Commun. Math. Phys. 262, 1–16 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1472-9
Communications in
Mathematical Physics
Double Products and Hypersymplectic Structures on R4n Adri´an Andrada, Isabel G. Dotti CIEM, FaMAF, Universidad Nacional de C´ordoba, Ciudad Universitaria, (5000) C´ordoba, Argentina. E-mail:
[email protected];
[email protected] Received: 16 January 2004 / Accepted: 26 July 2005 Published online: 9 December 2005 – © Springer-Verlag 2005
Abstract: In this paper we give a procedure to construct hypersymplectic structures on R4n beginning with affine-symplectic data on R2n . These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this construction are exhibited. 1. Introduction A hypersymplectic structure on a 4n-dimensional manifold M is given by (J, E, g), where J , E are endomorphisms of the tangent bundle of M such that J 2 = −1,
E 2 = 1,
J E = −EJ,
g is a neutral metric (that is, of signature (2n, 2n)) satisfying g(X, Y ) = g(J X, J Y ) = −g(EX, EY ) for all X, Y vector fields on M and the associated 2-forms ω1 (X, Y ) = g(J X, Y ),
ω2 (X, Y ) = g(EX, Y ),
ω3 (X, Y ) = g(J EX, Y )
are closed. Manifolds carrying a hypersymplectic structure have a rich geometry, the neutral metric is K¨ahler and Ricci flat and its holonomy group is contained in Sp(2n, R) ([8]). Moreover, the Levi Civita connection is flat, when restricted to the leaves of the canonical foliations associated to the product structure given by E (see [2]). Metrics associated to a hypersymplectic structure are also called neutral hyperk¨ahler (see [10]). Hypersymplectic structures have significance in string theory. In [14], N = 2 superstring theory is considered, showing that the critical dimension of such a string is 4 and
Both authors were partially supported by CONICET, ANPCyT, SECyT-UNC and ACC (Argentina).
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A. Andrada, I.G. Dotti
that the bosonic part of the N = 2 theory corresponds to self-dual metrics of signature (2, 2) (see also [5] and [9]). The quotient construction proved to be a powerful method to construct symplectic and hyperk¨ahler structures on manifolds. According to [8] this method cannot always be applied in the setting of hypersymplectic structures. Compact complex surfaces with neutral hyperk¨ahler metrics are biholomorphic to either complex tori or primary Kodaira surfaces and both carry non-flat neutral hyperk¨ahler metrics, by results of Kamada (see [10]). In higher dimensions, hypersymplectic structures on a class of compact quotients of 2-step nilpotent Lie groups were exhibited in [6] in their search of neutral Calabi-Yau metrics. The purpose of this paper is to give a procedure to construct hypersymplectic structures on R4n with complete and not necessarily flat associated neutral metrics. The idea behind the construction will be to consider the canonical flat hypersymplectic structure on R4n and then translate it by using an appropriate group acting simply and transitively on R4n . This group will be a double Lie group (R4n , R2n × {0}, {0} × R2n ) constructed from affine data on R2n . The most important feature achieved by this procedure is that the associated neutral metrics obtained will be complete and invariant by a 3-step nilpotent group of isometries (we note that homogeneity does not necessarily imply completeness in the pseudoriemannian setting.) The degree of nilpotency will be related to the flatness of the metric since we will show that the neutral metric is flat if and only if the group is at most 2-step nilpotent. Moreover, we provide explicit examples of 3-step nilpotent Lie groups admitting compact quotients and carrying invariant complete and non-flat hypersymplectic structures. The induced metric on the associated nilmanifold will be neutral K¨ahler, complete, non-flat and Ricci flat. The outline of this paper is as follows. In §2 we give to R4n a structure of a nilpotent Lie group. Starting with a fixed symplectic structure ω on R2n which is parallel with respect to a pair of affine structures we form the associated double Lie group (R4n , R2n × {0}, {0} × R2n ) and show that it is at most 3-step nilpotent. In §3 we consider canonical symplectic structures on R4n , constructed from the given ω on R2n and show they are invariant by the group constructed in §2. We analyze in §4 the geometry of the homogeneous metric obtained by using the double Lie group structure given to R4n to translate the standard inner product of signature (2n, 2n) on R2n ⊕ R2n . The resulting metric is hypersymplectic (hence Ricci flat), complete and not necessarily flat. Finally, in §5, we exhibit explicitly flat and non-flat complete neutral metrics on R4n which are also K¨ahler and Ricci flat. Complete flat hypersymplectic metrics are constructed on 2-step nilpotent Lie groups of dimension 8n (§5.1) carrying also a closed special form in the sense of ([6], Sect. 2) and thus, the procedure developed in [6] may be applied to produce non-flat neutral Calabi-Yau metrics on the associated Kodaira manifolds. In §5.2, complete non-flat hypersymplectic metrics are exhibited on R8 , where a particular example is given by 3 2 g = − (x1 + x3 ) + x2 − x4 (dx1 + dx3 )2 + 2(x1 + x3 )(dx1 + dx3 )(dx2 − dx4 ) 2 −x1 dx1 dx1 − dx1 dx2 + dx2 dx1 + x3 (dx1 dx3 + dx3 dx1 ) +(x1 + 2x3 ) dx3 dx3 − dx3 dx4 + dx4 dx3
with respect to coordinates x1 , . . . , x4 , x1 , . . . , x4 . Furthermore, metrics with similar properties can be obtained in higher dimensions.
Double Products and Hypersymplectic Structures on R4n
3
2. Group Structure on R4n The main goal of this section will be to attach a 3-step nilpotent Lie group to data (∇, ∇ , ω), where ∇, ∇ are affine structures on R2n compatible with a symplectic structure ω. We shall begin by recalling some definitions which will be used throughout this article. An affine structure (or a left symmetric algebra structure) on Rn is given by a connection ∇, that is, a bilinear map ∇ : Rn × Rn → Rn satisfying the following conditions: ∇x y = ∇y x, ∇x ∇y = ∇y ∇x
(1) (2)
for all x, y ∈ Rn . If ω is a non-degenerate skew-symmetric bilinear form on Rn , the affine structure ∇ is compatible with ω if ω(∇x y, z) = ω(∇x z, y),
x, y, z ∈ Rn .
(3)
We notice that affine structures ∇ on R2n compatible with ω satisfy a condition stronger than (2), namely, ∇x ∇y = 0,
x, y ∈ R2n .
(4)
The last equation follows from ω(∇x ∇y z, w) = ω(∇w x, ∇z y) = −ω(∇z ∇w x, y) = −ω(∇w ∇z x, y) = −ω(∇y w, ∇x z) = −ω(∇x ∇y z, w). Let ∇ and ∇ be two affine connections on R2n compatible with ω and assume furthermore that ∇ and ∇ satisfy the following compatibility condition: ∇x ∇y = ∇y ∇x
(5)
for all x, y ∈ R2n . From (5) and the compatibility of the connections with ω, we obtain the following: ∇x ∇y = ∇y ∇x
(6)
∇x ∇y = −∇y ∇x
(7)
and
for all x, y ∈ R2n . Indeed, (6) follows from ω(∇x ∇y z, w) = −ω(∇y z, ∇x w) = −ω(∇y ∇x w, z) = −ω(∇x ∇y w, z) = −ω(∇x z, ∇y w) = ω(∇y ∇x z, w), and (7) follows from ω(∇x ∇y z, w) = ω(∇y ∇x z, w) = ω(∇w y, ∇z x) = −ω(∇z ∇w y, x) = −ω(∇w ∇z y, x)=−ω(∇x w, ∇y z) =−ω(∇x ∇y z, w)=−ω(∇y ∇x z, w). We shall show in the next theorem that two affine structures ∇ and ∇ on R2n satisfying (5) and (6) give rise to a Lie group structure on the manifold R4n such that
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(R4n , R2n × {0}, {0} × R2n ) is a double Lie group. We recall that a double Lie group is given by a triple (G, G+ , G− ) of Lie groups such that G+ , G− are Lie subgroups of G and the product G+ × G− → G, (g+ , g− ) → g+ g− is a diffeomorphism (see [12]). The next result shows that the additional condition (3) of ∇ and ∇ with a fixed ω imposes restrictions on the Lie group obtained. Theorem 2.1. Let ∇ and ∇ be two affine structures on R2n compatible with a symplectic form ω and satisfying also (5). Then R2n × R2n with the product given by (x, x ) · (y, y ) = (x + α(x , y), β(x , y) + y ),
(8)
where x, x , y, y ∈ R2n and 1 α(x , y) = y + ∇y x − ∇y ∇y x , 2
1 β(x , y) = x − ∇x y − ∇x ∇x y 2
(9)
is a 3-step nilpotent double Lie group. Furthermore, the associated Lie bracket on its Lie algebra R2n ⊕ R2n is [(x, x ), (y, y )] = (∇y x − ∇x y , ∇x y − ∇y x ).
(10)
2n 2n 2n 2n 2n Proof. Let us set R2n + := R × {0} and R− := {0} × R . The maps α : R− × R+ −→ 2n 2n 2n 2n R+ and β : R− × R+ −→ R− satisfy the conditions
α0 = 1, αx (0) = 0, αx +y = αx ◦ αy , β0 = 1, βy (0) = 0, βx+y = βy ◦ βx
(11) (12)
2n for all x , y ∈ R2n − , x, y ∈ R+ , where we denote αx (y) := α(x , y) and βy (x ) := 2n 2n β(x , y) for x ∈ R− , y ∈ R+ . The above relations show that α is a left action of R2n − 2n 2n on R2n + and β is a right action of R+ on R− . These maps satisfy also the following compatibility conditions
αx (x + y) = αx x + αβx x y,
βx (x + y ) = βαy x x + βx y .
According to [12], the product given in (8) defines a Lie group structure on R4n such 2n that (R4n , R2n + , R− ) is a double Lie group. Note that the neutral element of this group structure is (0, 0) and the inverse of (x, x ) ∈ R4n is (α(−x , −x), β(−x , −x)). Let us determine now the associated Lie algebra. Linearizing the above actions, we 2n 2n 2n obtain representations µ : R2n − −→ End(R+ ) and ρ : R+ −→ End(R− ) given by d d (dαtx )0 (y), ρy (x ) = (dβty )0 (x ). µx (y) = dt 0 dt 0 For α and β given in (9), we obtain that (dαx )0 = 1 + ∇x ,
(dβy )0 = 1 − ∇y .
Hence, µx (y) = ∇x y,
ρy (x ) = −∇y x ,
showing that the bracket of its Lie algebra is the one given in (10).
Double Products and Hypersymplectic Structures on R4n
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If ad(x,x ) , x, x ∈ R2n stands for the transformation given by (10), then, using that ∇ and ∇ are torsion-free (see (1)) one has ad(x,x )
∇x −∇x = . −∇ ∇ x x
From (4) applied to both ∇ and ∇ , we obtain ad2(x,x )
∇x ∇x −∇x ∇x = −∇ ∇ ∇ ∇ x x x x
.
Finally, using (7), ad3(x,x ) = 0. Hence R2n ⊕ R2n is a 3-step nilpotent Lie algebra or R2n × R2n is a 3-step nilpotent Lie group, as claimed. We set the notation to be used in what follows. Since the construction of the Lie group structure on R4n in Theorem 2.1 depends on the affine structures ∇ and ∇ , we will denote this Lie group by N∇,∇ . The corresponding Lie algebra will be denoted n∇,∇ and the abelian Lie subalgebras R2n ⊕ {0}, {0} ⊕ R2n will be denoted n+ , n− , respectively. We note that (n∇,∇ , n+ , n− ) is a double Lie algebra, that is, n+ and n− are Lie subalgebras of n∇,∇ and n∇,∇ = n+ ⊕ n− as vector spaces. Remarks. (i) The construction of the Lie algebra in Theorem 2.1 can be made without requiring the affine structures to be compatible with a symplectic form. Indeed, if ∇ and ∇ are affine structures on Rm satisfying (5) and (6), then the bracket (10) defines a Lie algebra structure on Rm ⊕ Rm such that Rm ⊕ {0} and {0} ⊕ Rm are both abelian Lie subalgebras; hence Rm ⊕ Rm is 2-step solvable. Moreover, the centre of Rm ⊕ Rm is given by {(x, x ) ∈ Rm ⊕ Rm : ∇x = ∇x = 0, ∇x = ∇x = 0}.
(13)
(ii) If ∇ is any affine structure on Rm we denote by A the associative (and commutative) algebra obtained from Rm together with the product x.y = ∇x y, x, y ∈ Rm . In [3, 4],
aff(A) denoted the Lie algebra A ⊕ A with Lie bracket [(a, b), (c, d)] = (0, ad − bc) with a, b, c, d ∈ A. Note that if, in (i), we take ∇ = 0 then (5) and (6) trivially hold, obtaining in this case a semidirect product which coincides with aff(A). This family of algebras and various geometric properties were considered in [3, 4]. 3. Invariant Symplectic Structures on R4n In this section we show that n∇,∇ carries three symplectic structures, obtained from the symplectic form ω in R2n compatible with ∇ and ∇ . These forms, defined at the Lie
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algebra level, give rise to left-invariant symplectic forms on the corresponding Lie group N∇,∇ . Hence, R4n inherits symplectic structures which are invariant by this nilpotent group. First, we recall that a symplectic structure on a Lie algebra g is a non-degenerate skew-symmetric bilinear form ω satisfying dω = 0, where d ω(x, y, z) = ω(x, [y, z]) + ω(y, [z, x]) + ω(z, [x, y])
(14)
for x, y, z ∈ g. A given symplectic form ω on R2n allows us to define the following non-degenerate skew-symmetric bilinear forms on R2n ⊕ R2n : ω1 ((x, x ), (y, y )) := ω(x, y) + ω(x , y ), (15) ω2 ((x, x ), (y, y )) := −ω(x, y ) + ω(y, x ), ω ((x, x ), (y, y )) := ω(x, y) − ω(x , y ). 3 We show below that the above forms are closed with respect to the Lie bracket given in Theorem 2.1. Therefore, they define symplectic structures on n∇,∇ . Proposition 3.1. The 2-forms ω1 , ω2 and ω3 are closed on n∇,∇ . Proof. Since R2n ⊕ {0} and {0} ⊕ R2n are abelian subalgebras of n∇,∇ , it follows that the forms ωi , i = 1, 2, 3, are closed if and only if (dωi )((x, 0), (y, 0), (0, z )) = (dωi )((0, x ), (0, y ), (z, 0)) = 0 for all x, y, z, x , y , z ∈ R2n . But (dωi )((x, 0), (y, 0), (0, z )) = ωi ([(y, 0), (0, z )], (x, 0)) + ωi ([(0, z ), (x, 0)], (y, 0)) = ωi ((−∇y z , ∇y z ), (x, 0)) + ωi ((∇x z , −∇x z ), (y, 0)) and (dωi )((0, x ), (0, y ), (z, 0)) = ωi ([(0, y ), (z, 0)], (0, x ))+ωi ([(z, 0), (0, x )], (0, y )) = ωi ((∇z y , −∇z y ), (0, x ))+ωi ((−∇z x , ∇z x ), (0, y )). Using the expressions of ωi , i = 1, 2, 3 given in (15), we compute (dω1 )((x, 0), (y, 0), (0, z )) = (dω3 )((x, 0), (y, 0), (0, z )) = −ω(−∇y z , x) + ω(∇x z , y), x ), (0, y ), (z, 0)) (dω1 )((0, x ), (0, y ), (z, 0)) = −(dω3 )((0, = −ω(∇z y , x ) + ω(∇z x , y ) and
(dω2 )((x, 0), (y, 0), (0, z )) = ω(x, ∇y z ) − ω(y, ∇x z ), (dω2 )((0, x ), (0, y ), (z, 0)) = −ω(∇z y , x ) + ω(∇z x , y ).
Since ∇ and ∇ satisfy (1) and (3), we obtain that dωi = 0, i = 1, 2, 3. It follows from the definitions of the forms ωi , i = 1, 2, 3 that:
Double Products and Hypersymplectic Structures on R4n
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(i) the restrictions of ω1 and ω3 to n+ and n− are symplectic forms on these subalgebras; (ii) the Lie subalgebras n+ and n− are Lagrangian subspaces of n∇,∇ with respect to the symplectic form ω2 . Let the form ω on R2n be given by ω = e1 ∧ e2 + e3 ∧ e4 + · · · + e2n−1 ∧ e2n , where {e1 , . . . , e2n } is a fixed basis of R2n and {e1 , . . . , e2n } denotes the dual basis. Let us set ej := (ej , 0) and fj := (0, ej ), j = 1, . . . , 2n. Hence {e1 , . . . , e2n , f1 , . . . , f2n } is a basis of R2n ⊕ R2n and the forms ωi , i = 1, 2, 3, can be written as ω1 = e1 ∧ e2 + · · · + e2n−1 ∧ e2n + f 1 ∧ f 2 + · · · + f 2n−1 ∧ f 2n , ω2 = −e1 ∧ f 2 − · · · − e2n−1 ∧ f 2n + e2 ∧ f 1 + · · · + e2n ∧ f 2n−1 , ω3 = e1 ∧ e2 + · · · + e2n−1 ∧ e2n − f 1 ∧ f 2 − · · · − f 2n−1 ∧ f 2n . 3.1. Since n∇,∇ is a double Lie algebra, the endomorphism E given by E(x, y) = (x, −y) for x, y ∈ R2n is a product structure on n∇,∇ , that is, E 2 = 1 and E is integrable, in the sense that it satisfies the condition E[(x, x ), (y, y )] = [E(x, x ), (y, y )] + [(x, x ), E(y, y )] − E[E(x, x ), E(y, y )]. (16) for all x, x , y, y ∈ R2n . We note that the integrability of E is equivalent to n+ and n− , the eigenspaces corresponding to the eigenvalues ± of E, being Lie subalgebras of n∇,∇ . Moreover, since n+ and n− have equal dimension, E is a paracomplex structure on n∇,∇ . The symplectic form ω2 satisfies ω2 (E(x, x ), E(y, y )) = −ω2 ((x, x ), (y, y )) for all x, x , y, y ∈ R2n . Therefore, {n∇,∇ , E, ω2 } is an example of a parak¨ahler Lie algebra in the sense of Kaneyuki (see [11]). Another endomorphism on n∇,∇ related to its decomposition as a double Lie algebra is given by J (x, y) = (−y, x) for x, y ∈ R2n . The endomorphism J is a complex structure on n∇,∇ , that is, J 2 = −1 and J is integrable, i.e., it satisfies J [(x, x ), (y, y )] = [J (x, x ), (y, y )] + [(x, x ), J (y, y )] + J [J (x, x ), J (y, y )] (17) for all x, x , y, y ∈ R2n . We note that J E = −EJ , and therefore {J, E} is a complex product structure on n∇,∇ (see [3]). The symplectic form ω1 satisfies ω1 (J (x, x ), J (y, y )) = ω1 ((x, x ), (y, y )) for all x, x , y, y ∈ R2n . Hence, ω1 is a K¨ahler form on n∇,∇ .
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4. Induced Geometry on R4n In this section we analyze the properties of the metric on the manifold R4n obtained by left-translating by the Lie group N∇,∇ , the standard inner product of signature (2n, 2n) on R2n ⊕ R2n . We show that this metric is always complete and it is flat if and only if the Lie group N∇,∇ is 2-step nilpotent (see Theorem 4.2 and Theorem 4.3). Furthermore, this metric on R4n is hypersymplectic with respect to the structures J and E defined previously; in particular, it is neutral K¨ahler and Ricci-flat. Explicit examples will be given in subsequent sections. Let us define a bilinear form g on n∇,∇ by g((x, x ), (y, y )) = −ω(x, y ) + ω(x , y)
(18)
for all (x, x ), (y, y ) ∈ n∇,∇ . It is clearly symmetric and non-degenerate. With respect to the basis {e1 , . . . , e2n , f1 , . . . , f2n }, g can be written as g = 2 −e1 · f 2 − · · · − e2n−1 · f 2n + e2 · f 1 + · · · + e2n · f 2n−1 , where · denotes the symmetric product of 1-forms. Moreover, g satisfies the two following conditions: g(J (x, x ), J (y, y )) = g((x, x ), (y, y )), g(E(x, x ), E(y, y )) = −g((x, x ), (y, y ))
(19) (20)
for x, x , y, y ∈ R2n . Indeed, g(J (x, x ), J (y, y )) = g((−x , x), (−y , y)) = −ω(−x , y) + ω(x, −y ) = g((x, x ), (y, y )) and g(E(x, x ), E(y, y )) = g((x, −x ), (y, −y )) = ω(x, y ) − ω(x , y) = −g((x, x ), (y, y )). Thus, g is a Hermitian metric on n∇,∇ with respect to both structures J and E. We note that the subalgebras n+ and n− are both isotropic subspaces of n∇,∇ with respect to g and this metric has signature (2n, 2n). Moreover, it is easy to verify that the 2-forms ω1 , ω2 and ω3 can be recovered from g and the endomorphisms J and E. Indeed we have ω1 ((x, x ), (y, y )) = g(J (x, x ), (y, y )), (21) ω2 ((x, x ), (y, y )) = g(E(x, x ), (y, y )), ω ((x, x ), (y, y )) = g(J E(x, x ), (y, y )). 3 The endomorphisms J and E of n∇,∇ , as well as the 2-forms ω1 , ω2 and ω3 and the metric g can be extended to the group N∇,∇ by left translations. Hence, N∇,∇ is equipped with: (1) a complex structure J and a product structure E such that J E = −EJ ;
Double Products and Hypersymplectic Structures on R4n
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(2) a (pseudo) Riemannian metric g such that g(J (x, x ), J (y, y )) = g((x, x ), (y, y )), g(E(x, x ), E(y, y )) = −g((x, x ), (y, y )) for all x, x , y, y ∈ (T(N∇,∇ )); (3) three symplectic forms ω1 , ω2 and ω3 which satisfy (21). To summarize, we have obtained Theorem 4.1. The nilpotent Lie group N∇,∇ carries a left-invariant hypersymplectic structure given by the 3-tuple {J, E, g}. In particular, (N∇,∇ , J, g) is a (neutral) K¨ahler manifold and g is a Ricci-flat metric. Note also that E is a product structure on N∇,∇ and hence, there is a decomposition of T(N∇,∇ ) into the Whitney sum of two involutive distributions of the same rank which are interchanged by J . Remark. The leaves of both foliations given by E are Lagrangian submanifolds of the symplectic manifold (N∇,∇ , ω2 ). Hence, N∇,∇ is an example of a homogeneous parak¨ahler manifold (see [11]). 4.1. Curvature and completeness of g. Since g is left-invariant, the Levi-Civita connection ∇ g can be computed on left-invariant vector fields, i.e., on the Lie algebra n∇,∇ . After a computation one finds that ∇
g
(x,x )
∇x + ∇ x = 0
0 ∇x
+ ∇
x
.
(22)
One can verify, using the above expression of ∇ g , that J and E are parallel with respect to the Levi-Civita connection. We will show next that this connection need not be flat. If R denotes the curvature of ∇ g , it is easily seen (using (4)) that R((x, 0), (y, 0)) = R((0, x ), (0, y )) = 0. Moreover, R((x, 0), (0, y )) = ∇(x,0) ∇(0,y ) − ∇(0,y ) ∇(x,0) − ∇(−∇ y ,∇x y ) g
g
g
g
g
x
and using (22) together with (1), (5) and (7) one obtains ∇x ∇ 0 y R((x, 0), (0, y )) = 4 0 ∇x ∇y
= −4 ad[(x,0),(0,y )] .
Since R and the Lie bracket are skew-symmetric, one finally obtains R((x, x ), (y, y )) = −4 ad[(x,x ), (y,y )] ,
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thus showing that ∇ g will be flat if and only if N∇,∇ is 2-step nilpotent. Note also that R=0
if and only if
∇x ∇y = 0
(23)
for all x, y ∈ R2n . Summing up, we have shown Theorem 4.2. The following conditions are equivalent: (i) The Lie algebra n∇,∇ is 2-step nilpotent; (ii) ∇x ∇y = 0 for all x, y ∈ R2n ; (iii) The hypersymplectic metric is flat. We end this section studying the completeness of ∇ g . It follows from [7] that ∇ g will be complete if and only if the differential equation on n∇,∇ g
x(t) ˙ = adx(t) x(t)
(24) g
admits solutions x(t) ∈ g defined for all t ∈ R. Here adx means the adjoint of the transformation adx with respect to the metric g. It is easy to verify that the right-hand g g side of (24) is given by adx(t) x(t) = −∇x(t) x(t) for all t in the domain of x and thus we have to solve the equation g
x(t) ˙ = −∇x(t) x(t).
(25)
The curve x(t) on n∇,∇ can be written as x(t) = (a(t), b(t)), where a(t), b(t) ∈ R2n are smooth curves on R2n . Hence, using (22), Eq. (25) translates into the system a˙ = −∇a a − ∇b a, (26) b˙ = −∇a b − ∇b b. Let us differentiate the first equation of the system above. We have a¨ = −2∇a a˙ − ∇a b˙ − ∇b a˙ = 2∇a ∇a a + 2∇a ∇a b + ∇a ∇a b + ∇a ∇b b + ∇b ∇a a + ∇b ∇a b = 0, using (4), (5), (6) and (7). In the same fashion, we differentiate the second equation of (26) and obtain b¨ = −∇a b˙ − ∇b a˙ − 2∇b b˙ = ∇a ∇a b + ∇a ∇b b + ∇b ∇a a + ∇b ∇a b + 2∇b ∇a b + 2∇b ∇b b = 0, using again (4), (5), (6) and (7). Thus, there exist constant vectors A, B, C, D ∈ R2n such that a(t) = At + B,
b(t) = Ct + D.
The explicit solution of the system (26) with initial condition x(0) = (a0 , b0 ) is given by a(t) = (−∇a0 a0 − ∇a 0 b0 )t + a0 ,
b(t) = (−∇a0 b0 − ∇b 0 b0 )t + b0 .
Therefore, x(t) is defined for all t ∈ R and, in consequence, ∇ g is complete. Thus, we have obtained
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11
Theorem 4.3. Hypersymplectic metrics on n∇,∇ are always complete. Remark. The completeness of ∇ g could have been dealt with in the following manner when ∇ g is flat. In this case, from results in [15] we obtain that the completeness follows if the transformation (x, x ) → ∇ g (x,x ) (y, y ) is nilpotent for every (y, y ). But, using (22) one finds ∇y g ∇ (y, y ) = ∇ y
∇ y . ∇
(27)
y
Using (4) and (23) one shows that ∇ g (y, y )2 = 0 and hence the completeness follows. 5. Explicit Examples In this section we will consider particular cases of the constructions given previously. In the first one we give explicit affine structures ∇ and ∇ such that the resulting group is 8n-dimensional, 2-step nilpotent and with a 4n-dimensional center invariant by the complex structure. It also admits compact quotients, hence the associated nilmanifolds are Kodaira manifolds (see [6]). In this case the hypersymplectic metric obtained on the group is complete and flat, hence it is isometric to the standard one. On the other hand this Lie group carries a closed special form in the sense of ([6], Sect. 2) and thus, the procedure developed in [6] may be applied to produce non-flat K¨ahler Ricci-flat metrics on this family of Kodaira manifolds. In the second one we give a 3-parameter family ∇ and ∇ a,b,c on R4 satisfying all the requirements to obtain a 3-step nilpotent group structure on R8 . In this case the hypersymplectic metric obtained on R8 will be complete and non-flat. Moreover, the bracket relations will show that compact quotients can be obtained. This example can be generalized to higher dimensions, thus obtaining complete non-flat hypersymplectic metrics on R4n , n ≥ 2, invariant by a 3-step nilpotent Lie group. 5.1. Neutral K¨ahler Einstein metrics on Kodaira manifolds. Let us consider on R4n = span{e1 , . . . , e4n } the flat torsion-free connections ∇ and ∇ given by ∇ei ei = ei+1 , i odd, 1 ≤ i ≤ 2n, ∇ej ek = 0, otherwise and
∇e i ei = ei+1 , ∇e j ek = 0,
i odd, 2n + 1 ≤ i ≤ 4n, otherwise.
Clearly, ∇∇ = 0 = ∇ ∇.Also, it is easy to see that both connections are compatible with the standard symplectic form ω on R4n given by ω = e1 ∧e2 +e3 ∧e4 +· · ·+e4n−1 ∧e4n .
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We can form then the 8n-dimensional Lie algebra n∇,∇ as in previous sections. It has a basis {ei , fi : i = 1, . . . , 4n}, where ei = (ei , 0) and fi = (0, ei ) and the only non-zero Lie brackets are fi+1 , i odd, 1 ≤ i ≤ 2n, [ei , fi ] = −ei+1 , i odd, 2n + 1 ≤ i ≤ 4n. Let N∇,∇ denote the simply connected Lie group associated to the Lie algebra n∇,∇ . Since the structure constants are 0, 1 or −1, by Malcev’s theorem [13], there exists a discrete subgroup of N∇,∇ such that M := \N∇,∇ is compact. Using (13), we obtain that the centre z of n∇,∇ is given by z = span{ei , fi : i is even}, showing in particular that this Lie group is 2-step nilpotent and z is 4n-dimensional and stable under the action of J . Therefore, the nilmanifold M is an 8n-dimensional Kodaira manifold (see [6]). According to the results in §4, N∇,∇ carries a left-invariant hypersymplectic structure, which is flat since N∇,∇ is 2-step nilpotent (see Theorem 4.2). Besides, since the centre z is, with respect to ω1 , a Lagrangian subspace, the symplectic form ω1 on the Lie group induces a closed special 2-form on M . The method described in [6] can be applied in this case to produce Ricci-flat neutral K¨ahler metrics on M . We note that z is a special Lagrangian subspace of n∇,∇ with respect to the J -holomorphic form = (ω2 + iω3 )2n . This gives rise to special Lagrangian submanifolds on the symplectic manifold M . The Levi-Civita connection ∇ g of the hypersymplectic metric g on the group N∇,∇ is given by g ∇ei ei = ei+1 , i odd, 1 ≤ i ≤ 2n, ∇ g f = f , i odd, 1 ≤ i ≤ 2n, i+1 ei i g ∇ e = e i odd, 2n + 1 ≤ i ≤ 4n, i+1 , f i ∇ gi f = f , i odd, 2n + 1 ≤ i ≤ 4n, i+1 fi i g
and 0 in all the other possibilities. Note that we have the relations z = span{∇x y : x, y ∈ n∇,∇ } and ∇ g x = 0 for x ∈ z. Hence, Proposition 4.1 in [6] can be applied, obtaining neutral Calabi-Yau metrics on compact quotients of the cotangent bundle of N∇,∇ . 5.2. Complete, non-flat, neutral K¨ahler Einstein metrics on R8 . We will consider next R4 = span{e1 , . . . , e4 } equipped with two affine structures ∇ and ∇ given by 1 0 1 0 0 −1 0 1 ∇e1 = ∇e3 = , −1 0 −1 0 0 −1 0 1 0 0 0 0 0000 −1 0 −1 0 1 0 1 0 ∇e2 = , ∇e4 = , 0 0 0 0 0 0 0 0 −1 0 −1 0 1010 a 0 a 0 0 0 0 0 c a b −a −a 0 −a 0 and ∇e 1 = , ∇e 2 = , −a 0 −a 0 0 0 0 0 c −a −b + 2c a −a 0 −a 0
Double Products and Hypersymplectic Structures on R4n
∇e 3
a c = −a −b + 2c
0 a −a −b + 2c 0 −a −a −2b + 3c
0 a , 0 a
13
∇e 4
0 a = 0 a
0 0 0 0
0 a 0 a
0 0 0 0
for a, b, c ∈ R. One can verify easily that ∇ and ∇ satisfy Eq. (3) with respect to the symplectic form ω = e1 ∧ e2 + e3 ∧ e4 on R4 . In order to see that the compatibility condition (5) holds, we observe that
∇ej ∇e k
0 −b + c = 0 −b + c
0 0 0 −b + c 0 0 0 −b + c
0 0 0 0
(28)
for (j, k) = (1, 1), (1, 3), (3, 1), (3, 3) and ∇ej ∇e k = 0 otherwise. From results in §2, we may construct the 8-dimensional nilpotent Lie group N∇,∇ (see Theorem 2.1), whose underlying manifold is R4 × R4 . The group structure is as follows (x, x ) · (y, y ) = (x + α(x , y), β(x , y) + y ), where α and β are given, in terms of their components, by α1 (x , y) = y1 + a(y1 + y3 )(x1 + x3 ), α2 (x , y) = y2 + (by1 − ay2 + cy3 + ay4 )x1 − a(y1 + y3 )x2 + (cy1 − ay2 + (−b + 2c)y3 + ay4 ) x3 +a(y1 + y3 )x4 + 21 (−b + c)(y1 + y3 )2 (x1 + x3 ),
α3 (x , y) = y3 − a(y1 + y3 )(x1 + x3 ), α4 (x , y) = y4 + (cy1 − ay2 + (−b + 2c)y3 + ay4 ) x1 − a(y1 + y3 )x2 + ((−b + 2c)y1 − ay2 + (−2b + 3c)y3 + ay4 ) x3
β1 (x , y) = β2 (x , y) =
x1 x2
+a(y1 + y3 )x4 + 21 (−b + c)(y1 + y3 )2 (x1 + x3 ), − (x1 + x3 )(y1 + y3 ), + (x2 − x4 )(y1 + y3 ) + (x1 + x3 )(y2 − y4 )
− 21 (−b + c)(x1 + x3 )2 (y1 + y3 ),
β3 (x , y) = x3 + (x1 + x3 )(y1 + y3 ), β4 (x , y) = x4 + (x2 − x4 )(y1 + y3 ) + (x1 + x3 )(y2 − y4 ) − 21 (−b + c)(x1 + x3 )2 (y1 + y3 ). We know from §4 that N∇,∇ carries an invariant hypersymplectic structure whose associated neutral metric is complete (Theorem 4.3). Moreover, using (23) and (28), we may conclude that if b = c, then n∇,∇ is 2-step nilpotent and the hypersymplectic metric is flat, whereas if b = c, then n∇,∇ is 3-step nilpotent and the hypersymplectic metric is not flat. Note that taking a, b, c ∈ Q, the structure constants of n∇,∇ with respect to the canonical basis {e1 , . . . , e4 , f1 , . . . f4 } of n∇,∇ are rational, and thus there exists a discrete co-compact subgroup of N∇,∇ [13]. The complete non-flat hypersymplectic metric on the Lie group induces a metric with the same properties on the associated compact quotient.
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The Lie group N∇,∇ is diffeomorphic to R8 , hence there exists a global system of coordinates x1 , . . . , x4 , x1 . . . , x4 such that the left-invariant 1-forms dual to the basis {e1 , . . . , e4 , f1 , . . . , f4 } are given as follows e1 = 1 − a(x1 + x3 ) dx1 − a(x1 + x3 ) dx3 , e2 = (−bx1 + ax2 − cx3 − ax4 ) dx1 + 1 + a(x1 + x3 ) dx2 + −cx1 + ax2 − (−b + 2c)x3 − ax4 dx3 − a(x1 + x3 ) dx4 , e3 = a(x1 + x3 ) dx1 + 1 + a(x1 + x3 ) dx3 , e4 = −cx1 + ax2 − (−b + 2c)x3 − ax4 dx1 + a(x1 + x3 ) dx2 + −(−b + 2c)x1 + ax2 − (−2b + 3c)x3 − ax4 dx3 + 1 − a(x1 + x3 ) dx4 , f 1 = (x1 + x3 ) dx1 + (x1 + x3 ) dx3 + dx1 , f 2 = − 21 (−b + c)(x1 + x3 )2 − x2 + x4 dx1 − (x1 + x3 ) dx2 + − 21 (−b + c)(x1 + x3 )2 − x2 + x4 dx3 + (x1 + x3 ) dx4 + dx2 , f 3 = −(x1 + x3 ) dx1 − (x1 + x3 ) dx3 + dx3 , f 4 = − 21 (−b + c)(x1 + x3 )2 − x2 + x4 dx1 − (x1 + x3 ) dx2 + − 21 (−b + c)(x1 + x3 )2 − x2 + x4 dx3 + (x1 + x3 ) dx4 + dx4 . The K¨ahler form ω1 on n∇,∇ is ω1 = e1 ∧ e2 + e3 ∧ e4 + f 1 ∧ f 2 + f 3 ∧ f 4 and the hypersymplectic metric g is given by g = −e1 ⊗ f 2 + e2 ⊗ f 1 − e3 ⊗ f 4 + e4 ⊗ f 3 +f 1 ⊗ e2 − f 2 ⊗ e1 + f 3 ⊗ e4 − f 4 ⊗ e3 . In the particular case a = 0, b = 1, c = 0, we obtain that 3 2 g = − (x1 + x3 ) + x2 − x4 (dx1 + dx3 )2 2 +2(x1 + x3 )(dx1 + dx3 )(dx2 − dx4 ) − x1 dx1 dx1 − dx1 dx2 + dx2 dx1 + x3 (dx1 dx3 + dx3 dx1 ) +(x1 + 2x3 ) dx3 dx3 − dx3 dx4 + dx4 dx3 is a complete non-flat K¨ahler Ricci-flat neutral metric on R8 . 6. Final Comments and Questions We note that the complex structure J defined in n∇,∇ satisfies the condition [J x, J y] = [x, y] for all x, y ∈ n∇,∇ , which implies the integrability of J . An almost complex structure J on a Lie algebra g satisfying [J x, J y] = [x, y] for all x, y ∈ g is called abelian. We also observe that E satisfies a similar condition, [Ex, Ey] = −[x, y] for all x, y ∈ n∇,∇ , which implies the integrability of E. An almost product structure E on a Lie algebra g satisfying [Ex, Ey] = −[x, y] for all x, y ∈ g will be called abelian. Related to these notions we have the following characterization:
Double Products and Hypersymplectic Structures on R4n
15
Proposition 6.1. Let {J, E} be a complex product structure on the Lie algebra g and let (g, g+ , g− ) be the associated double Lie algebra, i.e., g+ and g− are the Lie subalgebras of g such that E|g+ = 1, E|g− = −1. Then the following assertions are equivalent: (i) J is an abelian complex structure. (ii) The Lie subalgebras g+ and g− are abelian. (iii) If A+ and A− denote the annihilators of g− and g+ , respectively, in g∗ , then 2 ∗ dA+ ⊂ A+ ⊗ A− , dA− ⊂ A+ ⊗ A− , where d : g∗ −→ g is given by (df )(x ∧ y) = −f ([x, y]). (iv) E is an abelian product structure. Proof. (i) ⇐⇒ (ii) Assume first that J is abelian. If x, y ∈ g+ then [x, y] ∈ g+ and [J x, Jy] ∈ g− since g+ and g− are subalgebras. But then [x, y] = [J x, J y] ∈ g+ ∩ g− = {0}, and thus [x, y] = [J x, J y] = 0 for all x, y ∈ g+ . Thus, g+ and g− are abelian. Conversely, suppose that g+ and g− are abelian. For u, v ∈ g+ or u, v ∈ g− , from the integrability of J we obtain [J u, v] + [u, J v] = J [u, v] − J [J u, J v] = 0. If x = x1 + x2 , y = y1 + y2 with x1 , y1 ∈ g+ , x2 , y2 ∈ g− , then J ([J x, Jy] − [x, y]) = [J x1 + J x2 , y1 + y2 ] + [x1 + x2 , J y1 + J y2 ] = [J x1 , y1 ] + [J x2 , y2 ] + [x1 , J y1 ] + [x2 , J y2 ] = ([J x1 , y1 ] + [x1 , J y1 ]) + ([J x2 , y2 ] + [x2 , J y2 ]) = 0, and thus J is abelian. (ii) ⇐⇒ (iii) Suppose first that g+ and g− are abelian. Take f ∈ A+ , which is the 2 A+ ⊕ A+ ⊗ A− (see [3]), so we only have annihilator of g− . It is known that df ∈ 2 A+ is zero. For x, y ∈ g+ , we have to see that the component of (df ) in (df )(x ∧ y) = −f ([x, y]) = 0, showing that df ∈ A+ ⊗ A− . The corresponding assertion for f ∈ A− follows in a similar manner. Conversely, if (iii) is valid, take f = f1 + f2 ∈ g∗ with f1 ∈ A+ , f2 ∈ A− , and x, y ∈ g+ or U, V ∈ g− . Then f ([x, y]) = −(df )(x ∧ y) = −(df1 )(x ∧ y) − (df2 )(x ∧ y) = 0, since df1 , df2 ∈ A+ ⊗ A− . Then [x, y] = 0 and both g+ and g− are abelian. (ii) ⇐⇒ (iv) This follows by a straightforward computation. If one of the conditions in the proposition above holds, we will say that the complex product structure {J, E} is abelian. We will say that a hypersymplectic structure is abelian when the underlying complex product structure is abelian. Using the previous proposition and results in [1] one can show that any Lie algebra carrying an abelian hypersymplectic structure is of the form given in Theorem 2.1, that is, a double product of two abelian Lie algebras endowed with compatible affine structures and symplectic forms. Furthermore, we showed in previous sections that in this case the Lie algebra is nilpotent and also the neutral metric is always complete and not necessarily flat. It would be of interest to know geometric properties of neutral
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metrics compatible with {J, E}, J and E abelian, without imposing the condition on the associated forms being closed. We also believe it is of interest to proceed as we did in this paper, carrying out the construction of double Lie groups from affine-symplectic data on another class of Lie groups (not necessarily abelian) and then understand the properties of the resulting hypersymplectic manifold. References 1. Andrada, A.: Hypersymplectic Lie algebras. To appear in J. Geom. Phys. 2. Andrada, A.: Estructuras producto complejas y m´etricas hipersimpl´ecticas asociadas. PhD thesis, FaMAF, Universidad Nacional de C´ordoba, December 2003 3. Andrada, A., Salamon, S.: Complex product structures on Lie algebras. Forum Math. 17, 261–295 (2005) 4. Barberis, M. L., Dotti, I.: Abelian complex structures on solvable Lie algebras. J. Lie Theory 14(1), 25–34 (2004) 5. Barret, J., Gibbons, G. W., Perry, M. J., Pope, C. N., Ruback, P.: Kleinian geometry and the N = 2 superstring. Int. J. Mod. Phys. A9, 1457–1494 (1994) 6. Fino, A., Pedersen, H., Poon, Y. S., Sørensen, M.: Neutral Calabi-Yau structures on Kodaira manifolds. Commun. Math. Phys. 248, 255–268 (2004) 7. Guediri, M.: Sur la compl´etude des pseudo-m´etriques invariantes a` gauche sur les groupes de Lie nilpotentes. Rend. Sem. Mat. Univ. Pol. Torino 52, 371–376 (1994) 8. Hitchin, N.: Hypersymplectic quotients. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 124 Suppl., 169–180 (1990) 9. Hull, C. M.: Actions for (2, 1) sigma-models and strings. Nucl. Phys. B 509(2), 252–272 (1998) 10. Kamada, H.: Self-dual K¨ahler metrics of neutral signature on complex surfaces. Tohoku Mathematical Publications, Number 24 (2002) 11. Kaneyuki, S.: Homogeneous symplectic manifolds and dipolarizations in Lie algebras. Tokyo J. Math. 15, 313–325 (1992) 12. Lu, J.-H., Weinstein, A.: Poisson Lie groups, dressing transformations and Bruhat decompositions. J. Diff. Geom. 31, 501–526 (1990) 13. Malcev, A. I.: On a class of homogeneous spaces. Reprinted in Amer. Math. Soc. Translations, Series 1, 9, 276–307 (1962) 14. Ooguri, H., Vafa, C.: Geometry of N = 2 strings. Nucl. Physics B 361, 469–518 (1991) 15. Segal, D.: The structure of complete left-symmetric algebras. Math. Ann. 293, 569–578 (1992) Communicated by G.W. Gibbons
Commun. Math. Phys. 262, 17–32 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1473-8
Communications in
Mathematical Physics
Semi-Focusing Billiards: Hyperbolicity Leonid A. Bunimovich1, , Gianluigi Del Magno2, 1
Southeast Applied Analysis Center, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. E-mail:
[email protected] 2 Centro di Ricerca Matematica “Ennio De Giorgi”, Scuola Normale Superiore, Piazza dei Cavalieri 3, 56100 Pisa, Italy. E-mail:
[email protected] Received: 9 August 2004 / Accepted: 16 June 2005 Published online: 24 November 2005 – © Springer-Verlag 2005
Abstract: In this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of R3 . We also prove that a related class of semi-focusing billiards has mixed dynamics, i.e., their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other. These billiards are the first rigorous examples of billiards in domains of R3 with divided phase space.
1. Introduction It is well known that the dynamics of a gas of elastically colliding particles (the hard balls or Boltzmann gas) can be reduced to a billiard in a domain with boundary formed by an union of (not necessarily disjoint) cylinders [S2]. The corresponding billiards are called semi-dispersing. An elegant mechanical model of nuclei has been recently introduced [P1, P2], where N point particles interact via an attracting potential which keeps the distances between any two particles less than a constant L (“diameter of a nucleus”). The particles move freely by inertia until the distance between a pair of particles equals L. At this moment, the two particles “collide” elastically. The potential of interaction is therefore a hard core potential as in the hard spheres gas, but the particles are located inside rather than outside the “core”. This model can be reduced to a billiard inside a domain where some smooth components of the boundary are pieces of convex outward cylinders as the interaction potential is attractive. In the hard sphere gas, instead, the repulsive potential generates boundary components which are convex inward cylinders. Because of this duality such billiards can be naturally called semi-focusing billiards.
The first author was partially supported by the NSF grant #0140165 and the Humboldt Foundation. The second author was partially supported by the FCT (Portugal) through the Program POCTI/FEDER.
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L.A. Bunimovich, G. Del Magno
It is well known that semi-dispersing cylindrical billiards in R3 are non-uniformly hyperbolic if the cylinders are orthogonal and their bases span R3 [Sz] (for more general results on the hyperbolic and ergodic properties of semi-dispersing cylindrical billiards, see the review [Si]). In this paper, we prove an analogous result (Theorem 1) for semifocusing billiards in R3 . Nowhere dispersing ergodic billiards in Rn with n ≥ 3 were constructed in [B-R1, B-R2, B-R3], but the corresponding billiard domains were nonconvex. Numerical studies suggested [P1, P2, P3] that convex hyperbolic billiards exist. Here we prove hyperbolicity for a class of three-dimensional billiard domains containing the one studied in [P3]. The ergodicity will be addressed in a future paper. These billiards, as well as the ones in [B-R1, B-R2, B-R3], can be viewed as higher-dimensional generalizations of two-dimensional stadia. More precisely, we consider billiards in domains with boundaries formed by flat faces and pieces of cylinders whose sections are absolutely focusing curves [B3, D]. Pieces of spheres instead of cylinders were used in [B-R1, B-R2, B-R3]. We also present the first rigorous examples (Theorem 2) of three-dimensional billiards with phase space which is an union of two sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other (divided phase space). 2. Generalities 2.1. Billiards in Rn . Let be an open and connected subset of Rn , n ≥ 3 such that ∂ consists of a finite number of hypersurfaces of class C 3 intersecting at most at their boundaries. Let T1 be the unit tangent bundle of which can be identified with × S n−1 , where S n−1 is the unit sphere in Rn . We will denote by {φt }t∈R the billiard flow inside acting on the space obtained from × S n−1 identifying the elements of ∂ × S n−1 according to the standard law of reflection: the angle of incidence equals the angle of reflection. For a precise definition of the billiard flow and the billiard map, see [C-F-S]. Denote by the collection of all unit vectors in Rn attached to the boundary of and pointing inward. The billiard map T : → is the first return map induced by the billiard flow on . If t : → R+ is the first return time to , then the billiard map is given by Tp = φt (p) p for any p = (q, v) ∈ . This map preserves the probability measure dµ = cv, n(q)dqdω, where dq is the Lebesgue measure on ∂, dω is the Lebesgue measure on S n−1 , n(q) is unit normal of ∂ at q pointing inside , and c is a normalizing constant. One of the characteristic features of general billiard maps is that they are not defined and smooth everywhere on . Let S + be the subset of , where T is not defined or fails to be C 1 . The set S − is defined similarly with T replaced by T −1 . S + (S − ) is called the − k−1 i − −i + singular set of T (T −1 ). For any k > 0, the sets Sk+ = ∪k−1 i=0 T S and Sk = ∪i=0 T S + − + =∪ − are the singular sets of T k and T −k . Finally, let S∞ k≥0 Sk and S∞ = ∪k≥0 Sk . All + − these sets have zero measure because µ(S ) = µ(S ) = 0 [C-F-S]. 2.2. The differential of T . We compute the differential of T with respect to an appropriate system of coordinates of the tangent spaces of . We follow [W4, W3]. Let π : → ∂ be the canonical projection given by π(q, v) = q for any (q, v) ∈ . For any p ∈ , denote by Lp and Vp , respectively, the tangent plane of ∂ at π(p) and the plane orthogonal to the vector p. The tangent space Tp can be naturally identified with Lp × Vp . Let P : Vp → Lp be the projection along p, and let I be the
Semi-Focusing Billiards: Hyperbolicity
19
identity operator on Vp . The operator P × I : Vp × Vp → Lp × Vp identifies Lp × Vp with Vp × Vp . Furthermore we identify Vp and VTp by transporting Vp parallel to itself up to π(Tp) and then by using the transformation Up : Vp → VTp which reflects Rn about the tangent plane LTp . After these identifications Dp T becomes a linear operator on Vp × Vp . In fact, Dp T is the composition of two linear maps on Vp × Vp , the first describes the free motion of the point particle from π(p) up to the point of reflection π(T p), and the second describes the reflection of the point particle at π(T p). Both maps can be represented as 2 × 2 matrices of linear operators on Vp . The first map has the following block form: I lI , (1) 0 I where l is the distance between two consecutive reflections, and I is the identity operator on Vp . Let Tp = (q , v ). The second map has the block form I 0 , (2) R I where R = 2v , n(q )P1∗ KP1 is a self-adjoint operator on Vp , P1 : Vp → LTp is the projection onto LTp along p, and K is the second fundamental form of ∂ evaluated at q . Note that P1∗ : LTp → Vp is the projection onto Vp along n(q ). Therefore, as a linear operator on Vp × Vp , the map Dp T has the block form I lI . (3) R I + lR Let ·, · be the Euclidean scalar product on Vp . For any u = (ξ, η) ∈ Vp × Vp and v = (ξ , η ) ∈ Vp × Vp , let ω(u, v) = ξ, η − ξ , η be the standard symplectic form on Vp × Vp . Note that the maps (1)-(3) are symplectic with respect to ω. 3. Semi-Focusing Billiards From now on, we will only consider billiards in domains of R3 . We start this section by introducing a special class of curves called absolutely focusing that were used to construct hyperbolic planar billiards [B3, D]. Next we construct a family of cylindrical surfaces having absolutely focusing curves as sections. These surfaces together with flat faces (connected and bounded subset of planes) form the boundary of our billiards. 3.1. Absolutely focusing curves. Consider a C 3 planar, strictly convex, simple and compact curve γ of R2 . Let Mγ = {(q, v) ∈ γ × R2 : v = 1 and v, n(q) ≥ 0}, where n(q) is the normal vector of γ at q pointing inside the convex hull of γ . The set Mγ is the billiard phase space over γ . Fix an orientation on γ , and call O the first endpoint of γ with respect to this orientation. For any z = (q, v) ∈ Mγ , let s = s(z) be the length of the subarc of γ with endpoints O and q, and let θ = θ(z) ∈ [0, π ] be the angle between v and the oriented tangent of γ at q. The pair (s, θ ) forms a system of coordinates for Mγ . If γ is parametrized by s, then r(s) > 0 and κ(s) > 0 denote the radius of curvature and the curvature of γ at s, respectively. We assume that 0 κ(s)ds ≤ π , where is
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the length of γ . The last condition together with the fact that the third derivative of γ is bounded (see [H]) imply that no trajectory can have infinitely many consecutive collisions with γ with the exception, perhaps, of the periodic trajectory connecting the endpoints of γ . An incoming ray to γ is said to be focused by γ if the infinitesimal family of rays parallel to the incoming ray focuses in linear approximation after leaving γ , i.e., after a complete series of consecutive collisions with γ . Definition 1. A curve γ as above is called absolutely focusing if all the incoming rays are focused by γ . Absolute focusing plays a crucial role for constructing hyperbolic billiards with at least one focusing component of the boundary. To ensure such billiards be hyperbolic, one needs to avoid having arbitrarily long focusing times for narrow beams of rays with a series of reflections along the boundary. Indeed, the focusing produces convergence (rather than divergence that is necessary for hyperbolicity) of nearby orbits. It is the mechanism of defocusing that produces hyperbolicity of billiards with focusing components. Defocusing means that an initially convergent beam of rays has enough time before the next collision with the boundary to become divergent. To make this happen, one needs to ensure first that parallel beams of rays are never formed after reflections from the focusing boundary. Clearly, it is in the neighborhood of such “parabolic” orbits that focusing times are not bounded and defocusing does not occur. If such parabolic orbits do not exist, then the focusing times are bounded and one only needs the free passes between reflections from any focusing component and any other component of the boundary to be sufficiently big to ensure defocusing. These two conditions, the absence of parallel beams after a series of reflections along focusing components and sufficiently long free passes after leaving focusing components produce hyperbolicity. 3.2. Cone fields for absolutely focusing curves. We refer the reader to the papers [W1, W2, D] for an introduction to invariant cone fields and their application to billiards. Consider an absolutely focusing curve γ . Let z = (q, v) ∈ Mγ . A vector u ∈ Tz Mγ corresponds to a smooth variation of z, i.e., a smooth family of directions containing z. We say that u focuses if the projection of the corresponding variation onto v ⊥ vanishes in linear approximation at some point q ∈ R2 . The distance between q and q taken with a positive or negative sign depending on whether or not q follows q along its trajectory is called the focusing time of u and is denoted by τ + (z, u). If (us , uθ ) are the components of u ∈ Tz Mγ , z = (s, θ ) ∈ Mγ with respect to the coordinates (s, θ ), then [W1, D] τ + (z, u) =
sin θ . κ(s) + uθ /us
Any absolute focusing curve admits an invariant cone field [D]. Such a cone field C = {C(z)}z∈Mγ for a curve γ is of the form C(z) = u ∈ Tz Mγ : 0 ≤ τ + (z, u) ≤ τC+ (z) for a suitable positive function τC+ : Mγ → R. We associate to C another function τC− : Mγ → R defined through the so-called Mirror Formula 1 τC+ (z)
+
1 τC− (z)
=
2κ(s) , sin θ
z = (s, θ ) ∈ Mγ .
(4)
Semi-Focusing Billiards: Hyperbolicity
21
If u ∈ C(z) such that τ + (z, u) = τC+ (z), then τC− (z) has the following geometrical meaning. Consider the variation associated to u. We obtain a new variation if we reverse the velocities of the variation associated to u and reflect them off γ . Then τC− (z) is the focusing time of this new variation. It follows that τC+ (z) + τC− (T z) ≤ t (z) for any z ∈ Mγ such that T z ∈ Mγ . Denote by τγ+,C the supremum of τC+ (z) over all z ∈ Mγ leaving γ and by τγ−,C the supremum of τγ−,C over all z ∈ Mγ entering γ . Let τγ ,C = max{τγ+,C , τγ−,C }. In this paper, we will only consider absolutely focusing curves γ with a cone field C for which τγ ,C is finite. We will implicitly make this assumption every time that we deal with focusing curves, and we will call such a C the cone field of γ . Some examples of such curves are described in the remaining part of this section. Consider a curve γ as before. For any z = (q, v) ∈ Mγ , let us denote by L(z) the ray emerging from q and parallel to v and by β(z) the length of the segment of L(z) contained in the osculating circle of γ at q. If z = (q, v) and z = (q , v ) are two elements of Mγ corresponding to consecutive collisions of a trajectory with γ , then t (z) is the length of the segment connecting q and q . Among the examples of absolutely focusing curves considered in this paper, there are the curves γ that verify the relation (W ) : β(z) + β(z ) ≤ 2t (z) for any two consecutive collisions z and z with γ [W2]. For C 4 curves, (W) is equivalent to d 2 r/ds 2 ≤ 0. Other examples of curves which are absolutely focusing are those that verify the relation (M) : β(z)(t (z) + t (z )) ≤ 2t (z)t (z ) for any two consecutive collisions z and z with γ . These curves were introduced in [M] and proved to be absolutely focusing in [C-M]. If a C 4 curve γ satisfies d 2 r 1/3 /ds 2 ≥ 0, then any sufficiently small subarc of γ verifies (M) [M]. Examples of curves that satisfy (W) or (M) are arcs of circles, cardioids, √ logarithmic spirals, the arcs of the ellipse given by x 2 /a 2 + y 2 /b2 = 1, |x| ≤ √a/ 2 for 0 < b < a and sufficiently small arcs of x 2 /a 2 + y 2 /b2 = 1, |x| ≥ b2 / a 2 + b2 for 0 < b < a containing one of the points x = ±a, y = 0. An example of an absolutely focusing arc which √ does not verify (W) and (M) is the half-ellipse x 2 /a 2 + y 2 /b2 = 1, x ≥ 0 with a/b < 2 [D, B3].
3.3. A class of semi-focusing billiards. We describe now the class of billiard tables considered in this paper. Let {e1 , e2 , e3 } be the canonical basis of R3 . For any q ∈ R3 , let (q1 , q2 , q3 ) be the components of q with respect to {e1 , e2 , e3 }. Definition 2. Let ai > 0 for i = 1, 2, 3. A box B with edges lying on the coordinate axes of R3 and having length 2a1 , 2a2 , 2a3 is the parallelepiped B = {q ∈ R3 : |qi | ≤ ai , 1 ≤ i ≤ 3}. Let 1 ≤ i ≤ 3. The sets Bi+ = {q ∈ B : qi = ai } and Bi− = {q ∈ B : qi = −ai } are the faces of B perpendicular to ei . Let Bi∞ = {q ∈ R3 : |qj | ≤ aj , j = i} be the infinite box obtained by stretching B to ∞ in the direction of ei . Definition 3. Let 1 ≤ i = j ≤ 3. An absolutely focusing curve γ lying on span(ei , ej ) and attached to the face Bi+ (Bi− ) of a box B is an absolutely focusing curve t ∈ [0, 1] → γ1 (t)ej + γ2 (t)ei such that γ1 (0) = −γ1 (1) = aj , γ2 (0) = γ2 (1) = ai (−ai ) and γ2 (t) ≥ ai (≤ −ai ) for any 0 ≤ t ≤ 1.
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Fig. 1. A semi-focusing billiard table
Definition 4. Let 1 ≤ i = j ≤ 3. A cylinder C attached to the face Bi+ (Bi− ) of a box B is a set of the form C = Bi∞ ∩ P −1 γ , where γ is an absolutely focusing curve lying on span(ei , ej ) and attached to Bi+ (Bi− ), and P : R3 → span(ei , ej ) is the orthogonal projection of R3 onto span(ei , ej ). The curve γ and the space N = span(ei , ej )⊥ are called section and axis of C, respectively. Let C1 and C2 be two cylinders attached to opposite faces Bi+ and Bi− of a box B such that their axes are orthogonal. Let be the union of B and the convex hulls of C1 and C2 . An example of a billiard in a domain is depicted in Fig. 1. Let ∂+ and ∂0 be the union of the two cylinders and the union of the faces of ∂, respectively. Let + = π −1 (∂+ ) and 0 = π −1 (∂0 ). 3.4. Spectrum and eigenvectors of R. We compute the eigenvalues and eigenvectors of the operator R when T p = (q , v ) is attached to a cylinder C ∈ ∂. Choose a system of Cartesian coordinates in R3 such that the origin coincides with the point q , the xz-plane coincides with the tangent plane LTp , the z-axis coincides with the axis of C and n(q) = e2 . Let p = (q, v). Using polar coordinates 0 ≤ ρ, 0 ≤ θ1 ≤ π, 0 ≤ θ2 ≤ 2π , we write v = (sin θ1 cos θ2 , sin θ1 sin θ2 , cos θ1 ) and v = (sin θ1 cos θ2 , − sin θ1 sin θ2 , cos θ1 ). The matrix of K(q ) with respect to the basis {e1 , e3 } is given by −κ(q ) 0 , 0 0 where κ(q ) is the curvature of the section γ of C at q . The operator R = 2v , n(q )P1∗ K (q )P1 is self-adjoint. Let k1 and k2 be its eigenvalues, and let w1 , w2 ∈ Vp be the corresponding normalized eigenvectors. One eigenvalue of R is equal to zero so that we may assume that k2 = 0 and w2 = P1−1 e3 , where P1−1 is the orthogonal projection of LTp onto Vp . We now compute w1 and k1 . It is easy to check that w1 , e3 = 0. Thus w1 = λ(−a2 , a1 , 0) ∈ R3 for some λ = 0, where ai = p, ei , i = 1, 2. A straightforward computation gives
Semi-Focusing Billiards: Hyperbolicity
23
Rw1 = −κ(q )
a12 + a22 a22
w1
so that k1 = −2κ(q )
sin θ1 . sin θ2
3.5. Condition on the distance between cylinders. Consider a domain . In this section, we formulate a condition on the distance between the cylinders of ∂ which guarantees the hyperbolicity of the billiard in . We start with some definitions. The notation here is as in the previous sections. Let C be a cylinder of ∂ with section γ . We define d(T p) =
sin θ2 , κ(q ) sin θ1
(5)
where, we recall, T p = (q , v ) and κ(q ) is the curvature of γ at q . Note that the non-zero eigenvalue k1 of the operator R is equal to −2/d(T p) and that d(T p) is the length of the segment of the trajectory of T p contained in the “half-osculating” cylinder of C at q , i.e., the cylinder tangent to C at q with circular section of radius (2κ(q ))−1 . Let C be the cone field of γ . If z = (q , θ2 ), then we define d ± (T p) =
τC± (z) . sin θ1
(6)
It follows immediately from the Mirror Formula (4) that d − (T p) + d + (T p) = 2
d − (T p)d + (T p) . d(T p)
(7)
Let C1 and C2 be the cylinders of ∂. If their sections are γ1 and γ2 , and C1 and C2 are the cone fields of γ1 and γ2 , then we impose the following condition on τγ1 ,C1 + τγ2 ,C2 < dist(C1 , C2 ).
(8)
Remark 1. When the sections of the cylinders are circles, this condition can be replaced by the condition that the distance between the cylinders is larger than the sum of the radii of the circles. Let [p1 , p2 ] denote the finite trajectory starting and ending at cylinders of ∂, where p1 , p2 ∈ + are its initial and final velocity vectors. If moreover l(p1 , p2 ) is the length of [p1 , p2 ], then it is not difficult to see that Condition (8) implies d + (p1 ) + d − (p2 ) < l(p1 , p2 )
(9)
for any trajectory [p1 , p2 ] for which p1 and p2 belong to distinct cylinders. This property is essential in producing the hyperbolic behavior of billiards in domains .
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L.A. Bunimovich, G. Del Magno
4. Hyperbolicity of Semi-Focusing Billiards In this section, we prove that the billiard map T in a domain satisfying (8) is hyperbolic, i.e., T has non-zero Lyapunov exponents µ-a.e. on . This can be done by constructing an eventually strictly invariant cone field for T [W1, L-W]. 4.1. Cone field. We define a cone field on the set + , and then we extend it to the set = + ∪ 0 by transporting the cones up to the flat faces of ∂ via DT . According to [L-W], to define a cone field on + , we need to specify a pair of transversal Lagrangian subspaces of Vp × Vp for all p ∈ + . Let W1 (p) = span ((w1 (p), 0), (w2 (p), 0)) , W2 (p) = span (w1 (p), −w1 (p)/d + (p)), (w2 (p), −w2 (p)/d + (p)) , where w1 (p) and w2 (p) are the eigenvectors of R(p) and 0 denotes the zero vector in Vp . Also d ± (p) are the quantities defined in (6), where τC± refers to the cone field of the section of the cylinder containing π(p). It is immediate to check that W1 and W2 are transversal and Lagrangian. We now define a quadratic form Q on + associated to W1 and W2 . Since Vp × Vp = W1 (p) ⊕ W2 (p), for every u = (ξ, η) ∈ Vp × Vp , we can write u = u1 + u2 , where u1 = (ξ + d + (p)η, 0) ∈ W1 , and u2 = (−d + (p)η, η) ∈ W2 . The quadratic form Q at p ∈ + is given by Qp (u) := ω(u1 , u2 ) = ξ, η + d + (p) η 2 for u = (ξ, η) ∈ Vp × Vp . The cone field C on + is defined as follows. For any p ∈ + , let C(p) = {u ∈ Vp × Vp : Qp (u) ≥ 0}. Note that C is piecewise continuous on + , because the function d + is piecewise continuous. To finish the construction of C, we extend it to the whole phase space by transporting the cones from + to 0 via Dp T . We recall some definitions from [L-W]. Let int C(p) = {u ∈ Vp ×Vp : Qp (u) > 0}. The map Dp T is monotone (strictly monotone) if Dp T C(p) ⊂ C(T p) (int C(T p)∪{0}) or, equivalently, if QTp (Dp T u) ≥ (>)Qp (u) for every 0 = u ∈ Vp × Vp . 4.2. Change of coordinates in Vp × Vp . To simplify the computations, we introduce a new system of coordinates in the spaces Vp × Vp , p ∈ + . For any p ∈ + , let us consider the set (ξ , η ) of coordinates of Vp × Vp defined by ξ = ξ + d + (p)η, η = η. In coordinates (ξ , η ), the map Dp T takes the form I 0 I d + (T p)I I (l(p, Tp) − d + (p))I , R I 0 I 0 I
(10)
Semi-Focusing Billiards: Hyperbolicity
25
where l(p, Tp) is the distance between π(p) and π(T p), and the quadratic form Q takes the expression Qp (u) = ξ , η for u = (ξ , η ) ∈ Vp × Vp . The next lemma is a direct consequence of the definition of (ξ , η ). Lemma 1. Let (ξ , η ) ∈ Vp × Vp , and (ξ , η ) = Dp T (ξ , η ). The operator Dp T is monotone (strictly monotone) if and only if ξ , η ≥ (>) ξ , η . In the rest of the paper, we will use coordinates (ξ , η ) unless otherwise specified. 4.3. Hyperbolicity. The cone field of two-dimensional stadium-like billiards is strictly invariant every time that the point particle leaves one boundary curve and bounces off another one. This property produces the hyperbolicity of the billiard. As we show in this section, the situation is more complicated for cylindrical billiards: two consecutive reflections at different cylinders are required in order to obtain strict invariance of the cone field. Denote by T1 the first return map on + induced by T . Let S + be the singular set of T1 , i.e., the set of elements of + , where T1 is not defined or fails to be C 1 . For every + −i + + + n > 0, let Sk+ = ∪k−1 i=0 T S . Since S ⊂ S∞ , we have µ(Sk ) = 0 for any k > 0. Theorem 1. The map T is hyperbolic. Proof. We show that the cone field C is eventually strictly invariant. Note that if p ∈ and Tp ∈ 0 , then Dp T C(p) = C(T p) by construction of C so that, in this case, the invariance of C is automatically satisfied. Furthermore the subset of consisting of elements whose trajectory never hits one of the two cylinders coincides with {(q, v) : q ∈ ∂0 and v ∈ ei⊥ } (recall that Bi− and Bi+ are the faces where the cylinders are attached), and therefore it has zero µ-measure. Thus in order to show that C is eventually strictly invariant, it is enough to check that the following properties are satisfied: P1. Dp0 T1 C(p0 ) ⊆ C(T1 p0 ) for all p0 ∈ + \ S + . P2. Given p−1 ∈ + \ S2+ , let p0 = T1 p−1 and p1 = T12 p−1 . For any p−1 ∈ + such that p0 and p1 are attached to different cylinders, we have Dp−1 T12 C(p−1 ) ⊂ int C(p1 ) ∪ {0}. These properties can be equivalently reformulated in terms of the quadratic form Q. P1 translates into QT1 p0 (Dp0 T1 u) ≥ Qp0 (u) for all u ∈ Tp0 + and p0 ∈ + \ S + , and P2 translates into QT 2 p−1 (Dp−1 T12 u) > Qp−1 (u) for all 0 = u ∈ Tp−1 + and 1
p−1 ∈ + \ S2+ such that p0 and p1 are attached to different cylinders. In order to prove P1 and P2, we need some lemmas concerning the matrix form of DT1 . Let p0 ∈ + and p1 = T1 p0 . By definition of T1 , there exists a positive integer m = m(p0 ) such that T m p0 = p1 and if m > 1, then T k p0 ∈ 0 for 1 ≤ k ≤ m − 1. Let p0,k = T k p0 for 0 ≤ k ≤ m. Note that p0,0 = p0 and p0,m = p1 . The map Dp0 T1 : Tp0 + → Tp1 + is equal to Dp0,m−1 T ◦ Dp0,m−2 T ◦ · · · ◦ Dp0,0 T , where Dp0,k T : Tp0,k → Tp0,k+1 for every 0 ≤ k ≤ m − 1. As explained in Sect. 2.2, for
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L.A. Bunimovich, G. Del Magno
every 0 ≤ k ≤ m − 1, the tangent space Tp0,k can be identified with Vp0,k × Vp0,k and, by using the maps Up0,0 , . . . , Up0,k (Up0,k reflects R3 about the plane Lp0,k+1 ), we can finally identify Vp0,k × Vp0,k with Vp0,0 × Vp0,0 = Vp0 × Vp0 . Doing so, all Dp0,k T and Dp0 T1 become linear maps on Vp0 × Vp0 . For every 0 ≤ k ≤ m, let p0,k = (qk , vk ). For every 0 ≤ k ≤ m − 1, let lk+1 be the length of the segment [p0,k , p0,k+1 ] and Rk+1 = 2vk+1 , n(qk+1 )P1∗ KP1 , where P1 and K are evaluated at p0,k+1 . As an operator on Vp0,k × Vp0,k , the block form matrix of Dp0,k T is given by I lk+1 I . Rk+1 I + lk+1 Rk+1 Let R˜ 1 = R1 and, for 1 ≤ k ≤ m − 1, let R˜ k+1 = −1 k Rk+1 k , where k = Up0,k−1 Up0,k−2 . . . Up0,0 . As a linear operator on Vp0,0 × Vp0,0 , Dp0,k T has the following block form: I lk+1 I . R˜ k+1 I + lk+1 R˜ k+1 From now on, we will think of Dp0 T1 as an operator on Vp0 × Vp0 . It is not difficult to see that the block form of Dp0 T1 is given by I l(p0 , p1 )I , (11) R˜ m I + l(p0 , p1 )R˜ m where l(p0 , p1 ) = l1 + · · · + lm is the length of the trajectory from p0 to p1 . In the computation of (11), we used the fact that Rk+1 is the zero-matrix as p0,k ∈ 0 for 0 ≤ k ≤ m − 2. Lemma 2. 1. Dp0 T1 admits the factorization J 0 I 0 I E I 0 0 J −1
F , I
where J, E, F are self-adjoint operators on Vp0 . 2. With respect to the basis {w1 (p0 ), w2 (p0 )} of Vp , J, E, F take the block form + d + (p1 ) 1) 0 2 0 − dd − (p (p1 ) d(p1 )d − (p1 ) , E= J = 0 1 0 0 and
l(p0 , p1 ) − d + (p0 ) − d − (p1 ) F = 0
0 . l(p0 , p1 ) − d + (p0 ) + d + (p1 )
+ ˜ ˜ Proof. By a straightforward computation, we obtain J = I +d (p1 )Rm , E = J Rm and F = l(p0 , p1 ) − d + (p0 ) I + d + (p1 )J −1 . The matrices J, E, F are self-adjoint. To finish the proof, we compute the entries of R˜ m with respect to the basis {w1 (p0 ), w2 (p0 )}, and we use Formula (7).
