the quasi-conformal action with , α complex as in i) leads to a differential operator realization of the action of G on these sections. When G = SU (2, 1) and k ≥ 3, as explained in [26], this action gives quaternionic discrete series representations; we explic7 This follows by specializing analysis performed in [36] of general “dual” quaternionic-Kähler spaces to symmetric spaces. It will be rederived below using purely group theoretic methods (in fact, this approach was a useful guide for the general analysis in [36]).
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itly compute the K -finite vectors of this submodule in the principal series. When G = G 2(2) we similarly study the K -finite vectors of the principal series and identify a natural subset which has the same K -type decomposition as the quaternionic discrete series. (v) Given a class in H 1 (Z, O(−k)), the Penrose transform produces a section of a vector bundle over the quaternionic-Kähler base G/K annihilated by some “quaternionic” differential operators [37,38,36]. For G = SU (2, 1) and k even, we argue formally but explicitly in Sect. 2.4.5 that computing the Penrose transform of is equivalent to evaluating a matrix element between and the lowest K type of the quaternionic discrete series. A similarly explicit understanding of the Penrose transform for k odd remains an open problem, which would require a proper prescription for dealing with the branch cuts in the formula of [36]. (vi) The quasi-conformal action on Z can be lifted to a tri-holomorphic action on the hyperkähler cone (or Swann space) S = R × G/M, locally isomorphic to the smallest nilpotent coadjoint orbit of G C . In particular, the action of G on S preserves the holomorphic symplectic form. The minimal representation of G can be viewed as the “holomorphic quantization” of S. We show explicitly in Sect. 2.6 for SU (2, 1) and 3.4.1 for G 2,2 that the leading differential symbols of the generators of the minimal representation are equal to the holomorphic moment maps of the action of G on S, and identify the corresponding semi-classical limit. (vii) We determine explicitly the lowest K -type of the minimal representation, generalizing the analysis of [39] to these two quaternionic groups. For G 2(2) , the lowest K -type in the real polarization (3.119) bears strong similarities to the result found in [39] for simply laced split groups, while the wave function in the complex (upside-down) polarization (3.119) is analogous to the topological string wave function. (Such a relation is not unexpected, given the results of [15], which showed that the holomorphic anomaly equations of the topological string can be naturally explained in terms of the minimal representation.) We show further that the semi-classical limit of the lowest K -type wave function yields the generating function for a holomorphic Lagrangian cone inside the holomorphic symplectic space S, invariant under the holomorphic action of G.8 It would be very interesting to formulate the hyperkähler geometry of S in terms of this Lagrangian cone, by analogy to special Kähler geometry. The organization of this paper is as follows: In Sects. 2 and 3, for the two rank 2 quaternionic groups G = SU (2, 1) and G = G 2(2) successively, we give explicit parametrizations of the quaternionic-Kähler homogeneous spaces G/K and their twistor spaces, provide explicit differential operator realizations of the principal series, quaternionic discrete series and minimal representations, and compute their spherical or lowest K -type vectors. Throughout this paper we work mostly at the level of the Lie algebras. We are not careful about the discrete factors in any of the various groups that appear. We emphasize concrete formulas even if they are formal, in the spirit of the early literature on twistor theory.
8 This Lagrangian cone was already instrumental in [35], in extending the real spherical vector found in [39] to the adelic setting.
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2. SU(2, 1) 2.1. Some group theory. The non-compact Lie group G = SU (2, 1) is defined as the group of unimodular transformations of C3 which preserve a given hermitian metric η with signature (+, +, −). It is convenient to choose ⎛ ⎞ 1 η = ⎝ 1 ⎠. (2.1) 1 The Lie algebra g = su(2, 1) consists of traceless matrices such that ηX + X † η = 0. This condition is solved by ⎛ ⎞ H + i J /3 E p − i E q , iE −2i J /3 −(E p + i E q )⎠ ≡ X i X i , X = ⎝F p + i Fq (2.2) −i F −(F p − i F q ) −H + i J /3 where {X i } = {H , J , E p , E q , E, F p , F q , F} are the real coefficients of the generators {X i } = {H, J, E p , E q , E,F p , Fq , F} in g. The latter are represented by anti-hermitian operators in any unitary representation. They obey the commutation relations consistent with the matrix representation above. In particular, E p , E q , E obey the Heisenberg algebra E p , E q = −2E, (2.3) while E, H, F generate a Sl(2, R) subalgebra [E, F] = H, [H, E] = 2E, [H, F] = −2F.
(2.4)
The center of the universal enveloping algebra U (g) is generated by the quadratic Casimir C2 =
1 2 3 2 1 1 H − J + (E p F p + F p E p + E q Fq + Fq E q ) + (E F + F E) 4 4 4 2
(2.5)
and cubic Casimir C3 =H 2 J + J 3 − E (F p2 + Fq2 ) + F(E 2p + E q2 )
(2.6)
+ (Fq E p − F p E q )H + (4E F − F p E p − Fq E q − 2)J. The Casimir operators take values [6]
1 4i C2 = p + q + ( p 2 + pq + q 2 ), C3 = ( p − q)( p + 2q + 3)(q + 2 p + 3) (2.7) 3 27 i ...i
in a finite-dimensional representation of SU (2, 1) corresponding to a tensor T j11... jqp with p upper and q lower indices (in particular, (C2 , C3 ) = (3, 0) and (4/3, 80i/27) for the adjoint and fundamental representations, respectively). It is convenient to continue to use the same variables ( p, q) defined in (2.7), no longer restricted to integers, to label the infinite-dimensional representations of SU (2, 1). Alternatively, one may define the “infinitesimal character” 1 1 1 x1 = − ( p + 2q + 3), x2 = ( p − q), x3 = (2 p + q + 3) 3 3 3
(2.8)
Quasi-Conformal Actions, Quaternionic Discrete Series and Twistors: SU (2, 1) and G 2(2)
J
J Fp+iFq
−2
Ep+iEq
+i
−1
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+1
+2
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J+
L0 −2i
−i
i
2i
L−
L+ S
Fp−iFq
−i
−i
Ep−iEq
J−
K+
Fig. 1. Root diagram of SU (2, 1) with respect to the mixed Cartan torus H, J (left) and the compact Cartan torus L 0 , J (right). The compact (resp. non-compact) roots are indicated by a white (resp. black) dot
with x1 + x2 + x3 = 0, such that the Weyl group of SU (2, 1) acts by permutations of (x1 , x2 , x3 ). In a unitary representation, either all xi are real, or one is real and the other two are complex conjugates [6]. In the former case one may choose to order x1 ≤ x2 ≤ x3 , corresponding to p ≥ −1, q ≥ −1. We shall be particularly interested in representations where p = q, such that the infinitesimal character is proportional to the Weyl vector (−1, 0, 1). The generators H, J generate a Cartan subalgebra of g. The spectrum of the adjoint action of Spec(J ) = {0, ±i}, Spec(H ) = {0, ±1, ±2}
(2.9)
shows that H and J are non-compact and compact, respectively. In fact, the generator H gives rise to a “real non-compact” 5-grading g = F|−2 ⊕ {F p , Fq }|−1 ⊕ {H, J }|0 ⊕ {E p , E q }|1 ⊕ E|2 ,
(2.10)
where the subscript denotes the eigenvalue under H . In this decomposition, each subspace is invariant under hermitian conjugation. Moreover J ⊕ {E, H, F} generate a U (1) × S L(2, R) (non-compact) maximal subgroup of G. The remaining roots arrange themselves into a pair of doublets of S L(2, R) with opposite charge under J ,
F p − i Fq E p − i E q F p + i Fq E p + i E q
(2.11)
as shown on the root diagram 1. The parabolic subgroup P = L N with Levi L = R × U (1) generated by {H, J } and unipotent radical N generated by {F p , Fq , F}, corresponding to the spaces with zero and negative grade in the decomposition (3.8), is known as the Heisenberg parabolic subgroup, and will play a central rôle in all constructions in this paper. For later purposes, it will be useful to introduce another basis of g adapted to a maximal compact subgroup K = SU (2) × U (1). We first go to a compact basis for the
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S L(2, R) factor generated by E, F, H : 1 1 1 (F − E), K 0 = (2L 0 + 3J ), L ± = − √ (E + F ± i H ) , (2.12) 2 4 2 2 1 1 K ± = − E p ± i E q + (F p ± i Fq ) , J± = − √ E p ∓ i E q − (F p ∓ i Fq ) . 4 2 2 L0 =
Then {L + , L 0 , L − } and {K + , K 0 , K − } make two S L(2, R) subalgebras, with compact Cartan generators L 0 and K 0 , respectively, (2.13) L 0 , L ± = ±i L ± , [L + , L − ] = −i L 0 , (2.14) K 0 , K ± = ±i K ± , [K + , K − ] = −i K 0 . Since Spec(L 0 ) = {0, ± 2i , ±i}, (L 0 , J ) now form a Cartan torus, and L 0 gives rise to a new 5-grading, g = L − |−i ⊕ {K − , J− }|− i ⊕ {L 0 , J }|0 ⊕ {J+ , K + }| i ⊕ L + |i , 2
2
(2.15)
where the subscript denotes the eigenvalue under L 0 . Unlike the 5-grading (2.10), in (2.15) hermitian conjugation exchanges the positive and negative grade spaces. Next, we perform a π/3 rotation of the root diagram, and define 3 1 (F − E − 3J ), S = (F − E + J ), 4 √ 4√ = ±2 2i L ± , J± 1 ,∓ 3 = 2K ∓ .
J3 = J± 1 ,± 3 2
2
2
2
(2.16) (2.17)
Then {J+ , J3 , J− } and S generate the compact subgroup K = SU (2) × U (1),9 J3 , J± = ±i J± , [J+ , J− ] = 2i J3 . (2.18) In particular, J3 induces the “compact 5-grading” J− |−i ⊕ {L − , K + }|− i ⊕ {J3 , S}|0 ⊕ {K − , L + }| i ⊕ J+ |i , 2
2
(2.19)
where the subscript now denotes the J3 eigenvalue. The remaining roots can then be arranged as a pair of doublets under SU (2) with opposite U (1) S charges,
J− 1 ,− 3 J 1 ,− 3 2 2 2 2 (2.20) J− 1 , 3 J 1 , 3 2 2
2 2
as shown on the second root diagram on Fig. 1. In order to represent the generators in the compact basis by pseudo-hermitian matrices, it is convenient to change basis and diagonalize the hermitian metric (2.1), ⎞ ⎛ 1 ⎛ ⎞ √ 0 √1 1 2 ⎟ ⎜ 2 η ≡ CηC t = ⎝ −1 ⎠ , C = ⎝ √1 0 − √1 ⎠ . (2.21) 2 2 1 0 1 0 9 To be precise, K = (SU (2) × U (1))/Z . We abuse notation by suppressing this Z in most of what 2 2 follows.
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The generators preserving the metric η can now be parametrized as ⎞ ⎛ i −J − − 2 (J 3 + S) 2J − 1 ,− 3 2 2 ⎜ 2J 1 3 iS −2J − 1 , 3 ⎟ ⎠, ⎝ , 2 2
−2J 1 ,− 3
J+
2
i 2 (J 3
2
− S)
where J 3 , S are real, while J ± = J ∗∓ , J ± 1 ,± 3 = J ∗ 2
2
(2.22)
2 2
∓ 21 ,∓ 23
, J ± 1 ,∓ 3 = J ∗ 2
2
∓ 21 ,± 32
. By
analogy with the S L(2, R) = SU (1, 1) case, we shall refer to the matrix C as a Cayley rotation.
2.2. Quaternionic symmetric space. We now describe the geometry of the quaternionicKähler symmetric space K \G = (SU (2)×U (1))\SU (2, 1). This space is well known in the string theory literature as the tree-level moduli space of the universal hypermultiplet (see e.g. [11,40–42] for some useful background). It is in the class of “dual quaternionic manifolds”, in the sense that it can be constructed by the c-map procedure [43,12] from the trivial zero-dimensional special Kähler manifold with quadratic prepotential F = −i(X 0 )2 /2. In order to parameterize this space, we use the Iwasawa decomposition of G, ˜
g = k · e−U H · eζ E p −ζ Eq · eσ E ,
(2.23)
where k is an element of the maximal compact subgroup K = SU (2)×U (1). In terms of the fundamental representation, this is g = k ·e Q K , where e Q K is the coset representative ⎛ −U ⎞ ⎛ ⎞ 1 ζ˜ + iζ i σ − 21 (ζ˜ 2 + ζ 2 ) e 1 ⎠·⎝ eQ K = ⎝ (2.24) 1 −(ζ˜ − iζ ) ⎠ . eU 1 The right-invariant g-valued 1-form is then ⎛ ⎞ −dU e−U (d ζ˜ + idζ ) ie−2U (dσ + ζ˜ dζ − ζ d ζ˜ ) ⎝ 0 θ = de Q K e−1 0 −e−U (d ζ˜ − idζ ) ⎠ . (2.25) QK = 0 0 dU Expanding θ on the compact basis of g, its 1-form components are
J−1,3 J 1,3 u¯ v¯ 2 2 2 2 =− , J − 1 ,− 3 J 1 ,− 3 vu 2
2
2
(2.26)
2
⎞ ⎞ ⎛ u¯ J− 3i 1 ⎝ J 3 ⎠ = − ⎝ i (v − v) ¯ ¯ ⎠ , S = − (v − v), 4 2 8 J+ u ⎛
(2.27)
where we defined the 1-forms 1 i u = − √ e−U (d ζ˜ + idζ ), v = dU − e−2U (dσ + ζ˜ dζ − ζ d ζ˜ ). 2 2
(2.28)
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The non-compact components (2.26) give the quaternionic viel-bein of the invariant metric, while the compact components (2.27) give the spin connection, with restricted holonomy SU (2) × U (1). The invariant metric on K \G is thus given by ds 2 = 2(u u¯ + v v) ¯
2 1 2 = 2(dU )2 + e−2U d ζ˜ 2 + dζ 2 + e−4U dσ + ζ˜ dζ − ζ d ζ˜ . 2
(2.29)
The group G acts on the coset space K \G by right multiplication, followed by a left action of the maximal compact K = SU (2) × U (1) so as to maintain the Iwasawa gauge k = 1 in (2.23). The metric is invariant under this action. This gives an action of SU (2, 1) by Killing vectors on K \G: E Q K = ∂σ ,
E pQ K = ∂ζ˜ − ζ ∂σ ,
E qQ K = −∂ζ − ζ˜ ∂σ ,
J Q K = ζ ∂ζ˜ − ζ˜ ∂ζ , Fp Q K Fq Q K F QK
H Q K = −∂U − ζ ∂ζ − ζ˜ ∂ζ˜ − 2σ ∂σ ,
3ζ 2 − ζ˜ 2 1 2U = −ζ˜ ∂U −(σ +2ζ ζ˜ )∂ζ + e + ∂ζ˜ + ζ e2U + (ζ 2 + ζ˜ 2 ) −σ ζ˜ ∂σ , 2 2
2 ˜2 2U 3ζ −ζ 2U 1 2 ˜ 2 ˜ ˜ = ζ ∂U −(σ − 2ζ ζ )∂ζ˜ − e + ∂ζ + ζ e + (ζ + ζ ) +σ ζ ∂σ , 2 2
2
1 1 = −σ ∂U + e2U + (ζ 2 + ζ˜ 2 ) −σ 2 ∂σ − ζ˜ e2U + (ζ 2 + ζ˜ 2 ) + σ ζ ∂ζ 2 2
1 + ζ e2U + (ζ 2 + ζ˜ 2 ) − σ ζ˜ .∂ζ˜ . (2.30) 2
The action of G on K \G also induces a representation of G on L 2 (K \G). The quadratic Casimir in this representation is proportional to the Laplace-Beltrami operator of the metric (2.29), while the cubic Casimir vanishes identically: C2 =
1 1 √ 1 = √ ∂i gg i j ∂ j , C3 = 0. 4 4 g
(2.31)
2.3. Twistor space and Swann space. The twistor space Z is a CP1 = U (1)\SU (2) bundle over the quaternionic-Kähler space (SU (2) × U (1))\SU (2, 1), which carries a Kähler-Einstein metric. The fibration is such that the SU (2) “cancels” [44], so that Z is an homogeneous (but not symmetric) space, Z = (U (1) S × U (1) J3 )\SU (2, 1).
(2.32)
Let H denote the subgroup U (1) S ×U (1) J3 . In the following we construct two canonical sets of complex coordinates on Z, adapted to two different Heisenberg algebras, and relate them to the coordinates U, ζ, ζ˜ , σ on the base and the stereographic coordinate z on the sphere.10 10 Of course, these “coordinates on Z” really cover only an open dense subset; they become singular at the (canonically defined) north and south poles of the CP1 fibers.
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2.3.1. Harish-Chandra coordinates. The complex structure on Z can be constructed by using the Borel embedding H \G → PC \G C , where PC is the parabolic (Borel) subgroup of the complexified group G C , generated by the positive roots J 1 ,± 3 , J+ and 2 2 Cartan generators J3 , S. To obtain complex coordinates from this embedding we go to the Cayley-rotated matrix representation (2.22), and perform a N¯ A N decomposition, ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 1 ab 1 pk ⎠ · ⎝ 1 q⎠ . (2.33) CeZ C −1 = ⎝ p˜ 1 ⎠ · ⎝ 1/a 2 1 a/b k˜ q˜ 1 The entries p, q, k in the upper-triangular matrix by construction provide holomorphic coordinates on Z. The lower-triangular and diagonal matrices are then expressed in terms of p, q, k and their complex conjugates p, ¯ q, ¯ k¯ by requiring that (2.33) is an element of SU (2, 1) rather than of its complexification: ¯ ¯ k k¯ − p p¯ + 1 p¯ − kq ( p ¯ q ¯ − k) k¯ p − p¯ q¯ p + q¯ p˜ = √ , k˜ = , q˜ = , (2.34a) √ k k¯ − p p¯ + 1 1 1/4 , b= , 1/4 ¯ ¯ (k k − p p¯ + 1) (k k − p p¯ + 1)1/4 1/4
a=
(2.34b)
where = 1 + k k¯ − q q¯ − k¯ pq − k p¯ q¯ + p pq ¯ q. ¯
(2.35)
These coordinates are adapted to the holomorphic action of the Heisenberg algebra generated by J− 1 ,± 3 , J− , in the sense that 2
2
J− 1 , 3 = −2(∂q + p∂k ), 2 2
J− 1 ,− 3 = 2∂ p , 2
2
J− = −∂k .
(2.36)
It is useful to record the action of the other generators in the compact basis, i 3i ( p∂ p + q∂q + 2k∂k ), S = ( p∂ p − q∂q ), 2 2 2 J+ = −kp∂ p − (k − pq)q∂q − k ∂k , J3 =
J 1 , 3 = −2 p ∂ p − 2(k − pq)∂q − 2kp∂k ,
(2.37a) (2.37b)
J 1 ,− 3 = −2k∂ p + 2q ∂q .
2
2
2 2
2
2
(2.37c)
A G-invariant metric on Z can be constructed in the usual way, by applying an L-invari−1 . In contrast to the case ant quadratic form on g to the g-valued 1-form θ = deZ · eZ of K \G, this quadratic form is not unique up to scalar multiple, but has parameters (α, β, γ ) ∈ R3 : 2 dsZ = α J + J − + β J 1 , 3 J − 1 ,− 3 + γ J 1 ,− 3 J − 1 , 3 . 2 2
2
2
2
2
(2.38)
2 2
These parameters can be fixed by requiring that the resulting metric on Z is EinsteinKähler. In particular, the Kähler potential must be proportional to the volume element. This uniquely fixes α = −2, β = γ = 4 (up to rescalings) and gives the Kähler potential 1 ¯ + |k − pq|2 − q q) K Z = log (1 + k k¯ − p p)(1 ¯ . (2.39) 2 This reproduces Eq. 7.28 in [45] upon identifying k = ζ /2, p = v, q = −u. Under the action of J 1 , 3 + J− 1 ,− 3 , J 1 ,− 3 + J− 1 , 3 , J+ + J− , K Z transforms by a Kähler 2 2 2 2 2 2 2 2 transformation K Z → K Z + f + f¯, with f proportional to p, q, k − 1 pq, respectively. 2
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2.3.2. Iwasawa coordinates. We now exhibit the twistor space Z as an S 2 fibration over K \G, such that the S 2 fiber over any point on the base is holomorphically embedded in Z. For this purpose, we recall that the complex structure on S 2 = U (1)\SU (2) can be constructed using the Borel embedding U (1)\SU (2) → BC \S L(2, C). Here S L(2, C) ⊂ G C is generated by J+ , J3 , J− and BC is the Borel subgroup generated by J+ , J3 . The embedding is simply obtained by starting with a coset representative e ∈ U (1)\SU (2) and viewing it instead as a representative in BC \S L(2, C) (this is consistent since BC ∩ SU (2) = U (1).) The same class in BC \S L(2, C) is also represented by exp z J− for some z ∈ C (with one exception corresponding to z = ∞), giving the desired complex coordinate. Now we use the same idea for Z. So we return to the original matrix representation (2.2), and parameterize Z by the coset representative eZ = e−¯z J+ (1 + z z¯ )−i J3 e−z J− e Q K √ ⎛1 √z 2 (1 + 1 + z z¯ ) 2 1 ⎜ 1 − √1 z¯ = √ ⎝ 2 √ 1 + z z¯ 1 √z 2 (1 − 1 + z z¯ ) 2
√ ⎞ 1 + z z¯ ) ⎟ − √1 z¯ ⎠ · eQ K , 2 √ 1 2 (1 + 1 + z z¯ )
(2.40)
1 2 (1 −
(2.41)
where e Q K is a representative for K \G in the Iwasawa decomposition (2.24). The coordinate z is then a stereographic coordinate on the S 2 fiber over each point of K \G. By Cayley rotating (2.40) and performing the Harish-Chandra decomposition (2.33), we can now relate the complex coordinates p, q, k and their complex conjugates to the coordinates U, ζ, ζ˜ , σ on the base and the coordinate z on the fiber: 4
p=
− 1, √ 2 + 2iσ − ζ˜ 2 + 2i 2eU z(ζ + i ζ˜ ) √ 2 2eU (iζ + ζ˜ ) − ζ 2 + ζ˜ 2 + 2iσ + 2 − 2e2U z , q= √ 2i 2z(ζ + i ζ˜ ) + 4eU √ 2 2eU z + 2(iζ + ζ˜ ) k= . √ 2 + 2iσ + 2e2U − ζ 2 − ζ˜ 2 + 2i 2eU z(ζ + i ζ˜ ) + 2e2U
− ζ2
(2.42a)
(2.42b)
(2.42c)
Rather than obtaining the metric on Z in the coordinates U, ζ, ζ˜ , σ, z, z¯ from the Kähler potential (2.39) by following the change of variables, we can simply decompose the −1 invariant form θZ = deZ eZ and plug into (2.38) using the values of α, β, γ that were determined above. The components of θZ along (U (1) × U (1))\SU (2, 1) read
J−1,3 J 1,3 1 u + zv v − z¯ u 2 2 2 2 , (2.43a) =− √ J − 1 ,− 3 J 1 ,− 3 2 1 + z z¯ v¯ − z u¯ u¯ + z¯ v¯ 2 2 2 2 J+ = −
1 1 + z z¯
J− = −
1 1 + z z¯
1 D z¯ d z¯ + u¯ − z¯ (v − v) ¯ + z¯ 2 u ≡ − , 2 1 + z z¯
1 dz + u + z(v − v) ¯ + z 2 u¯ 2
≡−
Dz , 1 + z z¯
(2.43b)
(2.43c)
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leading to the Kähler-Einstein metric 2 dsZ = |u|2 + |v|2 − 2
Dz D z¯ (1 + z z¯ )2
(2.44)
with signature (4, 2). The connection term in Dz is the projective SU (2) connection 1 1 Dz = dz − (A1 + i A2 ) + i A3 z − (A1 − i A2 )z 2 2 2 1 2 ¯ + z u, = dz + u + z(v − v) ¯ 2
(2.45)
where ¯ A1 = −(u + u),
A2 = i(u − u), ¯
A3 =
v − v¯ 2i
(2.46)
are the components of the SU (2) spin connection computed in (2.27). A basis of holomorphic (1,0) forms providing a holomorphic viel-bein of Z is given by the components of θZ with negative weight under J3 , √ 1 2Dz, u + zv, v¯ − z u¯ . (2.47) V=√ 1 + z z¯ The Kähler form can be written as ωZ = −
Dz D z¯ + i x a ωa , (1 + z z¯ )2
(2.48)
where x a is the unit length vector with stereographic coordinate z, x1 =
z + z¯ i(z − z¯ ) 1 − z z¯ , x2 = , x3 = , 1 + z z¯ 1 + z z¯ 1 + z z¯
(2.49)
and ωa are the quaternionic 2-forms on the base, ω1 =
1 1 1 (u v¯ − uv), ¯ ω2 = (u v¯ + uv), ¯ ω3 = (u u¯ − v v). ¯ 2i 2 2i
(2.50)
It may be checked that this triplet of 2-forms satisfies the constraints from quaternionic-Kähler geometry, dωi + i jk A j ∧ ωk = 0, d Ai + i jk A j ∧ Ak = 2ωi .
(2.51)
2.3.3. Complex c-map coordinates. While the complex coordinates p, q, k are adapted to the action of the generators J− 1 , 3 , J− 1 ,− 3 , J− , in the sequel it will be useful to 2 2 2 2 have complex coordinates ξ, ξ˜ , α adapted to the “non-compact” Heisenberg algebra E p , E q , E, i.e. such that the action of these generators takes the canonical form11 E p = ∂ξ˜ + ξ ∂α ,
E q = −∂ξ + ξ˜ ∂α ,
E = −∂α .
(2.52)
11 Since G acts holomorphically, we have abused notation by writing only the holomorphic part; the real vector fields would be obtained by adding the complex conjugates, in (2.52) and below.
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The change of variables can be found by diagonalizing the action of these generators in the p, q, k variables, leading to i k 2 − ( p + 1)qk + p + 1 −k 2 + ( p + 1)qk + p + 1 ξ = √ , ξ˜ = √ , 2( p + 1)(−k + pq + q) 2( p + 1)(−k + pq + q) i(q + kp − p 2 q) α= ( p + 1)(k − pq − q)
(2.53a) (2.53b)
or, conversely, √ i ξ 2 + ξ˜ 2 + 2iα + 2 ξ 2 + ξ˜ 2 − 2iα + 2 2i 2(ξ − i ξ˜ ) , q= . p=− , k=− √ ξ 2 + ξ˜ 2 − 2iα − 2 ξ 2 + ξ˜ 2 − 2iα − 2 2 2(ξ + i ξ˜ ) (2.54) The full action of G is then given by E QC = −∂α ,
E p QC = ∂ξ˜ + ξ ∂α ,
E q QC = −∂ξ + ξ˜ ∂α ,
H QC = −ξ˜ ∂ξ˜ − ξ ∂ξ − 2α∂α ,
(2.55b)
F p QC
(2.55c)
Fq QC F QC
J QC = −ξ˜ ∂ξ + ξ ∂ξ˜ , 1 1 2 ξ(ξ˜ + ξ 2 ) + 2α ξ˜ ∂α , = (3ξ 2 − ξ˜ 2 )∂ξ˜ + (α − 2ξ˜ ξ )∂ξ − 2 2 1 1 2 ξ˜ (ξ˜ + ξ 2 ) − 2α ξ˜ ∂α , = − (3ξ˜ 2 − ξ 2 )∂ξ + (α + 2ξ˜ ξ )∂ξ˜ − 2 2 1 2 1 ξ(ξ˜ + ξ 2 ) + 2α ξ˜ ∂ξ˜ − = ξ˜ (ξ˜ 2 + ξ 2 ) − 2αξ ∂ξ 2 2 1 2 2 2 2 ˜ − (ξ + ξ ) − 4α ∂α . 4
(2.55a)
(2.55d)
(2.55e)
In the new coordinate system ξ, ξ˜ , α, the Kähler potential (2.39) (after a Kähler transformation by f = ( p + 1)(k − q − pq)) is KZ =
2 1 1 log N4 = log (ξ − ξ¯ )2 + (ξ˜ − ξ¯˜ )2 + 4(α − α¯ + ξ ξ¯˜ − ξ¯ ξ˜ )2 . (2.56) 2 2
¯˜ α) Here we note that N4 (ξ, ξ˜ , α; ξ¯ , ξ, ¯ is the quartic distance function of quasi-conformal geometry. Since G acts by isometries on Z, it leaves the Kähler potential (2.56) invari¯˜ α) ant up to Kähler transformations. Equivalently, the quartic norm N4 (ξ, ξ˜ , α; ξ¯ , ξ, ¯ ¯ ˜ α), defined by (2.56) transforms multiplicatively by a factor f (ξ, ξ˜ , α) f¯(ξ¯ , ξ, ¯ where the holomorphic function f depends on the generator under consideration. In particular, the “quartic light-cone” N4 = 0 is invariant under the full action of SU (2, 1), which motivated the appellation “quasi-conformal action” in [25]. Such a coordinate system adapted to the holomorphic action of a Heisenberg group exists for any c-map space and was used heavily in [36]. Moreover, the result (2.56) agrees with a general formula for the Kähler potential given there.
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2.3.4. Twistor map. Combining the changes of variables (2.42) and (2.53), we find that the complex coordinates (ξ, ξ˜ , α) are expressed in terms of the Iwasawa coordinates U, ζ, ζ˜ , σ, z, z¯ as i 1 ξ = ζ − √ eU z + z −1 , ξ˜ = ζ˜ + √ eU z − z −1 , (2.57a) 2 2 1 (2.57b) α = σ + √ eU z(ζ + i ζ˜ ) + z −1 (−ζ + i ζ˜ ) . 2 Again (ξ, ξ˜ , α) is a holomorphic (rational) function of z, at fixed values of U, ζ, ζ˜ , σ , so that the twistor fiber over any point is a rational curve in Z. Conversely, e2U = −
N4 8[(ξ − ξ¯ )2 + (ξ˜ − ξ¯˜ )2 ]
,
(2.58a)
2 2(α − α)( ¯ ξ˜ − ξ¯˜ ) + (ξ − ξ¯ ) ξ 2 − ξ¯ 2 + ξ˜ 2 − ξ¯˜ ζ =
ζ˜ =
σ =
2[(ξ − ξ¯ )2 + (ξ˜ − ξ¯˜ )2 ]
,
(2.58b)
,
(2.58c)
2 ¯ ¯ 2 2 2 ¯ ˜ ¯ ˜ ˜ ˜ 2(α¯ − α)(ξ − ξ ) + (ξ − ξ ) ξ − ξ + ξ − ξ 2[(ξ − ξ¯ )2 + (ξ˜ − ξ¯˜ )2 ]
α + α¯ − 2
2 ¯ ¯ 2 2 2 ˜ ¯ ˜ ¯ ˜ ˜ (α − α¯ + ξ ξ − ξ ξ ) ξ − ξ + ξ − ξ 2[(ξ − ξ¯ )2 + (ξ˜ − ξ¯˜ )2 ]
,
(2.58d)
ξ − ξ¯ − i(ξ˜ − ξ¯˜ ) (ξ − ξ¯ )2 + (ξ˜ − ξ¯˜ )2 − 2i(α − α¯ + ξ ξ¯˜ − ξ¯ ξ˜ ) z= × . ¯ ¯ ˜ ˜ ξ − ξ + i(ξ − ξ ) (ξ − ξ¯ )2 + (ξ˜ − ξ¯˜ )2 + 2i(α − α¯ + ξ ξ¯˜ − ξ¯ ξ˜ ) (2.58e) These formulae are in agreement with the general results in [36]. 2.3.5. Swann space. The twistor space Z admits an important complex line bundle O(2), which can be constructed in two different ways. On the one hand, one may use Z = H \G, and obtain O(2) by induction from H with character exp i J3 – in other words, a section of O(2) is a function on G which transforms under H by this character.12 This definition makes it clear that O(2) admits a Hermitian structure. On the other hand, one may construct O(2) using the Borel embedding Z → PC \G C , and define O(2) as the pullback to Z of the line bundle determined by the character exp H of PC . This second definition makes it clear that O(2) is a holomorphic line bundle; it is 12 By definition, the character “exp i J ” takes the value exp i J on exp(J J + SS) ∈ H ; we will use 3 3 3 3 this notation for characters frequently, always implicitly with respect to some basis of the corresponding Lie algebra.
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what we will use in the discussion of quaternionic discrete series below. More generally, m define O(m) = O(2)⊗ 2 for m even; this is similarly a Hermitian holomorphic line bundle over Z. The Swann space S, also known as the hyperkähler cone over K \G, is the total space of O(−2) over Z; locally it could be parameterized by complex coordinates on Z plus one additional coordinate in the fiber of O(−2). It is naturally a hyperkähler manifold, with an SU (2) isometry rotating the complex structures into one another [46,45]. The circle in the unit circle bundle of O(−2) “cancels against” the U (1) in the denominator of (2.32). So this circle bundle is the homogeneous 3-Sasakian space U (1)\SU (2, 1), which can be parameterized by the coset representative e3S = eiφ J3 eZ .
(2.59)
S is then a real cone over this homogeneous space, S = R+ × U (1)\SU (2, 1).
(2.60)
−1 The right-invariant form θ3S = de3S e3S has J3 component J 3 = Dφ, where
i 1 z¯ dz − zd z¯ + 2(¯z u − z u) ¯ + (v − v)(z ¯ z¯ − 1) (2.61a) 1 + z z¯ 2 i = dφ + (2.61b) (¯z dz − zd z¯ ) + xa Aa , 1 + z z¯
Dφ = dφ +
while the other components are identical to those of θZ except for a rotation under U (1) J3 . The metric of the 3-Sasakian space is obtained by adding α J3 2 to (2.38), so as to enforce SU (2) (left) invariance. The metric on S is therefore 2 dsS2 = − dr 2 + r 2 Dφ 2 − dsZ , (2.62) with indefinite signature (4,4).
2.4. Quasi-conformal representations. 2.4.1. Principal series. An interesting family of “principal series” representations of G are obtained by induction from the parabolic subgroup P generated by {F, F p , Fq , H, J }, using the character χk = e−k H /2 of P for some k ∈ C. We now briefly recall the definition of induction, which we will use many times in this paper; see e.g. [47] for more details. The representation space of the induced representation consists of functions f on G which obey f (gp) = χk ( p) f (g). (2.63) These functions can also be thought of as representing sections of a homogeneous line bundle over P\G defined by the character χk . We represent them concretely by choosing specific representatives of P\G; a simple choice is to use elements of the opposite nilpotent radical N¯ generated by E p , E q , E, i.e. upper-triangular matrices.
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So to compute concretely the action of some E ∈ g, we act on the upper-triangular matrix ⎛ ⎞ 1 ζ˜ + iζ iσ − 21 (ζ 2 + ζ˜ 2 ) ⎝ ⎠ (2.64) 1 −ζ˜ + iζ 1 by E and then act from the left by a suitable X ∈ p to put the result back in upper triangular form. This gives a differential operator acting on (ζ, ζ˜ , σ ), to which we must add χk (X ) reflecting the twist by (2.63). The result is E QC = −∂σ , H
QC
F p QC Fq QC F QC
E p QC = ∂ζ˜ + ζ ∂σ ,
= −ζ˜ ∂ζ˜ − ζ ∂ζ − 2σ ∂σ − k,
E q QC = −∂ζ + ζ˜ ∂σ ,
(2.65a)
= −ζ˜ ∂ζ + ζ ∂ζ˜ , (2.65b) 1 1 ζ (ζ˜ 2 + ζ 2 ) + 2σ ζ˜ ∂σ − k ζ˜ , = (3ζ 2 − ζ˜ 2 )∂ζ˜ + (σ − 2ζ˜ ζ )∂ζ − 2 2 (2.65c) 1 1 ζ˜ (ζ˜ 2 + ζ 2 ) − 2σ ζ˜ ∂σ + kζ, = − (3ζ˜ 2 − ζ 2 )∂ζ + (σ + 2ζ˜ ζ )∂ζ˜ − 2 2 (2.65d) 1 2 1 ζ (ζ˜ + ζ 2 ) + 2σ ζ˜ ∂ζ˜ − ζ˜ (ζ˜ 2 + ζ 2 ) − 2σ ζ ∂ζ = (2.65e) 2 2 1 − (ζ˜ 2 + ζ 2 )2 − 4σ 2 ∂σ + kσ. 4 J
QC
The quadratic and cubic Casimirs are constants, C2 = −k(4 − k)/4, C3 = 0
(2.66)
corresponding to p = q = (k − 4)/2. If we choose k = 2 + is for s ∈ R, then this representation is infinitesimally unitary with respect to the L 2 inner product with measure dζ d ζ˜ dσ . 2.4.2. Quaternionic discrete series. In the last subsection we constructed the principal series representations using the action of G on appropriate sections of line bundles over P\G. There is a complex-analytic analogue of this construction, known as the quaternionic discrete series, which uses instead the action of G on the sheaf cohomology group H 1 (Z, O(−k)) [26], where O(−k) is the line bundle over Z defined in Sect. 2.3.5. Formally, the generators are obtained from (2.65) by replacing the real variables ζ, ζ˜ , σ on P\G by complex variables ξ, ξ˜ , α on PC \G C . For k = 0, they reduce to the holomorphic vector fields given in (2.55). The holomorphic functions f (ξ, ξ˜ , α) acted upon by these generators are to be interpreted as Cech representatives13 for classes in H 1 (Z, O(−k)) with respect to coverings by two open sets, in the spirit of the early literature on twistor theory. The resulting representation is formally unitary with respect to the inner product ¯˜ α) f1 | f2 = dξ d ξ˜ dα d ξ¯ d ξ¯˜ d α¯ e(k−4)K Z f 1∗ (ξ¯ , ξ, ¯ f 2 (ξ, ξ˜ , α). (2.67) Z
13 It is not obvious that one obtains all classes in H 1 (Z, O(−k)) in this way, but in most of our considerations we restrict ourselves to these, and indeed we simply write f ∈ H 1 (Z, O(−k)), where f is a section with singularities.
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Indeed, the invariant volume form on Z is e−4K Z |dξ d ξ˜ dα|2 (more generally −4 in the exponent would be replaced by −2d − 2, where d is the dimension of K \G) while the factor ek K Z comes from the Hermitian norm in O(−k) [36]. The formal inner product (2.67) however has to be interpreted with care, since f 1 and f 2 , being Cech representatives, are only well defined up to certain shifts by holomorphic functions; in order to get a well defined inner product, one must interpret (2.67) in a way that involves only contour integrals. Here we give only a formal heuristic account, which will be adequate for our purposes in Sect. 2.4.5. ¯ obtained by conWe begin by analytically continuing K Z to a function on Z × Z, sidering the holomorphic and antiholomorphic dependence independently. Then for any ¯ O(k ¯ − 4)) by f 1 ∈ H 1 (Z, O(−k)) we define fˆ1 ∈ H 1 (Z, ˆf 1 = dξ d ξ˜ dα e(k−4)K Z f 1 , (2.68) ¯ This construction is an anawhere the integral runs over some contour in Z × Z. logue of the “twistor transform” discussed in e.g. [48] for the case Z = CP3 . In [48] it is argued that this transform is involutive, fˆˆ = f ; we assume (and will make use of this fact in Sect. 2.4.5) that the same is true here.14 Complex conjugation gives f¯ˆ1 ∈ H 1 (Z, O(k − 4)), which can be paired with f 2 ∈ H 1 (Z, O(−k)) and contourintegrated, so that f | f = dξ d ξ˜ dα f¯ˆ f . (2.69) 1
2
1 2
This interpretation of (2.67) is still formal, since we did not specify the contours of integration. In the example we consider below there will be a natural choice. In general, however, it is difficult to make sense of this formal prescription, much less to check that it is positive definite; one instead checks the existence of a positive definite norm by a purely algebraic computation on a special basis of “elementary states” (in our context, these are the K -finite vectors). We do not perform such an analysis here, but rely on the results of [26]. There one finds that for k ≥ 2 the spaces H 1 (Z, O(−k)) are irreducible and unitarizable representations of G. For k ≥ 3 they belong to the discrete series and are called quaternionic discrete series representations. The representation at k = 2 is a limit of the quaternionic discrete series. 2.4.3. Quaternionic discrete series as subquotients of principal series. These quaternionic discrete series representations are expected to be obtained as subquotients of the principal series which we discussed above. To understand how this happens, we recall the simpler case of the unitary representations of S L(2, R). There the continuous principal series is realized in a space of sections of a line bundle Lk (k ∈ C) over the equator of CP1 = B+ \S L(2, C). If k ∈ Z, then Lk extends holomorphically over the whole of CP1 . The holomorphic and antiholomorphic discrete series are then obtained by restricting to sections which admit an analytic continuation from the equator over respectively the northern or southern hemisphere, and dividing out by the space of sections which can be continued over the whole of CP1 . 14 This is an analogue of the fact that on a Hermitian symmetric space the Kähler potential behaves as a reproducing kernel for holomorphic functions.
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In the quaternionic case the situation is similar. Firstly, the space P\G occurs as part of the boundary of Z in an appropriate sense; in terms of our coordinates (ξ, ξ˜ , α) on Z, this boundary is the locus where all coordinates become real (then they are identified with our coordinates on P\G by ξ → ζ , ξ˜ → ζ˜ , α → σ .) So for discrete values of k, one might expect to obtain a submodule of the principal series representation by considering those sections which are boundary values of holomorphic objects on Z, and then obtain a unitary representation as some quotient thereof.15 We will see this expectation realized in the algebraic discussion of the next subsection. 2.4.4. K -type decomposition. We now discuss the decomposition of the principal series representation under the maximal compact subgroup K = SU (2) × U (1), which we recall is generated by J± , J3 and S. We begin by constructing the spherical vector, invariant under K . For this purpose consider the action of K from the right on our coset representative for P\G, ⎛ ⎞ 1 ζ˜ + iζ iσ − 21 (ζ 2 + ζ˜ 2 ) ⎠. n=⎝ (2.70) 1 −ζ˜ + iζ 1 K acts on the three rows vi , preserving their Hermitian norms vi 2 . On the other hand, the action of P from the left mixes the rows. Since P is lower-triangular, though, its action on the top row is simple: it just acts by the character e H +i J /3 . Now consider the function −k/2
1 2 −k/2 2 2 2 2 2 ˜ ˜ = 1 + ζ + ζ + σ + (ζ + ζ ) (2.71) f K = v1
4 as an element in the principal series representation. By definition, to compute the action of k ∈ K on f K , we first transform n by k acting from the right, then act by a compensating element p ∈ P from the left to restore the form (2.70). This modifies (2.71) by a factor e−k H ( p)/2 . However, we also have to include the explicit factor ek H ( p)/2 from the definition of the principal series. So altogether we find that (2.71) is K -invariant. More generally, the highest weight vectors of SU (2) J (vectors annihilated by J+ ) are given by
f j,s
j− 1 s
1 + iσ − 21 (ζ 2 + ζ˜ 2 ) = 1 + iσ + 21 (ζ 2 + ζ˜ 2 ) 1 − iσ + 21 (ζ 2 + ζ˜ 2 ) −k/2 1 × 1 + ζ˜ 2 + ζ 2 + σ 2 + (ζ˜ 2 + ζ 2 )2 4 ζ˜ + iζ
3
j+ 1 s 3
(2.72)
with eigenvalues ji and si for J3 and S, respectively. In Sect. 2.6.1 below, we shall see that these states have a simple expression in terms of the moment maps of the action of G on the symplectization of P\G. For now, we 15 The notion of “boundary value” in this case is somewhat subtler than it was in the case of S L(2, R), because on Z we deal not with functions but with cohomology classes. The reason is that the structure of Z near the boundary is more complicated than that of the upper half-plane; it is not contained in any convex tube domain, essentially because of the circle parameterized by the phase of z, which winds around the boundary. The correct notion of boundary value in this case should involve integration over this circle, as described in e.g. [49].
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S
6
4−k k+8
k+6 10−k k 9/2
4−k k+6
k+4 8−k
k
6−k
10−k k+2
3 k+2
4−k k+4
6−k
k
6−k k+6 8−k k+2
10−k
6−k k+4
8−k
3/2 4−k k+2
k 4−k
k
4−k
k
6−k k+2
0
1
k
8−k k+4
2 6−k k+2 8−k k+4 k+2 6−k k+4
3
J
−3/2
Fig. 2. Structure of the module generated by the highest weights f j,s . The solid (resp. dotted) arrows denote the action of the raising (resp. lowering) operators, with coefficient proportional to the indicated function of k
observe that the highest weight states f j,s are mapped to each other by the action of the raising operators J 1 ,± 3 : 2
2
1 (3k + 6 j + 2s) f j+ 1 ,s+ 3 , 2 2 3√ 2 2 (3k + 6 j − 2s) f j+ 1 ,s− 3 . = 2 2 3
J 1 , 3 · f j,s = 2 2
J 1 ,− 3 · f j,s 2
2
(2.73)
Applying the lowering operators J− 1 ,± 3 gives a linear combination of the highest weight 2 2 state f j− 1 ,s± 3 and a descendant of f j+ 1 ,s± 3 : 2
2
2
2
(3 j − s)(6 + 6 j − 3k − 2s) 3k + 6 j + 2s J−1 · f j+ 1 ,s+ 3 , f j− 1 ,s+ 3 + √ 2 2 2 2 3(2 j + 1) 18 2(2 j + 1) √ (3 j +s)(6+6 j −3k +2s) 3k + 6 j − 2s f j−1 ,s−3 + 2 2 J−1 · f j+ 1 ,s− 3 . = 2 2 2 2 9(2 j +1) 3(2 j + 1) (2.74)
J− 1 , 3 · f j,s = 2 2
J− 1 ,−3 · f j,s 2
2
Using these equations, we may now study the structure of the module generated by f j,s and its descendants; it is pictured in Fig. 2. For generic k this module is irreducible and not manifestly unitarizable. When k is an integer, the situation is more interesting. For even integers k ≥ 2, there is an irreducible submodule generated by f j=(k−2)/2,s=0 . Its K -type decomposition coincides with that of the representation labeled by p = q = (k − 4)/2 in the parameterization of [6], namely, ∞
3m/2
m=k−2 s=−3m/2
[ j = m/2]s .
(2.75)
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We identify it as the quaternionic discrete series with index k (or limit discrete series for k = 2) [26]; in particular, it is unitarizable. It has no spherical vector unless k = 2. For (possibly negative) even integers k ≤ 2, we can similarly obtain the representation with p = q = −k/2, this time as a quotient instead of a submodule; namely, we divide out the submodule consisting of all states with 3 j − |s| < 3(k − 2)/2. The resulting representations are equivalent to the ones just discussed. In [26] one also finds quaternionic discrete series representations for odd k; one might wonder why we did not encounter those above. The answer is that strictly speaking they are not representations of G = SU (2, 1) but rather of its double cover. Correspondingly, they do not appear as subrepresentations of the principal series we considered here, but of a closely related “non-spherical principal series” representation of the double cover. 2.4.5. Matrix elements and Penrose transform. Suppose ρ is a spherical unitary representation of G, with spherical vector f K . Then ρ can be realized in the space of functions on K \G, by mapping any state f to the matrix element ϕ(e Q K ) = f |ρ(e−1 Q K ) f K .
(2.76)
ϕ is a function on K \G because the left action of k ∈ K on e Q K becomes a right action on e−1 Q K , hence a left action on f K , which is trivial because f K is spherical. Moreover, the ϕ so obtained obeys differential equations determined by ρ; in particular, it is an eigenfunction of the Laplacian on K \G with eigenvalue 2C2 (ρ). Even if ρ is not spherical one may still apply this construction replacing f K by any K -finite vector, and thus embed ρ into a space of sections of a homogeneous vector bundle over K \G, induced from the representation of K in which f K transforms. The sections so obtained have particularly good properties if f K is in the lowest K -type (e.g. for holomorphic discrete series representations they turn out to be holomorphic sections). We now apply this construction to the quaternionic discrete series representations, beginning with the special case k = 2. Using the Iwasawa decomposition (2.23) and the Baker-Campbell-Hausdorff formula e A e B = e B e A e[A,B] when [A, B] is central, ˜
˜
−(σ −ζ ζ )E e−1 · eζ Eq · e−ζ E p · e−U H QK = e
(2.77)
and ˜
˜
−(σ −ξ ζ +ξ ζ )∂α · e−ζ ∂ξ · e ρ(e−1 QK ) = e
−ζ˜ ∂ξ˜
·e
U (∂U +ξ ∂ξ +ξ˜ ∂ξ˜ +2α∂α +k)
.
(2.78)
Applying this ρ(e−1 Q K ) to the spherical vector f K = f 0,0 yields
(U, ζ, ζ˜ , σ ; ξ, ξ˜ , α) = e2U + (ξ˜ − ζ˜ )2 + (ξ − ζ )2 2 −1 . −2U 2 1 2 2 ˜ ˜ ˜ ˜ (σ +α+ ξ ζ −ξ ζ ) + (ξ − ζ ) +(ξ − ζ ) +e 4 (2.79) Since C2 (ρ) = −1, taking the inner product (2.76) between f and (2.79) yields an eigenfunction of the conformal Laplacian on K \G, Q K + 2 ϕ = 0. (2.80)
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Thus the transformed spherical vector (2.79) is the kernel for an integral operator which intertwines between the quaternionic discrete series and a subspace of the space of functions on K \G. In physics parlance, it is related to the boundary to bulk propagator for (2.80), upon taking the coordinates ξ , ξ˜ , α to be real — in that case they lie on a part of the boundary of Z, which may be identified with a boundary of K \G. (See [50] for an attempt to set up a bulk-boundary correspondence on this space.) Since K \G is a quaternionic-Kähler manifold there is an a priori different way to construct a function on K \G from f (ξ, ξ˜ , α) ∈ H 1 (Z, O(−2)), which is to apply the quaternionic Penrose transform [51,52]. Some details of this correspondence for quaternionic-Kähler spaces obtained by the c-map construction were worked out in [36], where a simple contour integral formula for the Penrose transform was established: Penrose[ f ] = ϕ(U, ζ, ζ˜ , σ ) = 2 e2U
dz f ξ(z), ξ˜ (z), α(z) . z
(2.81)
The resulting ϕ is annihilated by the conformal Laplacian Q K + 2. This agrees with (2.80) in the case of K \G, and in this context it can be checked directly by acting with Q K on the integrand of (2.81) and using (2.65):
2 z 1 2U z −1 f ξ(z), ξ˜ (z), α(z) = 0. (2.82) Q K + 2 − ∂z ∂z + z∂z + 2ie z∂α 2 2 Integration over z eliminates the total derivative, leading to (2.80). We now argue that these two constructions are in fact identical, i.e. f |ρ(e−1 Q K ) f K = Penrose[ f ]. According to our discussion of the inner product above, this is equivalent to dξ0 d ξ˜0 dα0 f = Penrose[ fˆ],
(2.83)
(2.84)
where = ρ(e−1 Q K ) f K ; then using (2.68) and the explicit form of K Z , (2.84) is equivalent to requiring that arises as the Penrose transform of 2 −1 1 (ξ, ξ˜ , α) = (α − α0 + ξ˜0 ξ − ξ˜ ξ0 )2 + (ξ − ξ0 )2 + (ξ˜ − ξ˜0 )2 . (2.85) 4 But this we can evaluate directly: the contour integral (2.81) defining Penrose[] has poles at z = z ± , where √ 2i 2eU (ζ − ξ0 ) − i(ζ˜ − ξ˜0 ) z+ = (2.86) 2i(σ − α0 + ξ˜0 ζ − ζ˜ ξ0 ) + (ζ − ξ0 )2 + (ζ˜ − ξ˜0 )2 − 2e2U and z − = −1/¯z + . The residue at z = z ± yields
= e2U + (ζ − ξ0 )2 + (ζ˜ − ξ˜0 )2 2 −1 (2.87) 1 (ζ − ξ0 )2 + (ζ − ξ0 )2 +e−2U (σ − σ0 + ξ˜0 ζ − ζ˜ ξ0 )2 + 4
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which indeed agrees with the formula (2.79) for ρ(e−1 Q K ) f K as desired. Similar considerations apply for other even values of k. In that case the Penrose transform gives a section of Sym k−2 (H ); for c-map spaces, in a natural trivialization of H , the formula is given in [36] as dz 1 m ϕm (U, ζ, ζ˜ , σ ) = 2 ekU z 2 f ξ(z), ξ˜ (z), α(z) , (2.88) z where m = −k + 2, . . . , k − 2 labels the 2k − 3 components of ϕ. This turns out to agree with the matrix element construction, where we now use the (2k −3)-dimensional lowest K -type of the quaternionic discrete series. Establishing a similar correspondence for k odd would require a better understanding of the branch cuts appearing in the contour integral (2.81). 2.5. Minimal representation. For any real Lie group G there is a notion of “minimal unitary representation” introduced in [53] and much studied thereafter (see e.g. [28] for a recent review.) For many G the minimal representation can be characterized as the unitary representation of smallest Gelfand-Kirillov dimension. For G = SU (2, 1) the situation is somewhat degenerate and there are many representations sharing this minimal dimension, including the holomorphic and antiholomorphic discrete series 16 . We focus here on the representation constructed explicitly in [25] by truncation of the minimal unitary realization of E 8(8) and whose structure parallels that of the minimal representation of higher rank groups. With a slight change of notation and normalization relative to this reference, the generators can be written as i 3 E p = i xu, F p = i∂u ∂x + u − ∂u u∂u , (2.89a) 2x i E q = x∂u , Fq = u∂x + i∂ 3 − iu∂u u , (2.89b) 2x u 2 i i i E = x 2 , F = ∂x2 + 2 u 2 − ∂u2 − 1 , (2.89c) 2 2 8x i 2 1 ∂u − u 2 , (2.89d) H = x∂x + , J = 2 2 acting on functions of two variables (u, x). Equivalently, by defining y = x 2 and x0 = xu, we reach a presentation analogous to the one used in [39] for split groups,17 i x3 i i x0 ∂02 + 2i y∂0 ∂ y + 02 + ∂0 , 2 2y 2 2 3x 1 x0 , E q = y∂0 , Fq = − y∂03 + 2x0 ∂ y + 0 ∂0 + 2 2y 2y
x02 i i 1 2 E = y, H = x0 ∂0 + 2y∂ y + , J = y∂0 − , 2 2 2 y E p = i x0 ,
Fp =
F = 2i y∂ y2 − i
(2.90a) (2.90b) (2.90c)
x04 3i x02 2 i 4 i x0 3i ∂0 + ∂ − y∂ + i∂ y + 2i x0 ∂ y ∂0 + . + 3 8y 2y 4y 0 8 0 8y (2.90d)
16 In fact, this is true for the entire quaternionic family SU (n, 2). 17 The split real form of SU (2, 1) is S L(3, R).
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Any minimal representation is annihilated by the Joseph ideal in the universal enveloping algebra of g. This means that the generators of the minimal representations satisfy certain quadratic identities, e.g. H 2 + 2(E F + F E) + J 2 + 1 = 0, E 2p + E q2 + 4J E = 0, 1 1 C2 (J ) + S 2 + = 0. 9 4
(2.91a) (2.91b) (2.91c)
These hold in addition to the Casimir identities 3 C2 = − , C3 = 0, 4
(2.92)
corresponding to the parameters ( p, q) = (− 21 , − 21 ) in the classification of [6]. The generators (2.89) or (2.90) are antihermitian with respect to the inner product f 1 | f 2 = y −1 d y d x0 f 1∗ (y, x0 ) f 2 (y, x0 ) = d x du f 1∗ (u, x) f 2 (u, x), (2.93) so this representation is unitary. For later reference, we note that the (non-normalizable) states
x02 1 1 2 y 2 exp ± (2.94) = x exp ± u 2y 2 are invariant under the nilpotent radical N generated by F, F p , Fq , and carry charges (3/2, ±i/2) under the Cartan generators (H, J ). 2.5.1. Induction from the maximal parabolic and deformation. The minimal representation, in fact a one-parameter deformation thereof, can be obtained by induction from the maximal parabolic subgroup Q ⊂ G C generated by {F, F p , Fq , H, J, E p + i E q }. For this purpose, decompose any element of G C into a product ⎛ ⎞ ⎛ ⎞ ∗00 1 z ia g = ⎝ ∗ ∗ ∗⎠ · ⎝ 1 0 ⎠ . (2.95) ∗∗∗ 1 Then induction from the character exp[τ (H + i J /3)] of Q gives the action of G on sections f (z, a) over Q\G C by first order differential operators, E p = ∂z + i z∂a , F p = −(ia + z 2 )∂z − az∂a + τ z,
(2.96a)
E q = −i∂z − z∂a , Fq = −(a + i z )∂z − iaz∂a + iτ z,
(2.96b)
2
E = ∂a , F = −az∂z − a ∂a + τ a, τ H = −z∂z − 2a∂a + τ, J = −i z∂z + i . 3 2
(2.96c) (2.96d)
Set ν = −(2τ + 3)/3. Passing from f (z, a) to f (u, x) by the intertwining operator 1 2 1 2 2 i 2 f (z, a) = dud x x 1+4ν e− 2 u +iux z+ 4 x z + 2 ax f (u, x) (2.97)
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the action of G on f (u, x) is given by a one-parameter deformation18 of the minimal representation (2.89), E (ν) p = E p,
E q(ν) = E q ,
(ν)
Ek
= Ek ,
(2.98a)
5 i H (ν) = H + ν, J (ν) = J − ν, 2 2 i 1 (ν) F p = F p + ν (3u + 5∂u ), Fq(ν) = Fq + ν (5u + 3∂u ), 2x 2x 2i i 2 (ν) 2 F = F + ν 2 3∂u + 10x∂x + 3(1 − u ) + 2 ν(ν − 1) 4x x
(2.98b) (2.98c) (2.98d)
with Casimirs 3 2 (ν − 1), C3 = i ν(1 − ν 2 ), 4
(2.99)
1 1 ( p, q) = − (1 − 3ν), − (1 + 3ν) 2 2
(2.100)
C2 = corresponding to
in the notation of [6]. The resulting representation is not obviously unitary for ν = 0, as the inducing character exp[τ (H + i J /3)] is in general not unitary. We shall however find evidence in the next section that it is unitarizable at ν = ±1. The annihilator of the ν-deformed minimal representation is deformed to H 2 + 2(E F + F E) + J 2 − 2iν J + (1 − ν 2 ) = 0, E 2p + E q2 + 4J E + 2iν E = 0, 1 1 C2 (J ) + S 2 + 3iν S + (1 − ν 2 ) = 0, 9 4 while the vectors invariant under the nilpotent radical N are deformed to
x02 x02 1−ν 1−4ν y 2 exp − , y 2 exp , 2y 2y
(2.101a) (2.101b) (2.101c)
(2.102)
carrying charges ( 23 (1 + ν), − 2i (1 + ν)) and ( 23 (1 − ν), 2i (1 − ν)) under (H, J ), respectively. 2.5.2. K -type decomposition. For completeness, we now review the K -type decomposition of the minimal representation, i.e. the decomposition under the maximal compact subgroup SU (2) × U (1), as discussed in [25]. 19 For this purpose, we change polarization to oscillator representation for both the u and x variable, i.e. define au , au† , N by 1 √ (u − ∂u ) = au† , 2
1 √ (u + ∂u ) = au , 2
Nu = au† au ,
(2.103)
18 The fact that the minimal representation is not isolated is a peculiarity of the A series. 19 We should note that the positive and negative grade generators we define in this paper are opposite to
those of [25].
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and similarly for x. The compact generator S is manifestly positive, 3 S= 2N x + 1 + x −2 Nu (Nu + 1) + 2Nu + 1 8
(2.104)
so the representation is of lowest-weight type. The positive grade generators in the 3-grading by S read √ 2 K + = au a x + (2.105a) (Nu + 1)au , x 1 1 (2.105b) L + = ax2 − 2 Nu (Nu + 1). 2 4x The only normalizable state annihilated by these two generators is ax† |0u,x , or, in the real polarization,
x02 1 2 1 2 1/2 f K = x exp − (u + x ) = y exp − y+ . (2.106) 2 2 y It is easy to check that this generator is a singlet of SU (2) J± ,J3 , but carries a non-zero charge −3i/2 under S. By acting with the raising operators K − and L − , we generate the complete K -type decomposition of the minimal representation, ∞ m m=0
2
− 3i2 (m+1)
,
(2.107)
where the term in the bracket is the spin of the SU (2) J representation, and the subscript indicates the S charge. This agrees with the K -type decomposition of the “ p-discrete” module at p = q = −1/2, as seen on Fig. 3, for k = 3. Let us now briefly discuss the ν-deformed minimal representation. It is easy to check that 1 f 0 = x 1−ν exp − (u 2 + x 2 ) (2.108) 2 is a singlet of SU (2) J , with charge − 3i2 (1 + ν) under S, and annihilated by the deformed generators K + , L + . Acting with deformed generators K − , L − produces a SU (2) J doublet with S3 = − 3i2 (2 + ν), 1 2 2 i 1 2 2 2−ν 2 exp − (u +x ) , f − 1 = √ (3+3ν −2y ) exp − (u +x ) . f 1 = −i u x 2 2 2 2 2 2 (2.109) For ν = −1, both of these states are annihilated by K + , L + , so generate a module corresponding to the semi-infinite line s = −3/2 j in the diagram on Fig. 3 for k = 4. Thus, the K -type decomposition of the ν-deformed minimal representation at ν = −1 has a ladder structure ∞ m m=1
2
− 3i2 m
.
(2.110)
Quasi-Conformal Actions, Quaternionic Discrete Series and Twistors: SU (2, 1) and G 2(2) y
y
k 2
y
k 3
4
4
4
2
2
2
j 1
2
1
2
j
4
3
1
-2
-2
-2
-4
-4
-4
y
k 4
j
4
3
195
y
k 5
k 6
2
y
4
4
3
k 8
7.5 4 5
2
2 2.5
j 1
2
j
j
4
3
1
2
3
4
2
5
4
6
8
-2.5 -2
-2
-5 -4 -7.5
-4
Fig. 3. K -type decomposition of the discrete series representations of SU (2, 1) in the ( j, y = 2/3s) plane, for low values of k = ( p − 4)/2 = (q − 4)/2. The quaternionic discrete series corresponds to the “ p + q discrete branch” in the terminology of [6]. The “ p-discrete” branch for k = 3 corresponds to the minimal representation, see Sect. 2.5.2. The second “ p-discrete branch” for k = 4, starting at ( j, s) = (1/2, ±3/2), corresponds to the deformed minimal representation at ν = ±1. For k = 5 and higher odd values of k, there are no “ p-discrete” or “q-discrete” branches, as the lowering operators J 1 ,± 3 map into forbidden regions, as 2
illustrated by the open links
2
As usual, we can use the lowest K -type to embed the minimal representation into the space of sections of a vector bundle on K \G, in this case a line bundle with −3/2 units of charge under S. For this purpose, as in (2.76), we let the coset representative e Q K act on the lowest K -type, and construct the intertwiner ˜
˜
(U, ζ, ζ˜ , σ ; u, x) = eU (x∂x + 2 ) · e−i ζ ux · eζ x∂u · e− 2 (σ −ζ ζ )x 1 1 3U − u 2 − (e2U + iσ )x 2 − = x exp 2 2 2 1
i
2
· f SU (2) (2.111) 1 ˜ x(2u + xζ )(ζ + i ζ ) . 2
The overlap f =
d x du f ∗ (u, x) (U, ζ, ζ˜ , σ ; u, x)
(2.112)
is then an eigenmode of the Laplacian twisted by S, Q K
1 2U 15 f = 0. − e ∂σ − 2 16
(2.113)
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2.5.3. As a submodule of the principal series representation. In this section, we investigate to what extent the minimal representation (or its ν-deformation) may be viewed as a submodule of the principal series representation. To that purpose, we first observe that the Casimirs (2.99) and (2.66) agree for k = 1, 3(ν = 0) or k = 0, 4 (ν = ±1). Second, Eq. (2.101b) in the annihilator of the deformed minimal representation, when expressed in terms of the generators of the quasi-conformal action, becomes C0 ≡ (∂ζ + ζ˜ ∂σ )2 + (∂ζ˜ − ζ ∂σ )2 − 2iν∂σ = 0.
(2.114)
The physicist will recognize C0 as the Hamiltonian of a charged particle on the plane (ζ, ζ˜ ), with a constant magnetic field proportional to i∂σ . The spectrum of C0 (for the usual L 2 norm on the plane) consists of the usual infinitely degenerate Landau levels. Defining ∇ = ∂ζ + ζ˜ ∂σ + i(∂ζ˜ − ζ ∂σ ),
(2.115a)
∇¯ = ∂ζ + ζ˜ ∂σ − i(∂ζ˜ − ζ ∂σ ),
(2.115b)
one may rewrite (2.114) as C0 = ∇ ∇¯ − 2i(ν + 1)∂σ = 0.
(2.116)
¯ This constraint The lowest Landau level corresponds to functions annihilated by ∇. commutes with the action of G for k = 0, ν = −1: this is evidently so for the positive root generators E p , E q , E (the former two being the generators of magnetic translations on the plane), and it suffices to check invariance under the action of the lowest root generator F,
i F, ∇¯ = − [3(ζ 2 + ζ˜ 2 ) − 2iσ ]∇¯ − k(ζ˜ + iζ ), 2
(2.117)
which indeed vanishes on the subspace annihilated by ∇¯ when k = 0. Solutions to ∇¯ = 0 are of the form 20 i 2 2 ˜ ˜ ˜ f (ζ, ζ , σ ) = g z ≡ ζ + iζ, a ≡ −σ + (ζ + ζ ) . (2.118) 2 It is straightforward to check that the quasi-conformal action reduces on this invariant subspace to the action (2.96) induced from the maximal parabolic at ν = −1. As far as the constraint C0 = 0 itself is concerned, one may check that it is invariant under the action of G at the values k = 1, ν = 0 appropriate for the undeformed minimal representation, as well as at k = 0, ν = −1 which we discussed above. Indeed, one may rewrite [F, C0 ] = −2σ C0 + 2(1 − k) (ζ 2 + ζ˜ 2 )∂σ − J + 2iν(ζ ∂ζ + ζ˜ ∂ζ˜ + k), (2.119) which is proportional to C0 for k = 1, ν = 0. From the point of view of the magnetic problem, this corresponds to non-normalizable states with energy below that of the 20 After Fourier transforming over σ , one recovers the usual form f = g (ζ˜ + iζ ) e− 2 K (ζ 2 +ζ˜ 2 )−i K σ of K 1
the lowest Landau level wave functions.
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lowest Landau level. In order to find the eigenmodes explicitly, it is useful to Fourier transform over ζ˜ , σ , ˜ f (ζ, ζ , σ ) = dp d K exp −i K σ − i p ζ˜ g(ζ, p, K ), (2.120) and redefine (P − 2K ζ )2 g(ζ, p, K ) = exp − h(ζ, P, K ), 4K
(2.121)
where P = p + K ζ . The constraint C0 = 0 becomes now an ordinary differential equation on h, ∂ζ2 + 2(P − 2K ζ )∂ζ − 2K (ν + 1) h(ζ, P, K ) ≡ 0. (2.122) For ν = −1, the solutions are
P2 P − 2K ζ h(ζ, P, K ) = h 1 (P, K ) + h 2 (P, K ) K −1/2 e− 2K erfi − √ . (2.123) 2K
Only the ζ -independent part h 1 (P, K ) obeys also (2.101a). Changing variables again by setting K = x 2 /2 = y/2 and P = −xu = −x0 , it is easy to check that the quasiconformal action on f (ζ, ζ˜ , σ ) at k = 0 gives precisely the deformed minimal representation (2.98) acting on h 1 (P, K ) at ν = −1. We conclude that the deformed minimal representation at ν = −1 can be embedded inside the principal series representation at k = 0 by
( p − K ζ )2 h( p + K ζ, K ). (2.124) f (ζ, ζ˜ , σ ) = dp d K exp −i K σ − i p ζ˜ − 4K The formula (2.124) can also be viewed as the matrix element ˜
f (ζ, ζ˜ , σ ) = f P | e−σ E−ζ Eq +ζ E p | h,
(2.125)
where f P is the P-covariant vector in the deformed minimal representation (2.102). For ν = 1, similar arguments show that the deformed minimal representation can be embedded inside the principal series representation at k = 0 via
( p − K ζ )2 h( p + K ζ, K ). (2.126) f (ζ, ζ˜ , σ ) = dp d K exp −i K σ − i p ζ˜ + 4K For other values of ν, the solution of (2.122) involves Hermite and hypergeometric functions,
P − 2K ζ 1 + ν 1 (P − 2K ζ )2 h = h 1 (P, K ) H− ν+1 − √ . + h 2 (P, K ) 1F1 , ; 2 4 2 2K 2K (2.127) It is worth noting that the formula (2.124) admits a simple generalization to all quaternionic groups, as we shall see for G 2(2) in (3.137) below.
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2.6. The minimal representation as a quantized quasi-conformal action.. We now explain how the minimal representation can be viewed as the quantization of the quasi-conformal action of G, or equivalently how the quaternionic discrete series arises as a semi-classical limit of the minimal representation. 2.6.1. Lifting the quasi-conformal action to the hyperkähler cone. As a first step, it is useful to “deprojectivize” the quasi-conformal action, i.e. lift it to an action on the hyperkähler cone S. For this purpose, we introduce an extra variable t and interpret the k-dependent quasi-conformal action as an action of functions of four variables fˆ(t, ξ, ξ˜ , α) = e−kt f (ξ, ξ˜ , α). We then implement the change of variables found in [36], v = e2t , v 0 = ξ e2t , w0 =
i 1 ξ˜ , w = (α + ξ ξ˜ ). 2 4i
(2.128)
The coordinates v , v 0 , w , w0 are complex coordinates on the Swann space S, such that the holomorphic symplectic form takes the Darboux form = dw ∧ dv + dw0 ∧ dv 0 .
(2.129)
The quasi-conformal action on O(−k) over Z now corresponds to the holomorphic action of G on S, restricted to the subspace of homogeneous functions of degree −k under the rescaling v I → µ2 v I , w I → w I .
(2.130)
Being tri-holomorphic isometries, the holomorphic vector fields generating the action of G preserve the holomorphic symplectic form . They can be represented by holomorphic moment maps X , such that the contraction ι X = d X : ! ! i i i (v 0 )2 E p = v 0 , E q = v w0 , E = − v , H = −2v w −v 0 w0 , J = −w02 − , 2 4 4 (v )2 ! ! ! ! !
i (v 0 )3 3(v 0 )2 w0 2 0 3 + 4(w ) v + 16v w w + 2v 0 w , Fp = 0 0 , F q = −2v (w0 ) + 2 4 (v ) 2v ! ! i 4 4 0 3 0 2 0 4 2 16 w (v . − 4w ) − 64v w w (v ) − 24(v w v ) + (v ) F= 0 0 0 16(v )3 ! (2.131) Returning to the variables t, ξ, ξ˜ , α, the holomorphic moment maps become, i i i i E p = e2t ξ, E q = e2t ξ˜ , E = − e2t , H = e2t α, 2 2 4 2 ! ! ! ! i 2t 2 i 2t 2 2 ξ(ξ + ξ˜ ) + 2α ξ˜ , F q = e ξ˜ (ξ + ξ˜ 2 ) − 2αξ , Fp = e 4 4 ! ! i 2t 2 i 2t 2 2 2 2 4α + (ξ + ξ˜ ) , J = e (ξ + ξ˜ 2 ). F= e 16 ! ! 4
(2.132)
Arranging these holomorphic moment maps into an element Q of g∗ , one may check that Q 2 = 0 in our matrix representation, i.e. that Q is valued in the minimal co-adjoint orbit.
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In particular, the holomorphic moment maps satisfy classical versions of the identities (2.91), H 2 + 4E F + J 2 = 0, E 2p + E q2 + 4J E = 0, ! !! ! ! ! !! 1 C 2 = C 3 = 0, C2 (J ) + S 2 = 0. ! ! 9! ! The holomorphic moment maps of the compact generators are given by
3i 2t 1 1 + ξ 2 + ξ˜ 2 + α 2 + (ξ 2 + ξ˜ )2 , e S= 4 ! 16
i 2t 1 2 3 2 2 2 2 ˜ ˜ 1 − 3ξ − 3ξ + α + (ξ + ξ ) , e J3 = 16 4 ! i 2t ξ ∓ i ξ˜ ξ 2 + ξ˜ 2 ± 2iα − 2 , J± = √ e ! 8 2 1 2t 2 J ± 1 ,± 3 = e (ξ + ξ˜ 2 )2 + 4α(α ± 2i) − 4 , 16 ! 2 2 i ξ 2 + ξ˜ 2 ± 2iα + 2 . J ± 1 ,∓ 3 = − √ e2t ξ ∓ i ξ˜ ! 2 2 8 2
(2.133a) (2.133b)
(2.134a) (2.134b) (2.134c) (2.134d) (2.134e)
It is interesting to note that the K -type (2.72) may be rewritten in terms of these moment maps, up to an overall numerical factor, as
1
1 J 1 , 3 j+ 3 s J 1 ,− 3 j− 3 s 1 2 2 2 2 ! f j,s = ! S− 2 k . (2.135) S S ! ! ! The covariance of (2.135) under K is then easy to see, using the identities (2.133) obeyed by the holomorphic moment maps, and the fact that G acts by Poisson brackets, X f = {X , f }. ! Moreover, the vanishing locus of the holomorphic moment maps associated to the compact generators S, J3 , J± consists of two branches ξ = ±i ξ˜ , α = ±i or equivalently, in terms of the variables on the hyperkähler cone,
(v 0 )2 1 v0 w = ± 1 − 2 , w0 = ± . 4 (v ) 2v
(2.136)
(2.137)
These relations define two lagrangian cones C± , with generating functions S± : they may be rewritten as
1 (v 0 )2 . (2.138) v + w = ∂v S± , w0 = ∂v 0 S± , S± (v , v 0 ) = ± 4 v By construction, C± are invariant under the holomorphic action of K , since the Poisson brackets with the constraints vanish on the constraint locus. As we shall see momentarily, S− describes the semi-classical limit of the lowest K -type (2.106) of the minimal representation.
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2.6.2. The classical limit of the minimal representation. We now return to the presentation (2.90) of the minimal representation acting on functions of y, x0 . The form (2.106) of the spherical vector suggests that a semi-classical limit exists as y, x0 are scaled simultaneously to infinity, if one restricts to wave functions of the form ˆ xˆ0 ) + O(1), f (y, x0 ) = exp [S(y, x0 )] , S(y, x0 ) = y S(
(2.139)
where xˆ0 = x0 /y is kept fixed in the limit y → ∞. Indeed, it is easy to check that, to leading order in this limit, the action of the infinitesimal generators on f produces ˆ ∂ ˆ S, ˆ xˆ 0 f + O(y), X · f = y X S, 0 !
(2.140)
for some (so far unspecified) function X . Changing variables to ! ˆ p0 = ∂x0 S(y, x0 ) = ∂xˆ0 S,
ˆ p y = ∂ y S(y, x0 ) = Sˆ − xˆ0 ∂xˆ0 S,
(2.141)
identifies the function X (y, x0 , p y , p0 ) as the leading differential symbol of the differential operator X in the!semi-classical limit (2.139). Further setting v = −2y, v 0 = 2x0 , w =
1 1 p y , w0 = − p 0 , 2 2
(2.142)
identifies X as the holomorphic moment map (2.131) associated to the tri-holomorphic action of G!on its hyperkähler cone S. Thus, we conclude that the minimal representation can be viewed as the quantization of the holomorphic symplectic manifold S. Furthermore, the semi-classical limit of the spherical vector (2.106) is given by the generating function S− of the Lagrangian cone S− defined in (2.138). One could also have given a real version of this construction, lifting the action of G on P\G to a real symplectic manifold by adding a single real coordinate. In that case we would say that the minimal representation arises by quantizing the real symplectic structure. Indeed, the real manifold so obtained is at least locally isomorphic to the minimal coadjoint orbit of G, so this makes contact with one of the standard ways of thinking about the minimal representation.
3. G 2(2) In this section, we describe the geometry of the quaternionic-Kähler space S O(4)\G 2(2) , and various associated unitary representations of G = G 2(2) . We will be somewhat briefer in this section since many of the constructions are parallel to ones we described for G = SU (2, 1) above. 3.1. Some group theory. It is convenient to represent the Lie algebra g of G = G 2(2) by the 7-dimensional matrix representation described in [54] (after some relabelings and change of normalization)
Quasi-Conformal Actions, Quaternionic Discrete Series and Twistors: SU (2, 1) and G 2(2)
⎛ Y0 Y−
−Y + 0
⎜ ⎜ ⎜ ⎜ ⎜ Y− ⎜ "0 ⎜ 2 ⎜− 2 E ⎜ " 3 q 1 √3 E p 1 ⎜ ⎜ 2F 2 ⎜ 3 p1 √3 F q1 ⎜ √ ⎜ ⎜ − 2E q0 √23 E q1 ⎝√ 2F p0 − √2 F p1 3
0 −Y +
" − 23 F q1 √2 F p 1 3 √ 2F p0
"
2 3 E p1 2 √ Eq 1 3
√ √ − 2F q0 2E p0 √2 F q √2 E p 1 − "3 1 "3 − 23 F p1 − 23 E q1
√ −Y 0 2E q0 √ 2E p0 H + 21 Y 0 −E − √1 Y + 0 2 √ 1 1 2F −F −H + 2 Y 0 0 −√ Y+ 2 " q0 − 23 E p1 √1 Y− 0 H − 21 Y 0 −E 2 " 2 √1 Y − − 3 F q1 0 −F − 21 Y 0 − H
201
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟≡ X i X i , ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
2
(3.1) where, as in (2.2), X i are real coordinates dual to the generators, to be represented by anti-hermitian operators in a given unitary representation. These matrices preserve the signature (4+ , 3− ) metric ηi j d xi d x j = −2d x1 d x3 + d x22 + 2d x4 d x7 − 2d x5 d x6 and the three-form d x123 − d x247 + d x256 +
√ √ 2d x167 + 2d x345 ,
(3.2)
(3.3)
thus providing an embedding of G 2(2) inside S O(3, 4). The generators satisfy the commutation relations consistent with the matrix representation (3.1), in particular the Heisenberg algebra E p I , E q J = −2δ JI E, (3.4) the S L(2, R) algebras (2.4) and [Y0 , Y± ] = ±Y± , [Y− , Y+ ] = Y0 .
(3.5)
The quadruplet (E p0 , E p1 , E q1 , E q0 ) transforms as a spin-3/2 representation under this Sl(2). The quadratic Casimir is 1 1 2 H + 2E F + 2F E + (Y02 − Y+ Y− − Y− Y+ ) C2 = 4 3 (3.6) 1 # + (E p I F p I + F p I E p I + E q I Fq I + Fq I E q I ), 4 I =0,1
normalized so that C2 (adjoint) = 4. There is also a degree 6 Casimir, corresponding to the trace of the sixth power of the matrix (3.1), which we shall not attempt to write. This basis is adapted to the maximal subgroup S L(2, R)short × S L(2, R)long , where the first factor is generated by {H, E, F} while the second is generated by {Y− , Y0 , Y+ }. The Cartan generators H, Y0 are non-compact, with spectrum Spec(Y0 ) = {0, ±1/2, ±1, ±3/2}, Spec(H ) = {0, ±1, ±2},
(3.7)
The non-compact generator H gives rise to the “real non-compact 5-grading” F|−2 ⊕ {F p I , Fq I }|−1 ⊕ {H, Y0 , Y± }|0 ⊕ {E p I , E q I }|1 ⊕ E|2 .
(3.8)
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Define a parabolic subgroup P = L N with Levi L = R × S L(2, R) generated by {H, Y0 , Y+ , Y− } and unipotent radical N generated by {Fq0 , Fq1 , F p0 , F p1 , F}, corresponding to the spaces with zero and negative grade in the decomposition (3.8). We call P the Heisenberg parabolic subgroup. Now we introduce a different basis adapted to the maximal compact subgroup SU (2) × SU (2). We first go to a compact basis for the S L(2, R) × S L(2, R) group: 1 L ± = − √ (E + F ± i H ) , 2 2 1 R0 = √ (Y+ + Y− ), 2 such that
1 (F − E), 2 √ 1 Y+ − Y− ∓ i 2Y0 , R± = 2
L0 =
L 0 , L ± = ±i L ± , R0 , R± = ±i R± ,
(3.9a) (3.9b)
[L + , L − ] = −i L 0 ,
(3.10a)
[R+ , R− ] = −i R0 .
(3.10b)
Moreover, we define the eigenmodes √ √ √ 2 K 1,3 = −(E p0 − i E q0 )−i 3(E p1 − i E q1 )+(F p0 −i Fq0 )+i 3(F p1 − i Fq1 ) , 2 2 4 √ √ 2 √ 3(E p0 + i E q0 ) + i(E p1 + i E q1 ) + 3(F p0 + i Fq0 ) + i(F p1 + i Fq1 ) , K 1,1 = 2 2 4 √ √ 2 √ 3(E p0 −i E q0 )−i(E p1 − i E q1 )− 3(F p0 −i Fq0 )+i(F p1 − i Fq1 ) , K 1 ,− 1 = 2 2 4 √ √ √ 2 −(E p0 +i E q0 )+i 3(E p1 +i E q1 )−(F p0 − i Fq0 )+i 3(F p1 + i Fq1 ) , K 1 ,− 3 = 2 2 4 (3.11) where the eigenvalues under (−i L 0 , −i R0 ) are indicated in subscript. Note that the hermiticity conditions are now L †± = −L ∓ ,
† R± = −R∓ ,
K −l0 ,−r0 = −K l†0 ,r0 .
(3.12)
The roots K ± 1 ,± 3 and K ± 1 ,∓ 1 are compact, as indicated on the diagram. The compact 2 2 2 2 Cartan generator L 0 gives rise to the “non-compact holomorphic 5-grading” L − |−2i ⊕ {K − 1 ,± 1 , K − 1 ,± 3 }|−i ⊕ {L 0 , R0 , R± }|0 ⊕ {K 1 ,± 1 , K 1 ,± 3 }|i ⊕ L + |2i . 2
2
2
2
2
2
2
2
(3.13) Now, we perform a π/3 rotation of the root diagram, and define J3 =
1 1 1 (L 0 + R0 ) = (F − E) + √ (Y+ + Y− ) 2 4 2 2
(3.14a)
S3 =
1 1 3 (3L 0 − R0 ) = (F − E) − √ (Y+ + Y− ) 2 4 2 2
(3.14b)
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203
as the new Cartan algebra. The new eigenmodes are now J− =
1 K 1 3, 2 − 2 ,− 2
J 1 ,− 3 = K − 1 , 3 , 2
2
1 3 3 K 1 , 3 , S− = K − 1 , 1 , S+ = K 1 ,− 1 , 2 2 2 2 2 2 2 2 2 $ √ 2 R+ , J 1 , 1 = K 1 , 1 , (3.15) = 2 2L + J 1 ,− 1 = 2 2 2 2 2 2 2 3
J+ = J1,3
2 2
2 2
together with their hermitian conjugates. They satisfy the SU (2) × SU (2) algebra,
J3 , J± = ±i J± , S3 , S± = ±i S± ,
[J+ , J− ] = 2i J3 ,
(3.16a)
[S+ , S− ] = 2i S3 .
(3.16b)
The subscript on J now denotes the eigenvalues under (−i J3 , −i S3 ). The matrix representation adapted to this√ “compact basis” is obtained from (3.1) by a Cayley rotation using the matrix C = e matrix representation ⎛
1 2 i(J 3
+ S3 )
⎜ ⎜ S− ⎜ " ⎜ ⎜ −2i 13 J 21 , 21 ⎜ " ⎜ ⎜−2i 23 J − 21 ,− 21 ⎜ ⎜ −2 J 1 3 ⎜ − 2 ,− 2 ⎜ ⎜ −J − ⎝ 0
π 2 4 (R+ +R− )
−S + 1 2 i(J 3
−2i
− S3 )
0 J−
"
1 3
−2J
−2J − 1 , 3 2 2 " −2 23 J − 1 , 1 2 2 " −2 13 J − 1 ,− 1 2
"
. The generators in the compact basis then have the
J 1 2
1 1 2 ,− 2
2i −2
,− 23
"
2 3
J
2 3
J
√
−i 2S −
−2
1 3
1 2
,− 21
⎞
−2 J 1 , 3 2 2 " −2 13 J 1 , 1 2
2
0 √
0
i 2S +
√
0 "
, 21
√ −i 2S +
i S3
2
1 2
J − 1 ,− 1 2
2
−2J − 1 ,− 3 2
2
i 2S − −i S 3 " 2 −2 3 J − 1 , 1 −2 J − 1 , 3 2 2 2 2 " " −2i 23 J − 1 ,− 1 −2i 13 J − 1 , 1 2
2
2
2
J+
0
0
−J +
−S −
− 21 i(J 3 + S 3 )
⎟ ⎟ ⎟ " ⎟ 1 ⎟ −2 3 J 1 , 1 −2 J 1 , 3 2 2 2 2 ⎟ " " ⎟ −2 23 J 1 ,− 1 2i 23 J 1 , 1 ⎟. 2 2 2 2 ⎟ " ⎟ −2 J 1 ,− 3 −2i 13 J 1 ,− 1 ⎟ 2 2 2 2 ⎟ ⎟ − 21 i(J 3 − S 3 ) S+ ⎠ (3.17)
The generator J3 gives rise to the “compact 5-grading” J− |−2i ⊕ {T− , R− , U− , L − }|−i ⊕ {J3 , S3 , S+ , S− }|0 ⊕ {T+ , R+ , U+ , L + }|i ⊕ J+ |2i , (3.18) where we denoted T+ = J 1 ,− 3 , U+ = J 1 , 1 . The various gradings we have described 2 2 2 2 may be seen in Fig. 4. We now consider the S L(3, R) and SU (2, 1) subalgebras of G 2(2) . The S L(3, R) subalgebra is generated by the long roots in the non-compact basis. Its quadratic Casimir reads 1 1 1 1 C2 [S L(3, R)] = H 2 + Y02 + (E p0 F p0 + F p0 E p0 + E q0 Fq0 + Fq0 E q0 )+ (E F + F E). 4 3 4 2 (3.19)
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Y0 Fq0
R0
+3/2
Ep0
Tp
Ep1
Sm
Yp
Rp
Fq1 −2
−1
+1
H
+2
F
E Fp1
Up
−i
−i/2
+i/2
Eq1
L0
+i
Lm
Lp Um
Ym Fp0
Jp
+3/2i
Sp Rm
−3/2
Eq0
Jm
−3i/2
Tm
Fig. 4. Root diagram of G 2(2) with respect to the split Cartan torus H, Y0 (left) and the compact Cartan torus L 0 , R0 (right). The compact (resp. non-compact) roots are indicated by a white (resp. black) dot. The long roots generate S L(3, R) (left) and SU (2, 1) (right) subgroups, respectively
The SU (2, 1) subalgebra is generated by the long roots in the compact basis. Its quadratic Casimir reads 1 C2 [SU (2, 1)] = − S32 − (J32 + J− J+ + J+ J− ) 3 (3.20) 1 1 + J 1 , 3 J− 1 ,− 3 + J− 1 ,− 3 J 1 , 3 + J− 1 , 3 J 1 ,− 3 + J 1 ,− 3 J− 1 , 3 . 2 2 2 2 2 2 2 2 2 2 2 2 8 2 2 2 2 8 3.2. Quaternionic symmetric space. The long SU (2) J± ,J3 endows K \G = (SU (2) × SU (2))\G 2(2) with a quaternionic-Kähler geometry. In order to describe its geometry, we perform the Iwasawa decomposition g = k · e Q K (slightly adapted from [55]) e Q K = τ2−Y0 · e
√
2τ1 Y+
· e−U H · e
−ζ 0 E q0 +ζ˜0 E p0
·e
√ √ − 3ζ 1 E q1 + 33 ζ˜1 E p1
· eσ E , (3.21)
where k is an element of the maximal compact subgroup. This decomposition defines coordinates (τ1 , τ2 , ζ 0 , ζ 1 , ζ˜0 , ζ˜1 , U, σ ) on K \G, where τ = τ1 + iτ2 is an element of the upper half-plane. The invariant g-valued one-form θ = de Q K · e−1 Q K may be expanded on the compact basis, leading to the quaternionic viel-bein ⎞ ⎛ ⎛ ⎞ J−1,3 J 1,3 u¯ v¯ 2 2 2 2 ⎜ J 1 1 J1 1 ⎟ ⎜ e¯ E¯ ⎟ ⎜ −2,2 2,2 ⎟ , (3.22) ⎟ = −⎝ ⎜J J E e⎠ ⎝ − 21 ,− 12 21 ,− 21 ⎠ v u J − 1 ,− 3 J 1 ,− 3 2
2
2
2
and the SU (2) × SU (2) spin connection ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ ⎞ √ ⎛ u¯ √ E¯ J− S − √ 3⎝ ⎝ J 3 ⎠ = − 1 ⎝ i (v − v) ¯ + 43 (e + e) ¯ ⎠ , ⎝ S3 ⎠ = −i 43 (v− v)+ ¯ 41 (e + e) ¯ ⎠, 4 2 2 J+ S u E + (3.23)
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where e−U u = √ 3/2 d ζ˜0 + τ d ζ˜1 + 3τ 2 dζ 1 − τ 3 dζ 0 , 2 2 τ2 i v = dU − e−2U (dσ + ζ˜0 dζ 0 + ζ˜1 dζ 1 − ζ 0 d ζ˜0 − ζ 1 d ζ˜1 ), √ 2 3 e=− dτ, 2τ2 e−U E = − √ 3/2 3d ζ˜0 + d ζ˜1 (τ + 2τ¯ ) + 3τ¯ (2τ + τ¯ )dζ 1 − 3τ τ¯ 2 dζ 0 . 2 6 τ2
(3.24a) (3.24b) (3.24c) (3.24d)
This form of the vielbein agrees with that of the c-map space with prepotential F = −(X 1 )3 / X 0 [43], as expected. The metric is then ds 2 = 2 u u¯ + v v¯ + e e¯ + E E¯ . (3.25) The right action of G on K \G is given by the vector fields E = ∂σ , = ∂ζ˜0 − ζ0 ∂σ , E q0 = −∂ζ0 − ζ˜0 ∂σ ,
E p0 √ = 3(∂ζ˜1 − ζ1 ∂σ ),
1 E q1 = √ (−∂ζ 1 − ζ˜1 ∂σ ), 3 0 1 H = −∂U − 2σ ∂σ − ζ ∂ζ 0 − ζ ∂ζ 1 − ζ˜0 ∂ζ˜0 − ζ˜1 ∂ζ˜1 ,
E p1
(3.26a) (3.26b) (3.26c) (3.26d)
1 Y+ = √ (∂τ1 + ζ0 ∂ζ 1 − 6ζ 1 ∂ζ˜1 − ζ˜1 ∂ζ˜0 ), 2
(3.26e)
1 Y− = √ (6τ1 τ2 ∂τ2 + 3(τ12 − τ22 )∂τ1 + 9ζ˜0 ∂ζ˜1 − 9ζ 1 ∂ζ 0 + 2ζ˜1 ∂ζ 1 ). 3 2
(3.26f)
The other negative roots are too bulky to be displayed. 3.3. Twistor space. The twistor space Z = (SU (2) S± ,S3 × U (1) J3 )\G 2(2)
(3.27)
can be parameterized by the coset representative eZ = e−¯z J+ (1 + z z¯ )−i J3 e−z J− e Q K .
(3.28)
As in Sect. 2.3.1, we can construct complex coordinates on Z using the Borel embedding: ⎛1 ⎞ ⎛∗ ∗ ⎞ ⎛1 ∗ ∗ ∗ ∗ ∗⎞ ∗ ∗
1
CeZ C
−1
⎜∗ ∗ ⎜ = ⎜∗ ∗ ⎝∗ ∗
1 1
1 ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ 1
⎟ ⎜ ⎟ ⎜ ⎟·⎜ ⎠ ⎝
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
⎟ ⎜ ⎟ ⎜ ⎟·⎜ ⎠ ⎝
∗ ∗ ∗ ∗
1 ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗⎟ ⎟ 1 ∗ ∗⎟. 1 ∗ ∗⎠ 1 1
(3.29)
The entries of the upper triangular matrix then provide a complex coordinate system such that the “compact Heisenberg algebra” J 1 ,± 1 /± 3 , J+ acts in a simple way, essentially 2
2
2
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by shifts. The Kähler potential in these coordinates (analogous to the bounded domain coordinates of S L(2, R)/U (1)) appears to be complicated; we do not consider them further here. To get a simpler form for the Kähler potential we construct complex coordinates (ξ 0 , ξ 1 , ξ˜0 , ξ˜1 , α) adapted to the “real” Heisenberg algebra, related to the coordinates on the base and the stereographic coordinate on the fiber by i eU i eU (3.30) ξ 0 = ζ 0 + √ 3/2 z + z −1 , ξ 1 = ζ 1 + √ 3/2 τ¯ z + τ z −1 , 2 2 τ2 2 2 τ2 i eU i eU ξ˜1 = ζ˜1 − √ 3/2 3τ¯ 2 z + 3τ 2 z −1 , ξ˜0 = ζ˜0 + √ 3/2 τ¯ 3 z + τ 3 z −1 , 2 2 τ2 2 2 τ2 i eU α = σ + √ 3/2 τ¯ 3 ζ 0 − 3τ¯ 2 ζ 1 − τ¯ ζ˜1 − ζ˜0 z + τ 3 ζ 0 −3τ 2 ζ 1 −τ ζ˜1 − ζ˜0 z −1 . 2 2 τ2 These coordinates are an example of the “canonical” coordinates of quasi-conformal geometries defined by Jordan algebras [25,31]. Such coordinates in fact exist for all c-map spaces [36], and (3.30) agrees with the “twistor map” derived in that context in [36]. To compare the two, recall that the Kähler potential on the special Kähler base (the upper half-plane in this case) is e−K = 8τ23 . The Kähler potential on Z is 1 1 K Z = log N4 = log I4 (ξ I − ξ¯ I , ξ˜ I − ξ¯˜ I ) + (α − α¯ + ξ I ξ¯˜ I − ξ¯ I ξ˜ I )2 , (3.31) 2 2 where 1 4 I4 (ξ, ξ˜ ) = (ξ 0 )2 ξ˜02 + 4(ξ 1 )3 ξ˜0 + 2ξ 0 ξ˜0 ξ 1 ξ˜1 − (ξ 1 )2 ξ˜12 − ξ˜13 ξ 0 3 27
(3.32)
is the quartic invariant of S L(2, R) in its spin-3/2 representation, and N4 is the quartic distance function of quasi-conformal geometry, which defines the “quartic light-cone”. In parallel to the discussion of Sect. 2.3.5, one can also define the homogeneous line bundles O(m) over Z, and the total space of O(−2) gives the Swann space S. 3.4. Quasi-conformal representations. We construct a principal series representation of G by induction from the parabolic P, with the character χk = e−k H /2 of P. The infinitesimal action of G in this representation is determined as in Sect. 2.4.1, this time using the decomposition of G as −ζ 0 E q0 +ζ˜0 E p0
g = p·e ⎛
·e
√ √ − 3ζ 1 E q1 + 33 ζ˜1 E p1
· eσ E ,
⎞ √ 1 0 0 0 0 2ζ˜0 ⎜ ⎟ 1 0 0 0 − 23 ζ˜1 ⎜ 0 ⎟ √ 0 √ ⎜ ⎟ 1 ⎜√ 0 ⎟ 0 1 0 − 2ζ 0 2ζ √ ⎜ ⎟ ˜0 1 σ − ζ 0 ζ˜0 − 1 ζ 1 ζ˜1 0 2ζ˜0 ζ 1 − 2 ζ˜ 2 ⎟ , = p · ⎜ 2ζ 1 2 ζ˜1 2 ζ 3 3 9 1 ⎟ ⎜ ⎜ 0 ⎟ 0 0 0 1 0 0 √ ⎜√ ⎟ ⎝ 2ζ 0 −2ζ 1 − 2 ζ˜1 0 2ζ 1 2 + 2 ζ 0 ζ˜1 1 σ + ζ 0 ζ˜0 + 1 ζ 1 ζ˜1 ⎠ 3 3 3 0 0 0 0 0 0 1 √ 2˜ 3 ζ1 −2ζ 1
(3.33)
(3.34)
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where p ∈ P. The generators act by H QC = −2σ ∂σ − ζ 0 ∂ζ 0 − ζ 1 ∂ζ 1 − ζ˜0 ∂ζ˜0 − ζ˜1 ∂ζ˜1 − k, E QC = −∂σ , √ QC QC E p0 = ∂ζ˜0 + ζ 0 ∂σ , E p1 = 3 ∂ζ˜1 + ζ 1 ∂σ , 1 QC ˜ ˜ −∂ E qQC = −∂ + ζ ∂ , E = + ζ ∂ √ 0 1 0 σ 1 σ , ζ ζ q1 0 3 √2 √ 1 1 0 1 3 1 QC QC ˜ ˜ Y+ = √ ζ ∂ζ 1 −3 2ζ ∂ζ˜1 − √ ζ1 ∂ζ˜0 , Y− = √ ζ0 ∂ζ˜1 −ζ ∂ζ 0 + ζ˜1 ∂ζ 1 , 3 2 2 2 1 0 3ζ ∂ζ 0 + ζ 1 ∂ζ 1 − 3ζ˜0 ∂ζ˜0 − ζ˜1 ∂ζ˜1 , Y0 QC = 2
3 1 QC 1 0˜ 1 02 ˜ 0 12 0 ˜ 1 2 0 ˜2 1 ˜ F = 2ζ +ζ ζ1 ζ +ζ ζ0 +σ ζ ∂ζ 0 + − ζ1 ζ +ζ ζ0 ζ − ζ ζ1 +σ ζ ∂ζ 1 3 9
1 2 2 1 2 1 0˜ ˜ 3 ˜ 1˜ 0 ˜2 ˜ ˜ ˜ ˜ ˜ ζ − ζ0 ζ ζ1 −ζ ζ0 +σ ζ0 ∂ζ˜0 + −6ζ0 ζ + ζ1 ζ −ζ ζ0 ζ1 +σ ζ1 ∂ζ˜1 + 3 27 1 −[I4 (ζ, ζ˜ ) + σ 2 ]∂σ − kσ, (3.35) where I4 (ζ, ζ˜ ) is the quartic polynomial defined in (3.32). The quadratic Casimir evaluates to 1 C2 = − k(6 − k). 4
(3.36)
This representation is unitary for k ∈ 3 + iR, with respect to the inner product f1| f2 =
dζ d ζ˜ dσ f 1∗ (ζ , ζ˜ , σ ) f 2 (ζ , ζ˜ , σ ).
(3.37)
Complexifying, we can also consider the holomorphic right action of G on sections of the line bundle O(−k) on the twistor space Z = PC \G C . The infinitesimal action is the complexification of the one above, where we replace the real variables ζ I , ζ˜ I , σ by complex variables ξ I , ξ˜ I , α. Naively this construction would require k ∈ Z, but in what follows, we will sometimes consider this action not only when k is integral but even when k ∈ 13 Z. Presumably this should be understood in terms of a triple cover of Z. This representation is formally unitary under the inner product f1| f2 =
¯˜ α) dξ d ξ˜ dα d ξ¯ d ξ¯˜ d α¯ e(k−6)K Z f 1∗ (ξ¯ , ξ, ¯ f 2 (ξ, ξ˜ , α). (3.38)
As discussed in Sect. 2.4.2 for G = SU (2, 1), we expect this formally unitary action on sections of O(−k) to yield a genuine unitary action on H 1 (Z, O(−k)) at least for k ≥ 3; for k ≥ 5 it should give the quaternionic discrete series of G 2(2) . Moreover, it should occur as a subquotient of the principal series for some k. We will discuss this further at the level of the K -finite vectors below.
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3.4.1. Lift to hyperkahler cone. Similar to the discussion of Sect. 2.6.1 for G = SU (2, 1), the action of G = G 2(2) on holomorphic sections of O(−k) over Z is equivalent to an action on holomorphic functions of homogeneity degree −k on the Swann space S. Introducing the complex coordinates on S, 1 (α + ξ 0 ξ˜0 + ξ 1 ξ˜1 ), (3.39) 4i √ √ i 1 i 3 ξ˜1 , (3.40) v 0 = 3 3ξ 0 e2t , w0 = − √ ξ˜0 , v 1 = − √ ξ 1 e2t , w1 = 2 6 3 3 v = e2t , w = −
it is straightforward to compute the holomorphic vector fields corresponding to the action of G C on S and determine their holomorphic moment maps. Expressing the result in terms of the coordinates ξ I , ξ˜ I , α on Z and t in the C∗ fiber, we find the holomorphic moment maps i i i H = e2t α, Y 0 = e2t (3ξ 0 ξ˜0 + ξ 1 ξ˜1 ), E = − e2t , 2 4 4 ! ! ! i 2t 2 i 2t 0 1 Y + = − √ e ξ˜1 + 9ξ˜0 ξ , Y − = √ e ξ ξ˜1 − 3(ξ 1 )2 , ! ! 2 2 6 2 √ 3i 3 2t i i i e ξ˜0 , E p1 = e2t ξ˜1 , E q0 = √ e2t ξ 0 , E q1 = − e2t ξ 1 , E p0 = − 2 2 2 ! ! ! ! 6 3 i 3 2 F p0 = √ e2t −2ξ 1 + ξ 0 ξ˜1 ξ 1 + αξ 0 + ξ 0 ξ˜0 , ! 6 3 i 2t 2 e 3ξ˜1 ξ 1 − 9αξ 1 − 9ξ 0 ξ˜0 ξ 1 − 2ξ 0 ξ˜12 , F p1 = 18 ! i F q0 = − √ e2t 2ξ˜13 + 27ξ˜0 ξ 1 ξ˜1 + 27ξ 0 ξ˜02 − 27α ξ˜0 , ! 6 3 1 i 2t 0˜ 12 1˜ ˜ ˜ F q = e ξ0 ξ ξ1 − 6ξ − ξ1 (3α + ξ ξ1 ) , 2 3 !1 2 2 1 0 ξ˜ 3 0 2 ξ˜ 2 2 ˜ ξ ξ ξ 2ξ α i 2t 3 1 0 − ξ 0 ξ˜0 ξ˜1 ξ 1 − + − F= e 2ξ˜0 ξ 1 + 1 . (3.41) 2 6 27 2 2 ! In order to compute the moment maps in the compact basis, which will be relevant in ¯ transforming homothe next section, it is convenient to change variables to (a, b, a, ¯ b) geneously under the compact generator R0 : 1 1 ¯ ¯ ξ1 = i(a − a¯ + 3b − 3b), ξ 0 = − (a + a¯ − b − b), 8 24 1 1 ¯ ¯ ξ˜1 = (a + a¯ + 3(b + b)), ξ˜0 = i(a − a¯ − b + b), 8 8 i ¯ ¯ . a∂a − a∂ ¯ a¯ − 3b∂b + 3b∂ R0 = b 2
(3.42) (3.43) (3.44)
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The holomorphic moment maps for the generators in the compact basis then read i e2t ¯ ¯ 1728(1+α 2 )−a 2 a¯ 2 −4(a 3 b+ a¯ 3 b)−1296b ¯ b−8) , J3 = b¯ + 27b2 b¯ 2 −18a a(b 27648 ! i e2t ¯ 1728(1+α 2 ) − a 2 a¯ 2 −4(a 3 b+ a¯ 3 b)+432b ¯ b¯ + 8) , S3 = b¯ + 27b2 b¯ 2 − 6a a(3b 9216 ! i ¯ + 8iα − bb) ¯ , J+ = − √ e2t 2a 3 + 9a a¯ b¯ + 27b(8 ! 1728 2 i ¯ , ¯ + 8iα + bb) S+ = √ e2t a a¯ 2 + 6a 2 b + 9a(8 ! 576 2 e2t ¯ 1728(1 − α 2 )+3456iα+a 2 a¯ 2 + 4a 3 b + 4a¯ 3 b+18a J 1,3 = − ab ¯ b¯ − 27b2 b¯ 2 , 6912 !2 2 i e2t 2 ¯ ¯ , J 1 1 = i√ e2t a 2 +3a¯ b¯ , J 1,1 = b) √ a a¯ + 6a¯ 2 b+9a(−8−8iα+b ,− !2 2 ! 2 2 24 3 288 6 i e2t 3 ¯ − 8iα + bb) ¯ . J 1 ,− 3 = (3.45) √ 2a + 9a a¯ b¯ − 27b(8 !2 2 864 2 Substituting (3.41) in (3.1), a tedious computation shows that the holomorphic moment map, seen as an element of gC , is nilpotent of degree 2. Thus, the Swann space S is isomorphic (at least locally) to the complexified minimal nilpotent orbit of G 2(2) . In particular, we note the classical identities √ √ E 2p1 + 3E p0 E q1 − 2 2E Y + = 0, ! √ ! ! √ !! E q21 − 3E q0 E p1 − 2 2E Y − = 0, ! ! ! !! 3E p0 E q0 + E p1 E q1 − 4E Y 0 = 0, ! ! ! ! !! 9C2 (L ) − C2 (R ) = 9C2 (J ) − C2 (S ) = 0. ! ! ! !
(3.46a) (3.46b) (3.46c) (3.46d)
3.4.2. Some finite K -types. We now discuss the finite K -types of the principal series representation. The full K -type decomposition can be obtained using Frobenius reciprocity21 ; here we are interested in the explicit construction of some specific states. As in Sect. 2.4.4, we can construct the spherical vector f K by noting that the Heisenberg parabolic subgroup P acting on the two-form e4 ∧ e6 gives a 1-dimensional representation, where ei is the i th row of the matrix in (3.1), and K preserves the norm of each row; so we take f K = e4 ∧ e6 −k/2 ,
(3.47)
e4 ∧ e6 2 = I˜4 + (I6 + α Iˆ4 ) − 2[1 + I2 + (α 2 − I4 )]2 .
(3.48)
where
21 The Frobenius reciprocity theorem implies that the number of occurrences of a K -type σ in the principal series we consider is equal to the number of singlets in the decomposition of σ under K ∩ M, where M is the centralizer in K of A, appearing in the Langlands decomposition P = M AN ; see e.g. [47].
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Here, I4 is the quartic polynomial (3.32), invariant under S L(2, R), while I2 , I˜4 , Iˆ4 , I6 are homogeneous polynomials of respective degree 2, 4, 4, 6 in ξ I , ξ˜ I , invariant under the maximal compact subgroup S O(2) ⊂ S L(2, R). In terms of the variables (3.42), 1 1 3 i 3 ¯ I2 = (a a¯ + 3bb), I˜4 = − ba + a¯ 3 b¯ , Iˆ4 = a b − a¯ 3 b¯ , (3.49a) 12 27 27 1 4(ba 3 + b¯ a¯ 3 ) + a¯ 2 a 2 − 27b¯ 2 b2 + 18a ab I4 = ¯ b¯ , (3.49b) 1728 1 3 ¯ I6 = − 2a¯ 3 a 3 + 54a¯ 2 a 2 bb¯ + 9(a a¯ + 3bb)(ba + b¯ a¯ 3 ) . (3.49c) 5832 Using (3.45), one easily recognizes that the spherical vector is related to the quadratic Casimirs of SU (2) J and SU (2) S simply by −k/4 −k/4 = C2 (S )/9 (3.50) e−kt f K = C2 (J ) ! ! which makes the K invariance manifest. Other K -types may be obtained by acting on f K with the non-compact generators J± 1 ,± 3 . For example, a set of J+ -highest weights are obtained by acting with symme2 2 trized products of the raising operators J 1 ,± 3 , J 1 ,± 1 , which transform in the spin-3/2 2 2 2 2 representation of SU (2) S (antisymmetric combinations of these operators lead instead to J+ -descendants); these generate the spectrum
n 3 . (3.51) ⊗ Sn 2 J 2 S n To see this spectrum appearing more explicitly, note that a class of (J+ , S+ )-highest weight states can be obtained by acting on f K with M1 = J 1 , 3 , 2 2
(3.52a)
√
M2 = J 12, 1 + i 3J 1 ,− 1 J 1 , 3 , 2 2
2
2
(3.52b)
2 2
2 M3 = J 12, 3 J 1 ,− 3 + i J 1 , 3 J 1 , 1 J 1 ,− 1 + √ J 13, 1 , 2 2 2 2 2 2 2 2 2 2 3 3 2 2 1 M4 = J 12, 3 J 12,− 3 + J 12, 1 J 12,− 1 + 2i J 1 , 3 J 1 , 1 J 1 ,− 1 J 1 ,− 3 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 4i 4 + √ J 1 , 3 J 13,− 1 + √ J 13, 1 J 1 ,− 3 , 3 3 2 2 2 2 3 3 2 2 2 2 which satisfy [Mi , M j ] ≡ 0,
M12 M4 +
4 3 M − M32 ≡ 0 27 2
(3.52c)
(3.52d)
(3.53)
when acting22 on states annihilated by J+ . A generating function for the (S, J ) spectrum obtained by acting with the Mi on f K is Tr z S q 2J =
1 − q 6 z3 , (1 − q z 3/2 ) (1 − q 2 z) (1 − q 3 z 3/2 )(1 − q 4 )
(3.54)
22 For example, the second equation of (3.53) follows from M 2 M + 4 M 3 − M 2 = 416 J 2 J 2 − 1 4 3 27 2 9 1,3 + 2 2 √ 8J 31 3 J 1 ,− 3 J+ − 56i J 21 3 J 1 , 1 J 1 ,− 1 J+ − 409 3 J 1 , 3 J 31 1 J+ . 3 2 2 2 2 2,2 2,2 2 2,2 2 2 2
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where the denominator corresponds to the action of Mi while the numerator reflects the constraint (3.53). This generating function indeed agrees with (3.51).23 As an example, the action of Mi on f K may be easily computed, ⎛ ⎞ S 3 J 1 , 3 − √i J 1 , 1 S + 3 !2 2! ⎠ fK , M1 f K = k ⎝ ! ! 2 2 (3.55a) C2 (J ) !
J + S+ M2 f K = k(3k − 2) ! ! (3.55b) fK , C2 (J ) !
J+ J 1,3 2 2 2 M3 f K = k(9k − 4) ! ! (3.55c) fK , C2 (J ) ! 2 J + M4 f K = k 2 (9k 2 − 4) (3.55d) fK , ! C2 (J ) ! where equalities hold up to unimportant numerical factors. Acting with the Mi on f K is not sufficient to construct all of the (J+ , S+ ) highest weight vectors. To see this explicitly, it is enough to note that there are other operators built from J± 1 , 3 , J± 1 , 1 which commute with S+ and J+ , for example 2 2
2 2
S 2+ P2 f K ≡ J 1 , 3 J− 1 , 1 − i J− 1 , 3 J 1 , 1 f K = k(k − 2) f K . (3.56) ! 2 2 2 2 2 2 2 2 C2 (J ) ! Nevertheless, we focus on (J+ , S+ )-highest weight states generated by the action of Mi only, of the form p1
− k − p − 1 ( p + p + p ) i p p p J 1 2 3 J + 3 S + 4 C2 (J ) 4 1 2 2 3 4 , f p1 , p2 , p3 , p4 = S 3 J 1 , 3 − √ J 1 , 1 S + , ! !2 2 !2 2 ! ! ! 3!2 2! (3.57)
where pi are integers related to the SU (2) J × SU (2) S spin by j=
1 1 ( p1 + p2 + 2 p3 ), s = (3 p1 + 3 p2 + 4 p4 ). 2 2
(3.58)
The set of such states is pictured in Fig. 5. Now we recall some results of [26]. First, the K -type decomposition of H 1 (Z, O(−k)) for k ≥ 3 is given by
∞ k−2+n 3 . (3.59) ⊗ Sn 2 2 S J n=0
For k ≥ 5 this gives a quaternionic discrete representation of G, which we expect to find as a submodule of the principal series. Indeed, when k − 2 is a multiple of 4, these K -types do appear in Fig. 5: they are the ones reached by acting with the Mi on f K and in particular using M4 at least 41 (k − 2) times. 23 A simple check is obtained by manipulating (3.54) to get % 2J = 1/(1 − q)4 , the partition n,S (2S + 1)q
function of four free bosons.
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S 10 9 8 7 6 5 4 3 2 1
1
3
2
4
5
6
7
8
J
Fig. 5. The states (3.57) obtained by acting with the Mi on f K . Multiplicities are indicated by the number of concentric circles. The radius of the circle indicates the number of powers of M4 that need to be applied to f K in order to reach the state. This figure also represents the set of highest weight vectors one could obtain by acting with the Mi on some other ground state; in this case the labels J and S are shifted by the quantum numbers of the ground state, and in some cases one gets only a subset of the states pictured
The paper [26] also describes a pattern of other representations which would be expected to appear as submodules of the principal series for general quaternionic groups. While that analysis does not directly apply to G = G 2(2) we describe its naive extrapolation here, and explain how the expected K -type decompositions can be naturally obtained by acting with the operators Mi on an appropriate lowest K -type. The representations in question are supposed to correspond to the orbits of S L(2, C) acting on the complexified spin- 23 representation (C4 ). Their K -type decompositions would be of the form ∞ k−2+n 2
n=0
⊗ An (X ),
(3.60)
J
% where A(X ) = ∞ n=0 An (X ) is the algebra of functions on an orbit X of the action of S L(2, C) on C4 , considered as a representation of SU (2) S . There are three examples: (i) A representation π1 at k = 1, corresponding to the orbit X defined by I4 = 0. The K -type decomposition is obtained by removing the contribution of the operator M4 from (3.54), and acting on a highest weight vector with (J, S) = (0, 21 ) instead of f K : this gives 1
Trπ1 z S q 2J = z 2
(1 − q
1 − q 6 z3 . z) (1 − q 3 z 3/2 )
z 3/2 ) (1 − q 2
(3.61)
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This decomposition is multiplicity-free. π1 has Gelfand-Kirillov dimension 4, and K -types contained in a wedge — see the black dots on Fig. 5 (shifted by S = 21 ). It appears on the list of unitary representations of G 2(2) in [8]. at k = 2/3, corresponding to the locus where I = d I = (ii) A representation π2/3 4 4 24 0. Its K -type decomposition can be obtained by acting on a highest weight vector with (J, S) = (0, 2) with the operators M2 and M1 , but imposing the requirement that M22 = 0. It is represented in Fig. 5 by the two leftmost “Regge trajectories” of slope 3 (shifted by S = 2). at k = 4/3, corresponding to the locus where I = d I = (iii) A representation π4/3 4 4 2 d I4 = 0. This is the minimal or “ladder” representation of G 2(2) , with K -type decomposition
∞ m m=0
2
J
3m + 2 ⊗ 2
.
(3.62)
S
The highest weight states can be obtained by acting on the highest weight of a K -type with J = 0, S = 1 by the operator M1 only; it is represented in Fig. 5 by the leading “Regge trajectory” of slope 3 (shifted by S = 1). The values of k at which these submodules appear can be checked by analyzing the invariance of certain elements in the enveloping algebra, as explained in the case of the minimal representation in Sect. 3.5.6 below.
3.5. Minimal representation. The minimal representation of G 2(2) , of functional dimension 3, was first constructed in [53], and further analyzed in [26]. As we just recalled, it can be obtained as a submodule of a degenerate principal series representation [26]. According to the orbit philosophy, it arises by quantizing the minimal nilpotent orbit of G 2(2) , or equivalently by holomorphic quantization of the Swann space S. We start by recalling two different realizations of the minimal representation by differential operators, the first one acting on functions f (y, x0 , x1 ) of real variables, the second on functions f (x, a, b) of complex variables. 3.5.1. Real polarization. In the real polarization used in [39], iy 1 , Y0 = − (x1 ∂1 + 3x0 ∂0 + 2) , H = 2y∂ y + x0 ∂0 + x1 ∂1 + 2, 2 2
√ i x13 2 E p0 = −3 3y∂0 , F p0 = − √ x0 ∂ y − 2 , y 3 3 i 2 = − √ x0 , Fq 0 = − √ −y∂13 + 27i 2 + x0 ∂0 + x1 ∂1 + y∂ y ∂0 , 3 3 3 3 4x 2 ∂1 + 4x1 2i + x0 ∂12 , E p1 = y∂1 , F p1 = 2x1 ∂ y + 1 3y 9
E =−
Eq 0
24 The correspondence between orbits and representations in this case is somewhat degenerate, because actually this orbit is the same as the minimal orbit we discuss next (unlike what happens for other G, where the orbit defined by d I4 = 0 is different from that defined by d 2 I4 = 0).
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4 1 E q 1 = i x1 , Fq 1 + x1 ∂1 + x0 ∂0 + y∂ y ∂1 , 3 3
x12 1 1 1 2 Y+ = √ (i y∂1 + 9x1 ∂0 ), Y− = − √ − x0 ∂1 − i , 3 y 2 2 2 2i 2 2 2 3x1 ∂1 + 6x1 ∂1 + 2 − 2i 2 + y∂ y + x0 ∂0 + x1 ∂1 ∂ y . F = − x0 ∂13 − 2 x13 ∂0 − 27 y 9y 6x 2 = − 1 ∂0 + 2i y
It is easy to check that the principal symbols associated to these generators agree with the holomorphic moment maps (3.41), upon identifying i i i v = −2y, v 0 = 2x0 , v 1 = 2x1 , w = − p y , w0 = p0 , w1 = p1 . 2 2 2 (3.63) The minimal representation in the real polarization is unitary under the inner product f1 | f2 =
d y d x0 d x1 f 1∗ (y, x0 , x1 ) f 2 (y, x0 , x1 ).
(3.64)
√ √ √ Alternatively, one may redefine q0 = −x0 / 27y, q1 = x1 / y, x = y [56], leading to the polarization used by [29,32], i E = − x 2, 2
i 3i 1 ∂x + F = − ∂x2 − I4 , 2 2x 18x 2 1 √ 3 2i 3q1 − 9q0 (q0 ∂0 + q1 ∂1 ) + q0 ∂x , E p0 = x∂0 , F p0 = 9x √ 1 4q1 + q12 ∂1 + 2i 3q0 ∂12 − 3q0 q1 ∂0 + q1 ∂x , E p1 = x∂1 , F p1 = 3x 1 √ 3 2 3∂1 + 9i(q0 ∂02 + q1 ∂0 ∂1 + 3∂0 ) + i∂0 ∂x , E q 0 = i xq0 , Fq0 = 9x √ 1 5i∂1 − iq1 ∂12 + 2 3q12 ∂0 + 3iq0 ∂0 ∂1 + i∂1 ∂x , E q 1 = i xq1 , Fq1 = 3x √ i 2 √ 1 Y+ = √ ∂1 + i 3q1 ∂0 , Y− = √ −iq12 + 3q0 ∂1 , 2 2 1 Y0 = − (3q0 ∂0 + q1 ∂1 + 2), H = x∂x + 2. (3.65) 2 where √ √ I4 (q , ∂ ) = 4 3q0 ∂13 + 4 3q13 ∂0 − 3iq12 ∂12 + 18iq0 q1 ×∂0 ∂1 + 9iq02 ∂02 + 3iq1 ∂1 + 27iq0 ∂0 − 8i.
(3.66)
In accord with irreducibility, the quadratic Casimir evaluates to a constant C2 = −14/9,
(3.67)
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which coincides with that of the quaternionic discrete representation for k = 4/3 or k = −14/3. In addition, the minimal representation is annihilated by the Joseph ideal, e.g. √ 4Y+2 + 3 3(E p1 Fq0 − E p0 Fq1 ) √ √ E 2p1 + 3E p0 E q1 − 2 2E Y+ √ √ E q21 − 3E q0 E p1 − 2 2E Y− 3E p0 E q0 + E p1 E q1 − 4E Y0
= 0,
(3.68a)
= 0,
(3.68b)
= 0, = 0.
(3.68c) (3.68d)
These last three identities can be shown to imply the holomorphic anomaly equations satisfied by the topological amplitude in the one-modulus model with prepotential F = −(X 1 )3 / X 0 [15]. Other identities will be discussed in Sect. 3.5.3 below. For later reference, we note that the vector 0 −2/3
f P (y, x , x ) = (x ) 0
1
(x 1 )3 exp −i yx 0
(3.69)
transforms as a one-dimensional representation of the Heisenberg parabolic P (more specifically, it is annihilated by Y+ , Y0 , Y− , F p I , Fq I , F and carries charge 4/3 under H ). In particular, it is invariant under the Weyl reflection S with respect to the root E and therefore under Fourier transform over x 0 , x 1 . The power (P 0 )−2/3 is consistent with the semi-classical analysis in [57]. 3.5.2. Complex polarization. It is also useful to consider a different realization of the minimal representation [30], on functions f (x, a † , b† ) such that
R0 = i
√ √ 1 † 3 † 1 i i a a − b b− , R+ = √ a † a † + 3 ab , R− = √ a 2 + 3 a † b† , 2 2 2 2 2
L0 =
i 2 i (x − ∂x2 ) + 2 I4 (a, b), 4 9x
i 4 L ± = − √ (∂x ∓ x)2 + 2 I4 . (3.70) 9x 4 2
Here 9 √ 3 9 41 I4 (a, b) = − 3 a 3 b + a 3† b† − Na Nb − Na2 + Nb2 − 3Na − , (3.71) 2 4 4 16 where a, b, a † , b† are bosonic oscillators with [a, a † ] = 1, [b, b† ] = 1,
Na = a † a,
Nb = b† b
(3.72)
so that a ≡ ∂a † , b ≡ ∂b† ; we shall refer to this presentation as the “upside-up” complex polarization. Alternatively, we could consider functions f (x, a † , b) and represent b† = −∂b : this will be termed as the “upside-down” complex polarization, and will turn out to be the most convenient one to compute the lowest K -type. Irrespective of the
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choice of polarization for the oscillator algebra, the generators in the non-compact basis read
i 4 (3.73) ∂x2 − 2 I4 , 2 9x √ x E p1 = − √ (a − a † ) + 3(b − b† ) , 2 2 x √ † E q0 = √ 3(a − a) − (b† − b) , 2 2
i E = − x 2, 2 i x √ † E p0 = √ 3(a + a) − (b† + b) , 2 2 √ ix E q1 = − √ (a † + a) + 3(b† + b) , 2 2
F =−
while the compact generators are given by
J+ J− S+ S−
Na − Nb + 21 i 2 a †3 =− ∂x − x − b− √ , 2 x 3 3 x
Na − Nb − 21 i 2 a3 † = ∂x + x + b + √ , 2 x 3 3 x
√ 1 5 i 3 2 a †2 b† 3 Na + Nb + 6 = ∂x − x − a−√ , 2 x 3 x
√ 1 1 i 3 2 a2b † 3 Na + Nb + 2 =− ∂x + x + a +√ . 2 x 3 x
(3.74a) (3.74b) (3.74c) (3.74d)
This presentation of the minimal representation is related to (3.63) by a Bogoliubov transformation which is the quantum version of the canonical transformation (3.42). It can be decomposed into (i) a Fourier transform with respect to x0 , f (y, x0 , x1 ) =
dp0 e−i p0 x0 /y f (y, p0 , x1 ),
(3.75)
(ii) a change of variables √ √ x 3 x √ (w − 3v), x1 = − 3w + v , p0 = − 18 2
y = x 2,
(3.76)
and finally (iii) a standard Bogoliubov transform f (x, v, w) =
√ 1 2 † 2 2 2 † exp [w + (b ) − v − a ] + 2(av − b w) f (x, a, b† ), 2 (3.77)
√ implementing the change from real to oscillator polarization a + a † = v 2, b + b† = √ of unitary transformation in fact implements w 2. One may check that this sequence √ the Cayley transform by C = e
π 2 4 (R+ +R− )
.
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3.5.3. K -type decomposition. Let us now discuss the K -type decomposition (3.62) of the minimal representation in more detail. We note that the correlation between the two spins is a straightforward consequence of the identity in the Joseph ideal25 9C2 (J ) − C2 (S) + 2 = 0.
(3.78)
The fact that the lowest K -type is a SU (2) J singlet and SU (2) S triplet26 is deeper, and will be further discussed below. We also note that the K -type decomposition (3.62) is consistent with the known decomposition of the minimal representation under S L(3, R) and SU (2, 1) [8]: indeed, the minimal representation is an irreducible representation in the non-spherical principal series of S L(3, R). The K -type decomposition of this representation, worked out in [5] (Eq. 7.8), [1] + [2] + [3]2 + [4]2 + [5]3 + [6]3 + [7]4 + [8]4 + · · ·
(3.79)
is consistent with the diagonal embedding of the maximal compact of S L(3, R) inside SU (2) J × SU (2) S . Under SU (2, 1), the minimal representation decomposes as a sum of three irreducible principal series representations of SU (2, 1) with infinitesimal characters (0, 13 , − 13 ), ( 13 , 0, − 13 ), ( 13 , − 13 , 0). These three representations correspond to the three supplementary series at p = q = −2/3 in the terminology of [6], and transform with different characters of the center Z3 of SU (2, 1). This is in agreement with the Casimirs, 8 C2 [S L(3, R)] = − , C3 [S L(3, R)] = 0, 9 8 C2 [SU (2, 1)] = − , C3 [SU (2, 1)] = 0, p = q = −2/3. 9
(3.80a) (3.80b)
Let us now further analyze (3.78), by rewriting them in terms of the generators of G acting in the minimal representation. Using 1 (J+ J− + J− J+ ) = −J3 (J3 ± i) − J∓ J± , 2 1 C2 (S) = −S32 − (S+ S− + S− S+ ) = −S3 (S3 ± i) − S∓ S± , 2
C2 (J ) = −J32 −
(3.81a) (3.81b)
we see that a normalizable eigenmode f of J3 , S3 satisfying the highest weight condition J+ f = S+ f = 0 necessarily has (J3 , S3 ) ∈ 2i N. From (3.78), we have 9J3 (J3 + i) − S3 (S3 + i) + 2 = 4(2R0 + i)( R˜ 0 + i) = 0,
(3.82)
where R0 =
1 (3J3 − S3 ), 2
1 R˜ 0 = (3J3 + S3 ). 2
(3.83)
25 A similar identity holds in the non-compact basis, 9C (L) − C (R) + 2 = 0. 2 2 26 This is an exception among quaternionic Lie groups; the lowest K -type of the minimal representation is
usually a singlet of the Levi factor of P, while carrying a non-zero SU (2) J spin [26].
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Thus it is either an eigenmode of R0 = −i/2, or of R˜ 0 = −i. The second condition is inconsistent with the highest weight condition, so R0 = −i/2. With similar arguments we conclude that J+ f = S+ f = 0
⇒
J+ f = S− f = 0
⇒
J− f = S+ f = 0
⇒
J− f = S− f = 0
⇒
i i i m, S3 = (3m + 2), R0 = − , (3.84a) 2 2 2 i i i J3 = m, S3 = − (3m + 2), R˜ 0 = − , (3.84b) 2 2 2 i i i (3.84c) J3 = − m, S3 = (3m + 2), R˜ 0 = , 2 2 2 i i i J3 = − m, S3 = − (3m + 2), R0 = . (3.84d) 2 2 2 J3 =
In the following, we analyze the consequences of these equations in various polarizations. 3.5.4. Lowest K -type in the complex polarization. It turns out that the form of the lowest K -type is simplest in the complex polarization, where the generator R0 takes a simple form
1 † 3 1 . (3.85) R0 = i a a − b† b − 2 2 2 The following linear combinations of (3.74) lead to x-independent equations: √ √ i i T++ = a 3J+ + bS+ = − (a †2 + 3ab)(R0 + ), x 2 √ † √ † † 2 i † 2 T−− = 3a J− + b S− = − x(a + 3a b )(R0 − ), 3 2 √ √ T+− = a † 3J+ − bS− , T−+ = a 3J− − b† S+ ,
(3.86a) (3.86b) (3.86c)
where the first two lines are consistent with (3.84a). Now we restrict to a highest weight S+ f = 0, which is a singlet of SU (2) J , i.e. J+ f = J− f . It turns out that it is convenient to work in the “downside-up polarization” where b = ∂b† , a † = −∂a . The constraint R0 = −i/2 requires 1 a3 f (x, a, b† ) = (b† )−1/3 f 1 (x, z), z = √ † . 3 3b
(3.87)
The constraint T++ is automatically obeyed, however T−+ leads to a second order ordinary differential equation in z only, 1 2 2 2 z∂z − 2(z − )∂z + (z + x − ) f 1 (x, z) = 0. (3.88) 3 3 The solution is f 1 (x, z) = x 1/3 z 1/6 e z
√ √ K 1/3 2i x z f 2 (x) + I1/3 2i x z f˜2 (x) .
(3.89)
In the following, we assume that normalizability forces f˜2 (x) = 0. It is one of the drawbacks of the complex polarization that normalizability is difficult to check – at any
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rate, it is straightforward to generalize the computation below to include both solutions. Requiring the action of S+ on (3.87) to vanish, we find
6x∂x − 12z∂z2 + 2(6z − 4)∂z + (1 − 6x 2 )
f 1 (x, z) = 0.
(3.90)
Combining this with (3.88), we can produce a first order partial differential equation
7 x∂x − 2z∂z + 2z + y − 6
2
f 1 (x, z) = 0
(3.91)
whose solution is f 1 (x, z) = x 7/6 e−x
2 /2+z
f 3 (x 2 z).
(3.92)
√ f 3 (u) = K 1/3 2i u .
(3.93)
This uniquely determines f 2 (x) = x −7/6 e−x
2 /2
,
It is now easy to check that the resulting function is annihilated by S+ , J+ , J− , J3 , and is an eigenmode of S3 = i. The remaining two states in the triplet may be obtained by acting with S− . Altogether, we find that the lowest K -type is √ K 1/3 2i x z , √ 2 = 2 3−1/4 x 3/2 a (b† )−1 e z−x /2 K 2/3 2i x z , √ 2 2 = − √ (ax/b† )3/2 e z−x /2 K 1/3 −2i x z . 3
f 0,1 = (ax 3 /b† )1/2 e z−x f 0,0 f 0,−1
2 /2
(3.94a) (3.94b) (3.94c)
The highest weights in the higher K -types can be obtained by acting with the raising operators J1/2,3/2 . For example, f 1,5 = 2 2
1 3 1 3/2 1/2 † −1 z−x 2 /2 x K 4 + i 3 4 a 2 x K 2 + 3 b† (2x 2 − 3)K 1 , (3.95) a (b ) e 3 3 3 3
where the argument of the Bessel function is as in f 0,1 . We note that semi-classically, all K -types behave as ⎡ 1 a3 2 x2 exp ⎣− + √ † + 3/4 i x 2 3 3b 3
(
⎤ a3 ⎦ . b†
(3.96)
As explained in (2.6.2), and further at the end of the next subsection, the argument of the exponential (or “classical action”) provides the generating function for a complex Lagrangian cone inside the hyperkähler cone S.
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3.5.5. Lowest K -type in the real polarization. According to the above, the lowest K -type should be a triplet of SU (2) S , singlet under SU (2) J . Thus we impose the conditions J+ = J3 = J− = S+ = 0.
(3.97)
Note in particular that the generator F p0 − E p0 of rotations in the (y, x0 ) plane, which was at the start of the KPW solution for the spherical vector in the split case, no longer annihilates the state, so we need a different strategy. Our approach is to find a linear combination of the operators J+ , J3 , J− , S+ which involves first order differential operators only. The one of interest is $ 3 y S3 − 3J3 + 3i (J+ − 2J− − S+ ), (3.98) 2 x0 which allows to rewrite S3 as a first order differential operator in three variables, S3 = α y ∂ y + α0 ∂0 + α1 ∂1 + β,
(3.99)
where
x1 27i 2 x0 −12x12 + 9y 2 , α0 = −9x1 − + αy = y i − 9 y , α1 = , x0 2x0 3 2x0 β=
x1 (−6y + x12 + i x0 x1 ) . x0 y
(3.100)
(3.101)
The standard way to solve this equation is to integrate the flow dy = αy , ds
d x0 = α0 , ds
d x1 = α1 , ds
df = (β − s3 ) f. ds
(3.102)
Using inspiration from the split case [39], it is easy to find one constant of motion along the flow, 2 2 2 2 2 3/2 y + 27 x0 + 3 x1 z= . (3.103) 2 2 y 2 + 27 x0 To find the second constant of motion, and integrate the flow completely, we go to polar coordinates in the (y, x0 ) plane, √ √ y = 2 r cos θ, x0 = 3 3 r sin θ. (3.104) We now change variables to f (y, x0 , x1 ) = r
−2/3
h(z, r, θ ) exp −2i
x0 x13 y(2x02 + 27y 2 )
.
(3.105)
The action of S3 on h(z, r, θ ) is still a first order differential operator, but now involving only two variables r, θ : 1/2
2/3 z 3r S3 = −i cot θ ∂θ − −1 ∂r . (3.106) √ sin θ r 2
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Again, the way to solve this equation is to integrate the flow 1/2
2/3 i dr 3r z dθ =− , =− −1 , √ ds tan θ ds sin θ r 2
dh = s3 . hds
221
(3.107)
The ratio of the first two equations gives dθ/dr , and produces a second constant of motion (in addition to z),
$ 2/3 z √ (1 − i eiθ )2 1 + i −1
t= (1 + i
eiθ )2
1−i
r 2
$
z √
r 2
2/3
.
(3.108)
−1
The third equation in (3.107) then constrains the θ dependence to h(z, r, θ ) = [cos θ ]−is3 h 1 (z, t),
(3.109)
where s3 is the eigenvalue under S3 , which we independently know to be s3 = i for the lowest K -type. We now express the action of the other generators J± , S+ , R0 on our ansatz
x0 x13 −2/3 f (y, x0 , x1 ) = cos θ r h 1 (z, r ) exp −2i . (3.110) y(2x02 + 27y 2 ) For this purpose, we express r, θ, z, t in terms of y, x0 , x1 using (3.103),(3.104) and
√
1 + " 3i 2x1 √ " 2 2x02 +27y 2 4 x0 2 2 x0 √ t = 1+ + 2x02 + 27y 2 , 2 27 y 27 y 1 − " 3i 2 2x1
(3.111)
2x0 +27y 2
and act with the original differential operators. We then set x1 = 0, and revert to z, t variables using $ √ 1 2 z 2 t 2 z. (3.112) x0 = 27(r − y ), r = √ , y = 2 1+t 2 The last two identities are only valid at x1 = 0, but that is sufficient to get the full action on h 1 (z, r ). We find, in particular, √ √ √ i t i 2t 3t z −3z +8 t ∂t +4 t 4t 2 ∂t2 −3z∂z − 3(t +1)z , R0 + = − 2 5/3 2 9(t + 1) z √ √ t J+ + J− = 2 3t z+3z+4 t ∂ −24 t z∂ ∂ −3(t−1)z . (3.113) t z t 18(t + 1)z 5/3 A semi-classical analysis of the z → ∞ limit gives the expansion √ √
2 t (1 + t)2 1/6 −z/2 5 1+ z −1 − h 1 (z, t) ∼ z e √ √ 36 t 3(1 + t)2 √ √
35 t 385 5005 t −85085 −3 −2 z + z + · · · . (3.114) + − √ 2+ √ 2+ 54(1+ t) 2592 3878(1+ t) 279936
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This suggests the ansatz √ (1 + t)2 h 1 (z, t) = z 1/6 h 2 (z) + h 3 (z) . √ t
(3.115)
Indeed, we find that all constraints reduce to two ordinary differential equations for h 2 and h 3 , (1 + 3z)h 2 + 6z(h 2 + 4h 3 + 2h 3 ) = 0, (4 + 15z)h 2 + (54z − 2)h 3 + 12z(2h 2 + 9h 3 ) = 0,
(3.116a) (3.116b)
which decouple into
(1 + 18z + 9z 2 )h 2 − 36z(h 2 + zh 2 ) = 0,
(3.117a)
(9z − 5)h 3 − 36z h 3 = 0.
(3.117b)
2
2
Each of them has two independent solutions, but we keep only the one decaying at z → ∞ in order to ensure normalizability, leading to √ √ (1 + t)2 2/3 2 π 1/3 z U 7 , 4 ,7 (z), h 1 (z, t) = (3.118) z K 1/3 (z/2) + √ 6 3 3 t which correctly reproduces the subleading terms in (3.114). In total, the final answer for the lowest K -type in the real polarization is √ 2 √ 2 π 1/3 −2/3 (1 + t) 2/3 f 0,1 (y, x0 , x1 ) =(cos θ ) r z U 7 , 4 ,7 (z) z K 1/3 (z/2) + √ 6 3 3 t (3.119)
x0 x13 × exp −2i . y(2x02 + 27y 2 ) In the semi-classical limit, where y, x0 , x1 are scaled uniformly to ∞, it reduces to √ 2 −2/3 (1 + t) f 0,1 (y, x0 , x1 ) ∼ r (3.120) z 1/6 cos θ e−S , √ t where S is the “classical action” 2 3/2 2x0 2x1 2 2 + + y 27 3 2i x0 x1 3 . 2 S= − 2 y 2x0 + 27y 2 2 2x270 + y 2
(3.121)
The same reasoning as in Sect. 2.6.2 implies that S is the generating function of a holomorphic Lagrangian cone inside the hyperkähler cone S, invariant under the maximal compact K . The precise identification of the variables y, x0 , x1 in the minimal representation and the complex coordinates on S is given in (3.63). One may indeed check that the holomorphic moment maps of the compact generators vanish identically on the locus w I = ∂v I S.
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√ √ After rescaling x0 → x0 27/2, x1 → x1 / 2, we find that S can be cast in the same form as in Eq. 4.72 of [39], S=
x0 1
X − i " F, 2 y x 2 + y2
(3.122)
0
where X is an Sp(2n + 4) extension of the usual Sp(2n + 2) symplectic section appearing in the special geometry description of the c-map, X = y, x0 , x1 ; y˜ , x˜ 0 , y˜ 0 , X 2 = y 2 + y˜ 2 + (x0 )2 + (x1 )2 + (x˜ 0 )2 + (x˜ 1 )2 , (3.123) x3 y˜ = ∂ y F, xˆ i = ∂x i F, F = √ " 1 . 3 3 y 2 + x02
(3.124)
It would be interesting to see whether such a Sp(2n + 4) invariant description also exists for non-symmetric c-map spaces, or even for general hyperkähler manifolds, and to investigate whether the lowest K -type of the minimal polarization bears any relation to the topological string amplitude of the corresponding magical supergravity theory as discussed in [15]. 3.5.6. As a submodule of the principal series. We consider the submodule of the principal series annihilated by the “holomorphic anomaly” relations √ √ C+ ≡ E 2p1 + 3E p0 E q1 − 2 2E Y+ = 0, (3.125a) √ √ (3.125b) C− ≡ E q21 − 3E q0 E p1 − 2 2E Y− = 0 in the Joseph ideal (see (3.68b),(3.68c)). In terms of the differential operator realization (3.35), these conditions reduce to − 3(P 1 )2 + P 0 Q 1 = 0, (Q 1 )2 + 9Q 0 P 1 = 0,
(3.126a) (3.126b)
where P I = ∂ζ˜ I − ζ I ∂σ ,
Q I = ∂ζ I + ζ˜ I ∂σ ,
(3.127)
are covariant derivatives commuting with E p I , E q I , analog of ∇, ∇¯ in (2.115). For k = 4/3, this subspace is invariant under the action of G; indeed, the commutators of the constraints with the lowest negative root F can be rewritten as
2 2 (3.128a) [F, C+ ] = −2 σ + D+ C0 + D0 C+ , 3 3
2 2 (3.128b) F, C− = −2 σ − D− C0 − D0 C− , 9 3 where D0 , D± is the S L(2) triplet D+ = 3(ζ 1 )2 + ζ 0 ζ˜1 ,
D0 = 3ζ 0 ζ˜0 + ζ 1 ζ˜1 ,
D− = (ζ˜1 )2 − 9ζ 1 ζ˜0 .
(3.129)
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At the level of differential symbols, the constraints (3.126) are solved by Q 0 = −(P 1 )3 /(P 0 )2 ,
Q 1 = 3(P 1 )2 /(P 0 ).
(3.130)
It is therefore natural to go to a polarization where P I and E act diagonally, f (ζ, ζ˜ , σ ) = dp d K exp −i K σ − i p I ζ˜ I g(ζ I , p I , K ).
(3.131)
In this polarization, the generators become 1 (3.132) (3 p 0 ∂ p0 + 3ζ 0 ∂ζ 0 + p 1 ∂ p1 + ζ 1 ∂ζ 1 + 4), 2 E p0 = −i( p 0 + K ζ 0 ), E q0 = −∂ζ 0 − K ∂ p0 , √ √ E p1 = −i 3( p 1 + K ζ 1 ), E q1 = −(∂ζ 1 + K ∂ p1 )/ 3, 1 1 Y+ = √ (6i p 1 ζ 1 + p 0 ∂ p1 + ζ 0 ∂ζ 1 ), Y− = − √ (9 p 1 ∂ p0 +2i∂ p1 ∂ζ 1 +9ζ 1 ∂ζ 0 ), . . . , 2 3 2 E = i K , Y0 =
while the constraints are C+ = i( p 0 − K ζ 0 )(∂ζ 1 − K ∂ p1 ) − 3( p 1 − K ζ 1 )2 , (3.133a) 1 (3.133b) C− = −3i( p 1 − K ζ 1 )(∂ζ 0 − K ∂ p0 ) + (∂ζ 1 − K ∂ p1 )2 . 3 An invariant set of solutions can be found by restricting to functions (P 1 − 2K ζ 1 )3 I I 0 0 −2/3 h(P I , K ), (3.134) exp i g(ζ , p , K ) = (P − 2K ζ ) 2K (P 0 − 2K ζ 0 ) where P I = p I + K ζ I . The quasi-conformal action on f (ζ I , ζ˜ I , σ ) at degree k = 4/3 restricts to an action on h(P I , K ) given by i E = − y, 2
i E p0 = − √ x 0 , 3 3
√ E q0 = 3 3y∂0 ,
E p1 = i x 1 ,
E q1 = −y∂1 ,
1 H = 2y∂ y + x 0 ∂0 + x 1 ∂1 + 2, Y0 = (3x 0 ∂0 + x 1 ∂1 + 2), 2 1 )2 1 (x 1 , Y− = √ i y∂12 + 9x 1 ∂0 , . . . , Y+ = − √ x 0 ∂1 + 3i y 3 2 2
(3.135)
where we have redefined
√ √ x 0 = 3 3P 0 , x 1 = − 3P 1 ,
y = −2K .
(3.136)
We recognize this as the minimal representation in the polarization (3.63), after applying a Weyl reflection S with respect to the highest root E, which has the effect of exchanging E p I with E q I and Y+ with Y− . We conclude that the minimal representation can be embedded into the principal series at k = 4/3, via f (ζ I , ζ˜ I , σ ) = d P I d K f P∗ ( p I − K ζ I , K ) h( p I + K ζ I , K ) (3.137) = f P |e
−σ E−ζ I E q I +ζ˜ I E p I
| f ,
(3.138)
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225
where 0 −2/3
f P (K , P , P ) = (P ) 0
1
(P 1 )3 exp −i 2K (P 0 )
(3.139)
is the P-covariant vector introduced in (3.69). Acknowledgements It is our pleasure to thank A. Waldron for collaboration at an initial stage of this project, and W. Schmid, P. Trapa, D. Vogan and M. Weissman for correspondence and discussions. M.G. and B.P. express their gratitude to the organizers of the program “Mathematical Structures in String Theory” that took place at KITP in the Fall of 2005, where this study was initiated. The research of B.P. is supported in part by the EU under contracts MTRN–CT–2004–005104, MTRN–CT–2004–512194, by ANR (CNRS–USAR) contract No 05–BLAN–0079–01. The research of A. N. is supported by the Martin A. and Helen Chooljian Membership at the Institute for Advanced Study and by NSF grant PHY-0503584. The research of M.G. was supported in part by the National Science Foundation under grant number PHY-0555605 and the support of the Monell Foundation during his sabbatical stay at IAS, Princeton, is gratefully acknowledged. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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34. Günaydin, M.: JHEP 0705, 049 (2007) 35. Kazhdan, D., Polishchuk, A.: Minimal representations: spherical vectors and automorphic functionals. In: Algebraic groups and arithmetic, Mumbai: Tata Inst. Fund. Res., 2004, pp. 127–198 36. Neitzke, A., Pioline, B., Vandoren, S.: JHEP 0704, 038 (2007) 37. Salamon, S.M.: Invent. Math. 67(1), 143–171 (1982) 38. Baston, R.J., Eastwood, M.G.: The Penrose transform. Its interaction with representation theory. Oxford Mathematical Monographs. New York: The Clarendon Press Oxford University Press, 1989 39. Kazhdan, D., Pioline, B., Waldron, A.: Commun. Math. Phys. 226, 1–40 (2002) 40. Antoniadis, I., Minasian, R., Theisen, S., Vanhove, P.: Class. Quant. Grav. 20, 5079–5102 (2003) 41. Anguelova, L., Rocek, M., Vandoren, S.: Phys. Rev. D70, 066001 (2004) 42. Roˇcek, M., Vafa, C., Vandoren, S.: Quaternion-Kahler spaces hyperkahler cones and the C-map. To apper In: Handbook of pseudo Riemannian geometry and supersymmetry, IRMA Lect. in Math. Phys., Hambarg: Euro. Math. Soc., 2008, available at http://arxiv.org/list/math.dy/06034, 2006 43. Ferrara, S., Sabharwal, S.: Nucl. Phys. B 332, 317 (1990) 44. Wolf, J.A.: J. Math. Mech. 14, 1033–1047 (1965) 45. de Wit, B., Roˇcek, M., Vandoren, S.: JHEP 02, 039 (2001) 46. Swann, A.: Math. Ann. 289(3), 421–450 (1991) 47. Knapp, A.W.: Representation theory of semisimple groups. Reprint of the 1986 original. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 2001. An overview based on examples 48. Eastwood, M.G., Ginsberg, M.L.: Duke Math. J. 48(1), 177–196 (1981) 49. Cordaro, P.D., Gindikin, S., Trèves, F.: J. Funct. Anal. 131(1), 183–227 (1995) 50. Britto-Pacumio, R., Strominger, A., Volovich, A.: JHEP 11, 013 (1999) 51. Baston, R.J.: J. Geom. Phys. 8(1–4), 29–52 (1992) 52. Salamon, S.M.: Annales Scientifiques de l’École Normale Supérieure (4) 19(1), 31–55 (1986) 53. Joseph, A.: Comm. Math. Phys. 36, 325–338 (1974) 54. Bodner, M., Cadavid, A.C.: Class. Quant. Grav. 7, 829 (1990) 55. Mizoguchi, S., Schroder, G.: Class. Quant. Grav. 17, 835–870 (2000) 56. Pioline, B., Waldron, A.: Phys. Rev. Lett. 90, 031302 (2003) 57. Pioline, B.: JHEP 0508, 071 (2005) Communicated by N.A. Nekrasov
Commun. Math. Phys. 283, 227–253 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0512-7
Communications in
Mathematical Physics
An Integrable Discretization of the Rational su(2) Gaudin Model and Related Systems Matteo Petrera, Yuri B. Suris Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, D-85747 Garching bei München, Germany. E-mail:
[email protected];
[email protected] Received: 27 July 2007 / Accepted: 20 November 2007 Published online: 22 May 2008 – © Springer-Verlag 2008
Abstract: The first part of the present paper is devoted to a systematic construction of continuous-time finite-dimensional integrable systems arising from the rational su(2) Gaudin model through certain contraction procedures. In the second part, we derive an explicit integrable Poisson map discretizing a particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the contraction procedures enable us to construct explicit integrable discretizations of the continuous systems derived in the first part of the paper. 1. Introduction The models introduced in 1976 by M. Gaudin [14] and carrying nowadays his name attracted considerable interest among theoretical and mathematical physicists, playing a distinguished role in the realm of integrable systems. The Gaudin models describe completely integrable classical and quantum long-range interacting spin chains. Originally the Gaudin model was formulated [14] as a spin model related to the Lie algebra sl(2). Later it was realized [15,20] that one can associate such a model with any semi-simple complex Lie algebra g and a solution of the corresponding classical Yang-Baxter equation [5,37]. Depending on the anisotropy of interaction, one distinguishes between XXX, XXZ and XYZ models. Corresponding Lax matrices turn out to depend on the spectral parameter through rational, trigonometric and elliptic functions, respectively. Both the classical and the quantum Gaudin models can be formulated within the r -matrix approach [34]: they admit a linear r -matrix structure, and can be seen as limiting cases of the integrable Heisenberg magnets [39], which admit a quadratic r -matrix structure. In the 80s, the quantum rational Gaudin model was studied by Sklyanin [38] and Jurˇco [20] from the point of view of the quantum inverse scattering method. Sklyanin studied the su(2) rational Gaudin models, diagonalizing the commuting Hamiltonians by means of separation of variables and underlining the connection between his procedure and the functional Bethe Ansatz. In [12] the separation of variables in the rational
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Gaudin model was interpreted as a geometric Langlands correspondence. On the other hand, the algebraic structure encoded in the linear r -matrix algebra allowed Jurˇco to use the algebraic Bethe Ansatz to simultaneously diagonalize the set of commuting Hamiltonians in all cases when g is a generic classical Lie algebra. We mention also the work of Reyman and Semenov-Tian-Shansky [34] who widely studied classical Hamiltonian systems associated with Lax matrices of the Gaudin-type in the context of a general group-theoretic approach. Some other relevant papers on the separability property of Gaudin models are [1,9, 10,17,21,39]. In particular, the results in [9,12] are based on the interpretation of elliptic Gaudin models as conformal field theoretical models (Wess-Zumino-Witten models). As a matter of fact, elliptic Gaudin models played an important role in establishing the integrability of the Seiberg-Witten theory [36] and in the study of isomonodromic problems and Knizhnik-Zamolodchikov systems [11,30,35]. Important recent work on (classical and quantum) Gaudin models includes: • In [10] the bi-Hamiltonian formulation of sl(n) rational Gaudin models has been discussed. A pencil of Poisson brackets has been obtained that recursively defines a complete set of integrals of motion, alternative to the one associated with the standard Lax representation. The constructed integrals coincide, in the sl(2) case, with the Hamiltonians of the bending flows in the moduli space of polygons in the euclidean space introduced in [22]. • In [18] an integrable time-discretization of su(2) rational Gaudin models has been proposed, based on the approach to Bäcklund transformations for finite-dimensional integrable systems developed in [24]. • Integrable q-deformations of Gaudin models have been considered in [4] within the framework of coalgebras. Also the superalgebra extensions of the Gaudin systems have been worked out, see for instance [7,13,29]. • The quantum eigenvalue problem for the gl(n) rational Gaudin model has been studied and a construction for the higher Hamiltonians has been proposed in [41]. • Recently a certain interest in Gaudin models arose in the theory of condensed matter physics. In fact, it has been noticed [2,33] that the BCS model, describing the superconductivity in metals, and the sl(2) Gaudin models are closely related. Finally, we mention the so-called algebraic extensions of Gaudin models, which has been studied in [26,27,31] with the help of a general and systematic reduction procedure based on Inönü-Wigner contractions. These extensions constitute also the subject of the present paper, with a slightly different derivation. Suitable algebraic and pole coalescence procedures performed on the Gaudin Lax matrices with N simple poles, provide various families of integrable models whose Lax matrices have higher order poles but share the linear r -matrix structure with the ancestor models. This technique can be applied for any simple Lie algebra g and whatever the dependence (rational, trigonometric, elliptic) on the spectral parameter is. The models characterized by a single pole of increasing order N and with g = su(2), will be called here the one-body su(2) tower. The base of the rational tower (corresponding to N = 2) is nothing but the Lagrange top, a famous integrable system of classical mechanics. The many-body counterpart of the Lagrange top is called a Lagrange chain, it is a homogeneous integrable chain of Lagrange tops with a long-range interaction. On the other hand, the first element of the elliptic one-body su(2) tower is a particular case of the (three-dimensional) Clebsch system, describing the motion of a free rigid body in an ideal incompressible fluid, see [32].
An Integrable Discretization of the Rational su(2) Gaudin Model
229
A systematic approach to algebraic extensions of Gaudin models appears independently in [8] and [26]. We remark that in [8] only sl(n) Gaudin models are considered and no r -matrix formulation is provided, as opposed to [26]. The present paper is devoted to the construction of an integrable time discretization of the rational su(2) Gaudin model and its one-body and many-body extensions. The theory of integrable maps got a boost when Veselov developed a theory of integrable Lagrangian correspondences [42], symplectic multi-valued transformations possessing many independent integrals of motion in involution. Since then the theory of integrable discretizations has been substantially developed; a systematic presentation of the state of the art is given in [40]. Let us mention the main common features of the discretizations found in the present paper: • They are genuine birational maps, not just correspondences. • They preserve an invariant Poisson structure but deform integrals, so that they are not Bäcklund transformations in the strict sense. However they can be interpreted as Bäcklund transformations for deformations of the original integrable systems. The paper is organized as follows. In Sect. 2 we recall the main features of the continuous-time rational su(2) Gaudin model in order to give a systematic construction of continuous-time one-body and many-body rational su(2) towers in Sect. 3. Section 4 is devoted to the explicit integrable time discretization of the rational su(2) Gaudin model. Then, in Sect. 5, suitable contraction procedures on the discrete Gaudin model allow us to provide integrable discrete-time versions of the whole one-body rational su(2) tower and of the Lagrange chain. In this context, the main goal is the derivation of continuous-time integrable systems and their discretizations: we say practically nothing about solving them. However, we always have in mind one of the motivations of integrable discretizations, namely the possibility of applying integrable Poisson maps for actual numerical computations. Finally, some concluding remarks are contained in Sect. 6. Let us present here our main results. Our departure point is the following Hamiltonian flow of the continuous-time rational su(2) Gaudin model: y˙ i = λi p + Nj=1 y j , yi ,
1 ≤ i ≤ N,
(1)
where yi ∈ su(2), p ∈ su(2) is a constant matrix, and pairwise distinct numbers λi are parameters of the model. This flow admits N independent integrals in involution: Hk = p, yk +
N yk , y j j=1 j=k
λk − λ j
,
1 ≤ k ≤ N,
(2)
where ·, · denotes the scalar product in su(2) R3 . An integrable explicit discretization of the flow (1) is given by −1 yi = (1 + ε λi p) 1 + ε Nj=1 y j yi 1 + ε Nj=1 y j (1 + ε λi p)−1 ,
(3)
with 1 ≤ i ≤ N . Here the hat denotes the shift t → t + ε in the discrete time εZ, where ε is a (small) time step. The map (3) is Poisson w.r.t. the Lie-Poisson brackets
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on ⊕ N su(2)∗ and has N independent and involutive integrals of motion assuring its complete integrability: N N yk , y j ε2 ε 1 + λk λ j p, p − Hk (ε) = p, yk + p, [ yk , y j ] . λk − λ j 4 2 j=1 j=k
j=1 j=k
They are O(ε)-deformations of the original ones, given in Eq. (2). A contraction of N simple poles to one pole of order N provides the integrable flow of the one-body rational su(2) tower,
z˙ i = [ z0 , zi ] + p, zi+1 , 0 ≤ i ≤ N − 1, (4) with the convention z N = 0. Its integrals of motion, (N )
Hk
1 zi , zk−i−1 , 2 k−1
= p, zk +
0 ≤ k ≤ N − 1,
(5)
i=0
are in involution w.r.t. the Lie-Poisson structure obtained through a (generalized) InönüWigner contraction of ⊕ N su(2)∗ , see Eq. (20). An integrable discretization of the flow (4) is given by the following map: zi = (1 + ε z0 ) zi (1 + ε z0 )−1 + ε[ p, zi+1 ] N −i−1 ε j j − −2 adp zi+ j , 0 ≤ i ≤ N − 1. 2
(6)
j=2
This map is explicit (one can compute zi successively, from i = N − 1 to i = 0), and Poisson w.r.t. the bracket (20), preserving therefore the Casimir functions of this bracket. Additionally, it has N independent integrals of motion in involution, assuring its complete integrability: (N )
Hk
(ε) = p, zk +
1 ε ε2 zi , zk−i−1 + p, [ z0 , zk ] + p, p zi+1 , zk−i , 2 2 8 k−1
k−1
i=0
i=0
with 0 ≤ k ≤ N − 1 (these integrals are O(ε)-deformations of (5)). To stress the importance of the flow (4), we note that its simplest instance, corresponding to N = 2, describes the dynamics of the three-dimensional Lagrange top in the rest frame: z˙ 0 = [ p, z1 ],
z˙ 1 = [ z0 , z1 ],
with z0 ∈ R3 being the vector of kinetic momentum of the body, z1 ∈ R3 being the vector pointing from the fixed point to the center of mass of the body, and p being the constant vector along the gravity field. The Lagrange top is a Hamiltonian system w.r.t. the Lie-Poisson bracket on e(3)∗ , with the Hamiltonian function (2)
H1
1 = p, z1 + z0 , z0 . 2
An Integrable Discretization of the Rational su(2) Gaudin Model
231 (2)
Its complete integrability is ensured by the second integral of motion H0 = p, z0 , (2) (2) and by the Casimir functions C0 = z0 , z1 and C1 = 21 z1 , z1 . The map (6) for N = 2 coincides with the integrable discretization of the Lagrange top found in [6]: z0 = z0 + ε[ p, z1 ] ,
z1 = (1 + ε z0 ) z1 (1 + ε z0 )−1 ,
with the deformed Hamiltonian function 1 ε H1(2) (ε) = p, z1 + z0 , z0 + p, [ z0 , z1 ] , 2 2 (all other integrals remain non-deformed in this case). A contraction of N = 2M simple poles to M double poles provides the integrable flow of the Lagrange chain,
M M ˙ i = p, ai + µi p+ k=1 1 ≤ i ≤ M. mk , mi , a˙ i = µi p+ k=1 mk , ai , m Here (mi , ai ) ∈ e(3)∗ and µi ’s are free parameters of the model. (In particular, for M = 1 and µ1 = 0, one recovers again the Lagrange top, upon the re-naming z0 → m1 and z1 → a1 .) The Lagrange chain possesses 2M independent integrals of motion in involution, given in Eqs. (42,43). An explicit discretization is given by −1
M m 1+ε m m ai , m i = (1+ε µi p) 1+ε M (1+ε µi p)−1 +ε p, j i j j=1 j=1 −1 M ai = (1+ε µi p) 1+ε M a 1+ε m m (1+ε µi p)−1 , j i j j=1 j=1 with 1 ≤ i ≤ M. Expressions for the integrals of motion of this Poisson map are given in Eqs. (57,58); they are O(ε)-deformations of the integrals of the continuous system. 2. The Continuous-Time Rational su(2) Gaudin Model The aim of this section is to give a terse survey of the main features of the continuoustime rational su(2) Gaudin model. In particular, we give its Lax representation along with the interpretation of the latter in terms of the (linear) r -matrix structure. For further details we refer to [14,15,20,34]. Let us choose the following basis of the linear space su(2): 1 0 −i 1 0 −1 1 −i 0 , σ2 = , σ3 = . σ1 = 2 −i 0 2 1 0 2 0 i We recall that the correspondence 1 R a = (a , a , a ) ←→ a = 2 3
1
2
3
−i a 3 −i a 1 − a 2 −i a 1 + a 2 i a3
= a α σα ∈ su(2),
is an isomorphism between (su(2), [ ·, · ]) and the Lie algebra (R3 , ×), where × stands for the vector product. (Here and below we assume the summation over the repeated Greek indices.) This allows us to identify vectors from R3 with matrices from su(2). We supply su(2) with the scalar product ·, · induced from R3 , namely a, b =
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− 2tr (ab) = 2 tr (ba† ), ∀ a, b ∈ su(2). The matrix multiplication and the commutator in su(2) are related by the following formula: 1 1 a b = − a, b 1 + [ a, b ], 4 2
∀a, b ∈ su(2).
(7)
In particular, if a, b = 0, then ab + ba = 0. The above scalar product allows us to identify the dual space su(2)∗ with su(2), so that the coadjoint action of the algebra becomes the usual Lie bracket with minus, i.e. ad∗b a = [ a, b ] = −adb a, with a, b ∈ su(2). We will denote by {yiα }3α=1 , 1 ≤ i ≤ N , the coordinate functions (in the basis σα ) on the i th copy of su(2)∗ in ⊕ N su(2)∗ . So, yi = yiα σα . In these coordinates, the Lie-Poisson bracket on ⊕ N su(2)∗ reads
γ β yiα , y j = −δi, j αβγ yi , (8) with 1 ≤ i, j ≤ N . Here δi, j is the standard Kronecker symbol and αβγ is the skewsymmetric tensor with 123 = 1. The bracket (8) possesses N Casimir functions Ci =
1 yi , yi , 2
1 ≤ i ≤ N.
(9)
Fixing their values, we get a symplectic leaf where the Lie-Poisson bracket is non-degenerate. It is a union of N two-dimensional spheres. The continuous-time rational su(2) Gaudin model is governed by the following rational Lax matrix from the loop algebra su(2)[ λ, λ−1 ]: LG (λ) = p +
N i=1
yi , λ − λi
(10)
where the λi ’s, with λi = λk , 1 ≤ i, k ≤ N , are complex parameters of the model, and p ∈ su(2) is a constant vector. This Lax matrix yields a completely integrable system on the Lie-Poisson manifold ⊕ N su(2)∗ . In particular, its spectral invariants are in involution. This can be demonstrated with the help of a linear r -matrix formulation. We quote the following result [20]. Proposition 1. The Lax matrix (10) satisfies the linear r -matrix relation
LG (λ) ⊗ 1, 1 ⊗ LG (µ) + r (λ − µ), LG (λ) ⊗ 1 + 1 ⊗ LG (µ) = 0, ∀ λ, µ ∈ C, (11)
with 1 r (λ) = − σα ⊗ σα . λ
(12)
The r -matrix (12) is equivalent to r (λ) = − /(2 λ), where is the permutation operator in C2 ⊗ C2 .
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233
The spectral invariants of LG (λ) are the coefficients of its characteristic equation det(LG (λ) − µ 1) = 0, which reads −µ2 =
N Hi 1 1 Ci . p, p + + 4 2 λ − λi (λ − λi )2 i=1
Here Ci are the Casimir functions given in Eq. (9), whereas the functions Hi = p, yi +
N yi , y j j=1 j=i
λi − λ j
,
1 ≤ i ≤ N,
(13)
are the independent and involutive Hamiltonians of the rational su(2) Gaudin model. We shall focus our attention on Hamiltonians obtained as linear combinations of the integrals Hi : N
ηi Hi =
i=1
N N 1 ηi − η j yi , y j + ηi p, yi . 2 λi − λ j
(14)
i=1
i, j=1 i= j
An important specialization of the Hamiltonian (14) is obtained considering ηi = λi , 1 ≤ i ≤ N . It reads HG =
N N 1 yi , y j + λi p, yi . 2 i, j=1 i= j
(15)
i=1
From the physical point of view it describes an interaction of su(2) vectors yi (spins in the quantum case) with a homogeneous and constant external field p. One verifies by a direct computation that the Hamiltonian flow generated by the integral (15) is given by y˙ i = λi p + Nj=1 y j , yi ,
1 ≤ i ≤ N.
(16)
Equation (16) admits the following Lax representation: (−) (+) L˙ G (λ) = LG (λ), MG (λ) = − LG (λ), MG (λ) ,
(17)
with the matrix LG (λ) given in Eq. (10) and (−)
MG (λ) =
N λi yi , λ − λi i=1
(+)
MG (λ) = λ p +
N i=1
yi .
(18)
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3. Contractions of Rational su(2) Gaudin Models 3.1. Contraction of the Lie-Poisson algebra ⊕ N su(2)∗ . The following statement allows one to get the generalized Inönü-Wigner contraction of the direct sum of N copies of su(2)∗ [19,26,43]. It shall enable us to construct the rational one-body su(2) tower in Subsect. 3.3. See also [26,27] for further details. Proposition 2. Consider the Lie-Poisson bracket (8) of ⊕ N su(2)∗ (R3 ) N with coorN −1 , given by dinates (y j ) Nj=1 , and a linear map (R3 ) N → (R3 ) N , (y j ) Nj=1 → (zi )i=0 zi = ϑ i
N
ν ij y j ,
0 ≤ i ≤ N − 1,
(19)
j=1
with pairwise distinct ν j ∈ C and 0 < ϑ ≤ 1 (contraction parameter). Then the bracket N −1 induced on (R3 ) N with coordinates (zi )i=0 under the map (19) is regular for ϑ → 0, and tends in this limit to γ
− i + j < N, αβγ z i+ j α β zi , z j = (20) 0 i + j ≥ N, with 0 ≤ i, j ≤ N − 1. We shall denote the Lie-Poisson algebra (20) by C N (su(2)∗ ). Proof. Using Eqs. (8) and (19) we get:
β z iα , z j
ϑ
=ϑ
i+ j
N
j νni νm ynα , ymβ
n,m=1
= −αβγ ϑ
i+ j
N
i+ j νn
γ yn
=
n=1
γ
−αβγ z i+ j O(ϑ)
i + j < N, i + j ≥ N.
The limit ϑ → 0 leads to (20). It is easy to check that the antisymmetric bracket (20) satisfies the Jacobi identity. The following N functions are Casimirs for the Lie-Poisson bracket (20): (N )
Ck
=
N −1 1 zi , z N +k−i−1 , 2
0 ≤ k ≤ N − 1.
(21)
i=k
We illustrate this construction by the cases of small N . For N = 2 the contracted bracket C2 (su(2)∗ ) reads
γ γ β β β z 0α , z 0 = −εαβγ z 0 , (22) z 0α , z 1 = −εαβγ z 1 , z 1α , z 1 = 0. This is the Lie-Poisson bracket of e(3)∗ = su(2)∗ ⊕s R3 . Its Casimir functions are (2)
C0 = z0 , z1 ,
(2)
C1 =
1 z1 , z1 . 2
(23)
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235
For N = 3 we get the contracted Lie-Poisson bracket C3 (su(2)∗ ):
γ γ γ β β β z 0α , z 0 = −εαβγ z 0 , z 0α , z 1 = −εαβγ z 1 , z 0α , z 2 = −εαβγ z 2 , (24a)
γ β β β z 2α , z 2 = 0. (24b) z 1α , z 1 = −εαβγ z 2 , z 1α , z 2 = 0, Its Casimir functions are 1 (3) (3) C0 = z0 , z2 + z1 , z1 , C1 = z1 , z2 , 2 The following result will be useful in the next sections.
(3)
C2 =
1 z2 , z2 . 2
Proposition 3. Let H, G be two involutive functions w.r.t. the Lie-Poisson brackets (8) , G are the corresponding functions on C N (su(2)∗ )) obtained from on ⊕ N su(2)∗ . If H H, G by applying the map (19) in the contraction limit ϑ → 0, then they are in involution w.r.t. the Lie-Poisson brackets (20). Proof. In the local coordinates {yiα }3α=1 , 1 ≤ i ≤ N , we have: 0 = {H, G} =
N N ∂ H ∂G α β ∂ H ∂G γ y = − , y y αβγ i j ∂ yiα ∂ y β ∂ yiα ∂ y β i i, j=1 i=1 j i
= −αβγ = −αβγ
N N −1 ∂G n+m n+m γ ∂H ϑ νi yi β α ∂z n ∂z m i=1 n,m=0 N −1 n,m=0 n+m
∂G γ ∂H z + O(ϑ), β n+m α ∂z n ∂z m
where the first term does not depend explicitly on the contraction parameter ϑ. Perform, G} = 0. ing the limit ϑ → 0 we get { H 3.2. Contraction of the Lie-Poisson algebra ⊕ N M su(2)∗ . The following proposition enables one to get a Lie-Poisson algebra given by the direct sum of M copies of C N (su(2)∗ ) directly from the Lie-Poisson algebra ⊕ N M su(2)∗ associated with a N M-body Gaudin model. Its specialization to M = 1 is equivalent to Proposition 2. Proposition 4. Consider the Lie-Poisson brackets of ⊕ N M su(2)∗ (R3 ) N M with the M , and a linear map (R3 ) N M → (R3 ) N M , (y ) → (z ), given by coordinates (y j ) Nj=1 j i,n zi,n = ϑ i
N
i νN (n−1)+ j y N (n−1)+ j ,
1 ≤ n ≤ M, 0 ≤ i ≤ N − 1,
(25)
j=1
with pairwise distinct ν j ∈ C and 0 < ϑ ≤ 1. Then the bracket induced on (R3 ) N M with coordinates (zi,n ) under the map (25) is regular for ϑ → 0, and tends in this limit to γ
−δ β n,m αβγ z i+ j,n i + j < N , α z i,n , z j,m = (26) 0 i + j ≥ N, with 0 ≤ i, j ≤ N − 1 and 1 ≤ n, m ≤ M. We shall denote the Lie-Poisson algebra (26) by ⊕ M C N (su(2)∗ ).
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M. Petrera, Y. B. Suris
Proof. Using Eqs. (8) and (25) we get:
β α z i,n , z j,m
ϑ
=ϑ
N
i+ j
j
i νN (n−1)+l ν N (m−1)+k
β
y Nα (n−1)+l , y N (m−1)+k
l,k=1 N
= −αβγ ϑ i+ j
γ
j
i νN (n−1)+l ν N (m−1)+k δn,m δl,k y N (n−1)+l .
l,k=1
= −δn,m αβγ ϑ i+ j =
N
l=1 γ −δn,m αβγ z i+ j,n
O(ϑ)
The limit ϑ → 0 leads to (26).
γ
i+ j
ν N (n−1)+l y N (n−1)+l i + j < N, i + j ≥ N.
The Lie-Poisson brackets (26) have N M Casimir functions of the form (21). A computation similar to the one in the proof of Proposition 3 leads to the following statement. Proposition 5. Let H, G be two involutive functions w.r.t. the Lie-Poisson brackets (8) , G are the corresponding functions on ⊕ M C N (su(2)∗ ) obtained on ⊕ N M su(2)∗ . If H from H, G by applying the map (25) in the contraction limit ϑ → 0, then they are in involution w.r.t. the Lie-Poisson bracket (26). 3.3. The rational one-body su(2) tower. Our aim is now to apply the map (19), in the contraction limit ϑ → 0, to the Lax matrix (10), in order to get a new rational Lax matrix governing the rational one-body su(2) tower. To do so a second ingredient is needed: as shown in [23,26] we have to consider the pole coalescence λi = ϑ νi , 1 ≤ i ≤ N . This pole fusion can be considered as the analytical counterpart of the algebraic one given by the map (19). Proposition 6. Consider the Lax matrix (10) with λi = ϑ νi , 1 ≤ i ≤ N . Under the map (19) and upon the limit ϑ → 0 the Lax matrix (10) tends to L N (λ) = p +
N −1 i=0
zi , λi+1
while the Lax equation (17) turns into (−) (+) L˙ N (λ) = L N (λ), M N (λ) = − L N (λ), M N (λ) ,
(27)
(28)
with (−) M N (λ)
=
N −1 i=1
zi , λi
(+)
M N (λ) = λ p + z0 .
The Lax matrix (27) satisfies the linear r -matrix relation (11) with the same r -matrix (12).
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237
Proof. The first part of Proposition 6 can be proved by applying the map (19) and the pole coalescence λi = ϑ νi , 1 ≤ i ≤ N , on Eqs. (10) and (18). We get N N N −1 yj 1 ϑ νj i ϑ→0 = y j + O(ϑ) −−−→ L N (λ), LG (λ) = p + λ − ϑ νj λ λ j=1
j=1 i=0
and N ϑ νj yj MG (λ) = λ − ϑ νj (−)
j=1
=
N N −2 ϑ ν j i+1 λ
j=1 i=0 (+)
MG (λ) = λ p +
N
ϑ→0
y j + O(ϑ) −−−→
N −2 i=0
zi+1 (−) = M N (λ), λi+1
(+)
yi = M N (λ).
i=1
The fact that the Lax matrix (27) satisfies the linear r -matrix relation (11) with the same r -matrix (12) requires a longer but straightforward computation. We refer to [26,27,31] for a detailed proof. The Hamiltonian flow described by the Lax equation (28) is given by
z˙ i = [ z0 , zi ] + p, zi+1 , 0 ≤ i ≤ N − 1,
(29)
with z N = 0, while the characteristic equation of the Lax matrix det(L N (λ) − µ 1) = 0 reads − µ2 =
N −1 N −1 (N ) (N ) 1 Hk 1 1 Ck p, p + + , 4 2 λk+1 2 λk+N +1 k=0
where the functions functions
(N ) Ck ,
k=0
0 ≤ k ≤ N − 1, are the Casimir functions (21), while the
Hk(N ) = p, zk +
1 zi , zk−i−1 , 2 k−1
(30)
i=0
are the N independent involutive Hamiltonians of the rational one-body su(2) tower. Notice that it is possible to obtain the integrals (30) using the map (19), in the contraction limit ϑ → 0, and the pole coalescence λi = ϑ νi , 1 ≤ i ≤ N , from the integrals (13). Let us fix i such that 0 ≤ i ≤ N − 1. We get N
ϑ i νki Hk =
k=1
N
ϑ i νki p, yk +
k=1
=
N
i i N 1 i−1 νk − ν j ϑ yk , y j 2 νk − ν j j,k=1 j=k
ϑ
i
νki p, yk
k=1
= p, zi +
i−1 N 1 + (ϑ νk )m (ϑ ν j )i−m−1 yk , y j 2 m=0 j,k=1 j=k
i−1 1 zm , zi−m−1 = Hi(N ) . 2 m=0
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In the above computation we have taken into account the polynomial identity νki − ν ij = (νk − ν j )
i−1
νkm ν i−m−1 . j
m=0
The contracted version of the Hamiltonian (15) is given by H1(N ) , namely the integral of motion generating the Hamiltonian flow given in Eq. (29), while the contracted N N (N ) version of the linear integral k=1 Hk = k=1 p, yk is given by H0 . Let us remark that the involutivity of the spectral invariants of the Lax matrix L N (λ) is indeed ensured thanks to the r -matrix formulation (11). Their involutivity can be proved also without using the r -matrix approach, just by referring to Proposition 3. 3.3.1. N = 2, the Lagrange top Fixing N = 2 in the formulae of the previous subsection we recover the well-known dynamics of the three-dimensional Lagrange top described in the rest frame [3,6,16,23,34,40]. In other words the Lagrange top is the first element of the rational one-body su(2) tower. The Lagrange case of the rigid body motion around a fixed point in a homogeneous field is characterized by the following data: the inertia tensor is given by diag(1, 1, I3 ), I3 ∈ R, which means that the body is rotationally symmetric w.r.t. the third coordinate axis, and the fixed point lies on the symmetry axis. The equations of motion (in the rest frame) are given by: z˙ 0 = [ p, z1 ],
z˙ 1 = [ z0 , z1 ],
(31)
where z0 ∈ R3 is the vector of kinetic momentum of the body, z1 ∈ R3 is the vector pointing from the fixed point to the center of mass of the body and p is the constant vector along the gravity field. An external observer is mainly interested in the motion of the symmetry axis of the top on the surface z1 , z1 = constant. A remarkable feature of the equations of motion (31) is that they do not depend explicitly on the anisotropy parameter I3 of the inertia tensor [6]. Moreover they are Hamiltonian equations w.r.t. the Lie-Poisson brackets on e(3)∗ , see Eq. (22). The Hamiltonian function that generates the equations of motion (31) is given by (2)
H1
1 = p, z1 + z0 , z0 , 2
(32)
and the complete integrability of the model is ensured by the second integral of motion H0(2) = p, z0 . These involutive Hamiltonians can be obtained using Eq. (30) with N = 2, namely considering the spectral invariants of the Lax matrix L2 (λ), see Eq. (27). The remaining two spectral invariants are given by the Casimir functions (23). 3.3.2. N = 3, the first extension of the Lagrange top Let us now consider the dynamical system governed by the Lax matrix (27) with N = 3. The Lie-Poisson brackets are explicitly given in Eqs. (24a,24b). According to Eq. (30) the involutive Hamiltonians are: (3)
H0
= p, z0 ,
(3)
H1
1 = p, z1 + z0 , z0 , 2
(3)
H2
= p, z2 + z0 , z1 .
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239
Looking at the brackets (24a,24b) and taking into account that z0 and z2 span respectively su(2)∗ and R3 , we may interpret them as the total angular momentum of the system and the vector pointing from a fixed point (which we shall take as (0, 0, 0) ∈ R3 ) to the centre of mass of a Lagrange top. Let us remark that z0 does not coincide with the angular momentum of the top due to the presence of the vector z1 . We think of z1 , whose norm is not constant, as the position of the moving centre of mass of the system composed by the Lagrange top and a satellite, whose position is described by z1 − z2 . (3) Here we are assuming that both bodies have unit masses. Notice that the integral H1 formally coincides with the physical Hamiltonian of the Lagrange top (32) where now the vector z0 is the angular momentum of system and the vector z1 describes the motion of the total centre of mass. (3) According to Eq. (29) the Hamiltonian flow generated by the integral H1 reads z˙ 0 = [ p, z1 ],
z˙ 1 = [ z0 , z1 ] + [ p, z2 ],
z˙ 2 = [ z0 , z2 ].
We see that the vector z1 does not rotate rigidly, though z2 does. 3.4. The rational many-body su(2) tower. The rational many-body su(2) tower may be constructed simply regarding the Lax matrix (27) as the local matrix of a chain of many, say M, copies of the Lie-Poisson structure C N (su(2)∗ ). Indeed the r -matrix formulation (11) ensures that the Lax matrix L M,N (λ) = p +
M N −1 k=1 i=0
zi,k , (λ − µk )i+1
with pairwise distinct poles µk of order N describes an integrable system defined on ⊕ M C N (su(2)∗ ) with the same r -matrix formulation (11). See [26,31] for further details. Let us consider the special case N = 2, namely the Lie-Poisson algebra given by ⊕ M e(3)∗ . The resulting integrable system has been called Lagrange chain in [26,28]. We now present a new derivation of such a system without using the r -matrix approach, but just considering the contraction procedure of a rational su(2) Gaudin model defined on ⊕2M su(2)∗ . According to Proposition 4 the contraction of the direct sum of 2M copies of su(2)∗ (i.e. N = 2) leads to the Lie-Poisson brackets on ⊕ M e(3)∗ . It is convenient to simplify the notation: z0,k = mk ,
z1,k = ak ,
1 ≤ k ≤ M.
(33)
We interpret mk = (m 1k , m 2k , m 3k ) ∈ R3 and ak = (ak1 , ak2 , ak3 ) ∈ R3 as, respectively, the angular momentum and the vector pointing from the fixed point to the center of mass of the k th top. The Lie-Poisson bracket on ⊕ M e(3)∗ is:
β
m αk , m j
γ
= −δk, j εαβγ m k ,
β
m αk , a j
γ
= −δk, j εαβγ ak ,
β akα , a j = 0,
(34)
with 1 ≤ k, j ≤ M. This bracket possesses 2M Casimir functions: (1)
Q k = mk , ak ,
(2)
Qk =
1 ak , ak , 2
1 ≤ k ≤ M.
(35)
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Using the notation introduced in Eq. (33), the Lax matrix of the Lagrange chain reads L M,2 (λ) = p +
M i=1
mi ai + λ − µi (λ − µi )2
.
(36)
Let us now consider a rational su(2) Gaudin model with 2M poles. We have to apply 2M : the map defined in Eq. (25) to the set of R3 vectors {yi }i=1 (z0 )i = mi = y2i + y2i−1 ,
(z1 )i = ai = ϑ (ν2i y2i + ν2i−1 y2i−1 ),
(37)
with 1 ≤ i ≤ M. Moreover we define the following pole coalescence: λ2i = µi + ϑ ν2i ,
λ2i−1 = µi + ϑ ν2i−1 ,
1 ≤ i ≤ M,
(38)
where the λi are the 2M parameters of the rational su(2) Gaudin model. Proposition 7. Consider the Lax equation (17) with the pole coalescence (38). Under the map (19) and upon the limit ϑ → 0 it tends to (−) (+) L˙ M,2 (λ) = L M,2 (λ), M M,2 (λ) = − L M,2 (λ), M M,2 (λ) , (39) with the matrix L M,2 (λ) given by Eq. (36) and (−)
M M,2 (λ) =
M i=1
1 λ − µi
µi mi +
λ ai λ − µi
,
(+)
M M,2 (λ) = λ p +
M
mi .
(40)
i=1
Proof. We have: LG (λ) = p +
M i=1
= p+
y2i−1 y2i + λ − µi − ϑ ν2i−1 λ − µi − ϑ ν2i
M y2i−1 + y2i i=1
λ − µi
M ϑ (ν2i y2i + ν2i−1 y2i−1 ) ϑ→0 + + O(ϑ) −−−→ L M,2 (λ). (λ − µi )2 i=1
(±)
A similar computation leads to the auxiliary matrices M M,2 in Eq. (40) starting from the ones in Eq. (18). The Hamiltonian flow described by the Lax equation (39) is given by
M M ˙ i = p, ai + µi p + k=1 a˙ i = µi p + k=1 m mk , mi , mk , ai ,
(41)
with 1 ≤ i ≤ M, while the characteristic equation det(L M,2 (λ) − µ 1) = 0 reads 1 1 − µ = p, p + 4 2 M
2
k=1
(1)
(2)
Qk Qk Sk Rk + + + 2 3 λ − µk (λ − µk ) (λ − µk ) (λ − µk )4
,
An Integrable Discretization of the Rational su(2) Gaudin Model (1)
241
(2)
where the functions Q k , Q k are the Casimir functions (35), and the functions Rk = p, mk +
M mk , m j j=1 j=k
µk − µ j
mk , a j − m j , ak ak , a j , + −2 (µk − µ j )2 (µk − µ j )3
M ak , a j ak , m j 1 , Sk = p, ak + mk , mk + + 2 µk − µ j (µk − µ j )2
(42)
(43)
j=1 j=k
are the 2M independent and involutive Hamiltonians of the Lagrange chain. Notice that, as in the su(2) rational Gaudin model, there is a linear integral given by M M k=1 Rk = k=1 p, mk . A possible choice for a physical Hamiltonian describing the dynamics of the model can be constructed considering a linear combination of the Hamiltonians Rk and Sk similar to the one considered in Eq. (14). We have: H M,2 =
M M M 1 (µk Rk + Sk ) = p, µk mk + ak + mi , mk . 2 k=1
k=1
(44)
i,k=1
It is easy to check that the integral (44) generates the Hamiltonian flow (41). If M = 1, the Hamiltonian (44) gives the sum of the two integrals of motion of the Lagrange top. We can construct the integrals of motion of the Lagrange chain also by using the Lie-Poisson map (25) with the pole coalescence (38) directly in the Hamiltonians (13), according to Ri = lim [ H2i + H2i−1 ], ϑ→0
Si = lim [ ϑ (ν2i H2i + ν2i−1 H2i−1 ) ]. ϑ→0
(45)
4. Discrete-Time Rational su(2) Gaudin Models The main goal of this section is the construction of an integrable Poisson map discretizing the Hamiltonian flow (16). We shall provide an explicit map approximating, for a small discrete-time step ε, the time ε shift along the trajectories of the equations of motion (16) generated by the Hamiltonian function (15). We have to remark that no Lax representation (hence no r -matrix formulation) has been found for this map. Its Poisson property and integrability will be proved by direct inspection. Proposition 8. The map −1 DεN : yi → yi = (1 + ε λi p) 1 + ε Nj=1 y j yi 1 + ε Nj=1 y j (1 + ε λi p)−1 , (46) with 1 ≤ i ≤ N and ε ∈ R, is Poisson w.r.t. the brackets (8) on ⊕ N su(2)∗ and has N independent and involutive integrals of motion assuring its complete integrability: N N yk , y j ε2 ε 1 + λk λ j p, p − Hk (ε) = p, yk + p, [ yk , y j ] , (47) λk − λ j 4 2 j=1 j=k
with 1 ≤ k ≤ N .
j=1 j=k
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M. Petrera, Y. B. Suris
Proof. Let us first notice that the map (46) reproduces at order ε the continuous-time Hamiltonian flow (16). The map (46) is the composition of two non-commuting conjugations: DεN = (DεN )2 ◦ (DεN )1 , where −1 , (DεN )1 : yi → yi∗ = 1 + ε Nj=1 y j yi 1 + ε Nj=1 y j
(48)
yi = (1 + ε λi p) yi∗ (1 + ε λi p)−1 , (DεN )2 : yi∗ →
(49)
with 1 ≤ i ≤ N . Notice that (DεN )1 ◦ (DεN )2 = (DεN )2 ◦ (DεN )1 . The Poisson property of the map DεN is a consequence of the Poisson property of the maps (DεN )1 and (DεN )2 . In fact (DεN )1 is a shift along a Hamiltonian flow on ⊕ N su(2)∗ w.r.t. the Hamiltonian Nj=k=1 y j , yk . On the other hand (DεN )2 is a shift along a N Hamiltonian flow on ⊕ N su(2)∗ w.r.t. the Hamiltonian k=1 p, λk yk∗ . Therefore the N N composition (Dε )2 ◦ (Dε )1 is a Poisson map w.r.t. the bracket (8). Let us now prove the complete integrability of the map (46). We show that the functions (47) are indeed integrals of the map (46). Their independence is clear, while their involution w.r.t. the brackets (8) is proved in Appendix 2. Notice that the maps (48), (49) imply, respectively, the following relations:
yi∗ , y∗j = yi , y j , p, y j = pi , y∗j ,
yi∗ +
ε ∗ ε
yi , y j = yi + y j , yi , 2 2 N
N
j=1
j=1
ε
ε yi , p = yi∗ + λi p, yi∗ , yi + λi 2 2
(50)
(51)
with 1 ≤ i, j ≤ N . The preservation of the functions (47) is demonstrated by the following computation: N N yk , yj ε2 ε 1 + λk λ j p, p − p, [ yk , yj] λk − λ j 4 2
k (ε) = p, yk + H
= p, yk∗ +
= p, yk +
j=1 j=k
j=1 j=k
j=1 j=k
j=1 j=k
N y∗ , y∗ N ε2 ε k j 1 + λk λ j p, p + p, [ yk∗ , y∗j ] λk − λ j 4 2
N yk , y j j=1 j=k
λk − λ j
N ε2 ε 1 + λk λ j p, p − p, [yk , y j ] = Hk (ε), 4 2 j=1 j=k
with 1 ≤ k ≤ N . Here we have used Eq. (51) in the first step and Eq. (50) in the second one.
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243
Using the discrete Hamiltonians (47) we can compute the discrete-time version of the Hamiltonian (15). It reads: HG (ε) =
N k=1
N N 1 ε2 λk λ j p, p λk Hk (ε) = p, λk yk + yk , y j 1 + 2 4 k=1
−
j,k=1 j=k
N ε (λk − λ j ) p, [ yk , y j ] . 4 j,k=1 j=k
Moreover we still have a linear integral given by the continuous-time case.
N k=1
Hk (ε) =
N
k=1 p, yk
, as in
5. Contractions of Discrete-Time Rational su(2) Gaudin Models Performing the contraction procedures presented in Subsects. 3.1 and 3.2 we can now construct the integrable discrete-time versions of the Hamiltonian flows (29) and (41) of the whole rational one-body su(2) tower and of the Lagrange chain. 5.1. The discrete-time one-body su(2) tower. The integrable Poisson map discretizing the flow (29) of the rational one-body su(2) tower is given in the following proposition. Proposition 9. The map εN : zi → D zi = (1 + ε z0 ) zi (1 + ε z0 )−1 − 2
N −i−1
−
j=1
εj j adp zi+ j , 2
(52)
with 0 ≤ i ≤ N − 1 and ε ∈ R, is Poisson w.r.t. the brackets (20) on C N (su(2)∗ ) and has N independent and involutive integrals of motion assuring its complete integrability: Hk(N ) (ε) = p, zk +
1 ε ε2 zi , zk−i−1 + p, [ z0 , zk ] + p, p zi+1 , zk−i , 2 2 8 k−1
k−1
i=0
i=0
(53) with 0 ≤ k ≤ N − 1. Proof. Let us construct the map (52) through the usual contraction procedure and the pole coalescence λi = ϑ νi , 1 ≤ i ≤ N , performed on the map (46). Consider the map (DεN )1 in Eq. (48). Using the map (19) and assuming λi = ϑ νi , 1 ≤ i ≤ N , we get: zi∗
=
N k=1
ϑ
i
νki
yk∗
=
N
−1 ϑ i νki 1 + ε Nj=1 y j yk 1 + ε Nj=1 y j
k=1
= (1 + ε z0 ) zi (1 + ε z0 )−1 ,
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with 0 ≤ i ≤ N − 1. Hence the contracted version of (DεN )1 is given by εN )1 : zi → z∗ = (1 + ε z0 ) zi (1 + ε z0 )−1 , (D i
0 ≤ i ≤ N − 1.
On the other hand, a direct computation, with the help of Eq. (7), yields the contracted version of the map (DεN )2 in Eq. (49): N
ϑ i νki yk =
k=1
N
ϑ i νki (1 + ε ϑ νk p) yk∗ (1 + ε ϑ νk p)−1
k=1
=
N
i+ j
ϑ i+ j νk
(−ε) j (1 + ε ϑ νk p) yk∗ p j
k=1 j≥0
=
zi∗
+2
N −i−1 j=1
εj j ∗ adp zi+ j + O(ϑ), 2
with 0 ≤ i ≤ N − 1. Performing the limit ϑ → 0 we have: εN )2 : z∗ → zi = zi∗ + 2 (D i
N −i−1 j=1
εj j ∗ adp zi+ j , 2
0 ≤ i ≤ N − 1.
εN )1 is easily verified to result in the map D εN given εN )2 ◦ (D Now the composition (D N in Eq. (52). The Poisson property of the map Dε is a consequence of the one of the map DεN in Eq. (46). Next, we construct, by contraction of the functions (47), the integrals of the Poisson map (52). We know that fixing ε = 0 in Eq. (47) we recover the Hamiltonians (13) of the continuous-time su(2) rational Gaudin model. Their contraction gives the Hamiltonians (30) of the continuous-time rational one-body su(2) tower. Therefore it is enough to perform the contraction procedure just on the two ε-dependent terms of the integrals (47). We have: N
ϑ i νki Hk (ε) = p, zi +
i−1 1 zm , zi−m−1 2 m=0
k=1
−
+
ε 4
N
(ϑ i νki − ϑ i ν ij ) p, [ yk , y j ]
j,k=1 j=k
N νki+1 ν j − ν i+1 ε2 j νk p, p ϑ i+1 yk , y j 8 νk − ν j j,k=1 j=k
= p, zi +
i−1 1 ε zm , zi−m−1 + p, [ z0 , zi ] 2 2 m=0
+ with 0 ≤ i ≤ N − 1.
ε2 p, p 8
i−1
N
m=0 j,k=1 j=k
(N )
(ϑ νk )m+1 (ϑ ν j )i−m yk , y j = Hi
(ε),
An Integrable Discretization of the Rational su(2) Gaudin Model (N )
The involutivity of the integrals {Hk
245
N −1 (ε)}k=0 is ensured thanks to Proposition 3.
Let us remark that the specialization to N = 2 of the map (52) gives the integrable time-discretization of the Lagrange top found by Bobenko and Suris in [6]. According to Eq. (52) it reads:
z0 = z0 + ε p, z1 ,
z1 = (1 + ε z0 ) z1 (1 + ε z0 )−1 .
(54)
The above explicit map approximates, for small ε, the time ε shift along the trajectories of the Hamiltonian flow (31). This distinguishes the situation from the map in [25], where Lagrangian equations led to correspondences rather than to maps. The map (54) is Poisson w.r.t. the bracket (22) on e(3)∗ and its complete integrability is ensured by the integrals of motion (2)
H0
(2)
= p, z0 ,
H1 (ε) =
1 ε z0 , z0 + p, z1 + p, [ z0 , z1 ] . 2 2
(55)
A remarkable feature of the map (54) is that it admits a Lax representation and the same linear r -matrix bracket (11) as in the continuous case, see [6] for further details. The Lax matrix of the map is a deformation of the Lax matrix of the Lagrange top.
5.2. The discrete-time Lagrange chain. The integrable Poisson map discretizing the flow (41) of the Lagrange chain is given in the following proposition. Proposition 10. The map −1
M m i = (1 + ε µi p) 1 + ε M m 1 + ε m m ai , (1 + ε µi p)−1 + ε p, j i j j=1 j=1 −1 M ai = (1 + ε µi p) 1 + ε M a 1 + ε m m (1 + ε µi p)−1 , j i j j=1 j=1
(56a) (56b)
with 1 ≤ i ≤ M and ε ∈ R, is Poisson w.r.t. the brackets (34) on ⊕ M e(3)∗ and has 2M independent and involutive integrals of motion assuring its complete integrability: ε Rk (ε) = p, mk − p, mk , M j=1 m j 2 M ak , a j mk , m j ε2 1 + µ + −2 µ p, p k j µk − µ j (µk − µ j )3 4 j=1 j=k
mk , a j + (µk − µ j )2
m j , ak ε2 2 ε2 2 1+ 1+ µ p, p − µ p, p , 4 k (µk − µ j )2 4 j (57)
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M. Petrera, Y. B. Suris
1 ε2 2 ε Sk (ε) = p, ak + mk , mk 1 + µk p, p − p, ak , M j=1 m j 2 4 2 M ak , m j ε2 + µk µ j p, p 1+ µk − µ j 4 j=1 j=k
+
ak , a j (µk − µ j )2
1+
ε2 2 µk p, p , 4
(58)
with 1 ≤ k ≤ M. Proof. Using the map (37) and the pole coalescence (38) in the map (48) with N = 2M we immediately obtain the contracted version of (Dε2M )1 . It reads −1 M ∗ ∗ mi∗ = y2i + y2i−1 = 1+ε M , j=1 m j mi 1 + ε j=1 m j −1 M ∗ ∗ ai∗ = ϑ (ν2i y2i + ν2i−1 y2i−1 )= 1+ε M , j=1 m j ai 1 + ε j=1 m j
(59a) (59b)
with 1 ≤ i ≤ M. The same procedure leads to the contracted version of (Dε2M )2 . It reads m i = y2i + y2i−1 = (1 + ε µi p) mi∗ (1 + ε µi p)−1 + ε p, (1 + ε µi p) ai∗ (1 + ε µi p)−1 + O(ϑ), ai = ϑ (ν2i y2i + ν2i−1 y2i−1 ) = (1 + ε µi p)
ai∗
(1 + ε µi p)
(60a) −1
+ O(ϑ).
(60b)
Performing the limit ϑ → 0 in Eqs. (60a,60b) and combining the resulting equations with the maps in Eqs. (59a,59b) we obtain the map (56a,56b). Its Poisson property is ensured thanks to the Poisson property of the map (46). The construction of the discrete Hamiltonians (57,58) is similar to the one done for the continuous-time Lagrange chain. They can be obtained through the following formulae by a straightforward computation: Ri (ε) = lim [ H2i (ε) + H2i−1 (ε) ], ϑ→0
Si (ε) = lim [ ϑ (ν2i H2i (ε) + ν2i−1 H2i−1 (ε)) ], ϑ→0
2M being the Hamiltonians (47). {Hi (ε)}i=1 Let us finally notice that the Hamiltonians (57,58) are in involution w.r.t. the brackets (34) thanks to Proposition 5.
The discrete-time version of the Hamiltonian (44) is given by H M,2 (ε) =
M [ µk Rk (ε) + Sk (ε) ] k=1
M M 1 ε2 µ j µk p, p = p, µk mk + ak + m j , mk 1 + 2 4 k=1
j,k=1
An Integrable Discretization of the Rational su(2) Gaudin Model
−
247
M M ε ε M (µk − µ j ) p, [ mk , m j ] − p, k=1 ak , j=1 m j 4 2 j,k=1 j=k
+
M M ε2 ε2 p, p p, p µk mk , a j + ak , a j . 4 8 j,k=1 j=k
Notice that we still have the linear integral
j,k=1 j=k
M k=1
Rk (ε) =
M
k=1 p, mk
.
6. Concluding Remarks We presented a systematic construction of finite-dimensional integrable systems sharing the same linear r -matrix bracket with the rational su(2) Gaudin model. The resulting one-body and many-body integrable systems are obtained through suitable algebraic contractions of the Lie-Poisson structure of the ancestor model. We called these families of integrable systems su(2) towers. The three-dimensional Lagrange top is the first element of the rational one-body su(2) tower. The many-body counterpart of the Lagrange top, called Lagrange chain, is also presented and its Lax representation is given. In the second part of the paper we derived an explicit integrable Poisson map discretizing a Hamiltonian flow of the rational su(2) Gaudin model, thus providing a new integrable discretization of such a model. Then, the contraction procedures enable us to construct integrable discrete-time versions of the of the rational su(2) tower and of the Lagrange chain. The main open problem connected with this work is to find Lax representations (and then their r -matrix interpretation) for all the integrable Poisson maps introduced here (actually the only case for which the Lax representation is known is the discrete-time Lagrange top considered in [6]). These structures will allow to avoid a brute force verification of the integrability, which we had to perform here. Of course, finding a Lax representation for the discrete-time rational su(2) Gaudin model would yield the corresponding results for all the contracted systems. Also the following problem deserves further investigations. It is well-known that the continuous-time rational Gaudin models, as well as the one-body and many-body towers [31] described in the present work, admit a multi-Hamiltonian formulation [10]. Finding a multi-Hamiltonian formulation of our discrete-time maps is an open challenge.
Appendix 1: Visualization As shown in Proposition 9, the integrable Poisson map (52) discretizing the rational one-body su(2) tower is explicit and can be easily iterated. We present here its visualization in the case N = 2 (Lagrange top), see Fig. 1 and N = 3 (the first extension of the Lagrange top), see Fig. 2. On Fig. 3 we provide a visualization of the integrable discrete-time evolution of the axes of symmetry of the tops of a Lagrange chain with M = 2, given by the map (56a,56b).
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Fig. 1. Discrete time Lagrange top: map (52) with N = 2. Typical plots of the motion of the symmetry axis z1 on the surface z1 , z1 =constant for various initial data. Observe the typical precession
Fig. 2. First extension of the discrete time Lagrange top: map (52) with N = 3. Typical plots of the symmetry axis of the top z2 on the surface z2 , z2 =constant and of the satellite motion z1 − z2
Appendix 2: Proof of the Involutivity of the Functions {Hk (ε)} kN= 1 N given in Eq. (47) in the following way: Let us write the functions {Hk (ε)}k=1
Hk (ε) = h 0k −
ε 1 ε2 h + p, p h 2k , 2 k 4
1 ≤ k ≤ N,
where h 0k = p, yk +
N yk , y j j=1 j=k
h 1k =
N j=1 j=k
λk − λ j
p, [ yk , y j ] ,
,
(61a)
(61b)
An Integrable Discretization of the Rational su(2) Gaudin Model
249
Fig. 3. Discrete time Lagrange chain: map (56a,56b) with M = 2. A typical plot of the evolution of the points a1 and a2 on the respective surfaces a1 , a1 =constant and a2 , a2 =constant
h 2k
N λk λ j = yk , y j . λk − λ j
(61c)
j=1 j=k
In the following computations we shall use the Lie-Poisson brackets (8). We have:
ε
{Hk (ε), Hi (ε)} = h 0k , h i0 − h 0k , h i1 + h 1k , h i0 2
ε2 + p, p h 0k , h i2 + h 2k , h i0 + h 1k , h i1 4
ε3 1 2 2 1 ε4 hk , hi + hk , hi p, p 2 h 2k , h i2 . − + 8 16
(62)
N We already know that h 0k , h i0 = 0, 1 ≤ k, i ≤ N , since the integrals h 0k k=1 are the ones of the continuous-time su(2) rational Gaudin model. Let us compute the remaining brackets in Eq. (62) (61a,61b,61c) k = i. Notice that in the using Eqs. and assuming brackets h 0k , h i1 + h 1k , h i0 and h 0k , h i2 + h 2k , h i0 we shall explicitly write the order of |p| = p, p 1/2 = p appearing in the computation. At order ε we have:
h 0k , h i1 + h 1k , h i0 β
= p βρσ
N N j=1 l=1 j=k l=i
O(|p|)
1 1 ρ σ α α ρ σ α α y y ,y y + y y ,y y λk − λ j k j i l λi − λl k j i l
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M. Petrera, Y. B. Suris N
= − p β βρσ ασ γ
1 γ γ ρ ρ (y y α y + y j yi ykα ) λk − λ j k j i
j=1 j=k
− p β βρσ ραγ
N
1 γ γ (y y σ y α + yi y σj ykα ) λi − λk k j i
j=1 j=k
− p β βρσ σ αγ
N j=1 j=k
1 γ ρ γ ρ (y y y α + yi y αj yk ). λi − λ j k j i
The above expression vanishes if we swap the indices α and γ in each second term in the three brackets. Then we have: N
α ρ σ ρ σ α α β h 0k , h i1 + h 1k , h i0 = p p yk , yi yl + yk yl , yi βρσ 2 O(|p| )
l=1 γ
ρ
= p α p β (βρσ ασ γ + βγ σ σ αρ ) yk yi , that vanishes due to the properties of the tensor αβγ . At order ε2 we get:
h 0k , h i2 + h 2k , h i0 O(|p|)
N N λk λ j α β β λi λl α β β yk , yi yl − p α y ,y y = pα λi − λl λk − λ j i k j l=1 l=i
= − p α αβγ
j=1 j=k
λi λk β γ λi λk γ β yi yk + yi yk , λi − λk λi − λk
that vanishes swapping the indices γ and β in the second term. Moreover,
h 0k , h i2 + h 2k , h i0
= − αβγ
N N j=1 l=1 j=k l=i
λi λl + λk λ j β β ykα y αj , yi yl (λk − λ j )(λi − λl )
N λ (λ2 − λ2 ) − λ (λ2 − λ2 ) − λ (λ2 − λ2 ) k i i k j i j j k
(λk − λ j )(λi − λk )(λi − λ j )
j=1 j=k
= − αβγ
O(|p|0 )
=
N
γ
γ
β
yk y αj yi
β
yk y αj yi .
j=1 j=k
On the other hand: N N N
γ γ β ρ µ h 1k , h i1 = p α p σ αβγ σρµ ykα y j , yi yl = p σ p σ αβγ yk y αj yi , j=1 l=1 j=k l=i
j=1 j=k
An Integrable Discretization of the Rational su(2) Gaudin Model
251
where we have used the properties of the tensor αβγ . Hence we get: p, p
h 0k , h i2 + h 2k , h i0 + h 1k , h i1 = 0.
At order ε3 we have:
h 1k , h i2 + h 2k , h i1 = − p β βρσ
N N λk λ j α α ρ σ λi λl ρ σ α α yk y j , yi yl + yk y j , yi yl λk − λ j λi − λl j=1 l=1 j=k l=i
= p β βρσ ασ γ
N λk λ j γ γ ρ ρ (y y α y + y j yi ykα ) λk − λ j k j i j=1 j=k
− p β βρσ ραγ
N λk λi γ γ (y y σ y α + yi y σj ykα ) λi − λk k j i j=1 j=k
− p β βρσ σ αγ
N λi λ j γ ρ γ ρ (y y y α + yi y αj yk ). λi − λ j k j i j=1 j=k
The above expession vanishes if we swap the indices α and γ in each second term in the three brackets. Finally, at order ε4 , we get:
N N
h 2k , h i2 = j=1 l=1 j=k l=i
= −αβγ
N j=1 j=k
λk λ j λi λl β β ykα y αj , yi yl (λk − λ j )(λi − λl )
β y αj yi
γ yk
λ2k λ j λi (λi − λk )(λk − λ j )
λk λ2j λi λk λ j λi2 − + (λi − λ j )(λk − λi ) (λi − λ j )(λk − λ j )
.
A direct computation shows that the expression in the square brackets vanishes. Acknowledgements. M.P. wishes to express his gratitude to F. Musso, for his constant help and support. M.P. is also grateful to G. Satta and O. Ragnisco for many interesting discussions. M.P. was partially supported by the European Community through the FP6 Marie Curie RTN ENIGMA (Contract number MRTN-CT-2004-5652) and by the European Science Foundation project MISGAM.
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Commun. Math. Phys. 283, 255–284 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0497-2
Communications in
Mathematical Physics
Global Solutions to the Three-Dimensional Full Compressible Magnetohydrodynamic Flows Xianpeng Hu, Dehua Wang Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA. E-mail:
[email protected];
[email protected] Received: 7 August 2007 / Accepted: 17 October 2007 Published online: 6 May 2008 – © Springer-Verlag 2008
Abstract: The equations of the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows are considered in a bounded domain. The viscosity coefficients and heat conductivity can depend on the temperature. A solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global variational weak solution to the three-dimensional full magnetohydrodynamic equations with large data is established. 1. Introduction Magnetohydrodynamics, or MHD, studies the dynamics of electrically conducting fluids and the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. The applications of magnetohydrodynamics cover a very wide range of physical areas from liquid metals to cosmic plasmas, for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics, and plasma physics. Astrophysical problems include solar structure, especially in the outer layers, the solar wind bathing the earth and other planets, and interstellar magnetic fields. The primary geophysical problem is planetary magnetism, produced by currents deep in the planet, a problem that has not been solved to any degree of satisfaction. Magnetohydrodynamics is of importance in connection with many engineering problems as well, such as sustained plasma confinement for controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, magnetohydrodynamic power generation, electro-magnetic casting of metals, and plasma accelerators for ion thrusters for spacecraft propulsion. Due to their practical relevance, magnetohydrodynamic problems have long been the subject of intense cross-disciplinary research, but except for relatively simplified special cases, the rigorous mathematical analysis of such problems remains open. In magnetohydrodynamic flows, magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself.
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There is a complex interaction between the magnetic and fluid dynamic phenomena, and both hydrodynamic and electrodynamic effects have to be considered. The set of equations which describe compressible viscous magnetohydrodynamics are a combination of the compressible Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. In this paper, we consider the full system of partial differential equations for the three-dimensional viscous compressible magnetohydrodynamic flows in the Eulerian coordinates ([19,20]): ρt + div(ρu) = 0, (ρu)t + div (ρu ⊗ u) + ∇ p = (∇ × H) × H + div, Et + div u(E + p) = div ((u × H) × H + νH × (∇ × H) + u + κ∇θ ) , Ht − ∇ × (u × H) = −∇ × (ν∇ × H), divH = 0,
(1.1a) (1.1b) (1.1c) (1.1d)
where ρ denotes the density, u ∈ R3 the velocity, H ∈ R3 the magnetic field, and θ the temperature; is the viscous stress tensor given by = µ(∇u + ∇uT ) + λ divu I, and E is the total energy given by 1 1 1 E = ρ e + |u|2 + |H|2 and E = ρ e + |u|2 , 2 2 2 with e the internal energy, 21 ρ|u|2 the kinetic energy, and 21 |H|2 the magnetic energy. The equations of state p = p(ρ, θ ), e = e(ρ, θ ) relate the pressure p and the internal energy e to the density and the temperature of the flow; I is the 3 × 3 identity matrix, and ∇uT is the transpose of the matrix ∇u. The viscosity coefficients λ, µ of the flow satisfy 2µ + 3λ > 0 and µ > 0; ν > 0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, κ > 0 is the heat conductivity. Equations (1.1a), (1.1b), (1.1c) describe the conservation of mass, momentum, and energy, respectively. It is well-known that the electromagnetic fields are governed by Maxwell’s equations. In magnetohydrodynamics, the displacement current can be neglected ([19,20]). As a consequence, Eq. (1.1d) is called the induction equation, and the electric field can be written in terms of the magnetic field H and the velocity u, E = ν∇ × H − u × H. Although the electric field E does not appear in the MHD system (1.1), it is indeed induced according to the above relation by the moving conductive flow in the magnetic field. There have been a lot of studies on magnetohydrodynamics by physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenges; see [3,4,6,7,10,16,15,20,25] and the references cited therein. In particular, the one-dimensional problem has been studied in many papers, for examples, [3,4,7,15,18,23,25] and so on. However, many fundamental problems for MHD are still open. For example, even for the one-dimensional case, the global existence of classical solutions to the full perfect MHD equations with large data remains unsolved when all the viscosity, heat conductivity, and magnetic diffusivity coefficients are constant, although the corresponding problem for the Navier-Stokes equations was solved in [17] a long time ago. The reason is that the presence of the magnetic field and its interaction with the hydrodynamic motion in the MHD flow of large oscillation cause serious difficulties. In
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this paper we consider the global weak solution to the three-dimensional MHD problem with large data, and investigate the fundamental problem of global existence. More precisely, we study the initial-boundary value problem of (1.1) in a bounded spatial domain ⊂ R3 with the initial data: (ρ, ρu, H, θ )|t=0 = (ρ0 , m 0 , H0 , θ0 )(x), x ∈ ,
(1.2)
and the no-slip boundary conditions on the velocity and the magnetic field, and the thermally insulated boundary condition on the heat flux q = −κ∇θ : u|∂ = 0, H|∂ = 0, q|∂ = 0.
(1.3)
The aim of this paper is to construct the solution of the initial-boundary value problem of (1.1)–(1.3) and establish the global existence theory of variational weak solutions. In Hu-Wang [16], we studied global weak solutions to the initial-boundary value problem of the isentropic case for the three-dimensional MHD flow, while in this paper we study the full nonisentropic case. We are interested in the case that the viscosity and heat conductivity coefficients µ = µ(θ ), λ = λ(θ ), κ = κ(θ ) are positive functions of the temperature θ ; and the magnetic diffusivity coefficient ν > 0 is assumed to be a constant in order to avoid unnecessary technical details. As for the pressure p = p(ρ, θ ), it will be determined through a general constitutive equation: p = p(ρ, θ ) = pe (ρ) + θ pθ (ρ)
(1.4)
for certain functions pe , pθ ∈ C[0, ∞) ∩ C 1 (0, ∞). The basic principles of classical thermodynamics imply that the internal energy e and pressure p are interrelated through Maxwell’s relationship: ∂e ∂p 1 ∂e ∂Q = 2 p−θ , = = cυ (θ ), ∂ρ ρ ∂θ ∂θ ∂θ where cυ (θ ) denotes the specific heat and Q = Q(θ ) is a function of θ . Thus, the constitutive relation (1.4) implies that the internal energy e can be decomposed as a sum: e(ρ, θ ) = Pe (ρ) + Q(θ ),
(1.5)
where
ρ
Pe (ρ) = 1
pe (ξ ) dξ, ξ2
θ
Q(θ ) = 0
cυ (ξ )dξ.
If the flow is smooth, multiplying Eq. (1.1b) by u and (1.1d) by H, and summing them together, we obtain d 1 1 1 2 2 2 ρ|u| + |H| + div ρ|u| u + ∇ p · u dt 2 2 2 = div · u + (∇ × H) × H · u + ∇ × (u × H) · H − ∇ × (ν∇ × H) · H. (1.6) Subtracting (1.6) from (1.1c), we obtain the internal energy equation: ∂t (ρe) + div(ρue) + (divu) p = ν|∇ × H|2 + : ∇u + div(κ∇θ ),
(1.7)
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using div(νH × (∇ × H)) = ν|∇ × H|2 − ∇ × (ν∇ × H) · H, and div((u × H) × H) = (∇ × H) × H · u + ∇ × (u × H) · H,
(1.8)
where : ∇u denotes the scalar product of two matrices (see (4.10)). Multiplying Eq. (1.1a) by (ρ Pe (ρ)) yields ∂t (ρ Pe (ρ)) + div(ρ Pe (ρ)u) + pe (ρ)divu = 0,
(1.9)
and subtracting this equality from (1.7), we get the following thermal energy equation: ∂t (ρ Q(θ )) + div(ρ Q(θ )u) − div(κ(θ )∇θ ) = ν|∇ × H|2 + : ∇u − θ pθ (ρ)divu. (1.10) We note that in [6], Ducomet and Feireisl studied, using the entropy method, the full compressible MHD equations with an additional Poisson’s equation under the assumption that the viscosity coefficients depend on the temperature and the magnetic field, and the pressure behaves like the power law ρ γ with γ = 53 for large density. We also remark that, for the mathematical analysis of incompressible MHD equations, we refer the reader to the work [11] and the references cited therein; and for the related studies on the multi-dimensional compressible Navier-Stokes equations, we refer to [8,9,14,22] and particularly [8,9] for the nonisentropic case. In this paper, we consider compressible MHD flow with more general pressure, and use the thermal equation (1.10) as in [8] instead of the entropy equation used in [6], thus the methods of this paper differ significantly from those in [6]. There are several major difficulties in studying the global solutions of the initial-boundary value problem of (1.1)-(1.3) with large data, due to the interaction from the magnetic field, large oscillations and concentrations of solutions, and poor a priori estimates available for MHD. To deal with the possible density oscillation, we use the weak continuity property of the effective viscous flux, first established by Lions [22] for the barotropic compressible Navier-Stokes system with constant viscosities (see also Feireisl [9] and Hoff [13]). More precisely, for fixed T > 0, assuming (ρn , b(ρn ), pn ) → (ρ, b(ρ), p) weakly in L 1 ( × (0, T )), (un , Hn ) → (u, H) weakly in L 2 ([0, T ]; W01,2 ()), we will prove that, for some function b, ( pn − (λ(θn ) + 2µ(θn ))divun ) b(ρn ) → p − (λ(θ ) + 2µ(θ ))divu b(ρ) weakly in L 1 ( × (0, T )), where f denote a weak limit of a sequence { f n }∞ n=1 in L 1 ( × (0, T )). To overcome the difficulty from the concentration in the temperature in order to pass to limit in approximation solutions, we use the renormalization of the thermal energy equation (1.10). More precisely, multiplying (1.10) by h(θ ) for some function h, we obtain, ∂t (ρ Q h (θ )) + div(ρ Q h (θ )u) − K h (θ ) = ν|∇ × H|2 h(θ ) + h(θ ) : ∇u − h(θ )θ pθ (ρ)divu − h (θ )κ(θ )|∇θ |2 ,
(1.11)
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where
θ
Q h (θ ) = 0
cυ (ξ )h(ξ )dξ,
θ
K h (θ ) =
κ(ξ )h(ξ )dξ.
0
The idea of renormalization was used in Feireisl [8,9], and is similar to that in DiPerna and Lions [5]. In addition, we also need to overcome the difficulty arising from the presence of the magnetic field and its coupling and interaction with the fluid variables. We organize the rest of this paper as follows. In Sect. 2, we introduce a variational formulation of the full compressible MHD equations, and also state the main existence result (Theorem 2.1). In Sect. 3, we will formally derive a series of a priori estimates on the solution. In order to construct a sequence of approximation solutions, a three-level approximation scheme from [16] for isentropic MHD flow will be adopted in Sect. 4. Finally, in Sect. 5, our main result will be proved through a vanishing viscosity and vanishing artificial pressure limit passage using the weak convergence method. 2. Variational Formulation and Main Result In this section, we give the definition of the variational solution to the initial-boundary value problem (1.1)–(1.3) and state the main result. First we remark that, as shown later, the optimal estimates we can expect on the magnetic field H and the velocity u are in H 1 -norms, which can not ensure the convergence of the terms |∇ × H|2 and : ∇u in L 1 of Eq. (1.10), or even worse, in the sense of distributions. In other words, the compactness on the temperature does not seem to be sufficient to pass to the limit in the thermal energy equation. Thus, we will replace the thermal energy equality (1.10) by two inequalities in the sense of distributions to be in accordance with the second law of thermodynamics. More precisely, instead of (1.10), we only require that the following two inequalities hold: ∂t (ρ Q(θ )) + div(ρ Q(θ )u) − K (θ ) ≥ ν|∇ × H|2 + : ∇u − θ pθ (ρ)divu,
(2.1)
in the sense of distributions, and E[ρ, u, , θ, H](t) ≤ E[ρ, u, θ, H](0) for
t ≥ 0,
(2.2)
with the total energy E[ρ, u, θ, H] =
1 1 ρ Pe (ρ) + Q(θ ) + |u|2 + |H|2 d x, 2 2
and K (θ ) =
θ
κ(ξ )dξ.
0
Now we give the definition of variational solutions to the full MHD equations as follows: Definition 2.1. A vector (ρ, u, θ, H) is said to be a variational solution to the initialboundary value problem (1.1)–(1.3) of the full compressible MHD equations on the time interval (0, T ) for any fixed T > 0 if the following conditions hold:
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• The density ρ ≥ 0, the velocity u ∈ L 2 ([0, T ]; W01,2 ()), and the magnetic field H ∈ L 2 ([0, T ]; W01,2 ()) ∩ C([0, T ]; L 2weak ()) satisfy the Eqs. (1.1a), (1.1b), and (1.1d) in the sense of distributions, and T (ρ∂t ϕ + ρu · ∇ϕ) d xdt = 0, 0
for any ϕ ∈ C ∞ ( × [0, T ]) with ϕ(x, 0) = ϕ(x, T ) = 0 for x ∈ ; • The temperature θ is a non-negative function satisfying T (ρ Q(θ )∂t ϕ + ρ Q(θ )u · ∇ϕ + K (θ ) ϕ) d xdt 0
≤
T 0
(θ pθ (ρ)divu − ν|∇ × H|2 − : ∇u)ϕ d xdt
for any ϕ ∈ C0∞ ( × (0, T )) with ϕ ≥ 0; • The energy inequality (2.2) holds for a.e t ∈ (0, T ), with 1 |m0 |2 1 ρ0 Pe (ρ0 ) + ρ0 Q(θ0 ) + + |H0 |2 d x; E[ρ, u, θ, H](0) = 2 ρ0 2 • The functions ρ, ρu, and H satisfy the initial conditions in the following weak sense: ess lim+ (ρ, ρu, H)(x, t)η(x) d x = (ρ0 , m 0 , H0 )η d x, t→0
for any η ∈ D() :=
C0∞ ().
Now we are ready to state the main result of this paper. Theorem 2.1. Let ⊂ R3 be a bounded domain of class C 2+τ for some τ > 0. Suppose that the following conditions hold: the pressure p is given by Eq. (1.4) where pe , pθ are C 1 functions on [0, ∞) and ⎧ ⎪ ⎨ pe (0) = 0, pθ (0) = 0, for all ρ > 0, pe (ρ) ≥ a1 ρ γ −1 , pθ (ρ) ≥ 0 (2.3) ⎪ γ ⎩ p (ρ) ≤ a ρ γ , 3 p (ρ) ≤ a (1 + ρ ) for all ρ ≥ 0, e 2 θ 3 with some constants γ > 23 , a1 > 0, a2 > 0, and a3 > 0; κ = κ(θ ) is a C 1 function on [0, ∞) such that κ(1 + θ α ) ≤ κ(θ ) ≤ κ(1 + θ α ),
(2.4)
for some constants α > 2, κ > 0, and κ > 0; the viscosity coefficients µ and λ are C 1 functions of θ and globally Lipschitz on [0, ∞) satisfying 0 < µ ≤ µ(θ ) ≤ µ, 0 ≤ λ(θ ) ≤ λ,
(2.5)
for some positive constants µ, µ, λ; ν > 0 is a constant; there exist two positive constants cυ , cυ such that 0 < cυ ≤ cυ (θ ) ≤ cυ ;
(2.6)
Global Solutions to Magnetohydrodynamic Flows
and finally, the initial data satisfy ⎧ ρ0 ∈ L γ (), ρ0 ≥ 0 on , ⎪ ⎪ ⎪ ⎨θ ∈ L ∞ (), θ ≥ θ > 0 on , 0 0 |m 0 |2 1 (), ⎪ ∈ L ⎪ ρ0 ⎪ ⎩ H0 ∈ L 2 (), divH0 = 0 in D ().
261
(2.7)
Then, the initial-boundary value problem (1.1)–(1.3) of the full compressible MHD equations has a variational solution (ρ, u, θ, H) on × (0, T ) for any given T > 0, and ⎧ ρ ∈ L ∞ ([0, T ]; L γ ()) ∩ C([0, T ]; L 1 ()), ⎪ ⎪ ⎪ 2γ ⎪ ⎪ 2 ([0, T ]; W 1,2 ()), ρu ∈ C [0, T ]; L γ +1 () , ⎪ ⎪ u ∈ L ⎪ 0 weak ⎪ ⎨ α+1 ∞ θ ∈ L ( × (0, T )), ρ Q(θ ) ∈ L ([0, T ]; L 1 ()), (2.8) ⎪ 2 ( × (0, T )), ρ Q(θ )u ∈ L 1 ( × (0, T )), ⎪ ∈ L θ p ⎪ θ ⎪ ⎪ α ⎪ ⎪ ln(1 + θ ) ∈ L 2 ([0, T ]; W 1,2 ()), θ 2 ∈ L 2 ([0, T ]; W 1,2 ()), ⎪ ⎪ ⎩ H ∈ L 2 ([0, T ]; W01,2 ()) ∩ C([0, T ]; L 2weak ()). Remark 2.1. In addition, the solution constructed in Theorem 2.1 will satisfy the continuity equation in the sense of renormalized solutions, that is, the integral identity T b(ρ)∂t ϕ + b(ρ)u · ∇ϕ + (b(ρ) − b (ρ)ρ)divu ϕ d xdt = 0 0
holds for any γ
b ∈ C 1 [0, ∞), |b (z)z| ≤ cz 2 for z larger than some positive z 0 , and any test function ϕ ∈ C ∞ ([0, T ] × ) with ϕ(x, 0) = ϕ(x, T ) = 0 for x ∈ . Remark 2.2. The growth restrictions imposed on κ, µ, λ, and cυ may not be optimal, and γ > 23 is a necessary condition to ensure the convergence of nonlinear term ρu ⊗ u in the sense of distributions. In particular, our result includes the case of constant viscosity coefficients, with the assumption that the coefficient λ ≥ 0. Remark 2.3. Our method also works for the case with nonzero external force f in the momentum equation. It is obvious that in our analysis the presence of the external force does not add any additional difficulty, and usually can be dealt with by using classical Young’s inequality under suitable assumptions on the integrability of the external force f . 3. A Priori Estimates To prove Theorem 2.1, we first need to obtain sufficient a priori estimates on the solution. The total energy conservation (2.2) implies 1 2 1 2 ρ Pe (ρ) + Q(θ ) + |u| + |H| d x 2 2 1 |m0 |2 1 ≤ ρ0 Pe (ρ0 ) + ρ0 Q(θ0 ) + (3.1) + |H0 |2 d x. 2 ρ0 2
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But the assumption (2.3) implies that there is a positive constant c such that ρ Pe (ρ) ≥ cρ γ ,
for any ρ ≥ 0.
Thus, (3.1) implies that ρ γ , ρ Q(θ ), 21 ρ|u|2 and 21 |H|2 are bounded in L ∞ ([0, T ]; L 1 ()). Hence, 2γ ∞ γ ∞ γ +1 [0, T ]; L () . ρ ∈ L ([0, T ]; L ()), ρu ∈ L Next, in order to obtain estimates on the temperature, we introduce the entropy θ ρ cυ (ξ ) pθ (ξ ) s(ρ, θ ) = dξ − Pθ (ρ), with Pθ (ρ) = dξ. ξ ξ2 1 1 If the flow is smooth and the temperature is strictly positive, then by direct calculation, using (1.1a) and (1.10), we obtain ∂t (ρs) + div(ρsu) + div
q θ
=
q · ∇θ 1 ν|∇ × H|2 + : ∇u − . θ θ2
Integrating (3.2), we get T κ(θ )|∇θ |2 1 2 d xdt ν|∇ × H| + : ∇u + θ2 0 θ = ρs(x, t) d x − ρs(x, 0) d x.
(3.2)
(3.3)
On the other hand, assumptions (2.3) imply, using Young’s inequality, |ρ Pθ (ρ)| ≤ c + ρ Pe (ρ) for some c > 0.
(3.4)
Moreover, we have ρ 1
θ
cυ (ξ ) dξ ≤ ρ Q(θ ) for all θ > 0, ρ ≥ 0, ξ
(3.5)
since
θ 1
cυ (ξ ) dξ ≤ 0, if 0 < θ ≤ 1, ξ
and
θ 1
cυ (ξ ) dξ ≤ ξ
θ 1
cυ (ξ )dξ = Q(θ ) − Q(1) ≤ Q(θ ), if θ > 1.
Assuming that ρs(·, 0) ∈ L 1 (), then from (3.3)–(3.5), using the assumption (2.4) and the estimates from (3.1), we get T α |∇θ 2 |2 + |∇ ln θ |2 d xdt ≤ C, 0
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which, combining Sobolev’s imbedding theorem, implies α
ln θ and θ 2 are bounded in L 2 ([0, T ]; W 1,2 ()).
(3.6)
Finally, we turn to the estimates on the velocity and the magnetic field. Indeed, integrating (1.10) over × (0, T ), we get T : ∇u + ν|∇ × H|2 d xdt 0
=
T 0
θ pθ (ρ)divu d xdt +
ρ Q(θ )(x, T )d x −
ρ Q(θ )(x, 0) d x. (3.7)
Noticing that, using Hölder inequality, one has θ pθ (ρ) L 2 () ≤ θ L 6 () pθ (ρ) L 3 () .