The next corollary is an immediate consequence of the previous lemma and Relation (9) (consequence of Condition (8)).
Semi-Focusing Billiards: Hyperbolicity
27
Corollary 1. E, F are positive semi-definite. Furthermore F is positive definite if and only if π(p0 ) and π(p1 ) belong to different cylinders. Lemma 3. 1. Property P1 is satisfied. 2. Let p0 ∈ + \ S + . Suppose that for some 0 = u ∈ Tp0 + , QT1 p0 (Dp0 T1 u) = Qp0 (u). Then there exist a, b ∈ R such that u = (aw2 (p0 ), bw1 (p0 )) if π(p0 ) and π(T1 p0 ) belong to the same cylinder, and u = (aw2 (p0 ), 0) if π(p0 ) and π(T1 p0 ) belong to distinct cylinders. Proof. Given (ξ , η ) ∈ Vp0 × Vp0 , let ξ ξ = D ∈ Vp0 × Vp0 . T p1 1 η η By Lemma 2, we have
ξ η
=
J (ξ + F η ) . J −1 E(ξ + F η ) + η
Since J is symmetric, we obtain
Qp1 (ξ , η ) = ξ + F η , E(ξ + F η ) + η
= Qp0 (ξ , η ) + F η , η + ξ + F η , E(ξ + F η ) ≥ Qp0 (ξ , η )
(12)
because E, F are positive semi-definite by Lemma 2. This proves the first part of the lemma. Let ξk , ηk , 1 ≤ k ≤ 2 be the components of ξ and η with respect to the basis {w1 (p0 ), w2 (p0 )}. Using the matrix form of E, F with respect to this basis (Lemma 2), one can easily check that the equality in (12) holds if and only if ξ1 = η2 = 0 when π(p0 ) and π(Tp0 ) belong to the same cylinder, and if and only if ξ1 = η1 = η2 = 0 when π(p0 ) and π(T p0 ) belong to different cylinders. In this last case, in fact, F is positive semi-definite. This concludes the proof of the second part of the lemma. Let Projp be the orthogonal projection onto Vp . Note that Up p = T p. Lemma 4. Up Projp = ProjTp Up . Proof. For every w ∈ R3 , we have Up Projp w = Up (w − w, pp) = Up w − w, pT p = Up w − Up w, T pT p = ProjTp Up w.
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L.A. Bunimovich, G. Del Magno
Lemma 5. P2 is satisfied. Proof. Let p−1 ∈ + \ S2+ , p0 = T1 p−1 and p1 = T12 p−1 . Let qi = π(pi ), i = −1, 0, 1. We study only the case q−1 , q0 ∈ C1 and q1 ∈ C2 , where C1 and C2 are the cylinders of ∂. The other cases can be studied similarly. We argue by contradiction. Suppose that there exists a vector 0 = u ∈ Vp−1 × Vp−1 such that Qp−1 (u) = Qp0 (Dp−1 T1 u) = Qp1 (Dp−1 T12 u).
(13)
The second part of Lemma 3 applied to the first equality of (13) (from the left) implies that there exist a, b ∈ R such that u = (aw2 (p−1 ), bw1 (p−1 )). By Lemma 2, we obtain aw2 (p−1 ) Dp−1 T1 u = , b bw1 (p−1 ) where b = −d − (p0 )/d + (p0 ) = 0. On the other hand, the second equality of (13) and Lemma 2 imply that there exists c ∈ R such that Up−1 Dp−1 T1 u = (cw2 (p0 ), 0) ∈ Vp0 × Vp0 . Thus aUp−1 w2 (p−1 ) = cw2 (p0 ), b bUp−1 w1 (p−1 ) = 0.
(14)
b
Since = 0, we have b = 0. Moreover, as u = 0, we have ac = 0. Since w2 (p1 ) and w2 (p0 ) are unit vectors and Up is an isometry, we have a = c. Let N1 and N2 be the unit vectors parallel to the axes of C1 and C2 , respectively. Of course, span(N1 , N2 ) = ei⊥ for some 1 ≤ i ≤ 3, where Bi− and Bi+ are the faces of the box to which C1 and C2 are attached. According to the results of Sect. 3.4, w2 (p−1 ) = Projp−1 N1 and w2 (p0 ) = Projp0 N2 . Using Lemma 4 and the fact that Up0 N1 = N1 , we obtain Up−1 w2 (p−1 ) = Up−1 Projp−1 N1 = Projp0 Up−1 N1 = Projp0 N1 . Thus the first equation of (14) becomes Projp0 N1 = Projp0 N2 . Since N1 and N2 are orthogonal, this equation is satisfied only if p0 ∈ span(N1 , N2 ). But span(N1 , N2 ) = ei⊥ so that p0 , ei = 0. This implies that π(p1 ) ∈ / C2 contradicting our assumption. The proof of Theorem 1 is complete.
Remark 2. This proof fails if at least one pair of opposite faces of the box of are skew parallelograms instead of rectangles, because, in this case, (9) is not verified even if (8) is. More precisely, as a consequence of the fact that a pair of faces of the box of is a skew parallelogram, one of the cylinders of ∂ has the axis which is not parallel to a coordinate axis. Suppose that C1 is such a cylinder. Then it is not hard to see that (essentially because in a parallelogram with non-orthogonal adjacent sides, the number of reflections of a trajectory starting from a vertical side and ending at the other is bounded above, and d ± is arbitrarily large for vectors having direction close to the axis ˆ ⊂ π −1 (C1 ) of positive measure such that i) every of a cylinder) there exists a set
Semi-Focusing Billiards: Hyperbolicity
29
ˆ leaves C1 and after a finite number of collisions with some flat faces of hits p1 ∈ C2 , and ii) if p2 ∈ π −1 (C2 ) is the vector corresponding to the collision with C2 , then ˆ l(p1 , p2 ) < d + (p1 ) for every p1 ∈ . Remark 3. By Theorems 4 and 4.4. of [D], small C 6 perturbations of C 6 absolutely focusing curves are still absolutely focusing, and the focusing times τγ±,C vary continuously with γ . The same property is valid for C 4 curves verifying d 2 r/ds 2 < 0 [W2]. Therefore if satisfies (8) and the sections of the cylinders are C 6 or C 4 verifying d 2 r/ds 2 < 0, then we see that small perturbations (in the proper class of smoothness) of the sections of the cylinders of ∂ produce new domains for which (8) remains valid. By Theorem 1, the billiards in these domains are hyperbolic. 5. Billiards with Divided Phase Space We say that a system has divided phase space if its phase space is the union of two disjoint sets of positive measure such that one has non-zero Lyapunov exponents almost everywhere and the other has zero-Lyapunov exponents almost everywhere. Two-dimensional semi-focusing billiards with divided phase space were constructed in [B4]. Their phase space consists of integrable regions surrounded by hyperbolic and ergodic regions. Using essentially the same ideas as in [B4], we construct, in this section, a class of semi-focusing billiards in R3 with divided phase space. Consider a domain as in Fig. 1 where the sections of the cylinders C1 and C2 are semi-circles and their axes are orthogonal. We also choose so that Condition (8) is satisfied. Pick one of the two cylinders, say C1 , and denote by H its convex hull. Now contract \ H uniformly along the direction of the axis of C2 in such a way to obtain a mushroom-like domain M as in Fig. 2. The cylinder C1 is the “hat” of the mushroom. We assume, although it is not necessary, that M is symmetric with respect to the plane which is perpendicular to the axis of C2 and contains the axis of C1 . Remark 4. More complex mushroom-like domains can be designed following [B4], like, for instance, domains consisting of cylinders with semi-ellipses as sections connected by rectangular boxes. The billiards in these domains have several hyperbolic and integrable regions of positive measure. Assembling together in a proper way countably many cylinders and boxes, even three dimensional billiards with countably many ergodic components can be constructed. We stick to the simple domain described above to avoid technical complications, and because the mechanism producing the divided phase space is the same in all these billiards. Theorem 2. The phase space of the billiard in a domain M is an union of two disjoint invariant subsets 1 and 2 of positive µ-measure such that ˜ such that µ() ˜ = µ(1 ), and the restriction of T to 1. 1 contains an invariant set ˜ is integrable (and hence T has zero Lyapunov exponents µ-a.e. on 1 ), 2. the restriction of T to 2 is hyperbolic. Proof. Let H = π −1 (∂H ∩∂). Let N be the unit vector parallel to the axis of C1 , and let q0 be a point lying on the axis of C1 . For every (q, v) ∈ H , let I1 (q, v) = v, N , and let I2 (q, v) = (q − q0 ) ∧ v, N /(1 − I12 (q, v))1/2 which gives the minimum distance of the line passing through q and parallel to v from the axis of C1 . It is not difficult
30
L.A. Bunimovich, G. Del Magno
Fig. 2. A 3-dim mushroom-like billiard table
to check that I1 and I2 are in involution (with respect to the symplectic form ω; see Subsect. 2.2) and are independent on an open and dense set of H . We show how this can be done on π −1 (C1 ) ⊂ H . The proof for H \ π −1 (C1 ) is omitted because it is similar. Given (q, v) ∈ π −1 (C1 ), let 0 ≤ θ1 , θ2 ≤ π be polar coordinates for v, where θ1 is the angle formed by v with N and π/2 − θ2 is the angle formed by the projection of v onto N ⊥ with n(q). Let r1 be the radius of the section of C1 . A simple computation shows that I1 (q, v) = cos θ1 and I2 (q, v) = −r1 sin θ1 cos θ2 for every (q, v) ∈ π −1 (C1 ). The map (s1 , s2 ) → (x = r1 cos(s2 /r1 ), y = r1 sin(s2 /r1 ), z = s1 ) is a smooth parametrization of C1 where the origin of the system of Cartesian coordinates (x, y, z) lies on the axis of C1 . In coordinates (s1 , s2 , θ1 , θ2 ), the symplectic form ω is given by sin θ1 (ds1 ∧ dθ1 + ds2 ∧ dθ2 ). It follows immediately that I1 and I2 are in involution and independent on int π −1 (C1 ). Let 1 = {p ∈ H : T k z ∈ H
∀k ∈ Z}
be the set of the vectors whose trajectory is “trapped” inside H . It follows from the symmetry of H that |I1 | and |I2 | are first integrals of T |1 . Let 2a be the length of the edge of B parallel to the x-axis. For any 0 ≤ α1 < 1 and a < α2 ≤ r1 , let (α1 , α2 ) = {p ∈ 1 : |I1 (p)| = α1 and |I2 (p)| = α2 }. Each of these sets is a T -invariant, smooth and compact submanifold of codimension two. Let ˜ = (α1 , α2 ). 0≤α1 <1 a<α≤r1
˜ ⊂ 1 , µ() ˜ > 0 and T | ˜ is integrable [A, T]. Thus all It follows immediately that ˜ are equal to zero. To finish the proof of the first part of Lyapunov exponents of T on ˜ = 0. Note that if z ∈ 1 \ , ˜ then the the theorem, we have to show that µ(1 \ )
Semi-Focusing Billiards: Hyperbolicity
31
projection of the trajectory of z onto the xy-plane is periodic. It is not difficult to see that this implies that z ∈ I2−1 (α2 ), where α2 = r1 cos(m/(2r1 )) for some m ∈ Q. Thus m ˜ ⊂ 1 \ , I2−1 r1 cos 2r1 m∈Q
˜ = 0 follows immediately from the fact that I −1 (α2 ) is a smooth hyperand µ(1 \ ) surface of . Let 2 = \ 1 . This set consists of vectors p ∈ such that T k p ∈ \ H for some k = k(p) ∈ Z. By the geometry of , we see that for µ-a.e. p ∈ \ H , there exists a n = n(p) > 0 such that T n(p) p ∈ π −1 (C1 ). On the other hand, since µ(2 ) > 0, µ-a.e. p ∈ 2 returns infinitely many times to 2 by the Poincar´e Recurrence Theorem. Thus the trajectory of µ-a.e. p ∈ 2 bounces back and forth between the cylinders C1 and C2 infinitely many times. Theorem 1 applied to T |2 implies that T |2 is hyperbolic. Acknowledgements. This paper was written while the second author was visiting the Instituto Superior T´ecnico in Lisbon and the Centro di Ricerca Matematica “Ennio de Giorgi” in Pisa whose hospitality G. D. M. acknowledges gratefully. The visit at the last institute was sponsored by the city of Lizzanello (Italy) which G. D. M. thanks warmly. G. D. M. would also like to thank P. Balint for helpful discussions. Finally the authors would like to thank an anonymous referee for valuable and constructive remarks.
References [A]
Arnold, V.: Mathematical methods of classical mechanics. Graduate Texts in Mathematics 60, Berlin-Heidelberg-New York: Springer-Verlag, 1989 [B1] Bunimovich, L.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65, 295–312 (1979) [B2] Bunimovich, L.: Many-dimensional nowhere dispersing billiards with chaotic behavior. Physica D 33, 58–64 (1988) [B3] Bunimovich, L.: On absolutely focusing mirrors. In: Ergodic theory and related topics, III (Gastrow, 1990), Lect. Notes Math. 1514, Berlin-Heidelberg-NewYork: Springer-Verlag, 1992, pp. 62–82 [B4] Bunimovich, L.: Mushrooms and other billiards with divided phase space. Chaos 11(4), 1–7 (2001) [B-R1] Bunimovich, L., Rehacek, J.: Nowhere dispersing 3D billiards with non-vanishing Lyapunov exponents. Commun. Math. Phys. 189, 729–757 (1997) [B-R2] Bunimovich, L., Rehacek, J.: On the ergodicity of many-dimensional focusing billiards, Classical and quantum chaos. Ann. Inst. H. Poincar´e Phys.Th´eor. 68(4), 421–448 (1998) [B-R3] Bunimovich, L., Rehacek, J.: How high-dimensional stadia look like. Commun. Math. Phys. 197(2), 277–301 (1998) [C-M] Chernov, N., Markarian, R.: Entropy of non-uniformly hyperbolic plane billiards. Bol. Soc. Bras. Mat. 23, 121–135 (1992) [C-F-S] Cornfeld, I., Fomin, S., Sinai, Ya.: Ergodic theory. New York: Springer-Verlag, 1982 [D] Donnay, V.: Using integrability to produce chaos: billiards with positive entropy. Commun. Math. Phys. 141, 225–257 (1991) [H] Halpern, B.: Strange billiard tables. Trans. Amer. Math. Soc. 232, 297–305 (1977) [L-W] Liverani, C., Wojtkowski, M.: Ergodicity in Hamiltonian systems. Dynamics Reported 4, Berlin-Heidelberg-New York: Springer-Verlag, 1995 [M] Markarian, R.: Billiards with Pesin region of measure one. Commun. Math. Phys. 118, 87–97 (1988) [P1] Papenbrock, T.: Collective and chaotic motion in self-bound many-body systems. Phys. Rev. C 61, 034602 (2000) [P2] Papenbrock, T.: Lyapunov exponents and Kolmogorov-Sinai entropy for a high-dimensional convex billiard. Phys. Rev. E 61, 1337–1341 (2000)
32 [P3] [S1] [S2] [Si] [Sz] [T] [W1] [W2] [W3] [W4]
L.A. Bunimovich, G. Del Magno Papenbrock, T.: Numerical study of a three-dimensional generalized stadium billiard. Phys. Rev. E 61, 4626–4628 (2000) Sinai, Ya.: Dynamical systems with elastic reflections. Russ. Math. Surv. 25, 137–189 (1970) Sinai,Ya.: Development of Krylov’s ideas. Princeton Series in Physics. Princeton, NJ: Princeton University Press, 1979 Simanyi, N.: Hard Ball Systems and Semi-Dispersive Billiards: Hyperbolicity and ergodicity. In: Hard Ball Systems and the Lorentz Gas, D. Sz´asz (ed.), Berlin: Springer, 2000, pp. 51–88 Sz´asz, D.: The K-property of “orthogonal” cylindric billiards. Commun. Math. Phys. 160, 581–597 (1994) Tabachinkov, S.: Billiards. Panor. Synth. 1, 1995 Wojtkowski, M.: Invariant families of cones and Lyapunov exponents. Erg. Th. Dynam. Syst. 5, 145–161 (1985) Wojtkowski, M.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commum. Math. Phys. 105, 391–414 (1986) Wojtkowski, M.: Measure theoretic entropy of the system of hard spheres. Erg. Th. Dynam. Syst. 8, 133–153 (1988) Wojtkowski, M.: Linearly stable orbits in 3-dimensional billiards. Commun. Math. Phys. 129(2), 319–327 (1990)
Communicated by G. Gallavotti
Commun. Math. Phys. 262, 33–50 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1474-7
Communications in
Mathematical Physics
Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension Gerhard Keller1 , Carlangelo Liverani2 1
Mathematisches Institut, Universit¨at Erlangen-N¨urnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany. E-mail:
[email protected] 2 Dipartimento di Matematica, II Universit`a di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy. E-mail:
[email protected] Received: 4 November 2004 / Accepted: 30 June 2005 Published online: 9 December 2005 – © Springer-Verlag 2005
Abstract: We prove the existence of a unique SRB measure for a wide range of multidimensional weakly coupled map lattices. These include piecewise expanding maps with diffusive coupling.
1. Introduction The field of expanding coupled map lattices has witnessed an impressive series of results since the late 1980’s. Starting with [7] numerous authors contributed to the exploration of ergodic and statistical properties of invariant measures for such systems, see e.g. [1–5,8–20,29,31,32,34,35]. In all these publications the single site maps are hyperbolic or expanding (local) diffeomorphisms of a smooth manifold, and the coupling is modeled by a “diffeomorphism” of the infinite-dimensional state space. Only a few publications used a different approach which allows to treat also piecewise expanding maps and such a common coupling like the diffusive nearest neighbour coupling [21–28, 33]. Yet, the state of the field is still far from satisfactory. One of the outstanding open problems is to substantiate rigorously the numerical picture of a phase transition given in [30]. The model considered in the aforementioned paper is a Z2 lattice of expanding Lasota-Yorke like maps, coupled by a diffusive nearest neighbour interaction. As the coupling parameter increases from zero the authors notice the transition from a situation in which only one invariant measure describes the statistical properties of the system to one in which two relevant invariant measures appear (a phase transition, indeed). After more than ten years no aspect of such a picture has been rigorously proven. In the present paper we prove the first (easier) part of the picture: the existence of only one “relevant” (that is SRB) measure for small coupling. The proof is surprisingly elementary. It combines the following key ideas: The essential part of this research was done during an ESF explorative workshop at the Max-PlanckInstitute for Mathematics, Bonn. We thank both institutions for their support.
34
G. Keller, C. Liverani
(i) The starting point is a Lasota-Yorke type inequality for coupled systems (cf. [24, 28]). (ii) The transfer operator of the uncoupled system is interpreted as a tensor product operator of the single site transfer operators (cf. [33]). This allows to make optimal use of the strong mixing properties of the single site systems. (iii) A “site-by-site” decoupling procedure allows to reduce the dynamics of the coupled system “locally” to dynamics of tensor-product type at the cost of only small errors (cf. [25, 26]). (iv) The aforementioned small errors are not controlled in the original system but in a huge extension of that system. This is the essential new idea of this paper. We believe that it has applications far beyond the present model; indeed, we expect it to be useful for all kinds of weakly coupled systems where the local dynamics can be described by linear operators with an isolated simple leading eigenvalue. Typical examples are high temperature stochastic Ising models or weakly coupled uniformly contractive iterated function systems. The plan of the paper is as follows: Section 2 details the model and describes the basic result obtained in the paper. Sect. 3 describes the already mentioned extension of the system and how to use it to get the main estimate of the paper. Sect. 4 contains the proof of the main theorem based on the results of Sect. 3. Finally, Sect. 5 contains the proof for the case of more general coupling, but with an extra simplifying assumption on the single site map. 2. The Model and the Result Given a compact interval I ⊂ R we will consider the phase space := I , where either = Zd or is a box in Zd .1 In the following we always assume I = [0, 1] and 0 = (0, . . . , 0) ∈ , as this can be done without loss of generality.2 We will have a single site dynamics given by the map τ : I → I . We assume τ to be a piecewise C 2 map from I to I with singularities at ζ1 , . . . , ζN−1 ∈ (0, 1) in the sense that τ is monotone and C 2 on each component of I \ {ζ0 = 0, ζ1 , . . . , ζN−1 , ζN = 1}. We assume that τ /(τ )2 is bounded and that inf |τ | > 2.3 Next, we define the unperturbed dynamics T0 : → by [T0 (x)]p := τ (xp ). To define the perturbed dynamics we introduce couplings : → of the form (x) := x + A (x). We call a (a1 , a2 )-coupling, if there are operators A , A : 1 () → 1 () with a1 = A 1 , a2 = A 1 (maximal column sum norm) such that for all k, p, q ∈ , |(A )p | ≤ 2||,
|(DA )qp | ≤ 2||A qp ,
|∂k (DA )qp | ≤ 2||A qp .
(2.1)
Here ∂k denotes the partial derivative with respect to xk . In addition, we say that has finite coupling range w > 0, if ∂p ,q = 0 whenever |p − q| > w. So A qp = A qp = 0 1 By box here, and in the following, we mean a hypercube. Of course much more general shapes can be considered by the same arguments, yet for shapes with too large a boundary problems may arise. To avoid all the related technicalities we confine ourselves to the above mentioned case. 2 The reader should be aware that there is nothing special about Zd , any other lattice (or graph) can be treated similarly, provided the number of different sites that can be reached from a given site along a path of length n grows at most subexponentially in n. 3 Under mild additional assumptions on τ also maps with 1 < inf |τ | ≤ 2 can be treated. The complications, which arise in the proof of a Lasota-Yorke type inequality, were overcome in [28], see also the discussions of this point in [22] and in [26, Footnote 14].
Uniqueness for the SRB for CML
35
when |p − q| > w. We say that a coupling has short range if it is not of finite range and there exist constants L > 0 and γ ∈ (0, 1) such that A qp + A qp ≤ Lγ |p−q| . Similarly, we say that a coupling has long range if it is neither finite range nor short range and there exists c > 0 such that A qp + A qp ≤ L|p − q|c . The diffusive nearest neighbor coupling used in [30], and in much of the numerical literature, is defined by (xq − xp ) (p ∈ ) , (2.2) [ (x)]p = xp + 2d |p−q|=1
and it is an example of a (1, 0)-coupling with range w = 1.4 The dynamics T : → that we wish to investigate is then defined as T := ◦ T0 , and, more precisely, we wish to investigate its invariant measures in some appropriate class. Let M() be the set of signed Borel measures on .5 To state the main result of the paper we need to introduce the concept of measures of bounded variation. Let I be the set of all boxes 1 ⊂ . For each 1 ∈ I we define 6 Var µ := sup
sup
p∈ |ϕ|C 0 () ≤1
Var 1 µ := sup
sup
µ(∂p ϕ),
p∈1 |ϕ|C 0 (I 1 ) ≤1
µ(∂p ϕ) .
(2.3)
It is easy to prove that the set B() := {µ ∈ M() : Var µ < ∞} consists of measures whose finite dimensional marginals are absolutely continuous with respect to Lebesgue and the density is a function of bounded variation [25]. In addition, such measures have finite entropy density with respect to Lebesgue [26, Corollary 5]. In fact, “Var” is a norm and, with this norm, B() is a Banach space.7 It is also useful to introduce the usual total variation norm on signed measures: |µ| :=
sup
|ϕ|C 0 () ≤1
µ(ϕ) .
(2.4)
Just like in [26, Sect. 3.3] one checks easily that |µ| ≤
1 Var µ . 2
(2.5)
As we are interested in studying observable invariant measures, we must restrict to a subclass of the class of all measures in order to make relevant statements. Clearly, M() is too large for our purposes, but on the other hand, in the case in which is infinite, B() is quite small. As usual in thermodynamics, it makes sense to require 4 Of course, if = Zd , then the sum in (2.2) can involve sites not in . To properly define the dynamics it is then necessary to supply some boundary conditions, that is to specify some fixed value for xq , q ∈ . 5 The topology that we use on is the product one. 6 Here, and in the following, we will consider C 0 (I 1 ) as a subspace of C 0 () by the obvious inclusion. Also the sup is restricted to functions derivable with respect to xp . 7 See [26] for a careful discussion of bounded variation in the present context and the relevant associated properties.
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G. Keller, C. Liverani
some condition on the growth of the relevant quantity with respect to the volume. Let Mv () be the closure of the set 1 d
{µ ∈ M() : ∀η > 0 sup e−η|1 | Var 1 µ < ∞} , 1 ∈I
with respect to the norm | · |. Clearly, Mv () consists of measures that can be uniformly approximated by measures with absolutely continuous finite dimensional marginals whose densities are functions of bounded variation with the variation growing less than exponentially in the size of the boxes. The results of this paper can be summarized, a bit loosely, as follow. Theorem 2.1. For each (a1 , a2 )-coupling of finite range w, there exists 0 > 0 such that, for each || < 0 , the dynamical system (, T ) has a unique invariant measure µ in Mv ().8 In addition, µ belongs to B(), is exponentially mixing both in time and in space, and it is the SRB measure of the system. The proof, which also makes precise the statement, can be found in Sect. 4. To obtain such a result we consider the dynamics acting directly on the measures via the linear operator T∗ µ(A) := µ(T−1 A) (for each measurable set A). The basic facts concerning the operator T∗ are detailed in the following lemma. Lemma 2.2 (Lasota-Yorke inequality). For each (a1 , a2 )-coupling, there exist 1 > 0, λ > 1, and a, b > 0 such that, for each || < 1 , the operator T∗ is well defined as an operator on B(). In addition, for each µ ∈ B() holds true |T∗ µ| ≤ |µ|,
Var(T∗n µ) ≤ aλ−n Var µ + b|µ| . This is the special case θ = 1 of Proposition 4 in [26] (see below for the meaning of θ ). Observe that the proof given there for = Z applies (only if θ = 1!) without changes to ⊆ Zd . From preceding experience it is also useful to consider larger Banach spaces: first define, for each θ ∈ (0, 1], a norm µθ := sup θ |1 | Var 1 µ 1 ∈I
(2.6)
on B(). Then we let B(, θ ) be the completion of B() with respect to this norm.9 Observe that µθ=1 = Var µ. The key estimate on which Theorem 2.1 relies is given in the following lemma whose proof is the content of the next section. Let B 0 () := {µ ∈ B() : µ(1) = 0}. Lemma 2.3. Recall that = I . For each (a1 , a2 )-coupling with finite range w, there exist σ ∈ (0, 1) and C, 2 > 0 such that, for all || < 2 , µ ∈ B 0 (), θ ∈ (0, 1), and n ∈ N holds true T∗n µθ ≤ Cσ n min{||, |e ln θ|−1 } Var µ. 8 If the coupling is defined only for nonnegative (as it is the case for the diffusive nearest neighbour coupling on ), this has to be understood as “for each ∈ [0, 0 ) . . . ” here and in the sequel. 9 Note that if || = ∞ and 0 < θ < 1, then B (, θ ) contains objects that are not signed measures, see [25, 26] for details.
Uniqueness for the SRB for CML
37
Finally, we wish to emphasize the power of the approach by showing the possibility of extending it to more general settings. Short range interactions can be treated in a spirit similar to the one used for the finite range. Nevertheless, the technical construction becomes inevitably more involved. For the long range case the situation looks still similar but one cannot expect an exponential convergence to the invariant measure, so one cannot simply rely on an estimate of the spectral radius of the covering dynamics and the story is bound to acquire an extra layer of complexity. To keep the technicalities to a minimum here we content ourself with the following result (proved in Sect. 5) concerning the short range case with an additional assumption on the single site map. Theorem 2.4. If the map τ is Lipschitz, then for each (a1 , a2 )-coupling of short range, there exists 0 > 0 such that, for each || < 0 , the dynamical system (, T ) has a unique invariant measure µ in Mv (). In addition, µ belongs to B(), is exponentially mixing both in time and in space, and it is the SRB measure of the system.
3. Lifting the System, Proof of Lemma 2.3 From now on we will suppress the dependence on in notations like B(). The basic idea of the present work is to define an extension of the linear system (T∗ , B) and to study its spectral properties instead of the ones of T∗ . To do so define Bp := {µ ∈ B : ∂p ϕ = 0 ⇒ µ(ϕ) = 0} . 0
Remark that Bp ⊂ B 0 . We can then define B := Xp∈ Bp and B := (B 0 ) , these are Banach spaces with the norm µ ¯ := supp∈ Var µp . 0
0
0
As T∗ (B 0 ) ⊆ B 0 , the (coupled) dynamics is easily lifted to B , namely T : B → B can be defined as (T µ) ¯ p := T∗ µp . However, only in the uncoupled case = 0 the 0
operator T 0 leaves the subspace B of B invariant. Since the invariance of this subspace - also under a suitable lift of T∗ when = 0 - is crucial for our approach, we need to proceed more carefully in choosing a suitable lift. To start with, let us consider some total ordering σ : N → Zd of Zd with the property:10 1
1
c−1 i d ≤ |σ (i)| ≤ ci d .
(3.1)
For each p = p + σ (0) and q = p + σ (i) in Zd one can then define the (partial) telescoping operators p,q acting on test functions,11 p,q ϕ(x) :=
ϕ(x) dxp · · · dxp+σ (i−1) −
ϕ(x) dxp · · · dxp+σ (i) .
Essentially p ∈ specifies the point from which one starts to telescope and q how far one is in the telescoping procedure. Note that p,q ϕ = 0 if ∂q ϕ = 0, and that p,q ϕ 10
For example, on a square lattice one can spiral out from zero on larger and larger squares. d Here ϕ ∈ C 0 (I Z ). This definition suffices in view of the identification already mentioned in Footnote 6. 11
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G. Keller, C. Liverani
does not depend on the variables contained inside a box of size c−2 |q − p| centered at p.12 We then define the lift : B 0 → B by (µ)q := ∗0,q µ, and the projection map P : B → B(θ ) by P (µ) ¯ :=
µp
p∈
which is well defined for θ ∈ (0, 1) even if is infinite.13 In fact, P B →B(θ) ≤ min{||, |e ln θ|−1 } ,
(3.2)
because Var 1 µp = 0 if p ∈ \ 1 . Observe also that, for each function ϕ depending only on finitely many variables and for each µ ∈ B 0 , P ((µ))(ϕ) = µ(ϕ) . In addition, it is easy to verify that P T = T∗ P .
(3.3)
As remarked before, since T B ⊂ B , we need some way to go back to the space B . 0 This is achieved via the (partially defined!) telescoping operator H¯ : B → B , (H¯ µ)q = ∗p,q µp . (3.4) p∈
Indeed, the infinite sum is not always well defined, but as we only consider finite range couplings, the operators m T ,m := H¯ T (m ≥ 1)
(3.5)
are always well defined on B , as the next lemma shows. Lemma 3.1. Let m ≥ 1. The linear operator T ,m : B → B is well defined, and T ,m µ ¯ ≤ C(wm)d sup Var(T∗m µp ) .
(3.6)
p∈
Proof. Let ϕ ∈ C 1 (). As noted above, ∂q (p,q ϕ)m = 0 for all q ∈ such2 that −2 |q − p| < c |q − p|. Consequently, ∂p (p,q ϕ) ◦ T = 0 provided |q − p| > c mw. Therefore, for µp ∈ Bp , ∗p,q T∗m µp (ϕ) = µp (p,q ϕ) ◦ Tm = 0 if |q − p| > c2 mw.
Indeed, if q ∈ , |q − p| < c−2 |q − p|, then, by (3.1), σ −1 (q − p) ≤ cd |q − p|d < c−d |q − p|d ≤ Hence q has already been integrated out in p,q . 13 Note that, on each local test function, the sum reduces to a finite sum. 12
σ −1 (q − p).
Uniqueness for the SRB for CML
39
It follows that for µ¯ ∈ B , µ¯ = (µp )p∈ ,
¯ q = (H¯ T µ) ¯ q= (T ,m µ) m
∗p,q T∗m µp
(3.7)
|q−p|≤c2 mw
is well defined, and T ,m µ ¯ = sup Var(H¯ T µ) ¯ q ≤ 2(2c2 mw)d sup Var(T∗m µp ) . m
q∈
p∈
Recall from (3.3) that P T = T∗ P . As P H¯ = P whenever these operators are well defined, it follows P T ,m = T∗m P .
(3.8)
So we can use T ,m : B → B as a covering dynamics for T∗m : B 0 → B 0 . In particular, we have the commuting diagrams m
H¯
T
B
−−→ B 0 −−→ −−−−−−−−−−−−→ B H¯ ◦Tm =:T ,m P
B 0 −−−−−−−− −−−−−→ B(θ ) ∗m T
and, more generally,
(3.9)
n T ,m
B −−−−→ B P
for all n ≥ 1.
B 0 −−− −→ B(θ ) ∗mn T
which makes sense since, by Lemma 3.1, T ,m is a bounded operator on B . To conclude we need to have a closer look at the operator T ,m . Recall that the sum in (3.7) is a finite sum as the interaction has finite range. Next, writing m = m1 + m2 , for each µ¯ ∈ B and each p ∈ , holds Var(T∗m µp ) ≤ a λ−m1 Var µp + b|T∗m2 µp |
(3.10)
with a = a(a + 21 b), see Lemma 2.2 and Eq. (2.5). In order to profit from the strong mixing properties of the single site operator we use a decoupling trick originally introp duced in a similar context in [33]: approximate by , where site p is decoupled from all other sites. To this end we introduce the following notation: let ι¯p : I → I p be the map (¯ιp (x))q = xq if q = p and (¯ιp (x))p = 0. Then define : I → I ,
xp if q = p p ( (x))q = (3.11) ( (¯ιp (x)))q if q = p.
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G. Keller, C. Liverani
Note that (Dp )qp = δqp .
(3.12)
This implies (p )∗ (Bp ) ⊆ Bp .
(3.13) p
It is then natural to define the decoupled dynamics T,p := ◦ T0 . Observe that m x) = τ m (x ). (T,p p p Here is a basic estimate for comparing different couplings. It is a variant of [26, Proposition 5], and we give its proof in the appendix. Lemma 3.2. The lemma consists of two parts: a) Let F, F˜ : → be two Lipschitz maps14 with Lipschitz constant L > 0 that are close in the following sense: There are constants K0 , K1 , K2 > 0 such that (i) q∈ supx |F˜q (x) − Fq (x)| ≤ K0 , (ii) q∈ supp∈ supx =p I |∂p F˜q (x =p , ξ ) − ∂p Fq (x =p , ξ )|dξ ≤ K1 , and (iii) sup{Var(Ft∗ ν) : 0 ≤ t ≤ 1, ν ∈ B(), Var ν ≤ 1} ≤ K2 ; Ft = t F˜ + (1 − t)F . Then, for each ν ∈ B, |F˜ ∗ ν − F ∗ ν| ≤ K2 (K0 + K1 ) Var ν .
(3.14)
b) For use in Sect. 5 we provide a variant of the above estimate: if assumptions (i) and (ii) are replaced by 1 (iv) q∈ supx |F˜q (x) − Fq (x)| 2 ≤ K3 for some K3 > 0, then, for each ν ∈ B, 1
|F˜ ∗ ν − F ∗ ν| ≤ 2K22 K3 Var ν .
(3.15)
As in [26, Lemma 8] one shows that the assumptions of part a) of this lemma are p satisfied for F = and F˜ = . Hence one can compute |∗ µ − (p )∗ µ| ≤ ||(8a1 + 2a2 + 4) Var µ
(3.16)
provided || < min{ 6a11 , 9a22 }. Using Lemma 2.2 it is then straightforward to show (compare the proof of [26, Theorem 6]) ∗m2 |T∗m2 µ − T,p µ| ≤ Cm2 || Var µ.
(3.17)
We are finally at the punch-line: let h be the invariant probability density of the single 1 m2 x) =p , τ m2 ξ ) dξ does not depend on xp , we have site map τ . As ψ(x) := 0 h(ξ )ϕ((T,p µp (ψ) = 0, so that
1 d ∗m2 m2 m2 m2 T,p µp (ϕ) = µp (ϕ ◦ T,p ) = µp (χ[0,xp ] (ξ )−xp h(ξ ))ϕ((T,p x) =p , τ ξ )dξ dxp 0
1 d m2 m2 L (χ[0,xp ] − xp h)(ξ )ϕ((T,p x) =p , ξ )dξ , = µp dxp 0 14 F : → is a “Lipschitz map”, if all F (x) are Lipschitz with respect to each coordinate x with q p uniformly bounded Lipschitz constants. This means in particular that all partial derivatives of all Fq exist s Lebesgue-a.e., are uniformly bounded and that Fq (x + sep ) − Fq (x) = 0 ∂p Fq (x + ξ ep )dξ .
Uniqueness for the SRB for CML
41
where L is the transfer operator of the single site map. This means that, calling σ0 the mixing rate for the single site map, ∗m2 |T,p µp | ≤ Cσ0m2 Var µp .
Combining this equation with (3.10) and (3.17) yields Var(T∗m µp ) ≤ a λ−m1 Var µp + bCm2 || Var µp + bCσ0m2 Var µp . 1
Setting σ1 := max{λ−1 , σ0 } 4 < 1 and m1 = m2 , there is, for m large enough, (m) > 0 such that for || < (m) holds Var(T∗m µp ) ≤ σ1m Var µp .
(3.18)
In view of Lemma 3.1 we conclude that, for each µ¯ ∈ B , T ,m µ ¯ ≤ C(mw)d σ1m µ ¯ . 1 At this point we can choose m large enough so that C(mw)d m σ1 =: σ < 1, whereby obtaining T ,m µ ¯ ≤ σ m µ ¯ .
(3.19)
We now conclude the argument by using Eq. (3.2) and (3.9), T∗pm µθ = P T ,m (µ)θ ≤ 2 min{||, |e ln θ |−1 }σ pm Var µ. p
By the usual trick of writing n = pm + q, q < m and Lemma 2.2 we finally have T∗n µθ ≤ C min{||, |e ln θ |−1 }σ n Var µ,
(3.20)
for each µ ∈ B 0 and θ ∈ (0, 1). This finishes the proof of Lemma 2.3. 4. Proof of Theorem 2.1 Having obtained the exponential estimate (3.20), the assertions of Theorem 2.1 can be proved along well known lines. For our convenience we follow once more [26]. The existence of a T -invariant probability measure µ ∈ B follows from a weak compactness argument as in [26, Theorem 4]. The uniqueness of such a measure µ in B is an immediate consequence of (3.20). In fact, it is also possible to obtain an explicit formula for the invariant measure. To do so let µ0 be a reference probability measure (for example the invariant measure of the uncoupled system). Then each measure µ can be represented as µ = µ(1)µ0 + (H (µ − µ(1)µ0 ))q . q∈
It is then natural to consider the Banach space C × B . In such a space a measure is represented by (µ(1), H (µ − µ(1)µ0 )). It is easily seen that one can define a covering dynamics by Sm (a, µ) := (a, T ,m µ + aH (T∗m µ0 − µ0 )).
42
G. Keller, C. Liverani
Setting ν¯ := H (T∗m µ0 − µ0 ), the equation Sm (1, µ) = (1, µ) has the unique solution µ = (Id − T ,m )−1 ν. This gives the following explicit representations of the invariant measure:15 µ = µ0 +
∞
n
(T ,m ν)q = µ0 +
q∈ n=0
∞
T∗nm (T∗m µ0 − µ0 ) = lim T∗n µ0 . n→∞
n=0
(4.1)
Remark 4.1. In particular, the above means that the linear operator Sm : C × B → C × B has a simple leading isolated eigenvalue 1 and a spectral gap. Uniqueness in Mv () follows by a standard approximation argument. Assume there exists an invariant measure µ˜ ∈ Mv (). By definition, given δ, η > 0, µ˜ can be approx
1
imated by a measure µδ,η such that |µ˜ − µδ,η | ≤ δ and Var µδ ≤ Cδ,η eη| | d for any ∈ I. Let ϕ be a function depending only on the variables belonging to a box 0 ∈ I, so that ϕ ◦ Tn depends only on the variables in the nw-neighborhood n of 0 . Then µ(ϕ) ˜ = T∗n (µ˜ − µδ,η )(ϕ) + T∗n (µδ,η − µ )(ϕ) + µ (ϕ).
(4.2)
We must prove that µ(ϕ) ˜ = µ (ϕ). As |T∗n (µ˜ − µδ,η )| ≤ |µ˜ − µδ,η | ≤ δ with an arbitrary δ > 0, it just remains to show that the second term can be made as small as we like by choosing η and n appropriate. Given a measure µ let µn be its marginal with respect to the box n . Given a measure µ on I n it is convenient to extend it to a measure µ on all I by simply tensoring it with the Lebesgue measure on the complement of n ; note that such an extension does not increase the bounded variation of the measure. With these conventions, T∗n (µδ,η − µ )(ϕ) = T∗n ([µδ,η,n ] − [µ,n ] )(ϕ).
(4.3)
Hence, by (3.20), 1 d
|T∗n (µδ,η − µ )(ϕ)| ≤ Cθ θ −|0 | σ n (1 + Cδ,η eη|n | ).
(4.4)
1
As |n | ≤ |0 | + (|0 | d + 2nw)d it follows that, for 2wη < | ln σ |, one can make this term arbitrarily small by choosing n large. Hence µ˜ = µ . Since in the case || < ∞ one has Mv () = M(), it follows that finite systems have only one measure absolutely continuous with respect to Lebesgue. If || = ∞ and ∈ I, let , (x∈ ) := (x∈ , 0 ∈ ). Then , is still an (a1 , a2 )-coupling with finite range w for the configuration space I and T, := , ◦ T0 gives a dynamics to which all our results apply. Calling µ its unique invariant measure absolutely continuous to Lebesgue one can prove, with essentially the same argument as before that, for each ϕ ∈ C 0 (),
lim |µ (ϕ) − [µ ] (ϕ)| = 0.
→
(4.5)
The exponential mixing in space and time can be obtained exactly as in [26]. Finally, our use of the term SRB for the measure µ is justified by the fact that µ enjoys the law of large number with respect to a vast class of initial measures related 15
Note that all the limits below make sense when the measures are applied to local function.
Uniqueness for the SRB for CML
43
to Lebesgue and, in addition, is stable under smooth random perturbations, that is the random perturbations have a unique invariant measure that converges to µ , see [26]. Normally, for finite systems, the criteria for defining the SRB measure are three (e.g., see [36]), the absolute continuity of the measure along the unstable manifolds, the law of large numbers with respect to Lebesgue for smooth observables, the stability with respect to random perturbations. In the infinite case the situation is a bit more subtle since measures tend to be not absolutely continuous one with respect to the other and, in our case, [15] shows that a lot of invariant measures can have marginals absolutely continuous with respect to Lebesgue. Yet, we have shown that, if some moderate regularity is required, then only one invariant measure with absolutely continuous marginals exists, moreover this is the limit obtained by truncating the system to a finite size as (4.5) shows. This together with the fulfillment of the other two requirements is, in our opinion, sufficient to attribute to µ the qualification of SRB. 5. Short Range Since now the range is infinite it is necessary to decompose the interaction according to space scales, the point being that the interaction on larger scales is smaller and smaller in the weak norm but no control is available on its variation, hence it is necessary to wait longer and longer times for the dynamics to act effectively on it. This forces a more complex bookkeeping mechanism which is reflected in the necessity of a larger covering Banach space that, with a slight abuse of notation, we will still call B . Let S be a positive integer to be fixed later. We define the Banach spaces B := {µ¯ := (µq,t,l ) : q ∈ ; t ∈ NS := N \ {1, . . . , S}; l ∈ {0, . . . , t}, µq,t,l ∈ Bq } ; together with the norm µ ¯ := sup q∈
ρ t α l Var µq,t,l + ρ −t α l−t |µq,t,l | ,
sup
t∈NS l∈{0,...,t}
(5.1)
for some constants α, ρ ∈ (0, 1) to be fixed later. Pictorially, one can imagine the above space as a collection of towers at each site q ∈ , where the tower t has height t and the index l denotes the l th floor in this tower. We can now define the lifted linear dynamics
if l > 0 µq,t,l−1 ¯ q,t,l := (T µ) (5.2) ∗m(s+1) ∗ µp,s,s if l = 0, {(p,s):τ (q−p,s)=t} p,q T where
τ (q − p, s) :=
0 if |q − p| ≤ s 2 + S |q − p| if |q − p| > s 2 + S .
Roughly speaking, within each tower s at site p the operator T pushes each measure one floor up, except for the measure at the top level, which is first transformed according to the dynamics of the whole tower and then distributed (by means of the telescoping operators ∗p,q ) to the ground levels of towers at sites q in the following way: If q is close to p (in the sense |q − p| ≤ s 2 + S) the corresponding measure is mapped to the
44
G. Keller, C. Liverani
tower of height t = 0, whereas if q is farther away from p, it is mapped to the tower of height |q − p|. To relate the dynamics of the linear system (T , B ) with that of the operator T∗m (for an integer m to be fixed later) we introduce the space Bw (θ ) as the completion of B with respect to the weak norm |µ|θ := sup
θ |1 | |µ(ϕ)| .
sup
1 ∈I |ϕ|C 0 (I 1 ) ≤1
Then we define
: B → B , 0
(µ)q,t,l =
∗0,q µ 0
if t = l = 0 otherwise ,
and P : B → Bw (θ ),
P µ¯ =
T∗ml µq,t,l .
q,t,l n
n
It is easy to check that P T = T∗mn P , and hence P T = T∗mn , for each n ∈ N, so that the linear system (T , B ) is indeed an extension of T∗m : B 0 → Bw (θ ).16 The following lemma is the main result of this section. Lemma 5.1. If τ is Lipschitz and is a short range coupling (see Sect. 2 for this terminology), then there exist σ ∈ (0, 1) and C, 2 > 0 such that, for all || < 2 , µ ∈ B 0 (), θ ∈ (0, 1), and n ∈ N holds true, |T∗n µ|θ ≤ Cσ n min{||, |e ln θ |−1 } Var µ.
(5.3)
Proof. As in the case of a finite coupling range, estimate (5.3) follows from the fact that T is a strict contraction on B . Namely, we will show that there are m ∈ N, σ ∈ (0, 1), and 2 > 0 such that for all µ¯ ∈ B holds, T µ ¯ = sup |(T µ) ¯ q,t,l |t,l ≤ σ m µ, ¯
(5.4)
q,t,l
where |µ|t,l := ρ t α l Var µ + ρ −t α l−t |µ|, see (5.1). The case l = 0 is easy: for all q and t we have ¯ q,t,l |t,l = |µq,t,l−1 |t,l = α|µq,t,l−1 |t,l−1 ≤ αµ ¯ ≤ σ m µ ¯ , |(T µ)
(5.5)
1
where m > 0 and σ ∈ (α m , 1) will be determined in the course of the proof. ¯ q,0,0 is given So we assume from now on that l = 0 and start with the case t = 0. (T µ) by the sum in (5.2) that ranges over indices p and s. We begin with the contributions for s = 0. Without loss of generality we may assume that m is even. By C we denote any constant that may depend on the “ingredients” of the system (like a, b, λ, etc.) but which Observe that, since s∈NS ,0≤l≤s ρ −s α −l does not converge, it is not true that P µ¯ ∈ B(θ ) for each µ¯ ∈ B , while P B →B (θ) ≤ (1 − ρ)−1 (1 − α)−1 |e ln θ |−1 . 16
w
Uniqueness for the SRB for CML
45
is independent of any constant that is to be fixed during the proof (i.e. S, α, ρ, m, σ, 1 . A crucial choice will be m = δS for some δ > 0 to be fixed later.) Then |∗p,q T∗m µp,0,0 |0,0
= Var(∗p,q T∗m µp,0,0 ) + |∗p,q T∗m µp,0,0 | ≤ 2 Var(T∗m µp,0,0 ) + 2|T∗m µp,0,0 | ∗m 2
≤ 2aλ− 2 Var(T m
∗m 2
µp,0,0 ) + 2(b + 1)|T
µp,0,0 | m
−m 2
≤ Cλ Var(µp,0,0 ) + C||m Var(µp,0,0 ) + Cσ02 Var(µp,0,0 ) m ≤ σ1 Var(µp,0,0 ) 1
(σ1 := max{λ−1 , σ0 } 4 )
≤ σ1m µ ¯
for sufficiently large m and || < (m), where we used essentially the same arguments that lead already to (3.18). Summing this over all p for which τ (q − p, 0) = t = 0 means that one has to sum over all p for which |q − p| ≤ S:
{p:τ (q−p,0)=0}
∗p,q T∗m µp,0,0
0,0
≤ CS d σ1m µ ¯ ≤
σm µ ¯ 2
(5.6)
for a suitable σ ∈ (σ1 , 1), provided m = δS is sufficiently large. Next we estimate the contributions for s = 0 to the sum in (5.2) when l = 0 and t = 0. In this case |∗p,q T∗m(s+1) µp,s,s |0,0 = Var(∗p,q T ∗m(s+1) µp,s,s ) + |∗p,q T ∗m(s+1) µp,s,s | ≤ 2 Var(T ∗m(s+1) µp,s,s ) + 2|T ∗m(s+1) µp,s,s | ≤ 2aλ−m(s+1) Var(µp,s,s ) + 2(1 + b)|µp,s,s | ≤ [2aλ−m(s+1) ρ −s α −s + 2(1 + b)ρ s ]|µp,s,s |s,s . As s > S and |q − p| ≤ s 2 + S in the case under consideration, we conclude ∗p,q T∗m(s+1) µp,s,s {(p,s):s∈NS \{0},τ (q−p,s)=0}
≤C
∞
0,0
(s 2 + S)d [2aλ−m(s+1) ρ −s α −s + 2(1 + b)ρ s ]µ ¯
s=S+1
¯ ≤ ≤ Cσ2S µ
σm µ ¯ 2
(5.7)
for suitable σ2 ∈ (ρ, 1) and σ ∈ (σ2 , 1), provided λ−m < αρ 2 and m = δS is sufficiently large. We finally turn to the case l = 0 and t = 0. For this we will need the following estimate: There are β ∈ (0, 1) and δ > 0 such that δ/2
|∗p,q T∗m(s+1) µp,s,s | ≤ Cm β |q−p| Var(µp,s,s ), provided m(s + 1) < δ|q − p|. The proof will be given below.
(5.8)
46
G. Keller, C. Liverani
Now, since t = 0, the condition τ (q − p, s) = t in the summation in (5.2) means that t = |q − p| > s 2 + S. In particular, as m = δS, we have m(s + 1) = δS(s + 1) < δ(s 2 + S) < δ|q − p| so that (5.8) is applicable. Therefore |∗p,q T∗m µp,s,s |t,0 = ρ t Var(∗p,q T∗m(s+1) µp,s,s ) + ρ −t α −t |∗p,q T∗m(s+1) µp,s,s | ≤ 2aρ t λ−m(s+1) Var(µp,s,s ) + 2bρ t |µp,s,s | + ρ −t α −t Cm β |q−p| Var(µp,s,s ) ≤ 2aρ t (ραλm )−s + 2bρ s ρ t + (ρα)−s−t Cm β |q−p| |µp,s,s |s,s . √ Hence, observing that t = |q − p| > S and s < t, ∗p,q T∗m µp,s,s ≤
{(p,s):τ (q−p,s)=t} √ t t d
Ct
t,0
ρ (ραλm )−s + ρ s ρ t + (ρα)−s−t Cm β t µ ¯
(5.9)
s=0
≤ Cσ3S µ ¯ ≤ σ m µ, ¯ 1
for suitable σ3 ∈ (ρ, 1) and σ ∈ (σ32δ , 1), provided αρ 2 > β and S = δ −1 m is sufficiently large. Putting together (5.5), (5.6), (5.7), and (5.9) yields T µ ¯ ≤ σ m µ ¯ for some m > 0 and σ ∈ (0, 1). This concludes the proof of Lemma 5.1.
m(s+1) by a map Proof of estimate (5.8). The basic idea of the proof is to approximate T T˜s = T˜,p,q,m(s+1) with the property that ∂p ((T˜s )q ) = 0 if |q − p| ≥ |q − p|. To this end recall the map ι¯p : I → I from Sect. 3, (¯ιp (x))q = xq if q = p and (¯ιp (x))q = 0 if q = p. Then define
m(s+1) (T (x))q ˜ (Ts (x))q = m(s+1) (T (¯ιp (x)))q
if |q − p| < c−2 |q − p| if |q − p| ≥ c−2 |q − p| .
Note first that (p,q ϕ)(T˜s (x)) is constant as a function of xp because c ≥ 1. It follows that ∗p,q T˜s∗ µ = 0 if µ ∈ Bp . Hence, recalling that µp,s,s ∈ Bp , we see that |∗p,q T∗m(s+1) µp,s,s | = |∗p,q (T∗m(s+1) − T˜s∗ )µp,s,s | ≤ 2|(T∗m(s+1) − T˜s∗ )µp,s,s | . The latter quantity can be bounded using Lemma 3.2b. To this end let us check the hypotheses of that lemma.
0 if |q − p| < c−2 |q − p| |(T˜s (x) − Tm(s+1) (x))q | ≤ m(s+1) |(DT )q p |∞ if |q − p| ≥ c−2 |q − p|.
To estimate the derivative notice that 0 ≤ |(D )q p | ≤ δq p + 2L||γ |q −p| so that √ √ 1 1 0 ≤ |(D )q p | 2 ≤ δq p + 2L||γ 2 |q −p| =: (Id + 2L||B)q p . Hence, setting λ+ := |τ |∞ , by the triangular inequality
Uniqueness for the SRB for CML
47
|(DTn )q p | ≤ λn+ {([Id +
2L||B]n )q p }2 .
Using a Cram´er type estimate as in [27] this leads to the following bound for K3 : let n = m(s + 1) and r = |q − p| so that n < δr. Then, for any t > 0, q
|(T˜s (x) − Tm(s+1) (x))q | 2
1
≤
n
λ+2 ([Id +
|q −p|≥c−2 r
≤
1
|q −p|≥c−2 r δ
δr
λ+2 ([Id + −2
≤ (λ+2 e−tc )r
2L||B]n )q p
([Id +
2L||B]δr )q p et|q −p|−tc
−2 r
(5.10)
2L||B]δr )q 0 et|q |
q ∈Zd δ 2
−2
≤ (λ+ ψ(t)δ e−tc )r =: β r = β |q−p| , √ 1 where ψ(t) := q ∈Zd (δq 0 + 2L||γ 2 |q | )et|q | . Clearly, |ψ(t)| < ∞ for t ∈ (0, 1 2 | ln γ |). Hence, if we fix such a t and choose δ > 0 sufficiently small, then β ∈ (0, 1). (These choices are uniform for in a neighbourhood of 0.) The proof that K2 can be taken to be some fixed constant (depending on m but not on p and q) is completely standard and it is left to the reader. Accordingly, Lemma 3.2 yields |∗p,q T∗m(s+1) µp,s,s | ≤ Cm β |p−q| Var(µp,s,s ) and that is (5.8). Lemma 5.1 and (5.3) are the equivalent of Lemma 2.3 and (3.19) which were the basic ingredients to prove Theorem 2.1 in the finite range case. These results can be used now in a similar way to obtain the corresponding result in the short range case. Proof of Theorem 2.4. The proof follows the one of Theorem 2.1, let us outline the main points. The uniqueness of the invariant measure in B follows trivially from Lemma 5.1. On the other hand, the approximation argument is now more subtle since one can no longer use the finite range property in (4.3). Nevertheless, using large deviation type estimates like in the proof of Eq. (5.8) one can show that (4.3) continues to hold if modulo another small error term. The same remarks apply to obtaining the spatial decay of correlation out of the temporal ones: again one has to treat explicitly very long range effect by showing that they produce a very small contribution. Finally, the reasons to call the above invariant measure SRB remain unchanged from the short range case.
48
G. Keller, C. Liverani
6. Appendix Proof of Lemma 3.2. It suffices to estimate (F˜ ∗ ν − F ∗ ν)(ϕ) for a test function ϕ with |ϕ|C 0 () ≤ 1: (F˜ ∗ ν − F ∗ ν)(ϕ) =
ϕ(F˜ x) − ϕ(F x) dν(x) =
1
=
0
=
q∈
1
0 q∈
1
0
∂ (ϕ(Ft x)) dt dν(x) ∂t
∂ ∂q ϕ(Ft x) Ft,q (x) dν(x) dt ∂t
Ft∗ (F˜q − Fq ) · ν (∂q ϕ) dt
so that |F˜ ∗ ν − F ∗ ν| ≤ K2 Var (F˜q − Fq ) · ν q∈
≤ K2
|F˜q − Fq |∞ + sup sup p∈ x =p
q∈
I
|∂p F˜q (x =p , ξ ) − ∂p Fq (x =p , ξ )|dξ Var ν
≤ K2 (K0 + K1 ) Var ν . This proves part a) of the lemma. The above estimate is, in some sense, too good for our needs in Sect. 5 where it may be hard to verify that K1 < ∞. It is then convenient to have a rougher estimate. To this end let us define the function R ∈ C 0 (R , ) by 0 R(x)q = xq 1
if xq < 0 if xq ∈ [0, 1] it xq > 1,
and let ϕ¯ := ϕ ◦ R. Next, define κ, κη : R → [0, ∞), κ(y) := max{1 − |y|, 0} and κη (y) := η−1 κ(η−1 y). For each 1 ∈ I, we introduce, for each η¯ = (ηq )q∈ , the 0 convolution operators Qη, ¯ 1 on C (): (Qη, ¯ 1 ϕ)(x) :=
R1 p∈ 1
κηp (xp − yp )ϕ(x ¯ ∈1 , y) dy ν(dx) .
Not surprisingly, the estimate holds: |Q∗η, ¯ 1 ν − ν| ≤
1 ηq Var ν . 3 q∈1
(6.1)
Uniqueness for the SRB for CML
49
In fact, ν(Qη, ¯ 1 ϕ − ϕ) = κηp (xp − yp )[ϕ(x ¯ ∈1 , y) − ϕ(x)]dy ¯ ν(dx) R1 p∈ 1
=
1
dt 0
=
1
dt 0
=
R 1
R1 p∈ 1
R 1
dz
κηp (xp − yp )
p∈1
1
dt 0
q∈1
κηp (zp )
d ϕ(x ¯ ∈1 , x∈1 + t (y − x∈1 ))dy ν(dx) dt
∂q ϕ(x ¯ ∈1 , x∈1 + tz)zq dz ν(dx)
q∈1
zq
κηp (zp )ν(∂q ϕ¯t,z ),
p∈1
where ϕt,z ¯ ∈1 , x∈1 + tz). From the above formula estimate (6.1) follows, (x) := ϕ(x because R |z|κη (z) dz = η3 . Accordingly, if q∈ ηq < ∞ we can define Qη¯ ϕ := lim Qη, ¯ 1ϕ . 1 →
Then 2 |F ∗ ν − F˜ ∗ ν| ≤ K2 ηq Var ν + sup ν((Qη¯ ϕ) ◦ F − (Qη¯ ϕ) ◦ F˜ ) 3 |ϕ|≤1 q∈
2 ηq Var ν + |ν| ηq−1 |Fq − F˜q |∞ . ≤ K2 3 q q 1
1
− 2 to get Now, for all the q for which |Fq − F˜q |∞ = 0, choose ηq = K2 2 |Fq − F˜q |∞ 1
|F ∗ ν − F˜ ∗ ν| ≤ 2K22 Var ν
1
1
2 |Fq − F˜q |∞ ≤ 2K22 K3 Var ν .
q
This finishes the proof of part b) of the lemma.
References 1. Baladi, V., Degli Eposti, M., Isola, S., J¨arvenp¨aa¨ , E., Kupiainen, A.: The spectrum of weakly coupled map lattices. J. Math. Pures Appl. 77, 539–584 (1998) 2. Baladi, V., Rugh, H.H.: Floquet spectrum of weakly coupled map lattices. Commun. Math. Phys. 220, 561–582 (2001) 3. Bardet, J.-B.: Limit theorems for coupled analytic maps. Probab. Th. Rel. Fields 124, 151–177 (2002) 4. Bricmont, J., Kupiainen, A.: Coupled analytic maps. Nonlinearity 8(3), 379–396 (1995) 5. Bricmont, J., Kupiainen, A.: High temperature expansions and dynamical systems. Commun. Math. Phys. 178(3), 703–732 (1996) 6. Bunimovich, L.A.: Coupled map lattices: one step forward and two steps back. In: Chaos, order and patterns: aspects of nonlinearity—the “gran finale” (Como, 1993). Phys. D 86(1-2), 248–255 (1995) 7. Bunimovich, L.A., Sinai, Ya.G.: Space-time chaos in coupled map lattices. Nonlinearity 1, 491–516 (1988)
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8. Fischer, T., Rugh, H.H.: Transfer operators for coupled analytic maps. Ergod. Th.& Dynam. Sys. 20, 109–143 (2000) 9. Gielis, G., MacKay, R.S.: Coupled map lattices with phase transition. Nonlinearity 13, 867–888 (2000) 10. Gundlach, V.M., Rand, D.A.: Spatio-temporal chaos: 1. Hyperbolicity, structural stability, spatiotemporal shadowing and symbolic dynamics. Nonlinearity 6, 165–200 (1991) 11. Gundlach, V.M., Rand, D.A.: Spatio-temporal chaos: 2. Unique Gibbs states for higher-dimensional symbolic systems. Nonlinearity 6, 201–214 (1993) 12. Gundlach, V.M., Rand, D.A.: Spatio-temporal chaos: 3. Natural spatio-temporal measures for coupled circle map lattices. Nonlinearity 6, 215–230 (1993) 13. Gundlach, V.M., Rand, D.A.: Spatio-temporal chaos (Corrigendum). Nonlinearity 9, 605–606 (1996) 14. J¨arvenp¨aa¨ , E.: A note on weakly coupled expanding maps on compact manifolds. Annales Academiæ Scientiarum Fennicæ Mathematica 24, 511-517 (1999) 15. J¨arvenp¨aa¨ , E., J¨arvenp¨aa¨ , M.: On the definition of SRB-measures for coupled map lattices. Commun. Math. Phys. 220(1), 1–12 (2001) 16. Jiang, M.: Equilibrium states for lattice models of hyperbolic type. Nonlinearity 8(5), 631–659 (1995) 17. Jiang, M.: Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations. Commun. Math. Phys. 193, 675-712 (1998) 18. Jiang, M.: Sinai-Ruelle-Bowen measures for lattice dynamical systems. J. Stat. Phys. 111(3-4), 863–902 (2003) 19. Jiang, M., Mazel, A.E.: Uniqueness and exponential decay of correlations for some two-dimensional spin lattice systems. J. Stat. Phys. 82(3-4), 797–821 (1996) 20. Jiang, M., Pesin, Ya.: Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations. Commun. Math. Phys. 193(3), 675–711 (1998) 21. Keller, G.: Coupled map lattice via transfer operators on functions of bounded variation. In: Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks 45, Amsterdam: North-Holland, 1996, pp. 71–80 22. Keller, G.: Mixing for finite systems of coupled tent maps. Tr. Mat. Inst. Steklova 216, pp. 320–326 (1997), Din. Sist. i Smezhnye Vopr.; translation in Proc. Steklov Inst. Math. 1997, no. 1, 315–332 (1997) 23. Keller, G.: An ergodic theoretic approach to mean field coupled maps. In: Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), Progr. Probab., 46, Basel: Birkh¨auser, 2000, pp. 183– 208 24. Keller, G., K¨unzle, M.: Transfer operators for coupled map lattices. Ergodic Theory Dynam. Systems 12(2), 297–318 (1992) 25. Keller, G., Liverani, C.: Coupled map lattices without cluster expansion. Discrete and Continuous Dynamical Systems 11, n.2,3, 325–335 (2004) 26. Keller, G., Liverani, C.: A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps. Lecture Notes in Physics 671, 115–151, (2005) 27. Keller, G., Zweim¨uller, R.: Unidirectionally coupled interval maps: between dynamics and statistical mechanics. Nonlinearity 15(1), 1–24 (2002) 28. K¨unzle, M.: Invariante Maße f¨ur gekoppelte Abbildungsgitter. Dissertation, Universit¨at Erlangen, 1993 29. Maes, Ch., van Moffaert, A.: Stochastic stability of weakly coupled map lattices. Nonlinearity 10, 715–730 (1997) 30. Miller, J., Huse, D.A.: Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice. Pys. Rev. E, 48, 2528–2535 (1993). 31. Pesin, Ya.B., Sinai, Ya.G.: Space-time chaos in chains of weakly interacting hyperbolic mappings. Adv.Sov.Math. 3, 165-198 (1991) ´ 32. H.H. Rugh, Coupled maps and analytic function spaces. Ann. Sci. Ecole Norm. Sup. (4) 35(4) 489–535 (2002) 33. Schmitt, M.: BV -spectral theory for coupled map lattices. Dissertation, Universit¨at Erlangen (2003). See also: Nonlinearity 17, 671–690 (2004) 34. Volevich, D.L.: Kinetics of coupled map lattices. Nonlinearity 4, 37–45 (1991) 35. Volevich, D.L.: Construction of an analogue of Bowen-Sinai measure for a multidimensional lattice of interacting hyperbolic mappings. Russ. Acad. Math. Sbornink 79, 347–363 (1994) 36. Young, L.-S.: What are SRB measures, and which dynamical systems have them. Dedicated to David Ruelle andYasha Sinai on the occasion of their 65th birthdays. J. Stat. Phys. 108(5-6), 733–754 (2002) Communicated by A. Kupiainen
Commun. Math. Phys. 262, 51–89 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1425-3
Communications in
Mathematical Physics
Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals Dario Martelli1 , James Sparks2 1 2
Department of Physics, CERN Theory Division, 1211 Geneva 23, Switzerland. E-mail:
[email protected] Department of Mathematics and Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02318, U.S.A. E-mail:
[email protected]
Received: 15 December 2004 / Accepted: 15 March 2005 Published online: 24 November 2005 – © Springer-Verlag 2005
Abstract: Recently an infinite family of explicit Sasaki–Einstein metrics Y p,q on S 2 × S 3 has been discovered, where p and q are two coprime positive integers, with q < p. These give rise to a corresponding family of Calabi–Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are K¨ahler quotients C4 //U (1), namely the vacua of gauged linear sigma models with charges (p, p, −p + q, −p − q), thereby generalising the conifold, which is p = 1, q = 0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold C3 /Zp+1 ×Zp+1 for all q < p with fixed p. We hence find that the Y p,q manifolds are AdS/CFT dual to an infinite class of N = 1 superconformal field theories arising as IR fixed points of toric quiver gauge theories with gauge group SU (N )2p . As a non–trivial example, we show that Y 2,1 is an explicit irregular Sasaki– Einstein metric on the horizon of the complex cone over the first del Pezzo surface. The dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its IR fixed point using the AdS/CFT correspondence. The value we obtain is a quadratic irrational number and, remarkably, agrees with a recent purely field theoretic calculation using a-maximisation. 1. Introduction and Summary The AdS/CFT correspondence [1] predicts that type IIB string theory on AdS5 × Y5 , with appropriately chosen self-dual five-form flux, is dual to an N = 1 four-dimensional superconformal field theory whenever Y5 is Sasaki–Einstein [2–5]. This latter condition may be defined as saying that the metric cone over Y5 ds 2 (C(Y5 )) = dr 2 + r 2 ds 2 (Y5 )
(1.1)
is Ricci-flat K¨ahler, i.e. Calabi–Yau. The superconformal field theory may be thought of as arising from a stack of D3-branes sitting at the tip of the Calabi–Yau cone. Notice
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that unless Y5 is the round metric on S 5 , appropriately normalised, the tip of the cone at r = 0 will be singular. It is a striking fact that, until very recently, the only Sasaki–Einstein five-manifolds that were known explicitly in the literature1 were precisely the round metric on S 5 and the homogeneous metric T 1,1 on S 2 × S 3 , or quotients thereof. For the five-sphere the Calabi–Yau cone is simply C3 and the dual superconformal field theory is the maximally supersymmetric N = 4 SU (N ) theory. For T 1,1 the Calabi–Yau cone is the conifold and the dual N = 1 superconformal field theory was given in [3, 5]. Due to the rather limited number of examples in the literature detailed tests of the AdS/CFT conjecture for more interesting geometries have been lacking2 . Indeed, one is restricted to quotients (orbifolds) of S 5 and T 1,1 . These have been extensively studied using orbifold techniques which by now are completely standard. For example, Klebanov and Witten argued that the field theory for T 1,1 may be obtained via a relevant deformation of the N = 2 orbifold S 5 /Z2 . However, this has changed drastically with the recent discovery [6] of a countably infinite class of explicit Sasaki–Einstein metrics on Y p,q ∼ = S 2 ×S 3 . These were initially found by reduction and T-duality of a class of supersymmetric M-theory solutions discovered in [7]. The family is characterised by two relatively prime positive integers p, q, with q < p. A particularly interesting feature of these Sasaki–Einstein manifolds is that there are countably infinite classes which are both quasi-regular and irregular. These terms are not to be confused with regularity of the metric: the metrics are all smooth metrics on S 2 × S 3 . Rather, they refer to properties of the orbits of a certain Killing vector field. Indeed, on any Sasaki–Einstein manifold Y there exists a canonically defined Killing vector field K, called the Reeb vector in the mathematics literature. The orbits of this Killing vector field may or may not close. If they close then there is a (locally free) U (1) action on Y and such Sasaki–Einstein manifolds are called quasi-regular. The geometries Y p,q with 4p 2 −3q 2 a square are examples of such manifolds. If the orbits of the Reeb vector field do not close the Killing vector generates an action of R on Y , with the orbits densely filling the orbits of a torus, and the Sasaki–Einstein manifold is said to be irregular. The geometries Y p,q with 4p 2 − 3q 2 not a square are the first examples of such geometries in the literature3 . Another interesting feature of these metrics is that the volumes are always given by a quadratic irrational number times the √ volume of the round metric on S 5 – recall a quadratic irrational is of the form a + b c, where a, b ∈ Q, c ∈ N. Moreover, the volumes are rationally related to that of S 5 if and only if the Sasaki–Einstein is quasi-regular. Recall that all four-dimensional N = 1 superconformal field theories possess an R-symmetry, commonly referred to as the U (1) R-symmetry. However, crucially this symmetry is not always a U (1) symmetry – this is true only if the R-charges of all the fields are rational. In general, this is not true, as exemplified by the recent work of [9]. In the latter reference it is shown that the exact R-symmetry of a superconformal field theory maximises a certain combination of t’Hooft anomalies atrial (R) = (9TrR 3 −3TrR)/32. 1 E. Calabi has constructed an explicit K¨ahler–Einstein metric on del Pezzo 6 – recall that this is the blow-up of CP2 at 6 points – with a certain symmetric configuration of the 6 blown-up points. The corresponding Sasaki–Einstein metric on #6(S 2 × S 3 ) is thus also explicit. This metric has apparently never been published. We thank S.–T. Yau for pointing this out to us. 2 Although one can still deduce some geometric information for the regular Sasaki–Einstein manifolds #l(S 2 × S 3 ), which are U (1) bundles over del Pezzo surfaces with l points blown up, l = 3, . . . , 8, even though the general metrics are not known explicitly. 3 Thus disproving a conjecture of Cheeger and Tian [8] that such examples do not exist. We thank the referee for drawing our attention to this reference.
Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals
53
The maximal value is then precisely the exact a central charge of the superconformal field theory. Since one is maximising a cubic with rational coefficients, the resulting R-charges are always algebraic numbers. Recall that in AdS/CFT the R-symmetry is precisely dual to the canonical Killing vector field K discussed above. Moreover, the central charge aY for the field theory dual to Y is inversely proportional to its volume. In particular, we have [10] aY vol(S 5 ) . = aS 5 vol(Y )
(1.2)
It is thus clearly of interest to identify the dual superconformal field theories for the Sasaki–Einstein manifolds Y p,q , so as to compare the exact results on both sides of the duality. In this paper we take the first substantial steps in this program by analysing in considerable detail the geometry of the manifolds Y p,q , and the associated Calabi–Yau cones. The results allow us to show that the metrics Y p,q are dual to a class of N = 1 superconformal field theories arising as IR fixed points of certain toric quiver gauge theories, with gauge group SU (N )2p . The case p = 2, q = 1 is somewhat special. This corresponds to the geometry with largest volume, and is an irregular metric. The dual field theory therefore has the smallest central charge within the family, and moreover is expected to be quadratic irrational. Rather surprisingly, we find that the metric Y 2,1 turns out to be an explicit metric on the horizon of the complex cone over the first del Pezzo surface. For this, the corresponding SU (N )4 quiver gauge theory and superpotential have already been identified [11]. We can then compute the central charge (1.2) and also the R-charges of the baryons for this theory using AdS/CFT, where the baryons correspond to D3-branes wrapped over 3-cycles whose metric cones are supersymmetric cycles (complex divisors) in the cone over Y 2,1 . The values we find are all quadratic irrational numbers. At first sight these results present a puzzle, as the central charge computed in [9, 12, 13] was found to be a rational number. However, a closer inspection of the quiver theory shows that the a-maximisation calculation is somewhat more subtle in this case4 . Indeed, using a-maximisation [9] applied to the quiver theory, the authors of [14] find a central charge, as well as R-charges, which agree perfectly with the values obtained using the geometrical results of this paper. This constitutes an extremely beautiful test of the AdS/CFT correspondence, as well as the general a-maximisation procedure advocated in [9]. Given the results presented here, in principle the duals to the remaining geometries, with general p and q, q < p, can be constructed using the “toric algorithm” of [11]. These will provide an infinite series of N = 1 superconformal field theories, whose central charges are generically quadratic irrational. It will be interesting to obtain these explicitly, and to compare the results of a-maximisation for these theories with the various geometrical results presented in this paper. However, we leave these calculations for future work. As a final point, we note that in [15] a generalisation of the metrics Y p,q to all dimensions was presented (see also references [16] and [17] for a generalisation of this generalisation). In particular there are countably infinite classes of supersymmetric solutions AdS4 × Y7 to M-theory, which will have three-dimensional CFT duals, where the metric Y7 is built using any positive curvature K¨ahler–Einstein metric in real dimension four [15]. These have been classified [18, 19]. For the case when the K¨ahler–Einstein 4 We are very grateful to M. Bertolini, F. Bigazzi, A. Hanany, K. Intriligator, and B. Wecht for discussions on this issue.
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manifold is toric, one has only three cases: CP2 , CP1 × CP1 , and dP3 , where the latter is the third del Pezzo surface. Using the techniques developed in this paper, one can show that for the first two cases the metric cones over Y7 are given by K¨ahler quotients C5 //U (1), and C6 //U (1)2 , respectively, where the various U (1) charges are, with appropriate definitions5 of the Chern numbers p and k, Q = (p, p, p, −3p + k, −k) and Q1 = (p, p, 0, 0, −2p + k, −k), Q2 = (0, 0, p, p, −2p + k, −k), respectively. Outline. The first point to note about the manifolds Y p,q , and their associated Calabi– Yau cones, is that they are all toric. This essentially means that there is an effective action of a torus T3 ∼ = U (1)3 on C(Y p,q ) which preserves the symplectic form of the cone and commutes with the homothetic R+ action. Indeed, this torus action is an isometry, and so also preserves the metric. The torus action and symplectic form then allow us to define a moment map, µ : C(Y p,q ) → R3 . The image in R3 is always a good convex rational polyhedral cone in R3 [20]. These terms will be explained more carefully later. However, roughly this is a convex cone formed by intersecting some number of planes through the origin. The moment map exhibits C(Y p,q ) as a T3 fibration over this moment cone, with the fibres collapsing over the faces, or facets, of the cone in a way determined by the normal vectors to the facets. We shall find explicitly that the moment cone for Y p,q is a four-faceted good strictly convex rational polyhedral cone. Having computed the moment cone for C(Y p,q ) we may then apply a Delzant theorem [21] for symplectic toric cones worked out recently in [20]. In physics terms, this takes the combinatorial data defining the moment cone and uses it to produce a gauged linear sigma model [22]. By construction the classical vacuum of the linear sigma model is precisely the Calabi–Yau cone one started with. More mathematically, this would be called a symplectic – or, more precisely, K¨ahler – quotient of Cd by a compact abelian group. The final result is: • The metric cones over Y p,q are explicit Calabi–Yau metrics for the U (1) gauged linear sigma model on C4 with charges (p, p, −p + q, −p − q), and zero Fayet–Illiopolous parameter. If we denote the vacuum of a linear sigma model by X = C4 //U (1), then it is easy to see that, rather generally, c1 (X) = 0 is equivalent to the charges of the U (1) gauge group summing to zero. Clearly this is true for the gauged linear sigma model above, and hence X is indeed topologically Calabi–Yau. In this process we lose precise information about the metric – in particular, the induced metric from C4 is not Ricci-flat. However, we have now gained an explicit description of the Calabi–Yau singularity. Indeed, by constructing invariant monomials one obtains an algebraic description of the singularity. One easily sees that this is the hypersurface up+q v p−q = x p+q y p−q in
C4 ,
(1.3)
where the monomials are given by p−q p z3 ,
u = z1
p+q p z4 ,
v = z2
p−q p z3 ,
x = z2
p+q p z4
y = z1
.
(1.4)
We may then give the toric diagram for the Calabi–Yau singularity. This may be realised as an integral polytope in R2 . Roughly, the four outward pointing primitive normal vectors that define the moment cone lie in a plane as a result of the Calabi–Yau 5
In particular, the definitions here are different from those in [15].
Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals
55
condition. Projecting these vectors onto this plane yields the vertices of the toric diagram for a minimal presentation of the singularity. We show that the resulting toric diagrams may all be embedded inside that of the orbifold C3 /Zp+1 × Zp+1 , where the two factors −1 −1 are generated by (ωp+1 , ωp+1 , 1), (ωp+1 , 1, ωp+1 ) ⊂ SU (3), respectively, where ωp+1 th is a (p + 1) root of unity. The vertices of the polytope are then (0, 0), (0, p + 1) and (p + 1, 0) (the position of the origin is irrelevant) and we show that the toric diagram for C(Y p,q ) lives inside this polytope for all q < p and fixed p. Geometrically, this means that the Calabi–Yau cone C(Y p,q ) may be obtained by (partial) toric crepant resolution of the orbifold [5, 23]. Also, as part of our general analysis, we find a class of supersymmetric submanifolds in the geometries C(Y p,q ). Specifically, we show that the cones over the special orbits of the cohomogeneity one action on Y p,q are calibrated submanifolds – in fact complex divisors – of the Calabi–Yau. Recall that D3-branes wrapped over the horizon 3-cycles are dual to baryons in the AdS/CFT correspondence [24, 25]. We compute the volumes of these submanifolds, and hence give a prediction for the R-charges of the corresponding baryons. Given the toric diagram for C(Y p,q ) there are methods to construct a superconformal field theory, whose Higgs branch is the toric variety X ∼ = C(Y p,q ), purely from the combinatorial data that defines X [11]. Indeed, the point is that the field theory for the orbifold C3 /Zp+1 × Zp+1 , in which the geometries are “embedded”, is known from standard orbifold techniques. The Calabi–Yau cones C(Y p,q ) are obtained by partial resolution, which amounts to turning on specific combinations of Fayet–Illiopolous parameters in the gauged linear sigma model. The field theories in question are then rather conventional quiver gauge theories with polynomial superpotentials. The number of nodes of the quiver is simply twice the area of the toric diagram, which is 2p for all q with fixed p. Rather surprisingly, we find that the toric diagram for Y 2,1 is precisely the same as that for the complex cone over the first del Pezzo surface. Recall that the latter is the blow-up of CP2 at one point, and that the complex cone over this is indeed a real cone over S 2 × S 3 . It follows that Y 2,1 , which is irregular, is an explicit Sasaki–Einstein metric on the horizon, or boundary, of this cone. This is interesting, since the higher del Pezzo surfaces, which are CP2 with 3 ≤ r ≤ 8 generic points blown up, admit K¨ahler–Einstein metrics [18, 19]. The complex cones then carry regular Sasaki–Einstein metrics. The case of one or two points blown up has always been something of a puzzle, since these del Pezzos do not admit K¨ahler–Einstein metrics and thus the Sasaki–Einstein metrics associated to the complex cones could not possibly be regular. We have thus resolved this puzzle, at least in the case of one blow-up. The quiver gauge theory dual to the complex cone over the first del Pezzo surface has been presented in the literature [11]. The AdS/CFT correspondence then predicts the exact central charge of this theory in the IR. Using the explicit metric Y 2,1 , the result we obtain is √ aS 5 13 13 + 46 vol(Y 2,1 ) 7.74 = = ∼ . aY 2,1 12 · 27 27 vol(S 5 )
(1.5)
Remarkably, this value coincides precisely with a recent application of a-maximisation [9] to the quiver gauge theory [14]. Moreover, we also find perfect agreement for the charges of (SU (2)F singlet) baryons in the gauge theory. The plan of the rest of the paper is as follows. In Sect. 2, after recalling some basic facts about Sasaki–Einstein geometry, we give a summary of the construction of the
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metrics Y p,q , and recall several of their features. Section 3 contains a review of symplectic toric geometry – in particular toric contact geometry – which we use extensively in the remainder of the paper. In Sect. 4 we compute the image of the moment map associated to the toric Calabi–Yau cones C(Y p,q ). In Sect. 5 we apply a Delzant construction to obtain a gauged linear sigma model (GLSM) description of the Calabi–Yau spaces. Moreover we analyse directly the structure of the moduli space of vacua of the GLSM in Sect. 5.3. In Sect. 6 the associated toric Gorenstein singularities are described. In Sect. 7 we demonstrate that Y 2,1 is an irregular metric on the horizon of the complex cone over the first del Pezzo surface, and exhibit an explicit (non-K¨ahler and non-Einstein) metric on the latter. Section 8 concludes with a comparison of the geometrical results obtained here with the results of a-maximisation applied to the quiver gauge theory corresponding to the complex cone over the first del Pezzo surface [14]. In Appendix A the techniques used in the paper, which perhaps are unfamiliar to many physicists, are applied to the familiar example of the conifold. 2. Sasaki–Einstein Metrics on S 2 × S 3 In this section we review the geometry of the recently discovered Sasaki–Einstein metrics on S 2 × S 3 [6]. There is an infinite family of such metrics, labeled by two coprime integers p > 1, q < p – we refer to these as Y p,q . Geometrically they are all U (1) principle bundles6 over an axially squashed S 2 bundle over a round S 2 . The integers label the twisting, or Chern numbers, of the U (1) bundle over the two two-cycles, with the constraint q < p arising as a regularity condition on the metric. The manifolds are all cohomogeneity one. The fact that they are all topologically S 2 × S 3 follows from a theorem of Smale [26] on the classification of five-manifold topology. In the following we first recall basic material about Sasakian–Einstein geometry and then turn to the metrics Y p,q .
2.1. Sasakian–Einstein geometry. A Sasaki–Einstein manifold may be defined as a complete positive curvature Einstein manifold7 whose metric cone is Ricci–flat K¨ahler, i.e. a Calabi–Yau cone. The structure of a Sasaki–Einstein manifold may thus be thought of as “descending” from the Calabi–Yau structure of its metric cone (1.1). In particular, contracting the Euler vector r∂/∂r, which generates the homothetic R+ action on the cone, into the K¨ahler form gives rise to a one-form on the base of the cone, Y . The dual of this is a constant norm Killing vector field – called the Reeb vector in the mathematical literature – which via the AdS/CFT correspondence is isomorphic to the R-symmetry of the dual field theory. The Killing vector defines a foliation of the Sasaki–Einstein manifold, and one finds that the transverse leaves have a K¨ahler–Einstein structure. More precisely, one can write the local form of the metric as follows: ds 2 (Y ) = ds42 +
1
3 dψ
+σ
2
,
(2.1)
6 This U (1) is not to be confused with the isometry generated by the Reeb vector. The latter is embedded non-trivially inside the torus defined by this U (1) and U (1) that rotates the axially squashed S 2 fibre. 7 We also require simply-connectedness. This is not strictly necessary. However, given this condition we can use a theorem which relates contact structures to the existence of globally-defined Killing spinors. The latter is the physical property that we wish our manifolds to possess.
Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals
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where ds42 is a local K¨ahler–Einstein metric. In particular we have that dσ = 2J4 , d4 = i3σ ∧ 4 ,
(2.2)
where J4 and 4 are the local K¨ahler and holomorphic (2, 0) form for ds42 , respectively. The Reeb Killing vector is given by K≡3
∂ . ∂ψ
(2.3)
Sasaki–Einstein manifolds may then be classified into three families, according to the global properties of the orbits of this Killing vector field: • If the orbits close, and moreover the associated U (1) action is free, the Sasaki–Einstein manifold is said to be regular. The length of the orbits are then all equal. One thus has a principle U (1) bundle over a four-dimensional base K¨ahler–Einstein manifold. • Suppose that the isotropy group x of at least one point x is non-trivial. Notice that x is necessarily isomorphic to Zm , for some integer m, since these are precisely the proper subgroups of U (1). The U (1) action is then locally free, meaning that the isotropy groups are all finite – note that the Killing vector cannot vanish anywhere since it has constant norm. The Sasaki–Einstein manifold is then said to be quasi–regular. In this case notice that the length of the orbit through x is 1/m times the length of the generic orbit. The quotient of any manifold by a locally free compact Lie group action is canonically an orbifold. One thus has a principle orbifold U (1) bundle, or orbibundle, over a K¨ahler–Einstein base orbifold. Moreover, the point x will descend to a Zm – orbifold point x in this base space. • If the orbits do not close, the Sasaki–Einstein manifold is said to be irregular. In this case one does not have a well-defined quotient space. Note that such a Sasaki–Einstein manifold necessarily has at least a U (1)d isometry group, d ≥ 2, with the orbits of the Killing vector filling out a dense subset of the orbits of the torus action. Indeed, the isometry group of a compact Riemannian manifold is always a compact Lie group. Hence the orbits of a Killing vector field define a one-parameter subgroup, the closure of which will always be an abelian subgroup and thus a torus. The dimension of the closure of the orbits is called the rank. Thus irregular Sasaki–Einstein manifolds have rank greater than 1. The five-dimensional regular Sasaki–Einstein manifolds are classified completely [27]. This follows since the smooth four-dimensional K¨ahler–Einstein metrics with positive curvature on the base have been classified by Tian and Yau [18, 19]. These include the special cases CP2 and S 2 × S 2 , with corresponding Sasaki–Einstein manifolds being the homogeneous manifolds S 5 (or S 5 /Z3 ) and T 1,1 (or T 1,1 /Z2 ), respectively. For the remaining metrics, the base is a del Pezzo surface obtained by blowing up CP2 at k generic points with 3 ≤ k ≤ 8 and, although proven to exist, the generic metrics are not known explicitly. We emphasise the lack of existence of K¨ahler–Einstein metrics on the del Pezzo surfaces with one or two points blown up, as this will play an important role later. This fact is actually rather simple to understand. It is a fairly straightforward calculation [28] to show that the Lie algebra H generated by holomorphic vector fields on a K¨ahler–Einstein manifold is a complexification of the Lie algebra generated by Killing vector fields, i.e. isometries. The latter is always a reductive algebra (meaning it is the sum of its centre together with a semi-simple algebra) but for the first and second del Pezzo surfaces the
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algebra H is not reductive. Clearly then H being reductive is always necessary. This is Matsushima’s Theorem [28]. One also requires that the anti-canonical bundle be ample, that is c1 > 0, otherwise the putative K¨ahler–Einstein metric would be indefinite. In complex dimension two, these necessary conditions are in fact sufficient for existence of a K¨ahler–Einstein metric [18, 19], and this leads to the list stated above. It was only recently [29–32] that quasi-regular Sasaki–Einstein metrics were shown to exist on #l(S 2 × S 3 ) with l = 1, . . . , 9. In particular, there are 14 known inhomogeneous Sasaki–Einstein metrics on S 2 × S 3 . We stress that the proof of this is via existence arguments, rather than giving explicit metrics. Specifically, one uses a modification of Yau’s argument to prove existence of K¨ahler–Einstein metrics on certain complex orbifolds, and then builds the appropriate U (1) orbibundle over these to obtain Sasaki–Einstein manifolds. One can also obtain quasi-regular geometries rather trivially by taking quotients of the explicit regular geometries discussed above by appropriate freely-acting finite groups. For example, one can take a freely-acting finite subgroup of SU (3) and quotient S 5 ⊂ C3 by the induced action. 2.2. The metrics Y p,q . We will now review, as well as work out some new, properties of the Sasaki–Einstein metrics Y p,q on S 2 × S 3 . These were presented in [6] in the following local form: ds 2 =
1 − cy 1 q(y) (dθ 2 + sin2 θ dφ 2 ) + dy 2 + (dψ − cos θ dφ)2 6 w(y)q(y) 9
+ w(y) [dα + f (y)(dψ − cos θdφ)]2 ≡ ds 2 (B) + w(y)[dα + A]2 ,
(2.4)
where 2(a − y 2 ) , 1 − cy a − 3y 2 + 2cy 3 q(y) = , a − y2 ac − 2y + y 2 c f (y) = . 6(a − y 2 )
w(y) =
(2.5)
For c = 0 the metric takes the local form of the standard homogeneous metric on T 1,1 . Otherwise, c can be scaled to 1 by a diffeomorphism. Henceforth we assume this is the case. The base B. The analysis of [6] first showed that the four dimensional space B can be made into a smooth complete compact manifold with appropriate choices for the ranges of the coordinates. In particular, for8 0
(2.6)
one can take the ranges of the coordinates (θ, φ, y, ψ) to be 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π , y1 ≤ y ≤ y2 , 0 ≤ ψ ≤ 2π so that the “base space” B is an axially squashed S 2 8 In the limit a → 1 the two positive roots become equal and y = 1 is a double root. In the case a = 1 the metric is locally that of the round metric on S 5 .
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bundle over a round S 2 . The latter is parametrised by θ , φ, with ψ being an azimuthal coordinate on the axially squashed S 2 fibre. This bundle is geometrically twisted, and may be thought of as the S 2 bundle over S 2 formed by taking the tangent bundle of the round two-sphere and adding a point at infinity to each fibre. Now, the inclusion map U (1) → SO(3) induces a map Z ∼ = π1 (U (1)) → π1 (SO(3)) ∼ = Z2 which is reduction modulo 2. Here we are thinking of U (1) as the group in which the transition functions of T S 2 take their values, and SO(3) as the structure group of the associated oriented S 2 bundle over S 2 . Since T S 2 has Chern number 2 ∼ = 0 mod 2, it follows that the S 2 bundle is trivial and thus the manifold B is topologically a product space, B ∼ = S 2 × S 2 . The range of y is fixed so that 1 − y > 0, a − y 2 > 0, w(y) > 0, q(y) ≥ 0. Specifically, yi are two zeroes of q(y), i.e. are two roots of the cubic Q(y) ≡ a − 3y 2 + 2y 3 = 0 .
(2.7)
If 0 < a < 1 there are three real roots, one negative (y1 ) and two positive, the smallest being y2 . The values y = y1 , y2 then correspond to the south and north poles of the axially squashed S 2 fibre. One may check explicitly that the metric is smooth here with the above identifications of coordinates. The circle fibration. It was shown in [6] that for a countably infinite number of values of a, with 0 < a < 1, one can now choose the period of α so as to describe a principle S 1 bundle over B. This is true if and only if the periods of dA are rationally related. Thus one requires P1 = p,
P2 = q
(2.8)
with the periods Pi , i = 1, 2, given by 1 Pi = 2π
dA ,
(2.9)
Ci
where C1 and C2 give the standard basis for the homology group of two-cycles on B∼ = S 2 × S 2 . In this case, one may take 0 ≤ α ≤ 2π ,
(2.10)
and the five-dimensional space is then the total space of an S 1 fibration over B ∼ = S 2 ×S 2 , with Chern numbers p and q over the two two-cycles. An explicit calculation shows that P1 3 = . P2 2(y2 − y1 )
(2.11)
Moreover, the function y2 (a) − y1 (a) is a monotonic increasing function of a, taking the range 0 < y2 (a) − y1 (a) < 3/2, thus implying a countably infinite number of solutions with 0 < q/p < 1. Furthermore, for any p and q coprime, the space Y p,q is topologically S 2 × S 3 – see [6]. This follows from a result of Smale on the classification of five-manifold topology.
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The volumes. One finds that =
q 3q 2 − 2p 2 + p(4p 2 − 3q 2 )1/2
(2.12)
and the volume of Y p,q is given by vol(Y p,q ) =
q 2 [2p + (4p 2 − 3q 2 )1/2 ] π3 3p 2 [3q 2 − 2p 2 + p(4p 2 − 3q 2 )1/2 ]
(2.13)
which is a quadratic irrational number times the volume π 3 of a unit round S 5 . We note that at fixed p the volume is a monotonic function of q, and is bounded by the following values: vol(T 1,1 /Zp ) > vol(Y p,q ) > vol(S 5 /Z2 × Zp ) .
(2.14)
The rational case, which is easily seen to correspond to quasi–regular manifolds, is described by p, q ∈ N, hcf(p, q) = 1, q < p, which are solutions to the quadratic diophantine 4p2 − 3q 2 = n2
(2.15)
for some n ∈ Z. The solutions to this were given in closed form in [6]. The isometry group. The isometry group of the metrics (2.4) is clearly locally SU (2) × U (1) × U (1), and in particular there are three commuting Killing vectors ∂/∂φ, ∂/∂ψ, and ∂/∂γ . Here we have defined α ≡ γ
(2.16)
so that the three generators have canonical period 2π . For us it will be important to note that the global form of the effectively acting isometry group depends on p and q. In particular, for both p and q odd it is SO(3) × U (1)2 , otherwise it is U (2) × U (1). This will be explained later in Sect. 4. Note that this is precisely analogous to the case of the Einstein manifolds known in the physics literature as T p,q . For these the effectively acting isometry group is shown [33] to be SO(3) × SU (2) when one integer is even, and SO(4) ∼ = (SU (2) × SU (2))/Z2 when both are odd. The latter of course includes the case of T 1,1 [3]. The local K¨ahler–Einstein structure. Employing the change of coordinates α = −β/6 − ψ /6, ψ = ψ one can [6] bring the metric (2.4) into the local Sasaki–Einstein form (2.1). In particular 1−y dy 2 1 (dθ 2 + sin2 θ dφ 2 ) + + w(y)q(y)(dβ + cos θ dφ)2 6 w(y)q(y) 36 (2.17) 1 2 + [dψ − cos θ dφ + y(dβ + cos θdφ)] . 9 The corresponding J4 and 4 , satisfying (2.2), can be taken as ds 2 =
1−y 1 sin θ dθ ∧ dφ + dy ∧ (dβ + cos θ dφ) , (2.18) 6 6 1−y w(y)q(y) 4 = (dθ + i sin θ dφ) ∧ dy + i (dβ + cos θdφ) , 6w(y)q(y) 6 J4 =
(2.19)
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while the Reeb Killing vector is given by K=3
1 ∂ ∂ − . ∂ψ 2 ∂γ
(2.20)
Note that this has compact orbits when is a rational number and corresponds to the quasi-regular class, by definition. This is true if and only if (2.15) holds. If is irrational the generic orbits do not close, but instead densely fill the orbits of the torus generated by [∂/∂ψ, ∂/∂γ ] and we thus fall into the irregular class. The rank of these metrics is thus equal to 2. Note that the orbits close only over the submanifolds given by y = y1 , y2 . These are precisely the special9 orbits of the cohomogeneity one action. The Killing spinors. To show that these manifolds admit globally defined Killing spinors one appeals to the following theorem [34]: every simply-connected spin Sasaki–Einstein manifold, where the latter is defined in terms of the existence of a certain contact structure, admits a solution to the Killing spinor equation. In particular we note that the dual one-form to K is given by 1 η = −2y(dα + A) + q(y)(dψ − cos θ dφ) (2.21) 3 which is globally-defined (the factor of q(y) is essential here). The contact structure is then easy to exhibit in terms of η for the manifolds Y p,q [6]. This theorem is the reason why one a priori requires hcf(p, q) = 1–however see below. The Calabi–Yau cones. It will be important for us to exploit the symplectic structure of the associated Calabi–Yau cones. Rather generally, the Calabi–Yau structure on the metric cone is specified by a K¨ahler (hence also symplectic) form J and a holomorphic (3, 0) form , which in terms of the four-dimensional K¨ahler–Einstein data read as follows: J = r 2 J4 + rdr ∧ ( 13 dψ + σ ), = eiψ r 2 4 ∧ dr + ir( 13 dψ + σ ) .
(2.22) (2.23)
In the specific case of C(Y p,q ), we have 1−y sin θ dθ ∧ dφ 6
1 1 2 + rdr ∧ (dψ − cos θdφ) − d(yr ) ∧ dα + (dψ − cos θ dφ) 3 6
J = r2
(2.24) and
1−y (dθ + i sin θdφ) 6w(y)q(y) ∧ dy − iw(y)q(y) dα + 16 (dψ − cos θdφ) ∧ dr − 2ir ydα + (y − 1) 16 (dψ − cos θ dφ) ,
=e r
iψ 2
(2.25)
9 The manifolds Y p,q are cohomogeneity one, meaning that the generic orbit under the action of the isometry group is codimension one. There are then always precisely two special orbits of higher codimension.
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where we used (2.18), (2.19) and have then rewritten the expressions in terms of the original coordinates. Note that this calculation shows that is invariant under ∂/∂α, namely L∂/∂α = i∂/∂α d + d(i∂/∂α ) = 0,
(2.26)
implying that the Killing spinors are also invariant. This explicitly checks that upon performing a T -duality along the α direction to Type IIA string theory, the number of preserved supersymmetries is unchanged. In fact, this is obvious given the original construction [7] of these metrics. Since we are guaranteed existence of Killing spinors by the theorem of [34], and since we have now shown that the spinors are independent of α, it follows that one may in fact take hcf(p, q) = h > 1 by taking a smooth quotient by Zh of the simply-connected Sasaki–Einstein manifold Y p/ h,q/ h . Since this is rather trivial, we take this as understood in the remainder of the paper. Complex coordinates. It is easy to introduce a (local) set of complex coordinates. To do so we seek three closed complex one-forms ηi such that ∧ ηi = 0. First, consider the following local one-forms obeying the latter property: 1 dθ + idφ, sin θ 1 1 η˜ 2 = dy − i(dα + (dψ − cos θ dφ)), w(y)q(y) 6
dr 1 3 η˜ = − i dα + (y − 1)(dα + (dψ − cos θ dφ)) , 2r 6 η1 =
(2.27)
where now = 2e r
iψ 3
Q(y) sin θ η1 ∧ η˜ 2 ∧ η˜ 3 . 3
(2.28)
Taking z1 = tan θ2 eiφ we immediately find η1 =
dz1 . z1
(2.29)
To obtain two more integrable one-forms one is free to consider linear combinations of the one-forms (2.27). Take 1 η2 = − cos θ η1 + η˜ 2 , 6 1 3 η = cos θ η1 − y η˜ 2 + η˜ 3 . 6
(2.30)
Notice that one can now simply drop the tildes in (2.28). Moreover the η2 , η3 are now closed and hence locally exact. In particular ηi =
dzi , 6zi
(2.31)
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i = 2, 3, with 1 −1 −1 −1 z2 = (y − y1 ) y1 (y2 − y) y2 (y3 − y) y3 e−6iα−iψ , sin θ z3 = r 3 sin θ Q(y) eiψ .
(2.32)
In terms of the {zi }, the three-form assumes a very simple form: =
1 dz1 ∧ dz2 ∧ dz3 . √ z1 z2 18 3
(2.33)
Supersymmetric cycles. In this subsection we will show that the cones over the submanifolds y = y1 , y2 , which recall are the special orbits of the cohomogeneity one action, are in fact divisors in the Calabi–Yau cone. This amounts to showing that they are calibrated with respect to the four-form 21 J ∧ J . We denote the three-submanifolds as i , i = 1, 2, respectively. Thus, we compute the pull–back of 21 J ∧ J to the four-cycles in the Calabi–Yau cone C(Y p,q ) specified by y = yi . The latter are in fact cones over the Lens spaces 1 ∼ = S 3 /Zp+q , 2 ∼ = S 3 /Zp−q . We shall show in detail that this is indeed the topology in Sect. 4. However, this fact can also be seen by computing the pull-back of the K¨ahler form to the four-submanifolds. Defining k = p + q, l = p − q, these are10
J |y=y1 J |y=y2
k 2 = y1 − r sin θ dθ ∧ dφ − rdr ∧ (d2γ − k cos θ dφ) , 2 l 2 r sin θ dθ ∧ dφ − rdr ∧ (d2γ + l cos θdφ) , = y2 2
(2.34) (2.35)
and are precisely the K¨ahler forms associated to cones over round Lens spaces S 3 /Zk and S 3 /Zl , respectively. Indeed, since γ has period 2π , the one-forms multiplying dr are precisely global angular forms (global connections) on the total spaces of circle bundles over S 2 with Chern numbers k and −l, respectively. The total spaces of such bundles are precisely S 3 /Zk and S 3 /Zl , respectively. From these expressions, one calculates 1 r 3 yi (1 − yi ) J ∧ J |y=yi = sin θ dθ ∧ dφ ∧ dγ ∧ dr . 2 3
(2.36)
Let us compare this with the volume form induced on i from the metric (2.4). This is given by vol =
√ r 3 w(yi )(1 − yi ) sin θ dθ ∧ dφ ∧ dγ ∧ dr . 6
(2.37)
Remarkably, since w(yi ) = 4yi2 at any root of the cubic (2.7) we see that this precisely agrees with (2.36). Thus we see that both C(1 ) = {y = y1 } and C(2 ) = {y = y2 } are divisors of C(Y p,q ), or in other words they are supersymmetric submanifolds. 10
Recall that y1 0 and y2 0.
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We may now write down the volumes of the i . Here one needs to use the explicit formulae for the roots of the cubic y1 and y2 in terms of p and q:
1 2p − 3q − 4p 2 − 3q 2 , y1 = 4p (2.38)
1 y2 = 2p + 3q − 4p 2 − 3q 2 . 4p One then easily calculates 2
q 2 (p + q) −2p + 3q + 4p 2 − 3q 2 2 vol(1 ) = 2 π ,
2 2 2 2 2 2p 3q − 2p + p 4p − 3q 2
q 2 (p − q) 2p + 3q − 4p 2 − 3q 2 2 vol(2 ) = 2 π .
2 2 2 2 2 2p 3q − 2p + p 4p − 3q
(2.39)
In particular, let us write down the volumes of i in the case of p = 2, q = 1: vol(1 ) =
√ π2 (31 + 7 13) , 108
vol(2 ) =
√ π2 (7 + 13) . 36
(2.40)
3. Moment Maps and Convex Rational Polyhedral Cones In the remainder of this paper it will be crucial for us that the Sasaki–Einstein manifolds Y p,q admit an effectively acting three-torus T3 = U (1)3 of isometries, which moreover is Hamiltonian. The latter means that the action preserves the symplectic form of the cone C(Y p,q ) and that one can use this to introduce a moment map. The torus is just the maximal torus in the isometry group, and the fact that the torus is half the dimension of the cone means that, by definition, the cones are toric. The image of the cone under the corresponding moment map generally belongs to a special class of convex rational polyhedral cones in R3 [35, 20] – these are simply convex cones formed by intersecting some number of planes through the origin. The normal vectors to these planes, or facets, are necessarily rational and describe which U (1) subgroup of T3 is vanishing over the corresponding codimension two submanifold of C(Y p,q ). This generalises the wellknown result in symplectic geometry that the image of the moment map for a compact toric symplectic manifold is always a particular type of convex rational polytope called a Delzant polytope. In this section we give a general review of symplectic toric geometry. This is mainly rather standard material from the point of view of a symplectic geometer – the reader who is familiar with this subject may therefore wish to skip this section. On the other hand, we hope that this will be a useful self-contained presentation of the material. 3.1. Moment maps for torus actions. In this subsection we give a general summary of moment maps, Hamiltonian torus actions, and symplectic toric manifolds, orbifolds and cones, together with the properties of their images under the moment maps, which are always particular types of rational polytopes (or polyhedral cones) in Rn . The case
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of compact manifolds [36, 37, 21] is rather standard in symplectic geometry, but the generalisation for orbifolds [38], and especially cones [35, 20], is quite recent. We begin by giving a general definition. Suppose that the torus Tn acts effectively – meaning that every non-trivial element moves at least one point – on a symplectic manifold M with symplectic form ω. We identify the Lie algebra of this torus, as well as its dual, with Euclidean n-space, so tn ≡ Lie(Tn ) ∼ = Rn , tn∗ ∼ = Rn . Then a moment n map for the torus action is simply a T –invariant map, µ : M → tn∗ ∼ = Rn ,
(3.1)
dµi = V i ω .
(3.2)
satisfying the condition
Here V i denotes the vector field on M corresponding to the basis vector ei in tn ∼ = Rn , and µi denotes the component of the map µ in the direction ei , i.e. µ = (µ1 , . . . , µn ). Clearly this moment map is unique only up to an additive integration constant. To see where this map comes from, suppose for simplicity that one has a U (1) action on a symplectic manifold M, generated by some vector field V , which moreover preserves the symplectic form. One then says that the U (1) action is symplectic. The latter means that LV ω = 0,
(3.3)
where L is the Lie derivative. Since ω is closed, this condition is just d(V ω) = 0 .
(3.4)
As long as the closed one-form V ω is trivial as a cohomology class, [V ω] = 0 ∈ H 1 (M; R), then one can “integrate” this equation to a function µ, which is precisely the moment map for the U (1) action. The action is then said to be Hamiltonian. For example, the U (1) which rotates one of the circles in T2 , with obvious symplectic form, is not Hamiltonian. Clearly, if H 1 (M; R) is trivial then all symplectic actions are in fact Hamiltonian. A symplectic toric manifold is then by definition a symplectic manifold of dimension 2n with an effective Hamiltonian torus action by Tn . It is by now a classic fact in symplectic geometry that, for a compact symplectic toric manifold M, the image of M under µ is a certain kind of convex rational polytope in Rn called a Delzant polytope [21]. Recall that a polytope is just the convex hull of some finite number of points in Rn . The codimension one hyperplanes that bound the polytope are called its facets. The symplectic toric manifold is then a torus fibration over this polytope, with the fibres collapsing in a certain way over the facets. More precisely, over an interior point of the polytope the fibre of the moment map (the inverse image of the point) is the whole torus Tn , but over the boundary facets this fibre collapses to Tn−1 ∼ = Tn /U (1). Such a U (1) subgroup is specified by a vector in the weight lattice n v ∈ Z of Tn , and this vector is in fact just the normal vector to the facet. Moreover the U (1) fixes a corresponding codimension two submanifold of M. To see this, consider the case where v = e1 = (1, 0, . . . , 0). Denote the corresponding vector field as V . Then over a codimension two fixed point set F ⊂ M we have that V = 0, and moreover F is itself symplectic toric with respect to the torus Tn−1 ∼ = Tn /U (1). In particular, the moment map µ restricted to F is constant in the direction corresponding to V , i.e. µ1 = c = constant. Then µF ≡ µ |F is a moment map for F with < µF , e1 >= c. This defines the hyperplane at x1 = c, where {xi }, i = 1, . . . , n are coordinates on
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Rn . The general case follows similarly. The normal vectors to the facets are thus all rational vectors. If two facets intersect over a codimension two face in Rn , then both the corresponding U (1)’s vanish, and the fibre over this face is a Tn−2 . Continuing in this way, the vertices of the polytope are precisely the points in M which are fixed under the entire torus action. The fact that the polytope is always convex follows from an argument using Morse theory [36, 37]. Delzant polytopes satisfy some additional conditions, as well as being rational: • simplicity – n edges meet at each vertex. • smoothness – for each vertex, the corresponding n edge vectors ui , i = 1, . . . , n form a Z–basis11 of Zn . The polytope data is sufficient to recover the original symplectic toric manifold. Moreover, the correspondence between Delzant polytopes and compact symplectic toric manifolds is one-to-one. Thus, to any Delzant polytope one can associate a corresponding symplectic toric manifold whose image under the moment map is precisely . The proof of this is by construction. This will be extremely important for us in Sect. 5 – in physics terms, the construction realises the manifold as the vacuum of a gauged linear sigma model [22]. We now briefly explain the above conditions. Assuming the first condition holds, the second condition avoids orbifold singularities. Indeed if the smoothness condition fails then Tn / < ui >∼ = is a non-trivial finite abelian group, where ui denotes the span of the ui over Z. In this case the corresponding point in M is an orbifold point with structure group . Indeed, there is a corresponding classification of symplectic toric orbifolds where the smoothness condition is dropped, and moreover one attaches to each facet a positive integer label [38]. This latter necessity can be seen by considering the weighted projective space CP1[k,l] . This is topologically a sphere, with neighbourhoods of the north and south poles replaced by orbifold singularities C/Zk and C/Zl , respectively. The quotient by the U (1) action which rotates around the equator is clearly just a line segment. Thus the orbifold information is completely lost when one takes the image under the moment map. To remedy this [38], quite generally, one associates to each facet a positive integer label m, such that the pre-image of any point in that facet has local orbifold structure group Zm . In the case at hand, the endpoints of the interval are assigned labels k and l, respectively. The first condition - simplicity - avoids even worse singularities than orbifold singularities. As we shall see, for symplectic toric cones this condition is not satisfied at the vertex corresponding to the apex of the cone, unless of course the cone is in fact an orbifold singularity.
3.2. Toric Calabi–Yau cones. This brings us to the generalisation of this theorem [35, 20] for symplectic toric cones, which is the case of interest for us. These may be regarded as non-compact symplectic toric manifolds with a homothetic action of R+ which commutes with the torus action and acts by rescaling the symplectic form. In fact, every symplectic toric cone is a cone over a toric contact manifold Y , and vice versa. In this case the moment map for the symplectic toric cone C(Y ) = R+ × Y may still be defined, away from the apex of the cone, and takes a special form. Define the one-form ηC = r∂/∂rω,
11 This means that the set { n u | n ∈ Z, i = 1, . . . , n} is precisely Zn . i i i i
(3.5)
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where r∂/∂r is the Euler vector, which generates the R+ action on the cone, and ω is the symplectic form. Identifying the base of the cone Y = C(Y ) |r=1 we may define the one-form12 η = ηC |r=1 . One then easily sees that 1 ω = rdr ∧ η + r 2 dη . 2
(3.6)
A straightforward calculation then shows that the moment map µ on the cone is given by µ, ei = ηC (V i )
(3.7)
for any basis vector ei of tn and corresponding vector field V i . Here ηC (V i ) just denotes the dual pairing between one-forms and vectors. The choice of integration constant makes this moment map transform homogeneously under the R+ homothetic action. It also ensures that the apex of the cone, at r = 0, is mapped to the origin of Rn . Let us now also assume13 that the symplectic toric cones are of Reeb type. This means that there is some element ζ ∈ tn ∼ = Rn such that µ, ζ is a strictly positive function on C(Y ). The image of the moment map is then a strictly convex rational polyhedral cone in Rn [35], which, moreover, is good in the sense of reference [20]. Recall that a rational polyhedral cone may be defined as a set of points in Rn of the form C = {x ∈ Rn | x, vi ≤ 0, i = 1, . . . , d},
(3.8)
where the rational vectors vi are the outward pointing normal vectors to the facets of the cone C. Here we may assume that the set {vi } is minimal, meaning that one cannot drop any vector vi from the definition without changing the cone, and also primitive – recall that a vector with integral entries is said to be primitive if it cannot be written as nv, where 1 = n ∈ Z and v is also a vector with integral entries. The requirement that this polyhedral cone is strictly convex means that it is a cone over a polytope. The “conelike" nature of the subspace (3.8) of course descends from the “conelike" nature of the cone we began with – the property that C(Y ) is invariant under a group R+ of homotheties will be inherited by the image under the moment map since by definition the moment map commutes with the R+ action. Clearly the simplicial condition will fail at the apex = origin of Rn unless d = n. Moreover, even in this case the smoothness condition will fail unless the edges span Zn . In this case, by an SL(n; Z) transformation of the torus, one can take this to be the standard basis, whence it is easy to see that the cone one started with is just R2n with its usual symplectic structure. This latter point brings up an issue worth stressing: one is of course free to make an SL(n; Z) transformation of the torus Tn resulting in a change of the basis ei . This will generate a corresponding SL(n; Z) transformation on the image under the moment map. Thus the polytopes and polyhedral cones are only unique up to such transformations. As shown in [20], the image of a symplectic toric cone under its moment map is also a good polyhedral cone. This means the following. Let F be a proper face of the cone C. Over this face there will be a corresponding torus TF ⊂ Tn which is collapsing to zero. For example, in the case that F is a facet, TF ∼ = U (1). For a face F of codimension m the torus is dimension m: dim TF = m. Now, the torus TF ⊂ Tn determines a lattice More precisely we embed Y in C(Y ) at r = 1 and then pull back ηC to Y to give η. The symplectic toric cones that are not of Reeb type are rather uninteresting: they are either cones over S 2 × S 1 , cones over principle T3 bundles over S 2 , or cones over products Tm × S m+2j −1 , m > 1, j ≥ 0 [20]. 12 13
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ZTF = ker(exp : tF → TF ) ⊂ Zn . We then require that the corresponding collection of normal vectors form an integral basis for this lattice, i.e. the collection of normal vectors span the lattice ZTF over Z. This condition may be regarded as a generalisation of Delzant’s conditions for symplectic toric manifolds to symplectic toric cones. In the particular case where the symplectic cone came from a Calabi–Yau cone, one has additional information. In particular, the Sasaki–Einstein metric on Y may be used to define the dual vector field K with η(K) = 1. This is called the Reeb vector in the language of contact geometry. Physically this is dual to the R-symmetry of the field theory. Then there is a corresponding Lie algebra element ζ ∈ tn , and we have µY , ζ = η(K) = 1 .
(3.9)
It follows that the image µY (Y ) lies in the above hyperplane, which is called the characteristic hyperplane [39]. In particular, note that the polytope one obtains by intersecting the polyhedral cone with the characteristic hyperplane will be rational if and only if ζ is rational. The latter condition is required precisely for quasi-regularity of the Sasaki-Einstein metric. Correspondingly, this is also the condition that the characteristic polytope satisfies for an orbifold polytope, and thus that the quotient of Y by the U (1) action generated by K gives an orbifold. Notice that one may then apply the modified Delzant construction of [38] to obtain a gauged linear sigma model describing this orbifold. In principle one could do this for our quasi-regular Sasaki-Einstein manifolds, although we will not pursue this here. 4. The Moment Map and Its Image In this section we explicitly construct the polyhedral cone corresponding to the image of C(Y p,q ) under its moment map. The Calabi–Yau cones on Y p,q are symplectic toric cones. In particular, the T3 action, which is the maximal torus of the isometry group, is Hamiltonian, and one can explicitly integrate the symplectic form (2.24) to obtain a moment map. Note in fact that (2.24) can be written as 1−y 1−y J = dφ ∧ d r 2 cos θ + dψ ∧ d −r 2 + dγ ∧ d r 2 y . (4.1) 6 6 The torus T3 is essentially generated by the Killing vectors ∂/∂φ, ∂/∂ψ, ∂/∂γ . However, one must be careful to ensure that the Killing vectors one takes really do form a basis for an effectively acting T3 . Since this is a slightly subtle point, we first explain a simpler example. A brief detour on Lens spaces. Let us consider the Lens spaces L(1, m) = S 3 /Zm , where we regard S 3 as a (squashed) Hopf S 1 fibration over a round two-sphere. The isometry groups of the latter may be analysed as follows. Embed the round sphere S 3 in R4 , and regard R4 ∼ = H as the space of quaternions. The isometry group of S 3 , preserving its orientation, is SO(4) ∼ = (SU (2)L × SU (2)R )/Z2 , where SU (2)L,R denote left and right actions by the unit quaternions Sp(1) ∼ = SU (2). Thus H q → aqb−1 , ∼ where (a, b) ∈ SU (2) × SU (2) = Spin(4). Notice that (−1, −1) acts trivially, i.e. the two SU (2) factors intersect precisely over the antipodal map. Thus, for a squashed three-sphere, meaning that one squashes the Hopf S 1 fibre relative to the base round S 2 , we see that the isometry group is U (2) ∼ = (SU (2) × U (1))/Z2 .
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However, suppose we now take a quotient of R4 ∼ = H on the right by Zm ⊂ U (1). One still has a left SU (2) action and a right U (1) action, where the latter now factors through a cyclic group of order m. For example, take m = 2, thus giving S 3 /Z2 ∼ = RP3 . ∼ In complex coordinates, H = C ⊕ C, this means (z1 , z2 ) ∼ (−z1 , −z2 ) which identifies antipodal points on the three-sphere. It follows that the centre of SU (2)L acts trivially and hence the effectively acting isometry group is SO(3) × U (1), where U (1) rotates the S 1 fibre with weight one - half the weight of U (1) ⊂ SU (2)R . It now follows that the isometry group for S 3 /Zm for all odd m is U (2), whereas for even m it is SO(3)×U (1)– it is precisely the even cases where Zm contains the antipodal map above. Clearly these Lens spaces have an isometric T2 action. Take m = 2r. From our discussion above, if V1 denotes the Killing vector that rotates the S 2 about its equator with weight one, and V2 denotes the Killing vector that rotates the S 1 fibre, also with weight one, then V1 , V2 do indeed form a basis for an effectively acting T2 . This is the obvious T2 in SO(3) × U (1). For m = 2r + 1 one needs to be more careful: the isometry group is U (2). For example, for r = 0 one has the unit chiral spin bundle of S 2 . As is well-known, a single rotation of S 2 will not result in the spinor coming back to itself: one needs to rotate twice. For an effective action one should thus take a basis e1 = V1 + 21 V2 , e2 = V2 . Here e1 is half the generator of the diagonal U (1) in SU (2) × U (1), and V2 generates the U (1) factor. Of course, one can use the basis e1 = V1 + m2 V2 , e2 = V2 quite generally in all cases. Indeed, recall that the choice of basis is unique only up to an SL(2; Z) transformation. For m = 2r even, this basis is just the SL(2; Z) transformation
1r (4.2) 01 of the basis {V1 , V2 }. The moment cone. After this brief digression, we return to the case of interest. First let us note from the results above that the isometry group of the base B is SO(3) × U (1). Indeed, for fixed y, y1 < y < y2 , we have a copy of S 3 /Z2 ∼ = RP3 , and the group SO(3) × U (1) acts with cohomogeneity one on B with fixed y as generic orbit. Thus, in particular, we may take a basis ∂/∂φ + ∂/∂ψ, ∂/∂φ for an effectively acting two-torus. For C(Y p,q ), one must also add the direction ∂/∂γ . However, here one must be careful to ensure the orbits of the vectors close, and that this torus then acts effectively, just as for the Lens spaces. One finds the following choice suffices: ∂ ∂ + , ∂φ ∂ψ l ∂ ∂ e2 = − , ∂φ 2 ∂γ ∂ e3 = . ∂γ e1 =
(4.3)
Recall that the submanifolds y = y1 , y = y2 of Y p,q are Lens spaces S 3 /Zk , S 3 /Zl , respectively, where recall k = p + q, l = p − q-the shift in e2 is then required precisely by the reasoning above. Note that one can replace l in the formula for e2 by anything congruent to l modulo two (for example, k)-this is just an SL(3; Z) transformation of the torus. Also note that for l even one can in fact take a basis ∂/∂φ, ∂/∂ψ, ∂/∂γ . The
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effectively acting isometry group is thus SO(3) × U (1) × U (1) in this case. For l odd this becomes U (2) × U (1). Let us now consider the moment map for C(Y p,q ). In terms of the basis e1 , e2 , e3 above one finds:
2 r r2 r2 (4.4) µ = (1 − y)(cos θ − 1), (1 − y) cos θ − l y, r 2 y . 6 6 2 Notice that this involves the generically irrational parameter . We will now describe the image of µ, and check that it is given by a good convex rational polyhedral cone in R3 , as predicted by the results of [35, 20]. First, note that the edges of the cone can be identified by fixing any non-zero value of r, say r = 1, and then finding the submanifolds which are fixed under some T2 ⊂ T3 action. Indeed, the edges of the cone, which generate it, are precisely the images of submanifolds in C(Y p,q ) over which some two-torus collapses. There are four such submanifolds at r = 1, given by the north (N) and south (S) poles of the base and fibre two-spheres: these are all copies of a circle – specifically, the fibre over the corresponding point on the base B. We denote the subspaces as follows: N N = {y = y2 , θ = 0}, N S = {y = y2 , θ = π }, SN = {y = y1 , θ = 0}, SS = {y = y1 , θ = π}. Then, using the useful relations14 1 − y1 = −3 ky1 , 1 − y2 = 3 ly2 ,
(4.5)
we find (at r = 1) µ(N N ) = y2 µ(N S) = y2 µ(SN ) = y1 µ(SS) = y1
(0, 0, 1), (−l, − l, 1), (0, − p, 1), (k, q, 1) .
(4.6)
Note that the irrational parameter has factored out and the vectors in (4.6) represent four lines which are spanned as r varies from 0 to infinity. Noting that y1 0 and y2 0 it is then easy to verify that these are the edges of a four-faceted polyhedral cone in R3 generated by: u1 = [0, p, −1],
u2 = [−k, −q, −1],
u3 = [0, 0, 1],
u4 = [−l, −l, 1] (4.7)
with outward-pointing primitive normals: v1 = [1, 0, 0],
v2 = [1, −2, −l],
v3 = [1, −1, −p],
v4 = [1, −1, 0] . (4.8)
As described above, these normals characterise codimension two fixed point sets in C(Y p,q ) over which a circle of the three-torus shrinks to zero size. The corresponding linear combination of Killing vectors in [∂/∂φ, ∂/∂ψ, ∂/∂γ ] should then have vanishing 14 One can derive these using the the explicit form for the periods P of Sect. 2, after using the cubic i equation (2.7).
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norm when restricted to the pre-image of the facet. Indeed, using the metric (2.4) it is straightforward to verify that the four Killing vectors V1 =
∂ ∂ + , ∂φ ∂ψ
V2 = −
∂ ∂ + , ∂φ ∂ψ
V3 =
∂ k ∂ − , ∂ψ 2 ∂γ
V4 =
∂ l ∂ + ∂ψ 2 ∂γ (4.9)
vanish on the submanifolds given by θ = 0, θ = π , y = y1 , and y = y2 respectively. Note that the normals obtained with the moment map use only the symplectic structure of the manifolds, whereas the norms of the Killing vectors are computed using the metrics. Let us now make the following observations about the normal vectors v1 , . . . , v4 : • {v1 , . . . , v4 } span Z3 over Z. Indeed it is trivial to see that v1 , v4 = E1 , E2 . The direction E3 is then obtained as a linear combination of v1 , v2 , v3 , v4 . Indeed, since hcf(l, p) = 1 by Euclid’s algorithm there are integers a, b ∈ Z such that al + bp = 1. • For each of the four edge vectors ui , i = 1, . . . , 4, the corresponding two normal vectors vi1 , vi2 , i1 = i2 ∈ {1234} with ui · vi1 = ui · vi2 = 0 satisfy {a1 vi1 + a2 vi2 | a1 , a2 ∈ R} ∩ Z3 = {a1 vi1 + a2 vi2 | a1 , a2 ∈ Z} .
(4.10)
The second condition is precisely the condition that the cone is good, in the sense of reference [20]. Indeed, this must be true since in [20] it is shown that the image of a symplectic toric cone under its moment map is always a good rational polyhedral cone. The first property does not generically hold, but is special to the geometries we are considering. As we will see later, it is related to the fact that the Sasaki-Einstein manifolds we began with are simply-connected. It will be useful to know the topology of the codimension two submanifolds. Let us denote them as Fi , where i = 1, . . . , 4, respectively. Explicitly we have F1 = {θ = 0}, F2 = {θ = π}, F3 = {y = y1 }, F4 = {y = y2 }. If we project out the γ direction, these are all copies of S 2 . The first two, F1 /U (1), F2 /U (1), are the two fibres of B = S 2 → S 2 over the north and south poles of the base S 2 , and so are representatives of the cycle C1 . The third and fourth, F3 /U (1), F4 /U (1), are the sections of the S 2 bundle at the south and north poles of the fibre S 2 , respectively15 . Since the γ direction describes a principle U (1) bundle over each of these spheres, the total spaces Fi will be Lens spaces L(1, m) ∼ = S 3 /Zm for various values of m ∈ Z. To see which Lens spaces one has, one can simply integrate the curvature two-form −1 dA over Fi /U (1) for each i = 1, . . . , 4. One finds F1 ∼ = F2 ∼ = S 3 /Zp ,
F3 ∼ = S 3 /Zk ,
F4 ∼ = S 3 /Zl .
(4.11)
Thus the facets of the polyhedral cone lift to cones over the above four Lens spaces. The latter two are calibrated submanifolds, as we saw in Sect. 2. 5. Gauged Linear Sigma Models In this section we begin by giving a brief review of gauged linear sigma models [22]. We then move on to describe Delzant’s construction [21] which from a polytope constructs a gauged linear sigma model whose vacuum manifold is precisely the symplectic toric manifold corresponding to . The construction also goes through for cones, 15
These were denoted S1 and S2 in [6].
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provided one starts with a good convex rational polyhedral cone [20]. We then use this method to construct the sigma model for the cone C(Y p,q ). Using this approach, turning on Fayet–Illiopolous parameters in the linear sigma model one (partially) resolves the conical singularity. As a check on our result, we explicitly show how one can recover the topology and group action on Y p,q from the linear sigma model description, thus closing the loop of arguments. This is summarised below: C(Y p,q )
moment map
−→
Delzant
vacuum
polyhedral cone ⊂ R3 −→ linear sigma model −→ C(Y p,q ).
5.1. A brief review. Let z1 , . . . , zd denote complex coordinates on Cd . In physics terms these will be the lowest components of chiral superfields i , i = 1, . . . , d. We may specify an action of the group Tr ∼ = U (1)r on Cd by giving the integral charge matrix i Q = {Qa | i = 1, . . . , d; a = 1, . . . , r}; here the a th copy of U (1) acts on Cd as 1
d
(z1 , . . . , zd ) → (λQa z1 , . . . , λQa zd ),
(5.1)
where λ ∈ U (1). We may then perform the K¨ahler quotient X = Cd //U (1)r by imposing the r constraints d
Qia |zi |2 = ta ,
a = 1, . . . , r,
(5.2)
i=1
where ta are constants, and then quotienting out by U (1)r . The resulting space has complex dimension n = d − r and inherits a K¨ahler structure, and thus also a symplectic structure, from that of Cd . In physics terms, the constraints (5.2) correspond to setting the D-terms of the gauged linear sigma model to zero to give the vacuum, where ta are Fayet–Illiopolous parameters – one for each U (1) factor. The quotient by Tr then removes the gauge degrees of freedom. Thus the K¨ahler quotient of the gauged linear sigma model precisely describes the classical vacuum of the theory. Note that the K¨ahler class of the quotient X depends linearly on the FI parameters ta , and moreover even the topology of the quotient will depend on these. Also observe that, setting all ta = 0, the resulting quotient will be a cone. One sees this by noting that zi → νzi , i = 1, . . . , d is a symmetry in this case, where ν ∈ R+ . The conical singularity is located at zi = 0, i = 1, . . . , d. It is also an important fact that c1 (X) = 0 is equivalent to the statement that the sum of the U (1) charges is zero for each U (1) factor. Thus d
Qia = 0,
a = 1, . . . , r .
(5.3)
i=1
This latter fact ensures also that the one-loop beta function is zero. The sigma model is then Calabi–Yau, although note that the metric induced by the K¨ahler quotient is not Ricci–flat.
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5.2. Delzant’s construction: from polytopes to gauged linear sigma-models. Let us first suppose we have a Delzant polytope which is the image of some compact symplectic toric manifold M under its associated moment map. One can reconstruct M from as follows. Let vi ∈ Zn , i = 1, . . . , d, denote the outward pointing primitive normal vectors to the facets of . For some λi ∈ R we may then write (5.4) = x ∈ Rn |< x, vi >≤ λi , i = 1, . . . , d . Consider now the linear map π : Rd → Rn which maps the standard basis vectors Ei of Rd to vi . Thus π(Ei ) = vi for each i = 1, . . . , d. From the Delzant properties of one easily sees that this map is surjective. The kernel has dimension r = d − n, and defines a corresponding torus Tr ⊂ Td . Now take Cd with its usual action by Td , and consider the moment map where we take the Fayet–Illiopoulos parameters to be ti = λi . From the induced action by Tr ⊂ Td above, we get an induced moment map for the Tr action. One may now take the symplectic reduction Cd //Tr , which is a symplectic manifold of complex dimension d − r = d − (d − n) = n. Moreover, this quotient also inherits an action of Tn = Td /Tr from that of Cd and is thus toric. In fact, it is not difficult to see that the image of Cd //Tr under its moment map, associated to Tn , is just . This is Delzant’s construction [21]. As a completely trivial example, consider the two-sphere S 2 with canonical U (1) action which rotates about the equator. The image of the moment map is just a line segment, with length proportional to the volume of the two-sphere. The outward pointing normal one-vectors are v1 = 1, v2 = −1. The kernel of the map π : Ei → vi is thus (1, 1), whence we see that S 2 is the symplectic reduction of C2 by U (1) with charges (1, 1)-the U (1) quotient is just the Hopf map S 3 → S 2 . There is a corresponding construction for compact symplectic toric orbifolds, which is a generalisation that takes into account that the normals may no longer form a Z–basis for Zn . This introduces finite subgroups which become local orbifold groups in the symplectic quotient [38]. A Delzant construction for cones. Recently a Delzant theorem has been proven for symplectic toric cones [20]. The language used is largely that of contact geometry – recall that a metric cone over a contact manifold is precisely a symplectic cone, and vice versa. The essential point is that the convex rational polyhedral cone one starts with must be good. This ensures that the symplectic quotient is smooth. Since the moment cones µ(C(Y p,q )) are all good cones, we may apply the theorem of [20]: one simply applies Delzant’s construction, as in the compact case, and sets all the Fayet–Illiopolous parameters to zero. Thus recall that the outward pointing primitive normal vectors were found to be v1 = [1, 0, 0],
v2 = [1, −2, −l],
v3 = [1, −1, −p],
v4 = [1, −1, 0] . (5.5)
By inspection the kernel is (p, p, −l, −k). Thus the Delzant theorem for cones gives • U (1) gauged linear sigma–model on C4 with charge vector Q = (p, p, −l, −k). As a preliminary check that this is indeed correct, notice that the charges sum to zero: p + p − l − k = 0, since k = p + q, l = p − q. It follows that the vacuum manifold X of this gauged linear sigma model is topologically Calabi–Yau, c1 (X) = 0, just as expected. Moreover, by turning on the Fayet–Illiopolous parameter t for the U (1) gauge field we will obtain orbifold resolutions of the cone.
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As interesting degenerate cases, consider p = 1, q = 0. This is the (resolved) conifold, which recall is the gauged linear sigma model on C4 with charges Q = (1, 1, −1, −1). Another important case is p = q = 1. This yields C times the linear sigma model on C3 with charges (1, 1, −2). The latter is OCP1 (−2). Taking t = 0 shrinks the CP1 to zero size, yielding the orbifold C2 /Z2 , which is also the A1 singularity. Thus the cone is C × (C2 /Z2 ). This has N = 2 rather than N = 1 supersymmetry. The horizons of these two spaces are thus T 1,1 and S 5 /Z2 . If one formally takes p = q = 1 and q = 0, one obtains Zp quotients of the cases above. In particular these will correspond to orbifolds (C2 /Z2 × C)/Zp and (conifold)/Zp respectively. It is interesting to note that these are consistent with the limiting volumes (2.14), although the metrics Y p,q are not valid in these limits. We can now use the results of [40] to perform further non-trivial checks. According to Theorem 1.1 of Ref. [40] we have the following general topological facts about the base Y of the symplectic toric cone C(Y ) we began with (provided it is of Reeb type): • π1 (Y ) ∼ = Zn /vi , is a finite abelian group. Recall that n = dim(Tn ) is the complex dimension of the cone C(Y ). • π2 (Y ) is a free abelian group of rank d − n, where d is the number of facets of the moment cone. We may now verify that these are indeed true for our examples Y p,q and their moment cones. In particular, for our polyhedral cones recall that the {vi } spanned Z3 over Z, and thus π1 (Y p,q ) is trivial, in agreement with the fact that Y p,q ∼ = S 2 × S 3 for all p, q. Moreover, we may now relax the condition that hcf(p, q) = 1. From the Gysin sequence for the U (1) fibration corresponding to ∂/∂γ , as in the appendix of [6], one sees that π1 (Y p,q ) ∼ = Zh , where h ≡ hcf(p, q). Since now hcf(l, p) = hcf(p, q) = h, Lerman’s theorem says that π1 (Y p,q ) ∼ = Zh , in agreement with the Gysin sequence calculation. For the second point in the theorem, since there are four normals, we also learn that π2 (Y p,q ) ∼ = Z, again in perfect agreement with the topology we started with. 5.3. The topology of the vacuum. In this subsection we verify that one can recover the topology of, as well as the action of the isometry group on, Y p,q correctly as the boundary, or horizon, of the linear sigma model (p, p, −l, −k). Of course, this is guaranteed by the Delzant theorem of [20]. Nevertheless, it is interesting to analyse the relation explicitly, since this sheds considerable light on the geometry and topology. Since this “hands on” approach is rather technical, the reader might well omit the remainder of this section. However, we will need the relation (5.9) between vectors on C4 and Y p,q in the next section. This section also constitutes a direct proof of the equivalence of the gauged linear sigma models with the Calabi–Yau cones, without using any theorems. A direct analysis of the topology. The point of this subsection is to show that the K¨ahler quotient C4 //U (1) is topologically the same as C(Y p,q ). This is far from obvious, but is nevertheless guaranteed by the general theorems we have used thus far. At z3 = z4 = 0 we have a finite sized CP1 , of size t/p, where recall that t is the FI parameter. We may thus introduce gauge invariant coordinates z = z1 /z2 , z = z2 /z1 which cover the open subsets U2 , U1 ⊂ CP1 , where Ui = {zi = 0, z3 = z4 = 0} ⊂ CP1 . On the overlap U2 ∩ U1 we have z = 1/z , thus making the Riemann sphere. However, for p > 1 this CP1 is a locus of Zp orbifold singularities in the K¨ahler quotient.
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Indeed, the subgroup Zp ⊂ U (1) stabilises the subspace (z1 , z2 , 0, 0) of C4 . The fact that we have a non-trivial isotropy subgroup means that this will descend to a locus of Zp orbifold singularities in the quotient space. To analyse this singularity, consider, for example, the subspace given by z1 = 0. Using a gauge transformation we may set z2 to be real and positive, which is thus the north pole of the base CP1 = S 2 . The action of Zp on (z3 , z4 ) is generated by (z3 , z4 ) → (z3 ωp−l , z4 ωp−k ), where ωp = e2πi/p generates q −q Zp . Note that this is equivalent to the anti-diagonal action (z3 , z4 ) → (z3 ωp , z4 ωp ). Thus if U (1)A ⊂ SU (2) ⊂ U (2) acts on C2 in the usual way, we have that the generator q ωp of Zp embeds in U (1)A as ωp . Notice that |z2 |2 ≥ t/p and that, for fixed |z2 |2 > t/p the D-term imposes that the coordinates z3 , z4 define an ellipsoid, which topologically is S 3 modulo the Zp action just discussed. Since q is prime to p this is the Lens space L(1, p). At |z2 |2 = t/p we have z3 = z4 = 0 and the Lens space collapses. Thus the subspace z1 = 0 is a copy of an Ap−1 singularity. Performing the quotient of the Lens space “at infinity” by U (1)A then gives a two-sphere, the map being the p th power of the anti-Hopf map16 . Clearly the same picture holds at all points in CP1 , not just at z1 = 0, as SO(3) acts as a symmetry. It follows that we have an Ap−1 fibration over this CP1 , which thus has a boundary which is a Lens space bundle over CP1 . In fact such a bundle structure of the metrics Y p,q was already noted in reference [6]. We may then quotient the boundary by U (1)A to obtain a space Bˆ that will be an S 2 bundle over the base CP1 = S 2 . To see what this bundle is we may introduce coordinates as follows. Suppose z2 = 0, giving the patch U2 on the base CP1 with coordinate z = z1 /z2 . In order to effectively go to the boundary of our space, we may set l|z3 |2 + k|z4 |2 = constant > 0. In particular, we cannot have both z3 and z4 zero. Suppose then that z3 = 0. We may now introduce the additional coordinate x2 = z¯ 4 /z3 z22 on the fibre. This is invariant under both the original U (1) action – the key point being that k + l = 2p – as well as U (1)A under which the fields have charges (0, 0, 1, −1). Similarly over U1 we have coordinate x1 = z¯ 4 /z3 z12 . The union of these two subspaces thus describes the bundle OCP1 (2). However, note that, due to the presence of the z¯ 4 s, the complex structure here is not inherited from the complex structure of C4 we started with. Since we are only interested in topology and group actions, this fact will not be important for the present discussion. Similarly, for z4 = 0 one has coordinates w2 = z3 z22 /¯z4 and w1 = z3 z12 /¯z4 . This describes OCP1 (−2). The intersection of these subspaces results in the gluing of the two C fibres together to create a Riemann sphere S 2 bundle over CP1 = S 2 –for example, x2 = 1/w2 on the overlap with z2 , z3 , z4 = 0. Thus we obtain precisely the same description as the manifold B discussed earlier: Bˆ ∼ = B. Topologically, the manifold Bˆ just described is the same thing as P(O⊕O(−2)) which is the second Hirzebruch surface F2 . However, due to the z¯ 4 s, the complex structure is not that inherited from C4 . Indeed, if one replaces z¯ 4 by z4 in the above coordinates, one precisely gets F2 , as one can see by analysing the linear sigma model for this manifold17 . 1 The fibre S 2 is thus perhaps best described as CP . Moreover, as explained in [6], as a real manifold F2 is actually just a product space S 2 × S 2 , i.e. the bundle is trivial. It remains to compute the twisting of U (1)A over this base B, which as we have just seen is naturally described as an S 2 bundle over S 2 with twist 2. Over the fibre 16 Note the distinction here with the diagonal subgroup U (1) of U (2). Quotienting by this is the Hopf D map, and moreover since this is a normal subgroup the quotient is also the group U (2)/U (1)D ∼ = SO(3). This SO(3) thus acts naturally on the projected space. 17 This is a U (1)2 model on C4 with charges Q = (1, 1, 2, 0), Q = (0, 0, 1, 1). 1 2
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S 2 , sitting at some point on the base S 2 = CP1 , the twisting of the U (1) is p, as is clear from the above discussion since the fibre sphere descended from the Lens space L(1, p) ∼ = S 3 /Zp . We now compute the U (1) twisting over the copies of S 2 at the south and north poles of the fibre S 2 –these are two sections of the S 2 bundle. Call them S1 and S2 , respectively, as in [6]. These are given by z3 = 0, z4 = 0, respectively, which give linear sigma models on C3 with weights (p, p, −k), (p, p, −l), respectively. The boundaries of these two spaces are Lens spaces L(1, k), L(1, l). To see this, note that S 1 /Zp ∼ = S 1 . Thus the boundaries are S 1 bundles over S 2 . The twisting in each case is easily seen to be k and l, respectively. We may now relate this to our earlier discussion. Recall that the canonical generators C1 , C2 of the second homology of S 2 × S 2 are related to the copies S1 , S2 of S 2 at the south and north poles of the fibre S 2 by 2C1 = S1 − S2 , 2C2 = S1 + S2 .