(3.8)
Thus, from assumption (2.3) and estimate (3.6), we have θ pθ (ρ) ∈ L 2 ( × (0, T )). The relation (3.7) together with (3.1), (3.8), gives rise to the estimate T : ∇u + ν|∇ × H|2 d xdt ≤ C(ρ0 , u0 , θ0 , H0 ). 0
The assumption (2.5), the fact ∇ × H L 2 = ∇H L 2 when divH = 0, and Sobolev’s imbedding theorem give that u, H are bounded in L 2 ([0, T ]; W01,2 ()). In summary, if ρs(·, 0) ∈ L 1 (), the system (1.1a), (1.1b), (1.1d), (1.10) with the initial-boundary conditions (1.3) and our assumptions (2.3)–(2.7) yield the following estimates: ⎧ ⎪ ρ Pe (ρ), ρ Q(θ ) are bounded in L ∞ ([0, T ]; L 1 ()); ⎪ ⎪ ⎪ ∞ γ ⎪ ⎪ ⎨ρ is bounded in L ([0, T ]; L ()), 2γ (3.9) ρu is bounded in L ∞ ([0, T ]; L γ +1 ()); ⎪ α ⎪ 2 1,2 ⎪ 2 ⎪ln θ and θ are bounded in L ([0, T ]; W ()); ⎪ ⎪ ⎩u, H are bounded in L 2 ([0, T ]; W 1,2 ()). 0 4. The Approximation Scheme and Approximation Solutions Similarly as Sect. 4 in [16] and Sect. 3 in [8], we introduce an approximate problem which consists of a system of regularized equations: ⎧ ρt + div(ρu) = ε ρ, ⎪ ⎪ ⎪ ⎪ β ⎪ (ρu) t + div (ρu ⊗ u) + ∇ p(ρ, θ ) + δ∇ρ + ε∇u · ∇ρ = (∇ × H) × H + div, ⎨ ∂t ((ρ + δ)Q(θ )) + div(ρ Q(θ )u) − K (θ ) + δθ α+1 (4.1) ⎪ ⎪ ⎪ = (1 − δ)(ν|∇ × H|2 + : ∇u) − θ pθ (ρ)divu, ⎪ ⎪ ⎩ Ht − ∇ × (u × H) = −∇ × (ν∇ × H), divH = 0,
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with the initial-boundary conditions ⎧ ⎪ ∇ρ · n|∂ = 0, ⎪ ⎪ ⎨ u|∂ = 0, ⎪∇θ |∂ = 0, ⎪ ⎪ ⎩H| = 0, ∂
ρ|t=0 = ρ0,δ , ρu|t=0 = m 0,δ , θ |t=0 = θ0,δ , H|t=0 = H0 ,
(4.2)
where ε and δ are two positive parameters, β > 0 is a fixed constant, and n is the unit outer normal of ∂. The initial data are chosen in such a way that ⎧ − 1 ⎪ ⎪ ρ0,δ ∈ C 3 (), 0 < δ ≤ ρ0,δ ≤ δ 2β ; ⎪ ⎪ ⎪ ⎪ ρ0,δ → ρ0 in L γ (), |{ρ0,δ < ρ0 }| → 0, as δ → 0; ⎪ ⎪
β ⎪ ⎪ ⎪ ⎨δ ρ0,δd x → 0, as δ → 0; m 0 , if ρ0,δ ≥ ρ0 , ⎪ m 0,δ = ⎪ ⎪ ⎪ 0, if ρ0,δ < ρ0 ; ⎪ ⎪ ⎪ 3 (), 0 < θ ≤ θ ⎪ ⎪ θ ∈ C 0,δ 0,δ ≤ θ ; ⎪ ⎪ ⎩θ → θ in L 1 () as δ → 0. 0,δ 0
(4.3)
Noticing that the terms ν|∇ × H|2 and : ∇u are nonnegative, and θ = 0 is a subsolution of the third equation in (4.1), we can conclude that, using the maximum principle, θ (t, x) ≥ 0 for all t ∈ (0, T ) and x ∈ . From Lemma 3.2 in [16] and Proposition 7.2 in [9], we see that the approximate problem (4.1)–(4.2) with fixed positive parameters ε and δ can be solved by means of a modified Faedo-Galerkin method (cf. Chap. 7 in [9]). Thus, we state without proof the following result (cf. Prop. 3.1 in [8]): Proposition 4.1. Under the hypotheses of Theorem 2.1, let β be large enough, then the approximate problem (4.1)–(4.2) has a solution (ρ, u, θ, H) on × (0, T ) for any fixed T > 0 satisfying the following properties: • ρ ≥ 0, u ∈ L 2 ([0, T ]; W01,2 ()), H ∈ L 2 ([0, T ]; W01,2 ()), the first equation in (4.1) is satisfied a.e. on × (0, T ), the second and fourth equations in (4.1) are satisfied in the sense of distributions on × (0, T ) (denoted by D ( × (0, T )), u and H are bounded in L 2 ([0, T ]; W01,2 ()), and, for some r > 1, 2γ
γ +1 ()), ρt , ρ ∈ L r ( × (0, T )), ρu ∈ C([0, T ]; L weak T T δ ρ β+1 d xdt ≤ c1 (ε, δ), ε |∇ρ|2 d xdt ≤ c2 ,
0
where c2 is a constant independent of ε.
0
(4.4)
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• The energy inequality T 1 1 a2 δ (−ψt ) ρ|u|2 + |H|2 + ργ + ρ β + (ρ + δ)Q(θ ) d xdt 2 2 γ −1 β −1 0 T +δ ψ : ∇u + ν|∇ × u|2 + θ α+1 d xdt 0 1 |m 0,δ |2 1 a2 δ γ β 2 ≤ + |H0 | + ρ + ρ + (ρ0,δ + δ)Q(θ0,δ ) d x 2 γ − 1 0,δ β − 1 0,δ 2 ρ0,δ T + ψ pb (ρ)divu d xdt 0
holds for any ψ ∈ C ∞ ([0, T ]) satisfying ψ(·, 0) = 1, ψ(·, T ) = 0, ψt ≤ 0 on , where, pe (ρ) has been decomposed as pe (ρ) = a2 ρ γ − pb (ρ), with pb ∈ C 1 [0, ∞), pb ≥ 0; • The temperature θ ≥ 0 satisfies that α
θ ∈ L α+1 ( × (0, T )), θ 2 ∈ L 2 ([0, T ]; W 1,2 ()), and the thermal energy inequality holds in the following renormalized sense: T (ρ + δ)Q h (θ )∂t ϕ + ρ Q h (θ )u · ∇ϕ + K h (θ ) ϕ − δh(θ )θ α+1 ϕ d xdt 0
≤
T 0
T + +ε
0 T
(δ − 1)h(θ )( : ∇u + ν|∇ × H|2 ) + h (θ )κ(θ )|∇θ |2 ϕ d xdt
h(θ )θ pθ (ρ)divuϕ d xdt −
0
(ρ0,δ + δ)Q h (θ0,δ )ϕ(x, 0) d x
∇ρ · ∇ ((Q h (θ ) − Q(θ )h(θ ))ϕ) d xdt,
(4.5)
for any function h ∈ C ∞ (R+ ) satisfying h(0) > 0, h non-increasing on [0, ∞), h (ξ )h(ξ ) ≥ 2(h (ξ ))2 ,
lim h(ξ ) = 0,
ξ →∞
for all ξ ≥ 0,
and any test function ϕ ∈ C 2 ( × [0, T ]) satisfying ϕ ≥ 0, ϕ(·, T ) = 0, ∇ϕ · n|∂ = 0. Remark 4.1. In fact, we have α
∇θ 2 L 2 (×(0,T )) ≤ c(δ).
(4.6)
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Comparing with the term ν|∇ × H|2 in (4.5), we need to pay more attention to the term : ∇u, because the later involves temperature-dependent coefficients and thus can not be dealt with by the standard weak lower semi-continuity. Indeed, the hypothesis (4.6) was imposed in [9] in order to make the function (θ, ∇u) → h(θ ) : ∇u convex, and, consequently, weakly lower semi-continuous (the stress tensor in [9] depends on ∇u only). In accordance with our new context, the following lemma is useful: Lemma 4.1. Let g(θ ) be a bounded, continuous and non-negative function from [0, ∞) to R. Suppose that θn and un are two sequences of functions defined on and θn → θ
a.e in ,
and un → u weakly in W 1,2 (). Then,
In particular,
g(θ )h(θ )|∇u|2 d x ≤ lim inf n→∞
g(θn )h(θn )|∇un |2 d x.
(4.7)
h(θ ) : ∇u d x ≤ lim inf n→∞
h(θn )(un ) : ∇un d x.
(4.8)
√ √ Proof.√First we show that g(θn )∇un converges weakly to g(θ )∇u in L 2 . Indeed, 2 since g(θn )∇un is uniformly bounded in L , it is enough to show g(θn )∇un φ d x → g(θ )∇u φ d x, for all φ ∈ C ∞ (). (4.9)
√
√ Since θn → θ a.e in , then g(θn )φ → g(θ )φ a.e in for all φ ∈ C ∞ (). Thus by Lebesgue’s dominated convergence theorem, we know that g(θn )φ → g(θ )φ in L 2 (), and (4.9) follows. Next, by virtue of Corollary 2.2 in [9], it is enough to observe that the function : (θ, ξ ) → h(θ )ξ 2 is convex and continuous on R+ × R. Computing the Hessian matrix of , we get 2 } = 2ξ 2 (h (θ )h(θ ) − 2(h (θ ))2 ) ≥ 0, det{∂θ,ξ
and 2 } = ξ 2 h (θ ) + 2h(θ ) ≥ 0, trace{∂θ,ξ
provided θ > 0 and h satisfies (4.6). Thus the Hessian matrix is positively definite; therefore is convex and continuous. Then, (4.7) is a direct application of Corollary 2.2 in [9].
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Finally, from (4.7) and the calculation: 2 3 µ(θ ) ∂u i ∂u j : ∇u = + + λ(θ )|divu|2 , 2 ∂ x j ∂ xi
(4.10)
i, j=1
(4.8) follows.
In the next two steps, in order to obtain the variational solution of the initial-boundary value problem (1.1)–(1.3), we need to take the vanishing limits of the artificial viscosity ε → 0 and artificial pressure coefficient δ → 0 in the approximate solutions of (4.1)(4.2). As seen in [8,9,16], the techniques used in those two procedures are rather similar. Moreover, in some sense, the step of taking ε → 0 is much easier than the step of taking δ → 0 due to the higher integrability of ρ. Hence we will omit the step of taking ε → 0 (readers can refer to Sect. 5 in [16] or Section 4 in [8]), and focus on the step of taking δ → 0. Thus, we state without proof the result as ε → 0 as follows. Proposition 4.2. Let β > 0 be large enough and δ > 0 be fixed, then the initial-boundary value problem (1.1)–(1.3) for full compressible MHD equations admits an approximate solution (ρ, u, θ, H) with parameter δ (as the limit of the solutions to (4.1)–(4.2) when ε → 0) in the following sense: • The density ρ is a non-negative function, and β
ρ ∈ C([0, T ]; L weak ()), satisfying the initial condition in (4.3). The velocity u and the magnetic field H belong to L 2 ([0, T ]; W01,2 ()). Eq. (1.1a) and (1.1d) are satisfied in D ( × (0, T )) and T δ ρ β+1 d xdt ≤ c(δ). 0
Moreover, ρ, u also solve Eq. (1.1a) in the sense of renormalized solutions; • The functions ρ, u, θ, H solve a modified momentum equation (ρu)t + div (ρu ⊗ u) + ∇( p(ρ, θ ) + δρ β ) = (∇ × H) × H + div, in
D ( × (0, T )).
(4.11)
Furthermore, the momentum 2γ γ +1 ρu ∈ C [0, T ]; L weak ()
satisfies the initial condition in (4.3); • The energy inequality T 1 2 δ 1 2 β ρ|u| + |H| + ρ Pe (ρ) + ρ + (ρ + δ)Q(θ ) d xdt (−ψt ) 2 2 β −1 0 T (4.12) +δ ψ : ∇u + ν|∇ × u|2 + θ α+1 d xdt 0 1 |m 0,δ |2 1 δ β ≤ + |H0 |2 + ρ0,δ Pe (ρ0,δ ) + ρ0,δ + (ρ0,δ + δ)Q(θ0,δ ) d x 2 β −1 2 ρ0,δ holds for any ψ ∈ C ∞ ([0, T ]) satisfying ψ(·, 0) = 1, ψ(·, T ) = 0, ψt ≤ 0;
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• The temperature θ is a non-negative function, and θ ∈ L α+1 ( × (0, T )), θ
α+1−ω 2
∈ L 2 ([0, T ]; W 1,2 ()), ω ∈ (0, 1], (4.13)
satisfying the thermal energy inequality in the following renormalized sense: T (ρ + δ)Q h (θ )∂t ϕ + ρ Q h (θ )u · ∇ϕ + K h (θ ) ϕ − δh(θ )θ α+1 ϕ d xdt 0
≤
T (δ − 1)h(θ )( : ∇u + ν|∇ × H|2 ) + h (θ )κ(θ )|∇θ |2 ϕd xdt 0
T +
0
h(θ )θ pθ (ρ)divuϕ d xdt −
(ρ0,δ + δ)Q h (θ0,δ )ϕ(0) d x,
(4.14)
for any admissible function h ∈ C ∞ (R+ ) satisfying (4.6) and any test function ϕ ∈ C 2 ( × [0, T ]) satisfying ϕ ≥ 0, ϕ(·, T ) = 0, ∇ϕ · n|∂ = 0. Remark 4.2. In Proposition 4.2, the second estimate in (4.13) can be explained as follows: Taking h(θ ) =
1 , ω ∈ (0, 1], ϕ(t, x) = ψ(t), 0 ≤ ψ ≤ 1, ψ ∈ D(0, T ), (1 + θ )ω
in (4.5), we obtain ω
κ(θε ) |∇θε |2 ψ d xdt ω+1 (1 + θ ) ε 0 T T ≤− (ρε + δ)Q h (θε )ψt d xdt + δ h(θε )θεα+1 ψ d xdt T
0
T + +ε
0 T
0
θε pθ (ρε )|divuε |ψ d xdt
0
|∇ρε · ∇ ((Q h (θε ) − Q(θε )h(θε ))ϕ) | d xdt.
Observing that T T α+1−ω 2 ∇(1 + θε ) 2 ψ d xdt ≤ c 0
0
κ(θε ) |∇θε |2 ψ d xdt, (1 + θε )ω+1
and T 0
θε pθ (ρε )|divuε |ψ d xdt
≤ c θε L 2 ([0,T ];L 6 ()) uε L 2 ([0,T ];W 1,2 ()) pθ (ρε ) L ∞ ([0,T ];L 3 ()) ,
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and, by the hypothesis (2.6), we have T ε |∇ρε · ∇[(Q h (θε ) − Q(θε )h(θε ))ϕ]| d xdt 0 Q(θε ) ∇θε ≤ cε ψ L ∞ ∇ρε L 2 2 ω+1 (1 + θε ) L α−1−ω ≤ cε ∇ρε L 2 (1 + θε ) 2 ∇θε 2 L α+1−ω 2 = cε ∇ρε L 2 ∇(1 + θε ) 2, L
where, the following property is used Q(θ )2 |∇θ |2 ≤ c(1 + θ )α |∇θ |2 . (1 + θ )ω+1 By Young’s inequality, Remark 4.1, and the energy inequality in Proposition 4.1, one has (1 + θε )
α+1−ω 2
∈ L 2 ([0, T ]; W 1,2 ()).
Thus, α+1−ω 2
θε
∈ L 2 ([0, T ]; W 1,2 ()),
since α−1−ω 2
θε
|∇θε | ≤ (1 + θε )
α−1−ω 2
|∇θε |.
In the next section, we shall take the limit of the other artificial term: the artificial pressure, as δ → 0. 5. The Limit of Vanishing Artificial Pressure In this section, we take the limit as δ → 0 to eliminate the δ-dependent terms appearing in (4.1), while in the previous section passing to the limit as ε → 0 has been done. Denote by {ρδ , uδ , θδ , Hδ }δ>0 the sequence of approximate solutions obtained in Proposition 4.2. In addition to the possible oscillation effects on density, the concentration effects on temperature is also a major issue of this section. To deal with these difficulties, we employ a variant of well-known Feireisl-Lions method [8,9,22] in our new context.
5.1. Energy estimates. The main object in this subsection is to find sufficient a priori estimates. First, our choice of the initial data (4.3) implies that, as δ → 0, 1 |m 0,δ |2 1 δ β 2 ρ + (ρ0,δ + δ)Q(θ0,δ ) d x + |H0 | + ρ0,δ Pe (ρ0,δ ) + 2 β − 1 0,δ 2 ρ0,δ 1 |m 0 |2 1 ρ0 Pe (ρ0 ) + ρ0 Q(θ0 ) + + |H0 |2 d x. → E[ρ, u, θ, H](0) = 2 ρ0 2
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Hence, from the energy inequality (4.12), we can conclude that ρδ is bounded in L ∞ ([0, T ]; L γ ()), √ ρδ uδ , Hδ are bounded in L ∞ ([0, T ]; L 2 ()),
(5.1) (5.2)
(ρδ + δ)Q(θδ ) is bounded in L ∞ ([0, T ]; L 1 ()), T β ρδ + θδα+1 d xdt ≤ c, δ
(5.3)
0
(5.4)
for some constant c, which is independent of δ. Now, we take ϕ(x, t) =
T − 21 t 1 , h(θ ) = T 1+θ
in (4.14) to obtain T
1−δ κ(θδ ) 2 2 (δ : ∇uδ + ν|∇ × Hδ | ) + |∇θδ | d xdt (1 + θδ )2 0 1 + θδ T T θδ ≤2 pθ (ρδ )divuδ d xdt + 2δ θδα d xdt 0 1 + θδ 0 − 2 (ρ0,δ + δ)Q 1 (θ0,δ ) d x + (ρδ + δ)Q 1 (θδ )(T ) d x,
(5.5)
where
θ
Q 1 (θ ) = 0
cυ (ξ ) dξ ≤ Q(θ ). 1+ξ
Using the estimates (5.3) and (5.4), we deduce from (5.5) that T 1−δ κ(θδ ) 2 2 (δ : ∇uδ + ν|∇ × Hδ | ) + |∇θδ | d xdt (1 + θδ )2 0 1 + θδ T ≤c 1+ pθ (ρδ )divuδ d xdt , 0
for some constant c which is independent of δ, and here the second term on right-hand side can be rewritten with the help of the renormalized continuity equation as T T pθ (ρδ )divuδ d xdt = ∂t (ρδ Pθ (ρδ )) d xdt. 0
0
By the hypothesis (2.3) and the estimate (5.1), one has T γ γ ∂t (ρδ Pθ (ρδ ))d xdt ≤ c() 1 + ρδ3 d x ≤ c() 1 + ρδ d x ≤ c(, T ). 0
Consequently, we can conclude that κ(θδ ) |∇θδ |2 ∈ L 1 ( × (0, T )), (1 + θδ )2
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which, combining with the hypothesis (2.4), gives us that α
∇ ln(1 + θδ ), ∇θδ2 are bounded in L 2 ( × (0, T )). Thus, combining with Sobolev’s imbedding theorem, we obtain that α
ln(1 + θδ ), θδ2 are bounded in L 2 ([0, T ]; W 1,2 ()).
(5.6)
Moreover, in view of the hypothesis (2.3), we get θδ pθ (ρδ ) is bounded in L 2 ( × (0, T )).
(5.7)
With (5.7) in hand, we can repeat the same procedure as above, taking now h(θ ) =
T − 21 t 1 , , ω ∈ (0, 1), ϕ(x, t) = (1 + θ )ω T
and finally we can get T 1−δ κ(θδ ) 2 2 (δ : ∇uδ + ν|∇ × Hδ | ) + ω |∇θδ | d xdt ≤ c, (5.8) ω (1 + θδ )1+ω 0 (1 + θδ ) for some constant c which is independent of δ. Letting ω → 0 and using the monotone convergence theorem, we deduce that uδ , Hδ are bounded in L 2 ([0, T ]; W01,2 ()).
(5.9)
Moreover, from (5.8), we have (1 + θδ )
α+1−ω 2
is bounded in L 2 ([0, T ]; W 1,2 ()),
for any ω ∈ (0, 1]. (5.10)
In particular, this implies that θδ is bounded in L 2 ([0, T ]; L 3α+2 ()).
(5.11)
Using Hölder inequality, we have α+ 43
θδ
α+ 23
L 1 (D) ≤ θδ
2
L 3 (D) θδ3
3 L 2 (D)
α+ 23
≤ θδ
2
L 3 (D) θδ L3 1 (D)
for any D ⊂ , which, together with (5.10), (5.3) and the hypothesis (2.6), yields α+ 4 θδ 3 d xdt ≤ c(d), {ρδ ≥d}
for any d > 0. In particular,
{ρδ ≥d}
for any d > 0.
θδα+1 d xdt ≤ c(d)
(5.12)
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5.2. Temperature estimates. In order to pass to the limit in the term K (θ ), our aim in this subsection is to derive uniform estimates on θδ in L α+1 ( × (0, T )). To this end, we will follow the argument in [9]. To begin with, we have {ρδ ≥d}
ρδ d x ≥ Mδ − d|| ≥
M − d||, 2
where Mδ denotes the total mass, Mδ =
ρδ d x,
independent of t ∈ [0, T ], and M=
ρ0 d x > 0.
On the other hand, Hölder inequality yields {ρδ ≥d}
ρδ d xdt ≤ ρδ L γ () |{ρδ ≥ d}|
γ −1 γ
.
Consequently, there exists a function = (d) independent of δ > 0 such that |{ρδ ≥ d}| ≥ (d) > 0 Fix 0 < d <
M 4||
for all t ∈ [0, T ],
if 0 ≤ d <
M . 2||
and choose a function b ∈ C ∞ (R) such that
b is non-increasing; b(z) = 0
for z ≤ d, b(z) = −1
if z ≥ 2d.
For each t ∈ [0, T ], let η = η(t) be the unique solution of the Neumann problem:
1 η = b(ρδ (t)) − || b(ρδ (t)) d x in ,
∇η · n|∂ = 0, η d x = 0,
(5.13)
where η = η(t) is a function of the spatial variable x ∈ with t as a parameter. Since the right-hand side of (5.13) has a bound which is independent of δ, there is a constant η such that η = η(t) ≥ η
for all δ > 0 and t ∈ [0, T ].
Now, we take ϕ(x, t) = ψ(t)(η − η), 0 ≤ ψ ≤ 1, ψ ∈ D(0, T ),
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as a test function in (4.14) to obtain
1 K h (θδ ) b(ρδ (t)) − b(ρδ (t)) ψ d xdt || 0 T ≤ 2 η L ∞ (×(0,T )) δh(θδ )θδ1+α + h(θδ )θδ pθ (ρδ )|divuδ | d xdt
T
+ ∇η L ∞ (×(0,T )) T +
0
0 T
0
(5.14) ρδ Q h (θδ )|uδ | d xdt
(ρδ + δ)Q h (θδ )(η − η)ψt − (ρδ + δ)Q h (θδ )ψ∂t η d xdt.
Next, taking h(θ ) =
1 (1 + θ )ω
for 0 < ω < 1, letting ω → 0, using Lebesgue’s dominated convergence theorem and the estimates (5.1), (5.3), (5.4), and (5.7), we obtain 1 K (θδ ) b(ρδ (t)) − b(ρδ (t)) d xdt || 0 T ≤c 1+ (ρδ + δ)θδ |∂t η| d xdt .
T
(5.15)
0
On the other hand, 1 K (θδ ) b(ρδ (t)) − b(ρδ (t)) d xdt || 0 1 K (θδ ) b(ρδ (t)) − b(ρδ (t)) d xdt = || {ρδ
T
where, by virtue of (5.12), the second integral on the right-hand side is bounded by a constant independent of δ. Furthermore, (2d) 1 1 |{ρδ ≥ 2d}| ≥ > 0. − b(ρδ ) d x ≥ − b(ρδ ) d x = || || {ρδ ≥2d} || || Thus, we obtain
1 K (θδ ) b(ρδ (t)) − || {ρδ
b(ρδ (t))
d xdt (5.16)
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X. Hu, D. Wang
Combining (5.15), (5.16) together, we get T K (θδ ) d xdt ≤ c 1 + (ρδ + δ)θδ |∂t η| d xdt {ρδ
0
(5.17)
with c independent of δ. Finally, since ρδ is a renormalized solution of Eq. (1.1a), we have, 1 (∂t η) = ∂t b(ρδ ) − ∂t b(ρδ ) d x || 1 (b (ρδ )ρδ − b(ρδ ))divuδ d x. = (b(ρδ ) − b (ρδ )ρδ )divuδ − div(b(ρδ )uδ ) + || Hence, ∂t η is bounded in L 2 ([0, T ]; W 1,2 ()). Consequently, using (5.17), we conclude that K (θδ ) d xdt ≤ c, c independent of δ {ρδ
which, together with (5.12) and the hypothesis (2.4), yields θδ is bounded in L 1+α ( × (0, T )).
(5.18)
5.3. Refined pressure estimates. Our goal now is to improve estimates on pressure. We follow step by step the argument of Sect. 5.1 in [16], that is, using the Bogovskii operator “div−1 [ln(1 + ρδ )]” as a test function for the modified momentum Eq. (4.11). Similarly to Lemma 5.1 in [16], we have the estimate β p(ρδ , θδ ) + δρδ ln(1 + ρδ ) d x ≤ c, (5.19)
and hence p(ρδ , θδ ) ln(1 + ρδ ) is bounded in L 1 ( × (0, T )).
(5.20)
Let us define the set Jkδ = {(x, t) ∈ (0, T ) × : ρδ (x, t) ≤ k}
for k > 0 and δ ∈ (0, 1).
In view of (5.1) and the hypothesis (2.3), there exists a constant s ∈ (0, ∞) such that for all δ ∈ (0, 1) and k > 0, s |{(0, T ) × − Jkδ }| ≤ . k We have the following estimate: T β δρδ d xdt = 0
β
Jkδ
δρδ d xdt + β
≤ T δk || + δ
T
β
×(0,T )−Jkδ
0
δρδ d xdt
β χ×(0,T )−J δ ρδ k
(5.21) d xdt.
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Then, by the Hölder inequality in Orlicz spaces (cf. [1]) and the estimate (5.19), we obtain δ
T 0
β
χ×(0,T )−J δ ρδ d xdt ≤ δ χ×(0,T )−J δ L N max{1, k
k
T 0
β
M(ρδ ) d xdt}
−1 T k β β max{1, 2(1 + ρδ )ln(1 + ρδ ) d xdt} ≤ δ N −1 s 0 (5.22) −1 T k β −1 ≤δ N max{1, (4ln2)T || + 4β ρδ ln(1 + ρδ ) d xdt} s 0 ∩{ρδ ≥1} −1 T β −1 k ≤ N max{δ, (4ln2)δT || + 4δβ ρδ ln(1 + ρδ ) d xdt}, s 0 where L M (), and L N () are two Orlicz Spaces generated by two complementary N-functions M(s) = (1 + s)ln(1 + s) − s,
N (s) = es − s − 1,
respectively. Due to (5.19), we know, if δ < 1, max{δ, (4ln2)δT || + 4δβ
T 0
β
ρδ ln(1 + ρδ ) d xdt} ≤ c,
for some c > 0 which is independent of δ. Combining (5.21) with (5.22), we obtain the following estimate T −1 β β −1 k δρδ d xdt ≤ T δk || + c N , s 0
where c does not depend on δ and k. Consequently T −1 β −1 k lim sup δρδ d xdt ≤ c N . s δ→0
0
(5.23)
The right-hand side of (5.23) tends to zero as k → ∞. Thus, we have T lim
δ→0 0
β
δρδ d xdt = 0,
which yields β
δρδ → 0 in D ( × (0, T )).
(5.24)
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X. Hu, D. Wang
5.4. Strong convergence of the temperature. Since ρδ , uδ satisfy the continuity equation (1.1a), then γ
ρδ → ρ in C([0, T ]; L weak ()), uδ → u weakly in L
2
(5.25)
([0, T ]; W01,2 ()),
(5.26)
and thus 2γ
ρδ uδ → ρu weakly-* in L ∞ ([0, T ]; L 1+γ ()),
(5.27)
where the limit functions ρ ≥ 0, u satisfy the continuity Eq. (1.1a) in D ( × (0, T )). Similarly, since ρδ , uδ , θδ , Hδ satisfy the momentum Eq. (4.11), we have 2γ 1+γ ()) ρδ uδ → ρu in C([0, T ]; L weak
and 6γ
ρδ uδ ⊗ uδ → ρu ⊗ u weakly in L 2 ([0, T ]; L 3+4γ ()). From the hypothesis (2.3), Eq. (1.1d), and the estimates (5.1), (5.9), we can assume pθ (ρδ ) → pθ (ρ) weakly in L ∞ ([0, T ]; L 3 ()), Hδ → H weakly in L 2 ([0, T ]; W01,2 ()) ∩ C([0, T ]; L 2weak ()), with divH = 0 in D ( × (0, T )). Hence, ρ, ρu, H satisfy the initial data (1.2). Due to the estimate (5.10), we can also assume θδ → θ weakly in L 2 ([0, T ]; W 1,2 ()), with θ ≥ 0 in D ( × (0, T )), since |∇θδ | ≤ (1 + θδ )
α−1−ω 2
|∇θδ |,
for ω ∈ (0, 1).
Thus θδ pθ (ρδ ) → θ pθ (ρ) weakly in L 2 ([0, T ]; L 2 ()), (∇ × Hδ ) × Hδ → (∇ × H) × H in D ( × (0, T )).
(5.28) (5.29)
∇ × (uδ × Hδ ) → ∇ × (u × H) in D ( × (0, T )), ∇ × (ν∇ × Hδ ) → ∇ × (ν∇ × H) in D ( × (0, T )).
(5.30) (5.31)
Similarly,
In view of (5.6) and the hypothesis (2.6), we can assume Q h (θδ ) → Q h (θ ) weakly in L 2 ([0, T ]; W 1,2 ()),
(5.32)
M(θδ ) → M(θ ) weakly in L ([0, T ]; W
(5.33)
2
for any M ∈ C 1 [0, ∞) satisfying the growth restriction α
|M (ξ )| ≤ c(1 + ξ 2 −1 ),
1,2
()),
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277
and, consequently, 6γ
(ρδ + δ)Q h (θδ ) → ρ Q h (θ ) weakly in L 2 ([0, T ]; L γ +6 ()),
(5.34)
since L γ →→ W −1,2 (), if γ > 23 . At this stage we need a variant of the celebrated Aubin-Lions Lemma (cf. Lemma 6.3 in [9]): Lemma 5.1. Let {θn }∞ n=1 be a sequence of functions such that 2 q ∞ 1 {θn }∞ n=1 is bounded in L ([0, T ]; L ()) ∩ L ([0, T ]; L ()), with q >
6 , 5
and assume that ∂t θn ≥ χn in D ( × (0, T )), where χn are bounded in L 1 ([0, T ]; W −m,r ()) for certain m ≥ 1, r > 1. Then {θn }∞ n=1 contains a subsequence such that θn → θ in L 2 ([0, T ]; W −1,2 ()). With this lemma in hand, we can show the following property: Lemma 5.2. Let h =
1 1+θ ,
then
(ρδ + δ)Q h (θδ ) → ρ Q h (θ ) in L 2 ([0, T ]; W −1,2 ()). Proof. Substituting h =
1 1+θ
into (4.14), we get
κ(θδ ) ∂t ((ρδ + δ)Q h (θδ )) ≥ − div(ρδ Q h (θδ )uδ ) + div ∇θδ 1 + θδ −δ
θδα+1 θδ − pθ (ρδ )divuδ , 1 + θδ 1 + θδ
in D ( × (0, T )). Since θδ pθ (ρδ )|divuδ | ≤ pθ (ρδ )|divuδ |, 1 + θδ we know that, in view of (2.3) and (5.1), θδ pθ (ρδ )divuδ is bounded in L 2 ([0, T ]; L r ()), 1 + θδ
for some r > 1,
and, consequently, θδ pθ (ρδ )divuδ is bounded in L 2 ([0, T ]; W −k,r ()), 1 + θδ
for all k ≥ 1.
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X. Hu, D. Wang
Similarly, by (5.18) δ
θδα+1 is bounded in L 2 ([0, T ]; W −k,r ()), 1 + θδ
for all k ≥ 1,
and div(ρδ Q h (θδ )uδ ) is bounded in L 1 ([0, T ]; W −1,r ()),
for some r > 1.
Next, by (2.4), we have α α κ(θδ ) |∇θδ | ≤ c(1 + θδ )α−1 |∇θδ | ≤ cθδ2 |∇θδ2 |, 1 + θδ
if θδ ≥ 1,
and κ(θδ ) |∇θδ | ≤ c(1 + θδ )α−1 |∇θδ | ≤ c|∇θδ |, 1 + θδ
if θδ ≤ 1,
thus, by (5.6) and (5.18), κ(θδ ) div ∇θδ is bounded in L 1 ([0, T ]; W −1,r ()), 1 + θδ
for some r > 1.
Finally, since Q h (θ ) ≤ Q(θ ), by (5.3), we deduce that (ρδ + δ)Q h (θδ ) ∈ L ∞ ([0, T ]; L 1 ()). Hence, combining Lemma 5.1 and (5.34) together, we have (ρδ + δ)Q h (θδ ) → ρ Q h (θ ) in L 2 ([0, T ]; W −1,2 ()). Lemma 5.2 and (5.33) imply (ρδ + δ)Q h (θδ )M(θδ ) → ρ Q h (θ ) M(θ ) in L 1 ( × (0, T )),
(5.35)
1 where h(θ ) = 1+θ . On the other hand, choosing M(θ ) = θ , then θ Q h (θ ) satisfies (5.33) since α > 2. Hence,
(ρδ + δ)Q h (θδ )θδ → ρθ Q h (θ ) weakly in L 1 ( × (0, T )).
(5.36)
Properties (5.35) and (5.36) imply θ Q h (θ ) = Q h (θ )θ, a.e. on {ρ > 0}, which yields θδ → θ in L 1 ( × (0, T )).
(5.37)
Global Solutions to Magnetohydrodynamic Flows
279
Indeed, we know that Q h (θ ) is strictly increasing and its derivative has upper bound, therefore its inverse Q −1 h (θ ) exists and has lower bound 1/cυ . Thus, T 0
|Q h (θ ) − Q h (θδ )|2 d xdt
≤ cυ = cυ
T 0
T
0
−1 (Q −1 h (Q h (θ )) − Q h (Q h (θδ )))(Q h (θ ) − Q h (θδ )) d xdt
(θ − θδ )(Q(θ ) − Q(θδ )) d xdt → 0,
as δ → 0.
Therefore, Q h (θδ ) → Q h (θ ), in L 2 ( × (0, T )),
as δ → 0,
Q h (θδ ) → Q h (θ ), a.e. in × (0, T ),
as δ → 0.
and, hence,
Because Q −1 h (θ ) is continuous, we deduce that −1 θδ = Q −1 h (Q h (θδ )) → θ = Q h (Q h (θ )), a.e. in × (0, T ),
as δ → 0,
which, combining Egorov’s theorem, Theorem 2.10 in [9], and the weak convergence of {θδ } to θ in L 1 ( × (0, T )), verifies (5.37). Finally, (5.37), together with (5.26) and Lemma 4.1, implies δ = µ(θδ )(∇uδ + ∇uδT ) + λ(θδ )divuδ I → = µ(θ )(∇u + ∇uT ) + λ(θ )divuI, (5.38) in D ( × (0, T )), and, T T h(θδ )δ : ∇uδ d xdt ≥ h(θ ) : ∇u d xdt. 0
0
(5.39)
5.5. Strong convergence of the density. In this subsection, we will adopt the technique in [9] to show the strong convergence of the density, specifically, ρδ → ρ in L 1 ( × (0, T )).
(5.40)
First, due to (5.20), Proposition 2.1 in [9] and (5.28), we can assume that p(ρδ , θδ ) → p(ρ, θ ) weakly in L 1 ( × (0, T )),
(5.41)
which, together with (5.24)–(5.31), implies that ρt + div(ρu) = 0, ∂t (ρu) + div(ρu ⊗ u) + ∇ p(ρ, θ ) = (∇ × H) × H + div, Ht − ∇ × (u × H) = −∇ × (ν∇ × H), divH = 0, in D ( × (0, T )).
(5.42) (5.43) (5.44)
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Similarly to Sect. 5.3 in [16], we use ϕi (x, t) = ψ(t)φ(x)Ai [Tk (ρδ )], ψ ∈ D(0, T ), φ ∈ D(), i = 1, 2, 3, as test functions for the modified momentum balance Eq. (4.11), where Ai can be expressed by their Fourier symbol as −1 −iξi F[·] , i = 1, 2, 3, Ai [·] = F |ξ |2 and Tk (ρ) are cut-off functions, Tk (ρ) = min{ρ, k}, k ≥ 1. A lengthy but straightforward computation shows that
∂ui ψφ − λ(θδ )divuδ )Tk (ρδ ) − 2µ(θδ ) δ Ri, j [Tk (ρδ )] ∂x j 0 T β ψ∂xi φ λ(θδ )divuδ − p(ρδ , θδ ) − δρδ Ai [Tk (ρδ )] d xdt = 0 T j ∂uδi ∂uδ + ψµ(θδ ) + ∂x j φAi [Tk (ρδ )] d xdt ∂ x j ∂ xi 0 T φρδ u iδ ψt Ai [Tk (ρδ )] + ψAi [(Tk (ρδ ) − Tk (ρδ )ρδ )divuδ ] d xdt − T
β ( p(ρδ , θδ ) + δρδ
d xdt
(5.45)
0
− + −
T
0 T 0
T
j
ψρδ u iδ u δ ∂x j φAi [Tk (ρδ )] d xdt j j ψu iδ Tk (ρδ )Ri, j [ρδ u δ ] − φρδ u δ Ri, j [Tk (ρδ )] d xdt ψφ(∇ × Hδ ) × Hδ · A[Tk (ρδ )] d xdt,
0
where the operators Ri, j = ∂x j Ai [v] and the summation convention is used to simplify notations. On the other hand, following the arguments of Sect. 5.3 in [16], one has T ∂ui ψφ p(ρ, θ ) − λ(θ )divu Tk (ρ) − 2µ(θ ) Ri, j [Tk (ρ)] d xdt ∂x j 0 T ψ∂xi φ λ(θ )divu − p(ρ, θ ) Ai [Tk (ρ)] d xdt = 0
∂ui ∂u j ∂x j φAi [Tk (ρ)] d xdt + ψµ(θ ) + ∂ x j ∂ xi 0 T φρu i ψt Ai [Tk (ρ)] + ψAi [(Tk (ρ) − Tk (ρ)ρ)divu] d xdt − T
−
0 T 0
ψρu i u j ∂x j φAi [Tk (ρ)] d xdt
(5.46)
Global Solutions to Magnetohydrodynamic Flows
T + −
0
T
0
281
ψu i Tk (ρ)Ri, j [φρu j ] − φρu j Ri, j [Tk (ρ)] d xdt ψφ(∇ × H) × H · A[Tk (ρ)] d xdt.
Now, following the argument in Sect. 5.3 in [16], the Div-Curl Lemma can be used in order to show that the right-hand side of (5.45) converges to that of (5.46), that is T ∂uδi lim ψφ ( p(ρδ , θδ ) − λ(θδ )divuδ )Tk (ρδ ) − 2µ(θδ ) Ri, j [Tk (ρδ )] d xdt δ→0 0 ∂x j (5.47) T ∂ui = ψφ ( p(ρ, θ ) − λ(θ )divu)Tk (ρ) − 2µ(θ ) Ri, j [Tk (ρ)] d xdt. ∂x j 0 Noting that T
∂ui ϕµ(θδ ) δ Ri, j [Tk (ρδ )] d xdt ∂x j 0 T ∂uδi ∂uδi = Ri, j ϕµ(θδ ) − ϕµ(θδ )Ri, j Tk (ρδ ) d xdt ∂x j ∂x j 0 T ϕµ(θδ )divuδ Tk (ρδ ) d xdt, + 0
for any ϕ ∈ D( × (0, T )), we have, using also (5.47), T
ϕ (( p(ρδ , θδ ) − (λ(θδ ) + 2µ(θδ ))divuδ )Tk (ρδ )) d xdt
lim
δ→0 0 T
=
0
ϕ
(5.48)
p(ρ, θ ) − (λ(θ ) + 2µ(θ ))divu Tk (ρ) d xdt,
since Lemma 4.2 in [8] and the strong convergence of the temperature give T ∂uδi ∂uδi Ri, j ϕµ(θδ ) − ϕµ(θδ )Ri, j Tk (ρδ ) d xdt ∂x j ∂x j 0 i T ∂ui ∂u Ri, j ϕµ(θ ) − ϕµ(θ )Ri, j Tk (ρ) d xdt. → ∂x j ∂x j 0 As in Sect. 5.4 of [16], we can conclude from (5.48) that there exists a constant c independent of k such that lim sup Tk (ρδ ) − Tk (ρ) L γ +1 ((0,T )×) ≤ c. δ→0+
This implies, in particular, that the limit functions ρ, u satisfy the continuity Eq. (1.1a) in the sense of renormalized solutions (cf. Lemma 5.4 in [16]). Finally, following the argument as in Sect. 5.6 in [16], (5.40) is verified.
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5.6. Thermal energy equation. In order to complete the proof of Theorem 2.1, we have to show that ρ, u, θ and H satisfy the thermal energy Eq. (1.10) in the sense of Definition 2.1. In view of (5.18) and (5.37), we have, as δ → 0, θδ → θ in L p ( × (0, T )),
for all 1 ≤ p < 1 + α.
Hence, by the Lebesgue’s dominated convergence theorem and the hypothesis (2.4), we know, as δ → 0, K h (θδ ) → K h (θ )
in L 1 ( × (0, T )).
By (5.25), (5.26) and (5.40), we have, as δ → 0, ρδ Q h (θδ )uδ → ρ Q h (θ )u, ρδ Q h (θδ ) → ρ Q h (θ ), h(θδ )θδ pθ (ρδ )divuδ → h(θ )θ pθ (ρ)divu, in D ( × (0, T )). Due to the strong convergence of the temperature (5.37) and (5.39)–(5.40), we can pass the limit as δ → 0 in (4.14) to obtain T (ρ Q h (θ )∂t ϕ + ρ Q h (θ )u · ∇ϕ + K h (θ ) ϕ) d xdt 0
≤−
T
+
0 T
0
(h(θ )( : ∇u + ν|∇ × H|2 )ϕ d xdt
(5.49)
h(θ )θ pθ (ρ)divuϕ d xdt −
since
T δ θδ1+α h(θδ )d xdt ≤ δ 0
{θδ ≤M}
ρ0 Q h (θ0 )ϕ(0) d x,
T θδ1+α d xdt + h(M)δ θδ1+α d xdt, 0
which tends to zero as δ → 0, because the first term on the right-hand side tends to zero for fixed M as δ → 0 while the second term can be made arbitrarily small by taking M large enough in view of (4.6) and (5.4). Next, taking h(θ ) =
1 , 0 < ω < 1, (1 + θ )ω
in (5.49), letting ω → 0, and using Lebesgue’s dominated convergence theorem, we get T (ρ Q(θ )∂t ϕ + ρ Q(θ )u · ∇ϕ + K (θ ) ϕ) d xdt 0 (5.50) T ≤
0
(θ pθ (ρ)divu − ν|∇ × H|2 − : ∇u)ϕ d xdt,
for any ϕ ∈ D(×(0, T )) and ϕ ≥ 0, since ρ Q h (θ ) ≤ ρ Q(θ ) belongs to L 1 (×(0, T )) by (5.1), (2.6), and (5.11).
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Finally, dividing (2.1) by 1 + θ and using Eq. (1.1a), we get q ∂t (ρ f (θ )) + div(ρ f (θ )u) + div 1+θ 1 q · ∇θ θ 2 ≥ − (ν|∇ × H| + : ∇u) − pθ (ρ)divu, 2 1+θ (1 + θ ) 1+θ in the sense of distributions, where f (θ ) = 0
θ
cν (ξ ) dξ. 1+ξ
Integrating the above inequality over × (0, T ), we deduce T 1 k(θ )|∇θ |2 2 (ν|∇ × H| + : ∇u) + d xdt (1 + θ )2 0 1+θ T θ pθ (ρ)|divu| d xdt. ρ f (θ ) d x + ≤ 2 sup 0 1+θ 0≤t≤T
(5.51)
By Hölder’s inequality, the estimates (5.1), (5.9), and the hypothesis (2.3), one has T T θ pθ (ρ)|divu| d x dt ≤ pθ (ρ)|divu| d x dt ≤ c. 0 1+θ 0 Similarly, by Hölder’s inequality, the assumption (2.6), and the estimates (5.1), (5.18), we have ρ f (θ ) d x ≤ c ρθ d x ≤ c.
Thus, (5.51) and the assumption (2.4) imply that α
ln(1 + θ ) ∈ L 2 ([0, T ]; W 1,2 ()), θ 2 ∈ L 2 ([0, T ]; W 1,2 ()). This completes our proof of Theorem 2.1. Acknowledgements. Xianpeng Hu’s research was supported in part by the National Science Foundation grant DMS-0604362. Dehua Wang’s research was supported in part by the National Science Foundation grants DMS-0244487, DMS-0604362, and the Office of Naval Research grant N00014-01-1-0446.
References 1. Admas, R.A.: Sobolev spaces, Pure and Applied Mathematics, Vol. 65. New York-London: Academic Press, 1975 2. Cabannes, H.: Theoretical Magnetofluiddynamics. New York: Academic Press, 1970 3. Chen, G.-Q., Wang, D.: Global solution of nonlinear magnetohydrodynamics with large initial data. J. Differ. Eqs. 182, 344–376 (2002) 4. Chen, G.-Q., Wang, D.: Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z. Angew. Math. Phys. 54, 608–632 (2003) 5. DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989) 6. Ducomet, B., Feireisl, E.: The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars. Commun. Math. Phys. 226, 595–629 (2006)
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7. Fan, J., Jiang, S., Nakamura, G.: Vanishing shear viscosity limit in the magnetohydrodynamic equations. Commun. Math. Phys. 270, 691–708 (2007) 8. Feireisl, E.: On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53, 1707–1740 (2004) 9. Feireisl, E.: Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications, 26. Oxford: Oxford University Press, 2004 10. Freistühler, H., Szmolyan, P.: Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves. SIAM J. Math. Anal. 26, 112–128 (1995) 11. Gerebeau, J.F., Bris, C.L., Lelievre, T.: Mathematical methods for the magnetohydrodynamics of liquid metals. Oxford: Oxford University Press, 2006 12. Goedbloed, H., Poedts, S.: Principles of magnetohydrodynamics with applications to laboratory and astrophysical plasmas. Cambridge: Cambridge University Press, 2004 13. Hoff, D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rat. Mech. Anal. 132, 1–14 (1995) 14. Hoff, D.: Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heatconducting fluids. Arch. Rat. Mech. Anal. 139, 303–354 (1997) 15. Hoff, D., Tsyganov, E.: Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z. Angew. Math. Phys. 56, 791–804 (2005) 16. Hu, X., Wang, D.: Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Submitted for publication 17. Kazhikhov, V., Shelukhin, V.V.: Unique global solution with respect to time of initial-boundary-value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977) 18. Kawashima, S., Okada, M.: Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A Math. Sci. 58, 384–387 (1982) 19. Kulikovskiy, A.G., Lyubimov, G.A.: Magnetohydrodynamics. Reading, MA: Addison-Wesley, 1965 20. Laudau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 2nd ed., New York: Pergamon, 1984 21. Lions, P.L.: Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press, 1996 22. Lions, P.L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press, 1998 23. Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Memoirs Amer. Math. Soc. 599, 1997 24. Novotný, A., Straškraba, I.: Introduction to the theory of compressible flow. Oxford: Oxford University Press, 2004 25. Wang, D.: Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM J. Appl. Math. 63, 1424–1441 (2003) Communicated by P. Constantin
Commun. Math. Phys. 283, 285–304 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0509-2
Communications in
Mathematical Physics
Forced Vibrations via Nash-Moser Iteration Jean-Marcel Fokam Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada. E-mail:
[email protected] Received: 31 July 2007 / Accepted: 9 December 2007 Published online: 4 July 2008 – © Springer-Verlag 2008
Abstract: We construct time periodic solutions for a cubic nonlinear wave equation with time-dependent forcing term. 1. Introduction In this paper we construct classical inhomogeneous time-periodic solutions for the following nonlinear wave equation with time-dependent forcing term f : u tt − u x x − v(x)u = u 3 + f (t, x).
(1.1)
Here v is an even real analytic potential and f is a trigonometric polynomial odd in time and space variables. We require the solution u to satisfy the Dirichlet boundary conditions, u(0, t) = u(π, t) = 0. Letting g(u) = u 3 , we denote the nonlinear term in (1.1) by G(u, f ) = g(u) + f, where f does not depend on u. The construction of the periodic solution here follows a Nash-Moser iteration scheme. We do not assume any smallness condition on f or on the right-hand-side of (1.1). The solution that we construct is smooth. Previous work on (1.1) either assumes a smallness condition on the right-hand side of (1.1) or proves the existence of a weak solution via critical point theory under the assumption that the time-frequency is rational. We describe those results later in this introduction. The method we follow in this paper originates in the work of Craig and Wayne [11]. They devised an efficient way to handle the small denominators problem (here the Current address: American University of Nigeria, Yola, Nigeria. E-mail:
[email protected]
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J.-M. Fokam
eigenvalues of the linearized operator accumulating at zero) arising from the linearized operator in equations of the type (1.1) for free vibrations ( f = 0) using a Nash-Moser iteration scheme and an estimate of Pöschel in [24]. Their work was later continued in higher dimension by Bourgain in [5]. Craig and Wayne in [11] constructed periodic solutions of (1.1) around a solution of the linearized problem. Here we show that the Nash-Moser iteration scheme can be used to construct periodic solutions for forced vibrations ( f = 0) when the time frequency is large and belongs to a set of positive measure which we construct. Choosing the frequency to be large has the effect of boosting the low Fourier-modes of the left-hand side of (1.1), compensating for the effects of the forcing term f in the right-hand side. This allows the application of the contraction mapping principle to give the existence of an approximate solution u 0 . Then the iterative scheme can be applied to construct the solution of (1.1). In [11] a combination of the Implicit Function Theorem (IFT) and non-resonance condition on v was used to prove existence of an approximate solution u 0 . Here since u = 0 is not a solution of (1.1) the forcing term and f is not an epsilon perturbation of g we do not know how the IFT can be used to find an approximate solution. Moreover it is well known that the convergence of Newton’s type iteration scheme is usually dependent on the start of a good approximate solution. Here we show in Lemma 3.1 that the Contraction Mapping Principle suffices to find a good approximate solution for all even analytic potentials v, without requiring a non-resonance condition, as long as the frequency is large. Then the other difficulty in the proof of the convergence of the iteration scheme is the upper bound on the measure of the excised set of frequencies during the iteration. This estimate depends on the variations of the eigenvalues-eigenvectors (ei , ψi ) of the Hamiltonian H (). Here we show that the singular set of (ei , ψi ) is at most countable. It is an improvement of the a.e. smooth results of [11] which is needed since there exists strictly increasing function with derivative equal to zero almost everywhere (see [19] p. 261) and hence, a better understanding of the singular set is needed to control the variations of the eigenvalues. As a consequence we show that < ψi , H ()ψi >= ei satisfies the Fundamental Theorem of Calculus and also this allows us to clarify the measurability of ∂ < ψi , H ()ψi > which was unclear. If v = 0 and the right-hand side is a small perturbation—that is, G(u, f ) is replaced by G(u, f )—then Rabinowitz in [25] proved the existence of a smooth periodic solution for sufficiently small. When v = 0 and is assumed to be rational, variational methods can be applied because the small denominators problem does not occur. In that case Tanaka [28] found critical points of some functional which are locally integrable. p The critical points in [28] belong to L loc [(0, π ) × R] hence their trace on the boundary is unclear. This contrasts with the critical points found by Rabinowitz for free vibrations ( f = 0) in [26] which are C ∞ [(0, π ) × (0, 2π )]. Plotnikov finds generalized periodic solutions as critical points of a functional he defines in [22]. He shows that these generalized periodic solutions belong to L p [(0, π ) × (0, 2π )] but their trace on the boundary is not well-defined. Berti and Bolle in [1] have found periodic solutions for nonlinear wave equations with irrational frequencies for free vibrations where f (x, t) does not appear in the nonlinearity. For forced vibrations, Brezis and Nirenberg ([8] Theorem 2) prove regularity of weak solutions equations of the type (1.1) for which one has an L ∞ a priori bound on the weak f) solution, the frequency is rational and ∂G(u, > 0. ∂u While a counterexample in [7] shows that even for the linearized equation existence is not necessarily guaranteed (see also Remark 2 below), neither is the analyticity of the periodic solution. In fact, Borel in his thesis in 1895, [6] had already noticed that there are real numbers α and periodic solutions u of
Forced Vibrations via Nash-Moser Iteration
287
u tt − α 4 u x x = f (t, x)
(1.2)
which are not analytic, while the forcing term f is analytic. We remark that KAM methods require some non-resonance conditions on the potentials v to ensure the convergence of the iteration scheme, see for instance [2]. Here no such condition is required for the scheme to converge. For some irrational the inverse of the linearized operator in (1.1) is bounded and fixed points methods can also provide a proof of existence of solutions (see Mawhin, Corollary 1 in [20], De La Llave, Theorem 2.1 in [14]), but require a uniform Lipschitz bound on g(u), hence g(u) grows at most linearly when applied to an equation of the type (1.1). Degree theoretical methods also require a linear growth condition on g(u) (see Brezis and Nirenberg [8]). 2. Statement of Results Rescaling (1.1) by the variable ξ = t we rewrite: 2 u ξ ξ − u x x − v(x)u = g(u) + f (ξ, x), where g(u) =
(2.3)
u3.
Theorem 2.1. Let f (t, x) an odd in space, and in time, forcing term, of frequency with ∈ N0 = [3L 0 , 10L 0 ], where L 0 > 0 is a large positive constant, analytic in a horizontal strip. There is a subset N ⊂ N0 such that (1.1) admits a solution for ∈ N . As L 0 tends to infinity the measure of the complement of N into N0 , |N0 \ N | → 0. Moreover, u is real analytic and admits an eigenfunction expansion of the form: u(x, ξ ) = u( ˆ j, k)ψ j (x)eikt ( j,k)∈N×Z
and |u(x, ξ )| ≤
1 L 1.5 0
.
(2.4)
Remark 1. Our method of proof works for any odd order polynomial g(x, u) with dependence on x, and for analytic polynomial of the form g(x, u) =
p
am (x)u m ,
m=3
where the lowest term in the expansion of g in u above is at least of order 3 in u. g must also preserve the oddness of the solution: g(−x, u(−x, ξ ) = −g(x, u(x, ξ )) and g(x, u(x, −ξ )) = −g(x, u(x, ξ )). If one wants to incorporate quadratic terms such as a(x)u 2 in g for instance, the oddness in time may not be preserved during the iteration procedure. The coefficients am (x) must also be bounded in a horizontal strip around the real axis in the x variables. The forcing term f here is finitely supported in Fourier space and odd in time and in space (like, for instance, sin x sin t) and our method of proof works as long as the forcing term f is analytic in x and ξ and bounded in some horizontal strip around the real line in x and ξ variables; and we define a norm: || f (x, ξ )||σ,∞ =
sup
| Im x|≤σ,| Im ξ |≤σ
| f (x, ξ )|.
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When f is not compactly supported in Fourier Space one has to add the remainder of f in (3.31). The fast decay of the remainder of f when || f ||σ,∞ < ∞ ensures the convergence of the iteration scheme. Remark 2. To illustrate the need to restrict the values of , we consider an example from [7]. We let v = 3, which is even, g = 0, which is odd in space and time variables, and f (x, t) = sin x sin t, which is odd in space and time variables as well, and set = 2. Then (1.1), which becomes u tt − u x x + 3u = sin x sin 2t, does not have any solution satisfying the periodicity and boundary conditions. The method in this paper applies to this case and it is therefore clear that one has to remove the “bad” values of if one is to expect a general existence theorem, hence the existence of solution for in a set of positive measure in the theorem (2.1). We can define the operator V () acting on sequences u( ˆ j, k) in the following way: [V ()][( j, k), ( j , k )] = (−2 k 2 + ω2j )δ(( j, k), ( j , k ).
(2.5)
We denote the right-hand side of (2.3) by W (u) + f , where W (u) is the sequence defined by π 2π W (u)( j, k) = ψ j (x)e−ikξ u 3 d xdξ, 0
0
where (ψ j , ω j ) are the eigenvectors-eigenvalues of the Sturm-Liouville operator ∂x x + v with Dirichlet boundary condition and fˆ is the representation of the function f (x, ξ ) in the eigenfunction basis {sin( j x)eikξ }. We seek an approximate solution u 0 which solves (2.6) for all ( j, k) such that | j| + |k| ≤ L 0 , where L 0 is a large constant. Let L n = 2n L 0 , where L 0 is a large constant depending on the potential v, and let Bn = {( j, k), | j| + |k| ≤ L n }, and Pn the projection on Bn . Define the function space H σ = {u( ˆ j, k), |u( ˆ j, k)|2 e2σ ( j+|k|) < ∞}, ( j,k)∈N×Z
and the operator norm, ||S||σ = max{sup x
|S(x, y)|eσ |x−y| , sup
y
y
|S(x, y)|eσ |x−y| }.
x
We can rewrite (1.1) in Fourier space as F(u, )( j, k) = V ()u( ˆ j, k) − W (u)( j, k) − f ( j, k) = 0 ∀ l = ( j, k). (2.6) The solution we construct in this paper is odd in time and odd in space so the mode k = 0 is irrelevant in the discussion. In this paper we use the spaces H σ above rather than the weighted Sobolev spaces defined in [10] (which have the advantage to be a Banach algebra) because we are dealing with the Dirichlet boundary conditions in analytic spaces, and oddness in x variables is not preserved when two eigenfunctions are multiplied. We first recall some properties of the spaces H σ used in [11] to construct the periodic solutions.
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Lemma 2.1. [11]. Let f (x, ξ ) be a function analytic in a strip | Im x| < σ , | Im ξ | < σ , odd in x, and let fˆ be its Fourier representation in the basis {ψ j eikξ }. Then we have the following properties: || f ||σ −γ ,∞ ≤
C ˆ || f ||σ , γ
(2.7)
C || f ||σ −γ ,∞ . γ
|| fˆ||σ −γ ≤
(2.8)
The nonlinearity we consider in this paper is cubic in u; a quick application of the preceding lemma then gives || u vw||σ −γ ≤
c ||uvw||σ −γ ,∞ γ
(2.9)
and c ||u|| ˆ σ ||v|| ˆ σ ||w|| ˆ σ. γ3
||uvw||σ −γ ,∞ ≤
(2.10)
In this paper following the notations in [11], we will use ||u||σ for the norm of the sequence u, ˆ ||u|| ˆ σ . We have ||(I − Pn )u||σ ≤ e−γ L n ||u||σ +γ
(2.11)
||Pn u||σ ≤ eγ L n ||u||σ −γ .
(2.12)
and
The operator norm is sub-multiplicative, ||ST ||σ ≤ ||S||σ ||T ||σ , and we have also the following estimate from [11]: ||DW (u)||σ −γ ≤
c ||Du g(u)||σ −γ /2,∞ , γ2
(2.13)
where DW (u)( j, k)( j , k ) =
π 0
2π
ψ j (x)eikξ ψ j e−ik ξ 3u 2 (x, ξ )d xdξ,
(2.14)
0
(Nn , ρn ) = { ∈ C such that | − 1 | ≤ ρn and 1 ∈ Nn }.
(2.15)
3. Construction of the Solution Construction of the approximate solution u 0 . The following lemma holds for ∈ (N0 , ρ0 ), where and N0 = { ∈ R, 3L 0 ≤ ≤ 10L 0 } with constants σ0 = σ2 , ρ0 = σ40 and σ = 2.
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In [11](Lemma 3.3) existence of an approximate solution was proved via the Implicit Function Theorem. Here u = 0 is not a solution and we do not have a particular solution to start with to apply the Implicit Function Theorem. Instead existence of the approximate solution here is proved via the Contraction Mapping Principle when the frequency is sufficiently large. We now start the iteration by solving (2.6) in a finite dimensional domain B0 and we seek u 0 such that F(u 0 , )( j, k) = V ()uˆ0 ( j, k) − W (u 0 )( j, k) − f ( j, k) = 0, l = ( j, k) ∈ B0 . u 0 odd in space and odd in time. (3.16) Lemma 3.1. Let C f = || f ||σ , then there exists a unique solution u 0 () of (3.16), analytic in , satisfying ||u 0 ||σ ≤ L −1.6 , the constant L 0 depending on C f . 0 Proof. The solution of the approximate problem in B0 is the fixed point of the operator S defined as follows: S(u) = P0 V ()−1 (P0 (u 3 + f )). This operator is a contraction from the ball centered at the origin and radius
1 L 1.6 0
in H σ
into the same ball: The sequence ω j satisfies the following asymptotic ω j = j 2 − v ∗ + d( j), where v ∗ denotes the average of v and d( j) ∈ l 2 (N)(see [11]). ∈ [3L 0 , 10L 0 ] and ( j, k) ∈ B0 , u is odd in time and space so the mode k = 0 is irrelevant in the discussion, 2 k 2 − ω2j ≥ 2 k 2 − ( j 2 − v ∗ + d( j)) ≥ 10L 20 − (2 j 2 + C(v)) ≥ L 20 .
(3.17)
For any analytic potential v we can find such large L 0 . Then since |V ()| ≥ L 20 and applying Lemma 2.1 with γ =
1 L0
we have
1 [e1 ||u 3 ||σ − 1 + || f ||σ ] L0 L 1.7 0 1 ≤ 1.7 [e1 L 40 ||u||3σ + || f ||σ ] L0 1 ≤ 1.6 . L0
||S(u)||σ ≤
(3.18)
Taking the respective norms and applying the inequalities (2.12) we have, ||S(u 1 ) − S(u 2 )||σ ≤
Now taking γ =
1 L0
1 γ L0 e (||(u 1 − u 2 )u 1 u 2 ||σ −γ + ||(u 1 − u 2 )u 2 u 2 ||σ −γ . L 1.6 0 +||(u 1 − u 2 )u 1 u 1 ||σ −γ ).
and applying (2.7) and (2.8) with ||u 1 ||σ , ||u 2 ||σ ≤
that ||S(u 1 ) − S(u 2 )||σ ≤
||u 1 − u 2 ||σ 2
1 L 1.6 0
we conclude
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and S is a contraction. Hence by the fixed point theorem there exists a unique solution of the problem denoted by u 0 (). Analyticity of u() in is deduced by applying the Implicit Function Theorem around any solution (0 , u(0 )) for 0 ∈ (N0 , ρ0 ) to the operator: F(u, )( j, k) = V ()u( ˆ j, k) − W (u)( j, k) − f ( j, k) = 0. In the previous lemma we are only able to obtain an estimate on ||u 0 ||σ as negative power of L 0 , rather than an exponential decay in L 0 because of the presence of the forcing term f . Now following Craig and Wayne [11], u()(x, ξ ) ∈ R for ∈ R. We now need to estimate by how much this approximate solution fails to be a solution: Lemma 3.2. u 0 is an approximate solution and satisfies the estimate ||F(u 0 )||σ −2γ ≤ ε0 where 0 = e−γ
L
0
.
Proof. ||V ()u 0 − (u 3 + f )||σ −2γ = ||(I − P0 )(V ()u 0 − u 3 − f )||σ −2γ ≤ Ce
γ
−γ L
(3.19)
= C0 , (3.20) σ0 = 100 . Having found u 0 we start the iteration. 2−νn L −α 0 , α = 1.1 and ν = 5 which “measures” 0
γ
is using 2.11, and the constant We define the sequence δn : δn = the tolerance of the small eigenvalues of the operator V () which needs to be inverted at each iterative step. Now ρn = ρ0 1280.Lδn+1 , where ρ0 , the constant defined at L2 l2 0 n+1 n+1
the beginning of this section, is the radius of analyticity of u n () and ln a sequence 1 nβ growing exponentially fast: ln = 30 2 , 0 < β < 1. Now the sequence n measures n 1 σ0 how fast the iterative scheme converges: n = 0k , with 1 < k < 2 and γn = 1+n 2 32 , here γn “measures” the loss of analyticity at every stage. We define the sequence σn+1 : σn+1 = σn − 5γn , and the sequence u n+1 = u n + vn , where the DW (u n (x, −ξ )) = DW (−u n (x, ξ )) = DW (u n (x, ξ )) vn = −(Pn+1 F (u n )Pn+1 )−1 Pn+1 F(u n ).
(3.21)
We define the Hamiltonian: Hn+1 = Pn+1 F (u n )Pn+1 ,
(3.22)
and we will prove that the estimate ||vn ||σn+1 ≤
n+1 CG 16((n+1)+1)
γn
δn+1
εn
(3.23)
is true for all n by induction. The estimate on the inverse of Hn+1 is proved in Theorem 3.2. We are now ready to state some properties of the sequences u n and vn . Since σ0 = σ0 ≤ σ − 2γ = σ0 − 2σ0 /100 and we have
σ 2,
Theorem 3.1. Let ∈ (Nn+1 , ρn+1 ) then u n+1 () satisfies the following estimate: ||F(u n+1 )||σn +1 ≤ εn+1 .
(3.24)
Proof. To apply the Nash-Moser method we need to invert Hn+1 at each iterative stage. Now by Taylor’s theorem,
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J.-M. Fokam
1 t
Pn+1 F(u n + vn ) = Pn+1 (F(u n ) + F (u n )vn ) + Pn+1 0
F (u n + τ vn )vn vn dτ dt,
0
and since vn satisfies the relation (3.21) then: 1 t Pn+1 F(u n + vn ) = Pn+1 F (u n + τ vn )vn vn dτ dt. 0
0
We need some estimate on vn so we use the definition (3.21), and taking norms: −1 ||vn ||σn+1 ≤ || − Hn+1 ||σn+1 ||Pn+1 F(u n )||σn+1 ,
||vn ||σn+1 ≤
n+1 CG 16((n+1)+1)
γn
δn+1
εn .
(3.25)
(3.26)
The estimate of −1 || − Hn+1 ||σn+1 ≤
n+1 CG 16((n+1)+1)
γn
(3.27)
δn+1
will be justified in Theorem 3.2. Then 1 t c ||Pn+1 6(u n + τ vn )vn vn dτ dt||σn −5γn ,∞ γn 0 0 C ≤ 4 (||u n ||σn −5γn +γn + ||τ vn ||σn −5γn +γn )||vn ||σn −5γn +γn ) γn ||vn ||σn −5γn +γn 1 n+1 ≤ e−k γ L 0 . (3.28) 2
||Pn+1 F(u n + vn )||σn+1 ≤
The inequality (3.28) follows from the fact that Ce(2n+2) ln C G e
(32n+30) ln
32(1+n 2 ) σ0
2−2ν(n+1) L −2α 0
e−k
n (2−k) σ0 L 100 0
<1
(3.29)
for large L 0 since 1 < k < 2 and k n (2 − k) grows faster than linear as n → ∞, ||(I − Pn+1 )F(u n + vn )||σn −4γn = ||(I − Pn+1 )(u n + vn )3 ||σn −4γn −L n+1 γn
||(u n + vn ) ||σn −4γn +γn ≤e 1 n+1 ≤ e−k γ L 0 . 2
(3.30)
3
(3.31)
The last estimate is equivalent to e
4 ln
32(1+n 2 ) σ0
L −4.5 e 0
−2n+1
σ0 L0 32(1+n 2 )
ek
n+1 σ0 L 100 0
<1
(3.32)
2 n+1 and for small n, L −4.5 dominates the which is satisfied since 1+n 2 grows faster than k 0 product for large L 0 . f is assumed to be a trigonometric polynomial so it did not appear in (3.31). n+1
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293
We now start proving the estimate (3.27). If ( j, k) is a site such that |ω2j −2 k 2 | < ds , then ( j, k) is called a singular site, the fixed the constant ds = 10−1 while ∈ N0 = [3L 0 , 10L 0 ]. Let C(S) be the neighborhood of size ln+1 of the singular site S in the lattice: C(S) = {( j, k) such that | j − j0 | + |k − k0 | ≤ ln+1 }, where ( j0 , k0 ) ∈ S. The proof follows [11] (Lemma 4.8) adapted to our case here where the large frequency depends on L 0 . Lemma 3.3. The singular sites are well separated when ln+1 ≤ L 0 . Let x = ( j, k) and x = ( j , k ) be distinct singular sites in Bn+1 \ Bn , then |x − x | ≥ 2ln+1 .
(3.33)
Proof. Let x = ( j, k) and x = ( j , k ) be two singular sites in Bn+1 \ Bn , that is: |ω2j − k 2 2 | ≤ ds , |ω2j − k 2 | ≤ ds . 2
Using the asymptotic of ω j we deduce: − ds + j 2 + v ∗ < k 2 2 < ds + j 2 + v ∗
(3.34)
(v ∗ denotes the average of v) and k 2 2min < k 2 2 < ds + j 2 + v ∗ .
(3.35)
Now if | j| ≤ L 0 and L 0 ≥ v ∗ then k 2 2min < k 2 2 < ds + j 2 + v ∗ < 2L 20 and with min ≥ 3L 0 implies k = 0 when x is a singular site. (This can also be seen from the fact that u is odd in time so the mode k = 0 is irrelevant, thus there are no singular site in ( j, k) ∈ B0 .) Necessarily | j| ≥ L 0 , since | j| + |k| ≥ L n . Hence for all singular sites x in Bn+1 \ Bn we have, since j is necessarily large and (3.34): 1 2 j ≤ k 2 2 ≤ 2 j 2 , 2
(3.36)
1 2 j ≤ ω2j ≤ 2 j 2 . 2
(3.37)
and
To prove that the singular sites are well separated we will argue by contradiction. If k and k are nonnegative: 1 |k + ω j | ≥ √ (|k| + | j|) using (3.37), 2 now ( j, k) is singular so |k − ω j ||k j + k| < ds ,
(3.38)
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J.-M. Fokam
and using (3.38) we now deduce: √ 2ds |k − ω j | ≤ . Ln Now we estimate the quantity, |(k − k ) − ( j − j )|: |(k − k ) − ( j − j )| ≤ |(k − ω j ) − (k − ω j )| + |(ω j − j)| + |(ω j − j )|, (3.39) but using (3.35) and (3.36) we also have: √ 2 k≤ | j| min and L n ≤ | j| + |k| ≤ | j| +
2 min
| j| ≤
√
2| j|.
Using (3.39) and the asymptotic of ω j from [11] we have, |ω j − j| ≤ c(v) j . (Note that the constant L 0 is chosen very large and L 0 c(v).) We continue from (3.39) to get |(k − k ) − ( j − j )| ≤
2 1 c + ≤ . L n cL n Ln
(3.40)
Now if k, k ≤ 0 then −k, −k ≥ 0, so we can use the same reasoning as above: | − k + ω j ||k + ω j | < ds , then 1 | − k + ω j | = −k + ω j ≥ √ (|k| + | j|) using (3.37) 2 and |k + ω j | ≤
ds ds ≤ . −k + ω j cL n
So once again we have an estimate similar to (3.40), |(k − k ) + ( j − j )| = |(k + ω j ) − (k + ω j ) − (ω j − j) + (ωj − j )| c ≤ (3.41) Ln and again a contradiction will follow. What remains now is the case k, k being of different sign. We show that in that case the singular are necessarily separated: from (3.36) we deduce √ √ (3.42) | j| ≤ 2max |k| ≤ 10 2L 0 |k|, and since x is not in Bn , L n ≤ | j| + |k| ≤ (1 + 14.5L 0 )|k|.
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295
Thus |k| ≥
2n L 0 1 n+1 1 (n+1)β 2 2 ≥ ≥ = ln+1 , 15L 0 30 30
(3.43)
where β is a positive number, 0 < β < 1, |k − k | = |k| + |k | ≥ 2ln+1 . Thus | j − j | + |k − k | ≥ 2ln+1 . Now is chosen in the interval [3L 0 , 10L 0 ] so if | j − j | + |k − k | ≥ 1 and | j − j | + |k − k | ≤ ln+1 ≤ L 0 we have the inequalities for nonnegative k, k : c (3.44) 1 ≤ |(k − k ) − ( j − j )| ≤ Ln and 1 ≤ |(k − k ) + ( j − j )| ≤
c Ln
(3.45)
for negative k, k . (The lower bound in (3.45) is L 0 when k = k , and if k = k then j = j since ( j, k) = ( j , k ), and the lower bound in the two previous estimates is 1.) This shows that as long as ln+1 ≤ L 0 we have a contradiction. For larger values of n the contradiction will follow when satisfies some diophantine condition which we describe now. Lemma 3.4. I = { ∈ N0 , |k − j| <
d , for ln ≤ | j| + |k| ≤ 4ln+1 } (| j| + |k|)τ
with d < 1, then 2 Meas I ≤ 16ln+1
2d . lnτ
Proof. Let ( j, k), be a site such that ln ≤ | j| + |k| ≤ 4ln+1 and 1 and 2 are two real numbers in I , and such that |k1 − j| <
d d and |k2 − j| < . τ (| j| + |k|) (| j| + |k|)τ |2 − 1 | ≤
2d . |k|(| j| + |k|)τ
We can suppose that |k| ≥ 1 since, if k = 0, |0 − j| < (|0|+|d j|)τ is impossible for d < 1 (this condition is required for ( j, k) = (0, 0)). As ln ≤ | j| + |k| we have |2 − 1 | ≤
2d . lnτ
2 sites that satisfy l ≤ | j| + |k| ≤ 4l There are at most 16ln+1 n n+1 so the 2 Meas I ≤ 16ln+1
2d , lnτ
Nn1 = Nn \{ ∈ N0 |k − j| < d(| j| + |k|)−τ ; ln ≤ | j| + |k| ≤ 4ln+1 }.
(3.46)
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J.-M. Fokam
For ∈ Nn1 we have |k − j| ≥
d . (| j| + |k|)τ
(3.47)
We now continue the proof of the separation of sites when ln+1 ≥ L 0 . If l N0 ≤ | j − j | + |k − k | ≤ 4ln+1 (where N0 is the largest integer such that ln+1 ≤ L 0 ) then for k, k nonnegative d τ 4τ ln+1
≤|
d j | + |k
− k |)τ
| ≤ |(k − k ) − ( j − j )| ≤
(| j − 2C c = n+1 = , 1 2 L0 (30ln+1 ) β L 0
c 2n L
0
where we used (3.40) for the upper estimate. Hence d β1 −τ l ≤1 C n+1 and as ln+1 ≥ L 0 with L 0 large, then choosing If k, k are negative then
1 β
greater than τ , we have a contradiction.
d c ≤ |(k − k ) + ( j − j )| ≤ n = τ ln+1 2 L0
c 1 β
,
ln L 0
where the upper estimate follows just as in (3.41), thus a contradiction, and if |( j − j ) + (k − k )| ≤ L 0 , we have a contradiction just as in (3.45). The sets we just excised at the n th stage were intervals. We denote Nn1 the union of open components from N0 which remain. We now have established that the singular sites are separated for ∈ Nn1 . Hn+1 (u n ) is then defined for ∈ Nn1 . We now estimate the measure of the set excised after all iterations are made. The excision here is made only after n ≥ N0 ≥ [ lnβL 0 ] so the estimate of the set of parameters excised is bounded by s N0 =
∞
2 16ln+1
n=N0 +1
≤
∞ 2d ≤ c 2β(2−τ )n lnτ n=N0 +1
) cL (2−τ 0
when we choose τ = 5, s N0 ≤ L −1 0 since we had l N0 +1 =
(3.48) 1 (N0 +1)β 30 2
≥ L 0.
To study the values of for which the eigenvalues of Hn+1 (u n ) may be small we will use a bump function to extend u n () to all of N0 . Then the Fundamental Theorem of Calculus will make it possible to study the variations of the eigenvalues. The bump function we use is the one defined in [9] from which we recall the following lemma:
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297
Lemma 3.5. [9]. Let R > 0 and ∈ Cd be a compact set. We construct a function χ ∈ C(Cd ) ∩ C ∞ (R d ) :→ [0, 1] with support in Y R () = ∪η0 ∈ {η ∈ Cd : ||η − η0 || ≤ R}, identically equal to one in Y R/2 and such that, ∀a ∈ Nd , sup |∂ηa χ | ≤ Rd
(|a| + 2)! . R |a|
We will use this lemma where is Nn1 , the union of finitely many disconnected intervals. The bump function defined by Chierchia in [9] is more than C ∞ , it is also analytic but at finitely many values of . This finer property of χd will be useful when studying the behavior of the eigenvalues of Hn+1 (u n ()). We define: un = u 0 +
n−1
χr vr ,
(3.49)
r =0
where χr is the Chierchia type bump function and where = Nr1 , defined in all of N0 . The bump function vanishes in N0 \ (Nr1 , ρr /2) without changing u r when ∈ Nn1 , and the corresponding Hamiltonian: Hn+1 (un ()) = Pn+1 [V () + DW (un ())]Pn+1 . We show in Lemma 3.8 that Nn+1 consists of disjoint intervals whose distance between the components is larger than 2ρn . So one can use a bump function which we denote by χn , in N0 such that χn u n is defined in all of N0 : χn () = 1 when ∈ N0 ∩ (Nn1 , ρn /4) χn () = 0 N0 \(Nn1 , ρn /2), where Nn1 is a union of sub-intervals of Nn . The bump function satisfies also the following estimate: c ∂χn |≤ . | ∂ ρn Lemma 3.6. For any ∈ (Nn , ρn ) we have ||u n ||σn −2γn ≤
1 L 1.5 0
,
(3.50)
and for any ∈ N0 ||un ||σn −2γn ≤
1 L 1.5 0
(3.51)
and ||
n+1 cL n+1 C G ∂ ||Pn+1 F(u n )||σn −2γn . (χn vn )||σn −2γn ≤ 16((n+1)+1) ∂ ρn γn δn+1
(3.52)
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J.-M. Fokam
Proof. ||u n ()||σn −2γn ≤ ||u 0 ||σ0 −2γ0 +
n−1
||v j ||σ j −2γ j
j=0
≤
1
(3.53)
L 1.5 0
if L 0 is large enough and ∈ Nn . For ∈ N0 gluing the bump functions χn does not worsen the estimates since |χ j | ≤ 1, so || u n ||σn −2γn ≤
1 L 1.5 0
,
∂ un ∂u 0 ||σn −2γn = || + || ∂ (χ j v j )||σn −2γn ∂ ∂ n−1 j=0
≤
c L 1.5 0
,
(3.54)
since j+1
(
2 j+1 L 0 C G
16( j+1)+1 ρjγj δj
ε j ).2− j L −2 0 ≤ 1.
j 0 The linear term in L 0 in −σ 100 L 0 k dominates all the other terms in ln L 0 in the exponential, and the other terms growing linearly in j or j ln j are dominated by k j . Therefore the estimates (3.53) and (3.54) are satisfied. We prove the estimate (3.52) on the derivative of vn , (which is supported in Fourier space in Bn+1 ): vn () 1 ∂vn (0 ) = d, (3.55) ∂ 2iπ C ( − 0 )2
where C is the circle of center 1 ∈ Nn and radius ρn and 0 ∈ (Nn , ρn /2). Making the polar change of variable, = 1 + ρn eiθ , estimate the norm ||
∂vn 16 (0 )( j, k)||2σn −2γn ≤ 2 sup |vn ()( j, k)|2 e2(σn −2γn )(| j|+|k|) ∂ ρn ∈(Nn ,ρn ) j,k
≤
16L 2n+1 ρn2
sup
∈(N n ,ρn )
||vn ()||2σn −2γn .
The Hamiltonian Hn+1 (un ()) is defined in all N0 and so will be its eigenvalues ei labeled by increasing order. For real , ei1 () ≤ ei2 () if i 1 ≤ i 2 .
(3.56)
We will show that they can vanish at most once in N0 and thus do not oscillate wildly in N0 , so just by excluding a neighborhood of the potential singular set Z i = {, ei () = },
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where −δn ≤ ≤ δn , we ensure that for the remaining values of |ei ()| ≥ δn . Denote PC(S) Hn+1 (u n ())PC(S) by HC(S) , where PC(S) is the projection on a singular region PC(S) . Lemma 3.7. The eigenvalues ei () of HC(S) labeled by increasing order are Lipschitz. Proof. Let denote by ω , ω = {1 , . . . , q } the set values of in N0 , where χ1 , . . . , χn are not analytic. There are only finitely many such values of . Away from these values of the operator Hn+1 (un ()) is analytic. Now for ∈ R the operator Hn+1 (un ()) is self-adjoint so we can apply the perturbation theory for self-adjoint analytic operators (see [27]) to conclude that for ∈ / {1 , . . . , q } then there exists a ball B(, r ) of center and radius r , and a labeling (local) of the eigenvalues denoted by e j such that the eigenvalues e j and the corresponding eigenvectors ψ j are analytic. We have at this point the local labeling of the eigenvalues e j , and the “global” valid in all of N0 labeling of the eigenvalues ei , which is done by labeling the eigenvalues by increasing order (see 3.56). It is known (see [27]) that such ei are continuous in . Let q1 and q2 be two q q consecutive values of in ω . Let m1 and m2 be two sequences such that q
q
q1 ≤ m1 ≤ m2 ≤ q2 q
q
with m1 converging to q1 as m tends to infinity and m2 converging to q2 . The q q interval [m1 , m2 ] is compact, so is covered by finitely many balls B(, r /2), q1 q2 ∈ [m , m ]. Now since the eigenvalues e j are analytic in each B(, r /2), they can intersect only finitely many times. Now since the eigenvalues of an operator at any value are independent of any labeling, the continuous eigenvalues ei inherit the analytical properties of the e j , as long as is not a point where the eigenvalues e j may cross. Given that the e j can only cross finitely many times in B(, r /2), one can then deduce that the continuous eigenvalues ei are analytic but at finitely many values of q q in [m1 , m2 ]. And we conclude that the set of values where ei may not be differentiable is at most countable. Now showing that the gradients of ei are bounded will imply the Lipschitz regularity of the eigenvalues ei . When ∈ N0 such that ei is analytic, we differentiate the Feynmann-Hellman (3.57) formula: ei () = i , HC(S) ( u n ())i ,
(3.57)
where i is a normalized eigenfunction (||i ()||l 2 (C(S)) = 1. {i ()( j, k)} j,k is the vector corresponding to the function in x, ξ variables as i ( j, k)ψ j (x)eikξ , (3.58) ( j,k)∈C(S)
where C(S) is a singular region, C(S) = {( j, k) such that | j − j0 | + |k − k0 | ≤ ln+1 }. In Lemma 3.3 the estimate n+1 (3.43) shows that the singular site ( j0 , k0 ) satisfies |k0 | ≥ 230 . Then if ( j, k) ∈ C(S) 1 nβ 2 ≥ 0 and we continue: we have: |k| − 60 ∂ ei = i , ∂ V ()i + i , D 2 W (un ())∂ u n ()i ln2 u n ()i ||l 2 (C(S)) ≤ −2 + ||D 2 W (un ())∂ 4 l2 ≤ −min n . 4
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J.-M. Fokam
The lower bound on ∂ ei is derived similarly since | − k 2 | ≤ 10L 0 L 2n+1 when ( j, k) ∈ L n+1 . Lipschitz regularity of the eigenvalues then follow. Now that i , HC(S) ( u n ()) i satisfies the Fundamental Theorem of Calculus we complete the argument on the variations of the eigenvalues. Lemma 3.8. If there exists an eigenvalue ei ( ) such that |ei ( )| ≤ δn for some then by excising all such that | − | ≤ L4δln2 all the remaining values of in N0 0 n satisfy |ei ()| ≥ δn Proof. The eigenvalues are Lipschitz in N0 so we can apply the fundamental theorem of calculus and deduce that: ei () − ei ( ) = ∂ω ei (ω)dω.
l2
Now ei is decreasing and satisfies ∂ω ei () ≤ −min 4n . If ≥ then we show that ei () ≤ −δn . Recalling that = ei ( ) we write: ei () − = ∂ω ei (ω)dω ≤ −( − )min ln2 ,
since we have excised a neighborhood of size | − | ≥
4δn L 0 ln2
from we have
δn , L 0 ln2
and hence ei () ≤ − ( − )min ln2 ≤ δn −
l2 4δn min n , 2 L 0 ln 4
and we conclude that: ei () ≤ −δn .
then ei () ≥ δn follows similarly. The eigenvalues should then remain If now ≤ nonsingular in a neighborhood of R. We now define Nn+1 to be the subset of Nn1 which remains after excising the possible singular values of for which |ei | ≤ δn . Nn+1 ⊂ R, and we investigate how the bounds we obtain on the real plane can be extended to the complex neighborhood (Nn+1 , ρn+1 ) of Nn+1 . We know that 1 ∈ Nn+1 is real and we have |ei (1 )| ≥ 2δn+1 .
(3.59)
We extend these estimates to (Nn+1 , ρn+1 ), using Neumann series as done in [11] in the discussion following Lemma 4.12. Here we check what the radius of analyticity ρn+1 has to be for the extension to be done. Lemma 3.9. For ∈ (Nn+1 , ρn+1 ) we have |ei ()| ≥ δn+1 . Proof. Let ∈ (Nn+1 , ρn+1 ) There exists 1 ∈ Nn+1 such that | − 1 | ≤ ρn+1 . We show that when ∈ (Nn+1 , ρn+1 ) then ||HC(S) (u n ()) − HC(S) (u n (1 ))||0 ≤
δn+1 , 2 8ln+1
(3.60)
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301
where HC(S) := PC(S) Hn+1 u n ()PC(S) and PC(S) denotes the projection on the singular region C(S). For |ξ | < δn+1 , HC(S) (, u n ) − ξ I will be shown to be invertible by using the Neumann series. We extend the bound (3.59) on the eigenvalues to the complex plane to prove that (HC(S) (, u n ) − ξ I )−1 exists whenever ∈ (Nn+1 , ρn+1 ). In the next lines we will denote H := HC(S) (u n ()) and H1 := HC(S) (u n (1 )). Formally we have (H − ξ I )−1 = [(−1)(H − ξ I )−1 (H − H1 )] p .(H1 − ξ I )−1 . p≥1
Then taking||0 norms with (3.60) we get p −1 −1 ||(H − ξ I )C(S) ||0 ≤ ||[(−1)(H − ξ I )−1 (H − H1 )]||0 ||(H1 − ξ I )C(S) ||0 . p≥1
For |ξ | ≤
δn 2,
H1 is has a basis of eigenfunctions, hence with (3.59) we have the inequality −1 |(H1 − ξ I )C(S) (i, j)| ≤
2 δn+1
.
(3.61)
Combining (3.61) with (3.59), the Neumann series converges thus −1 (H − ξ I )C(S) exists and for |ξ | ≤ 2δn+1 .
Hence any eigenvalue of H satisfies |ei ()| ≥
δn ≥ δn+1 2
as long as ∈ (Nn+1 , ρn+1 ). We estimate the measure of the set in excised to ensure that the Hamiltonians are nonsingular. Given that there are at most L 2n+1 singular sites in Bn+1 \ Bn , there are at most L 2n+1 singular regions. Now any singular region has diameter 4ln+1 sites and thus at most 4ln+1 eigenvalues ei . Now for each eigenvalue ei we excised a neighborhood of the potential singular value of of size Lδ0nln . We conclude that the measure of the set excised at the n th stage is bounded by L 2n+1 4ln+1 Lδ0nln Let S be the measure of the set excised after all the iterations are done, S=
∞
L 2n+1 4ln+1
n=1
≤ C L 1−α 0
∞
4δn L 0 ln
2(2−β−ν)n .
n=1
We choose α = 1.1 and ν = 5 so S ≤ cL −0.1 and Theorem 2.1 follows. 0 What needs to be checked now is that the Neumann series involved in the computation 1 βn of the inverse of Hn+1 () are convergent with the distance ln = 30 2 between the singular sites and the frequency in [3L 0 , 10L 0 ]. Let A0 = B0 ∪ A be a non-resonant part of the lattice, |V () A0 ( j, k)( j, k)| ≥ ds for ( j, k) ∈ A0 . Then we have the lemma:
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Lemma 3.10. Let ∈ (N0 , ρ0 ). Then we have: ||H A0 (u 0 ())−1 ||σ0 −γ0 ≤
c . ds
(3.62)
Proof. The proof is by Neumann series as in Theorem 5.1 in [11] with the a priori 1 estimates: ||u 0 ()||σ0 ≤ 1.55 and ||DW (u 0 ())||σ0 −γ0 ≤ γc2 ||3u 0 ()2 ||σ0 −γ0 . Then L0
0
for any subset of the lattice B = Bn+1 ∪ A we continue the proof by induction following the method introduced in [11] and we have the lemma:
Lemma 3.11. Let ∈ (Nn+1 , ρn+1 ). Then |HC(S) (u n ())−1 (x, y)| ≤
C e−(σn −3γn )|x−y| δn+1 γn14
and ||HC(S) (u n ())−1 ||σn −3γn ≤
C . δn+1 γn16
The proof follows Lemma 5.4 in [11] while taking into account the fact that here depends on L 0 : Theorem 3.2. Let ∈ (Nn+1 , ρn+1 ). Then H B (u n ()) (where B = Bn+1 ∪ A) is invertible and satisfies the estimate: ||H B−1 (u n ())||σn −5γn ≤
n+1 CG 16((n+1)+1)
δn+1 γn
.
(3.63)
The proof here again is by the same argument as Theorem 5.6 with large frequency . The next lemma is the analogue of Theorem 5.12 in [11]. Lemma 3.12. Let B = Bn+1 ∪ A. Then the correction vn when added to u n satisfies the estimate: ||G B (u n + vn )||σn+1 −γn+1 ≤
n+1 2C G 16((n+1)+1)
δn+1 γn
.
Proof. H B (u n + vn ) = V () B + DW B (u n + vn ). We will denote this operator by H B . Now vn is supported in Bn+1 and let R B (u n , vn ) be the operator defined by: 2π π R B (u n , vn )( j, k)( j , k ) = ψ j (x)eikξ ψ j e−ik ξ (2u n vn + vn2 )d xdξ. 0
0
Since u n and vn are supported in Bn+1 and Dvn (vn3 ) = 3vn2 and
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303
Du n (u 2n vn ) = 2u n vn , we can use the estimate (2.13) to continue. Since G B (u n + vn ) = G B (u n )(1 + R B G B (u n ))−1 and ||R B G B (u n )||σn+1 −γn+1 ≤
n+1 CC G
ε 16((n+1)+1) n
4 δ γn+1 n+1 γn
n CG CG 16((n+1)+1) δn+1 γn
≤
1 , 2
(3.64)
then ||G E (u n + vn )||σn+1 −γn+1 ≤
n+1 2C G 16((n+1)+1)
δn−1 γn −σ0
.
n
The estimate (3.64) is satisfied since for L 0 large, e 100 L 0 k dominates all the other terms: Ce(2n+2) ln C G e ×e2α ln L 0 e
4 ln
−σ0 100
32(1+(n+1)2 ) σ0
L0
kn
1 ≤ . 2
e
32(1+(n+1)) ln
32(1+(n+1)2 ) σ0
eν(n+1) ln 2
(3.65) (3.66)
Acknowledgements. The results of this paper were obtained in the Doctoral Thesis of the author in Austin. The author would like to thank Jim Kelliher for helpful comments. The author was supported in part by a Canadian Postdoctoral fellowship at McMaster University.
References 1. Berti, M., Bolle, P.: Periodic solutions of nonlinear wave equations with general nonlinearities. Commun. Math. Phys. 243(2), 315–328 (2003) 2. Bricmont, J., Kupiainen, A., Schenkel, A.: Renormalization group and the Melnikov problem for PDE’s. Commun. Math. Phys. 221(1), 101–140 (2001) 3. Coron, J.-M.: Periodic solutions of a nonlinear wave equation without the assumption of monotonicity. Math. Ann. 262(2), 273–285 (1983) 4. Bolle, P., Ghoussoub, N., Tehrani, H.: The multiplicity of solutions in non-homogeneous boundary value problems. (English. English Summary) Manus. Math. 101(3), 325–350 (2000) 5. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. 148(2), 363–439 (1998) 6. Borel, E.: Sur les equations ´ aux deriv ´ ees ´ partielles a coefficients constants et les fonctions non analytiques. C.R. Academie ´ Des Sciences de Paris 121, 933–935 (1895) 7. Brezis, H., Nirenberg, L.: Characterizations of the ranges of some nonlinear operators and applications to boundary value problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(2), 225–326 (1978) 8. Brezis, H., Nirenberg, L.: Forced vibrations for a nonlinear wave equation. Comm. Pure Appl. Math. 31(1), 1–30 (1978) 9. Chierchia, L.: A direct method for constructing solutions of the Hamilton-Jacobi equation. Meccanica 25, 246–252 (1990) 10. Craig, W.: Problemes ` de petits diviseurs dans les equations ´ aux deriv ´ ees ´ partielles. (In: French, with English French summary) [Small divisor problems in partial differential equations]. Panoramas et Synthses, 9. Paris: Societ ´ e´ Mathematique ´ de France, 2000 11. Craig, W., Wayne, G.: Newton’s method and periodic solutions of nonlinear waves equations. Commun. Pure Appl. Math. XLVI , 1409–1498 (1993) 12. Craig, W., Wayne, G.: Periodic solutions of nonlinear Schrodinger equation and the Nash-Moser method. Hamiltonian Mechanics (Torun 1993) Semanis, J. (ed.) NATO Adv. Dist. Ser. B Phys. 331, New York: Plenum, 1994, pp. 103–122 13. Lidskii, E.I., Shulmann, E.I.: Periodic solutions of the equation u tt − u x x + u 3 = 0. Funct. Anal. Appl. 22(4), 332–333 (1988)
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14. De La Llave, R.: Variational methods for quasi-periodic solutions of partial differential equations. Hamiltonian systems and celestial mechanics (Ptzcuaro, 1998), World Sci. Monogr. Ser. Math., 6, River Edge, NJ: World Sci. Publishing, 2000, pp. 214–228 15. Fokam, J.-M.: Periodic solutions of nonlinear waves equations. Master thesis, University of Cape Town, 1999 16. Kato, T.: Perturbation theory for linear operators. Reprint of the 1980 edition, Classics in Mathematics, Berlin: Springer-Verlag, 1995 17. Kuksin, S., Pöschel, J.: Invariant manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143, 149–179 (1996) 18. Hofer, H.: On the range of a wave operator with nonmonotone nonlinearity. Math. Nachr. 106, 327–340 (1982) 19. Mauldon, J.G.: Continuous functions with zero derivative almost everywhere. Quart. J. Math. Oxford (2) 17, 257–62 (1966) 20. Mawhin, J.: Periodic solutions of some semilinear wave equations and systems: a survey. Chaos, Solitons and Fractals 5(9), 1651–1669 (1995) 21. McKenna, P.J.: On solutions of a nonlinear wave equation when the ratio of the period to the length is irrational. Proc. Amer. Soc. 93(1), 59–64 (1985) 22. Plotnikov, P.I.: Existence of a countable set of periodic solutions on forced oscillations for a weakly nonlinear wave equation. Translation in Math. USSR-Sb. 64(2), 543–556 (1989) 23. Plotnikov, P.I., Yungermann, L.N.: Periodic solutions of weakly nonlinear wave equations with an irrational period to interval length. Transl. in Diff. Eq. 24(9), 1059–1065 (1988) 24. Pöschel, J.: On the Fröhlich-Spencer-estimate in the theory of Anderson localization. Manus. Math. 70(1), 27–37 (1990) 25. Rabinowitz, P.H.: Periodic solutions of nonlinear hyperbolic partial differential equations. Comm. Pure Appl. Math. 20, 145–205 (1967) 26. Rabinowitz, P.: Free vibrations for a semilinear wave equation. Commun. Pure Appl Math. 31(1), 31–68 (1978) 27. Rellich, F.: Perturbation theory of eigenvalues problems. New York-London-Paris: Gordon and Breach Science Publishers, 1969, pp. 47, 48 28. Tanaka, K.: Infinitely many periodic solutions for the equation: u tt − u x x ± |u| p−1 u = f (x, t). II. Trans. Amer. Math. Soc. 307(2), 615–645 (1988) Communicated by I.M. Sigal
Commun. Math. Phys. 283, 305–342 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0510-9
Communications in
Mathematical Physics
Torus n-Point Functions for R-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds Geoffrey Mason1, , Michael P. Tuite2 , Alexander Zuevsky2, 1 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA.
E-mail:
[email protected]
2 Department of Mathematical Physics, National University of Ireland, Galway, Ireland.
E-mail:
[email protected];
[email protected] Received: 7 August 2007 / Accepted: 6 November 2007 Published online: 29 May 2008 – © Springer-Verlag 2008
Abstract: We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay’s trisecant identity for elliptic functions. 1. Introduction This paper is one of a series devoted to the study of n-point functions for vertex operator algebras on Riemann surfaces of genus one, two and higher [T,MT1,MT2,MT3]. One may define n -point functions at genus one following Zhu [Z], and use these functions together with various sewing procedures to define n-point functions at successively higher genera [T,MT2,MT3]. In this paper we consider the genus one n-point functions for a Vertex Operator Superalgebra (VOSA) V with a real grading (i.e. a chiral fermionic conformal field theory). In particular, we compute all n-point functions for rank one and rank two fermion VOSAs. In the latter case, we consider n-point functions defined over an orbifold g-twisted module for a continuous V automorphism g generated by a Heisenberg bosonic state. We also consider the Heisenberg decomposition (or bosonization) of V and recover classical Frobenius elliptic versions of Fay’s generalized trisecant identity together with a new further generalization. The modular properties of the continuous orbifold n-point functions are also described. In his seminal paper, Zhu defined and developed a constructive theory of torus n-point functions for a Z-graded Vertex Operator Algebra (VOA) and its modules [Z]. Partial support provided by NSF, NSA and the Committee on Research, University of California, Santa Cruz. Supported by a Science Foundation Ireland Frontiers of Research Grant, and by Max-Planck Institut für Mathematik, Bonn.
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In particular, he described various recursion formulae where, for example, an n-point function is expanded in terms of n − 1-point functions and naturally occurring Weierstrass elliptic (and quasi-elliptic) functions. Indeed, one can prove the analytic, elliptic and modular properties of n-point functions for many VOAs from these recursion formulae (op.cit.). This technique has since been extended to include orbifold VOAs with a g-twisted module for a finite order automorphism g [DLM1], 21 Z-graded VOSAs [DZ1] and Z-graded VOSAs [DZ2]. Here we consider a further generalization to obtain recursion formulae for torus n-point functions for an R-graded VOSA. We consider n-point functions defined as the supertrace over the product of various vertex operators together with a general element of the automorphism group of the VOSA. The resulting recursion formula is expressed in terms of natural “twisted” Weierstrass elliptic functions periodic up to arbitrary multipliers in U (1). Such elliptic functions already appear in ref. [DLM1] for multipliers of finite order. Here, we give a detailed description of twisted Weierstrass elliptic functions (and associated twisted Eisenstein series) for general U (1) multipliers generalizing many results of the classical theory of elliptic functions. We consider two applications of the Zhu recursion formula. The first example is that of the rank one 21 Z-graded fermion VOSA. In this case, all n-point functions can be computed in terms of a single generating function. In particular, we obtain expressions for these n-point functions in a natural Fock basis in terms of the Pfaffian of an appropriate block matrix. The second example is that of the rank two fermion VOSA. As is well known, this VOSA contains a Heisenberg vector which generates a continuous automorphism g and for which a g-twisted module can be constructed [Li]. The Heisenberg vector can also be employed to define a “shifted” Virasoro with real grading [MN2,DM]. We demonstrate a general relationship between the n-point functions for orbifold g-twisted modules and the shifted VOSA. We next apply the recursion formula for R-graded VOSAs in order to obtain all continuous orbifold n-point functions. These are expressible in terms of determinants of appropriate block matrices in a natural Fock basis and can again be obtained from a single generating function. Decomposing the rank two fermion VOSA into Heisenberg irreducible modules as a bosonic Z-lattice VOSA (i.e. bosonization) we may employ results of ref. [MT1] to find alternative expressions for the n-point functions. In particular the generating function is expressible in terms of theta functions and the genus one prime form and we thus recover classical Frobenius elliptic function versions of Fay’s generalized trisecant identity. We also prove a further generalization of the elliptic Fay’s trisecant identity based on the n-point function for n lattice vectors. The paper concludes with a determination of the modular transformation properties for all rank two continuous orbifold n-point functions generalizing Zhu’s results for C2 -cofinite VOAs [Z]. The study of n-point functions has a long history in the theoretical physics literature and we recover a number of well known physics results here. Thus the Pfaffian and determinant formulas for the rank one and two fermion generating functions and the relationship between Fay’s generalized trisecant identity have previously appeared in physics [R1,R2,EO,RS,FMS,P]. However, it is important to emphasize that our approach is constructively based on the properties of a VOSA and that a rigorous and complete description of these n-point functions has been lacking until now. Thus, for example, no assumption is made about the local analytic properties of n-point functions as would normally be the case in physics. Similarly, other pure mathematical algebraic geometric approaches to n -point functions are based on an assumed local analytic structure [TUY]. Finally, apart from the intrinsic benefits of this rigorous approach, it is important to obtain a complete description of these n-point functions as the building blocks used in the construction of higher genus partition and n-point functions [T,MT2,MT3].
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The paper is organized as follows. We begin in Sect. 2 with a review of classical Weierstrass elliptic functions and Eisenstein series. We introduce twisted Weierstrass functions which are periodic up to arbitrary elements of U (1). We describe various expansions of these twisted functions, introduce twisted Eisenstein series and determine their modular properties. Section 3 contains one of the central results of this paper. We begin with the defining properties of an R-graded VOSA V . We define n -point functions as a supertrace over a V -module and describe some general properties. We then formulate a generalization of Zhu’s recursion formula [Z] to an R-graded VOSA module making use of the twisted Weierstrass and Eisenstein series. Sect. 4 contains a discussion of a VOSA containing a Heisenberg vector. We prove the general relationship between the n-point functions for a VOSA with a Heisenberg shifted Virasoro vector and g-twisted n-point functions where g is generated by the Heisenberg vector. In Sect. 5 we apply the results of Sect. 3 to a rank one fermion VOSA. In particular, we compute all n-point functions in terms of a generating function given by a particular n-point function. We also discuss n-point functions for a fermion number-twisted module. Sect. 6 contains a description of a rank two fermion VOSA. We make use of the results of Sect. 3 and Sect. 4 to compute all n -point functions for a g-twisted module where g is generated by a Heisenberg vector by means of a generating function. We next discuss the Heisenberg decomposition of this rank two theory—the bosonized theory. In particular, we derive an expression for the rank two generating function in terms of θ -functions and prime forms related to classical Frobenius elliptic function identities corresponding to the elliptic version of Fay’s generalized trisecant identity. A further generalization for Fay’s trisecant identity for elliptic functions is also discussed. Finally, we discuss the modular properties of all n-point functions for the rank two fermion VOSA. Properties of supertraces are recalled in the Appendix. We collect here notation for some of the more frequently occurring functions and symbols that will play a role in our work. Z is the set of integers, R the real numbers, C the complex numbers, H the complex upper-half plane. We will always take τ to lie in H, and z will lie in C unless otherwise noted. For a symbol z we set qz = exp(z), in particular, q = q2πiτ = exp(2πiτ ). 2. Some Elliptic Function Theory 2.1. Classical elliptic functions. We discuss a number of modular and elliptic-type functions that we will need. We begin with some standard elliptic functions [La]. The Weierstrass ℘-function periodic in z with periods 2πi and 2πiτ is 1 1 1 ℘ (z, τ ) = 2 + − 2 (1) z (z − ωm,n )2 ωm,n m,n∈Z (m,n)=(0,0)
=
1 + z2
(n − 1)E n (τ )z n−2 ,
(2)
n≥4,n even
for (z, τ ) ∈ C × H with ωm,n = 2πi(mτ + n). Here, E n (τ ) is equal to 0 for n odd, and for n even is the Eisenstein series [Se] E n (τ ) = −
r n−1 q r 2 Bn (0) + , n! (n − 1)! 1 − qr r ≥1
(3)
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G. Mason, M. P. Tuite, A. Zuevsky
where Bn (0) is the n th Bernoulli number (see (38) below). If n ≥ 4 then E n (τ ) is a holomorphic modular form of weight n on S L(2, Z). That is, it satisfies for all γ =
ab cd
E n (γ .τ ) = (cτ + d)n E n (τ ),
(4)
∈ S L(2, Z), where we use the standard notation γ .τ =
aτ + b . cτ + d
(5)
On the other hand, E 2 (τ ) is a quasimodular form [KZ] having the exceptional transformation law E 2 (γ .τ ) = (cτ + d)2 E 2 (τ ) −
c(cτ + d) . 2πi
(6)
We define Pk (z, τ ) for k ≥ 1 by
n − 1 (−1)k−1 d k−1 1 k Pk (z, τ ) = E n (τ )z n−k . P1 (z, τ ) = k + (−1) k−1 (k − 1)! dz k−1 z
(7)
n≥k
Then P2 (z, τ ) = ℘ (z, τ ) + E 2 (τ ) whereas P1 − z E 2 is the classical Weierstrass zeta function. Pk has periodicities Pk (z + 2πi, τ ) = Pk (z, τ ), Pk (z + 2πiτ, τ ) = Pk (z, τ ) − δk1 .
(8)
We define the elliptic prime form K (z, τ ) by [Mu] K (z, τ ) = exp(−P0 (z, τ )),
(9)
where P0 (z, τ ) = − log(z) +
1 k≥2
k
E k (τ )z k ,
(10)
so that P1 (z, τ ) = −
d 1 P0 (z, τ ) = − E k (τ )z k−1 . dz z
(11)
k≥2
K (z, τ ) has periodicities K (z + 2πi, τ ) = −K (z, τ ), K (z + 2πiτ, τ ) = −qz−1 q −1/2 K (z, τ ). We define the standard Jacobi theta function by1 e.g. [FK] a (z, τ ) = exp[iπ(n + a)2 τ + (n + a)(z + 2πib)], ϑ b
(12)
(13)
n∈Z
1 Note that the z dependence of the theta function is chosen so that the periods are 2πi and 2πiτ rather
than the standard periods of 1 and τ .
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309
with periodicities
a a ϑ (z + 2πi, τ ) = e2πia ϑ (z, τ ), b b a a (z, τ ). ϑ (z + 2πiτ, τ ) = e−2πib qz−1 q −1/2 ϑ b b
(14) (15)
We also note the modular transformation properties under the action of the standard 0 1 11 generators S = and T = of S L(2, Z) (with relations (ST )3 = −S 2 = 1) −1 0 01 a a −iπa(a+1) (z, τ + 1) = e ϑ (z, τ ), (16) ϑ b b + a + 21 1 z 2 a −b = (−iτ )1/2 e2πiab e−i z /4π τ ϑ ϑ − ,− (z, τ ). (17) b a τ τ K (z, τ ) can be expressed in terms of half integral theta functions as 1 ϑ K (z, τ ) = d dz ϑ
2 1 2
(z, τ ) 1 2 1 2
(0, τ )
−i ϑ = η(τ )3
1 2 1 2
(z, τ ),
(18)
where the Dedekind eta-function is defined by η(τ ) = q 1/24
∞
(1 − q n ).
(19)
n=1
2.2. Twisted elliptic functions. Let (θ, φ) ∈ U (1) × U (1) denote a pair of modulus one complex parameters with φ = exp(2πiλ) for 0 ≤ λ < 1. For z ∈ C and τ ∈ H we define “twisted” Weierstrass functions for k ≥ 1 as follows: (−1)k n k−1 qzn θ Pk (z, τ ) = , (20) φ (k − 1)! 1 − θ −1 q n n∈Z+λ
for q = q2πiτ , where
means we omit n = 0 if (θ, φ) = (1, 1).
Remark 1. (i) (20) was introduced in [DLM1] for rational λ, where it was denoted by Pk (φ, θ −1 , z, τ ). The alternative definition and notation used here is motivated by the modular and periodicity properties shown below and by the column vector notation for theta series. (ii) (20) converges absolutely and uniformly on compact subsets of the domain |q| < |qz | < 1 [DLM1]. (iii) For k ≥ 1, (−1)k−1 d k−1 θ θ (z, τ ) = (z, τ ). (21) Pk P1 k−1 φ φ (k − 1)! dz
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We now develop twisted versions of the standard results for the classical Weierstrass ℘-function reviewed above. A number of similar results appear in [DLM1]. However, the cases k = 1, 2 are treated separately there and only for rational λ, i.e. φ N = 1 for some positive integer N . The most canonical derivation of the periodic and modular properties of (20) for general λ follows from the following theorem: Theorem 1. For |q| < |qz | < 1 and φ = 1, φn θ (z, τ ) = Pk θm
φ z−ω
k ,
(22)
m θ θ Pk (z, τ ) = φn
k . φ n∈Z m∈Z z − ωm,n
(23)
m∈Z
whereas for θ = 1,
n∈Z
m,n
Remark 2. When both θ = 1 and φ = 1 then the double sums (22) and (23) are equal. For k ≥ 3, they are absolutely convergent and equal for all (θ, φ). In order to prove Theorem 1 it is useful to define the following convergent sum φn . (24) S(x, φ) = x − 2πin n∈Z
Clearly S(x + 2πi, φ) = φ S(x, φ), S(x, φ) = −S(−x, φ −1 ).
(25) (26)
We then have: Lemma 1. For φ = exp(2πiλ) with 0 ≤ λ < 1 we have S(x, φ) =
qxλ 1 δλ,0 + . 2 qx − 1
(27)
Proof. Both S(x, φ) and qxλ (qx − 1)−1 have simple poles at x = 2πin with residue φ n for all n ∈ Z. Furthermore, qxλ (qx − 1)−1 is regular at the point at infinity for 0 ≤ λ < 1. Thus S(x, φ) − qxλ (qx − 1)−1 is constant which from (25) and (26) must be given by 1 2 δλ,0 . We first prove Theorem 1 for the case k = 1 and φ = 1 (i.e. 0 < λ < 1). The double sum (22) is n qxλm φ , θm θ m S(xm , φ) = θm = z − ωm,n qxm − 1 m∈Z
n∈Z
m∈Z
m∈Z
using Lemma 1 for xm = z − 2πimτ with qxm = qz for m > 0 that qxm > 1 and hence qxλm qxm − 1
=
r ≤−1
q −m .
Since |q| < |qz | < 1 we find
qzr +λ (q −r −λ )m .
Torus n-Point Functions for R-graded Vertex Operator Superalgebras
311
Since θq −r −λ < 1 for r ≤ −1 we obtain
θ
m
n∈Z
m>0
φn z − ωm,n
=
qzr +λ
r ≤−1
=−
r ≤−1
(θq −r −λ )m
m>0 qzr +λ . 1 − θ −1 q r +λ
Similarly for m ≤ 0 we have qxm < 1, so that qxλm qxm − 1
=−
qzr +λ (q −r −λ )m .
r ≥0
Hence since θq r +λ < 1 for r ≥ 0 we find
θ
m
n∈Z
m≤0
φn z − ωm,n
=−
r ≥0
qzr +λ . 1 − θ −1 q r +λ
Altogether we obtain
θ
m
m∈Z
n∈Z
φn z − ωm,n
=−
r ∈Z
qzr +λ θ (z, τ ), = P 1 φ 1 − θ −1 q r +λ
proving (22) for k = 1. The result for k ≥ 2 follows after applying (21). In order to prove (23) it is useful to first consider the following double sum for φ = 1: 1 2 1 θ (z, τ ) = A θm φn − + . φ z − ωm,n z − ωm,n−1 z − ωm,n−2 m∈Z
n∈Z
By (25) we find A
θ θ m [S(xm , φ) − 2S(xm + 2πi, φ) + S(xm + 4πi, φ)] (z, τ ) = φ m∈Z θ (z, τ ). = (1 − φ)2 P1 φ
(28)
On the other hand, we have −8π 2 θ m n (z, τ ) = A θ φ . φ (z − ωm,n )(z − ωm,n−1 )(z − ωm,n−2 ) m∈Z
n∈Z
−3 This sum is absolutely convergent since the summand is O( ωm,n ) for |m|, |n| large. We may thus interchange the order of summation to find that, on relabelling,
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θ (z, τ ) becomes φ m φ θ −n
1 2 1 − + z − ω−n,m z − ω−n,m−1 z − ω−n,m−2 m∈Z n∈Z
1 2 1 1 m −n φ θ − + = − τ z − ωm,n z − ωm−1,n z − ωm−2,n m∈Z n∈Z 1 m = − φ S(xm , θ −1 ) − 2S(xm−1 , θ −1 ) + S(xm−2 , θ −1 ) , (29) τ m∈Z
where z 1 z = − , τ = − , ωm,n = 2πi(mτ + n), xm = z − 2πimτ . τ τ
(30)
Applying Lemma 1 with θ = exp(−2πiµ) for 0 ≤ µ < 1, it follows that S(xm , θ −1 ) − 2S(xm−1 , θ −1 ) + S(xm−2 , θ −1 ) µ
µ
µ qx qx qx 1 m−1 m−2 m −2 + . = (1 − 2 + 1). δµ,0 + 2 qxm − 1 qxm−1 − 1 qxm−2 −1
We may next repeat the arguments above leading to (28) to find that (29) becomes 1 1 z φ θ (z, τ ) = − A (1 − φ)2 P1 −1 . − ,− φ θ τ τ τ Comparing to (28), we find that for φ = 1, 1 z 1 φ θ P1 (z, τ ) = − − ,− P1 −1 . φ θ τ τ τ Considering this identity for (z , τ ) of (30) and using −1 θ θ P1 (z, τ ) = −P1 (−z, τ ), φ φ −1
(31)
(32)
(which follows from (22)) it is clear that (31) holds for all (θ, φ) = (1, 1). We may use (31) to prove (23) of Theorem 1 in the case k = 1. The double sum of (23) becomes on relabelling −n z 1 θ 1 φ m − ,− P1 −1 φ = − θ z − ω−n,m τ τ τ m∈Z n∈Z θ (z, τ ). = P1 φ The general result for k ≥ 2 follows from (21). Periodicity and modular properties now follow from Theorem 1. Thus we have
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θ (z, τ ) is periodic in z with periods 2πiτ and φ 2πi with multipliers θ and φ respectively.
Lemma 2. For (θ, φ) = (1, 1), Pk
Remark 3. Note that the periodicity in 2πi is determined by the second argument φ in contradistinction to the periodicity of the standard theta series (14). Periodicity for (θ, φ) = (1, 1) is given by (8). We now consider properties. Define the standard left action of the modu the modular ab lar group for γ = ∈ Γ = S L(2, Z) on (z, τ ) ∈ C × H with cd z aτ + b γ .(z, τ ) = (γ .z, γ .τ ) = , . (33) cτ + d cτ + d We also define a left action of Γ on (θ, φ), a b θ φ θ = γ. . φ θ c φd
(34)
Then we obtain: Proposition 1. For (θ, φ) = (1, 1) we have θ θ k (γ .z, γ .τ ) = (cτ + d) Pk (z, τ ). Pk γ . φ φ
(35)
Proof. Consider the case k = 1. It is sufficient to consider the action of the generators S, T of Γ , where S.(z, τ ) = (− τz , − τ1 ) and T.(z, τ ) = (z, τ + 1). Then for γ = S, (35) is given by (31) whereas for γ = T , the result follows directly from definition (7). It is straightforward to check the relations (ST )3 = −S 2 = 1 (using (32)) so that the result follows for k = 1. The general case follows from (21). Remark 4. (i) (35) is equivalent to Theorem 4.2 of [DLM1] for rational λ after noting Remark 1 (i) and (34). (ii) For γ = −I one finds −1 θ θ (z, τ ) = (−1)k Pk Pk (−z, τ ). (36) φ φ −1 We next introduce twisted Eisenstein series for n ≥ 1, defined by Bn (λ) 1 (r + λ)n−1 θ −1 q r +λ θ En (τ ) = − + φ n! (n − 1)! 1 − θ −1 q r +λ r ≥0
(r − λ)n−1 θq r −λ + , (n − 1)! 1 − θq r −λ (−1)n
(37)
r ≥1
where means we omit r = 0 if (θ, φ) = (1, 1) and where Bn (λ) is the Bernoulli polynomial defined by
qzλ 1 Bn (λ) n−1 = + z . qz − 1 z n! n≥1
In particular, we note that B1 (λ) = λ − 21 .
(38)
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Remark 5. (i) (37) was introduced in [DLM1] for rational λ where it was denoted by Q n (φ, θ −1, τ ). 1 (τ ) = E n (τ ), the standard Eisenstein series for even n ≥ 2, whereas (ii) E n 1 1 (τ ) = −B1 (0)δn,1 = 21 δn,1 for n odd. En 1 We may obtain a Laurant expansion analogous to (7). Proposition 2. We have n − 1 θ 1 θ k (z, τ ) = k + (−1) (τ )z n−k . En Pk φ φ k−1 z
(39)
n≥k
Proof. Consider (20) for k = 1: qzr +λ qz−r +λ θ (z, τ ) = − P1 − φ 1 − θ −1 q r +λ 1 − θ −1 q −r +λ r ≥0
=
r ≥1
qzλ qz − 1
−
qzr +λ
r ≥0
θ −1 q r +λ 1 − θ −1 q r +λ
θq r −λ 1 − θq r −λ r ≥1 1 θ (τ )z n−1 , = − En φ z +
qz−r +λ
n≥1
from (37) and (38). The general result then follows from (21). 1 (z, τ ) = 21 δk,1 + Pk (z, τ ) for k ≥ 1. Remark 6. For (θ, φ) = (1, 1) we have Pk 1 We also find Proposition 3. For φ = 1 then ⎡ ⎢ 1 θ (τ ) = Ek θm ⎢ ⎣ φ (2πi)k m∈Z
⎤ n∈Z (m,n)=(0,0)
⎥ φn ⎥, (mτ + n)k ⎦
(40)
whereas for θ = 1 ⎡ 1 n⎢ θ (τ ) = Ek φ ⎢ ⎣ φ (2πi)k n∈Z
⎤ m∈Z (m,n)=(0,0)
⎥ θm ⎥. k (mτ + n) ⎦
(41)
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315
Proof. Expand the sum of (22) for φ = 1 for k = 1 to find ⎡ ⎤ z ⎥ 1 φn 1 m⎢ θ ⎥. (z, τ ) = − P1 θ ⎢ ( )r −1 ⎣ r φ z 2πi 2πi (mτ + n) ⎦ m∈Z
r ≥1 n∈Z (m,n)=(0,0)
Comparing with (39) then (40) follows. Equation (41) similarly holds.
Remark 7. When both θ = 1 and φ = 1 then (40) and (41) are equal. For k ≥ 3, they are absolutely convergent and equal for all (θ, φ). For k ≥ 3, and (θ, φ) = (1, 1) we obtain the standard Eisenstein series (3). From Proposition 1 it immediately follows that θ Proposition 4. For (θ, φ) = (1, 1), E k is a modular form of weight k where φ θ θ (γ .τ ) = (cτ + d)k E k (τ ). Ek γ . φ φ
(42)
Remark 8. This is equivalent to Theorem 4.6 of [DLM1] for rational λ. Equation (42) also holds for (θ, φ) = (1, 1) for k ≥ 3, whereas E 2 is quasi-modular. It is useful to note the analytic expansions: θ 1 θ P1 + C (k, l)z 1k−1 z l−1 (z 1 − z 2 , τ ) = 2 , φ φ z1 − z2 k,l≥1 θ θ P1 (z + z 1 − z 2 , τ ) = D (k, l, z)z 1k−1 z l−1 2 , φ φ
(43) (44)
k,l≥1
where for k, l ≥ 1 we define k +l −2 θ θ (τ ), E k+l−1 (k, l, τ ) = (−1)l C φ φ k−1 k +l −2 θ θ (k, l, τ, z) = (−1)k+1 (τ, z). D Pk+l−1 φ φ k−1 We also note that (36) implies −1 θ θ (k, l, τ ) = −C C (l, k, τ ), φ φ −1 −1 θ θ (k, l, τ, z) = −D D (l, k, τ, −z). φ φ −1
(45) (46)
(47) (48)
Finally, we may also express the twisted Weierstrass functions in terms of theta series and the prime form as follows:
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Proposition 5. For (θ, φ) = (1, 1) with θ = exp(−2πiµ) and φ = exp(2πiλ) then λ + 21 ϑ (z, τ ) µ + 21 1 θ (z, τ ) = P1 , (49) 1 φ K (z, τ ) λ+ 2 ϑ (0, τ ) µ + 21 whereas
1 1 (z, τ ) = P1 1
d dz ϑ
2 1 2 1 2 1 2
(z, τ )
d dz ϑ
(0, τ )
1 . K (z, τ )
(50)
Proof. For (θ, φ) = (1, 1) the result follows by comparing the periodicity and pole structure of each expression using (14) and (15). For (θ, φ) = (1, 1) the result follows from (11) and (18). 3. n-Point Functions for R-Graded Vertex Operator Superalgebras 3.1. Introduction to vertex operator superalgebras. We discuss some aspects of Vertex Operator Superalgebra (VOSA) theory to establish context and notation. For more details see [B,FHL,FLM,Ka,MN1]. Let V be a superspace, i.e. a complex vector space V = V0¯ ⊕ V1¯ = ⊕α Vα with index label α in Z/2Z so that each a ∈ V has a parity (fermion number) p(a) ∈ Z/2Z. An R-graded Vertex Operator Superalgebra (VOSA) is a quadruple (V, Y, 1, ω) as follows: V is a superspace with a (countable) R-grading where V = ⊕r ≥r0 Vr for some r0 and with parity decomposition Vr = V0,r ¯ ⊕ V1,r ¯ . 1 ∈ V0,0 ¯ is the vacuum vector and ω ∈ V0,2 ¯ the conformal vector with properties described below. Y is a linear map Y : V → (EndV )[[z, z −1 ]], for formal variable z, so that for any vector (state) a ∈ V, Y (a, z) = a(n)z −n−1 . (51) n∈Z
The component operators (modes) a(n) ∈ EndV are such that a(n)1 = δn,−1 a for n ≥ −1 and a(n)Vα ⊂ Vα+ p(a) ,
(52)
for a of parity p(a). The vertex operators satisfy the locality property for all a, b ∈ V , (x − y) N [Y (a, x), Y (b, y)] = 0,
(53)
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for N 0, where the commutator is defined in the graded sense, i.e. [Y (a, x), Y (b, y)] = Y (a, x)Y (b, y) − (1) p(a) p(b) Y (b, y)Y (a, x). The vertex operator for the vacuum is Y (1, z) = I dV , whereas that for ω is Y (ω, z) =
L(n)z −n−2 ,
(54)
n∈Z
where L(n) forms a Virasoro algebra for central charge c, [L(m), L(n)] = (m − n)L(m + n) +
c (m 3 − m)δm,−n . 12
(55)
L(−1) satisfies the translation property Y (L(−1)a, z) =
d Y (a, z). dz
(56)
L(0) describes the R-grading with L(0)a = wt (a)a for weight wt (a) ∈ R and Vr = {a ∈ V |wt (a) = r }.
(57)
We quote the standard commutator property of VOSAs e.g. [Ka,FHL,MN1], [a(m), Y (b, z)] =
m j≥0
j
Y (a( j)b, z)z m− j .
(58)
Taking a = ω this implies for b of weight wt (b) that [L(0), b(n)] = (wt (b) − n − 1)b(n),
(59)
b(n)Vr ⊂ Vr +wt (b)−n−1 .
(60)
so that
In particular, we define for a of weight wt (a) the zero mode o(a) =
a(wt (a) − 1), for wt (a) ∈ Z 0, otherwise,
which is then extended by linearity to all a ∈ V .
(61)
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3.2. Torus n-point functions. In this section we will develop explicit formulas for the n-point functions for R-graded VOSA modules at genus one [Z,DLM1,MT1,DZ1]. Let (V, Y, 1, ω) be an R-graded VOSA. In order to consider modular-invariance of n-point functions at genus 1, Zhu introduced in ref. [Z] a second “square-bracket” VOA (V, Y [, ], 1, ω) ˜ associated to a given VOA (V, Y (, ), 1, ω). We review some aspects of that construction here. The new square bracket vertex operators are defined by a change of co-ordinates, namely Y [v, z] = v[n]z −n−1 = Y (qzL(0) v, qz − 1), (62) n∈Z c with qz = exp(z), while the new conformal vector is ω˜ = ω − 24 1. For v of L(0) weight wt (v) ∈ R and m ≥ 0, c(wt (v), i, m)v(i), (63) v[m] = m! i≥m
i
c(wt (v), i, m)x m
m=0
In particular we note that v[0] =
wt (v) − 1 + x = . i
wt (v)−1 i≥0
i
(64)
v(i).
We now define the torus n-point functions. Following (52) we let σ ∈ Aut(V ) denote the parity (fermion number) automorphism σ a = (−1) p(a) a.
(65)
Let g ∈ Aut(V ) denote any other automorphism which commutes with σ . Let M be a V -module with vertex operators Y M . We assume that M is stable under both σ and g, i.e. σ and g act on M [DZ1]. The n-point function on M for states v1 , . . . , vn ∈ V and g ∈ Aut(V ) is defined by2 FM (g; v1 , . . . vn ; τ ) = FM (g; (v1 , z 1 ), . . . , (vn , z n ); τ ) L(0) = STr M g Y M (q1 v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0)−c/24 ,
(66)
q = exp(2πiτ ), qi = exp(z i ), 1 ≤ i ≤ n, for auxiliary variables z 1 , . . . , z n and where STr M denotes the supertrace defined by STr M (X ) = T r M (σ X ) = T r M0¯ (X ) − T r M1¯ (X ).
(67)
In Appendix A we describe some basic properties of the supertrace. Taking g = 1 and all vi = 1 in (66) yields the partition function which we denote by (68) Z M (τ ) = FM (1; τ ) = STr M q L(0)−c/24 . We also denote the orbifold partition function for general g by Z M (g, τ ) = FM (g; τ ) = STr M gq L(0)−c/24 .
(69)
2 This n-point function would be denoted by T ((v , q ), . . . , (v , q ), (1, g), q) in the notation of [DLM1] n n 1 1 and [DZ1].
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319
For g = 1 (66) is defined by Zhu for a Z-graded VOA [Z]. For g of finite order, it is considered for Z -graded VOAs in ref. [DLM1], 21 Z-graded VOSAs in ref. [DZ1] and Z-graded VOSAs in ref. [DZ2]. Here we generalize these results to an R-graded VOSA for arbitrary g commuting with σ . For n = 1 in (66) we obtain the 1-point function denoted by Z M (g, v1 , τ ) = FM (g; (v1 , z 1 ); τ ) = STr M (go(v1 )q L(0)−c/24 ),
(70)
where o(v1 ) is the zero mode (61) and is independent of z 1 . We note the following useful result relating any n-point function to a 1-point function: Lemma 3. For states v1 , v2 , . . . , vn as above we have FM (g; (v1 , z 1 ), . . . , (vn , z n ); τ ) = Z M (g, Y [v1 , z 1n ].Y [v2 , z 2n ] . . . Y [vn−1 , z n−1n ].vn , τ ) = Z M (g, Y [v1 , z 1 ].Y [v2 , z 2 ] . . . Y [vn , z n ].1, τ ),
(71) (72)
where z i j = z i − z j . Proof. The proof follows Lemma 1 of ref. [MT1].
Every n-point function enjoys the following permutation and periodicity properties [Z,MT1]: Lemma 4. Consider the n-point function FM for states v1 , v2 , . . . , vn , as above, where each vi is of weight wt (vi ), parity p(vi ) and is a g-eigenvector for eigenvalue θi−1 . (i) If p(v1 ) + . . . + p(vn ) is odd then FM = 0. (ii) Permuting adjacent vectors, FM (g; (v1 , z 1 ), . . . , (vk , z k ), (vk+1 , z k+1 ), . . . , (vn , z n ); τ ) = (−1) p(vk ) p(vk+1 ) FM (g; (v1 , z 1 ), . . . , (vk+1 , z k+1 ), (vk , z k ), . . . , (vn , z n ); τ ). (iii) FM is a function of z i j = z i − z j and is non-singular at z i j = 0 for all i = j. (iv) FM is periodic in z i with period 2πi and multiplier φi = exp(2πiwt (vi )). (v) FM is periodic in z i with period 2πiτ and multiplier θi . Proof. (i) This follows from definition (67). (ii) Apply locality (53). (iii) FM is a function of z i j from (71). Suppose FM is singular at z n = y for some y = z j for all j = 1, . . . , n − 1. We may assume that z 0 = 0 by redefining z i to be z i − z 0 for all i. But from (72), Y [vn , z n ].1|z n =0 = vn is non-singular at z n j = 0. Applying (ii) the result follows for all z i j . (iv) This follows directly from the definition (66). (v) Using (iii) we consider periodicity of z n wlog. Under z n → z n + 2πiτ we have FM → Fˆ M , where Fˆ M = q −c/24 STr M (gY (q1L(0) v1 , q1 ) . . . Y (q L(0) qnL(0) vn , qqn )q L(0) ) = q −c/24 STr M (gY (q1
L(0)
v1 , q1 ) . . . q L(0) Y (qnL(0) vn , qn )),
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using q L(0) Y (b, z)q −L(0) = Y (q L(0) b, qz) (which follows from (59)). But L(0)
STr M (gY (q1
v1 , q1 ) . . . q L(0) Y (qnL(0) vn , qn ))
= (−1) p(vn ) STr M (Y (qnL(0) vn , qn )gY (q1
L(0)
L(0)
v1 , q1 ) . . . Y (qn−1 vn−1 , qn−1 )q L(0) )
= θn (−1) p(vn ) STr M (gY (qnL(0) vn , qn )Y (q1
L(0)
L(0)
= θn STr M (gY (q1
L(0)
v1 , q1 ) . . . Y (qn−1 vn−1 , qn−1 )q L(0) )
L(0)
v1 , q1 ) . . . Y (qn−1 vn−1 , qn−1 )Y (qnL(0) vn , qn )q L(0) ),
using g −1 Y (vn , qn )g = Y (g −1 vn , qn ) = θn Y (vn , qn ) and applying (ii) repeatedly. Thus Fˆ M = θn FM . 3.3. Zhu recursion formulas for n-point functions. We now prove a generalization of Zhu’s n-point function recursion formula [Z] for the n-point function (66) for an R-graded VOSA. We begin with the following lemma which follows directly from (58): Lemma 5. Suppose that u ∈ V is homogeneous of weight wt (u) ∈ R. Then for k ∈ Z and v ∈ V we have k Y (qzL(0) u(i)v, qz ). (73) u(k), Y (qzL(0) v, qz ) = qzk−wt (u)+1 i i≥0
Corollary 1. Suppose that u ∈ V is homogeneous of integer weight wt (u) ∈ Z. Then o(u), Y (qzL(0) v, qz ) = Y (qzL(0) u[0]v, qz ). (74) Similarly to Zhu’s Proposition 4.3.1 (op.cit.) we find Proposition 6. Suppose that v ∈ V is homogeneous of integer weight wt (v) ∈ Z with gv = v. Then for v1 , . . . , vn ∈ V , we have n
p(v, v1 v2 . . . vr −1 )FM (g; v1 ; . . . ; v[0]vr ; . . . vn ; τ ) = 0,
(75)
r =1
with p(v, v1 v2 . . . vr −1 ) of (147) in Appendix A. Let v be homogeneous of weight wt (v) ∈ R and define φ ∈ U (1) by φ = exp(2πiwt (v)).
(76)
We also take v to be an eigenfunction under g with gv = θ −1 v
(77)
g −1 v(k)g = θ v(k).
(78)
for some θ ∈ U (1) so that
Then we obtain the following generalization of Zhu’s Proposition 4.3.2 [Z] for the n-point function:
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321
Theorem 2. Let v, θ and φ be as as above. Then for any v1 , . . . vn ∈ V we have FM (g; v, v1 , . . . vn ; τ ) L(0) = δθ,1 δφ,1 STr M go(v)Y M (q1 v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0)−c/24 n θ (z − zr , τ ) p(v, v1 v2 . . . vr −1 )Pm+1 + φ r =1 m≥0
×FM (g; v1 , . . . , v[m]vr , . . . , vn ; τ ).
(79)
(The twisted Weierstrass function is defined in (20)). Proof. We have q c/24 FM (g; v, v1 , . . . vn ; τ ) L(0) = qz−k−1+wt (v) STr M g v(k)Y M (q1 v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0) . k∈Z
Thus we consider L(0) STr M g v(k)Y M (q1 v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0) L(0) = STr M g [v(k), Y M (q1 v1 , q1 ) . . . Y M (qnL(0) vn , qn )]q L(0) + p(v, v1 · · · vn )STr M g Y M (q1L(0) v1 , q1 ) . . . Y M (qnL(0) vn , qn )v(k)q L(0) n k k+1−wt (v) p(v, v1 . . . vr −1 ) q = i r r =1 i≥0 L(0) × STr M g Y M (q1 v1 , q1 ) . . . Y M (qrL(0) v(i)vr , qr ) . . . Y M (qnL(0) vn , qn )q L(0) + θq k+1−wt (v) STr M g v(k)Y M (q1L(0) v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0) , applying (59), (73), (78), (146) and Lemma 8 of Appendix A. Thus STr M g v(k)Y M (q1L(0) v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0) n k k−wt (v)+1 1 p(v, v . . . v ) q = 1 r −1 i r 1 − θq k+1−wt (v) r =1 i≥0 L(0) × STr M gY (q1 v1 , q1 ) . . . Y (qrL(0) v(i)vr , qr ) . . . Y (qnL(0) vn , qn )q L(0) , provided (θ, φ, k) = (1, 1, −1 + wt (v)). This implies FM (g; v, v1 , . . . vn ) is given by L(0) δθ,1 δφ,1 STr M go(v)Y M (q1 v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0)−c/24 k+1−wt (v) qr n k qz v(i)vr , . . . , vn ), p(v, v1 . . . vr −1 ) .FM (g; v1 , . . . + i 1 − θq k+1−wt (v) r =1
k∈Z
i≥0
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where the prime denotes the omission of k = −1 + wt (v) if (θ, φ) = (1, 1) and recalling (61). Now from (63) and (64) we find (k + 1 − wt (v))m k v(i) = v[m]. i m! i≥0
m≥0
The sum over k can then be computed in terms of a twisted Weierstrass function (20) for λ = wt (v)(modZ) as follows: k+1−wt (v) (k + 1 − wt (v))m qr qz 1 m! 1 − θq k+1−wt (v) k∈Z −1 1 θ m+1 (zr − z, τ ) − δθ,1 δφ,1 δm,0 = (−1) Pm+1 φ −1 2 1 θ (z − zr , τ ) − δθ,1 δφ,1 δm,0 , = Pm+1 φ 2 using (36). Thus we find FM (g; v, v1 , . . . vn , τ ) is given by δθ,1 δφ,1 STr M go(v)Y M (q1L(0) v1 , q1 ) . . . Y M (qnL(0) vn , qn )q L(0)−c/24 n θ (z − zr , τ )FM (g; v1 , . . . , v[m]vr , . . . , vn ; τ ) p(v, v1 v2 . . . vr −1 )Pm+1 + φ r =1 m≥0
1 − δθ,1 δφ,1 p(v, v1 . . . vr −1 )FM (g; v1 , . . . , v[0]vr , . . . , vn ; τ ). 2 n
r =1
θ to Remark 9. (i) Note that it is necessary for the V -grading to be real in order for Pk φ converge. Thus, VOSAs with C-grading such as those discussed in [DM] have divergent torus n-point functions. (ii) From Lemma 2 it follows that FM is periodic in z with periods 2πiτ and 2πi with multipliers θ and φ respectively in agreement with Lemma 4. Finally, it follows from (75) that the last sum is zero and hence (79) obtains.
Other standard recursion formulas can be similarly generalized. Thus Proposition 7. With notation as above, for any states v1 , . . . vn ∈ V, and for p ≥ 1 we have: FM (g; v[− p].v1 , . . . vn ; τ ) L(0)
= δθ,1 δφ,1 δ p,1 STr M (go(v)Y (q1 v1 , q1 ) . . . Y (qnL(0) vn , qn )q L(0)−c/24 ) θ m+1 m + p − 1 (τ )FM (g; v[m]v1 , . . . vn ; τ ) E m+ p + (−1) φ m m≥0 n m+ p−1 θ (z 1r , τ ) Pm+ p + p(v, v1 v2 . . . vr −1 )(−1) p+1 φ m r =2 m≥0
×FM (g; v1 , . . . v[m]vr , . . . vn ; τ ).
(80)
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Proof. Using (72) of Lemma 3 and associativity of VOSAs (e.g. [FHL]) we have: FM (g; (Y [v, z]v1 , z 1 ), . . . (vn , z n ); τ ) = Z M (g, Y [Y [v, z]v1 , z 1 ]Y [v2 , z 2 ] . . . Y [vn , z n ]1, τ ) = Z M (g, Y [v, z + z 1 ]Y [v1 , z 1 ]Y [v2 , z 2 ] . . . Y [vn , z n ]1, τ ) = FM (g; (v, z + z 1 ), (v1 , z 1 ), . . . (vn , z n ); τ ).
(81)
Expanding the LHS of (81) in z we find that FM (v[− p].v1 , z 1 ; . . . vn , z n ; g; τ ) is the coefficient of z p−1 . We can compare this to the expansion of the RHS in z from (79) of θ (z, τ ) Theorem 2. From (39) we find that for p ≥ 1, the coefficient of z p−1 in Pm+1 φ
θ is (−1)m+1 m+mp−1 E m+ p (τ ). Furthermore for r = 1 the coefficient of z p−1 in φ
θ θ Pm+1 (z + z 1r , τ ) is given by (−1) p+1 m+mp−1 Pm+ p (z 1r , τ ). Lastly, for p = 1 φ φ the first term of (79) also contributes. Thus the stated result follows. 4. Shifted VOSAs and Heisenberg Twisted Modules In this section we discuss the n-point functions for an orbifold g-twisted module for a VOSA, where g is a continuous symmetry generated by a Heisenberg vector. (For definitions and properties of twisted modules we refer the reader to refs. [Li,DLM1, DZ1]). In particular, we show below (Proposition 9) that every such g-twisted n-point function is related to an n-point function for the original VOSA but with a “shifted” Virasoro vector [MN2,DM]. This generalizes a similar result for partition functions found in [DM] and allows us to apply Theorem 2 in order to compute all such g-twisted n-point functions. The general relationship at the operator level between these shifted and twisted formalisms is discussed elsewhere [TZ]. A Heisenberg bosonic vector is an element h ∈ V0,1 ¯ such that [DM] 1. 2. 3. 4.
h(0) is semisimple with real eigenvalues. h is a primary vector so that L(n)h = 0 for all n ≥ 1. h(n)h = 0 for all n ≥ 0 except n = 1 for which h(1)h = ξh 1 for some ξh ∈ C. [h(m), h(n)] = ξh mδm,−n .
Remark 10. If the VOSA grading is non-negative and V0 = C1 then (2)–(4) follow automatically for all h ∈ V0,1 ¯ from (58). Given a Heisenberg vector h then h(0) generates a VOSA automorphism g = exp(2πi h(0)).
(82)
The order of g is finite iff the eigenvalues of h(0) are rational and otherwise is infinite. We can define [DLM2] and construct a g-twisted module in all cases as follows. We define [Li] ⎛ ⎞ h(n) (−z)−n ⎠. (83) ∆(h, z) = z h(0) exp ⎝− n n≥1
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Noting ∆(h, z)−1 = ∆(−h, z) one finds ∆(h, z)Y (v, z 0 )∆(−h, z) = Y (∆(h, z + z 0 )v, z 0 ).
(84)
This leads to: Proposition 8 ([Li]). Let (M, Y M ) be a V -module. Defining Yg (v, z) = Y M (∆(−h, z)v, z),
(85)
for all v ∈ V then (M, Yg ) is a g-twisted V -module3 . Note for v = ω we find ∆(−h, z)ω = ω − hz −1 + ξh z −2 /2 so that the (M, Yg ) grading is determined by L g (0) = L(0) − h(0) +
ξh . 2
(86)
We define the orbifold g-twisted n-point function for any automorphism f commuting with g and σ by FM (( f, g); v1 , . . . , vn ; τ ) = STr M f Yg (q1L(0) v1 , q1 ) . . . Yg (qnL(0) vn , qn )q L g (0)−c/24 .
(87)
We denote the orbifold g-twisted partition function by Z M (( f, g), τ ). For each Heisenberg element h we may also construct a VOSA (V, Y, 1, ωh ) with the original vector space and vertex operators but using a “shifted” conformal vector ([MN2,DM]) ωh = ω + h(−2)1. With Y (ωh , z) =
n∈Z
(88)
L h (n)z −n−2 we find L h (n) = L(n) − (n + 1)h(n),
(89)
ch = c − 12ξh .
(90)
and central charge
In particular, L h (−1) = L(−1) and the grading is determined by L h (0) = L(0) − h(0).
(91)
We denote the partition function for a V -module M with a h-shifted L h (0) by Z M,h (τ ). Following (66) the shifted n-point function is denoted by L (0) FM,h ( f ; v1 , . . . , vn ; τ ) = STr M f Y (q1 h v1 , q1 ) . . . Y (qnL h (0) vn , qn )q L h (0)−ch /24 , (92) 3 Note that we apply the definition of g-twisted module of ref. [Li] which corresponds to a g −1 -twisted module in refs. [DLM1] and [DM].
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where f commutes with g and σ . We denote the h-shifted partition function by Z M,h ( f, τ ). Comparing (86) and (91) we see that L g (0) −
c ch = L h (0) − , 24 24
so that ([DM]) Z M ((1, g), τ ) = Z M,h (1, τ ).
(93)
This relationship can be generalized to relate all orbifold g-twisted n-point functions to h-shifted n-point functions as follows: Proposition 9. Let M be a module for V and let g = exp(2πi h(0)) be generated by a Heisenberg state h. Then the n-point function for the orbifold g-twisted and the untwisted n-point function for M with shifted L h (0)-vertex operators are related as follows: FM (( f, g); v1 , . . . , vn ; τ ) = FM,h ( f ; U v1 , . . . , U vn ; τ ),
h(n) n where U = ∆(−h, 1) = exp and f commutes with g and σ . n (−1)
(94)
n≥1
Proof. First we prove ∆(−h, qz ) qzL(0) = qzL h (0) U. From (83) one finds using [L(0), h(n)] = −nh(n) that
h(n) ∆(−h, qz )qzL(0) = qz−h(0) exp (−qz )−n qzL(0) n n>0
h(n) −h(0) L(0) −n (−qz ) = qz qz exp exp(ad−z L(0) ) n n>0
h(n) L h (0) −n n (−qz ) qz = qzL h (0) U. exp = qz n n>0
Therefore from (85) Yg (qzL(0) v, qz ) = Y M (∆(−h, qz )qzL(0) v, qz ) = Y M (qzL h (0) U v, qz ). Thus the LHS of (94) is L(0) STr M f Yg (q1 v1 , q1 ) . . . Yg (qnL(0) vn , qn )q L g (0)−c/24 L (0) = STr M f Y M (q1 h U v1 , q1 ) . . . Y M (qnL h (0) U vn , qn )q L h (0)−ch /24 = FM,h ( f ; U v1 , . . . , U vn ; τ ).
(95)
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We conclude this section by showing that U maps between the square bracket vertex operators (62) of the original and shifted VOSAs. We let Y [v, z]h = Y (qzL h (0) v, qz − 1),
(96)
denote a square bracket vertex operator in the h-shifted VOSA. We then have Lemma 6. For v ∈ V we have U Y [v, z]U −1 = Y [U v, z]h .
(97)
Proof. Using associativity, (85) and (95) we obtain L(0)
Yg (q1 = = =
L(0)
v1 , q1 )Yg (q2
v2 , q2 ) L(0) L(0) Yg (Y (q1 v1 , q1 − q2 ) q2 v2 , q2 ) L(0) Yg (q2 Y [v1 , z 12 ]v2 , q2 ) Y (q2L h (0) U Y [v1 , z 12 ]v2 , q2 ).
On the other hand Yg (q1L(0) v1 , q1 )Yg (q2L(0) v2 , q2 ) L (0)
= Y (q1 h = Hence the result follows.
L (0)
U v1 , q1 )Y (q2 h
L (0) Y (q2 h Y [U v1 , z 12 ]h
U v2 , q2 )
U v2 , q2 ).
5. Rank One Fermion VOSA We begin with the example of the rank one “Neveu-Schwarz sector” fermion VOSA V = V (H, Z + 21 ) generated by one fermion [FFR,Li]. This is a 21 Z graded VOSA with H = Cψ for a fermion vector ψ of parity 1 and modes obeying [ψ(m), ψ(n)] = ψ(m)ψ(n) + ψ(n)ψ(m) = δm+n+1,0 .
(98)
The superspace V is spanned by Fock vectors of the form ψ(−k1 )ψ(−k2 ) . . . ψ(−km )1,
(99)
for integers 1 ≤ k1 < k2 < · · · km with ψ(k)1 = 0 for all k ≥ 0 so that V is generated by Y (ψ, z). The conformal vector is ω = 21 ψ(−2)ψ(−1)1 of central charge c = 21 for which the Fock vector (99) has L(0) weight 1≤i≤m (ki − 21 ) ∈ 21 Z. In particular, wt (ψ) = 21 . The partition function is η 1τ 1 1 L(0)− 48 − 48 n+ 12 2 Z V (τ ) = STr V q , (100) =q 1−q = η(τ ) n≥0
whereas for g = σ of (65) we find 1 1 1 Z V (σ, τ ) = STr V σ q L(0)− 48 = q − 48 1 + q n+ 2 = n≥0
η(τ )2
. (101) η(2τ )η 21 τ
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Let us next introduce the n-point function (66) for V where vi = ψ for all i = 1, . . . n: G n (g; z 1 , . . . , z n ; τ ) = FV (g; (ψ, z 1 ), . . . , (ψ, z n ); τ ),
(102)
which we will refer to as the generating function. We use the recursion formula (79) of Theorem 2 to compute G n . Since wt (ψ) = 21 we have φ = −1 from (76) and θ = 1 for g = 1 and θ = −1 for g = σ from (77 ). For n = 1, G 1 (g; z 1 ; τ ) = Z V (g, ψ, τ ) = 0 since o(ψ) = 0. For n = 2, (79) implies θ (z 12 , τ )FV (g; ψ[m]ψ; τ ). Pm+1 G 2 (g; z 1 , z 2 ; τ ) = 0 + −1 m≥0
Passing to the square bracket formalism (62) we find the same fermion commutator algebra as (98) obtains, namely [ψ[m], ψ[n]] = δm+n+1,0 . Thus it follows that ψ[m]ψ = δm,0 1 giving θ (z 12 , τ )Z V (g, τ ). G 2 (g; z 1 , z 2 ; τ ) = P1 −1
(103)
(104)
We may similarly compute G n for all n by repeated application of (79). It is easy to see that G n = 0 for n odd. For n even G n is expressed in terms of a Pfaffian which is totally antisymmetric in z i as expected from Lemma 4 (ii). Let us first recall the definition of the Pfaffian of an anti-symmetric matrix M = (M(i, j)) of even dimension 2m given by Pf(M) = εi1 j1 ...im jm M(i 1 , j1 )M(i 2 , j2 ) . . . M(i m , jm ), (105) Π
where the sum is taken over the set of all partitions Π of {1, 2, . . . , 2m} into pairs with elements {(i 1 , j1 ), (i 2 , j2 ) . . . (i m , jm )}, for i k < jk and i 1 < i 2 < · · · i m , and where εi1 j1 ...im jm is the Levi-Civita symbol. We also note that √ Pf(M) = det M. We then obtain: Proposition 10. For n even and g = 1 or σ we have G n (g; z 1 , . . . , z n ; τ ) = Pf(P)Z V (g, τ ), where P denotes the anti-symmetric n × n matrix with components θ (z i j , τ ), (1 ≤ i = j ≤ n), P(i, j) = P1 −1 for z i j = z i − z j with θ = 1 for g = 1 and θ = −1 for g = σ .
(106)
(107)
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Proof. We first note that P is anti-symmetric from (36) since θ = ±1. We prove the result by induction. For n = 2 the result is given in (104). For general n we apply (79) to obtain n θ r (z 1r , τ )G n−2 (g; z 2 , . . . , zˆr , . . . z n ; τ ) G n (g; z 1 , . . . z n ; τ ) = (−1) P1 −1 r =2
n ˆ V (g, τ ), = (−1)r P(1, r )Pf(P)Z r =2
where zˆr is deleted and Pˆ is the “cofactor” matrix obtained by deleting the 1st and r th rows and columns of P. The result (107) follows from the definition (105). G n enjoys the following analytic properties following Remark 1 (ii): Corollary 2. G n is an analytic function in z i and converges absolutely and uniformly on compact subsets of the domain |q| < qzi j < 1 for all z i j = z i − z j with i = j. We now show that all n-point functions can be computed from G n . Consider a V basis of square bracket Fock vectors denoted by Ψ [−k] = ψ[−k1 ]ψ[−k2 ] . . . ψ[−km ]1,
(108)
where k = k1 , k2 , . . . , km for integers 1 ≤ k1 < k2 < · · · km . We will determine an explicit formula for all n -point functions for such Fock vectors. Thus the 1-point m ki −1 zi in G n since function Z V (g, Ψ [−k], τ ) is the coefficient of i=1 G n (g; z 1 , . . . , z m ; τ ) = Z V (g, Y [ψ, z 1 ] . . . Y [ψ, z m ]1, τ ) Z V (g, ψ[−k1 ] . . . ψ[−km ]1, τ )z 1k1 −1 . . . z nkm −1 . = k1 ,...km ∈Z
θ −1 (z i j , τ ) given in (43). It follows that Z V (g, Ψ [−k], τ ) = 0 for m odd whereas for m even
Examining (106) we can explicitly find this coefficient from the expansion of P1
Z V (g, Ψ [−k], τ ) = Pf(C)Z V (g, τ ),
(109)
where C denotes the antisymmetric m × m matrix with (i, j) -entry θ (ki , k j , τ ), C(i, j) = C −1 (cf. (45)). C is antisymmetric from (47) since θ = ±1. We may similarly derive an expression for an arbitrary two-point function FV (1) (1) (2) (2) ((Ψ [−k(1) ], z 1 ), (Ψ [−k(2) ], z 2 ); g; τ ) for k(1) = k1 , . . . km 1 and k(2) = k1 , . . . km 2 . First consider the one-point function Z V (g, Y [Y [ψ, x1 ] . . . Y [ψ, xm 1 ]1, z 1 ].Y [Y [ψ, y1 ] . . . Y [ψ, ym 2 ]1, z 2 ]1, τ ). (110)
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−1 m 1 m 2 ki(1) −1 k (2) FV (g; (Ψ [−k(1) ], z 1 ), (Ψ [−k(2) ], z 2 ); τ ) is the coefficient of i=1 yj j j=1 x i in (110). By associativity (e.g. [FHL]) and using Y [1, z] = Id V we find (110) can be expressed as
Z V (g, Y [ψ, x1 + z 1 ] . . . Y [ψ, xm 1 + z 1 ].Y [ψ, y1 + z 2 ] . . . Y [ψ, ym 2 + z 2 ]1, τ ) = G n (g; x1 + z 1 , . . . , xm 1 + z 1 , y1 + z 2 , . . . , ym 2 + z 2 ; τ ). −1 m 1 m 2 ki(1) −1 k (2) yj j can then be extracted from the expansions The coefficient of i=1 j=1 x i (43) and (44). Thus the two point function vanishes for m 1 +m 2 odd, whereas for m 1 +m 2 even,
FV (g; (Ψ [−k(1) ], z 1 ), (Ψ [−k(2) ], z 2 ); τ ) = Pf(M)Z (g, τ ),
(111)
where M is the antisymmetric (m 1 + m 2 ) × (m 1 + m 2 ) block matrix (11) (12) D C , M= D(21) C(22) where for a, b ∈ {1, 2}, θ (a) (a) C(aa) (i, j) = C (ki , k j , τ ), (1 ≤ i, j ≤ m a ), −1 θ (ab) (ki(a) , k (b) D (i, j) = D j , τ, z a − z b ), (1 ≤ i ≤ m a , 1 ≤ j ≤ m b ), −1 (112) (using (46)). M is antisymmetric from (47) and (48). In a similar fashion we are lead to the general result: (a)
(a)
(a) Proposition 11. Let Ψ [−k(a) ] for a = 1 . . . n be n Fock vectors for k = k1 , . . . km a . Then the n-point function vanishes for odd a m a and for a m a even is given by
FV (g; (Ψ [−k(1) ], z 1 ), . . . (Ψ [−k(n) ], z n ); τ ) = Pf(M)Z (g, τ ),
(113)
where M is the antisymmetric block matrix ⎛ (11) (12) ⎞ C D . . . D(1n) ⎜ D(21) C(22) ⎟ ⎜ ⎟ M =⎜ . ⎟, .. ⎝ .. ⎠ . (n1) (nn) D ... C with C(aa) and D(ab) of (112). Equation (113) is an analytic function in z i and converges absolutely and uniformly on compact subsets of the domain |q| < qzi j < 1 for all z i j = z i − z j with i = j. We have also established Proposition 12. G n (g; z 1 , . . . , z n ; τ ) is a generating function for all n-point functions.
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We conclude this section by noting that we may also consider the “Ramond sector” σ -twisted module V (H, Z) for V (H, Z + 21 ). This is discussed in detail in [FFR,Li, DZ1,DZ2]. V (H, Z) decomposes into two irreducible σ -twisted modules which are interchanged under the induced action of σ . For either irreducible σ -twisted module Mσ the partition function is 1 STr Mσ q L(0)− 48 = 0, 1 1 η(2τ ) STr Mσ σ q L(0)− 48 = q 48 (1 + q n ) = . η(τ ) n≥0
We may similarly consider the generator of all σ -twisted n-point functions defined by G Mσ ,n (g; z 1 , . . . , z n ; τ ) = FMσ (g; (ψ, z 1 ), . . . , (ψ, z n ); τ ), for g = 1 or σ . This vanishes for all n for g = 1 and for n odd for g = σ . By applying a VOSA orbifold Zhu reduction formula of ref. [DZ1] we find as in Proposition 10 that Proposition 13. For n even we have
G Mσ ,n (σ ; z 1 , . . . , z n ; τ ) = Pf(P1
η(2τ ) −1 (z i j , τ )) , 1 η(τ )
(114)
for z i j = z i − z j . One can similarly describe analytic properties as in Corollary 2 and determine all σ -twisted n-point functions by expanding this generating function along the same lines as Proposition 11, though we do not carry this out here. 6. Rank Two Fermion VOSA 6.1. h-shifted and orbifold g-twisted n-point functions. In this section we consider the rank two fermion VOSA formed from the tensor product of two copies of the rank one fermion VOSA and hence is generated by two free fermions ψ1 = ψ ⊗1 and ψ2 = 1⊗ψ. We may therefore compute all the untwisted and σ -twisted n-point functions based on the last section. However, as is well known, this VOSA contains a bosonic Heisenberg state h = αψ ⊗ ψ (for α ∈ C) and we will compute all h-shifted and g-twisted n-point functions where g is generated by h as discussed in Sect. 4. It is convenient to introduce the off-diagonal basis ψ ± = √1 (ψ1 ± iψ2 ), where ψ ± -modes obey the commutation relations
2
[ψ + (m), ψ − (n)] = δm,−n−1 , [ψ ± (m), ψ ± (n)] = 0. (115) The VOSA V is generated by Y (ψ ± , z) = ψ ± (n)z −n−1 , where the vector space V n∈Z
is a Fock space with basis vectors of the form
ψ + (−k1 ) . . . ψ + (−ks )ψ − (−l1 ) . . . ψ − (−lt )1,
(116)
for 1 ≤ k1 < k2 < · · · ks and 1 ≤ l1 < l2 < · · · lt with ψ ± (k)1 = 0 for all k ≥ 0. We define the conformal vector to be 1 ω = [ψ + (−2)ψ − (−1)+ψ − (−2)ψ + (−1)]1, (117) 2
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whose modes generate a Virasoro algebra 1. Then ψ ± has L(0)-weight of central charge 1 1 1 1≤i≤s (ki − 2 ) + 1≤ j≤t (l j − 2 ). 2 and the Fock state (116) has weight The weight 1 parity zero space is V0,1 ¯ = Ca for (normalized) Heisenberg bosonic vector a = ψ + (−1)ψ − (−1)1,
(118)
with modes obeying [a(m), a(n)] = mδm,−n , and ω of (117) is nothing but the standard Heisenberg VOA conformal vector 1 a(−1)2 1. 2 Following Sect. 4 we define a one parameter family of Heisenberg vectors ω=
h = κa, κ ∈ R,
(119)
for which ξh = κ 2 . The shifted conformal vector (88) is then ωh = ω + κa(−2)1 with central charge ch = 1 − 12κ 2 from (90). Then ψ ± has L h (0) = L(0) − κa(0) weight wth (ψ ± ) = 21 ∓ κ and the Fock state (116) has L h (0) weight 1≤i≤s (ki − 21 − κ) + 1 1≤ j≤t (l j − 2 + κ). Noting that σ = eπia(0) and following Sect. 4, we can construct a σ g-twisted module for σ g = e2πi h(0) so that g = e2πiβa(0) ,
(120)
1 β=κ− . 2
(121)
φ = exp(2πiwth (ψ + )) = e−2πiβ .
(122)
for real β where
We also define φ ∈ U (1) by
Introduce the automorphism f = e2πiαa(0) , α ∈ R,
(123)
which commutes with g, σ . Then f ψ ± = θ ∓1 ψ ± for θ = e−2πiα ∈ U (1).
(124)
Finally, we denote the orbifold σ g-twisted trace by f (τ ) = Z V (( f, σ g), τ ). ZV g We find using Proposition 9 that 1 1 2 f Z V,h ( f, τ ) = Z V (τ ) = q κ /2−1/24 (1 − θ −1 q l− 2 −κ )(1 − θq l− 2 +κ ). (125) g l≥1
Note that Z V,h ( f, τ ) = 0 for (θ, φ) = (1, 1), i.e. (α, β) ≡ (0, 0) (mod Z).
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Remark 11. The RHS of (125) is related to a theta series via the Jacobi triple product formula as briefly reviewed below in Sect. 6.4. Hence Z V,h ( f, τ ) depends on α(mod Z) and β(mod Z) up to an overall α-dependent constant. We next consider general σ g-twisted and h-shifted n-point functions which are related via Proposition 9. As in the rank one case, it is sufficient to consider n-point functions for the generating states ψ ± only. To this end we define the h-shifted VOSA n-point generating function G 2n,h ( f ; x1 , . . . , xn ; y1 , . . . , yn ; τ ) = FV,h ( f ; (ψ + , x1 ), (ψ − , y1 ), . . . , (ψ + , xn ), (ψ − , yn ); τ ).
(126)
Remark 12. Note the choice of an alternating ordering of the operators with respect to the ± superscript here. We can also define a σ g-twisted n-point function denoted by f FV ((v1 , z 1 ) . . . , (vn , z n ); τ ) = FV (( f, σ g); (v1 , z 1 ) . . . , (vn , z n ); τ ), g with generating function f G 2n (x1 , . . . , xn ; y1 , . . . , yn ; τ ) g = FV (( f, σ g); (ψ + , x1 ), (ψ − , y1 ), . . . , (ψ + , xn ), (ψ − , yn ); τ ). Then noting that U ψ ± = ψ ± and applying Proposition 9 we find Lemma 7. f G 2n (x1 , . . . , xn ; y1 , . . . , yn ; τ ) = G 2n,h ( f ; x1 , . . . , xn ; y1 , . . . , yn ; τ ). σg These generating functions are totally antisymmetric in xi , y j as expected from Lemma 4 (ii) and can be expressed in terms of a determinant computed by means of our recursion formula (79). Due to the leading term on the RHS of (79), we consider the cases (θ, φ) = (1, 1) and (θ, φ) = (1, 1) separately. 6.2. n-point functions for (θ, φ) = (1, 1). Proposition 14. For (θ, φ) = (1, 1) we have G 2n,h ( f ; x1 , . . . , xn ; y1 , . . . , yn ; τ ) = detP. Z V,h ( f ; τ ), where P is the n × n matrix: θ (xi − y j , τ ) , (1 ≤ i, j ≤ n), P = P1 φ
(127)
(128)
with θ, φ of (124) and (122). Furthermore, G 2n,h is an analytic function in x i , y j and converges absolutely and uniformly on compact subsets of the domain |q| < qxi −y j < 1.
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Proof. We apply Theorem 2 directly with e2πiwt (ψ ) = φ and f ψ + = e2πiβ ψ + = θ −1 ψ + of (122) and (124). The Zhu recursion formula (79) results in a determinant similarly to the proof of Proposition 10. The region of analyticity follows as before. +
In order to describe general n-point functions, first note that [ψ + [m], ψ − [n]] = δm,−n−1 , [ψ ± [m], ψ ± [n]] = 0. Now introduce Ψ = Ψ [−k; −l] = ψ + [−k1 ] . . . ψ + [−ks ]ψ − [−l1 ] . . . ψ − [−lt ]1, (129) + + − − Ψh = Ψ [−k; −l]h = ψ [−k1 ]h . . . ψ [−ks ]h ψ [−l1 ]h . . . ψ [−lt ]h 1, (130) where k = k1 , . . . , ks and l = l1 , . . . , lt ; these denote Fock vectors (116) in the square bracket and h-shifted square bracket formalisms respectively. From Lemma 6 and using U ψ ± = ψ ± we have Ψ [−k; −l]h = U Ψ [−k; −l]. By expanding G 2n,h appropriately and following the same approach that lead to Proposition 11, we obtain a determinant formula for every n-point function as follows: Proposition 15. Consider n Fock vectors Ψ (a) = Ψ (a) [−k(a) ; −l(a) ] and Ψh(a) = Ψ (a) (a) = l (a) , . . . l (a) with a = 1 . . . n. Then [−k(a) ; −l(a) ]h for k(a) = k1(a) , . . . ks(a) ta a and l 1 for (θ, φ) = (1, 1) the corresponding n-point functions are non-vanishing provided n
(sa − ta ) = 0.
a=1
In this case they are given by f ((Ψ (1) , z 1 ), . . . , (Ψ (n) , z n ); τ ) FV g (1)
(n)
= FV,h ( f ; (Ψh , z 1 ), . . . , (Ψh , z n ); τ ) = detM. Z V,h ( f ; τ ), where M is the block matrix
⎛
⎞ C(11) D(12) . . . D(1n) ⎜ D(21) C(22) . . . D(2n) ⎟ ⎜ ⎟ M=⎜ . .. ⎟, .. ⎝ .. . . ⎠ (n1) (nn) ... C D
with (aa)
C
θ (a) (a) (ki , l j , τ ), (1 ≤ i ≤ sa , 1 ≤ j ≤ ta ), (i, j) = C φ
for sa , ta ≥ 1 with 1 ≤ a ≤ n and θ (ab) (ki(a) , l (b) D (i, j) = D j , τ, z ab ), (1 ≤ i ≤ sa , 1 ≤ j ≤ tb ), φ
(131)
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for sa , tb ≥ 1 with 1 ≤ a, b ≤ n and a = b. is the sign of the permutation associated with the reordering of ψ ± to the alternating ordering of (126) following Remark 12. Furthermore, the n-point function (131) is an analytic function in z a and converges absolutely and uniformly on compact subsets of the domain |q| < qzab < 1. Example. Consider the n-point function for n vectors Ψ = a for a = ψ + [−1]ψ − [−1]1 and (θ, φ) = (1, 1). Then f f ((a, z 1 ), . . . , (a, z n ); τ ) = det M.Z V (τ ), FV g g for ⎞ θ θ θ −E (τ ) P (z (z , τ ) . . . P , τ ) 1 1 12 1 1n ⎟ ⎜ φ φ φ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ P1 θ (z 21 , τ ) −E 1 θ (τ ) . . . P1 θ (z 2n , τ ) ⎟ ⎟. ⎜ φ φ φ M=⎜ ⎟ ⎟ ⎜ .. .. .. ⎟ ⎜ . . ⎟ ⎜ . ⎠ ⎝ θ θ (z n1 , τ ) (τ ) ... −E 1 P1 φ φ ⎛
For θ, φ ∈ {±1}, it follows from (36) that the diagonal Eisenstein √ terms vanish and that det M = 0 for odd n. Taking n even and recalling that Pf(M) = det M, we recover the square of the rank one generating function (106) for φ = −1 and the rank one σ -twisted generating function (114) for φ = 1. 6.3. n-point functions for (θ, φ) = (1, 1). We consider (α, β) = (0, 0) so that ( f, g) = (1, 1) and (θ, φ) = (1, 1) with κ = 21 (cf. Remark 11 ). We then have wth (ψ + ) = 0, wth (ψ − ) = 1 and ch = −2. For n = 1, Eq. (126) can be computed from (79) to give the (x, y independent) result: G 2,h (1; x, y; τ ) = FV,h (1; (ψ + , x), (ψ − , y); τ ) = STr V oh (ψ + )oh (ψ − )q L h (0)+1/12 + 0, where oh (v) = v(wth (v) − 1) from (61) and recalling Z V,h (1; τ ) = 0. Furthermore, oh (ψ + )oh (ψ − ) = ψ + (−1)ψ − (0) acts as a projection operator on V preserving those Fock vectors (116) containing an ψ + (−1) operator. Hence we find G 2,h (1; x, y; τ ) = q 1/12 (−q 0 )
(1 − q k−1 )
k≥2
(1 − q l ) = −η(τ )2 .
(132)
l≥1
We may proceed much as before to compute the generator G 2n,h to find: Proposition 16. For (θ, φ) = (1, 1) we have G 2n,h (1; x1 , . . . , xn ; y1 , . . . , yn ; τ ) = det Q.η(τ )2 ,
(133)
Torus n-Point Functions for R-graded Vertex Operator Superalgebras
335
where Q is the (n + 1) × (n + 1) matrix: ⎛ P1 (x1 − y1 , τ ) . . . P1 (x1 − yn , τ ) .. .. ⎜ . . Q=⎜ ⎝ P (x − y , τ ) P1 (xn − yn , τ ) 1 n 1 1 ... 1
⎞ 1 .. ⎟ .⎟ 1⎠
(134)
0
(P1 (z, τ ) as in (11)). Furthermore, G 2n,h is an analytic function in xi , y j and converges absolutely and uniformly on compact subsets of the domain |q| < qxi −y j < 1. Proof. We prove the result by induction. For n = 1 we obtain the result from (132). Assuming the result for n − 1, we apply the Zhu recursive formula (79) to find G 2n,h (1; x1 , . . . , xn ; y1 , . . . , yn ; τ ) − L(0) + L(0) − L(0)+1/12 ψ , q ) . . . Y (q ψ , q )Y (q ψ , q )q = STr V o(ψ + )Y (q yL(0) y x y n n 1 xn yn 1 +
n ˆ (−1)r −1 Q(1, r )det Q.η(τ )2 , r =1
ˆ denotes the matrix found from Q by deleting row 1 and column r . Next note where Q from Lemma 4 (ii) and (115) that G 2n,h vanishes for x1 = x2 so that − L(0) − L(0)+1/12 STr V o(ψ + )Y (q yL(0) ψ , q ) . . . Y (q ψ , q )q y y n 1 yn 1 =−
n ˆ (−1)r Q(2, r )det Q.η(τ )2 . r =2
Hence we find G 2n,h is given by n ˆ (−1)r −1 (Q(1, r ) − Q(2, r ))det(Q)η(τ )2 = det Q.η(τ )2 , r =1
on evaluating det Q after subtracting row 2 from row 1.
We may similarly obtain a determinant formula for all n-point functions along the same lines as Propositions 11 and 15. 6.4. Bosonization. As is well known, the rank two fermion VOSA V can be constructed as a rank one bosonic Z-lattice VOSA. V is decomposed in terms of the Heisenberg subVOA M generated by the boson a of (118) and its irreducible modules M ⊗ em for a(0) eigenvalue m ∈ Z (cf. [Ka]). In particular, the partition function Z V,h ( f ; τ ) and the generating function G n,h can be computed in this bosonic decomposition using the results of ref. [MT1], leading to the Jacobi triple product formula and Fay’s trisecant identity (for elliptic functions) respectively. We also describe a further new generalization of Fay’s trisecant identity for elliptic functions. The highest weight lattice vector for the irreducible module M ⊗ em is + ψ (−m)ψ + (1 − m) . . . ψ + (−1).1, m > 0, 1 ⊗ em = ψ − (m)ψ − (1 + m) . . . ψ − (−1).1, m < 0.
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Then the partition function is 2 f (τ ) = Z V,h ( f ; τ ) = Z V (−1)m e2πimα Tr M⊗em (q L(0)+κ /2−κm−1/24 ) g m∈Z e2πi(α+1/2)(β+1/2) −β + 21 ϑ (0, τ ), (135) = α + 21 η(τ ) in terms of the theta series (13). Comparing to (125) we obtain the standard Jacobi triple product formula. We can also compute the generating function G n,h (and hence all n-point functions) in the bosonic setting based on results of ref. [MT1]. We illustrate this with the 2-point function generator (126). Recall from (127) and (132) that ⎧ θ ⎨ (x − y, τ )Z V,h ( f ; τ ), (θ, φ) = (1, 1), P1 φ G 2,h ( f ; x; y; τ ) = (136) ⎩ (θ, φ) = (1, 1). −η(τ )2 , In the bosonic language we obtain: (−1)m e2πimα FM⊗em ,h (1; (1 ⊗ e+1 , x), (1 ⊗ e−1 , y); τ ) G 2,h ( f ; x; y; τ ) = m∈Z
=
(−1)m e2πimα exp(−κ(x − y))q κ
2 /2−κm
m∈Z
×FM⊗em (1; (1 ⊗ e+1 , x), (1 ⊗ e−1 , y); τ ), L (0)
noting that Y (qz h e±1 , qz ) = exp(∓κz)Y (qz of ref. [MT1] we obtain
L(0) ±1 e ,q
z ).
Using Propositions 4 and 5
q m /2 exp(m(x − y)) , η(τ ) K (x − y, τ ) 2
FM⊗em (1; (1 ⊗ e+1 , x), (1 ⊗ e−1 , y); τ ) =
where K is the prime form (9). Altogether, it follows that −β + 21 (x − y, τ ) ϑ α + 21 e2πi(α+1/2)(β+1/2) . G 2,h ( f ; x; y; τ ) = η(τ ) K (x − y, τ ) Comparing with (136) we confirm the identities (49) for (θ, φ) = (1, 1) and (18) for (α, β) = (0, 0), i.e. (θ, φ) = (1, 1). In a similar fashion we can compute the general generating function G 2n,h in the bosonic setting to obtain: Proposition 17.
n e2πi(α+1/2)(β+1/2) −β + 21 ϑ G 2n,h ( f ; x1 , . . . , xn ; y1 , . . . , yn ; τ ) = (xi − yi ), τ α + 21 η(τ ) i=1 K (xi − x j , τ )K (yi − y j , τ ) ×
1≤i< j≤n
1≤i, j≤n
K (xi − y j , τ )
.
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337
Comparing this to Proposition 14 for (θ, φ) = (1, 1) and Proposition 16 for (θ, φ) = (1, 1) we obtain the classical Frobenius elliptic function version of Fay’s Generalized Trisecant Identity [Fa]: Corollary 3. For (θ, φ) = (1, 1) we have −β + 21 n K (xi − x j , τ )K (yi − y j , τ ) ϑ (xi − yi ), τ 1 i=1 α+ 2 1≤i< j≤n det(P) = , K (xi − y j , τ ) −β + 21 ϑ (0, τ ) 1≤i, j≤n α + 21 (137) with P as in (128). For (θ, φ) = (1, 1),
n K i=1 (x i − yi ), τ
det(Q) = −
K (xi − x j , τ )K (yi − y j , τ )
1≤i< j≤n
, (138)
K (xi − y j , τ )
1≤i, j≤n
where Q is as in (134). We may generalize these identities using Propositions 4 and 5 of [MT1] again to consider the general lattice n-point function: Proposition 18. For integers m i , n j ≥ 0 satisfying r
mi =
i=1
s
n j,
j=1
we have FV ( f ; (1⊗em 1 , x1 ), . . . (1⊗em r , xr ), (1⊗e−n 1 , y1 ), . . . (1⊗e−n s , ys ); τ ) ⎞ ⎛ r s 2πi(α+1/2)(β+1/2) 1 e −β + 2 ⎝ ϑ = m i xi − n j yj, τ⎠ α + 21 η(τ ) i=1 j=1 K (xi − xk , τ )m i m k K (y j − yl , τ )n j nl ×
1≤i
1≤ j
K (xi − y j , τ )m i n j
.
1≤i≤r,1≤ j≤s
Comparing this to Proposition 15 we obtain a new elliptic generalization of Fay’s Trisecant Identity: Corollary 4. For (θ, φ) = (1, 1) we have s −β + 21 r ϑ m x − n y , τ i=1 i i j=1 j j α + 21 det(M) = . −β + 21 (0, τ ) ϑ α + 21 K (xi − xk , τ )m i m k K (y j − yl , τ )n j nl ×
1≤i
1≤i≤r,1≤ j≤s
1≤ j
K (xi − y j , τ )m i n j
,
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where M is the block matrix
⎞ D(11) . . . D(1s) ⎟ ⎜ M = ⎝ ... . . . ... ⎠, D(r 1) . . . D(r s) ⎛
with D(ab) the m a × n b matrix θ (i, j, τ, xa − yb ), (1 ≤ i ≤ m a , 1 ≤ j ≤ n b ), D(ab) (i, j) = D φ for 1 ≤ a ≤ r and 1 ≤ b ≤ s. A similar identity for (θ, φ) = (1, 1) generalizing (138) can also be described. 6.5. Modular properties of n-point functions. In this section we consider the modular properties of all n-point functions for the rank two fermion VOSA. Despite the fact the twisted sectors are neither rational or C2 -cofinite we obtain modular properties similar to those found in [Z,DZ1,DZ2]. It is convenient to employ the twisted n-point function formalism to describe these modular properties. We firstly consider the partition function f ab (τ ) and define a group action for γ = ∈ S L(2, Z) as follows: ZV g cd f f γ (τ ) = Z ZV (γ .τ ), (139) V γ. g g with
γ.
f g
=
f a gb , f c gd
(140)
and γ .τ as in (33). Remark 13. (i) (140) is equivalent to left matrix multiplication on α, β, aα + bβ α . = γ cα + dβ β (ii) In terms of the shifted VOA formalism, (139) reads Z V,h ( f ; τ ) γ = Z V,γ .h ( f a g b ; γ .τ ), with γ .h = (γ .β + 21 )a = ((cα + dβ) + 21 )a, recalling (119) and (121). 11 0 1 we can use the theta function and T = For S L(2, Z) generators S = 01 −1 0 modular transformation properties (16) and (17) and thereby find from (135) that f f f S(τ ) = ε Z (τ ), (141) ZV S V g g g f f f T (τ ) = ε Z (τ ), (142) ZV T V g g g
Torus n-Point Functions for R-graded Vertex Operator Superalgebras
where
339
1 1 = exp 2πi +β −α , 2 2 1 f = exp(πi(β(β + 1) + )). εT g 6
εS
f g
(143) (144)
One can check that the relations (ST )3 = −S 2 = 1 are satisfied so that Z V
f (τ ) is g
modular invariant as follows: Proposition 19. The partition function transforms under γ ∈ S L(2, Z) with multiplier f ∈ U (1), where εγ g f f f γ (τ ) = ε ZV Z (τ ), γ V g g g f f f generated from ε S and εT . with εγ g g g In order to discuss the modular properties of n-point functions we first define the left S L(2, Z) action f f FV ((v1 , z 1 ) . . . , (vn , z n ); τ ) γ = FV γ . g g ×((v1 , γ .z 1 ) . . . , (vn , γ .z n ); γ .τ ),
(145)
and γ .z as in (33). It is sufficient to consider the generating function: f transforms under γ ∈ S L(2, Z) Proposition 20. The generating function G 2n g f with weight n and multiplier εγ , that is g f f f n G 2n (x1 . . . xn ; y1 . . . yn ; τ ) γ = (cτ + d) εγ G 2n g g g ×(x1 . . . xn ; y1 . . . yn ; τ ). Proof. For (θ, φ) = (1, 1) we have f f = det(P) Z V (τ ), G 2n g g θ θ from Proposition 14. From Proposition 1 we have P1 (γ .z, γ .τ ) = (cτ +d)P1 φ φ (z, τ ). Hence using Proposition 19 the result follows. For (θ, φ) = (1, 1) we have f = det Q.η(τ )2 , G 2n g
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from Proposition 16 with Q as in (134). From (4) and (6) it follows that P1 (z, τ ) is quasi-modular: P1 (γ .z, γ .τ ) = (cτ + d)P1 (z, τ ) +
c z. 2πi
However, det Q is modular of weight n − 1 as follows. Subtract row 1 from rows 2 . . . n and then subtract col 1 from cols 2 . . . n to find det Q = det R, where for 2 ≤ i, j ≤ n, R(i, j) = P1 (xi − y j , τ ) + P1 (x1 − y1 , τ ) − P1 (xi − y1 , τ ) − P1 (x1 − y j , τ ), which is modular of weight 1. Hence the result follows.
The modular transformation properties for an arbitrary n-point function follows by appropriately expanding the generating function as before to find n-point functions for the Fock basis described in Proposition 15. We thus find Proposition 21. For n vectors va of wt[va ], a = 1, . . . , n, the n-point function trans f : forms under γ ∈ S L(2, Z) with weight K = a wt[va ] and multiplier εγ g f FV ((v1 , z 1 ), . . . , (vn , z n ); τ ) γ = g f f FV ((v1 , z 1 ), . . . , (vn , z n ); τ ). (cτ + d) K εγ g g This result is a natural generalization for continuous orbifolds of the rank two fermion VOSA of Zhu’s Theorem 5.3.2 for C2 -cofinite VOAs [Z]. 7. Appendix A: Parity and Supertraces A vertex operator Y (a, z) has parity p(a) ∈ {0, 1} if all its modes a(n) have parity p(a). Two operators A, B on V of parity p(A), p(B) have commutator defined by [A, B] = AB − p(A, B)B A, p(A, B) = (−1) p(A) p(B) . The commutator clearly obeys: [A, B] = − p(A, B)[B, A], and for B1 . . . Bn of parity p(B1 ), . . . p(Bn ) respectively we have [A, B1 . . . Bn ] n = p(A, B1 . . . Br −1 )B1 . . . Br −1 [A, Br ]Br +1 . . . Bn ,
(146)
r =1
where
p(A, B1 . . . Br −1 ) =
1 (−1) p(A)[ p(B1 )+...+ p(Br −1 )]
for r = 1 . for r > 1
(147)
Torus n-Point Functions for R-graded Vertex Operator Superalgebras
Let Vα =
$ r ≥r0
341
Vα,r denote the decomposition of Vα into L(0) homogeneous spaces
where r0 is the lowest L(0) degree. We assume that dim Vα,r is finite for each r, α. We define the Supertrace of an operator A by: STr(Aq L(0) ) = T r (σ Aq L(0) ) = T r V0¯ (Aq L(0) ) − T r V1¯ (Aq L(0) ) q r [T r V0,r = ¯ (A) − T r V1,r ¯ (A)]. r ≥r0
Clearly the supertrace is zero if A has odd parity. We then note the following: Lemma 8. Suppose that A is an operator on V of parity p(A) such that A : Vα,r → Vα+ p(A),r +s for some real s. Then for any operator B we have: STr(ABq L(0) ) = q s p(A, B) STr(B Aq L(0) ). Using (60) we find Corollary 5. For v homogeneous of weight wt (v) then STr(v(k) B q L(0) ) = p(v, B)q wt (v)−k−1 STr(Bv(k)q L(0) ).
(148)
We also have Corollary 6. If A : Vα,r → Vα+ p(A),r then for any operator B we have STr([A, B]q L(0) ) = 0.
(149)
References [B] [DLM1] [DLM2]
[DM] [DZ1] [DZ2] [EO] [Fa] [FFR] [FHL] [FK]
Borcherds, R.: Vertex algebras, kac-moody algebras and the monster. Proc. Natl. Acad. Sci. U.S.A. 83, 3068–3071 (1986) Dong, C., Li, H., Mason, G.: Modular-invariance of trace functions in orbifold theory and generalized moonshine. Commun. Math. Phys. 214, 1–56 (2000) Dong, C., Lin, Z., Mason, G.: On Vertex operator algebras as sl2 -modules. Arasu, K. T. (ed.) et al., In: Groups, difference sets, and the Monster. Proceedings of a special research quarter, Columbus, OH, USA, Spring 1993. Ohio State Univ. Math. Res. Inst. Publ. 4, Berlin: Walter de Gruyter, pp. 349–362 (1996) Dong, C., Mason, G.: Shifted vertex operator algebras. Math. Proc. Cambridge Philos. Soc. 141, 67–80 (2006) Dong, C., Zhao, Z.: Modularity in orbifold theory for vertex operator superalgebras. Commun. Math. Phys. 260, 227–256 (2005) Dong, C., Zhao, Z.: Modularity of trace functions in orbifold theory for Z-graded vertex operator superalgebras. http://arxiv.org/list/math.QA/0601571, 2006 Eguchi, T., Ooguri, H.: Chiral bosonization on a Riemann surface. Phys. Lett. B187, 127–134 (1987) Fay, J.D.: Theta functions on Riemann surfaces. Lecture Notes on Mathematics 352, New York:Springer-Verlag, 1973 Feingold, A.J., Frenkel, I.B., Reis, J.F.X.: Spinor construction of vertex operator algebras and (1) E 8 . Contemp. Math. 121, Providence, RI: Amer. Math. Soc., 1991 Frenkel, I., Huang, Y-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104(494), Providence, RI: Amer. Math. Soc., 1993 Farkas, H., Kra, I.M.: Theta constants, Riemann surfaces and the modular group. Graduate Studies in Mathematics, 37. Providence, RI: Ameri. Math. Soc., 2001
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[FLM] [FMS] [Ka] [KZ] [La] [Li] [MN1] [MN2] [MT1] [MT2] [MT3] [Mu] [P] [R1] [R2] [RS] [Se] [T] [TUY] [TZ] [Z]
G. Mason, M. P. Tuite, A. Zuevsky
Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. New York: Academic Press, 1988 Di Francesco, Ph., Mathieu, P., Sénéchal, D.: Conformal field theory. Graduate Texts in Contemporary Physics. New York: Springer-Verlag, 1997 Kac, V.: Vertex Operator Algebras for Beginners. University Lecture Series, Vol. 10, Providence, RI: Ameri. Math. Soc., 1998 Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms, The Moduli Space of Curves (Texel Island, 1994), Progr. in Math. 129, Boston: Birkhauser, 1995 Lang, S.: Elliptic functions. With an appendix by J. Tate. Second edition. Graduate Texts in Mathematics, 112. New York: Springer-Verlag, 1987 Li, H.: Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, Contemp. Math. AMS. 193, Providence, RI: Ameri. Math. Soc., 1996, pp. 203–236 Matsuo, A., Nagatomo, K.: Axioms for a Vertex Algebra and the locality of quantum fields. Math. Soc. Japan Memoirs 4, Tokyo: Math. Soc. of Japan, 1999 Matsuo, A., Nagatomo, K.: A note on free bosonic vertex algebras and its conformal vector. J. Alg. 212, 395–418 (1999) Mason, G., Tuite, M.P.: Torus chiral n-point functions for free boson and lattice vertex operator algebras. Commun. Math. Phys. 235, 47–68 (2003) Mason, G., Tuite, M.P.: On genus two Riemann surfaces formed from sewn tori. Commun. Math. Phys. 270, 587–648 (2007) Mason, G., Tuite, M.P.: The genus two partition function for free bosonic and lattice vertex operator algebras. http://arxiv.org/list/math.QA/07120628, 2007 Mumford, D.: Tata lectures on Theta I., Boston:Birkhäuser, 1983 Polchinski, J.: String Theory, Volumes I and II. Cambridge:Cambridge University Press, 1998 Raina A., K.: Fay’s trisecant identity and conformal field theory. Commun. Math. Phys. 122, 625–641 (1989) Raina, A.K.: An algebraic geometry study of the b-c system with arbitrary twist fields and arbitrary statistics. Commun. Math. Phys. 140, 373–397 (1991) Sen, S., Raina, A.K.: Grassmannians, multiplicative ward identities and theta function identities. Phys. Lett. B 203(3), 256–262 (1988) Serre, J-P.: A course in arithmetic. Berlin: Springer-Verlag, 1978 Tuite, M.P.: Genus two meromorphic conformal field theory. CRM Proceedings and Lecture Notes 30, 231–251 (2001) Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure. Math. 19, 459–566 (1989) Tuite, M.P., Zuevsky, A.: Shifting, twisting and intertwining Heisenberg modules. To appear Zhu, Y: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 283, 343–395 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0535-0
Communications in
Mathematical Physics
Superbosonization of Invariant Random Matrix Ensembles P. Littelmann1 , H.-J. Sommers2 , M. R. Zirnbauer3 1 Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany.
E-mail:
[email protected]
2 Fachbereich Physik, Universität Duisburg Essen, D-47048 Duisburg, Germany.
E-mail:
[email protected]
3 Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str.77, D-50937 Köln, Germany.
E-mail:
[email protected] Received: 8 August 2007 / Accepted: 8 November 2007 Published online: 24 June 2008 – © Springer-Verlag 2008
Abstract: ‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group Un , On , or USpn , giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner’s n-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type. 1. Introduction and Overview The past 25 years have seen substantial progress in the physical understanding of insulating and metallic behavior in disordered quantum Hamiltonian systems of the random Schrödinger and random band matrix type. A major role in this development, bearing especially on the metallic regime and the metal-insulator transition, has been played by the method of commuting and anti-commuting variables, or supersymmetry method for short. Assuming a Gaussian distribution for the disorder, this method proceeds by making a Hubbard-Stratonovich transformation followed by a saddle-point approximation (or elimination of the massive modes) to arrive at an effective field theory of the nonlinear sigma model type. This effective description has yielded many new results including, e.g., the level statistics in small metallic grains, localization in thick disordered wires, and a scaling theory of critical systems in higher dimension [6]. While the method has been widely used and successfully so, there exist some limitations and drawbacks. For one thing, the method works well only for systems with normaldistributed disorder; consequently, addressing the universality question for non-Gaussian
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P. Littelmann, H.-J. Sommers, M. R. Zirnbauer
distributions (like the invariant ensembles studied via the orthogonal polynomial method) has so far been beyond reach. For another, the symmetries of the effective theory are not easy to keep manifest when using the mathematically well-founded approach of Schäfer and Wegner [17]. A problem of lesser practical relevance is that the covariance matrix of the random variables, which is to be inverted by the Hubbard-Stratonovich transformation, does not always have an inverse [17]. In this paper we introduce a new variant of the supersymmetry approach which is complementary to the traditional one. Inspired by the method of bosonization of Dirac fermions, and following an appellation by Efetov and coworkers, we refer to the new method as ‘superbosonization’. As we will see, in order for superbosonization to be useful the distribution of the random Hamiltonians must be invariant under some symmetry group, and this group cannot be ‘too small’ in a certain sense. We expect the method to be at its best for random matrix ensembles with a local gauge symmetry such as Wegner’s n-orbital model [20] with gauge group K = Un , On , or USpn . Superbosonization differs in several ways from the traditional method of Schäfer and Wegner [17] based on the Hubbard-Stratonovich (HS) transformation: (i) the superbosonization field has the physical dimension of 1/energy (whereas the HS field has the dimension of energy). (ii) For a fixed symmetry group K , the target space of the superbosonization field is always the same product of compact and non-compact symmetric spaces regardless of where the energy parameters are, whereas the HS field of the Schäfer-Wegner method changes as the energy parameters move across the real axis. (iii) The method is not restricted to Gaussian disorder distributions. (iv) The symmetries of the effective theory are manifest at all stages of the calculation. A brief characterization of what is meant by the physics word of ‘superbosonization’ is as follows. The object of departure of the supersymmetry method (in its old variant as well as the new one) is the Fourier transform of the probability measure of the given ensemble of disordered Hamiltonians. This Fourier transform is evaluated on a supermatrix built from commuting and anti-commuting variables and thus becomes a superfunction; more precisely a function, say f , which is defined on a complex vector space V0 and takes values in the exterior algebra ∧(V1∗ ) of another complex vector space V1 . If the probability measure is invariant under a group K , so that the function f is equivariant with respect to K acting on V0 and V1 , then a standard result from invariant theory tells us that f can be viewed as the pullback of a superfunction F defined on the quotient of V = V0 ⊕ V1 by the group K . The heart of the superbosonization method is a formula which reduces the integral of f to an integral of the lifted function F. Depending on how the dimension of V compares with the rank of K , such a reduction step may or may not be useful for further analysis of the integral. Roughly speaking, superbosonization gets better with increasing value of rank(K ). From a mathematical perspective, superbosonization certainly promises to become a powerful tool for the investigation and proof of the universality of spectral correlations for a whole class of random matrix ensembles that are not amenable to treatment by existing techniques. Let us now outline the plan of the paper. In Sects. 1.1 and 1.2 we give an informal introduction to our results, which should be accessible to physicists as well as mathematicians. A concise summary of the motivation (driven by random matrix applications) for the mathematical setting of this paper is given in Sect. 2. In Sect. 3 we present a detailed treatment of the special situation of V1 ≡ 0 (the so-called Boson-Boson sector), where anti-commuting variables are absent. This case was treated in an inspiring paper by Fyodorov [8], and our final formula – the transfer of the integral of f to an integral of F – coincides with his. The details of the
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derivation, however, are different. While Fyodorov employs something he calls the Ingham-Siegel integral, our approach proceeds directly by push forward to the quotient V0 //K C . Another difference is that our treatment covers each of the three classical symmetry groups K = Un , On , and USpn , not just the first two. Section 4 handles the complementary situation V0 = 0 (the so-called FermionFermion sector). In this case the starting point is the Berezin integral of f ∈ ∧(V1∗ ), i.e., one differentiates once with respect to each of the anti-commuting variables, or projects on the top-degree component of ∧(V1∗ ). From a theoretical physicist’s perspective, this case is perhaps the most striking one, as it calls for the mysterious step of transforming the Berezin integral of f to an integral of the lifted function F over a compact symmetric space. The conceptual difficulty here is that many choices of F exist, and any serious theoretical discussion of the matter has to be augmented by a proof that the final answer does not depend on the specific choice which is made. Finally, Sect. 5 handles the full situation where V0 = 0 and V1 = 0 (i.e., both types of variable, commuting and anti-commuting, are present). Heuristic ideas as to how one might tackle this situation are originally due to Lehmann, Saher, Sokolov and Sommers [15] and to Hackenbroich and Weidenmüller [10]. These ideas have recently been pursued by Efetov and his group [7] and by Guhr [9], but their papers are short of mathematical detail – in particular, the domain of integration after the superbosonization step is left unspecified – and address only the case of unitary symmetry. In Sect. 5 we supply the details missing from these earlier works and prove the superbosonization formula for the cases of K = Un , On , and USpn , giving sufficient conditions of validity in each case. While it should certainly be possible to construct a proof based solely on supersymmetry and invariant-theoretic notions including Howe dual pairs, Lie superalgebra symmetries and the existence of an invariant Berezin measure, our approach here is different and more constructive: we use a chain of variable transformations reducing the general case to the cases dealt with in Sects. 3 and 4. 1.1. Basic setting. Motivated by the method of commuting and anti-commuting variables as reviewed in Sect. 2, let there be a set of complex variables Z ci with complex conjugates Z˜ ic := Z ci , where indices are in the range i = 1, . . . , n and c = 1, . . . , p. Let there also be two sets of anti-commuting variables ζei and ζ˜ie with index range i = 1, . . . , n and e = 1, . . . , q. (Borrowing the language from the physics context where the method is to be applied, one calls n the number of orbitals and p and q the number of bosonic resp. fermionic replicas.) It is convenient and useful to arrange the variables Z ci , ζei in the form of rectangular matrices Z , ζ with n rows and p resp. q columns. A similar arrangement as rectangular matrices is made for the variables Z˜ ic , ζ˜ie , but now with p resp. q rows and n columns. We are going to consider integrals over these variables in the sense of Berezin [1]. Let D Z , Z¯ ;ζ,ζ˜ := 2 pn
p n c=1 i=1
|dRe(Z ci ) dIm(Z ci )| ⊗ (2π )−qn
q n
∂2
e=1 j=1
∂ζe ∂ ζ˜ je j
, (1.1)
2 where the derivatives are left derivatives, i.e., we use the sign convention ∂ ˜ ζ˜ ζ = 1, ∂ζ ∂ ζ and the product of derivatives projects on the component of maximum degree in the anticommuting variables. The other factor is Lebesgue measure for the commuting complex
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variables Z . We here denote such integrals by f ≡ D Z , Z¯ ;ζ,ζ˜ f (Z , Z˜ ; ζ, ζ˜ ) Matn, p (C)
(1.2)
for short. The domain of integration will be the linear space of all complex rectangular n × p matrices Z , with Z˜ = Z † ∈ Mat p, n (C) being the Hermitian adjoint of Z . We assume that our integrands f decrease at infinity so fast that the integral f exists. In the present paper we will be discussing such integrals for the particular case where the integrand f has a Lie group symmetry. More precisely, we assume that a Lie group K is acting on Cn and this group is either the unitary group Un , or the real orthogonal group On , or the unitary symplectic group USpn . The fundamental K -action on Cn gives rise to a natural action by multiplication on the left resp. right of our rectangular matrices: Z → g Z , ζ → gζ and Z˜ → Z˜ g −1 , ζ˜ → ζ˜ g −1 (where g ∈ K ). The functions f to be integrated shall have the property of being K -invariant: f (Z , Z˜ ; ζ, ζ˜ ) = f (g Z , Z˜ g −1 ; gζ, ζ˜ g −1 ) (g ∈ K ). (1.3) We wish to establish a reduction formula for the Berezin integral f of such functions. This formula will take a form that varies slightly between the three cases of K = Un , K = On , and K = USpn . 1.1.1. The case of Un -symmetry. Let then f be an analytic and Un -invariant function of our basic variables Z , Z˜ , ζ, ζ˜ for Z˜ = Z † . We now make the further assumption that f extends to a GLn (C)-invariant holomorphic function when Z and Z˜ are viewed as independent complex matrices; which means that the power series for f in terms of Z and Z˜ converge everywhere and that the symmetry relation (1.3) for the extended function f holds for all g ∈ GLn (C), the complexification of Un . The rationale behind these assumptions about f is that they guarantee the existence of another function F which lies ‘over’ f in the following sense. It is a result of classical invariant theory [11] that the algebra of GLn (C)-invariant polynomial functions in Z , Z˜ , ζ , ζ˜ is generated by the quadratic invariants
( Z˜ Z )cc ≡ Z˜ ic Z ci , ( Z˜ ζ )ce ≡ Z˜ ic ζei , (ζ˜ Z )ec ≡ ζ˜ie Z ci , (ζ˜ ζ )ee ≡ ζ˜ie ζei .
Here we are introducing the summation convention: an index that appears twice, once as a subscript and once as a superscript, is understood to be summed over. How does this invariant-theoretic fact bear on our situation? To answer that, let F be
a holomorphic function of complex variables xcc , yee and anti-commuting variables σec ,
τce with index range c, c = 1, . . . , p and e, e = 1, . . . , q. Again, let us organize these
variables in the form of matrices, x = (xcc ), y = (yee ), etc., and write F as xσ c
e
c
e
. F(xc , ye ; σe , τc ) ≡ F τ y Then the relevant statement from classical invariant theory in conjunction with [16] is this: given any GLn (C)-invariant holomorphic function f of the variables Z , Z˜ , ζ, ζ˜ , it is possible to find a holomorphic function F of the variables x, y, σ, τ so that Z˜ Z Z˜ ζ F ˜ ˜ = f (Z , Z˜ ; ζ, ζ˜ ). (1.4) ζ Z ζζ
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To be sure, there exists no unique choice of such function F. Indeed, since the top degree
of the Grassmann algebra generated by the anti-commuting variables ζei and ζ˜ie is 2qn, any monomial in the matrix variables y of degree higher than qn vanishes identically
upon making the substitution yee = ζ˜ie ζei . In the following, we will use the abbreviated notation F = F(Q), where the symbol Q stands for the supermatrix built from the matrices x, σ, τ, y : xσ . (1.5) Q= τ y 1.1.2. Orthogonal and symplectic symmetry. In the case of the symmetry group being K = On the complex vector space Cn is equipped with a non-degenerate symmetric tensor δi j = δ j i (which you may think of as the Kronecker delta symbol). By definition, the elements k of the orthogonal group On satisfy the conditions k −1 = k † and k t δk = δ, where k t means the transpose of the matrix k. Let δ i j denote the components of the inverse tensor, δ −1 . In addition to Z˜ Z , Z˜ ζ , ζ˜ Z , ζ˜ ζ we now have the following independent quadratic K -invariants:
j (Z t δ Z )c c = Z ci δi j Z c , ( Z˜ δ −1 Z˜ t )c c = Z˜ ic δ i j Z˜ cj ,
(Z t δζ )c e = Z ci δi j ζe , (ζ˜ δ −1 Z˜ t )e c = ζ˜ie δ i j Z˜ cj , j
j (ζ t δζ )e e = ζei δi j ζe , (ζ˜ δ −1 ζ˜ t )e e = ζ˜ie δ i j ζ˜ je .
In the case of symplectic symmetry, the dimension n has to be an even number and Cn is equipped with a non-degenerate skew-symmetric tensor εi j = −ε j i . Elements k of the unitary symplectic group USpn satisfy the conditions k −1 = k † and k t εk = ε. If εi j = −ε j i are the components of ε−1 , the extra quadratic invariants for this case are
j (Z t ε Z )c c = Z ci εi j Z c , ( Z˜ ε−1 Z˜ t )c c = Z˜ ic εi j Z˜ cj , etc.
To deal with the two cases of orthogonal and symplectic symmetry in parallel, we introduce the notation β := δ for K = On and β := ε for K = USpn , and we organize all quadratic invariants as a supermatrix: ⎛
Z˜ Z ⎜ Z tβ Z ⎜ ⎝ ζ˜ Z −ζ t β Z
Z˜ β −1 Z˜ t Z˜ ζ Z t Z˜ t Z t βζ ˜ζ β −1 Z˜ t ζ˜ ζ −ζ t Z˜ t −ζ t βζ
⎞ Z˜ β −1 ζ˜ t Z t ζ˜ t ⎟ ⎟ ˜ζ β −1 ζ˜ t ⎠. −ζ t ζ˜ t
(1.6)
This particular matrix arrangement is motivated as follows. Let Q be the supermatrix (1.5) of before, but now double the size of each block; thus x here is a matrix of size 2 p × 2 p, the rectangular matrix σ is of size 2 p × 2q, and so on. Then impose on Q the symmetry relation Q = Tβ Q st (Tβ )−1 , where ⎛
⎛ ⎞ ⎞ 0 1p 0 −1 p ⎜1 0 ⎜1 0 ⎟ ⎟ , Tε = ⎝ p , Tδ = ⎝ p 0 −1q ⎠ 0 1q ⎠ 1q 0 1q 0
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P. Littelmann, H.-J. Sommers, M. R. Zirnbauer
and Q st means the supertranspose:
xt τ t . −σ t y t
Q st =
It is easy to check that the supermatrix (1.6) obeys precisely this relation QTβ = Tβ Q st . For the symmetry groups K = On and K = USpn – with the complexified groups being G = On (C) and G = Spn (C) – it is still true that the algebra of G-invariant holomorphic functions f of Z , Z˜ , ζ, ζ˜ is generated by the invariants that arise at the quadratic level. Thus, if f is any function of such kind, then there exists (though not uniquely so) a holomorphic function F(Q) which pulls back to the given function f : ⎞ ⎛ Z˜ Z Z˜ β −1 Z˜ t Z˜ ζ Z˜ β −1 ζ˜ t ⎜ Z t β Z Z t Z˜ t Z t βζ Z t ζ˜ t ⎟ ⎟ ˜ ˜ (1.7) F⎜ ⎝ ζ˜ Z ζ˜ β −1 Z˜ t ζ˜ ζ ζ˜ β −1 ζ˜ t ⎠ = f (Z , Z ; ζ, ζ ). −ζ t β Z −ζ t Z˜ t −ζ t βζ −ζ t ζ˜ t 1.2. Superbosonization formula. A few more definitions are needed to state our main result, which transfers the integral of f to an integral of F. In (1.2) the definition of the Berezin integral f was given. Let us now specify how we integrate the ‘lifted’ function F, beginning with the case of K = Un . There, the domain of integration will be D = D 0p × Dq1 , where D 0p is the symmetric space of positive Hermitian p × p matrices and Dq1 is the group of unitary q × q matrices, Dq1 = Uq . The Berezin superintegral form to be used for F(Q) is D Q := dµ D 0p (x) dµ Dq1 (y) (2π )− pq W1 ◦ Detq (x − σ y −1 τ ) Det p (y − τ x −1 σ ), (1.8) where the meaning of the various symbols is as follows. The Berezin form W1 is defined as the product of all derivatives with respect to the anti-commuting variables: W1 =
q p c=1 e=1
∂2 . ∂σec ∂τce
(1.9)
The symbol dµ Dq1 denotes a suitably normalized Haar measure on Dq1 = Uq and dµ D 0p means a positive measure on D 0p which is invariant with respect to the transformation X → g Xg † for all invertible complex p × p matrices g ∈ GL p (C). Our precise normalization conventions for these measures are defined by the Gaussian limits 2 2 p2 q2 t/π e−t Tr (x−Id) dµ D 0p (x) = 1 = lim t/π et Tr (y−Id) dµ Dq1 (y). lim t→+∞
D 0p
t→+∞
Dq1
Now assume that p ≤ n. Then we assert that the superbosonization formula vol(Un ) f = D Q SDet n (Q) F(Q) (1.10) vol(Un− p+q ) D holds for a large class of analytic functions with suitable falloff behavior at infinity. (In the body of the paper we state and prove this formula for the class of Schwartz functions, i.e., functions that decrease faster than any power. This, however, is not yet the
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optimal formulation, and we expect the formula (1.10) to hold in greater generality.) Here vol(Un ) := dµ Dn1 (y) is the volume of the unitary group, the integrands f and F are assumed to be related by (1.4), and SDet is the superdeterminant function, Det(x) xσ . SDet = τ y Det(y − τ x −1 σ ) It should be mentioned at this point that ideas toward the existence of such a formula as (1.10) have been vented in the recent literature [7,9]. These publications, however, do not give an answer to the important question of which integration domain to choose for Q. Noting that the work of Efetov et al. is concerned with the case of n = 1 and p = q 1, let us emphasize that the inequality p ≤ n is in fact necessary in order for our formula (1.10) to be true. (The situation for p > n is explored in a companion paper [3].) Moreover, be advised that analogous formulas for the related cases of K = On , USpn have not been discussed at all in the published literature. Turning to the latter two cases, we introduce two 2r × 2r matrices ts and ta : 0 1r 0 −1r , ta = , ts = 1r 0 1r 0 where r = p or r = q depending on the context. Then let a linear space Symb (C2r ) for b := s or b := a be defined by
Symb (C2r ) := M ∈ Mat 2r,2r (C) | M = tb M t (tb )−1 . Thus the elements of Symb (C2r ) are complex 2r × 2r matrices which are symmetric with respect to transposition followed by conjugation with tb . With this notation, we can rephrase the condition Q = Tβ Q st (Tβ )−1 for the blocks x resp. y as saying they are in Syms (C2 p ) resp. Syma (C2q ) for β = δ and in Syma (C2 p ) resp. Syms (C2q ) for β = ε. 0 × D 1 , where The domain of integration for Q will now be Dβ := Dβ, p β, q 0 + 2p Dδ, p = Herm ∩ Sym s (C ),
1 2q Dδ, q = U ∩ Sym a (C ),
in the case of β = δ (or K = On ), and 0 + 2p Dε, p = Herm ∩ Sym a (C ),
1 2q Dε, q = U ∩ Sym s (C ),
in the case of β = ε (or K = USpn ). Thus in both cases, β = δ and β = ε, the 0 and D 1 integration domains Dβ, p β, q are constructed by taking the intersection with the positive Hermitian matrices and the unitary matrices, respectively. The Berezin superintegral form D Q for the cases β = δ, ε has the expression D Q := dµ D 0 (x) dµ D 1 (y) W1 ◦ β, p
β, q
Detq (x − σ y −1 τ ) Det p (y − τ x −1 σ ) 1
(2π )2 pq Det 2 m β (1 − x −1 σ y −1 τ )
, (1.11)
where m δ = 1 and m ε = −1. The Berezin form W1 now is simply a product of derivatives w.r.t. all of the anti-commuting variables in the matrix σ . (The entries of τ are determined from those of σ by the relation Q = Tβ Q st (Tβ )−1 .) For β = δ one defines W1 =
p q
∂2
c=1 e=1
∂σec ∂σe+q
c+ p
⊗
p q
∂2 c+ p
c=1 e=1
∂σe
c ∂σe+q
,
(1.12)
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P. Littelmann, H.-J. Sommers, M. R. Zirnbauer
while for β = ε the definition is the same except that the ordering of the derivatives c+ p c in the second product has to be reversed. ∂/∂σe and ∂/∂σe+q It remains to define the measures dµ D 0 and dµ D 1 . To do so, we first observe that β, p
β,q
the complex group GL2 p (C) acts on Symb (C2 p ) by conjugation in a twisted sense: x → gxτb (g −1 ), τb (g −1 ) = tb g t (tb )−1 (b = s, a). A derived group action on the restriction to the positive Hermitian matrices is then obtained by restricting to the subgroup G ⊂ GL2 p (C) defined by the condition τb (g −1 ) = g † . This subgroup G turns out to be G GL2 p (R) for b = s and G GL p (H), the invertible p × p matrices whose entries are real quaternions, for b = a. In the sec1 2q tor of y, the unitary group U2q acts on Dβ, q = U ∩ Sym b (C ) by the same twisted conjugation, y → gyτb (g −1 ) (b = a, s). 0 and D 1 which are invariant Now in all cases, dµ D 0 and dµ D 1 are measures on Dβ, p β, q β, p β, q by the pertinent group action. Since the group actions at hand are transitive, all of our invariant measures are unique up to multiplication by a constant. As before, we consider a Gaussian limit in order to fix the normalization constant: t 2 p(2 p+m β ) lim t/π e− 2 Tr (x−Id) dµ D 0 (x) = 1. t→+∞
0 Dβ, p
β, p
The normalization of dµ D 1 is specified by the corresponding formula where we make β, q
the replacements p → q, and m β → −m β , and −Tr (x − Id)2 → +Tr (y − Id)2 . An explicit expression for each of these invariant measures is given in the Appendix. We are now ready to state the superbosonization formula for the cases of orthogonal and symplectic symmetry. Let the inequality of dimensions n ≥ 2 p be satisfied. We then assert that the following is true. Let the Berezin integral f still be defined by (1.2), but now assume the holomorphically extended integrand f to be G-invariant with complexified symmetry group G = On (C) for β = δ and G = Spn (C) for β = ε. Let K n = On in the former case and K n = USpn in the latter case. Then, choosing any holomorphic function F(Q) related to the given function f by (1.7), the integration formula vol(K n ) f = 2(q− p)m β D Q SDet n/2 (Q) F(Q) (1.13) vol(K n−2 p+2q ) Dβ holds true, provided that f falls off sufficiently fast at infinity. Thus the superbosonization formula takes the same form as in the previous case K = Un , except that the exponent n now is reduced to n/2. The latter goes hand in hand with the size of the supermatrix Q having been expanded by p → 2 p and q → 2q. Another remark is that the square root of the superdeterminant, n SDet n/2 (Q) = Det n (x ) Det (y − τ x −1 σ ),
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Table 1. Isomorphisms between integration domains and symmetric spaces K
D 0p
D˜ 0p
Dq1
D˜ q1
Un On USpn
Herm+ ∩ Mat p, p (C) Herm+ ∩ Syms (C2 p ) Herm+ ∩ Syma (C2 p )
GL p (C)/U p GL2 p (R)/O2 p GL p (H)/USp2 p
U ∩ Matq, q (C) U ∩ Syma (C2q ) U ∩ Syms (C2q )
Uq U2q /USp2q U2q /O2q
is always analytic in the sector of the matrix y. For the case of orthogonal symmetry this 1 = U∩Sym (C2q ) is isomorphic to the unitary skew-symmetric 2q ×2q is because Dδ, a q matrices and for such matrices the determinant has an analytic square root known as the 1 is the domain of definition Pfaffian. (In the language of random matrix physics, Dδ, q of the Circular Symplectic Ensemble, which has the feature of Kramers degeneracy.) In the case of symplectic symmetry, where the number n is always even, no square root is being taken in the first place. As another remark, let us mention that each of our integration domains is isomorphic to a symmetric space of compact or non-compact type. These isomorphisms Dq1 D˜ q1 and D 0p D˜ 0p are listed in Table 1. Detailed explanations are given in the main text. Let us also mention that the expressions (1.8) and (1.11) for the Berezin integration forms D Q can be found from a supersymmetry principle: each D Q is associated with one of three Riemannian symmetric superspaces in the sense of [22] (to be precise, these are the supersymmetric non-linear sigma model spaces associated with the random matrix symmetry classes AIII, B DI, and CII) and is in fact the Berezin integration form which is invariant w.r.t. the action of the appropriate Lie superalgebra gl or osp. We will make no use of this symmetry principle in the present paper. Instead, we will give a direct proof of the superbosonization formulas (1.10) and (1.13), deriving the expressions (1.8) and (1.11) by construction, not from a supersymmetry argument. Finally, we wish to stress that in random matrix applications, where n typically is a large number, the reduction brought about by the superbosonization formulas (1.10) and (1.13) is a striking advance: by conversion from its original role as the number of integrations to do, the big integer n has been turned into an exponent, whereby asymptotic analysis of the integral by saddle-point methods becomes possible. 1.3. Illustration. To finish this introduction, let us illustrate the new method at the example of Wegner’s n-orbital model with n orbitals per site and unitary symmetry. The Hilbert space V of that model is an orthogonal sum, V = ⊕i∈ Vi , where i labels the sites (or vertices) of a lattice and the Vi ∼ = Cn are Hermitian vector spaces of dimension n. The Hamiltonians of Wegner’s model are random Hermitian operators H : V → V distributed according to a Gaussian measure dµ(H ). To specify the latter, let i : V → Vi be the orthogonal projector on Vi . The probability measure of Wegner’s model is then given as a Gaussian distribution dµ(H ) with Fourier transform 1 e−i Tr (H K ) dµ(H ) = e− 2n i j Ci j Tr(K i K j ) , where K ∈ End(V ), and the variances Ci j = C j i are non-negative real numbers. We −1 observe that dµ(H ) is invariant under conjugation H → g H g by unitary transformations g ∈ i∈ U(Vi ); such an invariance is called a local gauge symmetry in physics.
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P. Littelmann, H.-J. Sommers, M. R. Zirnbauer
Let us now be interested in, say, the average ratio of characteristic polynomials: Det(E 1 − H ) dµ(H ) (Im E 0 > 0). R(E 0 , E 1 ) := Det(E 0 − H ) To compute R(E 0 , E 1 ) one traditionally uses a supersymmetry method involving the so-called Hubbard-Stratonovich transformation. In order for this approach to work, one needs to assume that the positive quadratic form with matrix coefficients Ci j has an inverse. If it does, then the traditional approach leads to the following result [4]: (yk − E 1 )n−1 dyk d xk n −1 R(E 0 , E 1 ) = e− 2 i j (C )i j (xi x j −yi y j ) DHS (x, y) , (xk − E 0 )n+1 2π/i k∈
where the integral is over xk ∈ R and yk ∈ iR. The factor DHS (x, y) is a fermion determinant resulting from integration over the anti-commuting components of the HubbardStratonovich field; it is the determinant of the matrix with elements n δi j − (C −1 )i j (xi − E 0 )(y j − E 1 ) . Notice that the integration variables xk and yk carry the physical dimension of energy. In contrast, using the new approach opened up by the superbosonization formula of the present paper, we obtain xk eiE 0 xk n d xk dyk − n2 i j Ci j (xi x j −yi y j ) R(E 0 , E 1 ) = e DSB (x, y) . yk eiE 1 yk 2π i xk yk k∈
Here the integral is over xk ∈ R+ and yk ∈ U1 (the unit circle in C). These integration variables have the physical dimension of (energy)−1 . The factor DSB (x, y) still is a fermion determinant, which now arises from integration over the anti-commuting variables of the superbosonization field; it is the determinant of the matrix with elements n δi j + Ci j xi y j . When both methods (Hubbard-Stratonovich and superbosonization) are applicable, our two formulas for R(E 0 , E 1 ) are exactly equivalent to each other. Please be warned, however, that this equivalence is by no means easy to see directly. From a practical viewpoint, the main difference between the two formulas is that one of them is expressed by the quadratic form of variances Ci j whereas the other one is expressed by the inverse of that quadratic form. A rigorous analysis based on the formula from Hubbard-Stratonovich transformation (or, rather, the resulting formula for the density of states) for the case of long-range Ci j and n = 1, was made in [4]. A similar analysis based on the formula from superbosonization has not yet been done. 2. Motivation: Supersymmetry Method Imagine some quantum mechanical setting where the Hilbert space is Cn equipped with its standard Hermitian structure. On that finite-dimensional space, let us consider Hermitian operators H that are drawn at random from a probability distribution or ensemble dµ(H ). We might wish to compute the spectral correlation functions of the ensemble or some other observable quantity of interest in random matrix physics.
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One approach to this problem is by the so-called supersymmetry method [6,21]. In that method the observables one wishes to compute are written in terms of Green’s functions, i.e., matrix coefficients of the resolvent operator of H , which are then expressed as Gaussian integrals over commuting and anti-commuting variables. The key object of the method is the characteristic function of dµ(H ), F (K ) = e−i Tr (H K ) dµ(H ), where the exact meaning of the Fourier variable K depends on what observable is to be computed. In the general case (with p bosonic and q fermionic ‘replicas’), defining the matrix entries of K with respect to the standard basis {e1 , . . . , en } of Cn by K e j = ei K ij , one lets K ij := Z ci Z˜ cj + ζei ζ˜ je , where Z ci , Z˜ cj and ζei , ζ˜ je are the commuting and anti-commuting variables of Sect. 1 and the summation convention is still in force. To compute, say, the spectral correlation functions of dµ(H ), one multiplies F (K ) by the exponential function
j exp iZ ci E cc Z˜ ic + iζe Fee ζ˜ je , E cc = E c δcc , Fee = Fe δee , (2.1) where the parameters E c and Fe have the physical meaning of energies, and one integrates the product against the flat Berezin integration form D Z , Z¯ ;ζ,ζ˜ over the real vector space defined by Z˜ ic = sgn(Im E c )Z ci (for c = 1, . . . , p and i = 1, . . . , n). The desired correlation functions are then generated by a straightforward process of taking derivatives with respect to the energy parameters at coinciding points. Note that for all g ∈ GLn (C) the exponential (2.1) is invariant under j
Z ci → g ij Z c ,
j j j j Z˜ ic → Z˜ cj (g −1 )i , ζei → g ij ζe , ζ˜ei → ζ˜e (g −1 )i .
Let us now pass to a basis-free formulation of this setup. For that we are going to think of the sets of complex variables Z ci and Z˜ ic as bases of holomorphic linear functions for the complex vector spaces Hom(C p , Cn ) resp. Hom(Cn , C p ), and we interpret the anti-commuting variables ζei and ζ˜ie as generators for the exterior algebras of the vector spaces Hom(Cq , Cn )∗ resp. Hom(Cn , Cq )∗ . Let V0 := Hom(C p , Cn ) ⊕ Hom(Cn , C p ), V1 := Hom(Cq , Cn ) ⊕ Hom(Cn , Cq ). If we now choose some fixed Hermitian operator H drawn from our random matrix ensemble, the exponential e−i Tr (H K ) is seen to be a holomorphic function on V0 with values in the exterior algebra ∧(V1∗ ). Under mild assumptions on dµ(H ) (e.g., bounded support, or rapid decay at infinity) the holomorphic property carries over to the integral exp(−iTr H K ) dµ(H ). The characteristic function F (K ) in that case is a holomorphic function F (K ) : V0 → ∧(V1∗ ), and so is the function resulting from F (K ) by multiplication with the Gaussian factor (2.1). We denote this product of functions by f for short.
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Combining V0 and V1 to a Z2 -graded vector space V := V0 ⊕ V1 , we denote the graded-commutative algebra of holomorphic functions V0 → ∧(V1∗ ) by AV . The main task in the supersymmetry method is to compute the Berezin superintegral of f ∈ AV . This task is rather difficult to carry out for functions f corresponding to a general probability measure dµ(H ). Let us therefore imagine that f has some symmetries. Thus, let a group G ⊂ GL(Cn ) be acting on Cn , and let ˜ = L g −1 ⊕ g L˜ (g ∈ G), g.(L ⊕ L) for L ∈ Hom(Cn , C p ) and L˜ ∈ Hom(C p , Cn ) be the induced action of G on V0 . We also have the same G-action on V1 , and the latter induces a G-action on ∧(V1∗ ). Now let the given probability measure dµ(H ) be invariant with respect to conjugation by the elements of (a unitary form of) such a group G. Via the Fourier transform, this symmetry gets transferred to the characteristic function F (K ), and also to the product of F (K ) with the exponential (2.1). Our function f then satisfies the relation f (v) = g. f (g −1 v) for all g ∈ G and v ∈ V0 and thus is an element of the subalgebra AVG ⊂ AV of G-equivariant holomorphic functions. Following Dyson [5] the complex symmetry groups G of prime interest in random matrix theory are G = GLn (C), On (C), and Spn (C). These are the complexifications of the compact symmetry groups Un , On , and USpn , corresponding to ensembles of Hermitian matrices with unitary symmetry, real symmetric matrices with orthogonal symmetry, and quaternion self-dual matrices with symplectic symmetry. To summarize, in the present paper we will be concerned with the algebra AVG of G-equivariant holomorphic functions f : V0 → ∧(V1∗ ), v → f (v) = g. f (g −1 v) (g ∈ G),
(2.2)
for the classical Lie groups G = GLn (C), On (C), and Spn (C), and the vector spaces V0 = Hom(Cn , C p ) ⊕ Hom(C p , Cn ), V1 = Hom(Cn , Cq ) ⊕ Hom(Cq , Cn ).
(2.3) (2.4)
Our strategy will be to lift f ∈ AVG to another algebra AW of holomorphic functions F : W0 → ∧(W1∗ ), using a surjective homomorphism AW → AVG . The thrust of the paper then is to prove a statement of reduction – the superbosonization formula – transferring the Berezin superintegral of f ∈ AVG to such an integral of F ∈ AW . The advantage of our treatment (as compared to the orthogonal polynomial method) is that it readily extends to the case of symmetry groups G ×G ×· · ·×G with direct product structure. This will make it possible in the future to treat such models as Wegner’s gauge-invariant model [20] with n orbitals per site and gauge group G. 2.1. Notation. We now fix some notation which will be used throughout the paper. If A and B are vector spaces and L : A → B is a linear mapping, we denote the canonical adjoint transformation between the dual vector spaces B ∗ and A∗ by L t : B ∗ → A∗ . We call L t the ‘transpose’ of L. A Hermitian structure , on a complex vector space A determines a complex anti-linear isometry c A : A → A∗ by v → v, ·. If both A and B carry Hermitian structure, then L : A → B has a Hermitian adjoint L † : B → A t † t ∗ ∗ † t defined by L † = c−1 A ◦ L ◦ c B . The operator (L ) : A → B is denoted by (L ) ≡ L. −1 Note L = c B ◦ L ◦ c A . Finally, if each of A and B is equipped with a non-degenerate
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pairing A × A → C and B × B → C, so that we are given complex linear isomorphisms α : A → A∗ and β : B → B ∗ , then there exists a transpose L T : B → A by β
Lt
α −1
L T : B → B ∗ → A∗ → A. To emphasize that this is really a story about linear operators rather than basis-dependent matrices, we use such notations as Hom(Cn , C p ) for the vector space of complex linear transformations from Cn to C p . In a small number of situations we will resort to the alternative notation Mat p, n (C). From here on in this article, Lie groups by default will be complex Lie groups; thus GLn ≡ GL(Cn ), On ≡ O(Cn ), and Spn ≡ Sp(Cn ), with n ∈ 2N in the last case. 3. Pushing Forward in the Boson-Boson Sector In this section, we shall address the special situation of V1 = 0, or fermion replica number q = 0. Thus we are now facing the commutative algebra AVG ≡ O(V )G of G-invariant holomorphic functions on the complex vector space V ≡ V0 = Hom(Cn , C p ) ⊕ Hom(C p , Cn ). In order to deal with this function space we will use the fact that O(V ) can be viewed as a completion of the symmetric algebra S(V ∗ ). Since the G-action on S(V ∗ ) preserves the Z-grading S(V ∗ ) = ⊕k≥0 Sk (V ∗ ) and is reductive on each symmetric power Sk (V ∗ ), one has a subalgebra Sk (V ∗ )G of G-fixed elements in Sk (V ∗ ) for all k. 3.1. G-invariants at the quadratic level. It is a known fact of classical invariant theory (see, e.g., [12]) that for each of the cases G = GLn , On , and Spn , all G-invariants in S(V ∗ ) arise at the quadratic level, i.e., S(V ∗ )G is generated by S2 (V ∗ )G . Let us therefore sharpen our understanding of these quadratic invariants S2 (V ∗ )G . 3.1.1. The case of G = GLn . All quadratic invariants are just of a single type here: they arise by composing the elements of Hom(C p , Cn ) with those of Hom(Cn , C p ). Lemma 3.1. S2 (V ∗ )G is isomorphic as a complex vector space to W ∗ = End(C p )∗ . Proof. Using the canonical transpose Hom(A, B) Hom(B ∗ , A∗ ) we have V Hom(Cn , C p ) ⊕ Hom((Cn )∗ , (C p )∗ ). For G = GLn there exists no non-zero G-invariant tensor in Cn ⊗ Cn or (Cn )∗ ⊗ (Cn )∗ . Therefore S2 (Hom(Cn , C p ))G = 0 and S2 (Hom((Cn )∗ , (C p )∗ ))G = 0, resulting in G S2 (V )G Hom(Cn , C p ) ⊗ Hom((Cn )∗ , (C p )∗ ) .
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The space of G-invariants in (Cn )∗ ⊗ Cn is one-dimensional (with generator ϕ i ⊗ ei given by the canonical pairing between a vector space and its dual). Since the action of G on C p is trivial, it follows that G S2 (V )G C p ⊗ (Cn )∗ ⊗ Cn ⊗ (C p )∗ C p ⊗ (C p )∗ End(C p ) ≡ W. The action of G = GLn on S2 (V ) and S2 (V ∗ ) is reductive. Therefore there exists a canonical pairing S2 (V )G ⊗ S2 (V ∗ )G → C and the isomorphism S2 (V )G → W dualizes to an isomorphism W ∗ → (S2 (V )G )∗ S2 (V ∗ )G . 3.1.2. The cases of G = On , Spn . Here Cn is equipped with a G-invariant non-degenerate bilinear form or, equivalently, with a G-equivariant isomorphism β : Cn → (Cn )∗ , which is symmetric for G = On and alternating for G = Spn . To distinguish between these two, we sometimes write β = δ in the former case and β = ε in the latter case. To describe S2 (V ∗ )G for both cases, we introduce the following notation. On U := C p ⊕ (C p )∗ we have two canonical bilinear forms: the symmetric form s(v ⊕ ϕ, v ⊕ ϕ ) = ϕ (v) + ϕ(v ), and the alternating form a(v ⊕ ϕ, v ⊕ ϕ ) = ϕ (v) − ϕ(v ). Definition 3.2. Let b = s or b = a. An endomorphism L : U → U of the complex vector space U = C p ⊕ (C p )∗ is called symmetric with respect to b if L = L T , i.e. if b(L x, y) = b(x, L y) for all x, y ∈ U . We denote the vector space of such endomorphisms by Symb (U ). Lemma 3.3. If U = C p ⊕ (C p )∗ , then the space of quadratic invariants S2 (V ∗ )G is isomorphic as a complex vector space to W ∗ , where W = Syms (U ) for G = On and W = Syma (U ) for G = Spn . Proof. We still have V Hom(C p , Cn ) ⊕ Hom((C p )∗ , (Cn )∗ ) but now, via the given complex linear isomorphism β : Cn → (Cn )∗ , we even have an identification V Hom(U, Cn ) U ∗ ⊗ Cn , U = C p ⊕ (C p )∗ . Also, letting Sym(V, V ∗ ) denote the vector space of symmetric linear transformations σ : V → V ∗ , σ (v)(v ) = σ (v )(v), there is an isomorphism S2 (V ∗ ) → Sym(V, V ∗ ) by ϕ ϕ + ϕϕ → v → ϕ (v)ϕ + ϕ(v)ϕ . This descends to a vector space isomorphism between S2 (V ∗ )G and Sym G (V, V ∗ ), the G-equivariant mappings in Sym(V, V ∗ ).