(5.6) (5.7)
We have just seen that the twisting over S1 and −S2 is k and l, respectively. This gives the Chern numbers over C1 and C2 to be (k + l)/2 = p, and (k − l)/2 = q, respectively. We thus precisely reproduce the topology of Y p,q described in Sec. 2. Moreover, the (not quite effectively acting) isometry group of the Sasaki–Einstein metrics is SU (2)×U (1)2 . The K¨ahler quotient above also has this isometry group – this is just the subgroup of U (4) that commutes with the original U (1) action. Relation between Killing vector fields. It is also now interesting to examine the codimension two fixed point sets of the linear sigma model (p, p, −l, −k) directly, and compare with our polyhedral cone for C(Y p,q ). Thus we now set t = 0. The codimension two fixed point sets are easily found: they are at zi = 0, for each i = 1, . . . , 4. Indeed, from our above discussion of the topology of the vacuum, these are precisely cones over the Lens spaces S 3 /Zp , S 3 /Zp , S 3 /Zk , S 3 /Zl , respectively. In terms of Y p,q , these are the submanifolds Fi , i = 1, . . . , 4, respectively. In particular, note that F3 /U (1) ∼ = S1 , F4 /U (1) ∼ = S2 . Thus we see explicitly that the topology of the subspaces {zi = 0} are the same as C(Fi ), respectively. The relation between the Killing vectors is also easy to make explicit. Let us denote ∂/∂θi as the U (1) that rotates the coordinate zi . Thus ∂/∂θi = 0 defines the codimension two submanifolds zi = 0. We find ∂ ∂ ∂ − , = ∂φ ∂θ1 ∂θ2 ∂ ∂ ∂ =l +k , 2p ∂ψ ∂θ3 ∂θ4 ∂ ∂ ∂ + . =− p ∂γ ∂θ3 ∂θ4 2
(5.8)
These require some explanation. We denote the weights of the ∂/∂θi as a row vector for convenience. Thus consider (1, −1, 0, 0). For t > 0 this precisely rotates the subspace z3 = z4 = 0, which is a copy of CP1 of size t/p, with weight two. Hence we identify this U (1) with 2∂/∂φ. Also, by construction, the ∂/∂γ direction is proportional to (0, 0, 1, −1) which, recall, we denoted U (1)A . However, the orbits of the vector (0, 0, −1, 1) actually wind p times around the circle fibre: recall the projection of this U (1) was the pth power of the anti-Hopf map. Hence this is p∂/∂γ . Finally, note
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that ∂/∂ψ rotates the fibre S 2 with weight one and does not act on the base S 2 . This determines the final vector, as one can see by analysing the action on the coordinates x1 , x2 , w1 , w2 introduced above. To make contact with the normal vectors discussed earlier, one must note that the Killing vector given by (p, p, −l, −k) acts trivially on the vacuum, by construction. Thus (p, −p, 0, 0) is equivalent to both (2p, 0, −l, −k) and (0, −2p, l, k). Thus we compute ∂ ∂ + ∂φ ∂ψ ∂ ∂ − + ∂φ ∂ψ ∂ k ∂ − ∂ψ 2 ∂γ ∂ l ∂ + ∂ψ 2 ∂γ
∂ ∂θ1 ∂ = ∂θ2 ∂ = ∂θ3 ∂ = ∂θ4
=
(5.9)
in perfect agreement with our earlier results: the vectors on the left hand side are precisely the Killing vectors (4.9) which fixed codimension two submanifolds of Y p,q . In particular, this means that the polyhedral cones for C(Y p,q ) and the linear sigma model with weights (p, p, −l, −k) are identical, and thus they are completely equivalent as symplectic toric cones, i.e. they are equivariantly symplectomorphic. We have shown this directly in this subsection, without appealing to any theorems. 6. Toric Gorenstein Canonical Singularities In this section we make contact with reference [5] by explaining the relation of the Calabi–Yau gauged linear sigma model (p, p, −l, −k) to so–called toric Gorenstein canonical singularities. The data required to define a toric Gorenstein canonical singularity of complex dimension n is a convex polygon on Rn−1 , all of whose vertices have integer coordinates. Given any such polygon one can reconstruct the toric singularity, as well as all of its toric crepant resolutions, as follows. Let {Vi | i = 1, . . . , d} denote all vectors in Rn−1 with integer coordinates and with the property that they lie within, or on the boundary of, the polygon. Marking these points gives the toric diagram D. Consider now the set of all linear relations among these vectors d
Qia Vi = 0
(6.1)
i=1
with integer coefficients Qia satisfying d
Qia = 0
(6.2)
i=1
for each a = 1, . . . r, where a labels the set of such linear relations. Clearly r = d − n. One now uses the matrix Qia as the charges of a linear sigma model on Cd with gauge group U (1)r . This is essentially a Delzant construction. The K¨ahler quotient
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X = Cd //U (1)r has complex dimension n = d − r. Setting all FI parameters to zero gives the toric singularity. Moreover, by turning on the FI parameters one obtains (partial) resolutions of the singularity – special values of the FI parameters will give rise to more singular spaces than the generic values. By including all the interior points Vi of the polygon, we have ensured that the linear sigma model reproduces all the toric crepant resolutions of the singularity. The sizes of the blow-ups are controlled by the FI parameters. However, this is not usually a very economical way of constructing the singularity – the minimal presentation, meaning the smallest possible d and thus least number of chiral superfields, arises by using only the vertices of the polygon18 . The toric diagram for the Calabi–Yau cone on Y p,q can be obtained as follows. Recall that the image of the moment map for C(Y p,q ) is a four-faceted polyhedral cone with primitive outward pointing normals v1 = [1, 0, 0],
v2 = [1, −2, −l],
v3 = [1, −1, −p],
v4 = [1, −1, 0] . (6.3)
Notice that these vectors lie in the plane at e1 = 1. Indeed, the normals belong to a plane in R2 precisely when the linear sigma model is Calabi–Yau. Thus we may project onto the e1 = 1 plane to obtain vectors [0, 0],
[−2, −l],
[−1, −p],
[−1, 0] .
We now shift the origin by [1, 0] and then make the SL(2; Z) transformation
l − 1 −1 l −1
(6.4)
(6.5)
to obtain vectors V1 = [l − 1, l],
V2 = [1, 0],
V3 = [p, p],
V4 = [0, 0]
(6.6)
respectively. This is a minimal presentation of the singularity. The pictures below display some examples with low values of p. It is interesting to note that the areas of these polygons are equal to p, independently of q. Indeed, for fixed p, varying q just slides the vertex V1 up and down the hypotenuse of the triangle that defines the orbifold C3 /Zp+1 × Zp+1 . Note that for (p, q) = (2, 1) the toric diagram is the same as that for the complex cone (canonical line bundle) over the first del Pezzo surface, as we discuss in detail in the following section.
Fig. 1. Toric diagram of Y 2,1 embedded in the orbifold C3 /Z3 × Z3
18 If these vectors do not span Zn−1 over Z one must in addition quotient the K¨ahler quotient by the finite group Zn−1 /Vi to correctly reproduce the singularity – this follows from our general discussion in Sect. 3.
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Fig. 2. Toric diagrams of Y 3,2 and Y 3,1 embedded in the orbifold C3 /Z4 × Z4
Let us also remark that the number of points inside the polygon is precisely p − 1. Each point corresponds to a normal vector to a plane in R3 . The total number of Fayet– Illiopolous parameters (K¨ahler parameters) is (4 − 3) + p − 1 = p, and by varying these one moves the planes in their normal directions so that they no longer intersect the origin. By assigning generic values one completely resolves the conical singularity. Indeed, these parameters roughly control the size of CP1 s. We thus learn that the Calabi–Yau cone, where all FI parameters are set to zero, has p collapsed two–spheres. Turning on the FI parameter t > 0 in the linear sigma model (p, p, −l, −k) partially resolves the singularity to an Ap−1 singularity fibred over CP1 , as discussed in the last section. Indeed, an Ap−1 singularity can be completely resolved by blowing up (p − 1) two-spheres – the metric is the p-centered Gibbons–Hawking metric. There are precisely (p − 1) FI parameters, giving 1 + (p − 1) = p in total. 7. The Complex Cone over F1 As noted above, the toric diagram we have found for Y 2,1 is the same as that for the complex cone over the first del Pezzo surface. We will refer to the latter as F1 and its complex Calabi–Yau cone as CC (F1 ). Here we elaborate on this point. In particular, it follows that we will inherit a metric on F1 from that on Y 2,1 , and we will write this down explicitly. Of course this metric will not be K¨ahler–Einstein. First we will use the toric data we have to deduce the Killing vector field on Y 2,1 corresponding to the complex cone direction. Adapting the metric to this direction, we shall indeed find a smooth metric on F1 . We label the five vertices of the toric diagram, including the blow-up mode corresponding to the interior point, as V1 = [0, 1],
V2 = [1, 0],
V3 = [2, 2],
V4 = [0, 0],
V5 = [1, 1] .
(7.1)
The last vector V5 is the additional blow-up vertex. A possible basis for the two charge vectors is given by Q1 = (1, 1, 0, −1, −1), Q2 = (0, 0, 1, 1, −2) .
(7.2)
We thus obtain a gauged linear sigma model on C5 with U (1)2 gauge group. Let us for the moment drop the last entry in these vectors. This gives a gauged linear sigma model on C4 with weights ˆ 1 = (1, 1, 0, −1), Q ˆ 2 = (0, 0, 1, 1) . Q
(7.3)
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Let us take each quotient in turn. The first quotient yields C × [OCP1 (−1)], since (1, 1, −1) is precisely OCP1 (−1). The former may also be regarded as OCP1 (0) ⊕ OCP1 (−1). The second row then projectivises this C2 = C ⊕ C bundle. This means one quotients each C2 fibre by the Hopf map C2 \ {0} → CP1 . The resulting space is the first Hirzebruch surface F1 = P(OCP1 (0) ⊕ OCP1 (−1)) .
(7.4)
This is also the same thing as CP2 blown up at a point19 . Indeed, CP2 may be obtained by taking O(1) → CP1 and gluing to its boundary, which is topologically S 3 , a ball in C2 . Blowing up the origin in C2 replaces it by a CP1 , which has local geometry OCP1 (−1). Equivalently one can describe this blowing up process as taking a connected sum with CP2 with reversed orientation: CP2 # − CP2 . We now have two copies of CP1 in the resulting space. In fact it is easy to see that these are two sections of F1 –this is precisely analogous to the topological construction of B. Note however that w2 (F1 ) = 0 and thus this is not a spin manifold. Adding back the fifth entry to the charge vectors (7.3) to give (7.2) then describes the canonical bundle over F1 –the charges sum to zero, meaning that the vacuum X (K¨ahler quotient) is topologically Calabi–Yau, c1 (X) = 0. This identifies the canonical bundle, or complex cone, over F1 . Consider now taking a different linear combination of charge vectors, corresponding to a change of basis for the T2 action. In particular, using an SL(2; Z) transformation we may take Q1 = (2, 2, −1, −3, 0), Q2 = (1, 1, 0, −1, −1) .
(7.5)
The first set of weights of course gives the gauged linear sigma model on C4 given by (2, 2, −1, −3) = (p, p, −l, −k), together with a factor of C. We may now effectively gauge away the second U (1). Indeed, this means ∂ 1 ∂ ∂ ∂ ∂ ∂ − , =− + + = ∂θ5 ∂θ4 ∂θ1 ∂θ2 ∂ψ 2 ∂γ
(7.6)
acting on the linear sigma model (2, 2, −1, −3) on C4 , and Y 2,1 , respectively. Here we have used the relations (5.9). Note that ∂/∂θ5 precisely rotates the complex line fibre over F1 . One can check explicitly that this Killing vector field on Y 2,1 is nowhere-vanishing. Indeed, its norm-squared is computed to be ∂ ∂θ
2 = F (y) ≡ q(y) + w(y) f (y) − 1 2 , 2 9 5
(7.7)
which is strictly positive. Here f (y) =
a − 2y + y 2 6(a − y 2 )
(7.8)
19 In the toric language, there is a nice way to understand this. In fact, it’s straightforward to compute the Delzant polytope for CP2 : this is an isosceles rectangular triangle. A toric blow-up is obtained by simply chopping off a vertex to give a rectangular trapezoid.
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is the function appearing in the local one-form A. Of course in this particular case a and take specific values. One finds √ 1 (1 − 1613 ), 2 √ y1 = 18 (1 − 13),
1 = √ , 2 13 −√5 y2 = 18 (7 − 13) .
a=
(7.9)
Let us summarise the situation. We have found that the metric Y 2,1 is an explicit irregular Sasaki–Einstein metric on the horizon of the complex cone CC (F1 ) over F1 , where the Killing vector field (7.6) rotates the complex cone direction. Crucially this is not the Reeb vector, whose generic orbits in fact don’t close. The quotient of the metric (2.4) by the U (1) action generated by (7.6) should be a metric on F1 . We will now explicitly compute this metric and verify that it is indeed a smooth metric on F1 . In order to perform the U (1) quotient of Y 2,1 , it is useful to first rewrite the metric adapted to the Killing vector field ∂/∂θ5 . Thus, let us change coordinates: ψ = θ5 ,
γ = −/2 − θ5 /2 .
(7.10)
It is then straightforward to compute the following expression for the metric : ds 2 =
1−y 1 w(y)q(y) 2 (dθ 2 + sin2 θ dφ 2 ) + dy 2 + (d + cos θ dφ)2 6 w(y)q(y) 36F (y) +F (y) [dθ5 − C]2 ,
(7.11)
where we have defined
1 q(y) C= w(y)(f (y) − 2 ) 2 d + + w(y)f (y)(f (y) − 2 ) cos θ dφ . F (y) 9 (7.12) The quotient by ∂/∂θ5 now simply gives the metric in the first line of (7.11), which again looks like a bundle over a base two-sphere. Let us now analyze regularity of this metric. First, notice that all the functions are positive semi-definite. So, as usual, one has to worry only about the smoothness conditions where the function q(y) vanishes, and then check that the resulting periodicities give a well-defined bundle-metric. Near such a zero yi , the “fibre metric”, i.e. the metric at fixed θ, φ, takes the form ds 2 (fibre) ≈
1 |yi ||y − yi | 2 2 dy 2 + d . 12|yi ||y − yi | 3F (yi )
(7.13)
Now, crucially, the following relations are true for any (p, q): F (y1 ) = (k − 1)2 2 y12 ,
F (y2 ) = (l + 1)2 2 y22 .
(7.14)
Introducing R = 2|y − yi |1/2 we find that for (k, l) = (3, 1)–and only for these values– the metric approaches
1 1 2 2 2 2 ds (fibre) ≈ (7.15) dR + R d 12|yi | 4 near the two zeros. We therefore obtain a smooth metric on R2 in this neighbourhood if and only if has period 4π. Indeed, one can see that this is the induced period for
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from the metric on Y 2,1 by examining the coordinate transformation (7.10): since ψ, γ and θ5 all have period 2π one can calculate the period of from the Jabobian of the coordinate transformation (7.10), which is −1/2. This indeed means that ∼ + 4π and moreover with this period we have that for fixed y, y1 < y < y2 , the resulting space is a squashed S 3 . These are then the generic orbits under the action of the isometry group U (2) on this manifold. We thus obtain an S 2 bundle over S 2 with twist one, which is topologically F1 , just as expected. Let us now label the two sections of F1 at y = y1 , y = y2 as H , E respectively. These are the hyperplane class and exceptional divisor of del Pezzo one, respectively. It is a simple exercise to compute the Chern numbers of the U (1) principle bundle, with coordinate θ5 , over these: dC dC = 3, = 1, (7.16) H 2π E 2π where, as ever, we have to use the cubic (2.7) and, in this particular case, k = 3, l = 1. Equations (7.16) give precisely the Chern numbers required so that the complex cone (or complex line bundle) defined by the U (1) bundle associated to θ5 is indeed Calabi– Yau. To see this, notice that the normal bundles of the two CP1 s corresponding to H, E inside F1 are topologically OCP1 (1) and OCP1 (−1), respectively, as is clear from our discussion of F1 above. Thus c1 (F1 ) restricted to the two cycles gives 1 + 2 = 3 and −1 + 2 = 1, respectively, where 2 = c1 (T S 2 ). The Chern numbers above for −dC precisely cancel these in the total space of the associated complex line bundle, thus giving a Calabi–Yau manifold. As shown at the end of Sect. 2.2, the cones over the U (1) bundles over H and E (which are the submanifolds y = y1 , y = y2 ) are divisors in the Calabi–Yau cone. Equivalently, the complex cones over the submanifolds H and E are divisors. Indeed, we already noted above the normal bundles to these submanifolds inside F1 , which translate into self-intersection numbers H · H = 1, E · E = −1. One can check that the metric on F1 is not Einstein. Thus, in particular it is not diffeomorphic to the Page metric on F1 [41], although it is rather similar in form. 8. New Non-Trivial AdS/CFT Predictions In this final section we discuss features of the gauge theory duals of the Sasaki–Einstein manifolds Y p,q , focusing in particular on Y 2,1 since a candidate dual is already known. In particular we may compare our geometrical results to the a-maximisation calculation20 presented in [14]. We find complete agreement with this field theory calculation, both for the central charge and for the SU (2)F singlet baryons of the theory. Let us first remark that, given a toric Gorenstein canonical singularity, an algorithm for constructing21 a quiver gauge theory that has the singularity as its Higgs branch has been developed in [11, 42] and subsequent works by these authors. This relies on the fact that any such singularity may be obtained by partial resolution of the orbifold C3 /Zp+1 × Zp+1 , and the field theory for the latter is known. In practise the algorithm requires a computer, even for relatively small p. However, the simple analytic expressions found in this paper suggest that all theories can be treated simultaneously. Indeed 20 Note that in [14] the central charge of the dP quiver gauge theory is also calculated, and found to 2 be quadratic irrational. 21 Note that, in earlier work, extending that of [3], the quiver gauge theories associated to some toric singularities were worked out in [43–45] without using these algorithms.
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2
4
3
83
Fig. 3. Quiver diagram associated to the complex cone over dP1
it is tempting to speculate that some members of the family could be related by deformations or connected via RG-flows. In particular, we can anticipate that, at fixed p, the parameter q will govern the matter content and superpotential of an SU (N )2p quiver. Recall also that at fixed p, the central charge a is a monotonic function of q which is bounded between the values corresponding to T 1,1 /Zp and (S 5 /Z2 )/Zp : a(T 1,1 /Zp ) < a(Y p,q ) < a(S 5 /Z2 × Zp ) ,
(8.1)
suggesting that the different q-theories might all be related to the same “parent” orbifold model. However, we will not pursue this direction any further in the present paper. Instead, we focus on Y 2,1 , where the dual quiver theory is already known. This instance already captures many of the essentially new features of these AdS/CFT duals. A quiver gauge theory for dP1 ∼ = F1 was obtained22 in [11] and is presented in Fig. 3. Let us briefly recall the notation of these diagrams. The nodes of the diagram represent different gauge group factors U (N ). Thus the gauge group for the theory is U (N )4 . An arrow from node i to node j represents a bifundamental field in the representation N⊗N, where the first factor denotes the anti-fundamental representation of the i th gauge group, and the second factor denotes the fundamental representation of the j th gauge group. We denote these fields as Xij . Thus the quiver diagram encodes the field content of the theory. One must also specify the superpotential. This is given by [42]: α β α β 3 α β X41 X13 − αβ tr X34 X23 X42 + αβ tr X12 X34 X41 X23 , (8.2) W = αβ tr X34 where αβ ∈ {±1} and α, β ∈ {1, 2} are indices of the non–abelian flavor symmetry group SU (2)F . Note that each term comes from a closed loop in the quiver. This allows one to construct gauge–invariant monomials, which may then appear in the superpotential. One is then particularly interested in the Higgs branch of such a theory. This arises by considering U (N) → U (1)N for each gauge group factor. One effectively considers the case N = 1 so that the gauge theory is an abelian theory–the case N > 1 will simply be given by N copies of the N = 1 case. The fields Xij have various charges under the U (1)4 gauge group. Setting the D-terms of the gauge theory to zero and dividing by the gauge group is, as we have discussed already in a different context in this paper, a K¨ahler 22
In this section we denote the first del Pezzo surface by dP1 .
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quotient construction, and the result is a toric variety (an overall U (1) decouples and is physically the centre of mass U (1) of the D3-branes). However, to get the vacuum of the theory one must also set the F -terms to zero, which means extremising the superpotential: dW =0. This gives a system of relations among the linear sigma model fields, which define hypersurfaces in the toric variety–the intersection of these define the Higgs branch of the theory, which is part of the moduli space of vacua. One can also get to this result by computing all invariant monomials in the fields, and then finding all relations among them, including those relations given by dW = 0. The slightly non-trivial fact is that this is indeed the complex cone over the first del Pezzo surface. We will not review this here, but instead refer the reader to the literature for details (see e.g. [23]). If the quiver gauge theory above is interpreted as living on a D3-brane, then this moduli space should be the geometry seen by the brane. For N > 1 one has N D3-branes in their Higgs phase, which is why one obtains N copies of the above moduli space. Let us now recall the flavour and R-symmetries of the theory. The superpotential above is manifestly invariant under the non-abelian flavour group SU (2)F , for which the α, β indices form a doublet. Crucially, there is also a non-anomalous U (1) × U (1) abelian flavour symmetry which is preserved in the IR. Taking this into account, the a-maximisation calculation applied to this theory [14] then gives the exact R-charges in the IR. For the sake of clarity, these are listed23 in Table 1. Recall that, as proposed in [9], the R-symmetry mixes with the abelian flavour symmetries maximising, among all such admissible R-symmetries, a certain combination of ‘t Hooft anomalies. The value of this combination of anomalies at the critical point is the exact central charge of the theory in the infra-red, and is given by the formula 3 (8.3) 3TrR 3 − TrR . a= 32 Substituting the values for the R-charges from Table 1 into (8.3), and comparing with (1.2) one finds a corresponding volume √ 13 13 + 46 3 π (8.4) 12 · 27 which precisely agrees with the volume of Y 2,1 (2.13) on setting p = 2, q = 1. Table 1. Exact R-charges computed from a-maximisation [14] Xij α X34 3 X34 α X41 α X23
X12 X13 X42
Rexact √ 13) √ −3 + 13 √ 4 (4 − 13) 3 √ 4 (4 − 13) 3 √ 1 3 (−17 + 5 13) √ −3 + 13 √ −3 + 13 1 (−1 + 3
Let us finally consider the baryons of the gauge theory. Recall that baryonic operators B of the gauge theory are dual to D3-branes wrapping supersymmetric cycles in the 23
We thank the authors of [14] for communicating the results of their calculation prior to publication.
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geometry. Their R-charges are related to the volumes of these supersymmetric cycles according to the general formula24 [25]
π 2 · vol() . (8.5) R[B] = · 3 2 vol(Y ) Recall we have shown in Sect. 2 that for each manifold Y p,q there are two supersymmetric 3-cycles, which are topologically Lens spaces 1 = S 3 /Zp+q and 2 = S 3 /Zp−q . We therefore expect that in each case there will be two types of baryonic operators B1 , B2 associated to them. Substituting for the volume (2.13) we can write down the general formula for the R-charges of the corresponding baryons in the Y p,q theory. These are given by the unlikely formulae: 1 2 2 2 2 R[B1 ] = 2 −4p + 2pq + 3q + (2p − q) 4p − 3q , 3q 1 R[B2 ] = 2 −4p 2 − 2pq + 3q 2 + (2p + q) 4p 2 − 3q 2 . (8.6) 3q Note that they are interchanged by changing the sign of q. Setting p = 2, q = 1 the formulae give √ √ R[B2 ] = 13 (−17 + 5 13) . (8.7) R[B1 ] = −3 + 13, These agree precisely with two of the four different R-charges listed in Table 1. Acknowledgements. We would like to thank M. Bertolini, F. Bigazzi, A. Hanany, K. Intriligator, E. Lerman, C. Vafa, D. Waldram, B. Wecht, and S.–T. Yau for discussions and e-mail correspondence. In particular we would like to thank E. Lerman for comments on a draft version of this paper. We are also grateful to the authors of [14] for earlier collaboration on related material, and especially for communicating their a-maximisation calculation. DM would like to thank the 2004 Simons Workshop on Mathematics and Physics, for hospitality at initial stages of this work. Part of this work was carried out whilst both authors were postdoctoral fellows at Imperial College, London. In particular DM was funded by a Marie Curie Individual Fellowship under contract number HPMF-CT-2002-01539, while JFS was supported by an EPSRC mathematics fellowship. At present JFS is supported by NSF grants DMS–0244464, DMS–0074329 and DMS–9803347.
A. The Conifold In this appendix we compute the moment cone, gauged linear sigma model and toric diagram for the conifold, C(T 1,1 ). Of course, many of these results are well-known in the physics literature–we include the discussion only as a simple illustration of the systematic techniques used in this paper, in the context of an example familiar to many physicists. The homogeneous Sasaki–Einstein metric on S 2 × S 3 is usually referred to as T 1,1 . The metric is particularly simple [47]: 1 1 ds 2 = (dθ12 + sin2 θ1 dφ12 + dθ22 +sin2 θ2 dφ22 ) + (dψ + cos θ1 dφ1 +cos θ2 dφ2 )2 . 6 9 (A.1) 24
We suppress the overall factors of N.
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Here θi , φi , i = 1, 2, are usual polar and axial coordinates on two round two-spheres, and ψ is a coordinate on a principle U (1) bundle over S 2 × S 2 . Here ψ has period 4π so that the Chern numbers over the two-spheres are both equal to one25 . In particular, 3∂/∂ψ is the Reeb vector so that this is a regular Sasaki–Einstein manifold–the base K¨ahler–Einstein manifold is just CP1 × CP1 . The symplectic form on the metric cone is ω=
1 2 r (sin θ1 dθ1 ∧ dφ1 + sin θ2 dθ2 ∧ dφ2 ) 6 1 − rdr ∧ (dψ + cos θ1 dφ1 + cos θ2 dφ2 ) . 3
(A.2)
Clearly we have three commuting Hamiltonian U (1)s generated by ∂/∂φi , i = 1, 2, and ∂/∂ψ. As in the main text, one must be careful to ensure that one picks a basis for an effectively acting T3 action when computing the moment map. If one fixes θ1 , φ1 on the first two-sphere, one obtains a copy of S 3 , written as a principle U (1) bundle over the second two-sphere. The effectively acting isometry group on this squashed S 3 is U (2), as discussed in the main text. Defining 2ν = ψ, so that ν has canonical period 2π , one can therefore take the following basis for the T3 action: ∂ + ∂φ1 ∂ + e2 = ∂φ2 ∂ . e3 = ∂ν e1 =
1 ∂ , 2 ∂ν 1 ∂ , 2 ∂ν
(A.3)
The corresponding moment map, homogeneous under rescaling of the cone, is now easily computed to be
1 2 1 1 µ = (A.4) r (cos θ1 + 1), r 2 (cos θ2 + 1), r 2 . 6 6 3 The image of the moment map µ : C(T 1,1 ) → R3 is a convex rational polyhedral cone generated by the four edge vectors: µ(N N ) = 13 (1, 1, 1), µ(N S) = 13 (1, 0, 1), µ(SN ) = 13 (0, 1, 1),
(A.5)
µ(SS) = 13 (0, 0, 1) . That is, the subspaces over which a T2 collapses are precisely the four subspaces N N = {θ1 = 0, θ2 = 0}, N S = {θ1 = 0, θ2 = π }, SN = {θ1 = π, θ2 = 0}, SS = {θ1 = π, θ2 = π }-these are all copies of the fibre circle over the corresponding point on the 25 One may also set ψ to have period 2π yielding T 1,1 /Z which is also a Sasaki–Einstein manifold. 2 In fact, this is the horizon manifold of the complex cone over F0 CP1 × CP1 . Note that one must be careful to ensure that the Killing spinors are well-defined on making such identifications.
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base S 2 × S 2 . The outward pointing primitive normal vectors to the cone are computed to be v1 = [1, 0, −1],
v2 = [0, 1, −1],
v3 = [0, −1, 0],
v4 = [−1, 0, 0] .
(A.6)
Notice that these indeed form a good cone, as defined in the main text. Also notice that the vectors {vi } span Z3 over Z. Lerman’s theorem then states that the base of the metric cone is simply-connected, which is of course correct. Moreover, the fact that there are four facets means that π2 (T 1,1 ) ∼ = Z, again correct. We may now apply the Delzant theorem. The kernel is trivially calculated to be (1, −1, −1, 1). Thus the theorem gives a U (1) gauged linear sigma model on C4 with charges (1, −1, −1, 1)–this is of course well-known to give the conifold. Turning on the FI parameter t > 0, t < 0 gives the two small resolutions of the conifold, related by the flop transition. We now apply the SL(3; Z) transformation 1 1 2 0 −1 −1 (A.7) 0 0 −1 to the torus T3 of symmetries. The normal vectors now read v1 = [−1, 1, 1],
v2 = [−1, 0, 1],
v3 = [−1, 1, 0],
v4 = [−1, 0, 0] .
(A.8)
These all lie in the plane at e1 = −1. Dropping this gives vectors in R2 : V1 = [1, 1],
V2 = [0, 1],
V3 = [1, 0],
V4 = [0, 0] .
(A.9)
The toric diagram may thus be embedded in the orbifold C3 /Z2 × Z2 and is presented below. We may also analyse the topology of the K¨ahler quotient directly, as in the main text. The D-term constraint reads |z1 |2 + |z2 |2 − |z3 |2 − |z4 |2 = t .
(A.10)
Setting t = 0 one obtains a singular space–the conifold. Defining gauge invariant coordinates u = z1 z3 , x = z1 z4 , y = z2 z3 , v = z2 z4 we have precisely one relation uv = xy in C4 , which is thus an equivalent definition of the conifold. At z3 = z4 = 0 we have a copy of CP1 = S 2 , of size t. On a patch in which z2 = 0 we may introduce a gauge invariant complex coordinate z = z1 /z2 . This patch covers a neighbourhood of the south pole at z1 = 0. Similarly the coordinate z = z2 /z1 covers a neighbourhood of the north pole at z2 = 0. Over the intersection of the patches we have the relation z = 1/z, thus making the Riemann sphere. Let us now turn to the remaining coordinates. Consider the subspace in which z2 = 0 and introduce gauge
Fig. 4. Toric diagram of the conifold embedded in the orbifold C3 /Z2 × Z2
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D. Martelli, J. Sparks
invariant coordinates x2 = z3 z2 , y2 = z4 z2 . Thus, over the open set U2 = {z2 = 0, z3 = z4 = 0} ⊂ CP1 , our subspace looks like a trivial rank two bundle C2 × U2 . Similarly, over U1 = {z1 = 0, z3 = z4 = 0} ⊂ CP1 we also have C2 × U1 , where the fibre is coordinatised by x1 = z3 z1 , y1 = z4 z1 . On the overlap U1 ∩ U2 we have the relation x1 = x2 (z1 /z2 ), y1 = y2 (z1 /z2 ). By definition, this gluing gives the bundle OCP1 (−1) ⊕ OCP1 (−1), which is the resolved conifold. The boundary, or horizon, of this manifold is an S 3 bundle over CP1 = S 2 , since S 3 is the boundary of C2 . There are various ways of seeing the topology of the horizon. One way is to projectivise the original bundle. Recall that to projectivise a rank two complex vector bundle, with transition functions in U (2), means that one replaces each C2 fibre with CP1 , and glues the fibres together across overlaps using the induced transition functions, which lie in U (2)/U (1)D ∼ = SO(3). Here SO(3) acts on the CP1 = S 2 fibre in the usual way. Since the transition functions of OCP1 (−1) ⊕ OCP1 (−1) are diagonal, the projectivisation is just the product CP1 × CP1 . The U (1) factor we projected out has unit winding over the fibre S 2 , since S 3 → S 2 is the Hopf map which has Chern number 1. The winding is also 1 over the base since we began with the sum of two copies of OCP1 (−1). Thus we see explicitly the topology of T 1,1 as the horizon manifold. References 1. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] 2. Kehagias, A.: New type IIB vacua and their F-theory interpretation. Phys. Lett. B 435, 337 (1998) 3. Klebanov, I.R., Witten, E.: Superconformal field theory on threebranes at a Calabi-Yau singularity. Nucl. Phys. B 536, 199 (1998) 4. Acharya, B.S., Figueroa-O’Farrill, J.M., Hull, C.M., Spence, B.: Branes at conical singularities and holography. Adv. Theor. Math. Phys. 2, 1249 (1999) 5. Morrison, D.R., Plesser, M.R.: Non-spherical horizons. I. Adv. Theor. Math. Phys. 3, 1 (1999) 6. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki-Einstein metrics on S 2 ×S 3 . Adv. Theor. Math. Phys. 8, 711 (2004) 7. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Supersymmetric AdS5 solutions of M-theory. Class. Quant. Grav. 21, 4335 (2004) 8. Cheeger, J., Tian, G.: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118(3), 493–571 (1994) 9. Intriligator, K., Wecht, B.: The exact superconformal R-symmetry maximizes a. Nucl. Phys. B 667, 183 (2003) 10. Gubser, S.S.: Einstein manifolds and conformal field theories. Phys. Rev. D 59 (1999) 025006 11. Feng, B., Hanany, A., He, Y.H.: D-brane gauge theories from toric singularities and toric duality. Nucl. Phys. B 595, 165 (2001) 12. Herzog, C.P., Walcher, J.: Dibaryons from exceptional collections. JHEP 0309, 060 (2003) 13. Herzog, C.P.: Exceptional collections and del Pezzo gauge theories. JHEP 0404, 069 (2004) 14. Bertolini, M., Bigazzi, F., Cotrone, A.: New checks and subtleties for AdS/CFT and a-maximization. JHEP 0412, 024 (2004) 15. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: A new infinite class of Sasaki-Einstein manifolds. http://arXiv:org/list/hep-th/0403038, 2004 16. Gauntlett, J.P., Martelli, D., Sparks, J. Waldram, D.: Supersymmetric AdS Backgrounds in String and M-theory. http://arXiv:org/list/hep-th/0411194, 2004 17. Chen, W., Lu, H., Pope, C.N., Vazquez-Poritz, J.F.: A Note on Einstein–Sasaki Metrics in D ≥ 7. http://arXiv.org/list/hep-th/0411218, 2004 18. Tian, G.: On K¨ahler–Einstein metrics on certain K¨ahler manifolds with c1 (M) > 0. Invent. Math. 89, 225–246 (1987) 19. Tian, G., Yau, S.T.: On K¨ahler–Einstein metrics on complex surfaces with C1 > 0. Commun. Math. Phys. 112, 175–203 (1987) 20. Lerman, E.: Contact toric manifolds. J. Symplectic Geom. 1(4), 785–828 (2003) 21. Delzant, T.: Hamiltoniens periodiques et images convexes de l’application moment. Bull. Soc. Math. France 116(3), 315–339 (1988)
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22. Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159 (1993) 23. Beasley, C., Greene, B.R., Lazaroiu, C.I., Plesser, M.R.: D3-branes on partial resolutions of abelian quotient singularities of Calabi-Yau threefolds. Nucl. Phys. B 566, 599 (2000) 24. Witten, E.: Baryons and branes in anti de Sitter space. JHEP 9807, 006 (1998) 25. Berenstein, D., Herzog, C.P., Klebanov, I.R.: Baryon Spectra and AdS/CFT Correspondence. JHEP 0206, 047 (2002) 26. Smale, S.: On the structure of 5-manifolds. Ann. Math. 75, 38–46 (1962) 27. Friedrich, Th., Kath, I.: Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator. J. Differ. Geom. 29, 263–279 (1989) 28. Matsushima, Y.: Sur la structure du groupe d’hom´eomorphismes analytiques d’une certaine vari´et´e kaehl´erienne. Nagoya Math. J. 11, 145–150 (1957) 29. Boyer, C.P., Galicki, K.: New Einstein metrics in dimension five. J. Differ. Geom. 57(3), 443–463 (2001) 30. Boyer, C.P., Galicki, K., Nakamaye, M.: On the Geometry of Sasakian–Einstein 5-Manifolds. Math. Ann. 325(3), 485–524 (2003) 31. Boyer, C.P., Galicki, K., Nakamaye, M.: Sasakian–Einstein structures on 9#(S 2 × S 3 ). Trans. Amer. Math. Soc. 354(8), 2983–2996 (2002) 32. Boyer, C.P., Galicki, K.: New Einstein metrics on 8#(S 2 × S 3 ). Differential Geom. Appl. 19(2), 245–251 (2003) 33. Wang, M.Y., Ziller, W.: Einstein metrics on principal torus bundles. J. Diff. Geom. 31, 215 (1990) 34. Boyer, C.P., Galicki, K.: 3-Sasakian Manifolds. Surveys Diff. Geom. 7, 123–184 (1999) 35. Falcao de Moraes, S., Tomei, C.: Moment maps on symplectic cones. Pacif. J. Math. 181(2), 357–375 (1997) 36. Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14, 1–15 (1982) 37. Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67, 491– 513 (1982) 38. Lerman, E., Tolman, S.: Hamiltonian torus actions on symplectic orbifolds and toric varieties. http://arXiv.org/list/dg-ga/9511008, 1995 39. Boyer, C.P., Galicki, K.: A Note on Toric Contact Geometry. J. Geom. and Phys. 35, 288–298 (2000) 40. Lerman, E.: Homotopy Groups of K-Contact Toric Manifolds. Trans. Amer. Math. Soc. 356(10), 4075–4083 (2004) 41. Page, D.N.: A compact rotating gravitational instanton. Phys. Lett. 79B(3), 235–238 (1978) 42. Feng, B., Franco, S., Hanany, A., He, Y.H.: Symmetries of toric duality. JHEP 0212, 076 (2002) 43. Dall’Agata, G.: N = 2 conformal field theories from M2-branes at conifold singularities. Phys. Lett. B 460, 79 (1999) 44. Fabbri, D., Fre’, P., Gualtieri, L., Reina, C., Tomasiello, A., Zaffaroni, A., Zampa, A.: 3D superconformal theories from Sasakian seven-manifolds: New nontrivial evidences for AdS(4)/CFT(3). Nucl. Phys. B 577, 547 (2000) 45. Ceresole, A., Dall’Agata, G., D’Auria, R., Ferrara, S.: M-theory on the Stiefel manifold and 3d conformal field theories. JHEP 0003, 011 (2000) 46. Intriligator, K., Wecht, B.: Baryon charges in 4D superconformal field theories and their AdS duals. Commun. Math. Phys. 245, 407 (2004) 47. Candelas, P., de la Ossa, X.C.: Comments On Conifolds. Nucl. Phys. B 342, 246 (1990) Communicated by G.W. Gibbons
Commun. Math. Phys. 262, 91–115 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1454-y
Communications in
Mathematical Physics
The Threshold Effects for the Two-Particle Hamiltonians on Lattices S. Albeverio1,2,3 , S.N. Lakaev4,6 , K.A. Makarov5 , Z.I. Muminov6 1 2 3 4
Institut f¨ur Angewandte Mathematik, Universit¨at Bonn, Germany. E-mail: [email protected] SFB 611, Bonn, BiBoS, Bielefeld - Bonn, Germany CERFIM, Locarno and USI, Switzerland Samarkand Division of Academy of Sciences of Uzbekistan, Uzbekistan. E-mail:
[email protected]. uni-bonn.de 5 Department of Mathematics, University of Missouri, Columbia, MO, USA. E-mail:
[email protected] 6 Samarkand State University, Samarkand, Uzbekistan. E-mail:
[email protected]
Received: 30 December 2004/ Accepted: 15 June 2005 Published online: 24 November 2005 – © Springer-Verlag 2005
Abstract: For a wide class of two-body energy operators h(k) on the d-dimensional lattice Zd , d ≥ 3, k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k = 0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups. 1. Introduction The main goal of the present paper is to give a thorough mathematical treatment of the spectral properties for the two-particle lattice Hamiltonians in dimensions d ≥ 3 with emphasis on new threshold phenomena that are not present in the continuous case (see, e.g., [4, 8, 13–15, 17] for relevant discussions and [9, 11, 16, 29] for the general study of the low-lying excitation spectrum for quantum systems on lattices). The kinematics of quantum quasi-particles on lattices, even in the two-particle sector, is rather exotic. For instance, due to the fact that the discrete analogue of the Laplacian or its generalizations (see (2.1) and (4.1)) are not rotationally invariant, the Hamiltonian of a system does not separate into two parts, one relating to the center-of-mass motion and the other one to the internal degrees of freedom. In particular, such a handy characteristic of inertia as mass is not available. Moreover, such a natural local substituter as the effective mass-tensor (of a ground state) depends on the quasi-momentum of the system and, in addition, it is only semi-additive (with respect to the partial order on the set of positive definite matrices). This is the so-called excess mass phenomenon for lattice systems (see, e.g., [15 and 17]): the effective mass of the bound state of an N -particle system is greater than (but, in general, not equal to) the sum of the effective masses of the constituent quasi-particles.
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The two-particle problem on lattices, in contrast to the continuous case where the usual split-off of the center of mass can be performed, can be reduced to an effective one-particle problem by using the Gelfand transform instead: the underlying Hilbert space 2 ((Zd )2 ) is decomposed as a direct von Neumann integral associated with the representation of the discrete group Zd by shift operators on the lattice and then, the total two-body Hamiltonian appears to be decomposable as well. In contrast to the continuous case, the corresponding fiber Hamiltonians h(k) associated with the direct decomposition depend parametrically on the internal binding k, the quasi-momentum, which ranges over a cell of the dual lattice. As a consequence, due to the loss of the spherical symmetry of the problem, the spectra of the family h(k) turn out to be rather sensitive to the variation of the quasi-momentum k. We recall that in the case of continuous Schr¨odinger operators in R3 one observes the emission of negative bound states from the continuous spectrum at so-called critical potential strength (see, e.g., [1, 14, 19, 26]). This phenomenon is closely related to the existence of generalized eigenfunctions, which are solutions of the Schr¨odinger equation with zero energy decreasing at infinity, but are not square integrable. These solutions are usually called zero-energy resonance functions and, in this case, the Hamiltonian is called a critical one and the Schr¨odinger operator is said to have a zero-energy resonance (virtual level). The appearance of negative bound states for critical (non-negative) Schr¨odinger operators under infinitesimally small negative perturbations is especially remarkable: it is the presence of zero-energy resonances in at least two of the twoparticle subsystems that leads to the existence of infinitely many bound states for the corresponding three-body system, the Efimov effect (see, e.g., [2, 13, 18, 23–25 , and 27]). It turns out that in the two-body lattice case there exists an extra mechanism for the bound state(s) to emerge from the threshold of the critical Hamiltonians which has nothing to do with additional (effectively negative) perturbations of the potential term. The role of the latter is rather played by the adequate change of the kinetic term which is due to the nontrivial dependence of the fiber Hamiltonians h(k) on the quasi-momentum k and is related to the excess mass phenomenon for lattice systems mentioned above. The main result of the paper is the (variational) proof of existence of the discrete spectrum below the bottom of the essential spectrum of the fiber Hamiltonians h(k) for all non-zero values of the quasi-momentum 0 = k ∈ Td , provided that the Hamiltonian h(0) has either a virtual level (in dimenstions three and four) or a threshold eigenvalue (in all dimensions d ≥ 3) (see Theorem 2). Apart from some technical smoothness assumptions upon the dispersion relation of normal modes εα (p), characterizing the free particles α = 1, 2, and as well as on smoothness assumptions (in the momentum representation) on the two-particle interactions (Hypothesis 1) the only additional assumption made (Hypothesis 2) is that the one-particle free Hamiltonians (in the coordinate representation) generate positivity preserving semi-groups exp(−t hˆ 0α ), t > 0, α = 1, 2. We remark that this property is automatically fulfilled for the standard Laplacian (discrete or continuous). The paper is organized as follows. In Sect. 2 we formulate the main hypotheses on the one-particle lattice systems in all dimensions d ≥ 1 and prove the basic inequality (see Lemma 1 below) for the dispersion relations that are conditionally negative definite. In Sect. 3 we introduce the concept of a virtual level for the lattice one-particle Hamiltonians in dimensions d = 3, 4, and develop the necessary background for our further considerations. In Sect. 4 we describe the two-particle Hamiltonians in both the coordinate and the momentum representation,
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introduce the two-particle quasi-momentum, and decompose the energy operator into the von Neumann direct integral of the fiber Hamiltonians h(k), thus providing the reduction to the effective one-particle case. In Sect. 5 we obtain efficient bounds on the location of the discrete spectrum for the two-particle fiber Hamiltonians in dimensions d ≥ 3 and prove the main result of this paper, Theorem 2, in the case where h(0) has either a threshold eigenvalue (d ≥ 3) or virtual level (d = 3, 4) at the bottom of its essential spectrum. In Appendix A, for readers’ convenience, we give a proof of Proposition 1 which is a “lattice” analogue of a result due to Yafaev [28] in the continuous case. In Appendix B we construct an explicit example of a one-particle discrete Schr¨odinger operator on the three-dimensional lattice Z3 that possesses both a virtual level and threshold eigenvalue at the bottom of its essential spectrum (cf., e.g., [1, 3 , and 12] for related discussions in the case of continuous Schr¨odinger operators). 2. The One-Particle Hamiltonian 2.1. Dispersion relations. The free Hamiltonian hˆ 0 of a quantum particle on the ddimensional lattice Zd , d ≥ 1, is usually associated with the following self-adjoint (bounded) multidimensional Toeplitz-type operator on the Hilbert space 2 (Zd ) (see, e.g., [15]): ˆ + s), ψˆ ∈ 2 (Zd ). ˆ (hˆ 0 ψ)(x) εˆ (s)ψ(x (2.1) = Here the series
s∈Zd
s∈Zd
εˆ (s) is assumed to be absolutely convergent, that is, {ˆε (s)}s∈Zd ∈ 1 (Zd ).
We also assume that the “self-adjointness” property is fulfilled εˆ (s) = εˆ (−s),
s ∈ Zd .
In the physical literature, the symbol of the Toeplitz operator hˆ 0 given by the Fourier series ε(p) = εˆ (s)ei(p,s) , p ∈ Td , (2.2) s∈Zd
being a real valued-function on Td , is called the dispersion relations of normal modes associated with the free particle in question (note that the Fourier coefficients of the funcd tion ε(p) differ from the coefficients εˆ (s) in (2.2) by the factor (2π ) 2 ). The one-particle free Hamiltonian is required to be of the form hˆ 0 = ε(−i∇), where ∇ is the generator of the infinitesimal translations. Under the mild assumption that vˆ ∈ ∞ (Zd ),
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ˆ where vˆ = {v(s)} ˆ s∈Zd is a sequence of reals, the one-particle Hamiltonian h, hˆ = hˆ 0 + v, ˆ describing the quantum particle moving in the potential field v, ˆ is a bounded self-adjoint operator on the Hilbert space 2 (Zd ). The one-particle Hamiltonian h in the momentum representation is introduced as ˆ h = F−1 hF, where F stands for the standard Fourier transform F : L2 (Td ) −→ 2 (Zd ), and Td denotes the three-dimensional torus, the cube (−π, π ]d with appropriately identified sides. Throughout the paper the torus Td will always be considered as an abelian group with respect to the addition and multiplication by real numbers regarded as operations on Rd modulo (2π Z)d . 2.2. Hamiltonians generating the positivity preserving semi-groups. The following important subclass of the one-particle systems is of certain interest (see, e.g., [6]). It is introduced by the additional requirement that the dispersion relation ε(p) is a realvalued continuous conditionally negative definite function and hence (i) ε is an even function, (ii) ε(p) has a minimum at p = 0. Recall (see, e.g., [21]) that a complex-valued bounded function ε : Td −→ C is called conditionally negative definite if ε(p) = ε(−p) and n
ε(pi − pj )zi z¯ j ≤ 0
(2.3)
i,j =1 d n for nany n ∈ N, for all p1 , p2 , .., pn ∈ T and all z = (z1 , z2 , ..., zn ) ∈ C satisfying i=1 zi = 0. It is known that in this case the dispersion relation ε(p) admits the (L´evy-Khinchin) representation (see, e.g., [5]) (ei(p,s) − 1)ˆε (s), p ∈ Td , ε(p) = ε(0) + s∈Zd \{0}
which is equivalent to the requirement that the Fourier coefficients εˆ (s) with s = 0 are non-positive, that is,
εˆ (s) ≤ 0,
s = 0,
s∈Zd \{0} εˆ (s) converges absolutely. In turn, this is also equivalent to the that the lattice Hamiltonian hˆ = hˆ 0 + vˆ generates the positivity preserving
and the series
requirement ˆ semi-group e−t h , t > 0, on 2 (Zd ) (see, e.g., [21] Ch. XIII). Following [6] we call the free Hamiltonians hˆ 0 = ε(−i∇) generating the positivity preserving semi-groups the generalized Laplacians. The following example shows that the standard discrete Laplacian is a generalized Laplacian in the sense mentioned above.
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Example 1. For the one-particle free Hamiltonian ˆ ˆ (hˆ 0 ψ)(x) = (−ψ)(x) =
ˆ ˆ + s)], [ψ(x) − ψ(x
x ∈ Zd ,
ψˆ ∈ 2 (Zd ),
|s|=1
the (Fourier) coefficients εˆ (s), s ∈ Zd , from (2.1) are necessarily of the form 2d, s = 0 εˆ (s) = −1, |s| = 1 0, otherwise. Hence, the corresponding dispersion relation ε(p) = 2
d
(1 − cos pi ),
p = (p1 , p2 , ... , pd ) ∈ Td ,
(2.4)
i=1
is a conditionally negative definite function. We need a simple inequality which will play a crucial role in the proof of the main results of the paper, Theorems 1 and 2. Lemma 1. Assume that the dispersion relation ε(p) is a real-valued continuous conditionally negative definite function on Td . Assume, in addition, that ε(0) is the unique minimum of the function ε(p). Then for all q ∈ Td \ {0} the inequality ε(p) + ε(q) >
ε(p + q) + ε(p − q) + ε(0), 2
a.e. p ∈ Td ,
(2.5)
holds. d Proof. Fix a q ∈ Td , q = 0. Then there exists an s0 ∈ Z \ {0} such that εˆ (s0 ) < 0 and cos(q, s0 ) = 1 (otherwise ε(q) = s∈Zd εˆ (s) = ε(0) which contradicts the hypothesis that ε(0) is the unique minimum of the function ε(·) on Td ). Since εˆ (s0 ) < 0 and cos(q, s0 ) = 1, the inequality
ε(p + q) + ε(p − q) F (p, q) ≡ ε(p) + ε(q) − − ε(0) 2 cos(p + q, s) + cos(p − q, s) = −1 εˆ (s) cos(p, s) + cos(q, s) − 2 s∈Zd cos(p + q, s0 ) + cos(p − q, s0 ) −1 ≥ 2ˆε (s0 ) cos(p, s0 ) + cos(q, s0 ) − 2 = 2ˆε (s0 ) (cos(p, s0 ) − 1)(1 − cos(q, s0 )) > 0, (p, s0 ) = 2nπ, n ∈ Z, (2.6) completes the proof.
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3. A Virtual Level and Threshold Eigenvalues In order to introduce the concept of a virtual level (threshold resonance) for the (lattice) energy operator h in dimensions three and four (d = 3, 4) we assume the following technical hypotheses that guarantee some smoothness of the dispersion relation ε(p) and the continuity of the Fourier transform d i(p,s) v(p) = (2π)− 2 v(s)e ˆ , d ≥ 3, (3.1) s∈Zd
of the interaction v. ˆ Hypothesis 1. Assume that the dispersion relation ε(p) is a continuous (periodic) realvalued function on Td with a unique (non-degenerate) minimum at the origin such that lim inf |p|→0
ε(p) − ε(0) > 0. |p|2
Assume, in addition, that v(p) is a continuous function on Td such that v(p) = v(−p),
p ∈ Td .
We remark that under Hypothesis 1 the sequence {v(s)} ˆ s∈Zd of the Fourier coefficients of the function v(p) is an element of 2 (Zd ) and then equality (3.1) should be d i(p,s) has the continuunderstood as follows: the 2 (Zd )-function (2π)− 2 s∈Zd v(s)e ˆ ous representative v(p). For λ ≤ ε(0) on the Banach space C(Td ), d ≥ 3, of continuous (periodic) functions on Td we shall consider the integral operator G(λ) with the (Birman-Schwinger) kernel function G(p, q; λ) = (2π)− 2 v(p − q)(ε(q) − λ)−1 , d
p, q ∈ Td .
(3.2)
Lemma 2. Let d ≥ 3. Assume Hypothesis 1. Then for λ ≤ ε(0) the operator G(λ) on C(Td ) given by (3.2) is compact. Proof. Given f ∈ L1 (Td ), for the function g introduced by − d2 g(p) = (2π) v(p − q)f (q)dq, Td
one has the estimates |g(p)| ≤ (2π)− 2 sup |v(p − q)|f L1 (Td ) d
(3.3)
p,q∈Td
and
d |g(p + ) − g(p)| =
(2π)− 2 (v(p + − q) − v(p − q))f (q)dq
Td
≤ (2π)
− d2
sup |v(t + ) − v(t)|f L1 (Td ) .
t∈Td
(3.4)
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Since for λ ≤ ε(0) and d ≥ 3, the function (ε(·) − λ)−1 , is integrable, the multiplication operator by the function (ε(·) − λ)−1 from C(Td ) into L1 (Td ) is continuous. Therefore, from (3.3) and (3.4) it follows that the image of the unit ball under G in C(Td ) consists of functions that are totally bounded and equicontinuous: v is continuous and, therefore, lim sup |v(t + ) − v(t)|f L1 (Td ) = 0.
||→0 t∈Td
An application of the Arzela-Ascoli Theorem then completes the proof.
Remark 1. Let d ≥ 3. Clearly (cf. [28]), the operator h has an eigenvalue λ ≤ ε(0), that is, Ker (h − λI ) = 0, if and only if the compact operator G(λ) on C(Td ) has an eigenvalue −1 and there exists a function ψ ∈ Ker (G(λ) + I ) such that the function f given by f (p) =
ψ(p) ε(p) − λ
a.e.
p ∈ Td ,
(3.5)
belongs to L2 (Td ). In this case f ∈ Ker (h − λI ). Moreover, if λ < ε(0), then dim Ker (h − λI ) = dim Ker (G(λ) + I )
(3.6)
and Ker (h − λI ) = {f | f (·) =
ψ(·) , ψ ∈ Ker (G(λ) + I )}. ε(·) − λ
In the case of a threshold eigenvalue λ = ε(0) equality (3.6) may fail to hold only if d = 3 or d = 4 (in dimensions d ≥ 5 the function f (p) given by (3.5) always belongs to L2 (Td ), cf. Lemma 3 below). In dimensions d = 3 or d = 4 equality (3.6) should be replaced by the inequality dim Ker (h − ε(0)I ) ≤ dim Ker (G(ε(0)) + I ). In order to discuss the threshold phenomena, that is, the case λ = ε(0), following [3 and 7] (see also [12] for a related discussion), under Hypothesis 1 we distinguish five mutually disjoint cases: Case I. −1 is not an eigenvalue of G(ε(0)), that is, 0 = dim Ker (h − λI ) = dim Ker (G(λ) + I ). Case II. −1 is a simple eigenvalue of G(ε(0)) and the associated eigenfunction ψ satisfies the condition ψ(·) ∈ / L2 (Td ), ε(·) − ε(0) that is, 0 = dim Ker (h − λI ) and dim Ker (G(λ) + I ) = 1.
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Case III. −1 is an eigenvalue of G(ε(0)) and any of the associated eigenfunctions ψ satisfies the condition ψ(·) ∈ L2 (Td ), ε(·) − ε(0) that is, 1 ≤ dim Ker (h − λI ) = dim Ker (G(λ) + I ). Case IV. −1 is a multiple eigenvalue of G(ε(0)) and at least one (up to a normalization) of the associated eigenfunctions ψ satisfies the condition ψ(·) ∈ / L2 (Td ), ε(·) − ε(0) that is, 2 ≤ dim Ker (G(λ) + I ) ≥ dim Ker (h − λI ) + 1. Case V. −1 is a multiple eigenvalue of G(ε(0)) and dim Ker (h − λI ) + 2 ≤ dim Ker (G(λ) + I ). Given the classification above, we arrive at the following definition of a virtual level in dimensions d = 3 or d = 4 (in dimensions d ≥ 5 Cases II, IV and V do not occur (see Remark 1)). Definition 1. Let d = 3, 4. In Cases II, IV and V the operator h is said to have a virtual level (at the threshold). Remark 2. Our definition of a virtual level is equivalent to the direct analogue of that in the continuous case (see, e.g., [1, 23, 25, 27, 28] and references therein). One can also introduce the concept of a virtual level in dimensions d = 1 and d = 2. However, due to additional threshold singularities of the Birman-Schwinger kernel in the momentum representation (cf. (3.2)) our approach is not directly applicable in low dimensions (d = 1, 2). Lemma 3. Let d ≥ 3. Assume Hypothesis 1 and suppose that ψ ∈ Ker (G(ε(0)) + I ). Then the function f (p) =
ψ(p) , ε(p) − ε(0)
p ∈ Td ,
d
belongs to the weak space Lw2 (Td ). Proof. Recall that f belongs to the weak Lq -space if sup t q mes{p | |f (p)| > t} < ∞. t>0
By Hypothesis 1 there exists a positive constant C such that ε(p) − ε(0) ≥ C|p|2 ,
p ∈ Td ,
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and then, since ψ(p) ∈ C(T3 ), mes{p ∈ Td | |f (p)| > t} = mes{p ∈ Td |
|ψ(p)| > t} ε(p) − ε(0)
≤ mes{p ∈ Td | ψC(Td ) > C|p|2 t} = O(t − 2 ) d
completing the proof.
as
t → ∞,
Corollary 1. If the Hamiltonian h has a virtual level and the corresponding function ψ(·) ψ , ψ ∈ Ker (G(ε(0)) + I ), is such that ε(·)−ε(0) ∈ / L2 (Td ), d = 3, 4, then (under Hypothesis 1) the function f (p) =
ψ(p) , ε(p) − ε(0)
p ∈ Td ,
d = 3, 4
(3.7)
r(d)
belongs to Lw (Td ), with r(d) =
3 2,
2,
d = 3, d = 4.
In particular, the function f given by (3.7) is the eigenfunction of the operator h associated with the eigenvalue ε(0) in the Banach space L1 (Td ), that is, hf = ε(0)f, and hence the following equation d ε(p)f (p) + (2π )− 2 v(p − q)f (q)dq = ε(0)f (p), Td
a.e. p ∈ Td ,
d = 3, 4
holds. Remark 3. A simple computation shows that the Fourier coefficients fˆ(s), s ∈ Zd , d = 3, 4, of the (integrable) function f solve the infinite system of homogeneous equations εˆ (s)fˆ(x + s) + (v(x) ˆ − ε(0))fˆ(x) = 0, x ∈ Zd , s∈Zd
and hence the equation (in the coordinate representation) hˆ fˆ = ε(0)fˆ has a solution fˆ, a threshold resonant state, that does not belong to 2 (Zd ) but vanishes at infinity, lim fˆ(s) = 0
|s|→∞
(by the Riemann-Lebesgue Theorem).
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Remark 4. If the dispersion relation ε(p) is known to be an even function, ε(p) = ε(−p), or, which is the same, the Fourier coefficients satisfy the condition εˆ (s) = εˆ (−s) ∈ R, s ∈ Zd , the Birman-Schwinger kernel G(p, q; λ) has the additional property that G(p, q; λ) = G(−p, −q; λ). Hence, if ψ ∈ Ker (G(λ) + I ), λ ≤ ε(0), so does the function ϕ(p) = ψ(−p). Therefore, at least one of the functions ψ ± ϕ is also an eigenfunction of G(λ) associated with the eigenvalue −1, and hence, without loss of generality one may assume that the ˜ operator G(λ) has an eigenfunction ψ˜ such that |ψ(·)| is an even function. To get finer results in dimensions d = 3, 4 (cf. [28]) we need an auxiliary scale of the Banach spaces B(µ), 0 < µ ≤ 1, of H¨older continuous functions on Td obtained by the closure of the space of smooth (periodic) functions f on Td with respect to the norm f µ = sup |f (t)| + ||−µ |f (t + ) − f (t)| . t,∈Td
Note that the spaces B(µ) are naturally embedded one into the other B(ν) ⊂ B(µ) ⊂ C(Td ), 0 < µ ≤ ν ≤ 1. If v ∈ B(κ) with κ > 21 for d = 3, and κ > 0 for d = 4, the following proposition, a variant of the Birman-Schwinger principle, is a convenient tool to decide whether the threshold ε(0) of the essential spectrum of h is an eigenvalue (resp. a virtual level ) for the operator h. Proposition 1. (cf. [28]). Let d = 3, 4. Assume Hypothesis 1. Assume, in addition, that v ∈ B(κ) with 1 , if d = 3 κ> 2 . (3.8) 0, if d = 4 Then the operator h−ε(0)I has a non-trivial kernel if and only if −1 is an eigenvalue of G(ε(0)) and one of the associated eigenfunctions ψ satisfies the condition ψ(0) = 0. In particular, the operator h has a virtual level if and only if −1 is an eigenvalue of G(ε(0)) and one of the associated eigenfunctions ψ satisfies the condition ψ(0) = 0. Proof. See Appendix A.
Remark 5. As it follows from Proposition 1, the eigensubspace of functions ψ associated with the eigenvalue λ = −1 of G(ε(0)) with the additional constraint ψ(0) = 0 is one-dimensional. This proves that Case V does not occur if (3.8) holds. In Case IV we have the coexistence of a (simple) virtual level and a (possibly multiple) threshold eigenvalue (see Appendix B for a concrete example of such a coexistence in Case IV in dimension three).
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Remark 6. It is known that for the continuous Schr¨odinger operators h = − + V (x) in dimension d = 3 with V ∈ L1 (R3 ) ∩ R, R the Rollnik class, Case V does not occur (see, [1] Lemma 1.2.3). It is also worth mentioning that if, in addition, the Schr¨odinger 3 operator h is non-negative, then under the L 2 -weak assumption on the potential V Cases III, and IV do not occur (there is no zero-energy eigenstate) (see, e.g., [10, 22 and 24]). 4. The Two-Particle Hamiltonian Reduction to the One-Particle Case 4.1. The coordinate representation. Throughout this section we assume that d ≥ 1. 0 of the system of two quantum particles α = 1, 2, with The free Hamiltonian H the dispersion relations εα (p), α = 1, 2, respectively, is introduced (as a bounded self-adjoint operator on the Hilbert space 2 ((Zd )2 ) 2 (Zd ) ⊗ 2 (Zd )) by 0 = hˆ 01 ⊗ I + I ⊗ hˆ 02 , H
(4.1)
with hˆ 0α = εα (−i∇),
α = 1, 2,
and I being the identity operator on 2 (Zd ). (in the coordinate representation) of the two-particle system The total Hamiltonian H is a self-adjoint bounded operator on the Hilbert with the real-valued pair interaction V space 2 ((Zd )2 ) of the form , =H 0 + V H where ψ)(x ˆ 1 , x2 ), ˆ 1 , x2 ) = v(x ˆ 1 − x2 )ψ(x (V
ψˆ ∈ 2 ((Zd )2 ),
with {v(s)} ˆ s∈Zd the Fourier coefficients of a continuous function v(p) satisfying Hypothesis 1. 4.2. The momentum representation. The transition to the momentum representation is performed by the standard Fourier transform F2 : L2 ((Td )2 ) −→ 2 ((Zd )2 ), where (Td )m denotes the Cartesian mth power of the three-dimensional cube Td = (−π, π )d : × · · · × Td, (Td )m = Td × Td m times
m ∈ N.
(4.2)
The two-particle Hamiltonian H in the momentum representation is then given by H = H0 + V, where (H 0 f )(k1 , k2 ) = (ε1 (k1 ) + ε2 (k2 ))f (k1 , k2 ),
f ∈ L2 ((Td )2 ),
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and V is the operator of partial integration given by − d2 (Vf )(k1 , k2 ) = (2π) v(k1 − k1 )δ(k1 + k2 − k1 − k2 )f (k1 , k2 )dk1 dk2 , (Td )2
f ∈ L2 ((Td )2 ). Here the kernel function is given by the Fourier series v(p) = (2π)−d/2 v(s) ˆ ei(p,s) ,
p ∈ Td ,
s∈Zd
and δ(p) denotes the Dirac delta-function. 4.3. Direct integral decompositions. The quasi-momentum. Denote by Uˆ s2 , s ∈ Zd , the unitary representation of the abelian group Zd by the shift operators on the Hilbert space 2 ((Zd )2 ): ˆ 1 , n2 ) = ψ(n ˆ 1 + s, n2 + s), (Uˆ s2 ψ)(n
ψˆ ∈ 2 ((Zd )2 ),
n1 , n2 , s ∈ Zd .
Via the Fourier transform F2 the unitary representation 2 = Uˆ s2 Uˆ t2 , Uˆ s+t
s, t ∈ Zd ,
induces the representation of the group Zd in the Hilbert space L2 ((Td )2 ) by unitary (multiplication) operators Us2 = F2−1 Uˆ s2 F2 , s ∈ Zd , (Us2 f )(k1 , k2 ) = exp − i(s, k1 + k2 ) f (k1 , k2 ), k1 , k2 ∈ Td , f ∈ L2 ((Td )2 ). (4.3) Given k ∈ Td , we define Fk as follows: Fk = {(k1 , k − k1 ) ∈(Td )2 : k1 ∈ Td , k − k1 ∈ Td }. Introducing the mapping π : (Td )2 → Td ,
π((k1 , k2 )) = k1 ,
we denote by πk , k ∈ Td , the restriction of π to Fk ⊂ (Td )2 , that is, πk = π|Fk .
(4.4)
We remark that Fk , k ∈ Td , is a three-dimensional manifold homeomorphic to Td . The following lemma is evident. Lemma 4. The mapping πk , k ∈ Td , from Fk ⊂ (Td )2 onto Td is bijective, with the inverse mapping given by (πk )−1 (q) = (q, k − q).
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Decomposing the Hilbert space L2 ((Td )2 ) into the direct integral 2 d 2 L ((T ) ) = ⊕L2 (Fk )dk k∈Td
yields the corresponding decomposition of the unitary representation Us2 , s ∈ Zd , into the direct integral 2 Us = ⊕Us (k)dk, k∈Td
with Us (k) = e−i(s,k) IL2 (Fk ) and IL2 (Fk ) being the identity operator on the Hilbert space L2 (Fk ). (in the coordinate representation) obviously commutes with the The Hamiltonian H group of translations, Uˆ s2 , s ∈ Zd , that is, =H Uˆ s2 , Uˆ s2 H
s ∈ Zd .
So does the Hamiltonian H (in the momentum representation) with respect to the group Us2 , s ∈ Zd , given by (4.3). Hence, the operator H can be decomposed into the direct integral ˜ H = ⊕h(k)dk (4.5) k∈Td
associated with the decomposition
L2 ((Td )2 ) =
k∈Td
⊕L2 (Fk )dk.