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Consider now Hom G (V, V ∗ ) Hom G (U ∗ ⊗ Cn , U ⊗ (Cn )∗ ). As a consequence of the G-action on U and U ∗ being trivial, one immediately deduces that Hom G (U ∗ ⊗ Cn , U ⊗ (Cn )∗ ) Hom(U ∗ , U ) ⊗ Hom G (Cn , (Cn )∗ ). The vector space Hom G (Cn , (Cn )∗ ) is one-dimensional with generator β. Because β is symmetric for G = On and alternating for G = Spn , it follows that Sym(U ∗ , U ), G = On , 2 ∗ G ∗ n n ∗ S (V ) Sym G (U ⊗ C , U ⊗ (C ) ) Alt(U ∗ , U ), G = Spn , where the notation Alt(U ∗ , U ) means the vector space of alternating homomorphisms A : U ∗ → U , i.e., ϕ(A(ϕ )) = −ϕ (A(ϕ)). Note that Sym(U ∗ , U ) Sym(U, U ∗ )∗ and Alt(U ∗ , U ) Alt(U, U ∗ )∗ by the trace form (A, B) → Tr (AB). Now let L ∈ Symb (U ). Since b = s is symmetric and b = a is alternating, the image of Symb (U ) in Hom(U, U ∗ ) under the mapping L → φ L defined by φ L (x)(y) := b(L x, y) is Sym(U, U ∗ ) for b = s and Alt(U, U ∗ ) for b = a. Moreover, since the bilinear form b is non-degenerate, this mapping is an isomorphism of C-vector spaces. Thus we have Sym(U ∗ , U ) Sym(U, U ∗ )∗ Syms (U )∗ , G = On , 2 ∗ G S (V ) Alt(U ∗ , U ) Alt(U, U ∗ )∗ Syma (U )∗ , G = Spn , which is the statement that was to be proved.
3.2. The quadratic map Q . Summarizing the results of the previous subsection, the vector space W of quadratic G-invariants in S(V ) is ⎧ ⎨ End(C p ), G = GLn , W = S2 (V )G = Syms (U p ), G = On , ⎩ Sym (U ), G = Sp , p a n where U ≡ U p = C p ⊕ (C p )∗ . For notational convenience, we will sometimes think of W = End(C p ) → End(C p ) ⊕ End((C p )∗ ) for the first case (G = GLn ) as the intersection Syms (U p )∩Syma (U p ) of the vector spaces W for the last two cases (G = On , Spn ). In the following we will repeatedly use the decomposition of elements w ∈ W as A B End(C p ) Hom((C p )∗ , C p ) . (3.1) ∈ w= Hom(C p , (C p )∗ ) End((C p )∗ ) C At Note that B and C are symmetric for the case of G = On and alternating for G = Spn . The case of G = GLn is included by setting B = C = 0. Note also the dimensions dim W = p 2 , p(2 p + 1), and p(2 p − 1) for G = GLn , On , and Spn , in this order. Our treatment below is based on the relationship of O(V )G with the holomorphic functions O(W ). To make this relation explicit, we now introduce a map Q : V = Hom(Cn , C p ) ⊕ Hom(C p , Cn ) → End(U p )
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by defining its blocks according to the decomposition (3.1) as L L˜ L β −1 L t . Q : L ⊕ L˜ → ˜ t ˜ ˜ t t L βL L L Recall that the G-equivariant isomorphism β : Cn → (Cn )∗ is symmetric for G = On , alternating for G = Spn , and non-existent for G = GLn in which case the off-diagonal blocks Lβ −1 L t and L˜ t β L˜ are understood to be zero. In all three cases this mapping Q ˜ for all g ∈ G. In the last two cases this is ˜ = Q(Lg −1 ⊕ g L) is G-invariant: Q(L ⊕ L) because g t βg = β by the very notion of what it means for β to be G-equivariant. Lemma 3.4. The G-invariant mapping Q : V → End(U ) is into W . Proof. Let G be one of the groups On or Spn and denote by L and L˜ the elements of Hom(Cn , C p ) resp. Hom(C p , Cn ). Introducing two isomorphisms ψ : V → Hom(Cn , U ), L ⊕ L˜ → L ⊕ L˜ t β, ψ˜ : V → Hom(U, Cn ), L ⊕ L˜ → L˜ ⊕ β −1 L t , ˜ The two maps ψ and ψ˜ are related by ˜ we have Q(v) = ψ(v)ψ(v) for v = L ⊕ L. ˜ β ψ(v) = ψ(v)t Tb , where Tb : U → U ∗ is the isomorphism given by x → b(x, ·). Note that Tb is symmetric for b = s and alternating for b = a. Using the relations above, one computes that −1 ˜ t ψ(v)t = Tbt ψ(v)(β t )−1 β ψ(v)T ˜ Q(v)t = ψ(v) b .
If parities σ (β), σ (Tb ) ∈ {±1} are assigned to β and Tb by β t = σ (β) β and Tbt = σ (Tb ) Tb , then σ (β) = σ (Tb ) by construction, and it follows that Q(v)t = Tb Q(v)(Tb )−1 . This is equivalent to saying that Q(v) = Q(v)T ∈ Symb (U ) = W , which proves the statement for the groups G = On , Spn . The remaining case of G = GLn is included as a subcase by the embedding End(C p ) → Syms (U p ) ∩ Syma (U p ). While the map Q : V → W will not always be surjective, as the rank of L ∈ Hom(Cn , C p ) is at most min(n, p), there is a pullback of algebras Q ∗ : O(W ) → O(V )G in all cases. Let us now look more closely at Q ∗ restricted to W ∗ , the linear functions on W . For this let {ei }, { f i }, {ec }, and { f c } be standard bases of Cn , (Cn )∗ , C p , and (C p )∗ , respectively, and define bases {Z ci } and { Z˜ ic } of Hom(C p , Cn )∗ and Hom(Cn , C p )∗ by ˜ = f i ( L˜ ec ), Z ci ( L)
Z˜ ic (L) = f c (L ei ),
where i = 1, . . . , n and c = 1, . . . , p. Also, decomposing w ∈ W ⊂ End(U ) as in
(3.1), define a set of linear functions xcc , y c c , and ycc on W by
xcc (w) = f c (Aec ),
y c c (w) = f c (B f c ),
ycc (w) = (Cec )(ec ).
Notice that y c c = ±y cc and ycc = ±yc c where the plus sign applies in the case of
G = On and the minus sign for G = Spn . The set of functions {xcc } is a basis of ∗ p ∗ W End(C ) for the case of G = GLn . Expanding this set by including the set
of functions {y c c , ycc }c≤c we get a basis of W ∗ for G = On . The same goes for G = Spn if the condition on indices c ≤ c is replaced by c < c .
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Lemma 3.5. The pullback of algebras Q ∗ : O(W ) → O(V )G restricted to the linear functions on W realizes the isomorphism of complex vector spaces W ∗ → S2 (V ∗ )G . Proof. Applying Q ∗ to the chosen basis of W ∗ we obtain the expressions
Q ∗ xcc = Z˜ ic Z ci ,
Q ∗ y c c = Z˜ ic β i j Z˜ cj ,
Q ∗ ycc = Z ci βi j Z c , j
where β i j = f i (β −1 f j ) and βi j = (βe j )(ei ). Now the p 2 functions Z˜ ic Z ci are linearly independent and form a basis of S2 (V ∗ )GLn . By including the p( p ± 1) linearly inde
j pendent functions Z˜ ic β i j Z˜ cj and Z ci βi j Z c we get a basis of S2 (V ∗ )G for G = On resp. Spn . Thus our linear map Q ∗ : W ∗ → S2 (V ∗ )G is a bijection in all cases.
Proposition 3.6. The homomorphism Q ∗ : O(W ) → O(V )G is surjective. Proof. Let C[W ] = S(W ∗ ) and C[V ]G = S(V ∗ )G be the rings of polynomial functions on W and G-invariant polynomial functions on V , respectively. Pulling back functions by the G-invariant quadratic map Q : V → W , we have a homomorphism Q ∗ : C[W ] → C[V ]G . This map Q ∗ is surjective because C[V ]G = S(V ∗ )G is generated by S2 (V ∗ )G and Q ∗ : W ∗ → S2 (V ∗ )G is an isomorphism. Our holomorphic functions are expressed by power series with infinite radius of convergence. Therefore the surjective property of Q ∗ : C[W ] → C[V ]G carries over to Q ∗ : O(W ) → O(V )G . In the sequel, we will establish a finer result, relating the integral of an integrable function Q ∗ F ∈ O(V )G along a real subspace of V to an integral of F ∈ O(W ) over a non-compact symmetric space in W . While Prop. 3.6 applies always, this relation between integrals depends on the relative value of dimensions and will here be developed only in the range n ≥ p (for G = GLn ) or n ≥ 2 p (for G = On , Spn ). We begin by specifying the integration domain in V. Using the standard Hermitian structures of Cn and C p, let a real subspace VR ⊂ V be defined as the graph of † : Hom(Cn, C p ) → Hom(C p, Cn ). Thus in order for L ⊕ L˜ ∈ V to lie in VR the linear transformation L˜ : C p → Cn has to be the Hermitian adjoint ( L˜ = L † ) of L : Cn → C p . Note that VR Hom(Cn, C p ). The real vector space VR is endowed with a Euclidean structure by the norm square ||L ⊕ L † ||2 := Tr(L L † ). Let then dvol VR denote the canonical volume density of this Euclidean vector space VR . Our interest will be in the integral over VR of f dvol VR for f ∈ O(V )G . To make sure that the integral exists, we will assume that f is a Schwartz function along VR . Note that the anti-linear bijection c p : C p → (C p )∗ , v → v, ·, determines a Hermi −1 tian structure on (C p )∗ by ϕ, ϕ := c−1 p ϕ , c p ϕ. The canonical Hermitian structure p p ∗ of U = C ⊕ (C ) is then given by the sum u ⊕ ϕ, u ⊕ ϕ = u, u +ϕ, ϕ . The following is a first step toward our goal of transferring the integral VR f dvol VR to an integral over a non-compact symmetric space in W .
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Lemma 3.7. The image of VR under the quadratic map Q lies in the intersection of W and the non-negative Hermitian operators. Thus ⎧ G = GLn , ⎨ Herm≥0 ∩ End(C p ), Q(VR ) ⊂ Herm≥0 ∩ Syms (C p ⊕ (C p )∗ ), G = On , ⎩ Herm≥0 ∩ Syma (C p ⊕ (C p )∗ ), G = Spn . Proof. In the first case this is immediate from Q(L ⊕ L † ) = L L † = (L L † )† ≥ 0. To deal with the other two cases we recall the expression L L † L β −1 L t . Q(L ⊕ L † ) = Lβ L † L L t We already know from Lemma 3.4 that Q(L ⊕ L † ) ∈ Symb (C p ⊕ (C p )∗ ), where b = s or b = a. The operator Q(L ⊕ L † ) is self-adjoint because (L t )† = L and β † = β −1 . It is non-negative because u ⊕ ϕ, Q(L ⊕ L † )(u ⊕ ϕ) = |L † u + β −1 L t ϕ|2 ≥ 0. Remark. The condition n ≥ p resp. n ≥ 2 p emerging below, can be anticipated as the condition for the Q-image of a generic element in VR to have full rank. 3.3. The symmetric space of regular K -orbits in VR . Recall that our groups G act on ˜ By the relation (L g −1 )† = gL † for unitary transfor˜ = L g −1 ⊕ g L. V by g.(L ⊕ L) mations g ∈ G, the G-action on V restricts to an action on VR by the unitary subgroup K = Un , On (R), or USpn , of G = GLn , On (C), resp. Spn . In this subsection we study the regular K -orbit structure of VR . For this purpose we identify VR Hom(Cn , C p ) by the K -equivariant isomorphism given by L ⊕ L † → L. 3.3.1. K = Un . Here and elsewhere let Hom (A, B) denote the space of homomorphisms of maximal rank between two vector spaces A and B. Lemma 3.8. If n ≥ p then Hom (Cn , C p )/Un GL p /U p (diffeomorphism). Proof. Since a regular transformation L ∈ Hom (Cn , C p ) is surjective, the space im(L † ) has dimension p. Thus the decomposition Cn = ker(L) ⊕ im(L † ) defines an element of the Grassmannian (U p × Un− p )\Un of complex p-planes in Cn . Fixing some unitary basis of im(L † ), we can identify the restriction L : im(L † ) → C p with an element of GL p . In other words, Hom (Cn , C p ) GL p ×U p (Un− p \Un ), which gives the desired statement by taking the quotient by the right Un -action.
3.3.2. K = On . To establish a similar result for the case of orthogonal symmetry, we need the following preparation. (Here and in the remainder of this subsection On ≡ On (R) means the real orthogonal group.) Recalling that we are given a symmetric isomorphism δ : Cn → (Cn )∗ , we associate with L ∈ Hom(Cn , C p ) an extended complex linear operator ψ(L) ∈ Hom(Cn , C p ⊕ (C p )∗ ) by ψ(L)v = (L v) ⊕ L δv.
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Lemma 3.9. The mapping ψ : L → L ⊕ (L ◦ δ) determines an On -equivariant isomorphism Hom(Cn , C p ) → Hom(Rn , R2 p ) of vector spaces with complex structure. Proof. Recall that L = c p ◦ L ◦ cn−1 , where c p : C p → (C p )∗ and cn : Cn → (Cn )∗ are the canonical anti-linear isomorphisms given by the Hermitian structures of C p resp. Cn . Writing T := cn−1 δ we have ψ(L) = L ⊕ (c p L T ). Because δ is a symmetric isomorphism, the anti-unitary operator T : Cn → Cn squares to T 2 = 1. Let Fix(T ) ⊂ Cn denote the real subspace of fixed points v = T v, and define on U = C p ⊕ (C p )∗ an anti-linear involution C by C(u ⊕ ϕ) := (c−1 p ϕ) ⊕ c p u. If Fix(C) ⊂ U denotes the real subspace of fixed points of C, then from v=T v
Cψ(L)v = L T v ⊕ c p Lv = ψ(L)v we see that the C-linear operator ψ(L) maps Fix(T ) Rn into Fix(C) R2 p . Thus we may identify ψ(L) with an element of Hom(Rn√, R2 p ). The correspondence L → ψ(L) is bijective and transforms multiplication by −1, L → iL, into ψ(L) → J ψ(L), where J : u + c p u → iu − ic p u is the complex structure of the real vector space R2 p Fix(C). By definition, the elements of the real orthogonal group On commute with T . Thus ψ(L) ◦ k = ψ(L k) for k ∈ On , which means that ψ is On -equivariant. As before, let U = C p ⊕ (C p )∗ be equipped with the Hermitian structure which is
−1 induced from that of C p by u ⊕ ϕ, u ⊕ ϕ = u, u + c−1 p ϕ , c p ϕ. Its restriction to Fix(C) R2 p is a Euclidean structure defining the real orthogonal group O2 p . Lemma 3.10. If n ≥ 2 p then Hom (Rn , R2 p )/On GL2 p (R)/O2 p . Proof. A regular linear operator L : Rn → R2 p determines an orthogonal decomposition Rn = ker(L) ⊕ im(L † ) into Euclidean subspaces of dimension n − 2 p, resp. 2 p and hence a point of the symmetric space (O2 p × On−2 p )\On . Therefore, arguing in the same way as in the proof of Lemma 3.8, we have an identification Hom (Rn , R2 p ) GL2 p (R) ×O2 p (On−2 p \On ). The desired statement follows by taking the quotient by On .
Remark. Although each of GL2 p (R) and O2 p has two connected components, their quotient GL2 p (R)/O2 p = GL+2 p (R)/SO2 p is connected. For later purposes note that the anti-unitary map C : U → U combines with the Hermitian structure of U to give the canonical symmetric bilinear form of U : C(u ⊕ ϕ), u ⊕ ϕ = ϕ (u) + ϕ(u ) = s(u ⊕ ϕ, u ⊕ ϕ ).
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3.3.3. K = USpn . In the final case to be addressed, we are given an alternating isomorphism ε : Cn → (Cn )∗ and hence an anti-unitary operator T := cn−1 ε : Cn → Cn which squares to T 2 = −1. Note n ∈ 2N. The Hermitian vector space Cn now carries the extra structure of a complex symplectic vector space with symplectic form ω(v, v ) := T v, v = T 2 v, T v = −ω(v , v). The symmetry group of the Hermitian symplectic vector space Cn is K = USpn . To do further analysis in this situation, it is convenient to fix some decomposition Cn = P ⊕ T (P) which is orthogonal with respect to the Hermitian structure of Cn and Lagrangian w.r.t. the symplectic structure. The latter means that P and T (P) are non-degenerately paired ∼ by ω, so that we have an isomorphism T (P) → P ∗ by T v → ω(T v, ·) = −v, ·. Writing U = C p ⊕ (C p )∗ we still define ψ : Hom(Cn , C p ) → Hom(Cn , U ) by ψ(L) = L ⊕ L ε = L ⊕ c p L T, and invoke the canonical Hermitian structure of U to determine the adjoint ψ(L)† . For future reference we note that the map L → ψ(L) is USpn -equivariant. Lemma 3.11. The decomposition Cn = ker ψ(L) ⊕ im ψ(L)† is a decomposition into Hermitian symplectic subspaces. Proof. By the definition of the operation of taking the Hermitian adjoint, the space im ψ(L)† is the orthogonal complement of ker ψ(L) in the Hermitian vector space Cn . Since U = C p ⊕ (C p )∗ is an orthogonal sum and c p : C p → (C p )∗ is a bijection, the condition 0 = ψ(L)v = L v ⊕ c p L T v implies that if v is in the kernel of ψ(L) then so is T v. Thus T preserves the subspace ker ψ(L). Being anti-unitary, the operator T then preserves also the orthogonal complement im ψ(L)† . It therefore follows that ω restricts to a non-degenerate symplectic form on both subspaces. Next, let an anti-unitary operator C : U → U with square C 2 = −1 be defined by C(u ⊕ ϕ) = (c−1 p ϕ) ⊕ (−c p u). The associated symplectic structure of U is given by the canonical alternating form: −C(u ⊕ ϕ), u ⊕ ϕ = ϕ (u) − ϕ(u ) = a(u ⊕ ϕ, u ⊕ ϕ ). A short computation shows that the complex linear operator ψ(L) : Cn → U satisfies the relation ψ(L) = Cψ(L)T −1 . Let us therefore decompose ψ(L) according to ψ(L) : P ⊕ P ∗ → C p ⊕ (C p )∗ . Recalling T 2 = −1 and the fact that the anti-unitary operator T exchanges the subspaces P and P ∗ , we then see that ψ(L) = Cψ(L)T −1 is already determined by its blocks α1 := ψ(L)| P→C p and α2 := ψ(L)| P ∗ →C p : α1 α2 . ψ(L) = −α¯ 2 α¯ 1
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Table 2. Meaning of the groups K n , Gp , Kp , K n, p for the three choices of G n Gn
Kn
Gp
Kp
K n, p
GLn (C) On (C) Spn (C)
Un On (R) USpn
GL p (C) GL2 p (R) GL p (H)
Up O2 p (R) USp2 p
Un− p On−2 p (R) USpn−2 p
This means that the matrix expression of ψ(L) with respect to symplectic bases of P ⊕ P ∗ and C p ⊕ (C p )∗ consists of real quaternions q ∈ H : 10 0 i 0 1 i 0 + q1 + q2 + q3 (q j ∈ R). q = q0 01 i 0 −1 0 0 −i Now assume that n ≥ 2 p and ψ(L) is regular. Then ψ(L) : im ψ(L)† → U is an isomorphism of Hermitian symplectic vector spaces. On expressing this isomorphism with respect to symplectic bases of im ψ(L)† and U , we can identify it with an element of GL p (H), the group of invertible p × p matrices with real quaternions for their entries. Note that another characterization of the elements g of GL p (H) as a subgroup of GL(U ) is by the equation Cg = g C. The subgroup of unitary elements in GL p (H) is the unitary symplectic group USp(U ) ≡ USp2 p . The rest of the argument goes the same way as before: a regular transformation ψ(L) is determined by a Hermitian symplectic decomposition Cn = ker ψ(L) ⊕ im ψ(L)† together with a GL p (H)-transformation from im ψ(L)† to U ; taking the quotient by the right action of USpn we directly arrive at the following statement. Lemma 3.12. If n ≥ 2 p then the space of regular USpn -orbits in the image of Hom(Cn , C p ) under ψ is isomorphic to GL p (H)/USp2 p . 3.4. Integration formula for K -invariant functions. Let us now summarize the results of the previous section. To do this in a concise way covering all three cases at once, we will employ the notation laid down in Table 2. Proposition 3.13. If rank(K p ) ≤ rank(K n ) so that K p ⊂ K n , the space of regular K n -orbits in Hom(Cn , C p ) is isomorphic to the non-compact symmetric space G p /K p . Motivated by this result, our next goal is to reduce the integral of a K n -invariant function on VR Hom(Cn , C p ) to an integral over G p /K p . To prepare this step we introduce some further notations and definitions as follows. First of all, let U p denote the Hermitian vector space ⎧ G n = GLn , ⎨ Cp U p = C p ⊕ (C p )∗ ; s G n = On , ⎩ C p ⊕ (C p )∗ ; a G = Sp . n n In the second case U p carries a Euclidean structure (on R2 p Fix(C) ⊂ U p ) by the symmetric form s, in the third case it carries a symplectic structure by the alternating form a. Then let us regard U p (assuming that, depending on the case, the inequality p ≤ n or 2 p ≤ n is satisfied) as a subspace of Cn with orthogonal complement Un, p , thereby fixing an orthogonal decomposition Cn = U p ⊕ Un, p . This decomposition is Hermitian, Euclidean, or Hermitian symplectic, respectively.
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Let now X p, n denote the vector space of structure-preserving linear transformations U p ⊕ Un, p → U p . Using the language of matrices one would say that ⎧ G n = GLn , ⎨ Mat p, n (C), G n = On , X p, n Mat 2 p, n (R), ⎩ Mat p, n/2 (H), G n = Spn . A special element of X p, n is the projector : U p ⊕ Un, p → U p on the first summand. By construction, the symmetry group of the kernel space ker( ) = Un, p is K n, p . Next, we specify our normalization conventions for invariant measures on the Lie groups and symmetric spaces at hand. For that purpose, let ψ denote the K n -equivariant isomorphism discussed in the previous subsection: ⎧ G n = GLn , ⎨ L, ψ : Hom(Cn , C p ) → X p, n , L → L ⊕ L δ, G n = On , ⎩ L ⊕ L ε, G n = Spn . To avoid making case distinctions, we introduce an integer m taking the value m = 0, +1, −1 for G = GLn , On , Spn , respectively. Then from Tr C p L L † = Tr (C p )∗ L L t we have the relation Tr C p L L † = (1 + |m|)−1 TrU p ψ(L)ψ(L)† , which transfers the Euclidean norm of the vector space VR Hom(Cn , C p ) to a corresponding norm on X p, n . In view of the scaling implied by this transfer, we equip the Lie algebra Lie(K p ) = Te K p with the following trace form (or Euclidean structure): Lie(K p ) → R,
A → A 2 := −(1 + |m|)−1 TrU p A2 ≥ 0.
The compact Lie group K p is then understood to carry the invariant metric tensor and invariant volume density given by this Euclidean structure on Lie(K p ). The same convention applies to the compact Lie groups K n and K n, p . Please note that these conventions are standard and natural in that they imply, e.g., vol(U1 ) = vol(SO2 ) = 2π . By the symbol dg K p we will denote the G p -invariant measure on the non-compact symmetric space G p /K p . In keeping with the normalization convention we have just defined, the restriction of dg K p to the tangent space To (G p /K p ) at o := K p is the Euclidean volume density determined by the trace form B → B 2 = (1+|m|)−1 TrU p B 2 , which is positive for Hermitian matrices B = B † . As a final preparation, we observe that the principal bundle G p → G p /K p is trivial in all cases. Recall also that the Euclidean vector space VR Hom(Cn , C p ) comes with a canonical volume form (actually, a density) dvolVR . Proposition 3.14. For V = Hom(Cn , C p ) ⊕ Hom(C p , Cn ) let f ∈ O(V )G n be a holo˜ = f (L h ⊕ h −1 L) ˜ for all h ∈ G n . morphic function on V with the symmetry f (L ⊕ L) Restrict f to a K n -invariant function fr on the real vector subspace VR Hom(Cn , C p ) by fr (L) := f (L ⊕ L † ). If fr is a Schwartz function, then vol(K n ) fr (L) dvol VR (L) = fr ◦ ψ −1 (g ) J (g) dg K p , vol(K n, p ) Hom(Cn ,C p )
G p /K p
where the Jacobian function J : G p /K p → R is given by J (g) = 2 p 2 for G n = GLn and J (g) = 22 p − pn |Det(g)|n for G n = On , USpn .
2 − pn
|Det(g)|2n
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Proof. Convergence of the integral on both sides of the equation is guaranteed by the requirement that the integrand fr be a Schwartz function. The first step is to transform the integral on the left-hand side to the domain X p, n with integrand (ψ −1 )∗ ( f dvol VR ). Of course the space X p, n of regular elements in X p, n has full measure with respect to (ψ −1 )∗ (dvol VR ). Now choose a section s of the trivial principal bundle π : G p → G p /K p and parameterize X p, n by the diffeomorphism φ : (G p /K p ) × (K n /K n, p ) → X p, n , (x, k K n, p ) → s(x) k −1 . Using φ, transform the integral from X p, n to (G p /K p ) × (K n /K n, p ). Right K n translations (as well as left K p -translations) are isometries of (ψ −1 )∗ (dvol VR ), and varying ψ(L) = s(x) k −1 we get δψ(L) = δs(x) k −1 − s(x) k −1 δk k −1 . Therefore, the pullback of dvol VR by ψ −1 ◦ φ is proportional to the product of invariant measures of G p /K p and K n /K n, p times a Jacobian j (x) which can be computed as the Jacobian of the map Ls(x) : X p, n → X p, n ,
L → s(x)L .
In the case of X p, n = Hom(Cn , C p ) this gives j (x) = |Det(s(x))|2n . In the other two cases the dimension of U p is doubled while the (real) dimension of X p, n stays the same; hence j (x) = |Det(s(x))|n . In all cases we may replace |Det(s(x))| by |Det(g)|, where g is any point in the fiber π −1 (x). Also, by the K n -invariance of the integrand f one has f ◦ ψ −1 (s(x) k −1 ) = f ◦ ψ −1 (g ) independent of the choice of g ∈ π −1 (x). This already proves that the two integrals on the left-hand and right-hand side are proportional to each other, with the constant of proportionality being independent of f . It remains to ascertain the precise value of this constant. Doing the invariant integral over K n /K n, p one just picks up the normalization factor of volumes vol(K n )/vol(K n, p ). 2 2 The remaining factor 2 p − pn or 22 p − pn in J (g) is determined by the following consideration. Decomposing the elements ξ ∈ k ≡ Lie(K n ) as A B k ∩ End(U p ) k ∩ Hom(Un, p , U p ) , kξ = ∈ k ∩ Hom(U p , Un, p ) k ∩ End(Un, p ) −B † D we have the norm square ξ 2 = (1 + |m|)−1 (−Tr A2 + 2 Tr B B † − Tr D 2 ). On the other hand, the differential of the mapping φ at (o, eK n, p ) ∈ (G p /K p ) × (K n /K n, p ) is A B → (H + A) ⊕ B ∈ X p, n ∩ End(U p ) ⊕ X p, n ∩ Hom(Un, p , U p ), H, −B † 0 which gives the norm square (H + A) ⊕ B 2 = (1 + |m|)−1 (Tr H 2 − Tr A2 + Tr B B † ). Thus the term Tr B B † gets scaled by a factor of two, and by counting the number of independent freedoms in B ∈ Hom(Un, p , U p ) we see that the Jacobian J (g) receives an extra factor of 2 p(n− p) for the case of G n = GLn and 22 p(n−2 p)/2 for G n = On , Spn . Remark. For n = 2 p and G n = On the space K n /K n, p = On (R) consists of two connected components and the volume factor means vol(K n )/vol(K n, p ) = vol(On (R)) = 2 vol(SOn (R)). On the other hand, for n > 2 p and the same case the volume factor is that of the connected space K n /K n, p = On (R)/On−2 p (R) = SOn (R)/SOn−2 p (R).
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To finish this section, we cast Prop. 3.14 in a form closer in spirit to the rest of paper. Recall that either we have G p = GL(U p ), or else G p ⊂ GL(U p ) is characterized by the commutation rule g C = Cg. Since C † = C −1 = ±C, all our groups G p are stabilized by the dagger operation. Thus there exists an involution θ : G p → G p , g → (g −1 )† , which is actually a Cartan involution fixing the elements of the maximal compact subgroup K p . The mapping γ : G p /K p → G p , g → gθ (g −1 ) = gg † , embeds the symmetric space into the group. To clarify the connection with the setting of Sect. 3.2, let us understand the image of this embedding as a subspace of W . Lemma 3.15. The Cartan embedding γ : G p /K p → G p ⊂ End(U p ) projected to the positive Hermitian operators in W ⊂ End(U p ) is a bijection. Proof. From ψ(L) = L in the first case, and ψ(L) = L ⊕ L β in the last two cases, we immediately see that the composition of mappings L ⊕ L † → ψ(L) → ψ(L)ψ(L)† = Q(L ⊕ L † ) ∈ Q(VR ) is the quadratic map Q : V → W (Sect. 3.2) restricted to VR , and since g ∈ G p arises from decomposing ψ(L) = g k −1 , the positive Hermitian operator gg † = ψ(L)ψ(L)† lies in Q(VR ) ⊂ W . Thus the embedding G p /K p → G p is into Herm+ ∩ W . It remains to be shown that γ : G p /K p → Herm+ ∩ W is one-to-one. In the case of + ∩ W has a unique positive G p = GL(U p ), every positive Hermitian √ operator √ −1h ∈ Herm √ √ √ Hermitian square root h, and h = h√θ ( h) = hk θ ( hk)−1 (k ∈ K p ). Thus there exists a unique inverse γ −1 (h) = h K p ∈ G p /K p . To deal with the other two cases we recall the relation Cb ·, · = ±b(·, · ), i.e., C ≡ Cb combines with the Hermitian structure of U p to give the bilinear form b, where b = s or b = a. This implies that the symmetric transformations w ∈ W = Symb (U p ) are characterized by the commutation rule Cb w = w † Cb . Indeed, Cb w ·, · = ±b(w ·, · ) = ±b(·, w · ) = Cb ·, w · = w † Cb ·, · , and hence W ∩ Herm are exactly the elements of End(U p ) that commute with Cb . The desired statement now follows from the definition G p = {g ∈ GL(U p ) | g Cb = Cb g} because the squaring map on Herm+ ∩ GL(U p ) remains a bijection when restricted to the set of fixed points Herm+ ∩ G p of the involution w → Cb w (Cb )−1 . In the sequel we will often use the abbreviations n := (1 + |m|)−1 n, Tr := (1 + |m|)−1 TrU p (m = 0, 1, −1 for G = GL, O, Sp). Let now D p := Herm+ ∩ W denote the set of positive Hermitian operators in W : ⎧ G n = GLn , ⎨ Herm+ ∩ End(C p ), D p = Herm+ ∩ Syms (C p ⊕ (C p )∗ ), G n = On , ⎩ Herm+ ∩ Sym (C p ⊕ (C p )∗ ), G = Sp . n a n
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D p is equipped with a G p -invariant measure dµ D p . In keeping with our general conventions, we normalize dµ D p so that (dµ D p )o agrees with the Euclidean volume density of the Euclidean vector space of Hermitian operators H ∈ Lie(G p ) with norm square H 2 = (1 + |m|)−1 Tr H 2 = Tr H 2 . The Cartan embedding g → gθ (g −1 ) on the Hermitian operators g = e H is the squaring map e H → e2H . Thus, pushing the G p invariant measure dg K p forward by the Cartan embedding we get 2−dim(G p /K p ) dµ D p . Now since ⎧ ⎨ p2 G n = GLn , dim(G p /K p ) = p(2 p + 1) G n = On , ⎩ p(2 p − 1) G = Sp , n n the following statement is a straightforward reformulation of Prop. 3.14. Corollary 3.16. Given f ∈ O(V )G n , and retaining the setup and the conditions of Prop. 3.14, define F ∈ O(W ) by Q ∗ F = f . Then
− p(n+m) vol(K n ) f dvol VR = 2 F(x) Det n (x) dµ D p (x). vol(K ) n, p VR Dp
In particular, since the function x → F(x) = e−Tr x pulls back to L → f (L) = † † −Tr e L L and the Gaussian integral e−Tr L L dvol VR (L) has the value π pn , we infer the formula
e−Tr x Det n (x) dµ D p (x) = (2π ) pn 2 pm vol(K n, p )/vol(K n ). (3.2) Dp
4. Lifting in the Fermion-Fermion Sector Having settled the case of V1 = 0 (or q = 0) we now turn to the complementary case where V0 = 0 (or p = 0). Thus, in the present section we consider V ≡ V1 = Hom(Cn , Cq ) ⊕ Hom(Cq , Cn ) C2qn , in which case our basic algebra AV becomes an exterior algebra of dimension 22qn : AV = ∧(V ∗ ) ∧(C2N ),
N = qn.
In the sequel, we will prove an analog of Prop. 3.14 for this situation.
4.1. Quadratic G-invariants. This subsection is closely analogous to Sect. 3.1, the main difference being that the role of the symmetric algebra S(V ∗ ) is now taken by the exterior algebra ∧(V ∗ ). It remains true [12] that for each of the classical reductive complex Lie groups G = GLn , On , and Spn , a basis of ∧2 (V ∗ )G is a generating system for ∧(V ∗ )G . Let us therefore make another study of these quadratic invariants. Recall that on the direct sum Uq := Cq ⊕ (Cq )∗ we have the canonical symmetric bilinear form s and the canonical alternating bilinear form a.
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Lemma 4.1. Let V = Hom(Cn , Cq ) ⊕ Hom(Cq , Cn ) carry the G-action induced from the fundamental action on Cn of G = GLn , On , or Spn . If Uq = Cq ⊕ (Cq )∗ , then the space of quadratic invariants ∧2 (V ∗ )G is isomorphic as a complex vector space to W ∗ , where W = End(Cq ) for G = GLn , W = Syma (Uq ) for G = On , and W = Syms (Uq ) for G = Spn . Proof. (Sketch). There is no conceptual difference from the proofs of Lemma 3.1 and Lemma 3.3, and we therefore give only a summary of the changes. In the case of G = GLn , all quadratic invariants still arise by composing the elements of Hom(Cq , Cn ) with those of Hom(Cn , Cq ). Thus ∧2 (V )G End(Cq ) and, since G acts reductively and ∧2 (V )G is paired with ∧2 (V ∗ )G , we have ∧2 (V ∗ )G End(Cq )∗ . In the other cases we use the isomorphism ∧2 (V ∗ ) → Alt(V, V ∗ ) given by ϕ ϕ − ϕϕ → v → ϕ (·)ϕ(v) − ϕ(·)ϕ (v) , which descends to an isomorphism ∧2 (V ∗ )G → Alt G (V, V ∗ ). Then, writing U ≡ Uq we make the G-equivariant identification V U ∗ ⊗ Cn and have G = On , Alt(U ∗ , U ), ∧2 (V ∗ )G Alt G (U ∗ ⊗ Cn , U ⊗ (Cn )∗ ) Sym(U ∗ , U ), G = Spn . This leads to the desired statement by the isomorphisms Alt(U ∗ , U ) Alt(U, U ∗ )∗ Syma (U )∗ and Sym(U ∗ , U ) Sym(U, U ∗ )∗ Syms (U )∗ . As before, let the elements w ∈ W be decomposed as A B Hom((Cq )∗ , Cq ) End(Cq ) . w= ∈ Hom(Cq , (Cq )∗ ) End((Cq )∗ ) C At Here B = −B t and C = −C t for the case of G = On , while B = +B t and C = +C t for G = Spn , and B = C = 0 for G = GLn . By simple counting, the dimensions of W are dim W = q 2 , q(2q − 1), and q(2q + 1) for G = GLn , On , and Spn , respectively. One can now reconsider the quadratic mapping Q : V → W defined in Sect. 3.2, with the twist that the elements of V in the present context are to be multiplied with each other in the alternating sense of exterior algebras. However, what matters for our purposes is not the mapping Q but the pullback of algebras Q ∗ : O(W ) → ∧(V ∗ )G . Let us now specify the latter at the level of the isomorphism Q ∗ : W ∗ → ∧2 (V ∗ )G . For this let {ei }, { f i }, {ec }, and { f c } be standard bases of Cn , (Cn )∗ , Cq , and (Cq )∗ , respectively, and define bases {ζci } and {ζ˜ic } of Hom(Cq , Cn )∗ and Hom(Cn , Cq )∗ by ˜ = f i ( L˜ ec ), ζ˜ c (L) = f c (L ei ), ζci ( L) i where L ∈ Hom(Cn , Cq ) and L˜ ∈ Hom(Cq , Cn ). Of course the index ranges are i = 1, . . . , n and c = 1, . . . , q. Then, decomposing w ∈ W ⊂ End(U ) into blocks
A, B, C as above, define a set of linear functions xcc , y c c , ycc : W → C by
xcc (w) = f c (Aec ),
y c c (w) = f c (B f c ),
ycc (w) = (Cec )(ec ).
Notice the symmetry relations y c c = ∓y cc and ycc = ∓yc c where the upper sign
applies in the case of G = On and the lower sign for G = Spn . The functions {xcc } ∗ q ∗ constitute a basis of W End(C ) for the case of G = GLn . Inclusion of the set
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of functions {y c c , ycc }c
functions y c c and ycc for c = c we get a basis of W ∗ for G = Spn . Recall that we are given a G-equivariant isomorphism β : Cn → (Cn )∗ which is symmetric (β = δ) for G = On and alternating (β = ε) for G = Spn . Lemma 4.2. The isomorphism W ∗ → ∧2 (V ∗ )G has a realization by
Q ∗ xcc = ζ˜ic ζci , Q ∗ y c c = ζ˜ic β i j ζ˜ jc , Q ∗ ycc = ζci βi j ζc , j
where βi j = (βe j )(ei ) are the matrix entries of β : Cn → (Cn )∗ , and β i j = f i (β −1 f j ).
Proof. All functions ζ˜ic ζci , ζ˜ic β i j ζ˜ jc and ζci βi j ζc are G-invariant in the pertinent cases.
The q 2 functions ζ˜ic ζci form a basis of ∧2 (V ∗ )GLn . By including the q(q − 1) functions
j ζ˜ c δ i j ζ˜ c and ζci δi j ζ for c < c , we get a basis of ∧2 (V ∗ )On . Replacing β = δ by β = ε i
j
j
c
and expanding the index range to c ≤ c we get a basis of ∧2 (V ∗ )Spn . Thus the linear operator Q ∗ takes one basis to another one and hence is an isomorphism. Next, we review some useful representation-theoretic facts about ∧(V ∗ )G . 4.2. G -irreducibility of AVG . Taking V ⊕ V ∗ C4N to be equipped with the canonical symmetric form s(v ⊕ ϕ, v ⊕ ϕ ) = ϕ (v) + ϕ(v ), one defines the Clifford algebra Cl(V ⊕ V ∗ ) to be the associative algebra generated by V ⊕ V ∗ ⊕ C with relations ww + w w = s(w, w )1
(w, w ∈ V ⊕ V ∗ ).
The linear span of the skew-symmetric quadratic elements (ww − w w) is closed under the commutator in Cl(V ⊕ V ∗ ) and is canonically isomorphic to the Lie algebra of the orthogonal group of the vector space V ⊕ V ∗ with symmetric bilinear form s. By exponentiating this Lie algebra inside the Clifford algebra, one obtains the spin group, a connected and simply connected Lie group denoted by Spin(V ⊕ V ∗ ) = Spin4N . Via their actions on V , the complex Lie groups G = GLn , On , and Spn , are realized as subgroups of Spin4N . The centralizer of G in Spin4N is another complex Lie group, G , called the Howe dual partner of G [12]. The list of such Howe dual pairs is G × G :
2q , On × Spin4q , Spn × Sp4q . GLn × GL
2q ⊂ Spin4q and GL 2q ⊂ Sp4q . Note that from On ⊂ GLn and Spn ⊂ GLn one has GL 2q is a double covering of GL2q (see below). In the case of n being odd, GL A few words of explanation concerning the pairs G × G are in order. In the case of the first pair one regards the vector space V ⊕ V ∗ as V ⊕ V ∗ Uq ⊗ Cn ⊕ (Uq )∗ ⊗ (Cn )∗ , Uq = Cq ⊕ (Cq )∗ , and the centralizer of G = GLn in Spin(V ⊕V ∗ ) is then seen to be G = GL(Uq ) ≡ GL2q (or a double cover thereof if n is odd). In the last two cases the G-equivariant isomorphism β : Cn → (Cn )∗ leads to an identification V ⊕ V ∗ (Uq ⊕ Uq∗ ) ⊗ Cn .
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The symmetric bilinear form s on V ⊕ V ∗ in conjunction with β induces a bilinear form on Uq ⊕ (Uq )∗ . For G = On this form is the canonical symmetric bilinear form s and one has the centralizer G = Spin(Uq ⊕ Uq∗ ; s) ≡ Spin4q . For G = Spn the induced form is the canonical alternating form a and one has G = Sp(Uq ⊕ Uq∗ ; a) ≡ Sp4q . Now the exterior algebra ∧(V ∗ ) carries the spinor representation of the Clifford algebra Cl(V ⊕V ∗ ). This is the representation which is obtained by letting vectors v ∈ V and linear forms ϕ ∈ V ∗ operate by contraction ι(v) : ∧k (V ∗ ) → ∧k−1 (V ∗ ) and exterior multiplication ε(ϕ) : ∧k (V ∗ ) → ∧k+1 (V ∗ ). By the inclusion G × G ⊂ Spin4N ⊂ Cl(V ⊕ V ∗ ) the spinor representation of the Clifford algebra gives rise to a representation on AV = ∧(V ∗ ) of each Howe dual pair G × G . It is known [12] that AV decomposes as a direct sum ⊕i (Ui ⊗Ui ) of irreducible G × G representations such that Ui U j and Ui U j for i = j. In particular, the representation of G on the algebra of G-invariants AVG = ∧(V ∗ )G is irreducible. Next we observe that each of our Howe dual groups G has rank 2q. Moreover, one can arrange for all of them to share the same maximal torus. This is the Abelian group T = (C× )q × (C× )q acting on V = V1 = Hom(Cn , Cq ) ⊕ Hom(Cq , Cn ) by diagonal transformations ˜ → (t1 L) ⊕ ( L˜ t2 ). (t1 , t2 ).(L ⊕ L) The induced action of elements H = (H1 , H2 ) of the Cartan algebra t = Lie(T ) = Cq ⊕ Cq on the spinor module is by operators (H1 )c [ι(eic ), ε( f ci )] + (H2 )c [ι(eci ), ε( f ic )] . (4.1) Hˆ = 21 c
Here {eic } means the standard basis of Hom(Cn , Cq ), and {eci } means the standard basis of Hom(Cq , Cn ), while { f ci } and { f ic } are the corresponding dual bases. The factor of 1/2 in front of the sum reflects the fact that the spinor representation is a “square root” representation. The zero-degree component ∧0 (V ∗ ) = C – the ‘vacuum’ in physics language – is stabilized by the action of these operators Hˆ . Applying H as Hˆ to 1 ∈ ∧0 (V ∗ ) we get H.1 = λ(H )1, λ(H ) =
q n (H1 )c + (H2 )c . 2 c=1
Note that the weight λ is integral for even n, but half-integral for odd n. (This is why the 2q .) We latter case calls for the group GL2q to be replaced by a double cover G = GL λ◦ln will denote the integrated weight or character by χ := e . 4.3. Berezin integral and lowest weight space. We are now going to think of the irreducible G -representation space AVG as an irreducible highest-weight module for the Lie algebra of G . To keep the notation simple we omit the prime and denote this Lie algebra by g := Lie(G ). Thus ⎧ ⎨ gl2q , G = GLn , G = On , g = o4q , ⎩ sp , G = Sp . 4q n
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The vacuum weight λ is a highest weight for the g-representation AVG . We emphasize this fact by making a change of notation AVG ≡ V (λ). k ∗ The spinor module comes with a Z-grading by the degree, ∧(V ∗ ) = ⊕2N k=0 ∧ (V ), 1 n where N = 2 dim V = qn. Since G is defined on C and acts on V , this grading carries over to the algebra AVG = V (λ): V (λ) = V (λ)k . k≥0
We denote the highest degree part by V (λ)top . The highest degree part ∧2N (V ∗ ) of the spinor module is a complex line stable under the symmetry group G. It is easy to check that G in fact acts trivially on ∧2N (V ∗ ), so V (λ)top = V (λ)2N = ∧2N (V ∗ ). Now there exists a canonical generator V ∈ ∧2N (V ) by the following principle. Since the trace form Hom(Cn , Cq ) ⊗ Hom(Cq , Cn ) → C,
A ⊗ B → Tr AB,
Hom(Cn , Cq )
and Hom(Cq , Cn ) are canonically is non-degenerate, the vector spaces dual to each other. If {e1 , . . . , e N } is any basis of Hom(Cn , Cq ), let { f 1 , . . . , f N } be the corresponding dual basis of Hom(Cq , Cn ). The exterior product V = f N ∧ e N ∧ · · · ∧ f 1 ∧ e1 then is independent of the choice of basis and only depends on how we order the two summands in V = Hom(Cn , Cq ) ⊕ Hom(Cq , Cn ). For definiteness, let us say that Hom(Cn , Cq ) is the first summand. We then have a canonical element V ∈ ∧2N (V ), and by evaluating the canonical pairing ∧2N (V ) ⊗ ∧2N (V ∗ ) → C with fixed argument V in the first factor, we get an identification V (λ)2N = ∧2N (V ∗ ) C. Definition 4.3. The projection π : V (λ) → V (λ)2N C is called the Berezin integral, and is here denoted by f → V [ f ]. Another way to view this projection is as follows. The vacuum ∧0 (V ∗ ) is the space of highest-weight vectors for g, whereas the top degree part ∧2N (V ∗ ) is the space of lowest-weight vectors. The latter are the weight vectors of weight −λ. Indeed, going from zero to top degree amounts to exchanging the operators ε and ι, and since the expression (4.1) is skew-symmetric in these, the weight changes sign. Now define the subgroup H ⊂ G to be the intersection of the stabilizer of V (λ)0 = 0 ∧ (V ∗ ) with the stabilizer of V (λ)2N = ∧2N (V ∗ ). For n ∈ 2N these are the groups ⎧ ⎨ GLq × GLq ⊂ G = GL2q , G = GLn , G = On , H = GL2q ⊂ G = Spin4q , ⎩ GL ⊂ G = Sp , G = Spn . 2q 4q If n is odd, we replace H by the double cover forced on us by the square root nature of the spinor representation or the highest weight λ being half-integral. Let us now specify how the Lie algebra Lie(H ) acts on the spinor module ∧(V ∗ ). (This will do as a temporary substitute for the more detailed description of the H -action on W ∗ = ∧2 (V ∗ )G given below.) In the first case, one has H = GL(Cq )×GL((Cq )∗ ) ≡ GLq × GLq and X = (A, D) ∈ Lie(H ) acts on ∧(V ∗ ) as Xˆ = 21 [ι(Aeic ), ε( f ci )] + 21 [ι(eci ), ε(D f ic )],
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where the notation of (4.1) is being used. Note that Xˆ .1 = n2 (Tr A + Tr D). In the last b=1,...,2q two cases, one uses V Hom(Cn , Uq ) and fixes a basis {eib }i=1,...,n of Hom(Cn , Uq ), i with dual basis { f b }. With these conventions, an element X of the Lie algebra of H = GL(Uq ) ≡ GL2q acts on ∧(V ∗ ) as Xˆ = 21 [ι(X eib ), ε( f bi )]. Note Xˆ .1 = n2 Tr X . By definition, the roots of H are the roots of g which are orthogonal to λ. Since all groups H are connected subgroups of G of maximal rank, they are in fact characterized by their root systems. Note also that all of our groups H are reductive. Furthermore, the character χ : T → C× extends to the character χ : H → C× , h → Detn/2 (h). Being orthogonal to the highest weight λ, the root system of H is orthogonal also to the lowest weight −λ. It follows that the space of lowest-weight vectors V (λ)2N is stable with respect to H : it is the one-dimensional representation of H corresponding to the reciprocal character χ −1 (h) = Det −n/2 (h). Since H is reductive and the T -weight space V (λ)2N = ∧top (V ∗ ) has dimension one, V (λ) decomposes canonically as a H -representation space: V (λ) = V (λ)2N ⊕ U, where U is the sum of all other H -subrepresentations in V (λ). From dim V (λ)2N = 1 we then infer that the space of H -equivariant homomorphisms Hom H (V (λ), V (λ)2N ) is one-dimensional. Now the Berezin integral π : V (λ) → V (λ)2N is a non-zero element of that space, and we therefore have the following result. Lemma 4.4. Hom H (V (λ), V (λ)2N ) = C π . 4.4. Parabolic induction. The g-representation V (λ) can be constructed in another way, as follows. Decompose g as g = g− ⊕h⊕g+ , where h = Lie(H ) and g± is the direct sum of the root subspaces of g corresponding to positive resp. negative roots not orthogonal to λ. Since the highest weight λ is the weight of the vacuum with generator 1 ∈ V (λ)0 , this implies that g+ .1 = 0 and g− .1 = V (λ)2 . Or, to put it in yet another way, g+ ⊂ g is the subspace of elements represented on the spinor module by operators of type ιι, while g− ⊂ g is the subspace of operators of type εε. Let p := h ⊕ g+ . (The notation is to convey that p can be viewed as the Lie algebra of a parabolic subgroup of G .) Since all roots of h are orthogonal to λ, the weight λ : t → C extends in the trivial way to a linear function λ : h → C ; the latter is the function λ(X ) = (n/2)Tr C2q X . We further extend λ trivially to all of p = h ⊕ g+ . Let U (p) be the universal enveloping algebra of p and denote by Vλ := V (λ)0 the one-dimensional U (p)-representation defined by X.vλ = λ(X )vλ for a generator vλ ∈ Vλ and elements X ∈ p. Then by the canonical left action of g on U (g), the tensor product M(λ) := U (g) ⊗U (p) Vλ is a U (g)-representation of highest weight λ and highest-weight vector m λ = 1 ⊗ vλ . This representation is called a generalized Verma module or the universal highest-weight g-representation which is given by parabolic induction from the one-dimensional representation Vλ of p. The module M(λ) has the following universal property.
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Lemma 4.5. Let W be any g-module with a vector w = 0 such that i) X.w = λ(X )w for all X ∈ p, and ii) U (g).w = W . Then there exists a surjective g-equivariant linear map M(λ) → W such that m λ → w. In particular, our irreducible g-representation V (λ) is of this kind. Thus there exists a surjective g-equivariant map p : M(λ) → V (λ). 4.5. H -structure of M(λ). The U (g)-representation M(λ) has infinite dimension and cannot be integrated to a representation of G . As we shall now explain, however, the situation is more benign for the subgroup H ⊂ G . From g/p g− we have an isomorphism of vector spaces U (g)⊗U (p) Vλ U (g− ). By making the identification as U (g− ) ⊗ Vλ M(λ), n ⊗ vλ → nm λ , we will actually get more, as follows. If α, β are any two roots such that α is orthogonal to λ and β is not, then α + β is not orthogonal to λ. From this one directly concludes that if h ∈ h and n ∈ g− , then [h, n] ∈ g− , i.e., g− (or g+ , for that matter) is normalized by h. This action of h on g− extends to an h-action on U (g− ): supposing that n = n 1 · · · nr ∈ U (g− ), where n i ∈ g− (i = 1, . . . , r ), we let ad(h)n := n 1 · · · n j−1 [h, n j ]n j+1 · · · nr , j
and by this definition we have the following commutation rule of operators in U (g): hn = ad(h)n + nh. If we now let U (h) act on M(λ) by the canonical left action and on U (g− ) ⊗ Vλ by h.(n ⊗ vλ ) := (ad(h)n) ⊗ vλ + λ(h)n ⊗ vλ , ∼
then we see that the identification U (g− ) ⊗ Vλ → M(λ) by n ⊗ vλ → nm λ is an isomorphism of U (h)-representations. Now every element in U (g− ) lies in an h-representation of finite dimension. Basic principles therefore entail the following consequence. Lemma 4.6. The representation of the Lie algebra h on M(λ) can be integrated to a representation of the Lie group H . In each of the three cases under consideration, g− is commutative and we can identify U (g− ) with the ring of polynomial functions C[W ] on a suitable representation W (g− )∗ of H . From g− g− .1 = V (λ)2 = ∧2 (V ∗ )G we have W = ∧2 (V )G , the subspace of G-fixed vectors in ∧2 (V ). The space W was described in Lemma 4.1 where we saw that W = End(Cq ), Syma (Uq ), and Syms (Uq ) for G = GL, O, and Sp, respectively. Note that in all cases W contains the identity element Id = IdCq or Id = IdUq . We now describe how the group H acts on W . In the last two cases, where H = GL(Uq ) ≡ GL2q (or a double cover thereof), we associate with each of the two bilinear forms b = s or b = a an involution τb : H → H by the equation b(τb (g)x, y) = b(x, g −1 y)
(x, y ∈ Uq ).
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The action of H on W = Symb (Uq ) is then by twisted conjugation, g.w = gwτb (g −1 ). In the notation of Sect. 1.1.2 we have τa (g −1 ) = ta g t (ta )−1 and τs (g −1 ) = ts g t (ts )−1 . Note that the group of fixed points of τb in H is the symplectic group Sp(Uq ) = Sp2q for b = a and the orthogonal group O(Uq ) = O2q for b = s. In the first case (G = GLn ) the group H is the subgroup of GL(Uq ) preserving the decomposition Uq = Cq ⊕ (Cq )∗ . Here again it will be best to think of the vector space W as the intersection of the vector spaces for the other two cases: W = End(Cq ) Syma (Uq ) ∩ Syms (Uq ). This means that we think of End(Cq ) as being embedded into End(Uq ) as z 0 End(Cq ) Hom((Cq )∗ , Cq ) , z → =: w. End(Cq ) → Hom(Cq , (Cq )∗ ) End((Cq )∗ ) 0 zt For an element g = (g1 , g2 ) ∈ H = GL(Cq ) × GL((Cq )∗ ) one now has τa (g1 , g2 )−1 = τs (g1 , g2 )−1 = (g2t , g1t ), and the action of H on W by twisted conjugation is given by z 0 0 g1 z g2t = . g.w = (g1 , g2 ). 0 zt 0 g2 z t g1t If we now define an involution τ0 by τ0 (g1−1 , g2−1 ) = (g2t , g1t ), then this action can be written in the short form g.w = gwτ0 (g −1 ). To sum up the situation, let τ = τ0 for G = GL, τ = τa for G = O, and τ = τs for G = Sp. Then the H -action on W always takes the form g.w = gwτ (g −1 ). In all three cases it is a well-known fact (see for example [12]) that the ring C[W ] is multiplicity-free as a representation space for H . It then follows that the universal highest-weight representation M(λ) C[W ] ⊗ Vλ is multiplicity-free. Lemma 4.7. Let V−λ = V (λ)2N be the one-dimensional H -representation associated to the character χ −1 = exp ◦(−λ) ◦ ln. Then the space Hom H (M(λ), V−λ ) of H -equivariant homomorphisms from M(λ) to V−λ has dimension one and is generated by π ◦ p, the composition of the projection p : M(λ) → V (λ) with the Berezin integral π : V (λ) → V−λ = V (λ)2N . Proof. Since p is surjective, π ◦ p is non-trivial and the space Hom H (M(λ), V−λ ) has at least dimension one. On the other hand, since M(λ) is multiplicity-free as an H -representation, the dimension of Hom H (M(λ), V ) cannot be greater than one for any irreducible representation V of H . Corollary 4.8. Let P : M(λ) = C[W ] ⊗ Vλ → V−λ be any non-trivial H -equivariant linear mapping. Then there exists a non-zero constant c P such that for every f ∈ V (λ) and any lift F ∈ p −1 ( f ) ⊂ C[W ] ⊗ Vλ one has P[F] = c P [ f ]. We are now going to realize P by integration over a real domain in W .
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4.6. Construction of H -equivariant homomorphisms. Let o ≡ Id denote the identity element of W . Then in all three cases the H -orbit H.o is open and dense in W and can be characterized as the complement of the zero set of a polynomial D : H.o = {w ∈ W | D(w) = 0}, where D will be the Pfaffian function for the case of G = On and will be the determinant function for G = GLn and G = Spn . Since D does not vanish on H.o, the map H.o → W ⊕ C, w → (w, D(w)−1 ) defines an inclusion, and one can view H.o as the zero set of a function on W ⊕ C : H.o = {(w, t) ∈ W ⊕ C | D(w) t − 1 = 0}. Hence the ring of algebraic functions on H.o, namely C[H.o], is the same as the ring C[W ⊕ C] factored by the ideal generated by the function (w, t) → D(w) t − 1. Let C(W ) be the field of rational functions on W . Thus an element r ∈ C(W ) can be expressed as a quotient r = f /g of polynomial functions f, g ∈ C[W ]. Denote by C[W ] D ⊂ C(W ) the subring of elements which can be written as a quotient r = f/D n , C[W ] D = w → f (w)/D n (w) | f ∈ C[W ], n ≥ 0 . We now identify C[H.o] with C[W ] D : an element in C[H.o] can be represented as a polynomial of the form (w, t) → f 0 (w)+ f 1 (w) t +· · ·+ f s (w) t s , where the f i ∈ C[W ], and the following map C[H.o] → C[W ] D then defines an isomorphism of rings: f1 fs f0 + + ··· + s. (w, t) → f 0 (w) + f 1 (w) t + · · · + f s (w) t s → 1 D D Our aim here is to construct an H -equivariant homomorphism M(λ) → V−λ . By the isomorphisms of H -representations: (V−λ )∗ Vλ and Vλ ⊗ Vλ V2λ , this is the same as constructing an H -invariant homomorphism M(λ) ⊗ (V−λ )∗ = C[W ] ⊗ V2λ → C to the trivial representation. For this purpose we identify the representation C[W ] ⊗ V2λ with the following subspaces of C[W ] D : ⎧ G = GLn , ⎨ Det −n C[W ] ⊂ C[W ]Det , C[W ] ⊗ V2λ Pfaff −n C[W ] ⊂ C[W ]Pfaff , G = On , ⎩ Det −n/2 C[W ] ⊂ C[W ]Det , G = Spn . In the first case, this identification is correct because g = (g1 , g2 ) ∈ H operates on the determinant function Det −n : W = End(Cq ) → C as (g. Det−n )(z) = Det −n (g1−1 z (g2t )−1 ) = Det n (g1 )Det n (g2 )Det −n (z), which is the desired behavior since Det n (g1 g2t ) agrees with the character χ (g1 , g2 )2 associated with the H -representation V2λ . In the other two cases we have (g. Det−n/2 )(w) = Det −n/2 (g −1 w τb (g)−1 ) = Det n (g)Det−n/2 (w) (b = a, s).
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Again, this is as it should be since Det n (g) = χ (g)2 is the desired character for V2λ . Here √ one should bear in mind that n is always an even number in the third case, and that Det = Pfaff in the second case. Recall now that o ≡ Id denotes the identity element in W , let Ho = {h ∈ H | h.o = o} be the isotropy group of o, and observe that H.o is isomorphic to H/Ho , ⎧ ⎨ (GLq × GLq )/GLq , G = GLn , G = On , H/Ho = GL2q /Sp2q , ⎩ GL /O , G = Spn . 2q 2q Then fix some maximal compact subgroup K ⊂ H such that the isotropy group K o ⊂ K is a maximal compact subgroup of the stabilizer Ho ⊂ H . Since Ho is a reductive group, we have natural isomorphisms C[W ] D = C[H/Ho ] = C[H ] Ho . The maximal compact subgroup K ⊂ H is Zariski dense, i.e., a polynomial function that vanishes on K also vanishes on H , so C[H ] = C[K ]. For the same reason, given any locally finitedimensional H -representation, the subspace of Ho -fixed points coincides with the K o -fixed points. Summarizing, we have i
C[W ] ⊗ V2λ → C[W ] D = C[H/Ho ] = C[H ] Ho = C[H ] K o = C[K ] K o . The benefit from this sequence of identifications is that on the space C[K ] there exists a natural and non-trivial K -invariant projection. Indeed, letting dk be a Haar measure on K , we may view f ∈ C[W ] D as a function on K and integrate: C[W ] D → C, f → f (k.o) dk. K
This projection is K -invariant, and its restriction to our space C[W ] ⊗ V2λ is still nontrivial. In the first case this is because Detn ∈ C[W ] and hence C 1 ⊂ Det−n C[W ]; in the last two cases Det n/2 ∈ C[W ] and hence C 1 ⊂ Det −n/2 C[W ]. We can now reformulate Cor. 4.8 to make the statement more concrete. Proposition 4.9. For each case G = GLn , On , or Spn , there is a choice of normalized Haar measure dk so that the following holds for all f ∈ ∧(V ∗ )G . If F ∈ C[W ] is any lift w.r.t. the identification and projection C[W ] C[W ] ⊗ Vλ → ∧(V ∗ )G then V [ f ] = F(k.o)Det −n (k) dk. K
Next, let us give another version of the ‘bosonization’ formula of Prop. 4.9 in order to get a better match with the supersymmetric formula to be developed below. For that, notice that K o = USp2q for G = On , K o = O2q (R) for G = Spn , and K o = Uq acting by elements (k, (k −1 )t ) for G = GLn . From this we see that Det −n (k) = 1 for k ∈ K o in all cases. We can therefore push down the integral over K to an integral over the orbit K .o K/K o . We henceforth denote this orbit by Dq := K .o. Writing y := kτ (k −1 ) for G = O, Sp we have Det −n (k) = Det −n/2 (kτ (k)−1 ) = Det −n/2 (y). Similarly, letting y := k1 (k2 )t for G = GL we have Det −n (k) = Det−n (k1 )Det −n (k2 ) = Det −n (y). Thus
the relation Det −n (k) = Det −n (y) always holds if we set n = (1 + |m|)−1 n, i.e., n = n for G = GL and n = n/2 for G = O, Sp. Let now dµ Dq denote a K -invariant measure on the K -orbit Dq . According to our general conventions, dµ Dq is normalized in such a way that (dµ Dq )o coincides with the
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Euclidean volume density on To Dq = W ∩ Lie(K ) which is induced by the quadratic form A → −Tr C p A2 for G = GL and A → − 21 TrU p A2 for G = O, Sp. We denote this by A → −Tr A2 for short. Pushing the Haar measure dk forward by K → K .o = Dq we obtain dµ Dq times a constant. Thus the formula of Prop. 4.9 becomes
V [ f ] = const × F(y) Det −n (y) dµ Dq (y). (4.2) Dq
To determine the unknown constant of proportionality, it suffices to compute both sides of the equation for some special choice of f (and a corresponding function F). If
we choose F(y) = eTr y , then f is simply a Gaussian with Berezin integral V [ f ] = 1. However, the integral on the right-hand side is not quite so easy to do. We postpone this computation until the end of the paper, where we will carry it out using a supersymmetric reduction technique based on relations developed below. To state the outcome, we recall the definition of the groups K n, p and let the sign of the positive integer p now be reversed; according to Table 2 of Sect. 3.4 this means that K n,− p = Un+ p , On+2 p (R), and USpn+2 p for G = GL, O, and Sp.
Lemma 4.10. Dq eTr y Det −n (y) dµ Dq (y) = (2π )−qn 2−qm vol(K n,−q )/vol(K n ). Remark. The similarity of this formula with Eq. (3.2) is not an accident; in fact, in Sect. 5.9 we will establish Lemma 4.10 by reduction to the latter result. Using Lemma 4.10 we can now eliminate the unknown constant of proportionality from (4.2). To state the resulting reformulation of Prop. 4.9, we will use the surjective mapping Q ∗ : O(W ) → ∧(V ∗ )G defined in Sect. 4.1. Theorem 4.11. For f ∈ ∧(V ∗ )G n , if F ∈ (Q ∗ )−1 ( f ) ∈ O(W ) is any holomorphic function in the inverse image of f , the Berezin integral f → V [ f ] can be computed as an integral over the compact symmetric space Dq K /K o : vol(K n )
V [ f ] = (2π )qn 2qm F(y) Det −n (y) dµ Dq (y), vol(K n,−q ) Dq where n = n for G n = GLn and n = n/2 for G n = On , Spn . 4.7. Shifting by nilpotents. In this last subsection, we derive a result which will be needed in the supersymmetric context of Sects. 5.8 and 5.9. Let p := To Dq be the tangent space of Dq = K .o at the identity Id = o. Since H = K C , Ho = K oC , Dq K /K o , and W is the closure of H/Ho , the tangent space p is a real form of W . More precisely, p = W ∩ Lie(K ). Linearizing τ (y) = τ (kτ (k)−1 ) = τ (k)k −1 = y −1 at y = o, we note that ξ ∈ p satisfies τ∗ (ξ ) = −ξ , where τ∗ is the differential τ∗ := dτ |o . Let now Dq be equipped with the canonical Riemannian geometry in which K acts by isometries and which is induced by the trace form −Tr on p. With each w ∈ W associate a complex vector field tw ∈ (Dq , C ⊗ T Dq ) by (tw f )(y) =
d f (y + tw) dt
t=0
.
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Lemma 4.12. The vector field tw has the divergence div(tw )(y) = −(q − m/2) Tr (y −1 w), where m = 0, +1, −1 for G = GL, O, Sp. Proof. For y ∈ Dq choose some fixed element k ∈ K so that y = kτ (k −1 ). Fixing an orthonormal basis {eα } of p, define local coordinate functions x α in a neighborhood of the point y by the equation y = k ex
α (y )e
α
τ (k −1 ).
By their construction via the exponential mapping, these are Riemann normal coordi nates centered at y. The Riemannian metric expands around y as α d x α ⊗ d x α + . . ., with vanishing corrections of linear order in the coordinates x α . Let ∂α = ∂/∂ x α and express the vector field tw in this basis as tw = twα ∂α . Differenα
tiating the equation k −1 (y + tw)τ (k) = e x (y +tw)eα with the help of the relation d α
x (y + tw) dt
t=0
= (Ltw x α )(y ) = twα (y ),
where Ltw is the Lie derivative w.r.t. the vector field tw , and then solving the resulting equation for twα , one obtains the following expansion of twα in powers of the x β : 1 twα (·) = −Tr (k −1 wτ (k)eα ) + x β (·)Tr k −1 wτ (k)(eα eβ + eβ eα ) + . . . . 2 Since the metric tensor is of the locally Euclidean form α d x α ⊗ d x α + . . ., the divergence is now readily computed to be Tr (k −1 wτ (k)eα2 ). div(tw )(kτ (k)−1 ) = (∂α twα )(kτ (k)−1 ) = α
eα2
The sum of squares is independent of the choice of basis eα . Making any convenient choice, a short computation shows that − eα2 = (q − m/2) Id, α
where m = 0, +1, −1 for G = GL, O, Sp. The statement of the lemma now follows by inserting this result in the previous formula and recalling y = kτ (k −1 ). 2 Remark. A check on the formula for the sum of squares − α eα is afforded by the relations −Tr eα2 = 1 and dimR p = dimC W = q 2 , 2q(q − 1/2), 2q(q + 1/2) for G = GL, O, Sp. Note also this: defined by the equation div(tw )dµ Dq = Ltw dµ Dq , the operation of taking the divergence does not depend on the choice of scale for the metric tensor. Therefore we were free to use a normalization convention for the metric which differs from that used elsewhere in this paper. Lemma 4.13. Let F : Dq → C be an analytic function, and let N0 = ⊕k≥1 ∧2k (C• ) be the nilpotent even part of a (parameter) Grassmann algebra ∧(C• ). Then for any w ∈ N0 ⊗ W one has F(y) dµ Dq (y) F(y + w) dµ Dq (y) = . q+m/2 (Id − y −1 w) Dq Dq Det
Superbosonization
379
Proof. Let tw be the vector field generating translations y → y + sw (s ∈ R). Since w is nilpotent, the exponential exp(sLtw ) of the Lie derivative Ltw is a differential operator s Lt of finite order. Applying it to the function F one has (e w F)(y) = F(y + sw). Now for any density on Dq the integral Dq Ltw vanishes by Stokes’ theorem for the closed manifold Dq . Therefore, partial integration gives F(y + sw) dµ Dq (y) = F(y) e−s Ltw dµ Dq (y) = F(y) Js (y) dµ Dq (y), Dq
Dq
Dq
where Js : Dq → C ⊕ N0 is the function defined by e−s Ltw dµ Dq = Js dµ Dq . We now set up a differential equation for Js . For this we consider the derivative d −s Ltw e dµ Dq = e−s Ltw −Ltw dµ Dq . ds By the relation Ltw dµ Dq = div(tw )dµ Dq we then get d Js dµ Dq = −e−s Ltw div(tw )dµ Dq = −e−s Ltw (div(tw )) Js dµ Dq . ds Using the expression for div(tw ) from Lemma 4.12 we obtain the differential equation d d log Js (y) = (q − m/2)Tr (w(y − sw)−1 ) = −(q − m/2) Tr log(y − sw). ds ds The solution of this differential equation with initial condition Js=0 = 1 is Js (y) =
Detq−m/2 (y) , Detq−m/2 (y − sw)
and setting s = 1 yields the statement of the lemma.
5. Full Supersymmetric Situation We finally tackle the general situation of V = V0 ⊕ V1 , where both V0 and V1 are non-trivial. The superbosonization formulas (1.10, 1.13) in this situation will be proved by a chain of variable transformations resulting in reduction to the cases treated in the two preceding sections. This proof has the advantage of being constructive.
5.1. More notation. To continue the discussion in the supersymmetric context we need some more notation. If V = V0 ⊕ V1 is a Z2 -graded vector space, one calls (V0 ∪ V1 )\{0} the subset of homogeneous elements of V . A vector v ∈ V0 \{0} is called even and v ∈ V1 \{0} is called odd. On the subset of homogeneous elements of V one defines a parity function | · | by |v| = 0 for v even and |v| = 1 for v odd. Whenever the parity function v → |v| appears in formulas and expressions, the vector v is understood to be homogeneous even without explicit mention. There exist two graded-commutative algebras that are canonically associated with k V = V0 ⊕ V1 . To define them, let T (V ) = ⊕∞ k=0 T (V ) be the tensor algebra of V , and let I± (V ) ⊂ T (V ) be the two-sided ideal generated by multiplication of T (V ) with all
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combinations v ⊗ v ± (−1)|v||v | v ⊗ v for homogeneous vectors v, v ∈ V . Then the graded-symmetric algebra of V = V0 ⊕ V1 is the quotient S(V ) := T (V )/I− (V ) S(V0 ) ⊗ ∧(V1 ), which is isomorphic to the tensor product of the symmetric algebra of V0 with the exterior algebra of V1 . The graded-exterior algebra of V is the quotient ∧(V ) := T (V )/I+ (V ) ∧(V0 ) ⊗ S(V1 ). Here we have adopted Kostant’s language and notation [14]. Recall that our goal is to prove an integration formula for integrands in AVG , the graded-commutative algebra of G-equivariant holomorphic functions f : V0 → ∧(V1∗ ) with V0 and V1 given in (2.3, 2.4). For that purpose we will view the basic algebra AV as a completion of the graded-symmetric algebra S(V ∗ ) = T (V ∗ )/I− (V ∗ ) S(V0∗ ) ⊗ ∧(V1∗ ). The latter algebra is Z-graded by S(V ∗ ) = ⊕k≥0 Sk (V ∗ ), where Sk (V ∗ )
k l=0
Sl (V0∗ ) ⊗ ∧k−l (V1∗ ) .
The action of G on V preserves the Z2 -grading V = V0 ⊕ V1 . Thus G acts on T (V ∗ ) while leaving the two-sided ideal I− (V ∗ ) invariant, and it therefore makes sense to speak of the subalgebra S(V ∗ )G of G-fixed elements in S(V ∗ ). It is a result of R. Howe – see Theorem 2 of [11] – that for each of the cases G = GLn , On , and Spn , the graded-commutative algebra S(V ∗ )G is generated by S2 (V ∗ )G . Hence, our attention once again turns to the subspace S2 (V ∗ )G of quadratic invariants.
5.2. Quadratic invariants. It follows from the definition of the graded-symmetric algebra S(V ∗ ) that the subspace of quadratic elements decomposes as S2 (V0∗ ⊕ V1∗ ) = S2 (V0∗ ) ⊕ ∧2 (V1∗ ) ⊕ (V0∗ ⊗ V1∗ ). So, since G acts on V0∗ and V1∗ we have a decomposition S2 (V0∗ ⊕ V1∗ )G = S2 (V0∗ )G ⊕ ∧2 (V1∗ )G ⊕ (V0∗ ⊗ V1∗ )G. To describe the components let us recall the notation Ur = Cr ⊕ (Cr )∗ for r = p, q. Lemma 5.1. S2 (V ∗ )G is isomorphic as a Z2 -graded complex vector space to W ∗ , where the even and odd components of W = W0 ⊕ W1 are W0 = W00 × W11 W1 =
⎧ ⎨ End(C p ) × End(Cq ), = Syms (U p ) × Syma (Uq ), ⎩ Sym (U ) × Sym (U ), p q a s
Hom(Cq , C p ) ⊕ Hom(C p , Cq ), Hom(Uq , U p ),
G = GLn , G = On , G = Spn ,
G = GLn , G = On , Spn .
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Proof. Writing S2 (V0∗ )G W00 and ∧2 (V1∗ )G W11 , the statement concerning the even part W0 ⊂ W is just a summary of Lemmas 3.1, 3.3, and 4.1. Thus what remains to be done is to prove the isomorphism (V0∗ ⊗ V1∗ )G W1∗ between odd components. Let us prove the equivalent statement (V0 ⊗ V1 )G W1 . In the case of G = GLn there are two types of invariant: we can compose an element L˜ ∈ Hom(C p , Cn ) with an element K ∈ Hom(Cn , Cq ) to form K L˜ ∈ Hom(C p , Cq ), or else compose K˜ ∈ Hom(Cq , Cn ) with L ∈ Hom(Cn , C p ) to form L K˜ ∈ Hom(Cq , C p ). This already gives the desired statement (V0 ⊗ V1 )GLn Hom(Cq , C p ) ⊕ Hom(C p , Cq ). In the cases of G = On , Spn we use Hom(Cn , Cr ) Hom((Cr )∗ , (Cn )∗ ) and the G-equivariant isomorphism β : Cn → (Cn )∗ to make the identifications V0 Hom(Cn , U p ), V1 Hom(Uq , Cn ). After this, the G-invariants in V0 ⊗ V1 are seen to be in one-to-one correspondence with composites L K˜ ∈ Hom(Uq , U p ), where K˜ ∈ Hom(Uq , Cn ) and L ∈ Hom(Cn , U p ). Remark. Defining Z2 -graded vector spaces C p|q := C p ⊕ Cq and U p|q := U p ⊕ Uq we could say that S2 (V )GLn End(C p|q ), while S2 (V )On S2 (U p|q ) and S2 (V )Spn ∧2 (U p|q ). We will not use these identifications here. 5.3. Pullback from AW to AVG . With W = W0 ⊕W1 as specified in Lemma 5.1, consider now the algebra AW of holomorphic functions F : W0 → ∧(W1∗ ). At the linear level we have the isomorphism of Lemma 5.1, which we here denote by Q ∗2 : W ∗ → S2 (V ∗ )G . This extends in the natural way to an isomorphism of tensor algebras Q ∗T : T (W ∗ ) → T S2 (V ∗ )G . Since Q ∗2 is an isomorphism of Z2 -graded vector spaces, Q ∗T sends the ideal I− (W ∗ ) ⊂
T (W ∗ ) generated by the graded-skew elements w ⊗ w − (−1)|w||w | w ⊗ w into the
ideal I− (V ∗ ) ⊂ T (V ∗ ) generated by the same type of element v ⊗ v − (−1)|v||v | v ⊗ v. Now, taking the quotient of T (V ∗ ) by I− (V ∗ ) is compatible with the reductive action of G, and it therefore follows that Q ∗T descends to a mapping Q ∗ : S(W ∗ ) → S(V ∗ )G. Because Q ∗ (W ∗ ) = Q ∗2 (W ∗ ) = S2 (V ∗ )G and S(V ∗ )G is generated by S2 (V ∗ )G , the map Q ∗ : S(W ∗ ) → S(V ∗ )G is surjective. The same holds true [18] at the level of our holomorphic functions AW and AVG : Proposition 5.2. The homomorphism of algebras Q ∗ : AW → AVG is surjective.
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5.4. Berezin superintegral form. For a Z2 -graded complex vector space such as our space V = V0 ⊕ V1 with dimensions dim V0 = 2 pn and dim V1 = 2qn, we denote by Ber(V ) the complex one-dimensional space Ber(V ) = ∧2 pn (V0∗ ) ⊗ ∧2qn (V1 ). Let now each of the Hermitian vector spaces V0 and V1 be endowed with an orienta˜ V0 ∈ ∧2 pn (V ∗ ) and a canonical generator tion. Then there is a canonical top-form 0 2qn ˜ V0 ⊗ V1 ∈ Ber(V ) is called the (flat) V1 ∈ ∧ (V1 ). Their tensor product V := Berezin superintegral form of V . Such a form V determines a linear mapping V : AV → (V0 , ∧2 pn (V0∗ )),
f → V [ f ],
from the algebra of holomorphic functions f : V0 → ∧(V1∗ ) to the space of top-degree holomorphic differential forms on V0 . Indeed, if v is any element of V0 , then by pairing f (v) ∈ ∧(V1∗ ) with the second factor V1 of V we get a complex number, and ˜ V0 results in an element of ∧2 pn (V ∗ ). subsequent multiplication by the first factor 0 In keeping with the approach taken in Sect. 3, we want to integrate over the real vector space V0,R defined as the graph of † : Hom(Cn , C p ) → Hom(C p , Cn ). For ˜ V0 | restricted to V0,R . (This this, let dvol V0,R denote the positive density dvol V0,R := | change from top-degree forms to densities is made in anticipation of the fact that we will transfer the integral to a symmetric space which in certain cases is non-orientable; see the Appendix for more discussion of this issue.) The Berezin superintegral of f ∈ AV over the integration domain V0,R is now defined as the two-step process of first converting the integrand f ∈ AV into a holomorphic function V1 [ f ] : V0 → C and then integrating this function against dvol V0,R over the real subspace V0,R : f → V1 [ f ] dvol V0,R . V0,R
Our interest in the following will be in this kind of integral for the particular case of G-equivariant holomorphic functions f : V0 → ∧(V1∗ ) (i.e., for f ∈ AVG ). 5.5. Exploiting equivariance. Recall from Sect. 3 the definition of the groups K n , K p , K n, p , and G p . Recall also that X p, n = ψ(Hom(Cn , C p )) denotes the vector space of structure-preserving linear transformations Cn → U p . To simplify the notation, let the isomorphism ψ now be understood, i.e., write ψ(L) ≡ L. The subset of regular elements in X p, n is denoted by X p, n . Taking ∈ X p, n to be the orthogonal projector Cn = U p ⊕ Un, p → U p we have the isomorphism ∼
G p ×(K p ×K n, p ) K n → X p, n , (g, k) → g k. Note that X p, n is a left G p -space and a right K n -space. Note also the relations k = k for k ∈ K p and k = 0 for k ∈ K n, p . Since the compact subgroup K n ⊂ G acts on V0,R , the given integrand f ∈ AVG restricts to a function f : X p, n → ∧(V1∗ ) which has the property of being K n -equivariant: f (L) = f (g k) = k −1 . f (g ) (k ∈ K n ).
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383
Now notice that since the action of G on ∧2qn (V1 ) is trivial, the Berezin form V1 is invariant under G and hence invariant under the subgroup K n : k( V1 ) = V1 ◦ k ∗ = V1 (k ∈ K n ). Consequently, applying V1 to the K n -equivariant function f we obtain V1 [ f (g k)] = V1 [k −1 . f (g )] = V1 [ f (g )], and this gives the following formula for the integral of f, vol(K n ) V1 [ f (L)] dvol V0,R (L) = V1 [ f (g )] J (g)dg K p , vol(K n, p )
(5.1)
G p /K p
as an immediate consequence of Prop. 3.14. Based on this formula, our next step is to process the integrand V1 [ f (g )]. 5.6. Transforming the Berezin integral. It will now be convenient to regard the odd vector space V1 = Hom(Cn , Cq ) ⊕ Hom(Cq , Cn ) for the case of G = GLn as V1 Hom(Cq , Cn ) ⊕ Hom((Cq )∗ , (Cn )∗ ) In the other cases, using the isomorphism β :
Cn
→
(Cn )∗
(G = GL). we make the identification
V1 Hom(Cq ⊕ (Cq )∗ , Cn ) = Hom(Uq , Cn ),
(G = O, Sp).
Following Sect. 3.4 we fix an orthogonal decomposition Cn = U p ⊕Un, p for G = On , Spn and Cn = C p ⊕Cn− p for G = GLn , which is Euclidean, Hermitian symplectic, and Hermitian, respectively, and let this induce a vector space decomposition V1 = V ⊕ V⊥ in the natural way. For G = On , Spn the summands are V = Hom(Uq , U p ), V⊥ = Hom(Uq , Un, p )
(G = O, Sp),
and in the case of G = GLn we have V = Hom(Cq , C p ) ⊕ Hom((Cq )∗ , (C p )∗ ),
V⊥ = Hom(Cq , Cn− p ) ⊕ Hom((Cq )∗ , (Cn− p )∗ )
(G = GL).
From the statement of Lemma 5.1 we see that V is isomorphic to W1 as a complex vector space (though not as a K p -space) in all three cases. The decomposition V1 = V ⊕ V⊥ induces a factorization ∧ (V1∗ ) ∧(V∗ ) ⊗ ∧(V⊥∗ )
(5.2)
of the exterior algebra of V1∗ . In all three cases (GL, O, Sp) the decomposition of V1 and that of ∧(V1∗ ) is stabilized by the group K p × K n, p . We further note that K p → K n acts trivially on ∧(V⊥∗ ) while K n, p → K n acts trivially on ∧(V∗ ). To compute V1 [ f (g )], we are going to disect the Berezin form V1 according to the decomposition (5.2). For this, recall that if V = A ⊕ A∗ is the direct sum of an N -dimensional vector space A and its dual A∗ , then there exists a canonical generator V = A⊕A∗ ∈ ∧2N (A ⊕ A∗ ) which is given by A⊕A∗ = f N ∧ e N ∧ . . . ∧ f 1 ∧ e1 for any basis {e j } of A with dual basis { f j } of A∗ . Note that A⊕A∗ = (−1) N A∗ ⊕A . The following statement is an immediate consequence of the properties of ∧.
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Lemma 5.3. If A, B, C are vector spaces and A = B ⊕ C then A⊕A∗ = B⊕B ∗ ∧ C⊕C ∗ . Now V1 and all of our spaces V⊥ and V are the direct sum of a vector space and its dual. Recall from Sect. 4.3 that V1 = Hom((Cq )∗ ,(Cn )∗ )⊕Hom(Cq ,Cn ) , and let V = Hom((Cq )∗ ,(C p )∗ )⊕Hom(Cq ,C p ) , V⊥ = Hom((Cq )∗ ,(Cn− p )∗ )⊕Hom(Cq ,Cn− p ) for G = GL, while in the case of G = O, Sp the corresponding definitions are V = Hom((Cq )∗ , U p )⊕Hom(Cq , U p ) , V⊥ = Hom((Cq )∗ , Un, p )⊕Hom(Cq , Un, p ) . Here the vector spaces Hom((Cq )∗ , U p ) and Hom(Cq , U p ) are regarded as dual to each other by the symmetric bilinear form s : U p × U p → C for G = O and the alternating bilinear form a : U p × U p → C for G = Sp. This means that for G = O we have V = Hom((Cq )∗ ,(C p )∗ )⊕Hom(Cq ,C p ) ∧ Hom((Cq )∗ ,C p )⊕Hom(Cq ,(C p )∗ ) , while the same Berezin form V for G = Sp has an extra sign factor (−1) pq due to the alternating property of a (cf. the sentence after the definition of W1 in Eq. (1.12)). The same conventions hold good in the case of V⊥ . For this we need only observe that the given symmetric or alternating bilinear form on Un, p induces such a form on V⊥ . Now, applying Lemma 5.3 to the present situation we always have V1 = V ∧ V⊥ . 5.6.1. Transformation of V . Recall the isomorphism of vector spaces V W1 , which we now realize as follows. Using the identifications V Hom(Uq , U p ) for the case of G = O, Sp and V Hom(Cq , C p ) ⊕ Hom((Cq )∗ , (C p )∗ ) for G = GL, we apply t ). ˜ g ∈ G p to v ∈ V to form gv, where gv for G = GL means gv = g.( L˜ ⊕L t ) = (g L)⊕(gL Note that the mapping (g, v) → gv has the property of being K p -invariant. Given this isomorphism g : V → W1 , let (g −1 )∗ : ∧(V∗ ) → ∧(W1∗ ), f → g. f , be the induced isomorphism preserving the pairing between vectors and forms. Then V1 [ f (g )] = g( V ) ∧ V⊥ [g. f (g )], so our next step is to compute g( V ). Here it should be stressed that we define the Berezin form W1 by the same ordering conventions we used to define V above. Lemma 5.4. Under the isomorphism V → W1 by v → gv the Berezin forms W1 and V are related by g( V ) = Detq (gg † ) W1 . Proof. Consider first the case of G = O, Sp, where V = Hom(Uq , U p ) and the same choice of polarization Hom(Uq , U p ) = Hom((Cq )∗ , U p ) ⊕ Hom(Cq , U p ) determines both V and W1 . Applying g ∈ G p to V ∈ ∧4 pq (V ) we obtain g( V ) = Det 2q (g) W1 . The groups at hand are G p = GL2 p (R), GL p (H), and Det(g) is real for these. Hence q Det 2q (g) = Det(g)Det(g) = Detq (gg † ). In the case of G = GL we have V = Hom((Cq )∗ ,(C p )∗ )⊕Hom(Cq ,C p ) . Transforming the second summand by L˜ → g L˜ for g ∈ GL p (C) we get the Jacobian Detq (g), transforming the first summand by L t → gL t we get Detq (g). Thus, altogether we obtain again g( V ) = Detq (g)Detq (g) W1 = Detq (gg † ) W1 .
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5.6.2. Bosonization of V⊥ . We turn to the Berezin form V⊥ for the factor ∧(V⊥∗ ) of the decomposition (5.2). Recall that the elements of this exterior algebra ∧(V⊥∗ ) are fixed under the action of K p . Since k = 0 for k ∈ K n, p , the K n -equivariance of f ∈ AVG implies that g. f (g ) (for any fixed g ∈ G p ) lies in ∧(W1∗ ) ⊗ ∧(V⊥∗ ) K n, p . For future reference, we are now going to record a (bosonization) formula for the Berezin integral V⊥ : ∧(V⊥∗ ) K n, p → C. For this notice that, since the action of K n, p is complex linear we have ∧(V⊥∗ ) K n, p = ∧(V⊥∗ )G n, p , where G n, p is the complexification of K n, p . From Table 2 of Sect. 3.4 we read off that G n, p = GLn− p (C), On−2 p (C), and Spn−2 p (C) for our three cases of GL, O, and Sp, respectively. The subalgebra ∧(V⊥∗ )G n, p is generated, once again, by ∧2 (V⊥∗ )G n, p , the quadratic invariants. Applying Lemma 4.1 with V ∗ ≡ V⊥∗ , G ≡ G n, p , and W ≡ W11 we get ∗ . ∧2 (V⊥∗ ) K n, p = ∧2 (V⊥∗ )G n, p W11
Now by the principles expounded in Sect. 4 we lift a given element f ⊥ ∈ ∧(V⊥∗ )G n, p to a holomorphic function F : W11 → C. To formalize this step, let P⊥∗ : O(W11 ) → ∧(V⊥∗ ) K n, p be the surjective mapping which was introduced in Sect. 4.1 and denoted by the generic symbol Q ∗ there. For F ∈ O(W11 ) we then have with n = n/(1 + |m|) the result V⊥ [P⊥∗ F]
qn(1− p/n ) qm
= (2π )
2
vol(K n, p ) vol(K n, p−q )
F(y) dµ Dq1 (y) Dq1
Det n − p (y)
,
(5.3)
as an immediate consequence of the formula of Thm. 4.11. Here we refined our notation by writing Dq1 for the compact symmetric spaces Dq of Sect. 4.6. The non-compact symmetric spaces D p introduced in Sect. 3.4 will henceforth be denoted by D 0p . 5.7. Decomposition of pullback. Recall from Sect. 5.3 that we have a pullback of gradedcommutative algebras Q ∗ : AW → AVG . To go further, we should decompose Q ∗ according to the manipulations carried out in the previous two subsections. This, however, will only be possible in a restricted sense, as some of our transformations require that the even part of w ∈ W be invertible. We start with a summary of the sequence of operations we have carried out so far. Recall that the elements of AW are holomorphic functions F : W0 → ∧(W1∗ ), where W1 and W0 = W00 × W11 were described in Lemma 5.1. Since our domain of integration will be D p ≡ D 0p G p /K p , given F ∈ AW let F1 denote F restricted to D 0p ⊂ W00 : F1 : D 0p × W11 → ∧(W1∗ ).
(5.4)
Now we use the Cartan embedding G p /K p → D 0p ⊂ G p by g → gθ (g −1 ) = gg † to pull back F1 in its first argument from D 0p to G p /K p . Applying also the mapping P⊥∗ : O(W11 ) → ∧(V⊥∗ ) K n, p we go to the second function F2 : G p /K p → ∧(W1∗ ) ⊗ ∧(V⊥∗ ) K n, p .
(5.5)
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In the next step, employing the isomorphism V → W1 , v → gv (pointwise for each coset g K p ∈ G p /K p ) we pull back F2 to a K p -equivariant function F3 : G p → ∧(V∗ ) ⊗ ∧(V⊥∗ ) K n, p .
(5.6)
Be advised that we are now at the level of the integrand F3 (g) = f (g ) of (5.1). In the final step, we pass to the unique extension of F3 to a K n -equivariant function K n −eqvt
F4 : X p, n −→ ∧(V1∗ )
(5.7)
by F4 (L) = F4 (g k) := F3 (g). Let us give a name to this sequence of steps. Definition 5.5. We denote by P ∗ the homomorphism of graded-commutative algebras taking F1 : D 0p × W11 → ∧(W1∗ ) to the K n -equivariant function F4 : X p, n → ∧(V1∗ ). The main point of this subsection will be to show that Q ∗ (restricted to D 0p × W11 ) is the composition of P ∗ with another homomorphism, S ∗ , which we describe next. Consider first the case of G = GL, where W = W0 ⊕ W1 and W0 = W00 ⊕ W11 = End(C p ) ⊕ End(Cq ), W1 = W01 ⊕ W10 = Hom(Cq , C p ) ⊕ Hom(C p , Cq ),
denote the subset of regular elements in W . On W := (W × W ) × and let W00 00 11 00 (W01 ⊕ W10 ) define a non-linear mapping S : W → W by
S(x, y ; σ, τ ) = (x, y + τ x −1 σ ; σ, τ ). This mapping is compatible with the structure of the graded-commutative algebra AW which is induced from the Z2 -grading W = W0 ⊕ W1 . Therefore, viewing the entries of σ and τ as anti-commuting generators, S determines an automorphism S ∗ : AW → AW
× W to ∧(W ∗ ). Adopting of the superalgebra AW of holomorphic functions from W00 11 1 the supermatrix notation commonly used in physics one would write x σ xσ =F (S ∗ F) . τ y τ y + τ x −1 σ Next consider the case of G = On , where W00 = Syms (U p ), W11 = Syma (Uq ), and W1 = Hom(Uq , U p ). Here we have the bilinear forms s on U p and a on Uq , and these determine an isomorphism Hom(Uq , U p ) → Hom(U p , Uq ), σ → σ T by s(σ v, u) = a(v, σ T u) (u ∈ U p , v ∈ Uq ). However, since σ and v in this definition are to be considered as odd and σ moves past v, the good isomorphism to use (the ’supertranspose’) has an extra minus sign: σ → σ sT := −σ T .
of W , define a mapping S : W → W
Restricting again to the regular elements W00 00
on W = (W00 × W11 ) × W1 by
S(x, y ; σ ) = (x, y + σ sT x −1 σ ; σ ).
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From x −1 ∈ Syms (U p ) and the definition of the transposition operation σ → σ sT via the bilinear forms s and a, it is immediate that σ sT x −1 σ ∈ W11 . Now for the same reasons as before, S determines an automorphism S ∗ : AW → AW . The definitions for the last case G = Spn are the same as for G = On but for the fact that the two bilinear forms s and a exchange roles. From here on we consider S ∗ to be restricted to the functions with domain D 0p × W11 . Lemma 5.6. The homomorphism of superalgebras Q ∗ : AW → AVG , when restricted to a homomorphism Q ∗ taking functions D 0p × W11 → ∧(W1∗ ) to K n -equivariant functions X p, n → ∧(V1∗ ), decomposes as Q∗ = P ∗ S∗. ∼
Proof. Since the isomorphism Q ∗ : W ∗ → S2 (V ∗ )G determines Q ∗ : AW → AVG , it suffices to check Q ∗ = P ∗ S ∗ at the level of the quadratic map Q : V → W . Let us write out the proof for the case of G = GL (the other cases are no different). Recall that the quadratic map Q : V → W in this case is given by ˜ ⊕ (K ⊕ K˜ ) → Q : (L ⊕ L)
L L˜ L K˜ . K L˜ K K˜
Now, fixing a regular element (L , L † ) ∈ X p, n , we have an orthogonal decomposition Cn = ker(L) ⊕ im(L † ), where im(L † ) C p and ker(L) Cn− p . Let L := L † (L L † )−1 L denote the orthogonal projection L : Cn → im(L † ). If we decompose K , K˜ as K = K (L) + K ⊥ (L),
K (L) = K L ,
K˜ = K˜ (L) + K˜ ⊥ (L),
K˜ (L) = L K˜ ,
then our homomorphism P ∗ is the pullback of algebras determined by the map P : X p, n × V1 → W , ((L , L † ), (K ⊕ K˜ )) →
L K˜ (L) L K˜ L L† L L† = . K (L)L † K ⊥ (L) K˜ ⊥ (L) K L † K (Id − L ) K˜
When the second map S : W → W is applied to this result, all blocks remain the same but for the W11 -block, which transforms as K (Id − L ) K˜ → K (Id − L ) K˜ + (K L † )(L L † )−1 (L K˜ ) = K K˜ . Thus S ◦ P agrees with Q on X p, n × V1 , which implies the desired result Q ∗ = P ∗ S ∗ . We now state an intermediate result en route to the proof of the superbosonization by f = Q ∗ F. We then do the folformula. Let f ∈ AVG and F ∈ AW be related lowing steps: (i) start from formula (5.1) for V1 [P ∗ S ∗ F] dvol V0,R ; (ii) transform the Berezin integral V1 [(P ∗ S ∗ F)(g )] by Lemma 5.4 for the part V and Eq. (5.3) for
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V⊥ ; (iii) use Cor. 3.16 to push the integral over G p /K p forward to D 0p by the Cartan embedding; (iv) use Detq (gg † ) = Detq (x). The outcome of these steps is the formula vol(K n ) V1 [ f ] dvol V0,R = 2(q− p)(n+m) π qn (2π )− pq(1+|m|) vol(K n, p−q ) " ! ×
D 0p
Dq1
W1 [S ∗ F(x, y)] Det p−n (y) dµ Dq1 (y) Detq+n (x) dµ D 0p (x).
(5.8)
Let us recall once more that n = n for G = GL and n = n/2 for G = O, Sp. 5.8. Superbosonization formula. We are now in a position to reap the fruits of all our labors. Introducing the notation (Sx∗ Det)(y) = Det(y + τ x −1 σ ) for G = GL and (Sx∗ Det)(y) = Det(y + σ sT x −1 σ ) for G = O and G = Sp, we note that the superdeterminant function SDet : D 0p × Dq1 → ∧(W1∗ ) is given by SDet(x, y) =
Det(x) . ((Sx∗ )−1 Det)(y)
We define a related function J : D 0p × Dq1 → ∧(W1∗ ) by J (x, y) =
Detq (x) Detq−m/2 (y) . ((Sx∗ )−1 Detq−m/2− p )(y)
Theorem 5.7. Let f : V0 → ∧(V1∗ )G be a G-equivariant holomorphic function which restricts to a Schwartz function along the real subspace V0,R . If F : W0 → ∧(W1∗ ) is any holomorphic function that pulls back to Q ∗ F = f , then vol(K n ) # V1 [ f ] dvol V0,R = 2(q− p)m vol(K n, p−q )
# W1 [(J · SDet n · F)(x, y)] dµ D 1 (y) dµ D 0 (x), × q p D 0p
Dq1
# V1 := 2 pn (2π )−qn V1 and # W1 := (2π )− pq(1+|m|) where n = n/(1 + |m|) ≥ p, and W1 are Berezin integral forms with adjusted normalization. Proof. We first observe that in the present context the formula of Lemma 4.13 can be written as F(x, y) Detq−m/2 (y) ∗ (S F)(x, y) dµ Dq1 (y) = dµ Dq1 (y). q−m/2 )(y) ∗ −1 Dq1 Dq1 ((Sx ) Det Our starting pointnow is Eq. (5.8). We interchange the linear operations of doing the ordinary integral Dq (...) dµ Dq and the Berezin integral W1 [...]. The inner integral over y is then transformed as
Dq1
(S ∗ F)(x, y) Det p−n (y) dµ Dq1 (y) =
F(x, y) Detq−m/2 (y) dµ Dq1 (y) Dq1
((Sx∗ )−1 Det n − p+q−m/2 )(y)
.
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The factor ((Sx∗ )−1 Det −n )(y) combines with the factor Det n (x) of the outer integral over x to give the power of a superdeterminant:
((Sx∗ )−1 Det −n )(y) Det n (x) = SDet n (x, y). Then, restoring the integrations to their original order (i.e., Berezin integral first, integral over y second) we immediately arrive at the formula of the theorem. Remark. The function J (x, y) is just the factor that appears in the definition of the Berezin measure D Q in Sect. 1.2. Using supermatrix notation, this is seen from the following computation: Detq (x) Detq−m/2 (y) Detq (x) Det p (y − τ x −1 σ ) = Detq−m/2− p (y − τ x −1 σ ) Detq−m/2 (1 − y −1 τ x −1 σ ) q p −1 Det (x) Det (y − τ x σ ) Detq (x − σ y −1 τ ) Det p (y − τ x −1 σ ) = = . Det −q+m/2 (1 − x −1 σ y −1 τ ) Det m/2 (1 − x −1 σ y −1 τ )
J (x, y) =
# V1 agrees with We also have adjusted the normalization constants, so that dvol V0,R ⊗ # W1 ◦ J agrees the Berezin superintegral form D Z , Z¯ ;ζ,ζ˜ of Eq. (1.1), and dµ D 0p dµ Dq1 ⊗ with D Q as defined in Eqs. (1.8, 1.11). Thus, assuming the validity of Thm. 4.11 we have now completed the proof of our main formulas (1.10) and (1.13). To complete the proof of Thm. 4.11 we have to establish the normalization given by Lemma 4.10. 5.9. Proof of Lemma 4.10. Lemma 4.10 states the value of the integral
eTr y Det −n (y) dµ Dq1 (y) Dq1
over the compact symmetric space Dq1 . To verify that statement, we are now going to compute this integral by supersymmetric reduction to a related integral,
e−Tr x Det n +q (x) dµ Dq0 (x), Dq0
over the corresponding non-compact symmetric space Dq0 . For that purpose, consider
Cn, q := W1 [(J · SDet n +q )(x, y)] eTr y−Tr x dµ Dq1 (y) dµ Dq0 (x), (5.9) Dq0
Dq1
(for each of the three cases G = GL, O, Sp) and first process the inner integral:
W1 [(J · SDet n +q )(x, y)] eTr y dµ Dq1 (y) =
Dq1
Dq1
$ W1
Det n +2q (x) Detq−m/2 (y)
((Sx∗ )−1 Det n +q−m/2 )(y)
= Det n +2q (x)
%
eTr y dµ Dq1 (y)
Dq1
W1 [Sx∗ (exp ◦Tr )(y)] Det −n (y) dµ Dq1 (y).
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Here, after inserting the definitions of SDet n +q and J for p = q, we again made use of the formula of Lemma 4.13, reading it backwards this time. The next step is to calculate the Berezin integral W1 of Sx∗ (exp ◦Tr )(y). By the definition of the shift operation Sx∗ this is a Gaussian integral. Its value is
W1 [S ∗ (exp ◦Tr )(y)] = eTr y Detq (x −1 ) in all three cases. Inserting this result into the above expression for Cn, q we get the following product of two ordinary integrals:
−Tr x n +q e Det (x) dµ Dq0 (x) × eTr y Det −n (y) dµ Dq1 (y). Cn, q = Dq0
Dq1
The first one is known to us from Eq. (3.2), while the second one is the integral that we actually want. The formula claimed for this integral in Lemma 4.10 is readily seen to be 2 equivalent to the statement that Cn, q = (2π )(1+|m|)q . Thus our final task now is to show 2 that Cn, q = (2π )(1+|m|)q . This is straightforward to do by the localization technique for supersymmetric integrals [19], as follows. To get a clear view of the supersymmetries of our problem, let us go back to our starting point: the algebra AVG of G-equivariant holomorphic functions V0 → ∧(V1∗ ) of the Z2 -graded vector space V = V0 ⊕ V1 for V0 = Hom(Cn , C p ) ⊕ Hom(C p , Cn ) and V1 = Hom(Cn , Cq ) ⊕ Hom(Cq , Cn ). There exists a canonical action of the Lie superalgebra gl p|q on C p|q , hence on V Hom(Cn , C p|q ) ⊕ Hom(C p|q , Cn ), and hence on the algebra AVG . To describe this gl p|q -action on AVG , let {E ia }, { E˜ ai }, {eib }, and {e˜bi } with index range i = 1, . . . , n and a = 1, . . . , p and b = 1, . . . , q be bases of Hom(Cn , C p ), Hom(C p , Cn ), Hom(Cn , Cq ), and Hom(Cq , Cn ), in this order. If {Fai }, { F˜ia }, etc., denote the corresponding dual bases, then the odd generators of gl p|q (the even ones will not be needed here) are represented on AVG by odd derivations dba = ε( f bi )δ(E ia ) + µ( F˜ia )ι(e˜bi ), d˜ab = ε( f˜ib )δ( E˜ ai ) − µ(Fai )ι(eib ), where the operators ε( f ), δ(v), µ( f ), and ι(v) mean exterior multiplication by the anti-commuting generator f , the directional derivative w.r.t. the vector v, (symmetric) multiplication by the function f , and alternating contraction with the odd vector v. Clearly, all of these derivations are G-invariant (for G = GLn , On , Spn ) and have vanishing squares (dba )2 = (d˜ab )2 = 0. Using the coordinate language introduced in Sect. 1.1 one could also write dba = ζbi
∂ ∂ ∂ ∂ + Z˜ ia b , d˜ab = ζ˜ib a − Z ai i . i ∂ Za ∂ζb ∂ ζ˜i ∂ Z˜ i
It will be of importance below that the flat Berezin superintegral form dvol V0,R ⊗ V1 is gl p|q -invariant, which means in particular that V1 [dba f ] dvol V0,R = V1 [d˜ab f ] dvol V0,R = 0 for any f ∈ AV with rapid decay when going toward infinity along V0,R . Superbosonization involves the step of lifting f ∈ AVG to F ∈ AW by the surjective G mapping Q ∗ : AW → AVG . Now, since W = S2 (V )G = T 2 (V )/I− (V ) and the Lie
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391
superalgebra gl p|q acts on T 2 (V ) by G-invariant derivations stabilizing I− (V ), we also have a gl p|q -action by linear transformations W → W . Realizing this action by derivations of AW we obtain a gl p|q -action on AW . In particular, there exist such derivations Dba and D˜ ab that for every F ∈ AW we have Q ∗ Dba F = dba Q ∗ F,
Q ∗ D˜ ab F = d˜ab Q ∗ F.
(In other words, our homomorphism of algebras Q ∗ : AW → AVG is gl p|q -equivariant.) For any positive integers p, q, n with n ≥ p consider now the Berezin superintegral
n W1 [J · SDet n · F] dµ Dq1 dµ Dq0 , I p, q [F] := D 0p
Dq1
which includes our integral Cn, q of interest as a special case by letting p = q and
F(x, y) = SDetq (x, y) eTr y−Tr x . n [F]; i.e., the Lemma 5.8. The odd derivations Dba and D˜ ab are symmetries of F → I p, q integrals of Dba F and D˜ ab F vanish, n a n ˜b I p, q [Db F] = I p, q [Da F] = 0 (a = 1, . . . , p ; b = 1, . . . , q),
for any integrand F ∈ AW such that Q ∗ F|V0,R is a Schwartz function. Proof. While some further labor would certainly lead to a direct proof of this statement, we will prove it here using the superbosonization formula of Thm. 5.7 in reverse. (Of course, to avoid making a circular argument, we must pretend to be ignorant of the constant of proportionality between the two integrals, which will remain an unknown until the proof of Lemma 4.10 has been completed. Such ignorance does not cause a problem here, as we only need to establish a null result.) Thus, applying the formula of Thm. 5.7 in the backward direction with an unknown constant, we have n a I p, q [Db F] = const × V1 [Q ∗ Dba F]dvol V0,R . We now use the intertwining relation Q ∗ Dba = dba Q ∗ of gl p|q -representations. The integral on the right-hand side is then seen to vanish because the integral form dvol V0,R ⊗ V1 n [D a F] = 0. By same argument also I n [D ˜ b F] = 0. is gl p|q -invariant. Thus I p, q p, q a b Thus we have 2 pq odd AW -derivations (or vector fields) Dba and D˜ ab which are symn . We mention in passing that for the cases of G = O and G = Sp metries of I p, n n q there exist further symmetries which promote the full symmetry algebra from gl p|q to osp2 p|2q . This fact will not concern us here. Let now p = q. Then there exists a distinguished symmetry q
D := Daa = D11 + D22 + . . . + Dq , which still satisfies D 2 = 0. Lemma 5.9. Viewed as a vector field on the supermanifold of functions Dq0 × Dq1 → ∧(W1∗ ), the numerical part of D vanishes at a single point o ≡ (Id, Id) ∈ Dq0 × Dq1 .
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Proof. We sketch the idea of the proof for G = GLn . In that case one verifies that D has the coordinate expression ∂ ∂ ∂ a + xba − yba D = σb + . ∂ xba ∂ yba ∂τba The second summand, the numerical part of D, is zero only when the coordinate functions xba and yba are equal to each other for all a, b = 1, . . . , q. Since Dq1 = Uq and Dq0 is the set of positive Hermitian q × q matrices, this happens only for x = Id ∈ Dq0 and y = Id ∈ Dq1 . The same strategy of proof works for the cases of G = On , Spn . We are now in a position to apply the localization principle for supersymmetric integrals [19]. Let F ∈ AW be a D-invariant function which is a Schwartz function on Dq0 . Choose a D-invariant function gloc : Dq0 × Dq1 → ∧(W1∗ ) with the property that gloc = 1 on some neighborhood U (o) ⊂ V (o) of o and gloc = 0 outside of V (o). (Such “localizing” functions do exist.) Then according to Theorem 1 of [19] we have n n Iq, q [F] = Iq, q [gloc F], n since Iq, q is D-invariant. (Although that theorem is stated and proved for compact supermanifolds, the statement still holds for our non-compact situation subject to the condition that integrands be Schwartz functions.) n [F] depends only on Taking V (o) to be arbitrarily small we conclude that F → Iq, q the numerical part of the value of F at o : n Iq, q [F] = const × num(F(o)).
To determine the value of the constant for G = GLn we consider the special function t
F = e− 2 (xa xb −ya yb +2σa τb ) . b a
b a
b a
2
n [F] = (2π )q num(F(o)) due An easy calculation in the limit t → +∞ then gives Iq, q to our choice of normalization for dµ Dq0 and dµ Dq1 . The same calculation for the cases 2
n [F] = (2π )2q num(F(o)). of G = On , Spn gives Iq, q These considerations apply to the integrand in Eq. (5.9) with num(F(o)) = 1. Thus 2 we do indeed get Cn, q = (2π )(1+|m|)q , and the proof of Lemma 4.10 is now finished.
6. Appendix: Invariant Measures In the body of this paper we never gave any explicit expressions for the invariant measures dµ D 0p and dµ Dq1 . There was no need for that, as these measures are in fact determined (up to multiplication by constants) by invariance with respect to a transitive group action, and this invariance really was the only property that was required. Nevertheless, we now provide assistance to the practical user by writing down explicit formulas for our measures (or positive densities) dµ D 0p and dµ Dq1 . For that purpose, we will use the correspondence between densities and differential forms of top degree. (Recall what the difference is: densities transform by the absolute value of the Jacobian, whereas top-degree differential forms transform by the Jacobian including sign.) Thus we shall give formulas for the differential forms corresponding to dµ D 0p and dµ Dq1 . This
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393
is a convenient mode of presentation, as it allows us to utilize complex coordinates for the complex ambient spaces as follows. Consider first the case of G = GLn (C), where D 0p = Herm+ ∩ End(C p ) and Dq1 = U ∩ End(Cq ). Then for r = p or r = q consider End(Cr ) and let z cc : End(Cr ) → C (with c, c = 1, . . . , r ) be the canonical complex coordinates of End(Cr ), i.e., the set of matrix elements with respect to the canonical basis of Cr . On the set of regular points of End(Cr ) define a holomorphic differential form ω(r ) by ω
(r )
= Det
−r
(z)
r &
dz cc ,
c, c =1
where z = (z cc ) is the matrix of coordinate functions. By the multiplicativity of the determinant and the alternating property of the wedge product, ω(r ) is invariant under transformations z → g1 z g2−1 for g1 , g2 ∈ GLr (C). The desired invariant measures (up to multiplication by an arbitrary normalization constant) are dµ D 0p ∝ ω( p)
Herm+ ∩End(C p )
, dµ Dq1 ∝ ω(q)
U∩End(Cq )
,
(6.10)
where we restrict ω(r ) as indicated and reinterpret dµ Dr• as a positive density on the orientable manifold Dr• (r = p, q). For example, for r = 1 we have ω(1) = z −1 dz. In this case we get an invariant positive density |d x| on the positive real numbers Herm+ ∩ C = R+ by setting z = e x with x ∈ R , and a Haar measure |dy| on the unit circle U ∩ C = U1 = S1 by setting z = eiy with 0 ≤ y ≤ 2π . Our normalization conventions for the invariant measures dµ Dr• are those described in Sect. 1.2. We turn to the cases of G = On (C) and G = Spn (C) and recall that the condition on elements M of the complex linear space Symb (C2r ) is M = tb M t (tb )−1 . On making the substitution M = L tb this condition turns into L = +L t for b = s, L = −L t for b = a, while the GL2r (C)-action on Symb (C2r ) by twisted conjugation becomes g.L = gL g t in both cases. Define the coordinate function z cc : Symb (C2r ) → C to be the function that assigns to M the matrix element of M(tb )−1 = L in row c and column c . We then have z cc = z c c for b = s and z cc = −z c c for b = a. As before let z = (z cc ) be the matrix made from these coordinate functions (where the transpose z t = z for b = s and z t = −z for b = a). Then let top-degree differential forms ω(r ; b) be defined locally on the regular points of Symb (C2r ) by & ω(r ; s) = Det −r −1/2 (z) dz cc , 1≤c≤c ≤r
ω
(r ; a)
= Det
−r +1/2
(z)
&
dz cc .
1≤c
These are invariant under pullback by L → gL g t , as the transformation behavior of Det −r ±1/2 is contragredient to that of the wedge product of differentials in both cases. We emphasize that this really is just a local definition so far, as the presence of the square root factors may be an obstruction to the global existence of such a form.
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0 and Now focus on the case of G = On (C). There, restriction to the domains Dδ, p 1 gives the differential forms ω( p; s) (q; a) Dδ, and ω . Both + 2 p 2q q Herm ∩Sym (C ) U∩Sym (C ) s
a
of these are globally defined. Indeed, we can take the factor L → Det −1/2 (L) in the first differential form to be the reciprocal of the positive square root of the positive Hermitian matrix M = L ts , and the square root L → Det 1/2 (L) appearing in the second form makes global sense as the Pfaffian of the unitary skew-symmetric matrix L t = −L. Reinterpreting these differential forms as densities we arrive at a GL2 p (R)-invariant 0 and a U -invariant measure on D 1 : measure on Dδ, 2q p δ, q dµ D 0 ∝ ω( p; s) δ, p
Herm+ ∩Syms (C2 p )
, dµ D 1 ∝ ω(q; a) δ, q
U∩Syma (C2q )
.
(6.11)
• are those of Sect. 1.2. Again, our normalization conventions for dµ Dδ,r
In the final case of G = Spn (C) the roles of ω(•; s) and ω(•; a) are reversed. This immediately leads to a good definition of dµ Dε,0 p for the non-compact symmetric space
0 . However, the remaining case of D 1 = U ∩ Sym (C2q ) is problematic because Dε, s p ε, q there exists no global definition of Det 1/2 on the unitary symmetric matrices. Thus the locally defined differential form ω(q; s) does not extend to a globally defined form 1 . (Please be advised that this is inevitable, as the compact symmetric space on Dε, q 1 Dε, q U2q /O2q lacks the property of orientability and on a non-orientable manifold any globally defined top-degree differential form must have at least one zero and therefore cannot be both non-zero and invariant in the required sense.) 1 . The Of course dµ Dε,1 q still exists as a density on the non-orientable manifold Dε, q discussion above is just saying that there exists no globally defined differential form corresponding to dµ Dε,1 q . Locally, we have dµ Dε,1 q ∝ ω(q; s) U∩Sym (C2q ) . s
Acknowledgements. This paper is the product of a mathematics-physics research collaboration funded by the Deutsche Forschungsgemeinschaft via SFB/TR 12. Note added in proof. After submission of this paper, it was brought to our attention that a bosonization formula for the fermion-fermion sector with Un symmetry had been developed by Kawamoto and Smit [13]. A supersymmetric generalization was suggested in Ref. [2].
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10. Hackenbroich, G., Weidenmüller, H.A.: Universality of Random-Matrix Results for Non-Gaussian Ensembles. Phys. Rev. Lett. 74, 4118–4121 (1995) 11. Howe, R.: Remarks on classical invariant theory. TAMS 313, 539–570 (1989) 12. Howe, R.: Perspectives on invariant theory: Schur duality, multiplicity free actions and beyond. In the Schur Lectures, Providence, RI: Amer. Math. Soc., 1995 13. Kawamoto, N., Smit, J.: Effective Lagrangian and dynamical symmetry breaking in strongly coupled lattice QCD. Nucl. Phys. B 192, 100–124 (1981) 14. Kostant, B.: Graded Lie theory and prequantization. Lecture Notes in Math. 570, 177–306 (1977) 15. Lehmann, N., Saher, D., Sokolov, V.V., Sommers, H.-J.: Chaotic scattering – the supersymmetry method for large number of channels. Nucl. Phys. A 582, 223–256 (1995) 16. Luna, D.: Fonctions differentiables invariantes sous l’operation d’un groupe reductif. Ann. Inst. Fourier 26, 33–49 (1976) 17. Schäfer, L., Wegner, F.: Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes. Z. Phys. B 38, 113–126 (1980) 18. Schwarz, G.: Lifting smooth homotopy of orbit spaces. Pub. Math. IHES 51, 37 (1980) 19. Schwarz, A., Zaboronsky, O.: Supersymmetry and localization. Commun. Math. Phys. 183, 463–476 (1997) 20. Wegner, F.J.: Disordered system with N orbitals per site: N = ∞ limit. Phys. Rev. B 19, 783–792 (1979) 21. Zirnbauer, M.R.: Supersymmetry methods in random matrix theory. In: Encyclopedia of Mathematical Physics, Vol. 5, Amsterdam: Elsevier, 2006 pp. 151–160 22. Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random matrix theory. J. Math. Phys. 37, 4986–5018 (1996) Communicated by J. Z. Imbrie
Commun. Math. Phys. 283, 397–415 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0507-4
Communications in
Mathematical Physics
Minimizing the Ground State Energy of an Electron in a Randomly Deformed Lattice Jeff Baker1 , Michael Loss2 , Günter Stolz1 1 University of Alabama at Birmingham, Department of Mathematics, Birmingham,
AL 35294-1170, USA. E-mail:
[email protected];
[email protected]
2 Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA.
E-mail:
[email protected] Received: 8 August 2007 / Accepted: 28 November 2007 Published online: 22 May 2008 – © Springer-Verlag 2008
Abstract: We provide a characterization ofthe spectral minimum for a random Schr¨odinger operator of the form H = − + i∈Zd q(x − i − ωi ) in L 2 (Rd ), where the single site potential q is reflection symmetric, compactly supported in the unit cube centered at 0, and the displacement parameters ωi are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible 2d -clusters of single site potentials. The main tool to prove this is a quite general phenomenon in the spectral theory of Neumann problems, which we dub “bubbles tend to the boundary.” How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schrödinger operator on this domain? For square or rectangular domains and reflection symmetric potentials, we show that the first Neumann eigenvalue is minimized when the potential sits in one of the corners of the domain and is maximized when it sits in the center of the domain. With different methods we also show a corresponding result for smooth strictly convex domains. 1. Introduction and Main Results 1.1. The Displacement Model. The one electron model of solid state physics describes the behavior of a single electron moving under the presence of an exterior force generated by the effective potentials of a fixed configuration of nuclei in a solid. Also disregarding electron-electron interactions, this results in the one-electron Schrödinger operator H = − + V in L 2 (Rd ), where − and V are the kinetic and potential energy of the particle, respectively. Typically one chooses a potential V that effectively models the characteristics of a particular solid. For example, one might use a periodic potential to model a crystal or
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other well ordered substance. While for a material containing a sufficient number of impurities, or disorder, one might use a random potential. In this paper we consider a potential generated by identical atoms or ions located at the points i + ωi , i ∈ Zd , i.e. Vω (x) = q(x − i − ωi ). i∈Zd
We refer to q as the single site potential and consider real valued single site potentials q ∈ L ∞ which are reflection symmetric, i.e. symmetric in each variable with the remaining variables fixed, and compactly supported in the unit cube 0 := (− 21 , 21 )d of Rd , i.e. supp q ⊂ [−r, r ]d ⊂ 0 , r < 1/2. We denote the collection of displacements by ω = {ωi }i∈Zd , where each ωi ∈ [−dmax , dmax ]d . Finally we choose r + dmax = 21 , which insures that adjacent single site potentials in the sum above do not overlap. For any possible collection of displacements ω, H (ω) := − + Vω
(1)
with domain H 2 (Rd ), the second order Sobolev space, defines a self adjoint operator in L 2 (Rd ). We will refer to the family H (ω) as the displacement model. As V is uniformly bounded with respect to ω, the spectrum of H (ω), σ (H (ω)), is uniformly bounded from below. The question we will address is the following: How can one characterize E 0 := inf inf σ (H (ω)), ω
(2)
i.e. the infimum of the ground state energy of H (ω) for all possible nuclear configurations ω? In particular, is there a minimizing configuration ωmin such that inf σ (H (ωmin )) = E 0 ,
(3)
and what does it look like? Our main result is the answer to this question: Theorem 1.1. A minimizing configuration ωmin for the ground state energy of H (ω) in the sense of (3) is given by ωimin = ((−1)i1 dmax , . . . , (−1)id dmax )
(4)
for all i = (i 1 , . . . , i d ) ∈ Zd . The energy minimizing potential Vωmin is 2-periodic in each coordinate and given by the densest possible cluster of the nuclei in the period cell (− 21 , 23 )d , namely all single site potentials within the cluster move as close to the center ( 21 , . . . , 21 ) of the period cell as possible, see Fig. 1. Thus, by Floquet-Bloch theory [20], E 0 is the lowest eigenvalue of −+ Vωmin restricted to L 2 ((− 21 , 23 )d ) with periodic boundary conditions. Due to the symmetry of the 2d -cluster this is the same as the ground state energy of − + q(x − (dmax , . . . , dmax )) on L 2 (0 ) with Neumann boundary conditions. For d = 1 Theorem 1.1 has been conjectured and partially proven in [17]. Our interest in this result is mostly motivated by the case where the displacements ωi , i ∈ Zd , are independent, identically distributed Rd -valued random variables with [−dmax , dmax ]d as the support of their common distribution µ. In this case we refer to
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Fig. 1. The support of Vωmin for d = 2 and radially symmetric f
H (ω) as the random displacement model, which is ergodic with respect to shifts in Zd . Thus its spectrum is almost surely deterministic, i.e. there exists ⊂ R such that σ (H (ω)) = for a.e. ω, e.g. [4]. Compared to other prominent models of random Schrödinger operators, e.g. the Anderson model or Poisson model, few rigorous results are known for the random displacement model. This is mostly due to the fact that H (ω) does not depend monotonically (in form sense) on the random parameters ωi . Even the structure of the almost sure spectrum is unclear. It can be said that σ (H (ω)), (5) = ω∈C per
where C per is the set of all configurations ω : Zd → supp µ which are periodic with respect to some sublattice of Zd . This follows by adapting the proof of the corresponding result for Anderson models, given e.g. in [14]. A consequence of Theorem 1.1 is Corollary 1.2. The infimum of the almost sure spectrum of the random displacement model is given by inf = E 0 = inf σ (H (ωmin )). Proof. This follows from (5) since ωmin ∈ C per and, by Theorem 1.1, inf σ (H (ωmin )) ≤ inf σ (H (ω)) for all other ω ∈ C per . Note that, at least for sign-definite q, the answer to the same question for the Anderson or Poisson model is quite straightforward and found by considerations involving not much more than minimizing the potential energy: For the Anderson model, inf is found by choosing all random couplings minimal. For the Poisson model one has inf = 0 if q ≥ 0 and inf = −∞ if q ≤ 0. In fact, the latter, with few exceptions, only requires that the negative part of q doesn’t vanish [1]. For the Anderson model with
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sign-indefinite q the description of or just inf causes difficulties similar to those for the random displacement model. Najar [18] has a result for this case in the small coupling regime, proven by perturbative arguments. What makes our result, as well as the techniques in its proof, rather interesting is that minimizing the spectrum in the displacement model requires an understanding of the interaction between kinetic and potential energy. Physically, one can understand our result best for the case of negative potential wells q. In this case the formation of clusters of 2d sites allows for states with low potential energy without sacrificing much kinetic energy. But we stress that Theorem 1.1 and Corollary 1.2 hold without any sign-restriction on q. For the multi-dimensional random displacement model it is not yet known if the spectrum is localized, in the sense of being pure point, near the bottom of the spectrum. This is in contrast to the situation for Anderson and Poisson models. For the Anderson model this is a long standing result, with the hardest case of Bernoulli distributed random couplings recently settled in [2]. The new type of multi scale analysis introduced in [2] was now also used to prove the corresponding fact for the Poisson model in arbitrary dimension [9,10]. For the one-dimensional displacement model, localization at all energies was proven in [3] and, with different methods and under more general assumptions, in [5]. The only available result on localization for the multi-dimensional displacement model is Klopp’s work [15], establishing the existence of a localized region for the semiclassical version −h 2 + Vω of (1) if h is sufficiently small. Theorem 1.1 should serve as a first step towards understanding the spectral type of − + Vω near inf by identifying the periodic configuration in which inf is attained. An important further step towards localization would be to quantify probabilistically how many other configurations have ground states close to inf , that is, to prove smallness of the integrated density of states (IDS) near inf (or a related finite volume property). To this end it is interesting to note that for d = 1 the configuration given by (4) (in this case “dimerization”) is only one of many minimizing periodic configurations. This will have interesting consequences for the IDS. In particular, one may not find the Lifshitz tail behavior familiar from Anderson and Poisson models and the exact asymptotics may depend strongly on the displacement distribution µ. However, we believe that, under suitable assumptions on q suggested by our proof of Theorem 1.1, in d ≥ 2 the configuration (4) is the unique periodic minimizer. We plan to investigate this further in a separate work.
1.2. Bubbles tend to the boundary. Theorem 1.1 amounts to optimizing the infinitely many parameters ωi , i ∈ Zd , with respect to minimizing the spectrum. Surprisingly, as will be shown in Sect. 3, its proof can be reduced to the following spectral optimization result in just one parameter. Theorem 1.3. Let q be as above, i.e. bounded, reflection symmetric, and supported in [−r, r ]d for some r < 21 . Let dmax = 21 −r and for a ∈ [−dmax , dmax ]d let HN0 (a) := − + q(x − a) in L 2 (0 ) with Neumann boundary conditions on ∂0 and denote the ground state energy of HN0 (a) by E 0 (a). Then we have the following alternative: Either (i) E 0 (a) is strictly maximized at a = 0 and strictly minimized in the 2d corners (±dmax , . . . , ±dmax ) of [−dmax , dmax ]d or
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(ii) E 0 (a) is identically zero. In this case the corresponding eigenfunction is constant outside of the support of q. In fact, we will show that in case (i) the function E 0 (a) is partially strictly decreasing away from the origin, i.e. that whenever all but one of the variables (a1 , . . . , ad ) are fixed, then E(a1 , . . . , ad ) is strictly decreasing for the remaining variable in [0, 1/2 − r ] and, by symmetry, strictly increasing in [−1/2 + r, 0]. A sufficient, but far from necessary condition for case (i) to hold is that q has fixed sign and does not vanish identically, as in this case E 0 (a) never vanishes. Case (ii) happens if the Neumann problem for − + q on the support of q has lowest eigenvalue 0. Non-vanishing q with this property are easily constructed. We find Theorem 1.3 quite interesting for its own sake, independent of its application to prove Theorem 1.1. It is a prototype of what seems to be a very general phenomenon appearing for Neumann problems on bounded domains, namely that “bubbles tend to the boundary”. To this end, we have the following result for general strictly convex smooth domains and smooth potentials, proven in Sect. 5 with a method very different from the one we use in Sect. 4 to prove Theorem 1.3. Consider an open, bounded domain D ⊂ Rd with smooth boundary. We shall assume that D is strictly convex. Let q(x) be any bounded smooth potential whose support is a subset of D. For a ∈ Rd let qa (x) := q(x − a). In D consider the Schrödinger operator H DN (a) = − + qa with Neumann boundary conditions on ∂ D (where restriction of qa to D is implied). We denote its ground state energy by E 0 (a). As shown in Lemma 2.1 of Sect. 2, E 0 (a) is continuous in a. Denote by G ⊂ Rd the collection of vectors a such that qa has its support also in D. Note that G is an open set. Theorem 1.4 (Strong minimum principle for E 0 ). If E 0 (a0 ) = inf a∈G E 0 (a) for some a0 ∈ G, then E 0 (a) is identically zero. In this case the wave function is constant outside the support of the potential. In other words if E 0 (a) does not vanish identically in G, then E 0 (a0 ) > inf a∈G E 0 (a) for all a0 ∈ G. The continuous function E 0 : G → R must assume its minimum. By Theorem 1.4, if E 0 (a) does not vanish identically, the minimum must be assumed on ∂G. In the same situation Theorem 1.3 gives the more precise result that the minimum is assumed in the corners of G = [−dmax , dmax ]d . For radially symmetric q and various types of domains D, the question of minimizing the first Dirichlet-eigenvalue of − + qa on D is well studied, see [12] for disks and regular polygons, or [11] for a more general class of domains which have a certain reflection property with respect to the symmetry axes of the potential. Common to all results for Dirichlet problems is that the maximizing and minimizing positions depend on the sign of the potential. For an obstacle q ≥ 0 (or “q = ∞”, meaning a hole in the domain marked by an additional Dirichlet boundary condition) the maximizing position is in the “center” of the domain, while the first eigenvalue is minimized when the obstacle is in contact with the boundary. The reverse is true for the case of a well q ≤ 0. As pointed out in [11], this is most easily understood, if not proven, by a perturbative argument: Consider − + λq on D with Dirichlet boundary condition. Its lowest eigenvalue E 0 (λ) satisfies q|u 0 |2 d x, (6) E 0 (0) = D
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where u 0 is the ground state eigenfunction of the Dirichlet Laplacian. Thus E 0 (λ) changes the most (least), if q is placed where u 0 is largest (smallest), which is near the center (boundary) of D. The sign of q determines the sign of E 0 and thus reverses the role of maximizer and minimizer. This motivation through first order perturbation theory fails for the Neumann problem. In this case (6) still applies, but the ground state of the Neumann Laplacian is constant and thus E 0 (0) is independent of the placement of q. This explains why the Neumann version of the problem is more subtle than the Dirichlet problem (roughly by one order of perturbation theory). Consequently, the methods used in [12] and [11] do not extend to give similar results for the Neumann case. An exception is a remark in [12] concerning infinite spherical obstacles in spherical domains. The only other work on the Neumann case, which we found in the literature, is [16], which gives perturbative and numerical results concerning the optimal configurations of small Dirichlet holes in planar domains for maximizing the first Neumann eigenvalue. We indeed use a second order perturbation theory formula as the starting point of the proof of Theorem 1.4, see (25) below. Our proof of Theorem 1.3 doesn’t use perturbation theory, but Floquet-Bloch theory, the variational characterization of ground states, and unique continuation of harmonic functions. Still, the result may be motivated by second order perturbation theory: In d = 2 (for simplicity) consider the Neumann problem − + λq on L 2 ((− 21 , 21 )2 , d xd y). The lowest eigenvalue E 0 (λ) satisfies the second order perturbation formula E 0 (0) = −2
(u 0 , qu k )2 k>0
Ek − E0
,
(7)
where E k and u k are the higher eigenvalues and eigenfunctions of the NeumannLaplacian, see Sect. 2.3. Considering only the leading term of (7), corresponding to E 1 = E 2 = π 2 , we get 4 − 2 π
q(x, y) sin(π x) d x d y
2
2 +
q(x, y) sin(π y) d x d y
,
which is negative, independent of the sign of q. If q = qa , with q0 reflection symmetric and of fixed sign, then both integrals are zero for a = 0, and both integrals become maximal (in absolute value) if a is located near one of the four corners (± 21 , ± 21 ). Again, this is independent of the sign of q.
2. Preliminaries Throughout this section ⊂ Rd will be open and bounded. The Neumann Laplacian N on is the unique selfadjoint operator whose quadratic form is −
|∇ f (x)|2 d x
for f in the domain H 1 ( ), the first order Sobolev space.
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2.1. Continuity of Eigenvalues. Assume that satisfies the H 1 -extension property, i.e. there exists a bounded operator E : H 1 ( ) → H 1 (Rd ) such that (E f )(x) = f (x) for all f ∈ H 1 ( ) and almost every x ∈ . Note that a sufficient condition for this is that has Lipschitz boundary, e.g. Theorem V.4.12 of [6]. Let q ∈ L ∞ (Rd ) be real-valued and define qa (x) = q(x − a) for all a ∈ Rd . Let N +q . H N (a) = − a Lemma 2.1. H N (a) has purely discrete spectrum consisting of eigenvalues E 0 (a) ≤ E 1 (a) ≤ . . . ≤ E n (a), counted with multiplicity, where all functions E n are continuous in a. Proof. Fix C > q ∞ . From the extension property of and boundedness of q it N + C)−1 and (H N (a) + C)−1 are compact, e.g. Theorem V.4.13 of [6]. follows that (− N Thus H (a) has purely discrete spectrum E 0 (a) ≤ E 1 (a) ≤ . . .. It remains to show that (H N (a)+C)−1 is norm-continuous in a. Continuity of the eigenvalues of (H N (a)+C)−1 , and thus the eigenvalues of H N (a), then follows from the min-max-characterization of eigenvalues. Without restriction, consider continuity at a = 0. For any p ∈ (d, ∞) with p ≥ 2 one has, e.g. Theorem 4.1 of [21], χ (qa − q)(− + C)−1/2 ≤ (2π )−d/ p χ (qa − q) p (| · |2 + C)−1/2 p → 0 as a → 0. (8) N + C)−1/2 As E : H 1 ( ) → H 1 (Rd ) is bounded, it follows that (− + C)1/2 E(− 2 2 d is bounded from L ( ) to L (R ). Combined with (8) this yields N + C)−1/2 → 0 as a → 0. χ (qa − q)(−
(9)
Norm-continuity of (H N (a) + C)−1 at a = 0 now follows from (9), boundedness of N + C)1/2 (H N (0) + C)−1/2 and the resolvent identity (− (H N (a) + C)−1 − (H N (0) + C)−1 = (H N (a) + C)−1 χ (qa − q)(H N (0) + C)−1 .
2.2. Positivity and non-degeneracy of the ground state. We will frequently use that for the domains considered by us and bounded potentials q the ground state energy E 0 of N +q is non-degenerate and that the corresponding eigenfunction can be chosen H = − strictly positive. This generally holds if, in addition to the assumptions from Sect. 2.1, N is non is connected. The latter guarantees that the ground state energy 0 of − N 2 degenerate (− ϕ = 0 implies that |∇ϕ| d x = 0, i.e. ∇ϕ ≡ 0 and thus ϕ constant by connectedness). Non-degeneracy and positivity of the ground state of H follows from the general theory of positivity preserving operators provided in Sect. XIII.12 and the following Appendix 1 of [20].
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2.3. Perturbation formulas. For completeness, let us briefly recall the derivation of the eigenvalue perturbation formulas which we use in our arguments. Most significantly, this will be the first and second order perturbation formulas (31) and (25) with respect to displacements of the potential in Sect. 5. However, they follow in the same way as the corresponding formulas for coupling constant dependence, e.g. (7), so we will focus on the latter. N+ Let and q satisfy the assumptions of the previous two subsections, H (λ) = − λq, E k = E k (λ) its eigenvalues ordered by E 0 < E 1 ≤ E 2 ≤ . . . and u k = u k (·, λ) corresponding real normalized eigenfunctions. Then E 0 (λ) = (u 0 , qu 0 )
(10)
and E 0 (λ) = −2
(u 0 , qu k )2 k>0
Ek − E0
.
(11)
The formulas (31) and (25) below follow with the same argument, using smoothness of q and differentiating separately with respect to each component of a (the extra term (u 0 , (q0 )u 0 ) in (25) does not appear in (11) as ∂λ2 (λq) = 0). Equation (10) is the classical Feynman-Hellmann formula, derived by using non-degeneracy of E 0 (and thus analyticity of E 0 and u 0 in λ) and the fact that (u 0 , ∂λ u 0 ) = 0. Differentiating (10) and using completeness of the u k we get E 0 (λ) = 2(∂λ u 0 , qu 0 ) =2 (qu 0 , u k )(∂λ u 0 , u k ),
(12)
k>0
noticing that the k = 0 term vanishes. Differentiating the eigenvalue equation −u 0 + λqu 0 = E 0 u 0 yields, for every k > 0, qu 0 = (E 0 − E k )∂λ u 0 + (u 0 , qu 0 )u 0 − (− + λq − E k )∂λ u 0 , and thus ∂λ u 0 =
qu 0 − (u 0 , qu 0 )u 0 + (− + λq − E k )∂λ u 0 . E0 − Ek
(13)
After noting that ((− + λq − E k )∂λ u 0 , u k ) = 0, (11) follows from inserting (13) into (12).
3. Theorem 1.3 Implies Theorem 1.1 Theorem 1.3 says that inf a E 0 (a) = E 0 (a min ), where a min corresponds to one of the 2d corners of the cube [−dmax , dmax ]d , say a min := (dmax , . . . , dmax ). Once we know this, then the central ideas of the proof of Theorem 1.1 are (i) Neumann bracketing to go from H (ω) to operators of the type HN0 (a) and (ii) extending the ground state of the minimizer HN0 (a min ) to Rd by repeated reflection.
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Fig. 2. The period cell of Vωmin in d = 2
Proof of Theorem 1.1. For any given configuration ω, the restriction of H (ω) to the unit cube centered at i ∈ Zd with Neumann boundary conditions is unitarily equivalent (via translation by i) to HN0 (ωi ), defined as in Theorem 1.3. Thus, by Neumann bracketing and Theorem 1.3, ⎛ inf σ (H (ω)) ≥ inf σ ⎝
⎞ HN0 (ωi )⎠
i∈Z D
≥ inf E 0 (a) : a ∈ [−dmax , dmax ]d
= E 0 (a min ). This holds for arbitrary configurations ω and thus, by (2), E 0 ≥ E 0 (a min ). Now consider ωmin = (ωimin )i∈Zd as given by (4). The corresponding potential Vωmin (x) =
q(x − i − ωimin )
i∈Zd
is 2-periodic in xi for each i. By Floquet-Bloch theory [20] the bottom of the spectrum per of H (ωmin ) = − + Vωmin is given by the smallest eigenvalue E 0 of its restriction to 1 3 d 2 0 := (− 2 , 2 ) with periodic boundary conditions, see Fig. 2.
On 20 the potential V ω is symmetric with respect to all hyperplanes xi = 1/2, per i = 1, . . . , d. Thus E 0 coincides with the smallest eigenvalue of the Neumann problem on 20 . Again by symmetry of the potential, the latter coincides with the smallest eigenvalue of the Neumann problem on 0 . As ω0min = (dmax , . . . , dmax ) = a min , this eigenvalue is E 0 (a min ). In summary we have shown that min
E 0 ≤ inf σ (H (ωmin )) = E 0 (a min ) ≤ E 0 . Thus E 0 = inf σ (H (ωmin )), which proves Theorem 1.1.
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Fig. 3. Periodic extension, q per (x), of q(x)
4. Proof of Theorem 1.3 This entire section is devoted to prove Theorem 1.3. Thus we work under the assumptions that q ∈ L ∞ is real-valued, non-vanishing, reflection symmetric and supported in [−r, r ]d , r < 1/2. Suppose that alternative (ii) of Theorem 1.3 is false. We will show that this implies that alternative (i) must hold. We begin by fixing all of the components of the displacement parameter except for one, which may be chosen to be the first, and consider the lowest Neumann eigenvalue as a function only of the first coordinate, i.e. E 0 (a) := E 0 (a, a2 , . . . , ad ). We note that E 0 (a) depends continuously on a (see Lemma 2.1) and that by symmetry we have E 0 (−a) = E 0 (a). For this reason we will restrict ourselves to the case a ∈ [0, 21 − r ] and show that E 0 (a1 ) > E 0 (a2 ) for 0 ≤ a1 < a2 ≤
1 2
− r.
(14)
As the same holds for E 0 as a function of each other coordinate, we conclude from this that E 0 has a strict maximum at the origin and strict minima at the corners (±dmax , . . . , ±dmax ), i.e. we are in the situation of alternative (i). As (a2 , . . . , ad ) will be kept fixed, we will use the (slightly sloppy) abbreviation q(x) := q(x1 , x2 − a2 , . . . , xd − ad ) for the rest of the section. For a scalar a and e1 = (1, 0, . . . , 0) we then write qa (x) = q(x − ae1 ), a notation to be used also for functions other than q. By ∂n we will denote the exterior normal derivative on the boundary of a given domain. Lemma 4.1. E 0 (0) > E 0 (a) for every a ∈ (0, 21 − r ]. Proof. Define the tube L := {x : |xi | < 21 , i = 2, . . . , d} and construct a periodic extension, q per (x), of the potential q(x) on L by q per (x) =
qi (x).
i∈Z
We consider the Neumann problem on L for − + q per (x), see Fig. 3. Let ψ denote the normalized Neumann ground state of − + q(x) on the unit cube 0 . Construct a new function on the tube L by periodically extending ψ on all of L. Symmetry of the potential implies that is a smooth solution of − (x) + q per (x) (x) = E 0 (0) (x)
(15)
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on all of L. Now multiply (15) by (x). Then over any unit cell C in L, periodicity of
implies we may integrate by parts without creating boundary terms. In particular, this holds for the unit cell a := 0 − ae1 , yielding |∇ (x)|2 + q per (x) 2 (x) d x. E 0 (0) = (16) a
Shifting to the right by a does not affect the result of Eq. (16), i.e. E 0 (0) = |∇ a (x)|2 + q per (x − ae1 ) a2 (x) d x 0 |∇ a (x)|2 + qa (x) a2 (x) d x. = 0
(17)
While a (x) does not satisfy Neumann boundary conditions on 0 , it is still in the form domain H 1 (0 ) of the Neumann operator − + qa (x) on 0 . Therefore, minimizing the right hand side of Eq. (17) over all normalized functions in H 1 (0 ) it is clear E 0 (0) ≥ E 0 (a). To show that indeed E 0 (0) is strictly greater than E 0 (a) it suffices to show that, when restricted to 0 , a is not equal to a multiple of the Neumann ground state eigenfunction corresponding to the potential qa on 0 . Suppose, for contradiction, that a was such a multiple. Then by construction the box B = (− 21 , − 21 + a) × (− 21 , 21 )d−1 is disjoint from the support of the potential and a satisfies the equation − a = E 0 (0) a
(18)
with Neumann conditions on the boundary of B. As a > 0, it is the ground state of the Neumann problem on B. Thus E 0 (0) = 0 and a must be constant on B. This entails that a is harmonic outside the support of the potential, and since it is constant on an open subset, by unique continuation of harmonic functions it must be constant everywhere outside the support of the potential. However this implies that alternative (ii) must hold, a contradiction. Thus E 0 (0) > E 0 (a). Lemma 4.2. For any positive integer n and a ∈ (0, 21 − r ], E 0 (na) > E 0 ((n + 1)a) so long as (n + 1) a is less than or equal to 21 − r . Proof. To keep notations simple, we first show this for n = 1. Again consider the tube L := {x : |xi | < 21 , i = 2, . . . , d}. Fix a with 0 < 2a ≤ 21 −r and consider a 2-periodic extension, w per , of the potential w(x) := qa (x) + q−a+1 (x)
(19)
on L given by w per (x) :=
w2i (x).
i∈Z
As before we consider the Neumann problem on L for − + w per (x), see Fig. 4, and let ψ denote the Neumann ground state of − + qa (x) on the unit cube 0 , normalized to ψ 2 = 21 . Construct a new function on all of L by 2-periodically extending ψ(x1 , x2 , . . . , xd ) + ψ(−x1 + 1, x2 , . . . , xd ).
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Fig. 4. Periodic extension, W per (x), of w(x) := wa (x) + w−a+1 (x)
Symmetry and periodicity of the potential implies that is a smooth solution of − (x) + w per (x) (x) = E 0 (a) (x) on all of L. Now we proceed in analogy to (16) and (17), this time considering cells of length 2 instead of unit cells and again shifting by a to the right. We find |∇ (x)|2 + w per (x) (x)2 d x E 0 (a) = =
{x∈L:− 12 −a<x1 < 23 −a} {x∈L:− 12 <x1 < 23 }
|∇ a (x)|2 + (q2a (x) + q1 (x)) a2 (x) d x.
(20)
As above we conclude that E 0 (a) is not smaller than the first Neumann eigenvalue E˜ of − + q2a + q1 on {x ∈ L : − 21 < x1 < 23 }. A further application of the argument in the last paragraph of the proof of Lemma 4.1 shows that a restricted to the unit 2 cell {x ∈ L : − 21 < x1 < 23 } is not a multiple of the ground state eigenfunction ˜ ˜ Thus E 0 (a) is strictly greater than E. corresponding to E. Imposing an additional Neumann condition at x1 = 21 can not increase the lowest eigenvalue, thus E 0 (a) > E˜ ≥ min{E 0 (2a), E 0 (0)} = E 0 (2a) by Lemma 4.1, which concludes the proof for n = 1. The crucial idea which allowed us to reduce the n = 1 claim to Lemma 4.1 was that the term q−a+1 in (19) was shifted back into the center of the cube {x ∈ L : 21 < x1 < 23 } in (20). The same mechanism can now be used to inductively prove the claim for all n. We can now readily complete the proof of (14): As a above was arbitrary, Lemma 4.2 implies that E 0 (a) is strictly decreasing on the set of all dyadic numbers in [0, 21 −r ]. As this set is dense in [0, 21 − r ] and E 0 (a) is continuous, then E 0 (a) is strictly decreasing on all of [0, 21 − r ]. Therefore assuming (ii) is false, (i) must be true. 5. Proof of Theorem 1.4 In Theorem 1.4 it is assumed that the domain and the potential are smooth. Thus we have by elliptic regularity that C ∞ (D) is a form core for the Neumann operator H DN (a) = ∞ − N D + qa . Moreover, the eigenfunctions are all in C (D) and have normal derivative zero on ∂ D. We call the eigenvalues E k (a), ordered and accounting for multiplicity, and the normalized eigenfunctions u k (x; a), k = 0, 1, . . .. We choose the ground state u 0 strictly
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positive and all other u k real. They form an orthonormal basis in L 2 (D). Thus, for any function f ∈ H 1 (D) we have that ∞
E k (u k , f )2 =
|∇ f (x)|2 + qa (x) f (x)2 d x ,
(21)
D
k=0
where (·, ·) denotes the inner product on L 2 (D). Note that f is not in the domain of the operator, just in the form domain. See [20] and [7]. Let N denote the outward normal vector field on ∂ D. It will be convenient to use that N can be extended to a smooth vector field in a neighborhood of ∂ D. To see this, first work in a neighborhood of a fixed point of the surface. Without loss (i.e. up to a rigid motion) we can choose this point to be the origin and the surface to be given by xd = f (x1 , . . . , xd−1 )
(22)
in a vicinity of the origin, where f is smooth, f (0, . . . , 0) = 0 and ∇ f (0, . . . , 0) = 0.
(23)
Thus at a point p = (x1 , . . . , xd−1 , f (x1 , . . . , xd−1 )) near 0 one has N ( p) =
(−∇ f (x1 , . . . , xd−1 ), 1) . |(−∇ f (x1 , . . . , xd−1 ), 1)|
This can be extended smoothly to x = (x1 , . . . , xd−1 , xd ) near 0 by N (x) = N (x1 , . . . , xd−1 , f (x1 , . . . , xd−1 )). We get a global extension of N to a neighborhood of ∂ D by using compactness of ∂ D and a standard partition of unity argument. In this neighborhood we define the matrix-valued smooth vector field K = (K i j ) = (∂i N j ). The restriction of K to ∂ D is the curvature matrix of the surface. Indeed, in the local coordinates used above, we have −(∂i ∂ j f )(0) if 1 ≤ i, j ≤ d − 1, ∂i N j (0) = 0 if i = d or j = d. We have assumed that D is strictly convex. This means that the Hessian of f is negative definite and thus at every p ∈ ∂ D the restriction of K ( p) to the tangent plane at p is positive definite. The following identity for E 0 (a) is the main technical ingredient into our proof. Here E 0 and ∇ E 0 refer to the a-derivatives of E 0 . Otherwise, all symbols such as ∂i , ∇, denote derivatives with respect to the spatial variable. B(·, ·) is the bilinear form B(u, v) = (u, v) − (u, v). Lemma 5.1 (Second order perturbation theory). The ground state energy satisfies the equation 2 i B(u k , ∂i u 0 ) E 0 − 4(u 0 , ∇u 0 )∇ E 0 = −2 ∇u 0 · K ∇u 0 d S − 2 . (24) Ek − E0 ∂D k =0
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Proof. We start with the second order perturbation theory formula 2 i (u 0 , (∂i qa )u k ) E 0 = (u 0 , (qa )u 0 ) − 2 , Ek − E0
(25)
k =0
see Sect. 2.3. Differentiating the eigenvalue equation yields (∇qa )u 0 = E 0 ∇u 0 − (− + qa )∇u 0 , and therefore (u k , (∇qa )u 0 ) = −(E k − E 0 )(u k , ∇u 0 ) + B(u k , ∇u 0 ) . Hence E 0 = (u 0 , (qa )u 0 ) − 2 −2
(26)
(E k − E 0 )(∇u 0 , u k )2 − 2(∇u 0 , u k ) · B(u k , ∇u 0 ) k =0
B(u k , ∂i u 0 )2 Ek − E0
i
k =0
which, using (26) once more, can be rewritten as E 0 = (u 0 , (qa )u 0 ) + 2 [B(u k , ∇u 0 ) + (u k , (∇qa )u 0 )] · (u k , ∇u 0 )
k =0
B(u k , ∂i u 0 )2 Ek − E0 k =0 = (u 0 , (qa )u 0 ) + 2 [B(u k , ∇u 0 ) + (u k , (∇qa )u 0 )] · (u k , ∇u 0 ) −2
i
k
− 2[B(u 0 , ∇u 0 ) + (u 0 , (∇qa )u 0 )] · (u 0 , ∇u 0 ) 2 i B(u k , ∂i u 0 ) −2 Ek − E0 k =0 = (u 0 , (qa )u 0 ) + 2(∇u 0 , (∇qa )u 0 ) + 2 B(u k , ∇u 0 ) · (u k , ∇u 0 ) k
− 2[B(u 0 , ∇u 0 ) + (u 0 , (∇qa )u 0 )] · (u 0 , ∇u 0 ) 2 i B(u k , ∂i u 0 ) −2 , Ek − E0 k =0
where finally the completeness relation of the u k was used. It is clear that (u 0 , (qa )u 0 ) + 2(∇u 0 , (∇qa )u 0 ) = 0 and using again (26) with k = 0 we can simplify further and get E 0 = 2 B(u k , ∇u 0 ) · (u k , ∇u 0 ) − 4(u 0 , (∇qa )u 0 ) · (u 0 , ∇u 0 ) k
−2
k =0
i
B(u k , ∂i u 0 )2 . Ek − E0
(27)
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Recall that B(u, v) = (u, v) − (v, u) and hence B(u k , ∇u 0 ) · (u k , ∇u 0 ) = [(u k , ∇u 0 ) − (u k , ∇u 0 )] · (u k , ∇u 0 ) . (28) k
k
Since u 0 ∈ C ∞ (D) we know that the vector ∇u 0 has square integrable components and hence (u k , ∇u 0 ) · (u k , ∇u 0 ) = (∇u 0 , ∇u 0 ) . k
The second term of (28) we write as ([− + qa ]u k , ∇u 0 ) · (u k , ∇u 0 ) − (qa u k , ∇u 0 ) · (u k , ∇u 0 ) . k
The second sum equals j
k j (∂ j u 0 , qa ∂ j u 0 )
E k (u k , ∂ j u 0 )2 =
k
while the first sum is
[|∇∂ j u 0 |2 + qa (x)(∂ j u 0 )2 ]d x D
j
since ∂ j u 0 is in the form domain. Collecting terms we find from (28) that B(u k , ∇u 0 ) · (u k , ∇u 0 ) = [(∂ j u 0 , ∂ j u 0 ) + ∇∂ j u 0 2 ] k
j
=
j
∂D
(∂ j u 0 )N · ∇(∂ j u 0 ) d S,
(29)
where Green’s identity was used. On an open neighborhood of ∂ D we have (∂ j u 0 )N · ∇(∂ j u 0 ) = (∂ j u 0 )∂ j N · ∇u 0 − (∂ j u 0 )(∂ j Ni )(∂i u 0 ) j
j
j,i
= ∇u 0 · ∇(N · ∇u 0 ) − ∇u 0 · K ∇u 0 ,
(30)
where K is the curvature matrix defined above. Using that N · ∇u 0 = 0 on ∂ D one has that the first term is ∇t u 0 ·∇t (N ·∇u 0 ) for points in ∂ D, where ∇t denotes the component of the gradient in the tangential directions. However, N · ∇u 0 = 0 and smoothness of ∂ D also implies ∇t (N · ∇u 0 ) = 0 and thus ∇u 0 · ∇(N · ∇u 0 ) = 0 on ∂ D. Thus (29) and (30) yield B(u k , ∇u 0 ) · (u k , ∇u 0 ) = − ∇u 0 · K ∇u 0 d S. ∂D
k
After substituting this and the first order perturbation formula ∇ E 0 = −(u 0 , (∇qa )u 0 ) into (27) we arrive at (24).
(31)
Lemma 5.2. Assume that q is a smooth potential with compact support in D. Assume that the Neumann ground state u 0 (x; a) for some fixed a0 ∈ G is constant on ∂ D. Then there exists an open neighborhood of a0 where E 0 (a0 ) ≥ E 0 (a).
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Proof. By shifting coordinates we assume that a0 = 0. We shall proceed by a trial function argument. Consider the problem with the shifted potential qa (x) := q(x − a). Denote by Da = D +a the shifted domain, i.e., the function u a (x) := u 0 (x −a; 0) solves −u a + qa u a = E 0 (0)u a , with a Neumann boundary condition on ∂ Da as well as being constant on ∂ Da . We shall construct a trial function φ in the following fashion. In the intersection of Da with D we set φ = u a and in D\Da we set φ to be a constant which equals the boundary value of u a . Note that φ ∈ H 1 (D). By the variational principle E 0 (a) ≤
D
|∇φ|2 + qa φ 2 d x . 2 D φ dx
For a ∈ G the right side equals D∩Da
|∇u a |2 + qa u a2 d x ≤ 2 D φ dx
Da
2 |∇u a |2 + qa u a2 d x u dx = E 0 (0) D 02 . 2 D φ dx D φ dx
(32)
In the case E 0 (0) = 0, this implies E 0 (a) ≤ 0, as was to be shown. Next we claim that for a sufficiently small 2 u dx D 0 <1 2 D φ dx if E 0 (0) > 0 and 2 u dx D 0 >1 2 D φ dx if E 0 (0) < 0. This yields the lemma for the remaining cases. If we denote by c the boundary value of u 0 we find that the claim follows once we show that in a vicinity of the boundary, u 0 < c for E 0 (0) > 0 and u 0 > c for E 0 (0) < 0. To see this, fix a point on the boundary, call it the origin and use the local coordinates (22) and (23) above. The normal vector at 0 is (0, . . . , 0, 1) and hence the normal derivative equals ∂d u 0 (0) = 0. Further, since u 0 is constant on the boundary we find by differentiating u 0 (x1 , . . . , xd−1 , f (x1 , . . . , xd−1 )) ≡ c that ∂i ∂ j u 0 (0) = −∂d u 0 (0)∂i ∂ j f (0) = 0 for i, j = 1, . . . d − 1. Hence ∂d ∂d u 0 (0) = u 0 (0) = −E 0 (0)u 0 (0) which is negative for E 0 (0) > 0 and positive for E 0 (0) < 0. As this holds at all points of the boundary, we get the required property of u 0 in a vicinity of the boundary. We remark that in the case E 0 (a0 ) = 0, the above proof actually gives the strict inequality E 0 (a0 ) > E 0 (a) for a close to a0 , i.e. E 0 (a0 ) is a strict local maximum. This is the case, for example, if q is sign-definite, and allows for a shorter argument in the following completion of the proof of Theorem 1.4.
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Proof of Theorem 1.4. Assume that E 0 attains its minimum value in G say at the point a0 . This entails that ∇ E 0 (a0 ) = 0 and E 0 (a0 ) ≥ 0. Using Lemma 5.1 we find that 2 i B(u k , ∂i u 0 ) E 0 (a0 ) = −2 ∇u 0 · K ∇u 0 d S − 2 . Ek − E0 ∂D k =0
The right side is non-positive, since D is convex. It cannot be strictly negative, since that would contradict the assumption that E 0 (a) has a local minimum at a0 . Thus the right side must vanish. Since D is strictly convex, the first term can vanish only if u 0 is constant on the boundary (recall that K is positive definite on the tangent space at each point of ∂ D and that ∇u 0 is a tangent vector). However, Lemma 5.2 shows that E 0 (a0 ) ≥ E 0 (a) in a neighborhood of a0 . Thus E 0 (a) = E 0 (a0 ) in this neighborhood. Since the set where E 0 (a0 ) = E 0 (a) is closed and open in G (the above argument applies to every a with E 0 (a0 ) = E 0 (a)) the function E 0 (a) must be constant. Assuming now that E 0 (a) is constant, u 0 is constant on the boundary and moreover, there must be equality in (32). This means that |∇u a |2 d x = 0 . Da \D
Hence u a must be constant in Da \D and therefore, for small a, 0 = −u a = E 0 (a)u a there. Thus, E 0 (a) ≡ 0. Since the support of qa is a subset of D (as a ∈ G), u a is constant in the non-empty open set Da \D, which is disjoint from the support of qa . Since it is harmonic outside the support of qa it must be constant there too. It follows that u 0 is constant outside the support of q. 6. Discussion: Extensions and Open Problems We conclude with some remarks about possible generalizations and open problems related to our main results. (i) Theorem 1.3 and its proof immediately generalize to the Neumann problem on an arbitrary rectangular box {x : |xi | < i , i = 1, . . . , d} rather than the unit cube 0 , which we chose to keep notations simple. This also gives a corresponding version of Theorem 1.1, where Zd is replaced by an arbitrary rectangular lattice. (ii) In Theorems 1.1 and 1.3 we also may replace the obstacle q by an reflection symmetric hole with Dirichlet boundary conditions, often interpreted as an infinite barrier. More precisely, let C ⊂ {x : |xi | < r, i = 1, . . . , d} be closed and reflection symmetric. Let Ca = C +a and HaN := − on \Ca with Neumann conditions on ∂ and Dirichlet conditions on ∂Ca . Then E 0 (a) is minimized when the hole is in a corner of 0 and inf σ (H (ω)) is minimized for a periodic configuration of 2d -clusters of holes. While we expect that Theorem 1.4 also extends to this situation, at least for holes with smooth boundary, our proof does not extend directly. (iii) Theorem 1.4 covers the situation of a radially symmetric potential (or Dirichlet hole) placed in a spherical domain, where all placements of q which touch the boundary are equivalent minimizing positions. However, in this case the methods of Lemma 5.2 may be used to also show that the maximal position occurs when the potential is centered in the domain. To see this, suppose D is a spherical domain centered at 0 and let q be a radially symmetric potential compactly supported in D, also centered at 0. Radial symmetry of
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the domain and potential then imply the ground state eigenfunction, u 0 , corresponding to E 0 (0) is radially symmetric. Thus it satisfies the conditions of Lemma 5.2. If E 0 (0) ≥ 0 one may show by reducing the problem to one dimension, or by using maximum principles, that outside the support of the potential u 0 ≤ c, where c denotes the value of u 0 on the boundary of D. Similarly if E 0 (0) ≤ 0, u 0 ≥ c outside the support of the potential. One may conclude using the arguments of Lemma 5.2 that E 0 (0) ≥ E 0 (a) for every a ∈ G. This then leads to the following alternative for the case of a spherical domain centered at 0 with radially symmetric potential: either E 0 (a) ≡ 0, in which case the corresponding eigenfunction is constant outside the support of the potential or E 0 (a) assumes a strict maximum at E 0 (0) and strict minima when the support of the potential touches the boundary. (iv) Theorems 1.3 and 1.4 are proven with very different methods and apply to mutually exclusive classes of domains (rectangular boxes vs strictly convex domains). It would be desirable to find a method of proof which covers both results, as this method would most likely also cover more general polygons and polyhedra. Particularly interesting cases would be equilateral triangles or hexagons, as they tile the plane and would lead to a corresponding extension of Theorem 1.1. (v) One can view Theorem 1.1 as a mechanism in which the nuclei of a solid selforganize into a simple periodic pattern, given a density condition (exactly one site per cube). It would be wrong, in our opinion, to see this as a model for crystallization since the regularity of the pattern is to a large extent determined by the density condition. Real crystallization, however, cannot be explained by the interaction of one electron with nuclei alone. It is a many-body effect and the nuclear repulsion and, more importantly, the Pauli exclusion principle play a role. Further, one needs sufficiently many electrons, e.g., a half filled band. Indeed, there have been results in this direction in [13] for the Falicov-Kimball model, a variant of the Hubbard model where the nuclei are treated classically and sit on a lattice where the electrons hop. Crystallization was shown in [13] for the half filled band. In our model, without the density condition, we expect that the nuclei would stick together. While this is an open question there is some evidence in this direction. For bosons it was shown in [13] that the nuclei indeed stick together. It would be interesting to consider an extension of our model, a continuous analog of the Falicov-Kimball model, in which one considers a finite periodic array of cubes on a torus. Assuming the same number of spinless fermions as the number of cubes and assuming one nucleus in each cube, it is not unreasonable to expect that in an energy minimizing configuration the nuclei sit at the center of each cube. This is an interesting open question. Needless to say that the methods in this paper have no bearing on this problem. For an overview of the Falicov-Kimball model the reader may consult [8]. Acknowledgements. The authors are indebted to Jean Bellissard for many useful discussions and suggestions which substantially improved this work. G. S. would also like to thank Michael Levitin, from whom he originally learned the ideas used in Sect. 3. He also acknowledges hospitality and support at the Isaac Newton Institute of the University of Cambridge, where part of this work was done, as well as partial support through NSF grant DMS 0245210. M. L. would like to acknowledge partial support through NSF grant DMS-0600037.
References 1. Ando, K., Iwatsuka, A., Kaminaga, M., Nakano, F.: The spectrum of Schrödinger operators with Poisson type random potential. Ann. Henri Poincaré 7, 145–160 (2006) 2. Bourgain, J., Kenig, C.: On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math. 161, 389–426 (2005)
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3. Buschmann, D., Stolz, G.: Two-Parameter Spectral Averaging and Localization for Non-Monotonic Random Schrödinger Operators. Trans. Amer. Math. Soc. 353, 635–653 (2001) 4. Carmona, R., Lacroix, J.: Sprectral Theory of Random Schrödinger operators. Basel: Birkhäuser, 1990 5. Damanik, D., Sims, R., Stolz, G.: Localization for one-dimensional, continuum, Bernoulli-Anderson models. Duke Math. J. 114, 59–100 (2002) 6. Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford: Clarendon Press, 1987 7. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, Providence, RI: Amer. Math. Soc., 1998 8. Freericks, J.K., Lieb, E.H., Ueltschi, D.: Segregation in the Falicov-Kimball model. Commun. Math. Phys. 227, 243–279 (2002) 9. Germinet, F., Hislop, P., Klein, A.: Localization for Schrödinger operators with Poisson random potential. J. Eur. Math. Soc. 9, 577–607 (2007) 10. Germinet, F., Hislop, P., Klein, A.: Localization at low energies for attractive Poisson random Schrödinger operators. In: Probability and Mathematical Physics, CRM Proceedings and Lecture Notes, Vol. 42, Amer. Math. Soc., 2007, pp. 153–165 11. Harrell, E.M., Kroger, P., Kurata, K.: On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue. SIAM J. Math. Anal. 33, 240–259 (2001) 12. Hersch, J.: The method of interior parallels applied to polygonal or multiply connected membranes. Pacific J. Math. 13, 1229–1238 (1963) 13. Kennedy, T., Lieb, E.H.: An itinerant electron model with crystalline or magnetic long range order. Phys. A 138, 320–358 (1986) 14. Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Spectral Theory and Mathematical Physics, Proceedings of Symposia in Pure Mathematics, Vol. 76, Part 2, Providence, Amer. Math. Soc., 2007, pp. 649–696 15. Klopp, F.: Localization for semiclassical continuous random Schrödinger operators. II. The random displacement model. Helv. Phys. Acta 66, 810–841 (1993) 16. Kolokolnikov, T., Titcombe, M.S., Ward, M.J.: Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps. Eur. J. Appl. Math. 16, 161–200 (2005) 17. Lott, J., Stolz, G.: The spectral minimum for random displacement models. J. Comput. Appl. Math. 148, 133–146 (2002) 18. Najar, H.: The spectrum minimum for random Schrödinger operators with indefinite sign potentials. J. Math. Phys. 47, 013515 (2006) 19. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. New York: SpringerVerlag, 1984 20. Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV, Analysis of Operators. New York: Academic Press, 1978 21. Simon, B.: Trace Ideals and their Applications. Cambridge: Cambridge University Press, 1979 22. Stollmann, P.: Caught by Disorder. Bound states in Random Media. Progress in Mathematical Physics, Vol. 20, Boston: Birkhauser, 2001 Communicated by B. Simon
Commun. Math. Phys. 283, 417–449 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0515-4
Communications in
Mathematical Physics
Large Time Asymptotics of Growth Models on Space-like Paths II: PNG and Parallel TASEP Alexei Borodin1 , Patrik L. Ferrari2 , Tomohiro Sasamoto3 1 California Institute of Technology, Mathematics 253–37, Pasadena, CA 91125, USA.
E-mail:
[email protected]
2 Weierstrass Institute (WIAS), Mohrenstr. 39, 10117 Berlin, Germany. E-mail:
[email protected] 3 Chiba University, Dept. of Mathematics and Informatics, 1-33 Yagot-cho, Inage,
Chiba 263-8522, Japan. E-mail:
[email protected] Received: 13 August 2007 / Accepted: 20 December 2007 Published online: 29 May 2008 – © Springer-Verlag 2008
Abstract: We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy1 process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update. 1. Introduction The main focus of this work is a stochastic growth model in 1 + 1 dimensions, called the polynuclear growth (PNG) model. It belongs to the KPZ (Kardar-Parisi-Zhang [19]) universality class and it can be described as follows (see Fig. 1). At time t, the surface is described by an integer-valued height function x → h(x, t) ∈ Z, x ∈ R, t ∈ R+ . It thus consists of up-steps () and down-steps (). The dynamics has a deterministic and a stochastic part: (a) up- (down-) steps move to the left (right) with unit speed and disappear upon colliding, (b) pairs of up- and down- steps (nucleations) are randomly added on the surface with some given intensity. The up- and down-steps of the nucleations then spread out with unit speed according to (a). The PNG model can be interpreted in several different ways, see [11] for a review. On a macroscopic scale the surface of the PNG model grows deterministically, i.e., limt→∞ t −1 h(ξ t, t) = H (ξ ) is a non-random function. However, on a mesoscopic scale fluctuations grow in time. This is called roughening in statistical physics and extensive numerical studies have been made [3]. Since the PNG model is in the KPZ universality class, the fluctuation of the surface height is expected to live on a t 1/3 scale and nontrivial correlations are to be seen on a t 2/3 scale. Therefore, to have an interesting large
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Fig. 1. Illustration of the PNG height and its dynamics. The bold vertical piece is a nucleation. The arrows indicate the movements of the steps. A Java animation of the PNG dynamics is available at [9]
time limit, we have to rescale the surface height as h(ut 2/3 , t) − t H (ut −1/3 ) . t 1/3
(1.1)
One of the initial conditions most natural and used for numerical simulations for PNG is the flat initial condition, i.e., h(x, 0) = 0 for all x ∈ R. We consider nucleations occurring with translation-invariant intensity. In other words, the nucleation events form a Poisson process with constant intensity in the space-time upper half-plane. We refer to the PNG model with such initial condition as flat PNG. In this case, by mapping the flat PNG to a point-to-line last passage directed percolation model it was proven [2,21,22] that the one-point distribution is, in the t → ∞ limit, the GOE Tracy-Widom distribution F1 , first discovered in random matrix theory [27]. However, no information on joint height distributions at several points has been previously known. New Results. The main results of this paper are precisely the computation and asymptotic analysis of these joint distributions. In particular, we prove the convergence of the height rescaled as in (1.1) to the Airy1 process in the t → ∞ limit (see Sect. 2.2 for a definition of the process). The Airy1 process has been discovered in the context of the asymmetric exclusion process [5–7,24]. Our result, stated in Theorem 6, is obtained by first determining an expression for the joint distributions for finite time t (Proposition 4) and then taking the appropriate scaling limit. Proposition 4 is actually just a particular case of Theorem 5, where we determine joint distributions along any space-like paths (as in the Minkowski diagram). Space-like paths are described later in detail; for now one can keep in mind the special case of fixed time t. The scaling limit is analyzed at this level of generality, thus Theorem 6 holds for any space-like paths. In contrast to previous works on the subject, our approach does not rely on the so-called RSK correspondence (RSK for Robinson-Schensted-Knuth), which was successfully applied for corner growth models, but does not seem to be well suited for the flat growth. Our real interest is the flat PNG model. However, one of the new key ingredients in our solution is a precise connection with the totally asymmetric simple exclusion process (TASEP) in discrete time with parallel update. In fact, to get the results for the flat PNG, we first consider a discrete time version of it, the Gates-Westcott dynamics [12,23]. This model is closely related to the TASEP in discrete time with parallel update and alternating initial conditions. The corresponding results for the TASEP are Theorem 1 for the joint distributions along space-like paths, and Theorem 3 for the convergence to the Airy1 process in the scaling limit. For the TASEP, the extreme situations of space-like paths are positions of different particles at a fixed time and positions of a fixed particle at different time moments (tagged particle). The space-like extension for TASEP is based on the previous paper [4].
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Previous works on PNG. Another type of initial conditions for the PNG model has been analyzed before. It is the corner growth geometry, where nucleations occur only inside the cone {|x| ≤ t}. The limit shape H is a semi-circle, and the model is called PNG droplet. In this geometry, the limit process has been obtained in [23]; it is known as Airy2 process (previously called simply Airy process). The approach uses an extension to a multilayer model (inherited from the RSK construction), see [16,23]. The multilayer method was also used in other related models [8,13,14,17,18,26]. Also, for the flat PNG it was used to connect the associated point process at a single position and the point process of GOE eigenvalues [10]. Results on the behavior for the PNG droplet along space-like paths can be found in [8]. For a very brief description of the previously known results on TASEP fluctuations see the introductions of [4,25]. Outline. In Sect. 2, we introduce our models and state the results. In Sect. 3, we give an expression of the transition probability of the discrete TASEP as a marginal of a determinantal signed point process. In Sect. 4 the Fredholm determinant expression for the joint distributions is obtained. The argument substantially relies on the algebraic techniques of [4]. In Sect. 5, we consider the scaling limit of the parallel TASEP. In Sect. 6, the continuous time PNG model is considered. In Sect. 7, we consider the scaling limit for the continuous PNG model. 2. Models and Results We start from the discrete time TASEP with parallel update. Then we will make the connection with a discrete version of the PNG, from which the continuous time PNG is obtained. 2.1. Discrete time TASEP with parallel update. We consider discrete time TASEP with parallel update and alternating initial conditions, i.e., particle i has initial position xi (0) = −2i, i ∈ Z. At each time step, each particle hops to its right neighbor site with probability p = 1 − q provided that the site is empty. The particle positions at time t is denoted by xi (t), i ∈ Z. The dynamics of a particle depends only on particles on its right. This fact allows us to determine the joint distributions of particle positions also for different times, but restricted to “space-like paths”. To define what we mean with “space-like paths”, we consider a sequence of couples (n i , ti ), where n i is the number of the particle and ti is the time when this particle is observed. On such couples we define a partial order ≺, given by (n i , ti ) ≺ (n j , t j ) if n j ≥ n i , t j ≤ ti , and the two couples are not identical.
(2.1)
A space-like path is a sequence of ordered couples, namely, S = {(n k , tk ), k = 1, 2, . . . |(n k , tk ) ≺ (n k+1 , tk+1 )}.
(2.2)
The reason of the name “space-like” will be clear in the large time limit, where everything becomes continuous. Then space-like is the same concept as in the Minkowski diagram. The border cases for space-like paths are fixed time (ti ≡ t, ∀i) and fixed particle number (n i ≡ n, ∀i). The next theorem proved in Sect. 4 concerns the joint distributions of particle positions.
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Theorem 1. Let the particle with label i start at xi (0) = −2i, i ∈ Z. Consider a space-like path S. For any given m, the joint distribution of the positions of the first m points in S is given by m xn k (tk ) ≥ ak = det(½ − χa(−) K χa(−) )2 ({(n 1 ,t1 ),...,(n m ,tm )}×Z) , È (2.3) k=1
where
(−) χa ((n k , tk ), x)
= ½(x < ak ). The kernel K t is given by
K ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ) = −φ ((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 )½[(n 1 ,t1 )≺(n 2 ,t2 )] ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ), +K
(2.4)
where ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ) K (1 − p)t1 −2n 1 −x1 −1 (1 + z)x2 +n 1 +n 2 = dz , 2π i 0 (−z)x1 +n 1 +n 2 +1 (1 + pz)t1 +t2 +1−(x1 +n 1 +n 2 )
(2.5)
and φ ((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 ) n 1 −n 2 −w (1 + pw)t1 −t2 1 dw , = 2π i −1 (1 + w)x1 −x2 +1 (1 + w)(1 + pw)
(2.6)
where 0 (resp. −1 ) is any simple loop, anticlockwise oriented, with 0 (resp. −1) being the unique pole of the integrand inside the contour. Remark. In the limit p → 0 under the time scaling by p −1 the discrete time TASEP converges to the continuous time TASEP, and Theorem 1 turns into a special case of Proposition 3.6 of [4], where a more general continuous time model called PushASEP was considered. 2.2. Airy1 process and scaling limit. Starting from Theorem 1 we can analyze large time limits. The limit process is the so-called Airy1 process introduced in [6,24], which we recall here. Definition 2 (The Airy1 process). Define the extended kernel, (ξ2 − ξ1 )2 1 K A1 (τ1 , ξ1 ; τ2 , ξ2 ) = − √ exp − ½(τ2 > τ1 ) 4(τ2 − τ1 ) 4π(τ2 − τ1 ) 2 (2.7) +Ai(ξ1 + ξ2 + (τ2 − τ1 )2 ) exp (τ2 − τ1 )(ξ1 + ξ2 ) + (τ2 − τ1 )3 . 3 The Airy1 process, A1 , is the process with m-point joint distributions at τ1 < τ2 < . . . < τm given by the Fredholm determinant, m È {A1 (τk ) ≤ sk } = det(½ − χs K A1 χs )L 2 ({τ1 ,...,τm }×R) , (2.8) k=1
where χs (τk , x) = ½(x > sk ).
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Fig. 2. An example of a space-like path π(θ ). Its slope is, in absolute value, at most 1
Theorem 1 allows us to analyze joint distributions of particle positions for situations spanning between fixed time and fixed particle number (the tagged particle problem). One way to parametrize such situations is via a space-like path. We thus consider an arbitrary smooth function π satisfying |π (θ )| ≤ 1 and π(θ ) + θ > 0,
(2.9)
see Fig. 2. The requirement π(θ ) + θ > 0 reflects t > 0. Then, we choose couples of (t, n) on {((π(θ ) + θ )T, (π(θ ) − θ )T ), θ ∈ R}, where T is a large parameter. The case of fixed time, say t = T , is obtained by setting π(θ ) = 1 − θ , while fixed particle number, say n = αT , by π(θ ) = α + θ with some constant α. From KPZ scaling exponents [19], we expect to see a nontrivial limit if we consider positions at distance of order T 2/3 . Thus, the focus on the region around θ T is given by θ T − uT 2/3 , i.e., setting θ − uT −1/3 instead of θ and, by series expansions, we scale time and particle number as t (u) = (π(θ ) + θ )T − (π (θ ) + 1)uT 2/3 + 21 π (θ )u 2 T 1/3 , n(u) = (π(θ ) − θ )T + (1 − π (θ ))uT 2/3 + 21 π (θ )u 2 T 1/3 .
(2.10)
The KPZ fluctuation exponent is 1/3, thus we expect to see fluctuations of particle positions on a scale of order T 1/3 . Therefore, we define the rescaled process T by xn(u) (t (u)) − (−2n(u) + vt (u)) . (2.11) −T 1/3 √ Here the mean speed of particles, v, is determined to be v = 1− q from the subsequent asymptotic analysis but can be known beforehand from the stationary measure for density 1/2 [15]. This process has a limit as T → ∞ given in terms of the Airy1 process. In Sect. 5 we prove u → T (u) =
Theorem 3. Let T be the rescaled process as in (2.11). Then lim T (u) = κv A1 (u/κh ),
T →∞
(2.12)
in the sense of finite dimensional distributions. The vertical (fluctuations) and horizontal (correlations) scaling coefficients are given by κv = (π(θ ) + θ )1/3 (1 − q)1/3 q 1/6 , κh =
(π(θ ) + θ )2/3 (1 − q)2/3 q −1/6 (π (θ ) + 1)(1 −
. √ q)/2 + 1 − π (θ )
(2.13) (2.14)
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Fig. 3. A surface growth model. For half-odd integer times this is equivalent to the discretized Gates-Westcott dynamics and for integer times to the discrete TASEP
Remark. A similar result for the PushASEP with alternating initial condition has been proved in Theorem 2.2 of [4].
2.3. TASEP and growth models. As mentioned in the Introduction, the discrete TASEP with parallel update is related to a surface growth model from which the polynuclear growth model in continuous time can be obtained as a limit. Let t ≥ 0 and x ∈ R denote the time and the one-dimensional space coordinate respectively, and let h t (x) be the height of the surface at time t and at position x. Let us introduce a dynamics of h t (x) as follows. Initially, at time t = 0, the surface is flat; h 0 (x) = 0, for all x ∈ R. Right after each integer time (t = 0+, 1+, 2+, . . .), there could occur a nucleation with width 0 and height 1 with probability q (0 < q < 1) independently at each integer position x such that t + x + h t (x) is even. Each nucleation is regarded as consisting of an upstep and a downstep and each upstep (resp. downstep) moves to the left (resp. right) with unit speed. This is a deterministic part of the evolution. When an upstep and a downstep collide, they merge together. See the solid line in Fig. 3 for an example until t = 2. The dynamics of the growth model, if we focus only on half-odd times (t = 21 , 23 , . . .),
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423
Fig. 4. Surface height and TASEP particle positions. An example for t = 4
is the same as one considered in [23], i.e., a discretized version of the Gates-Westcott dynamics [12]. It is known that in an appropriate q → 0 limit this growth model reduces to the standard continuous time PNG model [23]. To see the connection to the discrete TASEP, let us focus on integer times (t = 0, 1, 2, . . .) and positions (x ∈ Z) from now on and represent the surface as consisting of elementary upward slopes and downward slopes as indicated by dashed lines in Fig. 3. At t = 0, even (resp. odd) x’s are taken to be the center of the upward (resp. downward) slopes. Then the dynamics of the surface is described as follows: At each time step the surface grows upward by unit height deterministically and then each local maximum () of slope turns into a local minimum () independently with probability p ≡ 1 − q. If we interpret an upward (resp. a downward) slope as a site occupied by a particle (resp. an empty site), this is equivalent to the discrete time TASEP with parallel update under the alternating initial condition. The relation between the surface height h t (x) and the position of the TASEP particle is given by h t (x) ≤ H ⇔ x t−x−H (t) ≥ x
(2.15)
2
and is understood as follows. On the plot of the surface at some fixed time t, draw also the initial surface at h = t. See Fig. 4 for an expample. Then, from the correspondence between the growth model and the TASEP, the surface at time t can be regarded as the particle positions. In this plot particles move along the down-right direction as indicated. The left hand side of (2.15) is equivalent to the condition that the TASEP particle corresponding to (x, h = t − H ) has already reached x. Since the axis of the particle number n is in the down-left direction, the value of n corresponding to (x, h = t − H ) is (t − H −x)/2 . This consideration results in the relation (2.15). From the relation (2.15) the joint distributions of the height of the growth model is readily obtained through
È
m
i=1
{h ti (xi ) ≤ Hi } = È
m
{xn i (ti ) ≥ xi } ,
(2.16)
i=1
combined with Theorem 1. When q → 0, the TASEP particles move almost deterministically and the surface h t (x) grows slowly, when a particle decides not to jump (with probability q). The continuous √ time PNG model is obtained by taking q → 0 while setting space and time units to q/2 (the 2 is chosen to have nucleations with intensity 2 like in [23]). Denote by x and t the position and time variables in the continuous time PNG model. The PNG
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height function h PNG (x, t) is then obtained by the limit
√ h PNG (x, t) = lim h 2t/√q (−2x/ q). q→0
(2.17)
Here the minus sign on the right-hand side is there for convenience. The results below do not depend on this sign because of the symmetry of the model under consideration. The joint distribution of the surface height at time t is given as follows. Proposition 4. Consider m space positions x1 < x2 < . . . < xm . Then, the joint distribution at time t of the heights h PNG (xk , t), k = 1, . . . , m, is given by m
PNG h È (xk , t) ≤ Hk (2.18) = det(½ − χ H K tPNG χ H )2 ({x1 ,...,xm }×Z) , k=1
where the kernel is given by K tPNG (x1 , h 1 ; x2 , h 2 ) = −I|h 1 −h 2 | (2(x2 − x1 )) ½(x2 > x1 ) 2t + x2 − x1 (h 1 +h 2 )/2 + Jh 1 +h 2 2 4t2 − (x2 − x1 )2 ½(2t ≥ |x2 − x1 |), 2t − x2 + x1 (2.19) where In (x) and Jn (x) are the modified Bessel functions and the Bessel functions, see e.g. [1]. The last indicator function is obvious if one thinks about the PNG model. In fact, the height at position x at time t depends on events lying in the backward light cone of (x, t) on R × R+ . Thus, when |x2 − x1 | > 2t, the backwards light cones of (x1 , t) and (x2 , t) do not intersect in R × R+ , which implies that the two height functions are independent. The Fredholm determinant then splits into blocks. The result of Proposition 4 is actually a specialization of a more general situation which follows from the TASEP correspondence. In the TASEP, the space-like paths π we had for particle numbers and times become the paths (x, t) = (π(θ ) − 3θ, θ + π(θ )) .
(2.20)
The condition |π (θ )| ≤ 1 implies that ∂t/∂x ∈ [−1, 0], i.e., these are space-like paths as in special relativity oriented into the past. By the symmetry of the problem, one can consider also space-like paths locally oriented into the future, just looking at the process in the other direction. Denote by γ such a path on R × R∗+ , i.e., (x, t = γ (x)), then θ and π(θ ) are given by the relations θ = (γ (x) − x)/4, π(θ ) = (3γ (x) + x)/4, (2.21) and the joint distributions of the surface height along the path γ are expressed as in Theorem 5. This is proved in Sect. 6. Theorem 5. Let us denote tk = γ (xk ). Then, the joint distributions of h PNG (xk , tk ), k = 1, . . . , m, is given by m
PNG h È (xk , tk ) ≤ Hk = det(½ − χ H K PNG χ H )2 ({(x1 ,t1 ),...,(xm ,tm )}×Z) , k=1
(2.22)
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Fig. 5. A space-time path γ for continuous time PNG. T is proportional to the PNG time t
where the kernel is given by x2 − x1 + t1 − t2 |h 1 −h 2 |/2 K ((x1 , t1 ), h 1 ; (x2 , t2 ), h 2 ) = − x2 − x1 − t1 + t2 × I|h 1 −h 2 | 2 (x2 − x1 )2 − (t2 − t1 )2 ½{(t1 +x1 ,t1 )≺(t2 +x2 ,t2 )} (t1 + t2 ) + (x2 − x1 ) (h 2 +h 1 )/2 + Jh 1 +h 2 2 (t1 + t2 )2 − (x2 − x1 )2 (t1 + t2 ) − (x2 − x1 ) × ½{t1 +t2 ≥|x1 −x2 |)} , (2.23)
PNG
where In (x) and Jn (x) are the modified Bessel functions and the Bessel functions. The first term is present only when x2 − x1 ≥ t1 − t2 > 0 or x2 − x1 > t1 − t2 ≥ 0 due to (2.1). In the first term, for x2 > x1 , the condition x2 − x1 ≥ t1 − t2 is satisfied for tk = γ (xk ). Also, notice that when x2 − x1 → t1 − t2 , the first term of the kernel goes x |n| 1 + O(x |n|+1 ) for small x. to (2(x2 − x1 ))|h 1 −h 2 | /(|h 1 − h 2 |)! since I|n| (x) = |n|! 2 2.4. Scaling limit for the continuous PNG model. The last result of this paper is the large time behavior of the flat PNG. The large parameter denoted by T is proportional to time t. Using the function γ , we consider t = T γ (x/T ), see Figure 5. Since the system is translation invariant, we focus around the origin, i.e., we look at the PNG height at x(u) = uT 2/3 , (2.24) t(u) = γ (0)T + γ (0)uT 2/3 + 21 γ (0)u 2 T 1/3 . The surface height grows with the speed equal to 2. Thus, for large time t, the macroscopic height will be close to 2t. Fluctuations live on a T 1/3 scale. Consequently, we define the rescaled height process h PNG by T (u) = u → h PNG T
h PNG (x(u), t(u)) − 2t(u) . T 1/3
(2.25)
The large T (thus large time too) behavior of h PNG is given in terms of the Airy1 process T as stated below.
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Theorem 6. Let h PNG be the rescaled process as in (2.25). Then, in the limit of large T , T we have lim h PNG (u) = Sv A1 (u/Sh ), (2.26) T T →∞
in the sense of finite dimensional distributions. The scaling coefficients Sv and Sh are given by Sv = (2γ (0))1/3 , Sh = (2γ (0))2/3 = Sv2 . (2.27) For γ (x) = 1, i.e., fixed time, this was conjectured to hold in [6]. The proof of this theorem is given in Sect. 6. 3. Transition Probability for the Finite System Let G(x1 , . . . , x N ; t) denote the transition probability of the parallel TASEP with N particles starting at t = 0 at positions y N < · · · < y1 . This is the probability that the N particles starting from positions y N < · · · < y1 at t = 0 are at positions x N < · · · < x1 at t. Consider a determinantal signed point process on the set x = {xin , 1 ≤ i ≤ n ≤ N } by setting the measure N −1 n n+1 W N (x) = det(φ (xi , x j+1 ))0≤i, j≤n det(F−i+1 (x Nj − y N +1−i , t + 1−i))1≤i, j≤N , n=1
(3.1) ⎧ ⎪ ⎨1, y ≥ x, φ (x, y) = p, y = x − 1, ⎪ ⎩0, y ≤ x − 2,
where
the function Fn (x, t) defined by 1 F−n (x, t) = 2π i
0,−1
dw
wn (1 + (1 − q)w)t , (1 + w)n+x+1
(3.2)
(3.3)
with the paths 0,−1 being any simple loops anticlockwise oriented including 0, −1 and no other poles and we used the convention, x0n = −∞. The following proposition states that the one time transition probability of the TASEP is given as a marginal of the signed measure (3.1). Proposition 7. Let us set x1n = xn , n = 1, . . . , N . Then W N (x), G(x1 , . . . , x N ; t) =
(3.4)
D
where summation is over the variables in the set, n } D = {xin , 2 ≤ i ≤ n ≤ N |xin > xi−1
(3.5)
varying over Z. Note that W N (x) is actually symmetric with respect to permutations of variables with the same upper index, so the ordering in (3.5) is used for singling out the minimal x1n = min{xin , i = 1, . . . , n}.
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Remark. Similar representations for the transition probability of continuous time TASEP, discrete time TASEP with sequential update and PushASEP have been obtained in [4–6]. In the different parts of the proof of Proposition 7, we will use several properties of the function Fn , which are listed below. Lemma 8. Fn+1 (x, t) =
∞
Fn (y, t),
(3.6)
y=x
Fn (x, t + 1) = q Fn (x, t) + (1 − q)Fn (x − 1, t) = Fn (x, t) + (1 − q)Fn−1 (x − 1, t),
(3.7) (3.8)
(φ ∗ Fn )(x, t) = Fn+1 (x, t + 1), F−n (x, −n) = 0 for x < −n, n > 0, Fn (x, n) = 0 for x > n, n > 0, Fn (n, n) = (1 − q)n , n ≥ 0, F−n (−n, −n) = 1/(−q)n , n ≥ 0. Here “∗” represents the convolution: (φ ∗ f )(x) =
y
(3.9) (3.10) (3.11) (3.12) (3.13)
φ (x, y) f (y).
Proof of Lemma 8. These are proven by using the definition (3.2) and (3.3).
The first step in the proof of Proposition 7 is the following lemma. Lemma 9. Let us set
⎧ y−x , y ≥ x, ⎪ ⎨ν φν (x, y) = 1 − q, y = x − 1, ⎪ ⎩0, y ≤ x −2
(3.14)
and φν (−∞, y) = ν y . Then, for any antisymmetric function f (b1 , . . . , bn ), bn >...>b1 >b0 b0 :fixed
det(φν (ai , b j ))0≤i, j≤n · f (b1 , . . . , bn )
= gν (a1 , b0 )
bn >...>b1 >b0 b0 :fixed
det(φν (ai , b j ))1≤i, j≤n · f (b1 , . . . , bn ),
(3.15)
where an > · · · > a1 , a0 = −∞ and ⎧ ⎪ b ≥ a, ⎨0, b gν (a, b) = ν (1 − (1 − q)ν), b = a − 1, ⎪ ⎩ν b , b ≤ a − 2.
(3.16)
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A. Borodin, P. L. Ferrari, T. Sasamoto
Proof of Lemma 9. From the antisymmetry of f and of the determinant, (3.15) is equivalent to b1 ,...,bn >b0 b0 :fixed
det(φν (ai , b j ))0≤i, j≤n · f (b1 , . . . , bn )
= gν (a1 , b0 )
b1 ,...,bn >b0 b0 :fixed
det(φν (ai , b j ))1≤i, j≤n · f (b1 , . . . , bn ).
(3.17)
Since a basis of the antisymmetric functions is made of the antisymmetric delta functions and the relation to prove is linear in f , it is enough to consider
(−1)σ , if (b1 , . . . , bn ) = (bσ1 , . . . , bσn ) for some σ ∈ Sn , 0, otherwise (3.18) for fixed b1 , . . . , bn > b0 . Here Sn is the group of all permutations of {1, . . . , n}. For this special choice of f , the left-hand side of (3.17) is n! times the single determinant, f (b1 , . . . , bn ) =
⎡
ν b0 ⎢φν (a1 , b0 ) ⎢ det ⎢ .. ⎣ .
ν b1 φν (a1 , b1 ) .. .
... ...
φν (an , b0 )
φν (an , b1 )
...
⎤ ν bn φν (a1 , bn )⎥ ⎥ ⎥. .. ⎦ .
(3.19)
φν (an , bn )
We have the following three cases: (a) a1 ≤ b0 : the second row gives (ν b0 −a1 , . . . , ν bn −a1 ) which is proportional to the first row. Therefore in this case the LHS is zero. (b) a1 = b0 + 1: The second row is (1 − q, ν b1 −a1 , . . . , ν bn −a1 ). Subtracting ν a1 times the second row from the first row one obtains ν b0 (1 − (1 − q)ν) · det(φν (ai , b j ))1≤i, j≤n .
(3.20)
(c) a1 > b0 + 1: The first column is (ν b0 , 0, . . . , 0)t . Thus the determinant is ν b0 · det(φν (ai , b j ))1≤i, j≤n . The result in each case agrees with 1/n! times the RHS of (3.17) and hence the lemma is proved. Let N (x1 , . . . , x N ) denote the number of j’s s.t. x j − x j+1 = 1, j = 1, . . . , N − 1. Using the above lemma with ν = 1 in which case φν reduces to φ , we have the following result. Lemma 10. With x1n = xn , n = 1, . . . , N , one has D
W N (x) = q N (x1 ,...,x N ) det[F j−i (x N − j+1 − y N −i+1 , t + j − i)]1≤i, j≤N 1 , . . . , x N ; t). =: G(x
(3.21)
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Proof of Lemma 10. For simplicity, we denote f i (x) = F−i+1 (x − y N −i+1 , t − i + 1),
(3.22)
for i = 1, . . . , N . From the definitions (3.1), the LHS of (3.21) is written N −1
n >...>x n xnn >xn−1 1 n x1 :fixed,1≤n≤N
det(φ
(xin , x n+1 j+1 ))0≤i, j≤n
det( f i (x Nj ))1≤i, j≤N .
(3.23)
n=1
N , Applying Lemma 9 with ν = 1, n = N − 1, ai = xiN −1 , i = 1, . . . , N − 1, bi = xi+1 i = 0, . . . , N − 1 and
f (b1 , . . . , bn ) = det( f i (x Nj ))1≤i, j≤N ,
(3.24)
we obtain (3.23) =
g1 (x1N −1 , x1N ) ·
×
N −2
n >...>x n xnn >xn−1 1 x1n :fixed,1≤n≤N −1
det(φ
(xin , x n+1 j+1 ))0≤i, j≤n
n=1
det(φ (xiN −1 , x Nj+1 ))1≤i, j≤N −1 · det( f i (x Nj ))1≤i, j≤N .
x NN >x NN −1 >...>x1N x1N :fixed
(3.25) Heine’s identity, 1 det(ϕi (x j ))1≤i, j≤n det(ψi (x j ))1≤i, j≤n = det ϕi ∗ ψ j n! x ,...,x 1
1≤i, j≤n
,
(3.26)
n
allows us to rewrite the last summation in (3.25) as ⎡
f 1 (x1N ) ⎢ .. det ⎣ .
f N (x1N )
(φ ∗ f 1 )(x1N −1 ) .. .
(φ
∗
f N )(x1N −1 )
... ...
−1 ⎤ (φ ∗ f 1 )(x NN−1 ) ⎥ .. ⎦. .
(φ
∗
(3.27)
−1 f N )(x NN−1 )
We repeat the procedure up to a total of j − 1 times in column j and we get
(3.25) =
N −1
g1 (x1n , x1n+1 )
j−1
! "# $ N − j+1 det[φ ∗ . . . ∗ φ ∗ f i (x1 )]1≤i, j≤N .
(3.28)
n=1
The proof of the lemma is finished using (3.22), (3.9) and q N (x1 ,...,x N ) .
% N −1 n=1
g1 (x1n , x1n+1 ) =
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A. Borodin, P. L. Ferrari, T. Sasamoto
Proof of Proposition 7. We need to prove 1 , . . . , x N ; t). G(x1 , . . . , x N ; t) = G(x
(3.29)
This statement was also proved in [20] by the Bethe ansatz techniques. Our proof is by induction in t. We start by showing that the initial conditions agree, i.e., 1 , . . . , x N ; 0) = G(x1 , . . . , x N ; 0), that is, G(x q N (x1 ,...,x1 ) · det[F j−i (x N − j+1 − y N −i+1 , j − i)]1≤i, j≤N = 1
N
N
δxn ,yn .
(3.30)
n=1
We first show that LHS of (3.30) is zero if x N = y N . If x N ≤ y N − 1, since y N −i+1 ≥ y N + i − 1, one has x N − y N −i+1 < −i + 1, i = 1, . . . , N . Then, from (3.10) we have F1−i (x N − y N −i+1 , 1−i) = 0, i.e., the first column of LHS of (3.30) is zero. Similarly, if x N ≥ y N + 1, since x N − j+1 ≥ x N + j − 1, one has x N − j+1 − y N > j − 1, j = 1, . . . , N . Then, from (3.11) we have F j−1 (x N − j+1 − y N , j − 1) = 0, i.e., the first row of LHS of (3.30) is zero. This agrees with RHS of (3.30) also being zero if x N = y N . Now let us assume x N = y N . There are two cases. (a) y N −1 > y N + 1. In this case, since x N − y N −i+1 = y N − y N −i+1 < −i + 1, i = 2, . . . , N , one has F1−i (x N − y N −i+1 , 1−i) = 0, i = 2, . . . , N . Then the first column of LHS of (3.30) is (1, 0, . . . , 0)t and hence the determinant is equal to det[F j−i (x N − j+1 − y N −i+1 , j − i)]2≤i, j≤N . (b) y N −1 = y N + 1. First let us see that LHS of (3.30) is zero when x N −1 = y N −1 . We have x N −1 ≥ x N + 1 = y N + 1 = y N −1 . If x N −1 ≥ y N −1 + 1, we have x N − j+1 − y N ≥ x N −1 + j − 2 − (y N −1 − 1) ≥ j, for j = 2, . . . , N , and x N − j+1 − y N −1 ≥ j − 1, for j = 2, . . . , N . Then the first and the second row of LHS of (3.30) are both of the form, (∗, 0, . . . , 0) where ∗ represents an arbitrary number and hence the determinant is zero. Hence LHS of (3.30) is zero if x N −1 = y N −1 . On the other hand, when x N −1 = y N −1 , the upper-left 2 × 2 submatrix of the determinant is ' & ' & F0 (0, 0) 1 1−q F1 (1, 1) , (3.31) = −1/q 1 F−1 (−1, −1) F0 (0, 0) whose determinant is 1/q. Repeating the same procedure, at each step one has either case (a) or (b). The final result is that yk = xk , for k = 1, . . . , N , otherwise the determinant in LHS of (3.30) is zero. Moreover, when yk = xk , k = 1, . . . , N , denote by n 1 , n 1 + n 2 , . . . , n 1 + . . . + n % the values of j such that x j−1 − x j > 1. Then LHS of (3.30) is equal to m=1 Dn m with (3.32) Dn = det F j−i ( j − i, j − i) 1≤i, j≤n . Finally using (3.12), (3.13), we obtain an explicit form of the matrix. To compute its determinant it is enough to develop along the first row. The determinant of the (1, 1) minor is Dn−1 , while the one of the (1, 2) minor is (−1/q)Dn−1 because the minor is the same as the (1, 1) minor except the first column is multiplied by −1/q. All the other minors have determinant zero, because the first two columns are linearly dependent. Thus, Dn = 1 · Dn−1 − (1 − q)/(−q)Dn−1 , and since D1 = 1, it follows that Dn =
1 q n−1
.
This ends the part of the proof concerning initial conditions.
(3.33)
Large Time Asymptotics of Growth Models on Space-like Paths II
431
Next we prove that (3.29) holds for t + 1 if it does for t. Since this is true for t = 0, by induction it will be true for all t ∈ N. G satisfies the TASEP dynamics, thus G(x1 , . . . , x N ; t + 1) 1 , . . . , z N , t)w(z, x) G(z 1 , . . . , z N , t)w(z, x) = G(z = z
=
(3.34)
z
w(z, x)q N (z 1 ,...,z N ) det[F j−i (z N − j+1 − y N −i+1 , t + j − i)]1≤i, j≤N .
z
Here ⎧ ⎪ z n = z n−1 − 1, xn = z n , ⎨1, w(z, x) = vn , vn = q, z n < z n−1 − 1, xn = z n , ⎪ ⎩1 − q, z < z n=1 n n−1 − 1, x n = z n + 1, N
(3.35)
and in the second equality we have used the assumption of the induction. We rewrite 1 , . . . , x N ; t + 1) using (3.7) and (3.8) as follows. For k from 1 to N : G(x (a) if xk = xk+1 + 1, then we use (3.8) to the N + 1 − k th column. Then, the new term with the (1 − q) factor in front cancels out because it is proportional to its left column of the determinant. (b) if xk > xk+1 + 1, then we just use (3.7). With these replacements we get 1 , . . . , x N ; t + 1) (3.36) G(x N (x1 ,...,x N ) w(z, ˜ x)q det[F j−i (z N − j+1 − y N −i+1 , t + j − i)]1≤i, j≤N , = z
where ⎧ ⎪ xn = xn+1 + 1, z n = xn , ⎨1, w(z, ˜ x) = v˜n , v˜n = q, xn > xn+1 + 1, z n = xn , ⎪ ⎩1 − q, x > x n=1 n n+1 + 1, z n = x n − 1. N
(3.37)
Comparing (3.34) and (3.36), it is enough to show q N (z 1 ,...,z N ) w(z, x) = q N (x1 ,...,x N ) w(z, ˜ x).
(3.38)
This indeed holds and can be seen by checking case by case. We illustrate it using Fig. 6. First consider a block of particles, say m of them at time t + 1. There are two possibilities of reaching this situation in one time step, as indicated in Fig. 6 (a) and (b). The products of all the weights on the right and on the left are the same, i.e., (3.38) holds for a single block of particles. If two blocks of particles at time t + 1 are at distance at least 2, they are independent during one time step. We just have to check that (3.38) holds for two blocks at distance 2 at time t + 1. Case (a) is illustrated in (c) and the weights are unchanged for both blocks. Case (b) is illustrated in (d). This time, the q on the top line of the second block becomes a 1, but this is compensated by an extra factor q on the left.
432
A. Borodin, P. L. Ferrari, T. Sasamoto
(a)
(b)
(c)
(d)
Fig. 6. Graphical representation of (3.38). The dots represent empty places, while a line leaving/arriving to a point is an occupied position. In (a) and (b), on the left (resp. right) we indicate the weights different from 1 of LHS (resp. RHS) of (3.38). In (c) and (d) the bottom and top lines of two blocks at distance 2 at time t + 1 are represented, for the cases corresponding to (a) and (b) for the top block
4. Joint Distributions Along Space-like Paths Theorem 11. Let us consider particles starting from y1 > y2 > . . . and denote x j (t) the position of j th particle at time t. Take a sequence of particles and times which are spacelike, i.e., a sequence of m couples S = {(n k , tk ), k = 1, . . . , m | (n k , tk ) ≺ (n k+1 , tk+1 )}. The joint distribution of their positions xn k (tk ) is given by m xn k (tk ) ≥ ak = det(½ − χa K χa )2 ({(n 1 ,t1 ),...,(n m ,tm )}×Z) , È (4.1) k=1
where χa ((n k , tk ), x) = ½(x < ak ). Here K is the extended kernel with entries K ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ) = −φ ∗((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 ) +
n2
,t1 nn11−k (x1 )nn 22 ,t−k2 (x2 ),
k=1
(4.2) where φ ∗((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 ) (4.3) n 1 −n 2 w (1 + pw)t1 −t2 1 dw ½[(n1 ,t1 )≺(n2 ,t2 )] , = 2π i 0,−1 (1 + w)x1 −x2 +1 (1 + w)(1 + pw) n−1 the functions { n,t j } j=0 are given by
n,t j (x)
1 = 2π i
(1 + pw)t dw (1 + w)x−yn− j +1 −1
w (1 + w)(1 + pw)
j ,
(4.4)
where the contour −1 is any simple loop anticlockwise oriented and including −1 and n−1 no other poles. The functions {n,t j } j=0 are characterized by the two conditions: n,t k (x)ln,t (x) = δk,l , 0 ≤ k, l ≤ n − 1, (4.5) x∈Z
Large Time Asymptotics of Growth Models on Space-like Paths II
433
and n,t j (x) is a polynomial of degree j, j = 0, . . . , n − 1. Proof of Theorem 11. The statement is the analogue of Proposition 3.1 of [4]. The proof is also quite similar. We start with the analog of Theorem 4.1 of [4]. Proposition 12. Set t0 = 0 and n m+1 = 0. The joint distribution of particles from Theorem 11 is a marginal of a determinantal measure, obtained by summation of the variables in the set D = {xkl (ti ), 1 ≤ k ≤ l, 1 ≤ l ≤ n i , 0 ≤ i ≤ m} \ {x1n i (ti ), 1 ≤ i ≤ m}; the range of summation for any variable in this set is Z. Precisely,
È(xn (ti ) = x1n (ti ), (1 ≤ i ≤ m | xk (0) = yk (0), 1 ≤ k ≤ n 1 ) ) i
i
= const × × ×
m &
,t0 n 1 det nn11−l (xk (t0 ))
D
1≤k,l≤n 1
det[Tti ,ti−1 (xln i (ti ), xkn i (ti−1 ))]1≤k,l≤n i
i=1 ni
det[φ
' (xkn−1 (ti ), xln (ti ))]1≤k,l≤n
,
n=n i+1 +1
where 1 (1 + pw)ti −ti−1 Tti ,ti−1 (x, y) = dw , 2π i 0,−1 (1 + w)x−y+1 (1 + pw)(1 + w) 1 1 φ (x, y) = dw x−y+1 2π i 0,−1 (1 + w) w and φ (xnn−1 , y) = 1. The proof of this proposition is word-for-word repetition of the proof of Theorem 4.1 of [4] in the case when all parameters vi = 1 with the following replacements: PushASEP ϕ(x, y) Fn (x, a(t), b(t))
Parallel TASEP φ (x, y) n (x, t) := Fn (x, t + n) F
The role of Lemmas 4.3 and 4.4 of [4] used in the proof of Theorem 4.1 there is played by the identity n )(x, t) = F n+1 (x, t), (φ ∗ F cf. (3.9), and Proposition 7 above, respectively. As the next step, we would like to apply Theorem 4.2 of [4]. This cannot be done directly because the functions (4.43) of [4] are not well-defined (the series diverge), and we need to deform our measure to guarantee the convergence of all the series involved.
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A. Borodin, P. L. Ferrari, T. Sasamoto
Let v1 , . . . , vn be arbitrary nonzero complex numbers. Set ( j = 1, . . . , n) 1 2π i
φ j (x, y) =
1 2π i
n,t (x) = n− j
dw
v j −1,−1
−1
dw
j−1
(1 + p(v −1 1 j (1 + w) − 1))(1 + w) , x−y+1 (1 + w) v −1 j (1 + w) − 1
n vk−1 (1 + w) − 1 (1 + pw)t . (1 + w)x−y j +1 k= j+1 (1 + p(vk−1 (1 + w) − 1))(1 + w)
y
j−1
Also set φ j (x j , y) = v j (recall that x j are fictitious variables). Observe that as all n,t v j → 1 these functions converge to φ and n− j. Consider the following deformation of the measure from Proposition 12: ( ) n 1 ,t0 (x n 1 (t0 )) const × det n 1 −l k 1≤k,l≤n 1
×
×
m
det[Tti ,ti−1 (xln i (ti ), xkn i (ti−1 ))]1≤k,l≤n i
i=1
⎤
ni
det[φn (xkn−1 (ti ), xln (ti ))]1≤k,l≤n ⎦ .
(4.6)
n=n i+1 +1
n,t (x) are finitely supported, there are only finitely many sets Since all functions j of values of the variables {xkn }n≤n 1 , k≤n for which the above weights are nonzero. This implies that all the correlation functions are analytic in the parameters v1 , . . . , vn . Following [4], let us introduce additional notation. For any level n there is a number c(n) of terms det[T ] in (4.6) which involve the particles with upper index n (in other words c(n) is #{i | n i = n}). Let us denote the time moments involved in these factors n . Notice that t n = t n+1 , t n 1 = t , t n 1 = t , and t 0 = t 0 by t0n < · · · < tc(n) 0 1 1 0 0 c(0) = tm . c(n+1) 0 Let us now apply Theorem 4.2 of [4] to the measure (4.6). Since we are using the same notation (except here we have extra tilde over ’s), let us also use (4.36)–(4.40) 1 of [4]. The computation of the matrix [Mk,l ]nk,l=1 yields, cf. (4.59) in [4], Mk,l =
y vk
y∈Z
×
1 2π i
|w+1|=const
k
(1 + pw)tc(k) dw (1 + w) y−yl +1
n1
v −1 j (1 + w) − 1
j=l+1
(1 + p(v −1 j (1+w) − 1))(1+w)
n1 (1 + p(v −1 j (1 + w) − 1))(1 + w) j=k+1
v −1 j (1 + w) − 1 (4.7)
with the positive oriented integration contour going around points w = −1 and w = v j − 1, j = k + 1, . . . , n, but not other poles of the integrand. To make sure that we can move the sum inside the integral without making the sum divergent, and to guarantee the convergence of the sum in (4.9) below, we assume that |v1 | > |v2 | > · · · > |vn 1 |.
(4.8)
Then we obtain Mk,l = 0 for l < k, and Mk,k = 0 for any k. Thus, the matrix M is upper triangular and nondegenerate.
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Following Theorem 4.2 of [4], we now need to find the linear span of
n 1 n n (φ1 ∗ φ (tc(1) ,ta ) )(x10 , x), . . . , (φn ∗ φ (tc(n) ,ta ) )(xnn−1 , x) . We have (φk ∗ φ (tc(k) ,ta ) )(xkk−1 , x) = k
n
y
vk
y∈Z
×
n
(1 +
j=k+1
1 2π i
|w+1|=const1
p(v −1 j (1 + w) − 1))(1 + w) v −1 j (1 + w) − 1
(1 + pw)tc(k) −ta (1 + w) y−x+1 k
dw
= const · vkx ,
n
(4.9)
k n % v + p(v −v ) where the constant is given by (1+ p(vk −1))tc(k) −ta vkn−k−1 nj=k+1 j vk −vk j j . Remark that this constant diverges if v j = vk . n,t must be linear combinations of Thus, the corresponding biorthogonal functions j n,t are linear combinations of v1x , . . . , vnx . Equivalently, j
1 f k (x) = 2π i
dz(1 + z)
x
n (1 + p(v −1 j (1 + z) − 1))(1 + z) j=n−k
v −1 j (1 + z) − 1
,
(4.10)
k = 0, . . . , n − 1, with the integration contour going around the poles z = v j − 1, x , . . . , vx . j = n − k, . . . , n. Indeed, f k (x) is a linear combination of vn−k n Denote by G = [G k,l ]n−1 the Gram matrix k,l=0 n,t (x). f k (x) (4.11) G k,l = l x∈Z
Then we have n,t (x) = k
n−1
[G −1 ]k,l fl (x).
(4.12)
l=0
Theorem 4.2 of [4] implies that the correlation functions of the measure (4.6) are n,t ’s, n,t ’s, and determinantal, and the correlation kernel is expressed in terms of k l φn ’s. This statement can be analytically continued in the parameters v1 , . . . , vn varying in a small enough neighborhood of 1. Indeed, the only ingredients of the obtained formula that may not be analytic are the matrix elements of G −1 . Since M is triangular, G is yn−k +2 (1+ p(vn−k −1))t . So the diagonal elements also triangular. In particular, G k,k = vn−k of G are rational functions in v1 , . . . , vn not equal to zero at v1 = v2 = · · · = vn = 1. Thus, the matrix elements of G −1 are analytic around this point, and we can continue the result to v1 = · · · = vn = 1. Observe that for v1 = · · · = vn = 1, f k (x) is a degree k polynomial in x. Thus, n,t n,t k (x) are the unique polynomials of degrees deg k (x) = k that are biorthogonal to n,t l ’s. Theorem 11 holds for general fixed initial conditions. We want to apply it to the alternating initial condition. For that we first have to do the orthogonalization with the result given in the next lemma.
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Lemma 13. For initial conditions y j = −2 j, j = 1, . . . , n, we have n,t j (x) = and n,t j (x) =
1 2π i
1 2π i
0,−1
0
dz
dw
w j (1 + pw)t− j , (1 + w)x+2n− j+1
(4.13)
(1 + 2z + pz 2 )(1 + z)x+2n− j−1 , z j+1 (1 + pz)t− j+1
(4.14)
where, as before, p = 1 − q. In particular, n,t 0 (x) = 1. Proof of Lemma 13. The formula for nj is just obtained by substituting the initial conditions into (4.13). Now we prove that the orthonormality relation (4.5) holds. For k = 0, . . . , n − 1, the pole at w = 0 in kn is not present, and for x < −2n + k, kn (x) = 0 because the residue at −1 vanishes. Thus n,t j (x)kn,t (x) x∈Z
=
1 (2π i)2
×
0
∞ x=−2n+k
dz
−1
1+z 1+w
dw
x
(1 + 2z + pz 2 )(1 + z)2n− j−1 w k (1 + pw)t−k z j+1 (1 + pz)t− j+1 (1 + w)2n−k−1
,
(4.15)
where we have the constraint on the integration paths |1 + z| < |1 + w|. The last term (the sum) equals 1 + z −2n+k 1 + w . (4.16) 1+w w−z Now the pole at w = −1 has disappeared and instead of it there is a simple pole at w = z. Thus, the integral over w is just the residue at w = z, leading to
n,t n,t j (x)k (x) =
x∈Z
=
1 2π i 1 2π i
0
*
0
where we used the change of variable u =
dz
1 + 2z + pz 2 (1 + pz)2
z(1 + z) 1 + pz
k− j−1
duu k− j−1 = δ j,k , z(1+z) 1+ pz .
(4.17)
Lemma 13 together with Theorem 11 leads to the kernel for the alternating initial condition. Proposition 14. For y j = −2 j, j = 1, . . . , n, the kernel K in Theorem 11 is given by K ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ) = −φ ((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 )½[(n 1 ,t1 )≺(n 2 ,t2 )] ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ), +K
(4.18)
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437
where φ ((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 ) is given by (2.6) and w n 1 (1 + pw)t1 −n 1 +1 ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ) = 1 dw dz K (2π i)2 −1 (1 + w)x1 +n 1 +1 0 ×
1 (1 + z)x2 +n 2 (1 + 2z + pz 2 ) z n 2 (1 + pz)t2 −n 2 +2 (w − z)(w +
1+z 1+ pz )
.
(4.19) Here 0 (resp −1 ) is any simple loop, anticlockwise oriented, which includes the pole 1+z at z = 0 (resp. w = −1), satisfying {− 1+ pz , z ∈ 0 } ⊂ −1 and no point of 0 lies inside −1 . Proof of Proposition 14. We substitute (4.13) and (4.14) in the kernel (4.2). Since n,t over k to ∞. We can take the sum j (x) = 0 for j < 0, we can extend the sum + + + 1+ pw z(1+z) + inside the integrals if the integration paths satisfy + w(1+w) 1+ pz + < 1. Then we compute the geometric series and obtain ∞ 1 w n 1 (1 + pw)t1 −n 1 +1 ,t1 nn11−k (x1 )nn 22 ,t−k2 (x2 ) = dw dz (2π i)2 0,−1 (1 + w)x1 +n 1 +1 0 k=1
×
1 (1 + z)x2 +n 2 (1 + 2z + pz 2 ) z n 2 (1 + pz)t2 −n 2 +2 (w − z)(w +
1+z 1+ pz )
. (4.20)
At this point both simple poles w = z and w = −(1 + z)/(1 + pz) are inside the integration path 0,−1 , but the integrand does not have any pole anymore at w = 0. Thus we will drop the 0 in 0,−1 . Separating the contribution from the pole at w = z we get n2
,t1 ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 ) nn11−k (x1 )nn 22 ,t−k2 (x2 ) = K
k=1
+
1 2π i
0
1 + pz z
n 2 −n 1
(1 + z)n 2 +x2 −n 1 −x1 −1 .
(4.21)
Moreover, we also have φ ∗((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 ) = φ ((n 1 ,t1 ),(n 2 ,t2 )) (x1 , x2 ) 1 + pz n 2 −n 1 1 + (1 + z)n 2 +x2 −n 1 −x1 −1 . 2π i 0 z Thus the last two terms of (4.21) and (4.22) cancel out, leading to (4.19).
(4.22)
With Proposition 14 we almost obtained Theorem 1. What remains to do is to focus far enough into the negative axis, where the influence of the finiteness of the number of particles is not present anymore. There the kernel is equal to the kernel for the initial conditions yi = −2i, i ∈ Z. Proof of Theorem 1. The kernel for the flat case is obtained by considering the region satisfying x1 + n 1 + 1 ≤ 0, where the effect of the boundary in the TASEP is absent. Here the pole at w = −1 vanishes. Computing the residue at w = −(1 + z)/(1 + pz) in Proposition 14 gives the kernel (2.5) up to a factor (−1)n 1 −n 2 which we cancel by a conjugation of the kernel.
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5. Proof of Theorem 3 From Theorem 1 we have that È(xn (t) ≥ x) = det(½ − ½(−∞,x) K ½(−∞,x) ). We have such a situation but with x = y − sT 1/3 . With this change of variable, we get È(xn (t) ≥ y − sT 1/3 ) = det(½ − ½(s,∞) K Tresc ½(s,∞) ), where K Tresc (ξ1 , ξ2 ) = T 1/3 K (x1 − ξ1 T 1/3 , x2 − ξ2 T 1/3 ) (here we did not write explicitly the (n, t) entries). Taking into account the scaling (2.10), we thus have to analyze the rescaled kernel K Tresc (u 1 , ξ1 ; u 2 , ξ2 ) = T 1/3 K (n(u 1 ), t (u 1 )), x(u 1 ) − ξ1 T 1/3 ; (n(u 2 ), t (u 2 )), x(u 2 ) − ξ2 T 1/3 , (5.1) √ with x(u) = −2n(u)+vt (u), v = 1− q. In particular, we have to prove that, for u 1 , u 2 fixed, K Tresc (or a conjugate kernel of it) converges to the kernel κv−1 K A1 (κh−1 u 1 , κv−1 ξ1 , κh−1 u 2 , κv−1 ξ2 ) uniformly on bounded sets and has enough control (bounds) on the decay of K resc in the variables ξ1 , ξ2 such that also the Fredholm determinant converges. In order to have a proper limit of the kernel as T → ∞, we have to consider the conj conjugate kernel K T given by √ x1 −x2 q conj resc q n 1 −n 2 q −(t1 −t2 )/2 . K T (u 1 , ξ1 ; u 2 , ξ2 ) = K T (u 1 , ξ1 ; u 2 , ξ2 ) √ 1+ q (5.2) The new kernel does not change the determinantal measure, being just a conjugation of the old one. So, in the following we will determine the limit of (5.2) as T → ∞. Proposition 15. (Uniform convergence on compact sets) For u 1 , u 2 fixed, according to (2.10), set ) ( (5.3) xi = −2n(u i ) + vt (u i ) − ξi T 1/3 , n i = n(u i ), ti = t (u i ).
(5.4)
Then, for any fixed L > 0, we have lim K T (n 1 , x1 ; n 2 , x2 )T 1/3 = κv−1 K A1 (κh−1 u 1 , κv−1 ξ1 ; κh−1 u 2 , κv−1 ξ2 ) conj
T →∞
(5.5)
uniformly for (ξ1 , ξ2 ) ∈ [−L , L]2 , with the kernel K A1 given by (2.7) and the constants κv and κh given by (2.13). Proof of Proposition 15. First we consider the first term in (2.4). We thus consider (2.6) with the above replacements and conjugation. √ This term has to be considered only for u 2 > u 1 . The change of variable w = −1 + qz leads then to dz T 2/3 (g0 (z)−g0 (z c ))+T 1/3 (g1 (z)−g1 (z c )) T 1/3 e (5.6) 2π i 0 z √ with z c = (1 + q)−1 and √ √ g0 (z) = (u 2 − u 1 )(π (θ ) + 1) ln( q + (1 − q)z) − (1 − q) ln(z) √ q + (1 − q)z , + (u 2 − u 1 )(1 − π (θ )) ln √ z(1 − qz) √ √ g1 (z) = −(u 22 − u 21 ) 21 π (θ ) q ln(z) + ln(1 − qz) − (ξ2 − ξ1 ) ln(z). (5.7)
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The function g0 has a critical point at z = z c . The series expansions around z = z c are g0 (z) = g0 (z c ) + (u 2 − u 1 )κ1 (z − z c )2 + O((z − z c )3 ), √ g1 (z) = g1 (z c ) − (ξ2 − ξ1 )(1 + q)(z − z c ) + O((z − z c )2 ),
(5.8)
& ' √ 1− q √ √ 2 + 1 − π (θ ) . κ1 = q(1 + q) (π (θ ) + 1) (5.9) 2 To prove convergence of (5.6) we have to show that the contribution coming around the critical point dominates in the T → ∞ limit. We do it by finding a steep descent path1 for g0 passing by z = z c . Consider the path 0 = {ρeiφ , φ ∈ [−π, π )}. Then, on d 0 , dφ Re(ln(z)) = 0, √ q(1 − q)ρ sin(φ) d √ Re(ln( q + (1 − q)z)) = − √ , (5.10) dφ | q + (1 − q)z|2
where
√ qρ sin(φ) d √ Re(− ln(1 − qz)) = − . (5.11) √ dφ |1 − qz|2 Thus 0 is a steep descent path for g0 . Now we set ρ = z c . Then, the real part of g0 (z) is maximal at z = z c and strictly less than g(z c ) for all other points on 0 . Therefore, we can restrict the integration from 0 to 0δ = {z ∈ 0 ||z − z c | ≤ δ}. For δ small, the 2/3 error made is just of order O(e−cT ) with c > 0 (c ∼ δ 2 as δ 1). In the integral over 0δ we can use (5.8) to get √ √ (1 + q)T 1/3 2/3 2 1/3 dze T (u 2 −u 1 )κ1 (z−z c ) −T (ξ2 −ξ1 )(1+ q)(z−z c ) 2π i 0δ
and
×eO(T
2/3 (z−z
c)
3 ,T 1/3 (z−z
c)
2 ,(z−z
c ))
.
(5.12)
We use |e x − 1| ≤ |x|e|x| to control the difference between (5.12) and the same expression without the error terms. By taking δ small enough and the change of variable (z − z c )T 1/3 = W , we obtain that this difference is just of order O(T −1/3 ), uniformly for ξ1 , ξ2 in a bounded set. At this point we remain with (5.12) without the error terms. We extend the integration path to z c + iR and this, as above, gives an error of order 2/3 O(e−cT ). Thus we have (5.6) = O(e−cT , T −1/3 ) √ √ (1 + q)T 1/3 2/3 2 1/3 + dze T (u 2 −u 1 )κ1 (z−z c ) −T (ξ2 −ξ1 )(1+ q)(z−z c ) . 2π i z c +iR 2/3
(5.13)
Therefore, uniformly for ξ1 , ξ2 in bounded sets,
(ξ2 − ξ1 )2 1 lim (5.6) = √ exp − T →∞ 4(u 2 − u 1 )α 2 4π(u 2 − u 1 )α √ with α 2 = κ1 /(1 + q)2 = κv2 /κh . ,
(5.14)
1 For an integral I = t f (z) , we say that γ is a steep descent path if (1) Re( f (z)) is maximal at some γ dze z 0 ∈ γ : Re( f (z)) < Re( f (z 0 )) for z ∈ γ \ {z 0 } and (2) Re( f (z)) is monotone along γ \ {z 0 } except, if γ is
closed, at a single point where Re( f ) is minimal.
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Now we have to consider the second term√in (2.4). Notice that this time the restriction u 2 > u 1 does not apply. Set z c = −1/(1 + q). Then 1/3 conj (u 1 , ξ1 ; u 2 , ξ2 ) = −T K T 2π i
0
dz
e T f0 (z)+T eT
2/3
2/3
f 1 (z)+T 1/3 f 2 (z)+ f 3 (z)
f 1 (z c )+T 1/3 f 2 (z c )
(5.15)
with
& ' 1+z √ √ √ − (1 + q) ln(1 + (1 − q)z) + q ln(q) , f 0 (z) = (π(θ ) + θ ) (1 − q) ln −z √ √ f 1 (z) = (π (θ ) + 1) u 1 ((1 − q) ln(−z) + q ln((1 + (1 − q)z)/q)) √ −u 2 ((1 − q) ln(1 + z) − ln(1 + (1 − q)z)) (1 + z)(−z) , (5.16) + (1 − π (θ ))(u 1 − u 2 ) ln 1 + (1 − q)z f 2 (z) =
π (θ ) ( 2 √ √ u 1 (ln(1 + z) + q ln(−qz) − (1 + q) ln(1 + (1 − q)z)) 2 ) √ −u 22 ( q ln(1 + z) + ln(−z))
+ ξ1 ln(−qz/(1 + (1 − q)z)) − ξ2 ln(1 + z), f 3 (z) = − ln(−z(1 + (1 − q)z)).
(5.17)
The function f 0 has a double critical point at z = z c and the series expansions around z = z c of the f i ’s are given by f 0 (z) = 13 κ2 (z − z c )3 + O((z − z c )4 ), κ1 f 1 (z) = f 1 (z c ) + (u 2 − u 1 )(z − z c )2 + O((z − z c )3 ), q √ 1+ q f 2 (z) = f 2 (z c ) − (ξ1 + ξ2 ) √ (z − z c ) + O((z − z c )2 ), q √ √ f 3 (z) = ln((1 + q)/ q) + O((z − z c )),
(5.18)
with κ1 given in (5.9) and (π(θ ) + θ )(1 − q)(1 + κ2 = q
√ 3 q)
.
(5.19)
The leading contribution in the T → ∞ limit will come from the region around the double critical point. The first step is to choose for γ0 a steep √ descent path for f 0 . First we consider γ0 = {−ρeiφ , φ ∈ [−π, π )}, ρ ∈ (0, 1/(1 + q)]. √The only part in Re( f 0 (z)) which is not constant along γ0 is the term (π(θ ) + θ )(1 − q) multiplied by √ √ A(z) = ln |1 + z| − a ln |1 + (1 − q)z|, a = (1 + q)/(1 − q). Simple computations lead then to √ √ √ √ √ sin(φ)ρ q 2 + q − 4ρ(1 + q) cos(φ) + ρ 2 (2 − q)(1 + q)2 d . A(z) = − dφ |1 + z|2 |1 + (1 − q)z|2 (5.20)
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This expression is strictly less than zero along γ0 except at φ = 0, −π , provided that the last term is strictly positive for φ = 0, −π . This is easy to check because the last term reaches his minimum √ at cos(φ) = −1. Solving second degree√equations, we get that on ρ ∈ (0, 1/(1 + q)) it is strictly positive and at ρ = 1/(1 + q) is zero. Thus, the path γ0 is steep descent for f 0 . But close to the critical point, the steepest descent path leaves with an angle ±π/3. Therefore, consider for a moment γ1 = {z = z c + e−iπ sgn(x)/3 |x|, x ∈ R}. By symmetry we can restrict the next computations to x ≥ 0. We have to see that B(z) = ln |1 + z| − ln |z| − a ln |1 + (1 − q)z| is maximum at x = 0 and decreasing for x > 0. We have d x2 B(z) = − 2 2 dx 2|1 + z| |z| |1 + (1 − q)z|2 √ √ √ √ × 2q + 2(1 − q) q x − (1 − 3 q + q)(1 + q)2 x 2 + 2(1 − q)(1 + q)3 x 3 . (5.21) The term in the second line is always positive for all x ≥ 0. To see this, remark that it is a polynomial of third degree which goes to ∞ as x → ∞ and at x = 0 is already positive and has positive slope. Therefore one just computes its stationary points and, if reals, takes the right-most one. There, the term under consideration turns out to be positive, which concludes the argument. Consequently, γ1 is also a steep descent path. We choose a steep descent path 0 as follows. We follow γ1 starting from the critical point until we intersect it with γ0 , and then we follow γ0 . Since 0 is steep descent for f 0 , we can integrate only on 0δ = {z ∈ 0 ||z − z c | ≤ δ}. The error made by this cut is just of order O(e−cT ) for some c = c(δ) > 0 (with c ∼ δ 3 as δ → 0). Around the critical point we use the series expansions (5.18). Thus we have conj (u 1 , ξ1 ; u 2 , ξ2 ) = O(e−cT ) K T √ * 1 + q O((z−z c )4 T,(z−z c )3 T 2/3 ,(z−z c )2 T 1/3 ,(z−z c )) −T 1/3 dz √ e + 2π i 0δ q 1
×e 3
κ2 (z−z c )3 T +
√ q κ1 2 2/3 −(ξ +ξ ) 1+ 1/3 1 2 √q (z−z c )T q (u 2 −u 1 )(z−z c ) T
.