In the physical literature the parameter k, k ∈ Td , is called the two-particle quasi-momen˜ tum and the corresponding operators h(k), k ∈ Td , are called the fiber operators. ˜ 4.4. The two-particle dispersion relations. The fiber operators h(k), k ∈ Td , from the decomposition (4.5) are unitarily equivalent to the operators h(k), k ∈ Td , of the form h(k) = h0 (k) + v, where (h0 (k)f )(p) = Ek (p)f (p), d v(p − q)f (q)dq, f ∈ L2 (Td ) (vf )(p) = (2π)− 2 Td
and the two-particle dispersion relations Ek (p) = ε1 (p) + ε2 (k − p),
p ∈ Td ,
parametrically depend on the quasi-momentum k, k ∈ Td . The equivalence is given by the unitary operator uk : L2 (Fk ) → L2 (Td ), k ∈ Td , uk g = g ◦ (πk )−1 , with πk defined by (4.4).
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5. Spectral Properties of the Fiber Operators h(k) As we have learned from the previous section, the two-particle Hamiltonian H (up to unitary equivalence) can be decomposed into the direct integral ⊕h(k)dk, H k∈Td
where the fiber operators h(k) = h0 (k) + v can be considered as the one-particle Hamiltonians with the two-particle dispersion relations Ek (p) = ε1 (p) + ε2 (k − p),
p ∈ Td ,
(5.1)
with εα (p) the dispersion relations for the particles α = 1, 2. Under Hypothesis 1 the perturbation v of the operator h0 (k), k ∈ Td , is a HilbertSchmidt operator and, therefore, in accordance with the Weyl Theorem the essential spectrum of the operator h(k) fills in the following interval on the real axis: σess (h(k)) = [Emin (k), Emax (k)], where Emin (k) = min Ek (q), q∈Td
Emax (k) = max Ek (q). q∈Td
If the dispersion relations in the one-particle sector are conditionally negative definite, then so is the two-particle dispersion relation E0 (p) corresponding to the zero-value of the quasi-momentum k. Hence, under these assumptions, the Hamiltonian h(0) in the coordinate representation generates the positivity preserving semi-group e−th(0) , t > 0 (which is not necessarily true for the fiber Hamiltonians h(k) with k = 0: the function Ek (p) may not be even, and hence, not conditionally negative definite). Although the two-particle dispersion relations are not necessarily conditionally negative definite for nontrivial values of the quasi-momentum, they still satisfy some useful inequality, Lemma 5 below, analogous to that in Lemma 1 for the one-particle dispersion relations. Hypothesis 2. Assume Hypothesis 1 for both ε1 (p) and ε2 (p). Suppose that the dispersion relations εα (p), α = 1, 2, in the one-particle sectors are conditionally negative definite. We remark that the two-particle dispersion relation E0 (p) satisfies Hypothesis 1 if ε1 (p) and ε2 (p) do. Lemma 5. Assume Hypothesis 2. Then for any (fixed) k, q ∈ Td such that either k = q or q = 0, E0 (p) − E0 (0) + Ek (q) −
Ek (p + q) + Ek (q − p) > 0, 2
a.e. p ∈ Td .
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In particular, if k = 0, and p(k) is a (any) point where the function Ek (·) attains its minimal value, that is, Emin (k) = Ek (p(k)), the following inequality E0 (p) − Emin (0) + Emin (k) −
Ek (p + p(k)) + Ek (p(k) − p) > 0, 2
a.e. p ∈ Td ,
holds. Proof. Since |q|2 + |k − q|2 = 0 the claim is an immediate consequence of Lemma 1 and definition (5.1) of the two-particle dispersion relations: Ek (p + q) + Ek (q − p) − E0 (0) 2 ε1 (p + q) + ε1 (q − p) = ε1 (p) + ε1 (q) − − ε1 (0) 2 ε2 (k − q − p) + ε2 (k − q + p) +ε2 (p) + ε2 (k − q) − − ε2 (0) > 0, 2 a.e. p ∈ Td .
E0 (p) + Ek (q) −
From now and later on we will assume that d ≥ 3. Our first non-perturbative result shows that under Hypothesis 2 the discrete spectrum of the fiber operators h(k) under the variation of the quasi-momentum cannot be absorbed by the threshold. Theorem 1. Let d ≥ 3. Assume Hypothesis 2. Assume, in addition, that the dispersion relations εj (p), j = 1, 2, are twice differentiable functions. Denote by m(k), k ∈ Td , the lower bound of the operator h(k), m(k) = inf Spec (h(k)),
k ∈ Td .
Assume, in addition, that the lower edge m(0) = Emin (0) of the spectrum of the operator h(0) is an eigenvalue. Then Emin (0) − m(0) < Emin (k) − m(k),
k ∈ Td , k = 0,
Proof. Let 0 = f ∈ Ker (h(0) − m(0)I ) and hence d E0 (p)f (p) + (2π)− 2 v(p − q)f (q)dq = m(0)f (p), Td
d ≥ 3.
(5.2)
a.e. p ∈ Td .
By hypothesis the one-particle dispersion relations are conditionally negative definite functions. Then, as it can easily be seen from the definition of the two-body dispersion relation, the function E0 (p) corresponding to the zero-value of the quasi-momentum k is also conditionally negative definite. In particular, E0 (p) is an even function and, hence, by Remark 4, without loss of generality one may assume that the function |f (·)| is even. For k ∈ Td we introduce the trial L2 (Td )-function fk (p) = f (p − p(k)),
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where p(k) denotes the minimum point of the function Ek (p), that is, Ek (p(k)) = Emin (k) (if the minimum value of Ek (p) is attained in several points choose p(k) as any one of them arbitrarily). To prove (5.2) it is sufficient to establish the inequality
(k) = ([h(k) − (Emin (k) − Emin (0) + m(0))]fk , fk ) < 0,
k = 0.
(5.3)
One gets ([h(k) − (Emin (k) − Emin (0) + m(0))]fk )(p) = [Ek (p) − (Emin (k) − Emin (0) + m(0))]f (p − p(k)) − d2 +(2π) v(p − q)f (q − p(k))dq
(5.4)
Td
= [Ek (p) − (Emin (k) − Emin (0) + m(0))]f (p − p(k)) d +(2π)− 2 v(p − p(k) − q)f (q)dq Td
= [Ek (p) − Emin (k) − E0 (p − p(k)) + Emin (0)]f (p − p(k)). Using (5.4) one arrives at the representation E0 (p − p(k)) − Emin (0) − Ek (p) + Emin (k) |f (p − p(k))|2 dp,
(k) = − Td
k ∈ Td .
(5.5)
To check the basic inequality (5.3) we proceed as follows. Making the change of variable p → −p + 2p(k) in (5.5) and using the fact that the functions E0 (p) and |f (p)| are even, one obtains the representation E0 (p − p(k)) − Emin (0) − Ek (−p + 2p(k)) + Emin (k)
(k) = − Td
×|f (p − p(k))|2 dp.
(5.6)
Making again the change of variable p → p − p(k) in (5.5) and (5.6) and adding the results obtained we get
(k) = − F(k, p)|f (p)|2 dp, Td
where F(k, p) = E0 (p) − Emin (0) + Emin (k) −
Ek (p + p(k)) + Ek (p(k) − p) . 2
By Lemma 5, for any (fixed) k = 0, one concludes that F(k, p) > 0 for almost every
p ∈ Td , proving the basic inequality (5.3) and the claim follows. Our second non-perturbative result, the main result of the paper, provides sufficient conditions for the discrete spectrum of the whole family of fiber Hamiltonians h(k) with k = 0 to be non-empty.
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Theorem 2. Let d ≥ 3. Assume Hypothesis 2. Assume, in addition, that the dispersion relations εj (p), j = 1, 2, are twice differentiable functions. Suppose that the operator h(0) has either a threshold eigenvalue or a virtual level. Then, for all k ∈ Td \ {0} the discrete spectrum of the fiber Hamiltonian h(k) below the bottom Emin (k) of its essential spectrum is a non-empty set. Proof. The case where either h(0) has the discrete spectrum below the threshold or the bottom of its essential spectrum m(0), m(0) = Emin (0) = E0 (0),
(5.7)
is a (threshold) eigenvalue has been already treated in Theorem 1. Assume that d = 3 or d = 4 and suppose that h(0) has a virtual level at the bottom of its essential spectrum. Therefore, the equation G(E0 (0))ψ = −ψ,
ψ ∈ C(Td ),
has a nontrivial solution ψ ∈ C(Td ). As in the proof of Theorem 1, without loss of generality one may assume that the function |ψ(p)| is even. In particular, the equation − d2 v(p − q)f (q)dq = m(0)f (p), a.e. p ∈ Td , (5.8) E0 (p)f (p) + (2π) Td
has the L1 (Td )-solution (cf. Corollary 1) f (p) =
ψ(p) E0 (p) − E0 (0)
such that the function |f (·)| is even. 2 d Given k ∈ Td , introduce the sequence {fn,k }∞ n=1 of L (T )-functions fn,k (p) =
ψ(p − p(k)) E0 (p − p(k)) − E0 (0) +
1 n
.
By the dominated convergence theorem the sequence fn,k converges in the space L1 (Td ) as n → ∞ to the function fk fk (p) = f (p − p(k)),
p ∈ Td ,
with f (·) an integrable majorant. Under Hypothesis 2 this means that the sequence of functions [h(k) − Emin (k)]fk,n converges in L∞ (Td )-norm to the bounded function d v(p − q)fk (q)dq [Ek (p) − Emin (k)]fk (p) + (2π)− 2 Td
= [Ek (p) − Emin (k) − E0 (p − p(k)) + E0 (0)]f (p − p(k)), where we used (5.7), (5.8) and the representations ([h(k) − Emin (k)]fk,n )(p) = [Ek (p) − Emin (k)]fk,n (p) + (2π)− 2 d
Td
v(p − q)fk,n (q)dq.
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In particular, one concludes that the limit
(k) = lim ([h(k) − Emin (k)]fn,k , fn,k ),
k ∈ Td ,
n→∞
exists and is finite and, moreover, E0 (p − p(k)) − E0 (0) − Ek (p) + Emin (k)
(k) = − |ψ(p − p(k))|2 dp, (E0 (p − p(k)) − E0 (0))2
k ∈ Td .
Td
Next, exactly as it has been done in the proof of Theorem 1, one checks the inequality
(k) < 0,
k = 0.
(5.9)
It follows from (5.9) that there exists an n0 ∈ N such that ([h(k) − Emin (k)]fn0 ,k , fn0 ,k ) < 0,
k = 0,
proving the existence of the discrete spectrum of h(k) below its essential spectrum for k = 0. The proof is complete.
Remark 7. Let d = 3. The width w(k) of the essential spectrum band of the Hamiltonians h(k), w(k) = Emax (k) − Emin (k) may vanish for some values of the quasi-momentum k ∈ T3 . Therefore, the fiber Hamiltonians h(k) may have an infinite discrete spectrum for some values of the quasimomentum k even if the discrete spectrum of h(0) is empty. For instance, consider two (identical) particles on the lattice Z3 with the one-particle dispersion relations of the form ε1 (p) = ε2 (p) =
3
(1 − cos pi ),
p ∈ T3 .
(5.10)
i=1
Then if k0 = (π, π, π ) ∈ T3 we have the “strong degeneration" of the two-particle dispersion relation: Ek0 (p) = ε1 (p) + ε2 (k0 − p) = 6 holds for all p = (p1 , p2 , p3 ) ∈ T3 , which means that the essential spectrum of h(k0 ) is a one-point set, namely, ˆ 0 ) = Ek0 (−i∇)+ Spec ess (h(k0 )) = {6}. Therefore, in this case, the fiber Hamiltonian h(k vˆ has infinite discrete spectrum below the bottom of its essential spectrum, provided that 1 3 (i) vˆ = {v(s)} ˆ s∈Z3 ⊂ (Z ), 3 (ii) #{s ∈ Z |v(s) ˆ < 0} = ∞,
say. If, in addition, (iii) the C(T3 )-norm of the (continuous) function 3 i(p,s) v(p) = (2π)− 2 v(s)e ˆ s∈Z3
is small enough,
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ˆ then the discrete spectrum of the fiber Hamiltonian h(0) = E0 (−i∇) + vˆ corresponding to the zero value of the quasi-momentum, is empty (the Birman-Schwinger integral operator G(λ) on C(T3 ) with kernel function given by (cf. (3.2)) 3
G(p, q; λ) = (2π)− 2 v(p − q)(E0 (q) − λ)−1 , p, q ∈ T3 , λ∈ / [Emin (0), Emax (0)], is a contraction whenever vC(T3 ) is small enough. It is also worth mentioning that even a partial degeneracy of the two-particle dispersion relation Ek (·) for some values of the quasi-momentum k = 0 may generate a “rich” infinite discrete spectrum of the Hamiltonian h(k) outside the band [Emin (k), Emax (k)]. Remark 8. The study of the discrete spectrum of the fiber Hamiltonians above the edge of the essential spectrum follows the guidelines described above with obvious modifications, provided that dispersion relations εα (p), α = 1, 2, in the one-particle sector are conditionally positive definite functions A. Proof of Proposition 1 Assume without loss of generality that ε(0) = 0. “Only If Part.” Let f ∈ L2 (Td ), d = 3, 4, be an eigenfunction of the operator h(0) associated with a zero eigenvalue, that is, d v(p − q)f (q)dq, a.e. p ∈ Td (A.1) −ε(p)f (p) = (2π)− 2 Td
The same argument as in the proof of Lemma 2 shows that the equivalence class associated with the function f has a representative f˜ such that the function ψ(p) = ε(p)f˜(p)
(A.2)
is H¨older continuous, ψ ∈ B(κ). Hence the representative f˜ is continuous away from the origin and since from Hypothesis 1 it follows that lim inf p→0 ε(p)|p|−2 > 0, the following asymptotic representation ψ(0) + O(|p|−2+κ ), p → 0, f˜(p) = ε(p) holds. Since f˜ ∈ L2 (T3 ) and (3.8) holds, the H¨older continuous function ψ must vanish at the origin, that is, ψ(0) = 0. Comparing (A.1) and (A.2) one concludes that −1 is an eigenvalue of the operator G(ε(0)) on C(Td ), d = 3, 4, associated with the eigenfunction ψ with ψ(0) = 0. “If Part.” Assume that the operator G(ε(0)) has an eigenfunction ψ associated with the eigenvalue λ = −1, G(ε(0))ψ = −ψ
(A.3)
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S. Albeverio, S.N. Lakaev, K.A. Makarov, Z.I. Muminov
such that ψ(0) = 0. Following the strategy of the proof of Lemma 2 one gets that ψ ∈ B(κ). Introduce the function f (p) =
ψ(p) , ε(p)
p = 0.
Clearly, an argument as above shows that the following asymptotic representation f (p) = O(|p|−2+κ ),
p → 0,
holds. Since (3.8) holds, one proves that f ∈ L2 (Td ) and then (A.3) means that the operator h(0) has a nontrivial kernel, completing the proof. B. Coexistence of a Threshold Eigenvalue and a Virtual Level The main goal of this Appendix is to show by an explicit example that the Case IV is not empty. Example 2. Let hˆ λ,µ , λ, µ ∈ R, be the discrete Schr¨odinger operator of the form hˆ λ,µ = − + vˆλ,µ , where is the discrete Laplacian from Example 1 and µ, s = 0 vˆλ,µ (s) = λ2 , |s| = 1 0, otherwise. The Fourier transform of the interaction can be explicitly computed 3 1 v(p) = µ+λ cos pi , 3 (2π) 2 i=1 and, hence, for the Birman-Schwinger kernel one gets the representation 1 µ + λ 3i=1 cos(pi − qi ) G(p, q; 0) = , p, q ∈ T3 , (2π)3 ε(q)
(B.1)
where ε(q) is given by (2.4) and we have used the equality ε(0) = 0. Introduce the notations 1 1 dq cos qi dq a= , c = , i = 1, 2, 3, 3 3 (2π ) T3 ε(q) (2π) T3 ε(q) 1 1 sin2 qi dq cos2 qi dq s= , b = , i = 1, 2, 3, 3 3 (2π ) T3 ε(q) (2π) T3 ε(q) cos qi cos qj dq 1 d= , i, j = 1, 2, 3, i = j. 3 (2π ) T3 ε(q) We remark that since the function ε(q) = ε(q1 , q2 , q3 ) is invariant with respect to the permutations of its arguments q1 , q2 and q3 , the integrals c, s, b, d above do not
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depend on the particular choice of the indices i, j . A simple computation shows that the following relations: 1 , 6 b + 2d = 3c, a−c =
a =b+s
(B.2) (B.3)
1 2 and s = − (b − d) 6 3
(B.4)
hold. Lemma 6. a >
11 51
. In particular, c > 0.
Proof. We start with the representation π π π 4 4 dq dq dq = + π A − cos q π A + cos q A − cos q −4 −4 −π π π 4 4 dq dq + , + π A − sin q π A + sin q −4 −4
(B.5) |A| > 1,
which yields 1 dq dq 1 = 3 3 (2π ) T3 ε(q) 2(2π) T3 3 − cos q1 − cos q2 − cos q3 π 4 1 dq dq dq3 f (q1 , q2 , q3 ), = 1 2 3 2(2π ) T2 − π4
a=
(B.6)
where f (q1 , q2 , q3 ) =
1 1 + 3 − cos q1 − cos q2 − cos q3 3 − cos q1 − cos q2 + cos q3 1 1 + + . 3 − cos q1 − cos q2 + sin q3 3 − cos q1 − cos q2 − sin q3
Note that the function f is well defined on (−π, π ]3 \ {0}. One easily checks that for fixed q1 , q2 , the function f (q1 , q2 , q3 ) as a function of the argument q3 , q3 ∈ [− π4 , π4 ], attains its minimal value at the end points of the interval [− π4 , π4 ] and hence f (q1 , q2 , q3 ) >
2 3 − cos q1 − cos q2 −
√
2 2
+
2 3 − cos q1 − cos q2 +
√
2 2
, (B.7)
π π q3 ∈ (− , ). 4 4 Combining (B.6) and (B.7) proves the inequality 1 1 π √ dq dq + a> 1 2 3 (2π ) 2 T2 3 − 22 − cos q1 − cos q2 3+
√
2 2
1 − cos q1 − cos q2
.
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Applying the trick (B.5) two more times (first by getting rid of the variable q2 and then of q1 ) one arrives at the estimate 2 1 π 2 4 2 √ √ a> dq1 + + (2π )3 2 3 − cos q1 T 3 − 2 22 − cos q1 3 + 2 22 − cos q1 3 π 4 11 12 12 4 1 √ + √ + √ + √ = , > 3 (2π ) 2 51 3 − 3 22 3 − 22 3 + 22 3 + 3 22 completing the proof.
Corollary 2. The set
=
1 1 , s b−d
2a \ c
is nonempty. Proof. Assume to the contrary that = ∅, that is, 1 2a 1 = = . s b−d c
(B.8)
Solving (B.3), (B.4) and (B.8) simultaneously in particular yields a= which is impossible due to Lemma 6.
11 5 < , 24 51
Theorem 3. Assume that −λ ∈ and µ=−
1 + 3λc a+
λc 2
,
then the Hamiltonian hλ,µ has both a virtual level and a threshold eigenvalue. Proof. In accordance with Proposition 1 one needs to show that the integral operator G(0) given by (B.1) on the Banach space C(T3 ) has two eigenfunctions, ψ and ϕ associated with an eigenvalue −1: G(0)ψ = −ψ
and G(0)ϕ = −ϕ
such that ψ(0) = 0
and ϕ(0) = 0.
The space of all odd (resp. even) functions Co (T3 ) (resp. Ce (T3 )) is an invariant subspace for the integral operator G(0). The restrictions Go (respectively Ge ) of G(0) on the subspace Co (T3 ) (respectively Co (T3 )) have the kernel functions 3 λ sin pi sin qi , (2π)3 ε(q) i=1 1 µ + λ 3i=1 cos pi cos qi Ge (p, q) = . ε(q) (2π)3
Go (p, q) =
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The matrix of the restriction Go |S of Go onto its three-dimensional invariant subspaces S ⊂ Co (T3 ) spanned by the functions sin p1 , sin p2 , and sin p3 in the basis ei = sin pi , i = 1, 2, 3, is a diagonal matrix of the form λs 0 0 (B.9) Go |S = 0 λs 0 0 0 λs while the matrix of the restriction Ge |C of Ge onto its four-dimensional invariant subspaces C ⊂ Ce (T3 ) spanned by the functions 1, cos p1 , cos p2 , and cos p3 in the basis fi = cos pi , i = 1, 2, 3, f4 = 1 is given by λb λd λd λc λd λb λd λc . (B.10) Ge |C = λd λd λb λc µc µc µc µa From (B.10) it follows that if λ, µ and γ satisfy the relations
then
λb λd λd µc
λ(b + 2d + cγ ) = −1,
(B.11)
µ(3c + aγ ) = −γ ,
(B.12)
λd λb λd µc
λd λd λb µc
λc 1 1 λc 1 1 = . λc 1 1 µa γ γ
Given λ ∈ R, λ = − 2a c , solving Eqs. (B.11) and (B.12) with respect to µ and γ yields 1 − 3, λc 1 + 3λc γ (λ) 1 + 3λc =− µ(λ) = − , = 3c + aγ (λ) 3λc2 − 3λac − a a + λc 2 γ (λ) = −
(B.13) (B.14)
µ = µ(λ) satisfies (B.14) where we used (B.2) and (B.3). Therefore, if λ = − 2a c and 3 the operator G(0) has an eigenfunction ψ(p) = γ (λ) + i=1 cos pi and, moreover,
3
ψ(0) = γ (λ) + cos pi
i=1
p1 =p2 =p3 =0
= 0
as it follows from (B.13). Thus, the Hamiltonian hλ,µ(λ) has a virtual level. Next, from the matrix representation (B.9) for Go |S one gets that if λs = −1, then for any µ ∈ R the operator Go has a three-dimensional eigensubspace spanned by the functions sin pi , i = 1, 2, 3, associated with an eigenvalue −1 of multiplicity three. In particular, G(0)ϕ = −ϕ with ϕ(p) = sin p1 and hence ϕ(0) = 0.
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Similarly (cf. (B.10)), if λ(b − d) = −1, then for any µ ∈ R the operator Ge |C has two linearly independent eigenfunctions cos p1 − cos p2 and cos p1 − cos p3 associated with an eigenvalue −1 of multiplicity two. In particular, Gϕ = −ϕ with ϕ(p) = cos p1 − cos p2 , and hence ϕ(0) = 0. 1 Therefore, if λ = − 1s or λ = − b−d , then for any µ ∈ R the operator hλ,µ has an eigenvalue at the bottom of its (absolutely) continuous spectrum. 1 1 Taking −λ ∈ = s , b−d \ 2a (which is nonempty by Corollary 2) and c µ = µ(λ) = − 1+3λc λc one proves the coexistence of a virtual level and a threshold a+
2
eigenvalue for the Hamiltonian hλ,µ(λ) .
Remark 9. We were not able to find out whether the set is a one- or a two-point set and hence we cannot explicitly compute the multiplicity of the zero-energy eigenvalue (more information about numerical values of the integrals a, b, c, and d is needed). However, if contains two elements, then the Hamiltonian hˆ λ,− 1+3λc has a virtual level and a a+ λc 2
threshold eigenvalue of multiplicity two or three depending on the choice of −λ ∈ 1 or λ = − 1s respectively). (λ = − b−d If || = 1, it might happen that the Hamiltonian hˆ λ,− 1+3λc , −λ ∈ , has a virtual a+ λc 2
level and a threshold eigenvalue of multiplicity two, three or even five depending on which of the cases (i) (ii) (iii)
c 2a c 2a c 2a
= s = b − d, = b − d = s, = s = b − d.
takes place respectively. Acknowledgements. K. A. Makarov thanks F. Gesztesy and V. Kostrykin for useful discussions. He is also indebted to the Institute of Applied Mathematics of the University Bonn for its kind hospitality during his stay in the summer 2003. This work was also partially supported by the DFG 436 USB 113/4 Project and the Fundamental Science Foundation of Uzbekistan. S.N. Lakaev and Z.I. Muminov gratefully acknowledge the hospitality of the Institute of Applied Mathematics of the University Bonn. We are indebted to the anonymous referee for a number of constructive comments.
References 1. Albeverio, S., Gesztesy, F., Høegh-Krohn, R.: The low energy expansion in non-relativistic scattering theory. Ann. Inst. H. Poincar´e Sect. A (N.S.) 37, 1–28 (1982) 2. Albeverio, S., Høegh-Krohn, R., Wu, T.T.: A class of exactly solvable three-body quantum mechanical problems and universal low energy behavior. Phys. Lett. A 83, 105–109 (1971) 3. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. New York: Springer-Verlag, 1988; 2nd ed. (with an appendix by P. Exner), Chelsea: AMS, 2005 4. Albeverio, S., Lakaev, S.N., Muminov, Z.I.: Schr¨odinger operators on lattices. The Efimov effect and discrete spectrum asymptotics. Ann. Henri Poincar´e. 5, 743–772 (2004) 5. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic analysis on semigroups. Theory of positive definite and related functions. Graduate Texts in Mathematics, New York: Springer-Verlag, 1984. 289 pp. 6. Carmona, R., Lacroix, J.: Spectral theory of random Schr¨odinger operators. Probability and its Applications, Boston: Birkh¨auser, 1990
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7. Jensen, A., Kato, T.: Spectral properties of Schr¨odinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611 (1979) 8. Faria da Veiga, P.A., Ioriatti, L., O’Carroll, M.: Energy-momentum spectrum of some two-particle lattice Schr¨odinger Hamiltonians. Phys. Rev. E (3) 66, 016130, 9 pp. (2002) 9. Graf, G.M., Schenker, D.: 2-magnon scattering in the Heisenberg model. Ann. Inst. H. Poincar´e Phys. Th´eor. 67, 91–107 (1997) 10. Klaus, M., Simon, B.: Coupling constants thresholds in non-relativistic quantum mechanics. I. Short range two body case. Ann. Phys. 130, 251–281 (1980) 11. Kondratiev, Yu. G., Minlos, R.A.: One-particle subspaces in the stochastic XY model. J. Statist. Phys. 87, 613–642 (1997) 12. Kostrykin, V., Schrader, R.: Cluster properties of one particle Scr¨odinger operators. II. Rev. Math. Phys. 10, 627–682 (1998) 13. Lakaev, S.N.: The Efimov effect in a system of three identical quantum particles. Funct. Anal. Appl. 27, 166–175 (1993) 14. Lakaev, S.N.: Discrete spectrum and resonances of the one-dimensional Schr¨odinger operator for small coupling constants. Teoret. Mat. Fiz. 44, 381–386 (1980) 15. Mattis, D.C.: The few-body problem on a lattice. Rev. Mod. Phys. 58, 361–379 (1986) 16. Minlos, R.A., Suhov, Y.M.: On the spectrum of the generator of an infinite system of interacting diffusions. Commun. Math. Phys. 206, 463–489 (1999) 17. Mogilner, A.: Hamiltonians in solid state physics as multi-particle discrete Schr¨odinger operators: Problems and results. Adv. in Sov. Math. 5, 139–194 (1991) 18. Ovchinnikov, Yu. N., Sigal, I. M.: Number of bound states of three-particle systems and Efimov’s effect. Ann. Phys. 123, 274–295 (1989) 19. Rauch, J.: Perturbation theory for eigenvalues and resonances of Schr¨odinger Hamiltonians. J. Funct. Anal. 35, 304–315 (1980) 20. Reed, M., Simon, B.: Methods of modern mathematical physics. III: Scattering theory. New York: Academic Press, 1979 21. Reed, M., Simon, B.: Methods of modern mathematical physics. IV: Analysis of Operators. New York: Academic Press, 1979 22. Simon, B.: Large time behavior of the Lp norm of Schr¨odinger Semigroups. J. Funct. Anal. 40, 66–83 (1981) 23. Sobolev, A. V.: The Efimov effect. Discrete spectrum asymptotics. Commun. Math. Phys. 156, 127– 168 (1993) 24. Tamura, H.: The Efimov effect of three-body Schr¨odinger operators. J. Funct. Anal. 95, 433–459 (1991) 25. Tamura, H.: The Efimov effect of three-body Schr¨odinger operators: Asymptotics for the number of negative eigenvalues. Nagoya Math. J. 130, 55–83(1993) 26. Yafaev, D. R.: Scattering theory: Some old and new problems. Lecture Notes in Mathematics 1735 Berlin: Springer-Verlag, 2000, 169 pp. 27. Yafaev, D. R.: On the theory of the discrete spectrum of the three-particle Schr¨odinger operator. Math. USSR-Sb. 23, 535–559 (1974) 28. Yafaev, D. R.: The virtual level of the Schr¨odinger equation. J. Sov. Math. 11, 501–510 (1979) 29. Zhizhina, E. A.: Two-particle spectrum of the generator for stochastic model of planar rotators at high temperatures. J. Stat. Phys. 91, 343–368 (1998) Communicated by B. Simon
Commun. Math. Phys. 262, 117–135 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1385-7
Communications in
Mathematical Physics
PROP Profile of Poisson Geometry S.A. Merkulov Matematiska Institutionen, Stockholms Universitet, 10691 Stockholm, Sweden. E-mail:
[email protected] Received: 20 January 2005 / Accepted: 1 February 2005 Published online: 8 July 2005 – © Springer-Verlag 2005
Abstract: It is shown that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant (di) operads in spaces concentrated in degree zero. In particular, they admit natural infinity generalizations when one considers homotopy representations of the (di) operads in generic differential graded spaces. Poisson geometry provides us with a simplest manifestation of this phenomenon. 0. Introduction The first instances of algebraic and topological strongly homotopy, or infinity, structures have been discovered by Stasheff [St] long ago. Since that time infinities have acquired a prominent role in algebraic topology and homological algebra. We argue in this paper that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant PROPs P∞ in spaces concentrated in degree zero; in particular, they admit natural infinity generalizations when one considers homotopy representations of P∞ in generic differential graded (dg) spaces. The simplest manifestation of this phenomenon is provided by the Poisson geometry (or even by smooth germs of tensor fields!) and is the main theme of the present paper. Another example is discussed in [Mer2]. The PROPs P∞ are minimal resolutions of PROPs P which are graph spaces built from very few basic elements, genes, subject to simple engineering rules. Thus to a local geometric structure one can associate a kind of a code, genome, which specifies it uniquely and opens a new window of opportunities of attacking differential geometric problems with the powerful tools of homological algebra. Formal germs of geometric structures discussed in this paper are pointed in the sense that they vanish at the distinguished point. This is the usual price one pays for working with (di)operads without “zero terms” (as is often done in the literature). As structural equations behind the particular geometries we study in this paper are homogeneous, this
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restriction poses no problem: say, a generic non-pointed Poisson structure, ν, in Rn can be identified with the pointed one, ν, in Rn+1 , being the extra coordinate. We introduce in this paper a dg free dioperad whose generic representations in a graded vector space V can be identified with pointed solutions of the Maurer-Cartan equations in the Lie algebra of polyvector fields on the formal manifold associated with V . The cohomology of this dioperad can not be computed directly. Instead one has to rely on some fine mathematics such as Koszulness [GiKa, G] and distributive laws [Mar1, G]. One of the main results of this paper is a proof of Theorem 3.2 which identifies the cohomology of that dg free dioperad with a surprisingly small dioperad, Lie1 Bi, of Lie 1-bialgebras, which are almost identical to the dioperad, LieBi, of usual Lie bialgebras except that the degree of generating Lie and coLie operations differ by 1 (compare with Gerstenhaber versus Poisson algebras). The dioperad Lie1 Bi is proven to be Koszul. We use the resulting geometric interpretation of Lie1 Bi∞ algebras to give their homotopy classification (see Theorem 3.4.5) which is an extension of Kontsevich’s homotopy classification [Ko1] of L∞ algebras. As a side remark we also discuss graph and geometric interpretations of strongly homotopy Lie bialgebras using Koszulness of the latter which was established in [G]. 1. Geometry ⇒ PROP profile ⇒ Geometry ∞ Let P be an operad, or a dioperad, or even a PROP admitting a minimal dg resolution. Let PAlg be the category of finite dimensional dg P-algebras, and D(PAlg) the associated derived category (which we understand here as the homotopy category of P∞ -algebras, P∞ being the minimal resolution of P). For any locally defined geometric structure Geom (say, Poisson, Riemann, K¨ahler, etc.) it makes sense talking about the category of formal Geom-manifolds. Its objects are formal pointed manifolds (non-canonically isomorphic to (Rn , 0) for some n) together with a germ of formal Geom-structure at the distinguished point. Definition 1.1. The operad/dioperad/PROP P is called a PROP-profile, or genome, of a geometric structure Geom if • the category of formal Geom-manifolds is equivalent to a full subcategory of the derived category D(PAlg) , and • there is no sub-(di)operad of P having the above property. Definition 1.2. If P is a PROP-profile of a geometric structure Geom, then a generic object of D(PAlg) is called a formal Geom∞ -manifold. Presumably, Geom∞ -structure is what one gets from Geom by means of the extended deformation theory. Local geometric structures are often non-trivial and complicated creatures — the general solution of the associated defining system of nonlinear differential equations is not available; it is often a very hard job just to show existence of non-trivial solutions. Nevertheless, if such a structure Geom admits a PROP-profile, P = F ree(E)/I deal 1 , then Geom can be non-ambiguously characterized by its “genetic code”: genes are, by definition, the generators of E, and the engineering rules are, by definition, the generators of I deal. And that code can be surprisingly simple, as Examples 1.3–1.5 and illustrate. 1 Any operad/dioperad/etc. can be represented as a quotient of the free operad/dioperad/etc., F ree(E ) generated by a collection of m -left/n -right modules E = {E (m, n)}m,n≥1 , by an I deal. Often there exists a canonical, “common factors canceled out”, representation like this.
PROP Profile of Poisson Geometry
119 Table 1.
Genome P
generic representation of P∞ in Rn
generic representation of P∞ in a graded vector space V
smooth formal Hertling-Manin structure in Rn [HeMa]
smooth formal Hertling-Manin∞ structure in Vˆ [Mer1]
P is the G-operad
Genes: ◦?? , •??
Engineering rules: ◦? - ◦?? = 0 ◦< ◦= < = •? + •? + •;; = 0 •< CC •? •= ; < C ? = ; •? − ◦? − ◦;; = 0 FF•?? •== ;; ◦<< F P is the dioperad TF
??
Genes: ◦ , •?? Rules: •? + •? + •;; = 0 •< CC •? •= ; < C ? = ; ?? BB ◦ 77 •333 •<< @@@•PP q z DqD• zEz •99 = ◦ + ◦ + } ◦z + ◦z
smooth formal section of ⊗2 TRn (variants: of ∧2 TRn or of 2 TRn ) vanishing at 0
structure, (Vˆ , ð ∈ TVˆ ), of a smooth dg manifold together with a smooth section φ of ⊗2 TVˆ (variants: of ∧2 TVˆ or of 2 TVˆ ) vanishing at 0 and satisfying Lieð φ = 0.
P is the dioperad Lie1 Bi
?? << << FF 33 44 F 33 ◦ ◦? ◦? ◦ − ◦ = 0 ◦ − Rules: Genes: ◦ , •??
•? + •? + •;; = 0 •< CC •? •= ; < C ? = ; ?? ◦ 77 •333 •<< @@@•PP q •BB + ◦ + } ◦zz + DqD◦zzEz = 9 • ◦ 9
smooth formal Poisson structure in Rn vanishing at 0
structure, (V ⊕ V ∗ [1], ð), of a smooth dg manifold together with an odd symplectic form ωodd on V ⊕ V ∗ [1] such that the homological vector field ð is hamiltonian V ∗ [1] and vanishes on 0 ⊕
Notations: For a graded vector space V , Vˆ stands for the formal graded manifold (non-canonically) isomorphic to the formal neighbourhood of 0 in V , and TVˆ stands for the tangent bundle on Vˆ .
1.3. Hertling-Manin’s geometry and the G-operad. A Gerstenhaber algebra is, by definition, a graded vector space V together with two linear maps, ◦ : 2 V −→ V , a ⊗ b −→ a ◦ b
[ • ] : 2 V −→ V [1] a ⊗ b −→ (−1)|a| [a • b]
satisfying the identities, (i) a ◦ (b ◦ c) − (a ◦ b) ◦ c = 0 (associativity); (ii) [[a • b] • c] = [a • [b • c]] + (−1)|b||a|+|b|+|a| [b • [a • c]] (Jacobi identity); (iii) [(a ◦ b) • c] = a ◦ [b • c] + (−1)|b|(|c|+1) [a • c] ◦ b (Leibniz type identity).
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The operad whose algebras are Gerstenhaber algebras is often called the G-operad. It has a relatively simple structure, F ree(E)/I deal, with E spanned by two corollas, E = span ◦ = ◦?? , [ • ] = •?? and with engineering rules (i)–(iii). The minimal resolution of the G-operad has been constructed in [GetJo] and is often called a G∞ -operad. The derived category of Gerstenhaber algebras is equivalent to the category whose objects are isomorphism classes of minimal G∞ -structures on graded vector spaces V . Let (M, ∗) be the formal pointed graded manifold whose tangent space at the distinguished point is isomorphic to a vector space V , and let us choose an arbitrary torsion-free affine connection ∇ on M. With this choice a structure of G∞ algebra on a graded vector space V can be suitable described as • a degree 1 smooth vector field ð on M satisfying the integrability condition [ð, ð] = 0 and vanishing at the distinguished point ∗; (if ð has zero at ∗ of second order, then the G∞ -structure is called minimal); • a collection of homogeneous tensors,
µn1 ,... ,nk : TM⊗n1 ⊗TM⊗n2 ⊗ · · ·⊗ TM⊗nk → TM [k+1 − n1 −· · ·− nk ]
ni ,k≥1,ni +k≥2
satisfying an infinite tower of quadratic algebraic and differential equations. The first two floors of this tower read as follows: the data {µn }n≥1 (with µ1 := Lieð ) makes the tangent sheaf TM into a sheaf of C∞ algebras2 satisfying an “integrability” condition, [µ• , µ• ]G∞ = Lie𠵕,• for a certain bi-differential operator [ , ]G∞ whose leading term is just the usual vector field bracket of values of µ• . It is also required that each tensor µ•,... ,• : TM⊗• ⊗ · · · ⊗ TM⊗• → TM vanishes if the input contains at least one pure shuffle product, (v1 ⊗ · · · ⊗ vk )(vk+1 ⊗ · · · ⊗ vn ) :=
(−1)Koszul(σ ) vσ (1) ⊗ . . . ⊗ vσ (n) ,
Shuffles σ of type (k,n)
vi ∈ TM . A change of the connection ∇ alters the tensors µ•1 ,... ,•k , k ≥ 2, but leaves the homotopy class of the G∞ -structure on V invariant. If the vector space V is concentrated in degree 0, i.e. V Rn , then a G∞ -structure on V reduces just to a single tensor field µ2 : TM⊗2 → TM which makes the tangent sheaf into a sheaf of commutative associative algebras, and satisfies the differential equations, [µ2 , µ2 ]G∞ = 0. 2
C∞ stands for the minimal resolution of the operad of commutative associative algebras.
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The explicit form for the bracket [ , ]G∞ can be read off from the G∞ operad structural equations rather straightforwardly (see [Mer1] for details), [µ2 , µ2 ]G∞ (X, Y, Z, W ) = [µ2 (X, Y ), µ2 (Z, W )] −µ2 ([µ2 (X, Y ), Z], W ) − µ2 (Z, [µ2 (X, Y ), W ]) −µ2 (X, [Y, µ2 (Z, W )]) − µ2 [X, µ2 (Z, W )], Y ) +µ2 (X, µ2 (Z, [Y, W ])) + µ2 (X, µ2 ([Y, Z], W )) +µ2 ([X, Z], µ2 (Y, W )) + µ2 ([X, W ], µ2 (Y, Z)). The resulting geometric structure is precisely the one discovered earlier by Hertling and Manin [HeMa] in their quest for a weaker notion of Frobenius manifold; they call it an F -manifold structure on V . Hertling-Manin’s geometric structures arise naturally in the theory of singularities [He] and the deformation theory [Mer1].
1.4. Germs of tensor fields. A TF bialgebra is, by definition, a graded vector space V together with two linear maps, ?? 2 2 δ ≡ ◦ : V −→ ⊗ •? V , [ • ] ≡ ? : V −→ V [1] |a| a −→ a1 ⊗ a2 a ⊗ b −→ (−1) [a • b] satisfying the identities, (i) [[a • b] • c] = [a • [b • c]] + (−1)|b||a|+|b|+|a| [b • [a • c]] (Jacobi identity); (ii) δ[a • b] = a1 ⊗ [a2 • b] + [a • b1 ] ⊗ b2 + (−1)|a||b|+|a|+|b| ([b • a1 ] ⊗ a2 + b1 ⊗ [b2 • a]) (Leibniz type identity). 2 2 There are obvious ?? versions of the above notion with δ taking values in ∧ V and V , ◦ realizing either the trivial or sign representations of 2 . i.e. with the gene The dioperad whose algebras are TF bialgebras is denoted by TF. This quadratic dioperad is Koszul so that one can construct its minimal resolution using the results of [G, GiKa, Mar1]. It turns out that the structure of TF∞ -algebra on a graded vector space V is the same as a pair of collections of linear maps, µn : n V → V [1] n≥1 ,
and φn : n V → V ⊗ V n≥1 , satisfying a system of quadratic equations which are best described using a geometric language. Let M be the formal graded manifold associated to V . If {eα , α = 1, 2, . . . } is a homogeneous basis of V , then the associated dual basis t α , |t α | = −|eα |, defines a coordinate system on M. The collection of tensors {µn }n≥1 can be assembled into a germ, ð ∈ TM , of a degree 1 smooth vector field, ð :=
∞ 1 ∂ (−1) t α1 · · · t αn µα1 ,...β,αn β , n! ∂t n=1
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S.A. Merkulov
where =
n
|eαk |(1 +
k=1
k
|eαi |),
i=1
β
the numbers µα1 ...αn are defined by
µn (eα1 , . . . , eαn ) =
µα1 ...αβn eβ ,
and we assume here and throughout the paper summation over repeated small Greek indices. Another collection of linear maps, {φn }, can be assembled into a smooth germ, φ ∈ ⊗2 TM , of a degree zero contravariant tensor field on M, φ :=
∞ 1 ∂ ∂ (−1) t α1 · · · t αn φα1 ...αβn1 β2 β ⊗ β , n! ∂t ∂t n=1
where k n
= |eβ2 |(|eβ1 | + 1) +
|eαk ||eαi |
k=1 i=1 β β2
and the numbers µα1 ...αn1
are defined by
µn (eα1 , . . . , eαn ) =
1 β2 e µα1 ,...β,α ⊗ eβ2 . n β1
Proposition 1.4.1. The collections of tensors, µn : n V → V [1] n≥1 and φn : n V → V ⊗ V n≥1 , define a structure of TF∞ -algebra on V if and only if the associated smooth vector field ð and the contravariant tensor field φ satisfy the equations, [ð, ð] = 0 and Lieð φ = 0, where [ , ] stands for the usual bracket of vector fields and Lieð for the Lie derivative along ð. If V is finite dimensional and concentrated in degree zero, then a TF∞ -structure in V is just a germ of a smooth rank 2 contravariant tensor on V vanishing at 0.
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1.5. Poisson geometry and the dioperad of Lie 1-bialgebras. A Lie 1-bialgebra is, by definition, a graded vector space V together with two linear maps, δ : V −→ ∧2 V a −→ a1 ∧ a2
[ • ] : 2 V −→ V [1] a ⊗ b −→ (−1)|a| [a • b]
,
satisfying the identities, (i) (δ ⊗ Id)δa + τ (δ ⊗ Id)δa + τ 2 (δ ⊗ Id)δa = 0, where τ is the cyclic permutation (123) represented naturally on V ⊗ V ⊗ V (co-Jacobi identity); |b||a|+|b|+|a| [b • [a • c]] (Jacobi identity); (ii) [[a • b] • c] = [a • [b • c]] + (−1) ||a |a 1 2 | a ∧[a •b]+[a •b ]∧b −(−1)|b1 ||b2 | [a • (iii) δ[a •b] = a1 ∧[a2 •b]−(−1) 2 1 1 2 b2 ] ∧ b1 (Leibniz type identity). The dioperad whose algebras are Lie 1-bialgebras is denoted by Lie1 Bi. The superscript 1 in the notation is used to emphasize that the two basic operations ?? δ = ◦ , [ • ] = •?? have homogeneities which differ by 1. Similarly one can introduce the notion of Lie n-bialgebras: coLie algebra structure on V plus Lie algebra structure on V [−n] plus an obvious Leibniz type identity. Homotopy theory of Lie n-bialgebras splits into two stories, one for n even, and one for n odd. The even case (more precisely, the case n = 0) has been studied by Gan [G]. In this paper we study the odd case, more precisely, the case n = 1. The dioperad Lie1 Bi is Koszul. Hence one can use the machinery of [G, GiKa, Mar1] to construct its minimal resolution, the dioperad Lie1 Bi∞ . The structure of a Lie1 Bi∞ algebra on a graded vector space V is a collection of linear maps, µm,n : n V → ∧m V [2 − m] m≥1,n≥1 , satisfying a system of quadratic equations which can be described as follows. Let M be the formal graded manifold associated to V . If {eα , α = 1, 2, . . . } is a homogeneous basis of V , then the associated dual basis t α , |t α | = −|eα |, defines a coordinate system on M. For a fixed m the collection of tensors {µm,n }n≥1 can be assembled into a germ, m ∈ ∧m TM , of a smooth polyvector field (vanishing at 0 ∈ M), m :=
∞ n=1
1 ∂ ∂ (−1) t α1 · · · t αn µα1 ...αβn1 ...βm β ∧ · · · ∧ β , 1 m!n! ∂t ∂t m
where =
n
|eαk |(2 − m +
k=1
k
|eαi |) +
i=1 β ,... ,βm
1 and the numbers µα1 ,... ,α n
n k=1
(|eβk | + 1)
n
|eβi |
i=k+1
are defined by
µm,n (eα1 , . . . , eαn ) =
1 ,... ,βm e µα1 ,...β,α β1 ∧ · · · ∧ eβm . n
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Proposition 1.5.1. A collection of tensors, µm,n : n V → ∧m V [2 − m] m≥1,n≥1 , defines a structure of Lie1 Bi∞ -algebra on V if and only if the associated smooth polyvector field, m ∈ ∧• TM , := m≥1
satisfies the equation [ , ] = 0, where [ , ] stands for the Schouten bracket of polyvector fields. In particular, if V is concentrated in degree zero, then the only non-zero summand in is 2 ∈ ∧2 TM . Hence a Lie1 Bi∞ -algebra structure on Rn is nothing but a germ of a smooth Poisson structure on Rn vanishing at 0. 1.6. On the content of the rest. Section 2 is a reminder about PROPs, dioperads and Koszulness [G, GiKa, Mar1]. In Sects. 3 and 4 we prove Koszulness of the dioperads Lie1 Bi and TF, apply the machinery reviewed in Sect. 2 to give explicit graph descriptions of their minimal resolutions, Lie1 Bi∞ and TF∞ , prove Propositions 1.4.1 and 1.5.1 and introduce and study the notion of Lie1 Bi∞ morphisms. Section 5 is a comment on a geometric description of algebras over the dioperad of strongly homotopy Lie bialgebras, and their strongly homotopy maps. 2. PROPs and Dioperads [G] Let Sf be the groupoid of finite sets. It is equivalent to the category whose objects are natural numbers, {m}m≥1 , and morphisms are the permutation groups {m }m≥1 . A PROP P in the category, dgVec, of differential graded (shortly, dg) vector spaces is a functor P : Sf × Sf op → dgVec together with natural transformations, ◦A,B,C : P(A, B) ⊗ P(B, C) −→ P(A, C), ⊗A,B,C,D : P(A, B) ⊗ P(C, D) −→ P(A ⊗ B, C ⊗ D) and the distinguished elements IdA ∈ P(A, A) and sA,B ∈ P(A ⊗ B, B ⊗ A) satisfying a system of axioms [A] which just mimic the obvious properties of the following natural transformation, EV : (m, n) −→ H om(V ⊗n , V ⊗m ), canonically associated with an arbitrary dg space V . The latter fundamental example is called the endomorphism PROP of V . Given a collection of dg (m , n )-bimodules, E = {E(m, n)}m,n≥1 , one can construct the associated free PROP, F ree(E), by decorating vertices of all possible directed graphs with a flow by the elements of E and then taking the colimit over the graph automorphism group. The composition operation ◦ corresponds then to gluing output legs of one graph to the input legs of another graph, and the tensor product ⊗ to the disjoint union of graphs. Even for a small finite dimensional collection E the resulting free PROP can be a monstrous infinite dimensional object. The notion of dioperad was introduced by Gan [G] as a way to avoid that free PROP “divergence”. In the above setup, a free dioperad on E is built on graphs of genus zero, i.e. on trees. More precisely, a dioperad P consists of data:
PROP Profile of Poisson Geometry
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(i) a collection of dg (m , n ) bimodules, {P(m, n)}m≥1,n≥1 ; (ii) for each m1 , n1 , m2 , n2 ≥ 1, i ∈ {1, 2, . . . , n1 } and j ∈ {1, . . . , n1 } a linear map i ◦j
: P(m1 , n1 ) ⊗ P(m2 , n2 ) −→ P(m1 + m2 − 1, n1 + n2 − 1),
(iii) a morphism e : k → P(1, 1) such that the compositions e⊗I d
1 ◦i
I d⊗e
j ◦1
k ⊗ P(m, n) −→ P(1, 1) ⊗ P(m, n) −→ P(m, n) and P(m, n) ⊗ k −→ P(m, n) ⊗ P(1, 1) −→ P(m, n) are the canonical isomorphisms for all m, n ≥ 1, 1 ≤ i ≤ m and 1 ≤ j ≤ n. These data satisfy associativity and equivariance conditions [G] which can be read off from the example of the endomorphism dioperad EndV with EndV (m, n) = H om(V ⊗n , V ⊗m ), e : 1 → Id ∈ H om(V , V ), and the compositions given by i ◦j
: P(m1 , n1 ) ⊗ P(m2 , n2 ) −→ P(m1 + m2 − 1, n1 + n2 − 1) f ⊗g −→ (Id ⊗· · · ⊗ f ⊗· · · ⊗ Id)σ(Id ⊗· · ·⊗ g ⊗· · ·⊗ Id),
where f (resp. g) is at the j th (resp. i th ) place, and σ is the permutation of the set I = (1, 2, . . . , n1 + m2 − 1) swapping the subintervals, I1 ↔ I2 and I4 ↔ I5 , of the unique order preserving decomposition, I = I1 I2 I3 I4 I5 , of I into the disjoint union of five intervals of lengths |I1 | = i − 1, |I2 | = j − 1, |I3 | = 1, |I4 | = m2 − j and |I5 | = n1 − i. If P is a dioperad, then the collection of (m , n ) bimodules, P op (m, n) := (P(n, m), transposed actions of m and n ) , is naturally a dioperad as well. If P is a dioperad with P(m, n) vanishing for all m, n except for (m = 1, n ≥ 1), then P is called an operad. A morphism of dioperads, F : P → Q, is a collection of equivariant linear maps, F (m, n) : P(m, n) → Q(m, n), preserving all the structures. If P is a dioperad, then a P-algebra is a dg vector space V together with a morphism, F : P → EndV , of dioperads. We shall consider below only dioperads P with P(m, n) being finite dimensional vector spaces (over a field k of characteristic zero) for all m, n. The endomorphism dioperad of the vector space k[−p], p ∈ Z, is denoted by p. ⊗p ⊗p Thus p(m, n) is sgnn ⊗sgnm [p(n−m)], where sgnm stands for the one dimensional sign representation of m . Representations of the dioperad Pp := P ⊗ p in a vector space V are the same as representations of the dioperad P in V [p]. If P is a dioperad, then P := {sgnm ⊗P(m, n)[2−m−n]⊗sgnn } and −1 P :== {sgnm ⊗ P(m, n)[m + n − 2] ⊗ sgnn } are also dioperads.
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2.1. Cobar dual. If T is a directed (i.e. provided with a flow which we always assume in our pictures to go from the bottom to the top) tree, we denote by • V ert (T ) the set of all vertices, • edge(T ) the set of internal edges; det(T ) := ∧|edge(T )| spank (edge(T )); • Edge(T ) is the set of all edges, i.e. Edge(T ) := edge(T ) {input legs (leaves)} {output legs (roots)}; Det(T ) := ∧|Edge(T )| spank (Edge(T )); • Out (v) (resp. I n(v)) the set of outgoing (resp. incoming) edges at a vertex v ∈ V ert (V ). An (m, n)-tree is a tree T with n input legs labeled by the set [n] = {1, . . . , n} and m output legs labeled by the set [m] = {1, . . . , m}. A tree T is called trivalent if |Out (v) I n(v)| = 3 for all v ∈ V ert (T ). Let E = {E(m, n)}m,n≥1 be a collection of finite dimensional (m , n ) bimodules with E1,1 = 0. For a pair of finite sets, I, J ∈ Obj ects(Sf ), with |I | = m and |J | = n, one defines E(I, J ) := H omSf ([m], I ) × m E(m, n)) × n H omSf (J, [n]). The free dioperad, F ree(E), generated by E is defined by E(T ), F ree(E)(m, n) := (m,n)−trees T
where E(T ) :=
E(Out (v), I n(v)),
v∈V ert (T )
and the compositions i ◦j are given by grafting the j th root of one tree into the i th leaf of another tree, and then taking the “unordered” tensor product [MSS] over the set of vertices of the resulting tree. Let P = {P(m, n)}m,n≥1 be a collection of graded (m , n ) bimodules. We denote ¯ ¯ by P¯ the collection {P(m, n)}m,n≥1 given by P(m, n) := P(m, n) for m + n ≥ 3 ¯ ¯ and P(1, 1) = 0. The collection of dual vector spaces, P¯ ∗ = {P(m, n)∗ }m,n≥1 , is naturally a collection of (m , n )-bimodules with the transposed actions. We also set ¯ ¯ P ∨ = {P(m, n)∨ := sgnm ⊗ P(m, n)∗ ⊗ sgnn }. Let P be a graded dioperad with zero differential. The cobar dual of P is the dg dioperad DP defined by (i) as a dioperad of graded vector spaces, DP = −1 F ree(P¯ ∗ [−1]) = F ree( −1 P¯ ∗ [−1]); DP −i (m, n) with (ii) as a complex, DP is non-positively graded, DP(m, n) = m+n−3 i=0 the differential given by dualizations of the compositions • ◦• and edge contractions [G, GiKa], d
DP 3−m−n (m, n)→ || d P¯ ∨ (m, n) →
d
DP 4−m−n (m, n) → || d P¯ ∗ ⊗ Det(T )→
|edge(T )|=1
d
d
DP 3−m−n (m, n) →· · ·→ DP 0 (m, n) || || d d P¯ ∗ ⊗ Det(T )→· · ·→ P¯ ∗ ⊗ Det(T )
|edge(T )|=2
where the sums are taken over (m, n)-trees.
trivalent trees T
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Remark 2.2. The vector space DP is bigraded: one grading comes from the grading of P as a vector space and another one from trees as in (ii) just above. The differential preserves the first grading and increases by 1 the second one. The Z-grading of DP is always understood to be the associated total grading. In particular, degDP P¯ ∨ (m, n) = degVect (P¯ ∨ (m, n)[m + n − 3]). 2.3. Koszul dioperads. A quadratic dioperad is a dioperad P of the form P=
F ree(E) , I deal < R >
where E = {E(m, n)} is a collection of finite dimensional (m , n )-bimodules with E(m, n) = 0 for (m, n) = (1, 2), (2, 1), and the I deal in F ree(E) is generated by a collection, R, of three sub-bimodules R(1, 2) ⊂ F ree(E)(1, 2), R(2, 1) ⊂ F ree(E)(2, 1) and R(2, 2) ⊂ F ree(E)(2, 2). The quadratic dual dioperad, P ! , is then defined by P! =
F ree(E ∨ ) , I deal < R ⊥ >
where R ⊥ is the collection of the three sub-bimodules R ⊥ (i, j ) ⊂ F ree(E ∨ )(i, j ) which are annihilators of R(i, j ), (i, j ) = (1, 2), (2, 2), (2, 1). Clearly, DP 0 = F ree(E ∨ ) so that there is a natural epimorphism DP 0 −→ P ! . Its kernel is precisely Im d(DP −1 ). Hence H 0 (DP) = P ! . The quadratic operad P is called Koszul if the above morphism is a quasi-isomorphism, i.e. H i (DP) = 0 for all i < 0. In that case the operad DP ! provides us with a minimal resolution of the operad P and is often denoted by P∞ . Algebras over P∞ are often called strong homotopy P-algebras; their most important property is that they can be transferred via quasi-isomorphisms of complexes [Mar2].
2.4. Koszulness criterion. An (m, n)-tree T is called reduced if each vertex has • either an outgoing root or at least two outgoing internal edges, and/or • either an incoming leaf or at least two incoming internal edges. For a collection, E = {Em,n }m,n≥1 , of (m , n )-bimodules define another collection of (m , n )-bimodules as follows, F ree(E)(m, n) := E(T ). reduced (m,n)−trees
Let P be a quadratic dioperad, i.e. P = F ree(E)/I deal
for some generators E = {E(1, 2), E(2, 1)} and relations R = {R(1, 3), R(2, 2), R(3, 1)}. With P one can canonically associate two quadratic operads, PL and PR , such that PL =
F ree(E(1, 2)) F ree(E(2, 1)) op , PR = . I deal < R(1, 3) > I deal < R(2, 1) >
128
S.A. Merkulov op
Let us denote by PL PR the collection of (m , n )-bimodules given by if m = 1, n ≥ 1; PL (1, n) op op PL PR (m, n) := PL (m, 1) if n = 1, m ≥ 1; 0 otherwise. Theorem 2.4.1. [G, Mar1, MV]. A quadratic dioperad P is Koszul if the operads PL and PR are Koszul and op
P(i, j ) = F ree(PL PR )(i, j ) op
for (i, j ) = (1, 3), (2, 2), (3, 1). Moreover, in this case P(m, n)=F ree(PL PR )(m, n) for all m, n ≥ 1. 3. A Minimal Resolution of Lie1 Bi First we present a graph description of the dioperad Lie1 Bi; it will pay off when we discuss Lie1 Bi∞ . By definition (see Sect. 1.5), Lie1 Bi is a quadratic dioperad, Lie1 Bi =
F ree(E) , I deal < R >
where (i) E(2, 1) := sgn2 ⊗ 11 and E(1, 2) := 11 ⊗ 12 [−1], where 1n stands for the one dimensional trivial representation of n ; let δ ∈ E(2, 1) and [ • ] ∈ E(1, 2) be basis vectors; we can represent both as directed3 plane corollas, δ=
1?? 2
◦
1
[ • ] = •??
,
1
1
2
with the following symmetries, 1?? 2
2?? 1 ◦ = − ◦ 1
1
, 1
1
1
? ? •? = •? ; 2
2
1
(ii) the relations R are generated by the following elements, 1<< 2
3<< 1 2<< 3 ◦? 3 ◦? 2 ◦? 1 ◦ + ◦ + ◦ ∈ F ree(E)(3, 1),
•? + •? + •? ∈ F ree(E)(1, 3), = ? = ? = ? • = 3 • = 2 • = 1
1
2
3
1
2
3
1?? 2
2 1 2 1 ◦ 177 •33 277 •33 177 •33 277 •33 3 3 3 3 •9 − ◦ 2 + ◦ 2 − ◦ 1 + ◦ 1 ∈ F ree(E)(2, 2). 9
1 3
2
1
1
2
2
In all our graphs the direction of edges is chosen to go from the bottom to the top.
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Proposition 3.1. Lie1 Bi is Koszul. Proof. We have Lie1 BiL = Lie ⊗ {1} and Lie1 BiR = Lie, where Lie stands for the operad of Lie algebras and {m} := {m}(n) := sgn⊗m n [m(n − 1)] n≥1 for the endomorphism operad of k[−m]. As Lie is Koszul [GiKa], both the operads Lie1 BiL and Lie1 BiR are Koszul as well. Next, a straightforward analysis of all calculational schemes in Lie1 Bi represented by directed trivalent (i, j )-trees with i + j = 5 shows that they generate no new relations so that Lie1 Bi(i, j ) = F ree Lie ⊗ {1} Lieop (i, j ) for (i, j ) = (1, 3), (2, 2), (3, 1). Hence by Theorem 2.4.1, the dioperad Lie1 Bi is Koszul. Proposition 1.5.1 is a straightforward corollary of the following Theorem 3.2. The minimal resolution, Lie1 Bi∞ , of the dioperad Lie1 Bi can be described as follows: (i) As a dioperad of graded vector spaces, Lie1 Bi∞ = F ree(E), where the collection, E = {E(m, n)}, of one dimensional (m , n )-modules is given by sgnm ⊗ 1n [m − 2] if m + n ≥ 3; E(m, n) := 0 otherwise. (ii) If we represent a basis element of E(m, n) by the unique (up to a sign) planar (m, n)-corolla, 1KK2<· · · m−1sm KK
n−1 n
with skew-symmetric outgoing legs and symmetric ingoing legs, then the differential d is given on generators by I
1KK2<. . . m−1sm
d
KK
1 2
n−1
2 KKK<<. . . v I1 K< vv KKK<<. . . kKkK<•vH 5 5vH KK
J1
where σ (I1 I2 ) is the sign of the shuffle I1 I2 = (1, . . . , m). Proof. Claim (i) follows from the fact that Lie1 Bi! (m, n) = 1m ⊗ sgnn [1 − n] and Remark 2.2. Claim 2 is a straightforward though tedious graph translation of the initial term, d 1 ! ∗ (Lie1 Bi! )∨ (m, n) −→ (m,n)−trees T (Lie Bi ) ⊗ Det(T ), |edge(T )|=1
of Definition 2.1 of the differential d in DLie1 Bi! .
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S.A. Merkulov
3.3. A geometric model for Lie1 Bi∞ structures. Let V be a finite dimensional graded vector space. Then the graded formal manifold, M, modeled on the infinitesimal neighbourhood of 0 in the vector space V ⊕ V ∗ [1] has an odd symplectic form ω induced from the natural pairing V ⊗ V ∗ [1] → k[1]. In particular, the graded structure sheaf OM on M has a degree −1 Poisson bracket, { • }, such that {f • g} = (−1)|f ||g|+|f |+|g| {g • f } and the Jacobi identity is satisfied. The odd symplectic manifold (M, ω) has two particular Lagrangian submanifolds, L ⊂ M and L ⊂ M associated with, respectively, the subspaces 0 ⊕ V ∗ [1] ⊂ V ⊕ V ∗ [1] and V ⊕ 0 ⊂ V ⊕ V ∗ [1]. Proposition 3.3.1. A Lie1 Bi∞ algebra structure in a graded vector space V is the same as a degree two smooth function ∈ OM vanishing on L ∪ L and satisfying the equation { • } = 0. Proof. The manifold M is isomorphic to the total space of the shifted cotangent bundle, TM∗ [1], of the manifold M of Proposition 1.5.1. Hence smooth functions on M are the same as smooth polyvector fields on M, and the Poisson bracket { • } on M is the same as the Schouten bracket on M. 3.4. Lie1 Bi∞ morphisms. Let (V , {µm,n }) and (V , {µm,n }) be two Lie1 Bi∞ algebras. Definition 3.4.1. A Lie1 Bi∞ morphism F : V → V is, by definition, a symplectomorphism, F : (M, ω) → (M , ω ) such that F (L) ⊂ L , F (L) ⊂ L and F ∗ = . Thus a Lie1 Bi∞ morphism F : V → V is a pair of collections of linear maps, Fm,n : m V ⊗ ∧n V ∗ → V [−n] m≥1,n≥0 , ∗ F¯m,n : m V ⊗ ∧n V ∗ → V [1 − n] n≥0,m≥1 () satisfying the system equations, F ∗ (ω ) = ω and F ∗ = . In particular, the equation F ∗ (ω ) = ω says that the linear maps, ∗ ∗ F1,0 : (V , µ1,1 ) → (V , µ1,1 ) and F¯0,1 : (V ∗ , µ∗1,1 ) → (V , µ 1,1 ),
are morphisms of complexes, while the equation F ∗ (ω ) = ω says that the composition, ∗ F0,1 ◦ F¯1,0 : V −→ V
is the identity map. Definition 3.4.2. A Lie1 Bi∞ -morphism F : V → V is called a quasi-isomorphism if the morphisms of complexes ∗ ∗ F1,0 : (V , µ1,1 ) → (V , µ1,1 ) and F¯0,1 : (V ∗ , µ∗1,1 ) → (V , µ 1,1 )
induce isomorphisms in cohomology.
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Remark 3.4.3. One might get an impression that the notions introduced above make sense only for finite dimensional Lie1 Bi∞ algebras. However, this is no more than an artifact of the geometric intuition we tried to rely on in our definitions. In fact, everything above (and below) make sense for infinite dimensional representations as well. For example, one can replace () by Fm,n : m V → ∧n V ⊗ V [−n] m≥1,n≥0 , F¯m,n : m V ⊗ V [n − 1] → ∧n V n≥0,m≥1 , and reinterpret the equations defining the Lie1 Bi∞ morphism accordingly. For example, with this reinterpretation it is the morphism F1,0 ◦ F¯0,1 which is the identity map. 3.4.4. Contractible and minimal Lie1 Bi∞ -structures. Let V be a graded vector space and (M = V ⊕V ∗ [1], ω) the associated odd symplectic manifold (as in Sect. 3.3). There is a one-to-one correspondence between differentials, d : V → V , and quadratic degree 2 function, quad , on (M, ω), vanishing on L ∪ L and satisfying [ quad • quad ] = 0. If H ∗ (V , d) = 0, the associated data, (M, ω, quad ), is called a contractible Lie1 Bi∞ structure on V . A Lie1 Bi∞ structure, (M, ω, ), on V is called minimal if = 0 mod I 3 , where I is the ideal of the distinguished point, L ∩ L, in M. Put another way, the formal power series in some (and hence any) coordinate system on M begins with cubic terms at least. Theorem 3.4.5. (Homotopy classification of Lie1 Bi∞ -structures, cf. [Ko1, Ko2]). Each Lie1 Bi∞ algebra is isomorphic to the tensor product of a contractible Lie1 Bi∞ algebra and a minimal one. Proof. Let (M, ω, ) be the geometric equivalent of any given Lie1 Bi∞ algebra. To prove the statement we have to construct coordinates, (x a , y a , zα , ψa , φa , ξα ), on M such that a a (i) ω = (dx ∧ dψα + dy ∧ dφα + dzα ∧ dξα ,
a
α
ω2
ω1 ya
a (ii) L is given = zα = 0 while L is given by ψa = φa = ξα = 0, by x = a α (iii) = y ψa +(z , ξα ) for some formal power series (zα , ξα ) which begins a 2 1
with cubic terms at least. For then (M, ω, ) (M1 , ω1 , 1 ) × (M2 , ω2 , 2 ) with the first factor being a contractible Lie1 Bi∞ structure while the second factor is a minimal one. We shall establish existence of the above coordinates by induction. As the first step of the induction procedure we choose arbitrary linear coordinates, {t A }, A ∈ {1, . . . , dim V }, on V and the associated dual coordinates, {χA }, |χA | = −|t A | + 1, on V ∗ [1]. The odd symplectic form is given in these coordinates as ω = A A A dt ∧ dχA , L is given by t = 0 while L is given by χA = 0. Then, = CBA t A χB mod I 3 , A,B
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for some constants CBA which are nothing but the coefficients of the differential, d : V → V , associated to the quadratic bit of . As we work over the field of characteristic zero, we can choose a cohomological decomposition of V with respect to this differential, V = H (V , d) ⊕ B ⊕ B[−1], so that d vanishes on H (V , d) and B[−1] and, on the remaining summand B, it is equal to the natural isomorphism B → B[−1]. Let {zα } be some linear coordinates in H (V , d), {y a } linear coordinates on B and {x a }, |x a | = |y a | − 1, the associated (via the natural isomorphism) linear coordinates in B[−1]. Denote by (ξα , ψa , φa ) the coordinates on V ∗ [1] dual to (zα , x a , y a ). In the resulting coordinate system on M the conditions (i)–(ii) are satisfied, while the condition (iii) is satisfied modulo I 3 . Assume by induction that we have constructed a coordinate system on M in which conditions (i)–(iii) are satisfied modI N+1 . Then we have, = y a ψa + ≤N (zα , ξα ) + N+1 (x, y, z, ψ, φ, χ ) mod I N+2 . a polynomials of polynomial of degree N+1 degrees from 3 to N 1
The equation [ • ] = 0 mod I N+2 implies, δ N+1 = 0, where δ is the following differential on OM , δ : OM −→ O M f −→ [ a y a ψa • f ]. Let B ∈ OM be an arbitrary polynomial of degree N + 1 and with |B| = 1. It gives rise to a symplectomorphism, F : M → M, given as exp vB with the vector field vB defined by vB : OM −→ OM f −→ [B • f ]. One has, F ∗ =
y a ψa + ≤N (zα , ξα ) + N+1 + δB
mod I N+2 .
a
Thus N+1 (x, y, z, ψ, φ, χ ) is a δ-cycle defined up to a δ-coboundary. As cohomology of δ in OM is equal to k[[zα , ξα ]], one can always find N+1 such that it is a function of {zα , ξα } only. Corollary 3.4.6. If F : V → V is a Lie1 Bi∞ quasi-isomorphism, then there exists a Lie1 Bi∞ quasi-isomorphism G : V → V such that on the cohomology level [F1,0 ] = ¯ 0,1 ]∗ and [G1,0 ] = [F¯0,1 ]∗ . [G Proof. is exactly the same as the proof of an analogous statement for L∞ algebras in [Ko1].