(5.22)
We want to cancel the error terms. The difference between (5.22) and the same expression without the error terms is bounded using |e x − 1| ≤ e|x| |x|, applied to x = O(· · · ). Then, this error term becomes √ + T 1/3 * 1+ q + dz + √ O((z − z c )4 T, (z − z c )3 T 2/3 , (z − z c )2 T 1/3 , (z − z c )) 2π i 0δ q √ + 1+ q κ 1 c κ (z−z c )3 T +c2 q1 (u 2 −u 1 )(z−z c )2 T 2/3 −c3 (ξ1 +ξ2 ) √q (z−z c )T 1/3 + (5.23) ×e 1 3 2 + for some c1 , c2 , c3 depending on δ. As δ → 0, the ci → 1. Thus, for δ small enough, we have c1 > 0. By the change of variable (z − z c )T 1/3 = W we obtain that (5.23) is just of order O(T −1/3 ). Thus we have conj (u 1 , ξ1 ; u 2 , ξ2 ) = O(e−cT , T −1/3 ) K T √ √ * 1 + q 13 κ2 (z−z c )3 T + κq1 (u 2 −u 1 )(z−z c )2 T 2/3 −(ξ1 +ξ2 ) 1+√qq (z−z c )T 1/3 −T 1/3 dz √ e . (5.24) + 2π i 0δ q
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The extension of the path 0δ to a path going from eiπ/3 ∞ to e−iπ/3 ∞ accounts into an 1/3 error O(e−cT ) only. We do the change of variable Z = κ2 T 1/3 (z − z c ) and we define 1/3 √ 2/3 q κ2 κ q . (5.25) κv = √ , κh = 2 1+ q κ1 Then, conj (u 1 , ξ1 ; u 2 , ξ2 ) = κv−1 −1 lim K T →∞ T 2π i
*
1
γ∞
dZ e 3 Z
3 +(u
2 −u 1 )Z
2 κ −1 −(ξ +ξ )Z κ −1 1 2 v h
, (5.26)
where γ∞ is any path going from eiπ/3 ∞ to e−iπ/3 ∞. The proof ends by using the Airy function representation (A.5). Proposition 16. (Bound for the diffusion term of the kernel) Let n i , ti , and xi be defined as in Proposition 15. Then, for u 2 − u 1 > 0 fixed and for any ξ1 , ξ2 ∈ R, the bound √ x1 −x2 + + q + ((n 1 ,t1 ),(n 2 ,t2 )) + 1/3 (x1 , x2 )T q n 1 −n 2 q −(t1 −t2 )/2 + ≤ const e−|ξ1 −ξ2 | +φ √ 1+ q (5.27) holds for T large enough and const independent of T . Proof of Proposition 16. We start with (5.6). The difference now is that the contribution coming from large |ξ1 − ξ2 | can be of the same order as the one from g0 (z). We consider as path 0 = {ρeiφ , φ ∈ [−π, π )}. The difference is that now we choose ρ as follows. For an ε with 0 < ε 1 and √ z c = 1/(1 + q), ⎧ (ξ2 −ξ1 )T −1/3 1/3 ⎪ ⎨z c + (u 2 −u 1 )κ1 , if |ξ2 − ξ1 | ≤ εT , ρ = z c + (u −uε )κ , (5.28) if ξ2 − ξ1 ≥ εT 1/3 , 2 1 1 ⎪ ⎩ ε if ξ2 − ξ1 ≤ −εT 1/3 . z c − (u 2 −u 1 )κ1 , By (5.10) and (5.11), 0 is a steep descent path for g0 (z) plus the term proportional to ξ1 − ξ2 in g1 (z). So, integrating on 0δ = {z = ρeiφ , φ ∈ (−δ, δ)} instead of 0 we do 2/3 only an error of order O(e−cT ) times the value at φ = 0, for some c > 0. Thus LHS of (5.27) = e T
2/3 (g (ρ)−g (z ))+T 1/3 (g (ρ)−g (z )) 0 0 c 1 1 c
× O(e
−cT 2/3
T 1/3 )+ 2π i
* 0δ
dz T 2/3 (g0 (z)−g0 (ρ))+T 1/3 (g1 (z)−g1 (ρ)) e . z (5.29)
On 0δ , the ξi -dependent term in Re(g1 (z) − g1 (ρ)) is equal to zero. With the same procedure as in Proposition 15 one shows that the integral is bounded by a constant, uniformly in T . It remains to estimate the first factor in (5.29). With our choice (5.28), we need just series expansions of g0 and g1 around ρ. Namely, by (5.8) T 2/3 (g0 (ρ) − g0 (z c )) = (u 2 − u 1 )κ1 (ρ − z c )2 T 2/3 (1 + O(ρ − z c )), √ T 1/3 (g1 (ρ) − g1 (z c )) = (ξ1 − ξ2 )(1 + q)(ρ − z c )T 1/3 (1 + O(ρ − z c )) +O((ρ − z c )2 )T 1/3 .
(5.30)
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First consider the case |ξ2 − ξ1 | ≤ εT 1/3 . We replace ρ given in (5.28) into (5.30) and get that the sum of the two contributions in (5.30) is written √ q(ξ2 − ξ1 )2 1 + O(ε) + O(T −1/3 ) . − (5.31) (u 2 − u 1 )κ1 O(ε) comes from O(ρ − z c ), while the O(T −1/3 ) from O((ρ − z c )2 ). Then, by taking ε small enough and T large enough, we get (5.31) ≤ −|ξ2 − ξ1 | + const.
(5.32)
In the case, ξ2 −ξ1 > εT 1/3 , we also replace the appropriate ρ given in (5.28) into (5.30). We explicitly use the bound εT 1/3 < ξ2 − ξ1 to bound O((ρ − z c )2 ) ≤ (ξ2 − ξ1 )T −1/3 ε. Then, we obtain the following bound for the sum of the two contributions in (5.30), √ q (1 + O(ε)) ≤ −|ξ2 − ξ1 |, (5.33) |ξ2 − ξ1 |εT 1/3 O(T −1/3 ) − (u 2 − u 1 )κ1 by taking a fixed ε small enough and then T large enough. Finally, for ξ2 − ξ1 < εT 1/3 , the same result holds in a similar way. Proposition 17. (Bound for the main term of the kernel) Let n i , ti , and xi be defined as in Proposition 15. Let L > 0 fixed. Then, for given u 1 , u 2 and ξ1 , ξ2 ≥ −L, the bound + conj + +K (u 1 , ξ1 ; u 2 , ξ2 )+ ≤ const e−(ξ1 +ξ2 ) (5.34) T holds for T large enough and const independent of T . Proof of Proposition 17. For ξ1 , ξ2 ∈ [−L , L] it is the content of Proposition 15. Thus we consider ξ1 , ξ2 ∈ [−L , ∞)2 \ [−L , L]2 . Define ξ˜i = (ξi + 2L)T −2/3 > 0. Then we consider a slight modification of (5.15), namely 1/3 conj (u 1 , ξ1 ; u 2 , ξ2 ) = −T K T 2π i
˜
0
dz
e T f0 (z)+T
2/3
e T f˜0 (z c )+T
f 1 (z)+T 1/3 f˜2 (z)+ f 3 (z) 2/3
(5.35)
f 1 (z c )+T 1/3 f˜2 (z c )
with f 1 (z) and f 3 (z) as in (5.16)–(5.17), f˜2 (z) as f 2 (z) in (5.17) but with ξ1 and ξ2 replaced by −2L, and finally f˜0 (z) is set to be equal to f 0 (z) in (5.16) plus the term − ξ˜1 ln((1 + (1 − q)z)/(−qz)) − ξ˜2 ln(1 + z).
(5.36)
We also chose 0 = {−ρeiφ , φ ∈ [−π, π )}. In the proof of Proposition 15 we already proved that 0 is a steep descent path for f 0 for the values ρ ∈ (0, z c ]. Also, since ξ˜i > 0, Re((5.36)) is also decreasing while |φ| is increasing. The precise choice of ρ is −z c − ((ξ˜1 + ξ˜2 )/κ2 )1/2 , if |ξ˜1 + ξ˜2 | ≤ ε, (5.37) ρ= if |ξ˜1 + ξ˜2 | ≥ ε, −z c − (ε/κ2 )1/2 , for some small ε > 0 which can be chosen later. Let us define Q(ρ) = e
Re T ( f˜0 (−ρ)− f˜0 (z c ))+T 2/3 ( f 1 (−ρ)− f 1 (z c ))+T 1/3 ( f˜2 (−ρ)− f˜2 (z c ))
.
(5.38)
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Then, since 0 is a steep descent path for f˜0 , (5.35) = Q(ρ)O(e
−cT
−T 1/3 ) + Q(ρ) 2π i
* 0δ
˜
dz
e T f0 (z)+T
f 1 (z)+T 1/3 f˜2 (z)+ f 3 (z)
2/3
e T f˜0 (ρ)+T
2/3
f 1 (ρ)+T 1/3 f˜2 (ρ)
, (5.39)
where 0δ = {−ρeiφ , φ ∈ (−δ, δ)}, for a small δ > 0. The expansion around φ = 0 leads to (5.40) Re( f˜0 (−ρeiφ ) − f˜0 (−ρ)) = −γ1 φ 2 (1 + O(φ)) with γ1 =
√ √ √ √ q))ρ q(1 − q)(ρq + (1 − ρ)(2 + q)) 2(1 − ρ)2 (1 − ρ(1 − q))2 ξ˜1 (1 − q) ξ˜2 ρ + + 2 (1 − ρ)2 (1 − ρ(1 − ρ))2 (1 − ρ(1 +
(5.41)
which is strictly positive for ρ chosen as in (5.37). Also, Re( f 1 (−ρeiφ ) − f 1 (−ρ)) = γ2 φ 2 (1+O(φ)) for some bounded γ2 (we do not write it down explicitly since the precise formula is not relevant). Therefore, the last term in (5.39) is bounded by * δ 2 −1/3 )) const Q(ρ)T 1/3 dφe−γ φ T (1+O(φ))(1+O(T (5.42) −δ
with γ = γ1 + γ2 T −1/3 . By choosing δ small enough and independent of T , and then T large enough, the error terms can be replaced by 1/2, and the integral is then bounded by the one on R. Thus 1 (5.42) ≤ const Q(ρ) . (5.43) γ T 1/3 In the worse case, when γ → 0, which happens when ρ → z c , we have γ1 T 1/3 (ξ1 + ξ2 + 4L)1/2 ≥ (2L)1/2 , which dominates γ2 for L large enough. Therefore we have shown that + conj + +K (u 1 , ξ1 ; u 2 , ξ2 )+ ≤ Q(ρ)O(1). (5.44) T It thus remains to find a bound on Q(ρ). We have, by (5.18), (
Q(ρ) = e
) √ κ1 1 3 2 2/3 (1+O (−ρ−z )) ˜ ˜ 1+√ q c 3 κ2 (−ρ−z c ) −(ξ1 +ξ2 ) q (−ρ−z c )T + q (u 2 −u 1 )(−ρ−z c ) T .
(5.45)
In the case |ξ˜1 + ξ˜2 | ≤ ε, we then obtain Q(ρ) ≤ e
√ 1 1+ q −1/2 (ξ1 +ξ2 +4L)3/2 3 − √q κ2 (1+O (ε))+(ξ1 +ξ2 +4L)O (1)
≤ conste−(ξ1 +ξ2 ) (5.46)
for L 1, ε 1. Finally, when |ξ˜1 + ξ˜2 | ≥ ε, we have
Q(ρ) ≤ e
−(ξ1 +ξ2 +4L)
√ q 1/2 1/3 −1/2 1 1+ √ +O (1) 3− q ε T κ2
by first choosing ε > 0 small and then T large enough.
≤ e−(ξ1 +ξ2 )
(5.47)
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Proof of Theorem 3. The proof of Theorem 3 is the complete analogue of Theorem 2.5 in [5]. The results in Propositions 5.1,5.3,5.4, and 5.5 in [5] are replaced by the ones in Proposition 15, 16, and 17. The strategy is to write the Fredholm series of the expression for finite T and, by using the bounds in Propositions 16 and 17, see that it is bounded by a T -independent and integrable function. Once this is proven, one can exchange the sums/integrals and the T → ∞ limit by the theorem of dominated convergence. For details, see Theorem 2.5 in [5]. 6. Proof of Theorem 5 In this section we prove Theorem 5. By Theorem 1, the right hand side of (2.16) with n i = (ti − Hi − xi )/2 can be written as Fredholm determinant of the kernel
½(X i < xi )K ((ni , ti ), X i ; (n j , t j ), X j )½(X j < x j )
(6.1)
with K given in (2.4). By the change of variable X i = −h i + Hi + xi , one obtains the Fredholm determinant of the kernel
½(hi > Hi )K ((ni , ti ), Hi + xi − hi ; (n j , t j ), H j + x j − h j )½(h j > H j ).
(6.2)
With this preparation, we now go to the proof of Theorem 5. Proof of Theorem 5. We have to analyze the kernel (2.4) with entries Hi ti + xi 2ti 2xi , ti = √ , xi = − √ + Hi − h i ni = √ − q 2 q q
(6.3)
and take the limit q → 0 with h i , Hi fixed. The scaling of xi might look different from 2xi the one in (2.17) but, as we can see below, (6.3) with the last one replaced by xi = − √ q gives the same limiting kernel. As q → 0, the kernel does not have a well defined limit and, as usual, we first have to consider a conjugate kernel. More precisely, we define K q ((x1 , t1 ), h 1 ; (x2 , t2 ), h 2 ) = K ((n 1 , t1 ), x1 ; (n 2 , t2 ), x2 )q (x1 −x2 )/2
q n 1 −n 2 . (6.4) q (t1 −t2 )/2
What we have to prove is lim det(½ − χ H K q χ H ) = det(½ − χ H K PNG χ H ).
q→0
(6.5)
First we prove the pointwise convergence and then we obtain bounds allowing us to take the limit inside the Fredholm determinant. √ Consider the term coming from (2.6). By the change of variable w = −1 + qz, we get n 1 −n 2 √ √ 1 − qz dz ( q + (1 − q)z)t1 −t2 1 (6.6) √ 2π i 0 z z x1 −x2 z( q + (1 − q)z) and, by inserting (6.3), one obtains √ √ √ 1 dz ( q + (1 − q)z)(1 − qz) (t1 −t2 )/ q 2π i 0 z z (x1 −x2 )/√q √ √ q + (1 − q)z q + (1 − q)z (H1 −H2 )/2 1 × . √ √ (1 − qz)z (1 − qz)z z h 2 −h 1
(6.7)
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Consider q ≤ q0 for some q0 < 1 fixed. Then, we can fix the path 0 independent of q, and the q → 0 limit is easily obtained. It results in dz 1 1 −1 −1 lim (6.7) = e−(t1 −t2 )(z−z ) e(x2 −x1 )(z+z ) h −h 2 1 q→0 2π i 0 z z x2 − x1 + t1 − t2 |h 1 −h 2 |/2 = I|h 1 −h 2 | 2 (x2 − x1 )2 − (t1 − t2 )2 , x2 − x1 − t1 + t2 (6.8) where we applied (A.4). It is the turn √ of the term coming from (2.5). We do the change of variable z = −w/(w + q) and then we insert (6.3). The result is (t1 +t2 )/√q h 1 √ q +w w √ (w + q)h 2 √ √ w(1 + qw) 1 + qw 0 (x1 −x2 )/√q+(H2 −H1 )/2 w × . (6.9) √ √ (w + q)(1 + qw)
1 2π i
dw √ w(1 + qw)
If x2 − x1 > t1 + t2 , then for q small enough, the result is identically equal to zero, because the pole at w = 0 vanishes. If x2 > t1 + t2 , then the result is also zero, √x1 −√ because the residues at all other poles, q, 1/ q, and √ ∞ vanish. In the other case, when |x2 − x1 | < t1 + t2 , the apparent pole at w = − q is actually not there. So, we can choose a 0 independent of q ≤ q0 for some q0 < 1. Then, we can simply take the limit q → 0 of the integrand, which leads to dw h 1 +h 2 (t1 +t2 )(w−1 −w) (x2 −x1 )(w+w−1 ) 1 lim (6.9) = w e e q→0 2π i 0 w t1 + t2 + x2 − x1 (h 1 +h 2 )/2 = Jh 1 +h 2 2 (t1 + t2 )2 − (x2 − x1 )2 , t1 + t2 − x2 + x1 (6.10) where in the last step we made the change of variable w → 1/w and applied (A.3). To have convergence of the Fredholm determinants we still need some bounds for large values of h 1 , h 2 . For q small enough, say q ∈ [0, q0 ] for some q0 < 1, we can set in (6.7) 0 = {z, |z| = e} in the case h 2 ≥ h 1 , and 0 = {z, |z| = e−1 } in the case h 2 < h 1 . Then, we get the bound |(6.7)| ≤ C1 e−|h 2 −h 1 |
(6.11)
for some finite constant C1 independent of q. Moreover, in (6.9) we can choose 0 = {z, |z| = e−2 }, which leads to the bound |(6.9)| ≤ C2 e−(h 2 +h 1 )
(6.12)
with C2 < ∞ independent of q ≤ q0 . These two bounds are enough to have convergence of the Fredholm determinants. The strategy is exactly the same as in the proof of Theorem 3.
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7. Proof of Theorem 6 We analyze the kernel (2.23) with the scalings xi = u i T 2/3 , ti = γ (0)T + γ (0)u i T 2/3 +
(7.1) γ (0) 2
u i2 T 1/3 ,
(7.2)
h i = 2ti + ξi T 1/3 .
(7.3)
(See (2.24) and (2.25).) The strategy of the proof is the same as that for Theorem 3 and hence we only give the main differences. First we consider the first term in (2.23). From (6.8) it is rewritten in the form (5.6) with g0 (z), g1 (z) replaced by g0 (z) = (u 2 − u 1 ) γ (0)(z − 1/z − 2 ln z) + (z + 1/z) , g1 (z) =
γ (0) 2
(u 22 − u 21 )(z − 1/z − 2 ln z) − (ξ2 − ξ1 ) ln z.
(7.4) (7.5)
The critical point of g0 (z) is now z c = 1. The series expansions around z = z c are g0 (z) = g0 (z c ) + (u 2 − u 1 )(z − z c )2 + O((z − z c )3 ),
(7.6)
g1 (z) = g1 (z c ) − (ξ2 − ξ1 )(z − z c ) + O((z − z c ) ).
(7.7)
2
The steep descent path can be taken to be 0 = {eiφ , φ ∈ [−π, π )}. Then the same arguments as in the proof of Theorem 3 give the first term in (2.7). Next we consider the second term in (2.23). From (6.10) it is rewritten in the form (5.15) with f 0 (z), f 1 (z), f 2 (z) replaced by f 0 (w) = 2γ (0)(1/w − w + 2 ln w),
(7.8)
f 1 (w) = γ (0)(u 1 + u 2 )(1/w − w + 2 ln w) + (u 2 − u 1 )(w + 1/w), γ (0) 2 f 2 (w) = (u 1 + u 22 )(1/w − w + 2 ln w) + (ξ1 + ξ2 ) ln w. 2
(7.9) (7.10)
Their series expansions around z c are 2γ (0) (w − 1)3 + O((w − 1)4 ), 3 f 1 (w) = f 1 (z c ) + (u 2 − u 1 )(w − 1)2 + O((w − 1)3 ),
(7.12)
f 2 (w) = f 2 (z c ) + (ξ1 + ξ2 )(w − 1) + O((w − 1) ).
(7.13)
f 0 (w) = −
2
(7.11)
The steepest descent path is taken to be 0 = {ρeiφ , φ ∈ [−π, π )} with 0 < ρ < 1. Using these one obtains the second term in (2.7). The bounds for the diffusion terms and the main term of the kernel are also proved in the same way as those of Propositions 16 and 17.
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A. Some Integral Representations In this appendix we list some integral representations of the Bessel functions and the modified Bessel functions (we use the conventions of [1]). For n ∈ Z, Jn (2t) = I|n| (2t) = 1 2π i
0
1 2π i
1 2π i
dz et (z−z z zn
0
0
−1 )
dz et (z+z z zn
−1
dz eb(z−z ) ea(z+z z zn
−1 )
−1 )
,
(A.1)
,
(A.2)
=
b+a b−a
n/2
Jn 2 b2 − a 2 ,
−1 −1 1 dz eb(z−z ) ea(z+z ) a + b |n|/2 2 − b2 , 2 = I a |n| 2π i 0 z zn a−b * −1 3 2 dvev /3+av +bv = Ai(a 2 − b) exp(2a 3 /3 − ab), 2π i γ∞
(A.3) (A.4) (A.5)
where γ∞ is any path from eiπ/3 ∞ to e−iπ/3 ∞. Acknowledgements. A. Borodin was partially supported by the NSF grants DMS-0402047 and DMS-0707163. P.L. Ferrari is grateful to H. Spohn for useful discussions. The work of T. Sasamoto is supported by the Grantin-Aid for Young Scientists (B), the Ministry of Education, Culture, Sports, Science and Technology, Japan.
References 1. Abramowitz, M., Stegun, I.A.: Pocketbook of mathematical functions. Thun-Frankfurt am Main: Verlag Harri Deutsch, 1984 2. Baik, J., Rains, E.M.: Symmetrized random permutations. In: Random Matrix Models and Their Applications, Vol. 40, Cambridge: Cambridge University Press, 2001, pp. 1–19 3. Barabási, A.L., Stanley, H.E.: Fractal concepts in surface growth. Cambridge: Cambridge University Press, 1995 4. Borodin, A., Ferrari, P.L.: Large time asymptotics of growth models on space-like paths I: PushASEP. http://arxiv.org/abs/0707.2813, 2007 5. Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process. Int. Math. Res. Papers 2007, rpm002 (2007) 6. Borodin, A., Ferrari, P.L., Prähofer, M., Sasamoto, T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129, 1055–1080 (2007) 7. Borodin, A., Ferrari, P.L., Sasamoto, T.: Transition between Airy1 and Airy2 processes and TASEP fluctuations. Commun. Pure. Appl. Math. doi:10.1002/cpa.20234, 2007 8. Borodin, A., Olshanski, G.: Stochastic dynamics related to Plancherel measure. In: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics V. Kaimanovich, A. Lodkin, eds., Amer Math. Soc. Transl., Series 2, vol. 217, Providence, RI: Amer. Math. Soc., 2006, pp. 9–22 9. Ferrari, P.L.: Java animation of the PNG dynamics, http://www.wias-berlin.de/people/ferrari/homepage/ animations/RSKFinal.html, 2007 10. Ferrari, P.L.: Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Commun. Math. Phys. 252, 77–109 (2004) 11. Ferrari, P.L., Prähofer, M.: One-dimensional stochastic growth and Gaussian ensembles of random matrices. Markov Processes Relat. Fields 12, 203–234 (2006) 12. Gates, D.J., Westcott, M.: Stationary states of crystal growth in three dimensions. J. Stat. Phys. 88, 999–1012 (1995) 13. Imamura, T., Sasamoto, T.: Fluctuations of the one-dimensional polynuclear growth model with external sources. Nucl. Phys. B 699, 487–502 (2004)
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14. Imamura, T., Sasamoto, T.: Polynuclear growth model with external source and random matrix model with deterministic source. Phys. Rev. E 71, 041606 (2005) 15. Jockush, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem. http://arxiv.org/ abs/math/9801068, 1998 16. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123, 225–280 (2002) 17. Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003) 18. Johansson, K.: The arctic circle boundary and the Airy process. Ann. Probab. 33, 1–30 (2005) 19. Kardar, K., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) 20. Povolotsky, A.M., Priezzhev, V.B.: Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. (2006), P07002 21. Prähofer, M., Spohn, H.: Statistical self-similarity of one-dimensional growth processes. Physica A 279, 342–352 (2000) 22. Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1 + 1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (2000) 23. Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002) 24. Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005) 25. Sasamoto, T.: Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques. J. Stat. Mech. (2007), P07007 26. Sasamoto, T., Imamura, T.: Fluctuations of a one-dimensional polynuclear growth model in a half space. J. Stat. Phys. 115, 749–803 (2004) 27. Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996) Communicated by H. Spohn
Commun. Math. Phys. 283, 451–477 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0504-7
Communications in
Mathematical Physics
Random Wavelet Series Based on a Tree-Indexed Markov Chain Arnaud Durand Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail:
[email protected] Received: 14 August 2007 / Accepted: 17 October 2007 Published online: 9 May 2008 – © Springer-Verlag 2008
Abstract: We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Hölder exponent form a set with large intersection. 1. Introduction Wavelets emerged in the 1980s as a powerful tool for signal processing, see [20,21,28]. Since then, they have found many applications in this field, such as estimation, detection, classification, compression, filtering and synthesis, see e.g. [5,8,12,19,29,36]. In these papers, the coefficients are implicitly assumed to be independent of one another and the exposed methods are based on scalar transformations on each wavelet coefficient of the considered signal. Nonetheless, it was observed that the wavelet coefficients of many real-world signals exhibit some correlations. In particular, the large wavelet coefficients tend to propagate across scales at the same locations, see [32,33]. In image processing, this phenomenon is related to the fact that the contours of a picture generate discontinuities. Therefore, methods that exploit dependencies between wavelet coefficients should yield better results in the applications. In order to develop such methods, M. Crouse, R. Nowak and R. Baraniuk [10] introduced a simple probabilistic model allowing to capture correlations between wavelet coefficients : the Hidden Markov Tree (HMT) model, which we now briefly recall. Let us consider a Markov chain X indexed by the binary tree with state space {0, 1}. Basically, the Markov property enjoyed by X means that if the state X u of a vertex u is x ∈ {0, 1}, then the states of the two sons of u are chosen independently according to the Present address: California Institute of Technology, Applied and Computational Mathematics – MC 217-50, Pasadena, CA 91125, USA. E-mail:
[email protected]
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transition probabilities from x. Conditionally on the Markov chain X , the wavelet coefficients are independent and the wavelet coefficient indexed by a given dyadic interval λ is a centered Gaussian random variable whose variance is large (resp. small) if the state of the vertex of the binary tree that corresponds to λ is 1 (resp. 0). The correlations between the wavelet coefficients are thus given by the underlying Markov chain. Furthermore, the unconditional law of each coefficient is a Gaussian mixture. This property agrees with the observation that the wavelet coefficient histogram of a real-world signal is usually more peaky at zero and heavy-tailed than the Gaussian. Moreover, let us mention that the HMT model was used in image processing, see [9,11] for instance. In this paper, we investigate the pointwise regularity properties of the sample paths of a model of random wavelet series which is closely related to the HMT model. The regularity of a function at a given point is measured by its Hölder exponent, which is defined as follows, see [25]. Definition (Hölder exponent). Let f be a function defined on R and let x ∈ R. The Hölder exponent h f (x) of f at x is the supremum of all h > 0 such that there are two reals c > 0 and δ > 0 and a polynomial P enjoying ∀x ∈ [x − δ, x + δ]
| f (x ) − P(x − x)| ≤ c |x − x|h .
Since we are interested in local properties, it is more convenient to work with wavelets on the torus T = R/Z, so that the random wavelet series that we study throughout the paper is in fact a random process on T. This process is defined in Sect. 2 and is denoted by R. Let φ be the canonical surjection from R to T. Note that R ◦ φ is a one-periodic random function defined on R. The Hölder exponent of R at a point x ∈ T is then defined by h R (x) = h R◦φ (x), ˙ where x˙ is a real number such that φ(x) ˙ = x. Equivalently, h R (x) is the supremum of all h > 0 such that d(x , x) ≤ δ
=⇒
|R(x ) − P(x − x)| ≤ c d(x , x)h
for all x ∈ T, some positive reals c and δ and some function P on T such that P ◦ φ coincides with a polynomial in a neighborhood of zero. Here, d denotes the quotient distance on the torus T. Our main purpose is to perform the multifractal analysis of the sample paths of the random wavelet series R. This amounts to studying the size properties of the iso-Hölder set E h = {x ∈ T | h R (x) = h} (1) for every h ∈ [0, ∞]. More precisely, in Subsect. 3.2, we give the law of the mapping d R : h → dim E h , where dim stands for Hausdorff dimension. This mapping is called the spectrum of singularities of the process R. A remarkable property, due to the correlations between wavelet coefficients, is that the spectrum of singularities of the process R is itself a random function. None of the multifractal stochastic processes studied up to now, such as the usual Lévy processes [22], the Lévy processes in multifractal time [4] or the random wavelet series with independent coefficients introduced by J.-M. Aubry and S. Jaffard [2,23], enjoys this property. The random wavelet series based on multifractal measures studied by J. Barral and S. Seuret [3] do not satisfy it either, even though their wavelet coefficients exhibit strong correlations. This remark also holds for the wavelet series based on branching processes introduced by A. Brouste [6]. In this last model, for any dyadic interval λ, the wavelet coefficient indexed by λ is either Gaussian or zero depending on whether or not
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the vertex of the binary tree that corresponds to λ belongs to a certain Galton-Watson tree. This model can be seen as a particular case of the HMT model by assuming that the underlying Markov chain cannot map to the state 1 a vertex whose father is mapped to the state 0. Let us also mention that A. Brouste, in collaboration with geophysicists, used this model to study the surface roughness of certain rocks, see [7]. For certain values of the parameters of the model, the sample paths of the random wavelet series R enjoy the remarkable property that almost every one of them displays an oscillating singularity at almost every point of the torus. We refer to Subsect. 3.3 for details. Note that this property also holds for the models of random wavelet series studied by J.-M. Aubry and S. Jaffard [2,23]. Conversely, the random wavelet series considered by J. Barral and S. Seuret [3] and those introduced by A. Brouste [6] do not exhibit any oscillating singularity. Certain random sets related to the iso-Hölder sets E h enjoy a notable geometric property which was introduced by K. Falconer [17]. To be specific, we establish that, for certain values of h, the sets h = x ∈ T h R (x) ≤ h E (2) are almost surely sets with large intersection, see Subsect. 3.4. In particular, this implies that they are locally everywhere of the same size, in the sense that the Hausdorff dimenh ∩ V does not depend on the choice of the nonempty open subset V sion of the set E h are not altered by of the torus. This also implies that the size properties of the sets E taking countable intersections. In fact, the Hausdorff dimension of the intersection of countably many sets with large intersection is equal to the infimum of their Hausdorff dimensions. This property is somewhat counterintuitive in view of the fact that the intersection of two subsets of the torus with Hausdorff dimensions s1 and s2 respectively is usually expected to be s1 + s2 − 1 (see [18, Chap. 8] for precise statements). The occurrence of sets with large intersection in the theory of Diophantine approximation and that of dynamical systems was pointed out by several authors, see [14,16,17] and the references therein. Their use in multifractal analysis of stochastic processes is more novel and was introduced by J.-M. Aubry and S. Jaffard. Indeed, they established in [2] that sets with large intersection arise in the study of the Hölder singularity sets of certain random wavelet series. As shown in [15], such sets also appear in the study of the singularity sets of Lévy processes. The rest of the paper is organized as follows. In Sect. 2 we give a precise definition of the model of random wavelet series that we study. Our main results are stated in Sect. 3 and are proven in Sects. 4 to 7. 2. Presentation of the Process In order to define the process that we study, let us introduce some notations. Throughout the paper, N (resp. N0 ) denotes the set of positive (resp. nonnegative) integers and Λ is the collection of the dyadic intervals of the torus T, that is, the sets of the form λ = φ(2− j (k + [0, 1))) with j ∈ N0 and k ∈ {0, . . . , 2 j − 1}. The integer λ = j is called the generation of λ. Furthermore, let us consider a wavelet ψ in the Schwartz class (see [30]). For any dyadic interval λ = φ(2− j (k + [0, 1))) ∈ Λ, let Ψλ denote the function on T which corresponds to the one-periodic function x → ψ(2 j (x − m) − k). m∈Z
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Then, the functions 2 λ /2 Ψλ , together with the constant function equal to one on T, form an orthonormal basis of L 2 (T), see [34]. Recall that there is a one-to-one correspondence between the set Λ of all dyadic intervals of the torus and the set U = {∅} ∪
∞
{0, 1} j .
j=1
The set U is formed of the empty word ∅ and the words u = u 1 . . . u j of finite length j ≥ 1 in the alphabet {0, 1}. The integer u = j is called the generation of u. In addition, let ∅ = 0 and U ∗ = U \ {∅}. For every word u ∈ U ∗ , the word π(u) = u 1 . . . u u −1 is called the father of u. Then, the directed graph with vertex set U and with arcs (π(u), u), for u ∈ U ∗ , is a binary tree rooted at ∅. To be specific, the bijection from U to Λ is ⎛ ⎞ u u → λu = φ ⎝ u j 2− j + [0, 2− u )⎠ . j=1
Thus, for every λ ∈ Λ, there is a unique vertex u λ ∈ U such that λu λ = λ. In the following, X = (X u )u∈U denotes a {0, 1}-valued stochastic process indexed by the binary tree U. The σ -field Gu = σ (X v , v ∈ U \ (uU ∗ )) can be considered as the past before u, because the set uU ∗ is composed of all the descendants of the vertex u in the binary tree U, that is, words of the form uv with v ∈ U ∗ . Conversely, the future after u begins with its two sons u0 and u1. For any integer j ≥ 0, let ν0, j and ν1, j denote two probability measures on {0, 1}2 . From now on, we assume that the process X is a Markov chain with transition probabilities given by the measures ν0, j and ν1, j . This means that the following Markov condition holds: (M) For any vertex u ∈ U and any subset A of {0, 1}2 , P((X u0 , X u1 ) ∈ A | Gu ) = ν X u , u (A). Equivalently, the conditional distribution of the vector (X u0 , X u1 ) conditionally on the past Gu is the probability measure ν X u , u . Thus, the Markov condition satisfied by X is conceptually similar to that enjoyed by inhomogeneous discrete time Markov chains. The random wavelet series R that we study is then defined by 2−h u 1{X u =1} + 2−h u 1{X u =0} Ψλu R= u∈U
=
2−h λ 1{X u λ =1} + 2−h λ 1{X u λ =0} Ψλ ,
λ∈Λ
where 0 < h < h ≤ ∞. The mean value of every sample path of R vanishes. Moreover, for any λ ∈ Λ, the wavelet coefficient indexed by λ is ⎧ ⎨Cλ = 2−h λ if X u λ = 1 (3) ⎩C = 2−h λ if X = 0. λ uλ
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The coefficient Cλ should be considered as large in the first case and small in the second one (it even vanishes if h = ∞ and λ ≥ 1). The model that we study is thus defined in the same way as the HMT model, except that, conditionally on the underlying Markov chain, the wavelet coefficients are deterministic instead of Gaussian. Note that the Markov condition (M) implies that the probability measures ν1, j affect the propagation of the large wavelet coefficients of R across scales, while the probability measures ν0, j govern their appearance. In what follows, the influence of each of these two phenomena is reflected by the values of the parameters γ j = 2 ν1, j ({(1, 1)}) + ν1, j ({(1, 0), (0, 1)})
(4)
η j = 1 − ν0, j ({(0, 0)}),
(5)
and respectively. Indeed, for any integer j ≥ 0, γ j is the expected number of sons mapped to the state 1 of a vertex with generation j that is mapped to the state 1 and η j is the probability that a vertex with generation j that is mapped to the state 0 has at least one son mapped to the state 1. 3. Statement of the Results 3.1. Preliminary remarks. As shown by (3), for every dyadic interval λ ∈ Λ, the modulus of the wavelet coefficient Cλ of R is at most 2−h λ . A standard result of [34] then implies that R belongs to the Hölder space of uniform regularity C h (T). It follows that the Hölder exponent h R (x) defined in Sect. 1 is at least h for every point x ∈ T. As shown by Theorems 1 and 2 below, the Hölder exponent of R is actually greater than h at most locations. The Hölder exponent of R at any point of the torus is highly related to the size of the wavelet coefficients which are indexed by the dyadic intervals located around x. More precisely, it can be computed using the following proposition, which is a straightforward consequence of [26, Prop. 1.3]. Proposition 1. Let h ∈ (0, ∞) and x ∈ T. (a) If h R (x) > h, then ∃κ > 0 ∀λ ∈ Λ
|Cλ | ≤ κ (2− λ + d(x, xλ ))h .
(6)
(b) If (6) holds, then h R (x) ≥ h. By virtue of (3), for each dyadic interval λ ∈ Λ, the modulus of the wavelet coefficient Cλ of R is at least 2−h λ . According to Proposition 1a, the Hölder exponent of R is less than h everywhere. It follows that ∀x ∈ T
h ≤ h R (x) ≤ h.
(7)
As a consequence, for every sample path of the random wavelet series R and every h ∈ [0, h) ∪ (h, ∞], the iso-Hölder set E h defined by (1) is empty. Conversely, for each h ∈ [h, h], the set E h need not be empty and its size properties are described by Theorems 1 and 2 below.
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Before stating these theorems, we need to introduce some further notations related to the parameters γ j defined by (4) and governing the propagation of the large wavelet coefficients of the process R. To be specific, let j = inf{ j0 ≥ 0 | ∀ j ≥ j0 γ j > 0} and ⎧ ⎨θ = lim inf j→∞ ⎩θ = −∞
log γ j +...+log γ j j log 2
if j < ∞ if j = ∞.
Note that θ is at most one. Moreover, γ j being the expected number of sons mapped to the state 1 of a vertex with generation j that is mapped to the state 1, the parameter θ expresses the trend with which large wavelet coefficients propagate across scales. For any integer j ≥ 0, we also need to consider the number ςj = 2
∞ ν1,n ({1, 1}) n= j
γn
n
= j
,
γ
which naturally occurs in the study of a specific random fractal set related to the process R, see Lemma 6 below. Throughout the rest of the paper, we suppose that θ is less than one. This assumption implies that the large wavelet coefficients of R cannot propagate too much across scales. It is not very restrictive, in view of the fact that the decomposition of a typical real-world signal in a wavelet basis has very few large coefficients.
3.2. Law of the spectrum of singularities. Theorems 1 and 2 below give the law of the spectrum of singularities of the sample paths of the random wavelet series R. Recall that it is the mapping d R : h → dim E h , where E h is defined by (1) and dim stands for Hausdorff dimension. With a view to recalling the definition of Hausdorff dimension, let us first define the notion of Hausdorff measure on the torus. To this end, let D denote the set of all nondecreasing functions g defined on a neighborhood of zero and enjoying lim0+ g = g(0) = 0. Any function in D is called a gauge function. For every g ∈ D, the Hausdorff g-measure of a subset F of T is defined by Hg (F) = lim ↑ Hεg (F) with Hεg (F) = ε↓0
inf
F⊆ p U p |U p |<ε
∞
g(|U p |).
p=1
The infimum is taken over all sequences (U p ) p≥1 of sets with F ⊆ p U p and |U p | < ε for all p ≥ 1, where | · | denotes diameter (in the sense of the quotient distance on the torus). Note that Hg is a Borel measure on T, see [35]. The Hausdorff dimension of a nonempty set F ⊆ T is then defined by s
s
dim F = sup{s ∈ (0, 1) | HId (F) = ∞} = inf{s ∈ (0, 1) | HId (F) = 0},
Random Wavelet Series Based on a Tree-Indexed Markov Chain
457
where Ids denotes the function r → r s and with the convention that sup ∅ = 0 and inf ∅ = 1, see [18]. In addition, we agree with the usual convention that the empty set ∅ has Hausdorff dimension −∞. The pointwise regularity properties of the sample paths of the process R depend significantly on whether or not the series j 2 j η j converges, where η j is given by (5). If it converges, then the probabilities with which large wavelet coefficients can appear at a given scale are quite low, so that the sample paths of R are regular in their irregularity, as shown by the following result. In its statement, the function Φ j is the generating function of the cardinality of the set S j = {u ∈ U | u = j and X u = 1} .
(8)
This means that Φ j (z) = E[z #S j ] for any complex number z and any integer j ≥ 0. The functions Φ j may be calculated recursively in terms of the transition probabilities of the Markov chain X . Indeed, the Markov condition (M) implies that for any integer j ≥ 0, E[z #S j+1 | F j ] =
{0,1}2
z x0 +x1 ν1, j (dx)
#S j {0,1}2
z x0 +x1 ν0, j (dx)
2 j −#S j
,
where F j is the σ -field generated by the variables X u , for u ∈ U with u ≤ j. Theorem 1. Let us suppose that j 2 j η j < ∞ and θ < 1. (a) If θ < 0, then with probability one, h R (x) = h for all x ∈ T. Therefore, with probability one, for all h ∈ [0, ∞], if h = h d R (h) = 1 d R (h) = −∞ otherwise. (b) If θ ≥ 0, then with probability one, for all h ∈ [0, ∞], ⎧ d R (h) ∈ {−∞, θ } if h = h ⎪ ⎪ ⎨ d R (h) = 1 if h = h ⎪ ⎪ ⎩ otherwise. d R (h) = −∞ Moreover, if there is an integer j∗ ≥ 0 such that ν1, j ({(0, 0)}) = 0 for any j ≥ j∗ , then P(d R (h) = −∞) = Φ j∗ (0) ·
∞
j
(1 − η j )2 .
j= j∗
If not, P(d R (h) = −∞) is positive and it is equal to one if and only if j Φ j (0) = 1 γ = 0 or ς j = ∞ or lim inf j→∞ ∀ j ≥ j η j = 0. = j Remark 1. As θ < 1, Theorem 1 implies that the iso-Hölder set E h has full Lebesgue measure in the torus with probability one. Thus, the Hölder exponent of almost every sample path of the process R is h almost everywhere.
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Remark 2. The process R enables to provide a partial answer to a question raised by S. Jaffard in [24]. For many examples of random processes F, such as the usual Lévy processes [22], the Lévy processes in multifractal time [4] or several models of random wavelet series [2,23], even though the function x → h F (x) is random, the spectrum of singularities of F is a deterministic function. Of course, this property does not hold in general: consider for instance a fractional Brownian motion whose Hurst parameter follows a Bernoulli law. The random wavelet series R supplies a more elaborate example. Indeed, Theorem 1 shows that its spectrum of singularities may be random. Theorem 2 below indicates that this property still holds if the series j 2 j η j diverges. If the series j 2 j η j diverges, large wavelet coefficients can appear with a relatively large probability at each scale, which makes the sample paths of R very irregular in their irregularity, as shown by the following result. In its statement, h is defined by (1−h/ h) j 2 ηj = ∞ , h = inf h > 0 j
which is clearly greater than or equal to h. Theorem 2. Let us suppose that j 2 j η j = ∞ and θ < 1. (a) If h < h, then with probability one, for all h ∈ [0, ∞],
h if h < h ≤ h d R (h) = h/ h. d R (h) = −∞ if h < h or h >
(b) If h ≥ h, then with probability one, for all h ∈ [0, ∞], ⎧ ⎪ ⎨d R (h) = h/h if h < h < h if h = h d (h) = 1 ⎪ ⎩ R d R (h) = −∞ if h < h or h > h. h ≥ θ , then with probability one, (c) If h/ d R (h) = h/ h. h < θ , then with probability one, (d) If h/ d R (h) ∈ {h/ h, θ }. Moreover, P(d R (h) = θ ) is positive and it is equal to one if and only if either j j 2 η j /ς j+1 = ∞ or ν1, j ({(0, 0)}) = 0 for all j large enough. Remark 3. As θ < 1, Theorem 2 implies that the iso-Hölder set E min( h,h) has full Lebesgue measure in the torus with probability one. So, the Hölder exponent of almost every sample path of the random wavelet series R is min( h, h) almost everywhere. In addition, if h = h, then the set E h is almost surely equal to the whole torus, so that the Hölder exponent of R is almost surely h everywhere.
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Remark 4. For some values of the parameters, the spectrum of singularities of the random wavelet series R need not be concave. Therefore, this spectrum cannot be determined using the multifractal formalisms derived in Besov or oscillation spaces. We refer to [25] for details concerning these multifractal formalisms. Moreover, in general, the spectrum of singularities of R does not coincide with its large deviation spectrum (see e.g. [31] for a fuller exposition), that is, the mapping h → lim ↓ lim sup ε↓0
j→∞
1 log2 #{λ ∈ Λ | λ = j and 2−(h+ε) j ≤ |Cλ | ≤ 2−(h−ε) j }. j
Indeed, this last function clearly maps any real h ∈ / {h, h} to −∞. Remark 5. Recall owing to Theorem 1, the spectrum of singularities of R may be that, j η < ∞. Theorem 2 shows that this property still holds when random when 2 j j j spectrum of singularities of R is random if and only j 2 η j = ∞. Specifically, the j if h/h is less than θ , the sum j 2 η j /ς j+1 is finite and ν1, j ({(0, 0)}) is positive for infinitely many integers j ≥ 0. Let us give an explicit example of probability measures ν0, j and ν1, j for which all these conditions hold. Given a real a ∈ (0, 1) and an integer b ≥ 2, let p0 = 2−a and, log ( j+1) −blogb j ) for any integer j ≥ 1, let p j = 2−a(b b , where · denotes the floor function. Next, let us consider that the measures ν1, j are the products ⊗2 ν1, j = p j δ1 + (1 − p j )δ0 , where δ0 and δ1 are the point masses at zero and one, respectively. Clearly, ν1, j ({(0, 0)}) is positive for all j, the number j vanishes and θ = 1 − a < 1. For every integer n ≥ 0, let jn = bn+1 − 1. Observe that, for all n greater than some n 0 , the sum ς jn is at least 2a(1−1/b)( jn +1)−1 and there exists a real q jn −1 such that 2−( jn −1) ≤ q jn −1 ≤
1 (a(1−1/b)−1) jn 2 . jn 2
Furthermore, let q j−1 = 0 for every integer j ≥ 1 that is not of the form jn with n > n 0 . Then, let us consider that the probability measures ν0, j are given by ⊗2 . ν0, j = q j δ1 + (1 − q j )δ0 h is at least h/(a(1 − 1/b)). It is easy to check that the sum j 2 j η j diverges and that h is less than θ . Let us suppose that a < 1/(2 − 1/b). This assumption ensures that h/ Moreover, ∞ ∞ 2 jn −1 η jn −1 ≤ ς jn
n=n 0 +1
n=n 0 +1
∞
1 2a(1−1/b) jn ≤ 21−a(1−1/b) , 2 a(1−1/b)( j +1)−1 n j2 jn 2 j=1
which ensures the finiteness of the sum j 2 j η j /ς j+1 . Theorem 2 finally implies that the spectrum of singularities of the process R is random when the probability measures ν0, j and ν1, j are chosen as above.
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3.3. Oscillating singularities. Theorems 1 and 2 above give the law of the Hausdorff dimension of the iso-Hölder sets E h defined by (1). Each set E h is formed of the points at which the Hölder exponent of the process R is h. It is possible to provide a more precise description of the pointwise regularity properties of these points. Indeed, a given Hölder exponent h at a point x ∈ T can result from many possible local behaviors near x. For example, if h is not an even integer, then the cusp x → |x − x|h and the chirp x → |x − x|h sin
|x
1 − x|β
(9)
both have Hölder exponent h at x, in spite of the fact that their oscillatory behavior is completely different, see [26]. The oscillating singularity exponent was introduced in [1] in order to describe the oscillatory behavior of a function near a given point and thus to determine if a function behaves rather like a cusp or like a chirp in a neighborhood of a point. It is defined using primitives of fractional order. To be specific, for any t > 0, any locally bounded function f defined on R and any x ∈ R with h f (x) < ∞, let h tf (x) denote the Hölder exponent at x of the function (Id −∆)−t/2 (χ f ), where χ is a compactly supported smooth function which is equal to one in a neighborhood of x and (Id −∆)−t/2 is the operator that corresponds to multiplying by ξ → (1 + ξ 2 )−t/2 in the Fourier domain. Definition (Oscillating singularity exponent). Let f be a locally bounded function defined on R and let x ∈ R with h f (x) < ∞. The oscillating singularity exponent of f at x is ∂h tf (x) β f (x) = − 1 ∈ [0, ∞]. ∂t + t=0
If β f (x) > 0, then f is said to display an oscillating singularity at x. It is proven in [26] that if f is defined by (9), then h tf (x) = h + t (β + 1), so that β f (x) = β. As required, the oscillating singularity exponent enables to recover the parameter β which governs the oscillatory behavior of a chirp. Note that this exponent is not defined for points at which the Hölder exponent is infinite. The oscillating singularity exponent β R (x) of the random wavelet series R at any point x ∈ T such that h R (x) < ∞ is then defined in the natural way, that is, β R (x) is set to be equal to β R◦φ (x) ˙ for any real number x˙ enjoying φ(x) ˙ = x, where φ is the canonical surjection from R to T. The following result, which is proven in Sect. 7, gives the value of the oscillating singularity exponent of R at every point of the iso-Hölder set Eh . Proposition 2. For every h ∈ [h, h] and every x ∈ E h , h/h − 1 if h < h β R (x) = 0 if h = h < ∞. Remark 6. Proposition 2 ensures that the random wavelet series R displays an oscillating singularity at every point of the set E h , for any h ∈ (h, h). Moreover, it is necessary to assume the finiteness of h for h = h in the statement of Proposition 2 because the oscillating singularity exponent is not defined for points at which the Hölder exponent is infinite.
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Remark 7. In the case where h < h, Theorem 2 and Proposition 2 ensure that almost every sample path of the random wavelet series R displays an oscillating singularity at almost every point of the torus. Note that this remarkable property is also verified by the models of random wavelet series with independent coefficients which were studied in [2,23]. 3.4. Large intersection properties of the singularity sets. For certain values of h, the h defined by (2) are sets with large intersection, in the sense that they belong to sets E specific classes G g (T) of subsets of the torus. These classes are the transposition into the toric setting of the classes G g (R) of subsets of R which were introduced in [14] in order to generalize the original classes of sets with large intersection of K. Falconer [17]. Let us first recall the definition and the basic properties of the classes G g (R). To begin with, they are defined for functions g in a set denoted by D1 . This is the set of all gauge functions g ∈ D such that r → g(r )/r is positive and nonincreasing on a neighborhood of zero. For any g ∈ D1 , let εg denote the supremum of all ε ∈ (0, 1] such that g is nondecreasing on [0, ε] and r → g(r )/r is nonincreasing on (0, ε] and let Λg denote the set of all dyadic intervals of diameter less than εg , that is, sets of the form λ = 2− j (k + [0, 1)) with j ∈ N0 , k ∈ Z and |λ| < εg . The outer net measure associated with g is defined by ∀F ⊆ R
g
M∞ (F) = inf
(λ p ) p≥1
∞
g(|λ p |),
p=1
where the infimum is taken over all sequences (λ p ) p≥1 in Λg ∪{∅} such that F ⊆ p λ p . In addition, for g, g ∈ D1 , let us write g ≺ g if g/g monotonically tends to infinity at zero. We can now give the definition of the classes G g (R). Recall that a G δ -set is one that may be expressed as a countable intersection of open sets. Definition (Sets with large intersection in R). For any gauge function g ∈ D1 , the class G g (R) of sets with large intersection in R with respect to g is the collection of all g g G δ -subsets F of R such that M∞ (F ∩ U ) = M∞ (U ) for every g ∈ D1 enjoying g ≺ g and every open set U . The classes G g (T) are then defined in the natural way using the classes G g (R) and the canonical surjection φ from R to T. Definition (Sets with large intersection in T). For any gauge function g ∈ D1 , the class G g (T) of sets with large intersection in T with respect to g is the collection of all subsets F of T such that φ −1 (F) ∈ G g (R). The results of [14] show that the classes G g (T) of sets with large intersection in the torus enjoy the following remarkable properties. Theorem 3. For any gauge function g ∈ D1 , (a) the class G g (T) is closed under countable intersections; (b) every set F ∈ G g (T) enjoys Hg (F ∩ V ) = ∞ for every g ∈ D1 with g ≺ g and every nonempty open set V , and in particular dim F ≥ sg = sup{s ∈ (0, 1) | Ids ≺ g}; (c) every G δ -subset of T with full Lebesgue measure belongs to G g (T).
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h defined by (2) belong to certain classes G g (T) As previously announced, the sets E of sets with large intersection. More precisely, Proposition 9 in Sect. 6 yields the following result. Proposition 3. Let us assume that h is finite. Then, with probability one, for all h belongs to the class GIdh/h (T). h, h)), the set E h ∈ [h, min( Together with Theorem 3, this result implies that with probability one, for every h has infinite Hausdorff measure for every gauge function h, h)), the set E h ∈ [h, min( h/ h g ∈ D1 with g ≺ Id . This property comes into play in the proof of Theorem 2, because it enables to obtain a sharp lower bound on the Hausdorff dimension of the corresponding iso-Hölder set E h , see Sect. 6. 4. Preparatory Lemmas In this section, we establish several results that are called upon at various points of the proofs of Theorems 1 and 2. The Hölder exponent of the random wavelet series R at a given point x of the torus depends on the way large wavelet coefficients are located around x. To be specific, let S = {u ∈ U | X u = 1} =
∞
Sj,
j=0
where the sets S j are defined by (8). As shown by (3), the vertices in S correspond to the dyadic intervals indexing the large coefficients of the wavelet series R. In addition, for every u ∈ U, let u xu = φ(x˙u ) with x˙u = u j 2− j (10) j=1
and, for every real α > h, let L α = {x ∈ T | d(x, xu ) < 2−h u /α for infinitely many u ∈ S},
(11)
where d is the quotient distance on the torus. It is straightforward to check that α → L α is nondecreasing. The following lemma establishes a connection between the sets L α h defined by (1) and (2) respectively. and the sets E h and E Lemma 4. (a) For every h ∈ [0, h) ∪ (h, ∞], the set E h is empty. (b) For every h ∈ [h, h], h \ h = L α and E h = E Lα. E h<α≤h
h<α
Proof. Assertion a is due to the fact that the Hölder exponent of the process R is everywhere between h and h, as shown by (7). Assertion b follows from the observation that for any α ∈ (h, h] and any x ∈ T, x ∈ L α =⇒ h R (x) ≤ α x ∈ L α =⇒ h R (x) ≥ α.
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Indeed, let us assume that x ∈ L α . Then, there are infinitely many dyadic intervals λ = φ(2− j (k + [0, 1))) ∈ Λ, with j ∈ N0 and k ∈ {0, . . . , 2 j − 1}, such that Cλ = 2−h λ and d(x, xλ ) < 2−h λ /α , where xλ = φ(k2− j ). Proposition 1 ensures that h R (x) ≤ α. Conversely, let us assume that x ∈ L α . Then, for every dyadic interval λ ∈ Λ such that λ is large enough, if Cλ = 2−h λ , then d(x, xλ ) ≥ 2−h λ /α . Thus, whether Cλ = 2−h λ or Cλ = 2−h λ , we have |Cλ | ≤ (2− λ + d(x, xλ ))α , so that h R (x) ≥ α thanks to Proposition 1.
The proof of the following lemma is modeled on that of Proposition 1 in [27, Chap. 11]. Lemma 5. With probability one, for every α ∈ ( h, ∞), we have L α = T. Proof. Let us consider a real number α > h. Moreover, for every u ∈ U, let Bu be the open ball of T with center xu and radius 2−h u /α . It is straightforward to establish the following inclusion of events: ⎧ ⎫ ∞ ∞ ⎨ ⎬ {T = L α } ⊆ Bu . T = (12) ⎩ ⎭ j0 =0 j= j0
u∈S j−1 ∪S j
Then, let j0 and j be two integers such that j − 1 ≥ j0 ≥
1 − log2 (2h/α − 1) 1 − h/α
(13)
and let us assume that T cannot be written as the union over u ∈ S j−1 ∪ S j of the balls Bu . So, there is an integer k ∈ {0, . . . , 2 j − 1} such that the closed ball with center φ(k2− j ) and radius 2− j−1 is not included in this last union of balls. As a result, the point φ(k2− j ) cannot belong to the union over u ∈ S j−1 ∪ S j of the open balls with center xu and radius 2−h u /α − 2− j−1 . Therefore, ⎫ ⎧ j −1 ⎬ 2 ⎨ j,k j,k A j−1 ∩ A j , (14) Bu ⊆ T = ⎭ ⎩ u∈S j−1 ∪S j
k=0
where A j denotes, for each j ∈ { j − 1, j}, the event corresponding to the fact that the Markov chain X maps to 0 all the vertices of the set j,k
A j = {u ∈ {0, 1} j | d(xu , φ(k2− j )) ≤ 2−1−h j /α }. j,k
Owing to (13), there is a set A ⊆ A j−1 such that # A = (2(1−h/α) j − 3)/2 and j,k
such that the two sons of each vertex in A belong to A j . The Markov condition (M) then yields j,k
P(A j−1 ∩ A j ) ≤ P(∀u ∈ A X u = X u0 = X u1 = 0) ≤ ν0, j−1 ({(0, 0)})# A . j,k
j,k
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It follows that the left-hand side of (14) is included in an event of probability at most v j = 2 j exp(−(2(1−h/α) j − 3)/2η j−1 ). Thus, the event ⎧ ⎫ ∞ ⎨ ⎬ Bu T = ⎩ ⎭ u∈S j−1 ∪S j
j= j0
has probability at most lim inf j v j , which vanishes because there are infinitely many integers j ≥ j0 + 1 such that η j−1 ≥ 2(−1+2h/(α+h)) j , owing to the fact that α > h. Then, (12) implies that L α is almost surely equal to the whole torus and the result follows from the fact that α → L α is nondecreasing. Let α ∈ (h, ∞). Lemma 7 below gives a useful decomposition of the set L α defined by (11). The first set which comes into play in this decomposition is
where
L α = {x ∈ T | d(x, xu ) < 2−h u /α for infinitely many u ∈ S},
(15)
S = {u ∈ U ∗ | X u = 1 and X π(u) = 0}.
(16)
The set S is clearly included in S, so that L α is included in L α . Essentially, a point of L α also belongs to L α if infinitely many of the large wavelet coefficients Cλ of the process R that are close to it have appeared at scale λ , i.e. are such that Cπ(λ) is a small wavelet coefficient, where π(λ) is the smallest dyadic interval of the torus that strictly contains λ. The second set coming into play in the decomposition of L α is a set denoted by Θ and defined as follows. For every vertex u of the binary tree U, let uU denote the set of all words of the form uw with w ∈ U and let τu = {v ∈ uU | ∀ j ∈ { u , . . . , v } X v1 ...v j = 1}. The set τu is empty if X u = 0. Otherwise, τu is the largest subtree of U rooted at u and formed of vertices which are mapped to 1 by the Markov chain X . The boundary of τu is the set ∂τu = {ζ = (ζ j ) j≥1 ∈ {0, 1}N | ∀ j ≥ u ζ1 . . . ζ j ∈ τu }. For every sequence ζ = (ζ j ) j≥1 in {0, 1}, let x˙ζ =
∞
ζ j 2− j .
j=1
Then, let
Θ˙ =
˙ {x˙ζ } and Θ = φ(Θ),
(17)
u∈U ζ ∈∂τu
where φ is the canonical surjection from R to T. Essentially, a point of the torus belongs to Θ if it can be obtained as the intersection of a sequence of nested dyadic intervals indexing large wavelet coefficients of the process R that propagate across scales. The following lemma provides the law of the Hausdorff dimension of Θ. We refer to Sect. 3 for the definitions of the parameters appearing in its statement.
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Lemma 6. If θ < 0, then Θ is empty with probability one. If not, then with probability one, Θ is empty or has Hausdorff dimension θ and, in addition, • if there is a j∗ ≥ 0 such that ν1, j ({(0, 0)}) = 0 for any j ≥ j∗ , then P(Θ = ∅) = Φ j∗ (0) ·
∞
j
(1 − η j )2 ;
j= j∗
• if ν1, j ({(0, 0)}) > 0 for infinitely many integers j ≥ 0 and if j 2 j η j < ∞, then P(Θ = ∅) is positive and it is equal to one if and only if j Φ j (0) = 1 ς j = ∞ or lim inf γ = 0 or j→∞ ∀ j ≥ j η j = 0; = j j • if ν1, j ({(0, 0)}) > 0 for infinitely many integers j ≥ 0, if j 2 η j = ∞ and if θ > 0, then P(Θ = ∅) is less than one and it is equal to zero if and only if j 2 η /ς = ∞. j j+1 j Proof. The lemma is a straightforward consequence of Proposition 4 in [13], which ˙ and the observation that the provides the law of the Hausdorff dimension of the set Θ, ˙ sets Θ and Θ have the same Hausdorff dimension. The following lemma supplies a precise statement of the aforementioned decomposition of the set L α in terms of the sets L α and Θ. Lemma 7. For every α ∈ (h, ∞), we have L α = L α ∪ Θ. Proof. To begin with, it is easy to check that L α ⊆ L α . Next, let us consider a point x in Θ. Then, there are a vertex u ∈ U and a sequence ζ = (ζ j ) j≥1 ∈ ∂τu such that x = φ(x˙ζ ). For every integer j ≥ u , we have ζ1 . . . ζ j ∈ S and d(x, xζ1 ...ζ j ) ≤ 2− j < 2−h j/α . The point x thus belongs to L α . It follows that Θ is included in L α . Hence, L α contains both L α and Θ. Conversely, let us consider a point x in L α which does not belong to L α . Then, there is an integer j0 ≥ 0 such that d(x, xu ) ≥ 2−h u /α for every vertex u ∈ S with generation at least j0 . We may assume that j0 ≥ (log2 (2h/α − 1))/(h/α − 1). Let S = {u ∈ S | d(x, xu ) < 2−h u /α } and observe that the set S cannot contain any vertex of S with generation at least j0 . Since x ∈ L α , there exists a sequence (v n )n≥1 in S such that v n is increasing. A standard diagonal argument leads to a sequence ζ = (ζ j ) j≥1 in {0, 1} such that for every j ≥ 1, there are infinitely many integers n ≥ 1 enjoying ζ1 . . . ζ j = v1n . . . v nj . Let u = ζ1 . . . ζ j0 and let us consider two integers j ≥ j0 and n satisfying v n > j and ζ1 . . . ζ j = v1n . . . v nj . The vertex v n belongs to S and its generation is at least j0 , so that vn ∈ S \ S. Hence, π(v n ) belongs to S. Moreover, d(x, xπ(v n ) ) ≤ d(x, xv n ) + d(xv n , xπ(v n ) ) < 2−h v
n /α
+ 2− v
n
≤ 2−h π(v
n ) /α
,
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which ensures that π(v n ) ∈ S . By repeating this procedure v n − j times, one can prove that ζ1 . . . ζ j = v1n . . . v nj ∈ S . In particular, X ζ1 ...ζ j = 1 for every integer j ≥ j0 , so that ζ ∈ ∂τu . Furthermore, for any j ≥ j0 , d(x, φ(x˙ζ )) ≤ d(x, xζ1 ...ζ j ) + d(xζ1 ...ζ j , φ(x˙ζ )) ≤ 2−h j/α +
∞
ζ j 2− j .
j = j+1
Letting j → ∞, we obtain x = xζ . The point x thus belongs to Θ. 5. Proof of Theorem 1 In order to establish Theorem 1, let us assume that integer j ≥ 1, let
j
2 j η j < ∞ and θ < 1. For any
S ∩ {0, 1} j , Sj = where S is the set defined by (16). The Markov condition (M) implies that ! " " ! E # S j F j−1 = E E[X u0 + X u1 | Gu ] F j−1 u∈{0,1} j−1 X u =0
≤ 2 j−1
{0,1}2
(x0 + x1 ) ν0, j−1 (dx) ≤ 2 j η j−1 ,
(18)
where F j−1 is the σ -field generated by the variables X u , for u ∈ U such that u ≤ j −1. It follows that the set S j is nonempty with probability at most 2 j η j−1 . As the sum j j 2 η j converges, the Borel-Cantelli lemma ensures that, with probability one, there S is are at most finitely many integers j ≥ 1 such that S j = ∅. Consequently, the set almost surely finite. So, with probability one, for every real α > h, the set L α given by (15) is empty. By virtue of Lemmas 4 and 7, it follows that with probability one, for all h ∈ [0, ∞], ⎧ Eh = Θ if h = h ⎪ ⎪ ⎨ E h = T \ Θ if h = h ⎪ ⎪ ⎩ otherwise. Eh = ∅ Theorem 1 is then a direct consequence of Lemma 6. 6. Proof of Theorem 2 In order to prove Theorem 2, let us assume that with, observe that
j
2 j η j = ∞ and θ < 1. To begin
a.s. ∀h ∈ [0, h) ∪ (min( h, h), ∞]
E h = ∅,
(19)
owing to Lemmas 4 and 5. Thus, we may now restrict our attention to the case in which h is between h and min( h, h). The next result gives an upper bound of the Hausdorff dimension of the iso-Hölder set E h for any h ∈ [h, min( h, h)).
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Proposition 8. With probability one, h, dim Θ) and ∀h ∈ (h, min( h, h)) dim E h ≤ h/ h. dim E h ≤ max(h/ Proof. Owing to (18), the expectation of # S j is at most 2 j η j−1 for any integer j ≥ 1. Markov’s inequality then implies that # S j is greater than 2 j η j−1 j 2 with probability at most 1/j 2 . By virtue of the Borel-Cantelli lemma, it follows that with probability one, # S j is bounded by 2 j η j−1 j 2 for all j large enough. Thus, a.s. ∃ κ ≥ 1 ∀j ≥ 1
# Sj ≤ κ 2 j η j−1 j 2 .
(20)
Let us assume that the event of probability one on which (20) holds occurs and let h ∈ [h, min( h, h)) and s > h/ h. For α ∈ (h, s h) and ε > 0, the set L α defined by (15) is covered by the open balls with center xu and radius 2−h u /α , for u ∈ S such that 2−h u /α ≤ ε/2. Therefore,
s HεId ( Lα) ≤
# S j · (21−h j/α )s ≤ κ
j∈N 2−h j/α ≤ε/2
2 j η j−1 j 2 (21−h j/α )s .
j∈N 2−h j/α ≤ε/2
Since α/s < h, this last series converges so that the right-hand side tends to zero as ε → 0. Hence, the Hausdorff Ids -measure of the set L α vanishes. It follows # that with probability one, for all h ∈ [h, min( h, h)), the Hausdorff dimension of h<α≤h L α is at most h/h. The result then follows from the fact that Eh = Θ ∪
h, h)) E h ⊆ L α and ∀h ∈ (h, min(
h<α≤h
owing to Lemmas 4 and 7.
Lα,
h<α≤h
Lemma 6 and Proposition 8, together with the assumption that θ is less than one, imply that with probability one, for any h ∈ [h, min( h, h)), the iso-Hölder set E h has Lebesgue measure zero. Moreover, with probability one, this set is empty for every h ∈ [0, h) ∪ (min( h, h), ∞], owing to (19). As a result, with probability one, the set E min( h,h) has full Lebesgue measure in the torus. In particular, its Hausdorff dimension is equal to one. In order to give a lower bound on the Hausdorff dimension of the iso-Hölder set E h for every real h ∈ [h, min( h, h)), we shall treat two cases separately: h < ∞ and h = ∞. Let us first consider the case in which h is finite. The lower bound then follows h is a set with large intersection, as shown by the following result. from the fact that E Recall that the classes G g (T) of sets with large intersection in the torus are defined in Subsect. 3.4. Proposition 9. Let us assume that h < ∞. Then, with probability one, for every real h ∈ [h, min( h, h)),
h ∈ GIdh/h (T) and dim E h ≥ h/ E h.
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Proof. Lemma 5 shows that, with probability one, for every real α > h, the set L α defined by (11) is equal to the whole torus. Let us assume that the corresponding event occurs and let h ∈ [h, min( h, h)). For each α ∈ (h, h], we have α h/ h > h, so that L α h/ h = T. Thus, the set |x − p − x˙ | < 2−hh u /(α h) u φ −1 (L α ) = x ∈ R h/ h for infinitely many (u, p) ∈ S × Z , where x˙u is defined by (10), has full Lebesgue measure in R. Following the terminology of [14], the family ( p + x˙u , 2−hh u /(α h) )(u, p)∈S×Z is a homogeneous ubiquitous system in R and Theorem 2 in [14] implies that the set |x − p − x˙u | < 2−h u /α −1 φ (L α ) = x ∈ R for infinitely many (u, p) ∈ S × Z h/ h
h/ h
belongs to the class GId (R), so that the set L α belongs to the class GId (T). Furtherh is equal more, owing to Lemma 4 and the fact that α → L α is nondecreasing, the set E to the intersection over the integers n > 1/(h − h) of the sets L h+1/n . Each of these sets h/ h
belongs to the class GId (T), which is closed under countable intersections thanks to Theorem 3. Hence, with probability one,
∀h ∈ [h, min( h, h))
h ∈ GIdh/h (T). E
(21)
In order to establish the remainder of the proposition, let us begin by observing that h ∈ GIdh/h (T) with probability one, by virtue of (21) and Lemma 4. Theorem 3 Eh = E h with probability one. then implies that the Hausdorff dimension of E h is at least h/ Moreover, for every vertex u ∈ U, using a standard diagonal argument, one easily checks that ζ ∈∂τu
{x˙ζ } =
∞ j= u
x˙v + [0, 2− j ] .
v∈τu v = j
This ensures that the set Θ˙ defined by (17) is a Fσ -set, i.e. a set that may be expressed ˙ is a Fσ -set as well. In as a countable union of closed sets. Therefore, the set Θ = φ(Θ) addition, this set has Lebesgue measure zero with probability one, because of Lemma 6 and the assumption that θ is less than one. So, T \ Θ is almost surely a G δ -subset of T with full Lebesgue measure. This property, (20) and (21) thus simultaneously hold with probability one. Let us assume that the corresponding event occurs and let h ∈ (h, min( h, h)). Lemmas 4 and 7 imply that h \ Θ) \ Lα. Eh = ( E h<α
h/h h and In addition, (21) and Theorem 3 show that the class GId (T) contains the sets E T \ Θ respectively. Since this class is closed under countable intersections, it contains h \ Θ. Hence, by Theorem 3 again, H−Idh/h ·log ( E h \ Θ) = ∞. Furthermore, for the set E
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α ∈ (h, h) and ε > 0 small enough, the set L α is covered by the open balls with center xu and radius 2−h u /α for u ∈ S such that 2−h u /α ≤ ε/2. Thus, owing to (20), h −Idh/h ·log j − 1 log 2. (L α ) ≤ κ 2 j η j−1 j 2 (21−h j/α )h/h Hε α j∈N 21−h j/α ≤ε
Since α h/ h < h, this last series converges so that the right-hand side tends to zero as h/ h ε → 0. It follows that H−Id ·log ( L α ) = 0. The fact that α → L α is nondecreasing implies that ⎛ ⎞ h/ h H−Id ·log ⎝ Lα⎠ = 0 h<α
because the union can actually be taken on a countable subset of (h, h). Therefore, h/ h H−Id ·log (E h ) = ∞, so that the Hausdorff dimension of E h is at least h/ h. Let us now consider the case in which h is infinite. The following result provides a lower bound on the dimension of the iso-Hölder set E h , for every h ∈ [h, h). Proposition 10. Let us assume that h = ∞. Then, with probability one, for every real h ∈ [h, h), dim E h ≥ 0. To prove Proposition 10, we do not use the theory of sets with large intersection. Instead, we follow the main ideas of the proof of Lemma 9 in [22]. Specifically, for any h ∈ [h, h), we obtain a point yh in the set E h as the intersection of a sequence (Inh )n≥1 of nested closed sets and we show that, with probability one, the construction of this sequence is possible for all h ∈ [h, h). To this end, let us establish the following preparatory result. Lemma 11. With probability one: (a) For every u ∈ U, the set uU ∗ ∩ S is nonempty. (b) There is real κ ≥ 1 such that κ 2 j η j−1 j 2 . ∀ j ≥ 1 # Sj ≤ (c) If θ < 0, then the set Θ is empty. Conversely, if θ ≥ 0, then there is a real κ ≥ 1 such that ∀ j ≥ 0 #S j ≤ κ 2
(1−θ) j/2
j−1
γ .
= j
Proof. To begin with, observe that b directly follows from (20). So, we only need to prove a and c. In order to prove a, let u ∈ U. The Markov condition (M) implies that, for any integer j ≥ u , P(∀v ∈ uU v ≤ j =⇒ X v = 1) ≤
j−1 = u
ν1, ({(1, 1)}) ≤ 2
u − j
j−1 = u
γ .
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A. Durand
If u ≤ j, the right-hand side vanishes for j large enough. Otherwise, since θ is assumed to be less than one, its limit inferior as j → ∞ vanishes. Thus, a.s. ∃u ∈ uU X u = 0. Furthermore, for any vertex
u
(22) u ,
∈ U and any integer j > observe that ∀u ∈ u U X u = 0 ⊆ B uj ,
(23)
where B uj is the event corresponding to the fact that X u = 0 for any vertex u in u U with generation at most j. Due to the Markov condition (M), the conditional probability of the event B uj conditionally on the σ -field generated by the variables X v , for v ∈ U such that v < j, is equal to 1Bu (1 − η j−1 )2 (1 − η j−1 )2 that
j− u −1
j− u −1
j−1
, so that the probability of B uj is
times that of B uj−1 . Arguing by induction, one then readily verifies
j
u P(B uj ) = P(B u )
(1 − η−1 )2
− u −1
.
= u +1
Since 2 η = ∞, the preceding product tends to zero as j → ∞. By (23) and the fact that U is countable, it follows that with probability one, for all u ∈ U, X u = 0 =⇒ ∃u ∈ u U ∗ X u = 1. Thanks to (22), the set uU almost surely contains a vertex u enjoying X u = 0 and, because of the last assertion, the set u U ∗ almost surely contains a vertex u with X u = 1. We may assume that the generation of the vertex u is minimal. The set uU ∗ ∩ S, containing u , is thus nonempty. The fact that U is countable finally leads to a. Let us prove c. If θ < 0, then Lemma 6 ensures that the set Θ is almost surely empty. Conversely, let us assume that θ ≥ 0. Owing to the Markov condition (M), for any integer j ≥ 1, the conditional expectation of #S j conditionally on the σ -field generated by the variables X u for u ∈ U with u < j is at most 2 j η j−1 + γ j−1 #S j−1 . Arguing by induction on j, one can establish that ∀ j ≥ 0 E[#S j ] ≤
j−1
2
k=−1
k+1
ηk
j−1
γ ,
=k+1
with the convention that η−1 = 1. As γ j−1 vanishes, it follows that ⎛ ⎞ j−1 j−1 2k+1 ηk γ ⎠ . ∀ j ≥ j E[#S j ] ≤ ⎝ k k= j−1 = j γ = j
Moreover, the fact that h = ∞ and θ ∈ [0, 1) implies that for all k large enough, ηk ≤ 2−(7+θ)k/8 and
k = j
γ ≥ 2(−1+θ)(k+1)/8 .
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As a result, for some real c > 0 and every integer j ≥ j, Markov’s inequality yields ⎛ P ⎝#S j > 2(1−θ) j/2
j−1
⎞
" ! E #S j
γ ⎠ ≤
= j
2(1−θ) j/2
We conclude using the Borel-Cantelli lemma.
j−1 = j
≤ c 2−(1−θ) j/4 . γ
From now on, we assume that the event on which the statement of Lemma 11 holds occurs. For any h ∈ [h, h), let us build recursively a sequence (Inh )n≥1 of nested closed subsets of the torus which lead to a point of the set E h . For this purpose, let ρ hj = j2h j/ h
∞
2(1−h/ h) j η j −1 ( j )2
(24)
j = j+1
h = ∞, it is easy to check that (ρ hj ) j≥0 for any integer j ≥ 0. Since j 2 j η j = ∞ and is a sequence of positive reals which enjoys ∀ε > 0 ρ hj = o(2εj ) as j → ∞.
(25)
Moreover, for any vertex u ∈ U, let Buh be the open ball with center xu and radius h 2−h u / h and let 23 Bu be the open ball with center xu and radius 3 · 2− u −1 . Together with the sequence (Inh )n≥1 of nested closed sets, we build a nondecreasing sequence ( jnh )n≥0 of nonnegative integers. The construction of the set I1h and the integers j0h and j1h depends on whether or not θ is negative. h = ∞, the • Step 1, if θ < 0. Let us build the set I1h and the integers j0h and j1h . As κ such that series j 2(1−h/ h) j η j−1 j 2 converges, so there is an integer j0h ≥ 4 ∀j ≥
j0h
2 κ
j
2(1−h/ h) j η j −1 ( j )2 ≤
j = j0h
1 , 4
(26)
where κ is given by Lemma 11b. Furthermore, Lemma 11a shows that, for j ≥ j0h large enough, the set S jh ∪ . . . ∪ S j is nonempty. In addition, there is at least one 0
connected component, denoted by I j , of the complement in the torus of the balls Buh , S j , which has Lebesgue measure at least for u ∈ S jh ∪ . . . ∪ 0
1−
j
# S j 21−h j / h
j = j0h j j = j0h
3
≥ # S j
4 κ
j j = j0h
2j η
. j −1
( j )2
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A. Durand
Note that this inequality follows from Lemma 11b and (26). The component I j can contain the image under the canonical surjection φ of some closed subinterval of R with length ρ hj 2−h j/ h if 3 · 2h j/ h
ρ hj ≤ 4 κ
j
.
j = j0h
2 j η j −1 ( j )2
As h = ∞, the right-hand side tends to infinity exponentially fast as j → ∞. Thus, (25) implies that the preceding inequality holds for j large enough. Let j1h be the smallest integer such that S jh ∪ . . . ∪ S j h is nonempty and the inequality holds. Remark that, next 0 1 h to I h , there is a ball Bu with u ∈ S h ∪ ... ∪ S h . Hence, I h contains a set, denoted j1
j0
by I1h , of the form
j1
j1
φ(xu + 2−h u / h + [0, ρ hjh 2−h j1 / h ]) or φ(xu − 2−h u / h + [−ρ hjh 2−h j1 / h , 0]), h
h
1
1
with u ∈ S jh ∪ . . . ∪ S j h . Clearly, the intersection of the sets Bvh and I1h is empty for 0 1 S h. every vertex v ∈ S h ∪ ... ∪ j0
j1
• Step 1, if θ ≥ 0. Since θ ∈ [0, 1), there is an infinite subset J of N such that γ j . . . γ j−2 ≤ 2(5θ+1)( j−1)/6 for any integer j ∈ J . Together with Lemma 11c, this ensures that #S j−1 is bounded by κ 2(2+θ)( j−1)/3 for any j ∈ J . Let j0h denote an integer in J which is greater than 4 κ and is large enough to ensure both (26) and 3κ 2−(1−θ)( j0 −1)/3 ≤ h
1 . 4
(27)
By virtue of Lemma 11a, for any j ≥ j0h large enough, the set S j h ∪. . .∪ S j is nonempty. 0 In addition, there is at least one connected component, denoted by I j , of the complement h S jh ∪ . . . ∪ S j , which in T of the balls 23 Bu , for u ∈ S j h −1 , and the balls Buh , for u ∈ 0 0 has Lebesgue measure at least 1 − 3#S j h −1 2−( j0 −1) −
j
h
0
#S j h −1 + 0
# S j 21−h j / h
j = j0h
j
1/2
≥
# S j
κ2
j = j0h
2+θ 3
( j0h −1)
+ κ
j j = j0h
. 2j η
j −1
( j )2
This last inequality follows from Lemma 11b–c, along with (26) and (27). The component I j can contain the image under φ of some closed subinterval of R with length ρ hj 2−h j/ h if 2−1+h j/ h
ρ hj ≤ κ
h 2(2+θ)( j0 −1)/3
+ κ
j j = j0h
. 2j η
j −1
( j )2
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This inequality holds for j large enough, because of (25) and the fact that h is infinite. Let j1h denote the smallest integer for which S jh ∪ . . . ∪ S j h is nonempty and the h
0
1
inequality holds. Next to I j h , there is a ball 23 Bu with u ∈ S j h −1 or a ball Buh with 1 0 S j h . Thus, I j h contains a set, denoted by I1h , of the form u ∈ S jh ∪ . . . ∪ 0
1
φ(xu + 3 · 2
− j0h
1
+ [0, ρ hjh 2−h j1 / h ]) or φ(xu − 3 · 2− j0 + [−ρ hjh 2−h j1 / h , 0]), h
h
h
1
1
with u ∈ S j h −1 , or of the form 0
φ(xu + 2−h u / h + [0, ρ hjh 2−h j1 / h ]) or φ(xu − 2−h u / h + [−ρ hjh 2−h j1 / h , 0]), h
h
1
1
S j h . Note that with u ∈ S jh ∪ . . . ∪ 0
and that
3 h 2 Bv
1
∩
Bvh
I1h
= ∅ for every vertex v ∈ S jh ∪ . . . ∪ S jh 0
1
∩ I1h = ∅ for every vertex v ∈ S j h −1 . 0
• Step n + 1 for n ≥ 1. Steps 1 to n have supplied the sets I1h ⊇ . . . ⊇ Inh and the h and the integer j h . Because of (24) integers j0h ≤ . . . ≤ jnh . Let us build the set In+1 n+1 h h and the fact that jn ≥ j0 ≥ 4 κ , we have 2 κ
∀ j ≥ jnh + 1
ρ hjh 2−h jn / h h
j
2
(1−h/ h) j
η j −1 ( j )2 ≤
n
2
j = jnh +1
.
(28)
Let us consider a vertex v ∈ U enjoying v ≥ jnh + 1 and λv ⊆ Inh . Lemma 11a ensures that there is a vertex v ∈ vU ∗ ∩ S. As a result, the set Bvh ∩ Inh , containing the point xv , is nonempty and the set S jnh +1 ∪ . . . ∪ S j is nonempty for j large enough. In addition, there is at least one connected component, denoted by I j , of the complement in Inh of S jnh +1 ∪ . . . ∪ S j , which has Lebesgue measure at least the balls Buh , for u ∈ ρ hjh 2−h jn / h −
j
h
n
j = jnh +1 j
1+
j = jnh +1
# S j 21−h j / h
ρ hjh 2−h jn / h h
≥
# S j
$
%.
n
2 1 + κ
j j = jnh +1
2j η
j −1
( j )2
The inequality follows from Lemma 11b and (28). The component I j can contain the image under φ of some closed subinterval of R with length ρ hj 2−h j/ h if ρ hjh 2h( j− jn )/ h h
ρ hj
≤
$
2 1 + κ
%,
n
j j = jnh +1
2j η
j −1
( j )2
h which holds for j large enough because of (25) and the fact that h is infinite. Let jn+1 be the smallest integer such that this inequality holds and such that there exists a vertex v ∈ S jnh +1 ∪ . . . ∪ S j h with xv ∈ Inh . Observe that, next to I j h , there is a ball Buh with n+1
n+1
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A. Durand
h , of the u∈ S jnh +1 ∪ . . . ∪ S j h . As a consequence, I j h contains a set, denoted by In+1 n+1 n+1 form
φ(xu + 2−h u / h + [0, ρ hjh 2−h jn+1 / h ]) or φ(xu − 2−h u / h + [−ρ hjh 2−h jn+1 / h , 0]) h
h
n+1
n+1
h is with u ∈ S jnh +1 ∪ . . . ∪ S j h . Observe that the intersection of the sets Bvh and In+1 n+1 empty for every vertex v ∈ S j h +1 ∪ . . . ∪ Sh .
jn+1 n h h The sets I0 , I1 , . . . given by the preceding procedure form a decreasing sequence of h closed subsets of the torus. Moreover, the diameter of each set Inh is at most ρ hjh 2−h jn / h , n
which tends to zero as n → ∞ by virtue of (25). Thus, the intersection over n ≥ 1 of the sets Inh is a singleton {yh }. Lemma 12. The point yh belongs to the iso-Hölder set E h . Proof. Let α ∈ (h, h]. Owing to (25), there exists an integer n 0 ≥ 2 such that jnh0 −1 ≥
(log2 3)/(h/ h − h/α) and ρ hjh 2−h jn / h ≤ 2−h jn /α /3 for every integer n ≥ n 0 . Let h
h
n
us consider an integer n ≥ n 0 . The point yh belongs to Inh , so there exists a vertex S j h +1 ∪ . . . ∪ S jnh for which un ∈ n−1
d(yh , xu n ) ≤ 2−h u
n / h
+ ρ hjh 2−h jn / h < 2−h u h
n
n /α
.
The last inequality follows from the fact that n ≥ n 0 and u n ≤ jnh . As a result, the point yh belongs to the set L α defined by (11). By Lemma 4, this point thus belongs to h . the set E This proves the lemma in the case where h = h. We may therefore assume that h > h. Lemma 4, together with the fact that α → L α is nondecreasing, shows that it suffices to establish that yh ∈ L h . Let us assume that θ < 0. The point yh belongs to I1h , so it cannot belong to any ball Buh for u ∈ S jh ∪ . . . ∪ S j h . Moreover, for any integer n ≥ 1, the point yh belongs to 0 1 h S j h +1 ∪ . . . ∪ S h . It follows that yh I , so it cannot belong to any ball Buh for u ∈ n+1
jn+1
n
does not belong to any ball Buh with u ∈ S and u ≥ j0h . Hence, yh does not belong to the set L h defined by (15). Furthermore, Lemma 7 shows that L h = L h ∪ Θ and Lemma 11c implies that Θ is empty. It follows that yh ∈ L h . Let us assume that θ ≥ 0. In this case, the point yh does not belong to any ball Buh with u ∈ S and u ≥ j0h and does not belong to any closed dyadic interval λu with u ∈ S j h −1 . Therefore, the point yh cannot belong to the set L h . It cannot belong to the set 0 Θ either. Otherwise, there would exist a vertex u ∈ S and a sequence ζ = (ζ j ) j≥1 ∈ ∂τu enjoying yh = φ(x˙ζ ). If u ≤ j0h −1, then the vertex ζ1 . . . ζ j h −1 would belong to S j h −1 0
0
and index a closed dyadic interval containing yh . If u ≥ j0h , then the point yh would belong to the set λu and thus to the ball Buh , since h > h. In both cases, we would end up with a contradiction. Hence, yh does not belong to Θ. Lemma 7 finally ensures that yh ∈ L h . We have established that, with probability one, for any h ∈ [h, h), it is possible to build a point yh in the set E h . Proposition 10 is thus proven.
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Using Propositions 8, 9 and 10 together with the fact that E h ⊇ Θ by Lemmas 4 and 7, we finally obtain the following result. Proposition 13. With probability one, dim E h = max(h/ h, dim Θ) and ∀h ∈ (h, min( h, h)) dim E h = h/ h. Theorem 2 is then an immediate consequence of this result and Lemma 6.
7. Proof of Proposition 2 In order to prove Proposition 2, let h ∈ [h, h] and let x ∈ E h . Let us first assume that h < h and let us consider a real number β > h/h −1. Owing to Lemma 4, for any integer n ≥ 1 such that h + 1/n < (β + 1)h, there exists a dyadic interval λn ∈ Λ enjoying λn ≥ n, Cλn = 2−h λn and d(x, xλn ) < 2−h λn /(h+1/n) . Note that these intervals λn are such that d(x, xλn )1+β ≤ 2− λn . Proposition 3 in [1] then shows that β R (x) ≤ β. This inequality holds for any β > h/h − 1. Thus, β R (x) ≤ h/h − 1. Conversely, let us consider a real number β > β R (x). Owing to Proposition 3 in [1], there exists a sequence (λn )n≥1 of dyadic intervals of the torus such that d(x, xλn )1+β ≤ 2− λn for all n ≥ 1, 2− λn + d(x, xλn ) −−−→ 0 n→∞
and
log |Cλn | −−−→ h. + d(x, xλn )) n→∞
log(2− λn
As h < h, for infinitely many integers n ≥ 1, we have log2 |Cλn | log |Cλn | ≤ < h, − λn log(2− λn + d(x, xλn )) so that X u λn = 1. Thanks to Lemma 4, it follows that h ≤ (1 + β)h. Letting β → β R (x), we obtain β R (x) ≥ h/h − 1. Let us now suppose that h = h < ∞ and consider a real β > 0. As h > h, the point x does not belong to L h by virtue of Lemma 4. Hence, there exists an integer j0 ≥ 0 such that X u = 0 for any vertex u ∈ U enjoying u ≥ j0 and d(x, xu ) < 2− u . For any integer j ≥ j0 , there is a vertex u j ∈ U satisfying u j = j and d(x, xu j ) < 2− j . Observe that 2− λu j + d(x, xλu j ) −−−→ 0 and j→∞
log |Cλu j |
log(2
− λu j
+ d(x, xλu j ))
−−−→ h. j→∞
Proposition 3 in [1] then ensures that β ≥ β R (x). We conclude by letting β → 0. Acknowledgement. The author is grateful to Stéphane Jaffard for many useful comments.
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Commun. Math. Phys. 283, 479–489 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0508-3
Communications in
Mathematical Physics
Wegner Bounds for a Two-Particle Tight Binding Model Victor Chulaevsky1 , Yuri Suhov2 1 Département de Mathématiques et Informatique, Université de Reims, Moulin de la Housse,
B.P. 1039, 51687 Reims Cedex 2, France. E-mail:
[email protected]
2 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WB, UK Received: 21 August 2007 / Accepted: 19 November 2007 Published online: 29 May 2008 – © Springer-Verlag 2008
Abstract: We consider a quantum two-particle system on a lattice Zd with interaction and in presence of an IID external potential. We establish Wegner-type estimates for such a model. The main tool used is Stollmann’s lemma. 1. Introduction. The Results d This paper considers a two-particle Anderson tight binding model on lattice Z with (2) (ω) is a lattice Schrödinger operator (LSO) interaction. The Hamiltonian H = HU,V
of the form H 0 + U + V1 + V2 acting on functions φ ∈ 2 (Zd × Zd ): H φ(x) = H 0 φ(x) + [(U + V1 + V2 ) φ] (x) φ(y) + U (x) + 2j=1 V (x j ; ω) φ(x), = y: y−x=1
(1.1)
x = (x1 , x2 ), y = (y1 , y2 ) ∈ Zd × Zd . (1) (d) (1) (d) and y j = y j , . . . , y j stand for coordinate vectors of Here, x j = x j , . . . , x j the j th particle in Zd , j = 1, 2, and · is the sup-norm in Rd × Rd : (i) x = max max x j , x = (x1 , x2 ) ∈ Rd × Rd . j=1,2 i=1,...,d
The same notation, ·, is used for the sup-norm in Rd ; this should not lead to confusion. We will use boldface notations, like x, for points in Zd × Zd describing positions of the two-particle system. In Sect. 2, boldface notations are used for vectors in an auxiliary Euclidean space R p . Throughout this paper, the random external potential V (x; ω), x ∈ Zd , is assumed to be real IID, with a common distribution function F on R. It can be quite arbitrary,
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although in many applications it is assumed at least continuous. In any case, the continuity modulus of a given non-decreasing function F can be defined by s() (= s(F, )) := sup (F(a + ) − F(a − 0)), > 0, a∈R
(1.2)
where F(a − 0) := limδ↑0 F(a − δ). Naturally, s() ≤ 1 for any probability distribution function F. In physically interesting models, the interaction potential U cannot be completely arbitrary. It is usually assumed to be symmetric (U (x1 , x2 ) = U (x2 , x1 )) or even (x1 − x2 )), and sufficiently rapidly decaying translation invariant (U (x1 , x2 ) = U as x1 − x2 → ∞. However, in the present paper we treat only finite-volume particle systems, and such assumptions are no longer imperative for the operator H 0 + V (x1 ; ω)+ V (x2 ; ω) + U (x1 , x2 ) to be well-defined in a finite volume. It suffices, e.g., to assume U (x) to be locally finite, so that the operator of multiplication by the function U (x), restricted to Hilbert space 2 (Λ) with |Λ| := card Λ < ∞, is bounded. Even this assumption can be relaxed so as to include the case of hard-core interactions, where U (x1 , x2 ) = +∞ for (x1 , x2 ) with x1 − x2 ≤ r0 < ∞, and U (x) is (at least locally) bounded for all other x = (x1 , x2 ). The latter case (hard-core interactions) would require certain technical modifications of notations and arguments, while our main results given in Theorems 1, 2 and 3 below would essentially remain valid. We do assume, however, symmetry of the function U (x1 , x2 ), having in mind future applications to quantum systems under Bose-Einstein or Fermi-Dirac quantum statistics. Again, it is worth mentioning that such an assumption is not imperative for our results to be valid; only the general setup would need to be modified. We plan to address various possible generalisations in a separate paper. Therefore, below we always assume that U (x) is a fixed (non-random), symmetric, locally bounded function on Zd × Zd . The purpose of this paper is to establish the so-called Wegner-type estimates for H . More precisely, these estimates are produced for the eigen-values of a finite-volume (2) approximation HΛ = HΛ,U,V (ω) acting in 2 (Λ): HΛ φ(x) = HΛ0 φ(x) + (U + V1 + V2 )Λ φ (x) = φ(y) + U (x) + 2j=1 V (x j ; ω) φ(x), y∈Λ: y−x=1
(1.3)
x = (x1 , x2 ), y = (y1 , y2 ) ∈ Λ.
Here Λ ⊂ Zd × Zd is a finite set of cardinality |Λ|. For definiteness, we will focus on the case where Λ is specified as a Zd × Zd lattice parallelepiped Cartesian written as the (1) (d) ∈ Zd and product of two Zd lattice cubes centred at points u 1 = u 1 , . . . , u 1 (d) ∈ Zd : u 2 = u (1) 2 , . . . , u2
d
d
(i) (i) (i) (i) ∩ Zd × Zd . × −L 1 + u 1 , u 1 + L 1 × × −L 2 + u 2 , u 2 + L 2
i=1
i=1
(1.4) Here L := (L 1 , L 2 ) ∈ N2 . Notice that the above lattice subset is non-empty even for L = 0, and its diameter equals 2L.
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A set Λ of the form (1.4) will be called a box and denoted by ΛL (u), u = (u 1 , u 2 ) ∈ Zd × Zd , while the Zd lattice cubes figuring in (1.4) as the Cartesian factors will be denoted by Π1 ΛL (u) and Π2 ΛL (u):
d (i) (i) (1.5) Π j ΛL (u) = × −L j + u j , u j + L j ∩ Zd , j = 1, 2. i=1
We will also call cubes Π1 ΛL (u) and Π2 ΛL (u) the projections of ΛL (u) and set Π ΛL (u) = Π1 ΛL (u) ∪ Π2 ΛL (u).
(1.6)
The cardinality of box ΛL (u) is denoted by |ΛL (u)| and the cardinality of cube Π j ΛL (u) by Π j ΛL (u). Symbol P will stand for the probability distribution generated by random variables V (x; ω), x ∈ Zd . Symbol B [Π ΛL (u)] is used for the sigma-algebra generated by random variables ω → V (x; ω), x ∈ Π ΛL (u).
(1.7)
Remark. Working with projections of different sizes may appear artificial. Indeed, in this paper this only allows to make assertions of Theorems 1, 2 and 3 slightly more general. However, having in mind future applications to quantum systems under Bose-Einstein or Fermi-Dirac statistics, it is preferable to allow projections of boxes to be of different sizes. The spectrum Σ HΛL (u) of HΛL (u) is a random subset of R consisting of |ΛL (u)| (k) (k) points (not necessarily distinct) λΛL (u) (= λΛL (u) (ω)), k = 1, . . . , |ΛL (u)| (random eigen-values in volume ΛL (u), measurable with respect to B [Π ΛL (u)]). Given a value E ∈ R, we denote (k) (1.8) dist Σ HΛL (u) , E = min E − λΛL (u) : k = 1, . . . , |ΛL (u)| . Our first result in this paper is the so-called single-volume Wegner bound given in Theorem 1. Theorem 1. ∀ E ∈ R, L ∈ N2 , u ∈ Zd × Zd and > 0, P dist Σ HΛL (u) , E ≤ ≤ |ΛL (u)| min Π j ΛL (u) · s(2). j=1,2
(1.9)
Single-volume Wegner-type bounds were often used in the (single-particle) Anderson localisation theory; see, e.g., original papers by Fröhlich, Martinelli, Scoppola and Spencer [4], and by von Dreifus and Klein [3]. In Theorem 2 below we deal with a two-volume Wegner bound. This bound assesses the probability that the random spectra Σ HΛL (u) and Σ HΛL (u ) are close to each other, for a pair of boxes ΛL (u) and ΛL (u ) positioned away from each other. It is worth mentioning that, in the conventional, single-particle localisation theory, such a bound can be derived from its single-volume counterpart (e.g., for IID random potentials). See [4] and [3] for details. With N > 1 particles, it requires additional arguments. Indeed, an important feature of two-particle operators is that the potential W (u 1 , u 2 ; ω) = U (u 1 , u 2 )+ V (u 1 ; ω)+ V (u 2 ; ω) is a symmetric function of the pair (u 1 , u 2 ) ∈ Zd ×Zd . Namely, let S : Zd × Zd → Zd × Zd be the following map: S : (u 1 , u 2 ) → (u 2 , u 1 ).
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Then the potential energy operator W (x) = U (x) + V (x1 ) + V (x2 ) satisfies W (S(x)) ≡ W (x). As a consequence, the spectra of operators HΛ and HS(Λ) are identical, although the distance dist[Λ, S(Λ)] may be arbitrarily large. We define the distance between two spectra in a usual way: dist Σ HΛL (u) , Σ HΛL (u ) (1.10) (k) (k ) = min λΛL (u) − λΛ (u ) , 1 ≤ k ≤ |ΛL (u)| , 1 ≤ k ≤ ΛL (u ) . L
Theorem 2. ∀ L = (L 1 , L 2 ), L = (L 1 , L 2 ) ∈ N2 , u, u ∈ Zd × Zd with min u − u , u − S(u )} > 8 max{L 1 , L 2 , L 1 , L 2 and ∀ > 0, the following inequality holds: P dist Σ HΛL (u) , Σ HΛL (u ) ≤ ≤ |ΛL (u)| ΛL (u ) max max Π j ΛL (u ) s(2).
(1.11)
(1.12)
j=1,2 u ∈{u,u }
The assertions of Theorems 1 and 2 are proved in the next section of the paper, with the help of the so-called Stollmann’s lemma. They are useful in the spectral analysis of H and HΛL (u) . Throughout the paper, the symbol is used to mark the end of a proof. 2. Stollmann’s Lemma. Proof of Theorems 1 and 2 2.1. Stollmann’s lemma and its use. For the reader’s convenience, we provide here the statement of Stollmann’s lemma; see Lemma 2.1 below, cf. [6] and [7], Lemma 2.3.1. Let Γ be a non-empty finite set of cardinality |Γ | = p. We assume that Γ is ordered and identify it with the set {1, 2, . . . , p}. Consider the Euclidean space RΓ ∼ = R p with standard basis (e1 , . . . , e p ), and its positive orthant RΓ+ = q = (q1 , . . . , q p ) ∈ RΓ : q j ≥ 0, j = 1, . . . , p . We believe that the use of boldface notations for vectors q ∈ RΓ , in this section, should not lead to confusion. For a given probability measure µ on R, denote by µΓ the product measure µ×· · ·×µ on RΓ and by µΓ \{1} be the marginal product measure induced by µΓ on RΓ \{1} . Next, ∀ > 0 set s(µ, ) = sup µ ([a, a + ]). a∈R
(2.1)
Definition 2.1. A function Φ : RΓ → R is called diagonally-monotone (DM) if it satisfies the following conditions: (i) ∀ r ∈ RΓ+ and any v ∈ RΓ , Φ(v + r) ≥ Φ(v);
(2.2)
(ii) moreover, with vector e = e1 + . . . + e p ∈ RΓ , ∀ v ∈ RΓ and t > 0, Φ(v + te) − Φ(v) ≥ t.
(2.3)
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Lemma 2.1. Suppose function Φ : RΓ → R is DM. Then ∀ > 0 and any open interval I ⊂ R of length , µΓ { v : Φ(v) ∈ I } ≤ |Γ | · s(µ, ).
(2.4)
The proof of this lemma can be found in the book by P. Stollmann [7], as well as in his original paper [6]. In our situation, it is also convenient to introduce the notion of a DM operator family. As before, Γ is a finite set, |Γ | = p < ∞, identified with {1, . . . , p}, so that RΓ ∼ = Rp. Definition 2.2. Let H be a Hilbert space of a finite dimension m. A family of Hermitian operators B(v) : H → H, v ∈ RΓ , |Γ | = p < ∞, is called DM if (i) ∀ r ∈ RΓ+ ∀ v ∈ RΓ , B(v + r) ≥ B(v)
(2.5A)
(in the sense of quadratic forms). (ii) ∀ f ∈ H (B(v + t · e) f, f ) − (B(v) f, f ) ≥ t · f 2 .
(2.5B)
That is, ∀ f ∈ H with f = 1, the function Φ f : RΓ → R defined by Φ f (v) = (B(v) f, f ) is DM. The importance of Stollmann’s Lemma 2.1 in spectral theory of random operators is illustrated by the following two elementary observations. (1)
Remark 2.1. Suppose that B(v), v ∈ RΓ , is a DM operator family in H. Let E B(v) ≤ (m)
. . . ≤ E B(v) be the eigen-values of B(v). Then, by virtue of the variational principle, ∀ (k)
k = 1, . . . , m, v → E B(v) is a DM function. Remark 2.2. If B(v), v ∈ RΓ , is a DM operator family in H, and K : H → H is an arbitrary Hermitian operator, then the family K + B(v) is also DM. The arbitrariness of operator K in Remark 2.2 illustrates the power of Stollmann’s lemma. In the context of multi-particle lattice quantum systems, it allows to consider fairly general kinetic energy operators H 0 and non-random interactions U . For a single-particle tight binding model with non-IID random potential, similar results are presented in [1]. 2.2. Proof of Theorem 1. The proof is a straightforward application of Lemma 2.1 and Remarks 2.1 and 2.2, cf. the proof of Theorems 2.3.2 and 2.3.3 in [2]. In our situation, the set Γ is identified as the set of smallest cardinality among Π1 ΛL (u) and Π2 ΛL (u), with p = |Γ | = min {|Π1 ΛL (u)|, |Π2 ΛL (u)|}. (If both projections have equal cardinality, we can pick Γ = Π1 ΛL (u), for the sake of definiteness.) Vector v is identified with a collection {V (x; ω), x ∈ Γ } of sample values of the external potential; to stress this fact we will write v ∼ {V (x; ω), x ∈ Γ }.
(2.6)
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Next, probability measure µ represents the distribution of a single value, say V (0; · ), and product-measure µΓ is identified as PΠi ΛL (u) , where the value i ∈ {1, 2} is chosen so that Πi ΛL (u) has smallest cardinality. Further, the Hilbert space H in Remarks 2.1 and 2.2 is 2 (ΛL (u)), of dimension m = |ΛL (u)|, in which the action of operator HΛL (u) is considered. Given x = (x1 , x2 ) ∈ ΛL (u), we can write V (x1 ; ω) + V (x2 ; ω) = =
y∈ΠΛL (u)
c(x, y)V (y; ω) +
y∈Γ
c(x, y)V (y; ω)
y∈ΠΛL (u)\Γ
c(x, y)V (y; ω),
where c(x, y) is defined as a function of y ∈ Π ΛL (u), for every x ∈ ΛL (u), by c(x, y) = δ y,x1 + δ y,x2 , so that, obviously, ∀ x ∈ ΛL (u) ∀ y ∈ Γ , c(x, y) ≥ 1. Now we can re-write the external random potential V (x1 ; ω) + V (x2 ; ω) as follows: (x; ω), V (x1 ; ω) + V (x2 ; ω) = VΓ (x; ω) + V where VΓ (x; ω) =
c(x, y)V (y; ω)
y∈Γ
and, respectively, (x; ω) = V
c(x, y)V (y; ω)
y∈ΠΛL (u)\Γ
(x; ω) so that only the term VΓ (x; ω) is measurable with respect to B [Πi ΛL (u)], while V is measurable with respect to {V (y; ·), y ∈ Π ΛL (u)\Γ }, and, therefore, independent of {V (y; ·), y ∈ Πi ΛL (u)}. Next, we write + VΓ = K + VΓ , K = H0 + U + V , HΛL (u) = H 0 + U + V is B [Πi ΛL (u)]-independent. In other so that VΓ is B [Πi ΛL (u)]-measurable, and K is non-random. So, with v ∼ {V (y; ω), words, relative to the measure µΓ , operator K with operator K of Remark 2.2, while the role of operator family y ∈ Γ }, we identify K B(v) is played by multiplication operators VΓ (·; ω): B(v)φ(x) = VΓ (x; ω)φ(x), x ∈ ΛL (u), φ ∈ 2 (ΛL (u)).
(2.7)
The above lower bound c(x, y) ≥ 1, valid for any y ∈ Γ , implies that, with identification (2.6), Hermitian operators B(v) form a DM family. + VΓ is a DM Then we use Remark 2.2 (cf. (1.3)), and obtain that HΛL (u) = K (k) family. Next, owing to Remark 2.1, each eigen-value λΛL (u) , k = 1, . . . , |ΛL (u)|, is a
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DM function of the sample collection v ∼ {V (x; ω), x ∈ Γ }. Hence, by Lemma 2.1, ∀ k = 1, . . . , |ΛL (u)|, and owing to our choice of the projection of smallest cardinality, (k) (2.8) P E − λΛL (u) ≤ ≤ min Π j ΛL (u) s(F, 2). j=1,2
The final remark is that the probability in the LHS of Eq. (1.9) is ≤ the RHS of Eq. (2.8) times |ΛL (u)|. We will need the following elementary geometrical statement which we prove later. Lemma 2.2. Consider two boxes ΛL (u) and ΛL (u ) and suppose that min(u − u , u − S(u )) > 8 max{L 1 , L 2 , L 1 , L 2 }.
(2.9)
Then there are two possibilities (which in general do not exclude each other): (i) ΛL (u) and ΛL (u ) are ‘completely separated’, when Π ΛL (u) ∩ Π ΛL (u ) = ∅.
(2.10)
(ii) ΛL (u) and ΛL (u ) are ‘partially separated’. In this case one (or more) of the four possibilities can occur: (A) Π1 ΛL (u) ∩ Π2 ΛL (u) ∪ Π ΛL (u ) = ∅, (B) Π2 ΛL (u) ∩ Π1 ΛL (u) ∪ Π ΛL (u ) = ∅, (2.11) (C) Π1 ΛL (u ) ∩ Π ΛL (u) ∪ Π2 ΛL (u ) = ∅, (D) Π2 ΛL (u ) ∩ Π ΛL (u) ∪ Π1 ΛL (u ) = ∅. Pictorially, case (ii) is where one of the cubes Π j ΛL (u), Π j ΛL (u ), j = 1, 2, is disjoint from the union of the rest of the projections of ΛL (u) and ΛL (u ). We note that the use of the max-norm · is convenient here: it leads to the constant 8 (two times the number of projections Π j ΛL (u) and Π j ΛL (u ), j = 1, 2) which does not depend on dimension d. 2.3. Proof of Theorem 2. Owing to Lemma 2.2, boxes ΛL (u) and ΛL (u ) satisfy either (i) or (ii), i.e. they are either completely or partially separated. Passing to the proof of Theorem 2 proper, consider separately cases where boxes ΛL (u) and ΛL (u ) satisfy (i) or (ii). (i) ‘Complete separation’. Then we can write P dist Σ HΛL (u) , Σ HΛ L (u ) ≤ (2.12) = E P dist Σ HΛL (u) , Σ HΛL (u ) ≤ | B Π ΛL (u ) . (k )
Note first that, under conditioning in Eq. (2.12), the eigen-values λΛL (u ) , k = 1, . . . , ΛL (u ), forming the set Σ HΛ (u ) are non-random. Therefore, it makes sense to L use the following inequality: P dist Σ HΛL (u) ,Σ HΛL (u ) ≤ | B Π ΛL (u ) (2.13) ≤ |ΛL (u )| sup P dist Σ HΛL (u) , E ≤ , E∈R
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since there are |ΛL (u )| eigen-values λΛL (u ) (counting multiplicities). Then, by virtue of Theorem 1, P dist Σ HΛL (u) , E ≤ ≤ |ΛL (u)| min Π j ΛL (u) · s(2), j=1,2
(2.14)
implying bound (1.12). (ii) ‘Partial separation’. For example, assume case (A) where Π1 ΛL (u), is disjoint from the union of the rest of the projections of ΛL (u) and ΛL (u ): Π1 ΛL (u) ∩ Π2 ΛL (u) ∪ Π ΛL (u )) = ∅.
(2.15)
We then estimate the probability in the LHS of (2.13) with the help of the conditional expectation (k ) P λ(k) − λ L (u ) Λ
ΛL (u ) ≤ B Π Λ L (u) (k) (k ) = E P λΛL (u) − λΛL (u ) ≤ B Π2 ΛL (u) ∪ Π ΛL (u ) B Π ΛL (u ) . (2.16) Here B Π2 ΛL (u) ∪ Π ΛL (u ) is the sigma-algebra generated by the random variables ω → V (x; ω), x ∈ Π2 ΛL (u) ∪ Π ΛL (u ); owing to (2.15) it is independent of the sigma-algebra B [Π1 ΛL (u)] generated by the random variables ω → V (x; ω), x ∈ Π1 ΛL (u). We see that the argument used in the proof of Theorem 1 is still applicable; here, we take the product-measure PΠ1 ΛL (u) (which again is identified with the product-measure µΓ from Lemma 2.1, with |Γ | = p = |Π1 ΛL (u)|). This allows us to write (k) (k ) P λΛL (u ) − λΛL (u ) ≤ B Π2 ΛL (u) ∪ Π ΛL (u ) ≤ |Π1 ΛL (u)| s(F, 2) (2.17) and deduce the required bound for the conditional probability in the LHS of (2.16). If, instead of (2.15), we have one of the other disjointedness relations (B)-(D) in Eq. (2.11), then the argument is conducted in a similar fashion. Specifically, in case (B) we exchange projections Π1 ΛL (u) and Π2 ΛL (u) in the above argument. In cases (C) and (D), we should exchange u and u as compared to arguments in cases (A) and (B). This concludes the proof of Theorem 2.
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2.4. Proof of Lemma 2.2. Recall that we have two boxes, ΛL (u) and ΛL (u ), satisfying the condition (2.9): min u − u , u − S(u ) > 8 max{L 1 , L 2 , L 1 , L 2 }. Notice that this can be viewed as a lower bound for the distance in the factor space Zd × Zd /S, where S(u 1 , u 2 ) = (u 2 , u 1 ). Since diam ΛL (u) = 2L, diam ΛL (u ) = 2L , this implies that the union of the four coordinate projections, Π1 ΛL (u), Π2 ΛL (u), Π1 ΛL (u ), Π2 ΛL (u ) cannot be connected. Therefore, it can be decomposed into two or more connected components. Cases (A), (B), (C) and (D) in the statement of Lemma 2.2 correspond to the situation where one of these coordinate projections is disjoint with the three remaining projections. So, it suffices to analyse the case where each connected component of the union Π ΛL (u) ∪ Π ΛL (u )
(2.18)
contains exactly two coordinate projections. Furthermore, it suffices to show that the only possible case is (2.10). To do so, we have to exclude two remaining cases, namely, ⎧ ⎪ ⎨ Π1 ΛL (u) ∪ Π1 ΛL (u ) ∩ Π2 ΛL (u) ∪ Π2 ΛL (u ) = ∅, (2.19) Π1 ΛL (u) ∩ Π1 ΛL (u ) = ∅, ⎪ ⎩Π Λ (u) ∩ Π Λ (u ) = ∅, 2 L 2 L and
⎧ ⎪ ⎨ Π1 ΛL (u) ∪ Π2 ΛL (u ) ∩ Π1 ΛL (u ) ∪ Π2 ΛL (u) = ∅, Π1 ΛL (u) ∩ Π2 ΛL (u ) = ∅, ⎪ ⎩Π Λ (u ) ∩ Π Λ (u) = ∅. 1 L 2 L
(2.20)
First, observe that (2.19) contradicts the assumption that ΛL (u) and ΛL (u ) are disjoint. Indeed, in such a case, there exist lattice points v1 ∈ Π1 ΛL (u) ∩ Π1 ΛL (u ), v2 ∈ Π2 ΛL (u) ∩ Π2 ΛL (u ), so that
∃ (v1 , v2 ) ∈ [Π1 ΛL (u) × Π2 ΛL (u)] ∩ Π1 ΛL (u ) × Π2 ΛL (u ) = ΛL (u) ∩ ΛL (u ) = ∅,
which is impossible. The case (2.20) can be reduced to (2.19), by the symmetry S. Namely, let u = S(u ), then Π1 ΛL (u ) = Π2 ΛL (u ), Π2 ΛL (u ) = Π1 ΛL (u ). Now (2.20) reads as follows in terms of boxes ΛL (u) and ΛL (u ): ⎧ ⎪ ⎨ Π1 ΛL (u) ∪ Π1 ΛL (u ) ∩ Π2 ΛL (u ) ∪ Π2 ΛL (u) = ∅, Π1 ΛL (u) ∩ Π1 ΛL (u ) = ∅, ⎪ ⎩Π Λ (u ) ∩ Π Λ (u) = ∅. 2 L 2 L
(2.21)
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The same argument as above shows then that ΛL (u)∩ΛL (u ) = ∅, which is impossible, since, by virtue of (2.9), dist(u, S(u )) > 8 max L 1 , L 2 , L 1 , L 2 . This completes the proof.
3. Concluding Remarks Remarks made by the referees of this paper allowed us to establish a sharper version of Eq. (1.9) and, consequently, a sharper version of Eq. 1.12 which include a factor s(). The reader may compare the current assertion of Theorem 1 with its counterpart in the preliminary version of this paper on [arXiv:0708.2056]. In an earlier manuscript [2], we proved Wegner-type bounds for two-particle lattice systems under a much more restrictive assumption of analyticity of the distribution function F of the random external potential V . The proofs, which were more involved than in this paper, also required the amplitude of the potential V to be sufficiently big. However, both approaches revealed an interesting fact. Speaking informally, having more than one particle can only make Wegner type bounds stronger, not weaker, as one might suppose. Assertions of Theorems 1 and 2 in [arXiv:0708.2056] make this particularly clear. A Wegner-type bound for multi-particle systems, with an arbitrary number of particles, close to Theorem 1 (but not Theorem 2) has been independently obtained by W. Kirsch [5] using arguments which are closer in spirit to the original argument given by F. Wegner [8] than to Stollmann’s method. In fact, our Theorem 1 can be extended without difficulty to the general case of N ≥ 2 particles. Denoting as before N -particle configurations by x, y, etc., the Hamiltonian H (N ) (= HU,V ) reads H φ(x) = H 0 φ(x) + [(U + V1 + · · · + VN ) φ] (x) φ(y) + U (x) + Nj=1 V (x j ; ω) φ(x), = y: y−x=1
(3.1)
x = (x1 , . . . , x N ), y = (y1 , . . . , y N ) ∈ Zd × · · · × Zd , (1) (d) (1) (d) where x j = x j , . . . , x j , y j = y j , . . . , y j ∈ Zd , j = 1, . . . , N , and d d x = max max x (i) j , x = (x 1 , . . . , x N ) ∈ R × · · · × R . j=1,...,N i=1,...,d
A similar formula defines HΛL (u) ; cf. (1.3). We again assume that values V (x; ω), x ∈ Zd , are IID with a common distribution function F. Function U (x) is assumed to be locally bounded and symmetric on Zd × · · · × Zd . (As before, value +∞ can also be incorporated.) The statement of Theorem 1 does not change: given u = (u 1 , . . . , u N ), with (1) (d) u j = (u j , . . . , u j ) ∈ Zd , and L = (L 1 , . . . , L N ) ∈ N N , define Π j ΛL (u) as
d (i) (i) (3.2) Π j ΛL (u) = × −L j + u j , u j + L j ∩ Zd , j = 1, . . . , N . i=1
Then set ΛL (u) = Π1 ΛL (u) × · · · × Π N ΛL (u).
(3.3)
Wegner Bounds for a Two-Particle Tight Binding Model
489
Theorem 3. ∀ E ∈ R, L ∈ N N , u ∈ Zd × · · · × Zd and > 0, P dist Σ HΛL (u) , E ≤ ≤ |ΛL (u)| min Π j ΛL (u) · s(2). j=1,...,N
(3.4)
The proof is completely analogous to that of Theorem 1, based on the representation N
V (x j ; ω) =
c(x, y)V (y; ω).
y∈Γ
j=1
Here Γ is as before the set of smallest cardinality among Π1 ΛL (u), . . ., Π N ΛL (u) and c(x, y) is given as a function of y ∈ Γ , for every x ∈ ΛL (u), by c(x, y) =
N
δ y,x j ≥ 1.
j=1
The statement of an analog of Theorem 2 for a general N -particle case will be a subject of a forthcoming paper. We want to conclude by noticing that Theorem 1 can be further extended when ΛL (u) is replaced by a general lattice domain (in Zd × · · · × Zd ). We decided to focus on lattice parallelepipeds because it suffices for traditional applications (Anderson localisation). Acknowledgements. The authors thank the referees of this paper for a number of remarks and suggestions leading to improvements of the paper. We also thank F. Klopp who pointed at Stollmann’s lemma as a way to treat multiparticle Wegner-type estimates. VC thanks The Isaac Newton Institute and Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, for hospitality during visits in 2003, 2004 and 2007. YS thanks the Département de Mathematique, Université de Reims Champagne–Ardenne, for hospitality during visits in 2003 and 2006, in particular, for the Visiting Professorship in the Spring of 2003. YS thanks the IHES, Bures-sur-Yvette, France, for hospitality during numerous visits in 2003–2007. YS thanks the School of Theoretical Physics, Dublin Institute for Advanced Studies, for hospitality during regular visits in 2003–2007. YS thanks the Department of Mathematics, Penn State University, for hospitality during a Visting Professorship in the Spring, 2004. YS thanks the Department of Mathematics, University of California, Davis, for hospitality during a Visiting Professorship in the Fall of 2005. YS acknowledges the support provided by the ESF Research Programme RDSES towards research trips in 2003–2006.
References 1. Chulaevsky, V.: A simple extension of Stollmann’ lemma for correlated potentials. Preprint, Université de Reims, April 2006.; http://arXiv.org/abs/0705:2873, 2007 2. Chulaevsky, V., Suhov, Y.: Anderson localisation for an interacting two-particle quantum system on Z. http://arXiv.org/abs/0705:0657, 2007 3. von Dreifus, H., Klein, A.: A new proof of Localization in the Anderson Tight Binding Model. Commun. Math. Phys. 124, 285–299 (1989) 4. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: A constructive proof of localization in Anderson tight binding model. Commun. Math. Phys. 101, 21–46 (1985) 5. Kirsch, W.: A Wegner estimate for multi-particle random Hamiltonians. http://arXiv.org/abs/0704:2664, 2007 6. Stollmann, P.: Wegner estimates and localization for continuous Anderson models with some singular distributions. Arch. Math. 75, 307–311 (2000) 7. Stollmann, P.: Caught by disorder. Basel-Boston: Birkhäuser, 2001 8. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B. Condensed Matter 44, 9–15 (1981) Communicated by B. Simon
Commun. Math. Phys. 283, 491–505 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0513-6
Communications in
Mathematical Physics
Adiabatic Elimination in Quantum Stochastic Models Luc Bouten, Andrew Silberfarb Physical Measurement and Control 266-33, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA. E-mail:
[email protected] Received: 22 August 2007 / Accepted: 7 December 2007 Published online: 22 May 2008 – © Springer-Verlag 2008
Abstract: We consider a physical system with a coupling to bosonic reservoirs via a quantum stochastic differential equation. We study the limit of this model as the coupling strength tends to infinity. We show that in this limit the solution to the quantum stochastic differential equation converges strongly to the solution of a limit quantum stochastic differential equation. In the limiting dynamics the excited states are removed and the ground states couple directly to the reservoirs. 1. Introduction It is a frequent occurence in physics to have a system that spends a very limited amount of time in its excited states. This is, for instance, the case if the system is strongly coupled to a low temperature environment (e.g. the optical field). The strong coupling ensures that excitations above the ground levels of the system quickly dissipate into its environment. It is therefore reasonable to ask for a model in which the excited states are eliminated from the description.That is, we would like to have a description that only involves the ground states of a system and its environment. The procedure for going from the full model to the reduced model is called adiabatic elimination. We study adiabatic elimination in the context of quantum stochastic models [14] which arise by taking a weak coupling limit of QED (quantum electrodynamics) models [1,5,12], and are widely applicable to systems studied in quantum optics. Specifically, quantum stochastic models are the starting point for deriving master equations, filtering equations, and input-output relations. In the quantum optics community adiabatic elimination is a common technique, used, for instance, in atomic systems [2,6,10,22] and in cavity QED models [11,23] as well as in more recent work on quantum feedback [7,9,24]. Rigorous results have been demonstrated for adiabatic elimination outside of the quantum stochastic models we consider [3,10,19]. At present, however, apart from the work [13] on the elimination of a leaky cavity (using a Dyson series expansion to prove weak convergence), no rigorous results have been obtained on adiabatic
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elimination in the context of the quantum stochastic models introduced by Hudson and Parthasarathy [14]. We start by considering a family, indexed by a parameter k, of quantum stochastic differential equations (QSDE’s). The parameter k can be interpreted as the coupling strength between the system and its environment. The environment is modelled by a collection of bosonic heat baths in the vacuum representation. We assume that the coefficients of the QSDE are all bounded and satisfy the usual conditions guaranteeing a unique unitary solution [14]. We state further assumptions on the coefficients and show that under these assumptions the solution of the QSDE converges strongly to the solution of a limiting QSDE as k tends to infinity (Theorem 1). The limiting QSDE represents the adiabatically eliminated time evolution of the system. The heart of the proof is a technique introduced by T.G. Kurtz [16] that enables the application of the Trotter-Kato Theorem [21]. This allows us to prove strong convergence of the unitaries using convergence of generators of semigroups rather than convergence of a Dyson series expansion. Convergence is first shown on the vacuum vector of the bosonic reservoirs. We then extend this result to any possible vector in the Hilbert space of the reservoirs by sandwiching the unitaries with Weyl operators and using a density argument. The remainder of this article is organized as follows. In Sect. 2 we introduce the system coupled to n bosonic reservoirs in the vacuum representation. We state assumptions on the coefficients of the QSDE and present the main convergence theorem. In Sect. 3 we discuss four applications of the theorem in the context of examples from atomic physics and cavity QED. Section 4 presents the proof of the main convergence theorem. In Section 5 we discuss our results.
2. The Main Result Let H be a Hilbert space and let n be an element of N. Let F be the symmetric Fock space over Cn ⊗ L 2 (R+ ) ∼ = L 2 (R+ ; Cn ), i.e. F =C⊕
∞
L 2 (R+ ; Cn )⊗s m .
m=1
Physically, the Hilbert space H⊗F describes a system H coupled to n bosonic reservoirs (e.g. n decay channels in the quantized electromagnetic field). For f ∈ L 2 (R+ ; Cn ), we define the exponential vector e( f ) in F by e( f ) = 1 ⊕
∞ f ⊗m √ . m! m=1
Moreover, we define the coherent vector π( f ) to be the exponential vector e( f ) normalized to unity, i.e. π( f ) = exp(− 21 f 2 )e( f ). The vacuum vector is defined to be the exponential vector Φ = e(0) = 1 ⊕ 0 ⊕ 0 . . .. The expectation with respect to the vacuum vector is denoted by φ, i.e. φ is a map from B(F) (the bounded operators on F) to C, given by φ(W ) = Φ, W Φ for all W ∈ B(F). The interaction between the system and the bosonic reservoirs is modelled by a quantum stochastic differential equation (QSDE) in the sense of Hudson and Parthasarathy
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493
[14] of the form ij j (k) (k) (k) (k)† (k) (k) Si j d At + K (k) dt Ut , dUt = Si j − δi j dΛt + L i d Ai† t − Li
(1)
(k)
where U0 = I . We consistently use the convention that repeated indices that are not within parentheses are being summed (i and j run through {1, . . . , n}). The Hilbert space adjoint is denoted by a dagger † . We have indexed the equation with a positive number k, (k) and in the following we will be interested in the behaviour of Ut as k tends to infinity. We assume that the following conditions on the coefficients of the QSDE are satisfied. (k)
(k)
Assumption 1. For each k ≥ 0, the coefficients K (k) , Si j and L i (i, j ∈ {1, . . . , n}) of the quantum stochastic differential equation (1) are bounded operators on H. Furthermore, for each k ≥ 0, the coefficients satisfy the following relations: (k)† (k) Li ,
K (k) + K (k)† = −L i
(k) (k)†
Sil S jl
= δi j I,
(k)† (k) Sl j
Sli
= δi j I.
Hudson and Parthasarathy [14] show that under Assumption 1, the quantum stochastic (k) (k)† differential equation (1) has a unique unitary solution Ut , and, the adjoint Ut satisfies the adjoint of Eq. (1). Assumption 2. There exist bounded operators Y, A, B, Fi , G i and Wi j (independent of k) on H such that K (k) = k 2 Y + k A + B,
(k)
Li
= k Fi + G i ,
(k)
Si j = Wi j ,
for all i, j ∈ {1, . . . , n}. We define P0 as the orthogonal projection onto Ker(Y ). Let P1 = I − P0 be its complement in H. We use the following notation H0 = P0 H and H1 = P1 H. Physically, one should think of H0 as the ground states and of H1 as the excited states of the system. Assumption 3. There exists a bounded operator Y1−1 on H such that P1 Y1−1 = Y1−1 P1 and Y Y1−1 P1 Z P0 = P1 Z P0 ,
P0 X P1 Y1−1 Y = P0 X P1 ,
(2)
where Z = A, Fi† Wi j , ( j ∈ {1, . . . , n}) and X = A, B, Fi , G i , Wi j , G i† Wi j , Fi Y1−1 F j , Fi Y1−1 A, Fi Y1−1 Fl† Wl j , AY1−1 A, AY1−1 Fi , AY1−1 Fl† Wl j , (i, j ∈ {1, . . . , n}). Moreover, for all i, j ∈ {1, . . . , n} the following products are zero P0 Y P1 = P0 A P0 = Fi P0 = P0 (δil + Fi Y1−1 Fl† )Wl j P1 = 0. Note that the existence of Y1−1 satisfying the assumptions in Eq. (2) is immediate if Y maps H1 surjectively onto H1 and is therefore invertible on H1 . Definition 1. Suppose Assumption 2 and 3 hold. We define for all i, j ∈ {1, . . . , n} the following bounded operators on H: L i = G i − Fi Y1−1 A P0 , K = P0 B − AY1−1 A P0 , Si j = δil + Fi Y1−1 Fl† Wl j P0 .
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Assumption 4. For all i, j ∈ {1, . . . , n} the following products are zero: P1 L i = P1 Si j = 0. Lemma 1. Suppose that Assumption 1, 2, 3 and 4 hold. The operators in Definition 1 satisfy K + K † = −L i† L i , Sil S †jl = δi j P0 , Sli† Sl j = δi j P0 . (k)† (k)
Proof. By Assumptions 1 and 2 we have K (k) +K (k) = −L i L i , K (k) = k 2 Y +k A+B (k) and L i = k Fi + G i for all k ≥ 0. Moreover, Fi P0 = 0, by Assumption 3. Combining these results leads to − Fi† Fi = Y + Y † , − P1 Fi† G i P0 = P1 (A + A† )P0 , −
P0 G i† G i P0
(3)
= P0 (B + B † )P0 .
We then use Y Y1−1 A P0 = A P0 from Assumption 3 and L i from Definition 1 to derive −L i† L i = −P0 (G i† − A† Y1−1† Fi† )(G i − Fi Y1−1 A)P0 = P0 (B + B † )P0 − P0 A† Y1−1† (A + A† )P0 − P0 (A + A† )Y1−1 A P0 + P0 A† (Y1−1† + Y1−1 )A P0 = P0 (B + B † )P0 − P0 AY1−1 A P0 − P0 A† Y1−1† A† P0 = P0 (K + K † )P0 . By Definition 1
Si j = δil + Fi Y1−1 Fl† Wl j P0 .
Combining this with −Fi† Fi = Y + Y † from above, † δml + Fm Y1−1† Fl† δln + Fl Y1−1 Fn† Wn j P0 Sli† Sl j = P0 Wmi = P0 Wli† Wl j P0 = P0 δi j . Then use P0 δil + Fi Y1−1 Fl† Wl j P1 = 0 from Assumption 3 and P1 Si j P0 = 0 from Asumption 4 to derive † δm j + Fm Y1−1† F j† P0 Sil S †jl = P0 δin + Fi Y1−1 Fn† Wnl Wml = P0 δin + Fi Y1−1 Fn† δn j + Fn Y1−1† F j† P0 = δi j P0 . The operators given by Definition 1 are the coefficients of a QSDE on the Hilbert space H ⊗ F, ij j † dUt = Si j − δi j P0 dΛt + L i d Ai† − L S d A + K dt Ut , U0 = I. (4) i j t t i Lemma 1 implies that under Assumptions 1, 2, 3 and 4, Eq. (4) has a unique unitary solution on H [14], and, the adjoint Ut† satisfies the adjoint of Eq. (4). Moreover, Ut maps H0 to H0 . Note that Ut P1 = P1 .
Adiabatic Elimination in Quantum Stochastic Models
495 (k)
Theorem 1. Suppose Assumption 1, 2, 3 and 4 hold. Let Ut be the unique unitary solution to Eq. (1). Let Ut be the unique unitary solution to Eq. (4) where the coefficients (k) are given by Definition 1. Then Ut P0 converges strongly to Ut P0 , i.e. lim Ut(k) ψ = Ut ψ,
k→∞
∀ψ ∈ H0 ⊗ F.
We prove Theorem 1 in Sect. 4. 3. Examples We use the following definitions in the first two examples below. Let (|e , |g ) be an orthogonal basis of C2 . Define the raising and lowering operators in this basis as 0 1 0 0 , σ− = . σ+ = 0 0 1 0 Define the Pauli operators σx = σ+ + σ− , and define the projectors
σ y = −iσ+ + iσ− , Pe = σ+ σ− ,
σz = σ+ σ− − σ− σ+ ,
Pg = σ− σ+ .
Example 1 (A two-level atom driven by a laser). The Hilbert space for a two-level atom is H = C2 , with |e the excited state, and |g the ground state. Define the detuning ∆ ∈ R, the decay rate γ ≥ 0 and the complex amplitude α ∈ C. The QSDE for this system in the electric dipole and rotating wave approximations is [2] √ √ (k) dUt = k γ σ− d A†t − k γ σ+ d At − ikασ+ dt − ik ασ ¯ − dt
k2γ (k) (k) σ+ σ− dt − ik 2 ∆σ+ σ− dt Ut , U0 = I. − 2 Define the operators Y, A, B, F, G, W as Y = (−i∆ − γ /2)σ+ σ− , √ F = γ σ− , G = 0,
A = −iασ+ − iασ− , W = I.
B = 0,
This satisfies Assumptions 1 and 2, and P0 = Pg . We take Y1−1 = −(i∆ + γ /2)−1 σ+ σ− , and Assumption 3 holds by inspection. Definition 1 leads to the following coefficients: √ α γ |α|2 i∆ − γ /2 K =− Pg , Pg , Pg . L = −i S= i∆ + γ /2 i∆ + γ /2 i∆ + γ /2 (k)
Note that P1 L = P1 S = 0 satisfying Assumption 4. Theorem 1 then shows that Ut P0 converges strongly to Ut P0 , given by Pg √ √ dUt = −γ dΛt − iα γ d A†t + i α¯ γ d At − |α|2 dt Ut , U0 = I. i∆ + γ /2
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In the case that γ = 0 the two level atom decouples from the field. In this case we may explicitly calculate the ground state evolution as P0 e−i
kασ+ +k ασ ¯ − +k 2 ∆σ+ σ− t
e−ik ∆t/2 (Ω cos(kΩt) + ik∆ sin(kΩt)) , Ω 2
P0 =
2 with Ω = ∆2 k 2 + 4|α|2 . For k → ∞ this expression limits to ei|α| t/∆ which is the 2 solution to our eliminated differential equation dUt = i |α| ∆ Ut dt, U0 = I . Example 2 (Alkali atom). Now consider a system with Hilbert space H = C2 ⊗ C2 . Physically, the system represents an alkali atom with no nuclear spin coupled to a driving field on the S1/2 → P1/2 transition. We have four orthogonal states in this system corresponding to the atomic excited and ground states with angular momentum m z = ± 21 along the z-axis. We define a detuning ∆ ∈ R, a decay rate γ ≥ 0 and a magnetic field Bi ∈ R, i ∈ {x, y, z}. The system may emit into n = 3 independent dipole modes, Ait , where the modes are labelled by i ∈ {x, y, z}. The QSDE for this system in the dipole and rotating wave approximations is [2], 3k 2 γ √ √ (k) i dUt = k γ σ− ⊗ σi d Ai† Pe ⊗ I dt t − k γ σ+ ⊗ σi d At − 2 −i k 2 ∆Pe ⊗ I + I ⊗ Bi σi dt Ut(k) , U0(k) = I. Defining the operators Y, A, B, Fi , G i , Wi j as 3γ Pe ⊗ I, Y = −i∆ − A = 0, B = −i I ⊗ Bi σi 2 √ Fi = γ σ− ⊗ σi , G i = 0, Wi j = δi j , −1 satisfies Assumptions 1 and 2, and P0 = Pg ⊗ I . We take Y1−1 = −(i∆ + 3γ 2 ) Pe ⊗ I , and Assumption 3 holds by inspection. Define the eliminated coefficients as
γ K = −i Pg ⊗ Bi σi , L i = 0, Si j = Pg ⊗ δi j I − σi σ j . i∆ + 3γ 2 (k)
This satisfies Assumption 4. Theorem 1 then shows that Ut P0 converges strongly to Ut P0 , given by γ ij dUt = Pg ⊗ −i Bi σi dt − σi σ j dΛt Ut , U0 = I. i∆ + 3γ 2 In the following two examples we make use of a truncated harmonic oscillator. We have truncated the oscillator to satisfy the boundedness condition of Assumption 1 in the following two examples. Let N be an element in N such that N ≥ 2. The Hilbert space of the oscillator is C N . We choose an orthonormal basis (|0 , . . . , |N − 1 ) in C N . The annihilation operator b : C N → C N is given by √ b|n = n|n − 1 , n ∈ {1, . . . , N − 1}, and b|0 = 0. The creation operator is defined to be the adjoint b† .
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Example 3 (Gough and Van Handel [13]). Let h be a Hilbert space. We define H = h ⊗ C N . The Hilbert space h describes a system inside a cavity. We model the cavity as a truncated oscillator C N . Let E i j , i, j ∈ {0, 1} be bounded operators on h such that E i†j = E ji . Consider the following QSDE: (k)
dUt
=
γ k2 † √ √ (k) b bdt − i H (k) dt Ut , γ kbd A†t − γ kb† d At − 2
(k)
U0
= I. (5)
Here γ is a real parameter and H (k) is given by H (k) = k 2 E 11 b† b + k E 10 b† + k E 01 b + E 00 . Define operators Y, A, B, F, G, W as γ † b b, A = −i E 10 b† + E 01 b , Y = −i E 11 − 2 √ F = γ b, G = 0, W = I.
B = −i E 00 ,
This satisfies Assumptions 1 and 2 and P0 = Ih ⊗ |0 0|. Let N1−1 : H1 → H1 be the −1 −1 inverse of the restriction of b† b to H1 . Taking Y1−1 = − i E 11 + γ2 N1 P1 satisfies Assumption 3. Definition 1 leads to the following coefficients: K = −i E 00 P0 − E 01 L=
1 i E 11 +
√ −i γ E 10 P0 , i E 11 + γ2
γ 2
S=
E 10 P0 , (6)
i E 11 − γ2 P0 . i E 11 + γ2 (k)
These coefficients satisfy Assumption 4. Theorem 1 then shows that Ut P0 converges strongly to Ut P0 , where Ut is given by dUt = (S − P0 )dΛt + Ld A†t − L † Sd At + K dt Ut , U0 = I. Remark 1. Note that we consider a truncated oscillator, where [13] treats the full oscillator, and that we prove our result strongly, whereas [13] proves a weak limit. The convergence of the Heisenberg dynamics follows immediately from our strong result. Apart from these points, Example 3 reproduces the result in [13]. Care must be taken when directly comparing the limit equations, since the results in [13] are presented in (k) the interaction picture with respect to the cavity. Under our assumptions, we define Vt as the solution to
γ k2 † √ √ (k) (k) (k) † † b bdt Vt , γ kbd At − γ kb d At − V0 = I. d Vt = 2 The unitary in the interaction picture is then given by U˜ t(k) = Vt(k)† Ut(k) , where Ut(k) is given by Eq. (5). Note that due to Theorem 1, Vt(k) P0 converges strongly to Vt P0 , where Vt is given by d Vt = −2P0 dΛt Vt , V0 = I. This accounts for the sign difference between the coefficients in the equation for U˜ t presented in [13], and the coefficients in the equation for Ut given by Eq. (6).
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Example 4 (Duan and Kimble [8]). We again consider a system inside a cavity, described by the Hilbert space H = h ⊗ C N . The system inside the cavity is a three level atom, i.e. h = C3 . Let (|e , |+ , |− ) be an orthogonal basis in h. In this basis we define ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 1 (+) (−) σ+ = ⎝0 0 0⎠ , σ+ = ⎝0 0 0⎠ . 0 0 0 0 0 0 (±)
(±)†
(±) (±)
Moreover define σ− = σ+ and P± = σ− σ+ . The QSDE for a lambda system with one leg (+ ↔ e) resonantly coupled to the cavity, under the rotating wave approximation in the rotating frame, is, γ k2 † √ √ (k) dUt = γ kbd A†t − γ kb† d At − b bdt 2 + gk 2 σ+(+) b − σ−(+) b† dt + k σ+(−) α − σ−(−) α¯ dt Ut(k) , U0(k) = I. Here γ is a positive real parameter and α is a complex parameter. Note that we extend the model from [8] to allow driving on the uncoupled leg (− ↔ e) of the transition. Define operators Y, A, B, F, G, W as γ Y = − b† b + g σ+(+) b − σ−(+) b† , A = σ+(−) α − σ−(−) α¯ , B = 0, 2 √ F = γ b, G = 0, W = I. This satisfies Assumptions 1 and 2 and P0 = (|+ +| + |− −|) ⊗ |0 0|. We define the following subspaces of H: n ∈ {1, . . . , N − 1}, Hn = span {|+ ⊗ |n , |− ⊗ |n , |e ⊗ |n − 1 } , H N = span {|e ⊗ |N − 1 } . N Hn and that the subspaces Hn (n ∈ {1, . . . , N }) are all invariant Note that H1 = n=1 under the action of Y . On the subspaces Hn , n ∈ {1, . . . , N − 1}, Y is given by √ ⎞ ⎛ γn − 2 0 −g n γn 0 ⎠, Y | Hn = ⎝ 0 − 2 √ γ (n−1) g n 0 − 2 with respect to the basis (|+ ⊗|n , |− ⊗|n , |e ⊗|n−1 ). Moreover, Y | HN = − γ (N2−1) . The inverse is readily computed to be ⎛ γ (n−1) √ ⎞ 0 −g n 2 1⎝ 2d Y |−1 n ∈ {1, . . . , N − 1}, 0 0 ⎠, Hn = − √ γn γn d g n 0 2 where d =
γ 2 n(n−1) 4
−1 2 + g 2 n. Moreover, Y |−1 = H N = − γ (N −1) . We now define Y1
N Y |−1 P . This satisfies Assumption 3. Definition 1 leads to the following coefficients ⊕n=1 Hn 1
K =−
|α|2 γ P− ⊗|0 0|, 2g 2
L=−
γ α (+) (−) σ σ+ ⊗|0 0|, g −
S = P0 −2P− ⊗|0 0|.
Adiabatic Elimination in Quantum Stochastic Models
499 (k)
These operators satisfy Assumption 4. Theorem 1 then shows that Ut P0 converges strongly to Ut P0 , where Ut is given by dUt = (S − P0 )dΛt + Ld A†t − L † Sd At + K dt Ut , U0 = I. Note that the ground state system is a two-level system on which S acts as σz . 4. Proof of Theorem 1 Definition 2. Suppose Assumptions 1, 2, 3 and 4 hold. Let B(H) and B(H0 ) be the Banach spaces of all bounded operators on H and H0 , respectively. We define for all t ≥ 0 and k ≥ 0, (k) (k) Tt (X ) = id ⊗ φ Ut† XUt , X ∈ B(H), Tt (X ) = id ⊗ φ Ut† XUt , X ∈ B(H0 ), (k)
where Ut
and Ut are given by Eqs. (1) and (4), respectively. (k)
Note that Tt
(k)
is intentionally skew with respect to Ut and Ut . (k)
Lemma 2. For each k > 0, the families of bounded linear maps Tt (t ≥ 0) and Tt (t ≥ 0) given by Definition 2 are norm continuous one-parameter contraction semigroups with generators (k)
L (k) (X ) = K † X + X K (k) + L i† X L i , L i† X L i ,
X ∈ B(H),
L (X ) = K X + X K + X ∈ B(H0 ), (k) respectively. That is Tt = exp tL (k) and Tt = exp(tL ) for all t ≥ 0. †
(7)
(k)
Proof. We only prove the lemma for Tt . The proof for Tt can be obtained in an analogous way. Since the conditional expectation id ⊗ φ is norm contractive and Ut and Ut(k) are unitary, we have † (k) (k) † (k) Tt (X ) ≤ Ut XUt ≤ Ut Ut X = X , (k)
for all X ∈ B(H). This proves that Tt is a contraction for all t ≥ 0. An application of the quantum Itô rule [14], together with the fact that vacuum expectations of stochastic integrals vanish, shows that (k) (k) dTt (X ) = id ⊗ φ d Ut† XUt (k) (k) (k) = id ⊗ φ Ut† K † X + X K (k) + L i† X L i Ut dt = Tt L (k) (X ) dt, (k) for all X ∈ B(H). That is, Tt = exp tL (k) is a one-parameter semigroup with generator L (k) . Furthermore, L (k) is bounded (k) † (k) L (X ) ≤ K † + K (k) + L i L i X , (k)
which proves that Tt
is norm continuous.
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The proof of Theorem 1 relies heavily on the Trotter-Kato theorem [15,21] in combination with an argument due to Kurtz [16]. We have taken the formulation of the Trotter-Kato theorem from [4, Thm 3.17, p. 80], see also [17, Chap. 1, Sect. 6]. The formulation is more general than needed for the proof of Theorem 1. Theorem 2 (Trotter-Kato Theorem). Let B be a Banach space and let B0 be a clo(k) sed subspace of B. For each k ≥ 0, let Tt be a strongly continuous one-parameter contraction semigroup on B with generator L (k) . Moreover, let Tt be a strongly continuous one-parameter contraction semigroup on B0 with generator L . Let D be a core for L . The following conditions are equivalent: 1. For all X ∈ D there exist X (k) ∈ Dom L (k) such that lim X (k) = X,
k→∞
lim L (k) X (k) = L (X ).
k→∞
2. For all 0 ≤ s < ∞ and all X ∈ B0 , (k) lim sup Tt (X ) − Tt (X ) = 0. k→∞
0≤t≤s
(k)
Proposition 1. Let Tt and Tt be the one-parameter semigroups on B(H) and B(H0 ) defined in Definition 2, respectively. We have (k) lim sup Tt (X ) − Tt (X ) = 0, k→∞
0≤t≤s
for all X ∈ B(H0 ) and 0 ≤ s < ∞. Proof. The proof follows the line of the proof of [16, Theorem 2.2]. Lemma 2 shows that Tt(k) = exp tL (k) and Tt = exp(tL ) are norm continuous, and therefore also strongly continuous semigroups with generators given by Eq. (7). This means we satisfy the assumptions of the Trotter-Kato Theorem (Thm. 2) with D = B(H0 ) and Dom L (k) = B(H). We can write L (k) (X ) = L0 (X ) + kL1 (X ) + k 2 L2 (X ), X ∈ B(H), where (recall Assumption 2) L0 (X ) = K † X + X B + L i† X G i ,
L1 (X ) = X A + L i† X Fi ,
L2 (X ) = X Y.
Let X be an element in B(H0 ) and let X 1 and X 2 be elements in B(H). We define X (k) = X + k1 X 1 + k12 X 2 . Collecting terms with equal powers in k, we find L (k) X (k) = (L0 (X ) + L1 (X 1 ) + L2 (X 2 )) + k (L1 (X ) + L2 (X 1 )) + k 2 (L2 (X )) 1 1 + (L0 (X 1 ) + L1 (X 2 )) + 2 (L0 (X 2 )) . k k
Adiabatic Elimination in Quantum Stochastic Models
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Note that L2 (X ) = 0 as X ∈ B(H0 ) and P0 Y = 0. Using the existence of Y1−1 , we set X 1 = −L1 (X )Y1−1 P1 , X 2 = − (L0 (X ) + L1 (X 1 )) Y1−1 P1 . Using the properties of Y1−1 in Assumption 3, we obtain 1 1 lim L (k) X (k) = lim L (X ) + (L0 (X 1 ) + L1 (X 2 )) + 2 L0 (X 2 ) k→∞ k→∞ k k = L (X ) The proposition then follows from the Trotter-Kato Theorem. Note that for all v ∈ H0 , we can write Ut v ⊗ Φ = P0 Ut v ⊗ Φ. This leads to 2 2 (k) (k) (Ut − Ut )v ⊗ Φ = (Ut − P0 Ut )v ⊗ Φ (k) (k) = v, 2I − Tt (P0 ) − Tt (P0 )† v . Here we have used that id ⊗ φ is a positive map, i.e. it commutes with the adjoint. Using Proposition 1 and noting that L (P0 ) = 0 by Lemmas 2 and 1, we see that Theorem 1 holds for all vectors in H0 ⊗ F of the form ψ = v ⊗ Φ. We now need to extend this to all ψ ∈ H0 ⊗ F. Let f be an element in L 2 (R+ ; Cn ). Denote by f t the function f truncated at time t, i.e. f t (s) = f (s) if s ≤ t and f t (s) = 0 otherwise. Define the Weyl operator W ( f t ) as the unique solution to the following QSDE
1 i dW ( f t ) = f (t)i d Ai† − f (t) d A − f (t) f (t) dt W ( f t ), W ( f 0 ) = I. (8) i t i i t 2 Note that W ( f t ) is a unitary operator from F to F. Moreover, it is not hard to see that π( f t ) = W ( f t )Φ, see e.g. [20]. Often we will identify a constant α ∈ Cn with the constant function on R+ taking the value α (truncated at some large T ≥ 0 so that it is an element of L 2 (R+ ; Cn )). Definition 3. Let f be an element in L 2 (R+ ; Cn ). Suppose that Assumptions 1, 2, 3 and (k) 4 hold and let Ut and Ut be given by Eqs. (1) and (4), respectively. Define (k f )
(k)
(f)
= W ( f t )† Ut W ( f t ), Ut = W ( f t )† Ut W ( f t ), (k f ) ( f )† (k f ) , X ∈ B(H), XUt Tt (X ) = id ⊗ φ Ut (k f ) ( f )† (f) Tt (X ) = id ⊗ φ Ut , X ∈ B(H0 ). XUt
Ut
Definition 4. Let α be an element in Cn and let i be an element in {1, . . . , n}. Let (k) (k) K (k) , K , L i , L i , Si j and Si j be the coefficients of Eqs. (1) and (4). Define operators K (kα) , K (α) , L i(kα) and L i(α) by
K (α) = K + α¯ i (Si j − P0 δi j )α j + α¯ i L i − α j L i† Si j , K
(kα)
=K
(k)
(k) + α¯ i (Si j
(k) − δi j )α j + α¯ i L i
(α)
Li
(k)† − α j L i Si j ,
= L i + α j Si j , (kα)
Li
(k)
= Li
(k)
+ α j Si j .
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Note that with the coefficients given by Definition 4, applying the quantum Itô rule to Ut(kα) and Ut(α) , defined in Definition 3, gives ij j (α) (α) (α)† (α) dUt = Si j − δi j P0 dΛt + L i d Ai† Si j d At + K (α) dt Ut , t − Li (9) ij j (kα) (k) (kα) (kα)† (k) (kα) dUt = Si j − δi j dΛt + L i d Ai† Si j d At + K (kα) dt Ut , t − Li (α)
with U0
(kα)
= U0
= I.
Definition 5. Suppose that Assumptions 1, 2, 3 and 4 hold. Let α be an element in Cn (α) and let i be an element in {1, . . . , n}. Define operators A(α) , B (α) and G i by A(α) = A + Fi α¯ i − α j Fi† Wi j , B (α) = B + α¯ i (Wi j − δi j )α j + G i α¯ i − α j G i† Wi j , G i(α) = G i + α j Wi j . Lemma 3. Suppose Assumptions 1, 2, 3 and 4 hold. Let A, B, Y, Fi , G i , Wi j , K , L i and Si j for i, j ∈ {1, . . . , n} be the various operators occurring in Assumption 1, 2, 3 and 4. Let K (α) and L i(α) for i ∈ {1, . . . , n} be given by Definition 4 and let A(α) , B (α) and G i(α) for i ∈ {1, . . . , n} be given by Definition 5. Then (α)
Li K
(α)
(α)
= (G i
− Fi Y1−1 A(α) )P0 ,
= P0 B
(α)
−
A(α) Y1−1 A(α)
(10a) P0 ,
(α)
(10b) (α)
i.e. Definition 1 holds with A = A(α) , B = B (α) , G i = G i L i = L i and K = K (α) . Moreover, Assumptions 1, 2, 3 and 4 hold for the altered coefficients with P0 and Y1−1 unchanged. (α)
Proof. To show that Definition 1 holds for the altered coefficients, substitute G i and (α) A(α) from Definition 5, and L i from Definition 4 into Eq. (10a). This gives L i + α j Si j = L i + α j Wi j + α j Fi Y1−1 Fl† Wl j , P0 , which holds if we substitute Si j = Wi j + Fi Y1−1 Fl† Wl j P0 from Definition 1. Furthermore, substituting A(α) and B (α) from Definition 5, and K (α) from Definition 4 into Eq. (10b) gives α¯ i Si j α j + α¯ i L i − α j L i† Si j = P0 α¯ i Wi j α j P0 + P0 G i P0 α¯ i − α j P0 G i† Wi j P0 − P0 (Fi α¯ i + A)Y1−1 (A − α j Fl† Wl j )P0 + P0 AY1−1 A P0 . This holds if we can show that
Si j = P0 Wi j + Fi Y1−1 Fl† Wl j P0 , L i = P0 G i − Fi Y1−1 A P0 , L i† Si j = P0 G i† Wi j − AY1−1 Fi† Wi j P0 .
(11a) (11b) (11c)
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Equations (11a) and (11b) are satisfied by Assumption 4 as P1 L i = P1 Si j = 0. Note that Eq. (11c) holds if we can show L i† δil + Fi Y1−1 Fl† Wl j P0 = P0 G l† Wl j P0 − P0 AY1−1 Fl† Wl j P0 . Substituting L i from Definition 1, this becomes − P0 A† Y1−1† Fl† Wl j P0 + P0 G i† Fi Y1−1 Fl† Wl j P0 − P0 A† Y1−1† Fi† Fi Y1−1 Fl† Wl j P0 + P0 AY1−1 Fl† Wl j P0 = 0. Now recall that P0 (A + A† )P1 = −P0 G i† Fi P1 , and Y + Y † = −Fi† Fi (see Eq. (3)) by Assumptions 1, 2 and 3. Moreover, Y Y1−1 P1 Fl† Wl j P0 = P1 Fl† Wl j P0 by Assumption 3 which shows that Eq. (11c) is satisfied. We now show that Assumptions 1, 2, 3 and 4 hold for the altered coefficients, with P0 and Y1−1 unchanged. Assumption 1 holds for the altered coefficients since, by Definition (kα) (k) 3, we have Ut = W ( f t )† Ut W ( f t ) which is clearly unitary. By Assumption 2 for the original coefficients and Definition 4 and 5, we see that Assumption 2 holds for the altered coefficients. Assumption 3 on the altered coefficients is seen to hold by direct substitution of the coefficients in Definition 4 and 5, followed by application of Assumption 3 for the (α) original system. Assumption 4 holds if P1 L i = P1 L i + αi P1 Si j = 0, which follows from Assumption 4 on the original system. (kα)
Lemma 3 shows that Proposition 1 holds with Tt respectively.
(α)
and Tt
(k)
replacing Tt
and Tt ,
Corollary 1. Suppose that Assumption 1, 2, 3 and 4 hold. Let α be an element of Cn . We have (kα) (α) lim sup Tt (X ) − Tt (X ) = 0, k→∞
0≤t≤s
for all X ∈ B(H0 ) and 0 ≤ s < ∞. Proof of Theorem 1. Let t ≥ 0. Let f be a step function in L 2 ([0, t]; Cn ), i.e. there exists an m ∈ N and 0 = t0 < t1 < · · · < tm = t and α1 , . . . , αm ∈ Cn such that s ∈ [ti−1 , ti ) =⇒ f (s) = αi ,
∀i ∈ {1, . . . , m}.
The cocycle property of solutions to QSDE’s and the exponential property of the symmetric Fock space lead to (k f )
Tt
(kαm )
(X ) = Tt1
(f) Tt (X )
=
(kα )
1 . . . Tt−tm−1 (X ),
(α ) (α ) Tt1 m . . . Tt−t1m−1 (X ),
X ∈ B(H), X ∈ B(H0 ).
It is easy to see that Corollary 1 also holds for the difference of a finite product of maps (kαi ) (α ) Tti −ti−1 and a finite product of maps Tti −ti i−1 . This leads to (k f ) (f) lim Tt (X ) − Tt (X ) k→∞ (kα1 ) (α1 ) m) m) = lim Tt(kα X ∈ B(H0 ). . . . Tt−t (X ) − Tt(α . . . Tt−t (X ) = 0, 1 1 m−1 m−1 k→∞
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This immediately yields for all step functions f ∈ L 2 ([0, t]; Cn ) and v ∈ H0 , (k)
lim Ut v ⊗ π( f ) = Ut v ⊗ π( f ).
(12)
k→∞
Note that the step functions are dense in L 2 ([0, t]; Cn ). This means that Eq. (12) holds for all f ∈ L 2 ([0, t]; Cn ). Now note that for all f ∈ L 2 (R+ ; Cn ) and t ≤ s ≤ ∞, we have (e.g. [20]) (k)
(k f t )
W ( f s )† U t W ( f s ) = U t
,
( ft )
W ( f s )† U t W ( f s ) = U t
.
This means that the result in Eq. (12) is true for all f ∈ L 2 (R+ ; Cn ). We now have (k)
lim Ut ψ = Ut ψ,
k→∞
for all ψ in D = span{v ⊗ π( f ); v ∈ H0 , f ∈ L 2 (R+ ; Cn )}. Theorem 1 then follows from the fact that D is dense in H0 ⊗ F (e.g. [20]). 5. Discussion In this article we have studied adiabatic elimination in the context of the quantum stochastic models introduced by Hudson and Parthasarathy. We have shown strong convergence of a quantum stochastic differential equation to its adiabatically eliminated counterpart, under four assumptions. Physically, Assumption 1 enforces the unitarity of the initial QSDE model. Assumption 2 ensures an appropriate scaling in the coupling parameter k such that we can distinguish excited and ground states in our system. Assumptions 3 and 4 ensure the existence of a limit dynamics independent of k. Note that Assumption 4 specifically forbids any quantum jumps which terminate in an excited state, the presence of which would preclude the construction of a valid limit dynamics. (k) Although a Dyson series expansion for Ut (e.g. in terms of Maassen kernels [18]) would provide a lot of intuition for the results we have obtained (see [13] and [4, Chap. 5, Sect. 4]), we have chosen a proof along the lines of semigroups and their generators. An infinitesimal treatment has the advantage that it can exploit the existence of results such as the quantum Itô rule [14], the Trotter-Kato Theorem [15,21] and the technique due to Kurtz [16]. Acknowledgement. We thank Mike Armen, Ramon van Handel and Hideo Mabuchi for stimulating discussion. We especially thank Ramon van Handel for pointing out mistakes in an earlier version of this work. L.B. is supported by the ARO under Grant No. W911NF-06-1-0378. A.S. acknowledges support by the ONR under Grant No. N00014-05-1-0420.
References 1. Accardi, L., Frigerio, A., Lu, Y.G.: The weak coupling limit as a quantum functional central limit. Commun. Math. Phys. 131, 537–570 (1990) 2. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Atom-Photon Interactions: Basic Processes and Applications. New York: John Wiley & Sons, 1992 3. Davies, E.B.: Particle-boson interactions and the weak coupling limit. J. Math. Phys. 20, 345–351 (1979) 4. Davies, E.B.: One-parameter semigroups. London: Academic Press Inc, 1980 5. Derezinski, J., De Roeck, W.: Extended weak coupling limit for Pauli-Fierz operators. Commun. Math. Phys. 279(1), 1–30 (2008)
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6. Deutsch, I.H., Jessen, P.S.: Optical lattices. Advances in Atomic, Molecular and Optical Physics 37, 95 (1996) 7. Doherty, A.C., Parkins, A.S., Tan, S.M., Walls, D.F.: Motional states of atoms in cavity qed. Phys. Rev. A 47, 4804–4817 (1998) 8. Duan, L.-M., Kimble, H.J.: Scalable photonic quantum computation through cavity-assisted interaction. Phys. Rev. Lett. 92, 127902 (2004) 9. Dunningham, J.A., Wiseman, H.M., Walls, D.F.: Manipulating the motion of a single atom in a standing wave via feedback. Phys. Rev. A 55, 1398–1411 (1997) 10. Gardiner, C.W.: Adiabatic elimination in stochastic systems. i. formulation of methods and application to few variable systems. Phys. Rev. A 29, 2814–2822 (1984) 11. Gardiner, C.W., Zoller, P.: Quantum Noise. Berlin: Springer, 2000 12. Gough, J.: Quantum flows as Markovian limit of emission, absorption and scattering interactions. Commun. Math. Phys. 254, 489–512 (2005) 13. Gough, J., van Handel, R.: Singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode. J. Stat. Phys. 127, 575 (2007) 14. Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984) 15. Kato, T.: Remarks on pseudo-resolvents and infinitesimal generators of semigroups. Proc. Japan. Acad. 35, 467–468 (1959) 16. Kurtz, T.G.: A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12, 55–67 (1973) 17. Kurtz, T.G., Ethier, S.N.: Markov Processes: Characterization and Convergence. New York: John Wiley & Sons, Inc., 1986 18. Maassen, H.: Quantum Markov processes on Fock space described by integral kernels. In: L. Accardi, W. von Waldenfels, (eds.), QP and Applications II, Volume 1136 of Lecture Notes in Mathematics, Berlin: Springer, 1985, pp. 361–374 19. Papanicolaou, G.C.: Some probabilistic problems and methods in singular perturbations. Rocky Mnt. J. Math 6, 653–674 (1976) 20. Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Basel: Birkhäuser, 1992 21. Trotter, H.: Approximations of semigroups of operators. Pacific J. Math. 8, 887–919 (1958) 22. Walls, D.F., Milburn, G.J.: Quantum Optics. Berlin Heidelberg: Springer Verlag, 1994 23. Wiseman, H.M., Milburn, G.J.: Quantum theory of field-quadrature measurements. Phys. Rev. A 47, 642–662 (1993) 24. Wong, K.S., Collett, M.J., Walls, D.: Atomic juggling using feedback. Opt. Commun. 137, 269–275 (1997) Communicated by A. Kupiainen
Commun. Math. Phys. 283, 507–521 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0519-0
Communications in
Mathematical Physics
Intersection Theory from Duality and Replica E. Brézin1 , S. Hikami2 1 Laboratoire de Physique Théorique , Ecole Normale Supérieure, 24 rue Lhomond, 75231,
Paris Cedex 05, France. E-mail:
[email protected]
2 Department of Basic Sciences, University of Tokyo, Meguro-ku, Komaba,
Tokyo 153, Japan. E-mail:
[email protected] Received: 27 August 2007 / Accepted: 15 January 2008 Published online: 29 May 2008 – © Springer-Verlag 2008
Abstract: Kontsevich’s work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In this article we show that a duality between k-point functions on N × N matrices and N-point functions of k × k matrices, plus the replica method, familiar in the theory of disordered systems, allows one to recover Kontsevich’s results on the intersection numbers, and to generalize them to other models. This provides an alternative and simple way to compute intersection numbers with one marked point, and leads also to some new results. 1. Introduction After Witten’s celebrated conjectures [1,2] on the relation between intersection numbers on moduli spaces of curves and the KdV hierarchy, and Kontsevich’s proof [3], the literature on the subject, both from the point of view of mathematics or from its string theory relationship, has become considerable. We want to add here a new method, duality plus replica, which allows one to recover easily some results which are difficult to obtain by fancier methods, and provides some new results as well. In a previous article [4] we have used explicit integral representations for the correlation functions [5–7] for a Gaussian unitary ensemble (GUE) of random matrices M in the presence of an external matrix source. The probabililty distribution for N × N Hermitian matrices is 1 − N trM 2 −N trM A PA (M) = e 2 . (1) ZA From this representation we have obtained the correlation functions of the ‘vertices’ V (k1 , . . . , kn ) =
1 trM k1 trM k2 · · · trM kn . Nn
(2)
Unité Mixte de Recherche 8549 du Centre National de la Recherche Scientifique et de l’École Normale Supérieure
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E. Brézin, S. Hikami
As is well-known from Wick’s theorem (when the source A vanishes) such correlation functions are just the numbers of pairwise gluing of the legs of the vertex operators. The dual cells of these vertices are polygons, whose edges are pairwise glued. This generates orientable surfaces, discretized Riemann surfaces whose genus is related to the expansion in power of 1/N 2 . Okounkov and Pandharipande [8–10] have shown that the intersection numbers, computed by Kontsevich [3], may be obtained by taking a simultaneous large N and large ki limit. In our previous work we have used the exact integral representation valid for finite N of those vertex correlation functions, and obtained explicitly the scaling region for large ki and large N by a simple saddle-point. This led to a practical way to compute intersection numbers from a pure Gaussian model, much simpler than Kontsevich’s Airy matrix model. In this article we want to show that duality and replica may be used also to recover easily earlier results, to establish some new ones and give support to Witten’s conjecture. In order to make this article self-contained we have added appendices in which we rederive some of the steps leading to the representation that we are using: (i) explicit formulae, valid for arbitrary N and arbitrary source matrix A for the average U (s1 , . . . , sk ) = Tres1 M · · · Tresk M
(3)
which rely on the Itzykson-Zuber formula [11,12] (Appendix A); (ii) a duality representation for the average of characteristic determinants det(λ1 − M) · · · det(λk − M) in terms of another GUE integral, but in which the random matrices are k × k , whereas the initial problem involved N × N matrices (Appendix B) [13]. (This duality seems to be a simple reflection of the open string/closed string duality [14].) With the help of these two kinds of results we may proceed to the replica approach based on the simple relation lim
n→0
1 ∂ 1 [det(λ − B)]n = tr . n ∂λ λ−B
(4)
Therefore after applying these two steps we end up looking for an n goes to zero limit on matrix integrals whose size vanishes with n, a very different problem from the familiar large N limit. The Feynman graph representation of this matrix integral connects as usual with Riemann surfaces with marked points; letting the rank n go to zero selects the graphs of maximal genus for a given number of vertices. The article is organized as follows: In Sect. 2 we first recall the main formulae (i) for the explicit integral representation of k-point functions in a Gaussian plus source, N × N , Hermitian matrix integral, (ii) for the N-k duality. The derivations are reproduced in Appendices A and B respectively. Specializing to a constant matrix source, we are led to Kontsevich’s cubic model in the appropriate scaling limit. In Sect. 3 we derive the theorem (20) which gives explicitly the sourceless k-point functions in the zero-replica limit. This allows one to recover the known results for the intersection numbers, such as τ3g−2 g = (24)1g g! (g = 0,1,2,…). In Sect. 4 we show that an appropriate tuning of the matrix source may be used to generate generalizations of the Kontsevich model to higher powers than cubic,thereby providing generalized intersection numbers τm, j
Intersection Theory from Duality and Replica
509
for moduli of curves with ‘spin’ j. For instance from the quartic generalized model we find τ 8g−5− j , j g = 3
( g+1 1 3 ) , (12)g g! ( 2− j ) 3
where j = 0 for g = 3m + 1 and j = 1 for g = 3m. (For g = 3m + 2, the intersection numbers are zero.) 2. The Duality Plus Replica Strategy We first consider the average of products of characteristic polynomials, defined as 1 det(λα − M) A,M ZN k
Fk (λ1 , . . . , λk ) =
α=1
=
1 ZN
dM
k
N
det(λi · I − M)e− 2 trM
2 +NtrMA
,
(5)
i=1
where M is an N × N Hermitian random matrix, A a given Hermitian matrix, whose eigenvalues are (a1 , · · · , an ) and Z N the normalization constant of the probability measure (for A = 0). We have shown earlier [13] that this correlation function has also a dual expression. This duality interchanges N , the size of the random matrix, with k, the number of points in Fk , as well as the matrix source A with the diagonal matrix = diag(λ1 , . . . , λk ). This duality reads [13] Fk (λ1 , . . . , λk ) =
1 Zk
dB
N
N
[det(a j − i B)]e− 2 tr(B−i) , 2
(6)
j=1
where = diag(λ1 , . . . , λk ) and B is a k × k Hermitian matrix. (The normalization is now on GUE ensembles of Hermitian k × k matrices Z k = d Bexp(− 21 trB2 )). The derivation is reproduced in Appendix B. If we specialize this formula to a source A equal to the unit matrix, a trivial shift for the original N × N matrices M, which has the effect of making the support of Wigner’s semi-circle law, for the asymptotic density of eigenvalues of M, to lie between 0 and 2 rather than (–1,+1), the formula (6) involves det(1 − i B) N = exp[N trln(1 − i B)] N N = exp[−i N tr B + tr B 2 + i tr B 3 + · · ·]. 2 3
(7)
The linear term in B in (7), combined with the linear term of the exponent of (6), shifts by one. The B 2 terms in (6) cancel. In a scale in which the initials λk are close to one, or more precisely N 2/3 (λk − 1) is finite, the large N asymptotics of (6) is given by matrices B of order N −1/3 . Then the higher terms in (7) are negligible and we are left with terms linear and cubic in the exponent, namely N N 2 3 Fk (λ1 , . . . , λk ) = e 2 tr (8) d Bei 3 tr B +i N tr B(−1) .
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E. Brézin, S. Hikami
So finally for this edge behavior problem, in which the matrix B is of order N −1/3 , and λ − 1 of order N −2/3 we can rescale B and λ − 1 to get rid of N . The result is nearly identical to the matrix Airy integral, namely Kontsevich’s model [3], which gives the intersection numbers of moduli of curves. The original Kontsevich partition function was defined as 1 2 i 3 Z = d Be−trB + 3 trB , (9) Z 2 where Z = d Be−trB . The shift B → B +i, eliminates the B 2 term and one recovers (8) up to a trivial rescaling. Let us apply this to a one-point function. In view of the replica limit we specialize those formulae to [det(λ − M)]n A,M = [det(1 − i B)] N ,B ,
(10)
where B is an n × n random Hermitian matrix. For this edge problem we have chosen for the source matrix A, the N-dimensional unit matrix, whereas is a multiple of the n × n identity matrix : = diag(λ, . . . , λ). Note that the average in the l.h.s. of (10) is meant for the N × N GUE ensemble, whereas in the r.h.s. the average is performed on n × n Hermitian matrices with the weight (8) or (9). The strategy that we will use is thus to take the integral (9) for = λ × 1 and expand it in powers of the cubic term. The formulae that will be established, being exact for finite n, are easily continued in n and allow one to take the n → 0 limit. The method relies on explicit exact representations of Gaussian averages [7,15] in the presence of an external matrix source (in Appendix A the main steps of the derivation have been recalled). As a result we will obtain formulae for quantities such as 1 (trBk )l n→0 n lim
computed with the (normalized) Gaussian weight Z −1 exp(− 21 trB2 ) on n × n Hermitian matrices; thereby this procedure gives the values of the intersection numbers of the moduli of curves for a number of cases. 3. Evolution Operators and Replica Limit We will rely on explicit expressions for 1 tres1 B · · · tresk B (11) n for a probability measure on n × n Hermitian matrices in the presence of an Hermitian matrix source A, whose eigenvalues are a1 , . . . , an . The average is thus defined with the normalized weight U A (s1 , . . . , sk ) =
1 − 1 trB2 +trAB e 2 . Z Then one has (see Appendix A) for the one-point function PA (B) =
s2
1 e 2λ U A (s) = tresB = n ns
du su/λ u − aα + s e ( ) 2iπ u − aα
(12)
n
1
(13)
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511
the contour, in the complex u-plane, circling around the n poles aα . We have chosen the normalization U A (0) = 1. This formula simplifies for a vanishing external source (although a non-vanishing source simplifies the derivation) to s2
e 2λ U (s) = ns
s du su/λ (1 + )n . e 2iπ u
(14)
This representation leads to a simple continuation in n, and to an expansion in powers of n. For instance it gives s2
e 2λ lim U (s) = n→0 s
s du su/λ e log (1 + ). 2iπ u
(15)
This last contour integral reduces to the integral of the discontinuity of the logarithm, giving readily 2
lim U (s) =
n→0
s sinh ( 2λ ) 2
s ( 2λ )
,
(16)
and thus lim
n→0
1 4k! trBk = 2k k . n λ 4 (2k + 1)!
(17)
Note that in terms of Feynman diagrams with double lines, the limit n → 0 selects the diagrams with one single internal index all along the lines of the diagrams. Those diagrams correspond to a surface of maximum genus for a given number of vertices. The same strategy works for higher point-functions. The k-point function is given (for a vanishing source) by 1 tres1 B · · · tresk B n k k k si2 du i k (u i si /λ) si 1 k(k−1)/2 1 1 2λ = (−1) e e (1 + )n det . (18) 2iπ ui u i + si − u j
U (s1 , . . . , sk ) =
1
1
Again the continuation to non-integer n is straightforward and leads to lim U (s1 , . . . , sk ) = (−1)k(k−1)/2 e
n→0
×
k
si2 1 2λ
k k du i k (u i si /λ) si 1 e 1 log (1 + ) det . 2iπ ui u i + si − u j 1
(19)
1
The calculation of the contour integrals is more cumbersome, but all the integration can be done explicitly to the end and give the following theorem, which is a remarkably compact result (proof is given in Appendix C). Theorem. lim U (s1 , . . . , sk ) =
n→0
with σ = s1 + · · · + sk .
k λ σ si , 2 sinh σ2 2λ 1
(20)
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This function is of course the generating function for the n = 0 limit of n1 trBp1 . . . trBpk by expanding in the si ’s. Selecting the coefficients of equal powers for every si , for instance of (s1 . . . sk )3 , we find 1 33g−2 2−2g (6g − 4)! 4g − 2 3 4g−2 (21) lim (trB ) = g n→0 n λ6g−3 all other powers of limn→0 n1 (trB3 )k vanishing unless k = 2 (mod4). This leads to the intersection number of the moduli of curves with one marked point. Indeed, following Kontsevich, these numbers are given by 1 log Z = tl τl n
(22)
l
with tl = (−2)−(4l+2)/6
l−1 0
2 (2m + 1)( )2m+1 λ
(23)
(g = 0, 1, 2, . . .).
(24)
from which the above result (21) provides τ3g−2 g =
1 (24)g g!
These numbers agree with the values of the intersection numbers computed earlier by Kontsevich, Witten and others [1,3,16,17]. Clearly the method allows one to compute more than that. For instance one can derive as well 1 22p−2 2p − 1 lim (trB4 )2p−1 = 4p−2 (4p − 3)! . (25) p n→0 n λ These vertices appear in the higher Airy fuctions and they are related to the intersection numbers of Witten’s top Chern class [1,3]. 4. Application to Intersection Numbers of Top Chern Class (p-Spin Curves) Up to now we have considered an external source matrix A which was a multiple of identity. Let us now choose a source matrix in which half of the eigenvalues of A are −a and the other half +a. Then the asymptotic density of states has for support two disconnected segments of the real axis (for a1) with a gap in-between. In the limit in which a approaches one the gap closes, and for a1 the support is made of one single segment. For the critical closing gap case a = 1 the density of state ρ(λ) vanishes as λ1/3 at the center of the band [18,19]. We will show that the correlation functions for this critical case correspond to a higher Airy matrix model, related to the intersection numbers of Witten’s top Chern class (p = 3 spin curves). The formula (20) derived in the previous section allows us to compute also those intersection numbers for one marked point. With the help of the previous duality and replica method, we shall show that these numbers are given by the Fourier transform of the one point function, which is given by an Airy function [18].
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This may be extended to higher Airy cases ( p3) for the n-point functions. The details are left to a future publication. The partition function for the pth generalization of Kontsevich’s Airy matrix-model is defined as 1 1 tr(Bp+1 − p+1 ) − tr(B − )p ] Z= d Bexp[ (26) Z0 p+1 normalized by the Gaussian part of the integrand (after the shift B → B + which cancels the linear terms in B of the exponent): Z0 =
d Bexp[
p−1 1 tr j Bp−j−1 B]. 2
(27)
j=0
The ‘free energy’, i.e. the logarithm of the partition function, is the generating function of the generalized intersection numbers τm, j for moduli of curves with ‘spin’ j [3,20] By taking the external source ai = ±1 in the formula of the duality (6), and by expanding the exponent up to order B 4 in the large N limit, we obtain N N det(ai − i B) = [det(1 + B 2 )] 2 i=1
N
d Be− 4 tr B
=
4 −i N tr B
,
(28)
which coincides with the above partition function Z in the case p = 3 after trivial scalings. For p3, we also obtain the partition function Z in (26) with appropriate choice of the external source [18]. The ‘free energy’, i.e. the logarithm of the partition function, is the generating function of the generalized intersection numbers τm, j for moduli of curves with ‘spin’ j [3,20] d
F=
dm, j tm,m,j j , τm, j dm, j !
dm, j m, j
(29)
m, j
where [3] tm, j = (− p)
j− p−m( p+2) 2( p+1)
m−1
1
l=0
mp+j+1
(lp + j + 1)tr
.
According to Witten [2] the intersection numbers is given by Cw ( j1 , · · · , jn )ψ1m 1 · · · ψnm n τm n , jn · · · τm n , jn g = M¯ g,n
(30)
(31)
with the condition which relates, for given p, the indices to the genus g of the surface ( p + 1)(2g − 2 + n) =
n ( pm i + ji + 1). i=1
(32)
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The cohomology class Cw ( j1 , . . . , jn ) is Witten’s class (top Chern class), and ψ1 is the first Chern class. Witten conjectured that F is the string solution of the pth GelfandDikii hierarchy [2,21]. The partition function Z of (26) is a model for the (p,1)-quantum gravity, which is equivalent to the ( p − 1)-matrix model. By retaining only the leading term of the limit n → 0 limit, we restrict ourselves to surfaces with one marked point. Then the expansion of (29) is simply given by 1 τm, j g tm, j . F= n→0 n lim
(33)
m, j
In the leading order of the limit n → 0, the matrix is replaced by a scalar = λ · I . After a simple rescaling of the Gaussian weight, we obtain, for instance in the case of p = 3, 1 i 1 1 Z = (34) d Bexp[ trB4 + √ trB3 − tr2 B2 ], Z0 36 2 3 3 1 (35) Z 0 = d Bexp[− tr2 B2 ]. 2 In this p = 3 case, we obtain from (20), ( g+1 1 3 ) τ 8g−5− j , j g = , 3 (12)g g! ( 2− j )
(36)
3
where j = 0 for g = 1, 4, 7, 10, . . . and j = 1 for g = 3, 6, 9, . . .. For g = 2, 5, 8, . . ., the intersection numbers are zero. When the genus g is equal to three, the above formula gives τ6,1 g=3 = (12)13 3!3 which agrees with the value obtained earlier by Shadrin [22]. In the calculation of the intersection numbers for p = 3, in addition to (trB3 )p and (trB4 )q given here above in (21) and (25) , one needs to compute mixed averages of the type (trB4 )k (trB3 )l . Such averages are indeed required if one deals with the expansion of (34), but they are also contained in the explicit formula (20) for the generating function. For general p, from (20), we have for g = 1, τ1,0 g = 1 = p−1 24 . To derive this result, we simply need to compute trB4 g = 1 = 1 and (trB3 )2 g = 1 = 3 with t1,0 = −n/( pλ p+1 ). In the case p = 4, we obtain from (20) by simple calculations up to order g = 4 as τ1,0 g = 1 = 18 , τ3,2 g = 2 = 829·5! , τ6,0 g = 3 = 839·5! and τ8,2 g = 4 = 857·11 . ·5!·10 Thus we find that the expression of (20) for the n → 0 limit of U (s1 , · · · , sk ) is a generating function for the intersection numbers for the moduli space of one-marked point, p spin, curves. This provides thus a simple algorithm for obtaining these numbers. We have restricted this article to surfaces with one marked point. To go beyond one marked point, the analysis of higher orders in n is required. For two marked points, we need the order n 2 . However for genus one, we have two different kinds of intersection numbers, τ0 τ2 and τ12 (here p = 2). These two terms are distinguished by coupling to different combinations of the tl parameters, namely t0 t2 and t12 ; those combinations have different dependence (30). Thus we need a matrix which is no longer a multiple of the identity. If we go to higher orders in n with simply for a multiple of the iden1 tity matrix, we obtain the values of the sum τ0 τ2 + τ12 = 12 , instead of that of the individual terms. We hope to be able to discuss higher marked points by extending the present approach to such cases.
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Alternative approach. We shall now show that the duality (6) allows us to recover the same numbers from the one-point function of the closing gap problem. This is similar to the p = 2 case developed in previous articles [4,8]. The Fourier transform of the density of state, the one-point correlation function U A (s), is defined as 1 U A (s) = (37) dλeiλs trδ(λ − M) A , N which is known for arbitrary A and reduces for the choice ai = ±1 (closing gap case) to N N s 1 N s 2 /2 s du e (1 − ) 2 (1 + ) 2 e N su . U A (s) = (38) Ns 2πi 1−u 1+u In the range in which N is large and s of order N −1/4 , one can expand the integrand and the leading range is given by u’s of order N −1/4 as well. Then 1 − N s4 du −N su 3 − 3 N s 2 u 2 −N s 3 u 2 e 4 e U A (s) = Ns 2πi 1 du −N su 3 − N s 3 u 4 e = , (39) Ns 2πi where we have expanded the exponent up to order u 3 . The last equation has been obtained by the shift u → u − 2s . The contour integral over u becomes an integral over the range [−i∞, +i∞] in the large N limit. This integral is clearly related to an Airy function Ai(x), defined by 1 (3a)1/3 ∞ Ai[±(3a)− 3 x] = dtcos(at 3 ± xt). (40) π 0 We have thus U A (s) = with z = −
1
Ai(z)
(41)
N s(3N s) 3
8
N 1
1
1
s 3 . The expansion of the Airy function for small z gives the numbers
4·3 3 N 3
U A (s) =
1 1 3
2 3
( 23 )
(1 +
1 3 1·4 6 1·4·7 9 z + z + z + · · ·) 3! 6! 9!
N s(3N s) 3 1 2 · 5 7 2 · 5 · 8 10 2 z + z + · · ·) − (z + z 4 + 1 1 4! 7! 10! N s(3N s) 3 3 3 ( 13 )
(42)
From the expansion (42), we find the intersection numbers of (36) up to a scaling factor. The first series of (42), in powers of z 3k , corresponds to the spin j = 1 1·4·7 case : τ6,1 g=3 = (12)13 3!·3 , τ14,1 g=6 = (12)1·4 6 6!·32 , τ22,1 g=9 = (12)9 9!·33 , . . . The second series in (42) , in powers of z 3k+1 corresponds to the spin j=0 intersection num1 bers, τ1,0 g=1 = 12 , τ9,0 g=4 = (12)24 4!·3 , τ17,0 g=7 = (12)2·5 7 7!·32 , and τ25,0 g=10 = 2·5·8 . (12)10 10!·33
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5. Conclusion The present article, using both duality and replica, provides exact formulae for the intersection numbers of the moduli of curves on Riemann surfaces with one marked point and p-spin. The method may be extended, at least for low genera, to a higher number of marked points. It provides a relationship between Kontsevich’s Airy matrix model and Okounkov’s work of the edge of Wigner’s semi-circle for the p = 2 case. For the p = 3 case, it provides a link between a higher Airy matrix model to the critical gap closing model. Appendix A: Gaussian Averages in the Presence of an External Matrix Source For the sake of completeness we reproduce here the main steps of the derivation given in [7]. The probability distribution for Hermitian n × n matrices is thus PA (B) =
1 − λ trB2 +trAB e 2 . Z
(43)
Let us first compute the one point function U A (s) =
1 tresB . n
(44)
Using the Itzykson-Zuber formula [11,12] to integrate out the unitary degrees of freedom one obtains: n n n λ 2 1 1 U A (s) = dbi (B)e 1 (− 2 bi +ai bi ) esbi , (45) n Z (A) 1
1
in which Z is a normalization constant fixed such as U A (0) = 1 and (B) is the VanderMonde determinant of the eigenvalues (B) = i< j (bi − b j ). We now use repeatedly the trivial identity n
dbi (B)e
n
λ 2 1 (− 2 bi +ai bi )
= Cn e
n
1 2 1 2λ ai
(A)
(46)
1
in which Cn is a simple constant, with ai replaced by a˜i = ai + sδi,α in which α takes the successive values 1 to n. This gives n 1 s 2 saα /λ ( A˜α ) e 2λ e n
(A) 1 2 aα − aβ + s 1 s = e 2λ esaα /λ n aα − aβ α
U A (s) =
(47)
β=α
which may be replaced by the contour integral around the n points (a1 , . . . , an ) U A (s) =
1 s2 e 2λ ns
du su/λ u + s − aα e . 2iπ u − aα n
1
(48)
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Note that this is an exact formula; it simplifies of course for the pure Gaussian case aα = 0, although the derivation does require a source matrix in the intemediate steps. The derivation for the k-point functions follows exactly the same lines: the identity (46) requires now an a˜ which carries k-indices. The result of the calculation is similar: 1 tres1 B · · · tresk B n k k si2 1 du i k si u i /λ (u i − u j + si − s j )(u i − u j ) 1 2λ = e e i=1 ns1 · · · sk 2iπ (u i − u j + si )(u i − u j − s j )
U A (s1 , . . . , sk ) =
i=1
×
k n
(1 +
i=1 α=1
i< j
si ). u i − aα
(49)
One can simplify the expression by noticing the Cauchy determinant identity 1 i< j (x i − x j )(yi − y j ) n(n−1)/2 = (−1) det xi − y j i, j (x i − y j )
(50)
with xi = u i + si , yi = u i , namely det
(u i − u j + si − s j )(u i − u j ) 1 1 . = u i + si − u j s1 · · · sk (u i − u j + si )(u i − u j − s j )
(51)
i< j
In the n = 0 limit we need to consider only the connected part of U A (the disconnected ones vanishing with n, as is obvious from Feynman diagrams or explicit formulae) which correspond to connected permutations in the expansion of the determinant. Taking a vanishing source (aα = 0) the n = 0 limit gives immediately (19). Appendix B: Duality Let us consider the Gaussian average, with a matrix source A, of the product of characteristic determinants of the N × N Hermitian random matrices 1 det(λα − M) A,M ZN k
Fk (λ1 , . . . , λk ) =
α=1
=
1 ZN
dM
k
1
det(λi · I − M)e− 2 Tr(M−A)
2
(52)
α=1
2 in which Z N = d Mex p(−1/2TrM2 ) = 2N/2 (π )N /2 . The unitary invariance of the measure allows us to assume, without loss of generality, that A is a diagonal matrix with eigenvalues (a1 , . . . , a N ). We introduce now N × k complex Grassmannian variables (i = 1, 2, . . . , N and α = 1, 2, . . . , k) (ψ¯ iα , ψiα ) (i = 1, 2, . . . , N and α = 1, 2, . . . , k) with the normalization 1 0 ¯ . (53) = d ψdψ ¯ 1 ψψ
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Then one may write k
det(λi · I − M) =
d ψ¯ iα dψiα e
N
k
i, j=1
α ¯α α=1 ψi (λα −M)i j ,ψ j
,
(54)
i=1
and perform the Gaussian average 1 1 1 2 2 d Me− 2 TrM +TrMX = e 2 TrX . ZN
(55)
We deal here with a matrix X given by X pq = a p δ pq −
k
ψ¯ qα ψ pα
(56)
α=1
and thus TrX = TrA − 2 2
2
N k
ap ψ¯ pα ψpα −
p=1 α=1
γα,β γ β,α ,
(57)
α,β=1,...,k
where γα,β =
N
β ψ¯ iα ψi .
(58)
i=1
To make the notations more transparent let us denote by ‘Tr’ the trace on the initial space of N × N matrices and by ‘tr’ the trace in the new space of k × k matrices; the last term in (57) for TrX2 is thus −trγ 2 , the minus sign being due to the anticommuting nature of the psi’s. We then introduce an auxiliary Hermitian k × k matrix β so that 1 1 2 − 12 trγ 2 = (59) dβe− 2 trβ +itrγβ . e Zk This leads us to the representation 1 d ψ¯ iα dψiα Fk (λ1 , . . . , λk ) = Zk N k 1 2 ¯α α × dβe− 2 trβ +itrγ (β−i) e− i=1 α=1 ai ψi ψi
(60)
in which is the k × k diagonal matrix with eigenvalues (λ1 , . . . , λk ). One can now easily perform the integration over the Grassmanian variables since the exponent is quadratic in those variables. This yields the announced dual representation Fk (λ1 , . . . , λk ) =
1 Zk
1
dβe− 2 trβ
in which the matrices are now k × k.
2
N j=1
det[(λρ − a j )δρ,σ + iβρ,σ ]
(61)
Intersection Theory from Duality and Replica
519
Remark. This duality takes a more symmetric form if we choose a pure imaginary diagonal matrix A, and if we shift the matrix β by i in the final formula (61). Then the duality reads 1 ZN
k
dM
α=1
= (−i) N k
1 Zk
1
det(λi · I − M)e− 2 Tr(M−iA)
1
dβ e− 2 tr(β+i)
2
N
2
det(a j δρ,σ − βρ,σ ).
(62)
j=1
This duality may be extended to a two-matrix model when the measure is an exponential of a quadratic form in the two matrices M and M , namely exp(aT r M 2 +bT r M 2 + cT r M M ). Appendix C: Contour integrals in the n = 0 limit We return to the contour integral (19) which gives the n = 0 limit of the k-point function. Let us consider for simplicity the case of the two-point function. We are then dealing with the sum of two integrals: lim U (s1 , s2 ) = −e
s12 +s22 2λ
n→0
2 du i 2 (u i si /λ) s1 s2 [log (1 + ) + log (1 + )] e 1 2iπ u1 u2 1
1 × det . u i + si − u j
(63)
The continuation to n = 0 requires to take contours in the u 1 and u 2 planes which circle around the respective cuts [−s1 , 0] and [−s2 , 0]. We have to choose some well-defined contours on the two variables before we can write the integral (63) as a sum of two integrals. For instance we choose an integral over a large contour in the u 2 plane, and close to the cut in the u 1 plane. The disconnected term 1/s1 s2 of the determinant gives a vanishing contribution. Thus U (s1 , s2 ) = e
s12 +s22 2λ
2 du i 2 (u i si /λ) s1 s2 e 1 [log (1 + ) + log (1 + )] 2iπ u1 u2 1
×
1 . (u 1 + s1 − u 2 )(u 2 + s2 − u 1 )
(64)
The second integral, the one which involves log (1 + us22 ), vanishes for the choice of contours that we have made, since we can integrate over u 1 first and there are no poles inside the contour. For the first part, that with log (1 + us11 ), we integrate over u 2 first, pick up the two poles and find s12 +s22 1 du 1 u 1 σ/λ s1 2 U (s1 , s2 ) = e 2λ log (1 + ) (65) ( −s2 /λ − es1 s2 /λ ) e σ 2iπ u1 with σ = s1 + s2 . Since 0 s1 λ du 1 u 1 σ/λ e log (1 + ) = − d xe xσ/λ = − (1 − e−s1 σ/λ ), 2iπ u1 σ s1
(66)
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we end up with U (s1 , s2 ) = 4
λ σ s1 σ s2 sinh sinh . σ2 2λ 2λ
(67)
This calculation may be repeated for the k-point function, although the combinatorics becomes heavy. This yields the final result (theorem in (20)), lim U (s1 , s2 , . . . , sk ) =
n→0
k λ si σ 2 sinh , σ2 2λ
(68)
1
where σ = s1 + s2 + · · · + sk . It is straightforward to generalize this formula to the case of a non-zero external matrix source A with eigenvalues (a1 , . . . , an ). We assume that the density of eigenvalues ρ(x) =
n 1 δ(x − ai ) n
(69)
1
has a finite limit ρ0 (x) when n goes to zero. Up to now we have dealt with ρ0 (x) = δ(x), but we could take other examples such as n/2 eigenvalues equal to +a and n/2 equal to −a. Then we would deal with ρ0 (x) = 21 [δ(x − a) + δ(x + a)]. The derivation goes through now in the same way: we write n 1
(1 +
s s ) = en d xρ(x) log (1+ u−x ) u − aα
(70)
which may be expanded when n goes to zero provided, as we have assumed, that ρ(x) has a limit. The result is then simply lim U A (s1 , s2 , . . . , sk ) = lim U (s1 , s2 , . . . , sk ) d xρ0 (x)e xσ/λ , (71) n→0
n→0
in which the first factor U is given again by (68). Acknowledgements. We are grateful to Profs. Zagier and Kontsevich for their interest and suggestions.
References 1. Witten, E.: Two dimensional gravity and intersection theory on moduli space. Surv. Differ. Geom. 1, 243 (1991) 2. Witten, E.: Algebraic geometry associated with matrix models of two dimensional gravity. In: Topological methods in modern mathematics (Stony Brook, NY, 1991), Houston, TX: Publish or Perish, 1993, p.235 3. Kontsevich, M.: Intersection Theory on the moduli Space of Curves and Matrix Airy Function. Commun. Math. Phys. 147, 1 (1992) 4. Brézin, E., Hikami, S.: Vertices from replica in a random matrix theory. J. Phys. A 40, 13545 (2007) 5. Brézin, E., Hikami, S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479, 697 (1996) 6. Brézin, E., Hikami, S.: Extension of level-spacing universality. Phys. Rev. E56, 264 (1997) 7. Brézin, E., Hikami, S.: Spectral form factor in a random matrix theory. Phys. Rev. E55, 4067 (1997) 8. Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz numbers, and Matrix models, I. http:// arxiv.org/list/math.AG/0101147, 2001
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9. Okounkov, A.: Generating functions for the intersection numbers on moduli spaces of curves. Intern. Math. Research. Notices 18, 933 (2002) 10. Okounkov, A.: Random matrices and random permutations. Intern. Math. Research. Notices 20, 1043 (2000) 11. Harish-Chandra.: Proc. Nat. Acad. Sci. 42, 252 (1956) 12. Itzykson, C., Zuber, J.-B.: The planar approximation II. J. Math. Phys. 21, 411 (1980) 13. Brézin, E., Hikami, S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111–135 (2000) 14. Hashimoto, A., Min-xin Huang, Klemm, A., Shih, D.: Open/closed string duality for topological gravity with matter. JHEP 05, 007 (2005) 15. Kazakov, V.A.: External matrix field problem and new multicriticalities in (-2)-dimensional random surfaces. Nucl. Phys. B 354, 614 (1991) 16. Itzykson, C., Zuber, J.-B.: Combinatorics of the Modular Group II. The Kontsevich Integrals. Int. J. Mod. Phys. A7, 5661 (1992) 17. Liu, K., Xu, H.: New properties of the intersection numbers on moduli spaces of curves. Math. Res. Lett. 14, 1041–1054 (2007) 18. Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E57, 4140 (1998) 19. Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E58, 7176 (1998) 20. Jarvis, T.J., Kimura, T., Vaintrob, A.: Moduli Spaces of Higher Spin Curves and Integrable Hierarchies. Comp. Math. 126, 157–212 (2001) 21. Adler, M., van Moerbeke, P.: A matrix integral solution to two-dimensional Wp-gravity. Commun. Math. Phys. 147, 25–56 (1992) 22. Shadrin, S.: Geometriy of meromorphic functions and intersections on moduli spaces of curves. Int. Math. Res. Not. 38, 2051 (2003); Faber, C., Shadrin, S., Zvonkine, D.: Tautological relations and the r-spin Witten conjecture. http://arxiv.org/list/math/0612510, 2006 Communicated by M.R. Douglas
Commun. Math. Phys. 283, 523–542 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0514-5
Communications in
Mathematical Physics
A Sharpened Nuclearity Condition and the Uniqueness of the Vacuum in QFT Wojciech Dybalski Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany. E-mail:
[email protected] Received: 30 August 2007 / Accepted: 10 December 2007 Published online: 29 May 2008 – © The Author(s) 2008
Abstract: It is shown that only one vacuum state can be prepared with a finite amount of energy and it appears, in particular, as a limit of physical states under large timelike translations in any theory which satisfies a phase space condition proposed in this work. This new criterion, related to the concept of additivity of energy over isolated subsystems, is verified in massive free field theory. The analysis entails very detailed results about the momentum transfer of local operators in this model. 1. Introduction Since the seminal work of Haag and Swieca [1] restrictions on the phase space structure of a theory formulated in terms of compactness and nuclearity conditions have proved very useful in the structural analysis of quantum field theories [2–6] and in the construction of interacting models [7,8]. However, the initial goal of Haag and Swieca, namely to characterize theories which have a reasonable particle interpretation, has not been accomplished to date. While substantial progress was made in our understanding of the timelike asymptotic behavior of physical states [9–15], several important convergence and existence questions remained unanswered. As a matter of fact, it turned out that the original compactness condition introduced in [1] is not sufficient to settle these issues. Therefore, in the present article we propose a sharpened phase space condition, stated below, which seems to be more appropriate. We show that it is related to additivity of energy over isolated subregions and implies that there is only one vacuum state within the energy-connected component of the state space, as one expects in physical spacetime [16]. We stress that there may exist other vacua in a theory complying with our condition, but, loosely speaking, they are separated by an infinite energy barrier and thus not accessible to experiments. The convergence of physical states to the vacuum state under large timelike translations is a corollary of this discussion. A substantial part of this work is devoted to the proof that the new condition holds in massive scalar free field theory. As a matter of fact, it holds also in the massless case which will be treated
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elsewhere. These last results demonstrate that the new criterion is consistent with the basic postulates of local relativistic quantum field theory [17] which we now briefly recall. The theory is based on a local net O → A(O) of von Neumann algebras, which are attached to open, bounded regions of spacetime O ⊂ Rs+1 and act on a Hilbert space H. The global algebra of this net, denoted by A, is irreducibly represented on this space. Moreover, H carries a strongly continuous unitary representation of the Poincaré group ↑ Rs+1 L + (x, ) → U (x, ) which acts geometrically on the net α(x,) A(O) = U (x, )A(O)U (x, )−1 = A(O + x).
(1.1)
We adopt the usual notation for translated operators αx A = A(x) and functionals αx∗ ϕ(A) = ϕ(A(x)), where A ∈ A, ϕ ∈ A∗ , and demand that the joint spectrum of the generators of translations H, P1 , . . . , Ps is contained in the closed forward lightcone V + . We denote by PE the spectral projection of H (the Hamiltonian) on the subspace spanned by vectors of energy lower than E. Finally, we identify the predual of B(H) with the space T of trace-class operators on H and denote by T E = PE T PE the space of normal functionals of energy bounded by E. We assume that there exists a vacuum state ω0 ∈ T E and introduce the subspace T˚E = {ϕ − ϕ(I )ω0 | ϕ ∈ T E } of functionals with the asymptotically dominant vacuum contribution subtracted. The main object of our investigations is the family of maps E : T˚E → A(O)∗ given by E (ϕ) = ϕ|A(O) , ϕ ∈ T˚E .
(1.2)
Fredenhagen and Hertel argued in some unpublished work that in physically meaningful theories these maps should be subject to the following restriction: Condition C . The maps E are compact for any E ≥ 0 and double cone O ⊂ Rs+1 . This condition is expected to hold in theories exhibiting mild infrared behavior [19]. In order to restrict the number of local degrees of freedom also in the ultraviolet part of the energy scale, Buchholz and Porrmann proposed a stronger condition which makes use of the concept of nuclearity1 [19]: Condition N . The maps E are p-nuclear for any 0 < p ≤ 1, E ≥ 0 and double cone O ⊂ Rs+1 . This condition is still somewhat conservative since it does not take into account the fact that for any ϕ ∈ T˚E the restricted functionals αx∗ ϕ|A(O) should be arbitrarily close to zero apart from translations varying in some compact subset of Rs+1 , depending on ϕ. It seems therefore desirable to introduce a family of norms on L(T˚E , X ), where X is some Banach space, given for any N ∈ N and x1 , . . . , x N ∈ Rs+1 by x1 ,...,x N = sup
ϕ∈T˚E,1
N
21 (αx∗k ϕ) 2
, ∈ L(T˚E , X ),
(1.3)
k=1
1 We recall that a map : X → Y is p-nuclear if there exists a decomposition = into rank-one n n maps s.t. ν p := n n p < ∞. The p-norm p of this map is the smallest such ν and it is equal to zero for p > 1 [18]. Note that for any norm on L(X, Y ) one can introduce the corresponding class of p-nuclear maps. Similarly, we say that a map is compact w.r.t. a given norm on L(X, Y ) if it can be approximated by finite rank mappings in this norm.
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and the corresponding family of p-norms p,x1 ,...,x N , (see footnote 1). It is easily seen that if E satisfies Condition C , respectively N , then E is also compact, respectively p-nuclear, with respect to the above norms, and vice versa. Important additional information is contained in the dependence of the nuclear p-norms on N . In Sect. 2 we argue that the natural assumption is: Condition N . The maps E are p-nuclear w.r.t. the norms · x1 ,...,x N for any N ∈ N, x1 , . . . , x N ∈ Rs+1 , 0 < p ≤ 1, E ≥ 0 and double cone O ⊂ Rs+1 . Moreover, there holds for their nuclear p-norms lim sup E p,x1 ,...,x N ≤ c p,E ,
(1.4)
where c p,E is independent of N and the limit is taken for configurations x1 , . . . , x N , where all xi − x j , i = j, tend to spacelike infinity. Restricting attention to the case N = 1, it is easily seen that Condition N implies Condition N , but not vice versa. Our paper is organized as follows: In Sect. 2 we show that Condition N implies a certain form of additivity of energy over isolated subsystems and guarantees the physically meaningful vacuum structure of a theory. More technical part of this discussion is postponed to Appendix A. In Sect. 3 we recall some basic facts about massive scalar free field theory and its phase space structure. In Appendix B we provide a simple proof of the known fact that Condition N holds in this model. Section 4 contains our main technical result, namely the proof that Condition N holds in this theory as well. The argument demonstrates, in this simple example, the interplay between locality and positivity of energy which allows to strengthen Condition N . The paper concludes with a brief outlook where we apply our techniques to the harmonic analysis of translation automorphisms. 2. Physical Consequences of Condition N In this section we show that theories satisfying Condition N exhibit two physically desirable properties: a variant of additivity of energy over isolated subregions and the feature that only one vacuum state can be prepared given a finite amount of energy. Combining this latter property with covariance of a theory under Lorentz transformations we will conclude that physical states converge to the vacuum state under large timelike translations. The concept of additivity of energy over isolated subsystems does not have an unambiguous meaning in the general framework of local relativistic quantum field theory and we rely here on the following formulation: We introduce the family of maps N , given by E,x1 ,...,x N : T˚E → A(O)∗ ⊗ Csup (2.1) E,x1 ,...,x N (ϕ) = E (αx∗1 ϕ), . . . , E (αx∗N ϕ) , N denotes the space C N equipped with the norm z = sup where Csup k∈{1,...,N } |z k |. We 2 claim that a mild (polynomial) growth of the ε-contents N (ε) E,x1 ,...,x N of these maps 2 The ε-content of a map : X → Y is the maximal natural number N (ε) for which there exist elements ϕ1 , . . . , ϕN (ε) ∈ X 1 s.t. (ϕi ) − (ϕ j ) > ε for i = j. Clearly, N (ε) is finite for any ε > 0 if and only if the map is compact.
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with N , (when xi − x j , i = j, tend to spacelike infinity), is a signature of additivity of energy over isolated subregions. In order to justify this formulation we provide a heuristic argument: Given a functional ϕ ∈ T˚E,1 , we denote by E k the ‘local energy content’ of the restricted functional ϕ|A(O+xk ) . Additivity of energy should then imply that E 1 +· · ·+ E N ≤ E for large spacelike distances between the regions O + x1 , . . . , O + x N . This suggests that to calculate N (ε) E,x1 ,...,x N one should count all the families of functionals (ϕ1 , . . . , ϕ N ), ϕk ∈ T˚E k ,1 , E 1 + · · · + E N ≤ E, which can be distinguished, up to accuracy ε, by measurements in O + x1 , . . . , O + x N . Relying on this heuristic reasoning we write N (ε) E,x1 ,...,x N = #{ (n 1 . . . n N ) ∈ N∗×N | n 1 ≤ N (ε) E 1 , . . . , n N ≤ N (ε) E N , for some E 1 , . . . , E N ≥ 0 s.t. E 1 + · · · + E N ≤ E }, (2.2) where we made use of the fact that the number of functionals from T˚E k ,1 which can be discriminated, up to ε, by observables localized in the region O + xk is equal to the ε-content N (ε) E k of the map E k : T˚E k → A(O + xk ) given by E k (ϕ) = ϕ|A(O+xk ) . Anticipating that N (ε) E k tends to one for small E k we may assume that N (ε) E k ≤ 1 + c(ε, E)E k
(2.3)
for E k ≤ E. (This is valid e.g. in free field theory due to Sect. 7.2 of [20] and Proposition 2.5 (iii) of [21]). From the heuristic formula (2.2) and the bound (2.3) we obtain the estimate which grows only polynomially with N , N (ε) E,x1 ,...,x N ≤ #{ (n 1 . . . n N ) ∈ N∗×N | n 1 + · · · + n N ≤ N + c(ε, E)E } ≤ (N + 1)c(ε,E)E , (2.4) where the last inequality can be verified by induction in N . Omitting the key condition E 1 +· · ·+ E N ≤ E in (2.2) and setting E k = E instead, one would arrive at an exponential growth of N (ε) E,x1 ,...,x N as a function of N . Thus the moderate (polynomial) increase of this quantity with regard to N is in fact a clear-cut signature of additivity of energy over isolated subsystems. It is therefore of interest that this feature prevails in all theories complying with Condition N as shown in the subsequent theorem whose proof is given in Appendix A. Theorem 2.1. Suppose that Condition N holds. Then the ε-content N (ε) E,x1 ,...,x N of the map E,x1 ,...,x N satisfies lim sup N (ε) E,x1 ,...,x N ≤ (4eN )
c(E) ε2
,
(2.5)
where the constant c(E) is independent of N and the limit is taken for configurations x1 , . . . , x N , where all xi − x j , i = j, tend to spacelike infinity. Now let us turn our attention to the vacuum structure of the theories under study. In physical spacetime one expects that there is a unique vacuum state which can be prepared with a finite amount of energy. This fact is related to additivity of energy and can be derived from Condition N .
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Theorem 2.2. Suppose that a state ω ∈ A∗ belongs to the weak* closure of T E,1 for some E ≥ 0 and is invariant under translations along some spacelike ray. Then the following assertions hold: (a) If Condition C is satisfied, ω is a vacuum state. (b) If Condition N is satisfied, ω coincides with the vacuum state ω0 . s+1 ) s.t. supp f˜ ∩ V = ∅ and Proof. (a) We pick any A ∈ A(O), a test function + f ∈ S(R s+1 define the energy decreasing operator A( f ) = A(x) f (x)d x. Next, we parametrize the ray from the statement of the theorem as { λeˆ | λ ∈ R }, where eˆ ∈ Rs+1 is some spacelike unit vector, choose a compact subset K ⊂ R and estimate ∗ ω(A( f ) A( f ))|K | = dλ ω (A( f )∗ A( f ))(λe) ˆ K
dλ (A( f )∗ A( f ))(λe) ˆ = lim ϕn n→∞ K ∗ (2.6) P dλ (A( f ) A( f ))(λ e) ˆ P ≤ E . E K
In the first step we exploited invariance of the state ω under translations along the spacelike ray. In the second step we made use of local normality of this state, which follows from Condition C , in order to exchange its action with integration. Approximating ω by a sequence of functionals ϕn ∈ T E,1 , we arrived at the last expression. (Local normality of ω and existence of an approximating sequence can be shown as in [22] p. 49). Now we can apply a slight modification of Lemma 2.2 from [11], (see also Lemma 4.1 below), to conclude that the last expression on the r.h.s. of (2.6) is bounded uniformly in K . As |K | can be made arbitrarily large, it follows that ω(A( f )∗ A( f )) = 0
(2.7)
for any A ∈ A(O) and f as defined above. Since equality (2.7) extends to any A ∈ A, we conclude that ω is a vacuum state in the sense of Definition 4.3 from [23]. Invariance of ω under translations and validity of the relativistic spectrum condition in its GNSrepresentation follow from Theorem 4.5 of [23], provided that the functions Rs+1 x → ω(A∗ B(x)) are continuous for any A, B ∈ A. Since local operators form a norm-dense subspace of A, it is enough to prove continuity for A, B ∈ A(O) for any open, bounded region O. For this purpose we recall from [19] that Condition C has a dual formulation which says that the maps E : A(O) → B(H) given by E (A) = PE A PE are compact for any open, bounded region O and any E ≥ 0. Given any sequence of spacetime points xn → x, there holds A∗ (B(xn )− B(x)) → 0 in the strong topology and, by compactness of the maps E , PE A∗ (B(xn ) − B(x))PE → 0 in the norm topology in B(H). Now the required continuity follows from the bound |ω A∗ (B(xn ) − B(x)) | ≤ PE A∗ (B(xn ) − B(x))PE (2.8) which can be established with the help of the approximating sequence ϕn ∈ T E,1 . (b) We note that for any open, bounded region O, E ≥ 0 and ε > 0, Condition N allows for such N and x1 , . . . , x N , belonging to the spacelike ray, that 2N − 2 E x1 ,...,x N ≤ 3ε . 1
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For arbitrary A ∈ A(O)1 we can find ϕ ∈ T E,1 s.t. supk∈{1,...,N } |ω(A(xk ))−ϕ(A(xk ))| ≤ 3ε and |1 − ϕ(I )| ≤ 3ε . Next, we note that ε 3 N N ∗ 1 α ∗ ω(A) − α ∗ ϕ(A) + 1 α ϕ(A) − ϕ(I )α ∗ ω0 (A) + ε ≤ xk xk xk xk N N 3 k=1 k=1 1 ε ≤ sup |ω(A(xk )) − ϕ(A(xk ))| + 2N − 2 E x1 ,...,x N + ≤ ε, (2.9) 3 k∈{1,...,N }
|ω(A) − ω0 (A)| ≤ |ω(A) − ϕ(I )ω0 (A)| +
where in the second step we made use of the fact that both ω and ω0 are invariant under the translations x1 , . . . , x N and in the third step we used the Hölder inequality and the fact that 21 (ϕ − ϕ(I )ω0 ) ∈ T˚E,1 . We conclude that the states ω and ω0 coincide on any local operator and therefore on the whole algebra A. The above result is of relevance to the problem of convergence of physical states to the vacuum under large timelike translations. In fact, the following lemma asserts that the respective limit points are invariant under translations in some spacelike hyperplane. Lemma 2.3 (D.Buchholz, private communication). Suppose that Condition C holds. Let ω0+ be a weak* limit point as t → ∞ of the net {αt∗eˆ ω}t∈R+ of states on A, where eˆ ∈ Rs+1 is a timelike unit vector and ω is a state from T E for some E ≥ 0. Then ω0+ is invariant under translations in the spacelike hyperplane {eˆ⊥ } = {x ∈ Rs+1 | eˆ · x = 0}, where dot denotes the Minkowski scalar product. Proof. Choose x ∈ {eˆ⊥ }, x = 0. Then there exists a Lorentz transformation and y 0 , y 1 ∈ R\{0} s.t. eˆ = y 0 eˆ0 , x = y 1 eˆ1 , where eˆµ , µ = 0, 1, . . . , s form the 1
canonical basis in Rs+1 . We set v = yy 0 and introduce the family of Lorentz transforma˜ t , where ˜ t denotes the boost in the direction of eˆ1 with rapidity tions t = −1 arsinh( vt ). By the composition law of the Poincaré group, the above transformations composed with translations in timelike direction give also rise to spacelike translations
ˆ I )(0, −1 ) = (t e, ˆ I ), t e ˆ = t 1 + (v/t)2 eˆ + x. (2.10) (0, t )(t e, t t t We make use of this fact in the following estimate: |αt∗eˆ ω(A) − αt∗eˆ ω(A(x))| ≤ |ω(αt eˆ A) − ω(αt αt eˆ α−1 A)| t
∗ ω(A) − αt∗eˆ ω(A(x))|, + |αt t eˆ
(2.11)
where A ∈ A(O). The first term on the r.h.s. of (2.11) satisfies the bound ω(αt eˆ A) − ω αt αt eˆ α−1 A t
∗ ≤ |αt∗eˆ ω(A − α−1 A)| + |(ω − α ω)(αt eˆ α−1 A)| t t
t
∗ ω A(O ≤ PE (A − α−1 A)PE + sup ω − α +s e) ˆ A , (2.12) t t
s∈R+
is a slightly larger region than O. Clearly, t → I for t → ∞ and therefore where O αt → id in the point - weak open topology. Then the above expression tends to
Sharpened Nuclearity Condition and Uniqueness of the Vacuum in QFT
529
zero in this limit by the dual form of Condition C and the assumption that Lorentz transformations are unitarily implemented. (The argument is very similar to the last step in the proof of Theorem 2.2 (a). We note that the restriction on Lorentz transformations can be relaxed to a suitable regularity condition). The second term on the r.h.s. of (2.11) converges to zero by the dual variant of Condition C and the following bound:
∗ ∗ ω A t 1 + (v/t)2 eˆ + x − A(t eˆ + x) ω(A) − α ω(A(x))| ≤ |αt t eˆ t eˆ −1 ≤ PE A 1 + (v/t)2 + 1 (v 2 /t)eˆ − A PE . (2.13) ω0+ (A)
ω0+ (A(x))
Thus we demonstrated that = extends by continuity to any A ∈ A.
for any local operator A. This result
It follows from Theorem 2.2 (a) that all the limit points ω0+ are vacuum states under the premises of the above lemma. On the other hand, adopting Condition N we obtain a stronger result from Theorem 2.2 (b): Corollary 2.4. Let Condition N be satisfied. Then, for any state ω ∈ T E , E ≥ 0, and timelike unit vector eˆ ∈ Rs+1 , there holds lim α ∗ ω(A) t→∞ t eˆ
= ω0 (A), for A ∈ A.
(2.14)
We note that in contrast to previous approaches to the problem of relaxation to the vacuum [9,16] the present argument does not require the assumption of asymptotic completeness or asymptotic abelianess in time. To conclude this survey of applications of Condition N let us mention another physically meaningful procedure for preparation of vacuum states: It is to construct states with increasingly sharp values of energy and momentum and exploit the uncertainty principle. Let P( p,r ) be the spectral projection corresponding to the ball of radius r centered around point p in the energy-momentum spectrum. Then, in a theory satisfying Condition N , any sequence of states ωr ∈ P( p,r ) T P( p,r ) converges, uniformly on local algebras, to the vacuum state ω0 as r → 0, since this is the only energetically accessible state which is completely dislocalized in spacetime. This fact is reflected in the following property of the map E : Proposition 2.5. Suppose that Condition N is satisfied. Then, for any E ≥ 0 and p ∈ V + , there holds lim E |T˚ = 0, ( p,r )
r →0
(2.15)
where T˚( p,r ) = {ϕ − ϕ(I )ω0 | ϕ ∈ P( p,r ) T E P( p,r ) }. Proof. We pick A ∈ B(H), ϕ ∈ T˚( p,r ) and estimate the deviation of this functional from translational invariance |ϕ(A) − αx∗ ϕ(A)| = |ϕ(P( p,r ) A P( p,r ) ) − ϕ(P( p,r ) ei(P− p)x Ae−i(P− p)x P( p,r ) )| = |ϕ(P( p,r ) ei(P− p)x A(1 − e−i(P− p)x )P( p,r ) ) + ϕ(P( p,r ) (1 − ei(P− p)x )A P( p,r ) )| ≤ 2 ϕ A |x| r, (2.16)
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W. Dybalski
where in the first step we used invariance of ω0 under translations to insert the projections P( p,r ) and in the last step we applied the spectral theorem. Consequently, for any x1 , . . . , x N ∈ Rs+1 and open bounded region O, ϕ A(O) ≤
N 1 ∗ αxk ϕ A(O) + sup ϕ − αx∗k ϕ A(O) N k∈{1,...,N } k=1
1 ≤ √ N
N
21 αx∗k ϕ 2A(O)
+ 2 ϕ r
k=1
sup
k∈{1,...,N }
|xk |.
(2.17)
To conclude the proof of the proposition we restate the above inequality as follows: E |T˚
( p,r )
1 ≤ √ E x1 ,...,x N + 2r sup |xk |, N k∈{1,...,N }
(2.18)
and make use of Condition N . It is a consequence of the above proposition that lim E0 N (ε) E = 1 in any theory complying with Condition N , as anticipated in our heuristic discussion. Since N (ε) E ≥ 1 and it decreases monotonically with decreasing E, the limit exists. If it was strictly larger than one, we could find nets of functionals ϕ1,E , ϕ2,E ∈ T˚E,1 s.t. E (ϕ1,E − ϕ2,E ) >√ε for any E > 0. But fixing some E 0 > 0 and restricting attention to E ≤ E 0 / 2 we obtain ε < E (ϕ1,E − ϕ2,E ) ≤ 2 E 0 |T˚
√ (0, 2E)
.
(2.19)
The last expression on the r.h.s. tends to zero with E → 0, by Proposition 2.5, leading to a contradiction. Up to this point we discussed the physical interpretation and applications of the novel Condition N from the general perspective of local relativistic quantum field theory. In order to shed more light on the mechanism which enforces this and related phase space criteria, we turn now to their verification in a model. 3. Condition N in Massive Scalar Free Field Theory In this section, which serves mostly to fix our notation, we recall some basic properties of scalar free field theory of mass m > 0 in s space dimensions. (See [24] Sect. X.7). The single particle space of thistheory is L 2 (Rs , d s p). On this space there act the multiplication operators ω( p) = | p|2 + m 2 and p1 , . . . , ps which are self-adjoint on a suitable dense domain and generate the unitary representation of translations (U1 (x) f )( p) = ei(ω( p)x
0− p x)
f ( p),
f ∈ L 2 (Rs , d s p).
(3.1)
The full Hilbert space H of the theory is the symmetric Fock space over L 2 (Rs , d s p). By the method of second quantization we obtain the Hamiltonian H = d (ω), and the momentum operators Pi = d ( pi ), i = 1, 2, . . . , s defined on a suitable domain in H.
Sharpened Nuclearity Condition and Uniqueness of the Vacuum in QFT
531
The joint spectrum of this family of commuting, self adjoint operators is contained in the closed forward light cone. The unitary representation of translations in H given by U (x) = (U1 (x)) = ei(H x
0−P x)
(3.2)
implements the corresponding family of automorphisms of B(H) αx (·) = U (x) · U (x)∗ .
(3.3)
Next, we construct the local algebra A(O) attached to the double cone O, whose base is the s-dimensional ball Or of radius r centered at the origin in configuration space. To 1 r )], where tilde denotes the Fourier this end we introduce the subspaces L± = [ω∓ 2 D(O transform. (The respective projections are denoted by L± as well.) Defining J to be the complex conjugation in configuration space we introduce the real linear subspace L = (1 + J )L+ + (1 − J )L−
(3.4)
and the corresponding von Neumann algebra A(O) = { W ( f ) | f ∈ L} ,
(3.5)
i(a ∗ ( f )+a( f ))
and a ∗ ( f ), a( f ) are the creation and annihilation opewhere W ( f ) = e rators. With the help of the translation automorphisms αx introduced above we define local algebras attached to double cones centered at any point x of spacetime A(O + x) = αx (A(O)).
(3.6)
The global algebra A is the C ∗ -inductive limit of all such local algebras of different r > 0 and x ∈ Rs+1 . By construction, αx leaves A invariant. Now we turn our attention to the phase space structure of the theory. Let Q E be the projection on states of energy lower than E in the single particle space and β ∈ R. We 1 2 define operators TE,± = Q E L± , Tβ,± = e− 2 (β| p|) L± . It follows immediately from [25], p. 137 that these operators satisfy |TE,± | p 1 < ∞, |Tβ,± | p 1 < ∞ for any p > 0, where · 1 denotes the trace norm. We introduce their least upper bound T T = s- lim
n→∞
1 n n n n (|TE,+ |2 + |TE,− |2 + |Tβ,+ |2 + |Tβ,− |2 ) 4
2−n .
(3.7)
Proceeding as in [26], pp. 316/317 one can show that this limit exists and that the operator T satisfies (3.8) T n ≥ |TE,± |n and T n ≥ |Tβ,± |n for n ∈ N, T ≤ max( TE,+ , TE,− , Tβ,+ , Tβ,− ) ≤ 1, (3.9) T p 1 ≤ |TE,+ | p 1 + |TE,− | p 1 + |Tβ,+ | p 1 + |Tβ,− | p 1 for p > 0. (3.10) In particular T is a trace class operator. Since it commutes with the conjugation J , the orthonormal basis of its eigenvectors {e j }∞ 1 can be chosen so that J e j = e j . The corresponding eigenvalues will be denoted {t j }∞ 1 . Given any pair of multiindices µ = (µ+ , µ− ) we define the operator −
Bµ = a(Le)µ = a(L+ e)µ a(L− e)µ . +
(3.11)
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W. Dybalski
We recall, that for any f 1 , . . . , f n ∈ L 2 (Rs , d s p) there hold the so called energy bounds [19] which in the massive theory have the form n
a( f 1 ) . . . a( f n )PE = PE a ∗ ( f n ) . . . a ∗ ( f 1 ) ≤ (M E ) 2 f 1 . . . f n , (3.12) where M E = mE . Consequently, the operators Bµ are bounded on states of finite energy. We note the respective bound Bµ PE ≤ a(Q E Le)µ PE ≤ (M E ) ≤ (M E )
|µ| 2
Q E Le µ
|µ| 2
t µ,
(3.13)
−
where |µ| = |µ+ | + |µ− |, t µ = t µ t µ , {t j }∞ 1 are the eigenvalues of T and in the last step we made use of the fact that |Q E L± |2 ≤ T 2 . We will construct the expansion of E into rank-one maps with the help of the bounded linear functionals Sµ,ν : T˚E → C, given by +
Sµ,ν (ϕ) = ϕ(Bµ∗ Bν ).
(3.14)
In particular S0,0 = 0, since ϕ(I ) = 0 for any ϕ ∈ T˚E . It follows from (3.13) that the norms of these maps satisfy the bound |µ|+|ν| 2
t µt ν .
Sµ,ν ≤ M E
(3.15)
Clearly, we can assume that M E ≥ 1 as E ≡ 0 otherwise. Since Sµ,ν = 0 for |µ| > M E or |ν| > M E , the norms of the functionals Sµ,ν are summable with any power p > 0. In fact ⎛ ⎞2 ⎛ ⎞4 + pM pM Sµ,ν p ≤ M E ⎝ t pµ ⎠ ≤ M E ⎝ t pµ ⎠ E
µ,ν
E
µ:|µ|≤M E
⎛ =
pM ME E
[M E]
k=0
µ+ :|µ+ |=k
⎝
⎞4 t
pµ+
⎠ ≤
µ+ :|µ+ |≤M E pM ME E
[M ] E
4 T p k1
,
(3.16)
k=0
where in the last step we made use of the multinomial formula. With this information at hand it is easy to verify that Condition N holds in massive scalar free field theory [19,20]. Theorem 3.1. In massive scalar free field theory there exist functionals τµ,ν ∈ A(O)∗ such that there holds in the sense of norm convergence in A(O)∗ , E (ϕ) = τµ,ν Sµ,ν (ϕ), ϕ ∈ T˚E . (3.17) µ,ν
Moreover, τµ,ν ≤ 25M E for all µ, ν and
µ,ν
Sµ,ν p < ∞ for any p > 0.
We give the proof of this theorem in Appendix B.
Sharpened Nuclearity Condition and Uniqueness of the Vacuum in QFT
533
4. Condition N in Massive Scalar Free Field Theory At this point we turn to the main goal of this technical part of our investigations, namely to verification of Condition N in the model at hand. By definition of the nuclear p-norms and Theorem 3.1 there holds the bound ⎛ ⎞1 ⎛ ⎞1 p p p p p 5M E ⎝ E p,x1 ,...,x N ≤ ⎝ τµ,ν Sµ,ν x1 ,...,x N⎠ ≤ 2 Sµ,ν x1 ,...,x N⎠ . (4.1) µ,ν
µ,ν
Consequently, we need estimates on the norms Sµ,ν x1 ,...,x N whose growth with N can be compensated by large spacelike distances xi − x j for i = j. This task will be accomplished in Proposition 4.4. The argument is based on the following lemma which is a variant of Lemma 2.2 from [11]. Lemma 4.1. Let B be a (possibly unbounded) operator s.t. B PE < ∞, B ∗ PE < ∞ and B PE H ⊂ PE−m H for any E ≥ 0. Then, for any x1 , . . . , x N ∈ Rs+1 , there hold the bounds N ∗ (B ∗ B)(xk )PE ≤ (M E + 1) { PE [B, (a) PE k=1 B ]PE ∗ +(N − 1) supk1 =k2 PE [B(xk1 ), B (xk2 )]PE , (b) PE K d s x(B ∗ B)( x )PE ≤ (M E + 1) K d s x PE [B( x ), B ∗ ]PE , x − y | x, y ∈ K }. where K is a compact subset of Rs and K = { Proof. Part (b) coincides, up to minor modifications, with [11]. In the proof of part (a) the modifications are more substantial, so we provide some details. We will show, by induction in n, that there holds the following inequality: N (B ∗ B)(xk )Pnm ≤ n P(n−1)m [B, B ∗ ]P(n−1)m Pnm k=1 + (N − 1) sup P(n−1)m [B(xk1 ), B ∗ (xk2 )]P(n−1)m , k1 =k2
(4.2)
where Pnm is the spectral projection of H on the subspace spanned by vectors of energy lower than nm. It clearly holds for n = 0. To make the inductive step we pick ω( · ) = N (| · |), ∈ (Pnm H)1 and define Q = k=1 (B ∗ B)(xk ). Proceeding like in [11], with integrals replaced with sums, one arrives at N N ∗ ∗ ω(Q Q) ≤ ω((B B)( xk )) P(n−1)m [B( xl ), B ( xk )]P(n−1)m k=1
l=1
+ ω(Q) P(n−1)m Q P(n−1)m .
(4.3)
The sum w.r.t. l in the first term on the r.h.s. can be estimated by the expression in curly brackets in (4.2). To the second term on the r.h.s. of (4.3) we apply the induction hypothesis. Altogether ω(Q Q) ≤ nω(Q) P(n−1)m [B, B ∗ ]P(n−1)m ∗
+ (N − 1) sup P(n−1)m [B(xk1 ), B (xk2 )]P(n−1)m . k1 =k2
(4.4)
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W. Dybalski
Making use of the fact that ω(Q)2 ≤ ω(Q Q) and taking the supremum over states ω which are induced by vectors from Pnm H one concludes the proof of estimate (4.2). The statement of the lemma follows by choosing n s.t. (n − 1)m ≤ E ≤ nm. In order to control the commutators appearing in the estimates in Lemma 4.1 we need a slight generalization of the result from [27] on the exponential decay of vacuum correlations between local observables. Theorem 4.2. Let H be a self-adjoint operator on a Hilbert space H s.t. Sp H = {0} ∪ [m, ∞], m > 0 and there exists exactly one (up to a phase) eigenvector of H with eigenvalue zero. Let A, B be operators such that belongs to their domains and to the domains of their adjoints. If there holds (| [A, eit H Be−it H ] ) = 0 for |t| < δ,
(4.5)
then 1
|(|AB) − (|A)(|B)| ≤ e−mδ { A A∗ B B ∗ } 2 .
(4.6)
With the help of the above theorem we prove the desired estimate. Lemma 4.3. Let e ∈ L 2 (Rs , d s p) be s.t. e ≤ 1 and J e = e. Then there holds for any x ∈ Rs+1 , 0 < ε < 1 and any combination of ± signs |L± e|e−(β| p|) U (x)L± e| ≤ cε,β e−m(1−ε)δ(x) , 2
(4.7)
where cε,β does not depend on x and e. Here δ(x) = | x | − |x 0 | − 2r and r is the radius of the double cone entering into the definition of the projections L± . Proof. We define the operators φ+ (e) = a ∗ (L+ e) + a(L+ e), φ− (e) = a ∗ (iL− e) + a(iL− e) and their translates φ± (e)(x) = U (x)φ± (e)U (x)−1 . Since the projections L± commute with J and J e = e, these operators are just the fields and canonical momenta of massive scalar free field theory. Assume that δ(x) > 0. Then, by locality, φ± (e) and φ± (e)(x) satisfy the assumptions of Theorem 4.2. As they have vanishing vacuum expectation values, we obtain |L± e|U (x)L± e| = |(|φ± (e)φ± (e)(x))| ≤ e−mδ(x) .
(4.8)
Let us now consider the expectation value from the statement of the lemma. We fix some 0 < ε < 1 and estimate |L± e|e−(β| p|) U (x)L± e| √ ≤ (2 π β)−s 2
√
+ (2 πβ)
−s
2
δ(y +x)≥(1−ε)δ(x)
ds y e
− |y |2
δ(y +x)≤(1−ε)δ(x)
4β
|L± e|U (x + y)L± e| 2
ds y e
− |y |2 4β
|L± e|U (x + y)L± e|
2 √ − |y | d s y e 4β 2 ≤ e−m(1−ε)δ(x) + (2 π β)−s |y |≥εδ(x) 2 y| √ − |y |2 + m(1−ε)| −m(1−ε)δ(x) −s s ε d y e 4β ≤e 1 + (2 π β) .
(4.9)
Sharpened Nuclearity Condition and Uniqueness of the Vacuum in QFT
535
In the first step we expressed the function e−(β| p|) by its Fourier transform and divided the region of integration into two subregions. To the first integral we applied estimate (4.8). Making use of the fact that the second integral decays faster than exponentially with δ(x) → ∞ we arrived at the last expression which is of the form (4.7). Since cε,β > 1, the bound (4.9) holds also for δ(x) ≤ 0. 2
It is a well known fact that any normal, self-adjoint functional on a von Neumann algebra can be expressed as a difference of two normal, positive functionals which are mutually orthogonal [28]. It follows that any ϕ ∈ T E,1 can be decomposed as − − + + ϕ = ϕRe − ϕRe + i(ϕIm − ϕIm ),
(4.10)
± ± , ϕIm are positive functionals from T E,1 . This assertion completes the list of where ϕRe auxiliary results needed to establish the required estimate for Sµ,ν x1 ,...,x N .
Proposition 4.4. The functionals Sµ,ν satisfy the bound m 2 √ Sµ,ν 2x1 ,...,x N ≤ 32t µ t ν (M E )2M E e(β E) 1 + cε,β (N − 1)e− 2 (1−ε)δ(x) , (4.11) where {t j }∞ 1 are the eigenvalues of the operator T given by formula (3.7) and δ(x) = inf i = j δ(xi − x j ). The function δ(x), the parameter ε and the constant cε,β appeared in Lemma 4.3. + the set of positive functionals from T Proof. We denote by T E,1 E,1 . Making use of the definition of · x1 ,...,x N , decomposition (4.10) and the Cauchy-Schwarz inequality we obtain
Sµ,ν 2x1 ,...,x N = sup
N
ϕ∈T˚E,1 k=1
≤ 16 sup
+ ϕ∈T E,1
|Sµ,ν (αx∗k ϕ)|2 ≤ 16 sup
+ ϕ∈T E,1
N
N
|αx∗k ϕ(Bµ∗ Bν )|2
k=1
αx∗k ϕ(Bµ∗ Bµ )αx∗k ϕ(Bν∗ Bν )
k=1
≤ 16(M E )|µ| t 2µ PE
N
(Bν∗ Bν )(xk )PE ,
(4.12)
k=1
where in the last step we applied the bound (3.13). We can assume, without loss of generality, that ν = 0 and decompose it into two pairs of multiindices ν = ν a + ν b in such a way that |ν b | = 1. Since Bν = Bν a Bν b , we get PE
N k=1
(Bν∗ Bν )(xk )PE = PE
N (Bν∗b PE Bν∗a Bν a PE Bν b )(xk )PE k=1
≤ Bν a PE 2 PE
N
(Bν∗b Bν b )(xk )PE
k=1 |ν |
= M E a t 2ν a PE
N a ∗ (Le)ν b a(Le)ν b (xk )PE , k=1
(4.13)
536
W. Dybalski
where in the last step we used again estimate (3.13). Next, let g be the operator of multiplication by 21 (β| p|)2 in L 2 (Rs , d s p) and let G = d (g) ≥ 0 be its second quantization. Since one knows explicitly the action of G and H on vectors of fixed particle number, it is easy to check that 1
1
e G PE = PE e G PE ≤ PE e 2 (β H ) PE ≤ e 2 (β E) . 2
2
(4.14)
Making use of this fact, Lemma 4.1 (a) and Lemma 4.3 we obtain from (4.13) the following string of inequalities: N ∗ (Bν Bν )(xk )PE PE k=1 N |ν a | 2ν a G ∗ − 12 (β| p|)2 ν b −2G − 21 (β| p|)2 νb G ≤ M E t PE e a (e (xk )e PE Le) e a(e Le) k=1 N 1 1 2 2 2 |ν | ≤ M E a t 2ν a e(β E) PE a ∗ (e− 2 (β| p|) Le)ν b a(e− 2 (β| p|) Le)ν b (xk )PE k=1 |ν |
≤ M E a t 2ν a e(β E) (M E + 1) (Le)ν b |e−(β| p|) (Le)ν b 2
2
νb
+ (N − 1) sup |(Le) |e i = j
≤
2 |ν| 2M E t ν e(β E)
−(β| p|)2
νb
U (xi − x j )(Le) |
√
1 + (N − 1) cε,β sup e
− m2 (1−ε)δ(xi −x j )
i = j
,
(4.15)
where in the last step we made use of the estimate |L± e j |e−(β| p|) U (x)L± e j | ≤ e j ||Tβ,± |2 e j ≤ e j |T 2 e j = t 2j and the fact that t j ≤ 1 which follows from (3.9). Substituting inequality (4.15) to formula (4.12), estimating t 2µ ≤ t µ and recalling that Sµ,ν = 0 for |µ| > M E or |ν| > M E we obtain the bound from the statement of the proposition. 2
It is now straightforward to estimate the p-norms of the map E . Substituting the bound from the above proposition to formula (4.1) and proceeding like in estimate (3.16) we obtain E p,x1 ,...,x N √
≤ (4 2)(25 M E ) M E e
1 2 2 (β E)
[M ] E
4p p 2
T k1
1+
√
m
cε,β (N − 1)e− 2 (1−ε)δ(x)
1 2
.
k=0
(4.16) It is clear from the above relation that lim supδ(x)→∞ E p,x1 ,...,x N satisfies a bound which is independent of N . Consequently, we get Theorem 4.5. Condition N holds in massive scalar free field theory for arbitrary dimension of space s.
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5. Conclusion and Outlook In this work we proposed and verified in massive scalar free field theory the new Condition N . Since this phase space criterion encodes the firm physical principle that energy is additive over isolated subsystems, we expect that it holds in a large family of models. In fact, we will show in a future publication that massless scalar free field theory also satisfies this condition for s ≥ 3. We recall that this model contains an infinite family of pure, regular vacuum states which are, however, mutually energy-disconnected [16]. In view of Theorem 2.2 (b), this decent vacuum structure is related to phase space properties of this model, as anticipated in [19]. Apart from more detailed information about the phase space structure of massive free field theory, our discussion offers also some new insights into the harmonic analysis of translation automorphisms. First, we recall from [11] that in all local, relativistic quantum field theories there holds the bound sup
ϕ∈T E,1
p))|2 < ∞, d s p| p|s+1+ε |ϕ( A(
(5.1)
p)), restricted to for any ε > 0, uniformly in A ∈ A(O)1 . It says that the distribution ϕ( A( the domain { p | | p| ≥ δ} for some δ > 0, is represented by a square integrable function, but at p = 0 it may have a power like singularity which is not square integrable. It turns out, however, that in massive scalar free field theory this distribution has a milder behavior at zero than one might expect from (5.1). Making use of Lemma 4.1 (b) and going through our argument once again one can easily establish that there holds, uniformly in A ∈ A(O)1 , ˚ x ))|2 < ∞, d s x|ϕ( A(
sup
ϕ∈T E,1
(5.2)
where A˚ = A − ω0 (A)I . By the Plancherel theorem, we obtain sup
ϕ∈T E,1
˚ p))|2 < ∞, A( d s p|ϕ(
(5.3)
˚ p)) is represented by a square integrable function. Consequently, i.e. the distribution ϕ( A( p)) can deviate from square integrability only by a delta-like singularity at p = ϕ( A( 0. The above reasoning demonstrates the utility of phase space methods in harmonic analysis of automorphism groups [29]. One may therefore expect that they will be of further use in this interesting field. Acknowledgement. This work is a part of a joint project with Prof. D. Buchholz to whom I am grateful for many valuable suggestions, especially for communicating to me the proof of Lemma 2.3. Financial support from Deutsche Forschungsgemeinschaft is gratefully acknowledged. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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W. Dybalski
A. Proof of Theorem 2.1 The argument is based on the following abstract lemma: Lemma A.1. Let X and Y be Banach spaces, Sk ∈ X ∗ for k ∈ {1, . . . , N } and τ ∈ Y N given by be s.t. τ = 1. Then the ε-content of the map : X → Y ⊗ Csup (ϕ) = τ (S1 (ϕ), . . . , S N (ϕ)), ϕ ∈ X,
(A.1)
satisfies the bound N (ε) ≤ (4eN ) where 2 = supϕ∈X 1 (
N
k=1 |Sk (ϕ)|
27 π 22 ε2
,
(A.2)
2 ) 21 .
Proof. Fix ε > 0 and let J0 = {(n 1 + in 2 )ε | n 1 , n 2 ∈ Z}. For √ each k ∈ {1, . . . , N } and ϕ ∈ X 1 we choose Jk (ϕ) ∈ J0 so that |Sk (ϕ) − Jk (ϕ)| ≤ 2ε and |Jk (ϕ)| ≤ |Sk (ϕ)|. Define the set J = {J1 (ϕ), . . . , J N (ϕ) | ϕ ∈ X 1 } of all N -tuples appearing in this way. We claim that #J ≥ N (4ε) . In fact, assume that there are ϕ1 , . . . , ϕ K ∈ X 1 , K > #J, s.t. for i = j there holds 4ε < (ϕi ) − (ϕ j ) =
sup
k∈{1,...,N }
|Sk (ϕi ) − Sk (ϕ j )|.
(A.3)
ˆ depending on (i, j), that 4ε < |S ˆ (ϕi )− S ˆ (ϕ j )|. Consequently, Then there exists such k, k k by a 3ε-argument √ |Jkˆ (ϕi ) − Jkˆ (ϕ j )| ≥ |Skˆ (ϕi ) − Skˆ (ϕ j )| − 2 2ε > ε, (A.4) which shows that there are at least K different elements of J in contradiction to our assumption. 22 , assume In order to estimate the cardinality of the set J we define M = ε2 2
for the moment that 0 < M ≤ 2N and denote by VM (R) ≤ e2π R the volume of the M-dimensional ball of radius R. Then
√ 2N M 1≤ 2 VM (2 M) ≤ (4N e)8π M . (A.5) #J ≤ M n 1 ,...,n 2N ∈Z n 21 +···+n 22N ≤M
We note that each admissible combination of integers n 1 , . . . , n 2N contains at most M non-zero entries. Thus to estimate the above sum we pick M out of 2N indices and M consider √ the points (n i1 , . . . , n i M ) ∈ Z which belong to the M-dimensional ball of M. Each radius √ such point is a vertex of a unit cube which fits into a ball of radius √ As in M dimensions a cube 2 M (since M is the length of the diagonal of the cube). √ has 2 M vertices, there can be no more than 2 M VM (2 M) points (n i1 , . . . , n i M ) ∈ Z M satisfying the restriction n i21 +· · ·+n i2M ≤ M. In the case M ≥ 2N a more stringent bound (uniform in N ) can be established by a similar reasoning. For M = 0 there obviously holds #J = 1.
Sharpened Nuclearity Condition and Uniqueness of the Vacuum in QFT
539
Proof of Theorem 2.1. Fix 0 < p < 23 . Then Condition N provides, for any δ > 0, a decomposition of the map E into rank-one mappings n ( · ) = τn Sn ( · ), where τn ∈ A(O)∗ and Sn ∈ T˚E∗ , s.t. ∞
1
p
p n x1 ,...,x N
≤ (1 + δ) E p,x1 ,...,x N .
(A.6)
n=1
Assuming that the norms n x1 ,...,x N are given in descending order with n, we obtain the bound n x1 ,...,x N ≤
(1 + δ) E p,x1 ,...,x N . n 1/ p
(A.7)
Similarly, we can decompose the map E,x1 ,...,x N into a sum of maps n of the form n (ϕ) = n (αx∗1 ϕ), . . . , n (αx∗N ϕ) = τn Sn (αx∗1 ϕ), . . . , Sn (αx∗N ϕ) .
(A.8)
Now we can apply Lemma A.1 with τ = τn / τn and Sk ( · ) = τn Sn (αx∗k · ). From estimate (A.7) we obtain n 2 = sup (
N
ϕ∈T˚E,1 k=1
1
τn 2 |Sn (αx∗k ϕ)|2 ) 2 = n x1 ,...,x N ≤
(1 + δ) E p,x1 ,...,x N . n 1/ p
(A.9)
Substituting this inequality to the bound (A.2) we get N (ε)n ≤ (4eN )
27 π(1+δ)2 E 2p,x ,...,x 1 N ε2 n 2/ p
.
(A.10)
We conclude with the help of Lemmas 2.3 and 2.4 from [21] that the ε-content of the map E,x1 ,...,x N satisfies N (ε) E,x1 ,...,x N ≤
∞
N (εn )n
(A.11)
n=1
for any sequence {εn }∞ 1 s.t.
∞
ε n=1 εn ≤ 4 . We choose εn =
n ε 4 ∞
− 32p − 32p
, make use of
n 1 =1 n 1
the bounds (A.10) and (A.11), and take the infinum w.r.t. δ > 0. There follows N (ε) E,x1 ,...,x N ≤ (4eN )
211 π E 2p,x ,...,x 1 N ε2
(
∞
n=1 n
− 32p 3 )
.
(A.12)
With the help of Condition N we obtain the bound in the statement of Theorem 2.1.
540
W. Dybalski
B. Proof of Theorem 3.1 Since the expansion of E into rank-one maps which appears in Theorem 3.1 differs slightly from those which are considered in the existing literature [19,20], we outline here the construction. Proof of Theorem 3.1. First, we recall from [20], Sect. 7.2.B. that given any pair of multiindices µ = (µ+ , µ− ) and an orthonormal sequence of J -invariant vectors (e.g. {e j }∞ 1 ), there exist weakly continuous linear functionals φµ on A(O) s.t. 1
−
φµ (W ( f )) = e− 2 f e| f + µ e| f − µ , 2
+
(B.1)
which satisfy the bound 1
φµ ≤ 4|µ| (µ!) 2 ,
(B.2)
where µ! = µ+ !µ− !. These functionals can be constructed making use of the equality (|[a(e1 ), [. . . , [a(ek ), [a ∗ (ek+1 ), [. . . , [a ∗ (el ), W ( f )], . . .]) =e
− 21 f 2
k
en 1 |i f
n 1 =1
l
i f |en 2 .
(B.3)
n 2 =k+1
Next, we evaluate the Weyl operator on some ϕ ∈ T˚E , rewrite it in a normal ordered form and expand it into a power series ϕ(W ( f )) 1
= e− 2 f
2
+
m ± ,n ± ∈N0
−
+
i m +n +2m + − + − ϕ(a ∗ ( f + )m a ∗ ( f − )m a( f + )n a( f − )n ). (B.4) + − + − m !m !n !n !
Subsequently, we expand each function f ± in the orthonormal basis {e j }∞ 1 of J inva± ± . Then, making use of the riant eigenvectors of the operator T : f = ∞ e e | f j=1 j j multinomial formula, we obtain
a ∗ ( f + )m = +
µ+ ,|µ+ |=m +
m+! + + e| f + µ a ∗ (L+ e)µ + µ !
(B.5)
and similarly in the remaining cases. Altogether we get ϕ(W ( f )) =
i |µ+ |+|ν + |+2|µ− | µ!ν!
µ,ν
=
φµ+ν (W ( f ))ϕ(a ∗ (Le)µ a(Le)ν )
τµ,ν (W ( f ))Sµ,ν (ϕ),
(B.6)
µ,ν |µ+ |+|ν + |+2|µ− |
where τµ,ν ( · ) = i φµ+ν ( · ). We recall that in the massive case Sµ,ν = 0 if µ!ν! |ν| > M E or |µ| > M E . Consequently, for the relevant indices there holds τµ,ν ≤
4|µ|+|ν| (µ!ν!)
1 2
(µ + ν)! µ!ν!
1 2
5
≤ 2 2 (|µ|+|ν|) ≤ 25M E ,
(B.7)
Sharpened Nuclearity Condition and Uniqueness of the Vacuum in QFT
541
where we made use of the bound (B.2) and properties of the binomial coefficients. Now it follows from estimate (3.16) that for any p > 0, [M ] 4 E pM E p p 5 pM E p k τµ,ν Sµ,ν ≤ 2 ME T 1 . (B.8) µ,ν
k=0
In view of this fact and of weak continuity of the functionals τµ,ν , equality (B.6) can be extended to any A ∈ A(O). In other words E (ϕ)(A) = ϕ(A) = τµ,ν (A)Sµ,ν (ϕ), (B.9) µ,ν
what concludes the proof of the theorem.
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23. Araki, H.: Mathematical Theory of Quantum Fields. Oxford: Oxford Science Publications, 1999 24. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Part II: Fourier Analysis, Self-Adjointness. New York-San Francisco-London: Academic Press, 1975 25. Buchholz, D., D’Antoni, C., Longo, R.: Nuclear Maps and Modular Structures II: Applications to Quantum Field Theory. Commun. Math. Phys. 129, 115–138 (1990) 26. Buchholz, D., Jacobi, P.: On the Nuclearity Condition for Massless Fields. Lett. Math. Phys. 13, 313–323 (1987) 27. Fredenhagen, K.: A Remark on the Cluster Theorem. Commun. Math. Phys. 97, 461–463 (1985) 28. Sakai, S.: C ∗ -Algebras and W ∗ -Algebras. Berlin-Heidelberg-New York: Springer-Verlag, 1971 29. Arveson, W.: The harmonic analysis of automorphism groups. In: Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., 38, Providence, R.I.: Amer. Math. Soc., 1982, pp. 199–269 Communicated by Y. Kawahigashi
Commun. Math. Phys. 283, 543–578 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0571-9
Communications in
Mathematical Physics
Calibrations and T–Duality Florian Gmeiner1 , Frederik Witt2 1 NIKHEF, Kruislaan 409, 1098 SJ Amsterdam, The Netherlands. E-mail:
[email protected] 2 NWF I – Mathematik, Universität Regensburg, D-93040 Regensburg, Germany.
E-mail:
[email protected] Received: 31 August 2007 / Accepted: 19 March 2008 Published online: 23 July 2008 – © Springer-Verlag 2008
Abstract: For a subclass of Hitchin’s generalised geometries we introduce and analyse the concept of a structured submanifold which encapsulates the classical notion of a calibrated submanifold. Under a suitable integrability condition on the ambient geometry, these generalised calibrated submanifolds minimise a functional occurring as D–brane energy in type II string theories. Further, we investigate the behaviour of calibrated cycles under T–duality and construct non–trivial examples.
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . Generalised Geometries . . . . . . . . . 2.1 The generalised tangent bundle . . . 2.1.1 Basic setup. . . . . . . . . . . . 2.1.2 Twisting with an H –flux. . . . . 2.2 Generalised metrics . . . . . . . . . 2.3 G L × G R –structures . . . . . . . . 3. Generalised Calibrations . . . . . . . . . 3.1 Classical calibrations . . . . . . . . 3.2 Generalised calibrated planes . . . . 3.3 Generalised calibrated submanifolds 4. Branes in String Theory . . . . . . . . . 4.1 World–sheet point of view . . . . . 4.2 Target space point of view . . . . . 5. T–Duality . . . . . . . . . . . . . . . . . 5.1 The Buscher rules . . . . . . . . . . 5.2 Integrability . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .
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1. Introduction In this paper, we introduce and investigate a notion of structured submanifold for a subclass of Hitchin’s generalised geometries. The most basic examples are provided by Harvey’s and Lawson’s calibrated submanifolds [19], a notion which has been intensively studied in Riemannian geometry, see for instance [28] and references therein. Recall that on a Riemannian manifold (M, g), a differential form ρ of pure degree p is said to define a calibration, if for any oriented p–dimensional subspace L ⊂ Tx M with induced Riemannian volume form L , the inequality gx (ρx , L ) ≤ L g = 1 holds ( · g denoting the norm on p T ∗ M induced by g). Submanifolds whose tangent spaces meet the bound everywhere are said to be calibrated by ρ. Examples comprise Kähler manifolds with ρ the Kähler form, for which the calibrated submanifolds are precisely the complex submanifolds. More generally, there are natural calibrations for all geometries on Berger’s list, and calibrated submanifolds constitute a natural class of structured submanifolds for these. An important property of calibrated submanifolds is to be locally homologically volume–minimising if the calibration is closed. This links into string theory, where the notion of a calibrated submanifold can be adapted to give a geometric interpretation of D–branes in type I theory, extended objects that extremalise a certain energy functional (cf. for instance [16,17,32] or the survey [15]). On the other hand, geometries defined by forms are the starting point of generalised geometry as introduced in Hitchin’s foundational article [22]. The rôle of p–forms is now played by even and odd forms which we can interpret as spinor fields for the bundle T M ⊕ T ∗M endowed with its natural orientation and inner product which is contraction. The distinctive feature of this setup is that it can be acted on by both diffeomorphisms and 2–forms or B–fields B ∈ 2 (M): These map X ⊕ ξ ∈ T M ⊕ T ∗M to X ⊕ (X B/2 + ξ ) and spinor fields ρ ∈ S± ∼ = ev,od (M) to e B ∧ ρ = (1 + B + B ∧ B/2 + · · · ) ∧ ρ. ∼ In particular, this action preserves the natural bilinear form · , · on S = S+ ⊕ S− = ∗ (M) and transforms closed spinor fields into closed ones if B itself is closed. Following this scheme, a generalised calibration will be an even or odd form which we view as a T M ⊕ T ∗M–spinor field ρ. The rôle of the Riemannian metric is now played by a generalised Riemannian metric G, an involution on T M ⊕ T ∗M compatible with the inner product. Further, a generalised Riemannian metric also induces a norm ·G on S± . Both G and the norm transform naturally under B–fields. However, it is not altogether clear what the B–field action ought to be on a submanifold. The solution to this is, taking string theory as guidance, to consider pairs (L , F) consisting of an embedded submanifold j L : L → M and a closed 2–form F ∈ 2 (L). A closed B–field B then acts on this by (L , F) → (L , F + jL∗ B). With such a pair, we can associate the pure spinor field (cf. Sect. 3.2) τ L ,F = e F ∧ θ k+1 ∧ . . . ∧ θ n ∈ S|L , where the θ i ∈ 1 (M)|L define a frame of the annihilator N ∗L ⊂ T ∗M of L. The calibration condition we adopt is
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ρ, τ L ,F ≤τ L ,F G , which therefore makes essential use of the generalised Riemannian metric.1 The first important result we prove is this (cf. Theorem 3.6): • Any spinor field of the form ρ ev,od = e B ∧ Re [ L ⊗ R ]ev,od defines a calibration, where [ L ⊗ R ] is the complex form naturally associated with the bi–spinor L ⊗ R ∈ ⊗ , for the spinor bundle associated with T M. • For such a ρ, the pair (L , F) is calibrated if and only if A( L ) = c
e−B ∧ τ L ,F · R , e−B ∧ τ L ,F G
where A denotes the so–called charge conjugation operator, c a complex constant and · the usual Clifford action of T M on . This type of calibrations defines what we call a G L × G R –structure. Generalised G 2 –, Spin(7)– and SU (m)–structures [38–40], which are natural generalisations of the geometries of holonomy G 2 , Spin(7) and SU (m) to Hitchin’s setup, are examples of these. This result also substantially extends work of Dadok and Harvey [14,18] on classical calibrations defined by spinors. Further, calibrated pairs (L , F) locally minimise the functional E(L , F) = τ L ,F G L
if the calibration form is closed. As we will explain, this integral represents the so–called Dirac–Born–Infeld term of the D–brane energy. Considering an inhomogeneous equation of the form dρ = ϕ also accounts for the so–called Wess–Zumino term. We discuss the relationship of calibrated pairs (L , F) with D–branes in type II string theory as we go along. In particular, our definition of a generalised calibration embraces the cases discussed in [31,33,34] in the physics literature. The final part of the paper is devoted to T–duality. In string theory, this duality interchanges type IIA with type IIB theory and plays a central rôle in the SYZ– formulation of Mirror symmetry [36]. It is also of considerable mathematical interest (cf. for instance [7–9]). Locally, T–duality is enacted according to the Buscher rules [10] which have a natural implementation in generalised geometry. Our final result (Theorem 5.6) states that if the entire data are invariant under the flow of the vector field along which we T–dualise, the calibration condition is stable under T–duality (as it is expected from a physical viewpoint). We will use this to construct some non–trivial examples of calibrated pairs. Finally, we remark that our definition of a generalised calibration also makes sense for twisted generalised geometries [22], as we shall explain in the paper. The outline of the paper is this. Based on [22] and [23], we introduce the setup of generalised geometry in Sect. 2. We thereby considerably extend ideas developed in [40] by generalising the theory to G L × G R –structures in arbitrary dimension and by incorporating twisted structures. Section 3 gives the definition of a generalised calibration and calibrated pairs. We investigate their properties and discuss first examples. Section 4 makes the link with string theory. Finally, Sect. 5 is devoted to T–duality and gives further examples. 1 For various other notions of generalised submanifolds in generalised geometry, where no metric structure is involved, see for instance [3] in the mathematical and [12] and [41] in the physics literature.
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2. Generalised Geometries 2.1. The generalised tangent bundle. 2.1.1. Basic setup. Let M n be an n–dimensional manifold. The underlying geometric setup we will use throughout this paper is known under the name of generalised geometry and was introduced in Hitchin’s foundational paper [22]. Its key feature is to be acted on by so–called B–field transformations induced by 2 (M). To implement these, we need to pass from the tangent bundle T M to the bundle T M ⊕ T ∗M. We endow T M ⊕ T ∗M with (a) its canonical orientation and (b) the metric of signature (n, n) given by contraction, namely2 (X ⊕ ξ, X ⊕ ξ ) = X ξ = ξ(X ).
(1)
Therefore, this vector bundle is associated with a principal S O(n, n)–fibre bundle PS O(n,n) . As G L(n) ⊂ S O(n, n), PS O(n,n) is actually obtained as an extension of the frame bundle PG L(n) associated with T M. Now take B ∈ 2 (M) and think of it as a linear map B : T M → T ∗M sending X to the contraction X B = B(X, ·). Then we define the corresponding B–field transformation by 1 X ⊕ ξ ∈ T M ⊕ T ∗M → X ⊕ ξ + X B . 2 This transformation can be interpreted as an exponential: For a point q ∈ M, Tq M ⊕ Tq∗M, (· , ·)q is an oriented pseudo–Euclidean vector space of signature (n, n). Its symmetry group is S O(n, n), whose Lie algebra we may identify with so(n, n) ∼ = 2 (Tq M ⊕ Tq∗M) ∼ = 2 Tq M ⊕ Tq∗M ⊗ Tq M ⊕ 2 Tq∗M. Hence, Bq ∈ 2 Tq∗M becomes a skew–symmetric endomorphism of Tq M ⊕ Tq∗M under the identification ζ ∧ η(X ⊕ ξ ) = (ζ, X ⊕ ξ )ξ − (η, X ⊕ ξ )ζ = X (ζ ∧ η)/2. Applying the exponential map exp : so(n, n) → S O(n, n), we obtain exp(Bq )(X ⊕ ξ ) = (
∞ Bql l=0
l!
)(X ⊕ ξ ) = X ⊕ ξ +
1 X Bq 2
as Bq2 (X ⊕ ξ ) = Bq (X Bq )/2 = 0. With respect to the splitting Tq M ⊕ Tq∗M, we will also use the matrix representation
Id 0 Bq e = 1 . Id 2 Bq Further, T M ⊕ T ∗M admits a natural bracket which extends the usual vector field bracket [· , ·], the so–called Courant bracket [13]. It is defined by 1 [[X ⊕ ξ, Y ⊕ η]] = [X, Y ] + L X η − LY ξ − d(X η − Y ξ ), 2 2 Our conventions slightly differ from [22] and [38] which results in different signs and scaling factors.
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where L denotes the Lie derivative. If B is a closed 2–form, then exp(B) commutes with [[· , ·]]. The inclusion G L(n) S O(n, n) can be lifted, albeit in a non–canonical way, to Spin(n, n). It follows that the S O(n, n)–structure is always spinnable. In the sequel, we assume all base manifolds to be orientable, hence the canonic lift of the inclusion3 G L(n)+ → S O(n, n) induces a canonic spin structure PG L(n)+ ⊂ PSpin(n,n)+ with associated spinor bundle S(T M ⊕ T ∗M) = PG L(n)+ ×G L(n)+ S. In order to construct the spin representations S for Spin(n, n), we consider its vector representation Rn,n which we split into two maximal isotropic subspaces, Rn,n = W ⊕ W . Using the inner product g = g n,n , we may identify W with W ∗ . Under this identification, g(w , w) = w (w)/2, that is, we recover precisely (1). Now X ⊕ ξ ∈ W ⊕ W ∗ acts on ρ ∈ ∗ W ∗ via (X ⊕ ξ ) • ρ = −X ρ + ξ ∧ ρ. This action squares to minus the identity, and by the universal property of Clifford algebras, it extends to an algebra isomorphism
Cliff(n, n) ∼ = Cliff(W ⊕ W ∗ ) ∼ = End(∗ W ∗ ).
(2)
Our initial choice of a splitting into isotropic subspaces is clearly immaterial, as Cliff(n, n) is a simple algebra, hence the representation (2) is unique up to isomorphism. The spin representation of Spin(n, n) is therefore isomorphic with S = ∗ W ∗ and can be invariantly decomposed into the modules of chiral spinors S± = ev,od W ∗ . Moreover, after choosing some trivialisation of n W ∗ , we can define the bilinear form ρ, τ = [ρ ∧ τ ]n ∈ n W ∗ ∼ = R. · is the sign–changing Here, [·]n denotes projection on the top degree component and operator defined on forms of degree p by p = (−1) p( p+1)/2 α p . α Then (X ⊕ ξ ) • ρ, τ = (−1)n ρ, (X ⊕ ξ ) • τ , and in particular, this form is Spin(n, n)+ –invariant. It is symmetric for n ≡ 0, 3 mod 4 and skew for n ≡ 1, 2 mod 4, i.e. ρ, τ = (−1)n(n+1)/2 τ, ρ . Moreover, S + and S − are non–degenerate and orthogonal if n is even and totally isotropic if n is odd. 3 The notation G refers to the identity component of a given Lie group G. +
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We also have an induced action of the Lie algebra so(n, n). By exponentiation, B ∈ 2 W ∗ acts on ρ ∈ S± via e B • ρ = (1 + B +
1 B ∧ B + · · · ) ∧ ρ. 2
This exponential links into the B–field transformation on W ⊕ W ∗ via the 2-1 covering map π0 : Spin(n, n) → S O(n, n), namely Id 0 π0∗ (B) B 2B . (3) π0 (e Spin(n,n) ) = e S O(n,n) = e S O(n,n) = B Id On the other hand, the element
Alm ξ l ⊗ X m (ρ) =
Alm ξ l ⊗ X m ∈ gl(n) ⊂ so(n, n) acts on ρ via
1 m l 1 Al [ξ , X m ] • ρ = Tr(A) + A∗ ρ, 2 2
where A∗ ρ denotes the natural extension to forms of the dual representation of gl(n). Consequently, √ S± ∼ = ev,od W ∗ ⊗ n W as a G L(n)+ –space, and we obtain for the spinor bundle √ S± = S(T M ⊕ T ∗M)± ∼ = PG L(n)+ ×G L(n)√+ ev,od Rn∗ ⊗ n Rn = ev,od T ∗M ⊗ n T M. √ The choice of a trivialisation of the line bundle n T M, that is, of a nowhere vanishing n–vector field ν, induces thus an isomorphism between T M ⊕ T ∗M–spinor fields and even or odd differential forms. Put differently, it reduces the structure group of M from G L(n)+ to S L(n), for which the bundles S± and ev,od T ∗M are isomorphic. We shall denote this isomorphism by Lν to remind ourselves that it depends on the choice of the n–vector ν. Moreover, from a G L(n)+ –point of view, the bilinear √ form · , · takes values in the reals, as for ρ, τ ∈ S ∼ = ∗ W ∗ ⊗ n W , we have √ √ ρ, τ ∈ n W ∗ ⊗ n W ⊗ n W ∼ = R. In particular, using the isomorphism Lν , we get ν (τ )]n ∈ C ∞ (M). ρ, τ = ν [Lν (ρ) ∧ L 2.1.2. Twisting with an H –flux. More generally still, there is a twisted version of previously discussed setup which is locally modeled on T M ⊕ T ∗M, leading to notion of a generalised tangent bundle [23]. Let H ∈ 3 (M) be a closed 3–form (in physicists’ terminology, this is so–called H –flux). Choose a convex cover of coordinate neighbourhoods {Ua } of whose induced transition functions are sab : Uab = Ua ∩ Ub → G L(n)+ . Locally, H|Ua = d B (a) for some B (a) ∈ 2 (Ua ). Then we can define the 2–forms (a)
(b)
β (ab) = B|Uab − B|Uab ∈ 2 (Uab )
the the the M,
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which are closed (this will be of importance later). On Uab , we can use the coordinates provided by either the chart Ua or Ub , and consider β (ab) as a map from Uab → 2 Rn∗ ; we denote this map by βa(ab) or βb(ab) . Any two trivialisations are therefore related by (ab) ∗ β (ab) and similarly for all other trivialised sections. Out of the transition = sab βa b functions 0 sab ∈ G L(n)+ ⊂ S O(n, n)+ Sab = −1tr 0 sab of the bundle T M ⊕ T ∗M, we define new transition functions (ab)
σab = Sab ◦ e2βb
(ab)
= e2βa
◦ Sab : Uab → S O(n, n)+ .
These satisfy indeed the cocycle condition on Uabc = ∅, i.e. (ab)
σab ◦ σbc = Sab ◦ e2βb = Sab ◦ e
(bc)
◦ e2βb
(a)
(b)
◦ Sbc
(b)
(c)
2(Bb −Bb +Bb −Bb )
= Sab ◦ Sbc ◦ e
◦ Sbc
(a) (c) 2(Bc −Bc )
(ac)
= Sac ◦ e2βc −1 = σca .
(4)
Therefore, we can define a vector bundle, the generalised tangent bundle, by E = E(H ) = Ua × (Rn ⊕ Rn∗ )/ ∼σab , where two triples (a, p, X ⊕ ξ ) and (b, q, Y ⊕ η) are equivalent if and only if p = q and X ⊕ ξ = σab ( p)(Y ⊕ ξ ). A section t ∈ (E) can therefore be regarded as a family of smooth maps {ta : Ua → Rn ⊕ Rn∗ } transforming under sa = σab (tb ). As the notation suggests, the generalised tangent bundle depends, up to isomorphism, only on the closed 3–form H . Indeed, assume we are given a different convex cover {Ua } together with locally defined 2–forms B (a) ∈ 2 (Ua ) such that H|Ua = d B (a) . This results in (ab)
= S ◦ exp(2β a new family of transition functions σab ). Now on the intersection ab b (a) (a) (a) Va = Ua ∩ Ua , the locally defined 2–forms G = B|Va − B|Va are closed, and one a(a) ) to define a gauge transformation, i.e. readily verifies the family G a = exp(G = G a−1 ◦ σab ◦ G b σab
on Vab = ∅.
respectively are isomorphic. In particular, the bundles defined by the families σab and σab ∗ The sections of the bundle E(H ) relate to (T M ⊕ T M) via the map K : (T M ⊕ T ∗ M) → (E(H )) defined by (a)
Ka (X a ⊕ ξa ) = e2Ba (X a ⊕ ξa ) = X a ⊕ Ba(a) (X a ) + ξa . This is indeed well–defined as (ba)
σba Ka (X a ⊕ ξa ) = eβb
(a)
X b ⊕ Bb (X b ) + ξb = X b ⊕ Bb(b) (X b ) + ξb = Kb (Sba (X a ⊕ ξa )) .
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Since the transition functions σab take values in S O(n, n), the invariant orientation and inner product (· , ·) on Rn ⊕ Rn∗ render E an oriented pseudo–Riemannian vector bundle. Furthermore, twisting with the closed 2–forms β (ab) preserves the Courant bracket, which is therefore also defined on E(H ). Since any other choice of a trivialisation leads to a gauge transformation induced by closed B–fields, this additional structure does not depend on the choice of a trivialisation either. For the pseudo–Riemannian structure on E thus defined, we can also consider spinor fields. A canonic spin structure is given by (ab)
σab = Sab • eβb
(ab)
= eβa
• Sab ,
where Sab ∈ G L(n)+ ⊂ Spin(n, n)+ denotes the lift of Sab and where we exponentiate β (ab) to Spin(n, n)+ , so that π0 ◦ σab = σab . The even and odd spinor bundles associated with E are S(E)± = Ua × S± / ∼ σab . a
An E–spinor field ρ is thus represented by a collection of smooth maps ρa : Ua → S± with ρa = σab • ρb . In order to make contact with differential forms, let ν be a trivialisation of n T M which we think of as a family of maps νa = λa−2 ν0 ∈ n Rn with √ λa−2 ∈ C ∞ (M). It follows that λb = sab · λb . The map LνE : (S(E)± ) → ev,od (M) induced by (a)
LνEa : (ρa : Ua → S± ) → (e−Ba ∧ λa · ρa : Ua → ev,od Rn∗ ) is an isomorphism, and we usually drop the subscript E to ease notation. This transforms correctly under the action of the transition functions sab on ev,od T ∗M, as one can show by using the fact that ρa = σab • ρb . Indeed, over Uab we have (b) (b) ∗ ∗ sab (e−Bb ∧ λb · ρb ) = λa det sab · e−Ba ∧ sab ρb (a)
(ab)
= λa · e−Ba ∧ eβa = λa · e
(a) −Ba
• Sab • ρb
∧ ρa ,
∗ ◦ Lν . The operators Lν and K relate via or equivalently, Laν ◦ σab = sab b
Lν (K(X ⊕ ξ ) • ρ) = −X Lν (ρ) + ξ ∧ Lν (ρ) =: (X ⊕ ξ ) • Lν (ρ). In the same vein, the Spin(n, n)–invariant as above a globally defined form · , · induces ν n ν inner product on (S) by ρ, τ = ν [L (ρ) ∧ L (τ )] . Twisting with closed B–fields allows the definition of a further operator, namely dν : (S(E)± ) → (S(E)∓ ) , (dν ρ)a = λa−1 · da (λa · ρa ), where da is the usual differential applied to forms Ua → ∗ Rn∗ , i.e. (dα)a = da αa . This definition gives indeed rise to an E–spinor field, for (ab) βa ∗ σab • (dν ρ)b = λ−1 ∧ det sab · sab db (λb · ρb ) b ·e (ab) ∗ (λb · ρb ) = λa−1 · da eβa ∧ sab = (dν ρ)a .
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The operator dν squares to zero and therefore induces an elliptic complex on (S(E)± ). It computes the so–called twisted cohomology, where one replaces the usual differential d of de Rham cohomology by the twisted differential d H = d + H ∧. Proposition 2.1. Let ρ ∈ (S(E)). Then Lν (dν ρ) = d H Lν (ρ). The assertion follows from a straightforward local computation. Remark. The identification of S(E)± with even or odd forms depends on the choice of the local B–fields B (a) . However, any other choice leads, as we have seen, to a gauge (a) , which therefore preserves the closeness of the transformation by closed 2–forms G induced differential forms. 2.2. Generalised metrics. In addition to the 3–form flux H and a nowhere vanishing n–vector field ν ∈ (n T M), we now wish to build in a Riemannian metric to define generalised Riemannian metrics. Again, we first consider the bundle T M ⊕ T ∗M. Let us think of a Riemannian metric as a map g : T M → T ∗M, which is invertible by virtue of the non–degeneracy. With respect to the decomposition T M ⊕ T ∗M, we define the endomorphism 0 g −1 . G0 = g 0 The B–transform of G0 induced by B ∈ 2 (M) is G B = e2B ◦ G0 ◦ e−2B Id Id 0 0 g −1 ◦ ◦ = −B B Id g 0 −g −1 B g −1 = . g − Bg −1 B Bg −1
0 Id
To simplify notation, we will usually write G (unless we want to emphasise the B–field) and refer to G as the generalised metric induced by g and B. Note that G squares to the identity, and its ±1–eigenspaces V ± give a metric splitting, i.e. an orthogonal decomposition T M ⊕ T ∗M = V + ⊕ V − into maximally positive/negative definite subbundles V ± . The restriction of (· , ·) to V ± will be denoted by g± . Conversely, any metric splitting gives rise to an honest Riemannian metric g and B ∈ 2 (M), whose corresponding generalised metric has V± as ±1–eigenspaces [38]. In terms of structure groups, we note that the global decomposition of T M ⊕ T ∗M into V + ⊕ V − gives rise to a reduction from S O(n, n)+ to A+ 0 2B −2B ∼ | A± : V ± → V ± such that e S O(n, 0) × S O(0, n)e ={ 0 A − A∗± g± = g± and det A± = 1}
= S O(V + ) × S O(V − ),
where S O(n, 0) × S O(0, n) is the subgroup preserving the eigenspace decomposition of G0 .
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Definition 2.1. ([23,38]). A generalised (Riemannian) metric for the generalised tangent bundle E(H ) is the choice of a maximally positive definite subbundle V + . Put differently, a generalised metric induces a splitting of the exact sequence 0 → T ∗M → E → T M → 0.
(5)
We denote the lift of vector fields X ∈ (T M) to sections in (V + ) by X + . Locally, X + corresponds to smooth maps X a+ : Ua → Rn ⊕ Rn∗ with X a+ = σab (X b+ ), and X a+ = X a ⊕ Pa+ X a for linear isomorphisms Pa+ : Rn → Rn∗ . From the transformation rule on {X a+ }, we deduce ∗ Pb+ sba βa(ab) = Pa+ − sab
(where X a = sab (X b )). As above, the symmetric part ga = (Pa+ + Pa+tr )/2 is positive definite, and since β (ab) is skew–symmetric, the symmetrisation of the right-hand side vanishes. Hence 1 + ∗ ∗ ∗ (P − sab Pb+ sba + Pa+tr − sab Pb+tr sba ) = ga − sab gb sba = 0, 2 a so that the collection ga : Ua → 2 Rn∗ of positive definite symmetric 2–tensors patches together to a globally defined metric. Conversely, a Riemannian metric g induces a gen(a) eralised Riemannian structure on E(H ): The maps Pa = Ba + ga induce local lifts of T M to E which give rise to a global splitting of (5). Proposition 2.2. A generalised Riemannian structure is characterised by the datum (g, H ), where g is an honest Riemannian metric and H a closed 3–form. Remark. Of course, the negative definite subbundle V − also defines a splitting of (5). (a) The lift of a vector field X is then induced by X a− = X a ⊕ Pa− X a with Pa− = −ga + Ba . In the presence of a metric, we can pick the canonic n–vector field locally given by νg|Ua = (det g|Ua )−1/2 ∂x 1 ∧. . .∧∂x n . We shall write L for Lνg . From our original choice ν this differs by a scalar function, i.e. ν = e2φ νg for φ ∈ C ∞ (M), so that Lν = e−φ L. We then write Lφ = Lν and dφ = dν . For reasons becoming apparent later, φ is referred to as the dilaton field. A generalised metric also induces further structure on spinor fields. On every fibre Tq M ⊕ Tq∗M, the straight generalised metric G0 acts as a composition of reflections. Namely, if e1 , . . . , en is an orthonormal basis of Tq M, then Vq± is spanned by vk± = ek ⊕ ±g(ek ) and G0 = Rv − ◦ · · · ◦ Rvn− ∈ O(n, n), 1
0 to Pin(n, n) is given, up to a where Rv − denotes reflection along vk− . Hence its lift G k sign, by the Riemannian volume form 0 = − = v − • · · · • vn− G Vp 1 0 is therefore globally well–defined. Note that G0 preserves or on Vq− . The operator G 0 preserves reverses the natural orientation on T M ⊕ T ∗M if n is even or odd, so that G
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or reserves the chirality of T M ⊕ T ∗M–spinor fields accordingly. There is an alternative description of its action which will turn out to be useful. For this, we denote by J the natural isomorphism between Cliff(T M, g) and ∗ T ∗M. Recall that for any X ∈ T M and a ∈ Cliff(T M, g) of pure degree, J(X · a) = −X J(a) + X ∧ J(a), J(a · X ) = (−1)deg(a) (X J(a) + X ∧ J(a)) . Moreover, g J(a) = J( a · g ), where g denotes the Riemannian volume form on T M as well as its image in Cliff(T M, g) under J by abuse of notation. If ∼ is the involution defined by a ev,od = ±a ev,od , then g · a = a · g if n is odd, while g · a = ± a · g for n even. By computing L d1− • . . . • dn− • J(J−1 (ρ)) = ±L J(ωg · J−1 (ρ)) , for 0 ρ) is equal to g L(ρ) where the sign depends on n = rk T M, we deduce that L(G for n odd and ρ of even or odd parity. For a B–field transformed n even and ± g L(ρ) Riemannian metric with V ± = exp(2B)(D ± ), we conjugate G0 by exp(2B) and therefore D − by exp(B) in view of (3). Hence V − • ρ = e B • D − • e−B • ρ. Up to signs, therefore coincides with the –operator in [38]. The twisted case follows easily, for G instance if n is even, (a)
(a)
La (V − a • ρa ) = La (e Ba • D − a • e−Ba • ρa ) = D − a • La (ρa ) = ga L a (ρa ). Proposition 2.3. If V + defines a generalised Riemannian metric, then the action of = V − on S(E)± is given by G n even: g L(ρ) = (L(ρ)) , L(Gρ) =: G n odd: g L(ρ) where g is the Riemannian metric induced by V + . Note that 2 = (−1)n(n+1)/2 I d G
τ = (−1)n(n+1)/2 ρ, Gτ . and Gρ,
is an isometry for · , · and defines a complex structure on S(E) if In particular, G n ≡ 1, 2 mod 4. Moreover, given a generalised metric, we define the bilinear form
Q± (ρ, τ ) = ±(−1)m ρ, Gτ
on S(E)± , n = 2m, 2m + 1.
Then for n = 2m (and similarly for n = 2m + 1),
Q± (ρ, τ ) = ±(−1)m ρ, Gτ
∧ ) ]n = ±(−1)m νg [L(ρ) ∧ g L(τ
= g (L(ρ), L(τ )) ,
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α ev,od . We shall denote the associated where we used the general rule α ev,od = ±(−1)m norm on S± by · G . If the manifold is spinnable, the presence of a generalised metric also implies a very useful description of the complexified spinor modules S(E)± ⊗ C as the tensor product n (T M) ⊗ n (T M) of the complex spin representation n of Spin(n, 0). From standard representation theory, we have algebra isomorphisms n even: End ( ) Cliff C (T M, ±g) ∼ = Cliff(T M ⊗ C, g C ) = n odd: EndC ( n ) ⊕ End ( ) . (6) n n C C The module n is the space of (Dirac) spinors and, as a complex vector space, is [n/2] isomorphic with C2 . Further, n carries an hermitian inner product q for which q(a · , ) = q( , a · ), a ∈ Cliff(T M ⊗ C, g C ). By convention, we take the first argument to be conjugate–linear. Restricting the isomorphism (6) to the Spin groups of signature ( p, q) yields the complex spin representation Spin( p, q) → G L( p+q ). If p+q is even, then there is a decomposition p+q = p+q,+ ⊕ p+q,− into the irreducible Spin( p, q)–representations p+q,± , the so–called Weyl spinors of positive and negative chirality. Finally, in all dimensions, there exists a conjugate–linear endomorphism A of
n (the charge conjugation operator in physicists’ language) such that [37] A(X · ) = (−1)m+1 X · A( ) and A2 = (−1)m(m+1)/2 I d, n = 2m, 2m + 1. In particular, A is Spin(n)–equivariant. Moreover, A reverses the chirality for n = 2m, m odd. We define a Spin(n)–invariant bilinear form (which, abusing notation, we also write A) by A( , ) = q(A( ), ), for which A( , ) = (−1)m(m+1)/2 A(, ) and A(X · , ) = (−1)m A( , X · ) if n = 2m, 2m + 1. We can inject the tensor product n ⊗ n into ∗ Cn by sending L ⊗ R to the form of mixed degree [ L ⊗ R ](X 1 , . . . , X n ) = A ( L , (X 1 ∧ . . . ∧ X n ) · R ) . In fact, this is an isomorphism for n even. In the odd case, we obtain an isomorphism by concatenating [· , ·] with projection on even or odd forms, which we write as [· , ·]ev,od . Since this map is Spin(n)–equivariant, it acquires global meaning over M, and we use the same symbol for the resulting map n (T M) ⊗ n (T M) → ev,od (M) (referred to as the fierzing map in the physics’ literature). Next we define [·,· ]ev,od
L−1
[·, ·]G ev,od : ( n (T M) ⊗ n (T M)) −→ ev,od (M) ⊗ C −→ (S(E)± ⊗ C) . A vector field X acts on T M–spinor fields via the inclusion T M → Cliff(T M, g) and Clifford multiplication. On the other hand, we can lift X to sections X ± of V ± which act on E–spinor fields via the inclusion V ± → Cliff(T M ⊕ T ∗M) and Clifford multiplication. To see how these actions are related by the map [· , ·]G , we use the fact that the orthogonal decomposition of T M ⊕ T ∗M into V + ⊕ V − makes Cliff(T M ⊕ T ∗M)
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Cliff(V − ).4 The vector bundles isomorphic with the twisted tensor product Cliff(V + )⊗ ± (V , g± ) are isometric to (T M, ±g) via the lifts X ∈ (T M) → X ± ∈ (V ± ). By Y → X + • Y − to Cliff(T M ⊕ T ∗M), we get a further isomorphism extending X ⊗ Cliff C (T M, −g) ∼ Cliff C (T M, g)⊗ = Cliff C (V + ⊕ V − , g+ ⊕ g− ). Proposition 2.4. We have [X · L ⊗ R ]G = (−1)n(n−1)/2 X + • [ L ⊗ R ]G , [ ⊗ Y · ]G = −Y − • [ ⊗ ]G . L
R
L
R
Proof. Fix a point q ∈ Ua and an orthonormal basis e1 , . . . , en of Tq M. To ease notation, we drop any reference to q and Ua and assume first Ba = 0. By definition and the properties of A recalled above, q A(e j · L ), e K · R e K [e j · L ⊗ R ] = K
= (−1)n(n−1)/2+1
q A( L ), e j · e K · R e K
= (−1)
n(n−1)/2+1
q A( L ), (−e j e K + e j ∧ e K ) · R e K
= (−1)
n(n−1)/2
K
K
q A( L ), e j e K · R e j ∧ (e j e K ) j∈K
− q A( L ), e j ∧ e K · R e j (e j ∧ e K ) j∈ K
= (−1)n(n−1)/2 −e j +g(e j )∧ • [ L ⊗ R ]. Compounding with L−1 , we therefore get [X · L ⊗ R ]G0 = X + • [ L ⊗ R ]G0 . For non–trivial local B–fields B (a) , we have Ba • (X ⊕ g(X )) • e−Ba • L−1 [ L ⊗ R ]|Ua [X · L ⊗ R ]G |Ua = e
= π0 (e Ba ) (X ⊕ g(X )) • [ L ⊗ R ]G |Ua = X + • [ L ⊗ R ]G |Ua . The second assertion follows in the same fashion.
A first, but important observation is that E–spinor fields corresponding to bi–spinor fields are “self–dual” in the following sense: The element V − in Cliff(V + ⊕ V − ) g , so that corresponds to 1⊗ L ⊗ R ]G = (−1)m [ L ⊗ g · R ]G G[ 4 The twisted tensor product ⊗ of two graded algebras A and B is defined on elements of pure degree as b = (−1)deg(b)·deg(a ) a · a ⊗ b · b . b · a ⊗ a⊗
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for n = 2m and L ⊗ R ]G = (−1)m [ L ⊗ G[ g · R ]G for n = 2m + 1. By standard Clifford representation theory, the action of the Riemannian volume form g on chiral spinor fields is given by g · ± = ±(−1)m(m+1)/2 i m ± , g · = (−1)m(m+1)/2 i m+1 ,
if n = 2m, if n = 2m + 1.
(7)
From this we deduce the Corollary 2.5. Let L ,R ∈ n . (i) If n = 2m and L ,R are chiral, then for R ∈ ± , L ⊗ R ]G = ±(−1)m(m−1)/2 i m [ L ⊗ R ]G . G[ (ii) If n = 2m + 1, then L ⊗ R ]G = (−1)m(m−1)/2 i m+1 [ L G[ ⊗ R ]G . 2.3. G L × G R –structures. Within the generalised setting, further reductions can be envisaged. For the theory of generalised calibrations we are aiming to develop in Section 3, a particular class of generalised structures is of interest. These generalise special geometries defined by T M–spinor fields: Example. A G 2 –structure on a seven–fold M 7 is defined by a 3–form ϕ with the property that for all p ∈ M, ϕ p ∈ 3 T p∗M lies in the unique orbit diffeomorphic to G L(7)/G 2 . The structure group therefore reduces from G L(7) to G 2 . Since G 2 ⊂ S O(7), ϕ induces a Riemannian metric. Conversely, fix a Riemannian metric g. If the manifold is spinnable, we can consider the spinor bundle (T M) associated with Spin(7). Its spin representation is of real type, that is, it is the complexification of an irreducible real spin representation R ∼ = R8 . The unit sphere is diffeomorphic to Spin(7)/G 2 and therefore, a unit spinor field also defines a G 2 –structure. This links into the “form” definition via the fierzing map [· , ·], namely [ ⊗ ] = 1 − ϕ − g ϕ + g .
(8)
To make contact with generalised geometries, we note that x ∈ Cn → i x ∈ Cn extends to a Clifford algebra isomorphism Cliff(Rn , g) ⊗ C ∼ = Cliff(Rn , −g) ⊗ C. n Restricted to Spin(n, 0) ⊂ Cliff(R , g)⊗C, this gives an isomorphism a ∈ Spin(n, 0) → a ∈ Spin(0, n). Then Y = Y1 · · · · · Y2l ∈ Spin(0, n) acts on ∈ n by Y · , so that by Proposition 2.4, −
Y · R ]G = (−1)l [ L ⊗ Y • [ L ⊗ R ]G = Y − • [ L ⊗ R ]G . Corollary 2.6. [· , ·]G is Spin(n, 0) × Spin(0, n)–equivariant.
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Consider now a Riemannian spin manifold (M, g) and two subgroups G L ,R ⊂ Spin(n, 0) which stabilise the collection of spinor fields { L , j } and { R,k } respectively. R in Spin(0, n). The tensor product (T M) ⊗ (T M) We identify G R with its image G is associated with a Spin(n, 0) × Spin(0, n)–structure, and the collection of bi–spinor R , fields { L , j ⊗ R,k } induces a reduction to G L × G R (or more accurately to G L × G but we will usually omit this) inside Spin(n, 0) × Spin(0, n). By the previous corollary, [· , ·]G maps this bi–spinor field to a G L × G R ⊂ Spin(V + ) × Spin(V − )–invariant E–spinor field, where Spin(V + ) × Spin(V − ) comes from the generalised metric on E. For instance, taking up the previous example leads to the notion of Generalised G 2 –structures [38]. A generalised G 2 –structure can be defined in either of the following three ways: • by the datum (g, H, L , R ), where L , R are two real unit spinor fields which reduce the Spin(7)–structure PSpin(7) to the principal fibre bundles PG 2L and PG 2R respectively, • by a principal G 2L × G 2R –fibre bundle to which the Spin(7, 7)+ –principal fibre bundle associated with E reduces, • by an even or odd E–spinor field ρ whose stabiliser is invariant under a group conjugated to G 2L × G 2R . It is implicitly defined by L(ρ) = [ L ⊗ R ]ev,od . ev ) = ρ od . The even and odd spinor fields are related by G(ρ Generalised SU (3)–structures. [25,26] In dimension 6, we have an isomorphism Spin(6) ∼ = SU (4) under which the complex spin representations ± are isomorphic with the standard vector representations of SU (4) and its complex conjugate, namely C4 and C4 . The unit spheres are isomorphic with SU (4)/SU (3), so that SU (3) stabilises a spinor ± in both + and − , which are related by − = A( + ). A generalised SU (3)–structure is characterised by either of the following: • by the datum (g, H, L , R ), where L , R are two unit spinor fields which reduce the Spin(6)–structure PSpin(6) to the principal fibre bundles PSU (3) L and PSU (3) R respectively, • by a principal SU (3) L × SU (3) R –fibre bundle to which the Spin(6, 6)+ –principal fibre bundle associated with E reduces, • by a pair (ρ0 , ρ1 ) of E–spinor fields whose stabiliser is invariant under a group conjugated to SU (3) L × SU (3) R . The spinor fields are implicitly defined by L(ρ0 ) = [A( L ) ⊗ R ], L(ρ1 ) = [ L ⊗ R ]. Similarly, one can define generalised SU (m)–structures (see also [40]). Applying the operator A ⊗ A to ⊗ yields the other SU (3) L × SU (3) R –invariant pair ([ ⊗ A( )], [A( ) ⊗ A( )]). In the particular case (subsequently referred to as “straight”) where L = = R , we obtain L(ρ0 ) = [A( ) ⊗ ] = e−iω , L(ρ1 ) = [ ⊗ ] = 3,0 , where ω is the Kähler form and 3,0 the invariant (3, 0)–form of the underlying single SU (3)–structure.
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Generalised Spin(7)–structures [38]. In analogy with dimension 7, the two chiral Spin(8)–representations ± are of real type, and the stabiliser of a real chiral unit spinor field is isomorphic with Spin(7). Hence, a generalised Spin(7)–structure is characterised by either of the following: • by the datum (g, H, L , R ), where L , R are two real unit spinor fields which reduce the Spin(8)–structure PSpin(8) to the principal fibre bundles PSpin(7) L and PSpin(7) R respectively, • by a principal Spin(7) L ×Spin(7) R –fibre bundle to which the Spin(8, 8)+ –principal fibre bundle associated with E reduces, • by an E–spinor field ρ whose stabiliser is invariant under a group conjugated to Spin(7) L × Spin(7) R . It is implicitly defined by L(ρ) = [ L ⊗ R ]. The spinor field ρ is even or odd if the spinors are of equal or opposite chirality, reflecting the fact that there are two conjugacy classes of Spin(7) in Spin(8), each of which stabilises a unit spinor in R+ or R− respectively. The straight case induces an even spinor given by L(ρ) = [ ⊗ ] = 1 − + g , where is the g –selfdual Spin(7)–invariant 4–form. If ρ is odd, the principal Spin(7)L – and Spin(7)R –bundle intersect in a principal G 2 –bundle. In this case, a generalised Spin(7)–structure is locally the B–field transform of a G 2 –structure on M 8 . 3. Generalised Calibrations Next we define a notion of structured submanifold for G L × G R –structures as introduced above. In the case of a straight structure, we will recover the so–called calibrated submanifolds which were first considered in Harvey’s and Lawson’s seminal work [19]. We recall some elements of their theory first. Throughout Sects. 3.1 and 3.2, T will denote a real, oriented, n–dimensional vector space. 3.1. Classical calibrations. Let (T, g) be a Euclidean vector space and ρ ∈ k T ∗ . One says that ρ defines a calibration if for any oriented k–dimensional subspace L ⊂ T (subsequently also referred to as a k–plane), with induced Euclidean volume form L , the inequality (9) g(ρ, L ) ≤ g( L , L ) = 1 holds with equality for at least one k–plane. Such a plane is then said to be calibrated by ρ. We immediately conclude from the definition that ρ defines a calibration and calibrates L if and only if g ρ defines a calibration and calibrates L ⊥ . Moreover, let A ∈ G L(n). The calibration condition is G L(n)–equivariant in the following sense: ρ defines a calibration with respect to g if and only if A∗ ρ does so with respect to A∗ g. If L is calibrated for ρ, then so is A(L) for A∗ ρ. In particular, if ρ is G–invariant, then so is the calibration condition. Therefore, calibrated subplanes live in special G–orbits of the Grassmannian Grk (T ). A spinorial approach to calibrations was developed by Dadok and Harvey [14,18] for G 2 , Spin(7) and further geometries in dimension n = 8m. Consider the case of a G 2 – and Spin(7)–structure on T defined by the unit spinor . Then
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• any homogeneous component [ ⊗ ] p of [ ⊗ ] ∈ ∗ T ∗ defines a calibration form • a k–plane L is calibrated with respect to [ ⊗ ]k if and only if = L · . If (M, g) is a Riemannian manifold, then a k–form ρ ∈ k (M) is called a calibration if and only if ρ p defines a calibration on T p M for any p ∈ M. A k–dimensional submanifold j = jL : L → M is said to be calibrated, if T p j (L) is calibrated for every p ∈ j (L). If the calibration form is closed, then calibrated submanifolds are locally homologically volume–minimising. Here are examples of interest to us. Example. (i) SU (3)–structures. The powers ωl /l! of the Kähler form and Re , the real part of the complex volume form, define calibrations. The submanifolds calibrated by the former are complex (of real dimension 2l) and special Lagrangian for the latter (of dimension 3). Similar remarks apply to SU (m)–structures for arbitrary m. (ii) G 2 –structures. The G 2 –invariant 3–form ϕ defines a calibration form whose calibrated submanifolds are called associative. The 4–dimensional submanifolds calibrated by ϕ are referred to as coassociative. (iii) Spin(7)–structures. The Spin(7)–invariant 4–form defines a calibration form. The calibrated submanifolds are the so–called Cayley submanifolds.
3.2. Generalised calibrated planes. To carry this notion over to the generalised setup, one is naturally led to replace k–forms by even or odd T ⊕ T ∗ –spinors. But what ought to be the analogue of k–planes? A natural object on which B–fields act are the maximally isotropic subplanes of T M ⊕ T ∗M. These can be equivariantly identified with lines of pure spinors. By definition, purity means that the totally isotropic space Wτ = {x ⊕ ξ ∈ T ⊕ T ∗ | (x ⊕ ξ ) • τ = 0} is of maximal dimension, that is dim Wτ = n. The following result is classical. Proposition 3.1. ([11] Prop. III.1.9, p.140) A spinor τ ∈ S± is pure if and only if it can be written in the form τ = c · e F • , where c ∈ R=0 , F ∈ 2 T ∗ and = θ 1 ∧ . . . ∧ θ l with θ i ∈ T ∗ is a decomposable form in l T ∗ . Further, if Wτ denotes the isotropic subspace in T ⊕ T ∗ corresponding to the line in S± spanned by τ , then θ 1 , . . . , θ l is a basis of Wτ ∩ T ∗ . Definition 3.1. The number rk τ = n − dim(Wτ ∩ T ∗ ) is called the rank of τ . Remark. The rank of a pure spinor is invariant under B–field transformations. Indeed dim(Wτ ∩ T ∗ ) = dim e2B (Wτ ∩ T ∗ ) = dim(We B •τ ∩ T ∗ ), for exp(2B) acts trivially on T ∗ .
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In other words, we can write Wτ ∩ T ∗ = N ∗L, where N ∗L denotes the annihilator of L = Ann = {X ∈ T | X = 0}. Therefore, the decomposable part of a pure spinor is unique up to a scalar. However, the 2–form F is only determined up to a 2–form F with F ∧ = 0. Put differently, F belongs to N L2 = J (N ∗L)∩2 T ∗ , the subspace of 2–forms in the ∗ T ∗ –ideal J (N ∗L) generated by N ∗L. Using the metric g, we can identify the orthogonal complement N L2⊥ with the image of 2 L ∗ under the pull–back map induced by the orthogonal projection p L : T → L. Hence, modulo rescaling any pure spinor can be uniquely written as τ = exp(F0 ) • , where F0 = p ∗L (F) ∈ p ∗L (2 L ∗ ) for L = Ann . Definition 3.2. An isotropic pair (L , F) consists of a k–plane L ⊂ T and a 2–form F ∈ 2 L ∗ . To render the correspondence between pure spinors and isotropic pairs explicit, we make the following choices: Let again L denote the induced Euclidean volume element as well as its pull–back to T . We then define the pure spinor τ L = L , where the Hodge dual is taken with respect to T . The corresponding maximally isotropic subspace is W L = L ⊕ N ∗L. By equivariance, the pure spinor ∗
L ∈ S(−1)k , τ L ,F = e p L (F) • then corresponds to W L ,F = exp(2F)W L . To ease notation, we shall simply write F instead of p ∗L (F). For later reference, we let [τ L ,F ] = R>0 τ L ,F denote the half–line inside S(−1)k spanned by τ L ,F . We say that the isotropic pair (L , F) is associated with the pure spinor τ , if [τ L ,F ] = [τ ]. Corollary 3.2. There is a Spin(n, n)+ –equivariant 1-1–correspondence between • isotropic pairs (L , F), • half–lines of spinors [τ L ,F ], • maximally isotropic oriented subspaces W L ,F = e2F (L ⊕ N ∗L).
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The definition of a calibration is this. Definition 3.3. Let (g, B) define a generalised metric on T ⊕T ∗ . A chiral spinor ρ ∈ S± is called a generalised calibration if for any isotropic pair (L , F), • the inequality ρ, τ ≤τG holds for one (and thus for all) τ ∈ [τ L ,F ] and • there exists at least one isotropic pair (L , F) for which one has equality. Isotropic pairs (L , F) which meet the bound are said to be calibrated by ρ.
(11)
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We usually drop the adjective “generalised” and simply speak of a calibration. Condition (11) is clearly Spin(n, n)+ –equivariant. Hence ρ defines a calibration which calibrates (L , F) if and only if A • ρ defines a calibration which calibrates (L A , FA ) acts as an isometry for QG , ρ defines a associated with A • τ L ,F . Furthermore, as G calibration if and only if Gρ does. If (L , F) is calibrated for ρ, then the isotropic pair L ,F ] is calibrated for Gρ. (L G, FG) associated with (−1)n(n+1)/2 [Gτ Our first proposition states this calibration condition to be the formal analogue of (9) and makes the appearance of the generalised metric (g, B) explicit. To harmonise with notation used in Sect. 3.3, we let j = jL : L → T denote the injection of the subspace L into T . The pull–back j ∗ = j L∗ : p T ∗ → p L ∗ is then simply restriction to L. Proposition 3.3. A spinor ρ defines a calibration if and only if for any isotropic pair (L , F) with k = dim L, the inequality [e−F ∧ j ∗ ρ]k ≤ det ( j ∗ (g + B) − F) L or equivalently, g(e−F ∧ ρ, L ) ≤
det ( j ∗ (g + B) − F),
holds.5 The bound is met if and only if (L , F) is calibrated. Proof. Regarding ρ as an exterior form, it follows from the remarks above that [e−F ∧ j ∗ ρ]k = g(e−F ∧ ρ, L ) L = [e−F ∧ ρ ∧ L ]n L = [ρ ∧ (e F ∧ )]n L
L
= ρ, τ L ,F L . The result is now√ an immediate consequence of the technical lemma below which implies τ L ,F G L = det ( j ∗ (g + B) − F) L . Lemma 3.4. Let L be an oriented subspace and α ∈ 2 T ∗ . Then g(eα ∧ L , eα ∧ L ) L = det ( j ∗ (g + α)) L = det ( j ∗ (g − α)) L . Proof. Fix an oriented orthonormal
[k/2] j ∗ α = l=1 al e2l−1 ∧ e2l . Then
det ( j ∗ (g − α)) =
[k/2]
basis
e1 , . . . , ek
of
L
such
1 + al2
that
(12)
l=1
[k/2] 2 = 1 + al2 + al21 · al22 + · · · + a12 · · · a[k/2] . l=1
l1
5 In the latter inequality, the determinant is computed from the matrix of the bilinear form j ∗ (g + B) − F with respect to some orthonormal basis of L.
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On the other hand 1 ∗ l j α = l!
a j1 · . . . · a jl e2 j1 −1 ∧ e2 j1 ∧ . . . ∧ e2 jl −1 ∧ e2 jl ,
j1 <...< jl
so that g(
1 1 ∗ j α ∧ L , j ∗ α ∧ L ) = l! l!
j1 <...< jl
a 2j1 . . . a 2jl , 2l ≤ k.
Summing and taking the square root yields precisely (12).
Next we show that in analogy with the classical counterpart discussed in Sect. 3.1, the spinors inducing a G L × G R –structure define calibrations in the generalised sense. In particular, we will recover “classical” calibrated k–planes as a special case. First, we associate with any isotropic pair (L , F) an isometry R L ,F , the gluing matrix (this jargon stems from physics, cf. Sect. 4.1). To start with, let B = 0 and consider the isotropic subspace W L . We can think of this space as the graph of the anti–isometry PL : D + → D − induced by the orthogonal splitting T ⊕ T ∗ = D + ⊕ D − with isometries π0± : X ∈ (T, ±g) → X ± ∈ (D ± , g± ) (cf. Sect. 2.2). Indeed, we have 0 = (X ⊕ PL X, X ⊕ PL X ) = (X, X ) + (PL X, PL X ) = g+ (X, X ) + g− (PL X, PL X ). If we choose an adapted orthonormal basis e1 , . . . , ek ∈ L, ek+1 , . . . en ∈ L ⊥ , then the matrix of PL associated with the basis dl± = π0± (el ) = el ⊕ ±el of D ± is 0 I dk . (13) PL = 0 −I dn−k Pulling this operator back to T via π0± yields the isometry −1 ◦ PL ◦ π0+ : (T, g) → (T, g). R L = π0−
(14)
Its matrix representation with respect to an orthonormal basis adapted to L is just (13). In the general case, W L ,F is the graph of the anti–isometry PL ,F : V + → V − with associated gluing matrix R L ,F = π−−1 ◦ PL ,F ◦ π+ , where now π± : (T, ±g) → V ± = e2B D ± . Let F = F − j ∗ B. Changing, if necessary,
[k/2] the orthonormal basis on L such that F = l=1 fl e2l−1 ∧ e2 j , e1 ⊕ f 1 e2 , e2 ⊕ − f 1 e1 , . . . , ek+1 , . . . , en is a basis of e2B W L ,F by (10). Decomposing the first k basis vectors into the D ± –basis d± j yields 2(e2l−1 ⊕ fl e2l ) =
− + + ⊕ d− d2l−1 + fl d2l 2l−1 − fl d2l
− + + ⊕ f d− 2(e2l ⊕ − fl e2l−1 ) = − fl d2l−1 + d2l l 2l−1 + d2l ,
, l ≤ k,
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while 2el = dl+ ⊕ −dl− for l = k + 1, . . . , n. Written in the new D ± –basis, + + + , w+ = − f d + + = d2l−1 + fl d2l w2l−1 l 2l−1 + d2l , l ≤ k, 2l
− − − − w2l−1 = d2l−1 − fl d2l , w2l =
− − fl d2l−1 + d2l , l ≤ k,
wl+ = dl+ , l > p on D + , wl− = dl− , l > k on D − ,
the matrix R L ,F is just (13). The base of change matrix for dl+ → wl+ is given by the + + ) → (w + + block matrix A = (A1 , . . . , Ak , idn−k ), where for (d2l−1 , d2l 2l−1 , w2l ), l ≤ k, 1 fl . Al = − fl 1 The basis change wl− → dl− is implemented by B = (B1 , . . . , Bk , idn−k ), where for − − − − , d2l ) → (w2l−1 , w2l ), l ≤ k, (d2l−1 1 − fl . Bl = fl 1 Computing B ◦ (I dk , −I dn−k ) ◦ A−1 and pulling back to T via π± , we finally find 0 ( j ∗ (g − B) + F) ( j ∗ (g + B) − F)−1 (15) R L ,F = 0 −I dn−k with respect to some orthonormal basis adapted to L. Proposition 3.5. For any isotropic pair (L , F), the element τ L ,F −1 −B J e • ∈ Cliff(T, g) τ L ,F G lies in Pin(T, g). Moreover, its projection to O(T, g) equals the gluing matrix R L ,F .
[k/2] Proof. Again let e1 , . . . , en be an adapted orthonormal basis so that F = l=1 fl e2l−1 ∧ e2l . Applying a trick from [5], we define = arctan F arctan( fl )e2l−1 · e2l l
1 1 + i fl = ln e2l−1 · e2l ∈ spin(n) ⊂ Cliff(T, g) 2i 1 − i fl l · and show that J exp(arctan F) L = e−B •τ L ,F /τ L ,F G ∈ ∗ T ∗ , where exp takes values in Spin(T, g). Since the elements e2l−1 ·e2l , e2m−1 ·e2m commute, exponentiation yields earctan( fk )e2m−1 ·e2m earctan F = m
cos arctan( f m ) + sin arctan( f m ) e2m−1 · e2m = m
=
1
+
fm
e2m−1 · e2m 1 + f m2
1 + l fl e2l−1 · e2l + l
1 + f m2
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√ 2 where √ we used the classical identities cos arctan x = 1/ 1 + x , sin arctan x = x/ 1 + x 2 . By Lemma 3.4, we finally get
J(earctan F · L ) = J(earctan F ) ∧ J( L ) = e−B •
τ L ,F . τ L ,F G
The projection π0 : Pin(T, g) → O(T, g) via π0 gives indeed the induced gluing matrix: Firstly, π (arctan F) π0 earctan F · L = e S 0∗ ◦ π0 ( L ). O(T ) Now eπ0∗ (arctan( fl )e2l−1 ·e2l ) = e2 arctan( fl )e2l−1 ∧e2l = cos (2 arctan( fl )) + sin (2 arctan( fl )) e2l−1 ∧ e2l 1 − fl2 2 fl = + e2l−1 ∧ e2l 1 + fl2 1 + fl2 1 0 1 − fl2 0 −2 fl + = 2 fl 0 0 1 − fl2 1 + fl2 which yields the matrix ( j ∗ g + F)( j ∗ g − F)−1 , while the block −I dn−k in the gluing matrix is accounted for by the projection of the volume form. The fact that e−B • τ L ,F / τ L ,F G can be identified with an element of Pin(T, g) enables us to prove the following Theorem 3.6. Let L , R be two chiral unit spinors Spin(T )–represen of the complex tation n . The T ⊕ T ∗ –spinors ρ ev,od = e B •Re [ L ⊗ R ]ev,od satisfy ρ, τ L ,F ≤ τ L ,F G . Moreover, an isotropic pair (L , F) meets the boundary if and only if A( L ) = ±(−1)m(m+1)/2+k i m (e−B •
τ L ,F ) · R τ L ,F G
for n = 2m and R ∈ ± and A( L ) = (−1)m(m+1)/2 i m+1 (e−B •
τ L ,F ) · R τ L ,F G
for n = 2m + 1. In particular, ρ ev,od define a calibration. Proof. Since eF ∧ L = (eF L )∧ , we obtain e B • Re[ L ⊗ R ]ev,od , τ L ,F = Re[ L ⊗ R ]ev,od , eF ∧ L
= g(Re[ L ⊗ R ]ev,od , eF L ) = Re q(A( L ), e I · R )g(e I , eF L ) = Re q(A( L ), eF L · R ).
(16)
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On the other hand, eF L · R = (−1)k(n−k) (eF L ) · R = (−1)k(n−k) (e−F ∧ L ) · R = (−1)k(n−k) (eF ∧ L ) · g · R k(n−k) = (−1) det( j ∗ g − F)(e−B • τ L ,F ) · g · R . Recall from (7) the action of g on ± . Since e−B • τ L ,F / τ L ,F G ∈ Spin(T, g), we get τ L ,F eF L · R q = det( j ∗ g − F) e−B • g · R q = det( j ∗ g − F) τ L ,F G (where the norm is taken for the Hermitian inner product q on n and g on T respectively). As a consequence, (16) is less than or equal to τ L ,F G by the Cauchy– Schwarz inequality for the norm · q . Moreover, equality holds precisely if A( L ) = (−1)k(n−k) (e−B • τ L ,F ) · g · R . As there always exists a subspace L such that A( R ) = L · L , we can , j ∗ B) as a calibrated isotropic pair. Hence, choose (Lev,od ev,od B the spinor ρ defines a calibration. = e • Re [ L ⊗ R ] Example. Consider a G 2 – or Spin(7)–structure on T induced by the real unit spinor . Then (L , 0) defines a calibrated plane for Re [ ⊗ ]ev,od if and only if = L · = L · as g acts as the identity on n for n = 7, 8. This is precisely the condition found by Dadok and Harvey (cf. Sect. 3.1). In general, if L is a calibrated plane with respect to some classical calibration form arising as the degree k component of ρ = Re [ ⊗ ]ev,od , the isotropic pair (L , 0) is calibrated with respect to ρ in the sense of Definition 3.3 with B–field transform (L , j ∗ B). More examples will be given in Sects. 3.3 and 5.1. In particular, we will see that even in a “straight” setting with B = 0, Definition 3.3 is more general as we can have calibrated isotropic pairs (L , F) with non–trivial F. 3.3. Generalised calibrated submanifolds. We introduce the notion of a generalised calibrated submanifold next. Let M be a smooth manifold with a generalised Riemannian metric V + ⊂ E corresponding to (g, H ). Definition 3.4. (i) A spinor field ρ ∈ (S(E)± ) is called a generalised calibration if and only if ρ p ∈ S(E)± p defines a generalised calibration for every p ∈ M. (ii) An isotropic pair (L , F) for (M, g, H ) consists of an embedded oriented submanifold j = j L : L → M, together with a 2–form F ∈ 2 (L) such that dF + j ∗ H = 0.
(17)
(iii) An isotropic pair (L , F) is said to be calibrated by ρ if and only if for every point p ∈ j (L) ⊂ M and for one (and thus for any) τ p in the half–line of S(E)± p induced by the pure spinor field τ L ,F = L−1 (eF ∧ L ) ∈ S(E)±|L , we have ρ p , τ L ,F =τ p G. Here, we think of F as a section of 2 T ∗M| j (L) via the pull–back induced by orthogonal projection p L : T M| j (L) → T j (L).
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Again, we will drop the qualifier “generalised” and simply speak of a calibration form. Also, we will usually think of L as a subset of M and identify L with j (L). An easy corollary is the following “form criterion” for an isotropic pair to be calibrated. Proposition 3.7. An isotropic pair (L , F) for (M, g, H ) with k = dim L is calibrated if and only if g(e−F ∧ L(ρ), L ) ≤τ L ,F G or equivalently, [e−F ∧ j L∗ L(ρ)]k ≤
det( j L∗ g − F) L .
This follows from ρ, τ L ,F = [L(ρ) ∧ e−F ∧ L ]n = g(e−F ∧ L(ρ), L ), while n F L(τ L , eF ∧ L ). τ L ,F 2G = [L(τ L ,F ) ∧ G L ,F ) ] = g(e ∧ Calibrated isotropic pairs minimise an extension of the volume functional, namely the brane energy e−φ τ L ,F G − e−φ j L∗ [e−F ∧ L(γ )], Eφ,γ (L , F) = L
L
where γ is a spinor of suitable parity and φ ∈ C ∞ (M) a dilaton field. The terminology comes from the local description of the spinor norm τ L ,F G . Let C = L(γ ). These are the so–called Ramond–Ramond potentials (cf. Sect. 4.2). Over Ua , H|Ua = d Ba(a) , so that for a disk D ⊂ Ua ∩ L, the brane energy is just the sum of the two terms D
and
e−φ τ L ,F G =
D
e−φ j L∗ [e−F
D
det j L∗ (g − Ba(a) ) + F (a) D
∧ L(γ )] = D
e−φ j L∗ [e−F
(a) +B (a)
∧ C]
In physics, these integrals are known as the Dirac–Born–Infeld and the Wess–Zumino term of the D–brane energy (cf. (23) and (24)). Definition 3.5. (i) Two isotropic pairs (L , F) and (L , F ) of dimension k are said of dimension k + 1 with to be homologous, if there exists an isotropic pair ( L, F) |L = F and F |L = F . ∂ L = L − L , F (ii) We call an isotropic pair (L , F) locally brane–energy minimising for Eφ,γ if Eφ,γ (D, F|D ) ≤ Eφ,γ (D , F ) for any embedded disc D ⊂ L and homologous isotropic pair (D , F ) (with D not necessarily embedded into L).
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Theorem 3.8. Let ρ be a calibration with dφ ρ = dφ γ , i.e. d H e−φ L(ρ) = d H e−φ L(γ ).
(18)
Then any calibrated isotropic pair (L , F) is locally brane–energy minimising for Eφ,γ . ) be an isotropic pair such F Proof. With the notation of the previous definition, let ( D, |D = F and F |D = F . By Stokes’ theorem = D − D, F that ∂ D ∗ −F −φ e jD ∧ d e L(ρ − γ ) 0= H D ∗ −F jD ∧ e−φ L(ρ − γ ) = d e D −F |∂ D ∧ e −φ L(ρ − γ )], j∂∗D = [e D−D
so that
D
e−φ j D∗ e−F ∧ L(ρ − γ ) =
D
e−φ j D∗ e−F ∧ L(ρ − γ ) .
Since D ⊂ L and (L , F) is calibrated, j D∗ e−φ [e−F ∧ L(ρ)] = e−φ τ L ,F G D
D
by the previous lemma, while for D , the integral over the norm is greater then or equal to D j D∗ [e−φ e−F ∧ L(ρ)]. Remark. In [26], the equation dφ ρ = dφ γ was shown to describe type II string compactification on 6 or 7 dimensions which are governed by the democratic formulation of Bergshoeff et al. [6]. Examples of calibrations are provided by G L × G R –structures: The following proposition is an immediate consequence of Theorem 3.6. Proposition 3.9. If ρ ∈ S(E)± is a spinor field such that L(ρ) = [ L ⊗ R ] for (chiral) unit spinor fields L ,R ∈ (T M), then Re(ρ) defines a calibration form. Moreover, an isotropic pair is calibrated if and only if A( L ) = ±(−1)m(m+1)/2+k i m
L(τ L ,F ) · R , L(τ L ,F )g
(19)
for n = 2m and R ∈ ± , and L(τ L ,F ) · R , L(τ L ,F )g for n = 2m + 1, where L(τ L ,F )g = g L(τ L ,F ), L(τ L ,F ) . A( L ) = (−1)m(m+1)/2 i m+1
(20)
Remark. Equations (19) and (20) reflect the physical fact that calibrated pairs “break the supersymmetry”: The spinor field τ L ,F induces a linear relation between L and R (or the supersymmetry parameters in physicists’s language, cf. Sect. 4.1), so that over L we are left with only one independent supersymmetry parameter. On the tangent space level, the two G–structures PG L ,R inside the orthonormal frame bundle are related by the corresponding gluing matrix.
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Examples. By the previous proposition, the E–spinor fields which induce a generalised SU (3)–, G 2 – or Spin(7)–structure, define calibrations. For straight generalised SU (3)–, G 2 – and Spin(7)–structures (cf. Sect. 2.3), we recover some well–known examples coming from physics which involve a non–trivial F. We will construct further examples in Sect. 5 using the device of T–duality. (i) Straight SU (3)–structures: Let (M 6 , g, ) be a classical SU (3)–structure whose induced straight SU (3)× SU (3)–structure is given by L(ρ0 ) = e−iω and L(ρ1 ) = . Starting with the even spinor field, we have Re (L(ρ ev )) = 1 − ω2 /2. By Proposition 3.7, we find as calibration condition k 1 ∗ j L ω + F = det( j L∗ g − F) L , k!
(21)
which defines so–called B–branes. These wrap holomorphic cycles in accordance with results from [30,32]. Calibrating with respect to Re L(ρ od ) = Re() yields so–called A–branes defined by
e−F ∧ j L∗ Re()
k
=
det( j L∗ g − F) L .
The calibrated pairs are therefore odd–dimensional. For k = 3, jL∗ Re = L is the condition for a special Lagrangian cycle. For k = 5 we need a non–vanishing F and obtain jL∗ Re ∧ F =
det( j L∗ g − F) L
which is the condition found in [30] for a coisotropic A–brane. In the non–straight case, Re L(ρ od ) can also contain a 1– and 5–form part. For a related discussion on this aspect see Sect. 4 of [4]. (ii) Straight G 2–structures: Let (M 7 , g, ) be a classical G 2–structure whose induced straight G 2 ×G 2 –structure is given by L(ρ ev ) = 1−ϕ. The calibration condition reads e−F ∧ jL∗ (1 − ϕ) ≤
det( j L∗ g − F) L .
A coassociative submanifold satisfies jL∗ ϕ = L and is therefore calibrated if F = 0. For non–trivial F, we find F ∧ F/2 − jL∗ ϕ = det( j L∗ g − F) L . Now F ∧ F/2 = Pf(F) L and det( j L∗ g − F) = 1 − Tr(F 2 )/2 + det(F). Squaring this shows the equality to hold for an isotropic pair (L , F), where L is a coassociative submanifold and F a closed 2–form with 2Pf(F) = −Tr(F 2 )/2, that is, F is anti–self–dual (cf. [32]). (iii) Straight Spin(7)–structures: Let (M 8 , g, ) be a classical Spin(7)–structure whose induced straight Spin(7) × Spin(7)–structure is given by L(ρ) = 1 − + g . Again, Cayley submanifolds are calibrated for F = 0. Further, as in (ii), they remain so if turning on an anti–self–dual gauge field F (cf. [32]).
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4. Branes in String Theory We briefly interlude to outline how in type II string theory the calibration condition as presented above arises from both the world–sheet and the target space point of view. For a detailed introduction see for instance [27] or [35]. The material of this section is independent of the mainstream development of the paper. Generally speaking, D–branes arise as boundary states in the conformal field theory on the string worldsheet. In the supergravity limit, ignoring corrections of higher order in α (the string tension), they can be described as submanifolds in the ten dimensional target space which extremalise a certain energy functional. In the special case of type II string compactifications on Calabi–Yau manifolds, D–branes can be classified by the derived category of coherent sheaves and the Fukaya category for type IIB and IIA respectively. Both notions are connected by Mirror symmetry, which in the case of toroidal fibrations ought to be realised through T–duality [36].
4.1. World–sheet point of view. We start with a two–dimensional conformal field theory on the string world–sheet , parametrised by a space–like coordinate s and a time–like coordinate t. D–branes arise when considering open string solutions (i.e. where the string is homeomorphic to an open interval) with Dirichlet boundary conditions. Let us first consider the simplest case without background fields, delaying this issue to the end of this section. Varying the world–sheet action on (which depends on the embedding functions X µ , µ = 0, . . . , 9, taken as coordinates of the 10–dimensional target space), we find the boundary term ∂s X µ δ X µ . δ Iboundary = ∂
There are two kinds of solutions at the boundary of the worldsheet (s = {0, π }), namely ∂s X µ |s=0,π = 0 δ X µ = 0 ⇒ X µ |s=0,π = const.
(von Neumann boundary condition) (22)
or (Dirichlet boundary condition).
Choosing the von Neumann boundary condition for µ = 0, . . . , p and the Dirichlet boundary condition for µ = p + 1, . . . , d, we define open strings whose endpoints can move along p spacial dimensions (the zeroth coordinate parametrising time), thus sweeping out a p + 1–dimensional surface, a D(p)–brane. Since we are dealing with a world–sheet theory that is supersymmetric, the boundary conditions for the world–sheet fermions need to be taken into account, too. Without D–branes we start with (1, 1) worldsheet supersymmetry, generated by two spinorial parameters, L ,R . The supersymmetric partners of the coordinate functions X µ are repµ resented by the set of worldsheet fermions ψ L ,R . The boundary conditions for these read µ µ −ψ Lµ δψ L + ψ Rµ δψ R |s=0,π = 0, which can be solved by taking ψ Lµ = ±ψ Rµ . However, the fermionic boundary conditions are related to their bosonic counterpart by supersymmetry, so only one choice is effectively possible. Moreover, the supersymmetry transformations between fermions and bosons imply that the supersymmetry parameters are related by R = − L . Hence
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we are left with only one linearly independent supersymmetry parameter so that worldsheet supersymmetry is broken to (1, 0). By convention, a + occurs if the coordinate µ satisfies the von Neumann boundary condition. For directions satisfying the Dirichlet boundary conditions we then obtain a minus sign, i.e. ψ Lµ = −ψ Rµ . These boundary conditions can be suitably encoded in the gluing matrix R (14), relating ψ L and ψ R via ψ L = Rψ R , R being the diagonal matrix (with respect to the coordinates µ = 0, . . . , 9) defined by +1 in von Neumann and −1 in Dirichlet directions. As a consequence of the D–brane boundary conditions, the symmetry group on the branes reduces to S O(1, p). After quantisation, the zero mode spectrum of the open strings contains a vector multiplet of N = 1 supersymmetry, that is, an element of the S O( p − 1) vector representation plus its fermionic counterpart. The vector multiplet also transforms under a U (1) gauge symmetry. In particular, we obtain a gauge field F – the curvature 2–form coming from the U (1)–connection. More generally, we can take N branes on top of each other to obtain a U (N ) gauge group, reflecting the fact that we have to include the fields from strings stretching between different branes of one stack. In any case, including the field strength leads to a generalised gluing condition of the form (15). To see how this happens, consider again the boundary conditions (22). First note that the Dirichlet condition is equivalent to ∂t X µ = 0. Including a nontrivial gauge field, or more generally a two–form field–strength background F, which also includes the background B–field (cf. Sect. 4.2), changes the boundary conditions into mixed Dirichlet– and von Neumann–type as follows: ∂s X |∂ L + g −1 (∂t X F)|∂ L = 0, where g is the space–time metric and restriction to ∂ L indicates restriction to the boundary of the D–brane.
4.2. Target space point of view. For type II supergravity theory on M, that is, at the zero–mode level of the corresponding string theory, we have the following field content in the so–called closed string sector: • a metric g, a 2–form B (the B–field) and a scalar function φ (the dilaton) in the NS-NS (NS=Neveu–Schwarz) sector, • a mixed form C (the R-R–potentials) in the R-R (R=Ramond) sector, which is odd or even in type IIA or IIB respectively. In the limit of supergravity, D–branes become non–perturbative objects and belong thus to the background geometry where they can be conceived as submanifolds of the 10–dimensional target manifold M. In this picture, H is given locally as the field strength H = d B. On the other hand, the open string sector leads to an effective field theory on the world–volume of a D–brane. If L denotes the corresponding submanifold, this field theory is given by the Dirac–Born–Infeld action, IDBI (L , F) = −N e−φ det( j L∗ g − F), (23) L
where as above N denotes the number of branes in the stack and F = F − jL∗ B with F the field strength of the U (1) gauge theory living on the brane. In particular, d(F + jL∗ B) = 0. Moreover, D–branes act as sources for the R-R–fields and couple
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571
to these via µ L C ( p+1) , where µ is the brane tension. This term is referred to as the Wess–Zumino action and in general looks like (24) IZW (L , F) = N µ e−F ∧ j L∗ C. L
Note that type IIA and IIB supergravity are related under T–duality. Under this transformation, the metric and the B–field of type IIA and IIB are mixed according to the Buscher rules [10] (see the next section). Moreover, with respect to D–branes, T–duality exchanges the even and odd R-R–potentials and modifies the boundary conditions (14) in such a way that Dirichlet– and von Neumann–conditions are exchanged. This leads to an exchange of D–branes of odd and even dimension. Whether the dimension of a brane increases or decreases depends on the direction in which the T–duality transformation is carried out. If T–duality is performed along a direction tangent to the brane, the dimension of the transformed brane will decrease, while a transformation along a transverse direction increases the dimension. The mathematical implementation of this will occupy us next. 5. T–Duality 5.1. The Buscher rules. T–dual generalised Riemannian metrics. In this section we show how the Buscher rules arise in the generalised context. This parallels work in the generalised complex case [2,24]. Consider the vector space T ⊕ T ∗ ∼ = Rn ⊕ Rn∗ with a metric splitting V + ⊕ V − corresponding to (g, B). Further, consider a decomposition T = RX ⊕ V, where X is a non–zero vector. We define the 1–form θ by θ (X ) = 1, ker θ = V and obtain the element = X ⊕ −θ ∈ Pin(n, n). M Projecting down to O(n, n) gives
⎛
⎜ M = (I dV + θ † ) ⊕ (I d N ∗ R X + X † ) = ⎝
IdV
⎞
0 0
0 1
IdN∗ RX
1⎟ ⎠=
A C
B D
.
0
Here θ † : T → T ∗ sends X to θ , θ † (V) = 0, and X † : T ∗ → T sends θ to X , X † (N ∗ RX ) = 0. Definition 5.1. The T–dual generalised Riemannian metric is given by V + = M(V + ). In terms of the corresponding pair (g , B ), the transformation can be expressed in adapted coordinates as follows. Proposition 5.1. Extend X to a basis x1 , . . . , xn−1 , xn = X with xi ∈ V, i = 1, . . . , n−1, whose dual basis is x 1 , . . . , x n−1 , x n = θ . Let gkl and Bkl be the coefficients of g and B with respect to this basis. Then the coefficients of g and B are given according to the Buscher transformation rules (cf. also [20,21] or the appendix in [29]), namely =g − gkl kl
= gkn
= Bkl
= Bkn
1 gnn (gkn gnl − Bkn Bnl ), Bkl + g1nn (gkn Bln − Bkn gln ),
1 gnn Bkn , 1 gnn gkn .
= gnn
1 gnn ,
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Proof. The elements of V + are given by Y ⊕ P + Y = AX + B P + X ⊕ C X + D P + X for X ⊕ P + X ∈ V + , hence P + = (C + D P + )(A + B P + )−1 . Writing g α β B + P =g+B = , + β tr 0 α tr q we obtain (A + B P + )−1 =
and
Id −(α − β)tr /q
(C + D P ) = +
Consequently, P + =
g+B 0
g + B − (α + β)(α − β)tr /q −(α − β)tr /q
that is,
g + =
and
B
whence the assertion.
+
=
0 1/q
α+β 1
.
(α + β)/q 1/q
g − (αα tr − ββ tr )/q β tr /q
β/q 1/q
B + (αβ tr − βα tr )/q alpha tr /q
α/q 0
= g + + B + ,
,
T–dual spinors. For ρ ∈ S± , its T–dual spinor is • ρ. ρ = M If the collection of chiral spinors ρ1 , . . . , ρs defines a generalised G L × G R –structure, that is, their stabiliser in Spin(n, n)+ is G L × G R , the collection ρ1 , . . . , ρs induces the L × G R )M. In T–dual G L × G R –structure associated with the conjugated group M(G particular, equivariance implies the induced generalised Riemannian metric to be given ρ = (Gρ) . by (g , B ), whence G Remark. Note that T–duality reverses the parity of the spinor. This reflects the physical fact that T–duality exchanges type IIA with type IIB theory and hence maps odd R-R potentials into even R-R potentials (cf. Sect. 4). also maps pure spinors to pure spinors. More concretely, if τ is pure The operator M • τ is pure with annihilator M(Wτ ). Hence, T–dualwith annihilator Wτ , then τ = M ity associates to any isotropic pair (L , F) a uniquely determined T–dual isotropic pair is norm–preserving (L , F ) specified by the condition [τ ] = [τ L ,F ]. Further, M in the sense that for τ ∈ S± , we have τ G =τG. Therefore ρ, τ ≤τG
if and only if ρ , τ ≤τ G ,
(25)
and equality holds on the left hand side if and only if equality holds on the right hand side.
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Proposition 5.2. A spinor ρ ∈ S± defines a calibration for the generalised Riemannian • ρ defines a calibration for the T–dual metric (g, B) on T ⊕ T ∗ if and only if ρ = M generalised metric (g , B ). A (calibrated) isotropic pair (L , F) corresponds to a uniquely (calibrated) isotropic pair (L , F ). The rank of τ L ,F is given as follows: • If X ∈ L, then rk τ L ,F = rk τ L ,F − 1, • If X ∈ L, then rk τ L ,F = rk τ L ,F + 1. Proof. By virtue of the preceding discussion, only the last assertion requires proof. By definition of the rank (cf. Proposition 3.1), rk τ L ,F = n − dim M(W L ,F ) ∩ T ∗ = n − dim M W L ,F ∩ M(T ∗ ) = n − dim W L ,F ∩ (RX ⊕ N ∗ RX ) = n − dim W L ∩ (RX ⊕ N ∗ RX ) , where for the last line we used that X F ∈ N ∗L if and only if X F = 0. Now suppose that X ∈ L. In this case, L = RX ⊕ L ∩ V, hence we may choose a basis x1 , . . . , xn of T such that x1 = X , x2 , . . . , xk generates L ∩ V, and x2 , . . . , xn generates V. Let x 1 , . . . , x n denote the dual basis. Hence θ = x 1 and W L = L ⊕ N ∗L = {a 1 X +
k i=2
a i xi ⊕
n
bi x i }.
j=k+1
The intersection with RX ⊕ N ∗ RX is spanned by X, x k+1 , . . . , x n , so that rk τ L ,F = n − k − 1 = rk τ L ,F − 1. Next assume that X ∈ L. Then there exists a basis x1 = X, x2 , . . . , xn of RX ⊕ V such that L is spanned by cX + x2 , . . . , xk+1 (possibly c = 0). Again θ = x 1 , and W L = L ⊕ N ∗L = {a 1 (cX + x2 ) +
k+1 i=3
a i xi ⊕ b(θ − cx 2 ) +
n
bi x i }.
j=k+2
The intersection with RX ⊕ N ∗ RX is now spanned by x k+2 , . . . , x n , hence rk τ L ,F = n − k + 1 = rk τ L ,F + 1. Example. Consider a G 2 –structure on T 7 . As we learned from the examples in Sect. 3.3, an isotropic pair (L , F) consisting of a coassociative subplane L and an anti–self–dual 2–form F is calibrated by ρ = 1 − ϕ. Now choose a vector X ∈ L and a complement V so that T = RX ⊕ V. Then (L , F ) is a calibrated pair with L of dimension dim L − 1 = 3. On the other hand, choosing a vector X ∈ L yields a T–dual calibrated pair with a 5–dimensional plane L . In particular, we see that calibrated isotropic pairs of “exotic” dimension (in comparison to the straight case, e.g. 3 and 4 for G 2 ) can occur. 5.2. Integrability. In this section, we discuss a local version of T–duality over M = Rn (now seen as a manifold) endowed with a generalised metric (g, H = d B). The induced generalised tangent bundle E(H ) → Rn is equivalent as a vector bundle to T Rn ⊕T ∗ Rn , but inequivalent from a generalised point of view if H = 0, as this implies twisting with a non–closed B–field (cf. also Sect. 2.1.2). Fix a dilaton field φ so that ν = exp(2φ)νg .
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Further, let X be a nowhere vanishing vector field transversal to the n − 1–dimensional distribution V so that T p Rn = RX ( p) ⊕ V( p). This determines the 1–form θ and hence = X ⊕ −θ on S(E). T–duality induces a map M : ev,od (Rn ) → the operator M od,ev n ∼ (R ) = S∓ E(H ) (where the last isomorphism is induced by Lν , the isomorphism between spinors and differential forms associated with H = d B ) defined by • e B • ρ. M(α) = eB ∧ M
The T–dual dilaton field φ is defined by ν = exp(2φ )νg . Proposition 5.3. The dilaton transforms under the Buscher rule φ = φ − ln X . Proof. Let q = X 2 . It suffices to show that νg = q · νg . This is a tensorial identity and can be computed pointwise. At a given point p, we may assume that V is spanned by ∂x i ( p), i ≤ n − 1, and X ( p) = ∂x n for coordinates x 1 , . . . , x n . By Proposition 5.1, g ( p) = Atr ◦ g( p) ◦ A, where A ∈ G L(n) is given by 0 Id . A= (β − α)tr /q 1/q The claim follows from νg ( p) = det A−1 · νg ( p).
Example. Consider the spinor field ρ defined by Lνg (ρ) = [ ⊗ ]od , where comes from an ordinary G 2 –structure. The T–dual spinor field ρ induces also a generalised G 2 –structure and is therefore determined by Lνg (ρ ) = Lφ (ρ ) = e−φ [ L ⊗ R ]od . Under additional assumptions, T–duality also preserves closeness of spinor fields. Theorem 5.4. Assume that (g, B, φ) are X –invariant, that is LX g = 0, L X B = 0 and L X φ = 0. Further, assume that L X θ = 0. If ρ and ϕ are two E(H )–spinor fields such that L X Lν (ρ) and L X Lν (ϕ) = 0, then dν ρ = ϕ if and only if dν ρ = −ϕ , that is d H Lν (ρ) = Lν (ϕ) if and only if d H Lν (ρ ) = −Lν (ϕ ).
(26)
x = φ − Proof. Fix coordinates x 1 , . . . , x n so that ν = e2φx ∂x 1 ∧ . . . ∧ ∂x n with φ ln(det x g)/4. The subscript x reminds us that this quantity depends upon the choice of x = φ − ln(det x g )/ coordinates. It is invariant under T–duality in the sense that φ 4 = φx . By definition, dν ρ = ϕ is then equivalent to
de−φx ρ = e−φx ϕ. Let ρ = ρ0 + θ ∧ ρ1 , ϕ = ϕ0 + θ ∧ ϕ1
(27)
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be the decomposition into basic forms, that is, X ρ0,1 = 0 and X ϕ0,1 = 0. The assumptions imply that L X Lν (ρ) = e−φ exp(−B) ∧ L X (ρ) = 0, whence X dρ0,1 = 0. Analogously, X dϕ0,1 = 0 holds. Therefore, (27) is equivalent to x ∧ ρ0 + dρ0 + dθ ∧ ρ1 , ϕ1 = d φ x ∧ ρ1 − dρ1 . ϕ0 = −d φ
(28)
For the right hand side of (26), we find x ∧ ρ0 + dρ0 + dθ ∧ ρ1 , −ϕ1 = d φ x ∧ ρ1 − dρ1 , − ϕ0 = −d φ
(29)
where • ρ = −(X +θ ∧)(ρ0 + θ ∧ ρ1 ) = −ρ1 − θ ∧ ρ0 , ρ = ρ0 + θ ∧ ρ1 = M x = φ x , we deduce from this that (29) is equivalent to (28) and similarly for ϕ . As φ and thus to (27). Example. To illustrate the theorem we take up the previous example where in addition we assume: (i) the G 2 –invariant 3–form ϕ is closed and coclosed (ii) there exists a vector field X of non–constant length such that L X ϕ = 0. In particular, X defines a Killing vector field. Local examples of this exist in abundance, cf. for instance [1]. Since Lν (ρ) = [ ⊗ ]od is closed, so is the T–dual Lν (ρ ) = e−φ [ L ⊗ R ]od . Since and its T–dual. Hence ϕ is also co–closed, the same holds for the even spinor field G(ρ)
d H e−φ [ L ⊗ R ] = 0,
(30)
that is, the generalised G 2 –structure is integrable [25,38], where Eq. (30) was shown to encode the supersymmetry variations of type II supergravity. In particular, this implies H = 0, for φ = −ln X is not constant. Although we started with B = 0 and φ = 0, the Buscher rules imply that we acquire a generalised G 2 –structure with non–trivial B– and dilaton field on the T–dual side. Since being a calibration is a pointwise condition, it follows from (25) that this property is preserved under T–duality. Similarly, one would expect calibrated isotropic pairs to transform under T–duality in the vein of Proposition 5.2. However, two problems may occur: Firstly, if (L , F) is calibrated by ρ, the rank of τ L ,F is not constant in general, which prevents this spinor field to be induced by an isotropic pair. In view of Proposition 5.2, we either need to restrict L to the open subset for which X ∈ T p L, or to assume X ∈ (T L). Secondly, if there is a pair (L , F ) such that [τ L ,F ] = [τ L ,F ], it might not be isotropic, that is, we possibly have dF + j L∗ H = 0. This requires to rephrase the isotropy condition on the pair in terms of integrability conditions of the induced pure spinor field. Definition 5.2. Let E → M be a generalised tangent bundle and ν a trivialisation of n T M. A pure spinor field half–line in S(E)± is called integrable if and only if it admits a dν –closed representative of constant rank. By abuse of language, we also refer to the representative as integrable.
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Lemma 5.5. Let (L , F) be an isotropic pair for (Rn , H = d B). Then locally, [τ L ,F ] is induced by an integrable half–line, that is there exists an open set U ⊂ R N and an integrable pure spinor field τ ∈ S(E)|U with [τ|U ∩L ] = [τ L ,F |U ∩L ]. Furthermore, if L X B = 0 and X is a vector field either transversal to L, or contained in T L with L X F = 0, then τ can be constructed to satisfy L X Lν (τ ) = 0. Conversely, an integrable pure spinor field τ over Rn gives locally rise to a foliation into isotropic pairs (L r , Fr ) such that [τ|L r ] = [τ L r ,Fr ]. Proof. Let us start with the converse. Restricting τ to some open subset U , we can write Lν (τ ) = exp(Fτ )∧ with Fτ ∈ 2 (U ) and ω = θ 1 ∧. . .∧θ n−k , θ i ∈ T ∗ U . Closeness is equivalent to d = 0, (dFτ + H ) ∧ = 0.
(31)
By Frobenius’ theorem, the smooth distribution Ann = {Y ∈ X(U ) | Y = 0} is thus integrable. Hence, on U there exist coordinates y 1 , . . . , y n (possibly upon shrinking U ) such that for r ∈ Rn−k and p ∈ L r = { p ∈ U | y i ( p) = r i , i = k + 1, . . . , n}, one has T p L r = Ann ( p). Further by (31), dFτ + H ∈ N L3r , the space of 3–forms of the ideal in ∗ (Rn )|L r generated by N ∗L r ⊂ T ∗ Rn|L r . Consequently, defining Fr = j L∗r Fτ , dFr + j L∗r H = j L∗r (dFτ + H ) = 0. By construction, (Fτ |L r − Fr ) ∧ = 0, hence [τ|L r ] = [τ L r ,Fr ] for suitably oriented Lr . Now consider the isotropic pair (L , F). We can fix around p ∈ L a coordinate system y 1 , . . . , y n on U ⊂ Rn with L = {y k+1 = . . . = y n = 0}. Define the closed 2–form FL = (F + j L∗ B)|U ∩L , which we extend trivially to a 2–form F on U , that is, F( p) = F( p 1 , . . . , p n ) = FL ( p 1 , . . . , p k , 0, . . . , 0). Set F = F − B|U and let τ ∈ S(E) be determined by Lν (τ ) = eF ∧dy k+1 ∧. . .∧dy n . Then [τ|U ∩L ] = [τ L ,F |U ∩L ]. We claim τ to be dν –closed. Indeed, this is equivalent to (dF + H|U ) ∧ d x k+1 ∧ . . . ∧ d x n = 0,
(32)
hence to d F ∧ dy k+1 ∧ . . . ∧ dy n = 0. But this holds on L, hence on U for we extended F trivially. In particular, jL∗r (dF + H|U ) = 0 using the previous notation, so the pairs (L r , F L r ) are isotropic. If X is a vector field transversal to L, we carry out this construction with a coordinate system which in addition satisfies ∂ y n = X . Then Lν (τ ) does not depend on y n for L X F = 0, hence L X Lν (τ ) = 0. On the other hand, if X restricts to a vector field on L, we can fix a coordinate system with ∂ y 1 = X and L = {y k+1 = . . . = y n = 0}. Again, τ is X –invariant as L X F = 0. We are now in a position to prove Theorem 5.6. (i) The spinor ))± ) defines a calibration for (g, H = d B) if and only if ρ ∈ (S(E(H ρ ∈ S(E(H ))± does so for (g , H = d B ). Furthermore, if dφ ρ = dφ γ under the assumptions of Theorem 5.4, then the calibrated isotropic pairs for ρ and ρ extremalise the functionals Eφ,γ and Eφ ,−γ respectively.
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(ii) Under the assumptions of (i), let (L , F) be a calibrated isotropic pair. If X ( p) ∈ T p L for some p ∈ L, there exists a calibrated isotropic pair (L , F ) with • p ∈ L and dim L = dim L + 1, • [τ L ,F ( p)] = [τ L ,F ( p)]. On the other hand, if X ∈ (T L) and L X F = 0, there exists a calibrated isotropic pair (L , F ) with • p ∈ L and dim L = dim L − 1, • [τ L ,F ( p)] = [τ L ,F ( p)]. Remark. The dimension shift coincides with expectations from physics, cf. Sect. 4.2. Proof. By virtue of the preceding discussion and Theorem 5.4, only assertion (ii) requires proof. By the previous lemma, we can locally extend [τ L ,F ] to an integrable pure spinor field half–line [τ ] with L X Lφ (τ ) = 0 and dφ τ = 0 (on U , the domain of τ ). Consequently, dφ τ = 0 by Theorem 5.4. If X is transversal to L, then X ∈ T L r . Hence τ is of constant rank rk(τ ) + 1 and therefore integrable on U . Appealing to the converse of the lemma, we deduce the existence of the isotropic pair (L , F ). The rest follows analogously. Example. We saw that if (M, ϕ) is a classical G 2 –manifold with G 2 –invariant spinor , all calibrated submanifolds L give rise to calibrated isotropic pairs (L , F = 0). If L X ϕ = 0 and X is transversal or restricts to a vector field on L, we obtain a calibrated isotropic pair (L , F ) for the non–straight generalised G 2 –structure [ ⊗ ]od by choosing some distribution complementary to X with L X θ = 0, which locally is always possible. Acknowledgements. We wish to thank Claus Jeschek for his collaboration in the early stages of this project. The second author was supported as a member of the SFB 647 “Space.Time.Matter” funded by the DFG. He also would like to thank Paul Gauduchon and Andrei Moroianu for helpful discussions and creating a very enjoyable ambiance during his stay at École Polytechnique, Palaiseau. Further, he thanks the Max–Planck– Institut für Physik, München, for a kind invitation during the preparation of this paper.
References 1. Apostolov, V., Salamon, S.: Kähler reduction of metrics with holonomy G 2 . Commun. Math. Phys. 246(1), 43–61 (2004) 2. Ben–Bassat, O.: Mirror symmetry and generalized complex manifolds. J. Geom. Phys. 56, 533–558 (2006) 3. Ben–Bassat, O., Boyarchenko, M.: Submanifolds of generalized complex manifolds. J. Symp. Geom. 2(3), 309–355 (2004) 4. Benmachiche, I., Grimm, T.: Generalized N = 1 orientifold compactifications and the Hitchin functionals. Nucl. Phys. B 748, 200–252 (2006) 5. Bergshoeff, E., Kallosh, R., Ortin, T., Papadopoulos, G.: kappa–symmetry, supersymmetry and intersecting branes. Nucl. Phys. B 502, 149–169 (1997) 6. Bergshoeff, E., Kallosh, R., Ortin, T., Roest, D., Van Proeyen, A.: New formulations of D = 10 supersymmetry and D8 − O8 domain walls. Class. Quant. Grav. 18, 3359–3382 (2001) 7. Bouwknegt, P., Evslin, J., Mathai, V.: T–Duality: Topology Change from H –flux. Commun. Math. Phys. 249(2), 383–415 (2004) 8. Bunke, U., Rumpf, P., Schick, T.: The topology of T–duality for T n –bundles. Rev. Math. Phys. 18(10), 1103–1154 (2006) 9. Bunke, U., Schick, T.: On the topology of T–duality. Rev. Math. Phys. 17, 77–112 (2005) 10. Buscher, T.: A symmetry of the string background field equations. Phys. Lett. B 194, 59–62 (1987) 11. Chevalley, C.: The algebraic theory of spinors and Clifford algebras. Collected works, Vol. 2, Berlin: Springer (1996)
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12. Chiantese, S., Gmeiner, F., Jeschek, C.: Mirror symmetry for topological sigma models with generalized Kähler geometry. Int. J. Mod. Phys. A 21, 2377–2390 (2006) 13. Courant, T.: Dirac manifolds. Trans. Amer. Math. Soc. 319, 631–661 (1990) 14. Dadok, J., Harvey, R.: Calibrations and spinors. Acta Math. 170(1), 83–120 (1993) 15. Gauntlett, J., Martelli, D., Waldram, D.: Superstrings with intrinsic torsion. Phys. Rev. D 69, 086002 (2004) 16. Gutowski, J., Ivanov, S., Papadopoulos, G.: Deformations of generalized calibrations and compact non–Kahler manifolds with vanishing first chern class. Asian J. Math. 7(1), 39–79 (2003) 17. Gutowski, J., Papadopoulos, G.: AdS calibrations. Phys. Lett. B 462, 81–88 (1999) 18. Harvey, R.: Spinors and Calibrations. Perspectives in Mathematics Vol. 9, Boston: Academic Press, (1990) 19. Harvey, R., Lawson, H.: Calibrated geometries. Acta Math. 148, 47–157 (1982) 20. Hassan, S.: T–duality, space–time spinors and R-R fields in curved backgrounds. Nucl. Phys. B 568, 145–161 (2000) 21. Hassan, S.: S O(d, d) transformations of Ramond–Ramond fields and space–time spinors. Nucl. Phys. B 583, 431–453 (2000) 22. Hitchin, N.: Generalized Calabi–Yau manifolds. Quart. J. Math. Oxford Ser. 54, 281–308 (2003) 23. Hitchin, N.: Brackets, forms and invariant functionals. Asian J. Math. 10(3), 541–560 (2006) 24. Jeschek, C.: Generalized Calabi–Yau structures and mirror symmetry. http://arXiv.org/list/hep-th/ 0406046, 2004 25. Jeschek, C., Witt, F.: Generalised G 2 –structures and type IIB superstrings. JHEP 0503, 053 (2005) 26. Jeschek, C., Witt, F.: Generalised geometries, constrained critical points and Ramond–Ramond fields. http://arXiv.org/list/math.DG/0510131, 2005 27. Johnson, C.: D–branes. Cambridge: Cambridge University Press, 2003 28. Joyce, D.: The exceptional holonomy groups and calibrated geometry. In: Akbulut, S., Onder, T., Stern R.J. (eds.) Proceedings of the Gokerte ¨ Geometry-Topology Conference 2005. Somerville, MA: Intie Press, 2006, pp. 110–139 29. Kachru, S., Schulz, M., Tripathy, P., Trivedi, S.: New supersymmetric string compactifications. JHEP 0303, 061 (2003) 30. Kapustin, A., Li, Y.: Topological sigma–models with H–flux and twisted generalized complex manifolds. Adv. Theor. Math. Phys. 11(2), 261–290 (2007) 31. Koerber, P.: Stable D–branes, calibrations and generalized Calabi–Yau geometry. JHEP 0508, 099 (2005) 32. Marino, M., Minasian, R., Moore, G., Strominger, A.: Nonlinear instantons from supersymmetric p–branes. JHEP 0001, 005 (2000) 33. Martucci, L.: D–branes on general N = 1 backgrounds: Superpotentials and D–terms. JHEP 0511, 048 (2005) 34. Martucci, L., Smyth, P.: Supersymmetric D–branes and calibrations on general N = 1 backgrounds. JHEP 0511, 048 (2005) 35. Polchinski, J.: Lectures on D–branes. http://arXiv.org/list/hep-th/9611050, 1996 36. Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T–duality. Nucl. Phys. B 479, 243–259 (1996) 37. Wang, M.: Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7(1), 59–68 (1989) 38. Witt, F.: Generalised G 2 –manifolds. Commun. Math. Phys. 265(2), 275–303 (2006) 39. Witt, F.: Special metric structures and closed forms. DPhil thesis, University of Oxford, 2005 40. Witt, F.: Metric bundles of split signature and type II supergravity. In: Alekseevski, D., Baum, H. (eds.) “Recent developments in pseudo-Riemannian Geometry” ESI–Series on Mathematics and Physics, Zurich: European Math. Soc., 2006 41. Zabzine, M.: Geometry of D–branes for general N = (2, 2) sigma models. Lett. Math. Phys. 70, 211–221 (2004) Communicated by G.W. Gibbons
Commun. Math. Phys. 283, 579–611 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0596-0
Communications in
Mathematical Physics
Equilibrium States for Interval Maps: Potentials with sup ϕ − inf ϕ < h t op ( f ) Henk Bruin1 , Mike Todd1,2 1 Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK.
E-mail:
[email protected]; http://www.maths.surrey.ac.uk/
2 Departamento de Matemática Pura, Faculdade de Ciências da Universidade do Porto,
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal. E-mail:
[email protected]; http://www.fc.up.pt/pessoas/mtodd/ Received: 2 August 2007 / Accepted: 19 June 2008 Published online: 7 August 2008 – © Springer-Verlag 2008
Abstract: We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials ϕ with the ‘bounded range’ condition sup ϕ −inf ϕ < h top ( f ), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of Perron-Frobenius operators. We demonstrate that this ‘bounded range’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context. 1. Introduction Thermodynamic formalism is concerned with existence and uniqueness of measures µϕ that maximise the free energy, i.e., the sum of the entropy and the integral over the potential. In other words hν ( f ) + h µϕ ( f ) + ϕ dµϕ = P(ϕ) := sup ϕ dν : − ϕ dν < ∞ , X
ν∈Merg
X
X
where Merg is the set of all ergodic f -invariant Borel probability measures. Such measures are called equilibrium states, and P(ϕ) is the pressure. This theory was developed by Sinai, Ruelle and Bowen [Si,R,Bo] in the context of Hölder potentials on hyperbolic dynamical systems, and has been applied to Axiom A systems, Anosov diffeomorphisms and other systems too, see e.g. [Ba,K2] for more recent expositions. In this paper we are interested in smooth interval maps f : I → I with a finite number of critical points. More precisely, H will be the collection of topologically mixing (i.e., for each n 1, f n has a dense orbit) C 2 maps on the interval (or circle) such that all its periodic points are hyperbolically repelling and all its critical points This research was supported by EPSRC grant GR/S91147/01. MT was partially supported by FCT grant SFRH/BPD/26521/2006 and CMUP.
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are non-flat. The existence of critical points prevents such maps from being uniformly hyperbolic for the ‘natural’ potential ϕ = − log |D f |. Inducing schemes were used in [PeSe] to regain hyperbolicity and prove the existences of equilibrium states for −t log |D f | for a large interval of t, but very specific Collet-Eckmann unimodal maps f . In [BrT] we investigated −t log |D f | with t close to 1 for multimodal maps whose derivatives critical orbits satisfy only polynomial growth. Combining inducing schemes with ideas of so-called Hofbauer towers and infinite state Markov chains (as presented by Sarig [Sa1,Sa2,Sa3]), we proved the existence and uniqueness of equilibrium states within the class M+ = µ ∈ Merg : λ(µ) > 0, supp(µ) ⊂ orb(Crit) , where λ(µ) = log |D f |dµ is the Lyapunov exponent of µ. In fact the assumptions that we make on the potentials in this paper ensure that any equilibrium state must lie in this class, and hence it is no restriction to only consider measures there. Remark 1. Note that the function µ → h µ ( f ) is upper semicontinuous, cf. [BrK, Lemma 2.3]. Hence, if the potential is upper semicontinuous, then the free energy map µ → h µ ( f ) + ϕ dµ is upper semicontinuous too. As Merg is compact in the weak topology, this gives the existence of equilibrium states, but not uniqueness. In this work we want to use inducing schemes to prove existence and uniqueness of equilibrium states for “general” potentials. In this area, there are many results, in particular several papers by Hofbauer and Keller [H1,H2,HK] from the late 1970s. These results were inspired by Bowen’s exposition [Bo] for hyperbolic dynamical systems, and investigate what happens when hyperbolicity fails. Their main tool was the Perron-Frobenius operator, which even for non-uniformly expanding interval maps continues to have a quasi-compact structure for many potentials. In this paper we focus on what can be proved for these problems using inducing techniques. We then apply Sarig’s theory of countable Markov shifts. (A related application of that theory for multidimensional piecewise expanding maps can be found in [BuSa].) In [HK] two main sets of results are given, based on different regularity conditions for the potential; we will present them briefly in Sects. 1.1 and 1.2. At the same time we set out some definitions which will be used throughout the paper. In Sect. 1.4 we present our main results. 1.1. Potentials in BV . Given a function ϕ : I → R, we define the semi-norm · BV as ϕ BV := sup
sup
N −1
N ∈N 0=a0 <···
|ϕ(ak+1 ) − ϕ(ak )|.
We say that ϕ ∈ BV if ϕ BV < ∞. The following result is proved by Hofbauer and Keller in [HK]. Theorem 1 (Hofbauer and Keller). Let f ∈ H and ϕ ∈ BV . If sup ϕ − inf ϕ < h top ( f ),
(1)
then there exists an equilibrium state for ϕ. Moreover, the transfer operator defined by Lϕ g(x) := eϕ(y) g(y) y∈ f −1 (x)
is quasi-compact.
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Condition (1) stipulates that ϕ does not vary too much; similar conditions have been used by e.g. Denker and Urba´nski [DU] for rational maps on the Riemann sphere, and by Oliveira [O] for higher dimensional maps without critical points. We next state a similar result to Theorem 1 from [DKU,P]. Paccaut [P] also gives many interesting statistical properties for the equilibrium states. Theorem 2 (Paccaut). Suppose that ϕ satisfies (a) exp(ϕ) ∞ ∈ BV ; (b) n=1 supC∈Pn ϕ|C BV < ∞; (c) sup ϕ < P(ϕ). Then there exists a unique equilibrium state µϕ for ϕ. Note that condition (b) on ϕ is stronger than the condition ϕ ∈ BV , used in Theorem 1. It is also stronger than that in our results in Sect. 1.4. However, (1) implies condition (c). This follows since assuming (1), the measure of maximal entropy µh top ( f ) gives P(ϕ) h top ( f ) + ϕ dµh top ( f ) h top ( f ) + inf ϕ > sup ϕ. Condition (c) implies that any equilibrium state µ must have h µ ( f ) P(ϕ) − + sup ϕ > 0. Similarly, supposing (1), and using Ruelle’s inequality on Lyapunov exponents (ı. e., h µ ( f ) λ(µ), see [Ru1]), equilibrium states µ satisfy λ(µ) h µ ( f ) = P(ϕ) − ϕ dµ h top ( f ) + ϕ dµh top ( f ) − sup ϕ h top ( f ) − (sup ϕ − inf ϕ) > 0. (2) Hence P+ (ϕ) := supµ∈M+ {h µ ( f ) + supported on orb(Crit).
ϕ dµ} = P(ϕ), unless the equilibrium state is
1.2. Potentials with summable variations. The results that we want to present rely on a different approach to variation than that above, which is closer to symbolic dynamics. Let P1 be the partition of I into the maximal interval of monotonicity (the branch n−1 −i f (P1 ). With respect to this partition we define that partition) and write Pn = i=0 n th variation, Vn (ϕ) := sup
sup |ϕ(x) − ϕ(y)|,
Cn ∈Pn x,y∈Cn
In this context the following was proved in [HK]. Theorem 3 (Hofbauer and Keller). Let f ∈ H be C 3 and let ϕ be a potential so that (i) it has summable variations, i.e., n Vn (ϕ) < ∞; (ii) the following specification-like property holds: for every x ∈ I , there is k and an increasing sequence {n i }i such that ∪kj=1 f n i + j (Cn i [x]) = I, where Cn i [x] ∈ Pn i is the n i -cylinder containing x.
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Then there exists an equilibrium state for ϕ and the transfer operator Lϕ is quasi-compact. Property (ii) above is not automatic for interval maps, and it is stronger than the standard specification property which holds for all topologically transitive interval maps, see [Bl] and [Bu1]. For instance, the Fibonacci unimodal map, or more generally, every map with a persistently recurrent critical point (see e.g. [Br2]) fails this condition. In [DKU], Denker et al. replace the conditions of Theorem 3 to (i) P(ϕ) > sup ϕ and (ii) supn βn (ϕ) < ∞, where βn is defined in (5). Notice that the set of potentials with summable variations and the set BV have non-empty intersection, but neither is contained in the other, as the following examples demonstrate. Example 1. Let f (x) = 2x (mod 1) on [0, 1] be the doubling map. Clearly, the n-cylinders of f are dyadic intervals of length 2−n . The potential function ⎧ 0 if x = 0, ⎪ ⎪ ⎨ 1 −1 ϕ(x) := log x if x ∈ 0, 2 , ⎪ ⎪ ⎩ 1 if x ∈ 1 , 1 , log 2
2
1 is increasing and bounded, and has ϕ BV = log1 2 . However, Vn (ϕ) n log 2 , because 1 ϕ(2−n ) − ϕ(0) = n log 2 . So n Vn (ϕ) diverges. Note that ϕ is not Hölder either.
Example 2. For f as in Example 1, the potential function ψn (x), where ψn (x) := 4−n sin(4n+1 π x) · 1 ψ(x) := n 1
1 , 1 2n 2n−1
(x)
has ψ BV = n ψn BV = ∞, since ψn BV = 2. But Vn (ψ) 4 · 2−n , so it has summable variations. Note that this function is Lipschitz. 1.3. Lifting potentials to inducing schemes. An inducing scheme (X, F, τ ) over (I, f ) consists of an interval X ⊂ I containing a (countable) collection of disjoint subintervals X i , and inducing time τ : X → N such that τi := τ | X i is constant and F| X i := f τi | X i is monotone onto X . If µ F is an F-invariant measure, and X τ dµ F < ∞, then µ F can be projected to an f -invariant measure µ as in formula (3) below. Any measure µ that can be obtained this way is called compatible to the inducing scheme. See Sect. 2.1 for the precise definitions. Proposition 1 below gives a general way of constructing inducing schemes, which we will apply throughout the paper. In Sect. 2.2, we explain the procedure of lifting measures µ to Hofbauer tower ( Iˆ, fˆ), which is behind the construction in this proposition. The full proof of Proposition 1 is given in [BrT, Theorem 3 and Lemma 2]. Proposition 1. If µ ∈ M+ then it is compatible to some induced system (X, F, τ ) that ˆ corresponds ˆ Xˆ ) > 0. to a first return map to a set X on the Hofbauer tower, where µ( 1 So ˆ Xˆ τ d µˆ < ∞, and in addition, we can take X ∈ Pn for some n. Conversely, µ( ˆ X) if an inducing scheme (X, F, τ ) has a non-atomic F-invariant measure µ F such that τ dµ F < ∞, then it projects to an f -invariant measure µ ∈ M+ .
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Given a potential ϕ, the lifted potential for the inducing scheme (X, F, τ ) is given (x)−1 by (x) := τk=0 ϕ ◦ f k (x). If
Vn ( ) < ∞,
(SVI)
n
then we say that ϕ satisfies the summable variations for the induced potential condition, with respect to this inducing scheme. Lemmas 3 and 4 give general conditions on ϕ and/or the inducing scheme that imply (SVI).
1.4. Main results. After these preparations we can state our main results on the existence and uniqueness of equilibrium states, and analyticity of the pressure function. The existence of equilibrium states in Merg often follows by Remark 1, but the following theorem gives conditions for uniqueness of equilibrium states in M+ . Theorem 4. Let f ∈ H and ϕ be a potential such that sup ϕ − inf ϕ < h top ( f ) and Vn (ϕ) → 0. If the induced potentials corresponding to the inducing schemes given by Proposition 1 satisfies (SVI), then (a) there exists a unique equilibrium state µϕ ; (b) µϕ is compatible to an induced system with inducing time such that the tails µ ({τ > n}) decrease exponentially. (Here µ is the equilibrium state of the (x)−1 ψ ◦ f j (x) of ψ := ϕ − P(ϕ).) induced potential (x) = τk= j Note that Vn (ϕ) → 0 implies that ϕ can only have discontinuities at precritical points. Remark 2. If the tails µ ({τ > n}) decrease at certain rates, then one can deduce many statistical properties of the equilibrium state. For instance, exponential decay of correlations follows from exponential tails, see [Y], but for the Central Limit Theorem, Invariance Principles, e.g. [MN1] and large deviations [MN2], already polynomial tail behaviour suffices. Instead of a single potential, the thermodynamic formalism makes use of families tϕ of potentials. The occurrence of phase transitions is related to the smoothness of the pressure function t → P(tϕ). Using the technique in [BrT] we derive Theorem 5. Let f ∈ H and ϕ as in Theorem 4. Then the map t → P(−t ϕ) is analytic for t in a neighbourhood of [−1, 1]. We will not supply a proof of the above theorem, since it follows rather easily from [BrT, Theorem 5]. We will focus our attention on the following related theorem dealing with the potential −t log |D f |. This potential is unbounded, except for t = 0. We conclude that t → P(−t log |D f |) is analytic near t = 0, which is somewhat surprising as we do not require any of the summability conditions of the critical orbits of f used in [BrT]. Theorem 6. Let f ∈ H. There exist t1 < 0 < t2 so that the map t → P(−t log |D f |) is analytic for t ∈ (t1 , t2 ). In fact, for t ∈ (t1 , t2 ) there exists a unique equilibrium state with respect to the potential −t log |D f |.
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We next make a detailed study of an example by Hofbauer and Keller [HK, pp. 32–33] which applies ideas from [H1]. They used it to show the importance of the condition (1) for the quasi-compactness of the transfer operator. We use the example to test the restrictions of the inducing scheme methods, and we also show that (1) cannot simply be replaced by Hölder continuity of the potential by proving the following proposition, cf. [Sa2]. Proposition 2. For α ∈ (0, 1), consider the Manneville-Pomeau map f α : x → x + x 1+α (mod 1). For any b < − log 2, there exists a Hölder potential with sup ϕ − inf ϕ = |b| and which has the form ϕ(x) = −2αx α for x close to 0, which has no equilibrium state accessible from an inducing scheme given by Proposition 1. The remainder of this paper is organised as follows. In Sect. 2 we set out our main tools for generating inducing schemes and applying the theory of thermodynamic formalism. Section 3 contains the tail estimates of inducing schemes we use. In Sect. 4 we prove our main theorem on existence and uniqueness of equilibrium states. In Sect. 5 we show that a consequence of our results is an analyticity result for the pressure, with respect to the kind of potentials considered in [BrT]. In Sect. 6 we give examples, including that in Proposition 2, to show where these techniques break down. Finally in Sect. 7 we discuss the recurrence implied by compactness of the transfer operator, and we present conditions implying the recurrence of the potential ϕ. 2. Equilibrium States via Inducing 2.1. Inducing schemes. As in [BrT] we want to construct equilibrium state via inducing schemes. We say that (X, F, τ ) is an inducing scheme over (I, f ) if • X is an interval1 containing a (countable) collection of disjoint intervals X i such that F maps each X i homeomorphically onto X . • F| X i = f τi for some τi ∈ N := {1, 2, 3 . . . }. The function τ : ∪i X i → N defined by τ (x) = τi if x ∈ X i , is called the inducing time. It may happen that τ (x) is the first return time of x to X , but that is certainly not the general case. Given an inducing scheme (X, F, τ ), we say that a measure µ F is a lift of µ if for all µ-measurable subsets A ⊂ I , µ(A) =
1 F,µ
i −1 τ
i
k=0
µ F (X i ∩ f −k (A))
for
F,µ :=
τ dµ F .
(3)
X
Conversely, given a measure µ F for (X, F), we say that µ F projects to µ if (3) holds. Not every inducing scheme is relevant to every invariant measure. Let X ∞ = ∩n F −n (∪i X i ) be the set of points on which all iterates of F are defined. We call a measure µ compatible with the inducing scheme if • µ(X ) > 0 and µ(X \ X ∞ ) = 0, and • there exists a measure µ F which projects to µ by (3), and in particular F,µ < ∞. 1 Due to our assumption that f is topological mixing, we can always find a single interval to induce on, but a similar theory works for X a finite union of intervals.
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2.2. The Hofbauer tower. Let Pn be the branch partition for f n . The canonical Markov extension (commonly called Hofbauer tower) is a disjoint union of subintervals D = f n (Cn ), Cn ∈ Pn , called domains. Let D be the collection of all such domains. For completeness, let P0 denote the partition of I consisting of the single set I , and call D0 = f 0 (I ) the base of the Hofbauer tower. Then Iˆ = n 0 Cn ∈Pn f n (Cn )/ ∼, where f n (Cn ) ∼ f m (Cm ) if they represent the same interval. Let π : Iˆ → I be the inclusion map. Points xˆ ∈ Iˆ can be written as (x, D) if D ∈ D is the domain that xˆ belongs to and x = π(x). ˆ The map fˆ : Iˆ → Iˆ is defined as fˆ(x) ˆ = fˆ(x, D) = ( f (x), D ) if there are cylinder sets Cn ⊃ Cn+1 such that x ∈ f n (Cn+1 ) ⊂ f n (Cn ) = D and D = f n + 1 (Cn + 1 ). In this case, we write D → D , giving (D, →) the structure of a directed graph. It is easy to check that there is a one-to-one correspondence between cylinder sets Cn ∈ Pn and n-paths D0 → · · · → Dn starting at the base of the Hofbauer tower and ending at some terminal domain Dn . If R is the length of the shortest path from the base to Dn , then the level of Dn is level(Dn ) = R. Let IˆR = level(D)R D. Several of our arguments rely on the fact that the “top” of the infinite graph (D, →) generates arbitrarily small entropy. These ideas go back to Keller [K1], see also [Bu2]. It is also worth noting that the main information is contained in a single transitive part of Iˆ. Lemma 1. If I is a finite union of intervals, and the multimodal map f : I → I is transitive, then there is a closed primitive subgraph (E, →) of (D, →) containing a dense fˆ-orbit and such that I = π(∪ D∈E D). We denote the transitive part of the Hofbauer tower by Iˆtrans . For details of the proof see [BrT, Lemma 1]. Let i : I → D0 be the trivial bijection (inclusion) such that i −1 = π | D0 . Given a probability measure µ, let µˆ 0 := µ ◦ i −1 , and µˆ n :=
n−1 1 µˆ 0 ◦ fˆ−k . n
(4)
k=0
We say that µ is liftable to ( Iˆ, fˆ) if there exists a vague accumulation point µˆ of the sequence {µˆ n }n with µˆ ≡ 0, see [K1]. The following theorem is essentially proved there, see [BrK] for more details. Theorem 7. Suppose that µ ∈ M+ . Then µˆ is an fˆ-invariant probability measure on Iˆ, and µˆ ◦ π −1 = µ. Conversely, if µˆ is fˆ-invariant and non-atomic, then λ(µ) ˆ > 0. The strategy followed in [BrT] is to take the first return map to the appropriate set in the Hofbauer tower of (I, f ) and to use the same inducing time for the projected partition on the interval. Saying that an induced system (X, F, τ ) corresponds to a first ˆ τ ) on the Hofbauer tower means that if xˆ ∈ Xˆ ⊂ Iˆ, then τ ◦ π is the return map ( Xˆ , F, first return time of xˆ under fˆ to Xˆ .
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2.3. Pressure and recurrence. A topological, i.e., measure independent, way to define pressure was presented in [W]; with respect to the branch partition P1 , it is defined as 1 sup eϕn (x) , log n→∞ n x∈Cn
Ptop (ϕ) := lim
Cn ∈Pn
n − 1
where ϕn (x) := k = 0 ϕ ◦ f k (x). We say that the Variational Principle holds if P(ϕ) = Ptop (ϕ). If ϕ has sufficiently controlled distortion, then the sum of supx∈Cn eϕn (x) over all n-cylinders can be replaced by the sum of eϕn (x) over all n-periodic points, and thus we arrive at the Gurevich pressure w.r.t. cylinder set C ∈ P1 , PG (ϕ) := lim sup n→∞
1 log Z n (ϕ, C) n
for
Z n (ϕ, C) :=
eϕn (x) 1C (x).
f n x=x
If (I, f ) is topologically mixing and βn (ϕ) := sup
sup |ϕn (x) − ϕn (y)| = o(n),
Cn ∈Pn x,y∈Cn
(5)
then PG (ϕ) is independent of the choice of C ∈ P1 , as was shown in [FFY]. Since the branch partition is finite, potentials with bounded variations are bounded, and hence their Gurevich pressure is finite. If ϕ is unbounded above (whence Ptop (ϕ) = ∞) or the number of 1-cylinders is infinite (as may be the case for induced maps F and induced potential ), Gurevich pressure proves its usefulness. Suppose that (I, f, ϕ) is topologically mixing. For every C ∈ P1 and n 1, recall that we defined Z n (ϕ, C) := eϕn (x) 1C (x). f n x=x
Let Z n∗ (ϕ, C) :=
eϕn (x) 1C (x).
f n x=x,
f k x ∈C /
for
0
The potential ϕ is said to be recurrent if2 λ−n Z n (ϕ) = ∞ for λ = exp PG (ϕ).
(6)
n
Moreover, ϕ is called positive recurrent if it is recurrent and In some cases we will use the quantity Z 0 (ϕ) := sup eϕ(x) .
n
nλ−n Z n∗ (ϕ) < ∞. (7)
C∈P1 x∈C
Proposition 1 of [Sa1] implies that if ϕ has summable variations then for any C, Z n (ϕ, C) = O(Z 0 (ϕ)n ). Hence Z 0 (ϕ) < ∞ implies PG (ϕ) < ∞. 2 The convergence of this series is independent of the cylinder set C, so we suppress it in the notation.
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Although we do not assume that the potential ϕ has summable variations, it is important that the induced potential has summable variations, as we want to apply the following result which collects the main theorems of [Sa3]. We give a simplified version of the original result since we assume that each branch of the induced system (X, F) is onto X . We refer to such a system as a full shift. Theorem 8. If (X, F, ) is a full shift and n 1 Vn ( ) < ∞, then has an invariant Gibbs measure if and only if PG ( ) < ∞. Moreover the Gibbs measure µ has the following properties: (a) If h µ (F) < ∞ or − dµ < ∞ then µ is the unique equilibrium state (in particular, P( ) = h µ (F) + X dµ ); (b) The Variational Principle holds, i.e., PG ( ) = P( ). Note that an F-invariant measure µ is a Gibbs measure w.r.t. potential if there is K 1 such that for every n 1, every n-cylinder set Cn and every x ∈ Cn , µ(Cn ) 1 (x)−n P ( ) K . G K e n Using this theory, the following was proved in [BrT]. Proposition 3. Suppose that ψ is a potential with PG (ψ) = 0. Let Xˆ be the set used Proposition 1 to construct the correspondinginducing scheme (X, F, τ ). Suppose that the lifted potential has PG ( ) < ∞ and n 1 Vn ( ) < ∞. Consider the assumptions:
i < ∞ for := sup (a) i x∈X i (x); i τi e (b) there exists an equilibrium state µ ∈ M+ compatible with (X, F, τ ); (c) there exist a sequence {εn }n ⊂ R− with εn → 0 and measures {µn }n ⊂ M+ such that every µn is compatible with (X, F, τ ), h µn ( f ) + ψ dµn εn and PG ( εn ) < ∞ for all n. If any of the following combinations of assumptions holds: 1. (a) and (b); 2. (a) and (c); then there is a unique equilibrium state µ for (I, f, ψ) among measures µ ∈ M+ with µ( ˆ Xˆ ) > 0. Moreover, µ is obtained by projecting the equilibrium state µ of the inducing scheme and we have PG ( ) = 0. In the remaining part of this section, we give some technical results which connect different ways of computing pressure and Gurevich pressure. We use the following theorem of [FFY] to show the connection between PG (ϕ) ˆ and P+ (ϕ). Theorem 9. If (, S) is a transitive Markov shift and ψ : → R is a continuous function satisfying βn (ψ) = o(n) then PG (ψ) = P(ψ). ˆ = o(n), and ϕˆ is continuous in the symbolic metric on ( Iˆ, fˆ) then Corollary 1. If βn (ϕ) PG (ϕ) ˆ = P+ (ϕ).
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Proof. We show that the system ( Iˆtrans , fˆ, ϕ) ˆ satisfies the conditions of Theorem 9, where Iˆtrans is given below Lemma 1. For x, ˆ yˆ ∈ Pˆ with Pˆ ∈ Pˆ n , we have |ϕˆn (x) ˆ − ϕˆn ( yˆ )| = o(n), and Theorem 9 implies PG (ϕ) ˆ = P(ϕ). ˆ It remains to show that P(ϕ) ˆ = P+ (ϕ). By Theorem 7, any measure in M+ lifts to Iˆ. We also know that a countable-to-one factor map preserves entropy, provided the Borel sets are preserved by lifting, see [DoS]. For similar arguments, see [Bu2]. Suppose that {µˆ n }n is a sequence of fˆ-invariant measures such that h µˆ n ( f ) + ϕˆ d µˆ n → P(ϕ) ˆ as ˆ also. n → ∞. Then for the projections µn = µˆ n ◦ π −1 , h µn ( f ) + ϕ dµn → P(ϕ) So P+ (ϕ) P( ˆ On the other hand, let {µn }n ⊂ M+ be a sequence of measures such ϕ). that h µn ( f ) + ϕ dµn → P+ (ϕ) as n → ∞. Lifting these measures using Theorem 7, we get h µˆ n ( f ) + ϕˆ d µˆ n → P+ (ϕ), so P+ (ϕ) P(ϕ) ˆ as required. We next show that Gurevich pressure can be computed from cylinders of any order. Lemma 2. Let (, f ) be a topologically mixing Markov shift. If ϕ : → R satisfies βn (ϕ) = o(n), then PG (ϕ, C) = PG (ϕ, C ) for any two cylinders C, C of any order. Proof. Denote the Markov partition of Iˆ into domains D by D. Take D, D ∈ D such that C ⊂ D and C ⊂ D . By transitivity, there is a k-path C ⊂ D → · · · → D and a k -path C ⊂ D → · · · → D. Then for every n-periodic point x ∈ C, there
is a point x ∈ C such that f k (x ) ∈ Cn [x], the n-cylinder containing x. Therefore
f k + n (x ) ∈ C and f k +n+k (x ) ∈ x . It follows that eϕn + k + k (x ) eβn + (k + k ) sup ϕ eϕn (x) , whence
Z n (ϕ, C) e−βn −(k + k ) sup ϕ Z n + k + k (ϕ, C ). Therefore, using βn = o(n), we obtain for the exponential growth rate PG (ϕ, C) limn βnn + PG (ϕ, C ) = PG (ϕ, C ). Reversing the roles of C and C yields PG (ϕ, C) = PG (ϕ, C ).
2.4. Summable variations for the inducing scheme (SVI). In this section we give conditions on ϕ and under which (SVI) holds for the inducing scheme. Lemma 3. (a) If
nVn (ϕ) < ∞,
n
then (SVI) holds with respect to any inducing scheme. (b) Let ϕ be α-Hölder continuous and let (X, F, τ ) be an inducing scheme obtained from Proposition 1 that satisfies sup i
τ i −1
| f k (X i )|α < ∞.
k=0
Then (SVI) holds w.r.t. that inducing scheme.
(8)
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Proof. To prove (a), we apply [Sa1, Lemma 3, Part 1]. Note that the results in the chapter of [Sa1] containing this result are valid if (X, F, τ ) is a first return map, which is not true for our case. However, from Proposition 1, we constructed (X, F, τ ) to be isomorphic to a first return map on the Hofbauer tower, with potential ϕˆ = ϕ ◦ π . Since τ (x)−1 τ (x)−1 (x) = k = 0 ϕ ◦ f k (x) = k = 0 ϕˆ ◦ fˆk (x) ˆ for each xˆ ∈ π −1 (x), both the original system and the lift to the Hofbauer tower lead to the same induced potential. Therefore [Sa1, Lemma 3, Part 1] does indeed apply. Now to prove (b), note that F : ∪i X i → X is extendible, f τi −k : f k (X i ) → X has bounded distortion for each 0 k < τi . Consequently, also f k : X i → f k (X i ) has bounded distortion. Suppose that |ϕ(x) − ϕ(y)| Cϕ |x − y|α . Since (x) = τi −1 k k = 0 ϕ ◦ f (x) for x ∈ X i , we get for x, y ∈ X i , | (x) − (y)|
τ i −1 k=0 τ i −1 k =0 τ i −1
|ϕ ◦ f k (x) − ϕ ◦ f k (y)| Cϕ | f k (x) − f k (y)|α α
Cϕ K | f (X i )| ·
k =0
k
|x − y| |X i |
α
,
where K is the relevant Koebe constant for F. Thus the condition in (b) implies that the variation V1 ( ) is bounded. Because F is uniformly expanding, the diameter of n-cylinders of F decreases exponentially fast, so if x and y ∈ X i belong to the same n-cylinder, the above estimate is exponentially small in n, and summability of variations follows. The following lemma gives conditions on f , under which condition (b) can be used for Hölder potentials.3 We say that c ∈ Crit has critical order c if there is a constant C 1 such that C1 |x − c|c | f (x) − f (c)| C|x − c|c for all x; f is non-flat if c < ∞ for all c ∈ Crit. Lemma 4. Assume that f is a C 3 multimodal map with non-flat critical points, and let max := max{c : c ∈ Crit}. There exists K = K (#Crit, max ) such that if lim inf |D f n ( f (c))| K n
for all c ∈ Crit,
then formula (8) holds for every α > 0 and every inducing scheme obtained as in Proposition 1 on a sufficiently small neighbourhood of Crit. Proof. We will use several results of [BRSS]. First, Theorem 1 of that paper says that for any r > 1, we can find ε0 > 0 and K = K (#Crit, max , r ) such that if lim inf n |D f n ( f (c))| K for all c ∈ Crit, then the following backward contraction property holds: Given ε ∈ (0, ε0 ) and Uε := ∪c∈Crit B( f (c); ε) and s ∈ N, if W is a component of f −s (Uε ) with d(W, f (Crit)) < ε/r , then |W | < ε/r . Furthermore, see [BRSS, Prop. 3], we can find a nice set V := ∪c∈Crit Vc ⊂ f −1 (Uε/r ), where each Vc is an interval neighbourhood of c ∈ Crit and nice means that f n (∂ V ) ∩ 3 In Lemma 4 we take inducing schemes on a union of intervals. As in Sect. 2.1, transitivity implies that this result passes to any single sufficiently small interval.
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V = ∅ for all n ∈ N. It follows that if W is a component of f −s (V ) contained in V , then |W | r −1/max maxc∈Crit |Vc |. Proceeding by induction, and assuming that r > 2 is sufficiently large to control distortion effects (cf. [BRSS, Lemma 3]), we can draw the following conclusion. Let V0 be a component of f −n (V ), Vi := f i (V0 ) and let 0 = t0 < t1 < · · · < tk = n be the successive times that Vt ⊂ V . Then |Vt j | 2 j−k maxc∈Crit |Vc |. Additionally, Mañé’s Theorem implies that there are λ > 1 and C > 0 (depending on V and f only) such that |Vi | Cλ−(t j −i) |Vt j | for t j−1 < i < t j . Therefore n i=0
|Vi |α
k
C α λ−mα |Vt j |α
j=0 m 0
k C α − jα 2 max |Vc |α . c∈Crit 1 − λ−α j=0
This implies the lemma. 3. Tail Estimates for Inducing Schemes In the following lemma, we let Xˆ ⊂ Iˆtrans be a cylinder in π −1 (P N ) ∨ D compactly contained in its domain. This cylinder set corresponds to an N -path q: D → · · · → D N in Iˆ. The first return map to Xˆ is the induced system that we will use. The growth rate of paths in the Hofbauer tower is given by the topological entropy. Clearly, if we remove Xˆ from the tower, then this rate will decrease: we will denote it by h ∗top ( f ). If Xˆ is very small, then h ∗top is close to h top ( f ), so (1) implies that sup ϕ − inf ϕ < h ∗top for Xˆ sufficiently small. Note that we can in fact take Xˆ to be the type of set, a union of domains in Iˆ, considered in [Br1]. We will use this type of domain in Sect. 5. Proposition 4. Suppose that Vn (ϕ) → 0 and let ψˆ = ϕˆ − PG (ϕ, ˆ Xˆ ). If Xˆ ∈ Pˆ N is so small that sup ϕ − inf ϕ < h ∗top , ˆ Xˆ ) < C e−γ n . then there exist C, γ > 0 such that Z n∗ (ψ, ˆ of all n − 1-paths ˆ Xˆ ) by adding the weights eϕˆn−1 (x) Proof. We will approximate Z n∗ (ϕ, from fˆ( Xˆ ) to Xˆ in the Hofbauer tower with outgoing arrows from Xˆ removed. By removing these arrows we ensure that these paths will not visit Xˆ before step n, so ˆ Xˆ ) and not Z n (ϕ, ˆ Xˆ ). In considering n − 1-paths, we we indeed approximate Z n∗ (ϕ, ˆ Xˆ ˆ for xˆ = fˆn ( x) in the weight eϕˆn (x) ˆ ∈ Xˆ , so it only miss the initial contribution eϕ| ∗ ∗ ˆ ˆ ˆ Xˆ ) = ˆ X ) of Z n (ϕ, ˆ X ). Since Z n∗ (ψ, will not effect the exponential growth rate PG (ϕ, ˆ ˆ X ) Z ∗ (ϕ, ˆ e−n PG (ϕ, n ˆ X ), the proposition follows if we can show the strict inequality ∗ ˆ P (ϕ, ˆ X ) < PG (ϕ, ˆ Xˆ ). G
Remark. It is this strict inequality that is responsible for the discriminant D F [ϕ] in Sect. 5 being strictly positive.
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The rome technique. We will approximate the Hofbauer tower by finite Markov graphs, and use the following general idea of romes in transition graphs from Block ˆ Xˆ ). Let G be a finite graph where every edge i → j et al. [BGMY] to estimate Z n∗ (ϕ, has a weight wi, j , and let W = (wi, j ) be the corresponding (weighted) transition matrix. More precisely, wi, j is the total weight of all edges i → j, and if there is no edge i → j, then wi, j = 0. A subgraph R of G is called a rome, if there are no loops in G \ R. A simple path p of length l( p) is given by i = i 0 → i 1 → · · · → il( p) = j, where i, j ∈ R, but the l( p) intermediate vertices belong to G \ R. Let w( p) = k=1 wik−1 ,ik be the weight of p. The rome matrix Arome (x) = (ai, j (x)), where i, j run over the vertices of R, is given by ai, j (x) = w( p)x 1−l( p) , p
where the sum runs over all simple paths p as above. (Note that with the convention that x 0 = 1 for x = 0, Arome (0) reduces to the weighted transition matrix of the rome R.) The result from [BGMY] is that the characteristic polynomial of W is equal to det(W − x IW ) = (−x)#G −#R det(Arome (x) − x Irome ),
(9)
where IW and Irome are the identity matrices of the appropriate dimensions. In our proof, we will use k-cylinder sets as vertices in the graph G, and we will take w( p) = eϕˆl( p) (x) for some x belonging to the interval in Iˆ that is represented by the path p. Choice of the rome. Fix a large integer k. The partition Pˆ k is clearly a Markov partition for the Hofbauer tower, and its dynamics can be expressed by a countable graph (Pˆ k , →), ˆ Qˆ ∈ Pˆ k only if fˆ( P) ˆ ⊃ Q. ˆ Choose R k (to be determined where Pˆ → Qˆ for P, later). Given a domain D of level R, from all the R-paths starting at D, at most two (namely those corresponding to the outermost R-cylinders in D) avoid IˆR . Any other R-path from D has a shortest subpath D → · · · → D , where both D and D ∈ IˆR . Let us call the union of all points in Iˆ that belong to one of such subpaths the wig of IˆR . The vertices of the rome R are those cylinder sets Pˆ ∈ Pˆ k , Pˆ ⊂ Xˆ , that are either contained in domains D ∈ D of level < R, or that belong to the wig. We retain all arrows between two vertices in R. Let AR be the weighted transition matrix of R. For ˆ x) ˆ . Let ˆ choose xˆ ∈ Pˆ such that fˆ(x) ˆ and set w ˆ ˆ = eϕ( each arrow Pˆ → Q, ˆ ∈ Q, P, Q ρR be the leading eigenvalue of the weighted transition matrix. The pressure PG∗ (ϕ) ˆ is approximated (with error of order Vk (ϕ)) ˆ by log ρR . The graph (R, →) is a finite subgraph of the full infinite Markov graph (Pˆ k , →). We will construct two other finite graphs (G0 , →) and (G1 , →) both having R as a rome, and minorising respectively majorising (Pˆ k , →) in the following sense: For each path in (G0 , →), including those passing through Xˆ , we can assign a path in (Pˆ k , →) of comparable weight, and this assignment can be done injectively. Conversely, for each path in (Pˆ k , →), except those passing through Xˆ , we can assign a path in (G1 , →) of comparable weight, and this assignment can be done injectively. As R is a rome to both G0 and G1 , we can use the rome technique to compare the spectral radii ρ0 and ρ1 of their respective weighted transition matrices W0 and W1 . ∗ By the above minoration/majoration property, we can separate e PG (ϕ) from e PG (ϕ) by ρ0 and ρ1 , up to a distortion error. By refining the partition of the Hofbauer tower into
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k-cylinders, i.e., taking k large, whilst maintaining the majoration/minoration property, we can reduce the distortion error (relative to the iterate), and also show that ρ0 < ρ1 . This will prove the strict inequality PG∗ (ϕ) ˆ < PG (ϕ). ˆ The graph G0 . First, to construct G0 , we add the arrows Pˆ → Qˆ for each Pˆ ∈ Pˆ k ∩ Xˆ ˆ x) ˆ for some chosen ˆ ⊃ Q. ˆ The weight of this arrow is eϕ( and Qˆ ∈ Pˆ k such that fˆ( P) ˆ xˆ ∈ P. Let W0 be the weighted transition matrix of G0 . It follows that its spectral radius ˆ , up to an error of order e Vk (ϕ) ˆ . Furthermore, the number of is a lower bound for e PG (ϕ) n(h ( f ) − ε ) top R n-paths in R is at least e , where ε R → 0 as R → ∞, cf. [H2]. Since each arrow has weight at least einf ϕˆ , we obtain ˆ R ˆ + Vk (ϕ) ˆ ρ0 := ρ(W0 ) e PG (ϕ) . eh top ( f ) + inf ϕ−ε
(10)
Let L be such that f L (π( Xˆ )) ⊃ I . Let v = (v Pˆ ) P∈ ˆ Pk be the positive left unit eigenvector corresponding to the leading eigenvalue ρR of AR . Recall that for each R0 ∈ N and D ∈ Iˆ there are at most two R0 -paths from D leading to domains of level > R0 . Each such path corresponds to a subinterval of D adjacent to ∂ D, and although this subinterval may consist of many adjacent cylinder sets of Pˆ k , fˆ R maps them monotonically onto adjacent cylinder sets of Pˆk−R0 . Therefore D∈D
level(D)>R0
ˆ Pk ∩D Q∈
v Qˆ =
1 R0
ρR
ˆ level( Q)>R 0
ˆ Pk P∈
−R0 2esup ϕˆ R0 ρR
R0 v Pˆ (AR ) P, ˆ Qˆ
v Pˆ
ˆ Pk P∈
ˆ ˆ ϕ−h ˆ top ( f )) = 2esup ϕˆ R0 −R0 PG (ϕ) 2e R0 (sup ϕ−inf ,
independently of k. Since sup ϕˆ − inf ϕˆ − h top ( f ) < 0, we can take R0 so large, independently of k, that for every x ∈ I , 1 ˆ x . v Qˆ < min v Qˆ : Qˆ ∈ Pk ∩ fˆL ( Xˆ ), π( Q) (11) 2 ˆ ˆ ˆ Q∈Pk , π( Q)x ˆ level( Q)>R 0
The idea is now to offset all contributions of n-paths starting from level > R0 to Z n∗ (ϕ, ˆ Xˆ ) L by the contribution of n-paths starting in fˆ ( Xˆ ) to Z n (ϕ, ˆ Xˆ ). Let N L be such that there is an N -path from Xˆ to every Qˆ of level R0 . Then N vW0N v AR + e− inf ϕˆ N Qˆ Qˆ N + e− inf ϕˆ N κ v ˆ ρR Qˆ if level( Q) R0 , (12) N ˆ > R0 , ρ vˆ if level( Q) R Q
where κ := min{v Qˆ : Qˆ ∈ Pˆ k ∩ fˆL ( Xˆ )}/ max v Qˆ , and is a nonnegative square matrix with some 1s in the rows corresponding to Pˆ ∈ Pˆ k ∩ fˆL ( Xˆ ) in such a way that ˆ R0 has at least one 1. The fact the column corresponding to each Qˆ with level( Q)
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that κ > 0 uniformly in the order of cylinder sets k rests on the following claim, which is proved later on: min
ˆ Pˆ k ∩ fˆ L ( Xˆ ) Q∈
v Qˆ / max v Qˆ > 0 ˆ Pˆ k Q∈
uniformly in R and k.
(13)
By the choice of L, R0 (see (11)) and N , ˆ ˆ Pˆ k ,level( Q)>R Q∈ 0
ˆ
N eϕˆ N ( Q) v Qˆ (AR ) Q, ˆ Pˆ
1 2
v Qˆ (W0N ) Q, ˆ Pˆ
(14)
ˆ Pˆ k ∩ fˆ L ( Xˆ ) Q∈
ˆ R0 . When we apply W N to (12) once more, the components for each Pˆ with level( P) 0 ˆ R0 have increased by a factor ρ N + e− inf ϕˆ N κ, whereas by (14), the v Qˆ with level( Q) R ˆ > R0 combined amount to at most half the weight of the components v ˆ with level( Q) Q
components v Qˆ with Qˆ ∈ Pˆ k ∩ fˆL ( Xˆ ). Therefore, we can generalise (12) inductively to m N vW0N m v AR + e− inf ϕˆ N Qˆ Qˆ 1 − inf ϕˆ N m N ˆ R0 , (ρR + 2 e κ) v Qˆ if level( Q) (15) N m ˆ > R0 , ρR v Qˆ if level( Q) for all m 1. It follows that 1 1 − inf ϕˆ N m N ρR + e κ v Qˆ 2 ρ0N m ˆ level( Q)
R0
ˆ level( Q)
→α
R0
ˆ level( Q)
1 ρ0m N
vW0m N
Qˆ
w Qˆ
R0
for some α < ∞ and w the left unit eigenvector corresponding to the leading eigenvalue ρ0 of W0 . This implies that 1/N 1 N + e− inf ϕˆ N κ whence ρ0 > ρR + κ (16) ρ0 ρR 2 for some κ = κ (κ, N , ϕ, ˆ R0 ) > 0, uniformly in R R0 and k ∈ N. The graph G1 . For each Pˆ ∈ Pˆ k ∩ D where D has level R, consider all R-paths p : Pˆ → . . . → Qˆ that avoid IˆR ; these are not included in (R, →). From each D of level R, there at most 2 such R-paths avoiding IˆR , corresponding to R-cylinders in D. These two R-cylinders are contained in two k-cylinders in D. For each such k-cylinder ˆ choose Qˆ ∈ Pˆk ∩ IˆR and Pˆ (i.e., vertex in (Pˆ k , →)), and each Q ∈ Pk ∩ f R (π( P)), ˆ Assign attach an artificial R-path with R − 1 new vertices and a terminal vertex Q. R sup ϕ ˆ to this path. Therefore, if f is d-modal, the number of vertices weight w( p) = e added to IˆR is therefore no larger than 2d(R − 1). Call the resulting graph G1 and W1 its weighted transition matrix. Any n-path in the Hofbauer tower that leaves IˆR for at least R iterates can be mimicked by an n-path following one of the additional R-paths in G1 . But n-orbits visiting Xˆ are still
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left out. It follows that this time, the leading eigenvalue estimate exceeds the exponential growth rate of the contributions of all n-periodic orbits in the Hofbauer tower that avoid ˆ still needs to be taken into account, we get Xˆ . Since the error of order e Vk (ϕ) ∗
ˆ ˆ k (ϕ) . ρ1 := ρ(W1 ) e PG (ϕ)−V
(17)
On the other hand, we can use (9) to deduce that det(W1 − x IW1 ) = (−x)#G1 −#R det(A1 (x) − x IR ),
(18)
R sup ϕˆ x 1−R where the rome matrix A1 (x) equals AR , except for new entries w P, ˆ Qˆ e for the R-path added to the rome. These paths correspond to R-cylinders, at most 2 for each of the d domains of level R, and since R k, there are at most 2d paths with ˆ In other words, initial vertices Pˆ ∈ Pk , each with at most #Pk terminal vertices Q. 1−R R sup ϕ ˆ A1 (x) AR + x e 1 , where 1 is a square matrix with at most 2d non-zero ˆ and zeros otherwise. Formula (18) shows that rows (corresponding to initial vertices P) ρ1 is also the leading eigenvalue of A1 (ρ1 ). Although matrices AR and ρ11−R e R sup ϕˆ 0 depend both on R and k, at the moment we will only need k so large that 1 ∗ h top ( f ) − (sup ϕˆ − inf ϕ) Vk α := ˆ (19) 2
and hence suppress the dependence on k until it is needed again. We first give some estimates necessary to apply Lemma 6 below with U R = AR and V R = ρ11−R e R sup ϕˆ 1 . The ‘left’ matrix norm (which is the maximal row-sum) of 1 is 1 := supv1 =1 v1 1 = #Pk , and therefore (using also (17)) we obtain ρ 1−R esup ϕˆ 1 #Pk ρ11−R e R sup ϕˆ ∗
ˆ ˆ ˆ k (ϕ)) G (ϕ)+V #Pk e R(sup ϕ−P ∗
ˆ ϕ−h ˆ top ( f )+Vk (ϕ)) ˆ #Pk e R(sup ϕ−inf #Pk e−α R
for α > Vk (ϕ) ˆ as in (19). The entries of (Am ˆ Qˆ indicate the sum of the weights of all R ) P, ˆ For each Qˆ ∈ Pˆk , the sum m-paths from Pˆ to Q. ˆ ∩ f −m (Q)} esup ϕm |π( P)ˆ , esup ϕˆm | Pˆ #{components of π( P) ˆ π( Q)⊂Q
paths
ˆ Qˆ P→
m ηm which has exponential growth-rate ρR . Therefore the left matrix norm Am R ρR e for some η = η(R, k) with lim R→∞ η(R, k) = 0 for each fixed k. If v is the positive left eigenvector of A1 (ρ1 ), corresponding to ρ1 and normalised so that v 1 := i |vi | = 1, then
ρ1 = v ρ1 1 = v (A1 (ρ1 )m 1
1/m
= (AR + ρ11−R e R sup ϕˆ 1 )m 1/m 1/m ˜ ˜ ρR 1 + AR em η(R,k) → ρR eη(R,k) where η(R, ˜ k) comes from Lemma 6.
as m → ∞,
(20)
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Using (20) and (16) we obtain ˜ ˜ ρ1 ρR eη(R,k) eη(R,k) (ρ0 − κ ).
By claim (13), κ > 0 uniformly in R and k, and by Lemma 6, we can choose R large (and hence η(R, ˜ k) small) to derive that ρ1 < ρ0 . It follows by (10) and (17) that ∗
ˆ ˆ ˆ ˆ k (ϕ) k (ϕ) e PG (ϕ)+V ρ0 > ρ1 e PG (ϕ)−V ,
ˆ > PG∗ (ϕ) ˆ as required. so taking the limit k → ∞, we get PG (ϕ) Proof of Claim (13). We start with the uniformity in R, i.e., the level at which the Hofbauer tower is cut off. Recall that we assumed that Xˆ is so small that sup ϕ−inf ˆ ϕˆ < h ∗top . ∗ ˆ ˆ k (ϕ)−ε R (see (10)), because R The leading eigenvalue ρR of AR satisfies ρR e PG (ϕ)−V is a subgraph of the Hofbauer tower with Xˆ removed. For any r and any domain D ∈ Iˆ, ˆ Qˆ ∈ Pˆ k , where Qˆ is there are at most two r -paths ending outside Iˆr . Therefore if P, ˆ Q-entry ˆ contained in a domain D of level r , the P, of ArR is at most 2er sup ϕˆ . Thus we find for the left eigenvector v, r v ArR Qˆ 2er sup ϕˆ ρR v Qˆ = v Pˆ 2er sup ϕˆ . ˆ Pˆ k ∩D Q∈
It follows that
ˆ Pˆ k ∩D Q∈
ˆ Pˆ k P∈
∗
∗
ˆ ϕ−h ˆ top ( f )+Vk (ϕ)+ε ˆ ˆ ˆ ˆ R) k (ϕ)+ε R )) 2er (sup ϕ−inf G (ϕ)+V v Qˆ 2er (sup ϕ−P
ˆ Pˆ k ∩D Q∈
is exponentially small in r . There are at most 2d domains D of level r , which implies that level( Q)>r v ˆ Pˆ is exponentially small in r , and this is independent of R r , and of how (or whether) the Hofbauer tower is truncated. Next take r0 so large that level( Q)>r v < 21 irrespective of the way the Hofbauer ˆ 0 Pˆ tower is cut, and such that Xˆ belongs to a transitive subgraph of Iˆr0 . Therefore there is r such that for every domain D of level(D) r0 and every Qˆ ∈ Pˆ k ∩ D, there 0
ˆ Pˆ entry in Ar0 is at least er0 inf ϕˆ for every is an r0 -path from Qˆ to Xˆ . Hence the Q, R
−1 AR )r0 , we find Pˆ ∈ Pˆ k ∩ Xˆ . Since v = v(ρR ˆ Pˆ k ∩ Xˆ P∈
−r
v Pˆ ρR 0 er0 inf ϕˆ
ˆ Pˆ k ∩ Iˆr Q∈ 0
v Qˆ
1 −r0 r inf ϕˆ ρ e0 2 R
independently of R r0 . Now we continue with the uniformity in k. This is achieved by analysing the effect of splitting of vertices of the transition graph into new vertices, representing cylinders of higher order. We do this one vertex at the time. Let W be a weighted transition matrix of a graph G. Given a vertex g ∈ G, we can represent the 2-paths from g by splitting g as follows (for simplicity, we assume that the first row/column in W represents arrows from/to g): • If g →w1,b1 b1 , g →w1,b2 b2 , . . . , g →w1,bm bm are the outgoing arrows, replace g by m vertices g1 , . . . , gm with outgoing arrows g1 →w1,b1 b1 , g2 →w1,b2 b2 , . . . , gm →w1,bm bm respectively, where w1,b j represents the weight of the arrow.
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• Replace all incoming arrows c →wc,1 g by m arrows c →wc,1 g1 , c →wc,1 g2 , . . . , c →wc,1 gm , all with the same weight. • If g → g was an arrow in the old graph, this means that g1 will now have m outgoing arrows: g1 →w1,1 g1 , g1 →w1,1 g2 , . . . , g1 →w1,1 gm , all with the same weight. Lemma 5. If W has leading eigenvalue ρ with left eigenvector v = (v1 , . . . , vn ), then the weighted transition matrix W˜ obtained from the above procedure has again ρ as leading eigenvalue, and the corresponding left eigenvector is v˜ = (v1 , . . . , v1 , v2 , . . . vn ). m
times
Proof. Write W = (wi, j ) and assume that w1,1 = 0, and the other non-zero entries in the first row are w1,b2 , . . . , w1,bm . The multiplication v˜ W˜ for the new matrix and eigenvector becomes m
⎛
times
w1,1 ⎜ 0 ⎜ . ⎜ . ⎜ . ⎜ 0 ⎜ ⎜w ⎜ 2,1 (v1 , . . . , v1 , v2 , . . . vn ) ⎜ ⎜w ⎜ 3,1 m times ⎜ . ⎜ . ⎜ . ⎜ . ⎝ . . wn,1
. . . w1,1 0 . . . . . . 0 . . . 0 w1,b2 0 .. . ... 0 ... . . . w2,1 w2,2 . . . . . . . w3,1 w3,2 . . .. . .. .. . . . . . wn,1 wn,2 . . .
...
0 w1,bm 0 ...
..
. ...
0
⎞
⎟ ⎟ ⎟ ⎟ ... ⎟ ⎟ w2,n ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ ⎟ ⎟ .. ⎟ ⎠ . wn,n
A direct computation shows that this equals ρ v. ˜ Since v˜ is positive, it has to belong to the leading eigenvalue, so ρ is the leading eigenvalue of W˜ as well. The proof when w1,1 = 0 is similar. The effect of going from Pˆ k to Pˆ k for k > k is that by repeatedly applying Lemma 5, the entries v Pˆ for Pˆ ∈ Pˆ k have to be replaced by #( Pˆ ∩ Pˆ k ) copies of themselves which, ˆ ⊂ π( Q), ˆ then the when normalised, leads to the new unit left eigenvector v. ˜ If π( P)
ˆ Since I contains number of k -cylinders in Pˆ is less than the number of k -cylinders in Q. a finite number of k-cylinders, there is C = C(k) such that #( Pˆ ∩ Pk ) C#( Qˆ ∩ Pk ) ˆ Qˆ ∈ Pk and k > k. When passing from Pˆ k to Pˆ k , we also need to adjust the for all P, ˆ weight eϕ(x) for x ∈ Pˆ ∈ Pˆ k slightly, but this adjustment is exponentially small since
Vk (ϕ) ˆ → 0. It follows that min P∈ ˆ Pˆ k ∩ fˆ L ( Xˆ ) v Pˆ / max v Pˆ is uniformly bounded away from 0, uniformly in k . We finish this section with the technical result used in (20). Lemma 6. Let {Un }n , {Vn }n be positive square matrices such that ρn 1 is the leading eigenvalue of Un . Assume that there exist M < ∞, τ ∈ (0, 1) and a sequence {ηk }k∈N with ηk ↓ 0 as k → ∞ such that for all n, Un M, Unk ρnk ekηk
and Vn Mτ n ,
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Then there exists a different sequence {η˜ n }n∈N with η˜ n → 0 as n → ∞ such that (Un + Vn ) j (1 + e j η˜ n )ρn . j
In particular, the leading eigenvalue of
1 ρn (Un
+ Vn ) tends to 1 as n → ∞.
Remark 3. Although this lemma works for any matrix norm, we need it for U = supv1 =1 vU 1 , i.e., the maximal row-sum of Un . Note that we do not assume that all Un have the same size (although Un and Vn have the same size for each n). Proof. Note that Un + Vn is a positive matrix and so its leading eigenvalue is equal to the growth rate lim j→∞ 1j log (Un + Vn ) j . We have (Un + Vn ) j =
p
q
p
q
Un 1 Vn 1 . . . Un t Vn t ,
| p|+|q|= j
pi and |q| = qi . More where p = ( p1 , . . . , pt ), q = (q1 , . . . , qt ) and | p| = precisely, the sum runs over all t ∈ {1, . . . , j/2} and distinct vectors p, q with pi , qi > 0 (except that possibly p1 = 0 or qt = 0). Let us split the above sum into two parts. (i) If |q| > εj, then each of the above terms can be estimated in norm by Un | p| Vn |q| M j (τ n )εj = (Mτ εn ) j . Since there are at most 2 j such terms, this gives & & & & & & & p1 q 1 pt q t & Un Vn . . . Un Vn & (2Mτ εn ) j . & & | p|+|q|= j & & &
(21)
|q|>εj
(ii) If (q1 , . . . , qt ) satisfies |q| εj, then there are at most t − 1 |q| indices i with pi N and at least one index i with pi > N , where N < 1/(2ε) is to be determined p p later. The norm of each of these terms can be estimated by Un 1 · · · Un t M |q| τ n|q| , where the factors pi η p ρn e N i if pi > N , p Un i MN if pi N . So the product of all these factors is at most ρn eη N j M εj N . Using Stirling’s formula, we can derive that there are at most j
εj j t=0
t
(1−ε) j εj √ ' j 1 1 εj εj e εj εj ε 1−ε
possible terms of this form. Combining all this gives an upper bound of this part of & & & & & & √ & p1 q 1 pt q t & j Un Vn · · · Un Vn & e ε j ρn eη N j M εj N (Mτ n )εj . (22) & & & | p|+|q|= j & & |q|εj
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Adding the estimates of (21) and (22), we get (Un + Vn ) j (2Mτ εn ) j + e 1
√
εj
ρn eη N j M εN j (Mτ n )εj . j
1
Now √take N = n 4 and ε = n − 2 (so indeed N < 1/(2ε)) and n so large that Mτ n 2Mτ n 1. Then we get −1/4 −1/4 log M) j (Un + Vn ) j ρn 1 + e j (n +ηn1/4 +n . The lemma follows with η˜ n = (n −1/4 + ηn 1/4 + n −1/4 log M). 4. Proof of Theorem 4 The following is [BrT, Lemma 3]. Lemma 7. For every ε > 0, there are R ∈ N and η > 0 such that if µ ∈ Merg has entropy h µ ( f ) ε, then µ is liftable to the Hofbauer tower and µ( ˆ IˆR ) η. ˆ ˆ Furthermore, there is a set E, depending only on ε, such that µ( ˆ E) > η/2 and min D∈D∩ IˆR d( Eˆ ∩ D, ∂ D) > 0. The following lemma will allow us to implement condition (c) in Proposition 3. Lemma 8. There exist sequences {εn }n ⊂ R− with εn → 0 and {µn }n ⊂ M+ so that h µn ( f ) + ψ dµn εn . Moreover, there exists a domain Xˆ compactly contained in some D ∈ D so that µˆ n ( Xˆ ) > 0. Proof. First notice that by the definition of pressure, theremust exist sequences {εn }n ⊂ R− with εn → 0 and {µn }n ⊂ Merg so that h µn ( f ) + ψ dµn εn . By (2), there exists ε > 0 so that we can choose h µn ( f ) > ε and {µn }n ⊂ M+ . Now by Lemma 7, we can choose Xˆ compactly contained in some D ∈ D and a subsequence {n k }k with µˆ n k ( Xˆ ) > 0 for all k. Proof of Theorem 4. Take ψ := ϕ − P(ϕ). By the remark below (2) and Corollary 1, ˆ Notice that Vn (ϕ) → 0 implies that βn (ϕ) ˆ = o(n) we have P(ϕ) = P+ (ϕ) = PG (ϕ). and ϕˆ is continuous in the symbolic metric on ( Iˆ, fˆ). Take Xˆ ⊂ Iˆtrans compactly contained in its domain in the Hofbauer tower and satisfying ˆ Xˆ ) < the statement of Lemma 7. By Proposition 4, there are C, η > 0 such that Z n∗ (ψ, Ce−ηn . We denote the first return time to Xˆ by r Xˆ , the first return map to Xˆ by R Xˆ := fˆr Xˆ , and ˆ := ψr . We will shift these potentials, defining ψ S := ψ − S. the induced potential by
Xˆ ˆ S = − Sr ˆ . Since PG (ψ) ˆ = 0 and therefore Z n (ψ, ˆ Xˆ ) < eo(n) , we can Then
X estimate Z 0 from (7) for S > −η as ˆ ˆ S) = eψn (x)−nS Z n∗ (ψˆ − S, Xˆ ) Z 0 (
n r ˆ (x)=n X
n
C
ˆ Xˆ ) en(−S−η) Z n (ψ,
n
C
n
ˆ
en(PG (ψ)−S−η)+o(n) < ∞.
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599
ˆ = 0, this implies that PG (
ˆ S ) < ∞ for all S > −η. In fact, it also shows Since PG (ψ) ∗ ˆ S) < ∞ that (a) of Proposition 3 holds. We let S −η < 0 be minimal such that PG (
∗ for all S > S . We can prove precisely the same estimates for the map F = f τ , where τ = r Xˆ ◦π |−1 , ˆ X
and the potential = ϕτ . That is, for all S > S ∗ , PG ( S ) < ∞ and (a) of Proposition 3 holds. By Lemma 8, item (c) of Proposition 3 holds. Therefore, Case 2 of Proposition 3 implies that there exists a unique equilibrium state µψ with µˆ ψ ( Xˆ ) > 0. To show that µ is the unique equilibrium state over I , we assume that there is another equilibrium state µ . Let µˆ be the corresponding measure on Iˆ from Theorem 7. We now use the fact that µˆ is positive on cylinders. This follows firstly by the Gibbs properties of the measures obtained for (X, F, µ), and then by the transitivity of (I, f ) and ( Iˆtrans , fˆ). ˆ Xˆ ), µˆ ( Xˆ ) > 0. Thus there exists some cylinder Xˆ in the Hofbauer tower which has µ( We can use the above arguments to say that the corresponding inducing scheme (X , F , ) satisfies (a) of Proposition 3. But since µψ is an equilibrium state compatible with (X , F ), also (b) is satisfied. Therefore, Case 1 of Proposition 3 completes the proof of uniqueness. Finally we note that µ {τ > n} decays exponentially in n, since by the Gibbs property there is C 1 such that µ (X i ) C e i = C Z k∗ (ψ, Xˆ ). µ ({τ > n}) = τi >n
τi >n
k n
By Proposition 4, the latter quantity decays exponentially, as required. 5. Analyticity of the Pressure Function In this section we prove Theorem 6. Throughout, let ϕt = −t log |D f |. Let X ⊂ I and (X, F, τ ) be an inducing scheme on X , where F = f τ . As usual we denote the set of domains of the inducing scheme by {X i }i∈N . Define a tower over the inducing scheme as follows (see [Y]): =
i −1 ( τ( (X i , j),
i∈N j=0
with dynamics f (x, j) =
(x, j + 1) if x ∈ X i , j < τi − 1; (F(x), 0) if x ∈ X i , j = τi − 1.
) For i ∈ N and 0 j < τi , let i, j := {(x, j) : x ∈ X i } and l := i∈N i,l is called the l th floor. Define the natural projection π : → X by π (x, j) = f j (x). Note that (, f ) is a Markov system, and the first return map of f to the base 0 is isomorphic (X, F, τ ). Also, given ψ : I → R, let ψ : → R be defined by ψ (x, j) = ψ( f j (x)). Then the induced potential of ψ to the first return map to 0 is exactly the same as the induced potential of ψ to the inducing scheme (X, F, τ ). The differentiability of *the pressure functional can be expressed using directional d derivatives ds PG (ψ + sυ)*s=0 . For inducing scheme (X, F, τ ), let ψ and υ be the
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lifted potentials to . Suppose that for ψ : → R, we have βn (ψ ) = o(n). We define the set of directions with respect to ψ:
* * ∞ * * * Dir F (ψ) := υ : sup * υ dµ** < ∞, βn (υ ) = o(n), Vn (ϒ) < ∞, and µ∈M+ n=2 + ∃ε > 0 s.t. PG (ψ + sυ ) < ∞ ∀ s ∈ (−ε, ε) , where ϒ is the induced potential of υ. Let ψ S := ψ − S (and so S = − Sτ ). Set p ∗F [ψ] := inf{S : PG ( S ) < ∞}.4 If p ∗F [ψ] > −∞, we define the X -discriminant of ψ as D F [ψ] := sup{PG ( S ) : S > p ∗F [ψ]} ∞. Given a dynamical system (X, F), we say that a potential : X → R is weakly Hölder continuous if there exist C, γ > 0 such that Vn ( ) Cγ n for all n 0.
(23)
The following is from [BrT, Theorem 5]. Theorem 10. Let f ∈ H be a map with potential ϕ : I → (−∞, ∞]. Suppose that ϕ satisfies condition (5). Take ψ = ϕ − P(ϕ). Then D F [ψ] > 0 if and only if (X, F, µ ) has exponential tails. We are now ready to prove Theorem 6. In this, and the proofs in the sequel, we write An Bn if ABnn → 1 as n → ∞. We also write A dis B if there exists a distortion constant K ∈ [1, ∞) so that K1 A B K A. Proof of Theorem 6. We fix (X, F) as in Proposition 1. Lemma 5 implies that we have exponential tails for the equilibrium state associated to the constant potential ψ = −h top ( f ), i.e., there exist C, η > 0 such that µ−τ h top ( f ) {τi = n} Ce−ηn .
(24)
Hence Theorem 10 implies that we have positive discriminant. We can then apply the arguments of the proof of [BrT, Theorem 5] to show that for υ ∈ Dir (−h top ( f )), there exists ε > 0 such that t → P(−h top ( f ) + tυ) is analytic. Therefore, in order to ensure analyticity here we must prove − log |D f | ∈ ∞ Dir (−h top ( f )). It follows from [BrT, Lemma 7] that this potential has n=2 Vn * * (− log |D F|) < ∞, and [Pr] gives supµ∈M+ * log |D f | dµ* < ∞; so it only remains to prove that there exists ε > 0 such that PG ((−h top ( f ) − t log |D f |) ) < ∞ for t ∈ (−ε, ε). Since PG ((−h top ( f ) − t log |D f |) ) PG (−τ h top ( f ) − t log |D F|), by Abramov´s Theorem it suffices to bound PG (−τ h top ( f ) − t log |D F|). As in Sect. 2.3, Z 0 ( ) < ∞ implies PG ( ) < ∞. In the following calculation we use the fact that for 4 Note that we use the opposite sign for p ∗ [ψ] to Sarig. F
Equilibrium States for Potentials with sup ϕ − inf ϕ < h top ( f )
601
n(h top ( f )+ε) , see the discussion at all ε > 0 there exists Cε > 0 so that #{τ n} Cε e i = t h top ( f ). Using the Hölder inequality, (29). For 0 < t < 1, choose 0 < ε < 1−t
Z 0 (−τ h top ( f ) − t log |D F|)
dis
dis
e
−nh top ( f )
|X i |t
τi =n
n
e−t log |D Fi |
τi =n
n
e−nh top ( f )
, e
n
Cε1−t
−nh top ( f )
-t |X i |
(#{τi = n})1−t
τi =n
en(−h top ( f )+(1−t)(h top ( f )+ε))
n
= Cε1−t
en(−th top ( f )+(1−t)ε) < ∞.
n
(For further explanation of these calculations see [BrT, Sect. 5 ].) For t < 0, first notice that by the Gibbs property of µ−τ h top ( f ) , 1 = e−nh top ( f ) #{τi = n}. µ−τ h top ( f ) {τ = n} e−nh top ( f ) τi =n
Hence, by (24), e−nh top ( f ) #{τi = n} Ce−ηn .
(25)
Since |X i | |X |e−γ τi for γ := log sup |D f |, we have Z 0 (−τ h top ( f ) − t log |D F|)
1 −nh top ( f ) e |X i |t |X |t n τi =n −nh top ( f ) [e #{τi = n}]e−γ nt dis
n
C
e−n(tγ +η) < ∞,
n
if tγ + η > 0. Hence there exists ε > 0 so that − log |D f | ∈ Dir (−h top ( f ) − ε). It remains to show existence and uniqueness of equilibrium states. By (25), we have for t 0, using the Hölder inequality again, Z 0 (−t log |D F| − τ P(ϕt )) dis e−n P(ϕt ) e−t log |D Fi | dis e−n P(ϕt ) |X i |t τi =n
n
,
e−n P(ϕt )
n
C
|X i |
τi =n
#{τi = n}1−t
τi =n
n
n
-t
[e
−nh top ( f )
n
#{τi = n}]1−t e−n(P(ϕt )−(1−t)h top ( f ))
en((1−t)(h top ( f )−η)−P(ϕt )) .
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Since P(ϕt ) → h top ( f ) as t → 0, for all small t we have (1−t)(h top ( f )−η )−P(ϕt ) < 0. Hence Z 0 (−t log |D F| − τ P(ϕt )) < ∞ for small positive t. For t < 0, we use a similar computation as before: Z 0 (−t log |D F| − τ P(ϕt )) dis e−n P(ϕt ) e−t log |D Fi | τi =n
n
dis
e
n
< Cε
|X i |t
τi =n
n
−n P(ϕt )
e−n(P(ϕt )+tγ ) #{τi = n}
e−n(tγ +P(ϕt )−h top ( f )−ε) ,
n
where we use the fact that for all ε > 0 there exists Cε > 0 so that #{τi = n} Cε en(h top ( f )+ε) . Since P(ϕt ) > h top ( f ) we can ensure that tγ + P(ϕt )−h top ( f )−ε > 0 for all t close to zero. Hence Z 0 (−t log |D F| − τ P(ϕt )) is finite for all t close enough to zero. This implies that for t in a neighbourhood of 0, PG (−t log |D F| − τ P(ϕt )) < ∞. Similarly property (a) of Proposition 3 holds, and thus we can apply Case 2 of that proposition to get existence of an equilibrium state µ. This is the unique equilibrium state among those that can be lifted to (X, F). Following the argument in the proof of Theorem 4, we have that µ is the unique global equilibrium state as required. 6. Necessity of the Condition sup ϕ − inf ϕ < h t op ( f ) In this section we show the importance of the condition (1) for the existence and uniqueness of equilibrium states obtained by inducing methods. Hofbauer and Keller gave an example, originally in a symbolic setting [H1] and later in the context of the angle doubling map on the circle [HK], which showed that (1) is essential for their results on quasi-compactness of the transfer operator. In Sect. 6.1, we discuss how that example fits in with our inducing results. The Hofbauer and Keller example uses a non-Hölder potential, so it is natural to ask if it is really the lack of Hölder regularity which causes problems in obtaining equilibrium states. In Sect. 6.2, we provide an example of a family of Hölder continuous potentials which, if a member of the family violates (1), then the equilibrium state is not obtained from any inducing scheme with integrable inducing time. We note here that these Markov examples are often modelled by the renewal shift, see [Sa2] and [PeZ]. That approach uses a rather different partition to the one we use in this paper, and so does not elucidate our theory. However, the inducing schemes we use and the ones that [Sa2] and [PeZ] get from the renewal shift are the same. 6.1. Hofbauer and Keller’s example. As mentioned in Theorem 1, potentials ϕ ∈ BV satisfying sup ϕ − inf ϕ < h top ( f ) have equilibrium states; in fact Hofbauer and Keller [HK] show that this equilibrium state is absolutely continuous w.r.t. to a ϕ-conformal measure, and that the transfer operator is quasi-compact. They also present, for the angle doubling map f (x) = 2x (mod 1), a class of potentials ϕ to show that (1) is essential for these latter properties. This map f was inspired by an example based in [H1] based
Equilibrium States for Potentials with sup ϕ − inf ϕ < h top ( f )
603
Fig. 1. Summary of results in [H1]: Eq. (2.6) and Sect. 5.
on the full shift σ : {0, 1}N → {0, 1}N , showing that Hölderness of potentials is essential to obtain the results from [Bo]. We demonstrate how this class of examples fits into the framework of our paper. Fix K 0 and let b < 0. Let ϕ = ϕb,K =
∞
ak · 1(2−k−1 ,2−k ] ,
k=0
where
ak :=
b for 0 k < K , k+1 for k K . 2 log k+2
Also let sn = n−1 k=0 ak . Since the Dirac measure δ0 at the fixed point has free energy h δ0 ( f ) + ϕ(0) = 0, the pressure P(ϕ) 0. Figure 1 summarises the results of [H1] and the example in [HK] that are relevant for us. ) Define the inducing scheme (X, F) where X = ( 21 , 1] and F : n X n → X is 1 the first return map to X where for n 1, X n := 2 + 2−n−1 , 21 + 2−n . Notice that if we denote X ∞ = {x : #orb(x) ∩ X = ∞}, then µ(X ∞ ) = 1 for every measure in Merg \ {δ0 }. In [HK], it is important that b is chosen so that −b > h top ( f ) = log 2, but for our case we allow b to vary. Lemma 9. For all K 2 there exists b K < − log 2 such that • b > b K implies P(ϕb,K ) > 0 and there exists a unique equilibrium state which can be found from (X, F); • b b K implies P(ϕb,K ) = 0 and the unique equilibrium state is the Dirac measure δ0 on 0. This cannot be found from (X, F). Moreover, b K → − log 2 as K → ∞. Proof. Firstly, we compute nb if n K , K +1 sn = j+1 n−1 = K b + 2 log n+1 if n > K . K b + 2 log j=K j+2
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As in [HK], we can estimate
e
sn
=
n
K
e
nb
+e
Kb
n=1
bK K + 1 2 b 1−e + ebK (K + 1). <e n+1 1 − eb
(26)
n>K
For b < − log 2 the first term is strictly less than 1 for all K and the secondterm tends to zero as b → −∞. Hence if we fix K , then we can find b K such that n esn 1 for b b K (with equality if and only if b = b K ), and Fig. 1 shows that P(ϕ) = 0. Alternatively, by fixing b < − log 2 and taking K large enough we have P(ϕ) = 0, and − log 2 as K → ∞. A computation similar to (26) shows that in fact sb K → K +1 2 n C (n + 1)e n n>K (n + 1)( n+1 ) diverges. Whenever P(ϕ) = 0, Fig. 1 shows that δ0 is the unique equilibrium state. We next show what P(ϕ) = 0 or P(ϕ) > 0 imply for obtaining the equilibrium state from the inducing scheme. As usual, we set ψ := ϕ − PG (ϕ). Notice that Vn ( ) = 0, so clearly we have summable variations. Also notice that for x ∈ X n , ,n−1 . k + 1
(x) = sn − n P(ϕ) −n P(ϕ) + 2 log = −n P(ϕ) − 2 log(n + 1). k+2 k=0
Therefore Z 0 ( ) =
∞
∞
e | X n
dis
n=1
e−n P(ϕ)−2 log(n+1) =
n=0
∞ −n P(ϕ) e < ∞, (n + 1)2
(27)
n=0
because P(ϕ) 0. So as in Sect. 2.3 this means that PG ( ) < ∞. Thus Theorem 8 yields a Gibbs state µ . Similarly to the calculation above, we can show from the Gibbs property of µ that −
dµ
∞ −n P(ϕ) e log(n + 1) dis n=0
(n + 1)2
<∞
for P(ϕ) 0. Therefore, µ is an equilibrium state for (X, F). We also have τ dµ
∞ ne−n P(ϕ) dis n=1
(n + 1)2
< ∞ if P(ϕ) > 0, = ∞ if P(ϕ) = 0.
(28)
Therefore if P(ϕ) = 0, we cannot project this measure to the original system. In the limit K → ∞, the potential is ϕ(x) = b for x ∈ (0, 1] and ϕ(0) = 0. It is easy to see that the same results above hold in this case and that for ϕ− log 2,∞ the equilibrium states are δ0 and the measure of maximal entropy. We briefly summarise the conclusions of this example, in order to clarify how it fits in with paper. We fix K 2. Since ϕ is monotone, ϕ BV < ∞, the results stated in this but n supC∈Pn ϕ|C BV = n Vn (ϕ) = ∞. Therefore Theorem 2 does not apply for any value of b. • For b b K , we have P(ϕ) = 0 but (1) fails, so Theorems 1 and 4 do not apply. However, there exists a unique equilibrium state δ0 by [H1].
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• For b K < b − log 2, we have P(ϕ) > 0, but again Theorems 1 and 4 do not apply. However, there exists a unique equilibrium state by [H1]. Moreover, direct computations as in (27) and (28) allow us to use our inducing method and Case 2 of Proposition 3 to show that there exists a unique equilibrium state, which can be obtained from an inducing scheme. • For − log 2 < b < 0, Theorem 1 applies (since ϕ BV < ∞) and Theorem 4 applies because is piecewise constant (so (SVI) holds and in fact, is weakly Hölder continuous, see (23)). Both theorems produce the unique equilibrium state. In general, inducing schemes are used to improve the hyperbolicity of the map or properties of the potential (e.g. to obtain weak Hölder continuity). For this system (or for the Manneville-Pomeau map of Sect. 6.2 below), there are inducing schemes that produce the equilibrium state δ0 . For instance, one can take the original map itself, or the ‘unnatural’ system consisting of the left branch only, as an induced system. But to obtain nice properties for map or potential, one has to induce to a domain disjoint from 0, and none of these ‘natural’ inducing schemes produces δ0 as equilibrium state. For b b K we have D F [ϕ] = 0, since PG ( − Sτ ) = ∞ for all S < 0. If ϕ had summable variations, then the discriminant theorem [Sa2] would imply that ϕ is not ‘strong positive recurrent’, but can be either positive recurrent or null recurrent. The fact that we cannot project µ appears to suggest that ϕ is null recurrent. However, since the variations of ϕ are not summable we are not able to use this theory. However, in the following lemma we make a direct computation to show that indeed ϕ is null recurrent when b b K . Lemma 10. Fix K 2. If b b K then ϕ is null recurrent. Proof. Let C0 and C1 be the left and right cylinders in P1 . Rather than considering all n-periodic cycles, we will restrict ourselves to special ones, and show that these are sufficient to imply recurrence. For each n there is a cycle cycn := pnn , . . . , pn1 , where pn1 ∈ X n as defined above, f pnk = pnk−1 for n k 2 and f ( pn1 ) = pnn in fact it n−k is easy to compute pkn = 22n −1 . For x ∈ cycn , ϕn (x) = sn . This cycle features n − 1 times in the computation of Z n (ϕ, C0 ). Hence, / 0 2 K sn Kb Z n (ϕ, C0 ) ne (n − 1 − K ) ·e , n+1 so n Z n (ϕ, C0 ) n Cn = ∞. Recalling that PG (ϕ) = 0 for b b K , this implies that the potential is recurrent. Notice that pn1 is the only point in cycn that belongs to C1 . So using this point and cylinder C1 , the same computation implies that n n Z n∗ (ϕ, C1 ) = ∞, so ϕ is null recurrent. 6.2. The Manneville-Pomeau map. The Manneville-Pomeau map f α (x) = x + x 1+α (mod 1) with α ∈ (0, 1) is well-known to have zero entropy equilibrium states for the potential −t log |D f α | and appropriate values of t. See [Sa2] for an exposition of this theory and the relevant references. Supposing that α < log2 2 , for p1 < p2 < 1 and b < − log 2 we will use the potential ⎧ α if x ∈ [0, p1 ], ⎪ ⎨ −2αx b+2αp1α α (x − p1 ) − 2αp1 if x ∈ ( p1 , p2 ], ϕ(x) = ϕα, p1 , p2 ,b (x) := p −p ⎪ ⎩b 2 1 if x ∈ ( p2 , 1],
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as an example to show that (1) is sharp. (Note that ϕ has the same Hölder exponent as − log |D f α |.) Since h top ( f ) = log 2, condition (1) is violated whenever b − log 2. It turns out that as soon as this occurs, we can choose α, p1 , p2 so that no equilibrium states can be achieved from a ‘natural’ inducing scheme on an interval bounded away from the neutral fixed point 0. Thus (1) is sharp, even when the potential is Hölder. The conclusion of Proposition 2 proved below is that Hölder regularity of the potential is not sufficient to dispense with condition (1). Proof of Proposition 2. We will make a suitable choice for p1 , p2 later in the proof. Let y0 = 1 and define yn ∈ (0, yn−1 ) for n 1 such that f α (yn ) = yn−1 . From the recursive α ) we derive (cf. [dB]) relation yn = yn+1 (1 + yn+1 −1 1 1 1 α α 2α 3α 1 − yn+1 , 1 + yn+1 = = + yn+1 + Err yn+1 yn yn+1 yn+1 3α )| = O y 3α . Using u = y −α this becomes where |Err (yn+1 n n n+1 , , --α 1 1 1 u n = u n+1 1 − + + Err u n+1 u 2n+1 u 3n+1 , , -α(α + 1) 1 α = u n+1 1 − + + Err , u n+1 2u 2n+1 u 3n+1 * * * * α(α+1) 1 −2 1 1 * * = O . Therefore u where * Err u 3 −u = α + + Err u n+1 n 3 n+1 , * 2 u n+1 u n+1
n+1
and using telescoping series this leads to u n = αn +
α(α + 1) log n + Err 2
1 . n
Transforming back to the original coordinate yn , we find −1/α 1/α 1/α 1/α 1 α(α + 1) 1 1 1 1+ log n + Err = . yn = un α n 2n n2 Thus
−1 α(α + 1) −2 1+ log n + Err (n −2 ) n 2n −2 α(α + 1) + = log n + Err (n −3 ) n n2
ϕ(yn ) = −2αynα =
for yn < p1 where |Err (n −3 )| = O(n −3 ). For all n sufficiently large, the variations w.r.t. the branch partition satisfy Vn (ϕ) n2 (obtained on the n-cylinder set [0, yn ]), so ϕ does not have summable variations. However, since ϕ is monotone, ϕ BV < ∞. For any b < − log 2 we will choose K > N and p1 = K and p2 = y N depending on α and b. n N implies (replacing the convergent sum of the last given and higher order terms by a single constant B = B N which is bounded in N ), sn :=
sup
n−1
x∈(yn+1 ,yn ] k=0
ϕ( f k (x)) = N b − 2
n−1 1 + B. k
k=N
Equilibrium States for Potentials with sup ϕ − inf ϕ < h top ( f )
607
Clearly choosing N large enough we can make this error as small as we like. By the above, we have N ∞ 2 Nb N sn kb N b+B b 1−e + e N b+B (N + 1). e e +e e b n 1 − e n k=1 N +1 Hence, we can choose N so large that n esn 1 and hence by Fig. 1 we have P(ϕ) = 0. (Likewise N , K and find a critical value bα,N ,K where below this we can fix suitable α, value, n esn 1 and above it, n esn > 1.) We define F to be the first return map to X := (y1 , 1], so if xi ∈ (y1 , 1] is such that f α (xi ) = yi , then X i = (xi+1 , xi ] and τi = i. A straightforward computation shows that | X n is monotone and there is C 1 such that for large n, −2 log n − Cn | X n −2 log n + Cn ; in fact is weakly Hölder. As in Lemma 9, we can show that Z 0 ( ) < ∞, so PG ( ) < ∞, and there is a unique equilibrium state µ for (X, F, ) which also satisfies the Gibbs property. However, as in (28), the inducing time has τ dµ = ∞, as required. 7. Recurrence of Potentials Although not crucial for the main results of this paper, the question whether the potential is recurrent (see (6)) is of independent interest. In this section we give sufficient conditions for ϕ to be recurrent, and for the topological pressure and the Gurevich pressure to coincide. Recall that Theorems 1 and 3 gave conditions under which transfer operator Lϕ is quasi-compact. Let us first lay out an argument why this implies that ϕ is recurrent. Recall that quasi-compactness means that the essential spectrum σess is strictly less than the leading eigenvalue λ = exp(P(ϕ)), and there are only finitely many eigenvalues outside {|z| σess }, each with finite multiplicity. A result due to Baladi and Keller [BaK] says that this spectral gap implies that the dynamical ζ -function ⎛ ⎞ ∞ n z ζ (z) = exp ⎝ eϕn (x) ⎠ n n f (x)=x
n=1
is meromorphic on {|z| λ−1 }, with a pole at λ−1 whose multiplicity is the same as the multiplicity of the eigenvalue λ of Lϕ . The argument why this implies recurrence of the potential is somewhat implicit in [BaK]. Namely, there is a function g which is analytic on {|z| < λ−1 }∩{|z −λ−1 | < ε} such that g(λ−1 ) = 0 and ζ (z)/ζ (z) = g(z)/(z −λ−1 ) on this region. Hence lim z→λ−1 ζ (z)/ζ (z) = ∞. Direct computation gives ∞
n=1
f n (x)=x
1 n ζ (z) = z ζ (z) z
e
ϕn (x)
∞
1 n = z Z n (ϕ), z n=1
so recurrence follows. Proposition 5. Let f ∈ H and ϕ be a potential such that sup ϕ − inf ϕ < h top ( f ) . If Vn (ϕ) → 0 and e−βn = ∞, n
then ϕ is recurrent. (Here βn := βn (ϕ) is defined as in (5).)
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n Clearly βn k=1 Vk (ϕ), and Vn (ϕ) → 0 implies βn = o(n). The condition −βn = ∞ is stronger: it implies that β = o(log n) and is implied by V (ϕ) = e n n n O(n −(1+ε) ). It is well known that the Variational Principle holds for the potential ϕ = 0; in fact
h top ( f ) = P(0) = PG (0) = Ptop (0) = lim n
1 log laps( f n ), n
where laps( f n ) := #Pn is the lap number, i.e., the number of maximal intervals on which f n is monotone, see [MSz]. In fact, the lap number is submultiplicative: laps( f n+m ) laps( f n )laps( f m ). Therefore h top ( f ) = inf n n1 log laps( f n ) and eh top ( f )n laps( f n ) en(h top ( f )+εn ) ,
(29)
where εn → 0 as n → ∞. We will extend this idea to ergodic averages of more general potentials in Lemma 11. For J ∈ Pm , let ϕm (J ) = sup{ϕm (x) : x ∈ J } and top Z m (ϕ) := eϕm (J ) . J ∈Pm
For the remainder of this section we assume that (I, f ) is topologically mixing, i.e., for each m, (I, f m ) is topologically transitive. In order to prove recurrence of ϕ, we need the following lemma. Lemma 11. Let ϕ be a potential satisfying (1) and with βn (ϕ) = o(n). Then there exists η > 0 such that Z n (ϕ) ηe−βn e Ptop (ϕ)n for all n, and Ptop (ϕ) = PG (ϕ). Proof. Since f is topologically transitive, there is a collection of intervals permuted cyclically by f , such that for any interval J , there is n such that f n (J ) contains a component of this cycle. For simplicity, let us assume that this collection is just a single interval I . Since every m-cylinder set can contain at most one m-periodic point, Z m (ϕ) top top Z m (ϕ) for all m. Furthermore, Z m (ϕ) is submultiplicative, cf. (29), so Ptop (ϕ) := lim
m→∞
1 1 top top log Z m (ϕ) = inf log Z m (ϕ) < ∞. m m m
Therefore PG (ϕ) Ptop (ϕ) < ∞. Recall that every J ∈ Pm corresponds to a unique m-path D0 → D1 → · · · → Dm in the Hofbauer tower ( Iˆ, fˆ) leading from the base D0 of the tower to some terminal domain Dm . The level of Dm was defined as the length of the shortest path from the base to Dm . We say that the pre-level of J is pre-level(J ) = R. The topological entropy h top ( f ) is the exponential growth rate of the number of n-paths D0 → · · · → Dn in the Hofbauer tower, and the limit of the exponential growth rates of the number of n-paths within IˆR as R → ∞, see [H2] for the unimodal and [BBr, Sects. 9.3-9.4] for the general case. Therefore, by taking R sufficiently large, we can find γ > 0 and C0 ∈ (0, 1) such that the number of k-paths #{D0 → D1 → · · · → Dk : level(Dk ) R, 1 j k} C0 ek(h top ( f )−γ ) (30) for all k 1, and sup ϕ − inf ϕ < h top ( f ) − γ −
log 2 . R
(31)
Equilibrium States for Potentials with sup ϕ − inf ϕ < h top ( f )
609
Since (I, f ) is topologically transitive (and using our simplifying assumption), there
exists R depending on R, such that for each D ∈ D with level(D) R, f R (D) ⊃ I . This implies that every J ∈ Pm with pre-level(J ) R. contains a periodic point of period n := m + R . The idea is now for an arbitrary J ∈ Pm to extend the corresponding path by R arrows to find an n-periodic point p ∈ J . If pre-level(J ) R, then by the choice of R , this is indeed possible. We call such cylinder sets J type 1, and we can thus compare type 1 Zm (ϕ) to Z n (ϕ) as: type 1 (ϕ) = eϕm (J ) Zm J ∈Pm ,type 1
eβm e−R
inf
ϕ ϕn ( p)
e
eβm e−R
inf
ϕ
Z n (ϕ).
(32)
p= f n ( p)∈J
J ∈Pm
is type 1
If pre-level(J ) > R, then the existence of an n-periodic point in J cannot be guaranteed. We call such cylinder sets J type 2. Given such a type 2 cylinder set J , there is a maximal m < m such that pre-level(J ) = R for the m -cylinder J containing J . As we mentioned before, from any domain in the Hofbauer tower, there are at most two R-paths that are outside IˆR . Using this property repeatedly, we find that there are at most
log 2
2(m−m )/R = e(m−m ) R starting at Dm but otherwise outside IˆR . From Dm , there is at least one R -path leading back to some D ∈ IˆR , and using (31) and (30) we derive that
there are at least C0 e(m−m −R )(h top ( f )−γ ) ‘type 1’ m − m -paths from Dm . From this we conclude that the type 1 cylinders “sufficiently” outnumber the type 2 cylinders, and we can bound the contributions of type 2 cylinders in J by the contribution of type 1 cylinders in J as follows: log 2
eϕm (J ) e(m−m )(sup ϕ+ R ) J ⊂J ,type 2
×
1 −(m−m −R )(h top ( f )−γ ) −(m−m ) inf ϕ e e C0
1 e C0
(m−m )
eϕm (J )
J ⊂J ,type 1 log 2 sup ϕ−inf ϕ−h top ( f )+γ + R R (h top ( f )−γ )
1 R h top ( f ) e C0
e
eϕm (J )
J ⊂J ,type 1
eϕm (J ) .
J ⊂J ,type 1
Summing over all m and J ∈ Pm , we get type 2 (ϕ) Zm
1 R h top ( f ) type 1 e Z m (ϕ). C0 top
Now we combine this with (32) and the fact that {Z n (ϕ)}n is submultiplicative to obtain type 1 top top top top type 2 en Ptop (ϕ) Z n (ϕ) Z R (ϕ) · Z m (ϕ) Z R (ϕ) Z m (ϕ) + Z m (ϕ) 2 1 1 R h top ( f ) type 1 top Z R (ϕ) 1 + Z m (ϕ) e C0
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2 1 1 R h top ( f ) −R inf ϕ βm top e Z R (ϕ) 1 + e e Z n (ϕ) C0 2 1
top e R (h top ( f )−inf ϕ) eβm −βn eβn Z n (ϕ) = eβn Z n (ϕ) Z R (ϕ) C0 η
C0 e−R (h top ( f )−inf ϕ) eβn −βm . Since n − m = R , we can assume that top
for η =
2Z R (ϕ)
eβm −βn is bounded independently of m, so η > 0. This proves the first statement. In fact, since βn = o(n), we also find Ptop (ϕ) = PG (ϕ). −β Corollary 2. If sup ϕ − inf ϕ < h top ( f ) and n e n = ∞, then the potential ϕ is recurrent. Proof. Since ϕ is recurrent by definition if n λ−n Z n (ϕ) = ∞ for λ = e PG (ϕ) , this corollary is immediate from Lemma 11. The above ideas lead us to show that in our setting Ptop and PG are in fact the same. ˆ for every cylinder ˆ = PG (ϕ, ˆ C) Corollary 3. If sup ϕ − inf ϕ < h top ( f ), then Ptop (ϕ) ˆ in Iˆtrans . set C ˆ Proof. This is the same proof as Lemma 11 with J ∈ Pm replaced by Jˆ ∈ Pˆ m ∩ C. Acknowledgements. We would like to thank Ian Melbourne, Benoît Saussol, Godofredo Iommi, Sebastian van Strien and Neil Dobbs for fruitful discussions. We would also like to thank the LMS for funding the visit of Saussol. HB would like to thank CMUP for its hospitality. We also thank the referee for careful reading and constructive comments.
References [Ab] [Ba] [BaK] [BGMY] [Bl] [Bo] [BBr] [Br1] [Br2] [BrK] [BRSS] [BrT]
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de Bruyn, N.G.: Asymptotic methods in analysis. New York: Dover Publ., 1981 Buzzi, J.: Specification on the interval. Trans. Amer. Math. Soc. 349, 2737–2754 (1997) Buzzi, J.: Markov extensions for multi-dimensional dynamical systems. Israel J. Math. 112, 357–380 (1999) Buzzi, J., Sarig, O.: Uniqueness of equilibrium measures for countable markov shifts and multidimensional piecewise expanding maps. Ergod. Theory Dyn. Sys. 23, 1383–1400 (2003) Denker, M., Keller, G., Urba´nski, M.: On the uniqueness of equilibrium states for piecewise monotone mappings. Studia Math. 97, 27–36 (1990) Denker, M., Urba´nski, M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4, 103–134 (1991) Downarowicz, T., Serafin, J.: Fiber entropy and conditional variational principles in compact non–metrizable spaces. Fund. Math. 172, 217–247 (2002) Fiebig, D., Fiebig, U.-R., Yuri, M.: Pressure and equilibrium states for countable state markov shifts. Israel J. Math. 131, 221–257 (2002) Hofbauer, F.: Examples for the nonuniqueness of the equilibrium state. Trans. Amer. Math. Soc. 228, 223–241 (1977) Hofbauer, F.: Piecewise invertible dynamical systems. Probab. Theory Relat. Fields 72(3), 359–386 (1986) Hofbauer, F., Keller, G.: Equilibrium states for piecewise monotonic transformations. Ergod. Theory Dyn. Sys. 2, 23–43 (1982) Keller, G.: Lifting measures to markov extensions. Monatsh. Math. 108, 183–200 (1989) Keller, G.: Equilibrium states in ergodic theory. London Mathematical Society Student Texts 42, Cambridge: Cambridge University Press, 1998 Melbourne, I., Nicol, M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Commun. Math. Phys. 260, 131–146 (2005) Melbourne, I., Nicol, M.: Large deviations in nonuniformly hyperbolic dynamical systems. to appear in Trans. Amer. Math. Soc., doi:S0002-9947(08)04520-0, June 4, 2008 Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings. Studia Math. 67, 45–63 (1980) Oliveira, K.: Equilibrium states for non-uniformly expanding maps. Ergod. Theory Dyn. Sys. 23, 1891–1905 (2003) Paccaut, F.: Statistics of return times for weighted maps of the interval. Ann. Inst. H. Poincaré Probab. Statist. 36, 339–366 (2000) Pesin, Y., Senti, S.: Thermodynamical formalism associated with inducing schemes for one– dimensional maps. Moscow J. Math. 5, 3 669–678, 743–744 (2005) Pesin, Y., Zhang, K.: Phase transitions for uniformly expanding maps. J. Stat. Phys. 122, 1095–1110 (2006) Przytycki, F.: Lyapunov characteristic exponents are nonnegative. Proc. Amer. Math. Soc. 119, 309–317 (1993) Ruelle, D.: An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat. 9, 83–87 (1978) Ruelle, D.: Thermodynamic formalism. Reading MA: Addison Wesley, 1978 Sarig, O.: Thermodynamic formalism for Markov shifts. PhD. thesis, Tel–Aviv, 2000 Sarig, O.: Phase transitions for countable markov shifts. Commun. Math. Phys. 217, 555–577 (2001) Sarig, O.: Existence of gibbs measures for countable markov shifts. Proc. Amer. Math. Soc. 131, 1751–1758 (2003) Sinai, Y.: Gibbs measures in ergodic theory, (Russian). Usp. Mat. Nauk 27, 21–64 (1972); English translation: Russ. Math. Surv. 27, 21–69 (1972) Walters, P.: Some results on the classification of non-invertible measure preserving transformations. Lecture Notes in Math. 318, Berlin: Springer, 1973, pp. 266–276 Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)
Communicated by G. Gallavotti
Commun. Math. Phys. 283, 613–646 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0506-5
Communications in
Mathematical Physics
Spectral Theory for the Standard Model of Non-Relativistic QED J. Fröhlich1 , M. Griesemer2 , I. M. Sigal3, 1 Theoretical Physics, ETH–Hönggerberg, CH–8093 Zürich, Switzerland.
E-mail: [email protected]
2 Department of Mathematics, University of Stuttgart, D–70569 Stuttgart, Germany.
E-mail: [email protected]
3 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4.
E-mail: [email protected] Received: 6 September 2007 / Accepted: 19 November 2007 Published online: 22 May 2008 – © Springer-Verlag 2008
Abstract: For a model of atoms and molecules made from static nuclei and nonrelativistic electrons coupled to the quantized radiation field (the standard model of non-relativistic QED), we prove a Mourre estimate and a limiting absorption principle in a neighborhood of the ground state energy. As corollaries we derive local decay estimates for the photon dynamics, and we prove absence of (excited) eigenvalues and absolute continuity of the energy spectrum near the ground state energy, a region of the spectrum not understood in previous investigations. The conjugate operator in our Mourre estimate is the second quantized generator of dilatations on Fock space. 1. Introduction According to Bohr’s well known picture, an atom or molecule has only a discrete set of stationary states (bound states) at low energies and a continuum of states at energies above the ionization threshold. Electrons can jump from a stationary state to another such state at lower energy by emitting photons. These radiative transitions tend to render excited states unstable, i.e., convert them into resonances. Exceptions are the ground state and, in some cases, excited states that remain stable for reasons of symmetry (e.g. ortho-helium). In non-relativistic QED, the instability of excited states finds its mathematical expression in the migration of eigenvalues to the lower complex half-plane (second Riemannian sheet for a weighted resolvent) as the interaction between electrons and photons is turned on. Indeed, the spectrum of the Hamiltonian becomes purely absolutely continuous in a neighborhood of the unperturbed excited eigenvalues [5,7]. The ground state, however, remains stable [4,5,16]. The methods used to analyze the spectrum near unperturbed excited eigenvalues have either failed [7], or not been pushed far enough [5], to yield information on the nature of the spectrum of the interacting Hamiltonian in a neighborhood of the ground state energy. The purpose of this paper is Supported by NSERC under Grant NA 7901.
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J. Fröhlich, M. Griesemer, I. M. Sigal
to close this gap: we establish a Mourre estimate and a corresponding limiting absorption principle for a spectral interval at the infimum of the energy spectrum. It follows that the spectrum is purely absolutely continuous above the ground state energy. As a corollary we prove local decay estimates for the photon dynamics. In non-relativistic QED (regularized in the ultraviolet), the Hamiltonian, H , of an atom or molecule with static nuclei is a self-adjoint operator on the tensor product, N L 2 (R3 ; C2 ) and the H := Hpart ⊗ F, of the electronic Hilbert space Hpart = ∧i=1 2 3 2 symmetric (bosonic) Fock space F over L (R , C ; dk). It is given by H=
N
(−i∇xi + α 3/2 A(αxi ))2 + V + H f ,
(1)
i=1
where N is the number of electrons and α > 0 is the fine structure constant. The variable xi ∈ R3 denotes the position of the i th electron, and V is the operator of multiplication by V (x1 , . . . , x N ), the potential energy due to the interaction of the electrons and the nuclei through their electrostatic fields. In our units, V (x1 , . . . , x N ) is independent of α and given by V (x1 , . . . , x N ) = −
N M i=1 l=1
Zl 1 + . |xi − Rl | |xi − x j | i< j
The operator H f accounts for the energy of the transversal modes of the electromagnetic field, and A(x) is the quantized vector potential in the Coulomb gauge with an ultraviolet cutoff. In terms of creation- and annihilation operators, aλ∗ (k) and aλ (k), these operators are d 3 k|k|aλ∗ (k)aλ (k), Hf = λ=1,2
and A(x) =
λ=1,2
d 3k
κ(k) ik·x −ik·x ∗ ε (k) e a (k) + e a (k) , λ λ λ |k|1/2
(2)
where λ ∈ {1, 2} labels the two possible photon polarizations perpendicular to k ∈ R3 . The corresponding polarization vectors are denoted by ελ (k); they are normalized and orthogonal to each other. Thus, for each x ∈ R3 , A(x) = (A1 (x), A2 (x), A3 (x)) is a triple of operators on the Fock space F. The real-valued function κ is an ultraviolet cutoff and serves to make the components of A(x) densely defined self-adjoint operators. We assume that κ belongs to the Schwartz space, although much less smoothness and decay suffice. We emphasize that no infrared cutoff is used; that is, (physically relevant) choices of κ, with κ(0) = 0 (3) are allowed. The spectral analysis of H for such choices of κ is the main concern of this paper. Under the simplifying assumption that |κ(k)| ≤ |k|β , for some β > 0, the analysis is easier and some of our results are already known for β sufficiently large; see the brief review at the end of this introduction. The spectrum of H is the half-line [E, ∞), with E = inf σ (H ). The end point E is an eigenvalue if N − 1 < l Z j [6,16,20], but the rest of the spectrum is expected to be purely absolutely continuous (with possible exception as explained above). For a large interval between E and the threshold, , of ionization, absolute continuity has
Spectral Theory for the Standard Model of Non-Relativistic QED
615
been proven in [6,7]; but the nature of the spectrum in small neighborhoods of E and has remained open. There are further results on absolute continuity of the spectrum for simplified variants of H , and we shall comment on them below. Our first main result concerns the spectrum of H in a neighborhood of E. Under the assumptions that α is sufficiently small and that e1 = inf σ (Hpart ) is a simple and N xi + V , we show that σ (H ) is purely absoisolated eigenvalue of Hpart = − i=1 lutely continuous in (E, E + egap /3), where egap = e2 − e1 and e2 is the first point in the spectrum of Hpart above e1 . It follows, in particular, that H has no eigenvalues near E other than E. Our second main result concerns the dynamics of states in the spectral subspace of H associated with the interval (E, E + egap /3). If f ∈ C∞ 0 (R) with supp( f ) ⊂ (E, E + egap /3), then
B −s e−i H t f (H ) B −s = O(
1 ), t s−1/2
(t → ∞),
(4)
where B, is the second quantized dilatation generator on Fock space, that is, B = d (b),
b=
1 (k · y + y · k). 2
(5)
Here y = i∇k denotes the “position operator” for photons and B := (1 + B 2 )1/2 . Estimate (4) is a statement about the growth of B under the time evolution of states in the range of f (H ) B −s . Since growth of B requires that either the number of photons or their distance to the atom grows, (4) confirms the expectation that, asymptotically as time tends to ∞, the state of an excited atom or molecule relaxes to the ground state by emission of photons, provided the maximal energy is below the ionization threshold [10,14,25]. In the course of this process the atom or molecule (not including the photons that were radiated off) will eventually wind up, energetically, in a neighborhood of the ground state energy E. Hence the importance of understanding the spectrum of H and the dynamics generated by H in spectral subspaces of energies near E. We remark that the details of the form of interaction between matter and radiation as given in (1) and (2) are essential for our results to hold, but that our methods are applicable to other models of matter and radiation as well, and our corresponding results will be published elsewhere. Our approach to the spectral analysis of H is based on Conjugate Operator Theory in its standard form with a self-adjoint conjugate operator. Our choice for the conjugate operator is the second quantized dilatation generator (5). The hypotheses of conjugate operator theory are a regularity assumption on H and a positive commutator estimate, called Mourre estimate. Concerning the first assumption we show that s → e−i Bs f (H )ei Bs ψ is twice continuously differentiable, for all ψ ∈ H and for all f of class C0∞ on the interval (−∞, ) below the ionization threshold . Our Mourre estimate says that, if α is small enough, then E (H − E)[H, i B]E (H − E) ≥
σ E (H − E), 10
(6)
for arbitrary σ ≤ egap /2 and = [σ/3, 2σ/3]. As a result we obtain all the standard consequences of conjugate operator theory on the interval (E, E + egap /3) [23], in particular, absence of eigenvalues (Virial Theorem), absolute continuity of the spectrum, existence of the boundary values B −s (H − λ ± i0)−1 B −s
(7)
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for λ ∈ (E, E + egap /3), s ∈ (1/2, 1) (Limiting Absorption Principle), and their Hölder continuity of degree s − 1/2 with respect to λ. This Hölder continuity implies the local decay estimate (4). The idea to use conjugate operator theory with (5) as the conjugate operator is not new and has been used for instance in [7]. It is based on the property that [H f , i B] = H f and that H f is positive on the orthogonal complement of the vacuum sector. There is an obvious problem, however, with the implementation of this idea that discouraged people from using it in the analysis of the spectrum close to E: if α 3/2 W = H − (Hpart + H f ) denotes the interaction part of H , then [H, i B] = H f + α 3/2 [W, i B],
(8)
and the commutator [W, i B] has no definite sign. It can be compensated for by part of the field energy H f so that H f + α 3/2 [W, i B] becomes positive, but only so on spectral subspaces corresponding to energy intervals separated from E by a distance of order α 3 [7]. For fixed α > 0 no positive commutator, and thus no information on the spectrum is obtained near E = inf σ (H ). For this reason, Hübner and Spohn and, later, Skibsted, Derezi´nski and Jakši´c, and Georgescu et al. chose the operator 1 ˆ Bˆ = d (kˆ · y + y · k), 2
k kˆ = , |k|
or a variant thereof, as conjugate operator; see [9,13,19,24]. It has the advantage that, forˆ = N , the number operator, which is bounded below by the identity opermally, [H f , i B] ˆ ≥ 1 N, ator on the orthogonal complement of the vacuum sector. It follows that [H, i B] 2 for α > 0 small enough, and one may hope to prove absolute continuity of the energy spectrum all the way down to inf σ (H ). The drawback of Bˆ is that it is only symmetric, but not self-adjoint, and hence not admissible as a conjugate operator. Therefore Skibsted, and, later, Georgescu, Gérard, and Møller developed suitable extensions of conjugate operator theory that allow for non-selfadjoint conjugate operators [13,24]. Skibsted applied his conjugate operator theory to (1) and obtained absolute continuity of the energy spectrum away from thresholds and eigenvalues under an infrared (IR) regularization, but not for (3). For the spectral results of Georgescu et al. see the review below. Given this background, the main achievement of the present paper is the discovery of the Mourre estimate (6). We now sketch the main elements of its proof. 1. As an auxiliary operator we introduce an IR-cutoff Hamiltonian Hσ in which the interaction of electrons with photons of energy ω ≤ σ is turned off. It follows that Hσ is of the form Hσ = H σ ⊗ 1 + 1 ⊗ H f,σ , with respect to H = Hσ ⊗ Fσ , where Fσ is the symmetric Fock space over L 2 (|k| ≤ σ ; C2 ) and H f,σ is d (ω) restricted to Fσ . We show that the reduced Hamiltonian H σ does not have spectrum in the interval (E σ , E σ + σ ) above the ground state energy E σ = inf σ (Hσ ) = inf σ (H σ ). It follows that, for any ⊂ (0, σ ), E (Hσ − E σ ) = P σ ⊗ E (H f,σ ), where P σ is the ground state projection of H σ .
(9)
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617
2. We split B into two pieces B = Bσ + B σ , where Bσ and B σ are the second quantizations of the generators associated with the vector fields ησ2 (k)k and ησ (k)2 k, respectively. Here ησ , ησ ∈ C ∞ (R3 ) is a partition of unity, ησ2 + (ησ )2 = 1, with ησ (k) = 1 for |k| ≤ 2σ and ησ (k) = 1 for |k| ≥ 4σ . It follows that B σ = B σ ⊗ 1 with respect to H = Hσ ⊗ Fσ , and that [H, B σ ] = [H σ , B σ ] ⊗ 1. Thus (9) and the virial theorem, P σ [H σ , B σ ]P σ = 0, imply that E (Hσ − E σ )[H, i B σ ]E (Hσ − E σ ) = 0.
(10)
3. The first key estimate in our proof of (6) is the operator inequality E (Hσ − E σ )[H, i Bσ ]E (Hσ − E σ ) ≥
σ E (Hσ − E σ ) 8
(11)
valid for the interval = [σ/3, 2σ/3] and α 1, with α independent of σ . This inequality follows from [H f , i Bσ ] = d (ησ2 ω) ≥ H f,σ
(12)
E (Hσ − E σ )[α 3/2 H f + α 3/2 W, i Bσ ]E (Hσ − E σ ) ≥ O(α 3/2 σ ).
(13)
and from
Indeed, by writing H f = (1 − α 3/2 )H f + α 3/2 H f , combining (12) and (13), and using (9) we obtain E (Hσ −E σ )[H, i Bσ ]E (Hσ −E σ ) ≥ (1 − α 3/2 ) inf + O(α 3/2 σ ) E (Hσ −E σ ). (14) For = [σ/3, 2σ/3] and α small enough this proves (11). 4. The second key estimate in our proof of (6) is the norm bound
f (H − E) − f (Hσ − E σ ) = O(α 3/2 σ )
(15)
valid for smoothed characteristic functions f of the interval = [σ/3, 2σ/3]. The Mourre estimate (6) follows from (10), (11), from B = Bσ + B σ and from (15) if α 1, with α independent of σ . We conclude this introduction with a review of previous work closely related to this paper. Absolute continuity of (part of) the spectrum of Hamiltonians of the form (1), or caricatures thereof, was previously established in [2,4,6,7,13,19,24]. Arai considers the explicitly solvable case of a harmonically bound particle coupled to the quantized radiation field in the dipole approximation. Hübner and Spohn study the spin-boson model with massive bosons or with a photon number cutoff imposed. Their work inspired [24] and [13], where better results were obtained: Skibsted analyzed (1) and assumed that |κ(k)| ≤ |k|5/2 , while, in [13], |κ(k)| ≤ |k|β , with β > 1/2, is sufficient for a Nelson-type model with scalar bosons. The main achievement of [13] is that no bound on the coupling strength is required. Papers [6] and [7] do not introduce an infrared regularization but establish the spectral properties mentioned above only away from O(α 3 )-neighborhoods of the particle ground state energy and the ionization threshold.