PROP Profile of Poisson Geometry
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4. Minimal Resolution of the Operad TF By definition (see Sect. 1.4), TF is a quadratic dioperad TF =
F ree(E) , I deal < R >
where (i) E(2, 1) := k[2 ] ⊗ 11 and E(1, 2) := 11 ⊗ 12 [−1]; we represent two basis vectors of k[2 ] ⊗ 11 by planar corollas 1?? 2
2?? 1 ◦ and ◦ 1
1
and a basis vector of E(1, 2) by the symmetric corolla, 1
1
1
? ? •? = •? ; 2
2
1
(ii) the relations R are generated by the following elements, •? + •? ∈ F ree(E)(1, 3), •? + == ?3 == ?2 == ?1 • • •
1
2
3
1
2
3
1?? 2
2 2 1 1 ◦ 177 •33 177 •33 2 7 • • 7 77 2 3 3 •99 − ◦ 1 − ◦ 2 − 1 ◦ − 2 ◦ ∈ F ree(E)(2, 2).
1
2
2
1
2
1
Proposition 1.4.1 follows immediately from the following Proposition 4.1. The minimal resolution, TF∞ , of the dioperad TF can be described as follows: (i) As a dioperad of graded vector spaces, TF∞ = F ree(E), where the collection, E = {E(m, n)}, of (m , n )-modules is given by k[2 ] ⊗ 1n if m = 2, n ≥ 2; E(m, n) := 1n [−1] if m = 1, n ≥ 2; 0 otherwise. (ii) If we represent two basis elements of E(2, n) by planar (2, n)-corollas, and the basis element of E(1, n) by planar (1, n) corolla,
1<
2< 1 << 2 << 1 <<K <<K • • •
1 2
n−1
1 2
n−1
1 2
n−1
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with symmetric ingoing legs, then the differential d is given on generators by, 1 1
d
•
1 2
n−1
kk•H55H kkk 55HHH k k = s•
J1 1<
1<
<< 2 <<K • d sss
n−1
2
<< <H5 kkk•5H55HH k k kkk . . . H •
J1 2
1
1< 2 H5 << v •?S?S?SSS kkk•5H55HH v k ? v k S ss<•
J1
J1
Proof. Using criterion 2.4 it is easy to see that the dioperad TF is Koszul. Then the cobar dual DTF! provides the required minimal resolution. The rest is a straightforward calculation. 5. A Comment on Lie1 Bi∞ Algebras It was shown in [G] that the dioperad, LieBi∞ , of (usual) Lie bialgebras is Koszul so that its minimal resolution, LieBi∞ , can be constructed using the techniques reviewed in Sect. 2. Here we present its explicit graph description; in fact, we prefer to show LieBi∞ 1. Proposition 5.1. The dioperad LieBi∞ 1 can be described as follows. (i) As a dioperad of graded vector spaces, LieBi∞ 1 = F ree(E), where the collection, E = {E(m, n)}, of one dimensional (m , n )-modules is given by 1m ⊗ 1n [2m − 3] if m + n ≥ 3; E(m, n) := 0 otherwise. (ii) If we represent a basis element of E(m, n) by the unique space (m, n)-corolla, 1KK2<. . . m−1sm
KK
1 2
n−1
PROP Profile of Poisson Geometry
135
then the differential d is given on generators by, I
1KK2<. . . m−1sm
d
KK
1 2
n−1
2 KKK<<. . . v K< vv KKK<<. . . kKkK<•vH 5 5vH k KK
I1 I2 =(1,... ,m) J1 J2 =(1,... ,n) |I1 |≥0,|I2 |≥1 |J1 |≥1,|J2 |≥0
J1
Let V be a graded vector space, and let M be the graded formal manifold isomorphic to the neighbourhood of zero in V [1] ⊕ V ∗ [1]. The manifold M has a natural even symplectic structure ω induced from the pairing V [1] ⊗ V ∗ [1] → k[2]; it also has two particular Lagrangian submanifolds, L and L , modeled on the subspaces 0 ⊕ V ∗ [1] ⊂ V [1] ⊕ V ∗ [1] and, respectively, V ⊕ 0 ⊂ V [1] ⊕ V ∗ [1]. The symplectic form induces degree −2 Poisson bracket, { , }, on the structure sheaf, OM , of smooth functions on M. The following result has been independently obtained by Lyubashenko [Lyu]. Corollary 5.2. A LieBi∞ algebra structure in a graded vector space V is the same as a degree 3 smooth function f ∈ OM vanishing on L ∪ L and satisfying the equation {f, f } = 0. LieBi ∞ morphisms and quasi-isomorphisms are defined exactly as in 3.4.1 and 3.4.2; then an obvious analogue of Theorem 3.4.5 holds true. We omit the details. References [A] Adams, J.F.: Infinite loop spaces. Princeton NJ: Princeton University Press, 1978 [G] Gan, W.L.: Koszul duality for dioperads. Math. Res. Lett. 10, 109–124 (2003) [GetJo] Getzler, E., Jones, J.D.S.: Operads, homotopy algebra, and iterated integrals for double loop spaces. http://arxiv.org/list/hep-th/9403055, 1994 [GiKa] Ginzburg, V., Kapranov, M.: Koszul duality for operads. Duke Math. J. 76, 203–272 (1994) [He] Hertling, C.: Frobenius manifolds and moduli spaces for singularities. Cambridge: Cambridge University Press, 2002 [HeMa] Hertling, C., Manin, Yu.I.: Weak Frobenius manifolds. Intern. Math. Res. Notices 6, 277–286 (1999) [Lyu] Lyubashenko, V.: Private communication [Ko1] Kontsevich, M.: Deformation quantization of Poisson manifolds I. Lett. Math.Phys. 66, 157–216 (2003) [Ko2] Kontsevich, M.: Topics in algebra-deformation theory. Berkeley Lectures 1995 (Unpublished notes by A. Weinstein) [Mar1] Markl, M.: Distributive laws and Koszulness. Ann. Inst. Fourier, Grenoble 46, 307–323 (1996) [Mar2] Markl, M.: Homotopy algebras are homotopy algebras. http://arxiv.org/list/math.AT/9907138, 1999 [MSS] Markl, M., Shnider, S., Stasheff, J.D.: Operads in Algebra, Topology and Physics. Providence, RI: AMS, 2002 [MV] Markl, M., Voronov, A.A.: PROPped up graph cohomology. http://arxiv.org/list/math.QA/ 0307081, 2003 [Mer1] Merkulov, S.A.: Operads, deformation theory and F -manifolds. In: Frobenius manifolds, quantum cohomology, and singularities, eds. C. Hertling, M. Marcolli, Wiesbaden: Vieweg 2004 [Mer2] Merkulov, S.A.: Nijenhuis infinity and contractable dg manifolds. http://arxiv.org/list/ math.AG/040324, 2004 to appear in Compositio Math [St] Stasheff, J.D.: On the homotopy associativity of H -spaces, I II. Trans. Amer. Math. Soc. 108, 272–292 & 293–312 (1963) Communicated by A. Connes
Commun. Math. Phys. 262, 137–159 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1457-8
Communications in
Mathematical Physics
The Holevo Capacity of Infinite Dimensional Channels and the Additivity Problem M.E. Shirokov Steklov Mathematical Institute, 119991 Moscow, Russia. E-mail: [email protected] Received: 28 January 2005 / Accepted: 2 June 2005 Published online: 29 October 2005 – © Springer-Verlag 2005
Abstract: The Holevo capacity of an arbitrarily constrained infinite dimensional quantum channel is considered and its properties are discussed. The notion of optimal average state is introduced. The continuity properties of the Holevo capacity with respect to constraint and to the channel are explored. The main result of this paper is the statement that additivity of the Holevo capacity for all finite dimensional channels implies its additivity for all infinite dimensional channels with arbitrary constraints. 1. Introduction The Holevo capacity (in what follows, χ -capacity) of a quantum channel is an important characteristic defining the amount of classical information which can be transmitted by this channel using nonentangled encoding and entangled decoding, see e.g. [8, 10, 21]. For additive channels the χ -capacity coincides with the full classical capacity of a quantum channel. At present the main interest is focused on quantum channels between finite dimensional quantum systems. But having in mind possible applications, it is necessary to deal with infinite dimensional quantum channels, in particular, Gaussian channels. In this paper the χ -capacity for an arbitrarily constrained infinite dimensional quantum channel is considered. It is shown that despite nonexistence of an optimal ensemble in this case it is possible to define the notion of the optimal average state for such a channel, inheriting important properties of the optimal average state for finite dimensional channels (Proposition 1). A “minimax” expression for the χ -capacity is obtained and an alternative characterization of the image of the optimal average state as the minimum point of a lower semicontinuous function on a compact set is given (Proposition 2). The notion of the χ -function of an infinite dimensional quantum channel is introduced. It is shown that the χ -function of an arbitrary channel is a concave lower semicontinuous function with natural chain properties, having continuous restriction to any set of continuity of the output entropy (Propositions 3-4). This and the result in [12] imply
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continuity of the χ -function for Gaussian channels with power constraint (Example 1). For the χ -function the analog of Simon’s dominated convergence theorem for quantum entropy is also obtained (Corollary 3). The question of continuity of the χ -capacity as a function of channel is considered. It is shown that the χ -capacity is a continuous function of channel in the finite dimensional case while in general it is only lower semicontinuous (Theorem 1, Example 2). The above results make it possible to obtain the infinite dimensional version of Theorem 1 in [11], which shows equivalence of several formulations of the additivity conjecture (Theorem 2). The main result of this paper is the statement that additivity of the χ-capacity for all finite dimensional channels implies its additivity for all infinite dimensional channels with arbitrary constraints (Theorem 3). This is done in two steps by using several results (Lemma 5, Propositions 5 and 6). These results are also applicable to analysis of individual pairs of channels as it is demonstrated in the proof of additivity of the χ -capacity for two arbitrarily constrained infinite dimensional channels with one of them noiseless or with entanglement breaking (Proposition 7). 2. Basic Quantities Let H be a separable Hilbert space, B(H) be the set of all bounded operators on H with the cone B+ (H) of all positive operators, T(H) be the Banach space of all trace-class operators with the trace norm · 1 and S(H) be the closed convex subset of T(H) consisting of all density operators on H, which is a complete separable metric space with the metric defined by the trace norm. Each density operator uniquely defines a normal state on B(H) [1], so in what follows we will also for brevity use the term “state”. Note that convergence of a sequence of states to a state in the weak operator topology is equivalent to convergence of this sequence to this state in the trace norm [3]. In what follows log denotes the function on [0, +∞), which coincides with the logarithm on (0, +∞) and vanishes at zero. Let A and B be positive trace class operators. Let {|i} be a complete orthonormal set of eigenvectors of A. The entropy is defined by H (A) = − i i| A log A |i while the relative entropy — as H (A B) = 1 i i| (A log A − A log B + B − A) |i, provided ranA ⊆ ranB, and H (A B) = +∞ otherwise (see [16, 17] for a more detailed definition). The entropy and the relative entropy are nonnegative lower semicontinuous (in the trace-norm topology) concave and convex functions of their arguments correspondingly [29]. Arbitrary finite collection {ρi } of states in S(H) with corresponding set of probabil ities {πi } is called ensemble and is denoted by = {πi , ρi }. The state ρ¯ = i πi ρi is called the average state of the above ensemble. Following [12] we treat an arbitrary Borel probability measure π on S(H) as a generalized ensemble and the barycenter of the measure π defined by the Pettis integral ρπ(dρ) ρ(π ¯ )= S(H)
as the average state of this ensemble. In these notations the conventional ensembles correspond to measures with finite support. For an arbitrary subset A of S(H) we denote by PA the set of all probability measures with barycenters contained in A. 1
ran denotes the closure of the range of an operator in H.
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In analysis of the χ -capacity we shall use Donald’s identity [6, 19] n
πi H (ρi ρ) ˆ =
i=1
n
πi H (ρi ρ) ¯ + H (ρ ¯ ρ), ˆ
(1)
i=1
which holds for an arbitrary ensemble {πi , ρi } of n states with the average state ρ¯ and arbitrary state ρ. ˆ Let H, H be a pair of separable Hilbert spaces which we shall call correspondingly input and output space. A channel is a linear positive trace preserving map from T(H) to T(H ) such that the dual map ∗ : B(H ) → B(H) (which exists since is bounded [5]) is completely positive. Let A be an arbitrary closed subset of S(H). We consider a constraint on the input ensemble {πi , ρi }, defined by the requirement ρ¯ ∈ A. The channel with this constraint is called the A -constrained channel. We define the χ-capacity of the A-constrained channel as (cf.[9–11]) ¯ C(; A) = sup χ ({πi , ρi }), ρ∈ ¯ A
where χ ({πi , ρi }) =
(2)
πi H ((ρi )(ρ)). ¯
i
In [12] it is shown that the χ -capacity of the A-constrained channel can be also defined by ¯ H ((ρ)(ρ(π ¯ )))π(dρ), (3) C(; A) = sup π∈PA
S(H)
which means coincidence of the above supremum over all measures in PA with the supremum over all measures in PA with finite support. ¯ The χ -capacity C(; S(H)) of the unconstrained channel is also denoted by ¯ C(). The χ -function of the channel is defined by ¯ {ρ}) = sup πi H ((ρi )(ρ)). (4) χ (ρ) = C(; i
πi ρi =ρ
i
The χ-function of the finite dimensional channel is a continuous concave function on S(H) [11]. The properties of the χ -function of an arbitrary infinite dimensional channel are considered in Sect. 4. 3. The Optimal Average State It is a known fact that for an arbitrary finite dimensional channel and an arbitrary closed set A there exists an optimal ensemble {πi , ρi } on which the supremum in definition (2) of the χ -capacity is achieved [4, 22]. The image of the average state of this optimal ensemble plays an important role in the analysis of finite dimensional channels [11]. For a general infinite dimensional constrained channel there are no reasons for existence of an optimal ensemble (with a finite number of states). In this case it is natural
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to introduce the notion of optimal generalized ensemble (optimal measure) on which the supremum in the definition (3) of the χ-capacity is achieved. In [12] a sufficient condition for existence of an optimal measure for an infinite dimensional constrained channel is obtained and the example of the channel with no optimal measure is given. The aim of this section is to show that even in the case of nonexistence of an optimal generalized ensemble we can define the notion of “optimal average state”, inheriting the basic properties of the average state of the optimal ensemble for the finite dimensional constrained channel. Using this notion we can generalize some results of [11] to the infinite dimensional case. Definition 1. A sequence of ensembles {πik , ρik } with the average ρ¯ k ∈ A such that ¯ lim χ ({πik , ρik }) = C(; A)
k→+∞
is called the approximating sequence for the A-constrained channel . A state ρ¯ is called an optimal average state for the A-constrained channel if this state ρ¯ is a limit of the sequence of the average states of some approximating sequence of ensembles for the A-constrained channel . This definition admits that an optimal average state may not exist or may not be unique. If A is a compact set then the set of optimal average states for the A-constrained channel is nonempty. It turns out that in the case of the convex compact set A all optimal average states have the same image. ¯ Proposition 1. Let A be a convex compact subset of S(H) such that C(; A) < +∞. ¯ = (, A) for Then there exists the unique state (, A) in S(H ) such that (ρ) the arbitrary optimal average state ρ¯ for the A-constrained channel . The state (, A) is the unique state in S(H ) such that ¯ µj H ((σj ) (, A)) ≤ C(; A) j
for any ensemble {µj , σj } with the average state σ¯ ∈ A. Note that the second part of this proposition is a generalization of Proposition 1 in [11], where the version of the “maximal distance property” of an optimal ensemble [22] adapted to the case of a constrained channel is presented. Since the assumed compactness of the set A implies existence of at least one optimal average state, Proposition 1 is proved by combining the two following lemmas. ¯ Lemma 1. Let A be a convex set such that C(; A) < +∞ and ρ¯ be an optimal average state for the A-constrained channel . Then ¯ µj H ((σj )(ρ)) ¯ ≤ C(; A) j
for any ensemble {µj , σj } with the average state σ¯ ∈ A. Proof. The proof of this lemma is a generalization of the proof of Proposition 1 in [11]. Let {πik , ρik } be an approximating sequence of ensembles such that ρ¯ = limk→+∞ ρ¯ k , and let {µj , σj } be an arbitrary ensemble of m states with the average σ¯ ∈ A. Consider the mixture k k ηk = {(1 − η)π1k ρ1k , ..., (1 − η)πn(k) ρn(k) , ηµ1 σ1 , ..., ηµm σm },
η ∈ [0, 1]
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of the ensemble {µj , σj } with an ensemble {πik , ρik } of the above approximating sequence, consisting of n(k) states. We obtain the sequence of ensembles with the corresponding sequence of the average states ρ¯ηk = (1 − η)ρ¯ k + ησ¯ ∈ A converging to the state ρ¯η = (1 − η)ρ¯ + ησ¯ ∈ A as k → +∞. For arbitrary k we have χ
ηk
= (1 − η)
n(k)
πik H ((ρik )(ρ¯ηk )) + η
i=1
m j =1
µj H ((σj )(ρ¯ηk )).
(5)
¯ By assumption C(; A) < +∞ both sums in the right side of the above expression are finite. Applying Donald’s identity (1) to the first sum in the right side we obtain n(k)
πik H ((ρik )(ρ¯ηk )) = χ (0k ) + H ((ρ¯ k )(ρ¯ηk )).
i=1
Substitution of the above expression into (5) gives χ ηk = χ (0k ) + (1 − η)H ((ρ¯ k )(ρ¯ηk )) m +η µj H ((σj )(ρ¯ηk )) − χ (0k ) . j =1
Due to nonnegativity of the relative entropy it follows that m j =1
µj H ((σj )(ρ¯ηk )) ≤ η−1 χ ηk − χ 0k + χ 0k , η = 0. (6)
By definition of the approximating sequence we have ¯ lim χ 0k = C(; A) ≥ χ ηk k→+∞
(7)
for all k. It follows that
lim inf lim inf η−1 χ ηk − χ 0k ≤ 0. η→+0 k→+∞
(8)
Lower semicontinuity of the relative entropy [29] with (6),(7) and (8) imply m
µj H ((σj )(ρ)) ¯ ≤ lim inf lim inf
η→+0 k→+∞
j =1
m j =1
¯ µj H ((σj )(ρ¯ηk )) ≤ C(; A).
¯ Lemma 2. Let A be a set such that C(; A) < +∞ and ρ be a state in S(H ) such that ¯ µj H ((σj ) ρ ) ≤ C(; A) j
for any ensemble {µj , σj } with the average σ¯ ∈ A. Then for the arbitrary approximating sequence {πik , ρik } of ensembles ρ = limk→+∞ (ρ¯ k ).
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Proof. Let {πik , ρik } an approximating sequence of ensembles with the corresponding sequence of the average states ρ¯ k . By assumption we have ¯ πik H ((ρik ) ρ ) ≤ C(; A). i
Applying Donald’s identity (1) to the left side we obtain πik H ((ρik ) ρ ) = πik H ((ρik )(ρ¯ k )) + H ((ρ¯ k ) ρ ). i
i
From the two above expressions we have ¯ 0 ≤ H ((ρ¯ k ) ρ ) ≤ C(; A) −
πik H ((ρik )(ρ¯ k )).
i
But the right side tends to zero as k tends to infinity due to the approximating property of the sequence {πik , ρik }. Proposition 1 provides the following generalization of Corollary 1 in [11]. ¯ Corollary 1. Let A be a convex compact set such that C(; A) < +∞ then ¯ C(; A) ≥ χ (ρ) + H ((ρ)(, A)) f or arbitrary state ρ in A. Proof. Let {πi , ρi } be an arbitrary ensemble such that i πi ρi = ρ ∈ A. By Proposition 1, ¯ πi H ((ρi )(, A)) ≤ C(; A). i
This inequality and Donald’s identity πi H ((ρi )(, A)) = χ ({πi , ρi }) + H ((ρ)(, A)) i
complete the proof.
¯ Corollary 2. Let A be a convex compact set such that C(; A) < +∞. Then ¯ H ((ρ)(, A)) ≤ C(; A) f or arbitrary state ρ in A. For the arbitrary approximating sequence {πik , ρik } of ensembles lim (ρ¯ k ) = (, A).
k→+∞
The first assertion of the corollary directly follows from Proposition 1 while the second is proved by using Lemma 2. There exists another approach to the definition of the state (, A). For the arbitrary ensemble {µj , σj } with the average σ¯ ∈ A consider the lower semicontinuous function F{µj ,σj } (ρ ) = j µj H ((σj ) ρ ) on the set (A). The function F (ρ ) = supj µj σj ∈A F{µj ,σj } (ρ ) is also lower semicontinuous on the compact set (A) and, hence, achieves its minimum on this set. The following proposition asserts, in particular, that the state (, A) can be defined as the unique minimal point of the function F (ρ ).
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¯ Proposition 2. Let A be a convex compact set such that C(; A) < +∞. The χ-capacity of the A-constrained channel can be expressed as ¯ C(; A) = min sup µj H ((σj ) ρ ) , ρ ∈(A)
j
µj σj ∈A j
and (, A) is the only state on which the minimum in the right side is achieved. Proof. We will show first that ¯ sup µj H ((σj )(, A)) = C(; A). j
(9)
µj σj ∈A j
It follows from Proposition 1 that “≤” takes place in (9). Let {πik , ρik } be an approximating sequence. By Donald’s identity (1) we have
πik H ((ρik )(, A)) =
i
πik H ((ρik )(ρ¯ k )) + H ((ρ¯ k )(, A)).
i
¯ The first term in the right side tends to C(; A) as k tends to infinity due to the approximating property of the sequence {πik , ρik }, while the second one is nonnegative. This implies “≥” and, hence, “=” in (9). Let be a minimal point of the function F (ρ ) introduced before Proposition 2. By (9) it follows that ¯ sup µj H ((σj ) ) = F ( ) ≤ F ((, A)) = C(; A). j
µj σj ∈A j
By Proposition 1 this implies that = (, A).
Note that the expression for the χ -capacity in the above proposition can be considered as a generalization of the “mini-max formula for χ ∗ ” in [22] to the case of an infinite dimensional constrained channel. Remark 1. Propositions 1-2 and Corollaries 1-2 do not hold without assumption of convexity of the set A. To show this it is sufficient to consider the noiseless channel = Id and the compact set A, consisting of two states ρ1 and ρ2 such that H (ρ1 ) = H (ρ2 ) < ¯ +∞ and H (ρ1 ρ2 ) = +∞. In this case C(; A) = H (ρ1 ) = H (ρ2 ), the states ρ1 and ρ2 are optimal average states in the sense of Definition 1 with the different images (ρ1 ) = ρ1 and (ρ2 ) = ρ2 . Moreover the first assertion of Corollary 2 is false in the ¯ following extreme form: C(; A) < H ((ρ1 )(ρ2 )) = H (ρ1 ρ2 ) = +∞. 4. The χ -Function The function χ (ρ) on S(H) is defined by (4). It is shown in [12] that χ (ρ) = sup H ((σ )(ρ))π(dσ ), π∈P{ρ}
S(H)
(10)
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where P{ρ} is the set of all probability measures on S(H) with the barycenter ρ, and that under the condition H ((ρ)) < +∞ the supremum in (10) is achieved on some measure supported by pure states. Note that H ((ρ)) = +∞ does not imply χ (ρ) = +∞. Indeed, it is easy to construct a channel from a finite dimensional system into an infinite dimensional one such that H ((ρ)) = +∞ for any ρ ∈ S(H).2 On the other hand, by the monotonicity property of the relative entropy [18] πi H ((ρi )(ρ)) ≤ πi H (ρi ρ) ≤ log dim H < +∞ i
i
for the arbitrary ensemble {πi , ρi }, and hence χ (ρ) ≤ log dim H < +∞ for any ρ ∈ S(H). For the arbitrary state ρ such that H ((ρ)) < +∞ the χ -function has the following representation χ (ρ) = H ((ρ)) − Hˆ (ρ), where Hˆ (ρ) = inf
π∈P{ρ}
S(H)
H ((ρ))π(dρ) = inf i
πi ρi =ρ
(11)
πi H ((ρi ))
(12)
i
is a convex closure of the output entropy H ((ρ)) (this is proved in [24]).3 Note that the notion of the convex closure of the output entropy is widely used in the quantum information theory in connection with the notion of the entanglement of formation (EoF). Namely, in the finite dimensional case EoF was defined in [2] as the convex hull (=convex closure) of the output entropy of a partial trace channel from the state space of a bipartite system onto the state space of its single subsystem. In the infinite dimensional case the definition of EoF as the σ -convex hull of the output entropy of a partial trace channel is proposed in [7] while some advantages of the definition of EoF as the convex closure of the output entropy of a partial trace channel are considered in [24]. It is shown that the two above definitions coincide on the set of states with finite entropy of partial trace, but their coincidence for an arbitrary state remains an open problem. In the finite dimensional case the output entropy H ((ρ)) and its convex closure (=convex hull) Hˆ (ρ) are continuous concave and convex functions on S(H) correspondingly and representation (11) is valid for all states. It follows that in this case the function χ (ρ) is continuous and concave on S(H). In the infinite dimensional case the output entropy H ((ρ)) is only lower semicontinuous and, hence, the function χ (ρ) is not continuous even in the case of the noiseless channel , for which χ (ρ) = H ((ρ)). But it turns out that the function χ (ρ) for the arbitrary channel has properties similar to the properties of the output entropy H ((ρ)). Proposition 3. The function χ (ρ) is a nonnegative concave and lower semicontinuous function on S(H). If the restriction of the output entropy H ((ρ)) to a particular subset A ⊆ S(H) is continuous then the restriction of the function χ (ρ) to this subset A is continuous as well. For example, the channel : ρ → 21 ρ ⊕ 21 Tr(ρ)τ , where τ is a fixed state with infinite entropy. Note that the second equality in (12) holds under the condition H ((ρ)) < +∞, it is not valid in general (see Lemma 2 in [24] and the notes below). 2 3
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The proof of this proposition is based on the following lemma. Lemma 3. Let {πi , ρi } be an arbitrary ensemble of m states with the average state ρ and let {ρn } be an arbitrary sequence of states converging to the state ρ. There exists the sequence {πin , ρin } of ensembles of m states such that lim πin = πi ,
lim ρin = ρi ,
n→+∞
n→+∞
and ρn =
m
πin ρin .
i=1
Proof. Without loss of generality we may assume that πi > 0 for all i. Let D ⊆ H be −1 the support of ρ = m i=1 πi ρi and P be the projector onto D. Since ρi ≤ πi ρ we have 0 ≤ Ai ≡ ρ −1/2 ρi ρ −1/2 ≤ πi−1 I, where we denote by ρ −1/2 the generalized (Moore-Penrose) inverse of the operator ρ 1/2 (equal 0 on the orthogonal complement to D). 1/2 1/2 1/2 1/2 Consider the sequence Bin = ρn Ai ρn + ρn (IH − P )ρn of operators in B(H). Since limn→+∞ ρn = ρ = Pρ in the trace norm, we have lim Bin = ρ 1/2 Ai ρ 1/2 = ρi
n→+∞
in the weak operator topology. The last equality implies Ai = 0. Note that TrBin = TrAi ρn + Tr(IH − P )ρn < +∞ and hence lim TrBin = TrAi ρ = Trρi = 1.
n→+∞
Denote by ρin = (TrBin )−1 Bin a state and by πin = πi TrBin a positive number for each i, then limn→+∞ πin = πi and limn→+∞ ρin = ρi in the weak operator topology and hence, by the result in [3], in the trace norm. Moreover, m i=1
πin ρin =
m i=1
πi Bin = ρn ρ −1/2 1/2
m
πi ρi ρ −1/2 ρn
1/2
1/2
1/2
+ ρn (IH − P )ρn
= ρn .
i=1
Proof of Proposition 3. Nonnegativity of the χ -function is obvious. Let us show first the concavity property of the χ -function. Note that for a convex set of states with finite output entropy this concavity easily follows from (11). But to prove concavity on the whole state space we will use a different approach. Let ρ and σ be arbitrary states. By definition for arbitrary ε > 0 there exist ensembles {πi , ρi }ni=1 and {µj , σj }m states ρ and σ correspondingly such that j =1 with the average π H ((ρ )(ρ)) > χ (ρ) − ε and i i i j µj H ((σj )(σ )) > χ (σ ) − ε. Taking the mixture {(1 − η)π1 ρ1 , ..., (1 − η)πn ρn , ηµ1 σ1 , ..., ηµm σm },
η ∈ [0, 1]
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of the above two ensembles we obtain the ensemble with the average state (1−η)ρ +ησ . By using Donald’s identity (1) we have χ ((1 − η)ρ + ησ ) ≥ (1 − η) πi H ((ρi )((1 − η)ρ + ησ )) +η
i
µj H ((σj )((1 − η)ρ + ησ )) = (1 − η)
j
+(1 − η)H ((ρ)((1 − η)ρ + ησ )) + η
πi H ((ρi )(ρ))
i
µj H ((σj )(σ ))
j
+ηH ((σ )((1 − η)ρ + ησ ) ≥ (1 − η) +η
πi H ((ρi )(ρ))
i
µj H ((σj )(σ )) ≥ (1 − η)χ (ρ) + ηχ (σ ) − ε,
j
where nonnegativity of the relative entropy was used. Since ε can be arbitrary small the concavity of the χ -function is established. To prove lower semicontinuity of the χ -function we have to show lim inf χ (ρn ) ≥ χ (ρ0 )
(13)
n→+∞
for arbitrary state ρ0 and arbitrary sequence ρn converging to this state ρ0 . For arbitrary ε > 0 let {πi , ρi } be an ensemble with the average ρ0 such that πi H ((ρi )(ρ0 )) ≥ χ (ρ0 ) − ε. i
By Lemma 3 there exists the sequence of ensembles {πin , ρin } of fixed size such that lim πin = πi ,
n→+∞
lim ρin = ρi ,
n→+∞
By definition we have lim inf n→+∞ χ (ρn ) ≥ lim inf n→+∞ ≥
i
and
ρn =
m
πin ρin .
i=1
i
πin H ((ρin )(ρn ))
πi H ((ρi )(ρ0 )) ≥ χ (ρ0 ) − ε,
where lower semicontinuity of the relative entropy [29] was used. This implies (13) (due to the freedom of the choice of ε). The last assertion of Proposition 3 follows from the representation (11) and from lower semicontinuity of the function Hˆ (ρ) established in [24]. The similarity of the properties of the functions χ (ρ) and H ((ρ)) is stressed by the following analog of Simon’s dominated convergence theorem for quantum entropy [28], which will be used later. Corollary 3. Let ρn be a sequence of states in S(H), converging to the state ρ and such that λn ρn ≤ ρ for some sequence λn of positive numbers, converging to 1. Then lim χ (ρn ) = χ (ρ).
n→+∞
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Proof. The condition λn ρn ≤ ρ implies decomposition ρ = λn ρn + (1 − λn )ρn , where ρn = (1 − λn )−1 (ρ − λn ρn ) is a state. By concavity of the χ -function we have χ (ρ) ≥ λn χ (ρn ) + (1 − λn )χ (ρn ) ≥ λn χ (ρn ), which implies lim supn→+∞ χ (ρn ) ≤ χ (ρ). This and lower semicontinuity of the χ-function completes the proof. Example 1. Let H be a positive unbounded operator on the space H such that Tr exp (−βH ) < +∞ for all β > 0 and h be a positive number. In the proof of Proposition 3 in [12] continuity of the restriction of the output entropy H ((ρ)) to the subset Ah = {ρ ∈ S(H) | Tr (ρ)H ≤ h } was established.4 By Proposition 3 the restriction of the χ -function to the set Ah is continuous. As it is mentioned in [12], the above continuity condition is fulfilled for Gaussian channels with the power constraint of the form TrρH ≤ h, where H = R T R is the many-mode oscillator Hamiltonian with nondegenerate energy matrix and R are the canonical variables of the system. We shall use the following chain properties of the χ -function. Proposition 4. Let : S(H) → S(H ) and : S(H ) → S(H ) be two channels. Then χ◦ (ρ) ≤ χ (ρ)
and
χ◦ (ρ) ≤ χ ((ρ)) f or arbitrary ρ in S(H).
Proof. The first inequality follows from the monotonicity property of the relative entropy [18] and (4), while the second one is a direct corollary of the definition (4) of the χ function. 5. On Continuity of the χ -Capacity In this section the question of continuity of the χ -capacity as a function of channel is considered. Dealing with this question we must choose a topology on the set C(H, H ) of all quantum channels from S(H) into S(H ). This choice is essential only in the infinite dimensional case because all locally convex Hausdorff topologies on a finite dimensional space are equivalent. Let L(H, H ) be the linear space of all continuous linear mapping from T(H) into T(H ). We will use the topology on C(H, H ) ⊂ L(H, H ) generated by the topology of strong convergence on L(H, H ). Definition 2. The topology on the linear space L(H, H ) defined by the family of seminorms {ρ = (ρ)1 }ρ∈T(H) is called the topology of strong convergence. Since an arbitrary operator in T(H) can be represented as a linear combination of operators in S(H) it is possible to consider only seminorms · ρ corresponding to ρ ∈ S(H) in the above definition. Note that a sequence n of channels in C(H, H ) strongly converges to a channel ∈ C(H, H ) if and only if limn→+∞ n (ρ) = (ρ) for all ρ ∈ S(H). Due to the result in [3] the above limit may be in the weak operator topology. 4 The value Tr (ρ)H is defined as a limit of nondecreasing sequence Tr (ρ)Q H , where Q is n n the spectral projector of H corresponding to the lowest n eigenvalues [10].
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Theorem 1. Let A be an arbitrary closed and convex subset of S(H).5 ¯ In the case of finite dimensional spaces H and H the χ -capacity C(, A) is a con tinuous function on the set C(H, H ). If n is an arbitrary sequence of channels in C(H, H ), converging to some channel in C(H, H ), then there exists lim (n , A) = (, A).
n→∞
(14)
¯ In general the χ -capacity C(, A) is a lower semicontinuous function on the set C(H, H ) equipped with the topology of strong convergence. Proof. Let us first show lower semicontinuity of the χ-capacity. Let ε > 0 and λ be an arbitrary net of channels, strongly converging to the channel , and {πi , ρi } be an ensem¯ ble with the average ρ¯ such that χ ({πi , ρi }) > C(, A) − ε. By lower semicontinuity of the relative entropy [29], ¯ πi H (λ (ρi )λ (ρ)) ¯ ≥ πi H ((ρi )(ρ)) ¯ > C(, A) − ε. lim inf λ
i
i
This implies ¯ λ , A) ≥ C(, ¯ A). lim inf C( λ
It follows that ¯ n , A) ≥ C(, ¯ A) lim inf C( n→+∞
(15)
for an arbitrary sequence n of channels strongly converging to a channel . Now to prove the continuity of the χ -capacity in the finite dimensional case it is sufficient to show that for the above sequence of channels ¯ ¯ n , A) ≤ C(, A). lim sup C(
(16)
n→+∞
For an arbitrary A-constrained channel from C(H, H ) there exists an optimal ensemble consisting of m = (dim H)2 states (probably, some states with zero weights) [4, 22]. Let P be the compact space of all probability distributions with m outcomes. Consider the compact space6 PCm = P × S(H) × · · · × S(H),
m
consisting of sequences ({πi }m i=1 , ρ1 , ..., ρm ), corresponding to an arbitrary input ensemble {πi , ρi }m of m states. i=1 Suppose (16) is not true. Without loss of generality we may assume that ¯ n , A) > C(, ¯ A). lim C(
n→+∞
(17)
Let {πin , ρin }m i=1 be an optimal ensemble for the A-constrained channel n . By compactnk nk ness of PCm we can choose a subsequence ({πink }m i=1 , ρ1 , ..., ρm ) converging to some 5 6
Convexity of A is used only in the proof of (14). with product topology.
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m ∗ ∗ element ({πi∗ }m i=1 , ρ1 , ..., ρm ) of the space PC . By definition of the product topology m on PC it means that
lim πink = πi∗ ,
k→∞
lim ρ nk k→∞ i
= ρi∗ .
The average state of the ensemble {πi∗ , ρi∗ }m i=1 is a limit of the sequence of average states of the ensembles {πink , ρink }m and hence lies in A (which is closed by the assumption). i=1 By continuity of the quantum entropy in finite dimensional case we have ¯ nk , A) = lim χn ({π nk , ρ nk }) = χ ({πi∗ , ρi∗ }) ≤ C(, ¯ lim C( A), k→+∞
k→+∞
k
i
i
which contradicts (17). Comparing (15) and (16) we see that ¯ n , A) = C(, ¯ A). lim C( n→+∞
It follows that the above ensemble {πi∗ , ρi∗ }m i=1 is optimal for the A-constrained channel . Hence, there exists the optimal average state ρ¯ ∗ for the A-constrained channel which is a partial limit of the sequence {ρ¯ n } of the optimal average states for the A-constrained channels n . Suppose (14) is not true. Without loss of generality we may (by compactness argument) assume that there exists limn→∞ (n , A) = (, A) . By Proposition 1 this contradicts the previous observation. The assumption of finite dimensionality in the first part of Theorem 1 is essential. The following example shows that generally the χ -capacity is not a continuous function of a channel even in the stronger trace norm topology on the space of all channels. The example is a purely classical channel which has a standard extension to a quantum one. Example 2. Consider the Abelian von Neumann algebra l∞ and its predual l1 . Let q {n ; n = 1, 2, ...; q ∈ (0, 1)} be the family of classical unconstrained channels defined by the formula ∞ q n ({x1 , x2 , ..., xn , ...}) = {(1 − q) ∞ i=1 xi , q i=n+1 xi , qx1 , ..., qxn , 0, 0, ...} for {x1 , x2 , ..., xn , ...} ∈ l1 . Defining 0 ({x1 , x2 , ..., xn , ...}) = { ∞ i=1 xi , 0, 0, ...} we have ∞ ∞ q (n − 0 )({xi }∞ i=1 xi , i=n+1 xi , x1 , ..., xn , 0, 0, ...}1 i=1 )1 = q{− = q(|
∞
i=1 xi | + |
∞
i=n+1 xi | + |x1 | + · · · + |xn |)
≤ 3q{xi }∞ i=1 1 ,
q
hence n − 0 → 0 as q → 0 uniformly in n. q
To evaluate the χ -capacity of the channel n it is sufficient to note that q
H (n (any pure state)) = h2 (q) = −q log q − (1 − q) log(1 − q) and q
q
1 1 1 H (n (any state)) ≤ H (n ({ n+1 , n+1 , ..., n+1 , 0, 0...})) = q log(n + 1) + h2 (q).
n+1
¯ qn ) C(
= q log(n + 1), q ∈ (0, 1), n ∈ N. It follows by definition that Take arbitrary C such that 0 < C ≤ +∞ and choose a sequence q(n) such that q(n) limn→∞ q(n) = 0 while limn→∞ q(n) log(n+1) = C . Then we have limn→∞ n q(n) 0 0 ¯ ¯ n ) = C > 0 = C( − = 0 but limn→∞ C( ).
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Remark 2. The above example demonstrates harsh discontinuity of the χ -capacity in the infinite dimensional case. One can see that a similar discontinuity underlies Shor’s construction [27] allowing to prove equivalence of different additivity properties by using channel extension and a limiting procedure. 6. Additivity for Constrained Channels Let : S(H) → S(H ) and : S(K) → S(K ) be two channels with the constraints, defined by closed subsets A ⊂ S(H) and B ⊂ S(K) correspondingly. For the channel ⊗ we consider the constraint defined by the requirements ω¯ H := Tr K ω¯ ∈ A and ω¯ K := Tr H ω¯ ∈ B, where ω¯ is the average state of an input ensemble {µi , ωi }. The closed subset of S(H ⊗ K) consisting of states ω such that Tr K ω ∈ A and Tr H ω ∈ B will be denoted A ⊗ B. The application of the results of Sect. 3 to the A ⊗ B-constrained channel ⊗ is based on the following lemma. Lemma 4. The set A ⊗ B is a convex subset of S(H ⊗ K) if and only if the sets A and B are convex subsets of S(H) and of S(K) correspondingly. The set A ⊗ B is a compact subset of S(H ⊗ K) if and only if the sets A and B are compact subsets of S(H) and of S(K) correspondingly. Proof. The first statement of this lemma is trivial. To prove the second, note that compactness of the set A ⊗ B implies compactness of the sets A and B due to continuity of the partial trace. The proof of the converse implication is based on the following characterization of a compact set of states: a closed subset A of S(H) is compact if and only if for any ε > 0 there exists a finite dimensional projector Pε such that TrPε ρ > 1 − ε for all ρ ∈ A. This characterization can be deduced by combining results of [20] and [3] (see the proof of the lemma in [10]). Its proof is also presented in the Appendix of [12]. Let A and B be compact. By the above characterization for arbitrary ε > 0 there exist finite rank projectors Pε and Qε such that TrPε ρ > 1 − ε, ∀ρ ∈ A Since
ωH
∈ A and
ωK
and
TrQε σ > 1 − ε, ∀σ ∈ B.
∈ B for arbitrary ω ∈ A ⊗ B we have
Tr((Pε ⊗ Qε ) · ω) = Tr((Pε ⊗ IK ) · ω) − Tr(Pε ⊗ (IK − Qε )) · ω) ≥ TrPε ωH − Tr(IK − Qε )ωK > 1 − 2ε. The above characterization implies compactness of the set A ⊗ B.
The conjecture of additivity of the χ -capacity for the A-constrained channel and the B-constrained channel is [11, 12] ¯ ¯ C¯ ( ⊗ ; A ⊗ B) = C(; A) + C(; B).
(18)
Remark 3. Let ρ¯ and σ¯ be the optimal average states for the A-constrained channel and the B-constrained channel correspondingly. The additivity (18) implies that the state ρ¯ ⊗ σ¯ is an optimal average state for the A⊗B-constrained channel ⊗. Indeed, the tensor product of ensembles of the approximating sequence for the A-constrained channel with ensembles of the approximating sequence for the B-constrained channel provides (due to (18)) an approximating sequence of ensembles for the A ⊗ B-constrained channel ⊗ .
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The results of the previous sections make it possible to obtain the following infinite dimensional version of Theorem 1 in [11]. Theorem 2. Let : S(H) → S(H ) and : S(K) → S(K ) be arbitrary channels. The following properties are equivalent: (i) Eq. (18) holds for arbitrary subsets A ⊆ S(H) and B ⊆ S(K) such that H ((ρ)) < +∞ for all ρ ∈ A and H ((σ )) < +∞ for all σ ∈ B; (ii) inequality χ⊗ (ω) ≤ χ (ωH ) + χ (ωK )
(19)
holds for an arbitrary state ω such that H ((ωH )) < +∞ and H ((ωK )) < +∞; (iii) inequality Hˆ ⊗ (ω) ≥ Hˆ (ωH ) + Hˆ (ωK )
(20)
holds for an arbitrary state ω such that H ((ωH )) < +∞ and H ((ωK )) < +∞. Proof. (i) ⇒ (iii). Let ω be an arbitrary state with finite H ((ωH )) and H ((ωK )). The validity of (i) implies ¯ ¯ C¯ ⊗ ; {ωH } ⊗ {ωK } = C(; {ωH }) + C(; {ωK }). By Remark 3 the state ωH ⊗ ωK is the optimal average state for the {ωH } ⊗ {ωK } -constrained channel ⊗ . By Lemma 4 the set {ωH } ⊗ {ωK } is a convex compact subset of S(H ⊗ K). Noting that ω ∈ {ωH } ⊗ {ωK } and applying Corollary 1 we obtain ¯ ¯ {ωH }) + C(; {ωK }) χ (ωH ) + χ (ωK ) = C(; = C¯ ⊗ ; {ωH } ⊗ {ωK }
(21)
≥ χ⊗ (ω) + H (( ⊗ )(ω)(ωH ) ⊗ (ωK )). Due to H (( ⊗ )(ω)(ωH ) ⊗ (ωK )) = H ((ωH )) + H ((ωK )) − H (( ⊗ )(ω)) the inequality (21) together with (11) implies (20). (iii) ⇒ (ii). It can be derived from expression (11) for the χ -function and subadditivity of the (output) entropy. (ii) ⇒ (i). It follows from the definition of the χ -capacity (2) and inequality (19) that ¯ ¯ C¯ ( ⊗ ; A ⊗ B) ≤ C(; A) + C(; B). Since the converse inequality is obvious, there is equality here.
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The validity of inequality (19) for arbitrary ω ∈ S(H ⊗ K) seems to be substantially stronger than the equivalent properties in Theorem 2. This property is called subadditivity of the χ -function for the channels and . By using arguments from the proof of Theorem 2 it is easy to see that subadditivity of the χ-function for the channels and is equivalent to validity of Eq. (18) for arbitrary subsets A ⊆ S(H) and B ⊆ S(K). By using Proposition 6 below it is possible to show that properties (i)–(iii) in the above theorem are equivalent to subadditivity of the χ -function for the channels and having the following property: H ((ρ)) < +∞ and H ((σ )) < +∞ for arbitrary finite rank states ρ ∈ S(H) and σ ∈ S(K). We see later (Proposition 7) that the set of quantum infinite dimensional channels for which the subadditivity of the χ -function holds is nontrivial. Remark 4. By Theorem 1 in [11] the subadditivity of the χ -function for the arbitrary finite dimensional channels and is equivalent to validity of inequality (20) for the arbitrary state ω ∈ S(H ⊗ K), which implies additivity of the minimal output entropy inf
ω∈S(H⊗K)
H ( ⊗ (ω)) =
inf
ρ∈S(H)
H ((ρ)) +
inf
σ ∈S(K)
H ((σ ))
(22)
for these channels. This follows from the inequality H ( ⊗ (ω)) ≥ Hˆ ⊗ (ω) ≥ Hˆ (ωH ) + Hˆ (ωK ) ≥
inf
ρ∈S(H)
H ((ρ)) +
inf
σ ∈S(K)
H ((σ )),
(23)
valid for the arbitrary state ω ∈ S(H ⊗ K) for which inequality (20) holds. In contrast to this in the infinite dimensional case we can not prove the above implication (without some additional assumptions). The problem consists in the existence of pure states in S(H ⊗ K) with infinite entropies of partial traces, which can be called superentangled. To show this note first that the monotonicity property of the relative entropy [18] provides the following inequality H (ωH ) + H (ωK ) − H (ω) = H ω ωH ⊗ ωK ≥ H ⊗ (ω) ωH ⊗ ωK = H ωH + H ωK − H ( ⊗ (ω)), which shows that H (ωH ) = H (ωK ) < +∞ implies H ((ωH )) < +∞ and H ((ωK )) < +∞ for the arbitrary pure state ω ∈ S(H ⊗ K) with finite output entropy H ( ⊗ (ω)). By this and Theorem 2 the subadditivity of the χ -function for arbitrary infinite dimensional channels and implies validity of inequality (20) and hence validity of inequality (23) for all pure states ω such that H (ωH ) = H (ωK ) < +∞ and H ( ⊗ (ω)) < +∞. So if we considered only such states ω in the calculation of the minimal output entropy for the channel ⊗ , we would obtain that it is equal to the sum of inf ρ∈S(H) H ((ρ)) and inf σ ∈S(K) H ((σ )), but this additivity can be (probably) broken by taking into account superentangled states.
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7. Generalization of the Additivity Conjecture The main aim of this section is to show that the conjecture of additivity of the χ -capacity for arbitrary finite dimensional channels implies the additivity of the χ -capacity for arbitrary infinite dimensional channels with arbitrary constraints. It is convenient to introduce the following notation. The channel is – FF-channel if dim H < +∞ and dim H < +∞; – FI-channel if dim H < +∞ and dim H ≤ +∞. Speaking about the quantum channel without reference to FF or FI we will assume that dim H ≤ +∞ and dim H ≤ +∞. Let : S(H) → S(H ) be an arbitrary channel such that dim H = +∞ and Pn be a sequence of finite rank projectors in H increasing to IH and Hn = Pn (H ). Consider the channel (24) n (ρ) = Pn (ρ)Pn + Tr(IH − Pn )(ρ) τn from S(H) into S(Hn ⊕ Hn ) ⊂ S(H ), where τn is a pure state in some finite dimensional subspace Hn of H Hn . If dim H < +∞ we will assume that n = for all n. Note that for the arbitrary FI-channel the corresponding channel n is a FF-channel for all n. For arbitrary channel : S(K) → S(K ) we will consider the sequences n and n ⊗ of channels as approximations for the channels and ⊗ correspondingly. Despite the discontinuity of the χ -capacity as a function of a channel in the infinite dimensional case the following result is valid. Lemma 5. Let and be arbitrary channels. If subadditivity of the χ -function holds for the channel n defined by (24) and the channel for all n, then subadditivity of the χ -function holds for the channels and . Proof. The channel n can be represented as the composition n ◦ of the channel with the channel n : S(H ) → S(Hn ⊕ Hn ) defined by n (ρ ) = Pn ρ Pn + Tr(IH − Pn )ρ τn . Proposition 4 implies χn (ρ) = χn ◦ (ρ) ≤ χ (ρ),
∀ρ ∈ S(H),
∀n ∈ N.
Since lim n (ρ) = (ρ),
n→+∞
∀ρ ∈ S(H),
it follows from Theorem 1 that lim inf χn (ρ) ≥ χ (ρ), n→+∞
∀ρ ∈ S(H).
The two above inequalities imply lim χn (ρ) = χ (ρ),
n→+∞
∀ρ ∈ S(H).
(25)
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It is easy to see that n ⊗ (ω) = (Pn ⊗ IK ) · ( ⊗ (ω)) · (Pn ⊗ IK ) +τn ⊗ Tr H ((IH − Pn ) ⊗ IK ) · ( ⊗ (ω)) ,
∀ω ∈ S(H ⊗ K).
Hence lim n ⊗ (ω) = ⊗ (ω),
n→+∞
∀ω ∈ S(H ⊗ K),
and by Theorem 1 we have lim inf χn ⊗ (ω) ≥ χ⊗ (ω), n→+∞
∀ω ∈ S(H ⊗ K).
(26)
By the assumption χn ⊗ (ω) ≤ χn (ωH ) + χ (ωK ),
∀ω ∈ S(H ⊗ K),
∀n ∈ N.
This, (25) and (26) imply χ⊗ (ω) ≤ χ (ωH ) + χ (ωK ),
∀ω ∈ S(H ⊗ K).
Proposition 5. Subadditivity of the χ -function for all FF-channels implies subadditivity of the χ -function for all FI-channels. Proof. This can be proved by double application of Lemma 5. First, we prove the subadditivity of the χ-function for any two channels, when one of them is of FI-type while another is of FF-type. Second, we remove the FF restriction from the last channel. Now we will turn to channels with an infinite dimensional input quantum system. We will use the following notion of subchannel. Definition 3. The restriction of a channel : S(H) → S(H ) to the set of states with support contained in a subspace H0 of the space H is called the subchannel 0 of the channel , corresponding to the subspace H0 . It is easy to see that subadditivity of the χ -function for the channels and implies subadditivity of the χ -function for arbitrary subchannels 0 and 0 of the channels and . The properties of the χ -function established in Sect. 4 make it possible to prove the following important result. Proposition 6. Let : S(H) → S(H ) and : S(K) → S(K ) be arbitrary channels. Subadditivity of the χ -function for any two FI-subchannels of the channels and implies subadditivity of the χ -function for the channels and . Proof. It is sufficient to consider the case dim H = +∞, dim K ≤ +∞. Let ω be an dim K arbitrary state in S(H ⊗ K). Let {|ϕk }+∞ k=1 and {|ψk }k=1 be ONB of eigenvectors H K of the compact positive operators ω and ω such that the corresponding sequences of eigenvalues are nonincreasing. Let Pn = nk=1 |ϕk ϕk | and Qn = nk=1 |ψk ψk |. In the case dim K < +∞ we will assume Qn = IK for all n ≥ dim K. The nondecreasing sequences {Pn } and {Qn } of finite rank projectors converge to IH and to IK correspondingly in the strong operator topology. Let Hn = Pn (H) and Kn = Qn (K).
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Consider the sequence of states ωn = (Tr ((Pn ⊗ Qn ) · ω))−1 (Pn ⊗ Qn ) · ω · (Pn ⊗ Qn ), which are well defined for all n by the choice of the projectors Pn and Qn . Since obviously lim ωn = ω,
(27)
n→+∞
Proposition 3 implies lim inf χ⊗ (ωn ) ≥ χ⊗ (ω). n→+∞
(28)
The next part of the proof is based on the following operator inequalities: λn ωnH ≤ ωH ,
λn ωnK ≤ ωK ,
where
λn = Tr ((Pn ⊗ Qn ) · ω) .
(29)
Let us prove the first inequality. By the choice of Pn and due to suppωnH ⊆ Hn it is sufficient to show that λn ωnH ≤ Pn ωH . Let ϕ ∈ Hn . By the definition of a partial trace, ϕ|λn ωnH |ϕ =
dim K
ϕ ⊗ ψk |Pn ⊗ Qn · ω · Pn ⊗ Qn |ϕ ⊗ ψk
k=1
=
m
ϕ ⊗ ψk |ω|ϕ ⊗ ψk ≤
k=1
dim K
ϕ ⊗ ψk |ω|ϕ ⊗ ψk = ϕ|ωH |ϕ,
k=1
where m = min{n, dim K}. The second inequality is proved the same way. By using (27) and applying Corollary 3 due to (29) we obtain lim χ (ωnH ) = χ (ωH )
n→+∞
and
lim χ (ωnK ) = χ (ωK ).
n→+∞
(30)
For each n the {ωnH }-constrained channel and the {ωnK }-constrained channel can be considered as FI-subchannels of the channels and corresponding to the subspaces Hn and Kn . Hence by the assumption, χ⊗ (ωn ) ≤ χ (ωnH ) + χ (ωnK ),
∀n ∈ N.
This, (28) and (30) imply χ⊗ (ω) ≤ χ (ωH ) + χ (ωK ). It is known that additivity of the χ -capacity for all unconstrained FF-channels is equivalent to subadditivity of the χ -function for all FF-channels [11, 27]. By combining this with Proposition 5 and Proposition 6 we obtain the following extension of the additivity conjecture. Theorem 3. The additivity of the χ -capacity for all FF-channels implies additivity of the χ -capacity for all channels with arbitrary constraints.
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This theorem and Theorem 2 imply the following result concerning superadditivity of the convex closure of the output entropy for infinite dimensional channels. Note that in the case of a partial trace channel the convex closure of the output entropy coincides with the entanglement of formation (EoF). Corollary 4. If inequality (20) holds for all FF-channels and and all states ω then inequality (20) holds for all channels and and all states ω such that H ((ωH )) < +∞ and H ((ωK )) < +∞. Proof. The validity of inequality (20) for two FF-channels and and for all states ω is equivalent to subadditivity of the χ -function for these channels [11]. Hence the assumption of the corollary and Theorem 3 imply subadditivity of the χ -function for any channels, which, by Theorem 2, implies the validity of inequality (20) for all channels and and all states ω such that H ((ωH )) < +∞ and H ((ωK )) < +∞. Remark 5. By combining Shor’s theorem in [27] and Theorem 3 we obtain that additivity of the minimal output entropy (22) for all FF-channels implies additivity of the χ-capacity (18) for all channels with arbitrary constraints. But due to existence of superentangled states (see Remark 4) we can not show that it implies additivity of minimal output entropy for all channels. So, in the infinite dimensional case the conjecture of additivity of the minimal output entropy for all channels seems to be substantially stronger than the conjecture of additivity of the χ -capacity for all channels with arbitrary constraints. Note that in contrast to Proposition 5, Proposition 6 relates the subadditivity of the χ-function for the initial channels with the subadditivity of the χ -function for its FIsubchannels (not any FI-channels!). This makes it applicable for analysis of individual channels as it is illustrated in the proof of Proposition 7 below. We will use the following natural generalization of the notion of entanglement breaking a finite dimensional channel [15]. Definition 4. A channel : S(H) → S(H ) is called entanglement breaking if for an arbitrary Hilbert space K and for an arbitrary state ω in S(H ⊗ K) the state ⊗ Id(ω) lies in the closure of the convex hull of all product states in S(H ⊗ K), where Id is the identity channel from S(K) onto itself. Generalizing the result in [15] it is possible to show that a channel : S(H) → S(H ) is entanglement-breaking if and only if it admits the representation (ρ) = ρ (x)µρ (dx), X
where X is a complete separable metric space, ρ (x) is a Borel S(H )-valued function on X and µρ (A) = Tr(ρM(A)) for any Borel A ⊂ X, with M positive operator valued measure on X [14]. The following proposition is a generalization of Proposition 2 in [11]. Proposition 7. Let be an arbitrary channel. The subadditivity of the χ -function holds in each of the following cases: (i) is a noiseless channel;
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(ii) is an entanglement breaking channel; (iii) is a direct sum mixture (cf.[11]) of a noiseless channel and a channel 0 such that the subadditivity of the χ-function holds for 0 and (in particular, an entanglement breaking channel). Proof. In the proof of each point of this proposition for FF-channels the finite dimensionality of the underlying Hilbert spaces was used (cf.[26, 11]). The idea of this proof consists in using our extension results (Proposition 6 and Lemma 5). (i) Note that any FI-subchannel of an arbitrary noiseless channel is a noiseless FFchannel. Hence by Proposition 6 it is sufficient to prove the subadditivity of the χ -function for arbitrary noiseless FF-channel and the arbitrary FI-channel . But this can be done with the help of Lemma 5. Indeed, using this lemma with the noiseless FF-channel in the role of the fixed channel we can deduce the above assertion from the subadditivity of the χ -function for arbitrary two FF-channels with one of them noiseless (Proposition 2 in [11]). (ii) Note that any FI-subchannel of an arbitrary entanglement breaking channel is entanglement breaking. Hence by Proposition 6 it is sufficient to prove the subadditivity of the χ -function for an arbitrary entanglement breaking FI-channel and an arbitrary FI-channel . Similar to the proof of (i) this can be done with the help of Lemma 5, but in this case it is necessary to apply this lemma twice. First we prove the subadditivity of the χ -function for arbitrary entanglement breaking FIchannel and arbitrary FF-channel by noting that any FF-channel n , involved in lemma 5, inherits the entanglement breaking property from the channel and using the subadditivity of the χ -function for arbitrary two FF-channels with one of them entanglement breaking [26]. Second, by using the result of the first step we remove the FF restriction from another channel . (iii) Note that any FI-subchannel of the channel q = qId ⊕ (1 − q)0 has the same structure with FF-channel Id and FI-channel 0 . By the remark before Proposition 6 subadditivity of the χ -function for the channels 0 and implies subadditivity of the χ -function for their arbitrary subchannels. Hence by Proposition 6 it is sufficient to prove (iii) for the FI-channel q and the FI-channel . Let ω be a state in S(H ⊗ K) with dim H < +∞ and dim K < +∞. It follows that χId (ωH ) = H (ωH ) < +∞. By the established subadditivity of the χ-function for the FF-channel Id and the FI-channel and by the assumed subadditivity of the χ -function for the FI-channel 0 and the FI-channel we have χId⊗ (ω) ≤ χId (ωH ) + χ (ωK ) and χ0 ⊗ (ω) ≤ χ0 (ωH ) + χ (ωK ). Using this and Lemma 3 in [11]7 we obtain χq ⊗ (ω) ≤ qχId⊗ (ω) + (1 − q)χ0 ⊗ (ω) ≤ qχId (ωH ) + qχ (ωK ) + (1 − q)χ0 (ωH ) + (1 − q)χ (ωK ) = qH (ωH ) + (1 − q)χ0 (ωH ) + χ (ωK ) = χq (ωH ) + χ (ωK ), 7 This lemma implies that for arbitrary channels and from S(H) to S(H ) and to S(H ) 1 2 1 2 correspondingly one has
χq1 ⊕(1−q)2 ({πi , ρi }) = qχ1 ({πi , ρi }) + (1 − q)χ2 ({πi , ρi }) for the arbitrary ensemble {πi , ρi } of states in S(H) and arbitrary q ∈ [0; 1].
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where the last equality follows from the existence of the approximating sequence of pure state ensembles for the {ωH }-constrained FI-channel 0 . Acknowledgements. The author is grateful to A. S. Holevo for the idea of this work and for continued help, and to M.B.Ruskai for useful remarks. The author also acknowledges support from the QIS Program, the Newton Institute, Cambridge, where this paper was completed. The work was also partially supported by INTAS grant 00-738.
Note. After this paper had been completed it was found that the compactness assumption in Propositions 1 and 2 can be taken off. The generalized versions of these propositions and of Proposition 3 are presented in [25]. References 1. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. vol.I, New YorkHeidelberg-Berlin, Springer Verlag, 1979 2. Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed State Entanglement and Quantum Error Correction. Phys. Rev. A 54, 3824–3851 (1996) 3. Dell’Antonio, G.F.: On the limits of sequences of normal states. Commun. Pure Appl. Math. 20, 413–430 (1967) 4. Davies, E.B.: Information and Quantum Measurements. IEEE Trans.Inf.Theory 24, 596–599 (1978) 5. Davies, E.B.: Quantum theory of open systems. London, Academic Press, 1976 6. Donald, M.J.: Further results on the relative entropy. Math. Proc. Cam. Phil. Soc. 101, 363–373 (1987) 7. Eisert, J., Simon, C., Plenio, M.B.: The quantification of entanglement in infinite-dimensional quantum systems. J. Phys. A 35, 3911 (2002) 8. Holevo, A.S.: Quantum coding theorems. Russ. Math. Surv. 53, N6, 1295–1331 (1998) 9. Holevo, A.S.: On quantum communication channels with constrained inputs. http://arxiv.org/list/ quant-ph/9705054, 1997 10. Holevo, A.S.: Classical capacities of quantum channels with constrained inputs. Probability Theory and Applications. 48, N.2, 359–374 (2003) 11. Holevo, A.S., Shirokov, M.E.: On Shor’s channel extension and constrained channels. Commun. Math. Phys. 249, 417–430 (2004) 12. Holevo, A.S., Shirokov, M.E.: Continuous ensembles and the χ -capacity of infinite dimensional channels. Probability Theory and Applications 50, N.1, 98–114 (2005) 13. Holevo, A.S., Werner, R.F.: Evaluating capacities of Bosonic Gaussian channels. http://arxiv.org/list/ quant-ph/9912067, 1999 14. Holevo, A.S., Shirokov, M.E., Werner, R.F.: On the notion of entanglement in Hilbert space. Russ. Math. Surv. 60, N.2, 359–360 (2005) 15. Horodecki, M., Shor, P.W., Ruskai, M.B.: General Entanglement Breaking Channels. Rev. Math. Phys. 15, 629–641 (2003) 16. Lindblad, G.: Entropy, Information and Quantum Measurements. Commun. Math. Phys. 33, N.4, 305–322 (1973) 17. Lindblad, G.: Expectation and Entropy Inequalities for Finite Quantum Systems. Commun. Math. Phys. 39, N.2, 111–119 (1974) 18. Lindblad, G.: Completely Positive Maps and Entropy Inequalities. Commun. Math. Phys. 40, N.2, 147–151 (1975) 19. Ohya, M., Petz, D.: Quantum Entropy and Its Use. Texts and Monographs in Physics, Berlin: Springer-Verlag, 1993 20. Sarymsakov, T.A.: Introduction to Quantum Probability Theory. Tashkent, FAN: 1985, (In Russian) 21. Schumacher, B., Westmoreland, M.D.: Sending Classical Information via Noisy Quantum Channels. Phys. Rev. A 56, 131–138 (1997) 22. Schumacher, B., Westmoreland, M.D.: Optimal signal ensemble. Phys. Rev. A 63, 022308 (2001) 23. Shirokov, M.E.: On the additivity conjecture for channels with arbitrary constraints. http://arxiv.org/list/quant-ph/0308168, 2003 24. Shirokov, M.E.: On entropic quantities related to the classical capacity of infinite dimensional quantum channels. http://arxiv.org/list/quant-ph/0411091, 2004 25. Shirokov, M.E.: The Holevo capacity of infinite dimensional channels and the additivity problem. http://arxiv.org/list/quant-ph/0408009, 2004
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26. Shor, P.W.: Additivity of the classical capacity of entanglement breaking quantum channel. J.Math.Phys, 43, 4334–4340 (2002) 27. Shor, P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246, N.3, 453–472 (2004) 28. Simon, B.: Convergence theorem for entropy. Appendix in Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum mechanical entropy. J.Math.Phys. 14, 1938–1941 (1973) 29. Wehrl, A.: General properties of entropy. Rev. Mod. Phys. 50, 221–250 (1978) Communicated by M.B. Ruskai
Commun. Math. Phys. 262, 161–176 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1436-0
Communications in
Mathematical Physics
Standard Potentials for Non-Even Dynamics Huzihiro Araki Research Institute for Mathematical Sciences, Kyoto University, Sakyoku, Kyoto 606-8502, Japan. E-mail: [email protected] Received: 8 February 2005 / Accepted: 23 March 2005 Published online: 10 October 2005 – © Springer-Verlag 2005
Abstract: The notion of standard potentials is introduced for a general dynamics. This is a generalization of earlier works of Araki and Moriya which is restricted to even dynamics. Most formulae in the present analysis are√the same as the case of even dynamics: The time derivative of a local observable is i = −1 times the sum of commutators with all potentials, and application of the conditional expectation to the local algebra for a region I to a potential for a region J leave the potential unchanged if J ⊂ I and annihilate it otherwise (the standardness of the potential). However, the convergence condition for the potential takes a different form for the odd part of the potential. The equivalence of various characterizations of equilibrium states remain valid, except that the variational principle is out of the game for non-even dynamics because the translation invariance and non-evenness of dynamics are incompatible as is already known. 1. Introduction In recent works, Araki and Moriya [3] introduced the space of standard potentials in bijective correspondence with the space of all even ∗-derivations with the algebra of all local observables as their common domain. This enables the unique association of a standard potential with any even dynamics for which the domain of its generator contains all local observables (observables supported on finite regions). The standard potentials satisfy a very convenient condition that application of the conditional expectation for a region I selects exactly those potentials with its support contained in I (namely keep those unchanged and annihilate others) and a convergence condition which is natural in the sense that it is necessary and sufficient for the standard potential to be in bijective correspondence with the ∗-derivations. In the present work, we will generalize the above analysis to non-even dynamics. We will propose a space of standard potentials which is in bijective, real linear correspondence with the set of all ∗-derivations with the ∗ algebra of all local observables as their
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common domain. Under the assumption that the domain of the generator of the dynamics (which is a ∗-derivation) contains all local observables, we can associate a unique standard potential with the dynamics such that the time derivative of local operators is expressed in terms of the potentials by the traditional formula. The main features of the formulation of the standardness and convergence condition are as follows. The standardness turns out to be the same as the special case of even dynamics, but the convergence condition takes a new form for the odd part of potentials. This is due to different forms of even and odd parts of the commutant of a local algebra, which give rise to different forms of even and odd parts of the local Hamiltonian. In simpler terms, the local Hamiltonian of a finite region does not contain even potentials in the complementary region, but does contain odd potentials in the complementary region, a situation which complicates the form of the local Hamiltonian. The equivalence proof for various characterization of equilibrium states turns out to be the same as what is described in textbooks ([5]) because the variational principle is out of the game for non-even dynamics (for which translation invariance can not hold) and the proof for the equivalence of other characterizations are carried out just in terms of the local Hamiltonian without any need of the properties of potentials. In Sect. 2, we bring together notations and results about the algebra of observables with Fermionic grading, about the conditional expectations and about ∗-derivations. In Sect. 3, we derive the bijective correspondence between ∗-derivations and standard potentials, obtaining at the same time the defining properties of the standard potentials. In Sect. 4, we recall the main point of the proof of the equivalence of various characterizations of equilibrium states (excluding the variational principle for the reason already stated) so that it is clear that the same proof applies to the present situation. One of the characterization is the LTS (local thermodynamical stability) condition, for which two versions LTSM and LTSP (M and P for mathematical and physical) are introduced in [2]. Here we adopt the LTSM condition and present a physical reason for this choice. For this choice, the equivalence to other characterizations is proved in the same way as in [2]. On the other hand, we point out a shortcoming of the LTSP condition which is revealed by consideration of noneven dynamics. Note that its equivalence with other characterizations is proved only for even states under even dynamics in [2]. 2. Algebra and Conditional Expectations (A) Algebra. We consider a C ∗ algebra A for the description of the physical system on a lattice L = Zν . For a subset I of L, I c denotes the complement of I , |I | denotes the number of lattice sites in I when I is finite and the notation I ⊂⊂ L is used to indicate that I is a finite subset of L. The C ∗ algebra A has the following two structures: (Local Structure) For each lattice site i, there is a subalgebra A(i) of A, isomorphic to the full matrix algebra Md of all d × d matrices. For each subset I of L, there corresponds a C ∗ subalgebra A(I ) which is generated by all A(i) with i ∈ I . In particular, A = A(L). Due to the graded commutation relations below, A(I ) is isomorphic to the full matrix algebra of d |I | × d |I | matrices if I is finite. (Graded Structure) We consider an automorphism of the algebra A satisfying 2 = 1 and (A(i)) = A(i) for all i. (By an automorphism of a C ∗ -algebra, we mean an algebraic automorphism preserving also the adjoint ∗.) Then (A(I )) = A(I ) for all subsets I of L. Since 2 = 1, any A ∈ A can be split as a sum of even and odd elements, called the even and odd parts of A, uniquely:
Standard Potentials for Non-Even Dynamics
A = A+ + A− ,
163
A± = (A ± (A))/2 ∈ A± ,
(2.1)
where A± = {A ∈ A | (A) = ±A}.
(2.2)
A = A+ + A − .
(2.3)
Namely, we have
Correspondingly, we have the following splitting of all subalgebras: A(I ) = A(I )+ + A(I )− ,
A(I )± = A(I ) ∩ A± .