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2. Notations and Main Results This section describes in detail the class of Hamiltonians to which we shall apply our analysis, and it contains all our main results. For clarity and simplicity of the presentation of our techniques and main ideas, we shall restrict ourselves to a one-electron model where spin is neglected. Our analysis can easily be extended to the many electron model presented in the introduction, and spin may be included as well. The Hilbert space of our systems is the tensor product H = L 2 (R3 , d x) ⊗ F, where F denotes the symmetric Fock space over L 2 (R3 ; C2 ). The Hamiltonian H : D(H ) ⊂ H → H is given by H = 2 + V + H f ,
= −i∇x + α 3/2 A(αx),
(16)
where V denotes multiplication with a real-valued function V ∈ L 2loc (R3 ). We assume that V is −bounded with relative bound zero and that e1 = inf σ (− + V ) is an isolated eigenvalue with multiplicity one. The first point in σ (− + V ) above e1 is denoted by e2 and egap := e2 − e1 . The field energy H f and the quantized vector potential have already been introduced, formally, in the introduction. More proper definitions are H f := d (ω), the second quantization of multiplication with ω(k) = |k|, and A j (αx) = a(G x, j ) + a ∗ (G x, j ), where κ(k) G x (k, λ) := √ ελ (k)e−iαx·k , |k| and ελ (k), λ ∈ {1, 2}, are two polarization vectors that, for each k = 0, are perpendicular to k and to one another. We assume that ελ (k) = ελ (k/|k|). The ultraviolet cutoff κ : R3 → C is assumed to be a Schwartz-function that depends on |k| only. It follows that |G x (k, λ) − G 0 (k, λ)| ≤ α|k|1/2 |x||κ(k)|, ∂ G x (k, λ) ≤ α x |k|−1/2 f (k) |k| ∂|k|
(17) (18)
with some Schwartz-function f that depends on κ and ∇κ. For the definitions of the annihilation operator a(h) and the creation operator a ∗ (h), where h ∈ L 2 (R3 ; C2 ), we refer to [21,26]. The Hamiltonian (16) is self-adjoint on D(H ) = D(− + H f ) and bounded from below [18]. We use E = inf σ (H ) to denote the lowest point of the spectrum of H and to denote the ionization threshold
inf = lim ϕ, H ϕ , (19) R→∞
ϕ∈D R , ϕ =1
where D R := {ϕ ∈ D(H )|χ (|x| ≤ R)ϕ = 0}. Our conjugate operator is the second quantized dilatation generator B = d (b),
b=
1 (k · y + y · k), 2
(20)
Spectral Theory for the Standard Model of Non-Relativistic QED
619
where y = i∇k . By Theorem 8 of Sect. 4, the Hamiltonian H is locally of class C 2 (B) on (−∞, ). That is, the mapping s → e−i Bs f (H )ei Bs ϕ
(21)
is twice continuously differentiable, for every ϕ ∈ H and every f ∈ C0∞ (−∞, ). This makes the conjugate operator theory in the variant of Sahbani [23] applicable, and, in particular, it allows one to define the commutator [H, i B] as a sesquilinear form on ∪ K E K (H )H, the union being taken over all compact subsets K of (−∞, ). We are now prepared to state the main results of this paper. Theorem 1. Suppose that α 1. Then for any σ ≤ egap /2, E (H − E)[H, i B]E (H − E) ≥
σ E (H − E), 10
where = [σ/3, 2σ/3]. Given Theorem 1, the remark preceding it, and the fact that, by Lemma 16, ≥ E + egap /3 for α small enough, we see that both Hypotheses of Conjugate Operator Theory (Appendix B) are satisfied for = (E, E + egap /3). This implies that the consequences, Theorems 24 and Theorem 25, of the general theory hold for the system under investigation, and, thus, it proves Theorem 2 and Theorem 3 below. Alternatively, the first part of Theorem 2 can also be derived from Theorem 1 using Theorem A.1 of [7]. Theorem 2 (Limiting absorption principle). Let α 1. Then for every s > 1/2 and all ϕ, ψ ∈ H the limits lim ϕ, B −s (H − λ ± iε)−1 B −s ψ
ε→0
(22)
exist uniformly in λ in any compact subset of (E, E + egap /3). For s ∈ (1/2, 1) the map λ → B −s (H − λ ± i0)−1 B −s
(23)
is (locally) Hölder continuous of degree s − 1/2 in (E, E + egap /3). As a corollary from the finiteness of (22) one can show that B −s f (H )(H − f (H ) B −s is bounded on C± for all f ∈ C0∞ (R) with support in (E, E + egap /3). This implies H -smoothness of B −s f (H ) and local decay
B −s f (H )e−i H t ϕ 2 dt ≤ C ϕ 2 . z)−1
R
See [22], Theorem XIII.25 and its Corollary. From the Hölder continuity of (23) we obtain in addition a pointwise decay in time (cf. Theorem 25). Theorem 3. Let α 1 and suppose s ∈ (1/2, 1) and f ∈ C0∞ (R) with supp( f ) ⊂ (E, E + egap /3). Then
B −s e−i H t f (H ) B −s = O(
1 ), t s−1/2
(t → ∞).
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J. Fröhlich, M. Griesemer, I. M. Sigal
3. Proof of the Mourre Estimate This section describes the main steps of the proof of Theorem 1. Technical auxiliaries such as the existence of a spectral gap, soft boson bounds, and the localization of the electron are collected in Appendix A. The proof of Theorem 1 depends, of course, on an explicit expression for the commutator [H, i B]. By Lemma 29 and Proposition 10, we know that for f ∈ C0∞ (−∞, ), ei Bs − 1 f (H ) f (H )[H, i B] f (H ) = lim f (H ) H, s→0 s = f (H ) d (ω) − α 3/2 φ(ibG x ) · − α 3/2 · φ(ibG x ) f (H ), (24) where the limit is taken in the strong operator topology. Therefore we may identify [H, i B], as a quadratic form, with d (ω) − α 3/2 φ(ibG x ) · − α 3/2 · φ(ibG x ). One of our main tools for estimating (24) from below is an infrared cutoff Hamiltonian Hσ , σ as in Theorem 1, whose spectral subspaces for energies close to inf σ (Hσ ) are explicitly known (see Lemma 4). A second key tool is the decomposition of B into two pieces, Bσ and B σ . We now define these operators along with some other auxiliary operators and Hilbert spaces. As a general rule, we will place the index σ downstairs if only low-energy photons are involved, and upstairs for high-energy photons. The fact that this rule does not cover all cases should not lead to any confusion. Let χ0 , χ∞ ∈ C ∞ (R, [0, 1]), with χ0 = 1 on (−∞, 1], χ∞ = 1 on [2, ∞), and 2 2 ≡ 1. For a given σ > 0, we define χ (k) = χ (|k|/σ ), χ σ (k) = χ (|k|/σ ), χ0 + χ∞ σ 0 ∞ σ χ˜ (k) = 1 − χσ (k), and a Hamiltonian Hσ by Hσ = ( p + α 3/2 Aσ (αx))2 + V + H f ,
(25)
where p = −i∇x and Aσ (αx) = φ(χ˜ σ G x ). Let Fσ and F σ denote the symmetric Fock spaces over L 2 (|k| < σ ) and L 2 (|k| ≥ σ ), respectively, and let Hσ = L 2 (R3 ) ⊗ F σ . Then H is isomorphic to Hσ ⊗ Fσ , and, in the sense of this isomorphism, Hσ = H σ ⊗ 1 + 1 ⊗ H f,σ .
(26)
Here H σ = Hσ Hσ and H f,σ = H f Fσ . Next, we split the operator B into two pieces depending on σ . To this end we define new cutoff functions ησ = χ2σ , ησ = χ 2σ and cut-off dilatation generators bσ = ησ bησ , bσ = ησ bησ . Since ησ2 + (ησ )2 ≡ 1 and [ησ , [ησ , b]] = 0 = [ησ , [ησ , b]] it follows from the IMS-formula that b = bσ + bσ . Let Bσ = d (bσ ) and B σ = d (bσ ). Then B = Bσ + B σ . Theorem 8 implies that H is locally of class C 2 (B), C 2 (Bσ ) and C 2 (B σ ) on (−∞, ). By Lemma 16, − E ≥ (2/3)egap for α sufficiently small. It follows that (−∞, ) ⊃ (−∞, E + 2/3egap ) and hence, arguing as in (24), that [H, i Bσ ] = d (ησ2 ω) − α 3/2 φ(ibσ G x ) · − α 3/2 · φ(ibσ G x ),
(27)
[H, i B ] = d ((η ) ω) − α
(28)
σ
σ 2
3/2
σ
φ(ib G x ) · − α
3/2
σ
· φ(ib G x )
Spectral Theory for the Standard Model of Non-Relativistic QED
621
in the sense of quadratic forms on the range of χ (H ≤ E + egap /2), if α 1. Also H σ is of class C 1 (B σ ) and [H σ , i B σ ] = d ((ησ )2 ω) − α 3/2 φ(ibσ χ˜ σ G x ) · − α 3/2 · φ(ibσ χ˜ σ G x )
(29)
on χ (H σ ≤ E + egap /2)Hσ . As a further piece of preparation we introduce smooth versions of the energy cutoffs E (H − E) and E (Hσ − E σ ). We choose f ∈ C0∞ (R; [0, 1]) with f = 1 on [1/3, 2/3] and supp( f ) ⊂ [1/4, 3/4], so that f (s) := f (s/σ ) is a smoothed characteristic function of the interval = [σ/3, 2σ/3]. We define F = f (H − E),
F ,σ = f (Hσ − E σ ).
(30)
Finally, to simplify notations, we set dk := d 3k λ=1,2
and we suppress the index λ in aλ (k), aλ∗ (k), and G x (k, λ). Lemma 4. If α 1 and σ ≤ egap /2, then F ,σ = P σ ⊗ f (H f,σ ),
w.r.t. H = Hσ ⊗ Fσ ,
where P σ denotes the ground state projection of H σ . Proof. By Theorem 18 of Appendix A, H σ has the gap (E σ , E σ + σ ) in its spectrum if α 1. Since the support of f is a subset of (0, σ ), the assertion follows. Proposition 5. Let [H, i B σ ] be defined by (28). If α 1 and σ ≤ egap /2, then F ,σ [H, i B σ ]F ,σ = 0. Proof. From bσ = bσ χ˜ σ , Eqs. (28) and (29) it follows that [H, i B σ ] = [H σ , i B σ ] ⊗ 1 with respect to H = Hσ ⊗ Fσ . The statement now follows from Lemma 4 and the Virial Theorem P σ [H σ , i B σ ]P σ = 0, Proposition 26. Proposition 6. Let [H, i Bσ ] be defined by (27). If α 1 and σ ≤ egap /2, then F ,σ [H, i Bσ ]F ,σ ≥
σ 2 F . 8 ,σ
Proof. On the right hand side of (27) we move the creation operators a ∗ (ibσ G x ) to the left of and the annihilation operators a(ibσ G x ) to the right of . Since 3
[ j , a ∗ (ibσ G x, j )] + [a(ibσ G x, j ), j ] = 0 j=1
we arrive at [H, i Bσ ] = d (ησ2 ω) − 2α 3/2 a ∗ (ibσ G x ) · − 2α 3/2 · a(ibσ G x ).
(31)
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J. Fröhlich, M. Griesemer, I. M. Sigal
Next, we estimate (31) from below using only the fraction 2α 3/2 d (ησ2 ω) of d (ησ2 ω) at first. By completing the square we get, using (18), d (χσ2 ω) − a ∗ (ibσ G x ) · − · a(ibσ G x ) = ω χσ a ∗ − ω−1 · (ibχσ G x )∗ χσ a − ω−1 (ibχσ G x ) · dk −
3
n
n,m=1
≥ −const σ
3
(bχσ G x,n )∗ (bχσ G x,m ) m dk ω n x 2 n .
(32)
n=1
From (31) and (32) it follows that [H, i Bσ ] ≥ (1 − 2α 3/2 )d (ησ2 ω) − const α 3/2 σ
n x 2 n .
(33)
n
It remains to estimate F ,σ d (ησ2 ω)F ,σ from below and F ,σ n n x 2 n F ,σ from above. Using that F ,σ = P σ ⊗ f (H f,σ ), by Lemma 4, and d (ησ2 ω) ≥ H f,σ ,
f (H f,σ )H f,σ f (H f,σ ) ≥
we obtain F ,σ d (ησ2 ω)F ,σ ≥
σ 2 f (H f,σ ), 4
σ 2 F . 4 ,σ
(34)
Furthermore, by Lemma 17 and Lemma 15, sup x E [0,egap /2] (Hσ − E σ ) < ∞.
σ >0
(35)
Since E [0,egap /2] (Hσ − E σ )F ,σ = F ,σ the proposition follows from (33), (34), and (35). Proposition 7. Let F , F ,σ be given by (30). There exists a constant C such that for α 1 and σ ≤ egap /2, F − F ,σ ≤ Cα 3/2 σ. Proof. We begin with a Pauli-Fierz transformation Uσ effecting only the photons with |k| ≤ σ . Let Uσ = exp(iα 3/2 x · Aσ (0)), Aσ (αx) := φ(χσ G x ). Then H(σ ) := Uσ HUσ∗ 2 2 = p + α 3/2 A(σ ) (αx) + V + H f + α 3/2 x · E σ (0) + α 3 x 2 χσ κ 2 , 3
Spectral Theory for the Standard Model of Non-Relativistic QED
623
where A(σ ) (αx) := A(αx) − Aσ (0) and E σ (0) := −i[H f , Aσ (0)]. We compute, dropping the argument αx temporarily, H(σ ) − Hσ = 2α 3/2 p · (A(σ ) − Aσ ) + α 3 (A(σ ) + Aσ ) · (A(σ ) − Aσ ) 2 + α 3/2 x · E σ (0) + α 3 x 2 χσ κ 2 , 3
(36)
where (A(σ ) )2 − (Aσ )2 = (A(σ ) + Aσ ) · (A(σ ) − Aσ ) was used. Note that A(σ ) · Aσ = Aσ · A(σ ) . For later reference we note that A(σ ) (αx) − Aσ (αx) = Aσ (αx) − Aσ (0) = φ(χσ (G x − G 0 )) x · E σ (0) = φ(iωχσ G 0 · x).
(37) (38)
Step 1. Uniformly in σ ≤ egap /2,
(Uσ∗ − 1)F ,σ = O(α 3/2 σ ),
(α → 0).
(39)
Proof of Step 1. By the spectral theorem
(Uσ∗ − 1)F ,σ ≤ α 3/2 x · Aσ (0)F ,σ
= α 3/2 x · φ(χσ G 0 )F ,σ
≤ 2α 3/2 x · a(χσ G 0 )F ,σ + α 3/2 χσ G 0 · x F ,σ . The second term is of order α 3/2 σ as σ → 0, because, by assumption on G 0 , χσ G 0 = O(σ ), and because sup0<σ ≤egap /2 x F ,σ < ∞ by Lemma 17. The first term is of order α 3/2 σ as well, by Lemma 21 and Lemma 17. Step 2. Let F ,(σ ) := f (H(σ ) − E) = Uσ F Uσ∗ . Then, uniformly in σ ≤ egap /2,
F ,(σ ) − F ,σ = O(α 3/2 σ ),
(α → 0).
(40)
Step 1 and Step 2 complete the proof of the proposition, because F − F ,σ = Uσ∗ F ,(σ ) Uσ − F ,σ
= (Uσ∗ − 1)F ,σ + Uσ∗ F ,σ (Uσ − 1) + Uσ∗ F ,(σ ) − F ,σ Uσ .
Proof of Step 2. Let j ∈ C0∞ ([0, 1], R) with j = 1 on [1/4, 3/4] and supp( j) ⊂ [1/5, 4/5]. Let j (s) = j (s/σ ), so that f j = f , and let J = j (H − E) and J ,σ = j (Hσ − E σ ). We will show that
F ,(σ ) − F ,σ = O(α 3/2 σ 1/2 ),
(F ,(σ ) − F ,σ )J ,σ = O(α
3/2
σ ),
(41) (42)
and it will be clear from our proofs that (41) and (42) hold likewise with F and J interchanged. These estimates prove the proposition, because F ,(σ ) − F ,σ = F ,(σ ) J ,(σ ) − F ,σ J ,σ = F ,σ (J ,(σ ) − J ,σ ) + (F ,(σ ) − F ,σ )J ,σ + (F ,(σ ) − F ,σ )(J ,(σ ) − J ,σ ).
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J. Fröhlich, M. Griesemer, I. M. Sigal
To prove (41) and (42) we use the functional calculus based on the representation f (s) =
1 d f˜(z) , z−s
1 ∂ f˜ d f˜(z) := − (z)d xd y, π ∂ z¯
(43)
for an almost analytic extension f˜ of f that satisfies |∂z¯ f˜(x + i y)| ≤ const y 2 [8,17]. We begin with the proof of (42). From (30) and (43) we obtain (F ,(σ ) − F ,σ )J ,σ
1 1 −1 H(σ ) − Hσ − E + E σ J ,σ . =σ d f˜(z) z − (H(σ ) − E)/σ z − (Hσ − E σ )/σ (44) Since, by Lemma 22, |E − E σ | = O(α 3/2 σ 2 ), it remains to estimate the contributions of the various terms due to H(σ ) − Hσ as given by (36). To begin with, we note that
(A(σ ) − Aσ )J ,σ = O(ασ 2 ),
x · E σ (0)J ,σ = O(σ ). 2
(45) (46)
This follows from (37), (38), (17), and Lemma 21, as far as the annihilation operators in (45) and (46) are concerned. For the term due to the creation operator in (45) we use
a ∗ (χσ (G x − G 0 ))J ,σ ≤ a(χσ (G x − G 0 ))J ,σ + χσ (G x − G 0 ) J ,σ and χσ (G x − G 0 ) = O(|x|ασ 2 ), as well as supσ >0 |x|J ,σ < ∞. The operators p and A(σ ) + Aσ stemming from the first and second terms of (36) are combined with the first resolvent of (44): using Uσ∗ pUσ = p + α 3/2 Aσ (0) and Lemma 15 we obtain
(z − (H(σ ) − E)/σ )−1 p = (z − (H − E)/σ )−1 ( p + α 3/2 Aσ (0))
√ 1 + |z| , ≤ const |y| which is integrable with respect to d f˜(z). This proves that the first, second and third terms of (36) give contributions to (44) of order α 5/2 σ , α 4 σ , and α 3/2 σ , respectively. Since χσ κ 2 = O(σ 3 ), (42) follows. The proof of (41) is somewhat involved due to factors of x. We begin with F ,(σ ) − F ,σ = F ,(σ ) J ,(σ ) − F ,σ J ,σ = (F ,(σ ) − F ,σ )J ,σ + F ,(σ ) (J ,(σ ) − J ,σ ). The first term is of order α 3/2 σ by (42). The second one can be written as
−1 σ d f˜(z)R(σ ) (z)F ,(σ ) H(σ ) − Hσ − E + E σ Rσ (z),
(47)
with obvious notations for the resolvents. We recall that, by Lemma 22, |E − E σ | = O(α 3/2 σ 2 ). As in the proof of (42) we need to estimate the contributions due to the four
Spectral Theory for the Standard Model of Non-Relativistic QED
625
terms of H(σ ) − Hσ given by (36). We do this exemplarily for the second one and begin with the estimate
F ,(σ ) (A(σ ) + Aσ ) · (A(σ ) − Aσ )Rσ (z)
≤ F ,(σ ) x (A(σ ) + Aσ )
x −1 (A(σ ) − Aσ )(H f + 1)−1/2
(H f + 1)1/2 Rσ (z) . (48) For the second factor of (48) we use
x −1 (A(σ ) − Aσ )(H f + 1)−1/2 = x −1 φ(χσ (G x − G 0 ))(H f + 1)−1/2
≤ 2 sup x −1 χσ (G x − G 0 ) ω x
= O(ασ 3/2 ), which is of the desired order. In the first factor of (48) we use that Uσ commutes with x , A(σ ) , and Aσ , as well as Lemma 14, Lemma 15 and Lemma 17. We obtain the bound
F ,(σ ) x (A(σ ) + Aσ ) = F x (A(σ ) + Aσ )
≤ F x (H f + 1)1/2
(H f + 1)−1/2 (A(σ ) + Aσ )
≤ const F ( x 2 + H f + 1) < ∞. Finally, for the last factor of (48), Lemma 15 implies the bound
(H f + 1)1/2 Rσ (z) ≤ const
√ 1 + |z| , |y|
which is integrable with respect to d f˜(z). In a similar way the contributions of the other terms of (36) are estimated. It follows that (47) is of order O(α 3/2 σ 1/2 ) which proves (41). This completes the proof of Proposition 7. Proof of Theorem 1. Since (ησ )2 + ησ2 = 1 and bσ + bσ = b, it follows from (27) and (28) that C := d (ω) − α 3/2 φ(ibG x ) · − α 3/2 · φ(ibG x ) = [H, i Bσ ] + [H, i B σ ]. Thus Propositions 5 and 6 imply that F ,σ C F ,σ ≥
σ 2 F . 8 ,σ
We next replace F ,σ by F , using Proposition 7 and noticing that C F ,σ and F C are bounded, uniformly in σ . Since, by (24), C = [H, i B] on the range of F we arrive at F [H, i B]F ≥
σ 2 F + O(α 3/2 σ ). 8
After multiplying this operator inequality from both sides with E (H − E), the theorem follows.
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J. Fröhlich, M. Griesemer, I. M. Sigal
4. Local Regularity of H with Respect to B The purpose of this section is to prove that H is locally of class C 2 (B) in (−∞, ), where is the ionization threshold of H , and B is any of the three operators d (b), d (bσ ), d (bσ ) defined in Sect. 2. Some background on the concept of local regularity of a Hamiltonian with respect to a conjugate operator and basic criteria for this property to hold are collected in Appendix B. To prove a result that covers the three aforementioned operators we consider a class of operators B that contains all of them and is defined as follows. Let k → v(k) be a C ∞ -vector field on R3 of the form v(k) = h(|k|)k, where h ∈ C ∞ (R) such that s n ∂ n h(s) is bounded for n ∈ {0, 1, 2}. It follows |v(k)| ≤ β|k|,
for all k ∈ R3 ,
(49)
for some β > 0, and that partial derivatives of v times a Schwartz-function, such as κ, are bounded. We remark that the assumption that v is parallel to k is not needed if a representation of H free of polarization vectors is chosen. Let φs : R3 → R3 be the flow generated by v, that is, d φs (k) = v(φs (k)), ds
φ0 (k) = k.
(50)
Then φs (k) is of class C ∞ with respect to s and k, and by Gronwall’s lemma and (49), e−β|s| |k| ≤ |φs (k)| ≤ eβ|s| |k|,
for s ∈ R.
(51)
Induced by the flow φs on R3 there is a one-parameter group of unitary transformations on L 2 (R3 ) defined by (52) f s (k) = f (φs (k)) det Dφs (k). Since these transformations leave C0∞ (R3 ) invariant, their generator b is essentially self-adjoint on this space. From b f = id/ds f s |s=0 we obtain b=
1 (v · y + y · v), 2
(53)
where y = i∇k . Let B = d (b). The main result of this section is: Theorem 8. Let H be the Hamiltonian defined by (16) and let be its ionization threshold given by (19). Under the assumptions above on the vector-field v, the operator H is locally of class C 2 (B) in = (−∞, ) for all values of α. The proof, of course, depends on the explicit knowledge of the unitary group generated by B, and in particular on the formulas e−i Bs H f ei Bs = d (e−ibs ωeibs ) = d (ω ◦ φs ) e
−i Bs
A(x)e
i Bs
= φ(e
−ibs
G x ) = φ(G x,s )
(54) (55)
with G x,s given by (52). Another essential ingredient is that, by [15], Theorem 1,
x 2 f (H ) < ∞
(56)
for every f ∈ C0∞ (). We begin with four auxiliary results, Propositions 9, 10, 11, and 12.
Spectral Theory for the Standard Model of Non-Relativistic QED
627
Proposition 9. (a) For all s ∈ R, ei Bs D(H f ) ⊂ D(H f ) and
H f ei Bs (H f + 1)−1 ≤ eβ|s| . (b) For all s ∈ R, ei Bs D(H ) ⊂ D(H ) and
H ei Bs (H + i)−1 ≤ const eβ|s| . Proof. From e−i Bs H f ei Bs = d (e−ibs ω) = d (ω ◦ φs ) and (51) it follows that
H f ei Bs ϕ = d (ω ◦ φs )ϕ ≤ eβ|s| H f ϕ
for all ϕ ∈ F0 (C0∞ ), which is a core of H f . This proves, first, that ei Bs D(H f ) ⊂ D(H f ), and next, that the estimate above extends to D(H f ), proving (a). The Hamiltonian H is self-adjoint on the domain of H (0) = − + H f . Therefore the operators H (0) (H + i)−1 and H (H (0) + i)−1 are bounded and it suffices to prove (b) for H (0) in place of H . The subspace D( ) ⊗ D(H f ) is a core of H (0) . By (a) it is invariant w.r. to ei Bs and √
H (0) ei Bs ϕ ≤ ϕ + H f ϕ eβ|s| ≤ 2eβ|s| H (0) ϕ . As in the proof of (a), it now follows that ei Bs D(H (0) ) ⊂ D(H (0) ) and then the estimate above extends to D(H (0) ). Let Bs := (ei Bs − 1)/is. Then, by Proposition 9, [Bs , H ] is well defined, as a linear operator on D(H ). The main ingredients for the proof of Theorem 8 are Propositions 10 and 12 below. Proposition 10. (a) For all ϕ ∈ D(H ), i lim x −1 [H, Bs ]ϕ = x −1 d (∇ω · v)−α 3/2 φ(ibG x ) · − · φ(ibG x )α 3/2 ϕ. s→0
(b) sup x −1 [Bs , H ](H + i)−1 < ∞. 0<|s|≤1
Proof. Part (b) follows from (a) and the uniform boundedness principle. Part (a) is equivalent to the limit i lim x −1 s→0
1 −i Bs i Bs e He − H ϕ s
being equal to the expression on the right hand side of (a). By (54), for all ϕ ∈ D(H f ), 1 −i Bs 1 e H f ei Bs − H f ϕ = lim d (ω ◦ φs − ω)ϕ = d (∇ω · v)ϕ, s→0 s s→0 s lim
where the last step is easily established using Lebesgue’s dominated convergence Theorem. The necessary dominants are obtained from |s −1 (ω ◦ φs − ω)| ≤ |s|−1 (eβ|s| − 1)ω, by (51), and from the assumption ϕ ∈ D(d (ω)).
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J. Fröhlich, M. Griesemer, I. M. Sigal
It remains to consider the contribution due to Hint := 2α 3/2 A(αx) · p + α 3 A(αx)2 . Let G x,s := G x,s − G x . By (55), e−i Bs Hint ei Bs − Hint = 2α 3/2 φ( G x,s ) · p + α 3 φ( G x,s ) · φ(G x ) + α 3 φ(G x,s ) · φ( G x,s ), (57) a sum of three operators, each of which contains G x,s . By Lemma 13 at the end of this section, for each x ∈ R3 , 1 1 G x,s = G x,s − G x → −ibG x , s s
(s → 0)
(58)
in the norm · ω of L ω (R3 ) (see Appendix A), and sup x −1 bG x ω < ∞
(59)
x∈R3
by the assumptions on G x . Since the operators p(H f + 1)1/2 (H + i)−1 and H f (H + i)−1 are bounded by Lemma 15 and since, by Lemma 14, φ( f )(H f + 1)−1/2 ≤ f ω and
φ( f )φ(g)(H f + 1)−1 ≤ 8 f ω g ω for all f, g ∈ L 2 (R3 ), it follows from (57), (58), and (59) that 1 −i Bs e Hint ei Bs − Hint ϕ lim x −1 s→0 s −1 = x 2α 3/2 φ(−ibG x ) · p + α 3 φ(−ibG x ) · φ(G x ) + α 3 φ(G x ) · φ(−ibG x ) ϕ = −α 3/2 x −1 (φ(ibG x ) · + · φ(ibG x )) ϕ for all ϕ ∈ D(H ). Proposition 11. For all f ∈ C0∞ (), sup [Bs , f (H )] < ∞. 0<|s|≤1
Remark. By Proposition 27 this proposition implies that f (H ) is of class C 1 (B) for all f ∈ C0∞ (). Proof. Let F = f (H ) and let ad Bs (F) = [Bs , F]. If g ∈ C0∞ () is such that g ≡ 1 on supp( f ) and G = g(H ), then F = G F and hence ad Bs (F) = Gad Bs (F) + ad Bs (G)F. The norm of ad Bs (G)F is equal to the norm of its adjoint which is −F ∗ ad B−s (G ∗ ), where F ∗ = f¯(H ) and G ∗ = g(H ¯ ). It therefore suffices to prove that sup Gad Bs (F) < ∞
(60)
0<|s|≤1
for all f, g ∈ C0∞ (). To this end we use the representation f (H ) = d f˜(z)R(z), where R(z) = (z − H )−1 and f˜ is an almost analytic extension of f with |∂z¯ f˜(x +i y)| ≤ const|y|2 , cf. (43). It follows that Gad Bs (F) = d f˜(z)R(z)G[Bs , H ]R(z),
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which is well-defined by Proposition 9, part (b). Upon writing [Bs , H ] = x x −1 [Bs , H ]R(i)(i − H ) we can estimate the norm of the resulting expression for Gad Bs (F) with 0 < |s| ≤ 1, by −1
Gad Bs (F) ≤ sup x [Bs , H ]R(i)
g(H ) x |d f˜(z)| R(z)
(i−H )R(z) . 0<|s|≤1
(i − H )R(z) ≤ const 1 +
Since
1 , | Im(z)|
(61)
the integral is finite by choice of f˜. The factors in front of the integral are finite by Proposition 10 and by (56). Proposition 12. sup x −2 [Bs [Bs , H ]](H + i)−1 < ∞. 0<|s|≤1
Proof. By Definition of H , [Bs , [Bs , H ]] = [Bs , [Bs , H f ]] + α 3/2 [Bs , [Bs , p · φ(G x )]] + α 3 [Bs , [Bs , φ(G x )2 ]]. We estimate the contributions of these terms one by one in Steps 1–3 below. As a preparation we note that 1 ad Bs = iei Bs (W (s) − 1), (62) s 1 1 ad2Bs = − e2i Bs 2 (W (s) − 1)2 = −e2i Bs 2 (W (2s) − 2W (s) + W (0)) , (63) s s where W (s) maps an operator T to e−i Bs T ei Bs . In view of Eqs. (54), (55), we will need that for every twice differentiable function f : [0, 2s] → C, 1 | f (2s) − 2 f (s) + f (0)| ≤ sup | f (t)|. s2 |t|≤2|s|
(64)
Step 1. sup ad2Bs (H f )(H f + 1)−1 < ∞.
|s|≤1
By (63) and (54) 1 d (ω ◦ φ2s − 2ω ◦ φs + ω). (65) s2 Thus in view of (64) we estimate the second derivative of s → ω ◦ φs (k) = |φs (k)|. For k = 0, ad2Bs (H f ) = −e2i Bs
∂2 1 v(φs (k)) |φs (k)| = − φs (k), v(φs (k)) 2 + 2 ∂s |φs (k)| |φs (k)| 1 + φs (k)i vi, j (φs (k))φs (k) j . |φs (k)| i, j
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By assumption on v, vi, j ∈ L ∞ and |v(φs (k))| ≤ β|φs (k)| ≤ eβ|s| |k|. It follows that 1 |(ω ◦ φ2s − 2ω ◦ φs + ω) (k)| ≤ const eβ|s| ω(k), s2 which implies 1 d (ω ◦ φ2s − 2ω ◦ φs + ω)(H f + 1)−1 ≤ const eβ|s| . s2 By (65) this establishes Step 1. Step 2. sup sup x −2 ad2Bs (φ(G x ) · p)(H + i)−1 < ∞.
|s|≤1 x∈R3
Since p(H f + 1)1/2 (H + i)−1 is bounded, it suffices to show that sup x −2 ad2Bs (φ(G x ))(H f + 1)−1/2 < ∞.
|s|≤1, x
(66)
By Eq. (55) 1 1 (W (s) − 1)2 (φ(G x )) = 2 φ(G x,2s − 2G x,s + G x ), s2 s
(67)
and by (64) 1 −1/2 φ(G − 2G + G )(H + 1) x,2s x,s x f s2 2 −2 1 −2 ∂ ≤ x
G x,2s − 2G x,s + G x ω ≤ x 2 G x,s . 2 s ∂s ω
x −2
For k = 0 the function s → G x,s (k) is arbitrarily often differentiable by assumption on v and ∂ 1 G x,s (k) = (v · ∇k G x )s (k) + (div(v)G x )s (k), ∂s 2 ∂2 − 2 G x,s (k) = (v · ∇k )2 G x (k) + (div(v)v · ∇k G x )s s ∂s 1 1 div(v)2 G x . + (vi ∂i ∂ j v j )G x s + s 2 4
−i
(68) (69) (70)
i, j
By part (a) of Lemma 13 below, it suffices to estimate the L 2ω -norm of these four contributions with s = 0. By our assumptions on v, div(v) and vi ∂i ∂ j v j are bounded functions. This and the bound G x ≤ G 0 ω < ∞ account for the contributions of (70), and for the factor div(v) in front of the second term of (69). It remains to show that the L 2ω -norms of x −1 (v · ∇k )G x and x −2 (v · ∇k )2 G x are bounded uniformly in x. But this is easily seen by applying v · ∇k to each factor of G x (k, λ) = ελ (k)e−ik·x κ(k)|k|−1/2 and using that v · ∇ελ (k) = 0, v · ∇e−ik·x = −iv · xe−ik·x and that v · ∇|k|−1/2 is again of order |k|−1/2 by assumption on v.
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Step 3. sup x −2 ad2Bs (φ(G x )2 )(H f + 1)−1 < ∞.
|s|≤1, x
By the Leibniz-rule for ad Bs , ad2Bs (φ(G x )2 ) = ad2Bs (φ(G x )) · φ(G x ) + φ(G x ) · ad2Bs (ϕ(G x )) + 2ad Bs (φ(G x ))ad Bs (φ(G x )).
(71)
For the contribution of the first term we have x −2 ad2Bs (φ(G x )) · φ(G x )(H f + 1)−1
≤ x −2 ad2Bs (φ(G x ))(H f + 1)−1/2
φ(G x )(H f + 1)−1/2
which is bounded uniformly in |s| ≤ 1 and x ∈ R3 by (66) in the proof of Step 2. For the second term of (71) we first note that 1 (W (s) − 1)2 (φ(G x )) s2 1 = e2i Bs φ(G x,s ) 2 (W (s) − 1)2 (φ(G x )), s
φ(G x )ad2Bs (φ(G x )) = φ(G x )e2i Bs
and hence, by the estimates in Step 2, we obtain a bound similar to the one for the first term of (71) with an additional factor of e2β|s| coming from the use of Lemma 13. Finally, by (62) and (55),
G x,s − G x G x,2s − G x,s φ , ad Bs (φ(G x ))ad Bs (φ(G x )) = e2i Bs φ s s which implies that x −2 ad Bs (φ(G x ))ad Bs (φ(G x ))(H f + 1)−1 ≤
sup |s|≤2, x∈R3
2 x −1 ∂s G x,s ω .
This is finite by (68) and the assumptions on v and G x . Proof of Theorem 8. By Propositions 11 and 28 it suffices to show that sup ad2Bs ( f (H )) < ∞
(72)
0<s≤1
for all f ∈ C0∞ (). Let g ∈ C0∞ () with g f = f and let G = g(H ), F = f (H ). Then F = G F and hence ad2Bs (F) = ad2Bs (G F) = ad2Bs (G)F + 2ad Bs (G)ad Bs (F) + Gad2Bs (F). From Proposition 11 we know that sup0<s≤1 ad Bs (G) < ∞, and similarly with F in place of G. Moreover ∗ ad2Bs (G)F = F ∗ ad2B−s (G ∗ ).
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Thus it suffices to show that for all g, f ∈ C0∞ (), sup Gad2Bs (F) < ∞.
(73)
0<|s|≤1
To this end we use F = d f˜(z)R(z) with an almost analytic extension f˜ of f such that |∂z¯ f˜(x + i y)| ≤ const |y|4 . We obtain Gad2Bs (F) = 2 d f˜(z)R(z)G[Bs , H ]R(z)[Bs , H ]R(z) (74) + d f˜(z)R(z)G[Bs , [Bs , H ]]R(z). (75) Since, by (56), G x 2 < ∞ the norm of the second term is bounded uniformly in s ∈ {0 < |s| ≤ 1} by Proposition 12. In view of Proposition 10 we rewrite (74) (times 1/2) as d f˜(z)R(z)G x [Bs , H ]R(z) x −1 [Bs , H ]R(z) − d f˜(z)R(z)G [ x , [Bs , H ]R(z)] x −1 [Bs , H ]R(z). For the norm of the first integral we get the bound |d f˜(z)| R(z)
G x 2
x −1 [Bs , H ]R(i) 2 (i − H )R(z) 2 , which is bounded uniformly in s, by Proposition 10, the exponential decay on the range of G = g(H ) and by construction of f˜. The norm of the second term is bounded by |d f˜(z)| R(z) g(H ) x x −1 [ x , [Bs , H ]R(z)] x −1 [Bs , H ]R(z) . (76) The last factor is bounded by (i − H )R(z) , uniformly in s ∈ (0, 1], by Proposition 10. For the term in the third norm we find, using the Jacobi identity and [Bs , x ] = 0, that x −1 [ x , [Bs , H ]R(z)] = x −1 [Bs , [ x , H ]] R(z)+ x −1 [Bs , H ]R(z)[ x , H ]R(z), (77) where x 2 1 [ x , H ] = 2i ( p + A) + + . (78) x x x 3 Since (78) is bounded w.r.to H , the norm of the second term of (77), by Proposition 10, is bounded by (i − H )R(z) 2 uniformly in s. As for the first term of (77), in view of (78), its norm is estimated like the norm of x −1 [Bs , H ]R(z) in Proposition 10, which leads to a bound of the form const (i − H )R(z) . By (61) and by construction of f˜ it follows that (76) is bounded uniformly in |s| ∈ (0, 1]. We conclude this section with a lemma used in the proofs of Propositions 10 and 12 above. For the definition of L 2ω (R3 ) and its norm see Appendix A. Lemma 13. Let f → f s = e−ibs f on L 2ω (R3 ) be defined by (49), (50) and (52). Then
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(a) The transformation f → f s maps L 2ω (R3 ) into itself and, for all s ∈ R,
f s ω ≤ eβ|s|/2 f ω . (b) The mapping R → L 2ω (R3 ), s → f s is continuous. ˆ kˆ ∈ R3 , is continuously differentiable (c) For all f ∈ L 2ω (R3 ) for which |k| → f (|k|k), √ 2 3 on R+ and ω∂|k| f, ω∂|k| f ∈ L (R ), L 2ω − lim
s→0
1 1 ( f s − f ) = v · ∇ f + div(v) f. s 2
Remark. Statement (c) shows, in particular, that f ∈ D(b) and that −ib f = v · ∇ f + (1/2)div(v) f for the class of functions f considered there. Proof. (a) Making the substitution q = φs (k), dq = det Dφs (k)dk and using (51) we get
f s 2 = =
(|k|−1 + 1)| f (φs (k))|2 det Dφs (k) dk (|φ−s (q)|−1 + 1)| f (q)|2 dq ≤ eβ|s| f 2ω .
(b) For functions f ∈ L 2ω (R3 ) that are continuous and have compact support
f s − f ω → 0 follows from lims→0 f s (k) = f (k), for all k ∈ R3 by an application of Lebesgue’s dominated convergence theorem. From here, (b) follows by an approximation argument using (a). (c) By assumption on f , 1 f˜ := v · ∇ f + div(v) f ∈ L 2ω (R3 ). 2 Using that f s (k) − f (k) =
s
( f˜)t (k) dt,
k = 0
0
and Jensen’s inequality we get
s
−1
s 2 1 ˜ ˜ dk(|k| + 1) [ f t (k) − f (k)] dt s 0 2 1 s ˜ −1 ≤ dk(|k| + 1) f t (k) − f˜(k) dt s 0 1 s ˜ =
f t − f˜ 2 dt, s 0
( f s − f ) − f˜ 2ω =
−1
which vanishes in the limit s → 0 by (b).
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A. Operator and Spectral Estimates Let L 2ω (R3 , C2 ) denote the linear space of measurable functions f : R3 → C2 with 2
f ω = | f (k, λ)|2 (|k|−1 + 1)d 3 k < ∞. λ=1,2
Lemma 14. For all f, g ∈ L 2ω (R3 , C2 ),
a ( f )(H f + 1)−1/2 ≤ f ω ,
a ( f )a (g)(H f + 1)−1 ≤ 2 f ω g ω , where a may be a creation or an annihilation operator. The first estimate of Lemma 14 is well known, see e.g., [4]. For a proof of the second one, see [10]. Lemma 15 (Operator Estimates). Let cn (κ) = |κ(k)|2 |k|n−3 d 3 k for n ≥ 1. Then (i)
A(x)2 ≤ 8c1 (κ)H f + 4c2 (κ),
8 (ii) − c1 (κ)α 3 p 2 ≤ 2 p · A(αx)α 3/2 + H f , 3 (iii) p 2 ≤ 22 + 2α 3 A(αx)2 . If ±V ≤ εp 2 + bε for all ε > 0, and if ε ∈ (0, 1/2) is so small that 16εα 3 c1 (κ) < 1, then 1 (iv) 2 ≤ (H + bε + 8εα 3 c2 (κ)), 1 − 2ε 1 (v) Hf ≤ (H + bε + 8εα 3 c2 (κ)), 1 − 16εα 3 c1 (κ) 8c1 (κ) (H + bε + 8εα 3 c2 (κ)) + 4c2 (κ). (vi) A(x)2 ≤ 1 − 16εα 3 c1 (κ) Proof. Estimate (i) is proved in [16]. (ii) is easily derived by completing the square in creation and annihilation operators, and (iii) follows from 2α 3 p · A(αx) ≥ −(1/2) p 2 − 2α 3 A(αx)2 . From the assumption on V and statements (i) and (iii) it follows that H ≥ 2 − εp 2 − bε + H f ≥ (1 − 2ε)2 − 2εα 3 A(x)3 + H f − bε ≥ (1 − 2ε)2 + (1 − 16εα 3 c1 (κ))H f − 8εα 3 c2 (κ) − bε , which proves (iv) and (v). Statement (vi) follows from (i) and (v). Let E σ = inf σ (Hσ ) and let σ = lim R→∞ σ,R be the ionization threshold for Hσ , that is, σ,R =
inf
ϕ∈D R , ϕ =1
where D R = {ϕ ∈ D(Hσ )|χ (|x| ≤ R)ϕ = 0}.
ϕ, Hσ ϕ ,
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Lemma 16 (Estimates for E σ and σ ). With the above definitions 1. For all α ≥ 0, E σ ≤ e1 + 4c2 (κ)α 3 . 2. If c1 (κ)α 3 ≤ 1/8 then σ,R ≥ e2 − o R (1) − c1 (κ)α 3 C, (R → ∞), where C and o R (1) depend on properties of Hpart only. In particular σ ≥ e2 − c1 (κ)α 3 C uniformly in σ ≥ 0. Proof. Let ψ1 be a normalized ground state vector of Hpart , so that Hpart ψ1 = e1 ψ1 , and let ∈ F denote the vacuum. Then E σ ≤ ψ1 ⊗ , Hσ ψ1 ⊗ = e1 + α 3 ψ1 ⊗ , A(αx)2 ψ1 ⊗ ≤ e1 + 4c2 (κ)α 3 by Lemma 15. To prove Statement 2 we first estimate Hσ from below in terms of Hpart . By Lemma 15, Hσ = Hpart + 2 p · A(αx)α 3/2 + A(αx)2 α 3 + H f 8 ≥ Hpart − c1 (κ)α 3 p 2 . 3 Since p 2 ≤ 3(Hpart + D) for some constant D, it follows that Hσ ≥ Hpart (1 − 8c1 (κ)α 3 ) − 8c1 (κ)Dα 3 . By Perrson’s theorem, ϕ, (Hpart ⊗ 1)ϕ ≥ e2 − o R (1), as R → ∞, for normalized ϕ ∈ D R , with ϕ = 1, and by assumption 1 − 8c1 (κ)α 3 ≥ 0. Hence we obtain R,σ ≥ (e2 − o R (1))(1 − 8c1 (κ)α 3 ) − 8c1 (κ)Dα 3 = e2 − o R (1)(1 − 8c1 (κ)α 3 ) − 8c1 (κ)α 3 (e2 + D), which proves the lemma. Lemma 17 (Electron localization). For every λ < e2 there exists αλ > 0 such that for all α ≤ αλ and all n ∈ N, sup |x|n E λ (Hσ ) < ∞.
σ ≥0
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Proof. From [15, Theorem 1] we know that eε|x| E λ (Hσ ) < ∞ if λ + ε2 < σ . Moreover, from the proof of that theorem we see that sup eε|x| E λ (Hσ ) < ∞
σ ≥0
if R > 0 and δ > 0 can be found so that σ,R −
C˜ ≥ λ + ε2 + δ R2
(79)
holds uniformly in σ . Here C˜ is a constant that is independent of the system. Given λ < e2 , pick αλ > 0 so small that e2 − c1 (κ)αλ3 C > λ with C as in Lemma 16. It then follows from Lemma 16 that (79) holds true for some δ > 0 if R is large enough. Theorem 18 (Spectral gap). If α 1 then σ (Hσ Hσ ) ∩ (E σ , E σ + σ ) = ∅ for all σ ≤ (e2 − e1 )/2. Remark. Variants of this result are already known [3,12]. Proof. From [16] we know that inf σess (Hσ Hσ ) ≥ min(E σ + σ, σ ). On the other hand, by Lemma 16, σ − E σ ≥ e2 − e1 − α 3 (Cc1 (κ) + 4c2 (κ)) ≥ σ under our assumptions on α and σ . This proves that inf σess (Hσ Hσ ) ≥ E σ + σ. From Proposition 19, below, it follows that Hσ has no eigenvalues in (E σ , E σ + σ ). In order to complete the proof of Theorem 18, we need a further commutator estimate ˆ + α 3/2 x · φ(i bˆ χ˜ σ G 0 ), where and a corresponding Virial Theorem. We define B˜ = d (b) ˆ bˆ = (kˆ · y + y · k)/2 and kˆ = k/|k|, and begin with a formal computation of the commu˜ To this end we set σ = p + α 3/2 Aσ (αx) so that Hσ = 2σ + V + H f . tator [Hσ , i B]. It follows that ˜ = σ [σ , i B] ˜ + [σ , i B] ˜ σ + [H f , i B], ˜ [Hσ , i B] where and
˜ = N − α 3/2 x · φ(ωbˆ χ˜ σ G 0 ) [H f , i B]
˜ = σ , id (b) ˆ + σ , iα 3/2 x · φ(i bˆ χ˜ σ G 0 ) [σ , i B] = −α 3/2 φ(i bˆ χ˜ σ G x ) + α 3/2 φ(i bˆ χ˜ σ G 0 ) − 2α 3 Re χ˜ σ G x , x bˆ χ˜ σ G 0 = −α 3/2 φ(i bˆ χ˜ σ G x ) − 2α 3 Re χ˜ σ G x , x bˆ χ˜ σ G 0 . (80)
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˜ is our definition of this Here G x = G x − G 0 . The resulting expression for [Hσ , i B] commutator as a quadratic form on RanE (0,σ ) (Hσ − E σ ), where α 1 and 0 < σ ≤ egap /2 are assumed. The reason for the contribution α 3/2 x · φ(i bˆ χ˜ σ G 0 ) to the operator B˜ is that in Eq. (80) it leads to φ(i bˆ χ˜ σ G x ) rather than φ(i bˆ χ˜ σ G x ). The more regular behavior of G x (k) as k → 0 is essential to get estimates that hold uniformly in σ ∈ (0, egap /2). The following proposition completes the proof of Theorem 18. ˜ be defined as above and suppose that α 1 and Proposition 19. Let [Hσ , i B] 0 < σ ≤ egap /2. Then ˜ (0,σ ) (Hσ − E σ ) ≥ E (0,σ ) (Hσ − E σ )[Hσ , i B]E
1 E (0,σ ) (Hσ − E σ ), 2
˜ = 0. and moreover, if Hσ ϕ = Eϕ with E − E σ ∈ (0, σ ), then ϕ, [Hσ , i B]ϕ ˜ − N between spectral projections E (0,σ ) (Hσ − E σ ) Proof. We first show that [Hσ , i B] 3/2 is O(α ) as α → 0. To this end we set λ = (1/4)e1 + (3/4)e2 and prove Steps 1–3 below. Note that, by Lemma 16, E σ + σ ≤ λ for σ ≤ egap /2 and 2c2 (κ)α 3 ≤ egap /4. Step 1. sup E λ (Hσ )x · φ(ωbˆ χ˜ σ G 0 )E λ (Hσ ) < ∞.
σ >0
One has the estimate
E λ (Hσ )x · φ(ωbˆ χ˜ σ G 0 )E λ (Hσ ) ≤ E λ (Hσ )x
ωbˆ χ˜ σ G 0 ω (H f + 1)1/2 E λ (Hσ ) , where each factor is bounded uniformly in σ > 0. For the first one this follows from Lemma 17, for the second one from |ωbˆ χ˜ σ G 0 (k)| = O(|k|−1/2 ) and for the third one from supσ (H f + 1)1/2 (Hσ + 1)−1 < ∞, by Lemma 15. Step 2. sup E λ (Hσ )σ · φ(i bˆ χ˜ σ G x )E λ (Hσ ) < ∞.
σ >0
This time we use
E λ (Hσ )σ · φ(i bˆ χ˜ σ G x )E λ (Hσ )
≤ E λ (Hσ )σ sup x −1 bˆ χ˜ σ G x ω x (H f + 1)1/2 E λ (Hσ ) . x
Since κ(k) bˆ χ˜ σ G x (k, λ) = i ∂|k| + |k|−1 χ˜ σ (e−ik·x − 1) √ ελ (k) |k| = O( x |k|−1/2 ),
(k → 0),
while, as k → ∞, it decays like a Schwartz-function, it follows that sup x −1 bˆ χ˜ σ G x ω < ∞. x,σ
(81)
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The first factor of (81) is bounded uniformly in σ > 0 thanks to Lemma 15, and for the last one we have
x (H f + 1)1/2 E λ (Hσ ) ≤ x 2 E λ (Hσ ) + (H f + 1)E λ (Hσ ) , which, by Lemma 17 and Lemma 15, is also bounded uniformly in σ . Step 3. sup E λ (Hσ )σ · Re χ˜ σ G x , x · bˆ χ˜ σ G 0 E λ (Hσ ) < ∞. σ
This follows from estimates in the proof of Step 2. From Steps 1, 2, 3 and N ≥ 1 − P it follows that ˜ λ (Hσ ) ≥ E λ (Hσ )(1 − P )E λ (Hσ ) + O(α 3/2 ). E λ (Hσ )[Hσ , i B]E
(82)
In Steps 4, 5, and 6 below we will show that E (0,σ ) (Hσ − E σ )P E (0,σ ) (Hσ − E σ ) = O(α 3/2 ) as well. Hence the proposition will follow from (82). ⊥ =1− P Let Ppart be the ground state projection of − + V and let Ppart part . Recall that Ppart is a projection of rank one, by assumption on e1 = inf σ (− + V ). Step 4. ⊥ ⊗ P )E λ (Hσ ) = O(α 3/2 ).
(Ppart
Let H (0) denote the Hamiltonian H with α = 0 and let f ∈ C0∞ (R) with supp( f ) ⊂ ⊥ ⊗ (−∞, e2 ) and f = 1 on [inf σ ≤egap E σ , λ]. Then E λ (Hσ ) = f (Hσ )E λ (Hσ ), (Ppart P ) f (H (0) ) = 0 and f (Hσ ) − f (H (0) ) 1 1 = d f˜(z) 2α 3/2 p · Aσ (αx) + α 3 Aσ (αx)2 = O(α 3/2 ). z − Hσ z − H (0) It follows that
⊥ ⊥
(Ppart ⊗ P )E λ (Hσ ) = (Ppart ⊗ P ) f (Hσ ) − f (H (0) ) E λ (Hσ )
≤ f (Hσ ) − f (H (0) ) = O(α 3/2 ).
Step 5. Let Pσ denote the ground state projection of Hσ . Then
Ppart ⊗ P − Pσ = O(α 3/2 ). Since 1 − P ≤ N 1/2 we have ⊥ ⊗ P 1 − Ppart ⊗ P = 1 − P + Ppart ⊥ ≤ N 1/2 + Ppart ⊗ P ⊥ ⊗ P )P = O(α 3/2 ) by Step 4 and N 1/2 P = O(α 3/2 ) by Lemma 20. where (Ppart σ σ It follows that (1 − Ppart ⊗ P )Pσ = O(α 3/2 ). Hence, for α small enough, Pσ is of rank one and the assertion of Step 5 follows.
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Step 6. E (0,σ ) (Hσ − E σ )(1 ⊗ P )E (0,σ ) (Hσ − E σ ) = O(α 3/2 ). Since Pσ E (0,σ ) (Hσ − E σ ) = 0, it follows from Step 4 and Step 5 that
(1 ⊗ P )E (0,σ ) (Hσ − E σ ) = (1 ⊗ P − Pσ )E (0,σ ) (Hσ − E σ )
⊥ ≤ (Ppart ⊗ P − Pσ )E (0,σ ) (Hσ − E σ ) + (Ppart ⊗ P )E (0,σ ) (Hσ − E σ )
= O(α 3/2 ). ˜ = 0, for eigenvectors ϕ with In order to prove the Virial Theorem, ϕ, [Hσ , i B]ϕ energy E ∈ (E σ , E σ + σ ) we approximate B˜ with suitably regularized operators B˜ ε , ε > 0, that are defined on D(Hσ ), and converge to B˜ as ε → 0, in the sense that ˜ weakly as ε → 0. The Virial Theorem for [Hσ , i B˜ ε ] then implies [Hσ , i B˜ ε ] → [Hσ , i B] the asserted Virial Theorem. The infrared cutoff σ is crucial for this to work. For more details, see, e.g., [11], Appendix E. Lemma 20 (Ground state photons). Suppose Hσ Pσ = E σ Pσ , where σ ≥ 0, E σ = inf σ (Hσ ), and Pσ is the ground state projection of Hσ . Here Hσ =0 = H . Let Rσ (ω) = (Hσ − E σ + ω)−1 . Then (i) a(k)Pσ = −iα 3/2 1 − ω Rσ (ω) − 2Rσ (ω)(σ · k) + α Rσ (ω)k 2 x · G x (k)∗ Pσ −2α 3/2 Rσ (ω)k · G αx (k)∗ Pσ . There are constants C, D independent of σ, α ∈ [0, 1] such that C , |k|1/2 D (iii) xa(k)Pσ ≤ α 3/2 3/2 . |k| (ii) a(k)Pσ ≤ α 3/2
Proof. We suppress the subindex σ for notational simplicity. By the usual pull-through trick (H − E + ω(k))a(k)P = [H, a(k)]ϕ + ω(k)a(k)P = −α 3/2 2 · G x (k)∗ P. Since 2 = i[H, x] = i[H − E, x], and (H − E)ϕ = 0, we can rewrite this as iα −3/2 a(k)ϕ = R(ω) [(H − E)x − x(H − E)] G αx (k)∗ P = (1 − ω R(ω))(x · G x (k)∗ )P − R(ω)x[H, G αx (k)∗ ]P.
(83)
For the commutator we get [H, G x (k)∗ ] = ( · k)G x (k)∗ + G αx (k)∗ ( · k) = 2( · k)G x (k)∗ − αk 2 G x (k)∗ , and hence, using x( · k) = ( · k)x + ik, x[H, G x (k)∗ ] = 2( · k) − αk 2 x · G αx (k)∗ + 2ik · G x (k)∗ .
(84)
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From (83) and (84) we conclude that iα −3/2 a(k)P = 1 − ω R(ω) − 2R(ω)( · k) + α R(ω)k 2 x · G x (k)∗ P −2i R(ω)k · G x (k)∗ P. (ii) First of all supσ ≥0 x P < ∞ by Lemma 17 and |G x (k)| ≤ const|k|−1/2 by definition of G x (k). Since R(ω) ≤ |k|−1 and R(ω) ≤ const(1 + |k|−1 ) we find that for α, |k| ≤ 1. 1 − ω R(ω) − 2R(ω)( · k) + α R(ω)k 2 ≤ const This proves (ii). To estimate the norm of xa(k)P we use (i) and commute x with all operators in front of P so that we can apply Lemma 17 to the operator x 2 P. Since [x, R(ω)] = −2i R(ω)R(ω) the resulting estimate for xa(k)P is worse by one power of |k| than our estimate (i) for a(k)P . The following two lemmas are consequences of Lemma 20. Lemma 21 (Overlap estimate). Let P σ ⊗ f (H f,σ ) on Hσ ⊗ Fσ and χσ be defined as in Sect. 3. For every µ > −1 there exists a constant Cµ , such that for all α ∈ [0, 1], for all σ ∈ [0, egap /2] and for every function h x ∈ L 2 (R3 ), depending parametrically on the electron position x ∈ R3 , with |h x (k)| ≤ |k|µ x ,
a(χσ h x )P σ ⊗ f (H f,σ ) ≤ Cµ σ µ+3/2 x P σ .
Here x =
√ 1 + x 2.
Proof. Let ϕ ∈ Hσ ⊗ Fσ with ϕ = 1. By construction of χσ , σ a(χσ h x )P ⊗ f (H f,σ )ϕ = χσ (k)h x (k)a(k)P σ ⊗ f (H f,σ )ϕ dk σ ≤|k|≤2σ
+
|k|<σ
χσ (k)
h x (k) σ P ⊗ |k|1/2 a(k) f (H f,σ )ϕ dk. |k|1/2
Using |χσ h x (k)| ≤ |k|µ x , f (H f,σ ) ≤ 1, and the Cauchy-Schwarz inequality applied to the second integral we obtain
a(χσ h x )P σ ⊗ f (H f,σ )ϕ
µ σ ≤ |k| x a(k)P dk + σ ≤|k|≤2σ
|k|≤σ
1/2 |k|
2µ−1
dk
1/2
x P σ H f,σ f (H f,σ )ϕ . 1/2
The lemma now follows from Lemma 20 and H f,σ f (H f,σ ) ≤ σ 1/2 . Lemma 22. There exists a constant C such that |E − E σ | = Cα 3/2 σ 2 for all σ ≥ 0 and α ∈ [0, 1].
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Proof. Let ψ and ψσ be normalized ground states of H and Hσ respectively. Then, by Rayleigh-Ritz, E − E σ ≤ ψσ , (H − Hσ )ψσ , E σ − E ≤ ψ, (Hσ − H )ψ ,
(85) (86)
where H − Hσ = 2 − 2σ and 2 − 2σ = 2α 3/2 p · (A(αx) − Aσ (αx))
+α 3 [A(αx) + Aσ (αx)] · [A(αx) − Aσ (αx)].
(87)
To estimate the contribution due to (87) we note that [A(αx) + Aσ (αx)] · [A(αx) − Aσ (αx)] = [A(αx) + Aσ (αx)] · a(χσ G x ) + a ∗ (χσ G x ) · [A(αx) + Aσ (αx)] + 2 |G x (k)|2 χσ2 dk. (88) The last term in (88) is of order σ 2 and from Lemma 20 it follows that 1
a(χσ G x )ψσ , a(χσ G x )ψ ≤ Cα 3/2 |G x (k)| √ dk = O(α 3/2 σ 2 ). (89) |k| |k|≤2σ Moreover, by Lemma 15,
pψσ , [A(αx) + Aσ (αx)]ψσ ≤ const. It follows that the contributions of (87) to (85) and (86) are of order α 3/2 σ 2 and α 3 σ 2 . B. Conjugate Operator Method In this section we describe the conjugate operator method in the version of Amrein, Boutet de Monvel, Georgescu, and Sahbani [1,23]. In the paper of Sahbani the theory of Amrein et al. is generalized in a way that is crucial for our paper. For simplicity, we present a weaker form of the results of Sahbani with comparatively stronger assumptions that are satisfied by our Hamiltonians. The conjugate operator method to analyze the spectrum of a self-adjoint operator H : D(H ) ⊂ H → H assumes the existence of another self-adjoint operator A on H, the conjugate operator, with certain properties. The results below yield information on the spectrum of H in an open subset ⊂ R, provided the following assumptions hold: (i) H is locally of class C 2 (A) in . This assumption means that the mapping s → e−i As f (H )ei As ϕ is twice continuously differentiable, for all f ∈ C0∞ () and all ϕ ∈ H. (ii) For every λ ∈ , there exists a neighborhood of λ with ⊂ , and a constant a > 0 such that E (H )[H, i A]E (H ) ≥ a E (H ).
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Remarks. By (i), the commutator [H, i A] is well defined as a sesquilinear form on the intersection of D(A) and ∪ K E K (H )H, where the union is taken over all compact subsets K of . By continuity it can be extended to ∪ K E K (H )H. The following two theorems follow from Theorems 0.1 and 0.2 in [23] and assumptions (i) and (ii), above. Theorem 23. For all s > 1/2 and all ϕ, ψ ∈ H, the limit lim ϕ, A −s R(λ ± iε) A −s ψ
ε→0+
exists uniformly for λ in any compact subset of . In particular, the spectrum of H is purely absolutely continuous in . This theorem allows one to define operators A −s R(λ ± i0) A −s in terms of the sesquilinear forms ϕ, A −s R(λ ± i0) A −s ψ = lim ϕ, A −s R(λ ± iε) A −s ψ . ε→0+
By the uniform boundedness principle these operators are bounded. Theorem 24. If 1/2 < s < 1 then λ → A −s R(λ ± i0) A −s is locally Hölder continuous of degree s − 1/2 in . Theorem 25. Suppose assumptions (i) and (ii) above are satisfied, s ∈ (1/2, 1), and f ∈ C0∞ (). Then
1 −s −i H t −s (t → ∞).
A e f (H ) A = O s−1/2 , t Proof. For every f ∈ C0∞ (R) and all ϕ ∈ H, 1 e−i H t f (H )ϕ = lim e−iλt f (λ) Im(H − λ − iε)−1 ϕ dλ ε↓0 π
(90)
by the spectral theorem. Now suppose f ∈ C0∞ () and set F(z) = π −1 A −s Im(H − z)−1 A −s . Then (90) and Theorem 23 imply (91) A −s e−i H t f (H ) A −s ϕ = e−iλt f (λ)F(λ + i0)ϕ dλ. In this equation we replace H by H − π/t with t so large that f (· − π/t) has support in . Then it becomes (92) A −s e−i H t f (H − π/t) A −s ϕ = − e−iλt f (λ)F(λ + π/t + i0)ϕ dλ. Taking the sum of (91) and (92) and using f (H ) − f (H − π/t) = O(t −1 ), which may be derived from the almost analytic functional calculus, see (43), we get 2 A −s e−i H t f (H ) A −s + O(t −1 ) ≤ | f (λ)| F(λ + i0) − F(λ + π/t + i0) dλ = O(1/t s−1/2 ), where the Hölder continuity from Theorem 24 was used in the last step.
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For completeness we also include the Virial Theorem (Proposition 3.2 of [23]): Proposition 26. If λ ∈ is an eigenvalue of H and E {λ} (H ) denotes the projection onto the corresponding eigenspace, then E {λ} (H )[H, i A]E {λ} (H ) = 0. In the remainder of this section we introduce tools that will help us to verify assumption (i). To begin with we recall, from [1,23], that a bounded operator T on H is said to be of class C k (A) if the mapping s → e−i As T ei As ϕ is k times continuously differentiable for every ϕ ∈ H. The following propositions summarize results in Lemma 6.2.9 and Lemma 6.2.3 of [1]. Proposition 27. Let T be a bounded operator on H and let A = A∗ : D(A) ⊂ H → H. Then the following are equivalent. (i) T is of class C 1 (A). (ii) There is a constant c such that for all ϕ, ψ ∈ D(A),
(iii) lim inf s→0+
| Aϕ, T ψ − ϕ, T Aψ | ≤ c ϕ
ψ . 1 −i As e T ei As − T < ∞. s
Proof. If T is of class C 1 (A) then sups=0 s −1 (e−i As T ei As − T ) < ∞ by the uniform boundedness principle. Thus statement (i) implies statement (iii). To prove the remaining assertions we use that, for all ϕ, ψ ∈ D(A), 1 −i s ϕ, (e−i As T ei As − T )ψ = dτ ei Aτ Aϕ, T ei Aτ ψ − ei Aτ ϕ, T ei Aτ Aψ . s s 0 (93) Since the integrand is a continuous function of τ , its value at τ = 0, Aϕ, T ψ − ϕ, T Aψ , is the limit of (93) as s → 0. It follows that | Aϕ, T ψ − ϕ, T Aψ | = lim s −1 | ϕ, (e−i As T ei As − T )ψ | s→0+
≤ lim inf s −1 e−i As T ei As − T
ϕ
ψ . s→0+
(94)
Therefore (iii) implies (ii). Next we assume (ii). Then T D(A) ⊂ D(A) and [A, T ] : D(A) ⊂ H → H has a unique extension to a bounded operator ad A (T ) on H. The mapping τ → e−i Aτ ad A (T )ei Aτ ψ is continuous, and hence (93) implies that e−i As T ei As ψ − T ψ = −i
s
e−i Aτ ad A (T )ei Aτ ψ dτ
(95)
0
for each ψ ∈ H. Since the r.h.s is continuously differentiable in s, so is the l.h.s, and thus T ∈ C 1 (A).
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Let As = (ei As − 1)/is, which is a bounded approximation of A. Then 1 −i As i As e T e − T = −ie−i As ad As (T ). s
(96)
Hence, by Proposition 27, a bounded operator T is of class C 1 (A) if and only if lim inf s→0+ ad As (T ) < ∞. The following proposition gives an analogous characterization of the class C 2 (A). Proposition 28. Let A = A∗ : D(A) ⊂ H → H and let T be a bounded operator of class C 1 (A). Then T is of class C 2 (A) if and only if lim inf ad2As (T ) < ∞.
(97)
s→0+
Remark. This is a special case of [1, Lemma 6.2.3] on the class C k (A). We include the proof for the convenience of the reader. Proof. Since T is of class C 1 (A) the commutator [A, T ] extends to a bounded operator ad A (T ) on H and d (98) i e−i As T ei As ϕ = e−i As ad A (T )ei As ϕ ds for all ϕ ∈ H. By Proposition 27 the right-hand side is continuously differentiable if and only if | Aϕ, ad A (T )ψ − ϕ, ad A (T )Aψ | ≤ c ϕ
ψ ,
for ϕ, ψ ∈ D(A)
(99)
with some finite constant c. To prove that (99) is equivalent to (97), it is useful to introduce the homomorphism W (s) : T → e−i As T ei As on the algebra of bounded operators. By (95) s dτ1 W (τ1 )ad A (T ), (W (s) − 1)T = −i 0
and therefore
1 −i s 2 (W (s) − 1) T = dτ1 (W (s) − 1)W (τ1 )ad A (T ) s2 s2 0 s −1 s = 2 dτ1 dτ2 W (τ1 + τ2 )[A, ad A (T )] s 0 0
(100)
in the sense of quadratic forms on D(A), that is, ϕ, W (τ1 + τ2 )[A, ad A (T )]ψ := Aϕ, W (τ1 + τ2 )ad A (T )ψ − ϕ, W (τ1 + τ2 )ad A (T )Aψ for ϕ, ψ ∈ D(A). Since the right-hand side is continuous as a function of τ1 + τ2 , it follows from (100), as in the proof of Proposition 27, that 1 | ϕ, (W (s) − 1)2 T ψ | s→0+ s 2 1 ≤ lim inf 2 (W (s) − 1)2 T
ϕ
ψ . s→0+ s
| Aϕ, ad A (T )ψ − ϕ, ad A (T )Aψ | = lim
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Since, by (96), 1 (W (s) − 1)2 T = −e−2i As ad2As (T ), s2 condition (97) implies (99). Conversely, by (100) condition (99) implies that s −2 (W (s)− 1)2 T ≤ c for all s > 0, which proves (97). Lemma 29. Suppose that H is locally of class C 1 (A) in ⊂ R and that ei As D(H ) ⊂ D(H ) for all s ∈ R. Then, for all f ∈ C0∞ () and all ϕ ∈ H, ei As − 1 f (H )ϕ. f (H )[H, i A] f (H )ϕ = lim f (H ) H, s→0 s Proof. By Eq. 2.2 of [23], f (H )[H, i A] f (H ) = [H f 2 (H ), i A] − H f (H )[ f (H ), i A] − [ f (H ), i A]H f (H ), (101) where, by assumption, f (H ) and H f 2 (H ) are of class C 1 (A). Since, by (96) [T, i A]ϕ = −i lim ad As (T )ϕ s→0
of class C 1 (A), it follows from
for every bounded operator T ad As and the domain assumption As D(H ) ⊂ D(H ), that
(101), the Leibniz-rule for
f (H )[H, i A] f (H )ϕ = −i lim ad As (H f 2 (H )) − H f (H )ad As ( f (H )) − ad As ( f (H ))H f (H ) ϕ s→0
= −i lim f (H )ad As (H ) f (H )ϕ. s→0
References 1. Amrein, W., Boutet de Monvel, A., Georgescu, V.: C0 -groups, commutator methods and spectral theory for N -body Hamiltonians. Vol. 135 of Progress in Mathematics. Basel-Boston: Birkhäuser, 1996 2. Arai, A.: Rigorous theory of spectra and radiation for a model in quantum electrodynamics. J. Math. Phys. 24(7), 1896–1910 (1983) 3. Bach, V., Fröhlich, J., Pizzo, A.: Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field. Commun. Math. Phys. 264(1), 145–165 (2006) 4. Bach, V., Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2), 299–395 (1998) 5. Bach, V., Fröhlich, J., Sigal, I.M.: Renormalization group analysis of spectral problems in quantum field theory. Adv. Math. 137(2), 205–298 (1998) 6. Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999) 7. Bach, V., Fröhlich, J., Sigal, I.M., Soffer, A.: Positive commutators and the spectrum of Pauli-Fierz Hamiltonian of atoms and molecules. Commun. Math. Phys. 207(3), 557–587 (1999) 8. Davies, E.B.: The functional calculus. J. London Math. Soc. (2) 52(1), 166–176 (1995) 9. Derezi´nski, J., Jakši´c, V.: Spectral theory of Pauli-Fierz operators. J. Funct. Anal. 180(2), 243–327 (2001) 10. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math. 164(2), 349–398 (2001) 11. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré 3(1), 107–170 (2002)
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12. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Compton scattering. Commun. Math. Phys. 252(1–3), 415–476 (2004) 13. Georgescu, V., Gérard, C., Møller, J.S.: Spectral theory of massless Pauli-Fierz models. Commun. Math. Phys. 249(1), 29–78 (2004) 14. Gérard, C.: On the scattering theory of massless Nelson models. Rev. Math. Phys. 14(11), 1165–1280 (2002) 15. Griesemer, M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210(2), 321–340 (2004) 16. Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3), 557–595 (2001) 17. Helffer, B., Sjöstrand, J.: Équation de Schrödinger avec champ magnétique et équation de Harper. In: Schrödinger operators (Sønderborg, 1988), Volume 345 of Lecture Notes in Phys., Berlin: Springer, 1989, pp. 118–197 18. Hiroshima, F.: Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants. Ann. Henri Poincaré 3(1), 171–201 (2002) 19. Hübner, M., Spohn, H.: Spectral properties of the spin-boson Hamiltonian. Ann. Inst. H. Poincaré Phys. Théor. 62(3), 289–323 (1995) 20. Lieb, E.H., Loss, M.: Existence of atoms and molecules in non-relativistic quantum electrodynamics. Adv. Theor. Math. Phys. 7(4), 667–710 (2003) 21. Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1975 22. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1978 23. Sahbani, J.: The conjugate operator method for locally regular Hamiltonians. J. Operator Theory 38(2), 297–322 (1997) 24. Skibsted, E.: Spectral analysis of N -body systems coupled to a bosonic field. Rev. Math. Phys. 10(7), 989–1026 (1998) 25. Spohn, H.: Asymptotic completeness for Rayleigh scattering. J. Math. Phys. 38(5), 2281–2296 (1997) 26. Spohn, H.: Dynamics of charged particles and their radiation field. Cambridge: Cambridge University Press, 2004 Communicated by H.-T. Yau
Commun. Math. Phys. 283, 647–662 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0537-y
Communications in
Mathematical Physics
Generic Continuous Spectrum for Ergodic Schrödinger Operators Michael Boshernitzan, David Damanik Department of Mathematics, Rice University, Houston, TX 77005, USA. E-mail: [email protected]; [email protected]; URL: http://www.ruf.rice.edu/∼dtd3/ Received: 6 September 2007 / Accepted: 6 March 2008 Published online: 19 June 2008 – © Springer-Verlag 2008
Abstract: We consider families of discrete Schrödinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sampling function, the set of elements in the associated family of Schrödinger operators that have no eigenvalues is large in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus. 1. Introduction In this paper we study Schrödinger operators acting in 2 (Z) by (Hω ψ)(n) = ψ(n + 1) + ψ(n − 1) + Vω (n)ψ(n),
(1)
where the potential Vω is generated by some homeomorphism T of a compact metric space and a continuous sampling function f : → R as follows: Vω (n) = f (T n ω), ω ∈ , n ∈ Z.
(2)
If µ is a T -ergodic Borel probability measure, it is known that the spectrum and the spectral type of Hω are µ-almost surely independent of ω (see, e.g., CarmonaLacroix [18]). In general, however, the spectral type is not globally independent of ω (see Jitomirskaya-Simon [26] for counterexamples). Recall that (, T ) is called minimal if the T -orbit of every ω ∈ is dense in . In this case, a strong approximation argument shows that the spectrum of Hω does not depend on ω. Moreover, it is then also true, but much harder to prove, that the absolutely continuous spectrum of Hω does not depend on ω; see Last-Simon [29]. D. D. was supported in part by NSF grant DMS–0653720.
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Popular examples include shifts = Td , T ω = ω + α
(3)
= T2 , T (ω1 , ω2 ) = (ω1 + 2α, ω1 + ω2 )
(4)
and skew-shifts
on the torus. Here, we write T = R/Z. Operators with potentials generated by shifts on the torus are called quasi-periodic. In these examples, minimality of (, T ) holds if and only if the coordinates of α together with 1 are independent over the rational numbers. Moreover, in these cases, normalized Lebesgue measure on the torus in question is the unique T -ergodic Borel probability measure and hence one is particularly interested in identifying the spectrum and the spectral type of H for Lebesgue almost every ω. We refer the reader to Bourgain’s recent book [14] for the current state of the art, especially for these two classes of models. While the quasi-periodic case has been studied heavily for several decades and many fundamental results have been obtained, the case of the skew-shift is much less understood. It is of interest as it naturally arises in the study of the quantum kicked rotor model; compare [13]. The general expected picture is that, while the skew-shift model seems to be formally close to a quasi-periodic model, on the operator level one observes much different behavior – similar to that found for random potentials. Namely, while any spectral type occurs naturally for quasi-periodic models and their spectra have a tendency to be Cantor sets, it is expected that skew-shift models “almost always” have pure point non-Cantor spectrum. Thus, interest in Schrödinger operators generated by the skew-shift is also triggered by the question of how much one can reduce the randomness of the potentials until one fails to observe complete localization phenomena. Such a transition could occur at or near skew-shift models. To illustrate this point, let us recall some conjectures from [14, p. 114]. Consider the potential V (n) = λ cos(2π(ω1 + ω2 n + αn(n − 1))).
(5)
Then, one expects that for every λ = 0, Diophantine α ∈ T, and Lebesgue almost every ω ∈ T2 , the operator (1) with potential (5) has positive Lyapunov exponents,1 pure point spectrum with exponentially decaying eigenfunctions, and its spectrum has no gaps. Generally, one expects further that these properties persist even when the cosine is replaced by a sufficiently regular function f . There are very few positive results in this direction. For example, Bourgain, Goldstein, and Schlag proved a localization result for analytic f = λg and sufficiently large λ and Bourgain proved the existence of some point spectrum for (5) with small λ and certain (α, ω) ∈ T3 ; see [11,16]. Some results (that, however, do not determine the spectral type) assuming weaker regularity can be found in the paper [19] by Chan, Goldstein, and Schlag. Naturally, negative results that point out limitations to the scope in which the expected properties actually hold are of interest as well. One result of this kind is obtained in the work [1] by Avila, Bochi, and Damanik, where it is shown that for the skew-shift model (and generalizations thereof), the spectrum is a Cantor set for a residual set of continuous 1 Lyapunov exponents measure the averaged rate of exponential growth of the so-called transfer matrices. Since we will not need them in our study, we omit the exact definition.
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sampling functions. Another result showing that expected phenomena may not occur can be found in the paper [5] by Bjerklöv. He showed that even for (large) analytic sampling functions, the Lyapunov exponent may vanish. Further negative results, concerning the spectral type, will be established in the present paper. Namely, we will show under reasonably weak assumptions that for a generic continuous sampling function f , the point spectrum is empty. The present paper should be regarded as a companion piece to work by Avila and Damanik [2]. They proved that the absolutely continuous spectrum is generically empty. Putting these two results together, it follows that for a large class of ergodic Schrödinger operators, the generic spectral type is singular continuous. This will be discussed in more detail in Subsect. 4.4. A bounded potential V : Z → R is called a Gordon potential if there are positive integers qk → ∞ such that max |V (n) − V (n ± qk )| ≤ k −qk
1≤n≤qk
for every k ≥ 1.2 Equivalently, there are positive integers qk → ∞ such that ∀ C > 0 : lim max |V (n) − V (n ± qk )|C qk = 0. k→∞ 1≤n≤qk
Clearly, if V is a Gordon potential, then so is λV for every λ ∈ R. This condition is of interest because it ensures the absence of point spectrum. That is, if V is a Gordon potential, then the Schrödinger operator in 2 (Z) with potential V has no eigenvalues. In fact, it is even true that for every E ∈ C, the difference equation u(n + 1) + u(n − 1) + V (n) = Eu(n) has no non-trivial solution u with lim|n|→∞ u(n) = 0; compare [21,23,25]. For potentials generated by a shift on the circle, T : T → T, ω → ω +α, it is not hard to show that for f having a fixed modulus of continuity and α from an explicit residual subset of T (which depends only on the modulus of continuity and has zero Lebesgue measure), Vω is a Gordon potential for every ω. For the specific case of the cosine, this observation was used by Avron and Simon [3] to exhibit explicit quasi-periodic operators with purely singular continuous spectrum; see also [21]. Our goal is to exhibit a variety of situations, in the general context we consider, where the potentials satisfy the Gordon condition, either on a residual subset of or on a subset of that has full µ measure. Definition 1. A sequence {ωk }k≥0 in the compact metric space has the repetition property if for every ε > 0 and r > 0, there exists q ∈ Z+ such that dist(ωk , ωk+q ) < ε for k = 0, 1, 2, . . . , rq . Of course, this definition makes sense in any metric space; but we remark that the compactness of implies that the validity of the repetition property for any given sequence in is independent of the choice of the metric. The definition could be weakened by replacing the “for every r ” requirement by a suitable fixed value of r . The main results below extend to this modified notion. For simplicity, we work with this slightly stronger version. 2 Here, |V (n) − V (n ± q )| ≤ k −qk is shorthand for having both |V (n) − V (n + q )| ≤ k −qk and k k |V (n) − V (n − qk )| ≤ k −qk .
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Definition 2. We denote the set of points in the compact metric space whose forward orbit with respect to the homeomorphism T has the repetition property by P R P(, T ). That is, P R P(, T ) = {ω ∈ : {T k ω}k≥0 has the repetition property}. We say that (, T ) satisfies the topological repetition property (TRP) if P R P(, T ) is dense in . P R P(, T ) is always a G δ subset of and so, in particular, a dense G δ set in the closure of the orbit of any of its elements. In particular, if (, T ) is minimal and P R P(, T ) = ∅, then P R P(, T ) is a residual subset of . The following Theorems 1 and 2 demonstrate the usefulness of the repetition properties of systems (, T ) in establishing generic (topological and measure-theoretical) results on existence of “many” Gordon potentials. Theorem 1. Suppose (, T ) is minimal and satisfies (TRP). Then there exists a residual subset F of C() such that for every f ∈ F, there is a residual subset f ⊆ with the property that for every ω ∈ f , Vω defined by (2) is a Gordon potential. Combining this theorem with Gordon’s result, we find that Hω has purely continuous spectrum for topologically generic f ∈ C() and ω ∈ whenever (, T ) is minimal and satisfies (TRP). Remark. With more work, it is possible to remove the assumption of minimality in Theorem 1. We state and prove the version given above for the sake of simplicity. Let us now fix some T -ergodic measure µ. As mentioned above, given the general theory of ergodic Schrödinger operators (cf. [18]), it is a natural goal to identify the almost sure spectral type with respect to µ. By almost sure independence, it is sufficient to consider sets of positive µ measure. Thus, we are seeking a criterion that implies the applicability of Gordon’s lemma at least on a set of positive µ measure. Definition 3. We say that (, T, µ) satisfies the metric repetition property (MRP) if µ(P R P(, T )) > 0. The following result shows that (MRP) implies the desired Gordon property on a full measure set. Theorem 2. Suppose (, T, µ) satisfies (MRP). Then there exists a residual subset F of C() such that for every f ∈ F, there is a subset f ⊆ of full µ measure with the property that for every ω ∈ f , Vω defined by (2) is a Gordon potential. Naturally, the strongest repetition property, defined in terms of the size of the set P R P(, T ), that can hold is the following: Definition 4. We say that (, T ) satisfies the global repetition property (GRP) if P R P(, T ) = . Remark. One may wonder whether (GRP) implies a stronger statement for the associated potentials. Namely, when (GRP) holds, is it true that for generic continuous f , Vω is a Gordon potential for every ω ∈ ? In an earlier version of this manuscript we stated that we believe the answer is negative. One of the referees pointed out to us how this can
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be shown: Suppose that T is a shift on T with α having continued fraction approximants pk /qk satisfying lim supk→∞ qk−1 log qk+1 = 0 and f ∈ C(T) is such that every Vω is a Gordon potential. Then, f must be constant. Indeed, up to removing the periods of f , we must have lim inf h→0+ h −1 | f (ω + h) − f (ω)| = 0 for every ω, from which the claim follows. Detailed proofs of these two assertions will be given in Subsect. 4.2. Let us explore when (TRP), (MRP), and (GRP) hold for our two classes of examples. It will turn out that, in these examples, (TRP), (MRP), and (GRP) always hold or fail simultaneously. Assuming minimality of (, T ), it is clear that we always have the implications (GRP) ⇒ (MRP) ⇒ (TRP). One can ask whether some reverse implication holds in general. This turns out not to be the case, already for skew-products on higher dimensional tori; see the discussion in Subsect. 4.6. For minimal shifts of the form (3), the situation is particularly nice as the following result shows. Theorem 3. Every minimal shift T ω = ω + α on the torus Td satisfies (GRP), and hence also (TRP) and (MRP). As a consequence, we find that for every minimal shift on Td and a generic function f ∈ C(Td ), the operator Hω has empty point spectrum for almost every ω. This is especially surprising if a coupling constant is introduced. Generally, one expects pure point spectrum at large coupling, but our proof excludes point spectrum for all values of the coupling constant at once! This is discussed in more detail in Subsect. 4.3. We also want to mention that recent results indicate that point spectrum should become more and more prevalent as the dimension of the torus increases [12]. Our result on the absence of point spectrum, on the other hand, holds for all torus dimensions. Let us now turn to minimal skew-shifts of the form (4). Recall that α ∈ T is called badly approximable if there is a constant c > 0 such that c αq > q for every q ∈ Z \ {0}. Here, we write x = distT (x, 0) (= min{|x − p| : p ∈ Z}, where x denotes any representative in R). The set of badly approximable α’s has zero Lebesgue measure; see, for example, [28, Theorem 29 on p. 60]. In terms of the continued fraction expansion of α (cf. [28]), being badly approximable is equivalent to having bounded partial quotients. Theorem 4. For a minimal skew-shift T (ω1 , ω2 ) = (ω1 + 2α, ω1 + ω2 ) on the torus T2 , the following are equivalent: (i) (ii) (iii) (iv)
α is not badly approximable. (, T ) satisfies (GRP). (, T, Leb) satisfies (MRP). (, T ) satisfies (TRP).
Thus, for Lebesgue almost every α, the operator Hω generated by the corresponding skew-shift and a generic function f ∈ C(T2 ) has empty point spectrum for Lebesgue almost every ω. This is surprising given that the expected spectral type for operators generated by the skew shift is pure point. Again, one can introduce a coupling constant and absence of point spectrum then holds for all values of the coupling constant simultaneously. Let us also emphasize that, to the best of our knowledge, our result provides the
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first examples of Schrödinger operators with potentials defined by a minimal skew-shift that have empty point spectrum for Lebesgue almost every ω. In particular, the expected localization result for such operators will need a suitable regularity assumption for the sampling function. The paper is organized as follows. In Sect. 2, we establish the relation between the topological or metric repetition property for the underlying dynamical system and the Gordon property for the associated potentials when a generic continuous sampling function is chosen; that is, we prove Theorems 1 and 2. The validity of the topological, metric, or global repetition property for the two classes of examples is then explored in Sect. 3, where we prove Theorems 3 and 4. We conclude the paper with some further results and comments in Sect. 4. 2. Repetition Properties and Gordon Potentials In this section we prove Theorems 1 and 2. Proof of Theorem 1. By assumption, there is a point ω ∈ whose forward orbit has the repetition property. For each k ∈ Z+ , consider ε = k1 , r = 3, and the associated qk = q(ε, r ). This ensures qk → ∞. Take an open ball Bk around ω with radius small enough so that T n (Bk ), 1 ≤ n ≤ 4qk are disjoint and, for every 1 ≤ j ≤ qk , 3
T j+lqk (Bk )
l=0
is contained in some ball of radius 4ε. Define Ck =
f ∈ C() : f is constant on each set
3
T
j+lq
(Bk ), 1 ≤ j ≤ qk
l=0
and let Fk be the open k −qk neighborhood of Ck in C(). Notice that for each m, Fk k≥m
is an open and dense subset of C(). This follows since every f ∈ C() is uniformly 3 T j+lqk (Bk ) goes to zero, uniformly in j, as continuous and the diameter of the set l=0 k → ∞. Thus, F= Fk m≥1 k≥m
is a dense G δ subset of C().
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Consider some f ∈ F. Then, f ∈ Fkl for some sequence kl → ∞. Observe that for every m ≥ 1, q
kl
T j+qkl (Bkl )
l≥m j=1
is an open and dense subset of since T is minimal and qkl → ∞. Thus, q
f =
kl
T j+qkl (Bkl )
m≥1 l≥m j=1
is a dense G δ subset of . It now readily follows that for every f ∈ F and ω ∈ f , Vω is a Gordon potential. qkl Explicitly, since ω ∈ f , ω belongs to j=1 T j+qkl (Bkl ) for infinitely many l. For each such l, we have by construction that −qkl
max | f (T j ω) − f (T j+qkl ω)| < 2kl
1≤ j≤qkl
and −qkl
max | f (T j ω) − f (T j−qkl ω)| < 2kl
1≤ j≤qkl
This shows that Vω (n) = f (T n ω) is a Gordon potential.
.
Proof of Theorem 2. Notice that, by ergodicity, the assumption µ(P R P(, T )) > 0 implies µ(P R P(, T )) = 1. Let us inductively define a sequence of positive integers, {n i }i≥1 , and a sequence of subsets of , {i }i≥1 . Since µ(P R P(, T )) = 1 we can choose n 1 ∈ Z+ large enough so that3 1 = ω ∈ : there exists n ∈ [1, n 1 ) such that for k1 , k2 ∈ [1, n] with k1 ≡ k2 mod n, we have dist(T k1 ω, T k2 ω) < 1 obeys µ(1 ) > 1 − 2−1 . Once n i−1 and i−1 have been determined, we can define n i and i as follows. It is possible to find n i > max{n i−1 , 2i } such that i = ω ∈ : there exists n ∈ [n i−1 , n i ) such that for k1 , k2 ∈ [1, in] with k1 ≡ k2 mod n, we have dist(T k1 ω, T k2 ω) < 2−i 3 The definition of contains redundancies whose purpose is to motivate the definition of the subsequent 1 sets i .
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obeys µ(i ) > 1 − 2−i .
(6)
For i ≥ 5, we set m i = (in i )2 and choose (using the Rokhlin-Halmos Lemma [20, Theorem 1 on p. 242]) Oi ⊂ in a way that T j Oi , 1 ≤ j ≤ m i are disjoint and ⎛ ⎞ mi j µ⎝ T Oi ⎠ > 1 − 2−n i −1 . j=1
Next we further partition Oi into sets Si,l , 1 ≤ l ≤ si such that for every 0 ≤ u ≤ m i , diam(T u Si,l ) <
1 . ni
(7)
Choose K i,l ⊆ Si,l compact with
µ(K i,l ) > µ(Si,l ) 1 − 2−n i −1 .
Then, T u K i,l , 0 ≤ u ≤ m i , 1 ≤ l ≤ si are disjoint and their total measure is ⎛ ⎞ µ ⎝ T u K i,l ⎠ ≥ (1 − 2−n i −1 )2 > 1 − 2−n i .
(8)
u,l
We will now collect a large subfamily of {T u K i,l }. For each l, we let u run from 0 upwards and ask for the corresponding T u K i,l whether it has non-empty intersection with i . If it does, then there is a point ω and a corresponding n ∈ (n i−1 , n i ). We add the current T u K i,l to the subfamily we construct, along with the sets T u+h K i,l , 1 ≤ h ≤ in. Then we continue with T u+in+1 K i,l and do the same. We stop when we are within n i steps of the top of the tower. Let us denote the subfamily so constructed by Ki . By (6), (8), and T -invariance of µ, we have that ⎛ ⎞ ni µ⎝ T u K i,l ⎠ > 1 − 2−n i − 2−i − . (9) m i u T K i,l ∈Ki
The next step is to group each of these “runs” into arithmetic progressions. Notice that locally we have n consecutive points that are (i − 1) times repeated up to some small error. Notice that this extends to the entire set if we make the allowed error a bit larger. Let us group these in sets into n arithmetic progressions of length i. The union of each of these arithmetic progressions of sets will constitute a new set Ci,m . By the definition of i and (7), we have diam(Ci,m ) <
2 + 2−i . ni
(10)
We will also consider the sets C˜ i,m that are defined similarly, but with the first and the last set in the corresponding sequence of i sets deleted. We can now continue as before. Define Fi = { f ∈ C() : f is constant on each set Ci,m }
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and let Fi be the i −n i neighborhood of Fi in C(). Notice that for each m, Fi i≥m
is an open and dense subset of C(). This follows from (9) and (10) since every f ∈ C() is uniformly continuous. Thus, F= Fi m≥1 i≥m
is a dense G δ subset of C(). If f ∈ F, there is a sequence i k → ∞ such that f ∈ Fik . For each k, f is within −n i
i k k of being constant on each set Cik ,m . Recall that this set is the union of i sets in arithmetic progression relative to T . If we instead consider C˜ ik ,m , we can go forward and backward one period and hence, by construction, this is exactly the Gordon condition at this level. Thus, it only remains to show that almost every ω ∈ belongs to infinitely many C˜ ik ,m . This, however, follows from the measure estimates obtained above and the Borel-Cantelli Lemma. 3. Repetition Properties for Shifts and Skew-Shifts In this section we consider minimal shifts and skew-shifts and identify those cases that obey (TRP), (MRP), and (GRP). That is, we prove Theorems 3 and 4. Proof of Theorem 3. By assumption, the orbit of 0 ∈ Td is dense. In particular, we can define qk → ∞ such that T qk (0) is closer to 0 than any point T n (0), 1 ≤ n < qk . In particular, for every ε > 0 and every r > 0, there is k(ε, r ) such that for k ≥ k(ε, r ), T qk is a shift on Td with a shift vector of length bounded by ε. The repetition property now follows for the forward orbit of any choice of ω ∈ Td . Thus, (GRP) is satisfied. Remark. The proof only used that the shift on the torus is an isometry. Thus, the result extends immediately to any minimal isometry of a compact metric space. Proof of Theorem 4. Iterating the skew-shift n times, we find T n (ω1 , ω2 ) = (ω1 + 2nα, ω2 + 2nω1 + n(n − 1)α). Thus, T n+q (ω1 , ω2 ) − T n (ω1 , ω2 ) = (2qα, 2qω1 + q 2 α + 2nqα − qα).
(11)
(i) ⇒ (ii): Assume that α is not badly approximable. This means that there is some sequence qk → ∞, such that lim qk αqk = 0.
k→∞
(12)
Let (ω1 , ω2 ) ∈ T2 , ε > 0, and r > 0 be given. We will construct a sequence q˜k → ∞ so that for 1 ≤ n ≤ r q˜k , (2q˜k α, 2q˜k ω1 + q˜k2 α + 2n q˜k α − q˜k α)
(13)
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is of size O(ε). Each q˜k will be of the form m k qk for some m k ∈ {1, 2, . . . , ε−1 + 1}. It follows from (12) that in (13), every term except 2q˜k ω1 goes to zero as k → ∞, regardless of the choice of m k , and hence is less than ε for k large enough. To treat the remaining term, we can just choose m k in the specified ε-dependent range so that 2q˜k ω1 = m k (2qk ω1 ) is of size less than ε as well. Consequently, by (11), the orbit of (ω1 , ω2 ) has the repetition property. Since (ω1 , ω2 ) was arbitrary, it follows that (GRP) holds. (ii) ⇒ (iii): This is immediate. (iii) ⇒ (iv): This is immediate. (iv) ⇒ (i): Assuming (TRP), we see that there is a point ω such that {T n ω}n≥0 has the repetition property. In particular, by (11), we see that for every ε > 0, there are qk → ∞ so that 2q1 ω1 + qk2 α + 2nqk α − qk α < ε for 0 ≤ n ≤ qk .
(14)
Evaluating this for n = 0, we find that 2q1 ω1 + qk2 α − qk α < ε. Now vary n. Each time we increase n, we shift in the same direction by 2qk α. If ε > 0 is sufficiently small, it follows from the estimate (14) that we cannot go around the circle completely and hence we have 2nqk α = n2qk α for every 0 ≤ n ≤ qk . We find that αqk qεk , which shows that α is not badly approximable. 4. Further Results and Comments 4.1. Interval exchange transformations. Of course, it is interesting to explore the validity of the various repetition properties for other underlying dynamical systems. In this subsection we will briefly discuss interval exchange transformations as these dynamical systems have been studied in the context of ergodic Schrödinger operators before; see [22] for references. An interval exchange transformation is defined as follows. Let m > 1 be a fixed integer and denote m = {λ ∈ Rm : λ j > 0, 1 ≤ j ≤ m} and, for λ ∈ m ,
0 β j (λ) = j
j = 0, i=1 λi 1 ≤ j ≤ m,
I λj = [β j−1 (λ), β j (λ)), |λ| =
m
λi ,
i=1
I λ = [0, |λ|). Denote by Sm the group of permutations on {1, . . . , m}, and set λπj = λπ −1 ( j) for λ ∈ m and π ∈ Sm . With these definitions, the (λ, π )-interval exchange map Tλ,π is given by Tλ,π : I λ → I λ , x → x − β j−1 (λ) + βπ( j)−1 (λπ ) for x ∈ I λj , 1 ≤ j ≤ m.
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A permutation π ∈ Sm is called irreducible if π({1, . . . , k}) = {1, . . . , k} implies k = m. We denote the set of irreducible permutations by Sm0 . Let π ∈ Sm0 . Then, for Lebesgue almost every λ ∈ m , (I λ , Tλ,π ) is strictly ergodic. The minimality statement follows from Keane’s work [27]. The unique ergodicity statement was shown by Masur [30] and Veech [33]; for a simpler proof of this result, see [6,7,35]. When unique ergodicity holds, it is clear that the unique Tλ,π -invariant Borel probability measure must be given by the normalized Lebesgue measure on I λ , denoted by Leb. It is therefore natural to ask whether (I λ , Tλ,π , Leb) satisfies (MRP). We have the following result: Theorem 5. Let π ∈ Sm0 . Then for Lebesgue almost every λ ∈ m , (I λ , Tλ,π ) is strictly ergodic and (I λ , Tλ,π , Leb) satisfies (MRP). The proof of this theorem will rely on the following result due to Veech; see [34, Theorem 1.4]. Theorem 6 (Veech 1984). Let π ∈ Sm0 . For Lebesgue almost every λ ∈ m and every ε > 0, there are q ≥ 1 and an interval J ⊆ I λ such that l J = ∅, 1 ≤ l < q, (i) J ∩ Tλ,π l J , 0 ≤ l < q, (ii) Tλ,π is linear on Tλ,π q−1 l (iii) Leb( l=0 Tλ,π J ) > 1 − ε, q (iv) Leb(J ∩ Tλ,π J ) > (1 − ε)Leb(J ).
Proof of Theorem 5. Consider a λ from the full measure subset of m such that (I λ , Tλ,π ) is strictly ergodic and all the consequences listed in Theorem 6 hold. We claim that (I λ , Tλ,π , Leb) satisfies (MRP). λ of points ω ∈ I λ for which there exists For ε > 0 and r > 0, consider the set Iε,r k+q
k ω, T q ∈ Z+ such that dist(Tλ,π λ,π ω) < ε for k = 0, 1, 2, . . . , rq . By Theorem 6, we have λ ) r ε. Leb(I λ \ Iε,r
Thus, by Borel-Cantelli, the set j,r ∈Z+
I2λ− j ,r
has full Lebesgue measure. Since this set is contained in P R P(I λ , Tλ,π ), it follows that (I λ , Tλ,π , Leb) satisfies (MRP). Remarks. (a) The astute reader may point out that we have defined (MRP) only for homeomorphisms, and an interval exchange transformation is in general discontinuous. This, however, can be remedied in two ways. The first is to pass to a symbolic setting, where we code an orbit by the sequence of the exchanged intervals it hits. The standard shift transformation on this sequence space over a finite alphabet (of cardinality m) is then a homeomorphism and we can work in this representation; compare, for example, [27, Sect. 5]. Notice that in the strictly ergodic situation, the spectral consequences for the associated Schrödinger operator family are the same
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M. Boshernitzan, D. Damanik
because the unique invariant measure in the symbolic setting is the push-forward of the Lebesgue measure. The other way to circumvent this issue is to extend our result relating (MRP) to the almost sure absence of eigenvalues for the Schrödinger operators associated with certain discontinuous maps T . (b) In light of the previous remark, it is interesting to point out the following. The metric repetition property (MRP) is not invariant under the passage to the symbolic setting. Indeed, while we showed that every irrational rotation of the circle obeys (MRP), its symbolic counterpart (a Sturmian sequence, resulting from the symbolic coding of an exchange of two intervals) satisfies (MRP) if and only if the rotation number has unbounded partial quotients; see, for example, [10,23,31]. 4.2. Global repetition does not imply global Gordon. In this subsection we prove the following theorem, which we learned from one of the anonymous referees. Denote by M the set of real irrational numbers whose continued fraction approximants qpkk satisfy lim sup k→∞
log qk+1 < ∞. qk
(15)
Note that M is a set of full Lebesgue measure containing all Diophantine (and in particular algebraic) numbers. Theorem 7. Let α ∈ M. If f ∈ C(T) is such that Vω (n) = f (ω + nα) is a Gordon potential for every ω ∈ T, then f is constant. Since we know that (GRP) holds in the setting of the theorem above, it follows that the answer to the question whether (GRP) implies that for generic f , every Vω is a Gordon potential, is a resounding no. In the proof of Theorem 7 we will need the following simple lemma. Lemma 1. Suppose f ∈ C(T) satisfies | f (ω + h) − f (ω)| =0 h→0+ h for every ω ∈ T. Then f is constant. lim inf
(16)
Proof. Let f satisfying the assumption of the lemma be given. Assume that f is nonconstant. Then there are ω0 , ω1 ∈ T such that ω1 = ω0 + γ , γ ∈ (0, 1) and f (ω1 ) − f (ω0 ) = d > 0. Consider on the interval I = {ω0 + tγ : 0 ≤ t ≤ 1} ⊂ T the modified function d f˜(ω) = f (ω) − f (ω0 ) − (ω − ω0 ). γ Then, f˜(ω0 ) = f˜(ω1 ) = 0 and, by (16), we have lim inf h→0+
d f˜(ω + h) − f˜(ω) ≤− <0 h γ
(17)
for every ω in the interior of I . Assume that f˜ takes a strictly negative value on I . Then, by continuity, it attains a strictly negative minimum on I . At the point ω where the minimum is taken, we cannot have (17); contradiction. Thus, f˜ is non-negative on I . From this it follows in turn that (16) fails at ω0 ; contradiction. We conclude that f must be constant.
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Proof of Theorem 7. Let α ∈ T and f ∈ C(T) satisfy the assumptions of the theorem. We have to show that f is constant. So let us assume that f is non-constant and, without loss of generality,4 f has no non-integer period. Let ω ∈ T be given. By assumption, Vω is a Gordon potential and hence there are positive integers m k → ∞ such that ∀ C > 0 : lim
max |Vω (n) − Vω (n ± m k )|C m k = 0.
k→∞ 1≤n≤m k
In other words, ∀ C > 0 : lim
max | f (ω + nα) − f (ω + nα ± m k α)|C m k = 0.
k→∞ 1≤n≤m k
By passing to a subsequence if necessary, we can assume without loss of generality that m k α converges in T to a limit L(ω). Taking k → ∞, we see that f is L(ω)-periodic. Since f has no non-integer period, it follows that L(ω) = 0. By choosing the appropriate sign, we can further assume without loss of generality that m k α converges to L(ω) = 0 from the right. By well-known general lower bounds for the distance of m k α from zero, the statement (16) follows (for ω + α, to be exact). In this step we used the assumption (15). Since ω ∈ T was arbitrary, we have (16) at every point and hence, by the lemma, f is constant. 4.3. Uniformity in the coupling constant. One often introduces a coupling constant λ and considers potentials of the form Vω (n) = λ f (T n ω)
(18)
instead of (2). Since the regimes of small and large couplings can be regarded as small perturbations of simple models (the free Laplacian and a diagonal matrix, respectively), it is of especial interest to explore whether the spectral type of the limit model extends to the perturbation. As noted immediately after giving the definition of a Gordon potential above, the Gordon property is invariant under a variation of the coupling constant. That is, if Vω is a Gordon potential for one non-zero value of λ, it is a Gordon potential for all values of λ. In particular, the results on the absence of point spectrum above extend from potentials of the form (2) covered by our work to all potentials of the more general form (18). 4.4. Generic singular continuous spectrum. Our work is closely related in spirit to the paper [2] by Avila and Damanik. It follows from [2] that there exists a residual set Fsing ⊆ C() such that for every f ∈ Fsing , the operator (1) with potential (2) has purely singular spectrum for almost every ω ∈ , and moreover, for Lebesgue almost every λ ∈ R, the operator (1) with potential (18) has purely singular spectrum for almost every ω ∈ . It is well known that discrete one-dimensional Schrödinger operators with periodic potentials have purely absolutely continuous spectrum. Thus, the results obtained in [2] and the present paper are rather strong implementations of the philosophy that absolutely continuous spectrum requires the presence of perfect repetition, while a rather weak repetition property already ensures the absence of eigenvalues. 4 If f does have a non-integer period, we pass to a smaller 1-torus. That is, we “remove” the period of f . This adjustment does not alter the condition (15) on α.
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Explicitly, the following consequence is obtained by combining the results on the absence of eigenvalues and the results on the absence of absolutely continuous spectrum. Theorem 8. Suppose that (, T, µ) satisfies (MRP). Then, for f ’s from a residual subset of C(), the operator (1) with potential (2) has purely singular continuous spectrum for µ almost every ω ∈ , and moreover, the operator (1) with potential (18) has purely singular continuous spectrum for µ almost every ω ∈ and Lebesgue almost every λ ∈ R. Recall that by Theorems 3 and 4, Theorem 8 applies to all minimal shifts of the form (3) and minimal skew-shifts of the form (4), where α is not badly approximable (which is satisfied by Lebesgue almost every α). 4.5. A remark on one-frequency quasi-periodic models. Here we discuss a technical point that appears to be remarkable in comparison with earlier studies of Gordon potentials generated by shifts on T (a.k.a. rotations of the circle). There are three truly distinct classes of sampling functions that lead to technically very different theories on the level of Schrödinger operators: piecewise constant functions with finitely many discontinuities (a.k.a. codings of rotations), continuous functions, and smooth functions. Very roughly speaking, one-frequency quasi-periodic models with piecewise constant sampling functions seem to have, as a rule, purely singular continuous spectrum, whereas one-frequency quasi-periodic models with smooth sampling functions seem to have purely absolutely continuous spectrum for small coupling and pure point spectrum for large coupling; see, for example, [15,17,22] and references therein. Continuous sampling functions fall between these two classes and in some sense, it is natural to use an approximation of a given continuous function from either side (i.e., by piecewise continuous functions or by smooth functions) in the study of such potentials. While it had been expected that models with continuous sampling function behave similarly to models with smooth sampling functions, recent work (especially [2] and the present paper) has shown that in fact they behave generically like models with piecewise continuous step functions. The technical point we would like to make is the following. While [2] indeed used approximation by piecewise constant functions as a key tool in the proof, in this paper we did not! We were able to show a very general result: by Theorems 2 and 3 we see that for any irrational shift on T, a generic continuous sampling will almost surely generate a Gordon potential. It is clear from the proof that much stronger Gordon-type conditions (with more repetitions and smaller error estimates) can be obtained in the same way. This is surprising in the case of a badly approximable α. It can be shown, [10], that for any piecewise constant function with finitely many discontinuities (that is not globally constant), such Gordon-type repetition properties do not hold for badly approximable α. Consequently, the general result just described cannot be obtained by approximation with piecewise constant functions. Of course, for smooth functions, badly approximable α do not generate Gordon potentials either. Thus, for shifts on T by a badly approximable α, the generic Gordon property in the continuous category is a novel feature. 4.6. Some remarks on the repetition properties. It is a natural question whether there are any non-trivial relations between the topological, metric, and global repetition properties. For example, as was pointed out earlier, in all the examples we have considered
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up to this point in the present paper, the properties (TRP), (MRP), and (GRP) either hold or fail simultaneously, so one could ask if there is a general principle. To exhibit an example satisfying (MRP) but not (GRP), take α with unbounded partial quotients and consider the natural coding of the shift by α on T with respect to the partition [0, α) and [α, 1) of T. The resulting Sturmian subshift satisfies (MRP) by [10,23] and it does not satisfy (GRP) by [4, Prop. 2.1]. Similarly, there are examples satisfying (TRP) but not (MRP). Here we only describe a procedure that generates the desired examples; we refer the reader to [9], where complete proofs and a much more detailed discussion can be found. Consider skew-products in the spirit of (4) on higher dimensional tori. That is, let = Td and let, for example, T : → be given by T (ω1 , ω2 , ω3 , . . . , ωd ) = (ω1 + α, ω1 + ω2 , ω2 + ω3 , . . . , ωd−1 + ωd ), where α is irrational. Then, (, T ) is minimal and normalized Lebesgue measure is the unique invariant probability measure; see, for example, [24]. If d ≥ 4, (, T, Leb) does not satisfy (MRP) for any α. On the other hand, if α is sufficiently well approximated by rational numbers, (, T ) does satisfy (TRP). Again, see [9] for proofs of these two statements in a more general context. We conclude with an interesting connection between the metric repetition property and the notion of metric entropy. It follows from [32] (see also [8]) that if (, T ) is a subshift over a finite set A and µ is an ergodic probability measure such that (, T, µ) satisfies (MRP), then it has metric entropy zero. In view of Theorem 2, this observation is nicely in line with the general expectation that positive entropy suggests positive Lyapunov exponents and localization for the associated Schrödinger operators, at least in an almost-everywhere sense. Acknowledgements. We are indebted to the anonymous referees for useful comments which lead to several improvements of the paper.
References 1. Avila, A., Bochi, J., Damanik, D.: Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. http://arxiv.org/abs/0709.2667, 2007, to appear in Duke Math. J. 2. Avila, A., Damanik, D.: Generic singular spectrum for ergodic Schrödinger operators. Duke Math. J. 130, 393–400 (2005) 3. Avron, J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc. 6, 81–85 (1982) 4. Berthé, V., Holton, C., Zamboni, L.: Initial powers of Sturmian sequences. Acta Arith. 122, 315–347 (2006) 5. Bjerklöv, K.: Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent. Geom. Funct. Anal. 16, 1183–1200 (2006) 6. Boshernitzan, M.: A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52, 723–752 (1985) 7. Boshernitzan, M.: A condition for unique ergodicity of minimal symbolic flows. Ergod. Th. & Dynam. Sys. 12, 425–428 (1992) 8. Boshernitzan, M.: Quantitative recurrence results. Invent. Math. 113, 617–631 (1993) 9. Boshernitzan, M., Damanik, D.: The repetition property for sequences on tori generated by polynomials or skew-shifts. http://arxiv.org/abs/0708.3234, 2007, to appear in Israel J. Math. 10. Boshernitzan, M., Damanik, D.: Pinned repetitions in symbolic flows. In preparation 11. Bourgain, J.: On the spectrum of lattice Schrödinger operators with deterministic potential. J. Anal. Math. 87, 37–75 (2002) 12. Bourgain, J.: On the spectrum of lattice Schrödinger operators with deterministic potential II. J. Anal. Math. 88, 221–254 (2002)
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13. Bourgain, J.: Estimates on Green’s functions, localization and the quantum kicked rotor model. Ann. of Math. 156, 249–294 (2002) 14. Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Annals of Mathematics Studies, 158. Princeton: Princeton, NJ: University Press, 2005 15. Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. of Math. 152, 835–879 (2000) 16. Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for Schrödinger operators on Z with potentials given by the skew-shift. Commun. Math. Phys. 220, 583–621 (2001) 17. Bourgain, J., Jitomirskaya, S.: Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148, 453–463 (2002) 18. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Boston: Birkhäuser, 1990 19. Chan, J., Goldstein, M., Schlag, W.: On non-perturbative Anderson localization for C α potentials generated by shifts and skew-shifts. http://arxiv.org/list/math/0607302, 2006 20. Cornfeld, I., Fomin, S., Sina˘ı, Ya.: Ergodic Theory, Grundlehren der Mathematischen Wissenschaften 245, New York: Springer-Verlag, 1982 21. Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, Berlin: Springer-Verlag, 1987 22. Damanik, D.: Strictly ergodic subshifts and associated operators, In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proceedings of Symposia in Pure Mathematics 74, Providence, RI: Amer. Math. Soc. pp. 505–538 23. Delyon, F., Petritis, D.: Absence of localization in a class of Schrödinger operators with quasiperiodic potential. Commun. Math. Phys. 103, 441–444 (1986) 24. Furstenberg, H.: Strict ergodicity and transformation of the torus. Amer. J. Math. 83, 573–601 (1961) 25. Gordon, A.: On the point spectrum of the one-dimensional Schrödinger operator. Usp. Math. Nauk. 31, 257–258 (1976) 26. Jitomirskaya, S., Simon, B.: Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators. Commun. Math. Phys. 165, 201–205 (1994) 27. Keane, M.: Interval exchange transformations. Math. Z. 141, 25–31 (1975) 28. Khintchine, A.: Continued Fractions. Mineola, NY: Dover, 1997 29. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999) 30. Masur, H.: Interval exchange transformations and measured foliations. Ann. of Math. 115, 168–200 (1982) 31. Mignosi, F.: Infinite words with linear subword complexity. Theoret. Comput. Sci. 65, 221–242 (1989) 32. Ornstein, D., Weiss, B.: Entropy and data compression schemes. IEEE Trans. Inform. Theory 39, 78–83 (1993) 33. Veech, W.: Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. 115, 201–242 (1982) 34. Veech, W.: The metric theory of interval exchange transformations. I. Generic spectral properties. Amer. J. Math. 106, 1331–1359 (1984) 35. Veech, W.: Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation. Ergod. Th. & Dynam. Sys. 7, 149–153 (1987) Communicated by B. Simon
Commun. Math. Phys. 283, 663–674 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0581-7
Communications in
Mathematical Physics
A Geometric Construction of the Exceptional Lie Algebras F4 and E8 José Figueroa-O’Farrill Maxwell Institute and School of Mathematics, University of Edinburgh, Edinburgh, UK. E-mail: [email protected] Received: 6 September 2007 / Accepted: 28 March 2008 Published online: 18 July 2008 – © Springer-Verlag 2008
Abstract: We present a geometric construction of the exceptional Lie algebras F4 and E 8 starting from the round spheres S 8 and S 15 , respectively, inspired by the construction of the Killing superalgebra of a supersymmetric supergravity background.
1. Introduction The Killing–Cartan classification of simple Lie algebras over the complex numbers is well known: there are four infinite families An≥1 , Bn≥2 , Cn≥3 and Dn≥4 , with the range of ranks chosen to avoid any overlaps, and five exceptional cases G 2 , F4 , E 6 , E 7 and E 8 . Whereas the classical series ( A-D) correspond to matrix Lie algebras, and indeed their compact real forms are the Lie algebras of the special unitary groups over C (A), H (B) and R (C and D), the exceptional series do not have such classical descriptions; although they can be understood in terms of more exotic algebraic structures such as octonions and Jordan algebras. There is, however, a uniform construction of all exceptional Lie algebras (except for G 2 ) using spin groups and their spinor representations, described in Adams’ posthumous notes on exceptional Lie groups [1] and, for the special case of E 8 , also in [2]. This construction, once suitably geometrised, is very familiar to practitioners of supergravity. The purpose of this note is to present this geometrisation, perhaps as an invitation for differential geometers to think about supergravity. Indeed in supergravity there is a geometric construction which associates a Lie superalgebra to any supersymmetric supergravity background: typically a lorentzian spin manifold with extra geometric data and with a notion of privileged spinor fields, called Killing spinors. The resulting superalgebra is called the Killing superalgebra because it is constructed out of these Killing spinors and Killing vectors. The Killing superalgebra for general ten- and eleven-dimensional supergravities is constructed in [3,4]. In this note we will apply this construction not to supergravity backgrounds, but to riemannian manifolds without any additional structure. The relevant notion of Killing spinor is then
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that of a geometric Killing spinor: a nonzero section ε of the spinor bundle satisfying ∇X ε =
1 X · ε, 2
where X is any vector field and the dot means the Clifford action. We will apply this construction to the unit spheres S 7 ⊂ R8 , S 8 ⊂ R9 and S 15 ⊂ R16 and in this way obtain the compact real Lie algebras so9 , f4 and e8 , respectively. It is curious that these three spheres are linked by the exceptional Hopf fibration which defines the octonionic projective line, S7
/ S 15 S8
and it is natural to wonder whether their Killing superalgebras are similarly related. We will not answer this question here. This note is organised as follows. In Sect. 2 we briefly review the relevant notions of Clifford algebras, spin groups and their spinorial representations. In Sect. 3 we define the Killing superalgebra after introducing the basic notions of Killing spinors and Bär’s cone construction. In Sect. 4 we construct the Killing superalgebras of the round spheres S 7 , S 8 and S 15 and show that they are isomorphic to the compact real Lie algebras so9 , f4 and e8 , respectively. Finally in Sect. 5 we discuss some open questions motivated by the results presented here.
2. Spinorial Algebra In this section we start with some algebraic preliminaries on euclidean Clifford algebras and spinors in order to set the notation. We will be sketchy, but fuller treatments can be found, for example, in [5–7].
2.1. Clifford algebras and Clifford modules. Let V be a finite-dimensional real vector space with a positive-definite euclidean inner product −, −. The Clifford algebra C(V ) is the associated algebra with unit generated by V and the identity 1 subject to the Clifford relations v 2 = − v, v 1
(1)
for all v ∈ V . More formally, the Clifford algebra is the quotient of the tensor algebra of V by the two-sided ideal generated by the Clifford relations. Since the Clifford relations— having terms of degree 0 and degree 2—are not homogeneous in the natural grading of the tensor algebra, C(V ) is not graded but only filtered. The associated graded algebra is the exterior algebra ΛV , to which it is isomorphic as a vector space. Nevertheless since the terms in (1) have even degree, C(V ) is Z2 graded C(V ) = C(V )0 ⊕ C(V )1 ,
(2)
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Table 1. Clifford algebras Cn , where K(m) denotes the algebra of m × m matrices with entries in K. n Cn
0 R
1 C
2 H
3 H⊕H
4 H(2)
5 C(4)
6 R(8)
7 R(8) ⊕ R(8)
with vector-space isomorphisms C(V )0 ∼ = Λeven V and C(V )1 ∼ = Λodd V . These isomorphisms can be seen explicitly as follows. Relative to an orthonormal basis ei for V , the Clifford relations become ei e j + e j ei = −2δi j 1,
(3)
which shows that up to terms in lower order we may always antisymmetrise any product ei1 ei2 . . . eik in C(V ) without ever changing the parity. The Clifford algebra of Rn generated by 1 and ei subject to (3) is denoted Cn . As a real associative algebra with unit it is isomorphic to one or two copies of matrix algebras, as shown in Table 1 for n ≤ 7. The higher values of n are obtained by Bott periodicity Cn+8 ∼ = Cn ⊗ R(16), where R(16) is the algebra of 16 × 16 real matrices. Since matrix algebras have a unique irreducible representation (up to isomorphism), we can easily read off the irreducible representations of Cn from the table. We see, in particular, that if n is even there is a unique irreducible representation, which is real for n ≡ 0, 6 (mod 8) or quaternionic when n ≡ 2, 4 (mod 8); whereas if n is odd there are two inequivalent irreducible representations, which are real when n ≡ 7 (mod 8) and quaternionic when n ≡ 3 (mod 8), and form a complex conjugate pair for n ≡ 1 (mod 4). These two inequivalent Clifford modules are distinguished by the action of ω := e1 e2 · · · en , which for n odd is central in Cn . This element obeys ω2 = (−1)n(n+1)/2 1, whence it is a complex structure for n ≡ 1 (mod 4), in agreement with the table. The dimension of one such irreducible Clifford module, relative to either R if real or C if not, is 2 n/2 . We will use the notation M for the unique irreducible Clifford module in even dimension, M± for the irreducible Clifford modules for n ≡ 3 (mod 4). For n ≡ 1 (mod 4) we will let M denote the irreducible Clifford module on which ω acts like +i and let M denote the irreducible module on which ω acts like −i. 2.2. The spin group and spinor modules. The Clifford algebra C(V ) admits a natural Lie algebra structure via the Clifford commutator. The map Λ2 V → C(V ) given by 1 ei ∧ e j → − ei e j , 2
(4)
for i < j, induces a Lie algebra homomorphism ρ : so(V ) → C(V ). Moreover the action of so(V ) on V is realised by the Clifford commutator, so that if A ∈ so(V ) and v ∈ V , then A(v) = ρ(A)v − vρ(A) ∈ C(V ).
(5)
Exponentiating the image of ρ in C(V ) we obtain a connected Lie group called Spin(V ). The subspace V ⊂ C(V ) is closed under conjugation by Spin(V ) whence we obtain a map Spin(V ) → SO(V ), whose kernel is the central subgroup consisting of ±1. Restricting an irreducible Clifford module M (or M± ) to Spin(V ) we obtain a spinor module, which may or may not remain irreducible. Since Spinn ⊂ (Cn )0 ∼ = Cn−1 , we can immediately infer the type of spinor module from Table 1. If n ≡ 1 (mod 8),
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M∼ = S ⊗ C is the complexification of the unique irreducible real spinor module S, whereas if n ≡ 5 (mod 8), M ∼ = S, but S possesses a Spinn -invariant quaternionic structure, whence M ∼ = S as well. For n ≡ 3 (mod 4), M± ∼ = S. For odd n, the spinor module is real if n ≡ 1, 7 (mod 8) and quaternionic otherwise and its dimension (over R if real and over C otherwise) is again 2(n−1)/2 . For even n, the unique irreducible Clifford module decomposes (perhaps after complexification) into two inequivalent Spinn modules, called half- (or chiral) spinor modules. They are denoted S± if n ≡ 0 (mod 4) and S and S if n ≡ 2 (mod 4). They are real if n ≡ 0 (mod 8), quaternionic if n ≡ 4 (mod 8) and complex otherwise. If n ≡ 6 (mod 8) then it is the complexification of M which decomposes M ⊗ C ∼ = S ⊕ S. In all cases, the dimension, computed relative to the appropriate field for the type, is 2(n−2)/2 . 2.3. Spinor inner products. The Clifford algebra C(V ) has a natural antiautomorphism defined by − id V on V . On a given irreducible Clifford module M (or M± ) there always exists an inner product (−, −) which realises this automorphism; that is, such that (v · ε1 , ε2 ) = − (ε1 , v · ε2 ) , for all v ∈ V and εi ∈ M. It follows that (−, −) is Spin(V )-invariant; indeed, ei e j · ε1 , ε2 = − ε1 , ei e j · ε2 .
(6)
(7)
In positive-definite signature, (−, −) is either symmetric or hermitian, depending on the type of representation, and positive-definite [7]. The transpose of the Clifford action V ⊗ M → M relative to the above inner product on M and the euclidean inner product −, − on V , defines a map which we suggestively denote by [−, −] : M ⊗ M → V . Explicitly, we have that for all v ∈ V and εi ∈ M, [ε1 , ε2 ], v = (ε1 , v · ε2 ) .
(8)
3. The Killing Superalgebra In this section we will define the Killing superalgebra of a riemannian spin manifold admitting Killing spinors. 3.1. Spin manifolds. Let (M, g) be an n-dimensional riemannian manifold and let O(M) denote the bundle of orthonormal frames. It is a principal On -bundle. If the manifold is orientable, we can restrict ourselves consistently to oriented orthonormal frames. In this case, the subbundle SO(M) of oriented orthonormal frames is a principal SOn -bundle. The obstruction to orientability is measured by the first Stiefel–Whitney class w1 ∈ H 1 (M; Z2 ). If (M, g) is orientable one can ask whether there is a principal Spinn -bundle Spin(M) lifting the oriented orthonormal frame bundle SO(M); that is, admitting a bundle map Spin(M) → SO(M) covering the identity and restricting fibrewise to the natural homomorphism Spinn → SOn . The obstruction to the existence of such a lift is measured by the second Stiefel–Whitney class w2 ∈ H 2 (M; Z2 ) and, if it vanishes, the manifold (M, g) is said to be spin. Spin structures Spin(M) on M need not be unique: they are measured by H 1 (M; Z2 ) = Hom(π1 M, Z2 ), which we can understand as assigning a sign (consistently) to every noncontractible loop. In this section we
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will assume our manifolds to be spin and that a choice of spin structure has been made. The main examples in this note are spheres, which are spin—indeed, the total space of the spin bundle of S n is the Lie group Spinn+1 —and, since they are simply-connected, have a unique spin structure. If M is a Cn -module, then it is also a (perhaps reducible) Spinn -module and we may form the spinor bundle S(M) := Spin(M) ×Spinn M over M as an associated vector bundle to the spin bundle. Furthermore we have a fibrewise action of the Clifford bundle C(T M) on S(M). The spinor inner products globalise to give an inner product on S(M). The Levi-Cività connection on the orthonormal frame bundle of (M, g) induces a connection on Spin(M) and hence on any associated vector bundle. In particular we have a spin connection on S(M): ∇ : Γ (S(M)) → Ω 1 (M; S(M)), allowing us to write down interesting equations on spinors. One such equation is the Killing spinor equation, which is the subject of the next section. A classic treatise on this equation is [8]. 3.2. Killing spinors. Throughout this section we will let (M n , g) be a spin manifold with chosen spinor bundle S(M) on which we have a fibrewise action of the Clifford bundle C(T M) and a Spinn -invariant inner product which in addition satisfies Eq. (6). A nonzero ε ∈ Γ (S(M)) is said to be a (real) Killing spinor if for all vector fields X , ∇ X ε = λX · ε,
(9)
where λ ∈ R is the Killing constant. The origin of the name is that if εi , i = 1, 2, are Killing spinors, then the vector field V := [ε1 , ε2 ] defined by Eq. (8) is a Killing vector. Indeed, for all vector fields X, Y , g(∇ X V, Y ) = (∇ X ε1 , Y · ε2 ) + (ε1 , Y · ∇ X ε2 ) = λ (X · ε1 , Y · ε2 ) + λ (ε1 , Y · X · ε2 ) = −λ (ε1 , X · Y · ε2 ) + λ (ε1 , Y · X · ε2 ) ,
(by definition of ∇) (using Eq (9)) (using Eq. (6))
which is manifestly skewsymmetric in X, Y , whence we conclude that g(∇ X V, Y ) + g(∇Y V, X ) = 0, which is one form of Killing’s equation. 3.3. The cone construction. The problem of determining which riemannian manifolds admit real Killing spinors was the subject of much research until it was elegantly solved by Bär [9] via the cone construction. We will assume that the Killing constant λ has been set to ± 21 by rescaling the metric, if necessary. Let (M, g) denote the (deleted) cone over M, defined by M = R+ × M and g = dr 2 + r 2 g, where r > 0 is the coordinate on R+ . Bär observed that there is a one-to-one correspondence between Killing spinors on M and parallel spinors on the cone M. More precisely, if n = dim M is even, there is an
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isomorphism between Killing spinors on M with Killing constant ± 21 and parallel spinors on M; the choice of sign having to do with the choice of embedding Cn ⊂ Cn+1 . If on the other hand n is odd, then the space of Killing spinors on M with Killing constant ± 21 is isomorphic to the space of parallel half-spinors on M, the chirality depending on the sign of the Killing constant. Together with a theorem of Gallot [10] which says that the cone of a complete manifold is either flat or irreducible, the above observation reduces the problem of determining the complete riemannian manifolds admitting real Killing spinors to a holonomy problem which was solved by Wang in [11]. If (M, g) is not complete, its cone may be reducible, but if so it can be shown to be locally a product of subcones and applying Bär’s results to each of the subcones allows one to write local forms for the metrics on M in terms of (double) warped products [12]. For example, in the case of M = S n , the cone is M = Rn+1 \ {0}, but the metric extends smoothly to the origin. The space of parallel (half-)spinors on Rn+1 is isomorphic to the relevant (half-)spinor representation of Spinn+1 . 3.4. The Killing superalgebra. To a riemannian manifold admitting real Killing spinors we may associate an algebraic structure called the Killing superalgebra which extends the Lie algebra of isometries in the following way. The underlying vector space is k = k0 ⊕k1 , where k0 is the Lie algebra of isometries and k1 is the space of Killing spinors with λ = 21 . (There is a similar story for λ = − 21 .) The bracket on k consists of three pieces: the Lie bracket on k0 , a map k0 ⊗ k1 → k1 and a map k1 ⊗ k1 → k0 . Depending on dimension and signature, the latter map may be symmetric or antisymmetric, whence the resulting bracket might correspond (if the Jacobi identity is satisfied) to a Lie algebra or a Lie superalgebra. In the riemannian examples in this section we will recover Lie algebras, but in the lorentzian examples common in supergravity the similar construction leads to Lie superalgebras. Let us now define these maps. The map k1 ⊗ k1 → k0 is induced from the algebraic map [−, −] in Eq. (8), which explains the notation. As we saw before the image indeed consists of Killing vector fields. The map k0 ⊗ k1 → k1 is given by the spinorial Lie derivative of Lichnerowicz and Kosmann(-Schwarzbach) [13] and which we now define. If X is a vector field on M, then let A X : T M → T M denote the endomorphism of the tangent bundle defined by A X Y = −∇Y X , for ∇ the Levi-Cività connection. The vector field X is Killing if and only if A X is skewsymmetric relative to the metric; that is, if and only if A X ∈ so(T M). Let ρ : so(T M) → End(S(M)) denote the spin representation and define the spinorial Lie derivative along a Killing vector X by L X = ∇ X + ρ(A X ).
(10)
In fact, this Lie derivative makes sense on sections of any vector bundle associated to the orthonormal frame bundle provided that we substitute ρ by the relevant representation. For instance, on the tangent bundle itself, we have L X Y = ∇ X Y + A X Y = ∇ X Y − ∇Y X = [X, Y ], as expected. The spinorial Lie derivative satisfies the following properties for all Killing vectors X, Y , spinors ε, functions f and arbitrary vector fields Z : • L X is a derivation, so that L X ( f ε) = X ( f )ε + f L X ε;
(11)
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• X → L X is a representation of the Lie algebra of Killing vector fields: L X LY − LY L X = L[X,Y ] ;
(12)
• L X is compatible with Clifford multiplication: L X (Z · ε) = [X, Z ] · ε + Z · L X ε;
(13)
• and L X preserves the Levi-Cività connection: L X ∇ Z − ∇Y L X = ∇[X,Z ] .
(14)
It follows from Eqs. (13) and (14) that the Lie derivative of a Killing spinor along a Killing vector is again a Killing spinor. Indeed, let ε be a Killing spinor and let X be a Killing vector. We have for all vector fields Y that ∇Y L X ε = L X ∇Y ε − ∇[X,Y ] ε = λL X (Y · ε) − λ[X, Y ] · ε = λY · L X ε,
(using (14)) (since ε is Killing) (using (13))
as advertised. We define [−, −] : k0 ⊗ k1 → k1 by [X, ε] := L X ε. Of course, the existence of a bracket is not enough to conclude that k is Lie (super) algebra: one must also check the Jacobi identity. The Jacobi identity is the vanishing of a tensor in k ⊗ Λ3 k∗ . Since k = k0 ⊕ k1 and the bracket respects the Z2 grading, there are four components to the Jacobi identity. The component in k0 ⊗ Λ3 k0 vanishes due to the Jacobi identity of the Lie algebra k0 . The component in k1 ⊗ Λ2 k∗0 ⊗ k∗1 vanishes because of the fact that k1 is a representation of k0 ; indeed, this identity says that if X, Y ∈ k0 and ε ∈ k1 , then [X, [Y, ε]] − [Y, [X, ε]] = [[X, Y ], ε], which is precisely Eq. (12). The component in k0 ⊗ Λ2 k∗1 ⊗ k∗0 vanishes because the bracket k1 ⊗ k1 → k0 is k0 -equivariant. Indeed, if X ∈ k0 and εi ∈ k1 for i = 1, 2, then for all vector fields Y , g ([X, [ε1 , ε2 ]], Y ) = g (L X [ε1 , ε2 ], Y ) = Xg ([ε1 , ε2 ], Y ) − g ([ε1 , ε2 ], L X Y ) (since X is Killing) = X (ε1 , Y · ε2 ) − (ε1 , L X Y · ε2 ) = (L X ε1 , Y · ε2 ) + (ε1 , L X (Y · ε2 )) − (ε1 , L X Y · ε2 ) = (L X ε1 , Y · ε2 ) + (ε1 , Y · L X ε2 ) (using (13)) = g ([[X, ε1 ], ε2 ], Y ) + g ([ε1 , [X, ε2 ]], Y ) . The final component of the Jacobi identity lives in the k0 -invariant subspace of k1 ⊗Λ3 k∗1 . This identity does not seem to follow formally from the construction, but requires a caseby-case argument. In some cases it follows because there simply are no k0 -invariant tensors in k1 ⊗ Λ3 k∗1 , but this is not universal and in many cases one needs to perform an explicit calculation. Luckily, for the examples in this note, the representation-theoretic argument will suffice.
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3.5. Equivariance of the cone construction. In order to calculate or simply identify the Killing superalgebras it is often convenient to work in the cone. This requires understanding how to lift the calculation of the Lie derivative of a Killing spinor along a Killing vector to the cone. In [14] it is shown that the cone construction is equivariant under the action of the isometry group of (M, g). We will work at the level of the Lie algebra. Every Killing vector on (M, g) defines a Killing vector on the cone (M, g). Generically there are no other Killing vectors on the cone, except in the case when (M, g) is the round sphere and hence the cone is flat. Let X be a Killing vector on (M, g) and let X denote its lift to a Killing vector on the cone. Similarly let ε be a Killing spinor on (M, g) and let ε denote the parallel spinor on the cone to which it lifts. Then it is proved in [14] that L X ε = L X ε, which suggests a way to calculate the bracket [−, −] : k0 ⊗ k1 → k1 : • we lift the Killing vectors in k0 and the Killing spinors in k1 to Killing vectors and parallel spinors, respectively, on the cone; • we compute the spinorial Lie derivative there; and • we restrict the result to a Killing spinor on (M, g). Although somewhat circuitous, this procedure has the added benefit that the Lie derivative of a parallel spinor is an algebraic operation: L X ε = ρ(A X )ε. Since parallel spinors are determined by their value at any one point, we can work at a point and we see that the above formula corresponds to the restriction of the spin representation of son+1 to the subalgebra corresponding to the image of k0 in son+1 , acting on the subspace of the spinor module which is invariant under the holonomy algebra of the cone. For the case of the round spheres which will occupy us in this paper, the holonomy algebra is trivial and the isometries act linearly in the cone, whence A X = −∇ X is actually constant. Therefore the above action is precisely the standard action of k0 = son+1 on the relevant spinor module. There is no need to lift the bracket k1 ⊗ k1 → k0 to the cone, but it is possible to do this as well. The only point to notice is that in the cone we do not square parallel spinors to parallel vectors, but to parallel 2-forms, which are constructed out of the lifts of the Killing vectors on (M, g). 4. The Killing Superalgebras of S7 , S8 and S15 In this section we will exhibit the Killing superalgebras of some low-dimensional spheres S n , for n = 7, 8, 15, and will show that they are Lie algebras isomorphic to so9 , f4 and e8 , respectively. The strategy is to exploit the equivariance of the cone construction to show that these Killing algebras are isomorphic to the Lie algebras constructed in [1]. 4.1. k(S 7 ) ∼ = so9 . The isometry Lie algebra of the unit sphere in R8 is so8 , acting via linear vector fields on R8 which are tangent to the sphere. The 7-sphere admits the maximal number of Killing spinors of either sign of the Killing constant, which here is 8. Lifting them to the cone, we have so8 acting on the positive chirality spinor module
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S+ which is real and eight-dimensional. The Killing superalgebra is thus k = so8 ⊕ S+ with the following brackets: so8 ⊂ k is a Lie subalgebra, so8 ⊗S+ → S+ is the standard action and the map Λ2 S+ → so8 is the transpose of the previous map relative to the inner products on both vectors and spinors. The map is skewsymmetric as shown because the spinor inner product is symmetric. Therefore we will obtain a Lie algebra. Observe that triality says that Λ2 S+ ∼ = Λ2 V , so that this map is actually an isomorphism in this case. The Jacobi identity requires the vanishing of a trilinear map Λ3 k → k. The only component which is in question is the one in Λ3 S+ → S+ . Using the inner product on S+ we may identify this with an so8 -invariant element in S+ ⊗ Λ3 S+ , but it may be shown the only such element is the zero map. Indeed, letting S+ , S− and V have Dynkin indices [0001], [0010] and [1000], respectively, we find that Λ3 S+ is irreducible with Dynkin index [1010], corresponding to the 56-dimensional kernel of the Clifford multiplication V ⊗ S− → S+ . Finally, a roots-and-weights calculation shows that S+ ⊗ Λ3 S+ ∼ = [0020] ⊕ [0100] ⊕ [1011] ⊕ [2000], whence there is no nontrivial invariant subspace. The Lie algebra structure just defined on k is 36-dimensional and coincides with so9 . ∼ f4 . The isometry Lie algebra of the unit sphere in R9 is so9 , acting via 4.2. k(S 8 ) = linear vector fields on R9 which are tangent to the sphere. The 8-sphere admits the maximal number of Killing spinors of either sign of the Killing constant, which here is 16. Lifting them to the cone, we have so9 acting on the spinor module S which is real and sixteen-dimensional. The Killing superalgebra is k = so9 ⊕ S with the following brackets: so9 is a Lie subalgebra, so9 ⊗ S → S is the standard action of so9 on its spinor representation, and Λ2 S → so9 is the transpose of the standard action using the inner products on vectors and spinors. Since the spinor inner product is symmetric, the map is skewsymmetric as shown. This means that we will obtain a Lie algebra. The only nontrivial component of the Jacobi identity lives in the subspace of so9 -equivariant maps Λ2 S → S, or using the inner product, an so9 -invariant element of S ⊗ Λ3 S. However one can check that there are no such invariants. Indeed, since S has Dynkin index [0001], a roots-and-weights calculation shows that Λ3 S ∼ = [0101] ⊕ [1001],
(15)
where the representations on the right-hand side have dimensions 432 and 128, respectively. Indeed, [1001] is the kernel of the Clifford multiplication V ⊗S → S. Tensoring the first with S we obtain [0101] ⊗ [0001] ∼ = [0002] ⊕ [0010] ⊕ [0100] ⊕ [0102] ⊕[0110] ⊕ [0200] ⊕ [1002] ⊕ [1010] ⊕ [1100], whereas tensoring the second with S we obtain [1001] ⊗ [0001] = [0002] ⊕ [0010] ⊕ [0100] ⊕ [1000] ⊕ [1002] ⊕[1010] ⊕ [1100] ⊕ [2000]. It is plain that there are no invariants in either expression. The resulting Lie algebra has dimension 36 + 16 = 52 and can be shown [1] to be a compact real form of f4 . Unlike the case of so9 in Sect. 4.1, here Λ2 S → so9 is not an isomorphism: indeed Λ2 S ∼ = Λ2 V ⊕ Λ3 V .
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4.3. k(S 15 ) ∼ = e8 . The isometry Lie algebra of the unit sphere in R16 is so16 . The 15-sphere admits the maximal number of Killing spinors of either sign of the Killing constant, which here is 128. Lifting them to the cone, we have so16 acting on the spinor module S+ which is real and 128-dimensional. The Killing superalgebra is k = so16 ⊕S+ with the following brackets: Λ2 so16 → so16 is the Lie bracket, so16 ⊗ S+ → S+ is the action of so16 on its half-spinor representation and Λ2 S+ → so16 the transpose map using the inner products. As before, since the spinor inner product is symmetric, the map is skewsymmetric as shown. This means that we will obtain a Lie algebra. The resulting bracket can be seen to satisfy the Jacobi identity. Indeed, the only nontrivial component of the Jacobi identity defines an so16 -equivariant map Λ3 S+ → S+ . Since the inner product is non-degenerate on S+ , we can think of this as an so16 -invariant element of S+ ⊗ Λ3 S+ , but we can see that no such nontrivial element exists. Indeed, letting [00000001] denote the Dynkin index of S+ , we find that Λ3 S+ ∼ = [00001001] ⊕ [01000010] ⊕ [10000001], whence tensoring each of the modules in the right-hand side with S+ we obtain [00001001]⊗[00000001] = [00000011] ⊕ [00001000] ⊕ [00001002] ⊕ [00001100] ⊕ [00010011]⊕[00011000]⊕[00100002]⊕[00100100] ⊕ [01000011]⊕[01001000]⊕[10000002]⊕[10000100], [00000001] ⊗ [01000010] = [00000011] ⊕ [00001000] ⊕ [00100000] ⊕ [01000011] ⊕ [01001000]⊕[01100000]⊕[10000020]⊕[10000100] ⊕ [10010000] ⊕ [11000000], and [00000001] ⊗ [10000001] = [00000011] ⊕ [00001000] ⊕ [00100000] ⊕ [10000000] ⊕ [10000002]⊕[10000100]⊕[10010000]⊕[11000000]. In all cases we see that there is a nonzero invariant element. The resulting Lie algebra has dimension 120 + 128 = 248 and can be shown [1,2] to be isomorphic to the compact real form of e8 . Choosing iS+ instead of S+ , we obtain the maximally split real form of e8 which has been the focus of recent attention [15]. Notice that again Λ2 S+ → so16 is not an isomorphism, instead Λ2 S+ ∼ = Λ2 V ⊕ Λ6 V . This construction of e8 is also explained in [2, §6.A], where the nontrivial component of the Jacobi identity is proved combinatorially using Fierz identities. 5. Conclusion We have seen that a notion arising from supergravity, namely the Killing superalgebra, when applied in a classical context, yields a geometric construction of the exceptional Lie algebras of type F4 and E 8 . This was accomplished by using Bär’s cone construction to relate the Killing superalgebra to the well-known construction of these algebras using spin groups and their spinor representations. There are a number of things left to explore in relation to the construction presented in this paper, some of which we are actively considering:
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• Further riemannian examples? The three examples considered here are of the following general form: k = k0 ⊕ k1 , where k0 is a Lie subalgebra, k1 an k0 -module, where there are k0 -invariant positive-definite inner products on k0 and k1 and hence on k by declaring k0 and k1 to be orthogonal. The bracket Λ2 k1 → k0 is defined precisely by the condition that the resulting inner product on k be ad-invariant. All but one component of the Jacobi identity of k vanish. If Jacobi is satisfied, then we obtain a Lie algebra with a symmetric split and a positive-definite ad-invariant scalar product. This means we have a riemannian symmetric space and in fact the nontrivial Jacobi identity is the algebraic Bianchi identity for the would-be curvature tensor. We may therefore read off the possible such constructions from the list of symmetric spaces whose isotropy representation is spinorial, in which case the only examples are the above ones and the ones involving the exceptional Lie algebras E 6 and E 7 , about which there is more below. At any rate, we have looked explicitly at riemannian spheres in dimension ≤ 40 which could give rise to Lie algebras, and have checked that the nontrivial Jacobi identity cannot follow trivially from representation theory. It is therefore doubtful that other examples exist of precisely this construction in riemannian signature. • Killing superalgebras of “spheres” in arbitrary signature. Considering other signatures (and hence possibly also imaginary Killing spinors) might provide geometric realisations of Lie superalgebras. • A similar construction for the remaining exceptional Lie algebras. In the case of E 6 and E 7 , k0 also contains “R-symmetries” which do not act geometrically on the manifold. Understanding these cases should help to understand conformal Killing superalgebras. There does not seem to be a construction of G 2 using only spinors. • Of which structure on S 15 is E 8 the automorphism group? The existence of a Lie group is most naturally explained as automorphisms of some structure. The construction of E 8 out of the 15-sphere suggests that there ought to be some structure on S 15 of which E 8 is the automorphism group. This may also provide a simple proof of the Jacobi identity without resorting to Fierz or roots-and-weights combinatorics. I hope to report answers to some of these questions in the near future. Acknowledgements. I have benefited from giving several talks on this topic at the Dipartamento di Matematica “U. Dini” dell’Università degli Studi di Firenze, at the Departamento de Análisis Matemático de la Universidad de Alicante, and at the 18th North British Mathematical Physics Seminar held at the University of York. I am grateful to Dmitri Alekseevsky and Andrea Spiro for arranging the visit to Florence, to Salvador Segura Gomis for arranging the one to Alicante, and to Niall MacKay for organising the meeting in York. My interest in the Killing superalgebra has been nurtured through collaboration with a number of people, most recently, Emily Hackett-Jones, Patrick Meessen, George Moutsopoulos, Simon Philip and Hannu Rajaniemi, to whom I offer my thanks. Finally, the roots-and-weights calculations in Section 4 were performed using LiE [16], a computer algebra package for Lie group computations.
References 1. Adams, J.F.: Lectures on exceptional Lie groups, Chicago, IL: The University of Chicago Press, 1996 Z. Mahmud, M. Mimura, (eds.) 2. Green, M., Schwarz, J., Witten, E.: Superstring Theory. 2 vols., Cambridge: Cambridge University Press, 1987 3. Figueroa-O’Farrill, J.M., Meessen, P., Philip, S.: Supersymmetry and homogeneity of M-theory backgrounds. Class Quant. Grav. 22, 207–226 (2005) 4. Figueroa-O’Farrill, J.M., Hackett-Jones, E., Moutsopoulos, G.: The Killing superalgebra of ten-dimensional supergravity backgrounds. Class. Quant. Grav. 24, 3291–3308 (2007)
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5. 6. 7. 8.
Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3, 3–38 (1964) Lawson, H., Michelsohn, M.: Spin geometry. Princeton, NJ: Princeton University Press, 1989 Harvey, F.: Spinors and calibrations. London-New York: Academic Press, 1990 Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistor and Killing spinors on riemannian manifolds. No.108 in Seminarberichte. Berlin, Humboldt-Universität, 1990 Bär, C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993) Gallot, S.: Equations différentielles caractéristiques de la sphère. Ann. Sci. École Norm. Sup. 12, 235–267 (1979) Wang, M.: Parallel spinors and parallel forms. Ann Global Anal. Geom. 7(1), 59–68 (1989) Figueroa-O’Farrill, J.M., Leitner, F., Simón, J.: “Supersymmetric Freund–Rubin backgrounds,” In preparation Kosmann, Y.: Dérivées de Lie des spineurs. Annali di Mat. Pura Appl. (IV) 91, 317–395 (1972) Figueroa-O’Farrill, J.M.: On the supersymmetries of Anti-de Sitter vacua. Class. Quant. Grav. 16, 2043–2055 (1999) “248-dimension maths puzzle solved.” BBC News, March, 2007 available at http://news.bbc.co.uk/1/hi/ sci/tech/6466129.stm, 2007 van Leeuwen, M.A.A.: LiE, a software package for Lie group computations. Euromath Bull. 1(2), 83–94 (1994)
9. 10. 11. 12. 13. 14. 15. 16.
Communicated by G.W. Gibbons
Commun. Math. Phys. 283, 675–699 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0517-2
Communications in
Mathematical Physics
An Obstruction to Quantization of the Sphere Eli Hawkins Institute for Mathematics, Astrophysics, and Particle Physics, Radboud University, Nijmegen, The Netherlands. E-mail: [email protected] Received: 10 September 2007 / Accepted: 10 December 2007 Published online: 3 June 2008 – © The Author(s) 2008
Abstract: In the standard example of strict deformation quantization of the symplectic sphere S2, the set of allowed values of the quantization parameter h ¯ is not connected; indeed, it is almost discrete. Li recently constructed a class of examples (including S2) in which ¯h can take any value in an interval, but these examples are badly behaved. Here, I identify a natural additional axiom for strict deformation quantization and prove that it implies that the parameter set for quantizing S2 is never connected. 1. Introduction The standard geometric quantization construction for a manifold M with symplectic form ω uses a line bundle over M with curvature equal to ω/¯h. Such a line bundle ω exists if and only if the de Rham cohomology class [ 2π ] ∈ H2(M) is integral. Unless h ¯ the symplectic form is exact, this greatly restricts the allowed values of ¯h. At best, there is some maximum h ¯ 0, and the allowed values of ¯h are ¯h0, h ¯ 0/2, ¯h0/3, . . . , 0. At worst, there does not exist any ¯h satisfying this integrality condition; for example, this is the case with the Cartesian product of two symplectic spheres whose symplectic volumes differ by an irrational factor. Of course, this is just an obstruction to a particular construction. It says nothing about whether ¯h is so restricted for any quantization of M. A more general existential result was found by Fedosov [2] in his study of “asymptotic operator representations” (AOR’s) ω of formal deformation quantizations. Let θ = [ 2π ] + · · · ∈ ¯h−1H2(M)[[¯h]] be the h ¯ characteristic class of a formal deformation quantization. Under some assumptions about the trace, he showed that as h ¯ → 0, θ must asymptotically satisfy integrality conditions at the allowed values of ¯h. This is an interesting result, but there are many examples of quantization in which Fedosov’s assumptions are not true and h ¯ violates his integrality conditions. ˜ One way to construct such quantizations is to consider the universal covering space M. ˜ that is covariant under the action of π1(M), then If we can construct a quantization of M
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ω this should descend to a quantization of M. This implies that requiring [ 2π ] (or θ) to h ¯ be integral on M is too restrictive. Instead, we should only expect it to give an integral ˜ cohomology class on M. ˜ is generally smaller than H2(M). To be This is a weaker condition, because H2(M) ˜ = π2(M); so, ω gives an integral precise, for a path-connected manifold, H2(M) 2πh ¯ ˜ if it pairs integrally with π2(M), or equivalently, for any smooth cohomology class on M map ϕ : S2 → M, ϕ∗ ω ∈ Z. (1.1) S2 2π¯h
In particular, there is no integrality condition if ω pairs trivially with π2(M). There are many examples of quantization where ¯h is unrestricted by this condition. The noncommutative torus [14] is a quantization of the symplectic torus T2 (for which π2(T2) is trivial). Klimek and Lesniewski [6] constructed quantizations for higher genus Riemann surfaces (where π2 is also trivial). I constructed [5] such a quantization for any compact Kähler manifold where π2(M) pairs trivially with ω. Natsume, Nest, and Peter [11] constructed such a quantization for any symplectic manifold with π2(M) trivial and π1(M) an exact group. On the other hand, Li [10] has constructed a very large class of examples of quantizations for any Poisson (or even almost Poisson) manifold. In these examples, ¯h takes any value in an interval, so this violates all possible integrality conditions (not to mention integrability conditions). However, his examples are badly behaved. In particular, the quantum algebras are “much too big”; the classical algebra C0(M) is actually a subalgebra of each of the quantum algebras. Nevertheless, these examples satisfy a definition of “strict quantization” which had previously seemed very reasonable. So, the problem is not to understand how integrality conditions arise from the definition of quantization. The problem is to improve the definition by identifying good properties of quantization which do imply integrality conditions. The sphere S2 obviously plays a key role in the integrality condition (1.1), so the case of the symplectic sphere is absolutely fundamental. There is no way to understand integrality conditions without understanding the case of the symplectic sphere. That is what I am considering in this paper. In a similar vein, Rieffel [14, Thm. 7.1] showed that there does not exist any noncommutative SO(3)-equivariant product on C∞ (S2) that can be completed to a C∗ -algebra. In particular, there does not exist any equivariant strict deformation quantization of S2 according to his original definition. The Berezin-Toeplitz quantization of S2 is equivariant and satisfies a slightly weaker definition (quantization maps are not injective). With this weaker definition, Rieffel’s proof (adapted from a theorem of Wassermann [18]) actually shows that the Berezin-Toeplitz quantization is essentially the unique equivariant quantization of S2. That in turn implies that the set of values of h ¯ cannot be connected. As Rieffel writes, this theorem leaves open the question of “deformation quantizations which need not be invariant at all”. This is what I am addressing here, although with a slightly different notion of strict deformation quantization. As there is no one standard definition of strict deformation quantization, I begin in Sect. 2 by reviewing some of the various definitions that appear in the literature. I then motivate and present my definition of order n strict deformation quantization and explain why anything less than second order should not really be considered quantization.
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Sections 3 and 4 are purely motivational. In Sect. 3, I briefly review the algebraic index theorem which Fedosov used to obtain his integrality result. In Sect. 4, I summarize the standard Berezin-Toeplitz quantization construction. This shows explicitly the structure we might expect in the quantization of S2. In Sect. 5 I give the preliminary results which are not specific to S2. Some of these are minor technical lemmas, but Lemma 5.2 is the key tool and shows an integrality property of 2 × 2 matrices over a C∗ -algebra. Finally, in Sect. 6, I prove the main results. I show that for a second order strict deformation quantization of S2, the set of values of ¯h cannot be connected. For an infinite order strict deformation quantization of C∞ (S2), I show that h ¯ is asymptotically restricted by Fedosov’s integrality condition in terms of the characteristic class of the corresponding formal deformation quantization.
1.1. Notation. In this paper, I will use the notations on(¯h) and On(¯h) a great deal. These are always in the context of maps from ¯h ∈ I {0} to Banach spaces. Definition 1.1. a(¯h) = on(¯h) if lim h ¯ −n a(¯h) = 0,
h ¯ →0
and a(¯h) = On(¯h) if h ¯ −n a(¯h) is bounded for sufficiently small ¯h. More frequently, I write a(¯h) ≈ b(¯h) mod on(¯h) if a(¯h) − b(¯h) = on(¯h). I will always use ≈ in this sense. Any statement involving on(¯h) is really a statement about a limit, but I think this notation allows things to be written more clearly. It would really be more correct to write “a ∈ on(¯h)”, but I don’t want to stray too far from existing notational conventions. H2(M) is de Rham cohomology with real coefficients. Given an algebra A, the algebra of m × m-matrices over A is denoted Matm(A). The partial trace tr : Matm(A) → A takes the sum of the diagonal entries. The standard Pauli matrices σi ∈ Mat2(C) are, 01 0 −i 1 0 , σ2 := , σ3 := . σ1 := 10 i 0 0 −1 More importantly, these are related by σiσj = δij + ikijσk; that is, σ1σ2 = −σ2σ1 = iσ3, σ2 1 = 1, et cetera. The summation convention is used for repeated indices, even if both are superscripts.
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2. Definitions of Quantization In order to consider properties of quantization, it is necessary to first consider the various definitions of quantization. All of the definitions for C∗ -algebraic deformation quantization are variations on the ideas proposed by Rieffel [14–16]. Most of these involve a continuous field of C∗ -algebras, either as a given structure, or as something whose existence and uniqueness is supposed to be guaranteed by the axioms. All of the definitions include a parameter set I ⊆ R, such that 0 ∈ I is an accumulation point. This is the set of values for the parameter ¯h, and ¯h = 0 is the “classical” value. Rieffel’s definition is stated in terms of a fixed vector space A0 with an ¯h-dependent product, involution, and norm. Using these norms, the vector space can be completed to a family of C∗ -algebras A h ¯ . Instead of starting with a fixed vector space, we can instead start with a family of normed ∗ -algebras A h ¯ which are identified by bijective linear “quantization maps” Q h ¯ : A0 → A h ¯ . More generally, we can consider quantizations in which these are not bijections. Either way, each ∗ -algebra A h ¯ is to be completed to a C∗ -algebra A h . These are supposed to form a continuous field, uniquely defined by ¯ the requirement that for any a ∈ A0, ¯h → Q h ¯ (a) is a continuous section. Finally, some definitions start with the collection of C∗ -algebras or continuous field as a given structure which is not necessarily defined by the quantization maps. The third setting is the most general, so all of the definitions I am reviewing can be stated in these terms. Let I ⊆ R with 0 ∈ I an accumulation point. Let {A h ¯ }h ¯ ∈I be a collection of C∗ -algebras. Let A0 ⊂ A0 be a dense ∗ -subalgebra. Let Q h ¯ : A0 → A h ¯ for each h ¯ ∈ I be linear maps such that Q0 : A0 → A0 is the inclusion. Let A h ¯ ⊂ Ah ¯ be the subalgebra generated by Im Q h ¯ ≡ Qh ¯ (A0). In most cases, the collection of C∗ -algebras forms a continuous field, A, such that for any a ∈ A0, ¯h → Q h ¯ (a) defines a continuous section, denoted Q(a) ∈ Γ (I, A). I also refer to Q : A0 → Γ (I, A) as a quantization map. Compatibility with a continuous field structure implies that for any a, b, c ∈ A0, ∗ Q h and Q h ¯ (a) + Q h ¯ (b) ¯ (a)Q h ¯ (b) − Q h ¯ (c) are continuous functions of ¯h ∈ I. Conversely, these conditions imply [16] that the C∗ -algebras generated by each Im Q h ¯ form a unique continuous field such that Q(a) is a continuous section. This means that if we start with a given continuous field structure, then even if the quantization maps do not generate the given continuous field, they still generate a continuous subfield. If there is not a continuous field structure, then these continuity properties are at least assumed to hold at 0 ∈ I. With one exception to be discussed below, in all the definitions A0 is a commutative algebra of functions on a Poisson manifold, and the quantization maps are related to the Poisson bracket by the condition that ∀a, b ∈ A0, [Q h ¯ (a), Q h ¯ (b)] ≈ i¯hQ h ¯ ({a, b})
mod o1(¯h).
(2.1)
After that, there are several variations in the definition. The index set I ⊆ R may or may not be connected or closed. The quantization maps Q h ¯ may be injective. The image of Q h ¯ may be a subalgebra of A h ¯ , or it may be dense, or it may generate a dense subalgebra. The quantization maps may or may not be ∗ -linear. The table on the next page summarizes the substantive variations among some of the definitions in the literature.
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[16] [17] [8] [8] [8] [9] [9] [11] [5] [10]
[10]
Name Strict deformation quantization Operator deformation quantization Strict quantization Strict deformation quantization Continuous quantization Strict deformation quantization Semi-strict deformation quantization Strict quantization Strict deformation quantization Strict quantization Faithful Hermitian Strict deformation quantization
679 C
✓ ✓
I open interval
1-1
Alg
dense
✓
✓
✓
[0,)
✓
Im Q¯h Im Q¯h
✓
✓
✓
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓
✓
[0,)
A¯h
closed
A¯h
closed
∗
✓ ✓
✓
closed
✓
✓
C The C∗ -algebras may form a continuous field. Otherwise, continuity is only required at ¯h = 0. I Any additional assumptions on the index set I ⊆ R. 1-1 The quantization maps may be assumed to be injective Q¯h : A0 → A¯h . Alg The image Im Q¯h ⊆ A¯h may be a ∗ -subalgebra. dense If the image is a ∗ -subalgebra, then it may be dense in A¯h . Otherwise, Im Q¯h may be dense, or the ∗ -subalgebra A generated by it may be dense. ¯h ∗ The quantization maps Q¯h may be ∗ -linear, i.e., Q¯h (a∗ ) = Q¯h (a)∗ .
As this table should make clear, there is significant variation in both the definitions and terminology, and these are not correlated. There is no consistent feature of “strict deformation quantization” as opposed to “strict quantization”. In the absence of a definitive definition, I shall consider the merits of each of these possible axioms. First, I favor requiring a continuous field structure, because the idea of strict deformation quantization is to continuously deform from A0 to A h ¯ . I can see no aesthetic justification for requiring continuity at 0 without requiring it at all h ¯ ∈ I. Some authors require the index set I to be an interval. This is a natural requirement, given the idea of deformation. However, in practice it is not a good assumption for quantization. The Berezin-Toeplitz quantization of a compact Kähler manifold [1] is one of the best behaved examples of quantization, and in that case 1 I = 0, . . . , 1 3, 2, 1 which is very different from an interval. On the other hand, it is technically awkward to work with a continuous field over a pathological space, so I will assume that I ⊆ R is locally compact. The example of Berezin-Toeplitz quantization violates another of these tentative axioms: The quantization maps are not injective. This is unavoidable because the algebras Ah ¯ >0 are finite dimensional. Note that the continuous field structure implies a sort of asymptotic injectivity; for any a = 0 ∈ A0, (2.2) lim Q h ¯ (a) = a = 0 ¯h→ 0
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so a ∈ ker Q h ¯ for ¯h sufficiently small. However, this does not imply that any of the maps Q h ¯ for ¯h = 0 are actually injective. The most delicate issue is whether the image Im Q h ¯ should be an algebra. This is true in the Berezin-Toeplitz example, but it is not clear if this is true in all nice examples of quantization. I also do not need this for the main result of this paper. Given this uncertainty, I am favoring the weaker definition and not assuming this property here. I will refer to a quantization with this property as algebraically closed. If the ∗ -subalgebras A h ¯ ⊆ Ah ¯ generated by each Im Q h ¯ are dense, then the continuous field structure is completely fixed by the quantization maps. This would be a nice property, but I do not need it for the main result here. I am also inclined to de-emphasize the role of any one quantization map. Instead, I think that the existence of a large class of equally good quantization maps is important. The tentative axiom that Im Q h ¯ ⊂ Ah ¯ is a dense subalgebra should probably be replaced by some sort of irreducibility property. Finally, many definitions do not require the quantization map to be ∗ -linear. However, if the quantization map is at all well behaved with respect to the involution, then the ∗ -linear part a → 1 [Q(a) + Q(a∗ )∗ ] will also be a perfectly good quantization map. 2 For this reason, we can assume ∗ -linearity without any important loss of generality. These axioms are actually too weak, because they do not require sufficient regularity at ¯h = 0. First note that compatibility with the continuous field structure implies that ∀a, b ∈ A0, Qh ¯ (a)Q h ¯ (b) ≈ Q h ¯ (ab)
mod o0(¯h).
(2.3)
I mentioned above that one definition does not include Eq. (2.1). That is Rieffel’s definition [16] for strict deformation quantization of a (possibly) noncommutative algebra. In that case, one cannot assume that the commutator vanishes at h ¯ = 0. Instead, Rieffel requires that Qh ¯ (a)Q h ¯ (b) ≈ Q h ¯ [ab + i¯hC1(a, b)]
mod o1(¯h),
(2.4)
where C1 is a given Hochschild 2-cocycle. The cohomology class of C1 in H2(A0, A0) is required to be a “noncommutative Poisson structure” [20]; that is, to have vanishing Gerstenhaber bracket with itself. This definition is slightly awkward; it is more appropriate to require that C1 belongs to a given Hochschild cohomology class, rather than being a given cocycle. Equation (2.4) requires the product to be expandable to first order in ¯h. Equation (2.1) only requires this of the antisymmetric part of the product. In this way, Eq. (2.4) is a slightly stronger condition. However, the case of noncommutative A0 suggests that Eq. (2.4) is more natural. Equations (2.3) and (2.4) are the beginning of a sequence of possible conditions. Definition 2.1. A quantization map Q is of order n (for some n = 0, 1, 2, . . . ) ∀a, b ∈ A0 if there exists a degree n polynomial a ∗n b ∈ A0[¯h] such that n Qh ¯ (a)Q h ¯ (b) ≈ Q h ¯ [a ∗ b]
mod on(¯h).
A quantization map is of order ∞ if it is of order n for any finite n.
(2.5)
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The latter definition is consistent, because order n implies order m < n. This definition has two aspects. If the quantization is not algebraically closed but is of order n, then Im Q h ¯ n. So, this ¯ is approximately algebraically closed to order h is partly a weaker version of the algebraic closedness axiom. On the other hand, if the quantization is algebraically closed and the quantization maps are injective, then the quantization actually defines a variable product on A0: [Q h Q−1 ¯ (a)Q h ¯ (b)], h ¯
(2.6)
and the order n condition means precisely that this is an n times differentiable function of ¯h at 0; in fact, a ∗n b is the nth order Taylor expansion of (2.6) about 0. So, “order n” is both an algebraic closedness and a differentiability condition. The same aesthetic principle that suggests using a continuous field of C∗ -algebras implies that if we impose a differentiability condition at 0, we should impose such a condition everywhere. However, this condition will not be needed here, so I will leave that discussion to a future paper. Definition 2.2. Given a continuous field and a quantization map, Q : A0 → Γ (I, A), (n)
define ΓQ (I, A) to be the set of sections a ∈ Γ (I, A) for which there exist elements a0, . . . , an ∈ A0 such that n n a(¯h) ≈ Q h ¯ (a0 + a1 ¯h + · · · + an ¯h ) mod o (¯h).
(2.7)
(n)
Note that a quantization map Q is of order n if and only if ΓQ (I, A) ⊂ Γ (I, A) is a (n)
subalgebra. The algebra ΓQ (I, A) should be thought of as the set of sections of A that are n-times differentiable at ¯h = 0. The next lemma shows that Eq. (2.7) is the analogue of a Taylor expansion, with the quantization map playing the role of a flat connection. (n)
Lemma 2.1. Given a quantization map Q and a section a ∈ ΓQ (I, A), there exists a unique degree n polynomial in A0[¯h] satisfying Eq. (2.7) (n)
Proof. Existence is guaranteed by the definition of ΓQ (I, A). If uniqueness is violated then there exist a0, . . . , an ∈ A0 (not all 0) such that n n Qh ¯ (a0 + a1 ¯h + · · · + an ¯h ) = o (¯h).
Suppose that ak is the first nonzero term, then Qh ¯ + · · · + anh ¯ n−k) = on−k(¯h), ¯ (ak + ak+1 h but setting ¯h = 0 shows that ak = Q0(ak) = 0. By contradiction, this proves uniqueness. Li [10] has constructed a disturbing class of examples which satisfy a reasonable seeming definition of “strict quantization” but which really should not be considered as quantizations. One of the surprising features of his construction is that it applies not just to any Poisson manifold, but to any manifold with an antisymmetric bivector field (an “almost Poisson manifold”). The problem is that Eq. (2.1) does not imply that the bracket { · , · } satisfies the Jacobi identity.
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This is a little surprising, since a commutator always satisfies the Jacobi identity. What goes wrong is that the error in Eq. (2.1) is not controlled enough. If we apply Eq. (2.1) to a double commutator, we only get 2 [Q h ¯ (a), [Q h ¯ (b), Q h ¯ (c)]] ≈ −¯h Q h ¯ ({a, {b, c}})
mod o1(¯h)
which is vacuous. The problem is that the commutator of Q h ¯ (a) with an error term of order o1(¯h) is only guaranteed to be of order o1(¯h), because we know nothing more about that error term. This problem can be fixed by requiring the quantization map to be second order, as is implied by a more general result: Proposition 2.2. Given an order n quantization map, the degree n polynomial a ∗n b satisfying Eq. (2.5) is unique. There exist unique bilinear maps Cj : A0 × A0 → A0 such that ∀a, b ∈ A0 a ∗ b = ab + n
n
(i¯h)jCj(a, b).
(2.8)
j=1
This ∗n is an approximately associative product in the sense that ∀a, b, c ∈ A0, a ∗n (b ∗n c) ≡ (a ∗n b) ∗n c
mod h ¯ n+1.
Proof. The uniqueness of a ∗n b is implied directly by Lemma 2.1. So, it defines a map ∗n : A0 × A0 → A0[¯h]. To check that this is bilinear, observe that if a is multiplied by a constant, then multiplying a ∗n b by the same constant gives a degree n polynomial satisfying Eq. (2.5); the same thing works with sums. Since it is a bilinear map to polynomials, ∗n can be written out as a polynomial of bilinear maps, a ∗n b =
n
(i¯h)jCj(a, b).
j=0
However, for any quantization map Q, Q0 : A0 → A0 is multiplicative. Setting ¯h = 0 in Eq. (2.5) shows that the degree 0 term in a ∗n b must be ab. This shows that a ∗n b must be of the form Eq. (2.8). Applying Eq. (2.5) to the product of a, b, c ∈ A0 gives Q(a)[Q(b)Q(c)] ≈ Q[a ∗n (b ∗n c)] mod on(¯h). There is an analogous expression with the other arrangement of parenthesis. Subtracting shows that the associator satisfies, Q[a ∗n (b ∗n c) − (a ∗n b) ∗n c] = on(¯h). By Lemma 2.1, this shows that the coefficients of the associator vanish up to order ¯hn. This means that ∗n defines an associative product on A0[¯h]/¯hn+1.
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Corollary 2.3. If Q is of order 1, then C1 is a Hochschild cocycle, defining an element of H2(A0, A0). If A0 is commutative then {a, b} := C1(a, b) − C1(b, a) is a derivation in both arguments. Proof. This is a well known result. The general case was first proven in [3]. The Hochschild coboundary is the first order part of the associator of ∗1 , which vanishes by Proposition 2.2. Because ∗1 is associative to first order, the ∗1 -commutator with any a ∈ A0 is a derivation to first order. In the commutative case, that commutator is just i¯h {a, · }, and so the bracket is a derivation of the commutative product. The point is that this alone does not imply the Jacobi identity. The commutator does satisfy the Jacobi identity. The Jacobi identity for the bracket should appear at order h ¯ 2, 1 but if Q is only of order 1, then the error terms are of order o (¯h). Corollary 2.4. If Q h ¯ is of order 2, then the cohomology class of C1 has vanishing Gerstenhaber bracket [C1, C1] = 0 ∈ H3(A0, A0) (i.e., it is a noncommutative Poisson structure [20]). In the case of A0 commutative, the bracket { · , · } satisfies the Jacobi identity and is thus a Poisson bracket. Proof. Again, this is a standard result and was first proved in [3], where [C1, C1] is referred to as the first obstruction cocycle for a deformation. This comes from the vanishing of the second order term in the associator for ∗2 . Because ∗2 is associative to second order, its commutator satisfies the Jacobi identity to second order. In the commutative case, this commutator equals i¯h{ · , · } to first order. So, the order h ¯ 2 Jacobi identity for the ∗2 -commutator is just the Jacobi identity for the bracket { · , · }. It is normally assumed that a quantization of a manifold corresponds to a Poisson structure. However, if the quantization is not algebraically closed or of at least second order, then the Jacobi identity is really an unnecessary and artificially imposed condition. For this reason, it seems that we should always require a quantization to be of order at least 2. To summarize, here is the main definition I will be using in this paper: Definition 2.3. Let A0 be a ∗ -algebra and π ∈ H2(A0, A0) a real Hochschild cohomology class. An order n strict deformation quantization (I, A, Q) of (A0, π) consists of: a locally compact subset I ⊆ R with 0 ∈ I an accumulation point, a continuous field of C∗ -algebras A over I, and a ∗ -linear map Q : A0 → Γ (I, A) such that: (1) At 0 ∈ I this is an inclusion Q0 : A0 → A0 of A0 as a dense ∗ -subalgebra; (2) Q satisfies the order n condition (2.5); (3) if n ≥ 1, then the order 1 term in the expansion belongs to the given cohomology class C1 ∈ 1 2 π. ∗ Such a quantization is algebraically closed if each Im Q h ¯ ⊆ Ah ¯ is a -subalgebra. It is unital if all the algebras A0 and A h ¯ and the quantization maps Q h ¯ : A0 → A h ¯ are unital. This is an (order n) strict deformation quantization of a Poisson manifold M if C0(M) ⊆ A0 ⊆ Cb (M) and π is the Poisson structure.
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As noted above, existence of a second order quantization implies the vanishing of the Gerstenhaber bracket [π, π] = 0 ∈ H3(A0, A0). For quantization of a Poisson manifold, π ∈ H2(A0, A0) implies that A0 is closed under the Poisson bracket. This definition encompasses a spectrum of concepts. The weakest — a 0th order strict deformation quantization — is really just a continuous field with a preferred collection of sections. Anything less than second order should not really be considered quantization. The class of examples constructed by Li [10] can be seen as a demonstration of this. His construction applies to (nonintegrable) almost Poisson manifolds because the quantization map is not required to be second order. If a strict deformation quantization is of infinite order, then this means precisely that the product has an asymptotic expansion, which defines a formal deformation quantization. This formal product is not necessarily differential (i.e., the Cj’s may not be bidifferential). However, any formal deformation quantization is equivalent (isomorphic as a C[[¯h]]-algebra) to a differential one [7]. In the case of Berezin-Toeplitz quantization of a compact Kähler manifold, Bordemann, Meinrenken, and Schlichenmaier [1] have shown that such an asymptotic expansion exists. Thus that quantization is of infinite order and in particular of second order.
3. Algebraic Index Theorem Fedosov’s integrality results are stated in the context of formal deformation quantization and follow from the algebraic index theorem. A formal deformation quantization of a Poisson manifold [2,7,19] is an associative product on the space C∞ (M)[[¯h]] of formal power series that reduces to ordinary multiplication modulo ¯h and whose commutator is given to first order by the Poisson bracket. ¯ for the space C∞ (M)[[¯h]] with this product. Write A h It is also possible to consider just an order n formal deformation quantization — that is, an associative product on C∞ (M)[¯h]/¯hn+1. Proposition 2.2 shows that an order n strict deformation quantization determines an order n formal deformation quantization. Two formal deformation quantizations are equivalent if the algebras are isomorphic as C[[¯h]]-algebras. The isomorphism is a C[[¯h]]-linear map G : C∞ (M)[[¯h]] → C∞ (M)[[¯h]] intertwining the products. Kontsevich [7] calls this a gauge equivalence if G is the identity map modulo h ¯. A formal deformation quantization of a symplectic manifold (M, ω) determines [ω] plus terms of a characteristic class θ ∈ ¯h−1H2(M)[[¯h]] which always equals 2πh ¯ nonnegative degree. Formal deformation quantizations of (M, ω) are classified up to gauge equivalence by θ [12,19]. A formal deformation quantization of (M, ω) has a unique C[[¯h]]-linear trace (modulo normalization) [2] 1 ¯ Tr ¯h : A h → ¯h− 2 dim MC[[¯h]].
There is a natural choice of normalization. This trace is the setting for the algebraic index theorem. The statement of the theorem ∞ is slightly simpler if M is compact. Suppose that e0 = e2 0 ∈ Matm[C (M)] is an idemh ¯ potent matrix of smooth functions. Then there exists an idempotent e h ¯ ∈ Matm(A )
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such that e h ¯ . For any such idempotent, the algebraic index theorem [2,12] ¯ ≡ e0 mod h shows that its trace is ^ Tr h eθ ∧ A(TM) ∧ ch(e0). (3.1) ¯ eh ¯ = M
H• (M)
Here, ch e0 ∈ is the Chern character of the image vector bundle e0 (Cm × M). Note that this trace does not depend upon the choice of e h ¯ . It only depends upon the topology of M, the characteristic class θ, and the class of e0 in K0(M). Now, let M = S2 ⊂ R3 be the unit sphere in Euclidean space and take the induced ^ class is trivial volume form as symplectic form. Because this is 2-dimensional, the A and Eq. (3.1) simplifies to θ Tr h e = e ∧ ch(e ) = (1 + θ) ∧ (rk e0 + c1(e0)) 0 ¯ h ¯ S2 S2 = rk e0 θ+ c1(e0). S2
S2
In particular, the normalization of the trace is Tr ¯h 1 =
θ.
(3.2)
S2
The K-theory K0(S2) is generated by the classes of 1 and the Bott projection. Let xi be the Cartesian coordinates on S2 ⊂ R3. Let σi ∈ Mat2(C) be the Pauli matrices. The Bott projection eBott ∈ Mat2[C∞ (S2)] is 1 1 x3 x1 − ix2 i . (3.3) + (1 + σ x ) = eBott := 1 i 2 −x3 2 2 x1 + ix2 This has rank 1 and Chern number S2 c1(eBott ) = 1, so Tr h θ. (3.4) ¯ eh ¯ =1+ S2
Because rk eBott = 1, this is actually not the best choice of generator for K-theory. It would be better to use [eBott ]−[1] ∈ K0(S2), which actually lies in the reduced K-theory. This corresponds to Tr h ¯ eh ¯ − Tr ¯h 1 = 1. The class [eBott ] − [1] is the protagonist of this paper. Fedosov [2] has applied this algebraic index theorem to find restrictions on I for “asymptotic operator representations” (AOR’s) of a formal deformation quantization. Any infinite order strict deformation quantization (I, A, Q) of a symplectic manifold determines a formal deformation quantization, and if the algebras A h ¯ are realized as concrete C∗ -algebras A h ¯ ⊂ L(H), then Q is an AOR of the formal deformation quantization. Suppose that the operators Q h ¯ (f) ∈ A h ¯ ⊂ L(H) for f ∈ A0 are all trace class, and that the formal trace is the asymptotic expansion Tr Q h ¯ (f) ∼ Tr h ¯ f of the operator trace. The operator trace of an idempotent is always an integer, therefore for any idempotent, Tr ¯h e h ¯ must be the asymptotic expansion of an integer valued function on I {0}. The trouble is that these assumptions do not generally hold for infinite order strict deformation quantizations of symplectic manifolds. In some very well behaved examples
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of strict deformation quantization, there does exist a unique trace, but it is not integervalued on idempotents. For example, in the “noncommutative torus” quantization of T2, for irrational values of h ¯ , the set of traces of idempotents is dense in [0, 1] (see [13]). That explains why the noncommutative torus can be defined over I = R, violating Fedosov’s restrictions on ¯h. In order to go beyond Fedosov’s result and find general restrictions on I, we must not assume a priori that there exist any traces on a strict deformation quantization. Nevertheless, Fedosov’s ideas do give an important heuristic guideline.
4. Berezin-Toeplitz Quantization The Berezin-Toeplitz construction [1] is a fairly general and very well behaved quantization construction. It provides some hints to the possible structure of an arbitrary strict deformation quantization of S2. Let M be a compact, connected Kähler manifold with symplectic form ω and such ω ] ∈ H2(M) is integral. Then there exists a holomorphic that the cohomology class [ 2π line bundle L → M with a Hermitian inner product and curvature equal to ω. The N-fold tensor power L⊗N is also a holomorphic line bundle with inner product and has curvature Nω. Using the inner product and the symplectic volume form, we can construct a Hilbert space L2(M, L⊗N) from sections of L⊗N. The set of holomorphic sections of L⊗N is a finite dimensional Hilbert subspace HN ⊂ L2(M, L⊗N). Let ΠN : L2(M, L⊗N) → → HN be the orthogonal projection onto this subspace. The Hilbert space L2(M, L⊗N) is a module of the algebra C(M) of continuous functions. For any f ∈ C(M) and ψ ∈ HN, TN(f)|ψ := ΠNf|ψ defines a map TN : C(M) → L(HN) from functions to operators (matrices) on HN. This defines an algebraically closed, infinite order, strict deformation quantization of M as follows. The index set is 1 I := 0, . . . , 1 , , 1 . 3 2 The algebras are A0 := C∞ (M), A0 := C(M), A1/N := L(HN). The quantization maps are Q1/N = TN (restricted to A0). These quantization maps are actually surjective and define a unique continuous field structure on {A h ¯ }h ¯ ∈I. 2 3 Now consider the sphere. Let M = S ⊂ R be the standard unit sphere in Euclidean space with the symplectic form equal to the induced volume form. This means that ω = 4π S2
so 1 2 ω satisfies the integrality condition. With this choice, the Poisson brackets of the Cartesian coordinates (of the embedding) satisfy the standard su(2) relations. The downside of this normalization is that we should take L to have curvature 1 2 ω, and identify 2 ¯h = N . The sphere S2 is of course an SU(2)-symmetric space, and the quantization can be carried out equivariantly. The Hilbert space HN is N + 1-dimensional, carrying the spin- N 2 irreducible representation of SU(2).
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Let xi ∈ C∞ (S2) be the Cartesian coordinates of the embedding S2 ⊂ R3. The quantizations of these functions are i i 2 i Qh ¯ (x ) = TN(x ) = N+2 J ,
where Ji are the standard self-adjoint su(2)-generators. Again, let σi ∈ Mat2(C) be the Pauli matrices and eBott ∈ Mat2(A0) the Bott projec1 1 i tion (3.3). Applying the quantization map gives a matrix Q h ¯ (eBott ) = 2 + N+2 σiJ ∈ N+1 Mat2(A h ¯ ) with only two eigenvalues, 0 and N+2 . This is easily corrected to give a projection, N+2 1 + N+1 σiJi. e(¯h) = 2(N+1)
(4.1)
From this, we can explicitly compute the trace, Tr e(¯h) = N + 2. 2 2 In fact, by comparing the traces Tr 1 = N + 1 = h + 1 and Tr e(¯h) = N + 2 = h +2 ¯ ¯ with Eqs. (3.2) and (3.4), we can deduce that if a formal deformation quantization is extracted from this Berezin-Toeplitz quantization, then its characteristic class is S2 θ = 2 + 1. See [4] for a more general result. h ¯ The projection e(¯h) extends the Bott projection eBott , so the K-theory class [e(¯h)] − 0 2 [1] ∈ K0(A h ¯ ) extends [eBott ] − [1] ∈ K (S ). The usual trace on the matrix algebra A h ¯ gives a homomorphism [Tr] : K0(A h ) → Z ⊂ R, and the trace of this particular class is ¯ [Tr]([e(¯h)] − [1]) = 1. This trace (in general, any finite trace) factors through degree 0 cyclic homology via the Chern character as ch
Tr
0 −→ HC0(A h → C. [Tr] : K0(A h ¯ )− ¯ )−
The cyclic homology group HC0(A h ¯ ) is just the quotient of A h ¯ by the subspace spanned by commutators. The degree 0 part of the Chern character of an idempotent is represented by its partial trace: Definition 4.1. The partial trace tr : Matm(A h ¯ ) → Ah ¯ maps a matrix over A h ¯ to the sum of its diagonal entries. 1 The partial trace of our idempotent is tr e(¯h) = 1 + N+1 , and the Chern character ch0([e(¯h)] − [1]) is represented by 1 tr e(¯h) − 1 = N+1 ∈ Ah (4.2) ¯ . Although (4.2) is a multiple of 1 ∈ A h ¯ , this is not typical. Because the construction of e(¯h) was SU(2)-equivariant, tr e(¯h) is inevitably invariant, and thus a multiple of 1 since HN is irreducible. Imagine now that we forget about the SU(2) action and quantize S2 using a non equivariant Kähler structure. We can still construct e(¯h) by the same procedure, but now tr e(¯h) − 1 ∈ A h ¯ will (probably) not be a multiple of 1. However, for ¯h small enough, e(¯h) will have the same class in the K-theory K0(A h ¯ ) as the equivariant one (see [4]). This implies that the (complete) trace of tr e(¯h) − 1 is the same. So, applying : Ah the normalized trace Tr ¯ → C, a → (Tr a)/(N + 1) should give
e(¯h) − 1] = 1 Tr[tr N+1 for N sufficiently large.
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Now what can we deduce about the value of N from the existence of the trace on A h ¯ without actually using the trace? The trick is to note that the normalized trace on A h ¯ is a state. This implies that if a self-adjoint element a = a∗ ∈ A h ¯ is bounded by real a satisfies the same bound, α ≤ Tr a ≤ β. Applying numbers as α ≤ a ≤ β, then Tr this to tr e, we see that if α ≤ tr e(¯h) − 1 ≤ β then 1 α ≤ N+1 ≤ β.
So, if tr e is sufficiently close to a multiple of the identity (β − α 1 ¯ 2), then there will 2h be a unique value of N ∈ N fitting this bound. I will show in the next section (Lem. 5.2) that an integer can be extracted from a projection in this way quite generally. 5. General Results Before proceeding further, I will need a simple result about the spectrum of a sum of bounded operators. Lemma 5.1. If a, b ∈ A are self-adjoint elements of a C∗ -algebra, then the spectrum of the sum satisfies Spec(a + b) ⊆ Spec a + [− b , b ],
(5.1)
where the sum denotes the set of all sums of elements of the two sets. Proof. If λ ∈ Spec a, then (a − λ)−1 is contained in the commutative C∗ -subalgebra generated by a and 1 in the unitalization of A. That is the algebra C(Spec a) of continuous functions on the spectrum. The norm of (a − λ)−1 is the supremum over Spec a of the inverse of the distance to λ. Equivalently, it is the inverse of the infimum of the distance, −1 (a − λ)−1 = (dist[λ, Spec a]) . Now, suppose that b < dist[λ, Spec a]. This implies that
b
b 1 ≤ a − λ dist[λ, Spec a] < 1 and the power series, j ∞ 1 1 −b a−λ a−λ
(5.2)
j=0
converges to an element of A. Multiplying (5.2) by a + b − λ shows that this sum must equal the inverse of a + b − λ. In particular, that inverse exists and therefore λ ∈ Spec(a + b). Equivalently, λ ∈ Spec(a + b) implies dist[λ, Spec a] ≤ b . Since a and b are self-adjoint, Spec a, Spec(a + b) ⊂ R, and this gives Eq. (5.1). The next lemma is my most important tool for proving the main results.
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Lemma 5.2. Let A be any unital C∗ -algebra, and e = e2 = e∗ ∈ Mat2(A) a projection. If there exist real numbers α ≤ β ∈ R bounding the partial trace α ≤ tr(e) − 1 ≤ β then either α ≤ 0 ≤ β or there exists an integer k ∈ Z such that 1 ≤ β. α≤ k
Proof. If α ≤ 0 ≤ β, then the claim is true. If α, β < 0, then we could equivalently consider the complementary projection 1 − e instead. So, it is sufficient to prove the case when 0 < α, β. If we denote the entries of this matrix as Z X , e= X∗ 1 − W then the properties e = e∗ and e2 = e give relations among X, Z, W ∈ A, Z = Z∗ , W = W ∗ , ZX = XW, Z(1 − Z) = XX∗ , W(1 − W) = X∗ X.
(5.3a) (5.3b) (5.3c)
The relations (5.3) have several implications for the spectra of Z and W. First, Eqs. (5.3a) and (5.3c) imply that Spec Z, Spec W ⊂ [0, 1]. Second, suppose λ ∈ Spec Z, Spec W. By (5.3c) and (5.3b), W(1 − W) = X∗ X = X∗ (Z − λ)−1(Z − λ)X = X∗ (Z − λ)−1X(W − λ). Multiplying this equation by (W − λ)−1 on the right gives, X∗ (Z − λ)−1X = W(1 − W)(W − λ)−1. The left-hand side explicitly extends to a continuous function of λ ∈ Spec Z. However, the right-hand side only extends continuously to λ ∈ Spec W if λ = 0 or 1. This shows that Spec W ⊆ {0, 1} ∪ Spec Z. By a symmetrical argument, Spec Z ⊆ {0, 1} ∪ Spec W, and so Spec Z ∪ {0, 1} = Spec W ∪ {0, 1}.
(5.4)
Third, using the bounds on tr(e) − 1 = Z − W, Lemma 5.1 shows that, Spec Z ⊂ Spec W + [α, β].
(5.5)
Relation (5.5) implies that sup(Spec Z) − sup(Spec W) ≥ α > 0. If 1 ∈ Spec Z, then Eq. (5.4) would imply that sup(Spec W) = sup(Spec Z), therefore by contradiction 1 ∈ Spec Z. Combining relations (5.4) and (5.5) gives Spec Z ⊂ ({0, 1} ∪ Spec Z) + [α, β] = [α, β] ∪ (Spec Z + [α, β])
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since 1 + α > 1. Iterating this gives Spec Z ⊂
∞
[kα, kβ].
k=1
Now, 1 ∈ Spec Z implies that there exists k ∈ N such that kα ≤ 1 ≤ kβ.
There is no way to avoid requiring that A is unital in this theorem. Without a unit, it would be meaningless to compare an element of A with real numbers. Any idempotent matrix of functions can be extended to an idempotent over a formal deformation quantization. This next lemma is the analogous result for strict deformation quantization. Lemma 5.3. If (I, A, Q) is an order n ≥ 0 strict deformation quantization, and (n)
a = a∗ ∈ ΓQ (I, A) such that a(0) = a2(0) is a projection, then there exists a neighborhood I ⊆ I of 0 (in the relative topology of I) such that 1 dλ e := (5.6) 2πi λ − a|I ∗ where the contour is the circle of radius 1 2 centered at 1, defines a projection e = e =
(n)
e2 ∈ ΓQ (I , A).
Proof. The function a(¯h) − a2(¯h) is continuous and vanishes at ¯h = 0, therefore, (by local compactness of I) there exists a neighborhood I ⊆ I of 0 over which it is bounded by some c < 1 4 . The spectrum of a restricted to I is contained in Spec a|I ⊆ λ ∈ R |λ − λ2| ≤ c
1 √1 , 1 ∪ 1 , 1 + √1 . = − ⊆ λ ∈ R |λ − λ2| < 1 4 2 2 2 2 2
2
Γ (I , A)
So the contour does not intersect Spec a|I and e ∈ is well defined by (5.6). Because a|I is self-adjoint, e is automatically a projection. Because a(0) is already a projection and the contour encloses 1 but not 0, e(0) = a(0). This proves the lemma for n = 0, so now suppose that n ≥ 1. I need to show that (n) (n) e ∈ ΓQ (I , A). Any polynomial of a|I is automatically contained in ΓQ (I , A), so it is sufficient to show that e is approximated well enough by polynomials of a|I . Consider the polynomial, p(x) = 3x2 − 2x3. This has the properties that p(0) = 0, p(1) = 1, and p (0) = p (1) = 0. Iterating p gives polynomials whose derivatives vanish at 0 and 1 to any desired order. (n) Note that a − a2 ∈ ΓQ (I, A) vanishes at h ¯ = 0, so a − a2 = O1(¯h). Define e0 := a|I and iteratively define ej := p(ej−1) for j ≥ 1. For numbers x ∈ R such that |x − x2| < 1 4 , the polynomial p satisfies 2 2 p(x) − p2(x) = 3 + 4x − 4x2 x − x2 ≤ 4 x − x2 .
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So, 2 2 e ≤ 4 (¯ h ) (¯ h ) − e (¯ h ) ej(¯h) − e2 j−1 j j−1 2j j 2 4 ≤ 1 (¯ h ) − e (¯ h ) = O2 (¯h). e 0 0 4 By definition, e is constructed from e0 = a|I by the contour integral (5.6). However, the polynomial p maps the parts of Spec e0 above and below 1 2 to points above and below 1 , so the same contour integral with e in place of e gives the same projection, e. j 0 2 The contour integral is equivalent to applying the step function θ(x − 1 2 ) to ej, since this function is holomorphic on the spectrum. The polynomial ej − e2 j is one measure of the failure of ej to be a projection; the difference e − ej is another. To compare these, consider
1 1 1 1 1 1 1 1 − − + − ≥ − = − θ x − x x x x − x2 = x − 1 2 2 2 2 2 2 2 2 2 . The last term measures the difference between the identity and the step function. Applying this to ej shows, 2j e(¯h) − ej(¯h) ≤ 2 ej(¯h) − e2 j (¯h) = O (¯h). For 2j ≥ n + 1, this error is of order better than on(¯h), e ≈ ej mod on(¯h). (n)
The space of sections ΓQ (I , A) is an algebra and ej is just a polynomial of e0, therefore (n)
(n)
ej ∈ ΓQ (I , A) and so e ∈ ΓQ (I , A).
There is a fairly natural notion of equivalence of quantizations. Lemma 5.4. If (I, A, Q) is an order n strict deformation quantization and Q : A0 → (n)
ΓQ (I, A) such that Q0 = Q0, then (I, A, Q ) is also an order n strict deformation quantization, the quantization maps Q and Q determine gauge equivalent products on (n) (n) A0[¯h]/¯hn+1, and ΓQ (I, A) = ΓQ (I, A).
Proof. Note that this is vacuously true for n = 0, so we can assume n ≥ 1. For any (n) a ∈ A0, Q (a) ∈ ΓQ (I, A), so by definition there exists a degree n polynomial G(a) ∈ A0[¯h] such that Q (a) ≈ Q[G(a)]
mod on(¯h),
and this is unique by Lemma 2.1. It is elementary to check that G(a) depends C-linearly on a, so this defines a C[¯h]-linear map G : A0[¯h] → A0[¯h]. Let ∗n and ∗n be the (approximately associative) products determined by Q and Q in eq. (2.5). By definition ∀a, b ∈ A0[¯h], Q[G(a ∗n b)] ≈ Q (a ∗n b) ≈ Q (a)Q (b) ≈ Q[G(a)] Q[G(b)] ≈ Q[G(a) ∗n G(b)]
mod mod
on(¯h) on(¯h).
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Therefore G(a∗n b) ≡ G(a)∗n G(b) mod h ¯ n+1. Since the map G equals the identity n+1 and therefore a gauge equivalence between these modulo ¯h, it is invertible modulo ¯h n+1 products on A0[¯h]/¯h . (n)
By definition, any section a ∈ ΓQ (I, A) has a “Taylor expansion” (2.7) in terms ∈ A such that of Q. Since G is invertible modulo ¯hn+1, there also exist a0 , . . . , an 0 n a0 + · · · + anh ¯ n+1. ¯ n ≡ G(a0 + · · · + an ¯h ) mod h
This implies that n n Qh ¯ ) ≈ Qh ¯ [G(a0 + · · · + an ¯h )] ¯ (a0 + · · · + an h
n ≈ Qh ¯ (a0 + · · · + an ¯h ) ≈ a(¯h)
mod on(¯h) mod on(¯h),
(n)
(n)
i.e., this is an expansion of a in terms of Q . Therefore, ΓQ (I, A) = ΓQ (I, A). This shows in particular that these maps give cohomologous Hochschild cocycles C1 and C1 for A0. Thus (I, A, Q ) is a quantization for the same (noncommutative) Poisson structure on A0. Lemma 5.2 only applies to unital C∗ -algebras, so it is not directly applicable to an arbitrary strict deformation quantization of S2. Fortunately, an arbitrary strict deformation quantization (of a unital algebra) can always be made unital. Lemma 5.5. If (I, A, Q) is an order n ≥ 0 strict deformation quantization of a unital algebra A0 ⊂ A0, then there exists an order n unital strict deformation quantization (I , A , Q ) of A0, where I ⊆ I is a neighborhood of 0, A ⊆ A|I is a continuous subfield of C∗ -algebras, and (n)
Q : A0 → ΓQ (I , A) ∩ Γ (I , A ). Proof. 1 ∈ A0 is a projection, so by Lemma 5.3, we can correct Q(1) to a projection (n)
:= eA e for h ∈ I . These e ∈ ΓQ (I , A) given by Eq. (5.6). Define the algebras A h ¯ h ¯ ¯ form a continuous subfield, because e is a continuous projection. The projection e(¯h) . becomes the unit in A h ¯
(n)
Define V := e Q(1)e ∈ ΓQ (I , A ); this is invertible in this algebra. By construction, V ≈ e mod O1(¯h). So, V −1/2 ≈
n 1 − 2
j=0
j
(V − e)j mod On+1(¯h)
(n)
∈ ΓQ (I , A ). Define Q (a) := V −1/2Q(a)V −1/2. This satisfies Q (a) = e Q (a) = Q (a)e, so in fact Q(a) ∈ Γ (I , A ). On the unit, this gives Q (1) = V −1/2Q(1)V −1/2 = e, so is unital. Qh ¯ : A0 → A h ¯ (n)
(n)
For any a ∈ A0, Q (a) ∈ ΓQ (I , A) because V ∈ ΓQ (I , A).
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6. The Obstruction Now consider some strict deformation quantization of S2. As always, the dense subalgebra A0 ⊂ C(S2) must be closed under the Poisson bracket. I will also need to assume that it is closed under solution of the Poisson equation. In order to use the Bott projection, I will also require that the Cartesian coordinates xi ∈ C(S2) of the unit sphere S2 ⊂ R3 are contained in A0. The most natural choices of algebra A0 ⊂ C(S2) would be the algebra of smooth functions C∞ (S2) or the algebra of polynomials (in the coordinate functions xi). Both of these algebras satisfy all of the conditions above. In order to apply Lemma 5.2, we need a projection whose partial trace is sufficiently close to a multiple of the identity. That is the purpose of the next lemma. Let ∆ denote the Laplacian. Lemma 6.1. If (I, A, Q) is an order n ≥ 2, unital, strict deformation quantization of S2, such that A0 xi and ∀f ∈ C(S2), ∆f ∈ A0 =⇒ f ∈ A0,
(6.1)
(n)
then there exists a projection e ∈ ΓQ [I , Mat2(A)] defined over a neighborhood I ⊆ I of 0, such that e(0) = eBott and tr e ≈ 1 + ϑ−1 (n−2)
mod on(¯h),
2 where ϑ(n−2) ∈ h + C[¯h] is a degree n − 2 Laurent polynomial. ¯
Proof. The quantization maps naturally extend to matrices by linearity, giving a quantization of Mat2(A0). So, Lemma 5.3 applies and a projection e can be constructed by applying functional calculus to Q(eBott ). This satisfies e(0) = eBott . It is a little easier to work with the equivalent grading operator u := 2e − 1. As any 2 × 2 matrix, it can be decomposed in terms of Pauli matrices in the form, i u = τ + σix^ , i
(n)
i
where τ, x^ ∈ ΓQ (I , A). In this case, e(0) = eBott means that τ(0) = 0, and x^ (0) = xi. The identity u2 = 1 implies the commutation relations, i i j k i i j k τ x^ + x^ τ = − 2i ijk[x^ , x^ ] ≈ ¯h ¯ Q(xi) ≈ h ¯ x^ mod O2(¯h). 2 jkQ({x , x }) = h 2 Therefore τ ≈ 1 2 ¯h mod O (¯h) and
tr e = 1 + τ ≈ 1 + 1 ¯ mod O2(¯h). 2h (n)
Since tr e ∈ ΓQ (I , A), this implies that tr e ≈ 1 + 1 ¯ +h ¯ 2Q(g) mod o2(¯h) 2h
(6.2)
for some g ∈ A0. This shows that tr e equals a multiple of the identity to order O2(¯h). The next step is to improve u, so that tr e will equal a multiple of the identity to order o2(¯h).
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First, add to u some self-adjoint δ ∈ ΓQ [I , Mat2(A)] (to be chosen below), and then define u by “correcting” u + δ with functional calculus so that u 2 = 1. My choice of δ is δ := Q(f) − u Q(f)u = u[u, Q(f)] = 1 2 [u, [u, Q(f)]]
(6.3)
for some f ∈ A0. This vanishes at h ¯ = 0, so δ = O1(¯h). This δ was deliberately chosen such that it anticommutes with u, i.e., uδ = −δu. So, (u + δ)2 = 1 + δ2 and we can write u explicitly as u = (1 + δ2)−1/2(u + δ) 2 ≈ u+δ− 1 mod O3(¯h). 2 uδ
(6.4)
We need to compute tr u to second order in ¯h. Surprisingly, we only need the first 2 terms of (6.4), because the partial trace of the third term is actually of order O3(¯h). i The reason is that τ is scalar up to first order, so [τ, x^ ] = O3(¯h). Hence, tr u − tr u ≈ tr(Q(f) − uQ(f)u) = 1 2 tr[u, [u, Q(f)]]
mod o2(¯h)
i i = [x^ , [x^ , Q(f)]] + [τ, [τ, Q(f)]]
≈ −¯h2Q({xi, {xi, f}}) ≈ ¯h2Q(∆f)
mod o2(¯h).
Therefore, this will change the partial trace of the projection to tr e ≈ tr e + 1 ¯ 2Q(∆f) mod o2(¯h). 2h Now we have already seen that tr e ≈ 1 + 1 ¯ +h ¯ 2Q(g) mod o2(¯h). We need to 2h replace the function g with its average value, 1 c := gω. 4π S2 The function g − c integrates to 0 and is therefore (by the Hodge decomposition) in the image of the Laplacian. So, by the hypothesis (6.1), there exists f ∈ A0 such that −2∆f = g − c, and so
−1 2 2 tr e ≈ 1 + 1 − 4c 2 ¯h + c¯h ≈ 1 + h ¯
mod o2(¯h).
This proves the claim up to second order. To correct u from order n − 1 to order n, we can use exactly the same procedure, except with u + h ¯ n−2δ. It would have been simpler to equivalently state that tr e is approximated by a polynomial 1 + 1 2 ¯h + . . . . However, this awkward formulation of Lemma 6.1 in terms of ϑ(n−2) simplifies Theorem 6.2, and the notation ϑ is motivated by Theorem 6.3 below.
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Theorem 6.2. If (I, A, Q) is an order n ≥ 2, unital, strict deformation quantization of S2, such that A0 xi and ∀f ∈ C(S2) ∆f ∈ A0 =⇒ f ∈ A0, then there is no connected neighborhood of 0 in I and in particular I is not connected. 2 More precisely, there exists a degree n − 2 Laurent polynomial ϑ(n−2) ∈ h + C[¯h] such ¯ that ∀¯h = 0 ∈ I, dist [ϑ(n−2) (¯h), Z] = on−2(¯h).
(6.5)
This Laurent polynomial is unique modulo adding a constant integer. If the quantization is unital, then ϑ(n−2) is the same as in Lemma 6.1. Proof. The first claim will follow from (6.5). Note that (6.5) is true if and only if it is true for a neighborhood I ⊂ I of 0, so by Lemmas 5.4 and 5.5, it is sufficient to prove (6.5) for a unital quantization. So, assume the quantization is unital and let e be the projection and ϑ(n−2) the Laurent polynomial from Lemma 6.1. Define E(¯h) := tr(e) − 1 − ϑ−1 (n−2) . By Lemma 6.1, this is of order on(¯h). For any nonzero ¯h ∈ I, Lemma 5.2 says that there exists k ∈ Z such that 1 ϑ(n−2) (¯h)−1 − E(¯h) ≤ ≤ ϑ(n−2) (¯h)−1 + E(¯h). k Since E(¯h)|ϑ(n−2) (¯h)| = on−1(¯h), it converges to 0 and is less than 1 for h ¯ sufficiently small, and dist [ϑ(n−2) (¯h), Z] ≤ |ϑ(n−2) (¯h) − k| ≤
E(¯h)ϑ2 h) (n−2) (¯ 1 − E(¯h)|ϑ(n−2) (¯h)|
= on−2(¯h).
This proves the existence of ϑ(n−2) . 2 ∈ h + C[¯h] is another degree n − 2 Laurent polynomial Now, suppose that ϑ(n−2) ¯ satisfying Eq. (6.5). Then ϑ(n−2) − ϑ(n−2) ∈ C[¯h] and dist[ϑ(n−2) (¯h) − ϑ(n−2) (¯h), Z] = on−2(¯h).
(6.6)
Taking the limit ¯h → 0 shows that m := ϑ(n−2) (0) − ϑ(n−2) (0) ∈ Z. So, ϑ(n−2) (¯h) − ϑ(n−2) (¯h) − m = O1(¯h)
is smaller than 1 2 for ¯h sufficiently small, and thus (6.6) becomes (¯h) − m = on−2(¯h), ϑ(n−2) (¯h) − ϑ(n−2) but ϑ(n−2) − ϑ(n−2) is of degree n − 2, so this proves that ϑ(n−2) (¯h) = ϑ(n−2) (¯h) + m. 2 For n = 2, we can write ϑ(n−2) = h + c. So, for all sufficiently large integers k ∈ Z, ¯
−1 ∈ I. 2 k−c+ 1 2
Since 0 ∈ I is an accumulation point, this shows that I — or any neighborhood of 0 in I — cannot be connected.
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If the quantization was only first order, then we would still have that 1 tr e ≈ 1 + 1 2 ¯h mod o (¯h),
but this would not restrict the values of h ¯ at all. Even if the error is of order O2(¯h), this would be inadequate. In that case, 2/¯h is restricted to a sequence of intervals around integers, but the width of the intervals is only bounded (rather than convergent to 0), so the intervals may overlap. The next (and final) theorem connects Theorem 6.2 with Fedosov’s integrality condition. Theorem 6.3. If (I, A, Q) is an infinite order, strict deformation quantization of C∞ (S2) and θ ∈ h ¯ −1H2(S2)[[¯h]] is the characteristic class of the corresponding formal deformation quantization, then for any n ≥ 2 ∈ N the Laurent polynomial ϑ(n−2) in Lemma 6.1 and Theorem 6.2 is the truncation of ϑ := θ. S2
In other words, ϑ(¯h) is the asymptotic expansion of some map I {0} → Z. Proof. Let ∗ be the formal deformation quantization product obtained by asymptotically expanding Q(f)Q(g). Because H2(S2) is 1-dimensional, any formal deformation quantization of S2 is gauge equivalent to an SU(2)-equivariant formal deformation quantization. Let ∗ be such an equivalent equivariant formal product, and let G : C∞ (S2)[[¯h]] → C∞ (S2)[[¯h]] be the equivalence; that is, G(f ∗ g) = G(f) ∗ G(g). By equivariance, there exist R = 1 + . . . and η = ¯h + · · · ∈ C[[¯h]] such that xi ∗ xi = R2 and ij
[xi, xj]∗ = iη kxk. 2 However, by a minor rescaling, we can always choose ∗ such that R2 = 1 − 1 4η . 2 Let u0 := σixi = 2eBott − 1. The ∗ product with itself is u0 ∗ u0 = 1 + 1 4 η + ηu0. 1 1 So, (u0 + 2 η) ∗ (u0 + 2 η) = 1 and 1 eh ¯ := eBott + 4 η = 1 + 1 η. is a projection for the product ∗ . This has partial trace tr e h 2 ¯ ∈ C[[¯h]], the algebraic index theorem shows that Since tr e h ¯ θ Tr ¯h e h e ∧ ch eBott = 1 + θ ¯ = S2 S2
! 1 1 = 1 + 2 η Tr ¯h 1 = 1 + 2 η θ.
Therefore, S2 θ = 2η−1 = 2¯h−1 + . . . .
S2
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Now let G(n) be the truncation of G up to order ¯hn. The idea is that Q ◦ G(n) is an approximately equivariant quantization map. Define, e0 := Q[G(n)(eBott )] and observe that (n) e2 (eBott ∗ eBott )] mod on(¯h). 0 ≈ Q[G
So if we correct e0 to a projection e by functional calculus, then it is approximately mod on(¯h).
e ≈ e0 + 1 4η This has partial trace
−1
tr e = 1 +
1 2η
≈1+
S2
θ(n−2)
where θ(n−2) is the truncation of θ at degree ¯hn−2.
mod on(¯h),
Note that this reproves Lemma 6.1, but with different hypotheses. Lemma 6.1 applies to any order ≥ 2 quantization with minor restrictions on A0. Theorem 6.3 requires an infinite order quantization with A0 = C∞ (S2), although the proof could probably be extended to finite orders by generalizing the classification results for formal deformation quantization in the literature. I have retained Lemma 6.1 because the hypotheses are slightly more general and the proof is explicit and constructive. 7. Remarks Theorem 6.2 shows that I {0} is very close to being discrete. It is contained in a union of smaller and smaller intervals, and for an infinite order quantization, these intervals shrink faster than any power of h ¯. This seems to be the strongest possible result. Consider the following variant of the Berezin-Toeplitz quantization of S2. Let I := {0} ∪
∞
2 N
2 − e−N, N + e−N ,
N=0 2 2 and for ¯h ∈ [ N − e−N, N + e−N], A h ¯ := L(HN), and Q h ¯ := TN. This defines an infinite order, algebraically closed, strict deformation quantization of S2. The only peculiar feature of this quantization is that Q h ¯ = 0, but ¯ is locally constant away from h an additional axiom to rule that out would probably rule out the trivial quantization of a manifold with 0 Poisson structure. The Bott projection is not the only 2 × 2 projection over S2. In fact, a 2 × 2 projection on S2 is equivalent to a map from S2 to CP1 = S2. The degree of the map gives the Chern number of the projection. Suppose that we started with a 2 × 2 projection with Chern number s, and applied h)] = the Berezin-Toeplitz quantization of S2. Then the normalized trace would be Tr[e(¯ s/(N + 1). This is not always the reciprocal of an integer. So, how does Lemma 5.2 “know” that we started with the Bott projection?
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The answer is that eBott and 1 − eBott are the only nontrivial equivariant 2 × 2 projections on S2. Any other such projection is rather far from being equivariant. Consequently, if we quantize it, then tr e(¯h) is never close to a multiple of the identity; its spectrum is always too wide for Lemma 5.2 to be nontrivial. It would be nice to show that the quantization of S2 is essentially unique, i.e., that the algebras A h ¯ =0 are all simple matrix algebras, and the Berezin-Toeplitz maps define continuous sections of the continuous field. However, the present axioms are not strong enough to imply this. Given Lemma 6.1, the proof of Lemma 5.2 can be extended slightly to show that Ah ¯ contains a subalgebra isomorphic to Matk(C), where k is the closest integer to |ϑ(n−2) (¯h)|, and that these subalgebras form a continuous subfield of A with structure equivalent to that given by Berezin-Toeplitz quantization. This is what Rieffel essentially proved in the equivariant case. ∼ However, this does not show that A h ¯ = Matk(C). I believe that would require some additional irreducibility axiom. The idea would be to require that any “subquantization” of (I, A, Q) is essentially the whole thing. However, I do not know of a reasonable formulation of such an axiom. For example, the trivial quantization of R should not be ruled out by the axioms. In this case, I is an interval, A0 = A h ¯ = C0(R) and Q h ¯ is the identity map. This has proper subquantizations corresponding to bundles of open subintervals in R × I, although these are isomorphic to (I, A, Q). It is plausible that the missing axiom is simply the condition that the quantization ∗ be algebraically closed (i.e., Im Q h ¯ is a -subalgebra). Indeed, it may be that algebraic closedness would imply that I is disconnected for S2, without requiring the quantization to be second order. I only know that the technique I used in Lemma 6.1 requires a second order quantization. It would be interesting to find an alternative proof using algebraic closedness, or alternatively to find an example of a first order, algebraically closed, strict deformation quantization of S2 with I connected. The next major step should be to generalize Theorem 6.3 to general symplectic manifolds. The key to this is perhaps some generalization of Lemma 5.2 to larger matrices. Acknowledgements. I would like to thank Klaas Landsman and Marc Rieffel for their comments, and Ryszard Nest for encouraging me to investigate this question. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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6. Klimek, S., Lesniewski, A.: Quantum Riemann surfaces for arbitrary Planck’s constant. J. Math. Phys. 37(5), 2157–2165 (1996) 7. Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003) 8. Landsman, N.P.: Mathematical topics between classical and quantum mechanics. Springer Monographs in Mathematics. New York: Springer-Verlag, 1998 9. Landsman, N.P., Ramazan, B.: Quantization of Poisson algebras associated to Lie algebroids. In: Groupoids in analysis, geometry, and physics (Boulder, CO, 1999), Contemp. Math., 282, Providence, RI: Amer. Math. Soc., 2001, pp. 159–192 10. Li, H.: Strict quantizations of almost Poisson manifolds. Commun. Math. Phys. 257(2), 257–272 (2005) 11. Natsume, T., Nest, R., Peter, I.: Strict quantizations of symplectic manifolds. Lett. Math. Phys. 66 (1–2), 73–89 (2003) 12. Nest, R., Tsygan, B.: Formal versus analytic index theorems. Internat. Math. Res. Notices 1996(11), 557–564 (1996) 13. Rieffel, M.A.: C∗ -algebras associated with irrational rotations. Pacific J. Math. 93(2), 415–429 (1981) 14. Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122(4), 531–562 (1989) 15. Rieffel, M.A.: Lie group convolution algebras as deformation quantizations of linear Poisson structures. Amer. J. Math. 112(4), 657–685 (1990) 16. Rieffel, M.A.: Deformation quantization for actions of Rd . Mem. Amer. Math. Soc. 106(506) (1993) 17. Sheu, A.J.-L.: Quantization of the Poisson SU(2) and its Poisson homogeneous space—the 2-sphere. With an appendix by Jiang-Hua Lu and Alan Weinstein. Commun. Math. Phys. 135(2), 217–232 (1991) 18. Wassermann, A.: Ergodic actions of compact groups on operator algebras. III. Classification for SU(2). Invent. Math. 93(2), 309–354 (1988) 19. Weinstein, A.: Deformation quantization. Séminaire Bourbaki, Vol. 1993/94, Astérisque No. 227, Exp. No. 789, 5, 389–409 (1995) 20. Xu, P.: Noncommutative Poisson algebras. Amer. J. Math. 116(1), 101–125 (1994) Communicated by A. Connes
Commun. Math. Phys. 283, 701–727 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0573-7
Communications in
Mathematical Physics
Finsler Spinoptics C. Duval Centre de Physique Théorique , CNRS, Luminy, Case 907, F-13288 Marseille Cedex 9, France. E-mail: [email protected] Received: 11 September 2007 / Accepted: 8 April 2008 Published online: 29 July 2008 – © Springer-Verlag 2008
Abstract: The objective of this article is to build up a general theory of geometrical optics for spinning light rays in an inhomogeneous and anisotropic medium modeled on a Finsler manifold. The prerequisites of local Finsler geometry are reviewed together with the main properties of the Cartan connection used in this work. Then, the principles of Finslerian spinoptics are formulated on the grounds of previous work on Riemannian spinoptics, and relying on the generic coadjoint orbits of the Euclidean group. A new presymplectic structure on the indicatrix-bundle is introduced, which gives rise to a foliation that significantly departs from that generated by the geodesic spray, and leads to a specific anomalous velocity, due to the coupling of spin and the Cartan curvature, and related to the optical Hall effect. Contents 1. 2.
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Introduction . . . . . . . . . . . . . . . . . . . Finsler Structures: A Compendium . . . . . . . 2.1 Finsler metrics . . . . . . . . . . . . . . . . 2.1.1 An overview. . . . . . . . . . . . . . . 2.1.2 Introducing special orthonormal frames. 2.1.3 The non-linear connection. . . . . . . . 2.2 Finsler connections . . . . . . . . . . . . . 2.2.1 The Chern connection. . . . . . . . . . 2.2.2 The Cartan connection. . . . . . . . . . Geometrical Optics in Finsler Spaces . . . . . . 3.1 Finsler geodesics . . . . . . . . . . . . . . 3.2 Geometrical optics in anisotropic media . .
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3.2.1 The Fermat Principle. . . . . . . . . . . . . . . . . . . . 3.2.2 Finsler optics. . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The example of birefringent solid crystals. . . . . . . . . Geometrical Spinoptics in Finsler Spaces . . . . . . . . . . . . . 4.1 Spinoptics and the Euclidean group . . . . . . . . . . . . . . 4.1.1 Colored light rays. . . . . . . . . . . . . . . . . . . . . 4.1.2 The spinning and colored Euclidean coadjoint orbits. . . 4.2 Spinoptics in Finsler-Cartan spaces . . . . . . . . . . . . . . 4.2.1 Minimal coupling to the Cartan connection . . . . . . . 4.2.2 The Finsler-Cartan spin tensor. . . . . . . . . . . . . . . 4.2.3 Laws of geometrical spinoptics in Finsler-Cartan spaces. Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Geometry and optics have maintained a lasting relationship since Euclid’s Optics where light rays were first interpreted as oriented straight lines in space (or, put in modern terms, as oriented, non-parametrized, geodesics of Euclidean space). One can, withal, trace back the origin of the calculus of variations to Fermat’s Principle of least optical path. This principle has served as the basis of geometrical optics in inhomogeneous, isotropic, media and proved a fundamental mathematical tool in the design of optical (and electronic) devices such as mirrors, lenses, etc., and in the understanding of caustics, and optical aberrations. Although Maxwell’s theory of wave optics has unquestionably clearly superseded geometrical optics as a bonafide theory of light, the seminal work of Fermat has opened the way to wide branches of mathematics, physics, and mechanics, namely, to the calculus of variations in the large, modern classical (and quantum) field theory, Lagrangian and Hamiltonian or presymplectic mechanics. It should be stressed that Fermat’s Principle has, in essence, a close relationship to modern Finsler geometry, as it rests on a specific “Lagrangian” F(x, y) = n(x) δi j y i y j , (1.1) where y = (y i ) stands for the “velocity” of light, and n(x) > 0 for the value of the (smooth) refractive index of the medium, at the “location” x in Euclidean space. (Note that Einstein’s summation convention is tacitly understood throughout this article.) As a matter of fact, the function (1.1) is a Finsler metric, namely a positive function, homogeneous of degree one in the velocity, smooth wherever y = 0, and such that the Hessian gi j (x, y) = 21 F 2 y i y j is positive-definite (see Sect. 2.1). Although, this is a very special case of Finsler metric—it actually defines a conformally flat Riemannian metric tensor, viz., gi j (x) = n2 (x) δi j —, this fact is worth noting for further generalization. The geodesics of the Fermat-Finsler metric (1.1) are a fairly good mathematical model for light rays in refractive, inhomogeneous, and non-dispersive media—provided polarization of light is ignored! It has quite recently been envisaged to consider a general Finsler metric F(x, y) to describe anisotropy of optical media, as the Finsler metric tensor, gi j (x, y), depends, in general non-trivially, on the direction of the velocity, y, or “élément de support” in the sense of Cartan [18]. This enables one to account for the fact that [3,28]
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In an anisotropic medium, the speed of light depends on its direction, and the unit surface is no longer a sphere. (Finsler, 1969) The Fermat Principle has also been reformulated in the presymplectic framework in [16], and generalized in [17] to the context of anisotropic media. By the way, the regularity condition imposed, in the latter reference, amounts to demanding a Finsler structure. One thus takes for granted that oriented Finsler geodesics may describe light rays in anisotropic media. A Finslerian version of the Fermat Principle now states that the second-order differential equations governing the propagation of light stem from the geodesic spray of a (three-dimensional) Finsler space, (M, F), given by the Reeb vector field of the contact 1-form = ω3 ,
(1.2)
where ω3 = Fy i d x i is, here, the restriction to the indicatrix-bundle, S M = F −1 (1), of the Hilbert 1-form. See Sect. 3, and also [23]. Now, geometrical optics is, from a different standpoint, widely accepted as a semiclassical limit [14] of wave optics with “small parameter” the reduced wavelength λ (or λ/L, where L is some characteristic length of the optical medium, see, e.g., [13]). It has, however, recently been established on experimental grounds that trajectories of light beams in inhomogeneous optical media depart from those predicted by geometrical optics. See, e.g. [9,10,12], and [33,34], for several approaches to photonic dynamics in terms of a semi-classical limit of the Maxwell equations in inhomogeneous, and isotropic media, highlighting the Berry connection [8]. See also [24]. The so-called optical Hall effect for polarized light rays, featuring a very small transverse shift, orthogonal to the gradient of the refractive index, has, hence, received a firm theoretical explanation. From quite a different perspective, a theory of spinning light in an arbitrary threedimensional Riemannian manifold has been put forward as an extension of the Fermat Principle to classical, circularly polarized photons, in inhomogeneous, (essentially) isotropic, media. This theory of spinoptics, presented in [21], and [22], relies fundamentally on the Euclidean group, E(3), viewed at the same time as the group of isometries of Euclidean space, E 3 , and as the group of symmetries of classical states of free photons represented by Euclidean coadjoint orbits. Straightforward adaptation of the general relativistic prescription of minimal coupling [31,40,41] readily yielded a set of differential equations governing the trajectories of spinning light in inhomogeneous, and isotropic media described by a Riemannian structure. Also this formalism for spinoptics helped put the optical Hall effect in proper geometrical perspective, in agreement with [33]. The main purport of the present article is, as might be expected, to try and provide a fairly natural extension of plain geometrical optics—in non-dispersive, anisotropic, media described in terms of Finsler geodesics—to spinoptics, i.e., to the case of circularly polarized light rays carrying color and helicity in such general optical media. In doing so, one must unavoidably choose a linear Finsler connection from the start (see (1.3)), the crux of the matter being that there is, apart from the special Riemannian case, no canonical Finsler connection at hand. The challenge may, in fact, be accepted once we take seriously the Euclidean symmetry as a guiding principle, a procedure that can be implemented by considering the dipole approximation to ordinary geometrical optics, namely the spinning coadjoint orbits of the Euclidean group. This is the subject of Sect. 4 which contains the main results of this article, where the Finsler-Cartan connection prevailed definitely over other Finsler connections, as regard to the original, fundamental, Euclidean symmetry of the free model. Let us, however, mention that
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all resulting expressions for the foliations we end up with ultimately depend upon the Finsler metric tensor, the Cartan tensor, and the Chern curvature tensors only. The hereunder proposed principle of Finsler spinoptics (see Axiom 4.5, which can be understood as the prescription of minimal coupling to a Finsler-Cartan connection) thus amounts to consider, instead of (1.2), the following 1-form: = ω3 + λ ω12 ,
(1.3)
where λ is the (signed) wavelength, the ωab , with a, b = 1, 2, 3, representing the components of the Cartan connection associated with a three-dimensional Finsler manifold (M, F). The 1-form (1.3) might be considered as providing a deformation of the Hilbert 1-form driven by the wavelength parameter, λ. See Remark 4.6 below. The characteristic foliation of the novel 2-form σ = d is explicitly calculated, and leads to a drastic deviation from the Finsler geodesic spray, dictated by spin-curvature coupling terms which play, in this formalism, quite a significant rôle, as expressed by Theorem 4.12. Of course, the equations of spinoptics in a Riemannian manifold [21] are recovered, as a special case of those corresponding to a Finsler-Cartan structure. We assert that this foliation can be considered a natural extension of the Finsler geodesics spray to the case of spinoptics in Finsler-Cartan spaces. The paper is organized as follows. Section 2 provides a survey of local Finsler geometry. We found it necessary to offer a somewhat technical and detailed introduction of the objects pertaining to Finsler geometry, in particular to the various connections used throughout this article, to make the reading easier to non-experts. Emphasis is put on the Chern and Cartan connections, as these turn out to be of central importance in this study. This section relies essentially on the authoritative Reference [6]. In Sect. 3, we review the principles of geometrical optics, extending Fermat’s optics to the area of Finsler structures characterizing anisotropic optical media. Then, special attention is paid to the Hilbert 1-form in the derivation of the Finsler geodesic spray. The connection of the latter to the Fermat differential equations associated with conformally related Finsler structures is furthermore analyzed. Section 4 constitutes the major part of the article. It presents, in some details, the basic structures arising in the classification of the SE(3)-homogeneous symplectic spaces, which are interpreted as the seeds of spinoptics, namely the Euclidean coadjoint orbits labeled by color, and spin, according to the classic [39]. The core of our study consists in the choice of a special Finsler connection, namely the Finsler-Cartan connection, to perform minimal coupling of spinning light particles to a Finsler metric. This is done and explained in this section, in which the derivation of the characteristic foliation of our distinguished presymplectic 2-form d , see (1.3), is spelled out in detail. This completes the introduction of Finslerian spinoptics. Conclusions are drawn in Sect. 5, and perspectives for future work connected to the present study are finally outlined. 2. Finsler Structures: A Compendium 2.1. Finsler metrics. 2.1.1. An overview. A Finsler structure is a pair (M, F) where M is a smooth, n-dimensional, manifold and F : T M → R+ a given function whose restriction to
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the slit tangent bundle T M \ M = {(x, y) ∈ T M | y ∈ Tx M \{0}} is smooth, and (fiberwise) positively homogeneous of degree one, i.e., F(x, λy) = λF(x, y), for all λ > 0; one furthermore demands that the n × n Hessian matrix with entries 1 2 F gi j (x, y) = (2.1) 2 yi y j be positive-definite for all (x, y) ∈ T M \ M. The quantities gi j defined in (2.1) are (fiberwise) homogeneous of degree zero, and g = gi j (x, y)d x i ⊗ d x j
(2.2)
defines a sphere’s worth of Riemannian metrics [7] on each Tx M parametrized by the direction of y. We will put (gi j ) = (gi j )−1 . If π : T M\M → M stands for the canonical surjection, the metric (or fundamental) “tensor” (2.2) is, actually, a section of the bundle π ∗ (T ∗ M) ⊗ π ∗ (T ∗ M) → T M \ M. The distinguished “vector field” (the direction of the supporting element) u = ui
∂ yi i , , where u (x, y) = ∂ xi F(x, y)
(2.3)
is, indeed, a section of π ∗ (T M) → T M \ M such that g(u, u) = gi j u i u j = 1. There is, at last, another tensor specific to Finsler geometry, namely the Cartan tensor, C = Ci jk (x, y)d x i ⊗ d x j ⊗ d x k , where Ci jk (x, y) = 41 F 2 y i y j y k . As in [6], we will also use ad lib. the quantities Ai jk = F Ci jk ,
(2.4)
which are totally symmetric, Ai jk = A(i jk) , and enjoy the following property, viz., Ai jk u k = 0.
(2.5)
There is a wealth of Finsler structures, apart fromthe well-known special case of Riemannian structures (M, g) for which F(x, y) = gi j (x)y i y j . See, e.g., [6,7,38] for a survey, and for a list of examples of Finsler structures. We will review below, see (3.11)–(3.13), examples of Finsler structure associated with optical birefringence [3]. 2.1.2. Introducing special orthonormal frames. Having chosen a coordinate system (x i ) of M, we denote—with a slight abuse of notation—by ∂/∂ x i (resp. d x i ) the so-called transplanted sections of π ∗ (T M) (resp. π ∗ (T ∗ M)). Accordingly we will denote by ∂/∂ y i (resp. dy i ) the standard, vertical, sections of T (T M \ M) (resp. T ∗ (T M \ M)). Introduce now special g-orthonormal frames (e1 , . . . , en ) for π ∗ (T M), such that g(ea , eb ) = δab ,
(2.6)
for all a, b = 1, . . . , n, with, as the preferred element, the distinguished section en = u.
(2.7)
Recall that each ea lies in the fiber π ∗ (T M)(x,y) above (x, y) ∈ T M \ M. The local decomposition of these vectors is given by ea = eai
∂ , ∂ xi
(2.8)
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for all a = 1, . . . , n, where the matrix (eai ), defined at (x, y) ∈ T M \M, is nonsingular. We thus have eni = u i .
(2.9)
The dual frames, for π ∗ (T ∗ M), which we denote by (ω1 , . . . , ωn ), are such that = δba , for all a, b = 1, . . . , n. Accordingly, we have the local decomposition
ωa (eb )
ωa = ωia d x i ,
(2.10)
for all a = 1, . . . , n, where (ωia ) = (eai )−1 . The 1-form dual to en is the Hilbert form ω H = ωn
(2.11)
which, in view of (2.3), reads ω H = u i d x i , with u i = gi j u j = Fy i . The following proposition introduces the principal bundle of orthonormal frames above the slit tangent bundle of a Finsler manifold. Proposition 2.1. The manifold, SOn−1 (T M \ M), of special g-orthonormal frames for π ∗ (T M) is endowed with a structure of SO(n − 1)-principal bundle over T M \ M. If GL+ (M) stands for the bundle of positively oriented linear frames of M, it is defined by 1 SOn−1 (T M \ M) = −1 (0), where : π ∗ (GL+ (M)) → R 2 n(n+1) × Rn−1 is given by (x, y, (ea )) = ((g(x,y) (ea , eb ) − δab ), y − F(x, y)en ), where a, b = 1, . . . , n. The proof is straightforward as is that of the following statement. Corollary 2.2. Let S M = F −1 (1) denote the indicatrix-bundle of a Finsler manifold (M, F), and ι : S M → T M \ M its embedding into the tangent bundle of M. The pull-back SOn−1 (S M) = ι∗ (SOn−1 (T M \ M)) is a principal SO(n − 1)-bundle over S M. 2.1.3. The non-linear connection. Let us recall that the Finsler metric, F, induces in a canonical fashion a splitting of the tangent bundle of the slit tangent bundle π : T M\M → M of M as follows: T (T M \ M) = V (T M \ M) ⊕ H (T M \ M),
(2.12)
where the vertical tangent bundle is V (T M \ M) = ker π∗ . The fibers, V(x,y) , of that subbundle are spanned by the vertical local basis vectors ∂/∂ y i , with i = 1, . . . , n. Those, H(x,y) , of the horizontal subbundle H (T M \ M) are spanned by the horizontal local basis vectors ∂ δ j ∂ = i − Ni j, δx i ∂x ∂y
(2.13)
j
with i = 1, . . . , n, where the Ni are the coefficients of a non-linear connection canonically defined by 1 ∂G i 2 ∂yj in terms of the spray coefficients G j = 21 g jk (F 2 ) y k x l y l − (F 2 )x k . N ij =
(2.14)
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The horizontal vectors, δ/δx i , see (2.13), and the vertical vectors, ∂/∂ y i , form a local natural basis for T(x,y) (T M \ M) whose dual basis is given by d x i and δ y i , where δ y i = dy i + N ij d x j .
(2.15)
We now introduce, following (2.8), (2.10) the g-orthonormal frames that will be needed in the sequel. Definition 2.3. We will call an hv-frame any frame for T (T M\M), compatible with the splitting (2.12), namely [6] eˆa = eai
δ , δx i
eˆa¯ = eai F
∂ , ∂ yi
(2.16)
where a¯ = a + n, with a = 1, . . . , n. The associated dual basis reads then ωa = ωia d x i ,
ωa¯ = ωia
δ yi , F
(2.17)
with a = 1, . . . , n; see (2.10) and (2.15). Owing to the properties of the non-linear connection (2.14), the metric, F, is horizontally constant, δF = 0, δx i
(2.18)
for all i = 1, . . . , n. This entails that the vertical 1-form ωn¯ is exact, ωn¯ = d log F.
(2.19)
2.2. Finsler connections. Unlike the Riemannian case, there is no canonical linear Finsler connection on the bundle π ∗ (T M), which is required as soon as one needs to differentiate tensor fields, i.e., sections of the bundles π ∗ (T M)⊗ p ⊗ π ∗ (T ∗ M)⊗q . 2.2.1. The Chern connection. A celebrated example, though, is the Chern connection ω ji = i jk (x, y)d x k which is uniquely defined by the following requirements [6]: (i) it is symmetric: i jk = ik j , and (ii) it almost transports the metric tensor: dgi j −ωik g jk − ω jk gik = 2 Ci jk δ y k , where the δ y k are as in (2.15). The Chern connection coefficients turn out to yield the alternative expression of the non-linear connection, namely, N ij (x, y) = i jk y k .
(2.20)
The covariant derivative ∇ : (π ∗ (T M)) → (T (T M \M) ⊗ π ∗ (T M)) associated with the Chern connection is related to the ω ji via ∇ X ∂/∂ x j = ω ji (X )∂/∂ x i , for all X ∈ Vect(T M\M). In terms of the special g-orthonormal frames, we can write ∇ X eb = j ωba (X )eb , for all X ∈ Vect(T M \ M), where ωba = ωia (debi + ω ji eb ) denote the frame components of the Chern connection. The following theorem summarizes the defining properties of the Chern connection.
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Theorem 2.4. [6]. There exists a unique linear connection, (ωba ), on π ∗ (T M), named the Chern connection, which is torsionfree and almost g-compatible, namely
a = dωa − ωb ∧ ωba = 0
(2.21)
ωab + ωba = −2 Aabc ωc¯ ,
(2.22)
and
where a, b, c = 1, . . . , n. The corresponding connection coefficients are of the form δg jk δgkl δg jl 1 , (2.23) i jk = gil + − 2 δx j δx k δx l for all i, j, k = 1, . . . , n. The curvature, ( ab ), of the linear connection, (ωba ), is defined by the structure equations ba = dωba − ωbc ∧ ωca . It retains the form
ba =
1 a ¯ R ωb ∧ ωd + Pbacd ωb ∧ ωd , 2 b cd
(2.24)
where R ji kl = eai ωbj ωkc ωld Rbacd , reads R ji kl =
δ i jl δx k
−
δ i jk δx l
+ imk mjl − iml mjk ,
(2.25)
and, accordingly, P ji kl = −F
∂ ijk ∂ yl
.
(2.26)
The hv-curvature, P, enjoys the fundamental property P ji kl u l = 0.
(2.27)
Using Cartan’s notation [18] (see also [6,36]), we write the covariant derivative of, e.g., a section X of π ∗ (T M) as (∇ X )i = d X i + ω ji X j = X |i j d x j + X i j F −1 δ y j , where i, j = 1, . . . , n. In particular, it can be easily deduced from (2.20) that the covariant derivative of the unit vector u is given by (∇u)i = −u i d log F +
δ yi , F
(2.28)
so that (2.18) leads to u i| j = 0,
u i j = δ ij − u i u j ,
(2.29)
for all i, j = 1, . . . , n. Let us deduce from (2.28) a classical formula highlighting a special property of the Chern connection.
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Proposition 2.5. There holds ¯
ωna = h ba ωb ,
(2.30)
where the h ab = δab − δan δbn , with a, b = 1, . . . , n, are the frame-components of the “angular metric”. We end this section by a useful lemma (see Sect. 3.4 B in [6] for a proof). Lemma 2.6. The first Bianchi identities imply Ai j[k l] = Ai j[k u l] , where the square brackets denote skew-symmetrization. If we define the covariant derivative of the Cartan tensor (2.4) in the direction, u, of the supporting element by A˙ i jk = Ai jk|l u l ,
(2.31)
then, the same Bianchi identities lead to Pi jk = u l Pli jk = − A˙ i jk . 2.2.2. The Cartan connection. The Cartan connection is another prominent Finsler linear connection on π ∗ (T M) which is related to the Chern connection in a simple way. Definition 2.7. [5,6]. Let (M, F) be a Finsler structure with Cartan tensor (Aabc ), and let (ωab ) denote its canonical Chern connection. The frame components of the Cartan connection of (M, F) are defined by ωab = ωab + Aabc ωc¯ ,
(2.32)
where c¯ = c + n, with a, b, c = 1, . . . , n. The fundamental virtue of these connection 1-forms is the skewsymmetry ωba = 0, ωab +
(2.33)
= 0, where ∇ stands for that guarantees that the fundamental tensor is parallel, ∇g the covariant derivative associated with the Cartan connection. The Cartan connection a = dωa − ωb ∧ is not symmetric; its torsion tensor ωba , is nonzero. Indeed, a a b c ¯ a
= − ω ∧ Aabc ω , and, since = 0, it retains the form a = −Aabc ωb ∧ ωc¯ .
(2.34)
One easily proves the following result. Theorem 2.8. There exists a unique linear connection, ( ωba ), on π ∗ (T M) whose torsion, a = 0, as expressed by (2.33). ), is given by (2.34), and which is g-compatible, ∇g ( This connection is the Cartan connection (2.32). Other characterizations of the Cartan connection can be found in the literature, e.g, in [1,3,4]. Proposition 2.9. The torsion of the Cartan connection is such that n = 0.
(2.35)
From now on, and whenever possible, we will use frame indices, a, b, c, . . ., rather than local coordinate indices, i, j, k, . . ..
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a = d Proposition 2.10. [5]. The curvature ωba − ωbc ∧ ωca of the Cartan connection b is given by ab = ab + Aabc nc + Aabd|c ωc ∧ ωd¯ + Aead Abce ωc¯ ∧ ωd¯ ,
(2.36)
where ( ba ) denotes the Chern curvature 2-form (2.24) with (2.25, 2.26), and (Aabc ) the Cartan tensor (2.4). One has the decomposition ba =
1 a bacd ωc ∧ ωd¯ + 1 Q a ωc¯ ∧ ωd¯ , ωc ∧ ωd + P R 2 b cd 2 b cd
(2.37)
with bacd = Rbacd + Aabe R ecd , R bacd = Pbacd + Aabd|c − Aabe A˙ ecd , P bacd = 2 Aae[c Aed]b , Q
(2.38) (2.39) (2.40)
where we use the notation R ecd = Rne cd , and A˙ abc = Aabc|n (see (2.31)). 3. Geometrical Optics in Finsler Spaces 3.1. Finsler geodesics. Following Souriau’s terminology [39], we call evolution space the indicatrix-bundle S M = F −1 (1)
(3.1)
above M, as it actually hosts the dynamics given by a presymplectic structure; the latter will eventually be inherited from the Finsler metric on T M \ M. Denote, again, by ι : S M → T M \ M the canonical embedding. The fundamental geometric object governing the geodesic spray on S M is the 1-form = ι∗ ω H ,
(3.2)
i.e., the pull-back on S M of Hilbert 1-form ω H (see (2.11)). The direction of this 1-form defines a contact structure on the (2n − 1)-dimensional manifold S M, since ∧ (d )n−1 = 0. See [19,20,23]. The following lemma is classical. Lemma 3.1. The exterior derivative of the Hilbert 1-form is given by ¯
dω H = δ AB ω A ∧ ω B
(3.3)
with A, B = 1, . . . , n − 1. Remark 3.2. The exterior derivative of the Hilbert 1-form is independent of the choice of a linear connection; it depends only on the non-linear connection (2.12). The main result regarding Finsler geodesics can be stated as follows. See also [23] for a full account on the geometry of second order differential equations. Theorem 3.3. The geodesic spray of a Finsler structure (M, F) is the vector field X of S M uniquely defined by σ (X ) = 0, where σ = d .
(X ) = 1,
(3.4)
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Proof. Write X ∈ Vect(S M) in the form X = X A eˆ A + X n eˆn + X A eˆ A¯ + X n¯ eˆn¯ , see (2.16), using dummy indices A and A¯ = A+n, where A = 1, . . . , n−1. Since X ∈ Vect(T M\M) is tangent to S M iff X (F) = ωn¯ (X ) = 0, as is clear from (2.19), we have X ∈ ker(σ ) iff dω H (X ) + λωn¯ = 0, where λ ∈ R is a Lagrange multiplier. The latter equation readily ¯ ¯ ¯ yields, with the help of (3.3), δ AB (X A ω B − X B ω A ) + λωn¯ = 0, hence X A = X A = 0, n n ¯ n ¯ n ¯ and λ = 0. Then X = X eˆn + X eˆn¯ is actually tangent to S M if ω (X ) = X = 0, which leads to X ∈ ker(σ )
⇐⇒
X = X n eˆn
(3.5)
for some X n ∈ R. Thus, (S M, σ ) is a presymplectic manifold. The quotient S M/ ker(σ ) is the set of oriented Finsler geodesics, which (if endowed with a smooth structure) becomes a (2n − 2)-dimensional symplectic manifold, see [2]. We then find that (X ) = X n , and the constraints (3.4) express the fact that X is the Reeb vector field, and retains the form X = eˆn , which, in view of (2.7), we can write X = ui
δ . δx i
The vector field (3.6) of S M is the geodesic spray [6] of the Finsler structure.
(3.6)
3.2. Geometrical optics in anisotropic media. The geodesic spray, X , given by (3.6), integrates to a Finsler geodesic flow, ϕt , on the bundle S M via the ordinary differential equation dϕt (x, u)/dt = X (ϕt ((x, u)) for all t ∈ I ⊂ R. The latter translates as ⎧ i dx ⎪ ⎪ = ui , ⎨ dt (3.7) i ⎪ ⎪ ⎩ du = −G i (x, u), dt where the acceleration components (or spray coefficients) read G i = N ij y j (see [6]), for i = 1, . . . , n. The geodesic flow then defines geodesics per se, xt = π(ϕt (x, u)), of the base manifold, M, with initial data (x, u) ∈ S M. 3.2.1. The Fermat Principle. are said to be conforDefinition 3.4. [30]. Two Finsler structures (M, F) and (M, F) y) = n(x)F(x, y) for some n ∈ C ∞ (M, R∗+ ). mally related if F(x, is a Riemannian structure conformally related If F is a Riemannian structure, then F to F, since their metric tensors are such that gi j (x) = n2 (x) gi j (x). In this case, the geodesics of (M, F) may be interpreted as the trajectories of light in a medium, modeled on the Riemannian manifold (M, F), and endowed with a refractive index n. This is, in essence, the Fermat Principle of geometrical optics. be conformally related Finsler structures, i.e., Proposition 3.5. Let (M, F) and (M, F) y) = n(x)F(x, y) for a given n ∈ C ∞ (M, R∗+ ), called their relative be such that F(x, refractive index. Their geodesic sprays are related as follows: X = ui
δ δ 1 1 ∂n ∂ & X = u i i + 3 (gi j − 2u i u j ) j , i δx n δx n ∂ x ∂ yi
(3.8)
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where u i = y i /F, for i = 1, . . . , n. Putting x˙ i = n X (x i ), and y˙ i = n X (y i ), we obtain the equations of the geodesics of (M, F) in the following guise: ⎧ x˙ i = u i , ⎨ (3.9) ⎩ ∇u (n u)i = gi j ∂n , ∂x j where ∇u is the covariant derivative with reference vector u, defined, for all vector fields, v, along the curve with velocity u, by ∇u (v)i = v˙ i + ijk (x, u)u j v k . Proof. The Hilbert 1-forms are related by ω H = n ω H , and their exterior derivatives ¯ by σ = n σ + dn ∧ ω H . In other words σ = n δ AB ω A ∧ ω B + n A ω A ∧ ωn , where i n A = e A ∂i n. Reproducing the proof of Theorem 3.3, we will decompose X ∈ Vect S M as ¯ X n eˆn + X A eˆ A¯ + X n¯ eˆn¯ , with the same notation as before. Again X ∈ ker( σ) X= X A eˆ A + n ¯ iff d ωH ( X ) + λ ω = 0 for some λ ∈ R. (Note that, in view of (2.19), we have ¯ X A = 0, and X A = nA ωn¯ = ωn¯ + dn/n.) This equation readily leaves us with X n , for all A = 1, . . . , n − 1, together with λ = 0. At last, X is tangent to S M if ωn¯ ( X ) = 0, i.e., if X n¯ = −(nn /n) X n . Then, X is the n Reeb vector field for F if ω H ( X ) = 1, i.e., if X = 1/n. The geodesic spray of the is thus Finsler structure (M, F)
1 nA nn eˆn + eˆ ¯ − eˆn¯ , X= n n A n while that of the Finsler structure (M, F) reduces to X = eˆn by letting n = 1. We thus recover (3.8) via (2.16) and (2.9), and also by the following fact, viz., j j n A eˆ A¯ − nn eˆn¯ = (δ AB eiA e B − eni en )∂ j n F∂/∂ y i = (gi j − 2u i u j )(∂ j n/n)∂/∂ y i , since = n F = 1 on F S M. X (x i ) = u i , and also that Let us now derive Eq. (3.9); we first notice that x˙ i = n y˙ i = n X (y i ) = −N ij u j +(F/n)(gi j −2u i u j )∂ j n. Defining, as in, e.g., [6], the covariant derivative of the vector field v, with reference vector u, by the expression ∇u (v)i = v˙ i + ijk u j v k = v˙ i + N ij v j /F, see (2.20), enables us to compute the “geodesic acceleration” ∇u (n u). Since (n u)i = (n2 y)i , we get ∇u (n u)i = 2nn˙ y i + n2 ∇u (y)i = 2n u j ∂ j n y i + n2 ( y˙ i + N ij u j ) = 2n u j ∂ j n y i + n2 (F/n)(gi j − 2u i u j )∂ j n = gi j ∂ j n, and we are done.
Remark 3.6. The differential equations (3.9) generalize to the Finsler framework, the Fermat equations ruling the propagation of light in a Riemannian manifold, through a dielectric medium of refractive index n. 3.2.2. Finsler optics. It has originally been envisioned by Finsler himself (see, e.g., [3,28]) that the indicatrix Sx M = {u ∈ Tx M | F(x, u) = 1} of a Finsler structure (M, F) might serve as a model for the geometric locus of the “phase velocity” of light waves at a point x ∈ M. The fact that, in anisotropic optical media, the velocity of a (plane) light-wave specifically depends upon the direction of its propagation, prompted him to put forward a classical (as opposed to field-theoretical) model of geometrical optics in anisotropic, non-dispersive, media ruled by Finsler structures. Finsler geodesics have
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therefore consistently received the interpretation of light trajectories in such optical media. Let us mention, among many an example, an application of Finsler optics to dynamical systems engendered by Finsler billiards [26]. When specialized to Riemannian structures, e.g., to Fermat structures presented in Sect. 3.2.1, Finsler geodesics are nothing but plain Riemannian geodesics, regarded as light rays in (non homogeneous) isotropic media. See, e.g., [16,17,21,22]. This justifies the following principle of Finsler geometrical optics. Definition 3.7. The light rays in a non-homogeneous, anisotropic, optical medium described by a Finsler structure (M, F) are the oriented geodesics associated with the geodesic spray (3.6) of this Finsler structure. For example, birefringent media (solid or liquid crystalline media) can be described by a pair of Finsler metrics, namely, the ordinary (resp. extraordinary) metric Fo (resp. Fe ) attached to a (three-dimensional) manifold, M, representing the anisotropic optical medium. Those are respectively given, in the particular case of uniaxial crystals, in terms of a pair of Riemannian metrics a = ai j (x) d x i ⊗ d x j , and b = bi j (x) d x i ⊗ d x j on M, by [3,28] Fo (x, y) =
ai j (x) y i y j ,
ai j (x) y i y j Fe (x, y) = . bi j (x) y i y j
(3.10) (3.11)
The geodesics of the metric Fe are meant to describe extraordinary light rays, whereas those of the Riemannian metric, Fo , will merely lead to ordinary rays. Remark 3.8. Let us emphasize that (M, Fe ), where Fe is as in (3.11), is a Finsler structure if its fundamental tensor giej = Fe2
bi j 4 ci c j 2 ai j − + , a(y, y) b(y, y) b(y, y)3
(3.12)
j where ci = a(y, y) bi j (x)y j − b(y, y) ai j (x)y √definite. This is, indeed, the √ , is positive case if the quadratic forms a and b verify b/ 2 < a < b 2, everywhere on T M \ M.
The more complex case of biaxial optical media is also studied in [3], and gives rise to a pair of specific Finsler metrics, ai j (x) y i y j F ± (x, y) = , bi±j (x) y i y j
(3.13)
where a, b+ , and b− are Riemannian metrics characterizing the optical properties of the anisotropic medium. (Let us note that Remark 3.8 applies just as well for the metrics (3.13)).
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3.2.3. The example of birefringent solid crystals. Let us review how Finsler metrics of the form (3.13), or (3.10) and (3.11), arise in the particular case of anisotropic solid crystals. To that end, we revisit the original derivation [3] of the Minkowski norms that account for the propagation of light in anisotropic dielectric solids with principal (positive) velocities v1 , v2 , v3 . In the framework of Maxwell’s wave optics, the Fresnel equation of wave normals uˆ 21 ( u 2 − v22 )( u 2 − v32 ) + uˆ 22 ( u 2 − v32 )( u 2 − v12 ) + uˆ 23 ( u 2 − v12 )( u 2 − v22 ) = 0 expresses the dependence of the phase velocity, u, of a plane wave upon its direction of propagation, uˆ = u/ u , in such a medium; we denote, here, by · the norm on standard Euclidean space (R3 , · , · ). - Assuming, e.g., v1 > v2 > v3 , one solves the Fresnel equation for the norm of the phase velocity, viz., u 2 = A + B cos(θ ± θ ), where, θ and θ are the angles ˆ and the (oriented) optical axes e and e ; the between the direction of propagation, u, 1 2 1 2 2 scalars A = 2 (v1 + v3 ), and B = 2 (v1 − v32 ), as well as the vectors e , and e , are characteristic of the crystal [14]. The Minkowski norm, F, associated with each solution of the Fresnel equation is easily found [3] using Okubo’s trick that amounts to the replacement u y/F(y), insuring that F(u) = 1. Easy calculation leads us to u 2 = y 2 /F(y)2 = A + B y −2 e , ye , y ∓ e × y e × y , that is, to F ± (y) =
y 2 , A y 2 + B e , ye , y ∓ e × y e × y
(3.14)
where × denotes the standard Euclidean cross-product. This expression admits straightforward generalizations to the case of fluid crystals, Faraday-active media, etc., where the quantities A, B, e , and e become position-dependent; it ultimately leads to the expression (3.13) of a pair of Finsler metric for general biaxial media. - The case of uniaxial media is treated by assuming, e.g., v1 = v2 > v3 , which implies e = e (= e). The Minkowski norms (3.14) admit a prolongation to this situation and read F − (y) =
y , v1
F + (y) =
(3.15) y 2
v32 y 2 + (v12 − v32 )e, y2
.
(3.16)
Those correspond, respectively, to an ordinary Euclidean metric, Fo = F − , and to an extraordinary Minkowski metric, Fe = F + , again generalized by (3.10), and (3.11). - The last case, for which v1 = v2 = v3 , clearly leads to a single Euclidean metric, namely F(y) = y /v1 , that rules geometrical optics in isotropic media with refractive index n = 1/v1 .
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4. Geometrical Spinoptics in Finsler Spaces So far, the polarization of light has been neglected in the various formulations of geometrical optics. We contend that spinning light rays do, indeed, admit a clear cut geometrical status allowing for a natural extension of plain geometrical optics to the case of circularly polarized light rays (Euclidean photons) traveling in arbitrary non dispersive optical media. The touchstone of our viewpoint about geometrical optics for spinning light is the Euclidean symmetry of the manifold of oriented lines in (flat) Euclidean space. This fundamental symmetry will be taken as a guiding principle to set up a model that could describe the geometry of spinning light rays in quite general, crystalline and liquid, optical media. See [21] and [22] for a first approach to geometrical spinoptics in inhomogeneous, isotropic, media. We will therefore start with some elementary facts about the symplectic structure of the space of oriented lines in Euclidean space. The consideration of the generic coadjoint orbits of the Euclidean group will then be justified on physical grounds. Let us recall that, if we denote by Ad the adjoint action of a Lie group, and by Coad, its coadjoint action, then the orbits of the latter action inherit a canonical structure of symplectic manifolds. These homogeneous symplectic manifolds play a central rôle in mechanics and physics, where some of them may be interpreted as the elementary systems associated with the symmetry group under consideration [39]. The following construction is standard. Theorem 4.1. Let G be a (finite-dimensional) Lie group G with Lie algebra g. Fix µ0 ∈ g∗ and define the following 1-form µ0 = µ0 · ϑG ,
(4.1)
where ϑG is the left-invariant Maurer-Cartan 1-form of G. Then, σµ0 = dµ0 is a presymplectic 2-form on G which is the pull-back of the canonical Kirillov-KostantSouriau symplectic 2-form on the G-coadjoint orbit Oµ0 = {µ = Coad g (µ0 ) | g ∈ G} ∼ = G/G µ0 ,
(4.2)
where G µ0 is the stabilizer of µ0 ∈ g∗ . 4.1. Spinoptics and the Euclidean group. From now on we will confine considerations to three-dimensional configuration spaces to comply with the physical principles of geometrical optics. An oriented straight line, ξ , in Euclidean affine space (E 3 , · , · ) is determined by its direction, a vector u ∈ R3 of unit length, and an arbitrary point Q ∈ ξ . Having chosen an origin, O ∈ E 3 , we may consider the vector q = Q − O, orthogonal to u. The set of oriented, non parametrized, straight lines is thus the smooth manifold M = {ξ = (q, u) ∈ R3 × R3 | u, u = 1, u, q = 0},
(4.3)
i.e., the tangent bundle M ∼ = T S 2 of the round sphere S 2 ⊂ R3 , which has been recognized by Souriau [39] as a coadjoint orbit of the group, E(3), of Euclidean isometries, and inherits, as such, an E(3)-invariant symplectic structure. See also [27].
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Consider the group, SE(3) = SO(3) R3 , of orientation-preserving Euclidean isometries of (E 3 , · , · , vol), viewed as the matrix-group whose elements read R x g= , (4.4) 0 1 where R ∈ SO(3), and x ∈ R3 . The (left-invariant) Maurer-Cartan 1-form of SE(3) is therefore given by ω ω , ϑSE(3) = 0 0
(4.5)
where ω = R −1 d R, and ω = R −1 dx. Let µ = (S, P) denote a point in e(3)∗ , where e(3) = o(3) R3 is the Lie algebra of SE(3). We will use the identification o(3) ∼ = 2 R3 (resp. (R3 )∗ ∼ = R3 ) given by Sab = S ab (resp. Pa = P a ), where S ab + S ba = 0,
(4.6)
for all a, b = 1, 2, 3. The pairing e(3)∗ × e(3) → R will be defined by 1 (S, P) · ( ω, ω) = − Tr(S ω) + P, ω 2 =
(4.7)
1 ab S ωab + Pa ωa . 2
(4.8)
The coadjoint representation of SE(3), viz., Coad g µ ≡ µ ◦ Ad g−1 , is given by Coad g (S, P) = (R(S + x ∧ P)R −1 , RP). Clearly, C = P, P and C = (S ∧ P)/vol are coadjoint SE(3)-invariants. These are the only invariants of the Euclidean coadjoint representation, and fixing (C, C ) or (C = 0, C ), where C = 21 S ab Sab , yields a single coadjoint orbit [25,32,39]. 4.1.1. Colored light rays. Specializing the construction of Theorem 4.1 to the case G = SE(3), with C = p 2 and p > 0 together with C = 0, we can choose µ0 = (0, P0 ) and P0 = (0, 0, p). The invariant p is the color [39] of the chosen coadjoint orbit. The 1-form (4.1) then associated, via the pairing (4.8), and the Maurer-Cartan 1-form (4.5), with the invariant p is thus µ0 = P0 , ω, or µ0 = Pa ωa
(4.9)
= pω . 3
(4.10)
Straightforward calculation yields µ0 = p u i with u i = δi j where u = e3 is the third vector of the orthonormal, positively oriented, basis R = (e1 , e2 , e3 ). The 1-form (4.10) is, up to an overall multiplicative constant, p, equal to the canonical 1-form (3.2) on the sphere-bundle S E 3 , associated with the trivial Finsler structure √ 3 (E , F), with F(x, y) = y, y. Proposition 3.5 just applies, with n = 1, and conforms to Euclid’s statement that light, whatever its color, travels in vacuum along oriented geodesics of E 3 . Indeed, the exterior derivative of the 1-form µ0 of S E 3 reads σµ0 = p du i ∧ d x i . Its kernel, given by u j,
dxi ,
X ∈ ker(σµ0 )
⇐⇒
X = λ ui
∂ , ∂ xi
(4.11)
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Fig. 1
with λ ∈ R, yields the (flat Euclidean) geodesic spray (λ = 1). We will resort to generalizations of this particular construct of the geodesic foliation. The integral invariant, σµ0 , descends, as a symplectic 2-form, to the quotient M = S E 3 / ker(σµ0 ) described by ξ = (q, u), where q = x − uu, x. This is the content of Theorem 4.1 insuring that M ∼ = T S 2 ⊂ e(3)∗ is endowed with a canonical symplectic structure, namely (M, p du i ∧ dq i ). We note that the SE(3)-coadjoint orbit Oµ0 is an E(3)-coadjoint orbit. 4.1.2. The spinning and colored Euclidean coadjoint orbits. The generic SE(3)-coadjoint orbits are, in fact, characterized by the Casimir invariants C = p 2 , with p > 0 (color), and C = sp, where s = 0 stands for spin. We call helicity the sign of the spin invariant, ε = sign(s). The origin, µ0 , of such an orbit can be freely chosen so as to satisfy the constraints S ab Pb = 0, for all a = 1, 2, 3, and 21 S ab Sab = s 2 , together with Pa P a = p 2 . One may posit 0 −s 0 S0 = s 0 0 and P0 = (0, 0, p). (4.12) 0 0 0 The coadjoint orbit, Oµ0 , passing through µ0 = (S0 , P0 ) ∈ e(3)∗ is, again, diffeomorphic to T S 2 . It is endowed with the symplectic structure coming from the 1-form (4.1) on the group SE(3), which now reads µ0 = Pa ωa +
1 ab S ωab 2
= p ω3 + s ω12 ,
(4.13) (4.14)
where ω (resp. ω) stand for the flat Levi-Civita connection (resp. soldering) 1-form on the bundle, SE(3) ∼ = SO(E 3 ), of positively oriented, orthonormal frames of E 3 . Remark 4.2. The 1-form (4.13) is the central geometric object of the present study. Straightforward computation then leads to µ0 = pe3 , dx − se1 , de2 . The exterior derivative of µ0 is found [21,25,32,39] to be given by s σµ0 = p du i ∧ d x i − i jk u i du j ∧ du k , 2
(4.15)
where, again, u = e3 , and also i jk stands for the signature of the permutation {1, 2, 3} → {i, j, k}. This 2-form conspicuously descends to SR3 ∼ = R3 × S 2 . Its characteristic foliation is, verbatim, given by (4.11): spinning light rays in vacuum are nothing but oriented Euclidean geodesics. As shown in the sequel, things will change dramatically for such light rays in a refractive medium.
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Remark 4.3. Actually, “photons” are characterized by |s| = , where is the reduced Planck constant; right-handed photons correspond to s = +, and left-handed ones to s = −, see [39]. We will, nevertheless, leave the parameter s arbitrary when dealing with “spinning light rays”. The manifold of spinning light rays, Oµ0 = S E 3 / ker(σµ0 ) ∼ = T S 2 (see Fig. 1) is, just as before, parametrized by the pairs ξ = (q, u) and endowed with the “twisted” symplectic 2-form ωµ0 = p du i ∧ dq i − 2s i jk u i du j ∧ du k . Note that the union of two SE(3)-coadjoint orbits defined by the invariants ( p, s) and ( p, −s) is symplectomorphic to a single E(3)-coadjoint orbit. Remark 4.4. The SE(3)-coadjoint orbits of spin s, and color p, are symplectomorphic to Marsden-Weinstein reduced massless SE(3, 1)0 -coadjoint orbits of spin s, at given (positive) energy E = pc, where c stands for the speed of light in vacuum, see [21]. This justifies that the color, p, of Euclidean light rays corresponds, via reduction, to the energy of relativistic photons, or to the frequency of their associated monochromatic plane waves [22,25,39]. 4.2. Spinoptics in Finsler-Cartan spaces. With these preparations, we formulate the principles of geometrical spinoptics, with the premise that (i) Finsler structures should be considered a privileged geometric background for the description of inhomogeneous, anisotropic, optical media, (ii) the original Euclidean symmetry which pervades geometrical optics should be invoked as a guiding principle in any formulation of spin extensions of geometrical optics. 4.2.1. Minimal coupling to the Cartan connection Axiom 4.5. The trajectories of (circularly) polarized light, originating from an Euclidean coadjoint orbit Oµ0 ⊂ e∗ (3) with color p > 0, and spin s = 0, in a threedimensional Finsler manifold (M, F), are governed by the following 1-form on the principal bundle SO2 (S M) over evolution space S M = F −1 (1), namely µ0 = µ0 · ϑ,
(4.16)
where ϑ = ( ω, ω) is the “affine” Cartan connection, ω = ( ωba ) denoting the Cartan a connection (2.32), and ω = (ω ) the coframe (2.10). The characteristic foliation of the 2-form σµ0 = dµ0 yields the differential equations of spinoptics in a medium described by the considered Finsler structure. The 1-form (4.16) corresponds, mutatis mutandis, to the Euclidean 1-form (4.13). (In (4.16), and from now on, we simplify the notation and denote by ϑ the pull-back ι∗ ϑ on SO2 (S M) of the corresponding 1-form of SO2 (T M\M).) In Axiom 4.5, the replacement of the Euclidean group SE(3), see Fig. 1, by the principal bundle SO2 (S M), see Fig. 2, and of the Maurer-Cartan 1-form by the affine Cartan connection is akin to the so-called procedure of minimal coupling. We refer to [31] where the minimal coupling of a spinning particle to the gravitational field was originally introduced in the general relativistic framework. See also [40,41]. The choice of the Cartan connection is impelled by the fact that the group underlying Finsler-Cartan geometry is the Euclidean group, E(n), which is precisely the fundamental symmetry group of the symplectic model of free photons, for n = 3. Indeed, the “flat”
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Fig. 2
a = 0, n-dimensional Finsler-Cartan structure defined by both conditions a = 0, and b for all a, b = 1, . . . , n, is (locally) isomorphic to the Euclidean space, (E n , · , · ): torsionfreeness yields Aabc = 0, see (2.34), hence that (M, F) is Riemannian; zero curvature then entails local flatness, via (2.37), and (2.25). The Euclidean group, E(n), is then the group of automorphisms of the flat structure. Let us emphasize that the Cartan connection has been originally referred to as the “connexion euclidienne” in [18]. The 1-form (4.16) of SO2 (S M) thus reads µ0 = Pa ωa +
1 ab S ωab . 2
(4.17)
In view of the choice (4.12) of the moment µ0 = (S, P) ∈ e(3)∗ , viz., S ab = s AB δ aA δ bB and Pa = p δa3 ,
(4.18)
for all a, b = 1, 2, 3, where AB = 2δ1[A δ2B] , for all A, B = 1, 2, we find ω12 . µ0 = p ω3 + s
(4.19)
Remark 4.6. The 1-form µ0 differs from the Hilbert 1-form, ω H = ω3 , by a spin-term, ω12 . This term, canonically associated to a generic Euclidean coadjoint orbit, is new in the framework of Finsler geometry, and akin to the Berry connection [8]. See, e.g., [9,33]. Putting, in (4.19), s = ε for photons (where ε = ±1 is helicity), and p = k, where k = ε/λ is the wave number, we observe that s/ p = λ, so that Formula (4.19) indeed corresponds to (1.3), up to an overall constant factor. Proposition 4.7. The exterior derivative, dµ0 , of the 1-form (4.17) on SO2 (S M) descends to the evolution space, S M, as σµ0 = p h ab ωa¯ ∧ ωb +
1 1 (S) − Sab ωa¯ ∧ ωb¯ ,
2 2
(4.20)
where the h ab = δab − δa3 δb3 denote the frame-components of the angular metric, and (S) = ab S ab the spin-curvature coupling 2-form.
Proof. Let us start with the expression (4.19) of the 1-form µ0 . We have found, see ¯ (3.3), that dω3 = δ AB ω A ∧ ω B . With the help of the structure equations of the Cartan 12 + 12 + 12 − connection, we obtain d ω12 = ω1a ∧ ωa2 = ω13 ∧ ω32 = ω31 ∧ ω32 , using the property (2.33). We then resort to (2.30), and to the property (2.5) of the ¯ Cartan tensor (that is Aab3 = 0, for all a, b = 1, 2, 3), to find ω3A = ω A . This yields 12 − ω1¯ ∧ ω2¯ , or, equivalently, d AB − ω A¯ ∧ ω B¯ , for A, B = 1, 2. d ω12 = ω AB = Thus ¯
dµ0 = p δ AB ω A ∧ ω B +
s 1 (S) − AB ω A¯ ∧ ω B¯ ,
2 2
(4.21)
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AB . In order to prove (4.20), we simply use the frame (S) = s 12 = 1 s AB where 2 components, S ab , of the spin tensor given in (4.18). To complete the proof, it is enough to verify that dµ0 , given by (4.21), is an integral invariant of the SO(2)-flow generated by Z = e1i
∂ ∂e2i
− e2i
This is, indeed, the case since dµ0 (Z ) = 0.
∂ ∂e1i
.
(4.22)
4.2.2. The Finsler-Cartan spin tensor. Let us now give a construction of the spin tensor on the indicatrix-bundle, S M, that will be useful in the sequel. Lemma 4.8. The following 3-form of SO2 (S M), viz., Vol = ω1 ∧ ω2 ∧ ω3
(4.23)
is an integral invariant of the flow generated by the vector field, Z , given by (4.22). It descends to S M as vol = 16 voli jk (x, u) d x i ∧ d x j ∧ d x k , with voli jk (x, u) =
det (glm (x, u)) i jk ,
(4.24)
where i, j, k, l, m = 1, 2, 3. Proof. In view of (2.17), we clearly have Vol(Z ) = 0. Using, for example, the Chern connection, we find, with the help of the structure equations (2.21) that dVol = −ωaa ∧ ¯ Vol. Then, Eq. (2.22) readily implies dVol = Aaab ωb ∧ Vol. Again, (2.17) entails that (dVol)(Z ) = 0, thus Z ∈ ker(Vol) ∩ ker(dVol), proving that Vol is an SO(2)-integral invariant. Locally, we have vol = det(ωia ) d x 1 ∧ d x 2 ∧ d x 3 = det (gi j ) d x 1 ∧ d x 2 ∧ d x 3 , since (2.6) can be rewritten as ωia ωbj δab = gi j . This proves Eq. (4.24).
Let us now regard the Sab as the frame-components of a skew-symmetric tensor, S, on S M, with components Si j = Sab ωia ωbj . Then, owing to (4.18), we easily find Si j = s AB ωiA ω Bj = 2s ω[i1 ω2j] ; those turn out to be nothing but the components of the tensor S = s vol(eˆ3 ) = s ω1 ∧ ω2 . Whence Lemma 4.9. If we call a spin tensor, associated with the moment (4.18), the tensor S = s vol(u), ˆ
(4.25)
Si j = s voli jk u k ,
(4.26)
where uˆ = eˆ3 , see (2.16), then
for all i, j = 1, 2, 3, where the voli jk are as in (4.24).
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4.2.3. Laws of geometrical spinoptics in Finsler-Cartan spaces. We are now ready to determine the explicit expression of the characteristic foliation of the 2-form (4.20) that will provide us with the differential equations governing the trajectories of spinning light in a Finsler-Cartan background. Lemma 4.10. The spin-curvature coupling term for the Cartan connection retains the form (S) =
1 cd ωc ∧ ωd¯ + 1 Q(S) cd ωc¯ ∧ ωd¯ , R(S)cd ωc ∧ ωd + P(S) 2 2
(4.27)
abcd S ab , etc., and cd = R where R(S) cd = R(S)cd , R(S) cd = P(S)cd = 2(Acda|b − Aace A˙ ebd )S ab , P(S)
(4.28) (4.29)
cd = −2 Aaec Aebd S ab . Q(S)
(4.30)
cd , for the Cartan connection exactly cd , and P(S) Proof. The frame-components R(S) match their counterpart for the Chern connection with curvature tensors R ji kl and P ji kl given by (2.25) and (2.26) respectively; indeed, Eqs. (4.28), and (4.29) are derived, in a straightforward way, using (2.38), and (2.39), together with the total skew-symmetry (resp. symmetry) of the spin tensor, (resp. the Cartan tensor), i.e., S ab = S [ab] (resp. Aabc = A(abc) ). The constitutive equation (Eq. (3.4.11) in [6]) for the hv-components of the Chern curvature in terms of the covariant derivatives of the Cartan tensor (see (2.29), and ˙m ˙m (2.31)), reads Pabcd = Aacd|b − Abad|c − Acbd|a + Abam A˙ m cd − Aacm Abd + Abcm Aad . e We readily deduce that P(S)cd = Pabcd S ab = 2(Acda|b − Aace A˙ bd )S ab , proving the last part of Eq. (4.29). The last equation (4.30) is a trivial consequence of (2.40).
Lemma 4.11. There holds P(S)a3 = P(S)3a = 0, for all a = 1, 2, 3. Proof. We have P(S)c3 = 0, because of (2.27). Likewise, the relation P(S)3a = 0 stems from (4.29) since Aa3e = 0 (see (2.5)), and (e3 )i| j = 0 (see (2.29)).
We can now proclaim our main result. Theorem 4.12. The characteristic foliation of the 2-form σµ0 of S M, given by (4.20), is expressed as follows, viz., X ∈ ker(σµ0 ), X = X3
δ 1 j ui + S j i R(S)k u k 2s δx i
∂ 1 1 j j S j i p δk − P(S)k Sl k R(S)ml u m + 2 2s 2 ∂u i
(4.31)
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for some X 3 ∈ R, where the S ji = s vol j i k u k are as in (4.26), and
1 = s 1 − 2 Q(S)(S) , 4s
1 1 1 p 2 − p P(S)i j gi j + 2 P(S)ik P(S) jl S i j S kl = 2 8s
(4.32)
1 R(S)(S), (4.33) 4s i j Si j . with R(S)(S) = R(S)i j S i j , and Q(S)(S) = Q(S) The 2-form σµ0 endows S M \ (−1 (0) ∪ −1 (0)) with a presymplectic structure of rank 4; the foliation (4.31) leads to a spin-induced deviation from the geodesic spray (3.6), and, according to Axiom 4.5, governs spinoptics in a 3-dimensional Finsler-Cartan structure (M, F). +
Proof. Using Lemmas 4.10 and 4.11, we can rewrite our 2-form (4.20) of S M, in the guise of (4.21), as s ¯ ¯ ¯ σµ0 = + p δ AB ω A ∧ ω B − AB ω A ∧ ω B 2 1 1 + R(S) AB ω A ∧ ω B + R(S) A3 ω A ∧ ω3 4 2
(4.34)
1 1 ¯ A¯ B¯ + P(S) AB ω A ∧ ω B + Q(S) AB ω ∧ ω . 2 4 The proof of Theorem 3.3 is adapted to the new 2-form (4.34) we are dealing with. In particular, the vector fields X ∈ Vect(S M) will be written in the following form, ¯ ¯ ¯ X = X A eˆ A + X 3 eˆ3 + X A eˆ A¯ + X 3 eˆ3¯ . Then X ∈ ker(σµ0 ) iff σµ0 (X ) + λω3 = 0, where λ ∈ R is a Lagrange multiplier associated with the constraint F = 1 defining S M → T M \ M. We find ¯
¯
¯
¯
¯
¯
σµ0 (X ) + λω3 = + p δ AB (X A ω B − X B ω A ) − s AB X A ω B + λω3 +
1 1 R(S) AB X A ω B + R(S) A3 (X A ω3 − X 3 ω A ) 2 2
+
1 1 ¯ ¯ A¯ B¯ P(S) AB (X A ω B − X B ω A ) + Q(S) AB X ω , 2 2
so that X ∈ ker(σµ0 ) iff
1 1 1 ¯ A A 0 = p δ B − P(S) B X B + R(S) BA X B + R(S)3 A X 3 , 2 2 2
1 1 ¯ A X B, = p δ BA − P(S) BA X B + s BA − Q(S) B 2 2
(4.35) (4.36)
= R(S) A3 X A ,
(4.37)
= λ.
(4.38)
Finsler Spinoptics
723
abcd S ab S cd , and consider (4.18) to Put R(S)(S) = Rabcd S ab S cd , and Q(S)(S) =Q readily get R(S) AB =
1 AB = 1 Q(S)(S) R(S)(S) AB and Q(S) AB , 2s 2s
(4.39)
for all A, B = 1, 2. Note that we also have R(S)(S) = s 2 R ABC D AB C D , and Q(S)(S) = −2s 2 A AC E A B D F AB C D δ E F (see (2.40)). Plugging (4.39) into (4.36), we easily find ¯
XA =
1 A 1 B p δCB − P(S)CB X C , 2
(4.40)
for all A = 1, 2, where
1 AB C D . =s 1− Q ABC D 4
(4.41)
We then find, with the help of (4.40), and (4.35), the following relationship 1 DA X D = − R(S)3 A X 3 , (4.42) 2 where DA = −1 p δ BA − 21 P(S) AB C B p δ CD − 21 P(S) DC + 21 R(S) DA . We clearly have AB + B A = 0, hence AB = AB , and find, with some more effort,
1 1 1 2 AB AB C D = p − p P(S) AB δ + P(S) AC P(S) B D 2 8 1 + s R ABC D AB C D , 4
(4.43)
where is as in (4.41). Let us point out that Eq. (4.37) trivially holds true in view of the skew-symmetry of AB ; indeed, (4.42) implies R(S)3A X A = AB X A X B = 0. Equation (4.42) then leaves us with XA =
1 A R(S)3 B X 3 , 2 B
(4.44) ¯
for all A = 1, 2. Let us recall that the latter equation for X A completely determines X A , ¯ via (4.40), the components X 3 , and X 3 remaining otherwise arbitrary. Now, ¯3 ¯ X ∈ Vect(S M) if X (F) = 0, i.e., ω (X ) = X 3 = 0. We are, hence, left with only one arbitrary parameter, X 3 , to define the direction ker(σµ0 ) wherever = 0, and ¯ ¯ = 0. Thus X = X A eˆ A + X 3 eˆ A + X A eˆ A¯ , where X A , and X A are as in (4.44), and (4.40) 3 respectively, with X ∈ R.
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Introducing the unit supporting element, (u a = δ3a ), as well as the spin tensor, (Sb a = s BA δ aA δbB ), given in (4.18), we find X ∈ ker(σµ0 ) X = X3
1 (S R(S)u) A eˆ A . + eˆ3 2s A 1 1 p S − S P(S) S R(S)u + 2 eˆ A¯ 2s 2
(4.45)
for some X 3 ∈ R. To complete the calculation, we express (4.45) in terms of the coordinates u i = e3i of the distinguished element u = e3 , and those, S i j = gik g jl Skl , of the spin tensor S, see (4.25), where the Si j are as in (4.26). We also bear in mind that eˆa = eai δ/δx i , and eˆa¯ = eai ∂/∂u i , for all a = 1, 2, 3, as given by (2.16), on the principal bundle SO2 (S M) above the evolution space S M. The upshot of the computation is that the characteristic foliation (4.45) can be recast in the form (4.31); Eqs. (4.32), and (4.33) also provide alternative expressions for (4.41), and (4.43). At those points (x, u) ∈ S M, where = 0, or = 0, singularities of the foliation (4.45) do occur; they must be discarded to guarantee a well-behaved presymplectic structure of (generic) rank 4. The proof is now complete.
Remark 4.13. Let us choose, e.g., X 3 = 1 in (4.31) to define the generator, X , of the foliation ker(σµ0 ). The latter significantly deviates from a spray since the velocity, x, ˙ given by the horizontal projection of X , differs from the direction, u, of the supporting element, namely x˙ i = u i +
1 j S i R(S)k u k , 2s j
(4.46)
where x˙ i = X (x i ), for all i = 1, 2, 3. The occurrence of this anomalous velocity in the presence of curvature can be classically interpreted (see [21,22]) as the source of the optical Hall effect. Moreover, the vertical components of the generator X , namely those of the geodesic acceleration, depend linearly on the helicity, ε = sign(s). They, notably, lead to a splitting, à la Stern-Gerlach, of light rays with opposite helicities. Let us finish with the following corollary of Theorem 4.12 which help us recover the simpler equations of spinoptics in the Riemannian case, derived in [21]. Corollary 4.14. If the Finsler structure, (M, F), is Riemannian, the characteristic foliation of the 2-form σµ0 is spanned by the vector field δ ∂ 1 1 j k i X = ui + S R(S) u − R(S) j i u j i (4.47) k 2 j δx i 2 ∂u with 1 R(S)(S), 4 are the components of the Riemann curvature tensor. = p2 +
where the Ri jkl
(4.48)
Finsler Spinoptics
725
Proof. Suffice it to note that the Cartan tensor vanishes iff the Finsler structure is i jkl = 0. The curvature tensor Ri jkl in (4.31) then reduces Riemannian, hence Pi jkl = Q to the Riemann curvature tensor, see (2.25). The proof is completed by noticing that S jk S ki = s 2 (u i u j − δ ij ), a direct consequence of (4.26).
5. Conclusion and Outlook We have proposed a generalization of the Fermat Principle enabling us to describe spinning light rays in a general, non dispersive, optical medium, namely an inhomogeneous and anisotropic medium modeled on a Finsler manifold. The guideline for this extension has been provided by the Euclidean symmetry of the free system, viewed as a generic coadjoint orbit of the Euclidean group, SE(3). Interaction with the optical medium has been justified in terms of a minimal coupling of the model to the (affine) Cartan connection of the Finsler structure; the gist of the procedure lies in the fact that the affine Cartan connection takes, indeed, its values in the Lie algebra e(3) and, thus, couples naturally to the moment µ0 ∈ e(3)∗ defining the original coadjoint orbit (the classical states of the free Euclidean photon). The resulting presymplectic structure on (an open submanifold of) the indicatrix-bundle has been investigated. In particular the characteristic foliation of this structure has been worked out, and shown to yield a system of differential equations governing the trajectories of spinning light rays, associated with a vector field departing from the usual Finslerian geodesic spray. The geodesic acceleration of spinning light rays is due to the coupling of spin with the Finsler-Cartan curvature, which also engenders an anomalous velocity. The latter, already present in Riemannian spinoptics [21,22] has proved crucial in the geometrical interpretation of the brand new optical Hall effect, see, e.g., [11,33]. The consubstantial nature of this effect with the geometry of Euclidean coadjoint orbits is precisely what prompted the present study, and our endeavor to depart from the case of isotropic media by taking advantage of Finsler-Cartan structures. Although the characteristic foliation (4.31) of the above-mentioned presymplectic structure is of a formidable complexity, it is nevertheless a mandatory consequence of a minimal, geometrically justified, modification (1.3) of the Hilbert 1-form (1.2) of central importance in Finsler geometry. The future perspectives opened by this work are manifold. It would be desirable to linearize the differential equations of Finsler spinoptics in the case of weakly curved Finsler-Cartan manifolds, to account for weakly anisotropic optical media. This should lead to substantial simplifications, suitable for an explicit calculation of the geodesic deviation in several non trivial examples, such as those given by (3.10), and (3.11). Also, would it be of great importance to compare this linearized set of differential equations with the outcome of the calculations performed by a (short wavelength) semi-classical limit of the Maxwell equations in weakly anisotropic and inhomogeneous media [13]. The Fermat Principle has, most interestingly, been generalized, via a novel variational calculus, to the case of lightlike geodesics in Finsler spacetimes with a Lorentzian signature [35]. It would be worth investigating how that relativistic version of geometrical optics extends to spinoptics in relativistic Finsler spacetimes. Let us note that Randers Finsler metrics play, as discussed in [15], a prominent rôle in such a framework, corresponding to induced (instantaneous) Finsler metrics on the material body of the optical medium. At last, specific applications of the equations of Finsler-Cartan spinoptics should be explored in a number of other directions such as the Kerr, the Faraday effects, and the
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Cotton-Mouton effect responsible for plasma birefringence, see, e.g., [37], as well as the photonic Hall effect [42] in the presence of a magnetic field. In truth, the present study of Finsler spinoptics was a challenge, taken up from a purely geometric standpoint; one may, conceivably, expect it will provide further insights into modern trends of geometrical optics of anisotropic media. Acknowledgements. It is a great pleasure to thank J.-C. Alvares Paiva, S. Tabachnikov, and P. Verovic, for useful correspondence and enlightening discussions. Thanks are also due to P. Horváthy for valuable advice.
References 1. Abate, M., Patrizio, G.: Finsler Metrics – A Global Approach, LNM 1591, Berlin-Heidelberg-New York: Springer-Verlag, 1994 2. Alvarez Paiva, J.C.: Some problems on Finsler geometry. In: Handbook of differential geometry. Vol. II, Dillen, F., Verstraelen, L. (eds.), Amsterdam: Elsevier, 2005 3. Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Boston: Kluwer Academic Press, 1993 4. Bejancu, A., Farran, H.R.: A Geometric Characterization of Finsler Manifolds of Constant Curvature K = 1. Internat. J. Math. & Math. Sci. 23(6), 399–407 (2000) 5. Bao, D., Chern, S.-S., Shen, Z.: On the Gauss-Bonnet integrand for 4-dimensional Landsberg spaces. In: Finsler Geometry, Bao, D., Chern, S.-S., Shen, Z. (eds.), Contemporary Mathematics 196, Providence, RI: Amer. Math. Soc., 1996 6. Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. GTM 200, New York: Springer 2000 7. Bao, D., Robles, C.: Ricci and Flag Curvatures in Finsler Geometry. In: A Sampler of Riemann-Finsler Geometry, Bao, D., Bryant, R.L., Chern, S.-S., Shen, Z. (eds.), MSRI Publications 50, Cambridge: Cambridge University Press, 2004 8. Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A 392, 45–57 (1984) 9. Bliokh, K.Yu.: Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect. Phys. Rev. Lett. 97, 043901 (2006) 10. Bliokh, K.Yu., Bliokh, Yu.P.: Topological spin transport of photons: the optical Magnus Effect and Berry Phase. Phys. Lett. A 333, 181–186 (2004) 11. Bliokh, K.Yu., Bliokh, Yu.P.: Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect. Phys. Rev. E 70, 026605 (2004) 12. Bliokh, K.Yu., Bliokh, Yu.P.: Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of Electromagnetic Wave Packet. Phys. Rev. Lett. 96, 073903 (2006) 13. Bliokh, K.Yu., Frolov, D.Yu., Kravtsov, Yu.A.: Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium. Phys. Rev. A 75, 053821 (2007) 14. Born, M., Wolf, E.: Principles of optics, Cambridge: Cambridge University Press, 1999 15. Caponio, E., Javaloyes, M.A., Masiello, A.: Variational properties of geodesics in non-reversible Finsler manifolds and applications. http://arxiv.org/abs/math/0702323, 2007 16. Cariñena, J.F., Nasarre, N.: On the symplectic structure arising in geometric optics. Fortschr. Phys. 44, 181–198 (1996) 17. Cariñena, J.F., Nasarre, N.: Presymplectic geometry and Fermat’s principle for anisotropic media. J. Phys. A 29, 1695–1702 (1996) 18. Cartan, E.: Les espaces de Finsler. Paris: Hermann, 1934 19. Chern, S.-S.: Riemannian geometry as a special case of Finsler geometry. In: Finsler Geometry, Bao, D., Chern, S.-S., Shen, Z. (eds.), Contemporary Mathematics 196, Providence, RI: Amer. Math. Soc., 1996 20. Chern, S.-S.: Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction. Not. Amer. Math. Soc. 43(9), 959–963 (1996) 21. Duval, C., Horváth, Z., Horváthy, P.: Geometrical Spinoptics and the Optical Hall Effect. J. Geom. Phys. 57, 925–941 (2007) 22. Duval, C., Horváth, Z., Horváthy, P.: Fermat Principle for spinning light and the Optical Hall effect. Phys. Rev. D 74, 021701 (R) (2006) 23. Foulon, P.: Géométrie des équations différentielles du second ordre. Ann. Inst. Henri Poincaré 45(1), 1–28 (1986) 24. Gosselin, P., Bérard, A., Mohrbach, H.: Spin Hall effect of Photons in a Static Gravitational Field. Phys. Rev. D 75, 084035 (2007)
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25. Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press, 1984 26. Gutkin, E., Tabachnikov, S.: Billiards in Finsler and Minkowski geometries. J. Geom. Phys. 40, 277–301 (2002) 27. Iglesias, P.: Symétries et moment. Paris: Hermann, 2000 28. Ingarden, R.: On physical applications of Finsler geometry. In: Finsler Geometry, Bao, D., Chern, S.-S., Shen, Z. (eds.), Contemporary Mathematics 196, Providence, RI: Amer. Math. Soc., 1996 29. Kravtsov, Yu.A., Bieg, B., Bliokh, K.Yu.: Stokes-vector evolution in a weakly anisotropic inhomogeneous medium. J. Opt. Soc. Am. A 24, 3388–3403 (2007) 30. Knebelman, M.S.: Conformal geometry of generalized metric spaces. Proc. Nat. Acad. Sci. USA 15, 376–379 (1929) 31. Künzle, H.P.: Canonical dynamics of spinning particles in gravitational and electromagnetic fields. J. Math. Phys. 13, 739–744 (1972) 32. Marsden, J.-E., Ratiu, T.: Introduction to Mechanics and Symmetry, Berlin-Heidelberg-New York: Springer, 1999 33. Onoda, M., Murakami, S., Nagaosa, N. Hall Effect of Light. Phys. Rev. Lett. 93, 083901 (2004) 34. Onoda, M., Murakami, S., Nagaosa, N.: Geometrical Aspects in Optical Wavepackets Dynamics. Phys. Rev. E 74, 066610 (2006) 35. Perlick, V.: Fermat principle in Finsler spacetimes. Gen. Relativ. Gravit. 38(2), 365–380 (2006) 36. Rund, H.: The differential geometry of Finsler spaces. Berlin-Heidelberg-New York: Springer Verlag, 1959 37. Segre, S.E., Zanza, V.: Derivation of the pure Faraday and Cotton-Mouton effects when polarimetric effects in a tokamak are large. Plasma Phys. Control. Fusion 48, 1–13 (2006) 38. Shen, Z.: Landsberg Curvature, S-Curvature and Riemann Curvature. In: A Sampler of Riemann-Finsler Geometry, Bao, D., Bryant, R.L., Chern, S.-S., Shen, Z. (eds.), MSRI Publications 50, Cambridge: Cambridge University Press, 2004 39. Souriau, J.-M.: Structure des systèmes dynamiques, Paris: Dunod, 1970, ©1969; Structure of Dynamical Systems. A Symplectic View of Physics, translated by C.H. Cushman-de Vries, (R.H. Cushman, G.M. Tuynman, Translation eds.), Basel-Boston: Birkhäuser, 1997 40. Souriau, J.-M.: Modèle de particule à spin dans le champ électromagnétique et gravitationnel. Ann. Inst. Henri Poincaré 20A, 315–364 (1974) 41. Sternberg, S.: On the Role of Field Theories in our Physical Conception of Geometry. In: Proc. 2nd Bonn Conf. Diff. Geom. Meths. in Math. Phys, Lecture Notes in Mathematics, 676, Berlin-HeidelbergNew York: Springer-Verlag, 1978, pp. 1–80 42. van Tiggelen, B.A.: Transverse diffusion of light in Faraday-Active Media. Phys. Rev. Lett. 75, 422–424 (1995) Communicated by G.W. Gibbons
Commun. Math. Phys. 283, 729–748 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0536-z
Communications in
Mathematical Physics
Group Orbits and Regular Partitions of Poisson Manifolds Jiang-Hua Lu1 , Milen Yakimov2 1 Department of Mathematics, The University of Hong Kong, Pokfulam,
Hong Kong. E-mail: [email protected]
2 Department of Mathematics, University of California, Santa Barbara,
CA 93106, USA. E-mail: [email protected] Received: 14 September 2007 / Accepted: 13 February 2008 Published online: 5 July 2008 – © Springer-Verlag 2008
Abstract: We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras d with Poisson structures defined by Lagrangian splittings of d. In the special case of g ⊕ g, where g is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on L defined by arbitrary Lagrangian splittings of g ⊕ g. Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin–Drinfeld splittings as special cases.
1. Introduction Lie theory provides a rich class of examples of Poisson manifolds/varieties. In this paper, we study a class of Poisson manifolds of the form (D/Q, u,u ), where D is an even dimensional connected real or complex Lie group whose Lie algebra d is quadratic, i.e. d is equipped with a nondegenerate invariant symmetric bilinear form , ; the closed subgroup Q of D corresponds to a subalgebra q of d that is coisotropic with respect to , , and (u, u ) is a pair of complementary Lagrangian subalgebras of d. A Lie subalgebra l of d will be called Lagrangian if l⊥ = l with respect to , . We will call such a splitting d = u+u a Lagrangian splitting. The Poisson structure u,u is obtained from the r -matrix 1 ξi ∧ xi ∈ ∧2 d, 2 n
ru,u =
i=1
where {x1 , . . . , xn } and {ξ1 , . . . , ξn } are pairs of dual bases of u and u with respect to , . We refer the reader to § 2.2 for the precise definition of u,u .
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Let U and U be the connected subgroups of D with Lie algebras u and u respectively. Our first main result, see Theorem 2.7 and Proposition 2.13, is that when [q, q] ⊂ q⊥ ,
(1.1)
and U -orbits in
all intersections of U D/Q are regular Poisson submanifolds. In fact, if N (u) and N (u ) denote the normalizers of u and u in D respectively, we also show that all intersections of N (u) and N (u )-orbits in D/Q are regular Poisson submanifolds. Note that the condition (1.1) is a property of the coisotropic subalgebra q of (d, , ) and does not depend on the Lagrangian splitting d = u + u . Once this condition is verified for a given q, the above result provides “regular” partitions for the Poisson structures u,u on D/Q for any Lagrangian splitting d = u + u . Our second main result shows that the condition (1.1) is satisfied when d is reductive and q is the normalizer subalgebra in d of any Lagrangian subalgebra of (d, , ). In fact, we show in Proposition 3.3 that in this case [q, q] = q⊥ . Let L(d, , ) be the variety of Lagrangian subalgebras of (d, , ). All D-orbits in L(d, , ) are of the form D/N (l), where N (l) is the normalizer subgroup in D of an l ∈ L(d, , ), and every Lagrangian splitting d = u + u defines a Poisson structure u,u on L(d, , ). As a corollary of the second main result we obtain that, if d is an even dimensional reductive quadratic Lie algebra, then every non-empty intersection of an N (u)-orbit and an N (u )-orbit on L(d, , ) is a regular Poisson submanifold with respect to u,u In §4, we take the special case when d = g⊕g for a complex semi-simple Lie algebra g and (x1 , x2 ), (y1 , y2 ) = x1 , y1 − x2 , y2 , x1 , x2 , y1 , y2 ∈ g, where ., . is a nondegenerate invariant symmetric bilinear form on g whose restriction to a compact real form of g is negative definite. Lagrangian splittings of (g ⊕ g, , ) have been classified by Delorme [4]. In particular, one has the Belavin– Drinfeld splittings g ⊕ g = gdiag + l, where gdiag is the diagonal of g ⊕ g. For any l ∈ L(g ⊕ g), the N (l)-orbits in L(g ⊕ g) can be described by using results in [15]. For an arbitrary Lagrangian splitting g ⊕ g = l1 + l2 we prove that the intersection of each N (l1 ) and N (l2 )-orbit on L(g ⊕ g) is connected. Further, using [15], we compute the rank of all corresponding Poisson structures l1 ,l2 on L(g ⊕ g). This result extends the dimension formulas for symplectic leaves in the second author’s classification [22] of symplectic leaves of Belavin–Drinfeld Poisson structures on complex reductive Lie groups. Our result also generalizes the rank formulas of S. Evens and the first author for the standard Poisson structure on L(g ⊕ g). As have been shown in [7,8], all real and complex semi-simple symmetric spaces, as well as certain of their compactifications can be embedded into suitable varieties of Lagrangian subalgebras. Our results show that all such spaces carry Poisson structures and natural partitions into regular Poisson subvarieties. All manifolds and vector spaces in this paper, unless otherwise stated, are assumed to be either complex or real. A submanifold N of a Poisson manifold (M, π ) will be called a complete Poisson submanifold if it is closed under all Hamiltonian flows or equivalently it is a union of symplectic leaves of π . 2. The Poisson Spaces D/ Q Recall that a quadratic Lie algebra is a pair (d, , ), where d is a Lie algebra and , is an invariant symmetric nondegenerate bilinear form on d. Throughout this section, we
Regular Partitions of Poisson Manifolds
731
fix a quadratic Lie algebra (d, , ) and a connected Lie group D with Lie algebra d. For a subspace V of d, set V ⊥ = {x ∈ d | x, y = 0, ∀y ∈ V }.
(2.1)
2.1. Lagrangian splittings. A coisotropic (resp. Lagrangian, isotropic) subalgebra of d (with respect to , ) is by definition a Lie subalgebra q of d such that q⊥ ⊂ q (resp. q⊥ = q, q ⊂ q⊥ ). Definition 2.1. A Lagrangian splitting of d is a vector space direct sum decomposition d = u + u , where u and u are both Lagrangian subalgebras of d. The triple (d, u, u ) is also called a Manin triple [12]. Given a Lagrangian splitting d = u + u , for a subspace W ⊂ u, set W 0 = {ξ ∈ u | ξ, x = 0, ∀x ∈ W } = W ⊥ ∩ u .
(2.2)
We now recall how Lagrangian splittings give rise to Poisson Lie groups. Recall that a Poisson Lie group is a pair (G, π ), where G is a Lie group and π is a Poisson structure on G such that the group multiplication G × G → G is a Poisson map. When a (not necessarily closed) subgroup H of G is also a Poisson submanifold with respect to π , (H, π ) is itself a Poisson Lie group and is called a Poisson Lie subgroup of (G, π ). If (G, π ) is a Poisson Lie group, then π(e) = 0, where e ∈ G is the identity element. Let g be the Lie algebra of G, and let de π : g → ∧2 g be the linearization of π at e defined by (de π )(x) = (L x π )(e), where for x ∈ g, x is any local vector fields with x (e) = x and L x˜ π is the Lie derivative of π at e. Then (g, de π ) is a Lie bialgebra [12] called the tangential Lie bialgebra of (G, π ). Assume that d = u + u is a Lagrangian splitting. The bilinear form , induces a non-degenerate pairing between u and u . Define δu : u −→ ∧2 u : δu(x), y ∧ z = x, [y, z], δu : u −→ ∧2 u : δu (x), y ∧ z = x, [y, z],
x ∈ u, y, z ∈ u , x ∈ u , y, z ∈ u.
(2.3) (2.4)
Then (u, δu) and (u , δu ) are Lie bialgebras [12]. Associated to the splitting d = u + u we also have the r -matrix 1 ξ j ∧ x j ∈ ∧2 d, 2 n
Ru,u =
(2.5)
j=1
where {x1 , x2 , . . . , xn } and {ξ1 , ξ2 , . . . , ξn } are bases of u and u , respectively, such that xi , ξ j = δi j for 1 ≤ i, j ≤ n. It is easy to see that Ru,u is independent of the choice of the bases. Moreover, the Schouten bracket [Ru,u , Ru,u ] ∈ ∧3 d is given by [Ru,u , Ru,u ], a ∧ b ∧ c = 2a, [b, c],
a, b, c ∈ d.
(2.6)
Recall that D is a connected Lie group with Lie algebra d. Denote by U and U the connected subgroups of D with Lie algebras u and u , respectively. Let Rul ,u and Rur ,u
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be the left and the right invariant bi-vector fields on D with values Ru,u at the identity element. Set πuD,u := Rur ,u − Rul ,u . (2.7) The following fact can be found in [6,12]. Proposition 2.2. The bivector field πuD,u is a Poisson structure on D and (D, πuD,u ) is a Poisson Lie group. Both U and U are Poisson Lie subgroups of (D, πuD,u ). Let πU = πuD,u |U ,
πU = −πuD,u |U .
(2.8)
Then the tangential Lie bialgebras of the Poisson Lie groups (U, πU ) and (U , πU ) are respectively (u, δu) and (u , δu ). 2.2. The Poisson spaces D/Q. Assume that Q is a closed subgroup of D whose Lie algebra q is a coisotropic subalgebra of (d, , ). For an integer k ≥ 1, let χ k (D/Q) be the space of k-vector fields on D/Q. Then the left action of D on D/Q gives rise to the Lie algebra anti-homomorphism κ : d −→ χ 1 (D/Q) whose multi-linear extension ∧k d → χ k (D/Q) will be denoted by the same letter. Given a Lagrangian splitting d = u + u , define the bivector field u,u on D/Q by u,u := κ(Ru,u ),
(2.9)
recall (2.5). The following theorem is the main result for this subsection. Theorem 2.3. For every Lagrangian splitting d = u + u and every closed subgroup Q of D whose Lie algebra q is coisotropic in d, 1) u,u is a Poisson bi-vector field on D/Q; 2) all U and U -orbits in D/Q are complete Poisson submanifolds of (D/Q, u,u ). Proof. 1) The Lie algebra of the stabilizer subgroup of each point of D/Q is a coisotropic subalgebra of d. To prove that u,u is Poisson, it suffices to show that [Ru,u , Ru,u ] ∈ q ∧ d ∧ d for each coisotropic subalgebra q of d. This is equivalent to [Ru,u , Ru,u ], a ∧ b ∧ c = 0, ∀a, b, c ∈ q⊥ which follows from (2.6) because q⊥ is an isotropic subalgebra of d. Denote by κq : d → d/q the canonical projection and its induced map ∧2 d → ∧2 (d/q). The second part of Theorem 2.3 now follows from Lemma 2.4 below. Lemma 2.4. For every coisotropic subalgebra q of d, one has κq(Ru,u ) ∈ κq(∧2 u) ∩ κq(∧2 u ) .
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Proof. It is sufficient to show that κq(Ru,u ) ∈ κq(∧2 u). Let {x1 , x2 , . . . , xl } be a basis for u ∩ q. Extend it to a basis {x1 , x2 , . . . , xl , xl+1 , . . . , xn } of u. Let {ξ1 , . . . , ξn } be the dual basis of u with respect to , . It is easy to see that (u ∩ q)0 = Span{ξl+1 , ξl+1 , . . . , ξn } = pu (q⊥ ), recall (2.2), where pu : d → u is the projection along u. Choose y j ∈ u such that y j + ξ j ∈ q⊥ for l + 1 ≤ j ≤ n and write R
u,u
l n n 1 1 1 = ξj ∧ xj + (y j + ξ j ) ∧ x j − yj ∧ x j. 2 2 2 j=1
j=l+1
j=l+1
Since κq(x j ) = 0 for 1 ≤ j ≤ l and κq(y j + ξ j ) = 0 for l + 1 ≤ j ≤ n, we have κq(R
u,u
n 1 )=− κq(y j ) ∧ κq(x j ) ∈ κq(∧2 u). 2
(2.10)
j=l+1
In the setting of Theorem 2.3, the action map (D, πuD,u ) × (D/Q, u,u ) −→ (D/Q, u,u ) is easily seen to be Poisson. Thus (D/Q, u,u ) is a Poisson homogeneous space [5] of (D, πuD,u ). By part 2) of Theorem 2.3, each U and U -orbit in D/Q is a Poisson homogeneous space of (U, πU ) and (U , −πU ), respectively. 2.3. Rank of the Poisson structure u,u on D/Q. For d ∈ D set d = d Q ∈ D/Q. Consider the U -orbit U.d through d. Then (U.d, u,u ) is a Poisson homogeneous space of (U, πU ). Denote by ld the Drinfeld Lagrangian subalgebra of d, associated to the base point d of U· d, cf. [5]. It is defined as follows: identify Td (U· d) ∼ = u/(u ∩ Add q) 2 and regard u,u (d) as an element in ∧ (u/(u ∩ Add q)). For ξ ∈ (u ∩ Add q)0 , recall (2.2), let ιξ u,u (d) ∈ u/(u ∩ Add q) be such that ιξ u,u (d), η = u,u (d)(ξ, η) for all η ∈ (u ∩ Add q)0 . Then ld ⊂ d is given by ld = {x + ξ | x ∈ u, ξ ∈ (u ∩ Add q)0 , ιξ u,u (d) = x + u ∩ Add q}. If Rank u,u (d) denotes the rank of u,u at d, it is easy to see from the definition of ld that (2.11) Rank u,u (d) = dim(U· d) − dim(u ∩ ld ). Proposition 2.5. For any d ∈ D, the Drinfeld Lagrangian subalgebra ld is ld = Add q⊥ + u ∩ Add q.
(2.12)
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Proof. Since the stabilizer subalgebra of d at d is Add q, it is enough to prove that le = q⊥ + u ∩ q, where e is the identity element of D. Note that since q⊥ ⊂ q, (q⊥ + u ∩ q)⊥ = q ∩ (u ∩ q)⊥ = q ∩ (u + q⊥ ) = u ∩ q + q⊥ . Thus q⊥ + u ∩ q is a Lagrangian subspace of d. Since le is Lagrangian in d and u ∩ q ⊂ le , it is sufficient to show that q⊥ ⊂ le . For 1 ≤ i ≤ n and l + 1 ≤ j ≤ n, let xi ∈ u, ξi ∈ u and y j ∈ u be as in the proof of Lemma 2.4. If y + ξ ∈ q⊥ , for some y ∈ u, ξ ∈ u , then ξ = nj=l+1 λ j ξ j . Thus n y + ξ − j=l+1 λ j (y j + ξ j ) ∈ u ∩ q⊥ ⊂ l. The proposition will now follow if we show that y j + ξ j ∈ le for every l + 1 ≤ j ≤ n. By (2.10), u,u (e) = −
n 1 (y j + u ∩ q) ∧ (x j + u ∩ q) ∈ ∧2 (u/(u ∩ q)) ∼ = ∧2 Te D/Q. 2 j=l+1
Thus for each l + 1 ≤ j ≤ n, ιξ j u,u (e) =
n 1 1 yj − ξ j , yk xk + u ∩ q. 2 2 k=l+1
Since 0 = y j + ξ j , yk + ξk = ξ j , yk + ξk , y j for l + 1 ≤ k ≤ n and since x j ∈ u ∩ q for 1 ≤ j ≤ l, we have ιξ j u,u (e) =
n n 1 1 1 1 yj + ξk , y j xk + u ∩ q = y j + ξk , y j xk + u ∩ q 2 2 2 2 k=l+1
k=1
1 1 = y j + y j + u ∩ q = y j + u ∩ q, 2 2 we see that y j + ξ j ∈ le . Since u + u = d, U -orbits and U -orbits in D/Q intersect transversally. By Theorem 2.3, any such non-empty intersection is a Poisson submanifold of u,u . The following corollary gives the corank of u,u in U· d ∩ U· d at d for every d ∈ D. Corollary 2.6. For any d ∈ D, Rank u,u (d) = dim(U· d ∩ U· d) + dim(D/Q) − dim(U· d) − dim(u ∩ ld ), where ld is the Drinfeld Lagrangian subalgebra given by (2.12). Proof. The statement follows immediately from (2.11) and the fact that dim(U· d) + dim(U· d) − dim(D/Q) = dim(U· d ∩ U· d).
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2.4. First main theorem. Recall that a manifold with a Poisson structure of constant rank is called a regular Poisson manifold. Theorem 2.7. If q is a coisotropic subalgebra of d such that [q, q] ⊂ q⊥ , then for any closed subgroup Q of D with Lie algebra q and for any Lagrangian splitting d = u + u of d, the intersection of any U -orbit with any U -orbit in D/Q is a regular Poisson submanifold for the Poisson structure u,u . Proof. Let again e be the identity in D. Since [Add q, Add q] ⊂ (Add q)⊥ for any d ∈ D, it suffices to show that U· e ∩ U· e is a regular Poisson manifold of u,u . Let a ∈ U and a ∈ U be such that a = a ∈ U· e ∩ U· e. Then there exists b ∈ Q such that a = a b. Thus dim(u ∩ la ) = dim(u ∩ Ada (q⊥ + u ∩ q)) = dim(u ∩ Ada b (q⊥ + u ∩ q)) = dim(u ∩ Adb (q⊥ + u ∩ q)). Since [q, q] ⊂ q⊥ , we have [q, q⊥ +u∩q] ⊂ [q, q] ⊂ q⊥ ⊂ q⊥ +u∩q. Thus Q normalizes q⊥ + u ∩ q, and Adb (q⊥ + u ∩ q) = q⊥ + u ∩ q. Hence dim(u ∩ la ) = dim(u ∩ le ). It follows from Corollary 2.6 that the rank of u,u at a is the same as that at e. The special case of Theorem 2.7, when the Drinfeld subalgebras of all points of (D/Q, u,u ) integrate to closed subgroups of D, can be also proved using Karolinsky’s result [10]. Remark 2.8. Note that if q ⊂ d is a coisotropic subalgebra such that n(q) = q, where n(q) is the normalizer of q in d, then [q, q] ⊃ q⊥ . Indeed, if x ∈ [q, q]⊥ , then x, [q, q] = 0 which implies that [x, q], q = 0, so [x, q] ⊂ q⊥ ⊂ q. Thus x ∈ n(q) = q. This shows that [q, q]⊥ ⊂ q, so [q, q] ⊃ q⊥ . We thus conclude that, if q is coisotropic such that n(q) = q and [q, q] ⊂ q⊥ , then [q, q] = q⊥ . This remark will be used in §3.2. Corollary 2.9. Let L be a closed subgroup of D whose Lie algebra l ⊂ d is Lagrangian. Then for any Lagrangian splitting d = u + u , symplectic leaves of u,u in D/L are precisely the connected components of the intersections of U and U -orbits in D/L. Proof. Again it is enough to prove that U· e ∩ U· e is symplectic. By Corollary 2.6 and Theorem 2.7, the corank of u,u in U· e ∩ U· e is equal to dim(U· e) + dim(u ∩ l) − dim(D/L) = 0. Remark 2.10. Corollary 2.9 describes the symplectic leaves for a large class of Poisson homogeneous spaces. Indeed, by [5], every Poisson homogeneous space of (U, πU ) is of the form U/H , where H is a subgroup of U whose Lie algebra is u ∩ l for a Lagrangian subalgebra l of d. In the case when H = U ∩ L, where L is a closed subgroup of D with Lie algebra l, we can embed U/H into D/L as the U -orbit through e. This embedding is also Poisson. Thus the symplectic leaves of U/H are the connected components of the intersections of U/H with the U -orbits in D/L, i.e. with the (U , −πU ) Poisson homogeneous spaces inside (D/L , u,u ). We conclude this subsection with an example showing that the statement of Theorem 2.7 is incorrect if the condition [q, q] ⊂ q⊥ is dropped.
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Example 2.11. Let G a connected complex simple Lie group with a pair of opposite Borel subgroups B and B − . Set g = Lie G, T = B ∩ B − , and h = Lie T . Then d = g ⊕ h is a quadratic Lie algebra with the bilinear form (x1 , x2 ), (y1 , y2 ) = x1 , y1 − x2 , y2 ,
x1 , y1 , ∈ g, x2 , y2 ∈ h,
where ., . is a nondegenerate symmetric invariant bilinear form on g. Let D = G × T . Given a parabolic subgroup P ⊃ B of G, the Lie algebra q of Q = P × T is coisotropic but does not satisfy the condition q⊥ ⊂ [q, q]. The following subalgebras provide a Lagrangian splitting of g ⊕ h: u = {(x + h, h) | x ∈ n, h ∈ h},
u = {(x + h, −h) | x ∈ n− , h ∈ h},
where n and n− are the nilpotent radicals of Lie B and Lie B − . Under the identification (G × T )/(P × T ) ∼ = G/P the Poisson structure u,u corresponds to the Poisson structure π =κ f α ∧ eα , α∈ +
where, + is the set of positive roots of g corresponding to n, {eα } and { f α } are sets of root vectors of g, normalized by eα , f α = 1, and κ is the extension to ∧2 g of the infinitesimal action of g on G/P. It was shown in [9] that the partition of (G/P, π ) by T -orbits of leaves (which is a partition by regular Poisson submanifolds) coincides with Lusztig’s partition [17] of G/P. The strata of this partition are WP pr P Bw1 .B ∩ B − w2 .B , w1 ∈ W, w2 ∈ Wmax . (2.13) Here pr P : G/B → G/P denotes the standard projection, W the Weyl group of (G, T ), WP the set of the maximal length representatives of cosets in W/W P where, W P and Wmax is the parabolic subgroup of W corresponding to P. Under the identification (G × T )/(P × T ) ∼ = G/P the U and U -orbits on (G × T )/(P × T ) correspond respectively to the B and B − -orbits on G/P. The coarser partition of (2.13) by intersecting B and B − -orbits on (G/P, π ) is no longer a partition by regular Poisson submanifolds if P = B or G. This is easily seen by applying Theorem 4.10 below and the Poisson embedding [9, (1.10)]. 2.5. Intersections of N (u) and N (u )-orbits. For a Lagrangian splitting d = u + u , let N (u) and N (u ) be the normalizer subgroups of u and u in D, respectively. Both N (u) and N (u ) are closed subgroups of D, and sometimes N (u) and N (u )-orbits in a space D/Q are easier to determine than the U and U -orbits. This is the case for the examples considered in §4. In this subsection, we prove some facts on N (u) and N (u )-orbits. It is clear from Theorem 2.3 that for any closed subgroup Q of D with coisotropic Lie subalgebra q and for any Lagrangian splitting d = u + u , all N (u) and N (u )-orbits in D/Q are complete Poisson submanifolds with respect to the Poisson structure u,u , cf. Theorem 2.3. Recall the Poisson structure πuD,u on D from (2.7). Lemma 2.12. For any Lagrangian splitting d = u + u , the Poisson structure πuD,u on D vanishes at all points in N (u) ∩ N (u ). Consequently, for any closed subgroup Q of D with coisotropic Lie subalgebra q, N (u) ∩ N (u ) leaves the Poisson structure u,u on D/Q invariant.
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Proof. Let d ∈ N (u) ∩ N (u ). If {x1 , . . . , xn } and {ξ1 , . . . , ξn } is a pair of dual bases for u and u with respect to , , then so are {Add (x1 ), . . . Add (xn )} and {Add (ξ1 ), . . . , Add (xn )}. Thus n n D πu,u (d) = L d (ru,u ) − Rd (ru,u ) = Rd Add (ξi ) ∧ Add (xi ) − ξi ∧ xi = 0. i=1
i=1
Since the (D, πuD,u )-action on (D/Q, u,u ) is Poisson, N (u) ∩ N (u ) leaves u,u invariant. Next we generalize Theorem 2.7 to intersections of arbitrary N (u)-orbits and N (u )orbits in D/Q. Its proof is similar to the one of Theorem 2.7 and is left to the reader. Proposition 2.13. If q is a coisotropic subalgebra of d such that [q, q] ⊂ q⊥ , then for any closed subgroup Q with Lie algebra q and for any Lagrangian splitting d = u + u of d, the intersection of any N (u)-orbit with any N (u )-orbit in D/Q is a regular Poisson submanifold for the Poisson structure u,u . Although Proposition 2.13 provides a stronger result than Theorem 2.7, it is apriori possible that the geometry of the strata of the coarser partition from Proposition 2.13 is more complicated than that of the strata of the finer partition from Theorem 2.7. The next result, Proposition 2.15, shows that this is not the case. First we prove an auxiliary lemma. Lemma 2.14. Let d = u + u be any Lagrangian splitting of d. Assume that N (u) is connected. Then N (u) = (U ∩ N (u))o U , where (U ∩ N (u))o denotes the identity component of the group U ∩ N (u). Moreover, N (u) is a Poisson Lie subgroup of (D, π ). Proof. This is because (U ∩ N (u))o U is a connected subgroup of D with Lie algebra u ∩ n(u) + u which is equal to n(u) because d = u + u . Proposition 2.15. Let d = u+u be any Lagrangian splitting of d, and assume that N (u) and N (u ) are both connected. Let X be any Poisson space with a Poisson (D, πuD,u )action. Let x ∈ X be such that N (u)x ∩ N (u )x = ∅. Then the group N (u) ∩ N (u ) acts transitively on the set of intersections of U -orbits and U -orbits in N (u)x ∩ N (u )x. Proof. Using Lemma 2.14 we obtain N (u)x ∩ N (u )x =
(αU x) ∩ (U βx)
α∈(U ∩N (u))o ,β∈(U ∩N (u ))o
=
α(U x ∩ U βx)
α∈(U ∩N (u))o ,β∈(U ∩N (u ))o
=
αβ(U x ∩ U x).
α∈(U ∩N (u))o ,β∈(U ∩N (u ))o
We finish this section with a formula for the corank of u,u to be used in §4. As before Q ⊂ D is assumed to be a closed subgroup with coisotropic Lie subalgebra q. For any d ∈ D, let Corank u,u (d) = dim(N (u)· d ∩ N (u )· d) − Rank u,u (d) be the corank of u,u in N (u)· d ∩ N (u )· d at d ∈ D/Q.
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Lemma 2.16. In the above setting, for any d ∈ D, Corank u,u (d) = dim n(u ) − dim(D/Q) + dim n(u) − dim u − dim(n(u) ∩ Add q) + dim(u ∩ Add q) − dim(n(u ) ∩ Add q) + dim(u ∩ ld ), where ld = Add q⊥ + u ∩ Add q is the Drinfeld Lagrangian subalgebra as in (2.12). Proof. Since N (u)· d and N (u )· d intersect transversally, dim(N (u)· d ∩ N (u )· d) = dim(N (u)· d) + dim(N (u )· d) − dim(D/Q) = dim n(u) − dim(n(u) ∩ Add q) + dim n(u ) − dim(n(u ) ∩ Add q) − dim(D/Q). By (2.11), Rank u,u (d) = dim u − dim(u ∩ Add q) − dim(u ∩ ld ). The formula for Corank u,u (d) in Lemma 2.16 thus follows. 3. The Variety of Lagrangian Subalgebras Associated to a Reductive Lie Algebra 3.1. General case. Let (d, , ) be a 2n-dimensional quadratic Lie algebra and let D be a connected Lie group with Lie algebra d. We will denote by L(d) the variety of all Lagrangian subalgebras of d. It is an algebraic subvariety of the Grassmannian Gr(n, d) of n-dimensional subspaces of d. The group D acts on L(d) through the adjoint action. Fix a Lagrangian splitting d = u + u , recall Ru,u ∈ ∧2 d given by (2.5). Let again κ : d → χ 1 (L(d)) be the Lie algebra anti-homomorphism from d to the Lie algebra of vector fields on L(d), and define the bi-vector field u,u = κ(Ru,u ) on L(d). For l ∈ L(d), let N (l) and n(l) be respectively the normalizer subgroup of l in D and the normalizer subalgebra of l in d. Then the D-orbit in L(d) through l is isomorphic to D/N (l). Clearly, n(l) is coisotropic in d because it contains l. Thus it follows from Theorem 2.3 that u,u is a Poisson structure on L(d), see also [7]. The following proposition now follows immediately from Theorem 2.7. Proposition 3.1. Assume that (d, , ) is an even dimensional quadratic Lie algebra and D is a connected Lie group with Lie algebra d such that for every l ∈ L(d), [n(l), n(l)] ⊂ (n(l))⊥ . Then for any Lagrangian splitting d = u + u , the intersection of an N (u)-orbit and an N (u )-orbit in L(d) is a regular Poisson submanifold for the Poisson structure u,u . 3.2. Second main theorem: the case of a reductive Lie algebra. When (d, , ) is a reductive quadratic Lie algebra, we have the following second main theorem of the paper. Theorem 3.2. If D is a connected complex or real reductive Lie group and , is a nondegenerate symmetric invariant bilinear form on d = Lie D, then for any Lagrangian splitting d = u + u , the intersection of any N (u)-orbit and any N (u )-orbit in L(d) is a regular Poisson submanifold for the Poisson structure u,u on L(d). To prove Theorem 3.2 we need to check that in the setting of Theorem 3.2 the condition of Proposition 3.1 is satisfied. In fact, we prove a stronger statement.
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Proposition 3.3. If d is an even dimensional complex or real reductive Lie algebra and , is a nondegenerate symmetric invariant bilinear form on d, then for all Lagrangian subalgebras l of (d, , ), [n(l), n(l)] = (n(l))⊥ . The real case in Proposition 3.3 follows from the complex one. Indeed, let (d, , ) be a quadratic real reductive Lie algebra. Then (dC , , C ) is a quadratic complex reductive Lie algebra. Let l be a Lagrangian subalgebra of (d, , ), and let n(lC ) be the normalizer subalgebra of lC in dC . Then n(lC ) = (n(l))C . Assume the validity of Proposition 3.3 in the complex case. We get [n(l), n(l)] = [n(lC ), n(lC )] ∩ d = (n(lC ))⊥ ∩ d = ((n(l))C )⊥ ∩ d = (n(l))⊥ , where (.)⊥ denotes orthogonal complements in d and dC . This proves the real case in Proposition 3.3. To obtain the complex case in Proposition 3.3 we need the following result of Delorme [4]. Theorem 3.4 [Delorme]. Assume that (d, , ) is an even dimensional reductive quadratic Lie algebra. For each Lagrangian subalgebra l of (d, , ) the normalizer of the nilpotent radical n of l is a parabolic subalgebra p of d. In addition, p has a Levi subal¯ = [m, m] decomposes as m ¯ = m1 ⊕ m2 and for gebra m whose derived subalgebra m which there exists an isomorphism θ : m1 → m2 . If z denotes the center of m then θ ¯ ⊕ z + n, ¯θ +n⊂l⊂ m (3.1) m ¯ θ = {x + θ (x) | x ∈ m1 } ⊂ m1 ⊕ m2 . where m Proof of Proposition 3.3 in the complex case. Let l be a Lagrangian subalgebra of (d, , ) as in Theorem 3.4. First we claim that n(l) ⊂ p. Indeed, the normalizer of l lies inside the normalizer of the nilpotent radical n of l which is p: if y ∈ n(l), then for small t, exp(t ad y ) is an automorphism of l and thus of its nilpotent radical n. Taking the derivative at t = 0, we get that y normalizes n. Next we will show that θ ¯ ⊕ z + n. n(l) = m (3.2) θ ¯ ⊕ z + n is clear from (3.1). Define the subspace The inclusion n(l) ⊃ m ¯ ¯ − = {x − θ (x) | x ∈ m1 } ⊂ m. m ¯ θ we have the direct sum decomposition of m ¯ θ -modules Under the adjoint action of m ¯θ ⊕m ¯ − ⊕ z ⊕ n. p=m
θ ¯ ⊕ z + n, then there exists a nonzero Y = y − θ (y) ∈ m ¯ − which belongs If n(l) = m θ − θ ¯ normalizes m ¯ we get that adY (m ¯ ) = 0 and thus ad y (m1 ) = 0. This to n(l). Since m is a contradiction since m1 is semi-simple and y = 0. This completes the proof of (3.2). Repeating the proof with n(l) in the place of l, which also satisfies (3.1) as shown above, we get that n(l) coincides with its normalizer. ¯ θ + n. Because l is a Lagrangian subalgebra of Now (3.2) implies [n(l), n(l)] ⊂ m θ ¯ + n ⊂ n(l)⊥ . Thus, [n(l), n(l)] ⊂ n(l)⊥ . By Remark 2.8, (d, , ), it is clear that m ⊥ [n(l), n(l)] = n(l) .
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4. Ranks of Poisson Structures on the Variety L(g ⊕ g) 4.1. The quadratic Lie algebra (g ⊕ g, , ). Assume that g is a complex semi-simple Lie algebra and , is a fixed nondegenerate invariant symmetric bilinear form whose restriction to a compact real form of g is negative definite. Let d = g ⊕ g be the direct sum Lie algebra and let , be the bilinear form on d given by (x1 , x2 ), (y1 , y2 ) = x1 , y1 − x2 , y2 ,
x1 , x2 , y1 , y2 ∈ g.
(4.1)
In this section, we will study in more detail the Poisson structure l1 ,l2 on L(g ⊕ g) defined by an arbitrary Lagrangian splitting g ⊕ g = l1 + l2 . A classification of Lagrangian subalgebras of g ⊕ g was first obtained by Karolinsky [11]. It also follows from the more general results of Delorme [4], where Lagrangian splittings of an arbitrary reductive quadratic Lie algebra were classified. We will recall Delorme’s classification in § 4.2. Let G be the adjoint group of g. For l ∈ L(g ⊕ g) denote by N (l) the normalizer subgroup of l in G × G. Let g ⊕ g = l1 + l2 be an arbitrary Lagrangian splitting. By Theorem 3.2 the intersection of any N (l1 )-orbit and any N (l2 )-orbit in L(g ⊕ g) is a regular Poisson submanifold for the Poisson structure l1 ,l2 . Using results from [15], we will describe the N (l1 ) and N (l2 )-orbits in L(g ⊕ g) and will obtain an explicit formula for the rank of l1 ,l2 at an arbitrary l ∈ L(g ⊕ g). 4.2. Lagrangian splittings of (g ⊕ g, , ). Fix a Cartan subalgebra h of g and a choice + of positive roots in the set of all roots for (g, h). Let be the set of simple roots in + . For each α ∈ , let Hα ∈ h be such that x, Hα = α(x) for all x ∈ h. We will also fix a root vector E α for each α ∈ such that [E α , E −α ] = Hα . Following [21,8], we define a generalized Belavin–Drinfeld (gBD) triple to be a triple (S, T, d), where S and T are subsets of and d : S → T is a bijection such that Hdα , Hdβ = Hα , Hβ for all α ∈ S. For a subset S of , let S be the set of roots in the linear span of S. Set mS = h + gα , n S = gα , n− g−α S = + α∈ S
α∈ − S
α∈ + − S
− ¯ S = [m S , m S ] and and p S = m S + n S and p− S = m S + n S . We set m
¯ S = SpanC {Hα : α ∈ S }, z S = {x ∈ h | α(x) = 0, ∀α ∈ S}. hS = h ∩ m ¯ S and Then we have the decompositions h = z S + h S , m S = z S + m ¯ S + n S , p− ¯ S + n− pS = zS + m S = zS + m S. Recall that G denotes the adjoint group of g. The connected subgroups of G with Lie − − algebras p S , p− S , m S , n S and n S will be respectively denoted by PS , PS , M S , N S and N S− . Correspondingly we have the Levi decompositions PS = M S N S , PS− = M S N S− . Let Z S be the center of M S , and let χ S : PS → M S /Z S be the natural projection by first projecting to M S along N S and then to M S /Z S . We also denote by χ S the similar projection from PS− to M S /Z S . For a generalized Belavin–Drinfeld triple (S, T, d), let Lspace (z S ⊕ zT ) be the set of all Lagrangian subspaces of z S ⊕ zT with respect to the (nondegenerate) restriction of ¯S →m ¯ T be the unique Lie algebra isomorphism satisfying , to z S ⊕ zT . Let θd : m θd (Hα ) = Hdα , θd (E α ) = E dα , ∀α ∈ S.
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For every V ∈ Ls pace (z S ⊕ zT ), define ¯ S } + (n S ⊕ nT ) ⊂ p S ⊕ pT , l S,T,d,V = V + {(x, θd (x)) | x ∈ m lS,T,d,V lS,T,d,V
= V + {(x, θd (x)) | x ∈ = V + {(x, θd (x)) | x ∈
− ¯ S } + (n S ⊕ n− m T ) ⊂ p S ⊕ pT , − ¯ S } + (n− m S ⊕ nT ) ⊂ p S ⊕ pT .
(4.2) (4.3) (4.4)
It is easy to see that l S,T,d,V , lS,T,d,V , and lS,T,d,V are all in L(g ⊕ g). The subalgebras lS,T,d,V and lS,T,d,V are of course conjugate to ones of the type l S,T,d,V . Indeed, let W be the Weyl group of (g, h), and let w0 be the longest element in W . For A ⊂ , let W A be the subgroup of W generated by simple reflections with respect to roots in A, and let x A = w0 w0,A , where w0,A denotes the longest element of W A . Then it is easy to see that lS,T,d,V = Ad−1 (e,x˙ T ) l S,−w0 (T ), x T d,Ad(e,x˙ T ) V ,
(4.5)
Ad−1 (x˙ S ,e) l−w0 (S),T, d x S−1 ,Ad(x˙ S ,e) V ,
(4.6)
lS,T,d,V =
where x˙ T and x˙ S are representatives in G of x T and x S respectively. Denote also by θd the (unique) group isomorphism M S /Z S → MT /Z T induced by ¯S →m ¯ T . Corresponding to the subalgebras in (4.2)–(4.4), we define θd : m R S,T,d = {( p1 , p2 ) ∈ PS × PT | θd (χ S ( p1 )) = χT ( p2 )} ⊂ PS × PT , R S,T,d R S,T,d
= {( p1 , p2 ) ∈ PS × = {( p1 , p2 ) ∈
PS−
PT−
(4.8)
× PT .
(4.9)
| θd (χ S ( p1 )) = χT ( p2 )} ⊂ PS ×
× PT | θd (χ S ( p1 )) = χT ( p2 )} ⊂
PS−
(4.7)
PT− ,
One knows that R S,T,d , R S,T,d , and R S,T,d are all connected [8, Lemma 2.19]. Corresponding to (4.5) and (4.6), we have = Ad−1 R S,T,d (e,x˙ T ) R S,−w0 (T ), x T d ,
(4.10)
Ad−1 (x˙ S ,e) R−w0 (S),T, d x S−1 .
(4.11)
R S,T,d
=
, and R S,T,d will be denoted by r S,T,d , rS,T,d , and The Lie algebras of R S,T,d , R S,T,d r S,T,d respectively.
Proposition 4.1 [8]. Every (G × G)-orbit in L(g ⊕ g) passes through an l S,T,d,V for a unique generalized Belavin–Drinfeld triple (S, T, d) and a unique V ∈ Lspace (z S ⊕ zT ). The normalizer subgroup of l S,T,d,V in G × G is R S,T,d . Definition 4.2. For generalized Belavin-Drinfeld triples (Si , Ti , di ), i = 1, 2, let d −1 d2
S2 1
= {α ∈ S2 | (d1−1 d2 )n α is defined and is in S2 for n = 1, 2, . . .}.
A generalized Belavin–Drinfeld system is a pair of quadruples (S1 , T1 , d1 , V1 ) and (S2 , T2 , d2 , V2 ), where for i = 1, 2, (Si , Ti , di ) is a generalized Belavin–Drinfeld triple and Vi ∈ Lspace (z Si ⊕ zTi ), such that d −1 d2
= ∅; 1) S2 1 2) h1 ∩ h2 = {0}, where hi = Vi + {(x, θd (x)) | x ∈ h Si } ⊂ h ⊕ h for i = 1, 2.
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Theorem 4.3 [4, Delorme]. Every Lagrangian splitting of g ⊕ g is conjugate by an element in G × G to one of the form g ⊕ g = l1 + l2 , where l1 = lS1 ,T1 ,d1 ,V1
and
l2 = lS2 ,T2 ,d2 ,V2
(4.12)
for a generalized Belavin–Drinfeld system (S1 , T1 , d1 , V1 ), (S2 , T2 , d2 , V2 ). Let g ⊕ g = l1 + l2 be a Lagrangian splitting with l1 and l2 given in (4.12). By Theorem 3.2, any non-empty intersection of an N (l1 ) and an N (l2 )-orbit in L(g ⊕ g) is a regular Poisson subvariety for the Poisson structure l1 ,l2 . A classification of N (l1 ) and N (l2 )-orbits will be given in § 4.3. We now prove that every non-empty intersection of an N (l1 )-orbit and an N (l2 )-orbit in L(g ⊕ g) is smooth and irreducible. Let H be the connected subgroup of G with Lie algebra h. Proposition 4.4. For a Lagrangian splitting g⊕g = l1 +l2 with l1 and l2 given by (4.12), N (l1 ) ∩ N (l2 ) is a subtorus of H × H of dimension dim z S1 + dim z S2 . In particular, N (l1 ) ∩ N (l2 ) is connected. Proof. It follows from Proposition 4.1 that N (l1 ) = R S 1 ,T1 ,d1
and
N (l2 ) = R S2 ,T2 ,d2 .
(4.13)
For notational simplicity, let x1 = x T1 , x2 = x S2 . By (4.10) and (4.11), −1 ∩ Ad . R R N (l1 ) ∩ N (l2 ) = Ad−1 −1 (e,x˙1 ) S1 ,−w0 (T1 ), x1 d1 (x˙2 ,e) −w0 (S2 ), T2 , d2 x 2
Since
x2−1
∈ W −w0 (S2 ) and x1 ∈ −w0 (T1 )W , we can use [15, Theorem 2.5] to determine d −1 d2
N (l1 ) ∩ N (l2 ). Set N = N∅ . Since S2 1 = ∅, [15, Theorem 2.5] implies that N (l1 ) ∩ N (l2 ) ⊂ B × B and N (l1 ) ∩ N (l2 ) = (N (l1 ) ∩ N (l2 ))red (N (l1 ) ∩ N (l2 ))uni , where (N (l1 ) ∩ N (l2 ))red = N (l1 ) ∩ N (l2 ) ∩ (H × H ), (N (l1 ) ∩ N (l2 ))uni = N (l1 ) ∩ N (l2 ) ∩ (N × N ). Moreover, [15, Theorem 2.5] also tells us that −1 (N (l1 ) ∩ N (l2 ))uni ∼ = (N ∩ Ad−1 x˙2 (N−w0 (S2 ) )) × (N ∩ Ad x˙1 (N−w0 (T1 ) )). −1 It is easy to see that (N ∩ Ad−1 x˙2 (N−w0 (S2 ) )) × (N ∩ Ad x˙1 (N−w0 (T1 ) )) is the trivial group. Thus N (l1 ) ∩ N (l2 ) = N (l1 ) ∩ N (l2 ) ∩ (H × H ) consists of all (h 1 , h 2 ) ∈ H × H such that θd1 χ S1 (h 1 ) = χT1 (h 2 ), θd2 χ S2 (h 1 ) = χT2 (h 2 ),
which are equivalent to β
d β
h α1 = h d21 α , ∀α ∈ S1 and h 1 = h 22 , ∀β ∈ S2 .
(4.14)
Let = {α1 , α2 , . . . , αr } be the set of simple roots of g. Since G is the adjoint group of g, we can identify H × H with the torus (C× )2r by the map H × H −→ (C× )2r :
(h 1 , h 2 ) −→ (h α1 1 , h α1 2 , . . . , h α1 r , h α2 1 , h α2 2 , . . . , h α2 r ).
Regular Partitions of Poisson Manifolds
743
The conditions in (4.14) imply that the coordinates h α1 of h 1 for α ∈ S1 ∪ S2 are expressed in terms of coordinates of h 2 , and we have the extra conditions h d21 α = h 2d2 α ,
α ∈ S1 ∩ S2
(4.15) d −1 d2
for the coordinates of h 2 . To understand the conditions in (4.15), recall that S2 1 = ∅. Thus for every α ∈ S1 ∩ S2 , there is a unique integer n ≥ 1 and unique elements α (0) = α, α (1) , α (2) , . . . , α (n−1) ∈ S1 ∩ S2 such that d2 α (0) = d1 α (1) , d2 α (1) = d1 α (2) , . . . d2 α (n−2) = d1 α (n−1) and either d2 α (n−1) ∈ / T1 or d2 α (n−1) = d1 α (n) for some α (n) ∈ S1 but α (n) ∈ / S2 . Then the conditions in (4.15) are equivalent to h d21 α = h 2d2 α = h d22 α
(1)
= · · · = h d22 α
(n−1)
.
/ d1 (S1 ∩ S2 ), we see that the conditions (4.15) express h d21 α for every Since d2 α (n−1) ∈ β / d1 (S1 ∩ S2 ). We conclude that the set of α ∈ S1 ∩ S2 in terms of h 2 for some β ∈ (h 1 , h 2 ) ∈ H × H satisfying (4.14) is a subtorus of H × H with dimension equal to 2 dim H − |S1 ∪ S2 | − |S1 ∩ S2 | = 2 dim H − |S1 | − |S2 | = dim z S1 + dim z S2 . Corollary 4.5. For any Lagrangian splitting g ⊕ g = l1 + l2 , all N (l1 )-orbits and N (l2 )orbits in L(g ⊕ g) intersect transversally, and every such non-empty intersection is smooth and irreducible. Proof. Clearly n(l1 ) + n(l2 ) = g ⊕ g, where n(li ) is the Lie algebra of N (li ) for i = 1, 2. Since N (l1 ) ∩ N (l2 ) is connected, Corollary 4.5 follows from [20, Cor. 1.5]. 4.3. N (l1 ) and N (l2 )-orbits in L(g ⊕ g). Assume that g ⊕ g = l1 + l2 is a Lagrangian splitting with l1 and l2 given in (4.12). We now use results in [15] to describe N (l1 ) and N (l2 )-orbits in L(g ⊕ g). For A ⊂ , let W A and AW be respectively the sets of minimal length representatives in the cosets in W/W A and W A \W . For each w ∈ W , we fix a representative w˙ of w in the normalizer of H in G. Proposition 4.6. 1) Every N (l1 ) = R S 1 ,T1 ,d1 -orbit in L(g ⊕ g) is of the form R S 1 ,T1 ,d1 Ad(v˙1 ,v˙2 m 2 ) l S,T,d,V for a unique generalized Belavin-Drinfeld triple (S, T, d), a unique V ∈ Lspace (z S ⊕zT ), a unique pair (v1 , v2 ) ∈ W S × T1W , and some m 2 ∈ MT (v1 ,v2 ) with T (v1 , v2 ) = {α ∈ T | (v2−1 d1 v1 d −1 )n α is defined and is in T for n = 1, 2, . . .}. 2) Every N (l2 ) = R S2 ,T2 ,d2 -orbit in L(g ⊕ g) is of the form R S2 ,T2 ,d2 Ad(w˙ 1 m 1 ,w˙ 2 ) l S,T,d,V for a unique generalized Belavin-Drinfeld triple (S, T, d), a unique V ∈ Lspace (z S ⊕zT ), a unique pair (w1 , w2 ) ∈ S2W × W T , and some m 1 ∈ M S(w1 ,w2 ) with S(w1 , w2 ) = {α ∈ S | (w1−1 d2−1 w2 d)n α is defined and is in S for n = 1, 2, · · · }.
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Proof. In [15] we gave a description of the (R S1 ,T1 ,d1 , R S,T,d )-double cosets in G × G. The first part follows from (4.10), [15, Theorem 2.2], and the fact that −w0 (A)W = x A AW , ∀A ⊂ , see §4.2 for the definition of x A . Let σ : G × G → G × G : (g1 , g2 ) → (g2 , g1 ). Then using first part and the facts that
R S2 ,T2 ,d2 = σ RT ,S ,d −1 and R S,T,d = σ (RT,S,d −1 ) 2
2
2
we get part 2). Let O1 be an N (l1 ) = R S 1 ,T1 ,d1 -orbit and O2 an N (l2 ) = R S2 ,T2 ,d2 -orbit in L(g ⊕ g). By Proposition 4.6, we can assume that O1 = R S 1 ,T1 ,d1 Ad(v˙1 ,v˙2 m 2 ) l S,T,d,V ,
(4.16)
O2 =
(4.17)
R S2 ,T2 ,d2 Ad(w˙ 1 m 1 ,w˙ 2 ) l S,T,d,V ,
where (S, T, d, V ), v1 , v2 , w1 , w2 and m 1 and m 2 are as in Proposition 4.6. Let S1 (v1 , v2 ) = d1−1 v2 T (v1 , v2 ) = v1 d −1 T (v1 , v2 ) ⊂ S1 , S2 (w1 , w2 ) = w1 S(w1 , w2 ) =
d2−1 w2 d S(w1 , w2 )
⊂ S2 .
(4.18) (4.19)
In other words, S1 (v1 , v2 ) is the largest subset of S1 that is invariant under the partial map v1 d −1 v2−1 d1 : → , and S2 (w1 , w2 ) is the largest subset of S2 that is invariant under the partial map w1 d −1 w2−1 d2 : → . In order to compute the rank of l1 ,l2 at an l ∈ O1 ∩ O2 using Lemma 2.16, we need to compute the dimensions of various intersections of subalgebras in n(l1 ), n(l2 ) and n(l). Such intersections can be described using the following Proposition 4.8 which is derived from [15, Theorem 2.5]. Recall that for q ⊂ g ⊕ g, q⊥ = {x ∈ g ⊕ g | x, y = 0, ∀y ∈ q}. Clearly ¯ S }, r⊥ S,T,d = n S ⊕ nT + {(x, θd (x)) | x ∈ m − ¯ S1 }, r,⊥ S1 ,T1 ,d1 = n S1 ⊕ nT1 + {(x, θd1 (x)) | x ∈ m − ¯ S2 }. r,⊥ S2 ,T2 ,d2 = n S2 ⊕ nT2 + {(x, θd2 (x)) | x ∈ m
Notation 4.7. Let S (resp. S , S ) be the set of all subspaces of r S,T,d (resp. rS1 ,T1 ,d1 ,
,⊥ ,⊥ rS2 ,T2 ,d2 ) that contain r⊥ S,T,d (resp. r S1 ,T1 ,d1 , r S2 ,T2 ,d2 ). For a ∈ S, let Va = a ∩ (z S ⊕ zT ) and
X a = Va + {(x, θd (x)) | x ∈ zd −1 T (v1 ,v2 ) ∩ h S },
X a = Va + {(x, θd (x)) | x ∈ z S(w1 ,w2 ) ∩ h S }. For a ∈ S and a ∈ S , let Va = a ∩ (z S1 ⊕ zT1 ) Va = a ∩ (z S2 ⊕ zT2 )
and and
Ya = Va + {(x, θd1 (x)) | x ∈ z S1 (v1 ,v2 ) ∩ h S1 }, Ya = Va + {(x, θd2 (x)) | x ∈ z S2 (w1 ,w2 ) ∩ h S2 }.
We also set ⊥ ¯ S1 (v1 ,v2 ) ⊕ m ¯ d1 S1 (v1 ,v2 ) ). f1 = r,⊥ S1 ,T1 ,d1 ∩ Ad(v˙1 ,v˙2 m 2 ) r S,T,d ∩ (m
(4.20)
Regular Partitions of Poisson Manifolds
745
Proposition 4.8. For any a ∈ S1 and a ∈ S, one has the direct sum a ∩ Ad(v˙1 ,v˙2 m 2 ) a = (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)red + (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)nil , where (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)red = a ∩ Ad(v˙1 ,v˙2 m 2 ) a ∩ (m S1 (v1 ,v2 ) ⊕ md1 S1 (v1 ,v2 ) ) = Ya ∩ Ad(v˙1 ,v˙2 ) X a + f1 (direct sum), and (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)nil = a ∩ Ad(v˙1 ,v˙2 m 2 ) a ∩ (n S1 (v1 ,v2 ) ⊕ n− d1 S1 (v1 ,v2 ) ). The dimension of the latter is equal to l(v2 ) + dim(n ∩ Adv˙1 (n S )). Proof. Using (4.5) and [15, Theorem 2.5], one can see that a ∩ Ad(v˙1 ,v˙2 m 2 ) a ⊂ p S1 (v1 ,v2 ) ⊕ p− d1 S1 (v1 ,v2 ) and that a ∩ Ad(v˙1 ,v˙2 m 2 ) a = (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)red + (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)nil = a ∩ Ad(v˙1 ,v˙2 m 2 ) a ∩ (m S1 (v1 ,v2 ) ⊕ md1 S1 (v1 ,v2 ) )
+a ∩ Ad(v˙1 ,v˙2 m 2 ) a ∩ (n S1 (v1 ,v2 ) ⊕ n− d1 S1 (v1 ,v2 ) ).
The dimension formula for (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)nil also follows from [15, Theorem 2.5]. It now remains to show that (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)red = Ya ∩ Ad(v˙1 ,v˙2 ) X a + f1 as a direct sum. Clearly, Ya ∩ Ad(v˙1 ,v˙2 ) X a and f1 intersect trivially, and their sum is contained in (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)red . Suppose that (x, y) ∈ (a ∩Ad(v˙1 ,v˙2 m 2 ) a)red . Write x = x1 +x2 and y = y1 + y2 , where ¯ S1 (v1 ,v2 ) , y1 ∈ zd1 S1 (v1 ,v2 ) and y1 ∈ m ¯ d1 S1 (v1 ,v2 ) . Because (x, y) ∈ x1 ∈ z S1 (v1 ,v2 ) , x2 ∈ m a ⊂ rS1 ,T1 ,d1 , we have θd1 χ S1 (x1 ) + θd1 (x2 ) = χT1 (y1 ) + y2 . It follows from the direct sum decomposition zd1 S1 (v1 ,v2 ) = zT1 + zd1 S1 (v1 ,v2 ) ∩ hT1 that χT1 (y1 ) ∈ z S1 (v1 ,v2 ) ∩ hT1 . Similarly, χ S1 (x1 ) ∈ z S1 (v1 ,v2 ) ∩ h S1 . Thus θd1 χ S1 (x1 ) = χT1 (y1 ) and θd1 (x2 ) = y2 . Hence (x2 , y2 ) ∈ r,⊥ S1 ,T1 ,d1 ⊂ a , and therefore (x 1 , y1 ) ∈ (z S1 (v1 ,v2 ) ⊕ zd1 S1 (v1 ,v2 ) ) ∩ a = Ya . In the same way one shows that (x1 , y1 ) ∈ Ad(v˙1 ,v˙2 ) X a and (x2 , y2 ) ∈ Ad(v˙1 ,v˙2 m 2 ) r⊥ S,T,d . Thus (x 1 , y1 ) ∈ Ya ∩ Ad(v˙1 ,v˙2 ) X a and (x 2 , y2 ) ∈ f1 . Corollary 4.9. For any a, b ∈ S, a , b ∈ S , and a , b ∈ S , dim (a ∩ Ad(v˙1 ,v˙2 m 2 ) a) − dim(b ∩ Ad(v˙1 ,v˙2 m 2 ) b) = dim(Ya ∩ Ad(v˙1 ,v˙2 ) X a ) − dim(Yb ∩ Ad(v˙1 ,v˙2 ) X b ),
(4.21)
dim (a ∩ Ad(w˙ 1 m 1 ,w˙ 2 ) a) − dim(b ∩ Ad(w˙ 1 m 1 ,w˙ 2 ) b) = dim(Ya ∩ Ad(w˙ 1 ,w˙ 2 ) X a ) − dim(Yb ∩ Ad(w˙ 1 ,w˙ 2 ) X b ).
(4.22)
Proof. Let f1 be as in(4.20). We know from Proposition 4.8 that (a ∩ Ad(v˙1 ,v˙2 m 2 ) a)red = Ya ∩ Ad(v˙1 ,v˙2 ) X a + f1 is a direct sum. Replacing a by b and a by b, we get dim (a ∩ Ad(v˙1 ,v˙2 m 2 ) a) − dim(b ∩ Ad(v˙1 ,v˙2 m 2 ) b) = dim(a ∩ Ad(v˙1 ,v˙2 m 2 ) a)red − dim(b ∩ Ad(v˙1 ,v˙2 m 2 ) b)red = dim(Ya ∩ Ad(v˙1 ,v˙2 ) X a ) − dim(Yb ∩ Ad(v˙1 ,v˙2 ) X b ). Equation (4.22) is proved by using (4.21) and the map σ : g ⊕ g → g ⊕ g : (x, y) → (y, x).
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4.4. The rank of the Poisson structure l1 ,l2 on L(g ⊕ g). Theorem 4.10. Let g ⊕ g = l1 + l2 be a Lagrangian splitting as in (4.12) and let O1 and O2 be respectively an N (l1 ) and an N (l2 )-orbit in L(g ⊕ g) as in (4.16) and (4.17). Then the corank of l1 ,l2 in O1 ∩ O2 is equal to dim z S1 + dim z S2 + dim z S − dim(Y1 ∩ Ad(v˙1 ,v˙2 ) X 1 ) + dim(Z 1 ∩ Ad(v˙1 ,v˙2 ) X 1 ) − dim(Y2 ∩ Ad(w˙ 1 ,w˙ 2 ) X 2 ) + dim(Z 2 ∩ Ad(w˙ 1 ,w˙ 2 ) X ), where X1 = = X2 = = Y1 = = Y2 = = Z1 = = Z2 = =
(zd −1 T (v1 ,v2 ) ⊕ zT (v1 ,v2 ) ) ∩ r S,T,d z S ⊕ zT +{(x, θd (x)) | x ∈ zd −1 T (v1 ,v2 ) ∩ h S }, (z S(w1 ,w2 ) ⊕ zd S(w1 ,w2 ) ) ∩ r S,T,d z S ⊕ zT +{(x, θd (x)) | x ∈ z S(w1 ,w2 ) ∩ h S }, (z S1 (v1 ,v2 ) ⊕ zd1 S1 (v1 ,v2 ) ) ∩ rS1 ,T1 ,d1 z S1 ⊕ zT1 + {(x, θd1 (x)) | x ∈ z S1 (v1 ,v2 ) ∩ h S1 }, (z S2 (w1 ,w2 ) ⊕ zd2 S2 (w1 ,w2 ) ) ∩ rS2 ,T2 ,d2 z S2 ⊕ zT2 + {(x, θd2 (x)) | x ∈ z S2 (w1 ,w2 ) ∩ h S2 }, (z S1 (v1 ,v2 ) ⊕ zd1 S1 (v1 ,v2 ) ) ∩ lS1 ,T1 ,d1 ,V1 V1 + {(x, θd1 (x)) | x ∈ z S1 (v1 ,v2 ) ∩ h S1 }, (z S2 (w1 ,w2 ) ⊕ zd2 S2 (w1 ,w2 ) ) ∩ lS2 ,T2 ,d2 ,V2 V2 + {(x, θd2 (x)) | x ∈ z S2 (w1 ,w2 ) ∩ h S2 },
and X˜ = p(X 1 ∩Ad−1 (v˙1 ,v˙2 ) Z 1 )+{(x, θd (x)) | x ∈ z S(w1 ,w2 ) ∩h S } with p : h⊕h → z S ⊕zT being the projection along h S ⊕ hT . Proof. Let l ∈ O1 ∩ O2 be given by l = Ad(r1 ,r2 )(v˙1 ,v˙2 m 2 ) l S,T,d,V = Ad(r1 ,r2 )(w˙ 1 m 1 ,w˙ 2 ) l S,T,d,V ,
(4.23)
where (r1 , r2 ) ∈ N (l1 ) = R S 1 ,T1 ,d1 and (r1 , r2 ) ∈ N (l2 ) = R S2 ,T2 ,d2 . The formula for the corank of l1 ,l2 at l ∈ O1 ∩ O2 involves the Drinfeld subalgebra T (l) of g ⊕ g defined by T (l) = n(l)⊥ +l1 ∩n(l), cf. Proposition 2.5, where again n(l) is the normalizer subalgebra of l in g ⊕ g. We first compute T (l). Since ⊥ Ad(r1 ,r2 )(v˙1 ,v˙2 m 2 ) r⊥ S,T,d = n(l) ⊂ T (l) ⊂ n(l) = Ad(r1 ,r2 )(v˙1 ,v˙2 m 2 ) r S,T,d ,
∈ L(z S ⊕ zT ). On the other we know that T (l) = Ad(r1 ,r2 )(v˙1 ,v˙2 m 2 ) l S,T,d,V for some V hand, −1 T (l) = Ad(r1 ,r2 )(v˙1 ,v˙2 m 2 ) r⊥ S,T,d + r S,T,d ∩ Ad(v˙1 ,v˙2 m 2 ) l1 . −1 −1 ⊥ Thus l S,T,d,V = r⊥ S,T,d + r S,T,d ∩ Ad(v˙1 ,v˙2 m 2 ) l1 = X 1 ∩ Ad(v˙1 ,v˙2 ) Z 1 + r S,T,d , where the second identity comes from Proposition 4.8. Hence
= p(X 1 ∩ Ad−1 Z 1 ). V (v˙1 ,v˙2 )
Regular Partitions of Poisson Manifolds
747
Now by Lemma 2.16, the corank of l1 ,l2 in O1 ∩ O2 at the Lagrangian subalgebra l given by (4.23) is Corank l 1 ,l 2 (l) = dim z S1 + dim z S2 + dim z S
− dim(rS1 ,T1 ,d1 ∩ Ad(v˙1 ,v˙2 m 2 ) r S,T,d ) + dim(lS1 ,T1 ,d1 ,V1 ∩ Ad(v˙1 ,v˙2 m 2 ) r S,T,d )
− dim(rS2 ,T2 ,d2 ∩ Ad(w˙ 1 m 1 ,w˙ 2 ) r S,T,d ) + dim(lS2 ,T2 ,d2 ∩ Ad(w˙ 1 m 1 ,w˙ 2 ) l S,T,d,V ).
Applying Corollary 4.9, we get the desired formula for the corank of l1 ,l2 in O1 ∩ O2 . This completes the proof of Theorem 4.10. Example 4.11. Let gdiag = {(x, x) | x ∈ g}. A Lagrangian splitting of the form g ⊕ g = gdiag + l, where l ∈ L(g ⊕ g), is called a Belavin-Drinfeld splitting. Let G diag = {(g, g) | g ∈ G}. It is shown in [1] (see also [8, Cor. 3.18]) that every Belavin-Drinfeld splitting of g ⊕ g is conjugate by an element in G diag to a splitting of the form g ⊕ g = gdiag + lS2 ,T2 ,d2 ,V2 ,
(4.24)
where (S2 , T2 , d2 ) is a Belavin-Drinfeld triple in the sense that S2d2 = {α ∈ S2 | d2n α is defined and is in S2 for n = 1, 2, . . .} = ∅, and V2 ∈ Lspace (z S2 ⊕ zT2 ) is such that hdiag ∩ (V2 + {(x, θd2 (x)) | x ∈ h S2 } = 0. In other words, (4.24) is the special case of the splitting in (4.12) with l1 = gdiag . Keeping the notation as in Theorem 4.10, we have v2 = 1, and the corank of l1 ,l2 in O1 ∩ O2 in this special case simplifies to dim z S2 + dim z S − dim(Y2 ∩ Ad(w˙ 1 ,w˙ 2 ) X 2 ) + dim(Z 2 ∩ Ad(w˙ 1 ,w˙ 2 ) X ). When lS2 ,T2 ,d2 ,V2 = l0 := n− ⊕ n + h−diag , where n and n− are respectively the span by positive and negative root vectors and h−diag = {(x, −x) | x ∈ h}, the splitting g⊕g = gdiag +l0 is called the standard splitting of g⊕g [8]. In this case, N (l0 ) = B − × B, where B − = P∅− and B = P∅ are two opposite Borel subgroups, and in the notation of Theorem 4.10, w1 ∈ W and w2 ∈ W T . The corank of l1 ,l2 in O1 ∩ O2 in this special X ), where case further simplifies to dim(h−diag ∩ Ad(w˙ 1 ,w˙ 2 ) −1 X = {(Ad−1 v˙1 x, x) | x ∈ zT (v1 ) , θd χ S (Adv˙1 x) = χT (x)} + {(y, θd (y)) | y ∈ h S }.
This formula has been obtained in [8]. 4.5. The wonderful compactification of G. Recall [3] that the wonderful compactification G of G is the closure of the Lagrangian subalgebra gdiag inside L(g ⊕ g). Let g ⊕ g = l1 + l2 be a Lagrangian splitting with l1 and l2 given by (4.12). Then G is a Poisson submanifold of L(g ⊕ g) with respect to the Poisson structure l1 ,l2 because it is (G × G)-stable. In [16], to each of the above Lagrangian splittings we associated two partitions Pi , i = 1, 2, of G into finitely many smooth irreducible locally closed N (li )-stable subsets. The strata of Pi are indexed by the Weyl group elements in Proposition 4.6 and are obtained by putting together the N (li )-orbits corresponding to different continuous parameters. When l1 = gdiag , the subsets in P1 are the G diag -stable pieces introduced by Lusztig [18,19]. Each stratum of P1 and P2 is a Poisson submanifold of (G, l1 ,l2 ). Theorem 4.10 shows that the corank of l1 ,l2 at an l ∈ G in (N (l1 )· l) ∩ (N (l2 )· l) depends only on the stratum of P1 (or P2 ) to which l belongs.
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Acknowledgements. The second author would like to thank the University of Hong Kong for the warm hospitality during his visits in March 2004 and August 2005 when this work was initiated. The first author would like to thank UC Santa Barbara for her visit in July 2006 during which the paper took its final form. We would also like to thank Xuhua He for a key argument in the proof of Proposition 4.4. The first author was partially supported by HKRGC grants 703304 and 703405, and the second author by NSF grant DMS-0406057 and an Alfred P. Sloan research fellowship.
References 1. Belavin, A., Drinfeld, V.: Triangular equations and simple Lie algebras. Math. Phys. Rev. 4, 93–165 (1984) 2. Carter, R.W.: Finite groups of Lie type. Conjugacy classes and complex characters. Chichester: WileyInterscience Publ., 1993 3. De Concini, C., Procesi, C.: Complete symmetric varieties. In: Invariant theory (Montecatini, 1982), Lecture Notes in Math. 996, 1983, pp. 1–44 4. Delorme, P.: Classification des triples de Manin pour les algèbres de Lie réductives complexes, with an appendix by Guillaume and Macey. J. Algebra 246, 97–174 (2001) 5. Drinfeld, V.G.: On Poisson homogeneous spaces of Poisson-Lie groups. Theo. Math. Phys. 95(2), 226–227 (1993) 6. Etingof, P., Schiffmann, O.: Lectures on quantum groups. Cambridge, MA: International Press, 1998 7. Evens, S., Lu, J.-H.: On the variety of Lagrangian subalgebras, I. Ann. Sci. École Norm. Sup. 34(5), 631–668 (2001) 8. Evens, S., Lu, J.-H.: On the variety of Lagrangian subalgebras, II. Ann. Sci. École Norm. Sup. 39(2), 347–379 (2006) 9. Goodearl, K.R., Yakimov, M.: Poisson structures on affine spaces and flag varieties, II. http://arxiv.org/ list/math.QA/0509075, 2005 to appear in Trans. Amer. Math. Soc. 10. Karolinsky, E.A.: Symplectic leaves on Poisson homogeneous spaces of Poisson–Lie groups. Mat. Fiz. Anal. Geom. 2(3–4), 306–311 (1995) 11. Karolinsky, E.: A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups, Banach Center Publ. 51. Warsaw: Polish Acad. Sci., 2000 12. Korogodski, L., Soibelman, Y.: Algebras of functions on quantum groups, part I. AMS, Mathematical surveys and monographs, Vol. 56, Hovidence, RI: Amer. Math. Soc., 1998 13. Koszul, J.L.: Crochet de Schouten–Nijenhuis et cohomologie. Asterisque, ´ hors série, Soc. Math. France, Paris 257–271 (1985) 14. Lu, J.-H.: Poisson homogeneous spaces and Lie algebroids associated to Poisson actions. Duke Math. J. 86(2), 261–304 (1997) 15. Lu, J.-H., Yakimov, M.: On a class of double cosets in reductive algebraic groups. Int. Math. Res. Notices 13, 761–797 (2005) 16. Lu, J.-H., Yakimov, M.: Partitions of the wonderful group compactification. http://arxiv.org/list/math. RT/0606579. Transformation Groups 12(4), 695–723 (2007) 17. Lusztig, G.: Total positivity in partial flag manifolds. Repr. Theory 2, 70–78 (1998) 18. Lusztig, G.: Parabolic character sheaves I. Moscow Math J. 4, 153–179 (2004) 19. Lusztig, G.: Parabolic character sheaves II. Moscow Math J. 4, 869–896 (2004) 20. Richardson, R.: Intersections of double cosets in algebraic groups. Indagationes Mathematicae 3(1), 69–77 (1992) 21. Schiffmann, O.: On classification of dynamical r-matrices. Math. Res. Letters 5, 13–31 (1998) 22. Yakimov, M.: Symplectic leaves of complex reductive Poisson-Lie groups. Duke Math J. 112(3), 453–509 (2002) Communicated by L. Takhtajan
Commun. Math. Phys. 283, 749–768 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0516-3
Communications in
Mathematical Physics
Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Killing Fields Stefan Hollands1 , Stoytcho Yazadjiev2 1 School of Mathematics, Cardiff University, Wales, UK.
E-mail: [email protected]; [email protected]
2 Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 J. Bourchier Blvd.,
Sofia 1164, Bulgaria. E-mail: [email protected] Received: 19 September 2007 / Accepted: 23 November 2007 Published online: 24 June 2008 – © Springer-Verlag 2008
Abstract: We show that two stationary, asymptotically flat vacuum black holes in 5 dimensions with two commuting axial symmetries are identical if and only if their masses, angular momenta, and their “interval structures” coincide. We also show that the horizon must be topologically either a 3-sphere, a ring, or a Lens-space. Our argument is a generalization of constructions of Morisawa and Ida (based in turn on key work of Maison) who considered the spherical case, combined with basic arguments concerning the nature of the factor manifold of symmetry orbits. 1. Introduction A key theorem about 4-dimensional stationary asymptotically flat black holes is that they are uniquely determined by their conserved asymptotic charges—the mass and angular momentum in the vacuum case [3,31], and the mass, angular momentum and charge in the Einstein-Maxwell case [2,27]. But the corresponding statement is no longer true in higher dimensions; there are different vacuum solutions with the same mass, and angular momenta [6,29]. Nevertheless, it is an interesting open question whether an analogous statement might still hold true if a finite number of suitable further parameters associated with the solution is specified in addition to the mass and angular momenta. The purpose of this note is to show that this is indeed true in the special case of stationary, asymptotically flat vacuum black holes in five dimensions having two commuting axial1 symmetries with the property that the exterior of the spacetime contains no points whose isotropy group is discrete. All exact solutions found so far fall in this class. In fact, what we will show is that the solution is uniquely determined in terms of its mass, the two angular momenta, a collection of positive real numbers (“moduli”) and a collection of integer-valued vectors (“winding numbers”). These data—which we 1 By this we mean Killing fields whose orbits are periodic. In higher dimensions, the set of fixed points of such a symmetry is actually generically a higher-dimensional “plane”, rather than an “axis.” We nevertheless refer to the symmetries as axial, by analogy to the 4-dimensional case.
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shall collectively call the “interval structure”—give a measure of the lengths and relative positions of the different rotation “axis” and the horizon, as well as the structure of the symmetry orbits near these axis. Some aspects of our construction parallel previous considerations of Harmark [13,14] (see also [7] for a special case), but as we shall see, there are also some key differences. The interval structure is shown to be subject to certain constraints. We also show that it determines the topology of the horizon, which we show may be either be a 3-sphere S 3 , a ring S 2 × S 1 , or a Lens-space L( p, q). Our uniqueness proof uses a known σ -model formulation of the reduced Einstein equations in five dimensions due to Maison [25], which is analogous to a formulation previously found by Mazur [27] and used in his uniqueness proof in four dimensions. We combine this technique with an elementary analysis of the global structure of the orbit space of the symmetries. Our result generalizes a result of [26] for the special case of S 3 -horizon topology, which has a particularly simple interval structure. We expect that our considerations can be generalized to dimensions greater than five, but we have not attempted to do so. In five dimensions, it is not known whether an arbitrary stationary, asymptotically flat vacuum black hole solution will have two commuting axial Killing fields as we are assuming. In fact, the higher dimensional rigidity theorem [18] only guarantees the existence of one axial Killing field in such spacetimes in addition to the timelike Killing field. In this regard, the situation in five dimensions is very different from the analogous situation in 4 dimensions: Here the original rigidity theorem [5,8,15,16,28,30] also guarantees the existence of one axial Killing field. But this suffices in 4 dimensions to reduce the Einstein equation to the 2-dimensional σ -model equations [27], and this formulation may then be used to prove the uniqueness. By contrast, in five dimensions, two axial Killing fields are required to make the analogous argument. As we have said, however, only one axial Killing field appears to be generic. Our conventions and notations follow those of Wald’s textbook [33].
2. Stationary Vacuum Black Holes in n Dimensions Let (M, gab ) be an n-dimensional, analytic, asymptotically flat, stationary black hole spacetime satisfying the vacuum Einstein equations Rab = 0, where n ≥ 4. Let t a be the asymptotically timelike Killing field, £t gab = 0, which we assume is normalized so that lim gab t a t b = −1 near infinity. We denote by H = ∂ B the horizon of the black hole, B = M \ I − (J + ), with J ± the null-infinities of the spacetime, which are of topology R × ∞ , with ∞ a compact manifold of dimension n − 2.2 We assume that H is non-degenerate and that the horizon cross section is a compact connected manifold of dimension n − 2. Under these conditions, one of the following 2 statements is true: (i) If t a is tangent to the null generators of H then the spacetime must be static [32]. (ii) If t a is not tangent to the null generators of H , then the higher dimensional rigidity theorem [18] states that there exist N additional linear independent, mutually commuting Killing fields ψ1a , . . . , ψ Na , where N is at least equal to 1. These Killing fields generate periodic, commuting flows (with period 2π ), and there exists a linear combination K a = t a + 1 ψ1a + · · · + N ψ Na , i ∈ R
(1)
2 In 4 dimensions, may be shown to be an S 2 under suitably strong additional hypothesis. A discussion ∞ of the structure of null-infinity in higher dimensions is given in [19].
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so that the Killing field K a is tangent and normal to the null generators of the horizon H , and K a ψia = 0 on H .
(2)
Thus, in case (ii), the spacetime is axisymmetric, with isometry group G = R × U (1) N . From K a , one may define the surface gravity of the black hole by κ 2 = lim H (∇a f )∇ a f / f , with f = (∇ a K b )∇a K b the norm, and it may be shown that κ is constant on H [33]. In fact, the non-degeneracy condition implies κ > 0. In case (i), one can prove that the spacetime is actually unique, and in fact isometric to the Schwarzschild spacetime [22] when n = 4, for higher dimensions see [12]. In this paper, we will be concerned with case (ii). We restrict attention to the exterior of the black hole, I − (J + ), which we shall again denote by M for simplicity. We assume that the exterior M is globally hyperbolic. By the topological censorship theorem [9], the exterior M is a simply connected manifold (with boundary ∂ M = H ). To understand better the nature of the solutions, it is useful to bring the field equations into a form that exploits the symmetries of the spacetime. For this, one considers first the factor space Mˆ = M/G, where G is the isometry group of the spacetime generated by the Killing fields. Since the Killing fields ψia in general have zeros, the factor space Mˆ = M/G will normally have singularities. We will analyze the manifold Mˆ in detail in the next section for the case n = 5, N = 2. The full Einstein equations Rab = 0 on M imply a set of coupled differential equations for the metric on the open subsets (of dimension d = n − N − 1) of the factor space Mˆ corresponding to points in M that have a trivial isotropy subgroup3 . To understand these equations in a geometrical way, we note that the projection π : M → M/G = Mˆ defines a G-principal fibre bundle over these open subsets of Mˆ (we will call the union ˆ At each point x in a fibre over π(x) in the interior of these sets the “interior” of M). ˆ of M, we may uniquely decompose the tangent space Tx M into a subspace of vectors tangent to the fibres, and a space Hx of vectors orthogonal to the fibres. Evidently, the distribution of vector spaces Hx is invariant under the group G of symmetries, and hence forms a “horizontal bundle” in the terminology of principal fibre bundles [24]. According to one of the equivalent definitions of a connection in the theory of principal fibre bundles [24], a horizontal bundle is equivalent to the specification of a G-gauge connection Dˆ a on the factor space, whose curvature we denote by Fˆab . The horizontal bundle gives an isomorphism Hx → Tπ(x) Mˆ for any x, and this isomorphism may be used to uniquely construct a smooth covariant tensor field tˆab...c on the interior of Mˆ from any smooth G-invariant covariant tensor field tab...c on M. For example, the metric ˆ We let Dˆ a act on ordinary tensors gab on M thereby gives rise to a metric gˆ ab on M. tˆab...c as the connection of gˆ ab , with Ricci tensor denoted Rˆ ab . By performing the well-known “Kaluza-Klein” reduction of the metric gab on M, we can locally write the Einstein equations as a system of equations on the factor space Mˆ in terms of metric gˆ ab , the components Fˆ I ab , I = 0, 1, . . . , N of the curvature and the (N + 1) × (N + 1) Gram matrix field G I J GI J =
gab X aI X bJ ,
X aI
t a if I = 0, = ψia if I = i = 1, . . . , N .
3 The isotropy subgroup of a point x ∈ M is the subgroup {g ∈ G; g · x = x}.
(3)
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The resulting equations are similar in nature to the “Einstein-equations” on Mˆ for gˆ ab , I and the “scalar fields” G , see [4,23]. We will not coupled to the “Maxwell fields” Fˆab IJ write these equations down here, as we will not need them in this most general form. The equations simplify considerably if the distribution of horizontal subspaces Hx is locally integrable, i.e., locally tangent to a family of (n − N − 1)-dimensional subI = 0, and the dimensionally reduced manifolds. In that case, the connection is flat, Fˆab equations may be written as Dˆ a (r G −1 Dˆ a G) IJ = 0
(4)
1 Rˆ ab = Dˆ a Dˆ b log r − ( Dˆ a G −1 ) I J Dˆ b G I J . 4
(5)
together with
ˆ corresponding to points The equations are well-defined at points in the interior of M, with trivial isotropy subgroup. At such points, the matrix G is not singular, i.e., the Gram determinant r 2 = |det G|
(6)
does not vanish. Conversely, one may find stationary axisymmetric solutions to the Einstein equations by solving the above equations subject to appropriate boundary conditions on Mˆ which ensure that the metric gab reconstructed from gˆ ab and G I J is smooth. Taking the trace of the first equation, one finds that r is a harmonic function on the ˆ interior of M, Dˆ a Dˆ a r = 0 .
(7)
If Mˆ has the structure of a manifold with boundary (as we will prove in the next section for the situation considered in this paper), then on the boundary of Mˆ we have r = 0. We may divide the boundary into (i) a part corresponding to H where r = 0 by Eq. (2), and (ii) a part corresponding to various “axis,” where G I J has a null space and where consequently one or more linear combinations of the axial Killing fields vanish. For an asymptotically flat spacetime, the quantity r must be approximately equal in an asymptotically Minkowskian coordinate system to the corresponding quantity formed from N commuting axial Killing fields and ∂/∂t on exact Minkowski spacetime. Thus, in the region of Mˆ corresponding to a neighborhood of infinity of M, and away from the axis, r → ∞. By the maximum principle, r must therefore be in the range 0 < r < ∞ in the ˆ Thus, in this case, the fields (G −1 ) I J are globally defined on the interior interior of M. ˆ of M, and therefore likewise the dimensionally reduced Einstein equations. ˆ 3. The Factor Space M In this section, we analyze in some detail the factor space Mˆ = M/G in the case when the dimension of M is equal to five. To begin, we consider a somewhat simpler situtation in which we have a Riemannian 4-manifold (, h ab ) with an isometry group K = U (1) × U (1), which may be thought of as a spatial slice of our spacetime M. We denote the elements of the isometry group by k = (eiτ1 , eiτ2 ) with 0 ≤ τ1 , τ2 < 2π , and
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we denote the Killing vector fields generating the action of the respective U (1) factors by ψ1a respectively ψ2a , £ψ1 h ab = 0 = £ψ2 h ab . These vector fields commute, 0 = [ψ1 , ψ2 ]a = ψ1b Db ψ2a − ψ2b Db ψ1a .
(8)
We denote the action of a symmetry on a point x by k · x. As is common, we call the set Ox = {k · x | k ∈ K} the orbit of the point x, and we call Kx = {k ∈ K | k · x = x} the isotropy subgroup. As part of our technical assumptions, we assume that the action is such that there are no points with a discrete isotropy subgroup. It is elementary to show that if ψ1a , ψ2a respectively ψ˜ 1a , ψ˜ 2a are two pairs of commuting Killing fields generating such an action of K, then they must be related by a matrix of integers n ij , ψ˜ ia =
2 j=1
j ni
· ψ aj
,
n 11 n 21
n 12 n 22
∈ S L(2, Z) ⇔ det
n 11 n 21
n 12 n 22
= ±1 .
(9)
We denote the Gram matrix of the Killing fields by f i j = h ab ψia ψ bj . ˆ = /K can be analyzed by elementary The general structure of the orbit space means and is described by the following proposition: ˆ = /K is a 2-dimensional manifold with boundaries Proposition 1. The orbit space and corners, i.e., a manifold locally modelled over R × R (interior points), R+ × R (1-dimensional boundary segments) and R+ × R+ (corners). Furthermore, for each of the 1-dimensional boundary segments, the rank of the Gram matrix f i j is precisely 1, and there is a vector v = (v 1 , v 2 ) with integer entries such that f i j v j = 0 for each point of the segment. If vi respectively vi+1 are the vectors associated with two adjacent boundary segments meeting in a corner, then we must have 1 vi1 vi+1 (10) ∈ S L(2, Z) ⇔ det (vi , vi+1 ) = ±1 . 2 vi2 vi+1 On the corners, the Gram matrix has rank 0, and in the interior it has rank 2. Proof. At each point x ∈ , let Vx ⊂ Tx be the linear span of the Killing fields at x, which is tangent to Ox , the orbit through x. Thus, the orbit has the same dimension as a manifold as the vector space Vx . We let Hx be the orthogonal complement of Vx . Each point x ∈ must be in precisely one of the sets 0) S0 , the set of all points such that the dimension of Vx is 0. 1) S1 , the set of all points such that the dimension of Vx is 1. 2) S2 , the set of all points such that the dimension of Vx is 2. The set S2 is open because it coincides with the set of all points such that the smooth function det f is different from zero, and the set S0 is closed because it is the set of all points where the smooth function Tr f is zero. Evidently, if a point x is in Si , then the entire orbit Ox is in Si , too. We will now show how to construct a coordinate chart in a neighborhood of each orbit Ox by considering the different cases separately. Case 2: If x ∈ S2 , then the orbit Ox has dimension 2. In that case, the isotropy group of x can be at most discrete. However, this cannot be the case by assumption, so the isotropy group is in fact trivial, and this also holds for points in a sufficiently small open neighborhood of Ox . If we now choose a coordinate system {y1 , . . . , y4 } in near x such that
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(∂/∂ y1 )a and (∂/∂ y2 )a are transverse to Vx , then the surface of constant y1 = 0 = y2 meets each orbit precisely once sufficiently near x. Thus, {y3 , y4 } furnish the desired ˆ near x, showing that this space can be locally modelled over coordinate system of R × R near x. Case 1: For a point x ∈ S1 , the orbit Ox is one-dimensional, i.e., a loop, and there exists a linear combination s a = v 1 ψ1a + v 2 ψ2a
(11)
such that s a vanishes on Ox , or equivalently f i j v i = 0 there. Hence, k = (eiv τ , eiv τ ), 0 ≤ τ < 2π is in the isotropy subgroup Kx . Since Kx is a closed subgroup of the compact group K = U (1) × U (1), the ratio v 1 /v 2 must either be rational or Kx = K. The latter would mean that we are in fact in case 0, so we may chose v 1 , v 2 to be integers 2 1 with no common divisor. It then follows that both (e2πi/v , 1) and (1, e2πi/v ) are in the isotropy subgroup. Thus, if we follow the loop Ox by acting with (eiτ , 1) on x, then we are back to x for the first time after τ = 2π/u 1 , where u 1 is an integer with |u 1 | ≥ |v 2 |, and if we likewise follow the loop by acting with (1, eiτ ) on x, then we are back for the first time after τ = 2π/u 2 , where |u 2 | ≥ |v 1 |. The same holds for any other point in the orbit Ox . ˆ can be modelled over R+ × R near Ox , it is useful To show that the orbit space to construct a special coordinate system {y1 , . . . , y4 } near Ox . This coordinate system is designed in such a way that the action of K takes a particularly simple form. We let y4 = u 1 τ be the parameter along the orbit τ → (eiτ , 1) · x. The coordinates {y1 , y2 , y3 } measure the geodesic distance from the orbit within a suitable tubular neighborhood, and are defined as follows. First, we pick an orthonormal basis (ONB) {e˜1a , e˜2a , e˜3a } of Hx and Lie-drag it along the orbit to an ONB at each x(τ ), 0 ≤ τ < 2π/u 1 . In general, the ONB will not return to itself after we have gone through Ox once, i.e., after τ = 2π/u 1 , but only after we have gone through it u 1 -times. Consequently, by choosing the ONB at x appropriately, we may assume that 1
⎛ a⎞ ⎛ e ˜ cos(2π w 1 /u 1 ) sin(2π w 1 /u 1 ) 1 1 (e2πi/u , 1) · ⎝e˜2a ⎠ = ⎝ sin(2π w 1 /u 1 ) cos(2π w 1 /u 1 ) e˜3a x 0 0
⎞ ⎛ a⎞ 0 e˜1 0⎠ ⎝e˜2a ⎠ , e˜3a x 1
2
(12)
for some integer w1 . In order to obtain an ONB of each Hx(τ ) varying smoothly as we go around the loop Ox once (including at τ = 2π/u 1 ), we define ⎛ a⎞ ⎞ ⎛ a⎞ ⎛ e1 e˜1 cos(−w 1 τ ) sin(−w 1 τ ) 0 ⎝e2a ⎠ = ⎝ sin(−w 1 τ ) cos(−w 1 τ ) 0⎠ ⎝e˜2a ⎠ , e3a x(τ ) e˜3a x(τ ) 0 0 1
(13)
i.e., we undo the rotation. We now define a diffeomorphism from the solid tube B 2 × B 1 × S 1 into an open neighborhood of Ox by (y1 , y2 , y3 , y4 ) → Expx(τ ) (y1 e1a + y2 e2a + y3 e3a ) ,
(14)
where y4 = u 1 τ is a periodic coordinate with period 2π , and y1 , y2 , y3 are sufficiently small. (B 2 is a small open disk around the origin in R2 and B 1 a small interval around
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the origin in R1 .) This diffeomorphism defines the desired coordinates. By construction, the action of (eiτ , 1) is given in these coordinates by y1 y2 y3 y4
→ → → →
cos(w 1 τ )y1 + sin(w 1 τ )y2 , cos(w 1 τ )y2 + sin(w 1 τ )y1 , y3 , y4 + u 1 τ .
(15) (16) (17) (18)
The action of (eiv τ , eiv τ ) on these coordinates can be found as follows: First, the 1 2 action of (eiv τ , eiv τ ) leaves each point x(τ ) in the orbit Ox invariant, and it also maps each space Hx(τ ) to itself. Furthermore, since s a and hence Da s b is invariant under the action of (eiτ , 1), it follows that the component matrix of Da s b |x(τ ) in the ONB {e1a , e2a , e3a } commutes with the matrix in Eq. (12). Thus, it must be a rotation in the 1 2 plane spanned by e1a |x(τ ) , e2a |x(τ ) . Therefore, it follows that the action of (eiv τ , eiv τ ) is given by 1
2
y1 y2 y3 y4
→ → → →
cos(N τ )y1 + sin(N τ )y2 , cos(N τ )y2 + sin(N τ )y1 , y3 , y4
(19) (20) (21) (22)
for some integer N . The action of (1, eiτ ) on our coordinates may now be determined in the same way, and is given in terms of integers u 2 , w 2 by y1 y2 y3 y4
→ → → →
cos(−w 2 τ )y1 + sin(−w 2 τ )y2 , cos(−w 2 τ )y2 + sin(−w 2 τ )y1 , y3 , y4 − u 2 τ .
(23) (24) (25) (26)
Our arguments so far can be summarized by saying that {y1 , . . . , y4 } furnish a coordinate system covering a tubular neighborhood of the orbit Ox , with y4 a 2π periodic coordinate system going around the loop Ox once. The Killing fields ψ1a , ψ2a are given in terms of these coordinates by a 1
a
ψ1 w1 u l = , (27) ma −u 2 −w 2 ψ2a where the vector fields l a , m a generate the longitude, respectively the meridian, of the tori of constant R = (y12 + y22 )1/2 and constant y3 . They are given in terms of the coordinates by
∂ a ∂ a ∂ a a a l = , m = y1 − y2 . (28) ∂ y4 ∂ y2 ∂ y1 By the remarks at the beginning of this section, since m a and l a locally generate an action of K which has no points with discrete isotropy group, the determinant u 1 w 2 − u 2 w 1 must be ±1. In view of the definition of s a , Eq. (11), and the fact that s a = N m a , it also follows that u 1 v 1 − u 2 v 2 = 0, v 1 w 1 − v 2 w 2 = N .
(29)
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The first equation implies that u 1 = cv 2 and u 2 = cv 1 for some c. Since the modulus of u 1 is bigger or equal than that of v 2 (and the same with 1 and 2 reversed), we must have |c| ≥ 1. In view of the second equation, this implies that |N | = |c| = 1, and hence that u 1 = v 2 , u 2 = v 1 . The orbit space may now be determined. We have shown that the orbits of (eiτ , 1) and (1, eiτ ) have the structure of a Seifert fibration, times an interval for the coordinate y3 . The fibrations are characterized by the winding numbers (v 2 , w 1 ) and (−v 1 , −w 2 ) respectively. Thus, for example the first fibration is such that as the l a generator winds around v 2 -times, the generator m a winds around w 1 -times, and similarly for the other action. Thus, if we factor by the action of (eiτ , 1), we locally obtain the space R × (R2 /Zv 2 ), where Z p ⊂ U (1) is the cyclic subgroup of p elements whose action on R2 ∼ = C is generated by the phase multiplication z → e2πi/ p z. The factor R in the Cartesian product corresponds to the coordinate y3 , while the other factor to the coordinates y1 , y2 . We next factor by (1, eiτ ). Since the only nontrivial part of this action on R × (R2 /Zv 2 ) is a rotation in the cone R2 /Zv 2 , we may parametrize the orbits in a neighborhood of Ox ˆ locally has the structure by y3 and R = (y12 + y22 )1/2 . This shows that, in case (1), 1 R × R+ . On the edge locally defined by R = 0, we have v ψ1a + v 2 ψ2a = 0. [If we had first factored by the action of (1, eiτ ), we would have locally obtained the space R × (R2 /Zv 1 ). The rest would be analogous.] Case 0: If x ∈ S0 , then ψ1a = 0 = ψ2a at the point x, and the linear transformations Da ψ1b , Da ψ2b in the tangent space Tx can be viewed as elements of the Lie-algebra o(4) of O(4), defined with respect to the Riemannian metric h ab on Tx . Taking a derivative of Eq. (8) and evaluating at x, it follows that these linear transformations commute at x, (Da ψ1b )Db ψ2c − (Da ψ2b )Db ψ1c = 0 at x.
(30)
This means that, if we form the self-dual and anti-self-dual parts 1 Da ψ1b ± ab cd Dc ψ1d , 2
1 Da ψ2b ± ab cd Dc ψ2d , 2
(31)
then the self-dual part of Da ψ1b must be proportional to that of Da ψ2b at x, and similarly for the anti-self-dual parts, as one may see using the Lie-algebra isomorphism between o(4) and o(3) × o(3) corresponding to the decomposition into self-dual and anti-self-dual parts. Now pick an orthonormal tetrad {e1a , e2a , e3a , e4a } at x. Then the basis for the 3-dimensional spaces of self-dual and anti-self-dual skew 2-tensors on Tx are given by e1[a e2b] ± e3[a e4b] , e1[a e3b] ± e2[a e4b] and e1[a e4b] ± e2[a e3b] , respectively. Performing an O(4) rotation of the tetrad corresponds to two independent O(3)rotations of the respective basis of self-dual and anti-self-dual tensors, and vice versa. It follows that tetrad may be rotated if necessary so that the self-dual parts of Da ψ1b and Da ψ2b are proportional e1[a e2b] + e3[a e4b] , and the anti-self-dual parts are proportional to e1[a e2b] − e3[a e4b] . Therefore we may write, at x,
1 2
n1 n1 Da ψ1b 2e1[a e2b] = (32) Da ψ2b 2e3[a e4b] n 12 n 22 for some matrix n ij . Let us now pick Riemannian normal coordinates {y1 , y2 , y3 , y4 } centered at x corresponding to our choice of tetrad. Then, since the Killing fields ψ1a and
Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Killing Fields
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ψ2a are globally determined by the tensors Da ψ1b and Da ψ2b at the point x, it follows j a from Eq. (32) that ψia = n i · s j in an open neighborhood of x, where s1a = y1
∂ ∂ y2
a
− y2
∂ ∂ y1
a
, s2a = y3
∂ ∂ y4
a
− y4
∂ ∂ y3
a .
(33)
Since both sets of Killing fields sia and ψia have periodic orbits with period 2π , both the matrix n ij and the matrix v ij = (n −1 )ij must be integer valued. We now define R1 = (y12 + y22 )1/2 , R2 = (y32 + y42 )1/2 . These quantities are clearly invariant under ˆ the the action of K and in 1—1 correspondence with the orbits near Ox . This gives structure of R+ × R+ near the orbit Ox . On the edges locally defined by Ri = 0, we have vi1 ψ1a + vi2 ψ2a = 0. We have now constructed the desired coordinate systems in the above 3 cases, and ˆ it can be checked that the transition functions are smooth. Thus we have shown that has the structure of a manifold with boundaries and corners. The same technique of proof may be used to analyze the possible horizon topologies of stationary, asymptotically flat black hole spacetimes with an action of K = U (1) ×U (1) satisfying the hypothesis that there are no points with discrete isotropy group under K. Proposition 2. Under the above hypothesis, each connected component of the horizon cross section H must be topologically either a ring S 1 × S 2 , a sphere S 3 , or a Lens-space L( p, q), with p, q ∈ Z. Remark 1. The Lens-spaces L( p, q) (see e.g. [1, Paragraph 9.2]) are the spaces obtained by factoring the unit sphere S 3 in C2 by the group action (z 1 , z 2 ) → (e2πi/ p z 1 , e2πiq/ p z 2 ). The fundamental group of the Lens space is π1 (L( p, q)) = Z p , and q is determined only up to integer multiples of p. Since a Lens-space is a quotient of the positive constant curvature space S 3 by a group of isometries, it can carry a metric of everywhere positive scalar curvature, like the other possible topologies S 3 and S 2 × S 1 . Thus, the possible horizon topologies listed in Proposition 2 are of so-called “positive Yamabe type,” in accordance with a general theorem [11]. Proof. As a result of the rigidity theorem [18], we can find a horizon cross section H which is itself a Riemannian manifold with induced metric qab , of dimension 3, invariant under the group K = U (1) × U (1) of axial symmetries generated by ψ1a and ψ2a . By the same arguments as in the proof of Proposition 1, H divided by K is a 1dimensional manifold with boundary, i.e., a union of intervals, each of which corresponds to a connected component of H. We restrict attention to one connected component of H, whose space of orbits is a single interval. The end points of the interval correspond to 1-dimensional orbits where a linear combination of the axial Killing fields vanishes4 . We call these orbits Ox1 and Ox2 . They are closed loops. All other points of the interval correspond to non-degenerate orbits diffeomorphic to the 2-torus S 1 × S 1 . At x1 , an integer linear combination m a1 = v11 ψ1a + v12 ψ2a vanishes, while at x2 , an integer linear combination m a2 = v21 ψ1a + v22 ψ2a vanishes. As in the proof of Proposition 1, we may introduce a local coordinate system in tubular neighborhoods of Ox1 and Ox2 such 4 There cannot be points x in H where both ψ a and ψ a vanish, since D ψ b and D ψ b would otherwise a 1 a 2 1 2 be two commuting but not linearly dependent infinitesimal S O(3) rotations in the tangent space of x, which is impossible.
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that each neighborhood is diffeomorphic to a solid tube S 1 × B 2 . We denote the radial coordinates measuring the distance from the origin in each of the discs B 2 by R1 for the first tubular neighborhood, and by R2 for the second tubular neighborhood. By construction, the tori of constant R1 , respectively R2 , correspond to 2-dimensional orbits of K, i.e., interior points of the interval. In fact, R1 and R2 measure the distance of the interior point of the interval to the first respectively second boundary point. If m a1 , l1a are the meridian of a torus of constant r1 in the first tubular neighborhood (with the longitude going around the S 1 -direction in the cartesian product S 1 × B 2 ), and m a2 , l2a the corresponding quantities for the second tubular neighborhood, then as in Case 1 in the proof of Proposition 1, we have a 2
a 2
a
ψ1 v1 w11 l1 v2 w21 l2 = = . (34) a a ψ2 m1 m a2 −v11 −w12 −v21 −w22 We must now smoothly join the coordinate systems defining the tubular neighborhoods of Ox1 respectively Ox2 . Each tubular neighborhood is a solid torus B 2 × S 1 . Their boundaries (each diffeomorphic to a torus S 1 × S 1 ) must be glued together in such a way that the orbits of ψ1a and ψ2a match. In order to exploit this fact, we act with the inverse of the second matrix on Eq. (34), to obtain the relation m a1 = pl2a + qm a2 , where q = w21 v11 − w22 v12 ,
p = v11 v22 − v12 v21 = det (v1 , v2 ) .
(35)
This means that, while the meridian goes around the torus bounding the first tubular neighborhood once, it goes p-times around the longitude and q-times around the meridian of the torus bounding the second tubular neighborhood. These solid tubes have to be glued together accordingly. When p = 0 = q, the manifold thereby obtained is topologically a Lens space L( p, q) according to one of the equivalent definitions of this space. Note that q is defined in terms of the vectors v1 , v2 by the above equation up to an integer multiple sp, since the vectors w1 respectively, w2 are only defined up to integer multiples of v1 respectively v2 by the condition that the matrices in Eq. (34) have determinant ±1. However, the Lens L( p, q) and L( p, q + sp) are known to be equivalent, so the Lens space is determined uniquely by the pair (v1 , v2 ). If q = 0 modulo pZ, then p = ±1, and vice versa. In that case, we may similarly argue as above and show that H is topologically S 3 . Finally, if p = 0, then q = ±1 and vice versa, and we may argue as above to show that H is topologically S 2 × S 1 . Remark 2. The proof shows how the different topologies S 3 , S 2 × S 1 , L( p, q) are related to the kernel of the Gram matrix G i j = gab ψia ψ bj at the 2 boundary points of the interval I = H/K, i.e., the “interval-vectors” introduced in the next section: If we denote the integer-valued vectors in the kernel by v1 , v2 , and set p = det (v1 , v2 ), then the topology of H is S 2 × S 1 if p = 0, it is S 3 if p = ±1, and a Lens space L( p, q) otherwise. We finally consider in detail the orbit space Mˆ = M/G of a stationary, asymptotically flat, Lorentzian vacuum black hole spacetime (M, gab ) of dimension 5 with 2-dimensional axial symmetry group K = U (1) × U (1). The Killing field t a that is timelike near infinity corresponds to the isometry group R, so that the full symmetry group is G = K × R. As above, we assume that there are no points in the exterior of M whose isotropy subgroup Kx is discrete. We denote the exterior of the black hole again by M, so that M itself is a manifold with boundary ∂ M = H . We also assume that M is globally hyperbolic. First, we note that t a can nowhere be equal to a linear combination of the axial Killing fields. Indeed, letting Fτ be the flow of t a , if t a were a
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linear combination of the axial Killing fields at a point x ∈ M, then the Fτ -orbit through x would either be periodic (for a rational linear combination), or almost periodic (for an irrational linear combination). This would imply that there are closed (or nearly closed) Fτ -orbits. However, consider the intersection Sτ of ∂ J + (Fτ (x)) with J + . Evidently, on the one hand, Sτ must be bounded as τ varies, because the orbits Fτ are periodic, or almost periodic. On the other hand, near J + , the Killing field t a is timelike, so the sets Sτ are related by a time-translation, and hence cannot be bounded as τ varies. Thus t a cannot be tangent to a linear combination of the axial Killing fields at any point. Next, we show that the linear span Vx of ψ1a , ψ2a is everywhere spacelike. Indeed if there was a linear combination ξ a of the axial Killing fields that was timelike or null somewhere, then we could consider the timelike or null orbit of ξ a . This orbit must necessarily have a closure in M that is non-compact, again invoking the global causal structure of M. On the other hand, ξ a is a linear combination of axial Killing fields, so it must have either periodic or almost periodic orbits and its closure must hence be isometric to a compact factor group of K, a contradiction. Thus, we have now learned that Vx is spacelike for all x, and that t a is transverse to Vx for all x. This can now be used to determine the general structure of the orbit space ˆ To do this, we split the isometry group G = K × R into the subgroup R generated M. by t a , and the compact subgroup K generated by the axial Killing fields. Proceeding as in the proof of Proposition 1, we first consider the factor space M/K. Using that Vx are everywhere spacelike, it now follows that M/K is a 3-dimensional manifold with boundaries and corners (of dimension 2 and 1 respectively). We then factor in addition by the subgroup R. Since the action of R is nowhere tangent to the orbits of K, the action is free, and we find that Mˆ = (M/K)/R is a 2-dimensional manifold with boundaries and corners. Finally, we know that M is simply connected by the topological censorship theorem [9,10]. By standard arguments from homotopy theory, because G is connected, also the factor space Mˆ has to be simply connected. We summarize our findings in a proposition: Proposition 3. Let (M, gab ) be the exterior of a stationary, asymptotically flat, 5-dimensional vacuum black hole spacetime with isometry group G = K × R, as described above. Then the orbit space Mˆ = M/G is a simply connected, 2-dimensional manifold with boundaries and corners. Furthermore, in the interior, on the 1-dimensional boundary segments (except the piece corresponding to H ), and on the corners, the Gram matrix G i j = gab ψia ψ bj has rank precisely 2, 1 respectively 0. 4. Classification of 5-Dimensional Stationary Spacetimes We now consider again the reduced Einstein equations for a stationary black hole spacetime with n −3 commuting axial Killing fields. We assume that the action isometry group K generated by the axial symmetries is so that there are no points with discrete isotropy group. We also assume in this section that the infinity is metrically and topologically a sphere, ∞ = S n−2 . Then n − 3 commuting axial Killing fields are only possible when n = 4, 5 but not for dimensions n ≥ 6, because the compact part S O(n − 2) of the asymptotic symmetry group admits at most (n − 2)/2 mutually commuting generators5 . When n = 4, the rigidity theorem [5,8,16,30] guarantees the existence of at least one 5 If we assume a different topology and metric structure of , such as = T n−2 , then the spacetime ∞ ∞ may have n − 2 commuting axial Killing fields.
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more axial Killing field, so that the total number of Killing fields is at least 2. Thus, for n = 4 we are always in the situation just described. If n = 5, the higher dimensional rigidity theorem [18] also guarantees at least one more axial Killing field, but for a solution with precisely one extra axial Killing field, we would not be in the situation just described if such solutions were to exist. From now on, we take n = 5, and we postulate that the number of axial Killing fields is N = 2. We also assume that the axial symmetries have been defined so as to act like the standard rotations in the 12-plane, resp. 34-plane, in the asymptotically Minkowskian region. As explained in Proposition 3 in the last section, in that case the factor space Mˆ is a simply connected 2-dimensional manifold with boundaries and corners. As in 4 dimensions, one can show using Einstein’s equations and Frobenius’ theorem that the horizontal subspaces Hx orthogonal to the Killing fields are locally integrable [33], so the metric may be written as gab = (G −1 ) I J X I a X J b + π ∗ gˆ ab
(36)
away from points where G is singular, where π : M → Mˆ = M/G is the projection. Furthermore, using that det G is nowhere vanishing in the interior of Mˆ and negative near infinity, it follows that gˆ ab is a metric of signature (++), i.e., a Riemannian metric. The reduced Einstein equations for this metric are given by Eqs. (4) and (5). Since Mˆ is an (orientable) simply connected 2-dimensional analytic manifold with boundaries and corners, we may map it analytically to the upper complex half plane {ζ ∈ C; Im ζ > 0} by the Riemann mapping theorem. Furthermore, since r is harmonic, we can introduce a harmonic scalar field z conjugate to r (i.e., Dˆ a z = ˆ ab Dˆ b r ). Since an analytic mapping is conformal we also have ∂ζ ∂ζ¯ r = 0 = ∂ζ ∂ζ¯ z, and from this, together with the boundary condition r = 0 for Im ζ = 0, one can argue that ζ = z + ir by a simple argument involving the maximum principle [34]. In particular, r and z are globally defined coordinates, and the metric globally takes the form gˆ ab = e2ν(r,z) [(dr )a (dr )b + (dz)a (dz)b ] .
(37)
Since Eq. (4) is invariant under conformal rescalings of gˆ ab , and since a 2-dimensional metric is conformally flat, it decouples from Eq. (5). In fact, writing the Ricci tensor Rˆ ab of (37) in terms of ν, one sees that Eq. (5) may be used to determine ν by a simple integration. The coordinate scalar fields r, z on Mˆ are uniquely defined by the above procedure up to a global conformal transformation of the upper half plane, i.e., a fractional transformation of the form ζ →
aζ + b , a, b, c, d ∈ R, ad − bc = 1 , ζ = z + ir . cζ + d
(38)
We will now show how r, z can in fact be uniquely fixed by a suitable condition near infinity, up to a translation of z. For 5-dimensional Minkowski spacetime, the Killing fields ψ1a = (∂/∂φ1 )a and ψ2a = (∂/∂φ2 )a are rotations in the 12-plane and the 34-plane, and the coordinates r, z as constructed above are given in terms of inertial coordinates
by r = R1 R2 and z = 21 (R12 − R22 ), with R1 = x12 + x22 and R2 = x32 + x42 , as = arctan(x1 /x2 ) and φ2 = arctan(x3 /x4 ). The conformal factor is given by well as φ1 √ e2ν = 1/2 r 2 + z 2 . In the general case, we may pick an asymptotically Minkowskian coordinate system and we may define the quantities r, z on the curved, axisymmetric
Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Killing Fields
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spacetime under consideration so that they are approximately equal near infinity to the expressions in Minkowski spacetime as just given. In particular, we may achieve that 1 e2ν → √ 2 z2 + r 2
(39)
near infinity, which corresponds to r → ∞, as z is fixed or to z → ±∞ for r = 0. This condition fixes a = d = 1, c = 0 and hence leaves only the freedom of shifting z by a constant. Thus, in summary, the Einstein equations are reduced to the two decoupled equations (4) and (5) on the factor manifold Mˆ = {ζ = z + ir ∈ C; Im ζ > 0} with metric (37) and a preferred coordinate system (r, z) that is determined up to a translation of z. The function ν is determined by Eq. (5), subject to the boundary condition (39). So far, our construction is similar to well-known constructions leading to the uniqueness theorems in n = 4 spacetime dimensions (for a review, see [17]). In fact, the only apparent difference to 4 dimensions is that the matrix field G I J is a 3 × 3 field in 5 dimensions, while it is a 2 × 2 matrix field in 4 dimensions. In particular, all information about the topology of M and the horizon might seem to be lost. In 4 dimensions, the reduced Einstein equations may be used to prove that stationary metrics are unique for fixed mass and angular momentum. On the other hand, it is known that in 5 dimensions, solutions are not uniquely fixed by these parameters, and that there are even different possibilities for the topology of the horizon. Thus, one naturally wonders where those differences are encoded in the above formulation. To understand this point, we must remember that the 2-dimensional orbit space Mˆ is a manifold with boundaries and corners by Proposition 3. The line segments of the boundary correspond to the axis (i.e., the sets where a linear combination v 1 ψ1a + v 2 ψ2a vanishes), or to the factor space of the horizon, Hˆ = H/G. The corners—the intersections of the line segments—correspond to points where the axis intersect (i.e., where both Killing fields vanish simultaneously), or to points where the axis intersect the horizon H . In the realization of Mˆ as the upper complex half plane, the line segments of ∂ Mˆ correspond to intervals (−∞, z 1 ), (z 1 , z 2 ), . . . , (z k , z k+1 ), (z k+1 , ∞)
(40)
of the real axis forming the boundary of the upper half plane. Evidently, if the horizon is connected as we assume, precisely one interval (z h , z h+1 ) corresponds to the horizon. The other intervals correspond to rotation-axis, while the points z j correspond to the intersection points of the axis, except for the boundary points of the interval (z h , z h+1 ) representing the horizon. Above, we argued that the coordinate z is defined in a diffeomorphism invariant way in terms of the solution up to shifts by a constant. Consequently, the k positive real numbers l1 = z 1 − z 2 , l2 = z 2 − z 3 , . . . lk = z k − z k+1
(41)
are invariantly defined, i.e., are the same for any pair of isometric stationary black hole spacetimes of the type we consider. Thus, they may be viewed as global parameters (“moduli”) characterizing the given solution in addition to the mass m and the two angular momenta J1 , J2 . Furthermore, with each l j , there is associated a label which is either a vector v j = (v 1j , v 2j ) of integers such that the linear combination v 1j ψ1a + v 2j ψ2a vanishes, or vh = (0, 0) if we are on the horizon. The labels corresponding to the “outmost” intervals (−∞, z 1 ) and (z k+1 , ∞) must be (0, 1) respectively (1, 0), because this is the case for Minkowski spacetime, and we assume that our solutions are asymptotically flat.
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Also from Proposition 1, and the Remark 2 following the proof of Proposition 2, we have det (v j , v j+1 ) = ±1 if (z j−1 , z j ) and (z j , z j+1 ) are not the horizon det (vh−1 , vh+1 ) = p if (z h , z h+1 ) is the horizon and the parameter p = det (vh−1 , vh+1 ) is related to the different horizon topologies by Topology of H p 0 S2 × S1 ±1 S3 other L( p, q) The numbers {l j } and the assignment of the labels {v j } were also considered from a local perspective by [14]. However, we note that, in [14], neither the condition that v 1 , v 2 be integers, nor the determinant conditions for adjacent interval vectors and their relation to the horizon topology were obtained. Furthermore, the interval vectors considered in [14] have 3 components, rather than 2. We will call the data consisting of {l j } and the assignments {v j } the “interval structure”. For 4 dimensional black holes, there is only the trivial interval structure (−∞, z 1 ), (z 1 , z 2 ), (z 2 , ∞), with the middle interval corresponding to the horizon, and the first and third corresponding to single axis of rotation of the Killing field. Furthermore, the interval length l1 may be expressed in terms of the global parameters m, J of the solution. By contrast, in 5 dimensions, the interval structure can be non-trivial, and in fact differs for the Myers-Perry [29] and Black Ring [6] solutions. For these cases, the interval structure is summarized in the following table [14]: Myers-Perry BH Black Ring Flat Spacetime
Interval Lengths ∞, l1 , ∞ ∞, l1 , l2 , ∞ ∞, ∞
Vectors (Labels) (1, 0), (0, 0), (0, 1) (1, 0), (0, 0), (1, 0), (0, 1) (1, 0), (0, 1)
Horizon Topology S3 2 S × S1 —
The following interval structure would represent a “Black Lens” if such a solution would exist: Interval Lengths Vectors (Labels) Horizon Topology Black Lens ∞, l1 , l2 , ∞ (1, 0), (0, 0), (1, p), (0, 1) L( p, 1) Even for a fixed set of of asymptotic charges m, J1 , J2 the invariant lengths of the intervals l1 = z 1 − z 2 , l2 = z 2 − z 3 may be different for the different Black Ring solutions, corresponding to the fact that there exist non-isometric Black Ring solutions with equal asymptotic charges [6,7]. On an interval labeled “(1, 0)”, all components of G 1 j = G j1 , j = 1, 2 vanish but not the other ones, while on an interval labeled “(0,1)”, all components G 2 j = G j2 vanish. The vector (1, 0) hence corresponds to a ∂/∂φ1 -axis, while the vector (0, 1) corresponds to a ∂/∂φ2 -axis. Thus, we see that the interval structure enters the reduced field equations through the boundary conditions imposed upon the matrix field G i j . The horizon topology is also determined by the interval structure by Proposition 2, see also Remark 2 following that proposition. This is how the different topology and global nature of the solutions in 5 dimensions are encoded in the reduced ˆ Einstein equations on the upper half plane M. Clearly, since we have argued that the interval structure is a diffeomorphism invariant datum constructed from the given solution, two given stationary black hole solutions with 2 axial Killing fields cannot be isomorphic unless the interval structures and the masses and angular momenta coincide. The main purpose of this paper is to point out the following converse to this statement:
Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Killing Fields
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Theorem. Consider two stationary, asymptotically flat, vacuum black hole spacetimes of dimension 5, having two commuting axial Killing fields that commute also with the time-translation Killing field. Assume that both solutions have the same interval structure, and the same values of the mass m and angular momenta J1 , J2 . Then they are isometric. Proof. As in 4 spacetime dimensions, the key step in the argument is to put the reduced Einstein equations in a suitable form. Following [26] (see also [25]), this is done as follows in 5 dimensions. On M, we first define the two twist 1-forms: ω1a = abcde ψ1b ψ2c ∇ d ψ1e ,
(42)
ω2a =
(43)
abcde ψ1b ψ2c ∇ d ψ2e
.
Using the vacuum field equations and standard identities for Killing fields [33], one shows that these 1-forms are closed. Since the Killing fields commute, the twist forms are invariant under G, and so we may define corresponding 1-forms ωˆ 1a and ωˆ 2a on the interior of the factor space Mˆ = {ζ ∈ C; Im ζ > 0}. These 1-forms are again closed. Thus, the “twist potentials”
ζ χi = ωˆ i ζ dζ + ωˆ i ζ¯ d ζ¯ (44) 0
are globally defined on Mˆ and independent of the path connecting 0 and ζ . We introduce the 3 × 3 matrix field by ⎞ ⎛ (det f )−1 −(det f )−1 χ1 −(det f )−1 χ2 (45) = ⎝ −(det f )−1 χ1 f 11 + (det f )−1 χ1 χ1 f 12 + (det f )−1 χ1 χ2 ⎠ . −(det f )−1 χ2 f 21 + (det f )−1 χ2 χ1 f 22 + (det f )−1 χ2 χ2 Here f i j is the Gram matrix of the axial Killing fields,
G 11 G 12 f = . G 21 G 22
(46)
The matrix satisfies T = , det = 1, and is positive semi-definite, meaning that it may be written in the form = S T S for some matrix S of determinant 1. As a consequence of the reduced Einstein equations (4) and (5), it also satisfies the divergence identity Dˆ a [r −1 Dˆ a ] = 0
(47)
ˆ on M. Consider now the exterior of the two black hole solutions as in the statement of ˜ g˜ ab ). We denote the corresponding matrices the theorem, denoted (M, gab ) and ( M, ˜ and we use the same “tilde” notation to distinguish any defined as above by and , other quantities associated with the two solutions. Since the orbit spaces of the respective spacetimes can both be identified with the upper half-plane as analytic manifolds, we can identify the respective orbit spaces. Furthermore, one can show by reversing the constructions of the local analytic coordinate systems in the proof of Proposition 1 that the G-manifold M can be uniquely reconstructed from the interval structure, i.e., M as a
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manifold with a G-action is uniquely determined by the interval structure modulo diffeomorphisms preserving the action of G. Therefore, since the interval structures {l˜j , v˜ j } and {l j , v j } are the same, M and M˜ are isomorphic as manifolds with a G action, and ˜ and that t˜a = t a , ψ˜ a = ψ a for i = 1, 2. It follows we may hence assume that M = M, i i in particular that g˜ ab and gab may be viewed as being defined on the same analytic manifold, M, and we may also assume that r˜ = r and z˜ = z. Consequently, it is possible to combine the divergence identities (47) for the two solutions into a single identity on the upper complex half plane. This key identity [27,25] is called the “Mazur identity” and is given by: −1 ˜ Dˆ a (r Dˆ a Tr ) = r gˆ ab Tr JaT J , (48) b where ˜ −1 Dˆ a ˜ −1 − 1, Ja = ˜ − −1 Dˆ a . =
(49)
˜ the Mazur identity can be presented ˜ = S˜ T S, Using now the identities = S T S and in the form (50) Dˆ a (r Dˆ a Tr ) = r gˆ ab Tr NaT Nb , ˜ where Na = S −1 Ja S. The key point about the Mazur identity (50) is that on the left side we have a total divergence, while the term on the right-hand side is non-negative. This structure is now ˆ Using Gauss’ theorem, one finds exploited by integrating the Mazur identity over M.
(51) r Dˆ a Tr d Sˆ a = r gˆ ab Tr NaT Nb d Vˆ , ˆ ∂ M∪∞
Mˆ
where the integral over the boundary includes an integration over the “boundary at infinity”. If one can show that the boundary integral on the left side is zero, then it follows ˆ and hence that ˜ = −1 Dˆ a . Since this implies that ˜ −1 Dˆ a that N a vanishes on M, −1 ˆ ˜ ˜ if this holds true at one is a constant matrix on M, one concludes that = ˆ ˜ point of M. Using that the Gram matrices f i j and f i j become equal near infinity, and ˜ using that χ˜ i is equal to χi on the axis (see below), one concludes that is equal to ˆ on an axis near infinity, and hence equal everywhere in M. This can now be used as follows to show that the spacetimes are isometric. First, it ˜ = that χ˜ i = χi and that the Gram matrices of the axial immediately follows from Killing fields are identical for the two solutions, f˜i j = f i j . To see that the other scalar products between the Killing fields coincide for the two solutions, let αi = gab t a ψib , β = gab t a t b , and define similarly the scalar products α˜ i , β˜ for the other spacetime. One derives the equation Dˆ a [( f −1 )i j α j ] = r (det f )−1 ˆa b ( f −1 )i j Dˆ b χ j .
(52)
The right side does not depend upon the conformal factor ν, so since χ˜ i = χi and f˜i j = f i j , it also follows that α˜ i = αi up to a constant. That constant has to vanish, since it vanishes at infinity. Furthermore, from β = ( f −1 )i j αi α j − (det f )−1r 2
(53)
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we have β˜ = β. Thus, all scalar products of the Killing fields are equal for the two solutions, G˜ I J = G I J on the entire upper half plane. Viewing now the reduced Einstein equation (5) as an equation for ν respectively ν˜ , one concludes from this that ν˜ = ν. Thus, summarizing, we have shown that if the boundary integral in the integrated Mazur identity Eq. (51) could be shown to vanish, then G˜ I J = G I J , r˜ = r , z˜ = z and ν˜ = ν. Since t˜a = t a , ψ˜ ia = ψia it follows from Eqs. (36) and (37) that g˜ ab = gab . This proves that the two spacetimes are isometric, proving the theorem. Thus, to establish the statement of the theorem, one needs to prove that the boundary integral in (51) vanishes. For this, one has to analyze the behavior of the integrand r Dˆ a Tr near the boundary Im ζ = 0 (i.e., the horizon and the axis) and near the boundary at infinity, Im ζ → ∞ as Re ζ is kept fixed (i.e., spatial infinity). At this stage one has to use again that the asymptotic charges and the interval structure of the solutions are assumed to be identical. We divide the boundary region into three parts: (1) The axis, (2) the horizon, and (3) infinity. (1) On each segment (z j , z j+1 ) of the real line Im ζ = r = 0 representing an axis, we know that the null spaces of the Gram matrices f i j and f˜i j coincide, because we are assuming that the interval structures of both solutions are identical. Furthermore, from Eq. (44), and from the fact that ωˆ ai vanishes on any axis by definition, the twist potentials χi are constant on the real line outside of the segment (z h , z h+1 ) representing the horizon. The difference between the constant value of χi on the real line left and right to the horizon segment can be calculated as follows:
z h+1 χi (r = 0, z h ) − χi (r = 0, z h+1 ) = ωˆ iζ dζ + ωˆ i ζ¯ d ζ¯ zh
1 ∇[a ψb] i d S ab (2π )2 H
1 = ∇[a ψb] i d S ab = const. Ji . 3 (2π )2 S∞ =
The first equality follows from the definition of the twist potentials, the second from the defining formula for the twist potentials and the fact that the twist potentials are invariant under the action of the 2-independent rotation isometries each with period 2π (with H a horizon cross section), the third equation follows from Gauss’ theorem and the fact that ∇ a (∇[a ψb] i ) = 0 when Rab = 0, and the last equality follows from the Komar expression for the angular momentum in 5 dimensions. ˜ g˜ ab ). Because we assume The analogous expressions hold in the spacetime ( M, ˜ that Ji = Ji , we can add constants to χi , if necessary, so that χi = χ˜ i on the axis, and in fact that χi = χi − χ˜ i = O(r 2 ) near any axis. One may now analyze the contributions to the boundary integral coming from the axis using the expression Dˆ a Tr = Dˆ a (det f˜)−1 {−(det f ) + ( f −1 )i j χi χ j } + ( f −1 )i j f i j . (54) (v 1 , v 2 ),
We consider a particular axis interval with interval vector v = which by assumption is identical for the two solutions. We pick a second basis vector w = (w1 , w 2 ), and we denote by v∗ , w∗ the dual basis. We conclude that, on the given interval f i j = avi∗ v ∗j + bwi∗ w ∗j + 2cv(i∗ w ∗j) ,
˜ i∗ w ∗j + 2cv f˜i j = av ˜ i∗ v ∗j + bw ˜ (i∗ w ∗j) , (55)
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with a˜ = O(r 2 ) = a, b˜ = O(1) = b and c˜ = O(r ) = c. We insert this into Eq. (54), we use that χi = O(r 2 ) on the axis, and we use the detailed fall-off properties of the metric as well as the quantities χi , f i j , χ˜ i , f˜i j for large z, which are the same for any asymptotically flat solution to the relevant order. One finds that Dˆ a Tr is finite on the axis, so that the corresponding contribution to the line integral vanishes. The details of this analysis are in close parallel to the corresponding analysis of Ida et al. [26], who analyzed the situation for a special horizon topology and interval structure. (2) On the horizon segment, the matrices f i j , f˜i j are invertible, so Dˆ a Tr is regular, and the boundary integral over this segment vanishes. (3) Near infinity, one has to use the asymptotic form of the metric for an asymptotically flat spacetime in 5 dimensions. In an appropriate asymptotically Minkowskian coordinate system such that asymptotically ψ1a = (∂/∂φ1 )a and ψ2a = (∂/∂φ2 )a , it takes the form
2µa1 (R 2 + a12 ) 2 µ −3 sin θ + O(R ) dtdφ1 g = − 1 − 2 + O(R −2 ) dt 2 + R R4
2µa2 (R 2 + a22 ) µ 2 −3 −3 + cos θ + O(R ) dtdφ + 1 + + O(R ) × 2 R4 2R 2 2 R + a12 cos2 θ + a22 sin2 θ 2 × R d R 2 + (R 2 + a12 cos2 θ + a22 sin2 θ ) dθ 2 (R 2 + a12 )(R 2 + a22 )
+(R 2 + a12 ) sin2 θ dφ12 + (R 2 + a22 ) cos2 θ dφ12 , where µ, a1 , a2 are parameters proportional to the mass, and the angular momenta J1 , J2 of the solution. One must then determine the functions z, r as functions of R, θ near infinity using the reduced Einstein equations, subject to the boundary condition (39) near infinity. This then enables one to find asymptotic expansions for f i j , f˜i j , χi , χ˜ i in terms of the parameters J1 , J2 , m, J˜1 , J˜2 , m. ˜ Using that these parameters coincide for both solutions, one shows that the contribution to the boundary integral (50) vanishes. Again, the details of this argument only depend upon the asymptotics of the solution, but not the horizon topology, or interval structure. They are therefore identical to the arguments given in [26] for spherical black holes, see also [14, Sect. 4.3]. This completes the proof.
5. Conclusion In this paper we have considered 5-dimensional stationary, asymptotically flat, vacuum black hole spacetimes with 2 commuting axial Killing fields generating an action of U (1) × U (1). Under the additional hypothesis that there are no points with a discrete isotropy subgroup, we have shown that the black hole must have horizon topology S 3 , S 2 × S 1 , or L( p, q), and that each solution is uniquely specified by the asymptotic charges (mass and the two angular momenta), together with certain data describing the relative position and distance of the horizon and axis of rotation—the “intervalstructure,” defined in a somewhat different form first by [14]. Our proof mostly relied on the known σ -model formulation of the reduced Einstein equation [25,26], combined
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with basic arguments clarifying the global structure of the factor manifold of symmetry orbits. As we have already pointed out in the introduction, the case considered in this paper presumably does not represent the most general stationary, asymptotically flat black hole solution in 5 dimensions. It appears highly unlikely that our method of proof could be generalized to solutions with only one axial Killing field, if such solutions were to exist. On the other hand, we believe that our assumption that there are no points with a discrete isotropy group is only of a technical nature. Without this assumption, the orbit space will have singular points (“orbifold points”), rather than being an analytic manifold with boundary. Our analysis of the integrated divergence identity (51) then would also have to include the boundaries resulting from the blow ups of the orbifold points. It seems not unlikely that the proof could still go through if the nature of the discrete subgroups is identical for the two solutions. Thus, it appears that we need to specify in general at least (a) the mass and angular momenta (b) interval structure, and (c) a datum describing the position of the points with discrete isotropy subgroups in the upper half plane, together with the specification of the subgroups themselves. It is also interesting to ask how the parameters in the interval structure are related to other properties of the solution, such as the invariant charges, horizon area, or surface gravity. For example, by evaluating the horizon area for the metric (36), one finds that the interval length lh associated with the horizon is given by lh = κ A/4π 2 , but we do not know whether similar relations exist for the other interval lengths. It is also not clear whether all interval structures can actually be realized in solutions to the vacuum equations, nor whether the horizon topologies L( p, q) can be realized6 . Finally it would be interesting to see if the constructions of this paper can be generalized to include matter fields [20]. Acknowledgements. We would like to thank Troels Harmark for comments and Thomas Schick for discussions. S. Y. would like to thank the Alexander von Humboldt Foundation for a stipend, and the Institut für Theoretische Physik Göttingen for its kind hospitality. He also acknowledges financial support from the Bulgarian National Science Fund under Grant MUF04/05 (MU 408). Note added in proof. After this manuscript was posted, it was noted by P. Chrusciel that our analysis did not properly take into account points with discrete isotropy group. We have added a corresponding assumption to the hypothesis. We are grateful to him for sharing his insight with us and for extensive discussions.
References 1. Adams, C.C.: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots New York, W. H. Freeman, 1994 2. Bunting, G.L.: Proof of the uniqueness conjecture for black holes. PhD Thesis, Univ. of New England, Armidale, N.S.W., 1983 3. Carter, B.: Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971) 4. Cho, Y.M., Freund, P.G.O.: Non-Abelian gauge fields as Nambu-Goldstone fields. Phys. Rev. D 12, 1711 (1975) 5. Chru´sciel, P.T.: On rigidity of analytic black holes. Commun. Math. Phys. 189, 1–7 (1997) 6. Emparan, R., Reall, H.S.: A rotating black ring in five dimensions. Phys. Rev. Lett. 88, 101101 (2002) 7. Emparan, R., Reall, H.S.: Generalized Weyl solutions. Phys. Rev. D 65, 084025 (2002) 8. Friedrich, H., Racz, I., Wald, R.M.: On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon, Commun. Math. Phys. 204, 691–707 (1999) 9. Galloway, G.J., Schleich, K., Witt, D.M., Woolgar, E.: Topological censorship and higher genus black holes. Phys. Rev. D 60, 104039 (1999) 6 Solutions with horizon topology L(n, 1) have however been found in Einstein-Maxwell theory, see [21].
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10. Galloway, G.J., Schleich, K., Witt, D., Woolgar, E.: The AdS/CFT correspondence conjecture and topological censorship. Phys. Lett. B 505, 255 (2001) 11. Galloway, G.J., Schoen, R.: A Generalization of Hawking’s black hole topology theorem to higher dimensions. Commun. Math. Phys. 266, 571 (2006) 12. Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness and non-uniqueness of static black holes in higher dimensions. Phys. Rev. Lett. 89, 041101 (2002) 13. Harmark, T., Olesen, P.: On the structure of stationary and axisymmetric metrics. Phys. Rev. D 72, 124017 (2005) 14. Harmark, T.: Stationary and axisymmetric solutions of higher-dimensional general relativity. Phys. Rev. D 70, 124002 (2004) 15. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152–166 (1972) 16. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press, 1973 17. Heusler, M.: Black hole uniqueness theorems. Cambridge: Cambridge University Press, 1996 18. Hollands, S., Ishibashi, A., Wald, R.M.: A higher dimensional stationary rotating black hole must be axisymmetric. Commun. Math. Phys. 271, 699 (2007) 19. Hollands, S., Ishibashi, A.: Asymptotic flatness and Bondi energy in higher dimensional gravity. J. Math. Phys. 46, 022503 (2005) 20. Hollands, S., Yazadjiev, S., Work in progress 21. Ishihara, H., Kimura, M., Masuno, K., Tomizawa, S.: Black holes on Euguchi-Hanson space in fivedimensional Einstein Maxwell theory, Phys. Rev. D 74, 047501 (2006) 22. Israel, W.: Event horizons in static vacuum space-times. Phys. Rev. 164, 1776–1779 (1967) 23. Kerner, R.: Generalization of Kaluza-Klein theory for an arbitrary non-abelian gauge group. Ann. Inst. H. Poincare 9, 143 (1968) 24. Kobayshi, S., Nomizu, K.: Foundations of Differential Geometry I. New York: Wiley, 1969 25. Maison, D.: Ehlers-Harrison-type Transformations for Jordan’s extended theory of graviation. Gen. Rel. Grav. 10, 717 (1979) 26. Morisawa, Y., Ida, D.: A boundary value problem for five-dimensional stationary black holes, Phys. Rev. D 69, 124005 (2004) 27. Mazur, P.O.: Proof of uniqueness of the Kerr-Newman black hole solution. J. Phys. A 15, 3173–3180 (1982) 28. Moncrief, V., Isenberg, J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89, 387–413 (1983) 29. Myers, R.C., Perry, M.J.: Black holes in higher dimensional space-times. Annals Phys. 172, 304 (1986) 30. Racz, I.: On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Class. Quant. Grav. 17, 153 (2000) 31. Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975) 32. Sudarsky, D., Wald, R.M.: Extrema of mass, stationarity, and staticity, and solutions to the Einstein Yang-Mills equations. Phys. Rev. D 46, 1453–1474 (1992) 33. Wald, R.M.: General Relativity. Chicago: University of Chicago Press, 1984 34. Weinstein, G.: On rotating black holes in equilibrium in general relativity. Commun. Pure Appl. Math. 43, 903 (1990) Communicated by G.W. Gibbons
Commun. Math. Phys. 283, 769–794 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0518-1
Communications in
Mathematical Physics
Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter Vassili Gelfreich1 , Dmitry Turaev2 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.
E-mail: [email protected]
2 Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva 84105, Israel.
E-mail: [email protected] Received: 20 September 2007 / Accepted: 19 November 2007 Published online: 5 June 2008 – © Springer-Verlag 2008
Abstract: We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows and Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from linear to exponential in time. Our theory also applies to non-Hamiltonian systems with a first integral. 1. Setting the Problem Consider a Hamiltonian system H = H ( p, q, εt)
(1)
with ε small. It is natural to compare its dynamics with the frozen system H = H ( p, q, ν),
(2)
where ν is now treated as a constant parameter. The Hamiltonian H is a first integral of the frozen system but not of the non-autonomous system described by (1). Let ( p(t), q(t)) be a trajectory of (1) and H (t) ≡ H ( p(t), q(t), εt). Differentiating with respect to time and using the Hamilton equations we see that the rate of energy change is small ∂H H˙ (t) = ε ( p(t), q(t), εt) . ∂ν Adiabatic invariants play an important role in description of dynamics for this class of systems [10]. It is also known that if the frozen system is integrable, then under certain
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assumptions the energy H may oscillate at a bounded distance from its initial value for a very long time. On the other hand, in the case of chaotic dynamics in the frozen system the behaviour of the energy may be drastically different. Indeed, in the mid nineties Mather discovered that adding a time-periodic perturbation to the Hamiltonian of a uniformly hyperbolic flow creates orbits with an unbounded energy growth. Moreover, the energy on Mather’s trajectories tends to infinity linearly, i.e., it changes at a much faster rate than it could do if the unperturbed system were integrable. This result and its generalisations were studied in [4,6,7,9,16,23], where the reader can e.g. find more detailed discussion on the history of the problem. While some papers treat the problem of estimating the energy growth in nonautonomous Hamiltonian systems mostly as a simplified model for Arnold diffusion, we think it has an independent interest and a wide range of applications (see e.g. [13–15,26] where billiards with time-dependent boundaries were discussed in connection with Fermi acceleration). In our paper we establish that the existence of orbits of unbounded and rapid energy growth is a very general phenomenon, typical for practically arbitrary slow non-autonomous perturbation of a Hamiltonian system with chaotic behaviour. The construction we employ is different from most of those used by the previous authors and is applied to a wider class of systems. Thus, we do not use variational constructions, nor KAM theory, we do not build heteroclinic chains, and we do not assume any kind of periodicity for the time-dependence. Finally, we provide formulas for calculating the energy growth rates, and provide examples for which the growth rates vary from linear to exponential one. In short, the acceleration mechanism we discuss here is as follows. First, by saying that the frozen system has a chaotic behaviour, we mean that there exists h ∗ such that the frozen system has a uniformly-hyperbolic, compact, transitive, invariant set hν in every energy level H = h ≥ h ∗ for all ν ≥ 0. In every given energy level, the set hν is in the closure of a set of hyperbolic periodic orbits each of which has an orbit of a transverse heteroclinic connection to any of the others. This means that orbits of (2) may stay close to any of the periodic orbits for an arbitrary number of periods, then come close to another periodic orbit and stay there, and so on. Recall also that periodic orbits of (2) form families parametrized by the value of H and by ν. By a standard averaging procedure (see e.g. [3,12]), one establishes that for the orbits of the original system (1) close to a periodic family of (2) there exists an adiabatic invariant – a function J (h, ν) such that J (H (t), εt) stays almost constant for a very long time ε−1 . Now we take two periodic families, L a and L b , of the frozen system. For the orbits that stay near L a the value of Ja (H (t), εt) will remain almost constant while Jb (H (t), εt) may grow or decrease, and for the orbits that stay near L b we will have Jb nearly constant while Ja changes. In this paper we show that under some natural conditions one can arrange jumps between L a and L b in such a way that one of the functions Ja or Jb will always grow while the second one rests. Then the sum Ja (H (t), εt) + Jb (H (t), εt) may tend to infinity. Note that both Ja and Jb are monotonically increasing functions of h, therefore the unbounded growth of Ja (H (t), εt) + Jb (H (t), εt) implies, typically, the unbounded growth of H (t). These considerations do not depend on how the Hamiltonian depends on εt. Indeed, we treat the cases of periodic and non-periodic perturbations simultaneously, and the results hold true for periodic, quasiperiodic and other settings. We note that the invariant set hν with the desired properties exists, provided the frozen system has a family of hyperbolic periodic orbits, each one with a transversal homoclinic trajectory.
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Technically, for the most part of this paper we do not use the Hamiltonian structure of the system. Therefore, for a greater generality, instead of (1) we consider x˙ = G(x, εt) ,
(3)
where x ∈ Rm . The corresponding frozen system has the form x˙ = G(x, ν) .
(4)
We assume that a function H (x, ν) is an integral, i.e. ∂H (x, ν) · G(x, ν) ≡ 0. ∂x
(5)
We will continue calling H the energy. Assume that for all ν ≥ 0 in every energy level H = h ≥ h ∗ the frozen system (4) has a heteroclinic cycle composed of a pair of hyperbolic periodic orbits L a and L b , and a pair of transverse heteroclinic orbits, ab and ba ; the first corresponds to a transverse intersection of W u (L a ) and W s (L b ), while the second one corresponds to a transverse intersection of W u (L b ) and W s (L a ). We note that the set of all orbits that stay in a small neighbourhood of the heteroclinic cycle in a given energy level is a locally maximal, uniformly hyperbolic, compact, transitive, invariant set [20]. We denote this set by . It is well known that a hyperbolic periodic orbit continues in a unique way as a smooth function of parameters h and ν. The same holds true for a transverse heteroclinic. So, L a and L b , as well as the transverse heteroclinic orbits ab and ba , depend on h and ν in a smooth way. Let Tc (h, ν) (where c = a or c = b) be the period of the orbit L c (h, ν) : x = xc (t; h, ν). Let us consider the average of Hν over the periodic orbit L c : Tc ∂H 1 vc (h, ν) = (x, ν) dt . (6) Tc 0 ∂ν x=xc (t;h,ν) Theorem 1. Assume that the differential equation dh = max{va (h, ν), vb (h, ν)} dν
(7)
has a solution h(ν) such that h(ν) ≥ h ∗ for all ν ≥ 0 and h(ν) → +∞ as ν → +∞. Then given any h 1 ≥ h 0 ≡ h(0) there exists t1 > 0 such that for every sufficiently small ε there is a solution x(t) of system (3) such that H (x(0), 0) = h 0 and H (x(t), εt) = h 1 at a time t ≤ t1 /ε. In Sect. 1.2 we show that in the Hamiltonian setup (i.e. in the case where system (3) is Hamiltonian) equation (7) possesses tending to infinity solutions under very mild assumptions. Thus, for the case of periodic or quasiperiodic dependence of the Hamiltonian on εt we show (Proposition 2) that the boundedness of solutions of (7) is a codimension infinity event. Simple sufficient conditions for the unbounded energy growth are given for special classes of Hamiltonian systems in Sect. 4. Theorem 1 does not directly imply that system (3) has an orbit with unbounded energy. In order to prove the existence of such an orbit we need information on the behaviour of the system near the hyperbolic set hν at ν and h tending to infinity, i.e., for a non-compact set of values of h and ν. Therefore, certain uniformity assumptions are necessary. As they are quite technical, we postpone their precise statements till Sects. 2
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and 3. In short, in condition [UA1] we require that the set hν has a cross-section and that the right-hand sides of the corresponding Poincaré map, when written in the so-called “cross-form” [22], are uniformly bounded, along with their first derivatives, for all sufficiently large h and ν and all small ε. In condition [UA2] we require a certain uniformity for the times of the first return to the cross-section and for the change in the energy between two consecutive returns. In Sect. 4 we check these uniformity assumptions for several classes of examples. For a greater generality, we allow for the right-hand side of system (3) under consideration to depend explicitly on ε, i.e., the system takes the form x˙ = G(x, εt; ε) ,
(8)
with G depending on ε continuously. Thus, the frozen system (4) and the integral H will also depend on ε, as well as the functions va,b in (6). Theorem 2. Assume the uniformity assumptions [UA1] and [UA2] hold true. Consider a differential equation dh = max{va (h, ν), vb (h, ν)} − δβ(h, ν), dν
(9)
where the smooth function β is defined by condition (46). Suppose there exists δ > 0 such that Eq. (9) has, for all small ε, a solution h δ (ν) that satisfies h δ (ν) ≥ h ∗ for all ν and tends to +∞ as ν → +∞. Then for all sufficiently small ε system (8) has an orbit x(t) for which H (x(t), εt; ε) → +∞ as t → +∞. Theorem 2 above is an immediate corollary of the following comparison theorem. Theorem 3. Assume the uniformity assumptions [UA1] and [UA2] hold true, let δ > 0 and denote as h δ a solution of the differential equation (9). Then for all sufficiently small ε system (8) has a solution x(t) such that H (x(0), 0) = h δ (0) and H (x(t), εt) ≥ h δ (εt) for all t ≥ 0. The proof of Theorem 3 is given in Sects. 2 and 3. Note that the uniformity assumptions are automatically fulfilled for any compact set of h and ν, hence Theorem 1 is indeed extracted from Theorem 3 by modifying, if necessary, the equations outside a neighbourhood of the region H (x, ν) ∈ [h 0 , h 1 ] and ν ∈ [0, t1 ]. Note also that although the function β in Theorem 2 is defined in technical terms, in the examples which we consider in Sect. 4 this function is asymptotically (as h → +∞) of the same order as the functions va,b . Therefore, the contribution of the second term of Eq. (9) is not very important (recall that δ in (9) can be taken arbitrarily small). In other words, the energy growth rate is, essentially, given by the solution of Eq. (7). 1.1. Scheme of the proof. Let us now describe the scheme of the proof of Theorem 3. Consider the family L c : x = xc (t; h, ν) of the hyperbolic periodic orbits of the frozen system (4) (here, c = a or c = b). This is a three-dimensional invariant manifold of system (4). Importantly, this manifold is normally-hyperbolic, because of the hyperbolicity of the periodic orbits which comprise it. Therefore, it persists for all small ε [8] (the set of values of (h, ν) under consideration is not compact and therefore we also need our
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uniformity assumptions to make such claim). Thus, system (8) has an invariant manifold x = x˜c (τ ; h, ν; ε) such that x˜c = xc at ε = 0; moreover, x˜c depend periodically on the first argument τ (with the period depending on h, ν and ε). The dynamics on this manifold is close to the dynamics of the frozen system, therefore the evolution of h and ν is slow, while the first argument τ is a fast rotating phase. Hence, in the first order with respect to ε, the evolution of h and ν on the invariant manifold is described by the system averaged with respect to the fast time h˙ = εvc (h, ν),
ν˙ = ε ,
(10)
where vc is defined by Eq. (6). Therefore if a trajectory stays close to L c its energy changes following the equation dh = vc (h, ν) + h.o.t. dν As we see, for given values of h and ν the velocity of the change of h depends on the periodic orbit L c . We will prove that the full system has a trajectory which switches between small neighbourhoods of L a and L b , always choosing the periodic orbit which gives larger velocity at the moment; clearly, this is the trajectory which implements the optimal strategy for the acceleration. For this trajectory the rate of energy change is described by the differential equation: dh = max{ va (h, ν), vb (h, ν) } + h.o.t. dν Hence, Eq. (7) correctly describes the evolution of h along the trajectory of the fastest energy growth. The small δ term in (9) takes care of all higher order corrections (we can neglect this term in the framework of Theorem 1, where the time of acceleration is finite). As we see, in order to prove Theorem 3, we just need to construct an orbit which actually jumps between L a and L b in the above described way. In order to do this, we code the orbits of the frozen system that stay in a small neighbourhood of the heteroclinic cycle L a ∪ L b ∪ ab ∪ ba by sequences of a’s and b’s. Given any such sequence, the corresponding orbit depends smoothly on h and ν, i.e. we have a normally-hyperbolic invariant manifold corresponding to any of these sequences. Because of the uniform normal hyperbolicity, all of these manifolds persist for all small ε (we supply a proof in Sect. 2; as a matter of fact, our approach is similar to that of [21]). We repeat that every sequence of a and b is a valid code, i.e. for every itinerary of the jumps between L a and L b system (8) has an invariant manifold, orbits on which implement this particular itinerary. In particular, it has an invariant manifold for the orbits on which the growth of the energy is estimated from below by Eq. (9). The rigorous construction is in Sect. 3.
1.2. Adiabatic invariant revisited. In this subsection we discuss the meaning of Eq. (7) in the Hamiltonian setup and conditions which imply that all its solutions tend to infinity. This section is of independent interest and the proofs of our main theorems do not rely on its results. Note that Theorems 1, 2 and 3 do not assume that system (3) is Hamiltonian. However, as we will show in a moment, in the Hamiltonian case Eq. (7) indeed has a tending to infinity solution under almost no assumptions.
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We first recall that hyperbolic periodic solutions L c comprise, at every fixed ν, a one-parameter family parametrized by the energy h. Thus, they fill a certain twodimensional cylinder in the phase space. As usual in the theory of slow perturbations, we may introduce “action-angle” variables on this surface. The “action” is defined by Jc (h, ν) = p dq (11) Lc
in the case of the standard symplectic form. In a more general case, where the Hamiltonian system (2) is defined on a manifold with a symplectic form , let us assume that the symplectic form is exact, i.e. = dϑ, where ϑ is an 1-form. Then the action is defined as Jc (h, ν) = ϑ. (12) Lc
It is well-known that ∂ Jc (h, ν) = Tc (h, ν), ∂h
∂ Jc (h, ν) = − ∂ν
Tc 0
∂ H dt. ∂ν x=xc (t;h,ν)
(13)
In order to see this, note that, by definition of the action, the difference between the actions corresponding to two close closed curves L c (h, ν) and L c (h + h, ν + ν) is, essentially, the area of the surface spanned by these two curves. Therefore in the case of standard symplectic form we obtain Tc (h,ν)
∂ pc (t; h, ν) ∂qc (t; h, ν) q˙c (t; h, ν) − p˙ c (t; h, ν) dt, ∂ν ∂ν 0 Tc (h,ν) ∂ Jc ∂ pc (t; h, ν) ∂qc (t; h, ν) = − p˙ c (t; h, ν) dt. q˙c (t; h, ν) ∂h ∂h ∂h 0
∂ Jc = ∂ν
In the general case we have ∂ xc (t; h, ν) dt, x˙c (t; h, ν) , ∂ν 0 Tc ∂ Jc ∂ xc (t; h, ν) = dt. x˙c (t; h, ν) , ∂h ∂h 0 ∂ Jc = ∂ν
Tc
Taking into account the definition of the frozen Hamiltonian vector field we see (x, ˙ ·) = d H (·), which implies in the coordinates Tc ∂ Jc ∂ H (x, ν) ∂ xc (t; h, ν) = dt, ∂ν ∂ x ∂ν 0 x=xc (t;h,ν) Tc ∂ Jc ∂ H (x, ν) ∂ xc (t; h, ν) = dt. ∂h ∂x ∂h 0 x=xc (t;h,ν) Since H (xc (t; h, ν), ν) ≡ h for all h and ν due to the definition, these formulas imply (13) immediately.
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Let us consider the Hamiltonian system with one degree of freedom defined by the Hamiltonian function Jc (h, ν): h = −
∂ Jc (h, ν) , ∂ν
ν =
∂ Jc (h, ν) . ∂h
Taking into account (13), we conclude that h =
Tc 0
∂H (x, ν)|x=xc (t;h,ν) dt, ∂ν
ν = Tc (h, ν) ,
(14)
which coincides with Eq. (10) up to a time change. Consequently Jc (h, ν) is an integral of (10), i.e., Jc (h(ν), ν) = J (h(0), 0) for every its solution. This gives us a leading order model for an orbit of the full system which stays close to L c : the action Jc is an adiabatic invariant and the energy oscillates like a trajectory of a Hamiltonian system with one degree of freedom described by the Hamilton function Jc . Thus, when the orbit is close to the invariant manifold that corresponds to L a , the function Ja (H (t), εt) remains almost constant for a long time, while the evolution of Jb (H (t), εt) is, in the first order, described by the equation ∂ Jb ∂ Ja ∂ Jb ∂ Ja ˙ Ta Jb = ε{Jb , Ja } = ε − , (15) ∂h ∂ν ∂ν ∂h where the factor Ta is due to the change of the time variable. Analogously, when the orbit is near the invariant manifold that corresponds to L b , the function Jb (H (t), εt) remains nearly constant, while the evolution of Ja is, in the first order, given by ∂ Ja ∂ Jb ∂ Ja ∂ Jb − . (16) Tb J˙a = ε{Ja , Jb } = ε ∂h ∂ν ∂ν ∂h As we see from (15) and (16), by virtue of the anti-symmetricity of the Poisson bracket, if {Ja , Jb } is not identically zero one can always choose between L a and L b in such a way that one of the functions Ja or Jb will be increasing, while the other is constant. Thus, for an orbit of (1) that stays near the invariant manifold corresponding to L a when {Jb , Ja } > 0 and near the invariant manifold corresponding to L b when {Jb , Ja } < 0, we will have the “total action” J := Ja + Jb steadily growing with time (in the first order of our approximations). Since ∂ J/∂h ≡ Ta + Tb is always positive, the growth of J allows h(t) ≡ H (x(t), εt) to grow (we will make this statement more precise below; see Propositions 1 and 2). It is remarkable that the above described itinerary of the switching between L a and L b coincides with that employed in Eq. (7), due to the following relation: {Ja , Jb } = Ta Tb (va − vb ),
(17)
which directly follows from Eqs. (13) and (6) and implies, obviously, that the Poisson bracket changes its sign at the same time as (va − vb ) does. Note that vc defined by (6) has a simple geometrical meaning: since (13) implies ∂ Jc ∂ Jc vc (h, ν) = − (h, ν) (h, ν), (18) ∂ν ∂h
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vc (h, ν) describes the angle between the direction of the Hamiltonian vector field of Jc at a point (h, ν) and the direction of the ν-axis. Now, Eq. (7) can be interpreted in the following way. On the plane (h, ν) there are two Hamiltonian vector fields generated by the Hamiltonian functions Ja and Jb . A solution of (7) follows the level lines of Ja and Jb : at every point there are two level lines and the solution chooses the one which leads to larger h in the immediate future. Because of the monotone dependence of Jc on h, such choice implies that both functions Jc (h, ν) are non-decreasing along the solutions of Eq. (7). Indeed, by (13) and (6), if h(ν) is a solution of (7), then 1 d Jc (h(ν), ν) = max{va , vb } − vc ≥ 0. Tc dν Now we are ready to formulate a general criterion for the unbounded growth of the solutions of (7). Note that since ∂ Jc /∂h = 0, every level line of Jc (h, ν) is a graph of a certain function h of ν. We will say that a certain level line of Ja is asymptotic to a level line of Jb if the difference in h between these two lines tends to zero as ν → +∞. Proposition 1. Let the actions Ja and Jb be defined at (h ≥ h ∗ , ν ≥ 0). Assume that limh→+∞ J (h, ν) = +∞ uniformly for all ν ≥ 0. Suppose also that the actions Ja,b (h, ν) ∂J remain bounded from above and the periods Ta,b (h, ν) ≡ ∂ha,b (h, ν) remain bounded away from zero on any bounded set of values of h, uniformly for all ν ≥ 0. Under these assumptions, if none of the level lines of Ja is asymptotic to a level line of Jb , then every solution of Eq. (7) that starts with a sufficiently large h 0 stays in the region h ≥ h ∗ and, if defined for all ν > 0, tends to infinity as ν → +∞. Proof. Let c = a or c = b. Since Jc (h, ν) uniformly tends to infinity as h → +∞, for every finite value of Jc the corresponding level line is defined for all ν and the corresponding values of h remain uniformly bounded. Let be a level line which corresponds to the value of Jc greater than supν≥0 J (h ∗ , ν), so this line stays entirely above h = h ∗ (recall that J (h, ν) is an increasing function of h for a fixed ν). Since Jc is non-decreasing along the orbits of Eq. (7), any solution of (7) that starts above at ν = 0 remains above it for all ν ≥ 0, i.e. it remains above h = h ∗ . Hence, unless it tends to ∞ at some finite ν, it is defined for all ν ≥ 0. If h(ν) is such a solution, then the monotonicity of Jc (h(ν), ν) implies that there exists limν→+∞ Jc (h(ν), ν), finite or infinite. Now suppose that h(ν) does not tend to infinity as ν → +∞. Then there exists at least a sequence of values of ν = νk → +∞ such that the corresponding values of h(νk ) remain all bounded from above by the same constant. By assumption, the values of, say, Ja (h(νk ), νk ) also remain uniformly bounded for all k, therefore J¯a := limν→+∞ Ja (h(ν), ν) is finite. Moreover, the line h = h(ν) stays entirely below the level line Ja (h, ν) = J¯a , hence h(ν) is uniformly bounded for all ν. Since both curves Ja (h, ν) = J¯a and h = h(ν) stay in the region of bounded h, the value of ∂ Ja /∂h = Ta remains bounded away from zero between these curves. Therefore, the fact that Ja (h(ν), ν) → J¯a as ν → +∞ implies that the line h = h(ν) tends to the level line Ja (h, ν) = J¯a . As the same arguments are equally applied to the action Jb , we find that by assuming that h(ν) does not tend to infinity we obtain the existence of two level lines, Ja (h, ν) = J¯a and Jb (h, ν) = J¯b , that are asymptotic to each other.
In the case of periodic or quasiperiodic dependence of H on εt the periodic orbits L c of the frozen system do not necessarily depend periodically, or quasiperiodically, on ν. However, let us assume that L a and L b are periodic or quasiperiodic functions of ν, or at
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least the corresponding actions Ja,b and, hence, the periods Ta,b are (this is always the case in many settings, e.g. for the classes of systems we consider in the Examples Section). Then all the uniformity assumptions of Proposition 1 are automatically fulfilled. Moreover, two level lines of (quasi)periodic functions may be asymptotic to each other only if these lines coincide (this is, of course, a very rare event). Thus, we arrive at the following result. Proposition 2. Let the actions Ja (h, ν) and Jb (h, ν) be defined at (h ≥ h ∗ , ν ≥ 0), and let them depend periodically or quasiperiodically on ν. Assume that lim h→+∞ J (h, ν) = +∞ uniformly for all ν ≥ 0. Then, if Ja and Jb do not have a common level line, then every solution of Eq. (7) that starts with a sufficiently large h 0 stays in the region h ≥ h ∗ and tends to infinity as ν → +∞. As we see, conditions of Theorem 1 are almost always fulfilled if the system under consideration is Hamiltonian. Thus, the phenomenon of an unbounded energy growth in slowly perturbed chaotic Hamiltonian systems has a universal nature, practically independent of a particular perturbation shape, or of the structure of the frozen system. It is caused by some basic properties of Hamiltonian dynamics, namely by the existence of adiabatic invariants for slowly perturbed one-degree-of-freedom systems and by the fact that the adiabatic invariant is the Hamiltonian of the corresponding averaged motion. In other words, this phenomenon is a direct consequence of the Hamiltonian structure of the problem. This approach naturally extends onto slow-fast Hamiltonian systems with several slow degrees of freedom [25]. 2. Description of a Horseshoe and Normally-Hyperbolic Invariant Manifolds Consider the frozen system x˙ = G(x, ν; ε),
(19)
and assume that a function H (x, ν; ε) is an integral of system (19), i.e. ∂H · G(x, ν) ≡ 0 ∂x
(20)
(we suppress, notationally, the dependence on ε in the frozen system from now on). Let system (19) have a pair of saddle periodic orbits L a : x = xa (t; h, ν) and L b : x = xb (t, h, ν) at all ν ≥ ν ∗ (for some ν ∗ < 0) in every energy level H = h ≥ h ∗ . Take a pair of small smooth cross-sections, a and b , to L a and L b respectively. As L a and L b depend smoothly on h and ν, the cross-sections a,b can also be taken to depend smoothly on h and ν. Denote the Poincaré map on c near L c as cc (c = a, b); the Poincaré map is smooth and depends smoothly on h and ν. We assume that the frozen system has, at all ν ≥ ν ∗ in every energy level H = h ≥ h ∗ , a pair of heteroclinic orbits: ab ⊆ W u (L a ) ∩ W s (L b ) and ba ⊆ W u (L b ) ∩ W s (L a ). Let ab and ba be maps on a and on b defined by the orbits close to ab and ba , respectively; ab acts from some open set in a into an open set in b , while ba acts from an open set in b into an open set in a . There is a certain freedom in the definition of the maps ab and ba : each of these maps acts from a neighbourhood of one point of a heteroclinic orbit to a neighbourhood of another point of the same orbit, and different choices of the pairs of points lead to different maps. When a definite choice of the maps is made (we will do it in a moment), we find for every orbit that lies entirely
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in a sufficiently small neighbourhood of the heteroclinic cycle L a ∪ L b ∪ ab ∪ ba a uniquely defined sequence of points Mi ∈ a ∪ b such that Mi+1 = ξi ξi+1 Mi , where ξi = c if Mi ∈ c (c = a, b). i=+∞ is called the code of the orbit. The sequence {ξi }i=−∞ The periodic orbits L a and L b are saddle, and the intersections of the stable and unstable manifolds of L a and L b that create the heteroclinic orbits are transverse, by the assumption of the theorem. This implies (cf. [1]) that one can choose the maps ab and ba and define coordinates (u, w) in a and b in such a way that the following holds:
• In the given coordinates, c = Uc × Wc , where Ua,b and Wa,b are certain balls in Rm−1 (we assume that the dimension of the x-space equals 2m); so we may choose some constant R such that max diam Ua , diam Ub , diam Wa , diam Wb ≤ R. (21) • For each pair c and c the Poincaré map cc can be written in the “cross-form” [22]; namely, there exist smooth functions f cc , gcc : Uc × Wc → Uc × Wc such that a ¯ u, point M(u, w) ∈ c is mapped into M( ¯ w) ¯ ∈ c by the map cc if and only if ¯ u¯ = f cc (u, w),
w = gcc (u, w). ¯
• There exists λ < 1 such that ∂( f σ σ , gσ σ ) ≤λ<1 ∂(u, w) ¯
(22)
(23)
(where we define the norm in U × W as max{u, w}). Inequality (23) means that the set hν of all the orbits that lie entirely in a sufficiently small neighbourhood of the heteroclinic cycle L a ∪ L b ∪ ab ∪ ba in the energy level H = h at the given value of ν is hyperbolic, a horseshoe. Thus, one can show that hν is in one-to-one correspondence with the set of all sequences of a’s and b’s, i.e. for every i=+∞ there exists one and only one orbit in which has this sequence sequence {ξi }i=−∞ hν i=+∞ if and only if the as its code. Indeed, by (22), an orbit from hν has code {ξi }i=−∞ intersection points Mi (u i , wi ) of the orbit with the cross-section satisfy u i+1 = f ξi ξi+1 (u i , wi+1 ),
wi = gξi ξi+1 (u i , wi+1 ),
+∞ is a fixed point of the operator i.e. the sequence { (u i , wi ) }i=−∞ +∞ +∞ { (u i , wi ) }i=−∞ → { ( f ξi−1 ξi (u i−1 , wi ), gξi ξi+1 (u i , wi+1 ) }i=−∞ .
+∞ By (23), this operator is a contracting map of the space i=−∞ Uξi × Wξi , hence the i=+∞ existence and uniqueness of the orbit with the code {ξi }i=−∞ follows (see e.g. [20]). Moreover, as the fixed point of a smooth contracting map depends smoothly on parameters, the orbit depends smoothly on h and ν, so the derivatives of (u i (h, ν, ξ ), wi (h, ν, ξ )) with respect to (h, ν) are bounded uniformly for all i and ξ .
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It also follows from (21),(23) that (u i (h, ν, ξ (1) ) − u i (h, ν, ξ (2) ), wi (h, ν, ξ (1) ) − wi (h, ν, ξ (2) )) ≤ Rλn−|i| (24) +∞ , ξ (2) = {ξ (2) }+∞ for any two code sequences ξ (1) = {ξi(1) }i=−∞ i i=−∞ which coincide at (1)
(2)
|i| ≤ n (i.e. ξi = ξi at |i| ≤ n); the constants R > 0 and λ ∈ (0, 1) are given by (21) and (23) and are independent of ξ (1,2) . Let us now switch to the system with a slowly changing parameter ν = εt. This means that we augment system (19) by the equation ν˙ = ε,
(25)
while (19) remains unchanged. Although relation (20) still holds true, the conservation of energy no longer follows: indeed, by (19),(20),(25), d ∂H H (x(t), ν(t)) = ε (x(t), ν(t)). dt ∂ν
(26)
By continuity, for system (19),(25) the Poincaré maps cc : ∪h,ν c → ∪h,ν c are still defined at small ε. Denoting z = (h, ν), for any compact set of z values we may write the maps in the following form: ⎧ ¯ z, ε), w = gcc (u, w, ¯ z, ε) ⎨ u¯ = f cc (u, w, (27) ⎩ z¯ = z + εφ (u, w, ¯ z, ε), cc where f, g, φ are bounded along with the first derivatives and f, g satisfy (23). Clearly, any smooth transformation of the z-variables will not change the form of map (27). As the set of values of ν and h under consideration is not compact (we are interested in the behaviour of the system for ν and h tending to infinity), we need certain uniformity assumptions. We require the following: [UA1] For all h ≥ h ∗ and ν ≥ ν ∗ , one can introduce coordinates (u, w) on a and
b and define z = (α(h, ε), ν) with a smooth function α such that α (h) > 0, in such a way that for all small ε: (i) formula (27) holds for the Poincaré maps cc , and the functions f, g, φ along with the first derivatives are uniformly bounded and uniformly continuous with respect to ε, for all h ≥ h ∗ and ν ≥ ν ∗ ; (ii) estimate (23) holds with the constant λ < 1 the same for all h ≥ h ∗ , ν ≥ ν ∗ and all small ε; (iii) the diameter of the balls Uc and Wc is uniformly bounded, i.e. (21) holds with the constant R the same for all h ≥ h ∗ , ν ≥ ν ∗ and all small ε. It also does no harm to assume that φ ≡ 0 if h = h ∗ or ν = ν ∗ , i.e. the region {h ≥ h ∗ , ν ≥ ν ∗ } is invariant with respect to the Poincaré map. If this is not the case, then we can modify φ in a small neighbourhood of h = h ∗ and in a small neighbourhood of ν = ν ∗ : as we are interested in the orbits for which h → +∞, they will never enter the region of h close to h ∗ ; and ν = εt is a growing function of t anyway. Now we are ready to formulate the main technical result beneath Theorems 1–3. It has a general nature and has little to do with the Hamiltonian structure of the equations. Rather we notice that by fixing any code ξ and varying h and ν we obtain at ε = 0 a sequence of smooth two-dimensional surfaces, the i th surface is the set run,
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as h and ν change, by the point Mi of the uniquely defined orbit with the code ξ ; this sequence is invariant with respect to the corresponding Poincaré maps and is uniformly normally-hyperbolic — hence it persists at all ε sufficiently small. Lemma 1. Given any sequence ξ of a’s and b’s, there exists a uniquely defined sequence of smooth surfaces Li (ξ, ε) : (u, w) = (u i (z, ξ, ε), wi (z, ξ, ε))
(28)
ξi ξi+1 Li = Li+1 .
(29)
such that The functions (u i , wi ) are defined for all small ε and all h ≥ h ∗ , ν ≥ ν ∗ , they are uniformly bounded along with their derivatives with respect to z and satisfy (24). Proof. Take a sufficiently large K and consider any sequence of surfaces of form (28) with ∂(u i , wi ) ≤K (30) ∂z (we further suppress notationally the dependence of u i and wi of ξ and ε). Define functions ηi (z) and η¯ i (z) by the relations z = ηi + εφξi ξi+1 (u i (ηi ), wi+1 (z), ηi , ε),
(31)
η¯ i = z + εφξi ξi+1 (u i (z), wi+1 (η¯ i ), z, ε).
(32)
and As all the derivatives of φ, u i and wi+1 are uniformly bounded, Eqs. (31) and (32) define the functions ηi (z) and η¯ i (z) uniquely. By (27), the sequence of surfaces will satisfy (29) if and only if the sequence of functions {u i (z), wi (z)} is a fixed point of the operator F : {u i (z), wi (z)} → {u˜ i (z), w˜ i (z)} defined by u˜ i+1 (z) = f ξi ξi+1 (u i (ηi (z)), wi+1 (z), ηi (z), ε), w˜ i (z) = gξi ξi+1 (u i (z), wi+1 (η¯ i (z)), z, ε).
(33)
+∞ satisfying (30). Let K be the space of sequences of functions ψ = {u i (z), wi (z)}i=−∞ Endow K with the norm
ψ = sup max{u i (z), wi (z)}.
(34)
i,z
It is easy to see that F( K ) ⊂ K provided K is large enough, and that F is contracting on K , for all small ε. Indeed, let us check this claim at ε = 0. In this case we have ηi ≡ η¯ i ≡ z (see (31),(32)). Therefore, ∂ u˜ i+1 ∂ f ∂(u i , wi+1 ) ∂ f = + ∂z ∂(u, w) ∂z ∂z ∂g ∂(u i , wi+1 ) ∂g ∂ w˜ i = + , ∂z ∂(u, w) ∂z ∂z
Unbounded Energy Growth in Hamiltonian Systems
which gives
781
∂(u˜ i+1 , w˜ i ) ≤ λ ∂(u i , wi+1 ) + sup ∂( f, g) ∂z ∂z ∂z
(see (23)). Thus, for any K >
(35)
∂( f, g) 1 sup ∂z , 1−λ
we have F( K ) ⊂ K indeed. To prove the contractivity of F at ε = 0 just note that it follows immediately from (33),(23) that (1) (2) (1) (2) (u˜ i+1 − u˜ i+1 , w˜ i(1) − w˜ i(2) ) ≤ λ(u i(1) − u˜ i(2) , wi+1 − wi+1 ).
At ε = 0 inequalities (35),(36) change to ∂(u˜ i+1 , w˜ i ) ≤ (λ + O(ε)) ∂(u i , wi+1 ) + sup ∂( f, g) + O(ε) ∂z ∂z ∂z
(36)
(37)
and (1)
(2)
(1)
(u˜ i+1 − u˜ i+1 , w˜ i
(2)
(1)
− w˜ i ) ≤ (λ + O(ε)) (u i
(2)
(1)
(2)
− u˜ i , wi+1 − wi+1 ).
(38)
Hence, at all small ε the operator F remains a contracting map K → K . Thus, it has a fixed point in the closure of K in the norm (34). This gives us the existence of the invariant sequence of Lipshitz continuous invariant surfaces — the smoothness is standard (see e.g. Theorem 4.4 of [22]). Finally, the estimate (24) follows immediately from (38) and (21).
i=+∞ According to this lemma, for all sufficiently small ε, for every code ξ = {ξi }i=−∞ system (19),(25) in the space of (x, ν) has a smooth three-dimensional invariant manifold Mξ that corresponds to this code, i.e. the manifold depends continuously on ε and, at ε = 0, it is the union, over all h ≥ h ∗ , ν ≥ ν ∗ of the orbits with the code ξ (recall that for each h, ν we have exactly one such orbit). The intersection of Mξ with the +∞ . Thus, cross-section ∪h,ν ( a ∪ b ) is exactly the sequence of surfaces {Li (ξ, ε)}i=−∞ dynamics on Mξ is described by the Poincaré map on the cross-section. The Poincaré map is obtained by plugging u = u i (z, ε; ξ ), w = wi (z, ε; ξ ) into (27). Namely, z i is the sequence of the points of intersection with the cross-section of an orbit on the invariant manifold Mξ if and only if
z i+1 = z i + εφξi ξi+1 (u i (z i , ε; ξ ), wi+1 (z i+1 , ε; ξ ), z i , ε).
(39)
Recall that in our notations z is a vector of two components: y := α(h, ε) and ν. So we will write yi+1 = yi + εθξi ξi+1 (u i (z i , ε; ξ ), wi+1 (z i+1 , ε; ξ ), z i , ε), νi+1 = νi + ετξi ξi+1 (u i (z i , ε; ξ ), wi+1 (z i+1 , ε; ξ ), z i , ε),
(40)
i.e., θ and τ denote the two components of the function φ in (39). Note that for the codes ξ = a ω (i.e. ξi = a for all i) and ξ = bω we have u i+1 ≡ u i and wi+1 ≡ wi for all i. We denote u c (z, ε) := u i (z, ε; cω ) and wc (z, ε) := wi (z, ε; cω ) (where c = a or b). By construction, the manifold (u, w) = (u c (z, 0), wc (z, 0)) is
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the set of the intersection points of the periodic orbit L c : {x = xc (t; h, ν)} of the frozen system with the cross-section (we have one intersection point for every value of z = (α(h, ε), ν)). By (25), the function τ in (40) is just the time of one return onto the cross-section. Therefore, for the orbits on the manifold Mcω we have τcc |ε=0 = Tc (h, ν),
(41)
where Tc is the period of L c . Analogously, for the function θcc in (40) we have yi+1 − yi 1 τ = lim y˙ dt, θcc |ε=0 = lim ε→0 ε→0 ε 0 ε where y˙ is the time derivative of y by virtue of system (19),(25). As y = α(h), we find from (26),(25) that y˙ = εα (H )Hν , so
θcc |ε=0 = α (h)
0
Tc
∂H (x, ν) dt ∂ν x=xc (t;h,ν)
(42)
(43)
for the orbits on the manifold Mcω . 3. Proof of Theorem 3 Before proceeding to the proof we will formulate the second uniformity assumption. It is automatically satisfied for any compact set of values of z = (y, ν), hence for any compact set of values of h and ν. Denote ∂ , ρ (z, ε) = sup θ |α (H )Hν | τcc (u, w, ζ, ε) + (u, w, ζ, ε) cc ∂(u, w) u,w,c,c ,ζ (44) ∂ Tρ (z, ε) = sup τcc (u, w, ζ, ε) + ∂(u, w) τcc (u, w, ζ, ε) , u,w,c,c ,ζ where ζ runs centered at z ball of some small radius ρ – we take ρ as small as we want, but independent of ε. The supremum of |α (H )Hν | is taken over the piece of the orbit that starts on the cross-section at the point (u, w, ζ ) and continues until the next hit with the cross-section (i.e. τcc (u, w, ζ, ε) gives the length of the corresponding time-interval). Thus, by (42), ερ estimates the maximal change in y between the two intersections with the cross-section. In particular, ρ (z, ε) ≥ |θcc (u, w, ζ, ε)|. The function Tρ , obviously, estimates the return times to the cross-section. Recall that Tc (h, ν) denotes the period of the saddle periodic orbit L c (c = a, b) of the frozen system (19) for a given value of z = (α(h, ε), ν). Assume that the following holds.