(2.4)
(Graded Commutation Relations) For any two disjoint subsets I and J of L and for any x± ∈ A± (I ) and y± ∈ A± (J ), we assume the following graded commutation relations (σ, σ = + or −): xσ yσ = −yσ xσ if σ = σ = −, xσ yσ = yσ xσ otherwise.
(2.5) (2.6)
(Self-adjoint Unitary Implementer of ) Since each A(i) is isomorphic to a full matrix algebra, its automorphism is inner, namely there exists a unitary v(i) ∈ A(i) satisfying v(i)∗ Av(i) = (A) for all A ∈ A(i). Since a full matrix algebra has a trivial center, v(i)2 is a scalar eiα of modulus 1. By multiplying v(i) with e−iα/2 , we obtain a new v(i) which is self-adjoint unitary and implements on A(i). In the following, v(i) denotes a self-adjoint unitary in A implementing on A(i). Such v(i) is unique up to ±. The self-adjoint unitary implementer v(i) of on A(i) is even because (v(i)) = v(i)3 = v(i). Hence it commutes with all elements of A(I c ). Hence for any finite I , v(i) (2.7) v(I ) = i∈I
is a self-adjoint unitary implementer of on A(I ). For infinite I , is an outer automorphism of A(I ). (See [3] Corollary 4.20.) (Formulae for Intersections and Commutants of Subalgebras) As a consequence of the Local Structure and the Graded Commutation Relations, the following formulae for the commutants of the subalgebra A(I ) within A and a related formula are known and useful (see [3] Corollary 4.12, Theorems 4.17, 4.19 and [1] Theorems 2.2, 2.4, 2.5): A(Ij ) = A( Ij ). (2.8) j
j
If I is finite, then A(I ) = A(I c )+ + v(I )A(I c )− ,
(A(I )+ ) = A(I c ) + v(I )A(I c ).
(2.9)
If I is not finite, then A(I ) = A(I c )+ ,
(A(I )+ ) = A(I c ).
(2.10)
Here, v(I ) is the selfadjoint unitary implementer of on A(I ) introduced above. All the commutants in the above formulae are taken within A.
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(B) Conditional Expectations. (Product states) A state ϕ of A is called a product state if ϕ(xy) = ϕ(x)ϕ(y)
(2.11)
for any x ∈ A(I ) and y ∈ A(J ) for any disjoint subsets I and J . If we denote the restriction of ϕ to A(i) by ϕi , then the state ϕ is called the product state of ϕi , being unique for any given set of ϕi . A state ϕ is called even if it is invariant under , namely if ϕ((A)) = ϕ(A) for all A. A state is even if and only if it vanishes on all odd elements. Examples of even product states are the tracial state τ characterized by τ (AB) = τ (BA) for all A, B ∈ A and the vacuum state for Fermions. (Conditional Expectations) Fix an even product state ϕ of A. Then, for any subset I of ϕ ϕ L, there exists a map EI from any x ∈ A to EI (x) ∈ A(I ) satisfying and uniquely determined by ϕ
ϕ(a1 xa2 ) = ϕ(a1 EI (x)a2 )
(2.12)
for all a1 and a2 in A(I ). It has the following properties of a conditional expectation. ϕ
(1) EI is linear, positive and unital. (2) For any x ∈ A and b ∈ A(I ), ϕ
ϕ
EI (xb) = EI (x)b,
ϕ
ϕ
EI (bx) = bEI (x).
(2.13)
ϕ
(3) EI is a projection of norm 1 onto A(I ). (4) For any x ∈ A, ϕ
ϕ
EI ((x)) = (EI (x)),
(2.14)
ϕ
namely, EI commutes with . ϕ Furthermore, EI for different I mutually commute and satisfy the following formula: ϕ
ϕ
ϕ
ϕ
ϕ
EI EJ = EJ EI = EI ∩J .
(2.15)
ϕ
Here E∅ is to be interpreted as ϕ. In particular, ϕ
ϕ
EI EI c = ϕ.
(2.16) ϕ
(Continuous Dependence on Subsets) The continuous dependence of EI on the subset I is given by the following formula which holds for any x in norm topology if Kν → K: ϕ
ϕ
lim EKν (x) = EK , ν
where Kν → K for a net Kν means Kν = Kν , K= µ
ν≥µ
µ
(2.17)
(2.18)
ν≤µ
in which the second equality is the criterion for convergence of the net Kν . In particular, ϕ
lim EK = 1.
KL
(2.19)
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(C) ∗-derivations. A ∗-derivation δ is a mapping from a dense ∗-subalgebra Dom(δ) of A into A satisfying the following conditions: (1) It is a linear mapping. (2) For any A and B in Dom(δ), δ(AB) = δ(A)B + Aδ(B).
(2.20)
δ(A∗ ) = δ(A)∗ .
(2.21)
(3) For any A in Dom(δ),
Dom(δ) is called the domain of δ. Note that δ(1) = 0 for any derivation δ as is seen by setting A = B = 1 in the second defining condition above. We consider the C ∗ -dynamical system (A, αt ), where A is the C ∗ -algebra we have introduced and αt , t ∈ R, is a one-parameter group of automorphisms of A such that αt (A) is continuous as a function of t for each A ∈ A. Its generator δα is defined as the time derivative at time 0 by δα (A) = lim (αt (A) − A)/t t→0
(2.22)
as a mapping from A ∈ Dom(δα ) into A with its domain Dom(δα ) being the set of all A ∈ A such that the above limit exists. The domain Dom(δα ) is automatically a dense ∗-subalgebra of A. We make the following basic assumption: Assumption (1): Dom(δ) ⊃ A0 , where A(I ) A0 =
(2.23)
I ⊂⊂L
denotes the dense ∗-subalgebra of A consisting of all elements supported by finite subsets of the lattice (which we will call local operators). Then we are allowed to consider the restriction δα0 of the generator and it belongs to (A0 ) = {∗-derivations δ | Dom(δ) = A0 }.
(2.24)
3. Standard Potentials The aim of this article is to find the characterization of ϕ-standard potentials which are in bijective real linear correspondence with (A0 ) introduced above so that we can associate with every dynamics (satisfying Assumption (1)) a unique ϕ-standard potential. This will be done in this section.
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(A) Local Hamiltonian H (I ). Theorem 3.1. For a finite subset I of L, there exists a unique self-adjoint H (I ) ∈ A satisfying ϕ
ϕ
(1) EI c (H (I )+ ) = 0, EI c (v(I )H (I )− ) = 0, where H (I )± is the even and odd parts of H (I ), and (2) δ(A) = i[H (I ), A], for all A ∈ A(I ). Furthermore, for any two finite subsets I ⊂ J of L, the following relations hold: ϕ ϕ (3) H (I )+ = H (J )+ − EI c (H (J )+ ), H (I )− = H (J )− − v(I )EI c (v(I )H (J )− ). Proof. [α] Property (2). Because A(I ) for a finite I is isomorphic to a full matrix algebra by the assumed structure of our local algebras, the existence of such H (I ) is guaranteed by a result of [6]. More explicitly, a candidate for H (I ) is given as follows: Let uij , i, j = 1, ..., d |I | be a self-adjoint system of matrix units of A(I ), namely the uij are its partial unitaries satisfying uii = 1. (3.1) uij ukl = δj k uil , u∗ij = uj i , i
Define h=
(3.2)
uij hij ,
ij
hij =
uli δ(uj l ) − δij d −|I |
l
ulm δ(uml ).
(3.3)
lm
Then one can prove by a straightforward computation (Proof (1),(2), and (3) of Lemma 5.1 in [3]) that hij is in the commutant of A(I ) and h is a self-adjoint element which satisfies condition (2). [β] Property (1). Using h obtained in [α], define ϕ
ϕ
H (I ) = h − EI c (h+ ) − v(I )EI c (v(I )h− ),
(3.4)
where h± are the even and odd parts of h. Since the second and third terms on the right-hand side belong to A(I ) , they commute with any A ∈ A(I ) and H (I ) satisϕ fies condition (2). Because EI c is a projection commuting with , v(I ) is even and 2 v(I ) = 1, this H (I ) satisfies condition (1). [γ ] Self-adjointness. We now prove the self-adjointness of the local Hamiltonian H (I ) defined by (3.4). We have H (I )∗ = h − EI c (h+ ) − EI c (h− v(I ))v(I ) ϕ
ϕ
ϕ
ϕ
= h − EI c (h+ ) − v(I )EI c (h− v(I )) because h± and v(I ) are self-adjoint (for the first equality) and v(I ) ∈ A(I )+ commutes ϕ with EI c (h− v(I )) ∈ (A(I )+ ) (for the second equality). Finite linear combinations of elements of the form A+ B and A− B with A± ∈ A(I )± and B ∈ A(I c ) are dense in A. For such products, we have ϕ
EI c (v(I )A± B) = ϕ(v(I )A± )B, ϕ EI c (A± Bv(I ))
=
ϕ EI c (A± v(I )B)
(3.5) =
ϕ EI c (A± v(I ))B
= ϕ(A± v(I ))B.
(3.6)
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Since the state ϕ is even and A− v(I ) is odd, we have ϕ(v(I )A− ) = 0 = ϕ(A− v(I )).
(3.7)
Hence (3.5) and (3.6) both vanish for A− and hence are equal. On the other hand, since v(I )A+ = (A+ )v(I ) = A+ v(I )
(3.8)
due to the invariance of A+ ∈ A(I )+ , (3.5) and (3.6) are equal also for A+ . Thus ϕ
ϕ
EI c (v(I )h− ) = EI c (h− v(I ))
(3.9)
holds for a dense set of h− and hence for all h− , and we obtain the self-adjointness of H (I ). [δ] Uniqueness. Suppose there are two H (I ), say H1 (I ) and H2 (I ), satisfying condition (2) for the same δ and condition (1). Then set x = H1 (I ) − H2 (I ).
(3.10)
By condition (2), x commutes with any A ∈ A(I ) and hence by the formula for commutants, we have x+ ∈ A(I c )+ ,
x− ∈ v(I )A(I c )− .
(3.11)
Therefore, x+ and v(I )x− are in A(I c ). By condition (1), we have ϕ
x+ = EI c (x+ ) = 0,
ϕ
x− = v(I )EI c (v(I )x− ) = 0
(3.12)
This means H1 (I ) − H2 (I ) = x = 0 and the uniqueness is proved. Note that we have not used the self-adjointness of H (I ) for this uniqueness proof, which is entirely based on conditions (1) and (2). (So the self-adjointness of H (I ) can also be proved from δ(A)∗ = δ(A∗ ).) [] Property (3). Suppose that a family of selfadjoint H (I ) satisfying conditions (1) and (2) are found. Consider two finite subsets I ⊂ J of L and set ϕ
ϕ
h = H (J ) − EI c (H (J )+ ) − v(I )EI c (v(I )H (J )− ).
(3.13)
Since the last two terms belong to A(I ) , they commute with any A in A(I ) and [h, A] = [H (J ), A] = δ(A).
(3.14)
Furthermore, h satisfies condition (1) as is easily checked because the conditional expectation is a projection and v(I )2 = 1. Hence H (I ) = h by the uniqueness proof above. This shows the validity of condition (3). Definition 3.1. By H we denote the set of all functions H of finite subsets I of L with the selfadjoint value H (I ) = H (I )∗ ∈ A satisfying conditions (1) and (3) of the above theorem. Theorem 3.2. The mapping from H ∈ H to δ ∈ (A0 ), given by δ(A) = i[H (I ), A]
(3.15)
for all A ∈ A(I ) for any I ⊂⊂ L, gives a real linear bijection from H to (A0 ).
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H. Araki
Proof. For any H ∈ H, we show that the δ defined in the theorem belongs to (A0 ). First, if A belongs simultaneously to A(I ) and to A(J ), we have two definitions of δ(A) and we have to prove that they are the same for the sake of consistency of the definition of δ. If I ⊂ J , then the last two terms in the condition (3) belong to A(I ) . Hence they commute with A ∈ A(I ) and we obtain [H (I ), A] = [H (J ), A].
(3.16)
For a general I and J , the consistency problem arises for A ∈ A(I ) ∩ A(J ) = A(I ∩ J ).
(3.17)
Hence, we can apply the consistency result just obtained for the pairs I ∩ J ⊂ I and I ∩ J ⊂ J , obtaining [H (I ), A] = [H (I ∩ J ), A] = [H (J ), A].
(3.18)
Second, δ is a derivation due to its commutator definition and it is a *-derivation because H (I ) is selfadjoint. It is defined exactly on A0 . Finally, the bijectivity of the map follows from the preceding theorem. (B) Internal Energy U (I ). We define ϕ
U (I ) = EI (H (I )). ϕ By applying EI on the equation in ϕ ϕ the formulae EI v(I ) = v(I )EI and
(3.19)
the condition (3) for H , and using
ϕ ϕ E I EI c
ϕ
ϕ(H (J )+ ) = ϕ(EJ c (H (J )+ )) = 0,
= ϕ,
(3.20)
we obtain the following relation involving U (I ) and U (J ): ϕ
ϕ
U (I ) = EI (U (J )) − ϕ(H (J ))+ − v(I )ϕ(v(I )H (J )− ) = EI (U (J )),
(3.21)
where the last term in the middle expression vanishes due to the evenness of ϕ. ϕ On the other hand, by applying EJ on the same equation, and using the mutual ϕ ϕ commutativity of EK for various K’s as well as the commutativity of EJ with v(I ) ∈ A(I ) ⊂ A(J ) (we are considering the case I ⊂ J ), we obtain ϕ
ϕ
ϕ
EJ (H (I )) = U (J ) − EI c (U (J )+ ) − v(I )EI c (v(I )U (J )− ).
(3.22)
ϕ
Since limJ L EJ = 1, we obtain the following expression of H (I ) in terms of U (J )’s, which shows that we can go back to H (I )’s from U (J )’s: ϕ ϕ H (I ) = lim U (J ) − EI c (U (J )+ ) − v(I )EI c (v(I )U (J )− ) , (3.23) J L
where the limit exists. The last two terms on the right-hand side are in the commutant of A(I ) and commute with any A ∈ A(I ). So we obtain from the above expression of H (I ) the following expression for the time derivative at time 0 in terms of U (I ): δ(A) = lim i[U (J ), A]. J L
Since I is an arbitrary finite subset of L, this formula holds for all A ∈ A0 .
(3.24)
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(C) Standard Potentials (I ). We define the (ϕ-)standard potential (I ) by the following formula: (−1)|I |−|K| U (K). (3.25) (I ) = K⊂I
Note that |I | denotes the number of lattice sites in I and similarly for |K|. This defining equation can be solved for U (K) : (K). U (I ) =
(3.26)
K⊂I
The equivalence of the above two formulae can be shown by substitution of one expression into the other and by an easy combinatorial computation (see Eq.(5.17) of [3]). Substituting the expression of U (J ) in terms of (K) into the above expression of δ in terms of U (J ), we obtain the following expression of δ in terms of the standard potentials: i[ (K), A]. (3.27) δ(A) = i lim J L
K⊂J
The limit converges and the formula holds for all A ∈ A0 . The properties of standard potentials (I ) are described in the next theorem. Theorem 3.3. (1) The functions (I ) of finite subsets I of L with values in A derived above satisfy the following properties: ( -1) (I ) ∈ A(I ), (∅) = 0. ( -2) (I )∗ = (I ). ϕ ( -3) EJ ( (I )) = 0 if J ⊂ I, J = I . ( -4) For each finite subset I of L, the following net, with the index J ⊂⊂ L ordered by the set inclusion, is a Cauchy net (in the norm topology) of A. ϕ ϕ ( (K) − EI c ( (K)+ ) − v(I )EI c (v(I ) (K)− )). (3.28) K⊂J
(2) Under the property ( -1), the property ( -3) is equivalent to the following property: ϕ
( -3’) EJ ( (I )) is (I ) if I ⊂ J and 0 otherwise. Proof. Since U (K) ∈ A(K) follows from the defining equation of U (I ) by projection onto A(I ), ( -1) follows from the defining equation of the potential in terms of U (K). The second property follows from the self-adjointness of H (I ). ( -4) is the convergence of the expression for H (I ) in terms of U (I ), where the expression of U (I ) in terms of the potential is substituted. In order to show ( -3), we first show the property ( -3’) for a fixed I under the assumption that ( -3) holds for I . The case I ⊂ J is immediate because of (I ) ∈ A(I ) ⊂ A(J ) in that case. The case of the second alternative follows from ( -3) for the fixed I because I ⊂ J if and only if K = J ∩ I is a proper subset of I and hence ϕ
ϕ
ϕ
ϕ
ϕ
EJ ( (I )) = EJ EI ( (I )) = EJ ∩I ( (I )) = EK ( (I )) = 0.
(3.29)
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We show now ( -3) by mathematical induction on I as follows. Suppose that the property holds for (K) with any proper subset K of I . The property ( -3’) then holds for any (K) with the same K by the above argument. The property ( -3) for ϕ J ⊂ I, J = I is now to be proved. Apply EJ on the following relation: (I ) = U (I ) − (K). (3.30) K⊂I,K=I
By our earlier computation, the first term on the right hand side yields U (J ) while the sum becomes the sum over K ⊂ J due to the inductive assumption and its consequence ( -3’) for any proper subset K of I . Then the sum is U (J ). Hence the right hand side vanishes, which was to be proved. Part (2) of the theorem follows from the argument preceding the proof of the third property. The above theorem motivates the following definition: Definition 3.2. A function of a finite subset I of L with values in A is called a ϕ-standard potential if it satisfies the 4 properties ( − 1), ( − 2), ( − 3), and ( − 4) in the preceding theorem. The set of all such functions , will be denoted by P ϕ . Theorem 3.4. The mapping from ∈ P ϕ to δ ∈ (A0 ) defined by δ(A) = lim i[ (K), A], J L
(3.31)
K⊂J
where the limit exists for all A ∈ A0 , gives a real linear bijection from P ϕ to (A0 ). Proof. We define H (I ) = lim
J L
ϕ
ϕ
( (K) − EI c ( (K)+ ) − v(I )EI c (v(I ) (K)− )),
(3.32)
K⊂J
where the limit exists due to ( -4). For A ∈ A(I ), we have i[H (I ), A] = i lim
J L
[ (K), A] = δ(A)
(3.33)
K⊂J
because the last two terms in the defining Eq. (3.32) of H (I ) belongs to the commutant of A(I ). This proves the convergence of the defining equation of δ in the theorem. We now show that H defined above belongs to H, which will prove that δ belongs to (A0 ). By ( -2), H (I ) is self-adjoint. Condition (1) for elements of H is shown as follows: We have ϕ H (I )+ = lim ( (K)+ − EI c ( (K)+ )), (3.34) J L
H (I )− = lim
J L
K⊂J
K⊂J
ϕ
( (K)− − v(I )EI c (v(I ) (K)− )).
(3.35)
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Therefore we have ϕ
EI c ((H (I ))+ ) = 0, ϕ EI c (v(I )H (I )− )
(3.36)
=0
(3.37)
For condition (3), we have already seen above that i[H (I ), A] has an expression independent of I . This consistency implies condition (3) as we have seen before in the proof of the properties of local Hamiltonians. We now show the injectivity of the map. Suppose that 1 and 2 give the same δ. Let Hi (I ) be defined as above from i , i = 1, 2. By the uniqueness of the local ϕ Hamiltonians for a given δ, we have H1 (I ) = H2 (I ). By applying EI on this equality, we obtain the equality of the corresponding U and hence the equality 1 = 2 . This proves the injectivity of the map. The preceding theorem together with the construction of preceding that theorem guarantees the surjectivity of the map. We note that the even part H (I )+ can be written as (K)+ (3.38) H (I )+ = lim J L
K⊂J,K∩I =∅
because the second term on the right-hand side of Eq. (3.34) is (K)+ if K ⊂ I c and 0 otherwise. Thus (K)+ is canceled if and only if K ⊂ I c and the sum is limited to K ⊂ I c , namely to K satisfying K ∩ I = ∅. This form of H (I )+ in terms of potentials is the same as in [3]. On the other hand, the sum for H (I )− can not be written in the same form. 4. Equivalence of Characterization of Equilibrium States (A) Characterizations. We first introduce various characterizations of equilibrium states, about which we are going to prove equivalence. (1) (αt , β) KMS Condition. This is well known and we omit its definition. (2) (δ, β) dKMS Condition. For a ∗-derivation δ and a real number β, this is defined as the following conditions for a state ω: ω(A∗ δ(A)) ∈ iR,
(4.1)
−iβω(A∗ δ(A)) ≥ S(ω(AA∗ ), ω(A∗ A)).
(4.2)
Here S(x, y) is a function of two real variables x, y y log y − y log x, S(x, y) = ∞, 0,
∈ [0, ∞) given by the following: x > 0, y > 0 x = 0, y > 0 y = 0.
(4.3)
(3) (H, β) Gibbs condition. This requires preparation before stating the condition. First about modular automorphisms: For a von Neumann algebra M on a Hilbert space H with a cyclic and separating unit vector , a one-parameter group of automorphisms σt of M, called the modular automorphism group, can be characterized by the (σt , −1)-KMS condition for the vector state ω of M by the vector . If a state ω1 of a C ∗ -algebra A satisfies the (α, β)-KMS condition, and (H1 , π1 , 1 ) is the associated
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H. Araki
GNS triplet of a Hilbert space, a representation of A and a unit vector, then 1 is cyclic and separating for the von Neumann algebra M1 = π(A) and the associated modular automorphism σt1 satisfies σt1 (π(A)) = π(α−βt (A)). (The same statements hold for a positive linear form (instead of a state) and the corresponding vector .) Now about the perturbation theory of modular automorphisms: Let h be a selfadjoint element of M and δ be the generator of σt . There exists another (unique) one-parameter group of automorphisms σth , whose generator δ h has the same domain Dom(δ) as δ and satisfies δ h (A) = δ(A) + i[h, A]
(4.4)
for all A in Dom(δ). Furthermore, there exists another cyclic and separating vector h (not necessarily a unit vector) with a specific explicit construction, for which the modular automorphisms are exactly σth . Definition 4.1. A state ω of A is said to satisfy the (H, β) Gibbs condition if the following two conditions are satisfied: (1) The vector of the GNS triplet (H, π, ) for the state ω is separating for π(A) . (2) For every finite subset I , the perturbed modular automorphisms σth of π(A) with h = π(βH (I )) leave all A ∈ A(I ) invariant, where the unperturbed σt is the modular automorphism group of π(A) for its vector state by . (4) ( , β) LTS Condition. We formulate this condition for each finite subset I of L taking the system outside of I to be described by A(I ) (corresponding to the LTS-M condition in [2]) because of the reason explained below. Definition 4.2. A state ω satisfies the ( , β) LTS condition if, for each I ⊂⊂ L, the conditional free energy F˜I (ψ) = S˜I (ψ) − βψ(H (I ))
(4.5)
is maximal at ψ = ω among all states ψ which coincide with ω on A(I ) , where S˜I is the conditional entropy S˜I (ψ) =
lim
(S(ψJ ) − S(ψJ )) = −S(τI ⊗ ψ , ψ).
J ⊃I,J L
(4.6)
In the above definition, S(φ) denotes the entropy of a state φ, ψJ is the restriction of the state ψ to A(J ), ψJ is the restriction of the state ψ to A(I ) ∩ A(J ), ψ is the restriction of the state ψ to A(I ) ∩ A, τI is the tracial state of A(I ) and β is a real number. The physical reason for adopting the above LTS condition as a characterization of an equilibrium state is as follows: In thermodynamics, the Helmholtz free energy is supposed to take the minimum value for the equilibrium situation. We are considering (−β) times the Helmholtz free energy for the above defined conditional free energy (in order that the principle has a reasonable solution also for the case β ≤ 0) so that the minimality requirement now becomes the maximality requirement. When we formulate this maximality requirement, a problem is that both entropy and energy for the total (infinite) system are infinite. To solve this problem, we consider a certain subset, say (ω), of states, which contains the candidate state ω for an equilibrium state, such that we get a well-defined value for the difference of the free energy between two states in this subset, in particular for two states ω and ψ, for which we compare their free energy.
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The requirement that the free energy of ω minus the free energy of ψ be non-negative will then be a necessary condition for ω to be an equilibrium state. Thus, for each finite subset I of L, we take as the restricted set (ω) of states the set of all states ψ such that its restriction to A(I ) ∩ A is the same as that of ω. With this choice of the set of ψ for comparison with ω, the conditional entropy is defined by (4.6). Since ψ = ω by the above choice of (ω), we have S˜I (ω) − S˜I (ψ) =
lim
(S(ωJ ) − S(ψJ )).
J ⊃I,J L
(4.7)
For the energy part, for any energy operator H (such as H (J )) giving the time derivative δ for all A ∈ A(I ) by the commutator i[H, A] = δ(A)(= i[H (I ), A]),
(4.8)
we have H − H (I ) ∈ A(I ) ∩ A and hence ω(H ) − ψ(H ) = ω(H (I )) − ψ(H (I )).
(4.9)
Therefore the difference F˜I (ω) − F˜I (ψ) of the conditional free energy can be considered to be the difference of free energy of the states ω and ψ. According to the idea stated above, the non-negativity of this difference, which is our LTS condition, can be considered as a necessary condition for ω to be an equailibrium state. Although this LTS condition for each I is not a sufficient condition, the totality of this condition for all finite subsets I of L, which includes as large a subset I as one likes, can be considered as a replacement of the maximality of the Helmholtz free energy (modified by multiplication of −β) and is likely a sufficient condition. In fact, we have a proof of its equivalence with other characterizations of an equilibrium state. In the case of the LTSP condition introduced in [2], one takes (ω) to be the set of all states ψ which has the same restriction to A(I c ) as that of ω. While the choice of A(I c ) as the outside algebra looks to be physically a natural choice, this choice has the following decisive drawback. If we follow the traditional physical assumption that the total energy operator H gives rise to the time derivative δ(A) of an observable A through the formula δ(A) = i[H, A], then H can be replaced by H (I ) for all A ∈ A(I ) modulo operators in A(I ) ∩ A. The ambiguity up to the operator in the commutant can be eliminated by taking all ψ with the same restriction to the commutant as that of ω in the case of the LTS condition formulated above. In the case of the LTSP condition, however, this ambiguity makes it impossible to obtain an unambiguous expression for the comparison of the energy term because an addition of an operator in A(I ) ∩ A (which is different from A(I c )) to H (I ) (due to the ambiguity of the energy operator explained above) may produce a different change of the energy expectation values in states ψ and ω, under the LTSP condition that the restriction of ψ to A(I c ) is the same as that of ω. (5) Equivalence of Characterizations for Equivalent General Potentials. Theorem 4.1. Each of the (αt , β) KMS condition, the (δ , β) dKMS condition, the (H, β) Gibbs condition, and the ( , β) LTS condition is equivalent for equivalent general potentials = 1 and = 2 . Proof. For equivalent potentials, the ∗-derivation δ and the automorphism group αt are the same; the theorem trivially holds for the KMS and dKMS conditions.
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H. Araki
Since H (I ) for equivalent potentials differs only by operators in A(I ) , the LTS condition is also equivalent for equivalent general potentials. Finally, in the case of the Gibbs condition, let H1 (I ) and H2 (I ) be the local Hamiltonian for equivalent general potentials 1 and 2 , respectively. Then they give rise to the same ∗-derivation on A(I ) and hence their difference H belongs to A(I ) . Hence the perturbed modular automorphisms by hi = βπ(Hi (I )), i = 1, 2 are a perturbation of each other by ±(h1 − h2 ) ∈ π(A(I ) ). Hence, if A(I ) is elementwise invariant by one, then so it is by the other. This proves the equivalence of the Gibbs condition for equivalent potentials. (B) Mutual Equivalence. We now discuss mutual equivalence of the above 4 characterizations of equilibrium states, which can be summarized in the following diagram, in which arrows indicate the implications, about which we know proof. Gibbs condition o
/ KMS condition O
LTS condition
/ dKMS0 condition
The proof of equivalence is given for even dynamics in earlier works. ([2, 3, 1]). We will give a brief indication that the same proof works for non-even dynamics. In earlier works, another characterization called the variational principle is also discussed. It is not relevant to the present discussion because the translation invariance of dynamics (needed in its formulation) is not compatible with non-evenness of the dynamics. (1) KMS and dKMS conditions. The (αt , β) KMS condition is known to be equivalent to the (δ, β) dKMS condition for the generator δ = δα of the modular automorphism group (e.g. see Theorem 3.5.15 in [5]) and hence implies trivially the same condition when δ is taken to be the restriction δα0 of δα to A0 (this being indicated by the subscript 0 of the dKMS condition in the above diagram), under Assumption (1) earlier introduced for the dynamics. The converse implication (from δα0 to δα ) is given in the proof of Theorem 7.6 in [3] under the following assumption for the dynamics: Assumption (2): A0 is the core of δα . (2) KMS and Gibbs conditions. The implication of the (H, β) Gibbs condition from the (αt , β) KMS condition follows from the perturbation theory without going into the explicit form of H as follows. Condition (1) in the Gibbs condition is guaranteed by the perturbation theory. The basic reason for the validity of condition (2) for the Gibbs condition is the formula stating that the generator of the perturbed modular automorphisms differs from the generator of the unperturbed modular automorphisms by the commutator [ih, x] for x = π(A), A ∈ A(I ). If the state ω satisfies the (αt , β)-KMS condition, then its modular automorphism group coincides with the scaled dynamics α−βt , and its generator acts as the commutator π([−iβH (I ), A]) on A ∈ A(I ). Hence the generator of the perturbed modular automorphisms (in the above definition of the Gibbs condition) annihilate the same A. Hence (d/dt)(σth (π(A))) = 0 for all t, which implies condition (2) of the (H, β) Gibbs condition.
(4.10)
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The converse direction works in the same way. (3) From Gibbs to LTS conditions. Assume the (H, β) Gibbs condition for a state ω. Then any A ∈ A(I ) is invariant under the perturbed modular automorphisms and hence the h perturbed state ωh for h = βπ(H (I )) − c, c = log ωβπ(H (I )) (1), is the product state τI ⊗ ρ of the tracial state τI on A(I ) and the restriction ρ of the state ωh to A(I ) , h where c is the normalization constant for ω to be a state. Let the restriction of ω to A(I ) be denoted by ω . Consider an arbitrary state ψ with its restriction to A(I ) coinciding with ω . We are going to use the following formulae for relative entropy.
S(ωh , ψ) = −ψ(h ) + S(ω, ψ), S(τI ⊗ ρ, ψ) − S(τI ⊗ ω , ψ) = S(ρ, ω ),
(4.11) (4.12)
where one uses the relation that ω is the restriction of ψ in the second equation. We also use the formula for the perturbation:
ω = (ωh )−h (= (τI ⊗ ρ)−h ).
(4.13)
Then we have, for any state ψ1 with its restriction to A(I ) ∩ A coinciding with ω, S(ω, ψ1 ) = ψ1 (h ) + S(τI ⊗ ρ, ψ1 ) = −c + βψ1 (H (I )) − S˜I (ψ1 ) + S(ρ, ω ) = −c − F˜I (ψ1 ) + S(ρ, ω ).
(4.14) (4.15) (4.16)
Substituting ψ1 = ψ and ψ1 = ω, we obtain S(ω, ψ) = −c − F˜I (ψ) + S(ρ, ω ), 0 = −c − F˜I (ω) + S(ρ, ω ).
(4.18)
F˜I (ω) − F˜I (ψ) = S(ω, ψ)
(4.19)
(4.17)
Hence
which is positive (and not 0 unless ψ = ω) by the strict positivity of relative entropy. The above proof is essentially the same as in [4] but we repeat it due to the difference in the definition of conditional entropy and difference in the condition for the set of states in which the maximum is reached. (4) From LTS to dKMS conditions. The proof is as indicated in [2] on the basis of the proofs of Theorem (b) of [4] and of Theorem 2 in [7]. Acknowledgements. This work was completed during the author’s stay at the Erwin Schr¨odinger International Institute for Mathematical Physics in Wien. The author would like to acknowledge the financial support and stimulating atmosphere of the Institute.
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References 1. Araki, H.: Conditional expectations relative to a product state and the corresponding standard potentials. Commun. Math. Phys. 246, 113–132 (2004) 2. Araki, H., Moriya, H.: Local thermodynamical stability of Fermion lattice systems. Lett. Math. Phys. 62, 33–45 (2002) 3. Araki, H., Moriya, H.: Equilibrium statistical mechanics of Fermion lattice systems. Rev. Math. Phys. 15, 93–198 (2003) 4. Araki, H., Sewell, G. L.: KMS conditions and local thermodynamical stability of quantum lattice systems. Comm. Math. Phys. 52, 103–109 (1977) 5. Bratelli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics 2. Second edition, Berlin-Heidelberg-New York: Springer Verlag, 1996 6. Sakai, S.: On one-parameter subgroups of *-automorphisms on operator algebras and the corresponding unbounded derivations. Amer. J. Math. 98, 427–440 (1976) 7. Sewell, G. L.: KMS conditions and local thermodynamical stability of quantum lattice systems. II. Commun. Math. Phys. 55, 53–61 (1977) Communicated by Y. Kawahigashi
Commun. Math. Phys. 262, 177–208 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1459-6
Communications in
Mathematical Physics
On Eta-Einstein Sasakian Geometry Charles P. Boyer1 , Krzysztof Galicki2, , Paola Matzeu3 1
Department of Mathematics & Statistics, University of New Mexico, Albuquerque, NM 87131, USA. E-mail: [email protected] 2 Max-Planck-Institut f¨ ur Mathematik, 53111 Bonn, Germany 3 Dipartimento di Matematica e Informatica, Universit´a di Cagliari, Cagliari, Italy. E-mail: matzeu@ vaxca1.unica.it Received: 21 February 2005 / Accepted: 9 May 2005 Published online: 15 November 2005 – © Springer-Verlag 2005
Abstract: A compact quasi-regular Sasakian manifold M is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact K¨ahler orbifold Z. In the case when the transverse space Z is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures. 1. Introduction The purpose of this paper is to study a special kind of Riemannian metric on Sasakian manifolds. A Sasakian manifold M of dimension 2n + 1 with a Sasakian structure S = (ξ, η, , g) is said to be η-Einstein if the Ricci curvature tensor of the metric g satisfies the equation Ricg = λg + νη ⊗ η for some constants λ, ν ∈ R. These metrics were introduced and studied by Okumura [Oku62] and then named by Sasaki [Sas65] in his lecture notes in 1965. Okumura assumed that both λ and ν are functions, and then proved, similar to the case of Einstein metrics, that they must be constant when n > 1. Obviously ν = 0 reduces to the more familiar Sasakian-Einstein condition. In general, λ + ν = 2n and every Sasakian η-Einstein manifold is necessarily of constant scalar curvature s = 2n(1 + λ). It is well-known that Sasakian-Einstein metrics are necessarily positive with Einstein constant equal to dim(M)−1. On the other hand, on K¨ahler manifolds one naturally considers K¨ahler-Einstein metrics with an Einstein constant of any sign. As every Sasakian manifold comes with a characteristic foliation Fξ whose transverse geometry is K¨ahler, On leave from Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA. E-mail: [email protected]
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the notion of the basic first Chern class c1B leads one to define null, positive, or negative Sasakian structures in analogy with K¨ahler geometry. One simply asks that c1B either vanish or be represented by a basic positive or negative (1, 1)-form, respectively. In such a context Sasakian η-Einstein metrics are the natural analogues of K¨ahler-Einstein metrics. In the c1B > 0 case the transverse geometry is Fano and the η-Einstein condition implies further that the transverse geometry is K¨ahler-Einstein of positive scalar curvature. It was first observed by Tanno that a positive Sasakian manifold M with an η-Einstein metric g admits another metric g which is Sasakian-Einstein [Tan79]. The respective Sasakian structures S and S are related by the so-called ‘transverse’ or ‘Dhomothety’ transformation (see Sect. 3 for a precise definition). Tanno used this simple observation to show that a unit tangent bundle M = T1 S n of any n-sphere admits a homogeneous Sasakian-Einstein structure. Actually, transverse homotheties define a family of Sasakian η-Einstein structures for each scale a ∈ R+ . This family can be represented by the unique Sasakian-Einstein structure which then can be either “squashed” or “stretched”. It turns out that the squashed Sasakian η-Einstein structures can be used to define an associated Einstein-Weyl geometry in a canonical way [Nar93, Nar97, Nar98, Hig93, PS93]. For instance, in 3 dimensions the only compact simply connected Sasakian-Einstein manifold is the round S 3 and, hence, the squashed η-Einstein metric is simply the Berger metric. The latter is a well-known example of a non-trivial EinsteinWeyl manifold. It follows that Sasakian-Einstein geometry offers quite a powerful tool in constructing odd-dimensional examples of Einstein-Weyl manifolds. In particular, many compact spin 5-manifolds admit families of Einstein-Weyl structures. Similarly, both the standard as well as some exotic spheres in odd dimensions also admit EinsteinWeyl structures. The c1B = 0 case is an example of the so-called transverse Calabi-Yau structure. As a consequence of a transverse version [EKA90] of Yau’s famous theorem, every null Sasakian structure can be deformed to a Sasakian η-Einstein structure which is transverse Calabi-Yau. We exhibit interesting examples of such structures on certain compact Sasakian manifolds. Interestingly, some of these examples are also Einstein-Weyl as observed by Narita in [Nar93]. In the c1B < 0 case an orbifold version of the theorem of Aubin and Yau shows that every negative Sasakian structure S can be deformed to a Sasakian η-Einstein structure. As a consequence, negative Sasakian η-Einstein metrics can be easily found on many compact contact spin manifolds. Remarkably, it follows that (possibly all) homotopy spheres which bound parallelizable manifolds admit infinitely many inequivalent negative Sasakian η-Einstein structures. Hence, for example, S 5 admits both positive and negative Sasakian η-Einstein structures, but no null Sasakian structures. Both positive and negative Sasakian η-Einstein structures exist on many compact simply connected spin 5-manifolds, including #k(S 2 × S 3 ) for certain k (positive Sasakian structures exist for all k) and certain rational homology 5-spheres. Some of these manifolds, for example #20(S 2 × S 3 ) admit all three types of Sasakian η-Einstein structures. One additional interesting feature of the c1B < 0 case is that every negative Sasakian η-Einstein structure yields a family of Lorentzian Sasakian η-Einstein metrics [Bau00, Boh03]. The metrics in the family are related by a transverse homothety and, like in the positive case, there is a unique Lorentzian Sasakian-Einstein metric in the family with Einstein constant 1 − dim(M). Hence, for instance, S 5 admits infinitely many Lorentzian Sasakian-Einstein structures.
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By virtue of admitting real Killing spinors, Sasakian-Einstein manifolds have recently received a lot of attention in physics. The physicists’interest in five and seven-dimensional manifolds admitting real Killing spinors dates back to the early eighties, when KaluzaKlein models played a central role in the supergravity theory. Today’s renewed interest in these manifolds has to do with the so-called p-brane solutions in superstring theory. These p-branes, “near the horizon” are modeled by the pseudo-Riemannian geometry of the product adSp+2 × M, where adSp+2 is the (p + 2)-dimensional anti-de-Sitter space (a Lorentzian version of a space of constant sectional curvature) and (M, g) is a Riemannian manifold of dimension d = D − p − 2. Here D is the dimension of the original supersymmetric theory. In the most interesting cases of M2-branes, M5-branes, and D3-branes D equals to either 11 (Mp-branes of M-theory) or 10 (Dp-branes in type IIA or type IIB string theory). Any residual supersymmetry forces M to admit real Killing spinors. For example, for the case of D3-branes of string theory the relevant near horizon geometry is that of adS5 × M, where M is a Sasakian-Einstein 5-manifold. The D3-brane solution interpolates between adS5 × M and R3,1 × C(M), where the cone C(M) is a Calabi-Yau threefold. In its original version the Maldacena conjecture (also known as AdS/CFT duality) states that the ’t Hooft large n limit of N = 4 supersymmetric Yang-Mills theory with gauge group SU (n) is dual to type IIB superstring theory on adS5 × S 5 [Mal99]. This conjecture has now been examined for many known SasakianEinstein metric in dimension 5 and 7. All this has led to a remarkable discovery of new cohomogeneity one Sasakian-Einstein metrics [GMSW04a, GMSW04b]. In dimension 5 the new examples produce infinitely many toric Sasakian-Einstein metric on S 2 × S 3 . The homogeneous Sasakian-Einstein metric on S 2 × S 3 examined in the context of the AdS/CFT duality by Klebanov and Witten [KW99] emerges as a special case of this construction [MS05, MSY05]. These metrics are of equal if not greater interest from the geometric point of view. They provide the first examples of compact Sasakian-Einstein manifolds which are not quasi-regular, i.e., the space of leaves of the associated characteristic foliation is not a topologically nicely behaved object such as an orbifold. As a consequence they are also counterexamples to a conjecture made in 1994 by Cheeger and Tian [CT94]. Deformations of Sasakian-Einstein metrics that naturally lead to Einstein-Weyl structures are important in general relativity theory. Null and negative Sasakian η-Einstein manifolds have not perhaps received as much attention. Yet, both in physics and in mathematics, they often emerge in a somewhat implicit fashion. For example, null Sasakian manifolds are orbifold bundles over compact Calabi-Yau orbifolds. They are very natural odd-dimensional companions of the Calabi-Yau spaces. Hence, for example, 7-dimensional null Sasakian η-Einstein manifolds should be important in the study of Calabi-Yau 3-folds with cyclic orbifold singularities. In particular all problems regarding the famous Mirror Symmetry should have a translation in the language of null Sasakian η-Einstein 7-manifolds. Our article is organized as follows: In Sect. 2 we recall some basic properties of the transverse geometry including invariants of the characteristic foliation. Section 3 follows with fundamentals about certain deformations of Sasakian structures. In Sect. 4 we begin the study of η-Einstein metrics and their relation to the Sasakian Calabi problem. Here we also describe the non-spin obstruction to the existence of η-Einstein metrics. The following section describes some general properties of Sasakian η-Einstein manifolds, in particular, we describe the relation between negative Sasakian η-Einstein structures and Lorentzian Sasakian-Einstein structures. Many of the most interesting examples of positive, null, and negative Sasakian structures can be found on links of isolated
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hypersurface singularities which are discussed in Sect. 6. In Sects. 7 and 8 we use this fact to introduce a large set of examples of Sasakian η-Einstein structures. Finally, in the last section we discuss relations between Einstein-Weyl and Sasakian η-Einstein structures. 2. The Characteristic Foliation and Transverse Invariants Recall that an almost contact structure on a differentiable manifold M is a triple (ξ, η, ), where is a tensor field of type (1, 1) (i.e., an endomorphism of T M), ξ is a vector field, and η is a 1-form which satisfies η(ξ ) = 1 and ◦ = −I + ξ ⊗ η, where I is the identity endomorphism on T M. A smooth manifold with such a structure is called an almost contact manifold. A Riemannian metric g on M is said to be compatible with the almost contact structure if for any vector fields X, Y on M we have g(X, Y ) = g(X, Y ) − η(X)η(Y ). An almost contact structure with a compatible metric is called an almost contact metric structure. In case η is a contact form (ξ, η, , g) is said to be a contact metric structure on M. A contact metric structure (ξ, η, , g) is called K-contact if ξ is a Killing vector field of g and it is called Sasakian if the metric cone (C(M), dr 2 + r 2 g, d(r 2 η)) is K¨ahler. In this section we study the transverse geometry of the Riemannian foliation Fξ defined on M by the characteristic, or as it is called Reeb vector field ξ. The transverse geometry of Sasakian structures were discussed in Part 3 of Sasaki’s notes [Sas68] in the regular case. This predated the modern development of foliation theory which is the proper setting for the study of Sasakian geometry. We first make note of some wellknown properties. The foliation Fξ is one dimensional whose leaves are geodesics with respect to the Sasakian metric g, and this metric is bundle-like. Let (M, ξ, η, , g) be a Sasakian manifold, and consider the contact subbundle D = ker η. There is an orthogonal splitting of the tangent bundle as T M = D ⊕ Lξ ,
(1)
where Lξ is the trivial line bundle generated by the Reeb vector field ξ. The contact subbundle D is just the normal bundle to the characteristic foliation Fξ generated by ξ. It is naturally endowed with both a complex structure J = |D and a symplectic structure dη. Hence, (D, J, dη) gives M a transverse K¨ahler structure with K¨ahler form dη and metric gD defined by gD (X, Y ) = dη(X, J Y )
(2)
which is related to the Sasakian metric g by g = gD ⊕ η ⊗ η.
(3)
Recall, see e.g. [Ton97], that a smooth p-form α on M is called basic if ξ α = 0, p
Lξ α = 0,
(4) p
and we let B denote the sheaf of germs of basic p-forms on M, and by B the set of p p global sections of B on M. The sheaf B is a module under the ring, 0B , of germs of
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smooth basic functions on M. We let CB∞ (M) = 0B denote global sections of 0B , i.e. the ring of smooth basic functions on M. Since exterior differentiation preserves basic forms we get a de Rham complex d
p
p+1
· · · −−−−→ B −−−−→ B −−−−→ · · ·
(5)
whose cohomology HB∗ (Fξ ) is called the basic cohomology of (M, Fξ ). The basic cohomology ring HB∗ (Fξ ) is an invariant of the foliation Fξ and hence, of the Sasakian structure on M. When M is compact it is related to the ordinary de Rham cohomology H ∗ (M, R) by the long exact sequence, (see e.g. [Ton97]) jp
p
δ
p−1
p+1
· · ·−−−−→HB(Fξ )−−−−→H p(M, R) −−−−→HB (Fξ ) −−−−→HB (Fξ )−−−−→· · ·, (6) where δ is the connecting homomorphism given by δ[α] = [dη ∧ α] = [dη] ∪ [α], and jp is the composition of the map induced by ξ with the well known isomorphism H r (M, R) ≈ H r (M, R)ᑮ , where ᑮ, a torus, is the closure of the Reeb flow, and H r (M, R)ᑮ is the ᑮ-invariant cohomology defined from the ᑮ-invariant r-forms r (M)ᑮ . We also note that dη is basic even though η is not. Next we exploit the fact that the transverse geometry is K¨ahler. Let DC denote the complexification of D, and decompose it into its eigenspaces with respect to J, that is, DC = D1,0 ⊕ D0,1 . Similarly, we get a splitting of the complexification of the sheaf 1B of basic one forms on M, namely 0,1 1B ⊗ C = 1,0 B ⊕ B . p,q
We let EB splitting
denote the sheaf of germs of basic forms of type (p, q), and we obtain a rB ⊗ C =
p,q
EB .
(7)
p+q=r p,q
The basic cohomology groups HB (Fξ ) are fundamental invariants of a Sasakian structure which enjoy many of the same properties as the ordinary Dolbeault cohomology of a K¨ahler structure. The following theorem was given in [BGN03b]: Theorem 1. Let (M, ξ, η, , g) be a compact Sasakian manifold of dimension 2n + 1. Then we have (1) HBn,n (Fξ ) ≈ R. (2) The class [dη]B = 0 lies in HB1,1 (Fξ ). p,p (3) HB (Fξ ) > 0. 2p+1 (4) HB (Fξ ) has even dimension for p < [n/2]. (5) H 1 (M, R) ≈ HB1 (Fξ ). p,q (6) (Transverse Hodge Decomposition) HBr (Fξ ) = p+q=r HB (Fξ ). (7) Complex conjugation induces an anti-linear isomorphism p,q
q,p
HB (Fξ ) ≈ HB (Fξ ).
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(8) (Transverse Serre Duality) There is an isomorphism H p,q (Fξ ) ≈ H n−p,n−q (Fξ ). Theorem 1 gives rise to fundamental invariants [BGN03c] of the Sasakian structure, namely the basic Betti numbers and basic Hodge numbers: brB (Fξ ) = dim HBr (Fξ ),
p,q
p,q
hB (Fξ ) = dim HB (Fξ )
which are related to the basic Betti numbers by brB (Fξ ) = hp,q (Fξ ),
(8)
(9)
p+q=r
and satisfy p,q
q,p
hB (Fξ ) = hB (Fξ ),
p,q
n−p,n−q
hB (Fξ ) = hB
(Fξ ).
(10)
For a description of other transverse holomorphic invariants and relationships between them, we refer the reader to [BGN03b, BGN03a]. Here we only mention the basic 0,1 geometric genus pg (Fξ ) = h0,n B (Fξ ), and the basic irregularity q = q(S) = hB . We remark that q = 21 b1B (Fξ ) = 21 b1 (M) is actually a topological invariant by (5) of Theorem 1. In the case n = 2, things simplify much more, and all the basic Hodge numbers are given in terms of the three invariants q, pg (Fξ ) and h1,1 B (Fξ ). Recall now Definition 2. The characteristic foliation Fξ is said to be quasi-regular if there is a positive integer k such that each point has a foliated coordinate chart (U, x) such that each leaf of Fξ passes through U at most k times. If k = 1 then the foliation is called regular. If Fξ is not quasi-regular, it is said to be irregular. We use the term non-regular to mean quasi-regular but not regular. In the case of a compact quasi-regular Sasakian manifold M we know that the space of leaves is a compact Riemannian orbifold Z. But since the transverse geometry on M is K¨ahler, the orbifold must be K¨ahler. But actually in the quasi-regular case more is true. It follows [BG00] that M is the total space of a V-bundle over Z, and the curvature of the connection form η is precisely the pullback of the K¨ahler form on Z. Thus, Z satisfies an orbifold integrality condition which we now elaborate. This integrality condition ties Sasakian geometry on compact manifolds to projective algebraic geometry. Recall that a 2 (Z, Z). By compact K¨ahler orbifold (Z, ω) is called a Hodge orbifold if [ω] lies in Horb a theorem of Baily [Bai57] a Hodge orbifold is a polarized projective algebraic variety. We have Theorem 3. Let S = (ξ, η, , g) be a compact quasi-regular Sasakian structure on a smooth manifold of dimension 2n + 1, and let Z denote the space of leaves of the characteristic foliation. Then (1) The leaf space Z is a Hodge orbifold with K¨ahler metric h and K¨ahler form ω which 2 (Z, Z) in such a way that π : (S, g)−→(Z, h) defines an integral class [ω] in Horb is an orbifold Riemannian submersion. The fibers of π are totally geodesic submanifolds of S diffeomorphic to S 1 . (2) Z is also a Q-factorial, polarized, normal projective algebraic variety. Conversely, given a Hodge orbifold (Z, ω) one can construct a Sasakian structure on 2 (Z, Z). the total space of the circle bundle defined by the integral class [ω] in Horb
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3. Deformation Classes of Sasakian Structures Sasakian structures like K¨ahler structures are plentiful; in fact, the ‘moduli space’ is infinite dimensional. It is thus important to look for distinguished Sasakian metrics in a given deformation class, and η-Einstein metrics provide good examples. There are two essential types of deformations of Sasakian structures, those that deform the characteristic foliation (deformations of type I) and those that fix the characteristic foliation but deform the contact 1-form (deformations of type II). Deformations of Type I. Let us consider deformations of a Sasakian structure with a fixed transverse holomorphic structure, that is deformations that deform the characteristic foliation Fξ by deforming the vector field ξ. This type of deformation was first considered by Takahashi [Tak78] who considered deformations by adding an infinitesimal automorphism ρ ∈ ᑾᒒᒑ(ξ, η, , g). However, here we consider the more general deformations introduced in [GO98], and used in the 3-dimensional case by Belgun [Bel01]. Explicitly, we consider deformations of the contact form whose Reeb vector field is an infinitesimal automorphism of the CR structure, that is, we consider a positive function f and tensor fields η˜ = f η,
ξ˜ = ξ + ρ,
˜ = − ξ˜ ⊗ η˜
(11)
such that η( ˜ ξ˜ ) = 1,
ξ˜ d η˜ = 0,
˜ = 0. £ξ˜
(12)
It is clear from Eqs. 11 that η˜ is a contact 1-form, the contact subbundle D = ker η = ker η˜ ˜ ξ˜ = 0. Furthermore, if X is a section of D, then remains unchanged, and that ˜ 2 X = (X ˜ − η(X)) ˜ = (X − η(X)) ˜ = 2 X = −X. ˜ 2 = −I + ξ˜ ⊗ η, ˜ is an underlying almost contact structure of the Thus, ˜ so (ξ˜ , η, ˜ ) contact structure. Notice that f and ρ are related by f =
1 . 1 + η(ρ)
Next define the Riemannian metric g˜ by ˜ ⊗ I) ⊕ η˜ ⊗ η. g˜ = d η˜ ◦ ( ˜ This metric is compatible with the contact structure and ξ˜ is a Killing vector field of g. ˜ ˜ g) Thus, (ξ˜ , η, ˜ , ˜ is K-contact. In fact, we claim that it is Sasakian. To see this we need to ˜ g) check normality. But according to Proposition 3.2 of [BG01a] the structure (ξ˜ , η, ˜ , ˜ ˜ ˜ is normal if and only if the almost CR-structure |D is integrable and is invariant under ξ˜ . The latter conditions both hold by Eqs. 11 and 12. We have arrived at Theorem 4. Let (ξ, η, , g) be a Sasakian structure on M. Then the deformations of type I are deformations through Sasakian structures.
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The metrics g˜ and g are related by g˜ =
g − η˜ ⊗ ξ˜ g − ξ˜ g ⊗ η˜ + g(ξ˜ , ξ˜ )η˜ ⊗ η˜ + η˜ ⊗ η. ˜ 1 + η(ρ)
A very special case of type I deformations occur when ρ = cξ for some constant c. This does not deform the characteristic foliation, but only rescales the vector field ξ and contact 1-form η. It is convenient to write this transformation in the form ξ = a −1 ξ,
η = aη,
= ,
g = ag + a(a − 1)η ⊗ η,
where a is constant. These were called D-homothetic deformations by Tanno [Tan68]. We have also referred to them as transverse homotheties. Deformations of Type II. Now fix a Sasakian structure (ξ, η, , g) on M 2n+1 and consider deformations which stabilize the characteristic foliation Fξ . More explicitly we wish to deform (ξ, η, , g) through Sasakian structures that have the same fundamental basic cohomology class up to a scale. As in K¨ahler geometry this deformation space is infinite dimensional. We consider a deformation of the structure (ξ, η, , g) by adding to η a continuous one parameter family of 1-forms ζt that are basic with respect to the characteristic foliation. We require that the 1-form ηt = η + ζt satisfy the conditions η0 = η,
ζ0 = 0,
ηt ∧ (dηt )n = 0
∀ t ∈ [0, 1].
(13)
This last non-degeneracy condition implies that ηt is a contact form on M for all t ∈ [0, 1] which by Gray’s Stability Theorem [Gra59] belongs to the same underlying contact structure as η. Moreover, since ζt is basic, ξ is the Reeb (characteristic) vector field associated to ηt for all t. Now let us define t = − ξ ⊗ ζt ◦ , gt = dηt ◦ ( ⊗ It ) + ηt ⊗ ηt .
(14) (15)
In [BG03] it was proved that for all t ∈ [0, 1] and every basic 1-form ζt such that dζt is of type (1, 1) and such that (13) holds (ξ, ηt , t , gt ) defines a continuous 1-parameter family of Sasakian structures on M belonging to the same underlying contact structure as η. Theorem 5. Let (M, ξ, η, , g) be a Sasakian manifold. Then for all t ∈ [0, 1] and every basic 1-form ζt such that dζt is of type (1, 1) and such that (13) holds (ξ, ηt , t , gt ) defines a Sasakian structure on M belonging to the same underlying contact structure as η. Given a Sasakian structure S = (ξ, η, , g) on a manifold M, we defined ᑠ(ξ ) [BGN03b] to be the family of all Sasakian structures obtained by deformations of type II. Notice that any two Sasakian structures (ξ, η, , g) and (ξ , η , , g ) in ᑠ(ξ ) are homologous in the sense that [dη ]B = [dη]B . This definition was then extended to include transverse homotheties in [BGN03c]. We defined ᑠ(Fξ ) to be the set of all Sasakian structures whose characteristic foliation is Fξ . Clearly, we have ᑠ(ξ ) ⊂ ᑠ(Fξ ). For any Sasakian structure S = (ξ, η, , g) there is the “conjugate Sasakian structure” defined by S c = (ξ c , ηc , c , g) = (−ξ, −η, −, g) ∈ ᑠ(Fξ ). So fixing S we define ᑠ+ (Fξ ) = ᑠ(a −1 ξ ), a∈R+
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and ᑠ− (Fξ ) to be the image of ᑠ+ (Fξ ) under conjugation. It is then easy to see that we have a decomposition ᑠ(Fξ ) = ᑠ+ (Fξ ) ᑠ− (Fξ ). Two Sasakian structures S = (ξ, η, , g) and S = (ξ , η , , g ) on a smooth manifold M are said to be a-homologous if there is an a ∈ R+ such that ξ = a −1 ξ and [dη ]B = a[dη]B . More generally it is convenient to include both deformations of type I and II as well as deformations of the CR-structure. For a fixed Sasakian structure S = (ξ, η, , g) we denote by ᑠ(S) the space of all Sasakian structures that can be obtained from S by deformations through CR-structures that are compatible with the contact structure or deformations of type I and II above. The topology of ᑠ(S) is compact open C ∞ topology induced by the corresponding tensor fields. The underlying contact structure remains fixed in ᑠ(S). 4. η-Einstein Metrics and the Calabi Problem In this section we begin the main objects of study, namely η-Einstein metrics which were introduced and studied by Okumura [Oku62] in 1962. In particular, he studied the relation between the existence of η-Einstein metrics and certain harmonic forms, although the correct cohomological setting and connection to the Calabi problem was not realized until much later. Definition 6. A Sasakian structure S = (ξ, η, , g) on M is said to be Sasakian η-Einstein or just η-Einstein if there are constants λ, ν such that Ricg = λg + νη ⊗ η. One can obviously generalize this definition to any almost contact metric manifold (M, η, ξ, , g). In such generality one typically asks for λ, ν ∈ C ∞ (M) to be arbitrary functions. However, on a Sasakian or even only K-contact manifold the Ricci curvature tensor satisfies Ricg (ξ, X) = 2nη(X) and it easily follows [Oku62] that Lemma 7. Let M be a K-contact manifold of dimension 2n + 1 such that Ricg = λg + νη ⊗ η for some λ, ν ∈ C ∞ (M). Then if n > 1, λ, and ν are constants satisfying λ + ν = 2n. Remark 8. In the 3-dimensional case any K-contact manifold is automatically Sasakian. A Sasakian 3-manifold M is always η-Einstein in the wider sense of the definition which allows λ, ν to be functions on M. Hence, in this paper we define a Sasakian 3-manifold to be η-Einstein as in Definition 6. On the other hand, when M is a metric contact manifold one can study metrics for which Ricg = λg + νη ⊗ η, with λ, ν ∈ C ∞ (M) (see [Bla02]). An immediate consequence of the definition is Corollary 9. Every Sasakian η-Einstein manifold is of constant scalar curvature s = 2n(λ + 1). We briefly recall the basic first Chern class from [BGN03b]. Now the contact subbundle D is a complex vector bundle and thus has a first Chern class c1 (D) ∈ H 2 (M, Z). Consider the long exact sequence 6 together with the natural map H 2 (M, Z)−−→
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H 2 (M, R) whose kernel is the torsion part of H 2 (M, Z). From (5) of Theorem 1 we have H 2 (M, Z) δ
(16)
ι∗
0−→R −−→ HB2 (Fξ ) −−→ H 2 (M, R)−−→H 1 (M, R)−−→ · · · . As in (6) the map δ is given by δ(c) = c[dη], where c ∈ R. Now on a Sasakian manifold the vector bundle D1,0 is holomorphic with respect to the CR-structure, so we can compute the free part of c1 (D) = c1 (D1,0 ) from the transverse K¨ahler geometry in the usual way. That is c1 (D) can be represented by a basic real closed (1, 1)-form ρB . The class c1B = [ρB ] ∈ HB2 (Fξ ) is independent of the transverse metric and basic connection used to compute it, and depends only on the foliated manifold (M, Fξ ) with its CR-structure. It is described in [EKA90] and called the basic first Chern class of D there. Definition 10. A Sasakian structure (ξ, η, , g) is said to be positive (negative) if c1B is represented by a positive (negative) definite (1, 1)-form. If either of these two conditions is satisfied (ξ, η, , g) is said to be definite, and otherwise (ξ, η, , g) is called indefinite. (ξ, η, , g) is said to be null if c1B = 0. In analogy with common terminology of smooth algebraic varieties we see that a positive Sasakian structure is a transverse Fano structure, while a null Sasakian structure is a transverse Calabi-Yau structure. The negative Sasakian case corresponds to the canonical bundle being ample; we refer to this as a transverse canonical structure. Recall the transverse Ricci tensor RicTg of gT to be the Ricci tensor of the transverse metric gT . It is related to the Ricci tensor Ricg of g, Ric(X, Y ) = RicT (X, Y ) − 2g(X, Y ).
(17)
Now as usual define the Ricci form ρg and transverse Ricci form ρgT by ρg (X, Y ) = Ricg (X, Y ),
ρgT (X, Y ) = RicTg (X, Y )
(18)
for smooth sections X, Y of D. It is easy to check that these are anti-symmetric of type (1, 1) and 17 implies that they are related by ρgT = ρg + 2dη.
(19)
Thus, as in the usual case we have Lemma 11. The basic class 2πc1B ∈ HB1,1 (Fξ ) is represented by the transverse Ricci form ρgT . In [BGN03b] we used El Kacimi-Alaoui’s ‘Transverse Yau Theorem’ [EKA90] to prove the following converse to Lemma 11: Theorem 12. Let (M, ξ, η, , g) be a Sasakian manifold whose basic first Chern class c1B is represented by the real basic (1, 1) form ρ, then there is a unique Sasakian structure (ξ, η1 , 1 , g1 ) ∈ ᑠ(ξ ) homologous to (ξ, η, , g) such that ρg1 = ρ − 2dη1 is the Ricci form of g1 .
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In Problem 16 below we give another version of the Sasakian Calabi Problem. We are interested in certain invariance properties of c1B . Since the transverse Ricci form ρgT is invariant under scaling, we see that c1B is independent of the Sasakian structure in ᑠ(Fξ ). Now from the exact sequence (16) we have Proposition 13. Let (ξ, η, , g) be a Sasakian structure with underlying contact bundle D. Then c1 (D) is a torsion class if and only if there exists a real number a such that c1B = a[dη]B . In particular, (1) If c1 (D) is a torsion class then every Sasakian structure in ᑠ(S) is either definite or null. (2) If ᑠ(Fξ ) admits a Sasakian η-Einstein structure, then c1 (D) is a torsion class. Hence, a non-torsion c1 (D) is the obstruction to the existence of a Sasakian η-Einstein metric. In particular, in the case when λ > −2 there is a canonical variation to a Sasakian-Einstein metric [Tan79], so c1 (D) is an obstruction to the existence of a Sasakian-Einstein metric on M in the given ξ -deformation class [BG01a]. Since c1 (D) is an invariant of the complex vector bundle D, (2) gives an obstruction to the type of contact structure that can admit a compatible Sasakian η-Einstein metric. However, since c1 (D) is related to a topological invariant, namely the second Stiefel-Whitney class w2 (M), there are topological obstructions to the existence of Sasakian η-Einstein metrics. From the natural splitting T M = D ⊕ Lξ , where Lξ is the trivial real line bundle generated by ξ, one gets w2 (M) = w2 (T M) = w2 (D), which is the mod 2 reduction of c1 (D) ∈ H 2 (M, Z). Since a manifold M admits a spin structure if and only if w2 (M) = 0, Proposition 13 implies Theorem 14. Let M be a non-spin manifold (i.e., w2 (M) = 0) with H1 (M, Z) torsion free. Then M does not admit a Sasakian η-Einstein structure. Thus, Theorem 14 includes as a special case the well-known fact [BG00, Mor97] that a simply connected Sasakian-Einstein manifold is necessarily spin. Next we give an example of a Sasakian manifold that does not admit a Sasakian η-Einstein metric. Example 15. Consider the non-trivial S 3 -bundle over S 2 , denoted by X∞ in Barden [Bar65]. We represent X∞ as a non-trivial circle bundle over a Hirzebruch surface. Recall the Hirzebruch surfaces Sn [GH78] are realized as the projectivizations of the sum of two line bundles over CP1 , namely Sn = P O ⊕ O(−n) . They are diffeomorphic to CP1 × CP1 if n is even, and to the blow-up of CP2 at one 2 , if n is odd. We need to take n odd to get a non-spin circle point, which we denote as CP bundle. Now Pic(Sn ) ≈ Z⊕Z, and we can take the Poincar´e duals of a section of O(−n) and the homology class of the fiber as its generators. The corresponding divisors can be represented by rational curves which we denote by C and F, respectively satisfying C · C = −n,
F · F = 0,
C · F = 1.
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Let α and β denote the Poincar´e duals of C and F , respectively, and write the K¨ahler form ω so that [ω] = aα + bβ. According to [HS97] every K¨ahler class can be represented in this form with b = 2 and a > 0. Let π
S 1 −−→M −−→ Sn be the circle bundle over Sn whose first Chern class is [ω]. Since we want M simply connected, we need to choose gcd(a, b) = 1. It suffices to take n = 3, and [ω] = α +2β. Now c1 (D) = π ∗ c1 (Sn ) = −π ∗ c1 (K), where K is the canonical divisor of Sn . From [GH78] we have K = −2C − (n + 2)F. So in our case we have c1 (K) = −2α − 5β. From the Gysin sequence of the circle bundle we have ∪ [ω]
π∗
0−−→Z −−−−→ H 2 (Sn , Z) −−−−→ H 2 (M, Z)−−→0, giving c1 (D) = π ∗ (c1 (−K)) = π ∗ (2α + 5β) = π ∗ β. Hence, w2 (M) = 0 implying that M = X∞ by Barden’s Classification Theorem [Bar65]. Thus, we have produced a Sasakian structure on X∞ , but by Theorem 14 it cannot admit a Sasakian η-Einstein structure. We are interested in proving the existence of Sasakian η-Einstein metrics in a given deformation class of Sasakian structures. Problem 16 (Sasakian Calabi Problem). Given a manifold M with Sasakian structure S = (ξ, η, , g) and with a basic first Chern class c1B that is represented by either a positive definite, negative definite real basic (1, 1) form ρ T , or if c1B vanishes, does there exist a Sasakian structure S ∈ ᑠ(Fξ ) with an η-Einstein metric g ? The solution to this problem is, of course, given in local CR-coordinates (zi , z¯ i , x) on M by the “transverse Monge-Amp`ere equation” det(giTj¯ + φi j¯ ) det(giTj¯ )
= e−kφ+F ,
giTj¯ + φi j¯ > 0,
(20)
where g T is the transverse metric, φ and F are real basic functions, and φi j¯ are the ¯ = dζt with respect to the transverse coordinates (zi , z¯ ¯ ). The cases components of ∂ ∂φ j B when c1 is zero, positive or negative definite correspond to finding solutions of Eq. 20 with the constant k = 0, > 0, < 0, respectively. In the negative and zero cases we have the transverse version of theorems of Yau [Yau78] and Aubin [Aub82] which give Theorem 17. If the class [ρ T ] ∈ HB2 (Fξ ) is zero or can be represented by a negative definite (1, 1) form, then there exists a Sasakian structure S ∈ ᑠ(ξ ) with an η-Einstein metric g on M with λ = −2 in the first case and λ < −2 in the second. In other words, in complete analogy with the K¨ahlerian case, we see that there are no obstructions to solving the Sasakian or transverse Monge-Amp`ere problem in the negative or null case.
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5. Further Results on η-Einstein Metrics Let us now consider a Sasakian η-Einstein with Sasakian structure S = (ξ, η, , g) and η-Einstein constants (λ, ν). A simple calculation shows that D-homotheties preserve the η-Einstein condition. More precisely, we have Proposition 18. Let (M, ξ, η, , g) be a Sasakian η-Einstein manifold with constants (λ, ν). Consider a D-homothetic Sasakian structure S = (ξ , η , , g ) = (a −1 ξ, aη, , ag + a(a − 1)η ⊗ η). Then (M, S ) is also η-Einstein with constants λ + 2 − 2a λ + 2 − 2a , ν = 2n − . a a Proof. The proof follows from a simple formula relating Ricci curvature tensors of the Sasakian metric g , g related by a D-homothetic transformation [Tan68]: λ =
Ricg = Ricg − 2(a − 1)g + (a − 1)(2n + 2 + 2na)η ⊗ η.
Note that D-homotheties must preserve the sign of the Sasakian η-Einstein structure (invariance of basic Chern classes) and they are completely analogous to the usual homotheties of Einstein metrics. When (M, ξ, η, , g) is null then always (λ, ν) = (−2, 2n + 2). When (M, ξ, η, , g) is positive then λ > −2. But by applying a suitable D-homothety one can get any value of λ ∈ (−2, ∞). Similarly, when (M, ξ, η, , g) is negative λ < −2 and by applying a suitable D-homothety one can get any value of λ ∈ (−∞, −2). An easy computation shows that, in addition to the structure Theorem 3, in the ηEinstein case we can actually say more. Theorem 19. Let (M, ξ, η, , g) be a compact quasi-regular Sasakian η-Einstein manifold of dimension 2n + 1, and let Z denote the space of leaves of the characteristic foliation. Then the leaf space Z is a Hodge orbifold with K¨ahler-Einstein metric h with Einstein constant λ + 2. The case of λ + 2 > 0 is special as follows from Proposition 18. By applying a suitable D-homothety one can always choose λ = 2n which turns the η-Einstein metric into an Einstein metric. This observation is originally due to Tanno who used it to show that a unit tangent bundle of S n has a homogeneous Sasakian-Einstein structure. When n = 3 one gets a homogeneous Sasakian-Einstein metric on S 2 × S 3 [Tan79]. We have Theorem 20. Let (M, ξ, η, , g) be a compact quasi-regular Sasakian η-Einstein manifold of dimension 2n + 1, and let λ > −2. Let (Z, h) denote the space of leaves of the characteristic foliation with its K¨ahler-Einstein metric h. Then M admits a Sasakian-Einstein structure (ξ , η , , g ) ∈ ᑠ(Fξ ), D-homothetic to S, with g = λ+2 . αg + α(α − 1)η ⊗ η and α = 2n+2 It is useful therefore to think of positive Sasakian η-Einstein structure as a family parameterized by a positive constant a ∈ R+ . In addition, if we choose the SasakianEinstein structure in this family S1 as a reference point we will distinguished between two subfamilies. Definition 21. Let (M, ξ1 , η1 , 1 , g1 ) be a compact quasi-regular Sasakian-Einstein manifold of dimension 2n + 1, and let Sa = (ξa , ηa , a , ga ) be the Sasakian η-Einstein structure obtained from S1 by applying a D-homothetic transformation to S1 with constant a ∈ R+ . We say that Sa is stretched when a > 1(−2 < λ < 2n, ν > 0) and squashed when 0 < a < 1 (λ > 2n, ν < 0).