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[UA2] (i) There exist a constant C > 0 and a small ρ > 0 such that max{ρ (α(h), ν, ε), Tρ (α(h), ν, ε)} ≤ C min{Ta (h, ν), Tb (h, ν)} (45) for all h ≥ h ∗ and ν ≥ ν ∗ , and for all small ε. (ii) The functions 1 φcc (u c (z, ε), wc (z, ε), z, ε) (c = a, b) Tc (h, ν) are uniformly continuous with respect to ε and z = (y, ν) for all h = α −1 (y) ≥ h ∗ and ν ≥ ν ∗ . Denote as β(h, ν, ε) any smooth function such that β(h, ν, ε) ≥
ρ (α(h), ν, ε) . α (h) min{Ta (h, ν), Tb (h, ν)}
(46)
By construction, /Ta,b estimates the velocity of the change of y = α(h), therefore the function β estimates the velocity of the change of h. Indeed, in the examples we consider in Sect. 4, it is of the same order as va,b . Now we can prove Theorem 3. Take any independent of ε, arbitrarily large N and consider any code ξ such that for some i we have ξi = ξi+1 = . . . = ξi+N −1 = c (where c = a or b). By (24), (u i+ j (z, ε, ξ ) − u c (z, ε), wi+ j (z, ε, ξ ) − wc (z, ε, ξ )) ≤ Rλmin{ j,N −1− j} . (47) Take any orbit on the invariant manifold Mξ that corresponds to this code ξ , and let z i be the sequence of intersection points of the orbit with the cross-section ∪h,ν ( a ∪ b ). It follows from (47) and from the uniform ε-closeness of the right-hand side of (39) to identity that for all sufficiently small ε z i+N − z i − εN φcc (u c (z i , ε), wc (z i , ε), z i , ε) ≤ K (ρ (z i , ε) + Tρ (z i , ε)) (ε + (εN )2 ),
(48)
where K is a constant. By (48),(41) and the uniformity assumption, lim (νi+N − νi )/ε = N Tc (h i , νi ) + O(1),
ε→0
(49)
uniformly for all z. It also follows immediately that uniformly for all z, lim
ε→0
yi+N − yi = vˆc (z i ) + O(N −1 β(h i , νi )α (h i )), νi+N − νi
(50)
where we denote vˆc (z) = (see (40),(43),(45),(46),(6)).
θcc (z, cω , 0) ≡ α (h)vc (h, ν) Tc (h, ν)
(51)
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Thus, for every δ > 0 there exists N such that for all sufficiently small ε and for every code ξ such that [ξi ξi+1 . . . ξi+N −1 ] = c N the change in y after N iterations of the Poincaré map is greater than the change in the solution y of the equation dy = vˆc (y, ν) − δβα dν
(52)
on the interval ν ∈ [νi , νi+N ], with the initial conditions y(νi ) = yi . Indeed, as the function vˆc is uniformly bounded and uniformly continuous (according to [UA2]), we have for the solution y(ν) of (52), lim
ε→0
y(νi+N ) − y(νi ) = vˆc (z i ) − δβα , νi+N − νi
(53)
uniformly for all z, so the claim follows from the comparison of (53) with (50). It follows immediately that if N is taken sufficiently large, then for all sufficiently small ε, given any code sequence ξ built of length N blocks of equal symbols, i.e. ξ j N = ξ j N +1 = . . . = ξ j N +N −1 = c j , where {c j }+∞ j=−∞ is an arbitrary sequence of a’s and b’s, for every orbit in the invariant manifold Mξ , y j N > y(ν j N ) for all j > 0,
(54)
+∞ where {(yi , νi )}i=−∞ is the sequence of the intersection points of with the cross-section, and y(ν) is the solution of the equation
1 dy = vˆc j (y, ν) − δβ(h, ν)α (h) at ν ∈ [ν j N , ν( j+1)N ] dν 2
(55)
with the initial condition y(ν0 ) = y0 . Let us now construct a particular code sequence ξ ∗ by the following rule. Fix some (y0 , ν0 ) such that y0 > α(h ∗ ), ν0 > ν ∗ . At i < 0 we put ξi∗ = a. At i ≥ 0 we put ξ ∗j N = ξ ∗j N +1 = . . . = ξ ∗j N +N −1 = c∗j , where the symbols c∗j are defined inductively, ∗( j)
= ξi∗ at i < j N and as follows. Denote as ξ ∗( j) the code sequence such that ξi ∗( j) ξi = a at i ≥ j N . Denote as M j the invariant manifold with the code ξ ∗( j) . Let ∗( j) ∗( j) ∗( j) ∗( j) ∗( j) be the orbit on M j with the initial conditions z 0 = (y0 , ν0 ). Let z i = (yi , νi ) denote the i th point of intersection of the orbit ∗( j) with the cross-section. Define ∗( j) ∗( j) a if vˆa (z j N ) > vˆb (z j N ), ∗ (56) cj = ∗( j) ∗( j) b if vˆa (z j N ) ≤ vˆb (z j N ). By construction, the value of c∗j is completely determined by the segment of ξ ∗ with i < j N , so we indeed can inductively define ξ ∗ in this way. Let ∗ be the orbit on the manifold Mξ ∗ with the same initial values of (y0 , ν0 ), as we have chosen for the orbits ∗( j) , and let z i∗ be the points of the intersection of ∗ with the cross-section. As the code ξ ∗ coincides with the code ξ ∗( j) for all i < j N , it follows from (24) that (u i (z, ξ ∗ ) − u i (z, ξ ∗( j) ), wi (z, ξ ∗ ) − wi (z, ξ ∗( j) ) ≤ Rλ j N −i . Plugging this into (39) gives, uniformly for all j, ∗( j)
z ∗j N − z j N = O(ε)
Unbounded Energy Growth in Hamiltonian Systems
785
∗( j)
(we use here that z 0∗ − z 0 = 0). Therefore, it follows from the uniform continuity of v(z) ˆ that vˆc∗j (z ∗( j) ) is uniformly close to max{ vˆa (z ∗( j) ), vˆb (z ∗( j) ) } (see (56)). This implies (see (54),(55)) that y ∗j N > y(ν ∗j N ) for all j > 0,
(57)
where y(ν) denotes here the solution of the equation dy = max{vˆa (y, ν), vˆb (y, ν)} − δβ(h, ν)α (h) dν
(58)
with the initial condition y(ν0 ) = y0 . As the change in y between N intersections with the cross-section is O(εN ρ ), i.e. it is uniformly small in comparison with βα min{Ta , Tb } (see (46)), we find from (57) that for every point on the orbit ∗ the value of y is larger than the value of y for the solution of (58) at the same value of ν. Now recall that y = α(h) with an increasing function α. Thus, it follows from (58) that for every point on the orbit ∗ the value of h is larger than the value of h for the solution of dh 1 = max{vˆa (y, ν), vˆb (y, ν)} − δβ(h, ν, ε) dν α (h) at the same value of ν, which completes the proof of Theorem 3 (see (51)). 4. Examples 4.1. Non-autonomous perturbation of a geodesic flow. We begin with the Mather problem: a geodesic flow on an m-dimensional manifold (m ≥ 2), with the Hamiltonian Hg , subject to a non-autonomous perturbation V (q, t). Here q denotes position in the configuration space, i.e. V does not depend on momenta. Assume the uniform hyperbolicity for the geodesic flow (i.e. strictly negative curvature; recall that the uniform hyperbolicity implies that periodic trajectories are dense in the phase space of the geodesic flow [2]), and assume uniform boundedness and continuity for V and its first and second derivatives. The trajectories of the unperturbed geodesic flow are the same in every energy level, just the velocity of motion grows as the square root of the energy. Namely, the flow does not change with the following scaling of time, energy, and momenta: √ √ t → t/ s, H → H s, p → p s . (59) At the same time, this transformation changes the perturbation. If we let s = ε−2 , then the perturbation term V (q, t) is replaced by ε2 V (q, εt). Therefore at large energies, adding V (q, t) to Hg is, effectively, a small and slow perturbation of the geodesic flow. Thus, this example belongs to the class of systems (1). Theorem 4. Let L a : q = qa (t) and L b : q = qb (t) be two periodic trajectories of the geodesic flow H = Hg in the energy level Hg = 1. Denote 1 V¯c (ν) = Tc
Tc 0
V (qc (t), ν)dt,
(60)
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where Tc (c = a, b) is the period of L c . Assume 1 s d ¯ ¯ lim inf (Va (ν) − Vb (ν)) dν > 0. s→+∞ s 0 dν
(61)
Then the Hamiltonian system H = Hg + V (q, t)
(62)
has orbits for which H linearly grows to infinity. Proof. After the scaling transformation (59) with s = ε−2 the Hamiltonian function recasts as Hˆ = Hg ( p, q) + ε2 V (q, εt) ,
(63)
which has the form (1). We check that the assumptions of Theorem 3 are all satisfied. First we note that if ε is sufficiently small, the frozen system Hˆ = Hg + ε2 V (q, ν) has a heteroclinic cycle close to the heteroclinic cycle of the geodesic flow in every energy level Hˆ = h ≥ 1. Indeed, after the scaling transformation (59) with s = h, the system in the energy level Hˆ = h is transformed into ε2 H˜ = Hg + V (q, ν) h in the energy level H˜ = 1, i.e. it is uniformly close to the geodesic flow in the level Hg = 1. Therefore the heteroclinic cycle of the frozen system exists for all h ≥ 1. It is useful to note that according to (59) the periods of the periodic orbits L a (h, ν) and L b (h, ν) in the heteroclinic cycle behave as (g)
Ta,b (h, ν) = h −1/2 (Ta,b + O(ε2 )),
(64)
(g)
where Ta,b denotes here the ν-independent period of the corresponding orbit of the geodesic flow in the level Hg = 1. In order to apply Theorem 2 we have to check that Poincaré maps near the heteroclinic cycle satisfy the uniformity assumptions [UA1] and [UA2]. Let (u 0 , w0 ) be the coordinates for which the Poincaré map for the geodesic flow in the level Hg = 1 has the form which satisfies (22) and (23). Then we define coordinates (u, w) on the cross-section in the following way: ˆ (u, w) = (u 0 (q, p/ H ( p, q, ν)), w0 (q, p/ Hˆ ( p, q, ν))) (65) and z = (h, ν) with h = Hˆ ( p, q, ν). Now we need uniform estimates for the Poincaré map represented in these coordinates. Let us take any sufficiently large s and consider the part of the phase space that corresponds to s ≤ Hˆ ( p, q, ν) ≤ 2s.
(66)
Unbounded Energy Growth in Hamiltonian Systems
787
√ 2 The scaling (59) transforms the system to H˜ ≡ Hg + εs V (q, εt/ s) in the energy levels 1 ≤ H˜ ≤ 2. The scaled system is uniformly O(ε2 /s)-close to the geodesic flow in these energy levels. Therefore, in the coordinates √ √ (u, w) = (u 0 (q, p/ s), w0 (q, p/ s)), h = Hˆ ( p, q, ν), ν = εt, (67) the following formulas hold for the Poincaré map: u¯ = f cc (u, w) ¯ + O(ε2 / h), h¯ = h + O(ε3 / h 1/2 ),
w = gcc (u, w) ¯ + O(ε2 / h), ν¯ = ν + O(ε/ h 1/2 ) ,
(68)
uniformly for all h ≥ 1 and ν; the functions f, g in (68) define the Poincaré map for the geodesic flow. The equations for h¯ and ν¯ are obtained immediately from the fact that the time of return to the cross-section behaves as O(h −1/2 ) (see e.g. (64)), while the time derivative of Hˆ along the orbit is given by ε3 ∂ V /∂ν, i.e it is uniformly O(ε3 ). As s/ Hˆ is uniformly bounded and separated from zero, it is easy to check that the Poincaré map written in the coordinates (u, w) defined by (65) also has the form (68). Recall that (u, w) run over balls of finite radii by construction, so the validity of the uniformity assumptions (with α(h, ε) ≡ h) follows immediately from (68) and (64). From the last line of (68) we see that the function β defined by (46) is uniformly O(ε2 ). Now, according to Theorem 3, it remains to check that for some sufficiently small δ > 0 solutions of the equation h (ν) = max{va (h, ν), vb (h, ν)} − ε2 δ tend to infinity asymptotically linearly with time. Recall that vc is the average change in H along the periodic solution of the frozen system (6). As the frozen system is O(ε2 )-close to the geodesic flow in our case, we find that d ¯ Vc (ν) + O(ε4 ) dν (see (60),(64)). For small ε the O(ε4 )-term is absorbed by ε2 δ, so after scaling h we are left to examine the behaviour of solutions of h (ν) = max V¯a (ν), V¯b (ν) − δ. vc (h, ν) = ε2
By taking the integral of both parts we find ν d 1 ¯ ¯a − V¯b dν − 2δ ν , h(ν) − h 0 = V Va (ν) + V¯b (ν) − V¯a (0) − V¯b (0) + dν 2 0 i.e. condition (61) ensures the existence of linearly tending to +∞ solutions indeed.
Note that in (61) we take an integral of a non-negative function, therefore (61) is not very restrictive. For example, if V (q, t) is periodic or quasi-periodic in time, condition (61) is equivalent to V¯a (ν) − V¯b (ν) = const. (69) Thus, for a (quasi)periodic in t potential V (q, t) the only case where there may be no trajectories of unbounded energy is that when the average (60) of V (q(t), ν) is the same (up to a constant) function of ν for every periodic trajectory of the geodesic flow. Note also that we do not, in fact, need the hyperbolicity of the flow in the whole phase space. It is sufficient to have a locally-maximal, uniformly-hyperbolic, transitive, compact, invariant set in the energy level Hg = 1. Theorem 4 then holds true, provided the periodic orbits L a and L b belong to .
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4.2. Exponential energy growth. In the previous example we found trajectories with the energy growth which is asymptotically linear in time. Such estimate is essentially optimal in that case: because of the uniform boundedness of the time derivative of the perturbation V (q, t) there can be no trajectories with energy growing faster than linearly. In this section we describe a different class of perturbations for geodesic flows which have trajectories whose energy tends to infinity at a much faster rate. These are obtained by a “parametric” perturbation of the geodesic flow. Namely, consider the following Hamiltonian: 1 H = p g(q, t) p, (70) 2 where g −1 is the metric tensor. We assume that the corresponding curvature remains strictly negative for all t. We also assume the uniform boundedness and continuity of g with all the derivatives up to the second order. The scaling (59) with s = ε−2 changes the Hamiltonian to 1 H˜ = p g(q, εt) p . 2 We see that at large energies the original system belongs to the class of Hamiltonian systems with slowly varying parameter. By the assumed strict negativity of the curvature at every fixed t, the frozen system is hyperbolic in every energy level except for H = 0. Therefore, we may take a pair of saddle periodic orbits L a (h, ν) and L b (h, ν) and a heteroclinic cycle that contains them. As in the case of Theorem 4, the period of L a,b behaves as ∼ εh −1/2 , while the change of h during one period is given by √ h ∼ ε h . Indeed, by the scaling invariance of the frozen system, we find that Tc (h,ν) ∂H ( p, q, ν) dt ∂ν 0 ( p,q)=( pc ,qc )(t;h,ν) Tc (1,ν) 1√ = h pc (t; 1, ν)gν (qc (t; 1, ν), ν) pc (t; 1, ν)dt, 2 0 where ( p, q) = ( pc , qc )(t; h, ν) is the equation of the periodic orbit L c (h, ν). Like we did it in the proof of Theorem 4, by using the fact that the scaling (59) can map a neighbourhood of an arbitrarily high energy level into a neighbourhood of the energy level Hˆ = 1, we find that the Poincaré map has the following form (uniformly for all h and ν): √ √ u¯ = f cc (u, w, ¯√ ν) + O(ε/ h), w√= gcc (u, w, ¯ ν) + O(ε/ h), (71) h¯ = h + O(ε h), ν¯ = ν + O(ε/ h), where the functions f, g define the Poincaré map for the frozen geodesic flow in the energy level H = 1. It follows immediately from (71),(64) that the uniformity assumptions are fulfilled with α(h) = ln h, and β(h) = O(h). Thus, by Theorem 3, there exist orbits bounded from below by a solution of the equation (72) h (ν) = h(ν) max{vˆa (ν), vˆb (ν)} − δ ,
Unbounded Energy Growth in Hamiltonian Systems
where 1 vˆc (ν) = 2Tc (1, ν)
Tc (1,ν) 0
789
pc (t; 1, ν)gν (qc (t; 1, ν), ν) pc (t; 1, ν)dt.
(73)
Let us estimate the solutions of (72). Denote Tˆc (ν) ≡ Tc (1, ν) (i.e. this is the period of the orbit L c of the frozen system in the energy level H = 1). By the invariance of the frozen system 1 p g(q, ν) p 2 with respect to energy scaling, we find that √ Tc (h, ν) = Tˆc (ν)/ h. H=
(74)
Let us introduce the action variable (see Sect. 1.2) Tc (h,ν) Jc (h, ν) = pc (t; h, ν)q˙c (t; h, ν)dt. 0
As q˙ = g(q, ν) p in system (74), this gives us the following explicit formula for the action: Tc (h,ν) Jc (h, ν) = pc (t; h, ν) g(qc (t; h, ν), ν) pc (t; h, ν)dt 0 (75) √ = 2hTc (h, ν) = 2 h Tˆc (ν). Now recall that by general formula (13), ∂ ˆ (76) Jc (1, ν) = −vˆc (ν)Tˆc (ν) . ∂ν Equations (75) with h = 1 and (76) imply that in the case of Hamiltonian (70) there is a closed formula which expresses vˆc (ν) in terms of Tˆc (ν): vˆc (ν) = −2
d ln Tˆc (ν) . dν
(77)
Plugging this in (72) we find
d d d ˆ ˆ ˆ ˆ ln h(ν) = − (ln Ta (ν) + ln Tb (ν)) + (ln Ta (ν) − ln Tb (ν)) − δ, dν dν dν
which gives us ln h(ν) − ln h 0 = ln Tˆa (0)Tˆb (0) − ln Tˆa (ν)Tˆb (ν) +
ν 0
d ˆa (ν)/Tˆb (ν)) dν − δν. ln( T dν
As we see, solutions of (72) tend exponentially to infinity for all sufficiently small δ, provided 1 s d ˆ ˆ lim inf (78) ln(Ta (ν)/Tb (ν)) dν > 0, s→+∞ s 0 dν and the functions Tˆa,b are bounded away from zero and infinity uniformly for all ν ≥ 0. Thus, we arrive at the following result.
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Theorem 5. If the periods Tˆa (ν) and Tˆb (ν) of the periodic orbits L a,b of the frozen system in the energy level H = 1 are bounded away from zero and infinity for all ν ≥ 0, and if condition (78) is satisfied, then Hamiltonian system (70) has orbits for which H exponentially grows to infinity. Note that similar to the previous section, if g(q, t) is periodic or quasi-periodic in time, condition (78) is equivalent to Tˆa (ν)/Tˆb (ν) = const.
(79)
4.3. Time-dependent billiard-like potentials. Another example: Let D ∈ Rm , m ≥ 2, be a bounded region whose boundary is composed of a finite number of strictly concave (when looking from inside of D) smooth (m −1)-dimensional manifolds which intersect by non-zero angles. Let V0 (q) (q ∈ D) be a positive function such that V0 (q) → +∞ as q → ∂ D. Consider the Hamiltonian system m pi2 H= + V0 (q) + V1 (q, t), 2
(80)
i=1
where V1 (q, ν) is uniformly bounded, along with the first and second derivatives, for all q ∈ D and ν ∈ R. By scaling time, energy and momenta by the rule (59) with s = ε−2 , this system transforms into H=
m pi2 + ε2 V0 (q) + ε2 V1 (q, εt). 2
(81)
i=1
Thus, at large energies, it is a slow perturbation of the singular Hamiltonian m pi2 0 at q ∈ D, H = Hb = + +∞ at q ∈ D. 2 i=1
This defines a billiard in D: inertial motion inside D and reflection at the boundary. In order to ensure the actual closeness of (81) to a billiard with the standard reflection law (“the angle of reflection equals the angle of incidence”) we need certain assumptions (see [17,24]). Namely, let S ⊂ ∂ D be the set of “corner” points, i.e. those where ∂ D is not smooth (these are the points where the smooth boundary components intersect). Assume that there exists an open neighbourhood U of ∂ D\S such that V0 (q) = W (Q(q))
(82)
for all q ∈ U . Here Q(q) is the so-called pattern function: it is at least a C 2 -smooth function defined for all q ∈ U , its first derivative ∂ Q/∂q does not vanish in U , and the smooth boundary components of D are given by the equation Q = 0. The function Q thus defines the shape of the billiard region D. The function W defines the growth of the potential as the boundary is approached. We assume that W = 0 for all small Q and that its inverse function W −1 satisfies W −1 (hV )−→ 0 as h → +∞, C2
(83)
Unbounded Energy Growth in Hamiltonian Systems
791
on any interval V ≥ C > 0. Roughly speaking, by representing the potential in form (82) we achieve that its gradient (“the reaction force”) is, in the limit h = +∞, normal to the billiard boundary, which is an obvious necessary condition for the validity of the standard reflection law; condition (83) ensures the C 1 -closeness of Poincaré map for the smooth flow (81) at large h to the Poincaré map for the billiard flow, outside the set of singular trajectories, i.e. those which hit S or which are tangent to a smooth component of the billiard boundary at some point. As the boundary components are strictly concave, the billiard in D is dispersing. This implies [18] the hyperbolicity of the billiard flow (outside the set of singular trajectories); moreover, periodic orbits are dense in the phase space [5,11]. We call a billiard orbit regular, if it stays bounded away from the singularities, i.e. from the set of points in the phase space which correspond to corner or to a tangency to the billiard boundary. Theorem 6. Let L a : q = qa (t) and L b : q = qb (t) be two regular periodic trajectories of the billiard in D, corresponding to kinetic energy equal to 1. Denote Tc 1 ¯ V1 (qc (t), ν)dt, (84) Vc (ν) = Tc 0 where Tc (c = a, b) is the period of L c . Assume that condition (61) (or condition (69) in the case of periodic or quasiperiodic dependence of V of t) is fulfilled. Then system (80) has orbits for which H linearly grows to infinity. Proof. It follows from [18,19] that for any two regular periodic orbits L a and L b in the strictly dispersing billiard there exists a pair of transverse heteroclinic orbits ab and ba , which are also regular. Take a sufficiently small neighbourhood of L a ∪ L b ∪ ab ∪ ba in the intersection of the phase space with the level Hb = 1. The hyperbolic set of the orbits that stay in this neighbourhood consists of regular orbits only; as a whole, stays bounded away from the singularity. One can take two small cross-sections,
a and b , to the orbits L a and L b in the intersection of the phase space with the level Hb = 1, such that every orbit of returns to a ∪ b at a finite time; the corresponding Poincaré maps are smooth, as the orbits of undergo only regular collisions with the billiard boundary. We have the same picture in every other energy level because of the invariance of the billiard flow with respect to energy scaling. According to [17,24], under conditions (82),(83), at small ε a finite-time flow map of the smooth system (80) near a regular orbit of the billiard flow is close, along with the first derivatives, to the corresponding map for the billiard flow (while only an autonomous case was considered in the quoted papers, the results and proofs do not change for our case where a bounded and slow non-autonomous term ε2 V1 is added to the Hamiltonian). Therefore, for any compact interval of the energy values, the Poincaré maps defined on the cross-section a ∪ b by the smooth system (80) is close to the Poincaré map of the billiard flow. Now, applying the scaling transformation (59) exactly as we did it in the proof of Theorem 4, we find that the uniformity assumptions [UA1] and [UA2] are fulfilled with α(h) = h and the Poincaré map can be written in the form (68). The equation for h¯ in the last line of (68) is found from the relation ∂ ¯h = h + ε3 (85) V1 (q(t), εt)dt ∂ν (see (81)) where the integral is taken over the corresponding orbit of the smooth system. As the orbits of the smooth system are close to the orbits of the billiard after an appropriate time-parametrization [17], the integral in (85) tends, as ε → 0, to the integral over
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the limit billiard orbit, i.e. it remains uniformly bounded along with the first derivatives with respect to initial conditions for any compact set of energy values. Now, by using the scaling transformation (59) with s = h we find that the integral in (85) behaves as O(h −1/2 ), which gives Eq. (68) for h¯ indeed. After formula (68) is established for the Poincaré map, the rest of the proof follows exactly like in Theorem 4.
Note that like in Theorem 4, the conditions on the billiard domain D may be relaxed. We do not need the billiard to be dispersing; it is enough to have a locally-maximal, uniformly-hyperbolic, transitive, compact, invariant set composed of regular orbits. Theorem 6 holds true, provided the regular orbits L a and L b belong to . 4.4. Nonautonomous perturbation of a homogeneous potential. In the last example, we consider the Hamiltonian system H = T ( p) + V0 (q) + V1 (q, t) + V2 (q, t),
(86)
where T is a quadratic polynomial of momenta p ∈ R m (m ≥ 2), V0 is a degree d ≥ 3 homogeneous polynomial of the coordinates q ∈ Rm , V1 is a degree d − 1 homogeneous polynomial of q, and V2 is a polynomial of q of degree less than d − 1; the coefficients of V1 and V2 are smooth functions of time, uniformly bounded, along with the first derivative, for all t. By scaling time, energy, momenta and coordinates by the rule √ t → t/s 1/2−1/d , H → H s, p → p s, q → qs 1/d (87) with s = ε−2d/(d−2) this system transforms into 2
4
H = T ( p) + V0 (q) + ε d−2 V1 (q, εt) + O(ε d−2 ).
(88)
It is a small and slow perturbation of the homogeneous Hamiltonian H = T ( p) + V0 (q).
(89)
This system is invariant with respect to the scaling (87), hence its behaviour is the same in every energy level. Assume that system (89) has a locally-maximal, uniformlyhyperbolic, compact, transitive, invariant set in the energy level H = 1. Take any two periodic trajectories L a : {q = qa (t), p = pa (t)} and L b : {q = qb (t), p = pb (t)} from . Denote Tc 1 V¯c (ν) = V1 (qc (t), ν)dt (c = a, b). (90) Tc 0 By the scaling invariance, we obtain that a pair of saddle periodic orbits L a,b (h) exists in every energy level with h > 0; since the orbits belong to a transitive hyperbolic set, they are connected by transverse heteroclinic orbits ab and ba . As the frozen system for (88) is close to (89) (recall that d > 2), the former also possesses a heteroclinic cycle in every energy level. By applying scaling transformation (87) with s = h, one can immediately see that the change of the energy along an orbit of system (88) for one round near L c (h) is given by Tc d V1 (qc (t), ν)dt + . . . , (91) h¯ − h = ε d−2 h 1/2 0
Unbounded Energy Growth in Hamiltonian Systems
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while the return time to the cross-section behaves as ν¯ − ν = εh −
d−2 2d
(Tc + . . .),
(92)
where the dots stand for the terms which tend to zero as the distance to L c (h) diminishes and ε → 0. Now, like we did it in the examples above, by using the scaling transformation (87) one can check that the Poincaré map satisfies the uniformity assumptions [UA1] and [UA2] with 1
α(h) = h d and 2
1
β = O(ε d−2 h 1− d ). This, along with Eqs. (91),(92) implies that system (88) has, for any δ > 0, orbits for which the energy grows not slower than the solution of the equation 2 1 d ¯ d ¯ h (ν) = ε d−2 h 1− d max (93) Va (ν), Vb (ν) − δ dν dν (see Theorem 3). By scaling energy back in order to return to the original system (86): h → hε2d/(d−2) , we rewrite this equation as d ¯ d ¯ 1− d1 max h (ν) = h Va (ν), Vb (ν) − δ . dν dν This is solved as h(ν)1/d − h(0)1/d ν d 1 ¯ ¯ ¯ ¯ ¯ ¯ = Va (ν) + Vb (ν) − Va (0) − Vb (0) + dν (Va − Vb ) dν − 2δ ν . 2d 0 Thus, we arrive at the following result. Theorem 7. If (61) is fulfilled, then system (86) has orbits for which H grows to infinity as t d . Like in Theorems 4 and 6 above, in the case of periodic or quasiperiodic dependence of V1 of t, condition (61) reduces to (69). As we see, every time we have a chaotic Hamiltonian system which is invariant with respect to a scaling of energy, its non-autonomous perturbation creates orbits of growing to infinity energy, provided very non-restrictive conditions of type (61), (69) or (79) are fulfilled. The rate of the energy growth with time depends on how the perturbation term rescales, and is determined by solving the corresponding equation (7). Acknowledgements. The authors thank the Royal Society which provided support for D.T. to visit V.G. at the University of Warwick, where a substantial part of this work had been completed. D.T. also acknowledges the support by grants ISF 926/04, 273/07 and MNTI-RFBR 06-01-72023.
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References 1. Afraimovich, V.S., Shilnikov, L.P.: On critical sets of Morse-Smale systems. Trans. Moscow Math. Soc. 28, 179–212 (1973) 2. Anosov, D.V.: Geodesic flows on closed Riemanian manifolds of negative curvature. Proc. Steklov Math. Inst. 90 (1967) 3. Arnold, V.I.: Mathematical methods of classical mechanics. New York: Springer-Verlag, 1989 4. Bolotin, S., Treschev, D.: Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity 12, 365–388 (1999) 5. Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45, 105–152 (1990) 6. Delshams, A., de la Llave, R., Seara, T.M.: A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential generic geodesic flows on T2 . Commun. Math. Phys. 209, 353–392 (2000) 7. Delshams, A., de la Llave, R., Seara, T.M.: Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Adv. Math. 202, 64–188 (2006) 8. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971) 9. Kaloshin, V.: Geometric proofs of Mather’s connecting and accelerating theorems. London Math. Soc. Lecture Note Ser. 310, 81–106 (2003) 10. Kasuga, T.: On the adiabatic theorem for the Hamiltonian system of differential equations in the classical mechanics. I, II, III. Proc. Japan Acad. 37, 366–382 (1961) 11. Katok, A., Strelcin, J.M.: Invariant manifolds, entropy and billiards — Smooth maps with singularities, Lecture Notes in Mathematics 1222, New York: Springer-Verlag, 1980 12. Lebovitz, N.R., Neishtadt, A.: Slow evolution in perturbed Hamiltonian systems. Stud. Appl. Math. 92, 127–144 (1994) 13. Loskutov, A., Ryabov, A.B., Akinshin, L.G.: Properties of some chaotic billiards with time-dependent boundaries. J. Phys. A 33, 7973–7986 (2000) 14. Loskutov, A., Ryabov, A.: Particle dynamics in time-dependent stadium-like billiards. J. Stat. Phys. 108, 995–1014 (2002) 15. Oliffson-Kamphorst, S., Leonel, E.D., da Silva, J.K.L.: The presence and lack of Fermi acceleration in nonintegrable billiards. J. Phys. A: Math. Theor. 40, F887–F893 (2007) 16. Piftankin, G.N.: Diffusion speed in the Mather problem. Nonlinearity 19, 2617–2644 (2006) 17. Rapoport, A., Rom-Kedar, V., Turaev, D.: Approximating multi-dimensional Hamiltonian flows by billiards. Commun. Math. Phys. 272, 567–600 (2007) 18. Sinai, Ya.G.: Dynamical systems with elastic reflections: Ergodic properties of scattering billiards. Russ. Math. Sur. 25, 137–189 (1970) 19. Sinai, Ya.G., Chernov, N.I.: Ergodic properties of some systems of two-dimensional disks and threedimensional balls. Russ. Math. Sur. 42, 181–207 (1987) 20. Shilnikov, L.P.: On a Poincaré-Birkhoff problem. Math. USSR Sb. 3, 91–102 (1967) 21. Shilnikov, L.P.: On the question of the structure of the neighborhood of a homoclinic tube of an invariant torus. Soviet Math. Dokl. 9, 624–628 (1968) 22. Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V., Chua, L.O.: Methods of qualitative theory in nonlinear dynamics. Part I, Singapore: World Scientific, 1998 23. Treschev, D.: Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17, 1803–1841 (2004) 24. Turaev, D., Rom-Kedar, V.: Islands appearing in near-ergodic flows. Nonlinearity 11, 575–600 (1998) 25. Brännström, N., Gelfreich, V.: Drift of slow variables in slow-fast Hamiltonian systems. Physica D (in press) (2008) 26. Gelfreich, V., Turaev, D.: Fermi acceleration in non-autonomous billiards. J. Phys. A: Math. Theor. 41, 212003 (6pp) (2008) Communicated by G. Gallavotti
Commun. Math. Phys. 283, 795–851 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0524-3
Communications in
Mathematical Physics
The Integrals of Motion for the Deformed W-Algebra Wq,t ( gl N ). II. Proof of the Commutation Relations Takeo Kojima1 , Jun’ichi Shiraishi2 1 Department of Mathematics, College of Science and Technology, Nihon University,
Surugadai, Chiyoda-ku, Tokyo 101-0062, Japan. E-mail: [email protected]
2 Graduate School of Mathematical Science, University of Tokyo, Komaba,
Meguro-ku, Tokyo, 153-8914, Japan Received: 21 September 2007 / Accepted: 18 January 2008 Published online: 15 July 2008 – © Springer-Verlag 2008
Dedicated to Professor Tetsuji Miwa on the occasion of the 60th birthday Abstract: We explicitly construct two classes of infinitely many commutative opera tors in terms of the deformed W -algebra Wq,t (gl N ), and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal, since they can be regarded as elliptic deformations of local and nonlocal integrals of motion for the Virasoro algebra and the W3 algebra [1,2]. 1. Introduction This is a continuation of the papers [3,4], hereafter referred to as Part 1 [3] and Part 2 [4]. In Part 1 we constructed two classes of infinitely many commutative operators, in terms of the deformed Virasoro algebra. In Part 2 we announced conjectural formulae of two classes of infinitely many commutative operators, in terms of the deformed W algebra Wq,t (gl N ), which is the higher-rank generalization of Part 1 [3]. We call one of them local integrals of motion and the other nonlocal one, since they can be regarded as elliptic deformations of local and nonlocal integrals of motion for the Virasoro algebra and the W3 algebra [1,2]. In this paper we give proofs of the commutation relations of the integrals of motion for the deformed W algebra Wq,t (gl N ). Let us recall some facts about the soliton equation and its quantization. B. Feigin and E. Frenkel [5] considered the so-called local integrals of motion I (cl) for the Toda field theory associated with the root system of finite and affine type {I (cl) , H (cl) } P.B. = 0, where H (cl) = 21 (eφ(t) + e−φ(t) )dt is the Hamiltonian of the Toda field theory. They showed the existence of infinitely many commutative integrals of motion by a cohomological argumemnt, and showed that they can be regarded as the conservation laws for the generalized KdV equation. In [5] they constructed the quantum deformation of the local integrals of motion, too. In other words they showed the existence of quantum deformation of the conservation laws of the generalized KdV equation. After quantization the Gel’fand-Dickij bracket {, } P.B. for the second Hamiltonian structure of the generalized KdV, gives rise to the W N algebra. V.Bazhanov et.al [1,2] constructed a field
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theoretical analogue of the commuting transfer matrix T(z), acting on the irreducible highest weight module of the Virasoro algebra and the W3 algebra. They constructed this commuting transfer matrix T(z) as the trace of the monodromy matrix associated 2 ) and Uq (sl 3 ), and showed that the commuwith the quantum affine symmetry Uq (sl tation relation [T(z), T(w)] = 0 is a direct consequence of the Yang-Baxter relation. The coefficients of the asymptotic expansion of the operator log T(z) at z → ∞, produce the local integrals of motion for the Virasoro algebra and the W3 algebra, which reproduce the conservation laws of the generalized KdV equation in the classical limit cC F T → ∞. They call the coefficients of the Taylor expansion of the operator T(z) at z = 0, the nonlocal integrals of motion for the Virasoro algebra and the W3 algebra. They have an explicit integral representation of the nonlocal integrals in terms of the screening currents. The purpose of this paper is to construct the elliptic version of the integrals of motion given by Bazhanov et.al [1,2] and to construct its higher rank generalization. Bazhanov et.al’s construction is based on the free field realization of the Borel subalgebra B± of 2 ) and Uq (sl 3 ). By using this realization they construct the monodromy matrix as Uq (sl the image of the universal R-matrix R¯ ∈ B+ ⊗ B− , and make the transfer matrix T(z) as the trace of the monodromy matrix. The universal R-matrix R¯ of the elliptic quantum group does not exist in B+ ⊗ B− . Hence it is impossible to construct the elliptic deformation of the transfer matrix T(z) in the same manner as [1]. Our method of construction should be completely different from those of [1,2]. Instead of considering the transfer mtrix T(z), we directly give the integral representations of the integrals of motion In , Gn for the deformed W algebra Wq,t (gl N ). The commutativity of our integrals of motion are not understood as a direct consequence of the Yang-Baxter equation. They are understood as a consequence of the commutative subalgebra of the Feigin-Odesskii algebra [10]. The organization of this paper is as follows. In Sect. 2, we review the deformed W algebra, including free field realization, screening currents [6,8]. In Sect. 3, we give integral representations for the local integrals of motion In , and show the commutation relations : [Im , In ] = [Im∗ , In∗ ] = 0. Very precisely, in Part 2 [4], we only give the Laurent series representation of the local integrals of motion, which is useful for proofs of the commutation relation and Dynkinautomorphism invariance. In this section we show the integral representations and the Laurent series representation give the same local integrals of motion. In Sect. 4, we give explicit formulae for the nonlocal integrals of motion Gn , and show the commutation relations : [Gm , Gn ] = [Gm∗ , Gn∗ ] = [Gm , Gn∗ ] = 0, [Im , Gn ] = [Im∗ , Gn ] = [Im , Gn∗ ] = [Im∗ , Gn∗ ] = 0. We show the commutation relation [Im , Gn ] = 0 using Dynkin-automorphism invariance η(In ) = In and η(Gn ) = Gn , which will be shown in the next section. In Sect. 5, we give proofs of Dynkin-automorphis invariance : η(In ) = In , η(In∗ ) = In∗ , η(Gn ) = Gn , η(Gn∗ ) = Gn∗ . In Appendix we summarize the normal ordering of the basic operators. We would like to point out a different point between the case of the deformed Virasoro V irq,t =
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797
2 ) and its higher-rank generalization, Wq,t (gl Wq,t (gl N ), (N ≥ 3). Basically situations 2 ). Howof Wq,t (gl ), (N 3) are more complicated than those of V irq,t = Wq,t (gl N 2 ). ever, one thing of Wq,t (gl ), (N 3) is simpler than those of V ir = Wq,t (gl N q,t In the case of V irq,t = Wq,t (gl2 ), the integrals of motions In , Gn have singularity at s = N = 2. Hence we considered the renormalized limits for the integral of motions In , Gn in the last section of the paper [3]. In the case of Wq,t (gl N ), (N 3), the integrals of motions In , Gn do not have singularity at s = N 3. At the end of the Introduction, we would like to mention two important degenerating limits of the deformed W algebra. One is the CFT-limit [1,2] and the other is the classical limit [14]. In the CFT-limit V.Bazhanov et.al. [1], [2] constructed infinitely many integrals of motion for the Virasoro algebra, as we mentioned above. We give a comment on the CFT-limit in Sect. 4. In the classical limit, the deformed Virasoro algebra degenerates to the Poisson-Virasoro algebra introduced by E. Frenkel and N. Reshetikhin [14]. 2. The Deformed W -Algebra Wq,t ( gl N ) In this section we review the deformed W -algebra and its screening currents. We prepare the notations to be used in this paper. Throughout this paper, we fix generic three parameters 0 < x < 1, r ∈ C and s ∈ C. Let us set z = x 2u . Let us set r ∗ = r − 1. The symbol [u]r for Re(r ) > 0 stands for the Jacobi theta function [u]r = x
u2 r −u
Θx 2r (x 2u ) , Θq (z) = (z; q)∞ (qz −1 ; q)∞ (q; q)∞ , (x 2r ; x 2r )3
(2.1)
where we have used the standard notation (z; q)∞ =
∞
(1 − q j z).
(2.2)
j=0
We set the parametrizations τ, τ ∗ , x = e−π
√
−1/r τ
= e−π
√
−1/r ∗ τ ∗
.
(2.3)
The theta function [u]r enjoys the quasi-periodicity property [u + r ]r = −[u]r , [u + r τ ]r = −e−π
√
√
−1τ − 2π r −1 u
[u]r .
(2.4)
The symbol [a] stands for [a] =
x a − x −a . x − x −1
(2.5)
2.1. Free Field Realization. Let i (1 i N ) be an orthonormal basis in R N relative to the standard basis in R N relative to the standard inner product (, ). Let us set N N ¯i = i − , = N1 i be the weight j=1 j . We identify N +1 = 1 . Let P = i=1 Z¯ lattice. Let us set αi = ¯i − ¯i+1 ∈ P.
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T. Kojima, J. Shiraishi j
Let βm be the oscillators (1 j N , m ∈ Z − {0}) with the commutation relations ⎧ [(s−1)m] ⎨ m [(r[r−1)m] (1 i = j N ) m] [sm] δn+m,0 j [βmi , βn ] = . (2.6) ⎩ −m [(r −1)m] [m] x sm sgn(i− j) δ (1 i = j N ) n+m,0 [r m] [sm] We also introduce the zero mode operator Pλ , (λ ∈ P). They are Z-linear in P and satisfy [i Pλ , Q µ ] = (λ, µ), (λ, µ ∈ P).
(2.7) j
Let us intrduce the bosonic Fock space Fl,k (l, k ∈ P) generated by β−m (m > 0) over the vacuum vector |l, k : j
j
Fl,k = C[{β−1 , β−2 , · · · }1 j N ]|l, k ,
(2.8)
j
(2.9)
where βm |l, k = 0, (m > 0),
r r −1 l− k |l, k , Pα |l, k = α, r −1 r √ r i r −1 Q l −i r −1 r Q k |0, 0 . |l, k = e
(2.10) (2.11)
Let us set the Dynkin-diagram automorphism η by 2s
2s
2s
η(βm1 ) = x − N m βm2 , · · · , η(βmN −1 ) = x − N m βmN , η(βmN ) = x N (N −1)m βm1 , (2.12) and η(i ) = i+1 , (1 i N ). 2.2. The Deformed W -Algebra. In this section we give short review of the deformed W -algebra Wq,t (gl N ) [7–9]. Definition 1. We set the fundamental operator Λ j (z), (1 j N ) by ⎞ ⎛ √ r m − x −r m x −2 r (r −1)P¯ j j βm z −m ⎠ : (1 j N ). : exp ⎝ Λ j (z) = x m m=0
(2.13) Definition 2. Let us set the operator T j (z), (1 j N ) by T j (z) = : Λs1 (x − j+1 z)Λs2 (x − j+3 z) · · · Λs j (x j−1 z) : . (2.14) 1s1 <s2 <···<s j N
Proposition 1. The actions of η on the fundamental operators Λ j (z), (1 j N ) are given by 2s
2s
η(Λ j (z)) = Λ j+1 (x N z), (1 j N − 1), η(Λ N (z)) = Λ1 (x N −2s z).
(2.15)
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799
Proposition 2. The operators T j (z), (1 j N ) satisfy the following relations. f i, j (z 2 /z 1 )Ti (z 1 )T j (z 2 ) − f j,i (z 1 /z 2 )T j (z 2 )Ti (z 1 ) j−i+2k i k−1 z2 x 2l+1 f i−k, j+k (x − j+i )Ti−k (x −k z 1 )T j+k (x k z 2 ) ∆(x )× δ =c z1 k=1 l=1 − j+i−2k z2 x j−i k −k f i−k, j+k (x )Ti−k (x z 1 )T j+k (x z 2 ) , (1 i j N ), −δ z1 (2.16) n where we have used the delta-function δ(z) = n∈Z z . Here we set the constant c and the auxiliary function ∆(z) by (1 − x 2r )(1 − x −2r +2 ) (1 − x 2r −1 z)(1 − x 1−2r z) , ∆(z) = . (1 − x 2 ) (1 − x z)(1 − x −1 z) Here we set the structure functions, c=−
f i, j (z) = exp
(2.17)
∞ 1 (1 − x 2r m )(1 − x −2(r −1)m )(1 − x 2m Min(i, j) )(1 − x 2m(s−Max(i, j)) ) |i− j|m z m . (2.18) x m (1 − x 2m )(1 − x 2sm ) m=1
Above proposition is one parameter “s” generalization of [9]. The proof is given by the same manner. Example. For N = 2 the operators T1 (z), T2 (z) satisfy f 1,1 (z 2 /z 1 )T1 (z 1 )T1 (z 2 ) − f 1,1 (z 1 /z 2 )T1 (z 2 )T1 (z 1 ) = c(δ(x 2 z 2 /z 1 )T2 (x z 2 ) − δ(x 2 z 1 /z 2 )T2 (x −1 z 2 )),
(2.19)
f 1,2 (z 2 /z 1 )T1 (z 1 )T2 (z 2 ) = f 2,1 (z 1 /z 2 )T2 (z 2 )T1 (z 1 ),
(2.20)
f 2,2 (z 2 /z 1 )T2 (z 1 )T2 (z 2 ) = f 2,2 (z 1 /z 2 )T2 (z 2 )T2 (z 1 ).
(2.21)
Example. For N = 3 the operators T1 (z), T2 (z), T3 (z) satisfy f 1,1 (z 2 /z 1 )T1 (z 1 )T1 (z 2 ) − f 1,1 (z 1 /z 2 )T1 (z 2 )T1 (z 1 ) = c(δ(x 2 z 2 /z 1 )T2 (x z 2 ) − δ(x 2 z 1 /z 2 )T2 (x −1 z 2 )),
(2.22)
f 1,2 (z 2 /z 1 )T1 (z 1 )T2 (z 2 ) − f 2,1 (z 1 /z 2 )T2 (z 2 )T1 (z 1 ) = c(δ(x 3 z 2 /z 1 )T3 (x z 2 ) − δ(x 3 z 1 /z 2 )T3 (x −1 z 2 )), f 2,2 (z 2 /z 1 )T2 (z 1 )T2 (z 2 ) − f 2,2 (z 1 /z 2 )T2 (z 2 )T2 (z 1 )
(2.23) (2.24)
= c f 1,3 (1)(δ(x 2 z 2 /z 1 )T1 (x z 2 )T3 (x z 2 ) − δ(x 2 z 1 /z 2 )T1 (x −1 z 2 )T3 (x −1 z 2 )), f 1,3 (z 2 /z 1 )T1 (z 1 )T3 (z 2 ) = f 3,1 (z 1 /z 2 )T3 (z 2 )T1 (z 1 ), (2.25) f 2,3 (z 2 /z 1 )T2 (z 1 )T3 (z 2 ) = f 3,2 (z 1 /z 2 )T3 (z 2 )T2 (z 1 ),
(2.26)
f 3,3 (z 2 /z 1 )T3 (z 1 )T3 (z 2 ) = f 3,3 (z 1 /z 2 )T3 (z 2 )T3 (z 1 ).
(2.27)
m( j) , (1 Definition 3. The deformed W -algebra Wq,t ( gl N ) is defined by the generators T m( j) as j N , m ∈ Z) with the defining relations (2.16). Here we should understand T ( j) j (z) = m∈Z T m z −m , (1 j N ). the Fourier coefficients of the operators T
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2.3. Screening Currents. In this section we introduce the screening currents E j (z) and F j (z). Definition 4. We set the screening currents F j (z), (1 j N ) by
F j (z) = e
i
r −1 r Qα j
(x
⎛
( 2s N −1) j
z)
r −1 r
Pα j + r −1 r
⎞ 1 j Bm z −m ⎠ :, (1 j N − 1), × : exp ⎝ m
FN (z) = e
i
m=0
r −1 r QαN
⎛
× : exp ⎝
(x
2s−N
z)
r −1 r −1 r P¯ N + 2r
⎞
z
(2.28)
r −1 − r −1 r P¯1 + 2r
1 B N z −m ⎠ :, m m
(2.29)
m=0
We set the screening currents E j (z), (1 j N ) by √ r √ r r 2s −i r −1 Qα j − r −1 Pα j + r −1 E j (z) = e (x ( N −1) j z) ⎛ ⎞ 1 [r m] j Bm z −m ⎠ :, (1 j N − 1), (2.30) × : exp ⎝− m [(r − 1)m] m=0 √ r √ r √ r r r −i r −1 QαN E N (z) = e (x 2s−N z)− r −1 P¯ N + 2(r −1) z r −1 P¯1 + 2(r −1) , ⎛ ⎞ 1 [r m] B N z −m ⎠ : . × : exp ⎝− (2.31) m [(r − 1)m] m m=0
Here we have set 2s
Bm = (βm − βm )x − N j
j
= (x
BmN
−2sm
j+1
βmN
jm
, (1 j N − 1),
− βm1 ).
(2.32) (2.33)
The screening currents F j (z), E j (z) (1 j N − 1) have already been studied in [11–13]. We introduce the new screening current FN (z), E N (z), which can be regarded as “affinization” of screening currents F j (z), E j (z) (1 j N − 1). The following commutation relations are convenient for calculations: j
j
j+1
j
[βm , Bn ] = mδm+n,0
[r ∗ m] (−1+ 2s j)m N x , (1 j N ) [r m]
[βm , Bn ] = −mδm+n,0 [βm1 , BnN ] = −mδm+n,0 j
j
[Bm , Bn ] = mδm+n,0 j
j+1
[r ∗ m] (1+ 2s j)m x N , (1 j N − 1) [r m]
(2.35)
[r ∗ m] m x , [r m]
(2.36)
[r ∗ m] [2m] , (1 j N ) [r m] [m]
[Bm , Bn ] = −mδm+n,0
(2.34)
[r ∗ m] (−1+ 2s )m N x , (1 j N ). [r m]
(2.37) (2.38)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
801
Here we read BmN +1 = Bm1 . We summarize the commutation relations of the screening currents for N 3. Proposition 3. The screening currents F j (z), (1 j N ; N 3) satisfy the following commutation relations for Re(r ) > 0 :
u1 − u2 −
s s − 1 F j+1 (z 2 )F j (z 1 ), F j (z 1 )F j+1 (z 2 ) = u 2 − u 1 + N r N r (1 j N ), (2.39)
[u 1 − u 2 ]r [u 1 − u 2 + 1]r F j (z 1 )F j (z 2 ) = [u 2 − u 1 ]r [u 2 − u 1 + 1]r F j (z 2 )F j (z 1 ), (1 j N ), (2.40) Fi (z 1 )F j (z 2 ) = F j (z 2 )Fi (z 1 ), (|i − j| 2).
(2.41)
We read FN +1 (z) = F1 (z). The screening currents F j (z), (1 j N ; N 3) satisfy the following commutation relations for Re(r ) < 0 : s s u1 − u2 + 1 − F j (z 1 )F j+1 (z 2 ) = u 2 − u 1 + F j+1 (z 2 )F j (z 1 ), N −r N −r (1 j N ), (2.42) [u 1 − u 2 ]−r [u 1 − u 2 − 1]−r F j (z 1 )F j (z 2 ) = [u 2 − u 1 ]−r [u 2 − u 1 − 1]−r , F j (z 2 )F j (z 1 ), (1 j N ), (2.43) Fi (z 1 )F j (z 2 ) = F j (z 2 )Fi (z 1 ), (|i − j| 2). (2.44) We read FN +1 (z) = F1 (z). The screening currents E j (z), (1 j N ; N 3) satisfy the following commutation relations for Re(r ∗ ) > 0 : s s u1 − u2 + 1 − E j (z 1 )E j+1 (z 2 ) = u 2 − u 1 + E j+1 (z 2 )E j (z 1 ), N r∗ N r∗ (1 j N ), (2.45) [u 1 − u 2 ]r ∗ [u 1 − u 2 − 1]r ∗ E j (z 1 )E j (z 2 ) = [u 2 − u 1 ]r ∗ [u 2 − u 1 − 1]r ∗ E j (z 2 )E j (z 1 ), (1 j N ), (2.46) E i (z 1 )E j (z 2 ) = E j (z 2 )E i (z 1 ), (|i − j| 2).
(2.47)
We read E N +1 (z) = E 1 (z). The screening currents E j (z), (1 j N ; N 3) satisfy the following commutation relations for Re(r ∗ ) < 0 :
u1 − u2 −
s s −1 E j (z 1 )E j+1 (z 2 ) = u 2 − u 1 + N −r ∗ N −r ∗ E j+1 (z 2 )E j (z 1 ), (1 j N ), (2.48)
[u 1 − u 2 ]−r ∗ [u 1 − u 2 + 1]−r ∗ E j (z 1 )E j (z 2 ) = [u 2 − u 1 ]−r ∗ [u 2 − u 1 + 1]−r ∗ E j (z 2 )E j (z 1 ), (1 j N ), (2.49) E i (z 1 )E j (z 2 ) = E j (z 2 )E i (z 1 ), (|i − j| 2). (2.50)
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Proposition 4. The screening currents E j (z), F j (z), (1 j N ; N 3) satisfy the following commutation relation Re(r ) < 0 : [E j (z 1 ), F j (z 2 )] =
1 (δ(x z 2 /z 1 )H j (x r z 2 ) − δ(x z 1 /z 2 )H j (x −r z 2 )), x − x −1 (1 j N ), (2.51)
E i (z 1 )F j (z 2 ) = F j (z 2 )E i (z 1 ), (1 i = j N ).
(2.52)
Here we have set 2s
H j (z) = x (1− N )2 j e ⎛
−√i
× : exp ⎝−
2s
Qα j
rr ∗
(x ( N −1) j z)
− √ 1 ∗ Pα j + rr1∗ rr
⎞
1 [m] j Bm z −m ⎠ :, (1 j N − 1), m [r ∗ m]
(2.53)
m=0 −√i
H N (z) = x 2(N −2s) e ⎛
× : exp ⎝−
rr ∗
QαN
(x 2s−N z)
− √ 1 ∗ P¯ N + 2rr1 ∗ − √ 1 ∗ P¯1 + 2rr1 ∗ rr
z
⎞
rr
1 [m] B N z −m ⎠ : . m [r ∗ m] m
(2.54)
m=0
Proposition 5. The actions of η on the screenings F j (z), (1 j N ; N 3) are given by 2s
η(F j (z)) = F j+1 (z)(x N −1 )
∗ ∗ − rr Pα j+1 − rr
2s
η(FN −1 (z)) = FN (z)(x 1− N ) η(FN (z)) = F1 (z)(x
∗ P¯ N + r2r
r∗ r
(1− 2s N )(N −1)
)
r∗ r
, (1 j N − 2), 2s
(x 1− N )
∗ P¯1 + r2r
(x
∗ ∗ − rr P¯1 + r2r
1− 2s N
)
(2.55)
,
∗ ∗ − rr P¯2 + r2r
(2.56) .
(2.57)
Especially we have η(F1 (z 1 )F2 (z 2 ) · · · FN (z N )) = FN (z 1 )F1 (z 2 ) · · · F1 (z N ).
(2.58)
The actions of η on the screenings E j (z), (1 j N ; N 3) are given by 2s
η(E j (z)) = E j+1 (z)(x N −1 )
√r
r∗
Pα j+1 − rr∗
, (1 j N − 2), √r r r 2s )− r ∗ P¯ N + 2r ∗ (x 1− N ) r ∗ P¯1 + 2r ∗ , η(E N −1 (z)) = E N (z)(x √ √ r r r r 2s 2s η(E N (z)) = E 1 (z)(x (1− N )(N −1) )− r ∗ P¯1 + 2r ∗ (x 1− N ) r ∗ P¯2 + 2r ∗ . 1− 2s N
√r
(2.59) (2.60) (2.61)
Especially we have η(E 1 (z 1 )E 2 (z 2 ) · · · E N (z N )) = E 2 (z 1 ) · · · E N (z N −1 )E 1 (z N ).
(2.62)
Proposition 6. The screening currents F j (z), (1 j N ; N 3) and the fundamental operators Λ j (z), (1 j N ; N 3) commute up to delta-function δ(z) = m, z n∈Z
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
803
2s 2s z2 ∗ ∗ A j (x N j−r z 1 ), (1 j N −1), [Λ j (z 1 ), F j (z 2 )] = (−x r + x −r )δ x N j−r z1 (2.63) 2s 2s z2 ∗ ∗ A j (x N j+r z 2 ), (1 j N − 1), [Λ j+1 (z 1 ), F j (z 2 )] = (x r − x −r )δ x N j+r z1 (2.64) z2 ∗ ∗ A N (x −r z 2 ), [Λ N (z 1 ), FN (z 2 )] = (−x r + x −r )δ x −r +2s (2.65) z1 z2 ∗ ∗ (2.66) [Λ1 (z 1 ), FN (z 2 )] = (x r − x −r )δ x r A N (x r z 2 ). z1 Here we have set i
r∗ r
Qα j
√ − rr ∗ (P¯ j +P¯ j+1 )
−j
r∗
P +r
∗
x (zx ) r α j r ⎞ ⎛ 1 j j+1 (x r m βm − x −r m βm )z −m⎠ :, (1 j N −1), × : exp ⎝ m
A j (z) = e
m=0
A N (z) = e
i
r∗ r
QαN
√ − rr ∗ (P¯ N +P¯1 )
(2.67)
r∗ r
∗ P¯ N + r2r
∗ ∗ − rr P¯1 + r2r
x (zx 2s−N ) z ⎛ ⎞ 1 × : exp ⎝ (2.68) (x (r −2s)m βmN − x −r m βm1 )z −m ⎠ : . m m=0 2s 2s ∗ z2 ∗ B j (x N j+r z 2 ), (1 j N − 1), [Λ j (z 1 ), E j (z 2 )] = (−x r + x −r )δ x N j+r z1 (2.69) 2s 2s z ∗ ∗ 2 B j (x N j−r z 1 ), (1 j N − 1), [Λ j+1 (z 1 ), E j (z 2 )] = (x r − x −r )δ x N j−r z1 (2.70) z ∗ ∗ 2 B N (x r z 1 ), (2.71) [Λ N (z 1 ), E N (z 2 )] = (−x r + x −r )δ x r +2s z1 ∗ z2 ∗ B N (x −r z 2 ). (2.72) [Λ1 (z 1 ), E N (z 2 )] = (x r − x −r )δ x −r z1 Here we have set B j (z) = e
−i
√r
r∗
Qα j
⎛
x
√ − rr ∗ (P¯ j +P¯ j+1 )
(zx − j )
√r
−
r∗
Pα j + rr∗
⎞ 1 [r m] ∗m j ∗ m j+1 −m −r r × : exp ⎝− βm − x βm )z ⎠ :, (1 j N − 1), (x m [r ∗ m] m=0
(2.73)
804
T. Kojima, J. Shiraishi
B N (z) = e
−i
r∗ r
QαN
⎛
× : exp ⎝
√ rr ∗ (P¯ N +P¯1 )
x−
√r √r r r (zx 2s−N )− r ∗ P¯ N + 2r ∗ z r ∗ P¯1 + 2r ∗ ⎞
1 [r m] ∗ ∗ (x (−r −2s)m βmN − x r m βm1 )z −m ⎠ : . ∗ m [r m]
(2.74)
m=0
2.4. Comparison with another definition. At first glance, our definition of the deformed W -algebra is different from those in [7–9]. In this section we show they are essentially the same thing. Let us set the element Cm by Cm =
N
x (N −2 j+1)m βm . j
(2.75)
j=1 A (z) and This element Cm is η-invariant, η(Cm ) = Cm . Let us divide Λ j (z) into Λ DW j Z(z), A (z)Z(z), (1 j N ), Λ j (z) = Λ DW j
(2.76)
where we set A (z) = x Λ DW j
√ −2 r (r −1)P¯ j
⎛
⎞ x r m − x −r m j [m]x βm − : exp ⎝ Cm z −m ⎠ :, m [N m]x m=0
⎛ Z(z) = : exp ⎝
⎞
x r m − x −r m [m]x Cm z −m ⎠ : . m [N m]x
(2.77) (2.78)
m=0
Let us set T jDW A (z) =
A − j+1 A − j+3 A j−1 : ΛsDW (x z)ΛsDW (x z) · · · ΛsDW (x z) : . 1 2 j
1s1 <s2 <···<s j N
(2.79) Proposition 7. The bosonic operators T jDW A (z), (1 j N −1) satisfy the following relations: A (z /z )T DW A (z )T DW A (z ) − f DW A (z /z )T DW A (z )T DW A (z ) f i,DW 2 1 i 1 j 2 1 2 j 2 i 1 j j,i i k−1 x j−i+2k z 2 DW A (x − j+i )T DW A =c ∆(x 2l+1 ) × δ f i−k, j+k i−k z1 k=1 l=1 x − j+i−2k z 2 −k DW A k DW A j−i DW A k DW A −k × (x z 1 )T j+k (x z 2 )−δ )Ti−k (x z 1 )T j+k (x z 2 ) , f i−k, j+k (x z1
(1 i j N − 1), (2.80)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
805
where δ(z) = n∈Z z n . We should understand TNDW A (z) = 1, T jDW A (z) = 0, ( j > N ). Here we set the constant c and the auxiliary function ∆(z) in (2.17). Here we set the structure functions, A (z) = f (z)| f i,DW i, j s=N j ∞ 1 (1 − x 2r m )(1 − x −2(r −1)m )(1 − x 2m Min(i, j) )(1 − x 2m(N −Max(i, j)) ) |i− j|m z m . = exp x m (1 − x 2m )(1 − x 2N m ) m=1
(2.81)
Proposition 8. The operators T jDW A (z) and Z(z) commutes with each other. T jDW A (z 1 )Z(z 2 ) = Z(z 2 )T jDW A (z 1 ), (1 j N − 1).
(2.82)
Therefore three parameter deformed W -algebra T j (z) is realized as an extension of two parameter deformed W -algebra T jDW A (z) in [7–9]. Note that upon the specialization s = N we have j
[BnN , BmN ] = 0, [Bm , BnN ] = 0 for j = N .
(2.83)
Hence we can regard BmN = 0 and T j (z) = T jDW A (z), TNDW A (z) = 1. 3. Local Integrals of Motion In this section we construct the local integrals of motion In . We study the generic case : 0 < x < 1, r ∈ C and Re(s) > 0. gl N ). Let us set the function h(u) and h ∗ (u) by 3.1. Local Integrals of Motion for Wq,t ( h(u) =
[u]s [u + r ]s [u]s [u − r ∗ ]s , h ∗ (u) = , ∗ [u + 1]s [u + r ]s [u + 1]s [u − r ]s
(3.84)
where we have set z = x 2u . Definition 5. • We define In for regime Re(s) > 2 and Re(r ∗ ) < 0 by In =
···
n
dz j √ 2π −1z j C j=1
h(u k − u j )T1 (z 1 ) · · · T1 (z n ) (n = 1, 2, · · · ).
1 j
(3.85) ∗
Here, the contour C encircles z j = 0 in such a way that z j = x −2+2sl z k , x −2r +2sl z k ∗ (l = 0, 1, 2, . . .) is inside and z j = x 2−2sl z k , x 2r −2sl z k (l = 0, 1, 2, . . .) is outside for 1 j < k n. We call In the local integrals of motion for the deformed W algebra. The definitions of In for generic Re(s) > 0 and r ∈ C should be understood as analytic continuation.
806
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• We define In∗ for regime Re(s) > 2 and Re(r ) > 0 by In =
···
n
dz j √ C j=1 2π −1z j
h ∗ (u k − u j )T1 (z 1 ) · · · T1 (z n ) (n = 1, 2, · · · ).
1 j
(3.86)
Here, the contour C encircles z j = 0 in such a way that z j = x −2+2sl z k , x 2r +2sl z k (l = 0, 1, 2, . . .) is inside and z j = x 2−2sl z k , x −2r −2sl z k (l = 0, 1, 2, . . .) is outside for 1 j < k n. We call In∗ the local integrals of motion for the deformed W -algebra. The definitions of In∗ for generic Re(s) > 0 and r ∈ C should be understood as analytic continuation. The following is one of the Main Results of this paper. Theorem 1. The local integrals of motion In commute with each other [In , Im ] = 0 (m, n = 1, 2, · · · ).
(3.87)
The local integrals of motion In∗ commute with each other [In∗ , Im∗ ] = 0 (m, n = 1, 2, · · · ).
(3.88)
3.2. Laurent-Series Formulae. In this subsection we prepare another formulae of the local integrals of motion In . Because the integral contour of the definition of the local integrals of motion In is not annulus. i.e. |x − p z k | < |z j | < |x p z k |, the defining relations of the deformed W -algebra (2.16) should be used carefully. Hence, in order to show the commutation relations [Im , In ] = 0, it is better for us to deform the integral representations of the local integrals of motion In to another formulae, in which the defining relations of the deformed W -algebra (2.16) can be used safely. Let us set the auxiliary function s(z), s ∗ (z) by h(u) = s(z) f 11 (z), h ∗ (u) = s ∗ (z) f 11 (z), (z = x 2u ), where h(u), h ∗ (u) and f 11 (z) are given in the previous section. We have explicitly (z; x 2s )∞ (x 2s−2r z; x 2s )∞ (1/z; x 2s )∞ (x 2s−2r /z; x 2s )∞ × , ∗ ∗ (x 2s−2 z; x 2s )∞ (x −2r z; x 2s )∞ (x 2s−2 /z; x 2s )∞ (x −2r /z; x 2s )∞ (3.89) ∗ ∗ 2s 2s+2r 2s 2s 2s+2r 2s z; x )∞ /z; x )∞ (1/z; x )∞ (x ∗ (z; x )∞ (x s ∗ (z) = x −2r × 2s−2 . (3.90) 2s−2 2s 2r 2s 2s 2r (x z; x )∞ (x z; x )∞ (x /z; x )∞ (x /z; x 2s )∞ s(z) = x −2r
∗
Let us set the auxiliary functions gi, j (z) by fusion procedure gi,1 (z) = g1,1 (x −i+1 z)g1,1 (x −i+3 z) · · · g1,1 (x i−1 z), gi, j (z) = gi,1 (x − j+1 z)gi,1 (x − j+3 z) · · · gi,1 (x j−1 z),
(3.91)
where g11 (z) = f 11 (z) is the structure function of the deformed W -algebra defined in (2.16), ∗
f 1,1 (z) =
1 (x 2s−2 z; x 2s )∞ (x 2r z; x 2s )∞ (x −2r z; x 2s )∞ . ∗ 1 − z (x 2 z; x 2s )∞ (x 2r +2s z; x 2s )∞ (x 2s−2r z; x 2s )∞
(3.92)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
807
The structure functions f 1, j (z) and g1, j (z) have the following relations: g1, j (z) = ∆(x − j+2 z)∆(x − j+4 z) · · · ∆(x j−2 z) f 1, j (z). r +r ∗
(3.93)
−r −r ∗
z)(1−x z) Here ∆(z) is given by ∆(z) = (1−x(1−x z)(1−x . −1 z) Let us set the formal power series A(z 1 , z 2 , . . . , z n ) by A(z 1 , z 2 , . . . , z n ) = ak1 ,...,kn z 1k1 z 2k2 . . . z nkn .
(3.94)
k1 ,...,kn ∈Z
We define the symbol [· · · ]1,z 1 ...z n by [A(z 1 , z 2 , . . . , z n )]1,z 1 ...z n = a0,0,...,0 .
(3.95)
Let us set D = {(z 1 , . . . , z n ) ∈ Cn | k1 ,...,kn ∈Z |ak1 ,...,kn z 1k1 z 2k2 . . . z nkn | < +∞}. When we assume closed curve J is contained in D, we have [A(z 1 , z 2 , . . . , z n )]1,z 1 ...z n =
···
n J
dz j A(z 1 , z 2 , . . . , z n ). (3.96) √ 2π −1z j j=1
Let us set the auxiliary functions, s11 (z) = s(z), h 11 (z) = h(u),(z = x 2u ) and si,1 (z) = s1,1 (x −i+1 z)s1,1 (x −i+3 z) . . . s1,1 (x i−1 z), si, j (z) = si,1 (x − j+1 z)si,1 (x − j+3 z) . . . si,1 (x j−1 z), h i,1 (z) = h 1,1 (x h i, j (z) = h i,1 (x
−i+1
− j+1
−i+3
− j+3
z)h 1,1 (x z)h i,1 (x
z) . . . h 1,1 (x
i−1
z),
z) . . . h i,1 (x
j−1
z),
(3.97) (3.98)
and ∗ ∗ ∗ ∗ si,1 (z) = s1,1 (x −i+1 z)s1,1 (x −i+3 z) . . . s1,1 (x i−1 z), ∗ ∗ ∗ si,∗ j (z) = si,1 (x − j+1 z)si,1 (x − j+3 z) . . . si,1 (x j−1 z),
(3.99)
∗ (z) = h ∗1,1 (x −i+1 z)h ∗1,1 (x −i+3 z) . . . h ∗1,1 (x i−1 z), h i,1 ∗ ∗ ∗ (x − j+1 z)h i,1 (x − j+3 z) . . . h i,1 (x j−1 z). h i,∗ j (z) = h i,1
(3.100)
In what follows we use the notation of the ordered product: T1 (zl ) = T1 (zl1 )T1 (zl2 ) . . . T1 (zln ), (L = {l1 , . . . , ln |l1 < l2 < · · · < ln }). −→ l∈L
(3.101) Theorem 2. For Re(s) > N and Re(r ∗ ) < 0, the local integrals of motion In are written as ⎡ ⎤ In = ⎣ s(z k /z j )On (z 1 , z 2 , . . . , z n )⎦ . (3.102) 1 j
1,z 1 ...z n
808
T. Kojima, J. Shiraishi
For Re(s) > N and Re(r ) > 0, the local integrals of motion In∗ are written as ⎡
⎤
In∗ = ⎣
s ∗ (z k /z j )On (z 1 , z 2 , · · · , z n )⎦
1 j
.
(3.103)
1,z 1 ...z n
Here we set the operator On (z 1 , z 2 , . . . , z n ) by
On (z 1 , z 2 , . . . , z n ) =
α1 , α2 , α3 , . . . , α N 0 α1 +2α2 +3α3 +···+N α N =n
×
−→ (1) j∈ A Min
×
N
(−c)t−1
t−1
T2 (x −1 z j ) · · ·
−→ (2) j∈ A Min
t=1
×
T1 (z j )
t=1
j
gt,t
αt
∆(x 2u+1 )t−u−1
zk zj
(s)
s=1,...,N j=1,...,αs (s) (s) N ∪αs A(s) ={1,2,...,n} A j ⊂{1,2,...,n}, |A j |=s, ∪s=1 j=1 j (s) (s) (s) Min(A1 )<Min(A2 )<···<Min(Aαs ) t
Tt (x −1+t−2[ 2 ] z j ) · · · αt
t=1
j=1 (t) j1 =A j,1 ··· (t) jt =A j,t
1t
(t) j∈ A Min (u) k∈ A Min
N
TN (x −1+N −2[ 2 ] z j )
−→ (N ) j∈ A Min
N
Aj
−→ (t) j∈ A Min
u=1
N
t
δ
u=1 σ ∈St σ (1)=1 u =[ t ]+1 2
x 2 z jσ (u+1)
z jσ (u)
u t zk gt,u x u−t−2[ 2 ]+2[ 2 ] . zj
(3.104)
Here we have set the constant c and the function ∆(z) in (2.17). When the index set (t) A(t) j = { j1 , j2 , . . . , jt | j1 < j2 < · · · < jt }, (1 t N , 1 j αt ), we set A j,k = jk , (t)
(t)
(t)
(t)
and A Min = {A1,1 , A2,1 , . . . , Aαt ,1 }. Here we should understand z jσ (t+1) = z jσ (1) in the 2 x z jσ (t+1) . delta-function δ zj σ (t)
Example. We summarize the operators On very explicitly. O1 (z) = T1 (z), O2 (z 1 , z 2 ) = g1,1 (z 2 /z 1 )T1 (z 1 )T1 (z 2 ) − cδ(x 2 z 2 /z 1 )T2 (x −1 z 1 ), O3 (z 1 , z 2 , z 3 ) = g11 (z 2 /z 1 )g1,1 (z 3 /z 1 )g1,1 (z 3 /z 2 )T1 (z 1 )T1 (z 2 )T1 (z 3 )
(3.105) (3.106)
− cg1,2 (x −1 z 2 /z 1 )T1 (z 1 )δ(x 2 z 3 /z 2 )T2 (x −1 z 2 ) − cg1,2 (x −1 z 1 /z 2 )T1 (z 2 )δ(x 2 z 3 /z 1 )T2 (x −1 z 1 ) − cg1,2 (x −1 z 1 /z 3 )T1 (z 3 )δ(x 2 z 2 /z 1 )T2 (x −1 z 1 )
(3.107)
+ c ∆(x )(δ(x z 2 /z 1 )δ(x z 1 /z 3 ) + δ(x z 1 /z 2 )δ(x z 3 /z 1 ))T3 (z 1 ). 2
3
2
2
We should understand above as T j (z) = 0 for j > N .
2
2
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
809
3.3. Weakly sense equality. In order to show thorem, we introduce a “weak sense” equality Definition 6. We say the operators P(z 1 , z 2 , . . . , z n ) and Q(z 1 , z 2 , · · · , z n ) are equal in the “weak sense” if
s(z j /z i )P(z 1 , z 2 , · · · , z n ) =
1i< j n
s(z j /z i )Q(z 1 , z 2 , · · · , z n ). (3.108)
1i< j n
We write P(z 1 , z 2 , . . . , z n ) ∼ Q(z 1 , z 2 , . . . , z n ), showing the weak equality. For example δ(z 1 /z 2 ) ∼ 0 and
1 z 1 −z 2 δ(z 1 /z 2 )
∼ 0.
Proposition 9. The following relations hold in the weak sense for 1 j N
j j x −1+ j−2[ 2 ] w1 g1, j T1 (z 1 )T j (x −1+ j−2[ 2 ] w1 ) z1 j x 1− j+2[ 2 ] z 1 −1+ j−2[ 2j ] w1 )T1 (z 1 ) −g j,1 T j (x w1 2 2 j j−1 j x wσ (t+1) x wσ (t+1) 2t+1 ∼c × δ ∆(x ) δ wσ (t) wσ (t) t=1 t=1 σ ∈S σ ∈S t=1
j j σ (1)=1 t =[ 2 ]+1
× δ
j
x 2 j−2[ 2 ] w1 z1
T j+1 (x
j−2[ 2j ]
w1 ) − δ
j j σ (1)=1 t =[ 2 ]+1 j
x −2−2[ 2 ] w1 z1
T j+1 (x
j−2−2[ 2j ]
w1 ) . (3.109)
We should understand TN +1 (z) = 0 and wσ ( j+1) = wσ (1) in the delta-function 2 x wσ ( j+1) . δ wσ ( j) Proof. We explain the mechanism by the simplest case for N 3.
g1,2 (x −1 z 2 /z 1 )T1 (z 1 )T2 (x −1 z 2 ) − g2,1 (x z 1 /z 2 )T2 (x −1 z 2 )T1 (z 1 ) δ(x 2 z 3 /z 2 ) = g1,2 (x −1 z 2 /z 1 )c(δ(z 2 /z 1 ) − δ(x −2 z 2 /z 1 ))δ(x 2 z 3 /z 2 )T1 (z 1 )T2 (x −1 z 2 ) +∆(x z 1 /z 2 )δ(x 2 z 3 /z 2 )( f 1,2 (x −1 z 2 /z 1 )T1 (z 1 )T2 (x −1 z 2 ) − f 2,1 (x z 1 /z 2 ) T2 (x −1 z 2 )T1 (z 1 )).
(3.110)
Here we have used g1,2 (z) = ∆(z) f 1,2 (z) and ∆(z) − ∆(z −1 ) = c(δ(x z) − δ(x −1 z)), ∆(z) =
(1 − x 2r −1 z)(1 − x −2r +1 z) . (1 − x z)(1 − x −1 z) (3.111)
810
T. Kojima, J. Shiraishi
Using δ(z 1 /z 2 ) ∼ 0 and δ(x 2 z 1 /z 2 )δ(x 2 z 3 /z 2 ) ∼ 0 , ∆(x 3 ) = ∆(x −3 ), and the defining relation of the deformed W -algebra (2.16), we get this proposition. As the same manner as above, we have the following proposition. Proposition 10. The following relations hold in the weak sense for i, j 2 j−i j i x j−i−2[ 2 ] w1 gi, j Ti (x −1+i−2[ 2 ] z 1 )T j (x −1+ j−2[ 2 ] w1 ) z1 2 2 j i x z σ (t+1) x wσ (t+1) × δ δ z σ (t) wσ (t) t=1 t=1 σ ∈S σ ∈S j i σ (1)=1 t =[ 2 ]+1
j j σ (1)=1 t =[ 2 ]+1
j i σ (1)=1 t =[ 2 ]+1
j j σ (1)=1 t =[ 2 ]+1
i− j j i x i− j−2[ 2 ] z 1 ∼ g j,i T j (x −1+ j−2[ 2 ] w1 )Ti (x −1+i−2[ 2 ] z 1 ) w1 2 2 j i x z σ (t+1) x wσ (t+1) . × δ δ z σ (t) wσ (t) t=1 t=1 σ ∈S σ ∈S
(3.112)
We should understand T j (z) = 0 for j > N . Let us introduce Sn -invariance in the “weak sense”. Definition 7. We call the operator P(z 1 , z 2 , . . . , z n )Sn -invariant in the “weak sense” if P(z 1 , z 2 , · · · , z n ) ∼ P(z σ (1) , z σ (2) , · · · , z σ (n) ), (σ ∈ Sn ).
(3.113)
Example. The operator O2 (z 1 , z 2 ) = g11 (z 2 /z 1 )T1 (z 1 )T1 (z 2 ) − cδ(x 2 z 2 /z 1 )T2 (x −1 z 1 ) is S2 -invariant. Theorem 3. The operator On defined in Theorem 2 is Sn -invariant in the weak sense, On (z 1 , z 2 , · · · , z n ) ∼ On (z σ (1) , z σ (2) , · · · , z σ (n) ) (σ ∈ Sn ).
(3.114)
This theorem plays an important role in proof of the Main Theorem 1. We will show the above theorem in the next section. 3.4. Proof of Sn -Invariance for On (z 1 , . . . , z n ). In this section we give proof of Theo2 is summarized in [3]. By straightforward but tedious rem 3. Proof for the special case gl calculations we have the following proposition. Proposition 11. The following relation holds in the weak sense: g1,1 (z k /z j ) T1 (z j ) − (z 1 ↔ z 2 ) −→ 1 j M
1 j
∼
M
t=0 3 j3 < j4 <···< jt+2 M
(−1)t ct+1
t u=1
∆(x 2u+1 )t+1−u
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
×
g1,1
3 j
× Tt+2 (x
−1+t−2[ 2t ]
zk zj
z1)
811
t z1 g1,t+2 x −1+t−2[ 2 ] zj
3 j M j = j3 ,..., jt+2
t+2
σ ∈St+2 σ (1)=1
u=1 u =[ 2t ]+2 j1 =1, j2 =2
δ
x 2 z jσ (u+1)
T1 (z j )
−→ 3 j M j = j3 ,..., jt+2
− (z 1 ↔ z 2 ).
z jσ (u)
(3.115)
We should understand T j (z) = 0 ( j > N ). Proof of Theorem 3. In order to show Sn -invariance, it is enough to show the case of the permutations σ = (i, i + 1) for 1 i n − 1. Because of the cancellations, the difference On (. . . , z i , z i+1 , . . .) − On (. . . , z i+1 , z i , . . .) has simplification. We don’t have to consider every summation (s) in the definition of On . We consider the Aj
following N -cases for σ = (i, i + 1):
s=1,...,N j=1,...,αs
(1)
1 (1) {i, i + 1} ⊂ ∪αj=1 Aj ,
(2) A(2) J = {i, i + 1} for some J, (3) A(3) J = {i, i + 1, j3 |i + 1 < j3 } for some J, ········· (s) (s) A J = {i, i + 1, j3 , . . . , js |i + 1 < j3 < · · · < js } for some J, ········· (N ) (N ) A J = {i, i + 1, j3 , j4 , . . . , j N |i + 1 < j3 < j4 < · · · < j N } for some J. We have On (z 1 , . . . , z i , z i+1 , . . . , z n ) − On (z 1 , . . . , z i+1 , z i , . . . , z n ) = On (z 1 , . . . , z i , z i+1 , . . . , z n ) − On (z 1 , . . . , z i+1 , z i , . . . , z n ).
(3.116)
Here we have set
N
α1 ,α2 ,...,α N 0 α1 +2α2 +···+N α N =n
t=1
On (z 1 , . . . , z i , z i+1 , . . . , z n ) = ⎛ ⎜ ⎜ N ⎜ ⎜ ×⎜ + ⎜ (s) t=2 ⎜ A s=1,...,N ⎝ j j=1,...,αs (1) αs {i,i+1}⊂∪ j=1 Aj
(−c)
(s) Aj
s=1,...,N j=1,...,αs (t) A J ={i,i+1, j3 ,..., jt |i+1< j3 <···< jt } for some J
t−1
t−2
αt ∆(x
)
2u+1 t−u−1
u=1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −→ ⎟1s N ⎠
−→ (s) j∈A Min
s
Ts (x −1+s+[ 2 ] z j )
812
×
×
T. Kojima, J. Shiraishi αt N
t=1
j=1 (t) j1 =A j,1 ··· (t) jt =A j,t
N
t=1
t
δ
x 2 z jσ (u+1) z jσ (u)
u=1 σ ∈St σ (1)=1 u =[ t ]+1 2
gt,t
1 j
zk zj
1t
(t) j∈A Min (u) k∈A Min
u−t z k gt,u x u−t−2[ 2 ] . zj
(3.117)
Let us consider the formulae relating to the first term in On (· · · , z i , z i+1 , · · · )− On (· · · , z i+1 , z i , . . .). Let us start from αt t−2 N t−1 2u+1 t−u−1 ∆(x ) (−c) α1 2 and α2 ,...,α N 0 α1 +2α2 +···+N α N =n
×
t=1
×
αt N
t
δ
j=1 (t) j1 =A j,1 ··· (t) jt =A j,t
N
t=1
1 j
gt,t
zk zj
s
Ts (x −1+s+[ 2 ] z j )
−→ −→ 2s N j∈A(s) Min
z jσ (u)
u=1 σ ∈St σ (1)=1 u =[ t ]+1 2
t=2
T1 (z j )
−→ (1) j∈A Min i+1< j
x 2 z jσ (u+1)
(s) Aj
s=1,...,N j=1,...,αs (1) αs {i,i+1}⊂∪ j=1 Aj
T1 (z j ) · T1 (z i )T1 (z i+1 ) ·
−→ (1) j∈A Min j
×
u=1
1t
(t) j∈A Min (u) k∈A Min
u−t z k − (z i ↔ z i+1 ). gt,u x u−t−2[ 2 ] zj (3.118)
By using the weak # sense relations in Proposition 9 we change the ordering of −→ T1 (z i )T1 (z i+1 ) and T1 (z j ). We have (1) j∈A Min i+1< j
N
α1 +2α2 +···+N α N =n α1 2 and α2 ,...,α N 0
t=2
−
×
N s=1 s =t
×
(−c)
s−1
s−2
i+1< j3 <···< jt (1) j3 ,..., jt ∈A Min (1) {i,i+1}⊂A Min (s) {A j }
αs
∆(x
)
2u+1 s−u−1
(−c)
t−1
u=1
−→ (1) j∈A Min −{i,i+1}
t−2
αt +1 ∆(x
)
2u+1 t−u−1
u=1 t
T1 (z j )Tt (x −1+t−2[ 2 ] z i )
−→ −→ 2s N j∈A(s) Min
s
Ts (x −1+s−2[ 2 ] z j )
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
×
t
σ ∈St σ (1)=1
u=1 u =[ 2t ]+1 j1 =1, j2 =2
×
δ
x 2 z jσ (u+1) z jσ (u)
g1,1 (z k /z j )
×
N
s=2
(1) j∈A Min −{i,i+1} (s) k∈A Min
N
αs
N
gt,s x
δ
x 2 z jσ (u+1)
z jσ (u)
u−t gt,u x u−t−2[ 2 ] z k /z j
2t
Min (u) k∈A Min
(1)
j∈A Min −{i,i+1}
t zi g1,t x −1+t−2[ 2 ] zj
zi
s=2 j∈A(s) Min
gs,s (z k /z j )
j
zj s−t−2[ s−t 2 ]
s
j=1
s zk g1,s x −1+s−2[ 2 ] zj
u=1 σ ∈Ss (s) (s) σ (1)=1 u =[ 2s ]+1 j1 =A j,1 ,··· , js =A j,s
s=2
N s=2
j
×
813
− (z i ↔ z i+1 ).
(3.119)
(s)
We change the summation variables {A j } in
0t N −2
α1 +2α2 +···+N α N =n (s) {A j } s=1,...,N α1 2 and α2 ,...,α N 0 j=1,...,αs (1) {i,i+1}⊂A Min
(3.120)
i+1< j3 <···< jt+2 (1) j3 ,..., jt+2 ∈A Min
(s)
to the following {B j },
0t N −2
β1 +2β2 +···+Nβ N =n β1 ,β2 ,...,β N 0
(s) {B j } s=1,...,N j=1,...,βs (t) B J ={i,i+1, j3 ,..., jt+2 |i+1< j3 <···< jt+2 } for some J
.
(3.121)
Similtaneously let us use the weak sense relations in Proposition 10 on the commutation relations between Ti (z) and T j (w) for i, j 2 and make the ordering s T j (x −1+s−2[ 2 ] z j ), −→ −→ 1s N j∈B (s) Min
(s)
(s)
(s)
where B Min = {Min(B1 ), . . . , Min(Bαs )}. We have exactly the same summation of the second to N th terms of On (. . . , z i , z i+1 , . . .) − On (. . . , z i+1 , z i , . . .) up to signature. Now we have shown theorem for the gl N case. 3.5. Derivation of Laurent-Series Formulae. In this section we give the proof of Theorem 2. 3 case. We start from the integral Proof of Theorem 2. At first we give the proof for the gl representation In in (3.85). Let us pay attention to the poles z J1 = x −2 z 1 , (2 J1 n). We have
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T. Kojima, J. Shiraishi
In =
n
··· C(1)
−
n
···
n
1) C(J
J1 =2
×
dz j √ 2π −1z j j=1
j=1 j = J1
T1 (z j )
−→ 1 jn
1 j
dz j √ 2π −1z j
h(u k − u j ) C x −2 z
1
dz J √ 1 2π −1z J1
h(u k − u j )
1 j
T1 (z j ).
(3.122)
−→ 1 jn
Here we have set C(1) : |x −2 z k | < |z 1 | < |x 2−2s z k |, (2 k n) |x −2 z k | < |z j | < |x 2 z k |, (2 j < k n),
(3.123)
1 ) : |x −2 z k | < |z 1 | < |x 2−2s z k |, (2 k J1 − 1) C(J |x −2 z k | < |z 1 | < |x 2 z k |, (J1 + 1 k n), |x −2 z k | < |z j | < |x 2 z k |, (2 j < k n; j, k = J1 ).
(3.124)
Here C x −2 z 1 is a small circle which encircles x −2 z 1 anticlockwise. The region {(z 1 , z k ) ∈ C2 ||x −2 z k | < |z 1 | < |x 2−2s z k |} for 2 k J1 , are annulus. Hence the defining relations of the deformed W -algebra can be used. Let us change the ordering of T1 (z 1 ) and T1 (z k ) for 2 k J1 − 1, and take the residue of T1 (z 1 )T1 (z J1 ) at z J1 = x −2 z 1 . We have
···
n
1) C(J
=c
×
j=1 j = J1
dz j √ 2π −1z j
···
n
C(J1 )
2 j
j=1 j = J1
C x −2 z
dz j √ 2π −1z j
h 11 (u k − u j )
J 1 −1 j=2
1
dz J √ 1 2π −1z J1
h(u k − u j )
1 j
T1 (z j ) · T2 (x −1 z 1 ) ·
−→ 2 j J1 −1
T1 (z j )
−→ 1 jn
T1 (z j )
−→ J1 +1 jn
n 1 1 . h 12 u 1 − u j − h 21 u j − u 1 + 2 2 j=J1 +1
(3.125) Here we have set C(J1 ) : |x −2 z j | < |z 1 | < |x 4 z j |, (2 j n; j = J1 ), |x −2 z k | < |z j | < |x 2 z k |, (2 j < k n; j, k = J1 ).
(3.126)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
815
Let us pay attention to the poles at z J2 = x 2 z 1 , (2 J2 n; J2 = J1 ). We deform the RHS of (3.125) to the following:
···
c
C(J1 )(J1 )
×
n
h 11 (u k − u j )
n
J2 =2 J2 = J1
×
J 1 −1 j=2
2 j
−c
j=1 j = J1
dz j √ 2π −1z j
···
×
−→ 2 j J1 −1
T1 (z j )
−→ J1 +1 jn
j=J1 +1
j=1 j = J1 ,J2
T1 (z j ) · T2 (x −1 z 1 ) ·
−→ 2 j J1 −1
T1 (z j ) · T2 (x −1 z 1 ) ·
n 1 1 h 12 u 1 − u j − h 21 u j − u 1 + 2 2
n
1 )(J2 ) C(J
dz j √ 2π −1z j
Cx 2 z
1
dz J √ 2 2π −1z J2
T1 (z j )
−→ J1 +1 jn
h 11 (u k − u j )
J 1 −1 j=2
2 j
n 1 1 . h 12 u 1 − u j − h 21 u j − u 1 + 2 2 j=J1 +1
(3.127) Here we have set C(J1 )(J1 ) : |x −2+2s z j | < |z 1 | < |x 4 z j |, (2 j n; j = J1 ), |x −2 z k | < |z j | < |x 2 z k |, (2 j < k n; j, k = J1 ).
(3.128)
For 2 J2 < J1 n we set 1 )(J2 ) : |x −2+2s z j | < |z 1 | < |x 4 z j |, (J2 j J1 − 1), C(J |x −2 z j | < |z 1 | < |x 4 z j |, (2 j J2 − 1 or J1 + 1 j n), |x −2 z k | < |z j | < |x 2 z k |, (2 j < k n; j, k = J1 ).
(3.129)
For 2 J1 < J2 n we set 1 )(J2 ) : |x −2+2s z j | < |z 1 | < |x 4 z j |, (2 j J2 ; j = J1 ), C(J |x −2 z j | < |z 1 | < |x 4 z j |, (J2 + 1 j n), |x −2 z k | < |z j | < |x 2 z k |, (2 j < k n; j, k = J1 ).
(3.130)
The above formulae for this integrand C(J1 )(J2 ) holds for Re(s) > N 3. For N = 2 another treatment should be done. Let us study the first term dz j c ··· √ . C(J1 )(J1 ) j= J 2π −1 1
See the integral contour C(J1 )(J1 ). The region {(z 1 , z j ) ∈ C2 ||x −2+2s z j | < |z 1 | < |x 4 z j |} for j = J1 are annulus. Hence the defining relations of the deformed W -algebra
816
T. Kojima, J. Shiraishi
can be used. By using the weak sense relation in Proposition 9, we deform the first term to the following:
···
c
n
C(J1 )(J1 )
×
j=1 j = J1
dz j √ 2π −1z j
Let us study the second term −c ···
1 )(J2 ) C(J j= J1 ,J2
J2 =2,J1
h 12
j=2 j = J1
2 j
T1 (z j ) · T2 (x −1 z 1 )
−→ 2 jn j = J1
n
h 11 (u k − u j )
1 u1 − u j − . 2
dz j √ 2π −1z j
Cx 2 z
(3.131)
dz J . √ 2 2π −1z J2
1
1 )(J2 ). The region {(z 1 , z j ) ∈ C2 ||x −2+2s z j | < |z 1 | < See the integral contour C(J 4 |x z j |} for 2 j J1 , is an annulus for Re(s) > N = 3. Hence the defining relations of the deformed W -algebra can be used. Let us change the ordering of T1 (z J2 ) and T1 (z k ) and make the product of the operators T1 (z J2 )T2 (x −1 z 1 ) or T2 (x −1 z 1 )T1 (z J2 ). Let us take the residue of T1 (z J2 )T2 (x −1 z 1 ) and T2 (x −1 z 1 )T1 (z J2 ) at z J2 = x 2 z 1 by regarding the weak sense equation in Proposition 9. We have c2 ∆(x 3 )
n
···
C(J1 )(J2 )
J2 =2 J2 = J1
×
T1 (z j )
−→ J1 +1 jn j = J2
n
dz j √ 2π −1z j
j=1 j = J1 ,J2
h 11 (u k − u j )
2 j
T1 (z j ) · T3 (z 1 )
−→ 2 j J1 −1 j = J2
J 1 −1
n $ % h 13 u 1 − u j h 31 (u j − u 1 ).
j=2 j = J2
j=J1 +1 j = J2
(3.132) Here we have set C(J1 )(J2 ) : |x −4+2s z j | < |z 1 | < |x 4−2s z j |, (2 j n; j = J1 , J2 ), |x −2 z k | < |z j | < |x 2 z k |, (2 j < k n; j = J1 , J2 ).
(3.133)
This integral contur C(J1 )(J2 ) holds only for the N = 3 case. For the N ≥ 4 case another treatment should be done. The region {(z 1 , z j ) ∈ C2 ||x −4+2s z j | < |z 1 | < |x 4 z j |} is an annulus. We move T3 (z j ) to the right, and get c2 ∆(x 3 )
n J2 =2 J2 = J1
×
2 j
···
n
C(J1 )(J2 )
h 11 (u k − u j )
j=1 j = J1 ,J2
n j=2 j = J1 ,J2
dz j √ 2π −1z j
$ % h 13 u 1 − u j .
T1 (z j ) · T3 (z 1 )
−→ 2 jn j = J1 ,J2
(3.134)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
817
Summing up every term, we have ⎛ ⎜ In = ⎜ ⎝
× ×
C{A(1) ,A(2) ,A(3) ,A
T1 (z j )
−→ (1) j∈Ac ∪A Min
×
A(3) ={1, j,k} A(1) =A(2) =φ
c}
(2)
(1)
k∈A Min j∈A Min ∪Ac
(1)
(2)
(3)
j∈Ac ∪A Min ∪A Min ∪A Min
T2 (x −1 z j )
−→ (2) j∈A Min
+2!c2 ∆(x 3 )
A(2) ={1, j} A(1) =A(3) =φ
···
−c
A(1) ={1} A(2) =A(3) =φ
⎞
⎟ ⎟ ⎠ dz j √ 2π −1z j
T3 (z j )
−→ (3) j∈A M I n
1 h 12 u k − u j − 2 (3)
h 11 (u k − u j )
j
(1) j,k∈Ac ∪A Min
$ % h 13 u k − u j .
(1)
k∈A Min j∈A Min ∪Ac
(3.135) Here we have set Ac = {1, 2, . . . , n} − A(1) ∪ A(2) ∪ A(3) . We have set A(t) Min = { j1 } for A(t) = { j1 < j2 < · · · < jt }. Here we have set C{A(1) , A(2) , A(3) , Ac } by |x −2 z k | < |z 1 | < |x 2−2s z k |, (k ∈ Ac for A(1) = φ), |x −2+2s z k | < |z 1 | < |x 4 z k |, (k ∈ Ac for A(2) = φ), |x −4+2s z k | < |z 1 | < |x 4−2s z k |, (k ∈ Ac for A(3) = φ), |x −2 z k | < |z j | < |x 2 z k |, ( j < k; j, k ∈ Ac ).
(3.136)
# −2 z , and −→ T1 (z j ). Let us take the residue at z J = x Next we deform the part 2 j∈Ac continue similar calculations as above. We use the weak sense equations in Proposition 10 and change the ordering of T2 (x −1 w) and T3 (w), without taking residues. Now we 3 . The proof for gl have shown the theorem for gl N is similar. 3.6. Proof of [Im , In ] = 0. In this section we show the commutation relation [Im , In ] = 0. Proposition 12. The folloing theta identity holds: n n+m σ ∈Sm+n j=1 k=n+1
m n+m 1 1 = , h(u σ (k) − u σ ( j) ) h(u σ (k) − u σ ( j) ) σ ∈Sm+n j=1 k=m+1
(3.137) n n+m σ ∈Sm+n j=1 k=n+1
1 = h ∗ (u σ (k) − u σ ( j) )
m n+m σ ∈Sm+n j=1 k=m+1
1 . h ∗ (u σ (k) − u σ ( j) ) (3.138)
Here h(u) and h ∗ (u) are given in (3.84).
818
T. Kojima, J. Shiraishi
This theta identity was written in [10] without proof. We have shown this theta identity by induction, in [3]. Hence we omit details here. Proposition 13. The following weak sense equation holds:
On (z 1 , . . . , z n )Om (z n+1 , . . . , z n+m ) ∼
1 jn n+1kn+m
1 Om+n (z 1 , . . . , z n+m ). g11 (z k /z j ) (3.139)
Proof. This is a direct consequence of the following explicit folrmulae: On+m (z 1 , . . . , z n+m ) ∼ g11 (z k /z j )On (z 1 , . . . , z n )Om (z n+1 , . . . , z n+m ) 1 jn n+1kn+m
N
α,α2 ,...,α N 0 α1 +2α2 +···+N α N =n 22α2 +···+N α N n
t=1
+
(−c)
N
t=1
j
gt,t
Here the summation
∆(x
αt N
(s) (s) {L j } s=1,...,N ,{R j } s=1,...,N j=1,...,αs j=1,...,αs N (s) (s) s=2 |L 1 ||R1 |1
×
αt
t−1
zk zj
)
2u+1 t−u−1
u=1
×
t−1
t=1
j=1 (t) j1 =A j,1 ··· (t) jt =A j,t
δ
u=1 σ ∈St σ (1)=1 u =[ t ]+1 2
1t
(t) j∈A Min (u) k∈A Min
(s) (s) {L j } s=1,...,N ,{R j } s=1,...,N j=1,...,αs j=1,...,αs N (s) (s) s=2 |L 1 ||R1 |1
(s)
t
x 2 z jσ (u+1)
z jσ (u)
u t zk . gt,u x u−t−2[ 2 ]+2[ 2 ] zj
(3.140)
is taken over the conditions that
(s)
(s)
N s ∪αj=1 L j = {1, 2, . . . , m}, L i ∩ L j = φ, (i = j), ∪s=1 (s)
(s)
Min(L 1 ) < Min(L 2 ) < · · · < Min(L (s) αs ), (s)
(s)
N s ∪αj=1 R j = {m + 1, m + 2, . . . , m + n}, Ri ∪s=1
(s)
∩ R j = φ, (i = j),
). Min(R1(s) ) < Min(R2(s) ) < · · · < Min(Rα(s) s (s)
(s)
(s)
(s)
(3.141) (s)
Here we have set A j = L j ∪ R j . We have set A j,k = jk for A j = { j1 < j2 < (s)
(s)
(s)
(s)
· · · < js }, and A Min = {A1,1 , A2,1 , ·, Aαs ,1 }. We want to point out that every term has the delta-function δ(x 2 z k /z j ), (1 of the summation {L (s) } (s) s=1,...,N ,{R j } s=1,...,N j j=1,...,αs j=1,...,α # s j m, m + 1 k m + n). Dividing g11 (z k /z j ) on both sides and using 1 jn n+1kn+m
1/g11 (x −2 ) = 0, we have shown this proposition.
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
819
Proof of Theorem 1. At first we restrict ourself to the regime, Re(s) > N and Re(r ) < 0, in order to use the power series formulae of the local integrals of motion, In . In Proposition 3 we have shown for σ ∈ Sn ,
s(z k /z j )On (z 1 , . . . , z n ) =
1 j
s(z σ (k) /z σ ( j) )On (z σ (1) , . . . , z σ (n) ).
1 j
(3.142) Hence we have I n · Im ⎡ ⎢ =⎣
⎤
s(z k /z j )On (z 1 , . . . , z n )
1 j
1 (n + m)!
⎥ s(z k /z j )Om (z n+1 , . . . , z n+m )⎦
n+1 j
⎡ ⎢ =⎣
n n+m
σ ∈Sn+m j=1 k=n+1
1 h(u σ (k) − u σ ( j) )
1,z 1 ···z n+m
s(z k /z j )
1 j
× On+m (z 1 , . . . , z n+m )]1,z 1 ···z n+m .
(3.143)
Hence the commutation relation In · Im = Im · In is reduced to the theta identity in Proposition 12: n n+m σ ∈Sm+n j=1 k=n+1
m n+m 1 1 = . h(u σ (k) − u σ ( j) ) h(u σ (k) − u σ ( j) ) σ ∈Sm+n j=1 k=m+1
(3.144) Proof of the commutation relation [Im∗ , In∗ ] = 0 is given in a similar way. Here we omit the details for In∗ .
4. Nonlocal Integrals of Motion In this section we give explicit formulae of the nonlocal integrals of motion. We study the generic case : 0 < x < 1, Re(r ) = 0 and s ∈ C (resp. 0 < x < 1, Re(r ∗ ) = 0 and s ∈ C).
4.1. Nonlocal Integrals of Motion. We explicitly construct the nonlocal integrals of motion and state the main results for N 3. The result for N = 2 is summarized in [3]. Definition 8. • For the regime Re(r ) > 0 and 0 < Re(s) < N , we define a family of operators Gm , (m = 1, 2, . . .) by
820
T. Kojima, J. Shiraishi
Gm =
m ( N
(t)
dz j √ (t) 2π −1z j
t=1 j=1 C (1) ×F1 (z 1 ) · · · N
×
(2)
(N )
(1) (2) (N ) F1 (z m )F2 (z 1 ) · · · F2 (z m ) · · · FN (z 1 ) · · · FN (z m ) (t) (t) (t) (t) ui − u j u j − ui − 1 r
t=1 1i< j m
r
m m s (1) s ) u i(t) − u (t+1) u i − u (N + 1 − + j j N r N r t=1 i, j=1 i, j=1 ) ) ) ⎛ ⎞ ) m ) m ) m (2) ) ) (N ) (1) ) ×ϑ ⎝ u j )) u j )) · · · )) uj ⎠. ) j=1 ) j=1 ) j=1 N −1
(4.145)
Here we have set the theta function ϑ(u (1) |u (2) | · · · |u (N ) ) by ϑ(u (1) | · · · |u (t) + r | · · · |u (N ) ) = ϑ(u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ), (4.146) (1) (t) (N ) ϑ(u | · · · |u + r τ | · · · |u ) (4.147) = e−2πiτ +
2πi r
√ (u t−1 −2u t +u t+1 + r (r −1)Pαt )
ϑ(u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ),
ϑ(u (1) + k| · · · |u (N ) + k) = ϑ(u (1) | · · · |u (N ) ), (k ∈ C), η(ϑ(u (1) | · · · |u (N ) )) = ϑ(u (N ) |u (1) | · · · |u (N −1) ).
(4.148) (4.149)
Here the integral contour C is given by |x N z (t+1) | < |z i(t) | < |x −2+ N z (t+1) |, (1 t N − 1, 1 i, j m), j j 2s
2s
(4.150) 2s (1) |x 2− N z j |
<
(N ) |z i |
<
2s (1) |x − N z j |,
(1 i, j m).
(4.151)
For generic s ∈ C, the definition of Gn should be understood as an analytic continuation. We call the operator Gn the nonlocal integral of motion for the deformed W -algebra Wq,t ( gl N ). • For the regime Re(r ) < 0 and 0 < Re(s) < N , we define a family of operators Gm , (m = 1, 2, . . .) by Gm =
m ( N
(t)
dz j √ 2π −1z (t) j
t=1 j=1 C (1) ×F1 (z 1 ) · · ·
(2)
(N )
(1) (2) (N ) F1 (z m )F2 (z 1 ) · · · F2 (z m ) · · · FN (z 1 ) · · · FN (z m )
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra N
×
821
u i(t) − u (t) j
t=1 1i< j m
−r
(t) u (t) j − ui + 1
−r
m m s (1) s (t) (t+1) (N ) ui − u j ui − u j − 1 + − N −r N −r t=1 i, j=1 i, j=1 ) ) ) ⎞ ⎛ ) m ) m ) m (2) ) ) (N ) (1) )) ) ⎝ (4.152) ×ϑ uj ) u j ) · · · )) uj ⎠. ) ) ) j=1 j=1 j=1 N −1
Here we have set the theta function ϑ(u (1) |u (2) | · · · |u (N ) ) by ϑ(u (1) | · · · |u (t) + r | · · · |u (N ) ) = ϑ(u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ), (4.153) (1) (t) (N ) ϑ(u | · · · |u − r τ | · · · |u ) (4.154) = e−2πiτ −
2πi r
√ (u t−1 −2u t +u t+1 + r (r −1)Pαt )
ϑ(u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ),
ϑ(u (1) + k| · · · |u (N ) + k) = ϑ(u (1) | · · · |u (N ) ), (k ∈ C),
(4.155)
η(ϑ(u (1) | · · · |u (N ) )) = ϑ(u (N ) |u (1) | · · · |u (N −1) ).
(4.156)
Here the integral contour C is given by 2s
(t+1)
|x −2+ N z j
(t)
2s
(t+1)
| < |z i | < |x N z j
|, (1 t N − 1, 1 i, j m), (4.157)
|x
− 2s N
z (1) j |
<
|z i(N ) |
< |x
2− 2s N
z (1) j |,
(1 i, j m).
(4.158)
For generic s ∈ C, the definition of Gn should be understood as analytic continuation. We call the operator Gn the nonlocal integrals of motion for the deformed W -algebra Wq,t ( gl N ). • For Re(r ∗ ) > 0 and 0 < Re(s) < N , we define a family of operators Gm∗ , (m = 1, 2, . . .) by N m ( dz (t) j ∗ Gm = √ (t) t=1 j=1 ∗ 2π −1z j C (1) ×E 1 (z 1 ) · · · N
×
(2)
(N )
(1) (2) (N ) E 1 (z m )E 2 (z 1 ) · · · E 2 (z m ) · · · E N (z 1 ) · · · E N (z m ) (t) (t) (t) u i − u (t) u − u + 1 j j i ∗ ∗
t=1 1i< j m
r
r
m m s (1) s (t) (t+1) (N ) ui − u j ui − u j − 1 + − N r∗ N r∗ t=1 i, j=1 i, j=1 ) ) ) ⎛ ⎞ ) m ) m ) m ) (N ) ) (2) ) ) )···) ×ϑ ∗ ⎝ u (1) u uj ⎠. (4.159) j ) j ) ) ) j=1 ) j=1 ) j=1 N −1
822
T. Kojima, J. Shiraishi
Here we have set the theta function ϑ ∗ (u (1) |u (2) | · · · |u (N ) ) by ϑ ∗ (u (1) | · · · |u (t) + r | · · · |u (N ) ) = ϑ ∗ (u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ), (4.160) ϑ ∗ (u (1) | · · · |u (t) + r ∗ τ | · · · |u (N ) ) (4.161) 2πi
= e−2πiτ + r ∗
√ (u t−1 −2u t +u t+1 + r (r −1)Pαt ) ∗
ϑ (u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ),
ϑ ∗ (u (1) + k| · · · |u (N ) + k) = ϑ ∗ (u (1) | · · · |u (N ) ), (k ∈ C), ∗
η(ϑ (u
(1)
| · · · |u
(N )
∗
)) = ϑ (u
(N )
|u
(1)
| · · · |u
(N −1)
(4.162)
).
(4.163)
Here the integral contour C ∗ is given by | < |z i(t) | < |x N z (t+1) |, (1 t N − 1, 1 i, j m), |x −2+ N z (t+1) j j 2s
2s
(4.164) 2s |x − N z (1) j |
<
|z i(N ) |
<
2s |x 2− N z (1) j |,
(1 i, j m).
(4.165)
For generic s ∈ C, the definition of Gn should be understood as analytic continuation. We call the operator Gn the nonlocal integrals of motion for the deformed W -algebra Wq,t ( gl N ). • For Re(r ∗ ) < 0 and 0 < Re(s) < N , we define a family of operators Gm∗ , (m = 1, 2, . . .) by Gm∗
=
m ( N
(t)
dz j √ (t) 2π −1z j
t=1 j=1C ∗ ×E 1 (z 1(1) ) · · · N
×
(1) (2) (N ) E 1 (z m )E 2 (z 1(2) ) · · · E 2 (z m ) · · · E N (z 1(N ) ) · · · E N (z m ) (t) (t) (t) (t) ui − u j u j − ui − 1 ∗ ∗ −r
t=1 1i< j m
−r
m m s s (1) (N ) u i(t) − u (t+1) u + 1 − − u + j i j N −r ∗ N −r ∗ t=1 i, j=1 i, j=1 ) ) ) ⎛ ⎞ ) ) ) m m m ) ) ) (1) (2) (N ) ×ϑ ∗ ⎝ u j )) u j )) · · · )) uj ⎠. (4.166) ) ) ) j=1 j=1 j=1 N −1
Here we have set the theta function ϑ ∗ (u (1) |u (2) | · · · |u (N ) ) by ϑ ∗ (u (1) | · · · |u (t) + r | · · · |u (N ) ) = ϑ ∗ (u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ), (4.167) ϑ ∗ (u (1) | · · · |u (t)r ∗ τ | · · · |u (N ) ) (4.168) 2πi
= e−2πiτ − r ∗
√ (u t−1 −2u t +u t+1 + r (r −1)Pαt ) ∗
ϑ (u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ),
ϑ ∗ (u (1) + k| · · · |u (N ) + k) = ϑ ∗ (u (1) | · · · |u (N ) ), (k ∈ C), η(ϑ ∗ (u (1) | · · · |u (N ) )) = ϑ ∗ (u (N ) |u (1) | · · · |u (N −1) ).
(4.169) (4.170)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
823
Here the integral contour C ∗ is given by 2s
(t+1)
|x N z j
(t)
2s
(t+1)
| < |z i | < |x −2+ N z j
|, (1 t N − 1, 1 i, j m), (4.171)
|x
2− 2s N
(1) zj |
<
(N ) |z i |
< |x
− 2s N
(1) z j |,
(1 i, j m).
(4.172)
For generic s ∈ C, the definition of Gn should be understood as analytic continuation. We call the operator Gn the nonlocal integrals of motion for the deformed W -algebra Wq,t ( gl N ). We summarize explicit formulae for the integrand function ϑ(u (1) |u (2) | · · · |u (N ) ). Proposition 14. For α1 , α2 , . . . , α N ∈ C and Re(r ) > 0, we set √ ϑα (u (1) |u (2) | · · · |u (N ) ) = [u (1) − u (2) − rr ∗ P¯2 + α1 P¯1 + α2 P¯2 + · · · + α N P¯ N ]r √ ×[u (2) − u (3) − rr ∗ P¯3 + α1 P¯1 + α2 P¯2 + · · · + α N P¯ N ]r ×··· √ ×[u (N ) −u (1) − rr ∗ P¯1 + α1 P¯1 + α2 P¯2 + · · · + α N P¯ N ]r . (4.173) This theta function ϑα (u (1) |u (2) | · · · |u (N ) ) satisfies the conditions (4.146), (4.147), (4.148) and (4.149). The following are some of the Main Results. Theorem 4. The nonlocal integrals of motion Gn commute with each other, [Gm , Gn ] = 0, (m, n = 1, 2, . . .).
(4.174)
The nonlocal integrals of motion Gn∗ commute with each other, [Gm∗ , Gn∗ ] = 0, (m, n = 1, 2, . . .).
(4.175)
Theorem 5. The nonlocal integrals of motion Gn and Gn∗ commute with each other for regime 0 < Re(r ) and Re(r ∗ ) < 0, [Gm , Gn∗ ] = 0, (m, n = 1, 2, . . .).
(4.176)
Theorem 6. The local integrals of motion In , In∗ and nonlocal integrals of motion Gm , Gm∗ commute with each other, [In , Gm ] = 0, [In , Gm∗ ] = 0, [In∗ , Gm ] = 0, [In∗ , Gm∗ ] = 0, (m, n = 1, 2, . . .).
(4.177) (4.178)
Comment (CFT-limit). We would like to give some comments on relations between the elliptic integrals of motion and those of CFT. Our integrals of motion can be regarded as elliptic deformation of those for the Virasoro algebra and the W3 algebra. First, we would like to comment on the nonlocal integrals of motion Gm . We demand that together with x → 1, the parameter u tends to a limiting value in such a way that z = x 2u is fixed. We call this limit CFT-limit. The elliptic screening currents F j (z), E j (z) of this paper becomes those of the CFT [1,2], in the CFT-limit. The theta function in the integrand of
824
T. Kojima, J. Shiraishi
the nonlocal integrals of motion Gm , Gm∗ degenerates trivial scalars. Hence the nonlocal integrals of motion Gm of this paper becomes those of CFT in the CFT limit. We have checked this relation for N = 2, 3 cases. However there does not exist any paper on integrals of motion of CFT for the N 4 case, we have already obtained conjectural formulae of T-Q-operators for general W N -algebra. We checked that this degeneration holds for the general W N -algebra, in the CFT-limit. In the future we would like to report on these T-Q-operators of CFT at another place. Second, we would like to comment on the local integrals of motion Im . The limit of this case is more complicated. See details in [3]. However we have already known the general formulae for the elliptic version of the local integrals of motion Im ; unfortunately, only a few leading terms of the local integrals of motion for the Virasoro algebra and the W3 -algebra are written in [1,2]. We have checked that a few leading terms of our local integrals of motion I1 and I2 for 2 ) degenerate to the known results for the Virasoro algebra in the special limit [3]. Wq,t (gl 4.2. Proof of [Gm , Gn ] = 0. In this section we study the commutation relations [Gm , Gn ] = 0 for Re(r ) > 0. We omit details for other cases, because they are similar. Proposition 15. For Re(r ) > 0 we have ) ) ) ⎛ ⎞ ) ) ) m m m ) ) ) (1) (2) (N ) ϑα ⎝ ··· u σ1 ( j) )) u σ2 ( j) )) · · · )) u σ N ( j) ⎠ ) ) ) σ1 ∈Sm+n σ2 ∈Sm+n σ N ∈Sm+n j=1 j=1 j=1 ) ) ) ⎞ ) m+n ) m+n ) ) ) ) (1) (2) (N ) × ϑβ ⎝ u σ1 ( j) )) u σ2 ( j) )) · · · )) u σ N ( j) ⎠ ) j=m+1 ) j=m+1 ) j=m+1 ⎛
×
m+n
(t) (t+1) (t+1) (t) u σt (i) − u σt+1 ( j) − Ns u σt+1 (i) − u σt ( j) + 1 − r (t) (t) (t) (t) u σt (i) − u σt ( j) u σt ( j) − u σt (i) − 1 j=m+1
N m m+n t=1 i=1
r
s N r
r
) ) ) ⎛ ⎞ ) n ) n ) n ) (N ) ) (2) ) (1) ϑβ ⎝ = ··· u σ1 ( j) )) u σ2 ( j) )) · · · )) u σ N ( j) ⎠ ) ) ) σ1 ∈Sm+n σ2 ∈Sm+n σ N ∈Sm+n j=1 j=1 j=1 ) ) ) ⎛ ⎞ ) m+n ) ) m+n (1) ) m+n (2) ) ) (N ) ) ) ⎝ ×ϑα u σ1 ( j) ) u σ2 ( j) ) · · · )) u σ N ( j) ⎠ ) ) ) j=n+1 j=n+1 j=n+1 ×
(t) (t+1) (t+1) (t) u σt (i) − u σt+1 ( j) − Ns u σt+1 (i) − u σt ( j) + 1 − r (t) (t) (t) (t) u σt (i) − u σt ( j) u σt ( j) − u σt (i) − 1 j=n+1
N n m+n t=1 i=1
r
s N r
. (4.179)
r
ϑβ (u (1) |u (2) | · · · |u (N ) ) are given by Here ϑα (u (1) |u (2) | · · · |u (N ) ) and ϑα (u (1) | · · · |u (t) + r | · · · |u (N ) ) = ϑα (u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ), (4.180) ϑα (u (1) | · · · |u (t) + r τ | · · · |u (N ) ) (4.181)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
= e−2πiτ +
2πi r
825
√ (u t−1 −2u t +u t+1 + r (r −1)Pαt )+να,t ϑα (u (1) | · · · |u (t) | · · · |u (N ) ),
(1 t N ),
ϑβ (u (1) | · · · |u (t) + r | · · · |u (N ) ) = ϑβ (u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ), (4.182) ϑβ (u (1) | · · · |u (t) + r τ | · · · |u (N ) ) =
(4.183)
√ 2πi ϑβ (u (1) | · · · |u (t) | · · · |u (N ) ), e−2πiτ + r (u t−1 −2u t +u t+1 + r (r −1)Pαt )+νβ,t
(1 t N ).
Here να,t , νβ,t ∈ C, (1 t N ). Proof. In order to consider the elliptic function, we divide the above theta identity by ϑγ with νγ ,t ∈ C, (1 t N ): ϑγ (u (1) | · · · |u (t) | · · · |u (N ) ), (1 t N ), (4.184) ϑγ (u (1) | · · · |u (t) + r | · · · |u (N ) ) = (4.185) ϑγ (u (1) | · · · |u (t) + r τ | · · · |u (N ) ) = e−2πiτ +
2πi r
Let us set LHS(m, n) =
√ (u t−1 −2u t +u t+1 + r (r −1)Pαt )+νγ ,t ϑγ (u (1) | · · · |u (t) | · · · |u (N ) ),
···
K 1 ∪K 1c ={1,2,...,n+m} |K 1 |=m,|K 1c |=n
(1 t N ).
(4.186)
c ={1,2,...,n+m} K N ∪K N c |=n |K N |=m,|K N
) ) ) ) ⎞ ⎛ ⎞ ) ) ) ) ) ) ) ) (1) (N ) (1) (N ) ϑβ ⎝ ϑα ⎝ u j )) · · · )) u j ⎠ u j )) · · · )) uj ⎠ ) j∈K N ) j∈K c ) ) j∈K 1 j∈K 1c N ) ) ⎛ ⎞ × )m+n ) m+n (1) ) ) (N ) ϑγ ⎝ u j )) · · · )) uj ⎠ ) j=1 ) j=1 (t) (t+1) s (t+1) s (t) u i −u j ui +1− −uj + N r N r N c i∈K t j∈K t+1 i∈K t+1 j∈K tc × , (t) (t) (t) u i − u (t) u − u − 1 t=1 j j i ⎛
RHS(m, n) =
K 1 ∪K 1c ={1,2,...,n+m} |K 1 |=n,|K 1c |=m
i∈K t j∈K tc
···
r
r
(4.187)
c ={1,2,...,n+m} K N ∪K N c |=m |K N |=n,|K N
) ) ) ) ⎞ ⎛ ⎞ ) ) ) ) ) ) ) ) (1) (N ) (1) (N ) ϑα ⎝ ϑβ ⎝ u j )) · · · )) u j ⎠ u j )) · · · )) uj ⎠ ) ) ) ) c c j∈K 1 j∈K N j∈K 1 j∈K N ) ) ⎛ ⎞ × ) ) m+n (1) ) (N ) )m+n ϑγ ⎝ u j )) · · · )) uj ⎠ ) ) j=1 j=1 (t) (t+1) s (t+1) (t) s u i −u j ui +1− −u j + N r N r N c i∈K t j∈K t+1 i∈K t+1 j∈K tc × . (t) (t) (t) (t) ui − u j u j − ui − 1 t=1 ⎛
i∈K t j∈K tc
r
r
826
T. Kojima, J. Shiraishi (t)
(t)
(t)
(t)
Candidates of poles of both LHS(m, n) and RHS(m, n) are u i = u j and u i = u j +1 (t)
(t)
and ϑγ = 0. Let us show that the points u i = u j are regular. Take the residue of the (1)
(1)
LHS(m, n) at u 1 = u 2 . We have
1
Resu (1) =u (1) 1
×
+ (1) (1) (1) (1) (1) (1) (1) (1) [u 1 − u 2 ]r [u 2 − u 1 − 1]r [u 1 − u 2 ]r [u 1 − u 2 − 1]r ···
2
L 1 ∪L c1 ={3,4,...,n+m} K 2 ∪K 2c ={1,2,...,n+m} |L 1 |=m−1,|L c1 |=n−1 |K 1 |=m,|K 1c |=n
ϑα ( ×
1
(1)
uj |···|
j∈L 1 ∪{1}
c ={1,2,...,n+m} K N ∪K N c |=n |K N |=m,|K N
(N ) u j ) ϑβ (
m+n
(1)
uj |···|
j∈L c1 ∪{1}
j∈K N
ϑγ (
(1)
uj |···|
j=1
n+m
(N )
uj )
c j∈K N
(N )
uj )
j=1
(1) s (2) s u 1 − u (2) u i − u (1) 2 + j +1− N r N r i∈K 2 j∈K 2c × (1) (1) (1) (1) u1 − u j u j − u1 − 1 r
j∈L c1
r
(N ) s (1) s (1) (N ) ui − u2 + 1 − u1 − u j + N r N r c j∈K N j∈K N × (1) (1) (1) (1) ui − u2 u2 − ui − 1 r
j∈L 1
r
(1) s (2) s (2) (1) ui − u j + 1 − ui − u j + N r N r i∈L 1 j∈K 2c i∈K 2 j∈L c1 × (1) (1) ui − u j r
i∈L 1 j∈L c1
(N ) s (1) s (1) (N ) ui − u j + 1 − ui − u j + N r N r c j∈K N j∈L c1 i∈L 1 j∈K N × (1) (1) u j − ui − 1
(4.188)
r
i∈L 1 j∈L c1
(t) s (t+1) s u i − u (t+1) ui +1− − u (t) j j + N r N r N c i∈K t j∈K t+1 i∈K t+1 j∈K tc × = 0. (t) (t) (t) (t) ui − u j u j − ui − 1 t=2 r
i∈K t j∈K tc
r
Because the first term : Resu (1) =u (1) 1
2
1 (1) (1) (1) (1) [u 1 −u 2 ]r [u 2 −u 1 −1]r
+
1 (1) (1) (1) (1) [u 1 −u 2 ]r [u 1 −u 2 −1]r
= 0,
we have Resu (1) =u (1) LHS(m, n) = 0. Because LHS(m, n) is symmetric with respect 1
2
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra (1)
(1)
827
(1)
to variables u 1 , u 2 , . . . , u m+n , we have Resu (1) =u (1) LHS(m, n) = 0 for 1 i = i
j
j m + n. In the same manner as above, we conclude that points u i(t) = u (t) j of LHS(m, n) and RHS(m, n) for 1 t N , 1 i = j m + n are regular. Let us show LHS(m, n) = RHS(m, n) by induction for m + n. Candidates of poles are only u i(t) = u (t) j + 1, 1 t N and 1 i = j m + n. We assume 1 m < n without losing generality. (The case m = n is trivial.) At first we show the starting point 1 = m < n : LHS(1, n) = RHS(1, n) . By straightforward calculations, we have Resu (1) =u (1) +1 · · · Resu (N ) =u (N ) +1 LHS(1, n) 2 1 1 2 s (t+1) s (t) (t+1) (t) N u1 − u2 u1 +1− − u2 + N r N r Resu (t) =u (t) +1 = (t) (t) (t) (t) 2 1 u1 − u2 u2 − u1 − 1 t=1 r r s s (t) (t+1) (t+1) (t) N n+1 u u − u + 1 − − u + 1 1 j j N r N r × (t) (t) (t) (t) u1 − u j u j − u1 − 1 t=1 j=3 r r ) ) ⎛ ⎞ ) ) n+1 n+1 ) (1) (N ) (1) ) (N ) ϑα (u 1 | · · · |u 1 ) ϑβ ⎝ u j )) · · · )) uj ⎠ ) ) j=2 j=2 ) ) ⎛ ⎞ , (4.189) × ) n+1 ) n+1 (1) ) ) (N ) ϑγ ⎝ u j )) · · · )) uj ⎠ ) ) j=1 j=1 Resu (1) =u (1) +1 · · · Resu (N ) =u (N ) +1 RHS(1, n) 2 1 1 2 s (t+1) s (t) (t+1) (t) N u1 − u2 u1 +1− − u2 + N r N r Resu (t) =u (t) +1 = (t) (t) (t) (t) 2 1 u u − u − u − 1 t=1 1 2 2 1 r r s s (t) (t+1) (t+1) (t) n+1 u N ui +1− − u2 + i − u2 N r N r × (t) (t) (t) (t) u u − u − u − 1 t=1 i=3 2 r 2 i i r ) ) ⎛ ⎞ ) ) n+1 n+1 ) ) ) ⎜ (1) ) (1) (N ) (N ) ⎟ ϑα (u 2 | · · · |u 2 ) ϑβ ⎝ uj )···) uj ⎠ ) ) j=1 ) j=1 ) j =2 j =2 ) ) ⎛ ⎞ . (4.190) × ) ) n+1 n+1 ) ) (1) (N ) ϑγ ⎝ u j )) · · · )) uj ⎠ ) ) j=1 j=1 (t)
(t)
Upon specialization u 2 = u 1 +1, (1 t N ), we have =
(t)
(t+1)
(t+1) (t) +1− Ns ]r [u i −u 2 + Ns ]r (t) (t) (t) (t) [u i −u 2 ]r [u 2 −u i −1]r (1) (1) u 2 =u 1 +1
[u i −u 2
LHS(1, n) = Res
(t)
(t+1)
(t+1) (t) +1− Ns ]r [u 1 −u j + Ns ]r (t) (t) (t) (t) [u 1 −u j ]r [u j −u 1 −1]r
[u 1 −u j
. Hence we have Resu (1) =u (1) +1 · · · Resu (N ) =u (N ) +1 2
1
2
1
· · · Resu (N ) =u (N ) +1 RHS(1, n), using the periodic condition 2
1
(1) (N ) (1) (N ) ϑα (u 1 + k| · · · |u 1 + k) = ϑα (u 1 | · · · |u 1 ). Both LHS(1, n) and RHS(1, n) are
828
T. Kojima, J. Shiraishi (t)
(t)
(t)
symmetric with respect to u 1 , u 2 , . . . , u n+1 , we have Resu (1) =u (1) +1 · · · Resu (N ) =u (N ) +1 LHS(1, n) = Resu (1) =u (1) +1 · · · i1
j1
iN
jN
i1
j1
×Resu (N ) =u (N ) +1 RHS(1, n), iN
(4.191)
jN
for 1 i t = jt n + 1 and 1 t N . After taking the residues finitely many times, every residue relation which comes from LHS(1, n) = RHS(1, n), is reduced to the above (4.191). Hence we have shown the starting relations n > m = 1. Second, we show the general n > m 1. We assume the relation LHS(m −1, n−1) = RHS(m −1, n−1). (t) Let us take the residue at u (t) 1 = u 2 + 1, (1 t N ). We have Resu (1) =u (1) +1 · · · Resu (N ) =u (N ) +1 (LHS(m, n) − RHS(m, n)) 2 1 1 2 (t) N u u (t+1) − u (t+1) + 1 − Ns − u (t) 1 2 1 2 + r Resu (t) =u (t) +1 = (t) (t) (t) 2 1 u (t) u − u − u − 1 t=1 1 2 2 1 r
(t) (t+1) N m+n u (t+1) + 1 − Ns − u (t) u1 − u j 1 j + r × (t) (t) u (t) u (t) t=1 j=3 1 −uj j − u1 − 1
×
r
L 1 ∪L c1 ={3,4,...,n+m} L 2 ∪L c2 ={3,4,...,n+m} |L 1 |=m−1,|L c1 |=n−1 |L 2 |=m−1,|L c2 |=n−1
···
s N r
r
s N r
r
L N ∪L cN ={3,4,...,n+m} |L N |=m−1,|L cN |=n−1
) ) ) ) ⎞ ⎛ ⎞ ) ) ) ) ) ) ) ) )⎠ ⎝ )⎠ ) ) ) ) ϑβ ϑα ⎝ u (1) u (N u (1) u (N j )···) j j )···) j ) j∈L N ∪{1} ) j∈L c ∪{1} ) ) j∈L 1 ∪{1} j∈L c1 ∪{1} N ) ) ⎛ ⎞ × ) ) m+n m+n (1) ) ) (N ) ϑγ ⎝ u j )) · · · )) uj ⎠ ) j=1 ) j=1 ⎛
⎛
(t) s (t+1) s (t+1) (t) ui − u j ui +1− −uj + ⎜N N r N r ⎜ i∈L t j∈L ct+1 i∈L t+1 j∈L ct ⎜ ×⎜ (t) (t) (t) (t) ⎝t=1 ui − u j u j − ui − 1 r
i∈L t j∈L ct+1
r
(t) s (t+1) s ⎞ (t+1) (t) ui − u j ui +1− −uj + N r N r⎟ N ⎟ i∈L ct j∈L t+1 i∈L ct+1 j∈L t ⎟ = 0. − (t) ⎟ (t) (t) (t) ⎠ u u − u − u − 1 t=1 i j j i i∈L ct j∈L t+1
r
r
(4.192)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
829
We have already used the hypothesis for (m −1, n −1). Both LHS(m, n) and RHS(m, n) (t) (t) are symmetric with respect to u (t) 1 , u 2 , . . . , u m+n , we have Resu (1) =u (1) +1 · · · Resu (N ) =u (N ) +1 LHS(m, n) = Resu (1) =u (1) +1 · · · i1
j1
iN
jN
i1
j1
×Resu (N ) =u (N ) +1 RHS(m, n), iN
(4.193)
jN
for 1 i t = jt m + n and 1 t N . After taking the residues finitely many times, every residue relation which comes from LHS(m, n) = RHS(m, n), is reduced to the above (4.193). Hence we have shown LHS(m, n) = RHS(m, n) for n > m 1. Now let us show the commutation relation [Gm , Gn ] = 0. Proof of Theorem 4. We show [Gm , Gn ] = 0 for Re(r ) > 0 and 0 < Re(s) < N . Others are shown in a similar way. We use the integral representation of the nonlocal integrals of motion. The following operators in the integrand of the nonlocal integrals of motion satisfy the Sn -invariance. For σ1 , σ2 , . . . , σ N ∈ Sm+n , we have (1)
(1)
(2)
(2)
(N )
(N )
F1 (z σ1 (1) ) · · · F1 (z σ1 (m+n) )F2 (z σ2 (1) ) · · · F2 (z σ2 (m+n) ) · · · FN (z σ N (1) ) · · · FN (z σ N (m+n) ) ×
N
(t)
(t)
u σt (i) − u σt ( j)
t=1 1i< j m+n (1)
(1)
(t) (t) u σt ( j) − u σt (i) − 1 r
r
(2)
(2)
(N )
(N )
= F1 (z 1 ) · · · F1 (z m+n )F2 (z 1 ) · · · F2 (z m+n ) · · · FN (z 1 ) · · · FN (z m+n ) N (t) (t) × u i(t) − u (t) u − u − 1 . j j i r
t=1 1i< j m+n
r
(4.194)
Hence we have Gm · Gn =
N m+n ( t=1 j=1
(t)
dz j (1) (1) (2) (2) F (z ) · · · F1 (z m+n )F2 (z 1 ) · · · F2 (z m+n ) · · · √ (t) 1 1 2π −1z j
(N ) FN (z 1(N ) ) · · · FN (z m+n ) N (t) (t) (t) (t) ui − u j u j − ui − 1
×
N −1 m+n t=1 i, j=1
×
r
t=1 1i< j m+n
s (t) (t+1) ui − u j +1− N r
1 m+n m m+n m (N ) s s (1) (N ) (1) +1 ui − u j + u j − ui − N r N r i=1 j=m+1
×
r
i=m+1 j=1
1 m s ) u i(1) − u (N + j N r
i, j=1
m+n i, j=m+1
s ) u i(1) − u (N + j N r
830
T. Kojima, J. Shiraishi
×
1 ((m + n)!) N
×ϑα (
m j=1
(1)
···
σ1 ∈Sm+n
u σ1 ( j) | · · · |
m
σ N ∈Sm+n (N )
u σ N ( j) )ϑβ (
j=1
m+n j=m+1
(1)
u σ1 ( j) | · · · |
m+n
(N )
u σ N ( j) )
j=m+1
s (t+1) s (t) (t+1) (t) m m+n N u σt+1 (i) − u σt ( j) + u σt (i) − u σt+1 ( j) + 1 − N r N r. × (t) (t) (t) (t) u σt (i) − u σt ( j) u σt ( j) − u σt (i) − 1 t=1 i=1 j=m+1 r
r
(4.195) Therefore we have the following theta function identity as a sufficient condition of the commutation relations Gm · Gn = Gn · Gm : ) ) ) ⎛ ⎞ ) m ) m ) m ) ) ) ) ⎠ ) ) ) ··· ϑα ⎝ u (1) u (2) u (N σ1 ( j) ) σ2 ( j) ) · · · ) σ N ( j) ) j=1 ) j=1 ) σ1 ∈Sm+n σ2 ∈Sm+n σ N ∈Sm+n j=1 ) ) ) ⎛ ⎞ ) m+n ) ) m+n (1) ) m+n (2) ) ) (N ) ×ϑβ ⎝ u σ1 ( j) )) u σ2 ( j) )) · · · )) u σ ( j) ⎠ ) j=m+1 N ) j=m+1 ) j=m+1 (t) (t+1) (t+1) (t) s m m+n N u σt+1 (i) − u σt ( j) + Ns u σt (i) − u σt+1 ( j) − N r r × (t) (t) (t) (t) u u − u − u − 1 t=1 i=1 j=m+1 σt (i) σt ( j) r σt ( j) σt (i) )r ) ) ⎛ ⎞ ) n ) n ) n (1) ) (2) ) ) (N ) = ··· ϑβ ⎝ u σ1 ( j) )) u σ ( j) )) · · · )) u σ ( j) ⎠ ) j=1 N ) j=1 2 ) σ1 ∈Sm+n σ2 ∈Sm+n σ N ∈Sm+n j=1 ) ) ) ⎛ ⎞ ) m+n ) ) m+n (1) ) m+n (2) ) ) (N ) ×ϑα ⎝ u σ1 ( j) )) u σ ( j) )) · · · )) u σ ( j) ⎠ ) j=n+1 N ) j=n+1 2 ) j=n+1 (t) (t+1) (t+1) (t) s n m+n N u σt+1 ( j) − u σt (i) + 1 − Ns u σt (i) − u σt+1 ( j) − N r r × . (4.196) (t) (t) (t) (t) u u − u − u − 1 t=1 i=1 j=n+1 σt (i) σt ( j) σt ( j) σt (i) r
r
This is a special case να,t = νβ,t = 0, (1 t N ) of the theta identity in Proposition 15. Now we have shown the theorem. 4.3. Proof of [Im , Gn ] = 0. In this section we give the proof of the commutation relation [Im , Gn ] = 0. The fundamental operators Λ j (z) and F j (z) commute almost everywhere: 2s 2s r∗ −r ∗ j−r z 2 N A j (x −r + N j z 1 ), [Λ j (z 1 ), F j (z 2 )] = (−x + x )δ x z1 2s 2s r∗ −r ∗ j+r z 2 N A j (x r + N j z 1 ). [Λ j+1 (z 1 ), F j (z 2 )] = (x − x )δ x z1 Hence, in order to show the commutation relations, the remaining task for us is to check whether delta-function factors cancel out or not. The Dynkin-automorphism invariance
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
831
η(Im ) = Im , η(Gm ) = Gm , which we will show later, plays an important role in the proof of this commutation relation [Im , Gn ] = 0. Proof of Theorem 6. For a while we consider the regime: 0 < Re(s) < N , 0 < Re(r ) < 1. At first we show the simple case, [I1 , Gn ] = 0, for the reader’s convenience. Using the Leibnitz rule of adjoint action [A, BC] = [A, B]C + B[A, C] and the invariance η(I1 ) = I1 , we have [I1 , Gn ] = (x
−r ∗
r∗
−x )
N −1 n n N t=1 j=1 u=1 k=1 C(t, j)
(t)
(t)
dz k(u) (1) F (z ) · · · √ (u) 1 1 2π −1z k (t)
2s
(t)
× · · · Ft (z 1 ) · · · Ft (z j−1 )At (x −r + N t z j )Ft (z j+1 ) · · · Ft (z n(t) ) · · · FN (z n(N ) ) N
×
(t)
(t)
ui − u j
(t) (t) u j − ui − 1 r
t=1 1i< j n N −1
r
n n s (1) s ) u i(t) − u (t+1) u i − u (N + 1 − + j j N r N r
t=1 i, j=1
i, j=1
) ) ) ⎞ ) n ) n ) n ) (N ) ) ) (1) (2) ×ϑ ⎝ u j )) u j )) · · · )) uj ⎠ ) ) ) j=1 j=1 j=1 ⎛
∗
∗
+(x r − x −r )
n n N −1 N t=1 j=1 u=1 k=1 C (t, j)
(t)
(t)
dz k(u) (1) F (z ) · · · √ (u) 1 1 2π −1z k
(t)
2s
(t)
× · · · Ft (z 1 ) · · · Ft (z j−1 )At (x r + N t z j )Ft (z j+1 ) · · · Ft (z n(t) ) · · · FN (z n(N ) ) N
×
(t)
t=1 1i< j n N −1
(t)
ui − u j
(t) (t) u j − ui − 1 r
r
n n s (1) s (N ) u i(t) − u (t+1) u + 1 − − u + j i j N r N r
t=1 i, j=1
i, j=1
) ) ) ⎞ ) n ) n ) n ) ) ) (1) (2) (N ) ×ϑ ⎝ u j )) u j )) · · · )) uj ⎠ ) ) ) j=1 j=1 j=1 ⎛
⎛ n n N +η ⎝
(u)
j=1 u=1 k=1 C
dz k (1) (N −1) F (z ) · · · F2 (z n(1) ) · · · FN (z 1 ) √ (u) 2 1 2π −1z k
· · · FN (z n(N −1) ) (N )
(N )
(N )
(N )
×F1 (z 1 ) · · · F1 (z j−1 )[I1 , F1 (z j )]F1 (z j+1 ) · · · F1 (z n(N ) )
832
T. Kojima, J. Shiraishi N
(t) u i(t) − u (t) u (t) j j − ui − 1 r
t=1 1i< j n
×
N −1
r
n n s (1) s (t) (t+1) (N ) ui − u j ui − u j + +1− N r N r
t=1 i, j=1
i, j=1
) ) ) ⎛ ⎞⎞ ) ) ) n n n ) ) ) (N ) (1) (N −1) ⎠⎠ ×ϑ⎝ u j )) u j )) · · · )) uj . ) j=1 ) j=1 ) j=1
(4.197)
Here we have set (t)
2s
(t)
2s
2s
(t)
(t) C(t, j) : |x 4r −2+ N t z j+1 |, . . . , |x 4r −2+ N t z m | < |z j | < |x −2r +2− N t z 1 |, . . . ,
|x −2r +2− N t z (t) j−1 |, 2s
(t−1)
2s
|x 2r +2− N z k 2s N
(t+1)
|x z k
(t)
(t)
| < |z j | < |x
−2+ 2s N
(t)
2s
(t−1)
2s
| < |z j | < |x 2r − N z k (t+1)
zk
|, (1 k m),
|, (1 k m),
(4.198)
(t)
2s
2s
(t)
(t) | < |z j | < |x −4r +2− N t z 1 |, . . . , C (t, j) : |x 2r −2+ N t z j+1 |, . . . , |x 2r −2+ N t z m 2s
(t)
|x −4r +2− N t z j−1 |, − N (t−1) zk |, (1 k m), |x 2− N z k(t−1) | < |z (t) j | < |x 2s
2s
2s
(t+1)
|x −2r + N z k
(t)
2s
(t+1)
| < |z j | < |x −2r −2+ N z k
(t)
|, (1 k m).
(4.199)
(t)
Let us change the variable z j → x 2r z j in the first term of (4.197). Using the periodicity of the integrand, we deform the first term to the second term of (4.197): N n
dz k(u) (1) (t) (t) F (z ) · · · Ft (z 1 ) · · · Ft (z j−1 ) √ (u) 1 1 2π −1z k
u=1 k=1 I (t, j) (t) ×At (x −r z (t) j )Ft (z j+1 ) · · · N
×
Ft (z n(t) ) · · · FN (z n(N ) )
(t)
(t)
ui − u j
t=1 1i< j n
(t) (t) u j − ui − 1 r
r
n s (1) s (N ) ui − u j + +1− N r N r t=1 i, j=1 i, j=1 ) ) ) ⎞ ⎛ ) n ) n ) n ) (N ) ) (2) ) (1) ⎝ u j )) u j )) · · · )) uj ⎠ ) ) ) j=1 j=1 j=1
=
N −1
n
(t) ui
N n u=1 k=1 I (t, j)
(t+1) −uj
(u)
dz k (1) (t) (t) F (z ) · · · Ft (z 1 ) · · · Ft (z j−1 ) √ (u) 1 1 2π −1z k
ϑ
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra (t)
833
(t)
×At (x r z j )Ft (z j+1 ) · · · Ft (z n(t) ) · · · FN (z n(N ) ) N
×
(t) (t) (t) (t) ui − u j u j − ui − 1 r
t=1 1i< j n
r
n n s (1) s (N ) u i(t) − u (t+1) u + 1 − − u + j i j N r N r t=1 i, j=1 i, j=1 ) ) ) ⎞ ⎛ ) ) ) n n ) n (N ) (1) )) (2) )) ⎝ uj ) u j ) · · · )) uj ⎠. ) ) ) j=1 j=1 j=1 N −1
ϑ
(4.200)
Hence we have η([I1 , Gn ]) ∗
∗
= (x −r − x r )
n n N
(u)
j=1 u=1 k=1 C(N , j)
dz k (1) F (z ) · · · F2 (z n(1) ) · · · √ (u) 2 1 2π −1z k
(N −1) FN (z 1 ) · · · FN (z n(N −1) ) 2s (N ) (N ) (N ) (N ) ×F1 (z 1 ) · · · F1 (z j−1 )A1 (x −r + N z j )]F1 (z j+1 ) · · · N
×
F1 (z n(N ) )
(t) (t) (t) (t) ui − u j u j − ui − 1 r
t=1 1i< j n
r
n n s (1) s (t) (t+1) (N ) ui − u j ui − u j + +1− N r N r t=1 i, j=1 i, j=1 ) ) ) ⎛ ⎞ ) ) ) n n ) n (N ) (1) )) (2) )) ⎝ ×ϑ uj ) u j ) · · · )) uj ⎠ ) ) ) j=1 j=1 j=1 N −1
∗
∗
−(x −r − x r ) (N −1)
FN (z 1
n n N
(u)
j=1 u=1 k=1 C (N , j)
dz k (1) F (z ) · · · F2 (z n(1) ) · · · √ (u) 2 1 2π −1z k
) · · · FN (z n(N −1) )
(N )
(N )
2s
(N )
(N )
×F1 (z 1 ) · · · F1 (z j−1 )A1 (x r + N z j )]F1 (z j+1 ) · · · F1 (z n(N ) ) N
×
t=1 1i< j n
(t) (t) (t) (t) ui − u j u j − ui − 1 r
r
n n s (1) s (N ) u i(t) − u (t+1) u + 1 − − u + j i j N r N r t=1 i, j=1 i, j=1 ) ) ) ⎛ ⎞ ) ) ) n n ) n (N −1) (N ) )) (1) )) ⎝ ⎠. ×ϑ uj ) u j ) · · · )) uj ) ) ) j=1 j=1 j=1 N −1
(4.201)
834
T. Kojima, J. Shiraishi
Here we have set ) ) 4r −2+ N (N ) −2r +2− N (N ) C(N , j) : |x 4r −2+ N z (N z m | < |z (N z 1 |, j+1 |, . . . , |x j | < |x 2s
2s
2s
(N )
2s
· · · , |x −2r +2− N z j−1 |, (N −1)
2s
|x 2r +2− N z k 2s N
(1)
(N )
(N )
|x z k | < |z j | < |x C (N , j) : |x
2r −2+ 2s N
(N −1)
z j+1
(N −1)
2s
| < |z j | < |x 2r − N z k −2+ 2s N
|, . . . , |x
|, (1 k m),
(1)
z k |, (1 k m),
2r −2+ 2s N
(4.202)
(N −1) zm |
) −1) −4r +2− N (N −1) z1 |, . . . , |x −4r +2− N z (N < |z (N j | < |x j−1 |, 2s
(N −1)
2s
|x 2− N z k |x
−2r + 2s N
2s
(N )
(N −1)
2s
| < |z j | < |x − N z k
) z k(1) | < |z (N j | < |x
−2r −2+ 2s N
|, (1 k m),
z k(1) |, (1 k m).
(4.203)
Using the periodicity of the integtrand, we have [I1 , Gn ] = 0. For the second, we consider the commutation relation [Im , Gn ] = 0. By using the invariance of the local integrals of motion, η(Im ) = Im , we have [Im , Gn ] = (x
−r ∗
r∗
−x )
N −1 m n n N
(u)
t=1 i=1 j=1 u=1 k=1 C(t, j)
) ) ) ⎞ ⎛ ) ) ) n ) n (N ) ) n (2) ) (1) u j )) u j )) · · · )) uj ⎠ ×⎝ ) ) ) j=1 j=1 j=1 N
×
(t) (t) (t) (t) ui − u j u j − ui − 1 r
t=1 1i< j n N −1
dz k ϑ √ (u) 2π −1z k
r
n n s (1) s (t) (t+1) (N ) ui − u j ui − u j + +1− N r N r
t=1 i, j=1
i, j=1
⎛
⎜m ⎜ dwk ) ⎫ ⎧) ×⎜ √ 2s ⎜ ⎨)) x −r + N t z (t) )) ⎬ 2π −1w j ) k ⎝k=1 I ∩ ) <1 ) ⎭ wi 1− ⎩)) )
+
×
x
m
) ⎫ ⎧) 2s ⎨)) x −r + N t z (t) )) ⎬ j ) k=1 I ∩ )) )>1⎭ wi ⎩) )
dwk √ 2π −1wk
1 −r + 2s N t z (t) x j
wi
r − 2s Nt (t)
wi
zj
1−
r − 2s t x N wi (t) zj
(t)
⎞ ⎟ ⎟ ⎟ ⎠ (t)
h(vk − vl )T1 (w1 ) · · · T1 (wi−1 )F1 (z 1 ) · · · Ft (z j−1 )
1k
(t)
(t)
×At (x −r + N t z j )Ft (z j+1 ) · · · FN (z n(t) )T1 (wi+1 ) · · · T1 (wm )
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
∗
∗
−(x −r − x r )
835
N −1 m n n N
(u)
t=1 i=1 j=1 u=1 k=1 C (t, j)
) ) ) ⎞ ) ) ) n n ) ) ) (1) (2) (N ) u j )) u j )) · · · )) uj ⎠ ×⎝ ) ) ) j=1 j=1 j=1 ⎛
n
N
×
dz k ϑ √ (u) 2π −1z k
(t) (t) (t) (t) ui − u j u j − ui − 1
r
t=1 1i< j n N −1
r
n n s (1) s (t) (t+1) (N ) ui − u j ui − u j + +1− N r N r
t=1 i, j=1
i, j=1
⎛
⎜m ⎜ dwk 1 ) ⎫ ⎧) ×⎜ √ 2s ⎜ ⎨)) x r + N t z (t) )) ⎬ 2π −1w r + 2s N t z (t) j ) k x ⎝k=1 I ∩ ) j )<1⎭ wi 1− ⎩)) wi )
+
×
x
m
) ⎫ ⎧) 2s ⎨)) x r + N t z (t) )) ⎬ j )>1 k=1 I ∩ )) ) ⎭ wi ⎩) )
−r − 2s Nt
dwk √ 2π −1wk 1−
⎞
wi
⎟ ⎟ ⎟ −r − 2s ⎠ N t wi x (t)
zj
(t)
zj
h(vk − vl )T1 (w1 ) · · · T1 (wi−1 )F1 (z 1(t) ) · · · Ft (z (t) j−1 )
1k
N
×
(t) (t) (t) (t) ui − u j u j − ui − 1
r
t=1 1i< j m N −1
n n s (1) s ) u i(t) − u (t+1) u i − u (N + 1 − + j j N r N r
t=1 i, j=1
×
r
m k=1 I
i, j=1
dwk √ 2π −1wk
h(vk − vl )T1 (w1 ) · · · T1 (wi−1 )
1k
) ×F2 (z 1(1) ) · · · F2 (z n(1) ) · · · FN (z 1(N −1) ) · · · FN (z n(N −1) )F1 (z 1(N ) ) · · · F1 (z (N j−1 ) (N )
(N )
×[T1 (wi ), F1 (z j )]F1 (z j+1 ) · · · F1 (z n(N ) )T1 (wi+1 ) · · · T1 (wm )).
(4.204)
836
T. Kojima, J. Shiraishi
Here C(t, j), C (t, j) are given in the same manner as in the proof of [I1 , Gn ] = 0. Because the integral contour I is not an annulus, we have used m m dwk 1 dwk wi /z + I ∩{|z/wi |<1} wk 1 − z/wi I ∩{|z/wi |>1} wk 1 − wi /z k=1
k=1
(t)
(t)
instead of δ(z/wi ) = m∈Z (z/wi )m . Let us change the variable z j → x 2r z j upon the conditions 0 < Re(s) < N , 0 < Re(r ) < 1. Using periodicity of the integrands, we have [Im , Gn ] = 0, in the same manner as [I1 , Gn ] = 0. Other commutation relations : [Im , Gn∗ ] = [Im∗ , Gn ] = [Im∗ , Gn∗ ] = 0 are shown in a similar way. 4.4. Proof of [Gm , Gn∗ ] = 0. In this section we give the proof of the commutation relation [Im , Gn ] = 0. The fundamental operators E j (z) and F j (z) commute almost everywhere: [E j (z 1 ), F j (z 2 )] =
1 (δ(x z 2 /z 1 )H j (x r z 2 ) − δ(x z 1 /z 2 )H j (x −r z 2 )). x − x −1
Hence, in order to show the commutation relations, the remaining task for us is to check whether delta-function factors cancel out or not. Proof of Theorem 5. We consider the regime Re(r ) > 0 and Re(r ∗ ) < 0. Using the Leibnitz rule of adjoint action and the commutation relations of screening currents E j (z), F j (z), we have [Gm∗ , Gn ]
=
N n n m ( m N N t=1 i=1 j=1 q=1 k=1 p=1
(t) ×Bi, j
l=1 l =t
(t)
Ci, j
dz (q)k dw ( p)l √ √ (q) (q) 2π −1z k 2π −1wl
(q)
(t)
x −r z i ; {z j } 1q N , {w (q) } 1kn
−
n n m ( N m N N t=1 i=1 j=1 q=1 k=1 p=1
×Bi,(t)j
l=1 l =t
(t)
Ci, j
(q)
Here we have set (t) Bi, j
1q N 1l = jm
(q) z; {z j } 1q N
, {w (q) }
1kn
=
dz (q)k dw ( p)l √ √ (q) (q) 2π −1z k 2π −1wl
x r z i(t) ; {z j } 1q N , {w (q) } 1kn
1q N 1l = jm
1q N 1l = jm
1 (1) (1) (1) F1 (z 1 ) · · · F1 (z n(1) )E 1 (w1 ) · · · E 1 (wm )··· x − x −1 (t)
(t)
(t)
(t)
× · · · Ft (z 1 ) · · · Ft (z i−1 )E t (w1 ) · · · E t (w j−1 ) (t)
(t)
(t) ×Ht (z)E t (w j+1 ) · · · E t (wm )Ft (z i+1 ) · · · Ft (z n(t) ) · · · (N )
(N )
(N ) × · · · E 1 (w1 ) · · · E 1 (wm )F1 (z 1 ) · · · F1 (z n(N ) )
.
(4.205)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra N
×
(q)
(q)
u k − ul
(q) (q) ul − u k − 1 r
q=1 1k
r
n n s (1) s (q) (q+1) (N ) u k − ul u k − ul + +1− N r N r
q=1 k,l=1
k,l=1
N
×
837
(q) (q) vk − vl
−r
q=1 1k
(q) (q) v − v − 1 l k ∗
−r ∗
n n s s (q) (q+1) vk − vl vk(1) − vl(N ) + +1− N −r ∗ N −r ∗ q=1 k,l=1 k,l=1 ) n ) m ) n ) ) m ) n m ) ) ) ) (1) )) (2) )) (1) )) (2) )) ) ) (N ) (N ) ) ×ϑ uk ) uk ) · · · ) uk vk ) vk ) · · · ) vk ϑ ) ) ) ) ) ) ) ) N −1
k=1
k=1
k=1
k=1
k=1
∗ v j =u− r2
k=1
.
(4.206) (t)
(t)
Here the contours Ci, j and Ci, j are characterized by (t)
2s
(t−1)
Ci, j : |x 2− N z k |x
−2−2r + 2s N
|x
−2r +3− 2s N
2s
(t+1)
|, |x N z k
(t+1)
zk
(t)
2s
(t−1)
| < |z i | < |x −2r − N z k
|,
|, (1 k n)
wl(t−1) |, |x −2r +1+ N wl(t+1) | < |z (t) | < |x −1− N wi(t−1) |, 2s
2s
|x −3+ N wi(t+1) |, (1 l m), 2s
2s
(q+1)
(q)
(q+1)
2s
|x N zl | < |z k | < |x −2+ N zl |, (1 q N ; (q, k) = (t, i), (q, l) = (t − 1, i)), 2s
(q+1)
(q)
2s
(q+1)
|x N wl | < |wk | < |x −2+ N wl |, (1 q N ; (q, k) = (t, j), (q, l) = (t − 1, j)), C
(t)
:
(4.207)
2s 2s 2s |x 2r +2− N z k(t−1) |, |x 2r + N z k(t+1) | < |z i(t) | < |x − N z k(t−1) |, 2s (t+1) |x −2+ N z k |, (1 k n), 2s 2s 2s (t−1) (t+1) (t) (t−1) |x 3− N wl |, |x 1+ N wl | < |z i | < |x 2r −1− N wl |, 2s (t+1) |x 2r −3+ N wl |, (1 l m), 2s (q+1) 2s (q+1) (q) |x N zl | < |z k | < |x −2+ N zl |,
(1 q N ; (q, k) = (t, i), (q, l) = (t − 1, i)), 2s
(q+1)
(q)
2s
(q+1)
|x N wl | < |wk | < |x −2+ N wl |, (1 q N ; (q, k) = (t, j), (q, l) = (t − 1, j)). (t)
Let us change the variable z i (q) {wk }
1q N 1k = jm
) and the contour
(t)
(4.208) (t)
(q)
→ x 2r z i of the integrand B (t) (x −r z i ; {z k } 1q N , (t) Ci, j .
1kn
Using the periodic condition of the theta function
838
T. Kojima, J. Shiraishi (t)
(q)
(q)
[u + r ]r = −[u]r , we have B (t) (x r z i ; {z k } 1q N , {wk } 1kn
Ci,(t)j . Hence we have N N n m ( q=1 k=1 p=1
=
l=1 l =t
(t)
Ci, j
l=1 l =t
) and the contour
dz (q)k dw ( p)l (q) (t) −r (t) (q) x B z ; {z } , {w } 1q N 1q N √ √ j i (q) (q) i, j 1 k n 1l = j m 2π −1z k 2π −1wl
N N n m ( q=1 k=1 p=1
1q N 1k = jm
(t)
Ci, j
dz (q)k dw ( p)l (q) (t) r (t) (q) x z i ; {z j } 1q N , {w } 1q N . B √ √ (q) (q) i, j 1 k n 1l = j m 2π −1z k 2π −1wl (4.209)
Therefore we have shown the commutation relation [Gm∗ , Gn ] = 0. Generalization to the generic parameter 0 < Re(r ) < 1 and s ∈ C should be understood as analytic continuation.
5. Dynkin-Automorphism Invariance In this section we consider the Dynkin-automorphism invariance of the integrals of motion.
5.1. Dynkin-Automorphism Invariance. The integrals of motion are invariant under the action of the Dynkin-automorphism. Theorem 7. The local integrals of motion In , In∗ are invariant under the action of Dynkin-automorphism η, η(In ) = In , η(In∗ ) = In∗ , (n ∈ N).
(5.210)
Theorem 8. The nonlocal integrals of motion Gn , Gn∗ are invariant under the action of Dynkin-automorphism η, η(Gn ) = Gn , η(Gn∗ ) = Gn∗ , (n ∈ N).
(5.211)
This theorem plays an important role in the proof of the commutation relation [Im , Gn ] = 0.
5.2. Proof of Dynkin-Automorphism Invariance η(In ) = In . In this section we show Dynkin-Automorphism Invariance η(In ) = In , by using Laurent series formulae In = # η p ,ηq [ j
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
839
for 0 J K N , 0 p J − 1, 0 q K − 1, η p ,ηq
h J,K (z) =
−q J − p K
2s
h 11 (x −K +J +2(k− j)+ N (q− p) z)
j=1 k=1 J
×
K
2s
h 11 (x −K +J +2(k− j)+ N (q− p) z)
j=J − p+1 k=K −q+1
×
J −p
K
2s
h 11 (x −K +J +2(k− j)+ N (q− p)−2s z)
j=1 k=K −q+1 J
×
K −q
2s
h 11 (x −K +J +2(k− j)+ N (q− p)+2s z).
(5.212)
j=J − p+1 k=1 η0 ,η0
Here we have set h 1,1 (z) = h(u) for z = x 2u . We use the notation h J,K (z) = h id,id J,K (z) = h J,K (z). Proof of Theorem 7. Let us study the invariance of I2 = [s(z 2 /z 1 )O2 (z 1 , z 2 )]1,z 1 z 2 . The action of η is given by % $ η [h 1,1 (z 2 /z 1 )T1 (z 1 )T1 (z 2 )]1,z 1 ,z 2 = [h 1,1 (z 2 /z 1 )T1 (z 1 )T1 (z 2 )]1,z 1 ,z 2 + [h 1,1 (z 2 /z 1 )(Λ1 (x −s z 1 ) −Λ1 (x s z 1 ))
N
Λ j (x s z 2 )]1,z 1 ,z 2
j=2
+[h 1,1 (z 2 /z 1 )
N
Λ j (x s z 1 )(Λ1 (x −s z 2 ) − Λ1 (x s z 2 ))]1,z 1 ,z 2 .
(5.213)
j=2
By using the relation h 1,1 (z) − h 1,1 (x 2s z) = c11 (δ(x 2 z) − δ(x 2r −2+2s z)), c11 = −
(x 2 ; x 2s )
∞
(x 2r −2 ; x 2s )
∞
(x 2s−2 ; x 2s )
∞
(5.214)
(x 2s−2r +2 ; x 2s )
∞
(x 2r −4 ; x 2s )∞ (x 2s ; x 2s )∞ (x 2s ; x 2s )∞ (x 2s−2r +4 ; x 2s )∞
,
we have [h 1,1 (z 2 /z 1 )Λ1 (x −s z 1 )
N
Λ j (x s z 2 )]1,z 1 z 2 = [h 1,1 (z 2 /z 1 )Λ1 (z 1 )
j=2
+cs11 (x −2 )[δ(x 2 z 2 /z 1 ) : Λ1 (x −2s z 1 )
N
Λ j (z 2 )]1,z 1 z 2
j=2 N
Λ j (x −2 z 1 ) :]1,z 1 z 2 ,
(5.215)
j=2
[h 1,1 (z 2 /z 1 )
N
Λ j (x s z 1 )Λ1 (x −s z 2 )]1,z 1 z 2 = [h 1,1 (z 2 /z 1 )
j=2
−cs11 (x −2 )[δ(x 2−2s z 2 /z 1 ) : Λ1 (x −2 z 1 )
N
Λ j (z 1 )Λ1 (z 2 )]1,z 1 z 2
j=2 N j=2
Λ j (z 1 ) :]1,z 1 z 2 .
(5.216)
840
T. Kojima, J. Shiraishi
Here we have used δ(x 2r −2+2s z 2 /z 1 )Λ1 (x −s z 1 )Λ j (x s z 2 ) = δ(x 2r −2 z 2 /z 1 )Λ j (x s z 1 )Λ1 (x −s z 2 ) = 0. (5.217) Summing up every term, we have $ % η [h 1,1 (z 2 /z 1 )T1 (z 1 )T1 (z 2 )]1,z 1 ,z 2 = [h 1,1 (z 2 /z 1 )T1 (z 1 )T1 (z 2 )]1,z 1 ,z 2 + cs(x −2 )[δ(x 2 z 2 /z 1 )η(T2 (x −1 z 1 ))]1,z 1 z 2 − cs(x −2 )[δ(x 2 z 2 /z 1 )T2 (x −1 z 1 )]1,z 1 z 2 . (5.218) Summing up every term of η([s(z 2 /z 1 )O2 (z 1 , z 2 )]1,z 1 z 2 ), we conclude η(I2 ) = I2 . Next we study η(I3 ) = I3 . We use the weak sense equation for the basic operator Λ j (z), z2 z1 Λ j (z 1 )Λi (z 2 ) − g1,1 Λi (z 2 )Λ j (z 1 ) g1,1 z1 z2 2 x z2 : Λi (z 2 )Λ j (z 1 ) :, (i < j). ∼ cδ (5.219) z1 We have ⎛⎡ ⎜ η ⎝⎣
⎤
⎞
h 1,1 (z k /z j )T1 (z 1 )T1 (z 2 )T1 (z 3 )⎦
⎟ ⎠
1 j
⎡ =⎣
1,z 1 z 2 z 3
h 1,1 (z k /z j )T1 (z 1 )T1 (z 2 )T1 (z 3 )
1 j
−cs(x −2 )h 1,2 (x −1 z 2 /z 1 )T1 (z 1 )T2 (x −1 z 2 )δ(x 2 z 3 /z 2 ) −cs(x −2 )h 1,2 (x −1 z 1 /z 2 )T1 (z 2 )T2 (x −1 z 1 )δ(x 2 z 3 /z 1 ) −cs(x −2 )h 1,2 (x −1 z 1 /z 3 )T1 (z 3 )T2 (x −1 z 1 )δ(x 2 z 2 /z 1 ) +cs(x −2 )h 1,2 (x −1 z 2 /z 1 )T1 (z 1 )η(T2 (x −1 z 2 ))δ(x 2 z 3 /z 2 ) id,η
+cs(x −2 )h 1,2 (x −1 z 1 /z 2 )T1 (z 2 )η(T2 (x −1 z 1 ))δ(x 2 z 3 /z 1 ) id,η
+cs(x −2 )h 1,2 (x −1 z 1 /z 3 )T1 (z 3 )η(T2 (x −1 z 1 ))δ(x 2 z 2 /z 1 ) id,η
+c2 s(x −2 )2 s(x −4 )∆(x 3 )δ(x 2 z 1 /z 2 )δ(x 2 z 3 /z 1 )η2 (T3 (z 1 )) +c2 s(x −2 )2 s(x −4 )∆(x 3 )δ(x 2 z 1 /z 3 )δ(x 2 z 2 /z 1 )η2 (T3 (z 1 )) +c2 s(x −2 )2 s(x −4 )∆(x 3 )δ(x 2 z 2 /z 1 )δ(x 2 z 3 /z 2 )η2 (T3 (z 2 )) (5.220) 2 −2 2 −4 3 2 2 2 2 +c s(x ) s(x )∆(x )(δ(x z 1 /z 2 )δ(x z 3 /z 1 ) + δ(x z 1 /z 3 )δ(x z 2 /z 1 ))T3 (z 1 ) −c2 s(x −2 )2 s(x −4 )∆(x 3 )(δ(x 2 z 1 /z 2 )δ(x 2 z 3 /z 1 ) +δ(x 2 z 1 /z 3 )δ(x 2 z 2 /z 1 ))η(T3 (z 1 ))]1,z 1 z 2 z 3 , η([cs(x −2 )h 1,2 (x −1 z 2 /z 1 )T1 (z 1 )T2 (x −1 z 2 )δ(x 2 z 3 /z 2 )]1,z 1 z 2 z 3 ) = [cs(x −2 )h 1,2 (x −1 z 2 /z 1 )T1 (z 1 )η(T2 (x −1 z 2 ))δ(x 2 z 3 /z 2 ) id,η
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
841
+c2 s(x −2 )2 s(x −4 )∆(x 3 )δ(x 2 z 3 /z 2 )δ(x 2 z 2 /z 1 )η2 (T3 (z 2 ))]1,z 1 z 2 z 3 ,
(5.221)
η([δ(x z 1 /z 2 )δ(x z 3 /z 1 )T3 (z 1 )]1,z 1 z 2 z 3 ) 2
2
= [δ(x 2 z 1 /z 2 )δ(x 2 z 3 /z 1 )η(T3 (z 1 ))]1,z 1 z 2 z 3 .
(5.222)
Summing up every term of η([s(z 2 /z 1 )s(z 3 /z 1 )s(z 3 /z 2 )O3 (z 1 , z 2 , z 3 )]1,z 1 z 2 z 3 ), we have η(I3 ) = I3 . We consider the case of general In . The action of η is given by ⎛⎡ ⎜ η ⎝⎣
⎞
⎤
1 j
1,z 1 ···z n
×
(1) α1 =α1 (1) (2) (1) (2) α2 ,α2 0, α2 +α2 =α2 (1) (2) (3) (1) (2) (3) α3 ,α3 ,α3 0, α3 +α3 +α3 =α3 ··· (1) (2) (N ) (1) (2) (N ) α N ,α N ,...,α N 0, α N +α N +···+α N =α N
×
(t,q) {A j }
⎟ ⎠=
h(z k /z j )T1 (z 1 )T1 (z 2 )T1 (z 3 ) · · · T1 (z n )⎦
α1 ,α2 ,...,α N 0 α1 +2α2 +···+N α N =n
(q) 1qt N , 1 jαt
(q) αt (t,q) (t,q) (t,q) N ∪t Aj ⊂{1,2,...,n}, |A j |=t, ∪t=1 q=1 ∪ j=1 A j (t,q) (t,q) Min(A1 )<Min(A2 )<···<Min(A (t) (t,q) ) αs
(3,3) (3,3) (3,3) Aj ⊂A j , |A j |=2 (4,3) (4,3) (4,4) (4,4) (4,3) (4,4) Aj ⊂A j , Aj ⊂A j , |A j |=3, | A j |=2 ··· (N ,4) (N ,4) (N ,N ) (N ,N ) (N ,3) (N ,4) (N ,N ) (N ,3) (N ,3) , Aj ⊂A j ,··· , A j ⊂A j , |A j |=N −2, | A j |=N −3, ··· ,| A j |=2 Aj ⊂A j
[ N2 ] t
[ N 2−1 ] t
(2u−1)
(2u)
u=1 α2t+1 × (−1) t=1 u=1 α2t + t=1 αt N t−2 t−1 t−1 2u+1 t−u−1 −2u t−u × ∆(x ) s(x ) c
t=2
u=1
⎡
N t ⎢ ⎢ ×⎢ ⎣
t=1 q=1
×
u=1
ηq−1 ,ηq−1
h t,t
(z k /z j )
j
t u
ηq−1 ,η p−1
h t,u
1t
×
×
⎧ ⎪ ⎪ ⎨
⎪ ⎪ ⎩ −→ (1,1) j∈A Min ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
−→ (N ,1) j∈A Min
T1 (z j )
⎫⎧ ⎪ ⎪ ⎪ ⎨ ⎬⎪ ⎪ ⎪ ⎪ ⎩ ⎭⎪
T2 (x −1 z j )
−→ (2,1) j∈A Min
N
TN (x −1+N −2[ 2 ] z j ) · · ·
−→ (2,2) j∈A Min
−→ (N ,N ) j∈A Min
u
t
(x u−t−2[ 2 ]+2[ 2 ] z k /z j ) ⎫ ⎪ ⎪ ⎬
η(T2 (x −1 z j )) · · · ⎪ ⎪ ⎭
N
η N −1 (TN (x −1+N −2[ 2 ] z j ))
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
842
T. Kojima, J. Shiraishi
(q) 2 2 2 α N 2 x z j2 × δ z j1
q=1 j=1
×
t=3 q=1
(q)
δ⎝
t−q∈2Z
×
x 2 zk
δ⎝
×
q−2
δ
⎞ ⎛
q+1 τ ([ 2 ])
⎠δ⎝
t−q σ ([ 2 ]+2)
⎞ ⎛
x 2 z kτ (1) zj
x 2 z kτ (u−1)
u=[ q+3 2 ]
z kτ (u)
x 2 z jσ (u+1)
δ
z jσ (u)
u=1 σ ∈St σ (1)=1 u =[ t ]+1 2
⎞
x2z j
⎠δ⎝
t−q σ ([ 2 ]+3)
z kτ (1)
t−q+2
⎤ ⎥ ⎥ ⎥ ⎦
⎠ ⎞
t−q σ ([ 2 ]+3)
zk
q−2
δ
δ
x 2 z kτ (u) z kτ (u−1)
u=[ q+3 2 ]
x2z j
t−q σ ([ 2 ]+2)
t−q+1∈2Z
x 2 z jσ (u+1)
q+1 τ ([ 2 ])
⎠
[ q−1 2 ]
δ
u=2
x 2 z kτ (u)
z jσ (u)
u=1 (t,q) (t,q) σ ∈St−q+2 t−q k1 ,k2 ,...,kq−2 ∈A j −A j σ (1)=1 u =[ 2 ]+2 k1
zj ⎛
t
j=1 (t,q) j1 = A j,1 ··· (t,q) jt−q+2 = A j,t−q+2
⎛
j=1 (t,q) j1 =A j,1 ··· (t,q) jt =A j,t
t=3 q=3
×
αt
N t
(q)
αt
τ ∈Sq−2
[ q−1 ] 2 x 2 z kτ (u−1) δ z kτ (u) u=2
z kτ (u−1)
.
(5.223)
1,z 1 ···z n (t,q)
(t,q)
Here we have set A Min = {Min(A1
(t,q)
), Min(A2
), · · · , Min(A
(t,q)
(q,t)
αt
)} for q = 0, 1,
(t,q) (t,q) (t,q) (t,q) and have set A Min = {Min( A1 ), Min(A2 ), · · · , Min( A (q,t) )} for q = 2, . . . , t. Here we have set
(u,t) A j,1 ,
(u,t) A j,2 , . . . ,
(u,t) A j,u
such that
(u,t) Aj
αt
(u,t)
(u,t)
= {A j,1 < A j,2 < · · · < (u,t) (u,t) = { A j,1 < A j,2 < · · · <
(u,t) (u,t) (u,t) (u,t) A j,u }, and have set A j,1 , · · · , A j,N +2−t such that A j (u,t) A j,N +2−t }. We give the action of η for the more general case. We prepare notations. Let us set β1 , β2 , . . . , β N 0 such that β1 + 2β2 + 3β3 + · · · + Nβ N = n. Let us set the subset (t) (t) N ∪αt B (t) = B j ⊂ {1, 2, . . . , n}, (1 t N , 1 j βt ) such that |B j | = t, ∪t=1 j=1 j {1, 2, . . . , n} and Min(B1(t) ) < Min(B2(t) ) < · · · < Min(Bα(t)t ). Let us set the index (t) (t) B j,k = jk for B j = { j1 , j2 , . . . , jt | j1 < j2 < · · · < jt }, (1 t N , 1 j αt ),
(t) (t) (t) = {B1,1 , B2,1 , · · · , Bα(t)t ,1 }. The action of η is given by following: and B Min
⎛⎡ ⎜⎢ ⎜⎢ N ⎜⎢ T1 (z j ) T2 (x −1 z j ) · · · TN (x −1+N −2[ 2 ] z j ) η ⎜⎢ ⎜⎢ −→ −→ −→ ⎝⎣ (1) (2) (N ) j∈B Min
j∈B Min
j∈B Min
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
×
N
(−c)
t=1
×
βt
t−1
t−1
∆(x
)
2u+1 t−u−1
u=1
t=1
j
gt,t
zk zj
1t
(t) j∈B Min (u) k∈B Min
=
βt N t=2
N
843
j=1 (t) j1 =B j,1 ··· (t) jt =B j,t
gt,u x
t
δ
x 2 z jσ (u+1)
z jσ (u)
u=1 σ ∈St σ (1)=1 u =[ t ]+1 2
⎤
⎞
⎥ ⎥ ⎥ ⎥ zj ⎥ ⎦
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
u−t−2[ u2 ]+2[ 2t ] z k
1,z 1 ···z n
(t,q) (1) α1 = α1 {A j } (q) 1qt N , 1 jαt (1) (2) (1) (2) α2 ,α2 0, α2 +α2 = α2 (q) αt (t,q) (t,q) (t,q) N ∪t (1) (2) (3) (1) (2) (3) Aj ⊂{1,2,...,n}, |A j |=t, ∪t=1 α3 ,α3 ,α3 0, α3 +α3 +α3 = α3 q=1 ∪ j=1 A j ··· (t,q) (t,q) (t,q) ) Min(A1 )<Min(A2 )<···<Min(A (t) (N ) (1) (2) (N ) (1) (2) α N ,α N ,...,α N 0, α N +α N +···+α N = α N αs
α1 ,α2 ,...,α N 0 α1 +2α2 +···+N α N =n
×
(3,3) (3,3) (3,3) Aj ⊂A j , |A j |=2 (4,3) (4,3) (4,4) (4,4) (4,3) (4,4) Aj ⊂A j , Aj ⊂A j , |A j |=3, | A j |=2 ··· (N ,4) (N ,4) (N ,N ) (N ,N ) (N ,3) (N ,4) (N ,N ) (N ,3) (N ,3) , Aj ⊂A j ,··· , A j ⊂A j , |A j |=N −2, | A j |=N −3, ··· ,| A j |=2 Aj ⊂A j
× (N −1) {B j }1 jβ (N ) {B j }1 jβ
N −1 N
(N ,2) ⊂{A j }
(N ,1) ⊂{A j }
(2) (t,t) {B j }1 jβ ⊂{ A j } (t) 2 3t N ,1 jαt
(2) 1 jα N
(3) (t,t−1) {B j }1 jβ ⊂{ A j } (t−1) 3 4t N ,1 jαt ··· (N −2) (N ,3) {B j }1 jβ ⊂{ A j } (3) N −2 1 jα N
(1) 1 jα N
N −1 (2u−1) [ 2 ] t (2u) N + t=1 u=1 α2t u=1 α2t+1 + j=2 β j
[ N2 ] t
× (−1) αt N t−2 t−1 t−1 2u+1 t−u−1 −2u t−u × ∆(x ) s(x ) c t=1
t=2
u=1
⎡
t N ⎢ ⎢ ×⎢ ⎣
u=1
t=1 q=1
ηq−1 ,ηq−1
h t,t
j
u
t
× (x u−t−2[ 2 ]+2[ 2 ] z k /z j )
(z k /z j )
t u
ηq−1 ,η p−1
h t,u
1t
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
−→ (1,1) j∈A Min
⎫⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬⎨ ⎬ −1 −1 T1 (z j ) T2 (x z j ) η(T2 (x z j )) · · · ⎪ −→ ⎪ ⎪ ⎪ ⎪ −→ ⎩ (2,1) ⎭⎪ ⎭ (2,2) j∈A Min
j∈A Min
844
×
T. Kojima, J. Shiraishi
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
−→ (N ,1) j∈A Min
q=1 j=1
×
t=3 q=1
δ⎝
δ⎝
t−q+1∈2Z
×
⎞ ⎛ q+1 τ ([ 2 ])
zj ⎛
⎠δ⎝
x 2 z kτ (1)
2
zj
z kτ (1)
t−q+2
⎠δ⎝
⎠ ⎞
t−q σ ([ 2 ]+3)
zk
q−2 u=[ q+3 2 ]
x2z j
t−q σ ([ 2 ]+2)
]
⎞
t−q σ ([ 2 ]+3)
δ
q+1 τ ([ 2 ])
⎠
u=2
x 2 z jσ (u+1) z jσ (u)
τ ∈Sq−2
[ q−1 ] 2 x 2 z kτ (u) x 2 z kτ (u−1) δ δ z kτ (u−1) z kτ (u) u=2
[ q−1 2 ]
⎪ ⎪ ⎭
z jσ (u)
u=1 σ ∈St σ (1)=1 u =[ t ]+1 2
x2z j
⎞ ⎛
⎤ q−2 2 x z kτ (u−1) ⎥ δ ⎦ z kτ (u) q+3
u=[
δ
x 2 z jσ (u+1)
u=1 (t,q) (t,q) σ ∈St−q+2 t−q k1 ,k2 ,...,kq−2 ∈A j −A j σ (1)=1 u =[ 2 ]+2 k1
t−q σ ([ 2 ]+2)
t
j=1 (t,q) j1 = A j,1 ··· (t,q) jt−q+2 = A j,t−q+2
x 2 zk
j=1 (t,q) j1 =A j,1 ··· (t,q) jt =A j,t
(q)
t−q∈2Z
×
⎛
(q)
αt
αt
N t t=3 q=3
N
η N −1 (TN (x −1+N −2[ 2 ] z j ))
−→ (N ,N ) j∈A Min
(q) 2 2 2 α N 2 x z j2 × δ z j1
×
N
TN (x −1+N −2[ 2 ] z j ) · · ·
⎫ ⎪ ⎪ ⎬
δ
x 2 z kτ (u)
z kτ (u−1)
.
(5.224)
1,z 1 ···z n
We note that differences between Eqs. (5.223) and (5.224) are only the signature and the restriction condition (N −1) {B j }1 jβ (N ) {B j }1 jβ
N −1 N
(N ,2) ⊂{A j }
(N ,1) ⊂{A j }
(2) 1 jα N
(1) 1 jα N
Hence, summing up every term of η([ we have shown η(In ) = In .
#
.
(2) (t,t) {B j }1 jβ ⊂{ A j } (t) 2 3t N ,1 jαt ··· (N −2) (N ,3) {B j }1 jβ ⊂{ A j } (3) N −2 1 jα N
1 j
s(z k /z j )On (z 1 , z 2 , . . . , z n )]1,z 1 ···z n ),
5.3. Proof of Dynkin-Automorphism Invariance η(Gn ) = Gn . In this section we show Dynkin-automorphism invariance η(Gn ) = Gn .
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
845
Proof of Theorem 8. For the reader’s convenience, we explain η(G1 ) = G1 first. We have the action of η as follows: ⎛
⎞
⎜N ( ⎟ ⎜ dz (t) F1 (z (1) )F2 (z (2) ) · · · FN (z (N ) )ϑ(u (1) |u (2) | · · · |u (N ) ) ⎟ ⎜ ⎟ η⎜ √ ⎟ (t) N −1 ⎜ ⎟ 2π −1z C s s ⎝t=1 ⎠ (t) (t+1) (1) (N ) u −u u −u +1− + N r N r =
t=1
N (
η(ϑ(u (1) |u (2) | · · · |u (N ) )) √ −1 2π −1z (t) N s (N ) s (2) s − u (1) + 1− u (t) − u (t+1) + 1− u u − u (1) + N r N r N r dz (t)
t=1 C
t=2
× F2 (z (1) )F3 (z (2) ) · · · FN (z (N −1) )F1 (z (N ) ).
(5.225)
Here we have used η(F1 (z 1 )F2 (z 2 ) · · · FN (z N )) = F2 (z 1 ) · · · FN (z N −1 )F1 (z N ). Let us change variables u (1) → u (N ) , u (2) → u (1) , u (2) → u (1) , . . . , u (N ) → u (N −1) , and move F1 (z (1) ) from the right to the left. We have N ( t=1 C
dz (t) F1 (z (1) )F2 (z (2) ) · · · FN (z (N ) )η(ϑ(u (2) |u (3) | · · · |u (N ) |u (1) )) . √ N −1 2π −1z (t) s s u (t) − u (t+1) + 1 − u (1) − u (N ) + N r N r t=1
(5.226) We conclude η(G1 ) = G1 from the theta property η(ϑ(u (2) |u (3) | · · · |u (N ) |u (1) )) = (1) ϑ(u (1) |u (2) | · · · |u (N ) ). Let us show η(Gm ) = Gm . After changing the variables u j → ) (2) (1) (3) (2) (N ) −1) → u (N , we have η(Gm ) as follows: u (N j , uj → uj , uj → uj ,··· uj j N m (
dz (t) j (1) (2) (N −1) (1) (2) (N −1) F (z ) · · · F2 (z m )F3 (z 1 ) · · · F3 (z m ) · · · FN (z 1 ) · · · FN (z m ) √ (t) 2 1 C 2π −1z t=1 j=1 j ) ) ) ⎛ ⎛ ⎞⎞ ) m ) ) m m (2) ) ) (N ) (1) )) (N ) (N ) ⎝ ⎝ × F1 (z 1 ) · · · F1 (z m )η ϑ uj ) u j )) · · · )) u j ⎠⎠ (5.227) ) j=1 ) ) j=1 j=1 N
×
(t) (t) (t) (t) ui − u j u j − ui − 1 r
t=1 1i< j m N −1
r
m m m s (N ) s (1) s (t) (t+1) (1) (2) ui − u j ui − u j + 1 − ui − u j + +1− N r N r N r
t=2 i, j=1
i, j=1
.
i, j=1
Here we have used (1)
(N −1)
(1) η(F1 (z 1 ) · · · F1 (z m ) · · · FN −1 (z 1
(N )
(N −1) (N ) ) · · · FN −1 (z m )FN (z 1 ) · · · FN (z m ))
(1) (2) = F2 (z 1(1) ) · · · F2 (z m )F3 (z 1(2) ) · · · F3 (z m ) · · · FN (z 1(N −1) ) · · · (N )
(N −1) (N ) FN (z m )F1 (z 1 ) · · · F1 (z m ).
(5.228)
846
T. Kojima, J. Shiraishi (1)
(2)
(2)
(3)
(N −1)
Let us change the variables u j → u j , u j → u j , · · · u j (N )
(N )
(N )
(N )
→ uj ,uj
(1)
→ uj ,
and move F1 (z 1 ) · · · F1 (z m ) from the right to the left. We have m ( N
(t)
t=1 j=1 C
dz j (1) (2) (N ) (1) (2) (N ) F (z ) · · · F1 (z m )F2 (z 1 ) · · · F2 (z m ) · · · FN (z 1 ) · · · FN (z m ) √ (t) 1 1 2π −1z j
N
×
(t) (t) (t) (t) ui − u j u j − ui − 1
r
t=1 1i< j m
r
m m s (1) s (t) (t+1) (N ) ui − u j ui − u j + +1− N r N r t=1 i, j=1 i, j=1 ) ) ) ) ⎛ ⎛ ⎞⎞ ) m ) m ) ) m m (3) ) ) (N ) ) (2) ) (1) × η ⎝ϑ ⎝ u j )) u j )) · · · )) u j )) u j ⎠⎠ . ) j=1 ) j=1 ) ) j=1 j=1 N −1
(5.229)
We conclude η(Gm ) = Gm from η(ϑ(u (1) | · · · |u (N ) )) = ϑ(u (N ) |u (1) )| · · · |u (N −1) ). Proof of η(Gm∗ ) = Gm∗ is given in the same manner as above. A. Normal Ordering We summarize the normal orderings of the basic operators. For generic Re(s) > 0, r ∈ C, we have Λi (z 1 )Λi (z 2 ) = :: (1−z 2 /z 1 )
Λi (z 1 )Λ j (z 2 ) = ::
(x 2 z 2 /z 1 ; x 2s )∞ (x 2r +2s−2 z 2 /z 1 ; x 2s )∞ (x 2s−2r z 2 /z 1 ; x 2s )∞ , (x 2s−2 z 2 /z 1 ; x 2s )∞ (x 2r z 2 /z 1 ; x 2s )∞ (x 2−2r z 2 /z 1 ; x 2s )∞ (A.230)
(x 2 z 2 /z 1 ; x 2s )∞ (x −2r z 2 /z 1 ; x 2s )∞ (x 2r −2 z 2 /z 1 ; x 2s )∞ , (x −2 z 2 /z 1 ; x 2s )∞ (x 2r z 2 /z 1 ; x 2s )∞ (x 2−2r z 2 /z 1 ; x 2s )∞ (A.231)
Λ j (z 1 )Λi (z 2 ) = ::
(x 2+2s z 2 /z 1 ; x 2s )∞ (x 2s−2r z 2 /z 1 ; x 2s )∞ (x 2s+2r −2 z 2 /z 1 ; x 2s )∞ , (x 2s−2 z 2 /z 1 ; x 2s )∞ (x 2s+2r z 2 /z 1 ; x 2s )∞ (x 2s+2−2r z 2 /z 1 ; x 2s )∞ (A.232) 2s
Λ j (z 1 )F j (z 2 ) = :: x −2r
∗
(1 − x r −2+ N j z 2 /z 1 ) 2s
(1 − x −r + N j z 2 /z 1 )
,
(A.233)
2s
F j (z 1 )Λ j (z 2 ) = ::
(1 − x 2−r − N j z 2 /z 1 ) 2s
(1 − x r − N j z 2 /z 1 )
,
(A.234)
2s
Λ j+1 (z 1 )F j (z 2 ) = :: x 2r
∗
(1 − x 2−r + N j z 2 /z 1 ) 2s
(1 − x r + N j z 2 /z 1 )
,
(A.235)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
847
2s
F j (z 1 )Λ j+1 (z 2 ) = ::
(1 − x r −2− N j z 2 /z 1 )
Λ1 (z 1 )FN (z 2 ) = :: x 2r FN (z 1 )Λ1 (z 2 ) = ::
∗
(A.236)
(1 − x 2−r z 2 /z 1 ) , (1 − x r z 2 /z 1 )
(A.237)
(1 − x r −2 z 2 /z 1 ) , (1 − x −r z 2 /z 1 )
Λ N (z 1 )FN (z 2 ) = :: x −2r FN (z 1 )Λ N (z 2 ) = ::
,
2s
(1 − x −r − N j z 2 /z 1 )
∗
(A.238)
(1 − x r −2+2s z 2 /z 1 ) , (1 − x −r +2s z 2 /z 1 )
(A.239)
(1 − x 2−r −2s z 2 /z 1 ) , (1 − x r −2s z 2 /z 1 )
(A.240)
2s
Λ j (z 1 )E j (z 2 ) = :: x
2r
(1 − x −r −1+ N j z 2 /z 1 ) 2s
(1 − x r −1+ N j z 2 /z 1 )
,
(A.241)
2s
E j (z 1 )Λ j (z 2 ) = ::
(1 − x r +1− N j z 2 /z 1 ) 2s
(1 − x −r +1− N j z 2 /z 1 )
,
(A.242)
2s
Λ j+1 (z 1 )E j (z 2 ) = :: x
−2r
(1 − x r +1+ N j z 2 /z 1 ) 2s
(1 − x −r +1+ N j z 2 /z 1 )
,
(A.243)
2s
E j (z 1 )Λ j+1 (z 2 ) = ::
(1 − x −r −1− N j z 2 /z 1 )
Λ1 (z 1 )E N (z 2 ) = :: x −2r E N (z 1 )Λ1 (z 2 ) = ::
,
(A.244)
(1 − x r +1 z 2 /z 1 ) , (1 − x −r +1 z 2 /z 1 )
(A.245)
2s
(1 − x r −1− N j z 2 /z 1 )
(1 − x −r −1 z 2 /z 1 ) , (1 − x r −1 z 2 /z 1 )
(A.246)
∗
(1 − x −r −2+2s z 2 /z 1 ) Λ N (z 1 )E N (z 2 ) = :: x , ∗ (1 − x r +2s z 2 /z 1 ) 2r
(A.247)
∗
E N (z 1 )Λ N (z 2 ) = ::
(1 − x r +2−2s z 2 /z 1 ) , ∗ (1 − x −r −2s z 2 /z 1 )
(A.248)
2s
x (1− N )2 j E j (z 1 )F j (z 2 ) = :: 2 , z 1 (1 − x z 2 /z 1 )(1 − x −1 z 2 /z 1 )
(A.249)
2s
F j (z 1 )E j (z 2 ) = ::
x (1− N )2 j , 2 z 1 (1 − x z 2 /z 1 )(1 − x −1 z 2 /z 1 )
(A.250)
848
T. Kojima, J. Shiraishi 2s
2s
E j (z 1 )F j+1 (z 2 ) = :: x ( N −1) j z 1 (1 − x −1+ N z 2 /z 1 ), 2s
(A.251)
2s
F j+1 (z 1 )E j (z 2 ) = :: x ( N −1)( j+1) z 1 (1 − x 1− N z 2 /z 1 ), 2s
(A.252)
2s
E j+1 (z 1 )F j (z 2 ) = :: x ( N −1)( j+1) z 1 (1 − x −1+ N z 2 /z 1 ), 2s
(A.253)
2s
E j (z 1 )F j+1 (z 2 ) = :: x ( N −1) j z 1 (1 − x 1− N z 2 /z 1 ).
(A.254)
For Re(r ∗ ) > 0 we have ∗
2r ∗
E j (z 1 )E j (z 2 ) = :: z 1r (1 − z 2 /z 1 )
(x −2 z 2 /z 1 ; x 2r )∞ , ∗ ∗ (x 2r +2 z 2 /z 1 ; x 2r )∞ z2 2r ∗ ) ∞ z1 ; x
2s
E j (z 1 )E j+1 (z 2 ) = :: (x
2s N −j
E j+1 (z 1 )E j (z 2 ) = :: (x
2s N − j−1
z1)
(x 2r −2+ N
r r∗
2s
∗
z2 2r ∗ ) ∞ z1 ; x
2s
z1)
2r
B j (z 1 )E j (z 2 ) = :: (x − j z 1 ) r ∗
E j (z 1 )B j (z 2 ) = :: (x
( 2s N −1) j
)
(x 2r − N 2s
(x − N z 2 /z 1 ; x 2r )∞ ∗
(x −r
∗ −2+ 2s N
(x 3r
∗ +2+ 2s N
2r r∗
j
∗
z 2 /z 1 ; x 2r )∞
∗ +2− 2s N
z1)
− rr∗
,
(A.257)
∗
(x 3r
j
(x r
(x −r
(A.258)
∗
z 2 /z 1 ; x 2r )∞ ∗
∗ + 2s ( j+1) N
,
(A.259) ∗
z 2 /z 1 ; x 2r )∞
∗ −2+ 2s ( j+1) N
(x r
,
z 2 /z 1 ; x 2r )∞
j
(x 3r
r
(A.256)
z 2 /z 1 ; x 2r )∞
∗ −2− 2s N
1
( 2s N −1)( j+1)
j
(x −r
B j (z 1 )E j+1 (z 2 ) = :: x r (1− N ) (x − j z 1 )− r ∗
E j+1 (z 1 )B j (z 2 ) = :: (x
,
(x N −2 z 2 /z 1 ; x 2r )∞ r r∗
(A.255)
∗ − 2s ( j+1) N
∗
z 2 /z 1 ; x 2r )∞
,
∗
z 2 /z 1 ; x 2r )∞
∗ +2− 2s ( j+1) N
(A.260)
∗
z 2 /z 1 ; x 2r )∞
,
(1 j N − 2), r
1
E N (z 1 )B N −1 (z 2 ) = :: (x 2s−N z 1 )− r ∗ (1− N )
B j (z 1 )E j−1 (z 2 ) = :: x
−r (1− N1 )
(x
−j
z1)
r
E j−1 (z 1 )B j (z 2 ) = :: (x ( N −1)( j−1) )− r ∗ 2s
(A.261) ∗
− rr∗
(x r
(x r
∗ +2+ 2s ( j−1) N
(x −r
(x 3r
(2 j N − 1),
∗
(x r −2s z 2 /z 1 ; x 2r )∞ , ∗ ∗ (x −r +2−2s z 2 /z 1 ; x 2r )∞
∗ + 2s ( j−1) N
∗ − 2s ( j−1) N
∗
z 2 /z 1 ; x 2r )∞ ∗
z 2 /z 1 ; x 2r )∞
,
(A.263)
∗
z 2 /z 1 ; x 2r )∞
∗ −2− 2s ( j−1) N
(A.262)
∗
z 2 /z 1 ; x 2r )∞
, (A.264)
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
E N (z 1 )B1 (z 2 ) = ::
− r∗ (1− N1 ) z1 r
849
∗
∗
(x 3r z 2 /z 1 ; x 2r )∞ . ∗ ∗ (x r −2 z 2 /z 1 ; x 2r )∞
(A.265)
For Re(r ∗ ) < 0, we have ∗
(x 2 z 2 /z 1 ; x −2r )∞ , ∗ −2 ∗ −2r (x z 2 /z 1 ; x −2r )∞
2r ∗
E j (z 1 )E j (z 2 ) = :: z 1r (1 − z 2 /z 1 )
E j (z 1 )E j+1 (z 2 ) = :: (x
2s N −j
z1)
(x −2r
r r∗
z2 −2r ∗ ) ∞ z1 ; x
2s
∗
(x N z 2 /z 1 ; x −2r )∞ (x −2r
r
2s
E j+1 (z 1 )E j (z 2 ) = :: (x N − j−1 z 1 ) r ∗
E j−1 (z 1 )H j (z 2 ) = :: (x
∗ −2+ 2s N
( 2s N −1)( j−1)
2
2s
2
(A.267)
,
(A.268)
z2 −2r ∗ ) ∞ z1 ; x
2s
∗
(x −r
− r1∗
∗ −2+ 2s N
(x −r
∗ + 2s N
∗
z 2 /z 1 ; x −2r )∞ ∗
z 2 /z 1 ; x −2r )∞
∗
∗
1
H j (z 1 )E j+1 (z 2 ) = :: (x ( N −1) j z 1 )− r ∗
(x −r
(A.269)
(A.270)
∗
(x −r +2 z 2 /z 1 ; x −2r )∞ , ∗ ∗ (x −r −2 z 2 /z 1 ; x −2r )∞
H j (z 1 )E j (z 2 ) = :: (x ( N −1) j z 1 ) r ∗
,
∗
(x −r +2 z 2 /z 1 ; x −2r )∞ , ∗ ∗ (x −r −2 z 2 /z 1 ; x −2r )∞
E j (z 1 )H j (z 2 ) = :: (x ( N −1) j z 1 ) r ∗
2s
,
(x 2− N z 2 /z 1 ; x −2r )∞
z1)
2s
∗ − 2s N
(A.266)
∗ −2+ 2s N
(x −r
∗ + 2s N
(A.271)
∗
z 2 /z 1 ; x −2r )∞ ∗
z 2 /z 1 ; x −2r )∞
.
(A.272)
For Re(r ) > 0 we have F j (z 1 )F j (z 2 ) = :: x
2r ∗ r
(1 − z 2 /z 1 ) r∗
2s
F j (z 1 )F j+1 (z 2 ) = :: (x N − j z 1 )− r
2s
(x N z 2 /z 1 ; x 2r )∞
F j+1 (z 1 )F j (z 2 ) = :: (x N − j−1 z 1 )− r
A j (z 1 )F j (z 2 ) = :: (x
−j
2s
z1)
2r ∗ r
(A.273)
2s
(x 2r −2+ N z 2 /z 1 ; x 2r )∞
r∗
2s
(x 2 z 2 /z 1 ; x 2r )∞ , (x 2r −2 z 2 /z 1 ; x 2r )∞ ,
(A.274)
,
(A.275)
2s
(x 2r − N z 2 /z 1 ; x 2r )∞ 2s
(x 2− N z 2 /z 1 ; x 2r )∞ 2s
(x −r +2+ N j z 2 /z 1 ; x 2r )∞ 2s
(x 3r −2+ N j z 2 /z 1 ; x 2r )∞
F j (z 1 )A j (z 2 ) = :: (x ( N −1) j z 1 )
2r ∗ r
,
(A.276)
2s
(x −r +2− N j z 2 /z 1 ; x 2r )∞ 2s
(x 3r −2− N j z 2 /z 1 ; x 2r )∞
,
(A.277)
850
T. Kojima, J. Shiraishi
A j (z 1 )F j+1 (z 2 ) = :: x −r
F j+1 (z 1 )A j (z 2 ) = :: (x
∗ (1− 1 N
)
r∗
(x − j z 1 )− r
( 2s N −1)( j+1)
z1)
r∗
A j (z 1 )F j−1 (z 2 ) = :: x
r ∗ (1− N1 )
F j−1 (z 1 )A j (z 2 ) = :: (x
(x
−j
( 2s N −1)( j−1)
2s
(x r +2− N ( j+1) z 2 /z 1 ; x 2r )∞
∗
− rr (1− N1 ) (x
z1)
(A.278)
(A.279)
2s
(x 3r + N ( j−1) z 2 /z 1 ; x 2r )∞
∗
∗
,
(x 3r −2s z 2 /z 1 ; x 2r )∞ , (x r +2−2s z 2 /z 1 ; x 2r )∞
− rr
− rr
,
2s
(1− N1 )
z1)
2s
(x −r + N ( j+1) z 2 /z 1 ; x 2r )∞
(x 3r − N ( j+1) z 2 /z 1 ; x 2r )∞
∗
− rr
(1 j N − 2), FN (z 1 )A N −1 (z 2 ) = :: (x 2s−N z 1 )− r
2s
(x r −2+ N ( j+1) z 2 /z 1 ; x 2r )∞
2s
(x r +2+ N ( j−1) z 2 /z 1 ; x 2r )∞
,
(A.280)
,
(A.281)
2s
(x r −2− N ( j−1) z 2 /z 1 ; x 2r )∞ 2s
(x −r − N ( j−1) z 2 /z 1 ; x 2r )∞
r −2 z
2r 2 /z 1 ; x )∞ , (x −r z 2 /z 1 ; x 2r )∞
FN (z 1 )A1 (z 2 ) = :: z 1
(A.282)
2s
2s
1
F j−1 (z 1 )H j (z 2 ) = :: (x ( N −1)( j−1) z 1 ) r 2s
2
2s
2
F j (z 1 )H j (z 2 ) = :: (x ( N −1) j z 1 )− r H j (z 1 )F j (z 2 ) = :: (x ( N −1) j z 1 )− r
(x r −2+ N z 2 /z 1 ; x 2r )∞ 2s
(x r + N z 2 /z 1 ; x 2r )∞
,
(A.283)
(x r +2 z 2 /z 1 ; x 2r )∞ , (x r −2 z 2 /z 1 ; x 2r )∞
(A.284)
(x r +2 z 2 /z 1 ; x 2r )∞ , (x r −2 z 2 /z 1 ; x 2r )∞
(A.285)
2s
H j (z 1 )F j+1 (z 2 ) = :: (x
( 2s N −1) j
z1)
1 r
(x r −2+ N z 2 /z 1 ; x 2r )∞ 2s
(x r + N z 2 /z 1 ; x 2r )∞
,
(A.286)
For Re(r ) < 0 we have F j (z 1 )F j (z 2 ) = :: x
2r ∗ r
(1 − z 2 /z 1 )
F j (z 1 )F j+1 (z 2 ) = :: (x
2s N −j
F j+1 (z 1 )F j (z 2 ) = :: (x
2s N − j−1
z1)
∗
− rr
z1)
(x −2 z 2 /z 1 ; x −2r )∞ , (x 2−2r z 2 /z 1 ; x −2r )∞
(A.287)
2s
(x −2r + N z 2 /z 1 ; x −2r )∞ 2s
(x −2+ N z 2 /z 1 ; x −2r )∞ ∗
− rr
,
(A.288)
2s
(x −2r +2− N z 2 /z 1 ; x −2r )∞ 2s
(x − N z 2 /z 1 ; x −2r )∞
.
(A.289)
Acknowledgements. We would like to thank Prof. B. Feigin, Prof. M.Jimbo and Mr. H. Watanabe for useful discussions. We would like to thank Prof. V. Bazhanov, Prof. A. Belavin, Prof. P. Bouwknegt, Prof. S. Duzhin, Prof. E. Frenkel, Prof. V. Gerdjikov, Prof. K. Hasegawa, Prof. P. Kulish, Prof. V. Mangazeev, Prof. K. Takemura
Integrals of Motion for Deformed Wq,t ( gl N )-Algebra
851
and Prof. M. Wadati for their interest in this work. T.K. is partly supported by a Grant-in Aid for Young Scientist B (18740092) from JSPS. J.S. is partly supported by a Grant-in Aid for Scientific Research C (16540183) from JSPS.
References 1. Bazhanov, V., Lukyanov, S., Zamolodchikov, Al.: Integral Structure of Conformal Field Theory, Quantum KdV Theory and Thermodynamic Bethe Ansatz. Commun. Math. Phys. 177(2), No.2, 381–398 (1996) 2. Bazhanov, V., Hibberd, A., Khoroshkin, S.: Integrable Structure of W3 Conformal Field Theory. Nucl. Phys. B 622, 475–547 (2002) 3. Feigin, B., Kojima, T., Shiraishi, J., Watanabe, H.: The Integrals of Motion for the Deformed Virasoro Algebra. http://arxiv.org/abs/0705.0427, 2007 4. Feigin, B., Kojima, T., Shiraishi, J., Watanabe, H.: The Integrals of Motion for the Deformed W -Algebra Wq,t (sl N ). Proceedings for Representation Theory 2006, Atami, Japan, p. 102–114 (2006) [ISBN49902328-2-8], available at http://arxiv.org/list/0705.0627, 2007 5. Feigin, B., Frenkel, E.: Integrals of Motion and Quantum Groups, Lecture Notes in Mathematics 1620 Integral Systems and Quantum Groups, Berlin: Springer, 1995 6. Shiraishi, J., Kubo, H., Awata, H., Odake, S.: A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions. Lett. Math. Phys. 38, 647–666 (1996) 7. Awata, H., Kubo, H., Odake, S., Shiraishi, J.: Quantum W N Algebras and Macdonald Polynomials. Commun. Math. Phys. 179, 401–416 (1996) 8. Feigin, B., Frenkel, E.: Quantum W -Algebra and Elliptic Algebras. Commun. Math. Phys. 178, 653–678 (1996) 9. Odake, S.: Comments on the Deformed W N Algebra. In APCTP-Nankai Joint Symposium on “Lattice Statistics and Mathematical Physics 2001”, Tianjin China, River Edge, NJ: World Scientific, 2002 10. Feigin, B., Odesskii, A.: A Family of Elliptic Algebras. Internat. Math. Res. Notices 11, 531–539 (1997) (1) 11. Asai, Y., Jimbo, M., Miwa, T., Pugai, Ya.: Bosoniztion of Vertex Operators for An−1 face model. J. Phys. A Math. Gen. 29, 6595–6616 (1996) 12. Feigin, B., Jimbo, M., Miwa, T., Odesskii, A., Pugai, Ya.: Algebra of Screening Operators for the Deformed Wn Algebra. Commun. Math. Phys. 191, 501–541 (1998) 13. Kojima, T., Konno, H.: The Ellptic Algebra Uq, p (sl N ) and the Drinfeld Realization of the Elliptic Quantum Group Bq,λ (sl ). Commun. Math. Phys. 239, 405–447 (2003) N 14. Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W -algebras. Commun. Math. Phys. 178, 237–264 (1996) 15. Frenkel, E.: Deformations of the KdV hierarchy and related soliton equations. Internat. Math. Res. Notices 2, 55–76 (1996) Communicated by Y. Kawahigashi
Commun. Math. Phys. 283, 853–860 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0538-x
Communications in
Mathematical Physics
Heat Kernel Coefficients for Two-Dimensional Schrödinger Operators Yuri Berest1, , Tim Cramer2 , Farkhod Eshmatov3 1 Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA.
E-mail: [email protected]
2 Department of Mathematics, Yale University, New Haven, CT 06520, USA.
E-mail: [email protected]
3 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA.
E-mail: [email protected] Received: 2 October 2007 / Accepted: 11 February 2008 Published online: 12 June 2008 – © Springer-Verlag 2008
Abstract: In this note, we compute the Hadamard coefficients of algebraically integrable Schrödinger operators in two dimensions. These operators first appeared in [BL] and [B] in connection with Huygens’ principle, and our result completes, in a sense, the investigation initiated in those papers. Let L = −n + V be a Schrödinger operator on Rn (n ≥ 1) with C ∞ -smooth potential V = V (x) defined in some open domain ⊆ Rn . Recall that the heat kernel of L is the solution + (x, ξ, t) ∈ C ∞ ( × × R1+ ) of the initial value problem ∂ + L + (x, ξ, t) = 0, lim + (x, ξ, t) = δ(x − ξ ), (1) t→0+ ∂t where δ(x − ξ ) is the Dirac delta-function on with support at ξ . It is well known (see, e.g., [R], Sect. 3.2.1) that + (x, ξ, t) has an asymptotic expansion of the form 2 ∞ e−|x−ξ | /4t ν + (x, ξ, t) ∼ Uν (x, ξ ) t 1+ as t → 0+, (2) (4π t)n/2 ν=1
with coefficients Uν (x, ξ ) ∈ C ∞ ( × ) determined by the following transport equations (x − ξ, ∂x ) Uν (x, ξ ) + ν Uν (x, ξ ) = −L[Uν−1 (·, ξ )](x), ν = 1, 2, . . . .
(3)
The system (3) has a unique solution {Uν (x, ξ )}∞ ν=0 if one sets U0 (x, ξ ) ≡ 1 and requires each Uν (x, ξ ) to be bounded in a neighborhood of the diagonal x = ξ . Following [G], we will refer to {Uν (x, ξ )} as the Hadamard coefficients of the operator L. In general, calculating the Hadamard coefficients for a given potential V is a difficult problem. Of special interest are potentials, for which the heat kernel expansion (2) is Berest’s work partially supported by NSF grant DMS 04-07502.
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Y. Berest, T. Cramer, F. Eshmatov
finite, i. e. the sum in the right-hand side of (2) has only finitely many nonzero terms. In this case, formula (2) yields not only a “short-time” asymptotics, but an exact analytic representation for + (x, ξ, t) valid for all t ∈ R1+ . Such potentials are usually called Huygens’ potentials in view of an important role they play in the theory of Huygens’ principle (see, e. g., [BV], Ch. I). The problem of describing all Huygens potentials goes back to Hadamard’s classical treatise [H] and still remains open in all dimensions, except for n = 1 (see [L]) and n = 2 (see [B]). In dimension one, these potentials coincide with the well-known AdlerMoser potentials [AM], which are rational solutions of the Korteweg-de Vries hierarchy of nonlinear integrable PDE’s (see [S] or [BV], Ch. 3, Sect. 3.2); apart from the original works [LS] and [L], their Hadamard coefficients have been studied recently in [Gr,I1,I2] and [Ha]. In the present paper, we will deal with the two-dimensional Huygens potentials, which have received so far much less attention (see, however, [CFV]). Our main result (Theorem 2) provides simple and explicit formulas for the Hadamard coefficients of the corresponding Schrödinger operators. It is surprising that these formulas do not seem to have analogues in dimension one: in a sense, the two-dimensional case is simpler than the one-dimensional one! We begin by recalling the main result of [BL] and [B]. Let k = (k0 , k1 , . . . , km ) be a finite, strictly increasing sequence of integers, with k0 = 0 , and let ϕi be real numbers given one for each integer ki , with ϕ0 = 0 . Passing to the polar coordinates (x1 , x2 ) = (r cos ϕ, r sin ϕ) in R2 , we associate to these data the following potential: Vk (x1 , x2 ) = −
2 ∂2 log Wr [χ0 , χ1 , . . . , χm ], r 2 ∂ϕ 2
(4)
where χi := cos(ki ϕ +ϕi ) , i = 0, 1, . . . , m , and Wr [χ0 , χ1 , . . . , χm ] is the Wronskian of the set {χi } taken with respect to the variable ϕ. As all ki ’s are integers, Vk is a single-valued rational function on R2 , homogeneous of degree −2, whose analytic continuation to C2 has singularities along certain lines passing through the origin. For example, for k = (0, 1, 3, 4) , with all ϕi ’s being 0, we have Vk (x1 , x2 ) =
12 (49 x14 + 28 x12 x22 − x24 ) x22 (7x12 + x22 )2
.
In general, (4) depends on both the choice of ki ’s and the choice of ϕi ’s, though we suppressed the latter from our notation. Theorem 1 ([BL,B]). A (locally) smooth function V on R2 , which is homogeneous of degree −2, is a Huygens potential if and only if V = Vk (x1 , x2 ) for some integer sequence k = (ki ) and real numbers (ϕi ). Remark. The “if” part of Theorem 1 was first proven in [BL] and then reproven by a different method, together with the “only if” part, in [B] (see loc. cit., Theorem 1.1). The assumption that V is homogeneous can be relaxed and replaced by a weaker condition that V and all the Hadamard coefficients of V are algebraic functions (see [CFV]). The main result of this paper can now be encapsulated in
Heat Kernel Coefficients for Two-Dimensional Schrödinger Operators
855
Theorem 2. The Hadamard coefficients of L = −2 + Vk with potential (4) are given, in terms of polar coordinates x = (r cos ϕ, r sin ϕ) and ξ = ( cos φ, sin φ) , by the following formulas: Uν (x, ξ ) =
m (−2)ν ci i (ϕ) i (φ) Tk(ν) (cos(ϕ − φ)), ∀ ν ≥ 0, i (r )ν
(5)
i=0
where i :=
Wr [χ0 , χ1 , . . . , χi−1 , χi+1 , . . . , χm ] , Wr [χ0 , χ1 , . . . , χm ]
ci :=
m
(ki2 − k 2j ),
(6)
j=0 j =i (ν)
and TN (z) := cos(N arccos z) is the N th Chebyshev polynomial, with TN (z) being its derivative of order ν with respect to z . Remark. It follows immediately from (5) that Uν (x, ξ ) ≡ 0 for ν > km , implying that Vk is a Huygens potential. For k of length one, i. e. k = (0, N ) , formula (4) yields the Calogero-Moser potential of dihedral type I2 (N ), and in this special case the coefficients (5) have already been found in [BL] (see loc. cit., Sect. IV, Example 1). There are several ways to prove Theorem 2. Perhaps, the most straightforward one is to use a differential recurrence relation between the Hadamard coefficients of operators Lk and Lk˜ , where k˜ = (k, km+1 ) is obtained by adding one integer on top of k (see [BL], (82)). This method requires double induction (in m and ν) and leads to rather unwieldy calculations. Here, we will offer a more illuminating argument based on the remarkable fact that the coefficients Uν (x, ξ ) appear not only in fundamental solutions of the heat equation but also in its elliptic and hyperbolic counterparts1 . Instead of the Cauchy problem (1), we will consider L[G( · , ξ )](x) = δ(x − ξ ), G( · , ξ ) ∈ D (),
(7)
where D () denotes the space of distributions on C ∞ -functions with compact support in ⊆ Rn . Of course, unlike the heat kernel, G(x, ξ ) is not uniquely determined by (7), but only up to adding smooth functions from Ker(L) . The problem is now to describe the singularities of G(x, ξ ) . In modern language, the solution to this classical problem is given in terms of Riesz distributions, and as in the hyperbolic case (see, e.g., [D]), it depends on whether n is even or odd. Specifically, if n ≥ 2 is even, G(x, ξ ) has the following asymptotics “in smoothness” (cf. [Ba], Sect. 3, (1.4)): G(x, ξ ) ∼
∞ ν=0
U˜ ν (x, ξ ) Sν− n−2 (γ ),
(8)
2
where γ = γ (x, ξ ) is the square of the Euclidean distance between x and ξ , Sλ (t) := t+λ / (λ + 1) is the family of Riemann-Liouville distributions on R1 , depending analytically on the parameter λ ∈ C , and Sλ := d Sλ /dλ are the adjoint distributions of Sλ with respect to λ (see [GS], Ch. VI, Sect. 2). The coefficients U˜ ν (x, ξ ) in (8) are smooth 1 For an excellent survey on this classical subject we refer the reader to [Ba].
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Y. Berest, T. Cramer, F. Eshmatov
functions, which satisfy, up to rescaling factor −1/4 , the same transport equations (3) as the heat kernel coefficients Uν (x, ξ ) . Like in the heat kernel case, this can be verified by direct calculation, substituting (8) into (7). By uniqueness of the regular solution of (3), we thus have 1 ν ˜ Uν (x, ξ ) = − Uν (x, ξ ) for all ν ≥ 0. (9) 4 Now, let n = 2 and assume that V = V (x1 , x2 ) is (locally) analytic, as are our potentials (4). In this case, the distributions Sλ appear in (8) only with non-negative integer λ’s, and for such λ’s they can be easily calculated: d Sλ (t) 1
= t+ν log t + Cν t+ν , ν = 0, 1, 2, · · · , (10) Sν (t) := dλ λ=ν ν! where Cν ∈ R are some constants, with C0 = 0. Substituting (9) and (10) in (8), we get G(x, ξ ) ∼ W (x, ξ ) log γ + . . . , where “ . . . ” denote smooth functions, which do not contribute to singularities of G(x, ξ ), and W (x, ξ ) :=
∞ ν=0
1 Uν (x, ξ ) γ ν . (−4)ν ν!
(11)
Since V is analytic, the series (11) uniformly converges in a neighborhood of the diagonal x = ξ . This was already observed by Hadamard, who called W (x, ξ ) the logarithmic term of the “elementary solution” G(x, ξ ). He also showed that W (x, ξ ) is the (unique) analytic solution of the Goursat problem L[W ( · , ξ )](x) = 0, W |γ =0 = 1,
(12)
with the boundary condition given on the (complex) characteristics of L (see [H], Ch. III, Sect. 46). Now, to prove Theorem 2 we will simply solve (12) for the operator L with potential (4), using the method of separation of variables, and then recover the coefficients Uν (x, ξ ) by expanding the solution in the vicinity of γ = 0 as in (11). First, we introduce a one-dimensional Schrödinger operator L by writing L in terms of polar coordinates (cf. [B], Sect. 3): L=−
∂2 1 1 ∂ + 2 L. − 2 ∂r r ∂r r
(13)
Explicitly, L = −∂ 2/∂ϕ 2 + Vk (cos ϕ, sin ϕ) , where Vk is given by (4). Next, we prove Lemma 1. If = (ϕ) satisfies L[] = k 2 with some integer k ≥ 0, then (ϕ) Tk [ (r/ + /r )/2 ] ∈ Ker(L) for any = 0.
Heat Kernel Coefficients for Two-Dimensional Schrödinger Operators
857
Proof. Changing the variables (r, ϕ) → (z, ϕ) in (13), with z := (r/ + /r )/2 , yields 1 ∂2 ∂ L = 2 (1 − z 2 ) 2 − z +L . r ∂z ∂z Since Tk (z) is an eigenfunction of (1 − z 2 )d 2/dz 2 − zd/dz with eigenvalue −k 2 , the claim is obvious.
Lemma 2. The functions i defined in (6) satisfy L[i ] = ki2 i , i = 0, 1, . . . , m, m ci i (ϕ) i (φ) cos(ki (ϕ − φ)) = 1.
(14) (15)
i=0
Proof. Equations (14) simply say that i ’s are eigenfunctions of L with eigenvalues ki2 . This follows directly from Crum’s classical theorem (see [C], p. 124). The identity (15) seems more interesting: we could not find it in the literature, so we will prove it in detail by induction on m. The case m = 1 is straightforward. Writing Sm (ϕ, φ) for the left-hand side of (15), we now fix m ≥ 1 and assume that Sm (ϕ, φ) ≡ 1 for all sequences of integers (k0 = 0, k1 , . . . , km ) and reals (ϕ0 = 0, ϕ1 , . . . , ϕm ) of length m . To make the induction step we add km+1 ∈ Z , km+1 > km , and ϕm+1 ∈ R to these sequences and consider Sm+1 (ϕ, φ) :=
m+1
˜ i (φ) cos(ki (ϕ − φ)), ˜ i (ϕ) c˜i
(16)
i=0
˜ i and c˜i are defined by formulas (6) with m replaced by m + 1. By Crum’s where ˜ i ’s are eigenfunctions of an operator L˜ obtained from L by applying the Theorem, Darboux transformation (cf. [B], (3.29)): 2 2 → L˜ = Am ◦ A∗m + km+1 , L = A∗m ◦ Am + km+1
where ˜ −1 ◦ Am := m+1
∂ ∂ϕ
˜ m+1 , ◦
A∗m
˜ m+1 ◦ := −
∂ ∂ϕ
(17)
˜ −1 . ◦ m+1
(18)
˜ m+1 ] = 0 . On the other hand, we have It is immediate from (18) that A∗m [ ˜ i ] = −i for all i = 0, . . . , m. A∗m [ In fact, a trivial calculation shows that (19) is equivalent to W Wi,m+1 Wi ∂ = , 2 ∂ϕ Wm+1 Wm+1
(19)
(20)
where W denotes the Wronskian of the set {χ0 , χ1 , . . . , χm+1 }, and Wi , Wi,m+1 are the Wronskians of this set with functions χi and {χi , χm+1 } being omitted. Now, to prove (20) consider the linear system m i=0
(k)
(k)
χi yi = χm+1 , k = 0, 1, . . . , m.
(21)
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Y. Berest, T. Cramer, F. Eshmatov
By Cramer’s Rule, the solution to this system is given by yi = (−1)m+i Wi /Wm+1 . On
m (k) the other hand, differentiating both sides of (21) with respect to ϕ yields i=0 χi
m yi = 0 for k = 0, 1, . . . , m − 1 and i=0 χi(m) yi = W/Wm+1 for k = m . Solving
2 , which is equivalent to m+i these equations for yi , we get yi = (−1) W Wi,m+1 /Wm+1 (20). ˜ i ] = −( L˜ − k 2 ) ˜ i . Hence It follows from (19) that Am [i ] = −Am A∗m [ m+1 2 ˜ i for all i = 0, . . . , m. − ki2 ) Am [i ] = (km+1
(22)
Now, applying Am and A∗m to (16), we can formally relate Sm+1 (ϕ, φ) to Sm (ϕ, φ). In fact, a straightforward calculation using (19) and (22) shows A∗m [Sm+1 ( · , φ)] + A∗m [Sm+1 (ϕ, · )] = Am [Sm ( · , φ)] + Am [Sm (ϕ, · )],
(23)
where we omit the variables on which the differential operators act. Since Sm (ϕ, φ) ≡ 1 by induction assumption, we can regard (23) as a first order PDE for the function Sm+1 (ϕ, φ). Specifically, substituting Sm = 1 into (23) yields ∂ ∂B ∂B ∂B ∂B ∂ Sm+1 + Sm+1 = − + + + , (24) ∂ϕ ∂φ ∂ϕ ∂φ ∂ϕ ∂φ ˜ m+1 (φ)] . This PDE can be easily integrated: changing the ˜ m+1 (ϕ) where B := log [ variables x := (ϕ + φ)/2 , t := (ϕ − φ)/2 and F := (Sm+1 − 1) e−B transforms (24) to the equation ∂ F/∂ x = 0 , which shows that F depends only on t. It follows that ˜ m+1 (ϕ) ˜ m+1 (φ), Sm+1 (ϕ, φ) = 1 + F(ϕ − φ)
(25)
where F is a differentiable function of one variable defined on (0, 2π ). Now, if we replace km with km+1 in the sum Sm and repeat the above argument, adding km to the partition (k0 = 0, k1 , . . . , km−1 , km+1 ) , then, instead of (25), we get ˜ m (ϕ) ˜ m (φ), Sm+1 (ϕ, φ) = 1 + G(ϕ − φ)
(26)
where G is another differentiable function on (0, 2π ). Comparing (25) and (26) shows that F = G ≡ 0. Indeed, if one of these functions (F say) is nonzero, then by continuity F(ϕ − φ) = 0 in an open subset of (0, 2π ) × (0, 2π ) , and in that subset we have G(ϕ − φ)/F(ϕ − φ) = h(ϕ) h(φ),
(27)
˜ m+1 / ˜ m . Differentiating both sides of (27) with respect to ϕ and φ and where h := adding the results yields h (ϕ) h(φ) + h(ϕ) h (φ) = 0 . Whence, letting ϕ = φ , we see ˜ m+1 is a multiple of ˜ m , which is imposthat h must be constant. This means that ˜ ˜ ˜ sible, since m and m+1 are nonzero eigenfunctions of L corresponding to different 2 and k 2 eigenvalues (km m+1 respectively). Thus F ≡ 0 , and therefore Sm+1 (ϕ, φ) ≡ 1, finishing the induction.
Now, combining the results of Lemmas 1 and 2, we see at once that W :=
m i=0
ci i (ϕ) i (φ) Tki [ (r/ + /r )/2 ]
(28)
Heat Kernel Coefficients for Two-Dimensional Schrödinger Operators
859
is a solution to (12). Indeed, by (14) and Lemma 1, each summand of (28) lies in the kernel of L, and hence L[W ] = 0 by linearity of L. On the other hand, in polar coordinates γ (x, ξ ) = r 2 + 2 − 2 r cos(ϕ − φ) , so 1 r γ + = + cos(ϕ − φ), (29) 2 r 2 r and therefore Tki [ (r/ + /r )/2 ] = cos(ki (ϕ − φ)) on γ = 0 for all i = 0, 1, . . . , m . The boundary condition for W follows then from (15). Now, by [B], Lemma 3.1, Uν (x, ξ ) are homogeneous functions of x and ξ , which can be written in terms of the polar coordinates as Uν (x, ξ ) =
1 σν (ϕ, φ), (r )ν
ν ≥ 0.
(30)
Clearly, there is at most one expansion of W (x, ξ ) of the form (11) with coefficients (30). To find this expansion, we take the obvious Taylor formulas (see (29)) ∞ γ ν (ν) 1 Tki [ (r/ + /r )/2 ] = Tki (cos(ϕ − φ)), (31) ν! 2 r ν=0
substitute (31) into (28) and reorder summations. Collecting coefficients under the different powers of γ gives then the desired formulas (5). In the end, we note that our operators L are examples of algebraically integrable Schrödinger operators (see [CV]). Such Schrödinger operators possess special eigenfunctions called the Baker-Akhiezer functions. Like fundamental solutions above, the Baker-Akhiezer functions have asymptotic expansions, with coefficients satisfying (up to rescaling factor 1/2) the same transport equations (3). This was originally discovered in the special case of Calogero-Moser potentials in [BV1], but the argument of [BV1] applies to any homogeneous operator (see, e.g., [CFV]). In combination with Theorem 2, this yields Theorem 3. The Baker-Akhiezer function of L = −2 + Vk (x1 , x2 ) is given by k m 1 BA (x, ξ ) = Uν (x, ξ ) e(x, ξ ) , 2ν ν=0
where (x, ξ ) := x1 ξ1 + x2 ξ2 and Uν (x, ξ ) are the same coefficients as in (5). References [AM] [Ba] [B] [BL]
Adler, M., Moser, J.: On a class of polynomials connected with the Korteweg de Vries equation. Commun. Math. Phys. 61, 1–30 (1978) Babich, V.M.: Hadamard’s ansatz, its analogues, generalizations and applications. St.-Petersburg Math. J. 3, 937–972 (1992) Berest, Y.: Solution of a restricted Hadamard problem on Minkowski spaces. Comm. Pure Appl. Math. 50, 1019–1052 (1997) Berest, Y., Loutsenko, I.: Huygens’ principle in Minkowski spaces and soliton solutions of the Korteweg-de Vries equation. Commun. Math. Phys. 190, 113–132 (1997)
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Communicated by L. Takhtajan
Commun. Math. Phys. 283, 861 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0592-4
Communications in
Mathematical Physics
Erratum
Nucleation of Instability of the Meissner State of 3-Dimensional Superconductors Peter Bates1 , Xing-Bin Pan2 1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
E-mail: [email protected]
2 Department of Mathematics, East China Normal University, Shanghai 200062,
People’s Republic of China. E-mail: [email protected] Received: 3 March 2008 / Accepted: 3 March 2008 Published online: 2 August 2008 – © Springer-Verlag 2008
Commun. Math. Phys. 276(3), 571–610 (2007)
In the above article, a condition on the domain is missing. In the following lines, the word “simply-connected domain” should be replaced by “simply-connected domain without holes”: Theorem 1, Proposition 2.1, Corollary 2.2, Lemma 2.3, Theorem 4.1, Theorem 5.1, Lemma 7.1, Lemma 7.3, Theorem 7.4, line 27 and line 30 on p. 577 (before (2.1)); footnote 8 on p. 577; line 4 on p. 578 (before (2.3)); line 15 on p. 605. The reason for this is that on line 4 of p. 578, we quoted a result by Bolik-Wahl [BW]; see (2.3): ∇BC α () ¯ ≤ C(, α) div BC α () ¯ + curl BC α () ¯ + ν × BC 1+α (∂) , which requires the domain to have no holes; see [BW], Theorem 2.1. In the proof of Proposition 2.1 in our paper, when k ≥ 1, we used the inequality (2.3). Since, in the proof of our main theorems, we used Proposition 2.1, the domain should have no holes. In the proof of (2.2) we also require the domain to have no holes. Reference [BW] Bolik, J., von Wahl, W.: Estimating ∇u in terms of div u, curl u, either (ν, u) or ν × u and the topology. Math. Methods Appl. Sci. 20, 737–744 (1997) Communicated by P. Constantin
The online version of the original article can be found under doi:10.1007/s00220-007-0335-y.