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Note that positive squashed Sasakian η-Einstein manifolds can have scalar curvature of any sign. There is a unique metric in the family Sa which makes λ = −1 in which case the Sasakian manifold becomes scalar-flat. In the negative case there is another interesting structure related to the Sasakian η-Einstein metric. Recall [Bau00, Boh03] the following Definition 22. Let (M, g) be a Lorentzian manifold of dimension 2n + 1 and let ξ be a time-like Killing vector field such that g(ξ, ξ ) = −1. We say that M is Sasakian if (X) = ∇X ξ satisfies the condition (∇X )(Y ) = g(X, ξ )Y + g(X, Y )ξ and Sasakian-Einstein if, in addition, g is Einstein. Equivalently, we can require the metric cone C(M) = (R+ ×M, −dt 2 +r 2 g, d(r 2 η)) to be pseudo-K¨ahler of signature (2, 2n). In the Sasakian-Einstein case the cone metric is pseudo-Calabi-Yau and the Einstein constant must equal to −2n. This is in complete analogy with the Riemannian case. Geometries of this type are interesting as they provide examples of the so-called twistor spinors on Lorentzian manifolds. As a simple consequence we get Proposition 23. Let (M, ξ, η, , g) be a compact quasi-regular Sasakian η-Einstein manifold of dimension 2n + 1, and let λ < −2. Let (Z, h) denote the space of leaves of the characteristic foliation with its K¨ahler-Einstein metric h. Then M admits a Lorentzian Sasakian-Einstein structure such that ξ = a −1 ξ, where a =
−g = ag + a(a − 1)η ⊗ η,
λ+2 2+2n .
Combining this with Theorem 17 we have Corollary 24. Every negative Sasakian manifold admits a Lorentzian Sasakian-Einstein structure. 6. η-Einstein Structures on Links Let w = (w0 , . . . , wn ) be any positive vector in Rn+1 , that is wi > 0 for all i = 0, . . . , n. The vector field corresponding to w is denoted by ξw , and the corresponding 1-form by ηw . Let (x0 , . . . , xn , y0 , . . . , yn ) be the coordinates on the unit sphere S 2n+1 in R2n+2 . A family of Sasakian structures Sw = (ξw , ηw , w , gw ) is now defined by ξw =
n i=0
n wi (yi ∂xi − xi ∂yi ),
ηw = n
i=0 wi
i=0 xi dyi , ((x i )2 + (y i )2 )
(21)
with w and gw further determined by ξw and ηw . We shall refer to (ξw , ηw , w , gw ) as the weighted Sasakian structure on S 2n+1 , and we denote the unit sphere with this Sasakian structure by Sw2n+1 . Note that the standard Sasakian-Einstein structure (Hopf fibration) corresponds to setting w = (1, . . . , 1). The deformed structures (ξw , ηw , w , gw ) are not all distinct. The Weyl group W(SU (n + 1)) = Nor(ᑮn+1 )/ᑮn+1 acts as outer automorphisms on the maximal torus, and it is well-known that W(SU (n+1)) = n+1 , the permutation group on n + 1 letters. So for any σ ∈ n+1 the Sasakian structures Sw = (ξw , ηw , w , gw ) and (ξσ (w) , ησ (w) , σ (w) , gσ (w) ) are isomorphic. Thus, we shall
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frequently take the weights to be ordered such that w0 ≤ w1 ≤ · · · ≤ wn . It is important to note that all the Sasakian structures Sw belong to the same underlying contact structure on S 2n+1 , namely the standard one. Consider the affine space Cn+1 together with a weighted C∗ -action given by (z0 , . . . , zn ) → λ · z = (λw0 z0 , . . . , λwn zn ),
(22)
where λ ∈ C∗ , and the weights wj are positive integers. Definition 25. A polynomial f ∈ C[z0 , . . . , zn ] is said to be weighted homogeneous of degree d if there are positive integers w0 , w1 , . . . , wn , d such that f (λ · z) = f (λw0 z0 , . . . , λwn zn ) = λd f (z0 , . . . , zn ) = λd f (z). We shall assume that w0 ≤ w1 ≤ · · · ≤ wn and that gcd(w0 , . . . , wn ) = 1 unless otherwise stated. The zero set Vf of a non-degenerate weighted homogeneous polynomial f, is just the weighted affine cone Cf . If the origin is an isolated singularity in Cf , then it is the only singularity of Cf . We shall assume this from now on. In this case we can take = 1 so that S2n+1 (z0 ) becomes the unit sphere S 2n+1 centered at the origin with its weighted Sasakian structure Sw . In this case our link Lf = Cf ∩ S 2n+1
(23)
is a smooth manifold of dimension 2n − 1 which by the Milnor Fibration Theorem is (n − 2)-connected. One can easily see that every link is either positive, negative, or null. More precisely, we have [BGK05] Theorem 26. Let f be a non-degenerate weighted homogeneous polynomial of degree d and weight vector w. The Sasakian structure (ξw , ηw , w , gw ) on S 2n+1 induces by restriction a Sasakian structure, denoted by Sw,f = Sw |Lf on the link Lf . Furthermore, let |w| = w0 + · · · + wn . Then (1) Sw,f is positive when d − |w| < 0, (2) Sw,f is null when d − |w| = 0, (3) Sw,f is negative when d − |w| > 0. Example 27. Let f (z) = z0a0 + · · · + znan . Then the link Lf is called the Brieskorn-Pham link and we will denote it by L(a) = L(a0 , . . . , an ). It can be seen that the weighted degree of the Brieskorn-Pham polynomial is d = lcm(a0 , . . . , an ) and the weights are wj = d/aj . Then by Theorem 26,
(1) L(a) is positive when i a1i > 1,
1 (2) L(a) is null when i ai = 1,
(3) L(a) is negative when i a1i < 1. Theorem 26 together with Theorem 17 provide a rich source of η-Einstein metrics in the non-positive case where there are no obstructions. Summarizing we have Theorem 28. Let Sw,f be null or negative. Then Lf admits a Sasakian structure S with η-Einstein metric g in the same deformation class as Sw,f .
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In the Fano case there are obstructions to the existence of such structures. However, as every η-Sasakian metric can be deformed to a Sasakian-Einstein one such obstruction is completely equivalent to the obstructions preventing existence of the orbifold K¨ahler-Einstein metric on the transverse space Zf . Existence of K¨ahler-Einstein metrics on various Fano orbifolds can be often established by the continuity method (cf. [BGK05] and references therein). We finish this section by reviewing very briefly how one determines the topology of the links. The procedure is that of Milnor and Orlik [MO70] and involves computing the Alexander polynomial of the link. For more details we refer to [MO70] and our previous work [BG01b, BGN03b]. Milnor and Orlik give a combinatorial formula for the (n−1)st Betti number of a (2n − 1)-dimensional link, viz. ui1 · · · uis bn−1 (Lf ) = (−1)n+1−s , (24) vi1 · · · vis lcm(ui1 , . . . , uis ) where the quotients d/wi are written in the irreducible form ui /vi , and the sum is taken over all the 2n+1 subsets {i1 , . . . , is } of {0, . . . , n}. In the case of determining homology spheres or more generally rational homology spheres it is more efficient to use the Brieskorn Graph Theorem. We refer to [BGK05] for details. Another important result proven in [BG01b] concerns 5-dimensional manifolds only. It says that if the weights (w0 , w1 , w2 , w3 ) satisfy the “well-formedness” condition gcd(wi , wj , wk ) = 1 for all distinct i, j, k, then the 5-dimensional link Lf has no torsion. 7. Sasakian Structures in Dimension 3 Three dimensional Sasakian geometry is well understood, culminating in the recent uniformization theorem due to Belgun [Bel00] which we state precisely below. Geiges [Gei97] showed that a compact 3-manifold admits a Sasakian structure if and only if it is diffeomorphic to one of the following: (1) S 3 / with ⊂ ᑣ0 (S 3 ) = SO(4). R)/ , where is universal cover of SL(2, R) and ⊂ ᑣ0 (SL(2, R)). (2) SL(2, (3) Nil3 / with ⊂ ᑣ0 (Nil3 ). Here is a discrete subgroup of the connected component ᑣ0 of the corresponding isometry group with respect to a ‘natural metric’, and Nil3 denotes the 3 by 3 nilpotent real matrices, otherwise known as the Heisenberg group. These are three of the eight model geometries of Thurston [Thu97], and correspond precisely to the compact Seifert bundles with non-zero Euler characteristic [Sco83]. For further discussion of the isometry groups we refer to [Sco83]. We refer to the three model geometries above as 2 type, and nil geometry, respectively. spherical, SL The purpose of this section then is to describe some examples represented as links of isolated hypersurface singularities. First note that in dimension 3 the basic cohomology group H 2 (Fξ ) is 1-dimensional so we cannot get indefinite forms other than zero. Hence Proposition 29. Let M be a compact 3-dimensional Sasakian manifold. Then M is is either positive, negative, or null. These 3 types of Sasakian structures correspond precisely to the 3 model geometries above. Belgun’s Theorem uniformizes these three cases. We prefer to rephrase Belgun’s theorem in terms of the earlier work of Tanno [Tan69] on constant -sectional curvature. In dimension three the -sectional curvature is determined by one function, namely
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H (X) = K(X, X). Thus, in analogy with 3-dimensional Einstein geometry we have (see also [Gui02]) Proposition 30. A three dimensional Sasakian manifold is η-Einstein if and only if it has constant -sectional curvature. Proof. The if part, which holds in all dimensions, follows easily from Tanno’s classification theorem [Tan69]. To prove the only if part we choose local orthonormal bases X, X, ξ for the Sasakian structure S = (ξ, η, , g). Now in three dimensions the Ricci curvature and the sectional curvature are related by [Pet98] Ricg (X, X) = K(X, X) + K(X, ξ ) = K(X, X) + 1, Ricg (X, X) = K(X, X) + K(X, ξ ) = K(X, X) + 1, Ricg (ξ, ξ ) = K(X, ξ ) + K(X, ξ ) = 2. So if g is η-Einstein we have Ricg = αg + (2 − α)η ⊗ η, and this gives K(X, X) = α − 1.
Now if a Sasakian structure S = (ξ, η, , g) has constant -sectional curvature c, then it can be transformed to one of the cases c = 1, −3, or −4 by a transverse homothety. The Uniformization Theorem can now be stated as: Theorem 31 (Uniformization [Bel00]). Let M be a 3-dimensional compact manifold admitting a Sasakian structure S = (ξ, η, , g). Then (1) If S is positive, M is spherical, and there is a Sasakian metric of constant -sectional curvature 1 in the same deformation class as g. 2 type, and there is a Sasakian metric of constant (2) If S is negative, M is of SL -sectional curvature −4 in the same deformation class as g. (3) If S is null, M is nil, and there is a Sasakian metric of constant -sectional curvature −3 in the same deformation class as g. In the positive case the universal cover M˜ is S 3 with its standard round sphere metric. In this case one generally needs both type I and II deformations. In the negative case M˜ = B 2 × R, the product of the unit 2-ball with R; whereas, in the null case M˜ = R3 . In both of these cases the metrics have constant -sectional curvature, but not constant sectional curvature. Moreover, deformations of type I do not exist in both the negative and null cases except for D-homotheties. Clearly, these ‘standard metrics’ are all η-Einstein. More generally in any dimension constant -sectional curvature metrics are η-Einstein. In the positive case deformations of type I are definitely needed. Theorem 31 does not hold without them. A counterexample is the weighted Sasakian structure on S 3 with weights w = (p, q), p = 1. The transverse space of the characteristic foliation is the weighted projective line P1C (p, q) which is an example of a Fano orbifold which does not admit an orbifold metric of constant curvature. More generally, the existence of K¨ahler-Einstein metrics on weighted projective spaces of arbitrary dimension is obstructed by the non-vanishing of the Futaki character [ACGTF04].
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Remark 32. A Sasakian-Einstein structure on a 3-Sasakian manifold does not have to be a part of the 3-Sasakian structure. The simplest example when this is the case is the lens space Zk \S 3 . Consider the unit 3-sphere S 3 Sp(1) as the unit quaternion σ ∈ H. Such a sphere has two 3-Sasakian structures generated by the left and the right multiplication. Consider the homogeneous space Zk \S 3 , where the Zk -action is given by the multiplication from the left by ρ ∈ Sp(1), ρ k = 1. The quotient still has the “right” 3-Sasakian structure. But it also has a “left” Sasakian structure (the centralizer of Zk in Sp(1) is an S 1 and it acts on the coset from the left). This left Sasakian structure is actually regular while none of the Sasakian structures of the right 3-Sasakian structure can be regular unless k = 1, 2. Example 33. Consider the standard metric on S 3 −→S 2 of constant sectional curvature equal to 1. This metric is Sasakian-Einstein and 3-Sasakian. Applying D-homothety to this metric produces a family of η-Einstein U(2)-invariant metrics which can be written as σ12 + σ22 + aσ32 , where σ1 , σ2 and σ3 are the standard left-invariant 1-forms on S 3 SU (2) and a is a nonzero constant. The metrics were first considered by Berger. We will return to this example in the last section as it is the most basic Einstein-Weyl manifold. Let us now consider some examples of Sasakian structures on 3-manifolds represented as links of isolated hypersurface singularities defined by weighted homogeneous polynomials. Example 34. Positive Sasakian 3-manifolds. Belgun’s Theorem 31 says that every positive Sasakian 3-manifold is a spherical space form S 3 / , where is a finite subgroup of SU (2). These are precisely (1) (2) (3) (4) (5)
= Zp the cyclic group of order p, = D∗m a binary dihedral group where m is an integer greater than 2, = T∗ the binary tetrahedral group, = O∗ the binary octahedral group, = I∗ the binary icosahedral group.
Orlik [Orl70] shows that all of these groups can be realized by links of weighted homogeneous polynomials, and that this exhausts all cases with |w| > d. It should be noted, however, that not all lens spaces L(p, q) occur, but only L(p, 1). The table below indicates the subgroup ⊂ SU (2) together with a representative polynomial and the corresponding Dynkin diagram of the singularity. Diagram Ap−1 Dm E6 E7 E8
Subgroup Zp D∗m T∗ O∗ I∗
Polynomial p z0 + z12 + z22 z02 z1 + z1m + z22 z04 + z13 + z22 z03 + z13 z0 + z22 z05 + z13 + z22
Notice that L(5, 3, 2) = S 3 /I∗ is the famous Poincar´e homology sphere.
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Example 35. Null Sasakian 3-manifolds. Up to covering these are all non-trivial circle bundles over a complex torus, and it is known [Mil75] that all such bundles can be constructed from the three dimensional Heisenberg group H R3 which can be represented as a subgroup of GL(3, R) by considering matrices of the form
1 A = 0 0
x 1 0
z y ; x, y, z ∈ R. 1
(25)
We also define subgroups Hk ⊂ H of matrices for which x, y, z are integers divisible by some positive integer k. It is easy to see that the quotient manifold Mk = H /Hk is a circle bundle over a torus with Chern number ±k. The fibration π : Mk −→T 2 is explicitly given by π(A) = (x mod k, z mod k).
(26)
Now Sasaki [Sas65] showed that R3 has a natural null Sasakian η-Einstein metric with constants (λ, ν) = (−2, 4), although the connection with the Heisenberg group wasn’t observed until later. This metric is g = dx 2 + dy 2 + (dz − y dx)2 .
(27)
The Sasakian structure S = (ξ, η, , g) on H is defined by the formulas
η = dz − y dx ∂ ∂ = ∂x ⊗ dy − + y ∂z
ξ= ∂ ∂y
∂ ∂z
⊗ dx.
(28)
Now one can easily check that the tensor fields defining S are invariant under the action of H on itself. Hence, S defines a Sasakian structure on the compact quotient manifolds Mk . Milnor [Mil75] observed that the integral homology can easily be computed, viz. H1 (Mk , Z) = Hk /[Hk , Hk ] = Z ⊕ Z ⊕ Zk . However, as indicated by the table below only the cases k = 1, 2, 3 can be realized as links of isolated hypersurface singularities of weighted homogeneous polynomials. There are precisely three null Sasakian 3-manifolds that can be represented by links of isolated hypersurface singularities of weighted homogeneous polynomials [Orl70]. These are realized by arbitrary weighted homogeneous polynomials of degrees 6, 4 and 3, and they exhaust all possibilities that satisfy |w| = d. The table below gives the weights, degrees, and number of monomials of the three polynomials. manifold weight vector degree # of monomials M1 (1, 2, 3) 6 7 M2 (1, 1, 2) 4 9 M3 (1, 1, 1) 3 10
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Example 36. Negative Sasakian 3-manifolds. These are up to covering circle bundles over Riemann surfaces of genus g > 1. There are infinitely many examples which can be represented as links of isolated hypersurface singularities defined by weighted homogeneous polynomials. Milnor proves [Mil75] that in the case of Brieskorn 3-manifolds R)/ , where SL(2, R) is the universal cover of SL(2, R) and L(a0 , a1 , a2 ) = SL(2, is a discrete subgroup completely determined by the triple (a1 , a2 , a3 ). Perhaps the most interesting are the integral homology spheres. In this case a theorem of Saeki [Sae87] says that the link of an isolated hypersurface singularity defined by a weighted homogeneous polynomial has the same knot type as a Brieskorn link. Now Breiskorn’s graph Theorem [Bri66] implies that a Brieskorn link L(a) is a homology sphere if and only if gcd(ai , aj ) = 1 for all i = j = 0, 1, 2. So clearly 3-dimensional homology spheres are quite numerous, and all but one, the Poincar´e homology sphere L(5, 3, 2), have an infinite fundamental group. More, generally for any link of an isolated hypersurface singularity defined by a weighted homogeneous polynomial, Orlik [Orl70] proved that π1 is infinite if and only if d ≥ |w|, and π1 is infinite nilpotent if and only if d = |w| which is the Euclidean or null Sasakian case. Obviously, for integral homology spheres π1 must be perfect, i.e. [π1 , π1 ] = π1 , where [π1 , π1 ] denotes the commutator subgroup. A rather large class of examples can be treated by considering Brieskorn polynomials of the form f = z06k±1 + z13 + z22 . Since d − |w| = 6k − 6 ± 1 ≥ 0, π1 (L(6k ± 1, 3, 2)) is infinite in all cases except the Poincar´e homology sphere S 3 /I ∗ = L(5, 3, 2). For example, L(7, 3, 2) is the quo R) of SL(2, R) by a co-compact discrete subgroup tient of the universal cover SL(2, ⊂ SL(2, R), and there is a covering of L(7, 3, 2) by the total space of a nontrivial circle bundle over a Riemann surface of genus g = 3 [Mil75]. More generally, except for the Poincar´e homology sphere, L(6k ± 1, 3, 2) has a finite covering by a manifold that is diffeomorphic to a nontrivial circle bundle over a Riemann surface of some genus g > 1. We end this section with a brief discussion of how one can distinguish integral homology spheres in three dimensions. This can be done by the Casson invariant λ which roughly speaking counts the number of irreducible representations of π1 in SU (2). More explicitly for a Brieskorn homology sphere L(p, q, r) Fintushel and Stern [FS90] proved that λ(L(p, q, r)) equals − 21 times the number of conjugacy classes of irreducible representations of π1 (L(p, q, r)) into SU (2). Furthermore, they showed how λ(L(p, q, r)) can be computed from the Hirzebruch signature of the parallelizable manifold V (p, q, r) whose boundary is the link L(p, q, r). Explicitly, λ(L(p, q, r)) =
sig V (p, q, r) . 8
Now Brieskorn [Bri66] had computed the signature of V (6k − 1, 3, 2) to be −8k, giving λ(L(6k−1, 3, 2)) = −k. Thus, L(6k−1, 3, 2) gives an infinite sequence of inequivalent homology spheres. Moreover, it is known that λ(L(7, 3, 2)) = −1, and since k = 1 from the previous sequence is the Poincar´e homology sphere, which is clearly distinct, we see that L(7, 3, 2) is also distinct from the others. For further discussion and references we refer the reader to the recent book [Sav02].
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8. η-Einstein Metrics in Dimension 5 and Higher In this section we give a large number of examples of manifolds in dimension five and higher that admit Sasakian η-Einstein metrics. For null and negative Sasakian structures the proof of existence amounts to presenting examples as links of weighted homogeneous polynomials and then invoking Theorem 28. In the positive or Fano case the existence of Sasakian η-Einstein metrics is equivalent to the existence of Sasakian-Einstein metrics which has been studied quite extensively [BGN03a, BGK05, BGKT05, Kol04b]. The list of spin 5-manifolds which admit many inequivalent families of Sasakian-Einstein metrics is now long and keeps growing. For example, S 5 has at least 68 inequivalent Sasakian-Einstein metrics realized as Brieskorn-Pham links. The example below gives a subfamily. Example 37. Consider sequences of the form a = (2, 3, 7, m). By explicit calculation, the corresponding link L(a) gives a Sasakian-Einstein metric on S 5 if 5 ≤ m ≤ 41 and m is relatively prime to at least two of 2, 3, 7. This is satisfied in 27 cases. Recently, Koll´ar has shown that there are many inequivalent Sasakian-Einstein metrics on all k-fold connected sums of S 2 × S 3 [Kol04b]. On the other hand Boyer and Galicki showed that [BG03] Theorem 38. The links L(3, 3, 3, k)k > 2, L(2, 4, 4, p), p > 2 and L(2, 3, 6, m), m > 4 all admit Sasakian-Einstein structures. The above links are rational homology 5-spheres under suitable assumptions on k, p, m. For example L(3, 3, 3, k) is a rational homology sphere with H2 (L, Z) = Zk ⊕ Zk as long as k is prime to 3. Since any L(3, 3, 3, k) is spin, by Smale’s theorem [Sma62], the second homology group completely determines the diffeomorphism type of L(3, 3, 3, k). In the null case there appear to be only a finite number of manifolds admitting such structures. We begin with a non-existence result in dimension 5. Theorem 39. Let S be a null Sasakian structure on a compact 5-manifold with H1 (M, Z) = 0. Then 2 ≤ b2 (M) ≤ 21 and H2 (M, Z) is torsion free. Furthermore, if b2 (M) = 21 then S is regular and M is diffeomorphic to #21(S 2 × S 3 ), and M/Fξ is a K3 surface. Proof. The lower bound on b2 (M) is immediate from Corollary 1.10 of [BGN03a], while the upper bound as well as the last statement follow from Corollary 81 of [Kol04a]. That H2 (M, Z) is torsion free follows from Proposition 80 of [Kol04a]. We mention that the lower bound in Theorem 39 holds with the weaker condition H1 (M, R) = 0. We now have Corollary 40. S 5 and S 2 ×S 3 or any quotient by a finite group do not admit null Sasakian structures. Compact 5-manifolds with null Sasakian structures can be easily obtained from the list of 95 orbifold K3 surfaces. Example 41. In 1979 Reid [Rei80] produced a list of 95 weighted K3 surfaces given as well-formed hypersurfaces in weighted projective space P(w0 , w1 , w2 , w3 ). His result
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generalizes the standard construction of the K3 surface as a quartic in CP(3). With each example of Reid one can consider the associated 5-dimensional link Lf . By a result of Boyer and Galicki [BG01b] all Lf are diffeomorphic to some k-fold connected sum of S 2 ×S 3 , where k = b2 (Lf ) can be computed in each case by the Milnor-Orlik procedure [MO70]. For instance L(4, 4, 4, 4) #21(S 2 × S 3 ) is regular and it is a circle bundle over the K3-surface X4 ⊂ P3 . No.3 on Reid’s list in the table of [IF00] is also a BP link L(6, 6, 6, 2) #21(S 2 × S 3 ). The degree 6 surface in X6 ⊂ P(1, 1, 1, 3) given by vanishing of f (z) = z06 + z16 + z26 + z32 is easily seen to be non-singular and as a result we also get a regular null Sasakian structure on a K3-surface and regular null Sasakian structure on the corresponding circle bundle. However, X4 and X6 cannot be the same as complex algebraic varieties and L(4, 4, 4, 4) and L(6, 6, 6, 2) are not equivalent as a Sasakian manifold. In all, there are 11 more examples with are Brieskorn-Pham links. All Reid’s examples have 3 ≤ b2 (Lf ) ≤ 21. Hence, for instance, the link Lf with f (z) = z02 + z12 + z2 z0 z1 z3 + z22 z3 + z23 z0 + z33 must be diffeomorphic to #3(S 2 × S 3 ). No. 14 on the list, is a BP link L(2, 3, 12, 12) and it is diffeomorphic to #20(S 2 × S 3 ). It is interesting that b2 (L) = 17 does not occur on Reid’s list. But other examples can be considered. For instance, Fletcher gives a list of 84 examples of codimension 2 complete intersections [IF00]. On the other hand, the only codimension 3 weighted complete intersection is the intersection of 3 quadrics in P5 . By writing a simple Maple program we have computed the second Betti number of all 95 orbifold K3 surfaces, and we find Corollary 42. #k(S 2 × S 3 ) admit both positive and null Sasakian η-Einstein structures for 3 ≤ k ≤ 21 and k = 17. The question which of the simply connected compact spin 5-manifolds of Smale’s list admit null Sasakian structures is equivalent to classifying all orbifold K3-surfaces. This is still open. An interesting question is whether k = 2 or 17 can occur. In higher dimensions there are many, though probably finite, null Sasakian structures, and they will all admit η-Einstein null structures. Interesting examples include the over 6000 Calabi-Yau orbifolds in complex dimension 3 [CLS90]. In the negative case even just Brieskorn-Pham links produce already very many examples. The simplest ones are the Fermat hypersurfaces of degree d in P3 . For d = 4 it gives #21(S 2 × S 3 ) as discussed in Example 41 above. For d ≥ 5 we get negative Sasakian η-Einstein metrics on #k(S 2 × S 3 ) for k = (d − 2)(d 2 − 2d + 2) + 1. These begin with k = 52 and grow rapidly. To obtain negative Sasakian η-Einstein structures we turn to some different examples. First, such structures exist on rational homology spheres, and in particular on S 5 . Example 43. Consider L(k, k, k + 1, p). Such a link is negative as long as k, p > 3 or with k = 3 and p > 12. Moreover, L(k, k, k + 1, p) is a 5-sphere as long as p is prime to both k and k + 1 which can easily be arranged. Proposition 44. Every negative Sasakian structure on S 5 is non-regular and there exist infinitely many inequivalent negative Sasakian η-Einstein structures. Hence, S 5 admits infinitely many different Lorentzian Sasakian-Einstein structures.
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Proof. The existence of infinitely many non-regular negative Sasakian structures on S 5 follows immediately from Example 43. These lie in different deformation classes, and from Theorem 17 each deformation class has a negative η-Einstein metric. To prove the first statement we assume that S 5 has a regular negative Sasakian structure. Then we have a circle bundle S 5 −→X with c1 (X) negative, π1 (X) = {1}, and b2 (X) = 1. This is impossible by a theorem of Yau, cf. Theorem V.1.1 of [BPVdV84]. It is rather easy to produce examples of negative Sasakian η-Einstein structures on #k(S 2 × S 3 ); however, we haven’t been able to find one nice sequence which does this, or even a sequence that works for odd k. The examples below give the essential ideas. Example 45. This is Example 43 with p = k +1. Now the BP link is L(k, k, k +1, k +1) and it is easy to see that the induced Sasakian structures are negative as long as k ≥ 4. Here the Milnor-Orlik method gives b2 (L(k, k, k + 1, k + 1)) = k(k − 1). This gives negative Sasakian η-Einstein structures on #k(S 2 × S 3 ) for infinitely many k, but with larger and larger gaps as k grows. For low values of k, for example, k = 4 and 5, we get negative Sasakian structures on #12(S 2 × S 3 ), and #20(S 2 × S 3 ), respectively. Example 46. Consider the BP links L(p, q, r, pqr) such that p, q, r ∈ Z+ are pairwise relatively prime. The transverse space is a surface Xpqr ⊂ P(pr, qr, pq, 1) and it is automatically well-formed, i.e., it has only isolated orbifold singularities. Hence, L(p, q, r, pqr) is diffeomorphic to #k(S 2 × S 3 ), where k = b2 (L(p, q, r, pqr)) = (pqr − pq − pr − qr − 1) + p + q + r. Now, L(p, q, r, pqr) is negative when the term in parenthesis is positive, i.e. when pq + pr + qr + 1 < pqr. To simplify we can consider L(2, 3, r, 6r) with r prime to both 3 and 2. L(2, 3, 7, 42) is one of the examples on Reid’s list. But from r > 7 we get infinitely many examples with second Betti number b2 = 2(r − 1). For example for r = 11, we see that L(2, 3, 11, 66) is diffeomorphic to #20(S 2 × S 3 ). Notice that both Examples 45 and 46 produce η-Einstein metrics on #k(S 2 × S 3 ) for even k only. We haven’t yet found a nice series that gives metrics for odd values of k. However, we can obtain η-Einstein metrics for odd k if we consider the more general weighted homogeneous polynomials. Example 47. Here we give a few examples only to illustrate the method. A much more extensive list can be generated with the aid of a computer a la [JK01, BGN03b]. We begin with two inequivalent ‘twin’ Sasakian η-Einstein metrics on S 2 × S 3 . Consider the weighted homogeneous polynomials z021 z1 + z15 z2 + z23 z0 + z32 ,
z021 z1 + z15 z0 + z23 z1 + z32 .
Both have degree 316 and both give links diffeomorphic to S 2 × S 3 . The weight vectors are w = (13, 43, 101, 158) and w = (11, 61, 85, 158), respectively. So |w| − d = −1 giving negative Sasakian η-Einstein metrics on S 2 × S 3 . Similarly, the polynomials z020 + z13 z3 + z23 z1 + z32 z0
z010 + z14 z2 + z22 z3 + z32 z0
have degree 40 and 159, respectively. They give negative Sasakian η-Einstein metrics on #7(S 2 × S 3 ) and #2(S 2 × S 3 ).
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We have Corollary 48. #7(S 2 × S 3 ), #12(S 2 × S 3 ) and #20(S 2 × S 3 ) all admit positive, null, and negative Sasakian η-Einstein structures. It is easy to construct examples of negative Sasakian η-Einstein metrics on homotopy spheres in arbitrary odd dimensions. Example 49. Let the integers ri , i = 1, . . . , 2m be any pairwise prime positive integers. Then the (4m+1)-dimensional link L(2, 2r1 , . . . , 2r2m , a) is diffeomorphic to the standard sphere if a ≡ ±1 mod 8 and to the Kervaire sphere if a ≡ ±3 mod 8. The link will be negative when i r1i < a−2 a which is easily satisfied for ri ’s large enough. Corollary 50. In any dimension 4m + 1 both the standard and the Kervaire spheres admit infinitely many inequivalent negative Sasakian η-Einstein structures. In dimension 4m + 3 one can construct examples of negative Sasakian structures existing on all homotopy spheres which bound parallelizable manifolds. That is to say, all BP links that by Brieskorn Graph Theorem are homeomorphic to a sphere of dimension 4m + 3 and are negative should easily contain all possible oriented diffeomorphism types. We have checked it in dimension 7 with the following Theorem 51. All 28 oriented diffeomorphism types of homotopy 7-spheres admit negative Sasakian structures. Hence, they all admit Lorentzian Sasakian-Einstein structures. Proof. Consider L(k, k, k, k +1, p), where p is prime to both k and k +1. The link L is a homotopy 7-sphere and it is negative for k and p large enough. Using the computer codes of [BGKT05], it is easy to check that such links realize all 28 oriented diffeomorphism types. Similarly, one should be able to show that the homotopy spheres in dimensions 11 and 15 that bound parallelizable manifolds admit negative Sasakian structures. Remark 52. Many of the Sasakian η-Einstein structures admit large moduli spaces. This is particularly true for BP type polynomials. We refer the reader to our previous work [BGN03b, BGK05, BGKT05]. 9. Positive η-Einstein Structures and Einstein-Weyl Geometry In this chapter we investigate relations between Einstein-Weyl structures and η-Einstein metrics. The first insight into this relation was provided by Swann and Pedersen [PS93] and by Higa [Hig93] who studied Einstein-Weyl geometries of circle bundles over positive K¨ahler-Einstein manifolds. Later Narita in a series of papers made the connection to η-Einstein geometry more explicit [Nar93, Nar97, Nar98]. We begin with some basic facts about Einstein-Weyl geometry (see [CP99] for a review of the subject). Definition 53. A Weyl structure on a manifold M of dimension n ≥ 3 is defined by a pair W = ([g], D), where [g] is a conformal class of Riemannian metrics and D is the unique torsion-free connection preserving [g]. The connection D is called the Weyl connection.
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Then, for every Riemannian metric g in the conformal class, there exists a 1-form θ , uniquely determined by D and g, such that, for every X, Y ∈ X − (M), DX Y = ∇X Y + θ (X)Y + θ (Y )X − g(X, Y )θ,
(29)
where ∇ denotes the Levi-Civita connection of g and θ # the dual vector field of θ with respect to g. In particular we have Dg = −2θ ⊗ g.
(30)
The above condition is invariant under Weyl transformation, i.e., g = e2f g,
θ = θ + df,
f ∈ C ∞ (M)
(31)
Hence, choosing a Riemannian metric in [g] we sometimes abuse the terminology and refer to the pair W = (g, θ ) as a Weyl structure. It is known that on a compact, conformal manifold of dimension at least 3, the conformal class [g] contains a unique (up to homothety) metric g such that the corresponding form θ is g-co-closed (cf. [Gau95]). This metric is the Gauduchon metric or the Gauduchon gauge. Definition 54. A conformal manifold (M, [g], D) is called Einstein-Weyl if RicD (X, Y ) + RicD (Y, X) = g(X, Y ) for some smooth function on M. Clearly the Einstein-Weyl condition on the symmetrized Ricci tensor of D is conformally invariant. A Weyl structure is said to be closed (resp. exact) if the Weyl connection is locally (resp. globally) the Levi-Civita connection of a compatible metric. Hence a closed Einstein-Weyl structure admits local (but not necessarily global) compatible Einstein metrics. On compact manifolds of dimension at least 3 with a closed but not exact EinsteinWeyl structure the form θ of the Gauduchon metric g is g parallel: ∇θ = 0 [Gau95]. In particular, θ is closed, hence g-harmonic. On an almost contact metric manifold (M, ξ, η, , g) it is natural to ask whether M has an Einstein-Weyl structure W = (g, θ ) such that θ = f η, for some function f ∈ C ∞ (M), and whether the metric g compatible with the almost contact structure is the Gauduchon metric of [g]. In fact, if (M, ξ, η, , g) is an almost contact manifold then, for any 1-form θ , the Weyl structure W = (g, θ ) on M is connected to S = (ξ, η, , g) by several fundamental relations. The existence of Einstein-Weyl structures on almost contact metric manifolds have been considered by several authors [Mat02, Mat00, Nar97, Nar98]. Here we review and then extend some of the known results. Let (M, ξ, η, , g) be a (2n + 1)-dimensional, n ≥ 1, almost contact metric manifold. If W = (g, θ ) is a Weyl structure on M, for every X, Y ∈ X − (M), the Ricci tensor RicD of D and Ricg of ∇ are related by the following equation [Hig93]: RicD (X, Y ) = Ricg (X, Y ) − 2n(∇X θ )(Y ) + (∇Y θ )(X) +(2n − 1)θ (X)θ (Y ) + (δθ − (2n − 1)|θ |2 )g(X, Y ),
(32)
where δθ , |θ | are the codifferential and the point-wise norm of θ with respect to g. The following local characterizations of Einstein-Weyl structures can be found in [Hig93].
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Proposition 55. Let W = (g, θ ) be a Weyl structure on an almost contact metric manifold (M, ξ, η, , g) of dimension 2n + 1 ≥ 3. Then W = (g, θ ) is an Einstein-Weyl structure if and only if there exists a smooth function σ on M such that: 1 − 2n ((∇X θ )(Y ) + (∇Y θ )(X)) + (2n − 1)θ (X)θ (Y ) + Ricg (X, Y ) = σ g(X, Y ), 2 (33) for every X, Y ∈ X − (M). Proposition 56. Let (M, ξ, η, , g) be an almost contact metric manifold of dimension 2n + 1 ≥ 3 and θ a 1-form on M. Consider the Weyl structures W ± defined by W ± = (g, ±θ ). Then both W + and W − are Einstein-Weyl if and only if (g, θ ) satisfies the following two equations for every X, Y ∈ X − (M): (∇X θ )(Y ) + (∇Y θ )(X) +
Ricg (X, Y ) −
2 δθ g(X, Y ) = 0, 2n + 1
(34)
s 2n − 1 2 g(X, Y ) = |θ| g(X, Y ) − (2n − 1)θ (X)θ (Y ), (35) 2n + 1 2n + 1
where s is the scalar curvature of g. First of all, as a direct consequence of (33), we can state Proposition 57. Let (M, ξ, η, , g) be a K-contact metric manifold of dimension 2n + 1 ≥ 3 and suppose that M admits an Einstein-Weyl structure W = (g, θ ) with θ = f η for some non-constant function f on M. Then X(f ) = 0 for every vector field X on M such that η(X) = 0 and the Ricci tensor of M is given by Ricg = σ g − (2n − 1)(f 2 − ξ(f ))η ⊗ η. Proof. We recall, at first, that the Ricci tensor of a K-contact manifold always satisfies the relation Ricg (X, ξ ) = 2nη(X). Then, for every X horizontal vector field on M (here and in the following we shall call horizontal any vector field on M orthogonal to ξ ) and Y = ξ , (33) becomes (∇X f η)(ξ ) + (∇ξ f η)(X) = 0,
(36)
which easily implies the first part of the proposition. A straightforward computation proves the second part of the assert. Corollary 58. If a K-contact manifold M of dimension at least 5 admits an EinsteinWeyl structure W = (g, f η) with f ∈ C ∞ (M), then M is η-Einstein. In particular, in such case, σ and f 2 − ξ(f ) must be constant. The following example generalizes the 3-dimensional case of Example 35. Example 59. Let H (n) be the Heisenberg Lie group 1 xt z H (n) = 0 1 y ; x, y ∈ Rn , z ∈ R , 0 0 1
(37)
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and let g be the following left invariant metric on H (n), g = dx · dx + dy · dy + (dz − x · dy)2 . The Sasakian structure S = (ξ, η, , g) on H (n) is defined by the formulas ∂ η = dz − x · dy, ξ = ∂z
∂ ∂ i = i ( ∂x i + y ∂z ) ⊗ dy i − ∂y∂ i ⊗ dx i .
(38)
(39)
One can easily check that this is a null Sasakian η-Einstein structure on H (n) [Sas65]. Moreover, a transverse homothety of S by a constant a gives a null Sasakian η-Einstein ¯ g) structure S¯ = (ξ¯ , η, ¯ , ¯ which admits the Einstein-Weyl structure W = (g, ¯ f η), ¯ with f = α tan(z + c), α, c ∈ R, provided that a = α1 , and α > 0 such that α 2 = 2n+2 2n−1 . This generalizes to arbitrary odd dimension the Einstein-Weyl structure described in [Nar97, Nar98]. In addition, one can get compact examples by taking quotients of H (n) by discrete subgroups. From the formula (33) we can also deduce that, if ξ is a Killing vector field on the almost contact metric manifold (M, ξ, η, , g) and M admits an Einstein-Weyl structure W + = (g, µη) for some constant µ, then M admits the Einstein-Weyl structure W − = (g, −µη) too. More precisely, applying Proposition 56 to the K-contact manifolds, we obtain the following Theorem 60. Let (M, ξ, η, , g) be a K-contact manifold of dimension 2n + 1 ≥ 3 and θ a 1-form on M. Suppose that both W ± = (g, ±θ ) are Einstein-Weyl structures on M. Then θ is co-closed, but not closed, so g is the Gauduchon metric, the dual vector field θ # to θ is a Killing vector field, and either (1) θ (ξ ) = 0 or (2) θ = µη with µ ∈ R. Proof. In fact, if both W ± = (g, ±θ ) are Einstein-Weyl structures on (M, ξ, η, , g), the 1-form θ satisfies both equations of Proposition 56. In particular, substituting ξ for Y in (35) we get 2nη(X) −
s 2n − 1 2 η(X) = |θ| η(X) − (2n − 1)θ (X)θ (ξ ). 2n + 1 2n + 1
(40)
Then the relation θ (X)θ (ξ ) = 0
(41)
is true for every horizontal vector field X. Now define the set S = {x ∈ M|θ (X)x = 0 ∀ horizontal X}. S is closed, and by (41) θ (ξ ) = 0 on the open set M\S. Then putting X = Y = ξ in (34) implies that δθ = 0 on M\S. But then again (34) implies (∇X θ )(Y ) + (∇Y θ )(X) = 0 on M\S for any X, Y ∈ X − (M). This means that the dual vector field θ # is a Killing field on M\S. Consequently, if θ were closed on M\S, it would also be parallel with respect to the Levi-Civita connection ∇ of g. In particular for every X ∈ X − (M) we have 0 = (∇X θ )(ξ ) = −θ ((X))
(42)
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implying that θ identically vanishes on M\S. This proves that θ cannot be closed on M\S. On the other hand, the set S0 where θ (ξ ) = 0 is open and a subset of S by (41). Thus, θ = µη on S for some µ ∈ C ∞ (S). So, for X, Y horizontal vector fields (34) becomes µ((∇X η)(Y ) + (∇Y η)(X)) +
2 δθ g(X, Y ) = 0 2n + 1
(43)
on S. But the first term vanishes since ξ is a Killing vector field. This implies that δθ = 0 on S and hence, on all of M. But then (34) gives (∇X θ )(Y ) + (∇Y θ )(X) = 0, on all M for all X, Y ∈ X − (M) which means that θ # is a Killing vector field, and also implies that µ is a constant. This completes the proof of the theorem. We now discuss several corollaries of Theorem 60. First combining our results with [Gau95] we have Corollary 61. Let (M, ξ, η, , g) be a compact K-contact manifold of dimension 2n + 1 ≥ 3 and θ a 1-form on M. Given the Weyl structure W = (g, θ ), then the metric g is the Gauduchon metric of the conformal class [g] if and only if M admits both W ± = (g, ±θ ) as Einstein-Weyl structures. Corollary 62. Let (M, ξ, η, , g) be a K-contact manifold of dimension 2n + 1 ≥ 5. Then (M, ξ, η, , g) admits an Einstein-Weyl structure W = (g, θ ) with θ = µη, µ ∈ R, if and only if M is η-Einstein with Einstein constants λ, ν such that ν < 0. Proof. If (M, ξ, η, , g) admits an Einstein-Weyl structure with 1-form θ = µη for some constant µ, then from (33) we obtain Ricg = σg − (2n − 1)µ2 η ⊗ η,
(44)
which proves the only if part of the assertion. On the contrary, if we suppose that the Ricci tensor of M is given by Ricg = λg + νη ⊗ η,
(45)
where ν < 0, (33) is satisfied for σ = λ and θ = µη where µ is a constant such that ν µ2 = − 2n−1 . In the case of Sasakian structures Theorem 60 can be improved: Theorem 63. A Sasakian manifold (M, ξ, η, , g) of dimension 2n + 1 ≥ 5 admits both the Einstein-Weyl structures W + = (g, θ ) and W − = (g, −θ) for some 1-form θ if and only if is η-Einstein with Einstein constants (λ, ν) such that ν < 0. Proof. From the proof of Theorem 60 we know that, if (M, ξ, η, , g) admits both W ± as Einstein-Weyl structures, then for the 1-form θ the relation θ(X)θ(ξ ) = 0 holds, for every horizontal vector field X on M. On the other hand, since on a Sasakian manifold the Ricci tensor satisfies the equality Ricg ((X), (Y )) = Ricg (X, Y ) − 2nη(X)η(Y ), substituting in (35) (X) for X and (Y ) for Y with X, Y horizontal vector fields, we get Ricg (X, Y ) −
s 2n − 1 2 g(X, Y ) = |θ | g(X, Y ) − (2n − 1)θ ((X))θ ((Y )), 2n + 1 2n + 1 (46)
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where s is the scalar curvature of g. The comparison between (35) and this last equation implies that θ must also obey the equation θ ((X))θ ((Y )) = θ (X)θ(Y ),
(47)
for all horizontal vector fields X, Y on M. Now, arguing as in the proof of Theorem 60, if we suppose θ (ξ ) = η(θ # ) = 0 and consider X = Y = θ # in (47), we obtain |θ | = 0 so that θ identically vanishes on M. Finally, Corollary 62 implies the theorem. Remark 64. We remark that, by taking Corollary 61 into account, in the case of a compact Sasakian manifold (M, ξ, η, , g), the above Theorem 63 provides the necessary and sufficient condition for g to be the Gauduchon metric of the conformal class [g] for some Einstein-Weyl structure on M. Finally, the following corollary follows easily from the previous analysis. Corollary 65. Let (ξ, η, , g) be a Sasakian-Einstein structure on a manifold M. Then M admits a pair of Einstein-Weyl structures W ± = (g, ±µη) obtain by squashing the Sasakian-Einstein structure to an η-Einstein structure with constants (λ, ν) and ν < 0. ν In this case µ2 = − 2n−1 . In particular, all homotopy spheres of dimension 4n + 1, 7, 11 and 15 that bound parallelizable manifolds admit many pairs of Einstein-Weyl structures. The above corollary combined with recent results of [BGN03b, BGK05, BGKT05, BG03, Kol04b, Kol04a] establishes the existence of Einstein-Weyl structures on many other compact simply connected spin manifolds in odd dimensions. Acknowledgements. We thank Gang Tian for answering several questions. We would also like to thank J´anos Koll´ar for numerous discussions and for explaining some of his recent results [Kol04a] to us, and to Liviu Ornea for pointing out reference [GO98]. CPB and KG were partially supported by the NSF under grant number DMS-0203219. The authors would also like to thank Universit`a di Roma “La Sapienza” for partial support where discussions on this work initiated. The second named author would like to thank Max-Planck-Institute in Bonn for hospitality and support. Much of the work on this article was done during his stay there.
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Communicated by G.W. Gibbons
Commun. Math. Phys. 262, 209–236 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1458-7
Communications in
Mathematical Physics
A Proof of Crystallization in Two Dimensions Florian Theil University of Warwick, Mathematics Institute, Coventry, CV7 4AL, UK. E-mail: [email protected] Received: 1 March 2005 / Accepted: 15 May 2005 Published online: 9 December 2005 – © Springer-Verlag 2005
Abstract: Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type E({y}) = V (|y(x) − y(x )|), where y(x) ∈ R2 is the position of 1≤x<x ≤N
particle x and V (r) ∈ R is the pair-interaction energy of two particles which are placed at distance r. Due to the Mermin-Wagner theorem it can’t be expected that at finite temperature this system exhibits long-range ordering. We focus on the zero temperature case and show rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E∗ : lim 1 min E({y}) N→∞ N y:{1...N }→R2
= E∗ ,
where E∗ ∈ R is the minimum of a simple function on [0, ∞). Furthermore, if suitable Dirichlet- or periodic boundary conditions are used, then the minimizers form a triangular lattice. To the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the Euclidean symmetry group consisting of rotations and translations. 1. Introduction The purpose of this paper is to provide a mathematically rigorous analysis for the asymptotic behavior of the ground state energy (minimum energy) of many-particle systems where the interaction is governed by pair-potentials: E({y}) :=
1 V (|y(x) − y(x )|). 2 x,x ∈XN x=x
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Here, the finite set XN indexes the individual particles, #XN = N ∈ N is the number of particles and y(x) ∈ Rd specifies the position of particle x. We will focus on the thermodynamic limit where N tends to infinity. One of the most interesting cases is when V is given by the Lennard-Jones potential VLJ (r) = r −12 − r −6 . Extensive numerical studies in the case d = 2 (see [1] and the references therein) suggest that the ground states (energy minimizers) approximate a translated, rotated and dilated copy of the triangular lattice 1 2 1 √ A2 = Z2 ⊂ R2 2 0 3 and the ground state energy divided by N converges to a constant as N → ∞. In the case d = 1 it is known that this is true for rather general (both discrete and continuous) systems, see [2–4] and the references therein. The first aim of this paper is to establish sufficient conditions on the potential V such that for d = 2 the ground state energy per particle converges to a value that can be found by minimizing over periodic particle configurations; those potentials will be referred to as 2d-crystalline. Radin proved in [5] that the class of 2d-crystalline potentials is not empty by characterizing the set of ground states for a special piecewise-affine potential 25 VRadin which assumes finite and nonzero values in [1, 24 ) and has its absolute minimum at 1. We will address the question whether Radin’s result persists if the system is generic, i.e. non-integrable in the sense that for finite numbers N the ground states are not translated, rotated and dilated subsets of A because of the presence of boundary layers or a small number of lattice defects. It is expected that a more analytical understanding of the underlying mathematical structures will also indicate how to attack the three-dimensional case which cannot be treated with Radin’s method since there are several non-equivalent lattices which are natural candidates for the ground state: fcc (face-centered cubic), bcc (body-centered cubic) and hcp (hexagonally close packed) to name only the most classical multi-lattices. The selection of the correct lattice is necessarily based on the contributions of second-nearest neighbor interactions, but the methods developed so far are not capable of dealing with short-range and long-range interactions simultaneously. We present a new method, based on recently discovered rigidity estimates that covers a natural class of potentials which also allows for power-law behavior at infinity and 0. For the sake of notational simplicity we will assume wlog that the interaction potential is normalized in the sense that limr→∞ V (r) = 0 and min V (r|ξ |) = V (|ξ |) = −6. (1) r≥0
ξ ∈A2 \{0}
ξ ∈A2 \{0}
Theorem 1.1 (Ground state energy). There exists a constant α0 ∈ (0, 13 ) such that for each α ∈ (0, α0 ) and every normalized potential V : [0, ∞) → [0, ∞] which has the properties V ∈ C 2 (1 − α, ∞) and 1 for r ∈ [0, 1 − α], α V (r) ≥ 1 for r ∈ (1 − α, 1 + α), V (r) ≥ −α for r ∈ 1 + α, 43 , |V (r)| ≤ αr −7 for r ∈ 43 , ∞ , V (r) ≥
(2) (3) (4) (5)
A Proof of Crystallization in Two Dimensions
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the identity lim
min
N→∞ y:XN
→R2
1 N
V (|y(x) − y(x )|) = −3
{x,x }⊂XN
holds. Note that the assumptions on V allow both for hard-core interactions (V (r) = +∞ if r ∈ [0, ρ0 ] for some ρ0 ∈ [0, 1 − α]) and soft-core interactions (V (r) < ∞ for all r ∈ [0, ∞)). The second aim of this paper is to characterize the actual ground states. This is a delicate problem due to the possible formation of surface layers. Rigorous results can be obtained in simplified situations where suitable boundary conditions are used. In two cases we obtain that the ground states are translated copies of the triangular Bravais lattice A2 . By abuse of notation we identify for L ∈ N the quotient √
space A2 mod LA2 with the representatives A2 ∩ L U , where U = 21 02 13 Q, and Q = [0, 1) × [0, 1) ⊂ R2 is the semi-open unit cell. Theorem 1.2 (Ground states with periodic boundary conditions). There exists a constant α0 ∈ (0, 13 ) such that for every pair of numbers α ∈ (0, α0 ), L ∈ N, every potential V (·) which satisfies the assumptions of Theorem 1.1 and every ground state ymin : A2 → R2 of per
EL ({y}) :=
V (|y(x) − y(x )|),
x∈A2 ∩L U x ∈A2 \{x}
subject to y ∈ YL := {y : A2 → R2 | y(x) − y(x ) = x − x if x − x ∈ LA2 }, per
there exists a translation vector τ ∈ R2 such that {ymin (x) + τ | x ∈ A2 } = A2 . Corollary 1.3 (Stability against compactly supported perturbations). Let the assumptions of Theorem 1.2 be satisfied and assume that A ⊂ A2 is an arbitrary but finite set. If the configuration ymin : A2 → R2 is a ground state of EA ({y}) :=
V (|y(x) − y(x )|)
{x,x }⊂A2 {x,x }∩A=∅
subject to the constraint Dir = {y : A2 → R2 | y(x) = x for all x ∈ A2 \ A}, y ∈ YA
then {ymin (x) | x ∈ A2 } = A2 .
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3
2
1
0
−1
−2
−3 0.5
1
1.5
2
2.5
3
Fig. 1. Plot of the potential V (r) = r −12 + tanh(4r − 13 2 ) − 1 (left) and the associated energies per particle (right) f (·, Z2 ) (dotted line) and f (·, A2 ) (solid line)
The normalization (1) does not affect the scope of the results, as for general interaction potentials it can be shown without trouble that if the function V∗ (r) = 16 ξ ∈A2 V (r|ξ |) does not admit a minimizer then the ground state can not be approximated by rotated, dilated and translated copies of subsets of A2 . On the other hand if r∗ ∈ (0, ∞) minimizes V∗ and V∗ (r∗ ) = 0 then we can simply replace V (r) by the rescaled potential −1 r V∗ (r∗ ) V ( r∗ ). Even in two dimensions for “natural” interaction potentials V the ground states might not converge to a triangular lattice as N → ∞. For example, if V (r) = r −12 + tanh(4r − 13 2 ) − 1, then the triangular lattice A2 leads to a higher energy per particle than the square lattice Z2 , in particular the ground states will not resemble a triangular 2 lattice. This follows from the simple observation that minr f (r, Z ) < minr f (r, A2 ), where f (r, L) = ξ ∈L\{0} V (r|ξ |), cf. Fig. 1. Notation. The energy per nearest neighbor bond in a homogeneously deformed triangular lattice A2 is given by the renormalized potential V∗ (r) :=
1 6
V (r|ξ |).
ξ ∈A2 \{0}
It coincides up to a multiplicative constant with the function f (r, A2 ) defined above. We will use the shorthand e({x, x }) for V (|y(x ) − y(x )|) and e∗ ({x, x }) for V∗ (|y(x) − y(x )|). The letter “C” denotes a generic positive constant which does not depend on α or #XN = N as long as α ∈ [0, 13 ]. The actual value of C may change from line to line. B(η, r) denotes the closed ball centered at η ∈ R2 with radius r. Subsets of XN are mostly denoted by calligraphic letters. Throughout the proof we assume that yN : XN → R2 is a ground state of E and depends on N , i.e. it globally minimizes the energy in the given class of admissible configurations. The main focus of the analysis is to characterize the properties of the state yN for fixed N , the asymptotic behavior as N → ∞ is only discussed towards the end. For this reason the subscript N is omitted in most cases. Note that the minimization property of yN is used only in two places: Lemma 2.2 and at the end of the proofs of Theorem 1.2 and Corollary 1.3. The other arguments are only based on the bound (13). By {x, x } ⊂ XN we mean sets with precisely 2 elements.
A Proof of Crystallization in Two Dimensions
213
As the main objective of this paper is to explain the method, which hasn’t been used before, we do not make any effort to optimize the constants. 2. Proof of Theorem 1.1 2.1. Outline. It is easy to see that 1 N→∞ y:XN →R2 N lim
min
e({x, x }) ≤ −3
{x,x }⊂XN
by choosing the trial configuration y(x) = (x) where : N → A2 is a reasonable bijection which does not create too much surface energy. To show the much harder direction 1 lim min e({x, x }) ≥ −3, (6) N→∞ y:XN →R2 N {x,x }⊂XN
we split the set of all pairs P := {p ⊂ X | #p = 2} into two parts: short-range and ˙ long-range P = S(yN )∪L(y N ), where
S(yN ) := {x, x } ∈ P | |yN (x) − yN (x )| − 1 ≤ α is the index set for the short-range interactions and L(yN ) := P \ S(yN ) contains the long-range interactions. By construction this splitting is induced by the actual ground state yN . For each x ∈ X we define the discrete neighborhood N (x) := {x ∈ X | {x, x } ∈ S(yN )} ∪ {x}, again this set depends on yN although the dependency is not shown. The defects ∂X(yN ) := {x ∈ XN | #N (x) = 7} are those particles whose neighborhood cannot be mapped bijectively onto A2 ∩ B(0, 1). Based on simple, combinatorial considerations we will later (Proposition 2.3) establish the estimate #S(yN ) ≤ 3N − 21 #∂X(yN )
(7)
which shows that the set of short-range interactions grows like N , not N 2 . The splitting ˙ P = S(yN )∪L(y N ) is adequate if we can show that the following inequality holds: e(p) ≥ e∗ (p). (8) p∈P
p∈S (yN )
The claim (6) follows immediately from (7) together with (8). Inequality (8) links the energy of an arbitrary configuration to the renormalized potential V∗ which provides the energy per bond of a homogeneously dilated triangular lattice. The crucial point is that the left-hand side of (8) is a complicated sum with 21 N (N − 1) terms whereas the right-hand side contains at most 3N terms. We will actually prove a slightly more complicated version of (8), namely 2 1 e(p) ≥ −#S(yN ) + |yN (x) − yN (x )| − 1 + 41 #∂X(yN ), (9) C p∈P
{x,x}∈S (yN )
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where C > 0 is independent of V , α and N . In order to establish the validity of (9) we will demonstrate that away from the defect set ∂X(yN ) there are big patches which can be identified with simply connected subsets of A2 . The statement of the theorem follows since the definition of V∗ , (7) and (9) imply (6). The estimates quantify the errors caused by neglecting two different types of energy contributions: I) omitted individual bonds and II) distortion of the bonds. Errors of type I will be absorbed into the term which is proportional to #∂X(yN ) in the right-hand side of (9). Errors of type II require a more subtle handling. Using the geometric rigidity bounds from Proposition 4.1 they will be absorbed into the sum over S(yN ) in the right-hand side of (9). This approach requires us to work with three different spaces and three different maps between these spaces. reference configuration
u
-
A2 J ] J J J J J J J J label space
y(XN )
deformed configuration
y
XN
2.2. Combinatorial considerations. We collect several estimates of the interaction potential V and the renormalized potential V∗ which are required later. Lemma 2.1. There exists α0 > 0 such that for all α ∈ (0, α0 ) and all V satisfying (1)(5), 1 , 2 V (r) ≥ −2 for all r ≥ 0, min
r∈[1−α,1+α]
V∗ (r) ≥
|V (r)| ≤ αr
−5
for all r ≥
(10) (11) 4 3
(12)
holds. Proof. The claim follows directly from assumptions (3), (5) and the definition of V∗ . The growth and decay estimates for the potential V result in a lower bound for the distance between particles in the ground state. Lemma 2.2 (Minimum distance). There exists a number α0 ∈ (0, 13 ) such that for each α ∈ (0, α0 ) and each V satisfying (1)- (5) all ground states yN : XN → R2 of E(·) satisfy the estimate min |y(x) − y(x )| > 1 − α,
x=x
(13)
A Proof of Crystallization in Two Dimensions
215
i.e. the minimum distance between the particle positions is at least 1 − α. Proof. Define M := maxη∈R2 #(y(X) ∩ B(η, 21 (1 − α))). The result is proven if we can show that M = 1. We can assume wlog that the maximum is achieved for η = 0. Set BM = B(0, 21 (1 − α)) and A = y −1 (BM ). By assumption (2) we have e(p) ≥ p⊂A, p∈P
− 1). As we could move the positions y(A) to infinity in such a way that the mutual distances diverge, we obtain the estimate 1 e({x, x }) ≤ − 2α M(M − 1). (14) 1 2α M(M
x∈A x ∈X\A
Let now n(k) := #y −1 ((k+1)BM \kBM ). As α ∈ [0, 13 ] the decay estimate (12) provides for each k ≥ 9 the following lower bound on the interaction energy between the particles indexed by A and those particles whose positions are contained in (k + 1)BM \ kBM : −5 e({x, x }) ≥ −n(k)Mα((1 − α) k−1 (15) 2 ) . x∈A y(x )∈(k+1)BM \kBM
Similarly, for k ∈ {2, . . . , 8} the lower bound (11) yields the estimate e({x, x }) ≥ −2n(k)M.
(16)
x∈A y(x )∈(k+1)BM \kBM
There exists a universal constant C > 0 such that the ring with outer radius r and thickness min{ 21 , r} can be covered with Cr translated copies of B(0, 13 ), this implies that n(k) ≤ CMk. It follows from (14) together with estimates (15) and (16) that 8 ∞ 2 k−1 −5 1 ≤ − 2α 2k + α ((1 − α) 2 ) M(M − 1). (17) −CM k=2
k=9
As both sums can be bounded by a constant which does not depend on α as long as α ∈ [0, 13 ], C does not depend on α or M, and M is a positive integer, inequality (17) can only hold with sufficiently small α if M = 1. This proves the claim. Proposition 2.3. There exists α0 > 0 such that for all α ∈ (0, α0 ) and all configurations y : X → R2 which satisfy (13) estimate (7) holds. Furthermore, #N (x) ≤ 7 for all x ∈ X.
(18)
Proof. The proof that (13) implies (18) follows from elementary geometric considerations. We only show that (18) implies (7). Let n(x) = #N (x) − 1 be the number of neighboring pairs which contain x, then #S =
1 1 n(x) = 2 2 x∈X
= 3#X − which proves (7).
1 2 #∂X,
x∈X\∂X
n(x) +
(18) 1 n(x) ≤ 3(#X − #∂X) + 25 #∂X 2 x∈∂X
216
F. Theil
2.3. Reference configurations and equilateral simplices. The proof is organized around the concept of “equilateral simplices”. An equilateral simplex T is a subset of X with three elements such that there exists a neighborhood ωT ⊃ T and a “continuous” imbedding T : ωT → A2 such that T (T ) ⊂ A2 is an equilateral simplex with side length λ in the usual sense, cf. Definition 2.6. The ratio of the actual bond lengths |y(x) − y(x )| for {x, x } ⊂ T and the reference lengths λ is used to account for changes of length and volume. This idea allows us to realize the central step in the proof where the energy contribution of long-range interactions is approximated by renormalized short range interaction energies up to an error which can be controlled. More precisely, instead of summing the individual pair interaction energies in an uncontrolled way we add the energies of equilateral simplices. The interaction energy of each equilateral simplex can be approximated by its volume. Using the additivity of the volume form we can construct an approximation of the interaction energy of equilateral simplices with side length λ by modifying the interaction energy of the simplices with side length 1. Definition 2.4 (Discrete imbeddings). Let y : X → R2 satisfy (13), ω ⊂ X \ ∂X(y) and : ω → A2 be a discrete map. is called y-continuous if |(x) − (x )| ≤ 1 for all {x, x } ∈ S(y) such that {x, x } ⊂ ω.
(19)
A discrete y-continuous map is called orientation preserving with respect to the configuration y if det(y(x2 ) − y(x1 ), y(x3 ) − y(x1 )) · det((x2 )−(x1 ), (x3 )−(x1 )) ≥ 0
(20)
holds for all {x1 , x2 , x3 } ⊂ ω such that {x1 , x2 }, {x1 , x3 }, {x2 , x3 } ∈ S(y). A y-continuous map which is also injective is called discrete imbedding. Throughout the paper “discrete imbedding” means “discrete orientation preserving imbedding”. Remark 2.5. In fact discrete imbeddings satisfy the stronger relation |(x) − (x )| = 1 if and only if {x, x } ∈ S(y), provided that {x, x } ⊂ ω. This follows from the definition and the obvious fact that for each η ∈ A2 we have that #{η ∈ A2 | |η − η | ∈ (0, 1]} = 6. We say that A ⊂ A2 is a sub-lattice if A + η − η = A for all {η, η } ⊂ A. For a sub-lattice A the lattice parameter is given by λ = min{|η − η | |η, η ∈ A, η = η }. A sub-lattice A is D6 -invariant if γ A = A for all γ ∈ D6 , the maximal subgroup of O(2) which leaves A2 invariant. For each λ ∈ R we define m(λ) ∈ N to be the number of different D6√ -invariant sub-lattices of A2 with lattice parameter λ divided by λ2 e.g. m(1) = 1, m( 7) = 2. The variable m(λ) is introduced for bookkeeping purposes only, it appears in intermediate expressions but is never evaluated or estimated explicitly. A simple geometric consideration shows that 2 1 k = 2λ2 . (21) m(λ) = 16 #(A2 ∩ {|η| = λ}) = 16 # k ∈ Z2 k · 1 2 The countable set = {λ > 0 | m(λ) = 0} is the set of distances. With this definition we obtain an equivalent formula for the renormalized potential: m(λ)V (λs). (22) V∗ (s) = λ∈
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217
Definition 2.6 (Equilateral simplices). Let T ⊂ X such that #T = 3 and λ ∈ . We say that T is an equilateral simplex with side length λ for the configuration y satisfying (13) and write shortly T ∈ Tλ (y) if either λ = 1 and p ∈ S for all p ⊂ T such that p ∈ P, or if λ > 1, B(z, 20λ) ∩ y(∂X) = ∅ and there exists a patch ωT ⊂ X \ ∂X with T ⊂ ωT and a discrete imbedding T : ωT → A ⊂ A2 with the properties
where z =
1 3
|(x) − (x )| = λ for all {x, x } ⊂ T , B(z, 5λ) ∩ y(X) ⊃ y(ωT ) ⊃ B(z, 3λ) ∩ A2 ,
(23) (24)
y(x).
x∈T
The definition is motivated by the following lemma which provides for each equilateral simplex T ∈ Tλ (y) a quantitative link between λ and the side length of a deformed simplex y(T ) ⊂ R2 . Lemma 2.7. There exists a pair of constants α0 , K > 0 such that for all α ∈ (0, α0 ), if λ ∈ , the configuration y satisfies (13) and T ∈ Tλ (y), then for all {x, x } ⊂ T the estimate 1 |y(x) − y(x )| − 1 ≤ Kα (25) λ holds. Proof. See appendix.
Definition 2.6 is brought to life by Proposition 2.8 which asserts that for any pair x, x ∈ X with the property that y(∂X) ∩ B( 21 (y(x) + y(x )), 28|y(x) − y(x )|) = ∅ we can find an equilateral simplex T which contains x and x . In other words, the absence of defects provides enough rigidity so that we can embed defect-free regions into the triangular lattice A2 . Proposition 2.8. There exists a number α0 > 0 such that for all α ∈ (0, α0 ), all configurations y satisfying (13) and all pairs {x1 , x2 } ⊂ X the set T (x1 , x2 ) = {T ∈ ∪λ∈\{1} Tλ (y) | {x1 , x2 } ⊂ T } satisfies the following assertions: (1) #T (x1 , x2 ) ≤ 2 and there exists a unique number λ ∈ such that T (x1 , x2 ) ⊂ Tλ (y). (2) If B ∩ y(∂X) = ∅, where B = B( 21 (y(x1 ) + y(x2 )), 28|y(x1 ) − y(x2 )|), then #T (x1 , x2 ) = 2. (3) Conversely, if #T (x1 , x2 ) ≤ 1, then y(∂X) ∩ B = ∅. Proof. See appendix.
The central motivation for the introduction of the scaling parameter λ is that it provides us a natural method to pass information about the deformation on large scales to small scales. In particular, the area covered by the big deformed simplices can be computed up to a surface error by summing the area of the small simplices that lie within the big simplices.
218
F. Theil
Proposition 2.9. There exists a pair of constants C, α0 > 0 such that for every α ∈ (0, α0 ), all configurations y : X → R2 satisfying (13) and every λ ∈ the following estimates are true: 1 0 ≤ #T1 − m(λ) #Tλ (y) ≤ Cλ2 #∂X, 1 0≤ meas(conv(y(S))) − 2 λ m(λ) S∈T1 meas(conv(y(T ))) ≤ Cλ2 #∂X, ×
(26)
(27)
T ∈Tλ (y)
#{T ∈ Tλ (y) | ωT ⊃ {x}} ≤ Cλ2 m(λ) for all x ∈ X. Proof. See appendix.
(28)
The strength of this result is based on the fact that just one mild assumption on the minimum inter-particle distance of arbitrary N -particle configurations implies that interesting large-scale geometric objects such as equilateral simplices enjoy the natural scaling estimates up to errors which are bounded by the number of defects, a purely local concept. We use the notion of equilateral simplices to define a further splitting of the set of short range interactions S(y) and long-range interactions L(y). For j = 0, 1, 2 let Sj (y) := {p ∈ S(y) | there exists precisely j simplices T ∈ T1 such that p ⊂ T }, Lj (y) := {p ∈ L(y) | there exists λ ∈ \ {1} and precisely j simplices T ∈ Tλ (y) such that p ⊂ T }. Proposition 2.8.1 implies that P = 2j =0 (Sj (y) ∪ Lj (y)). The partitioning of the set of pairs induces a partitioning of the energy p∈P
e(p) =
1 2
e(p) +
λ∈ T ∈Tλ p⊂T ,#p=2
+
p∈L0
1 e(p) + e(p) 2
p∈S0
e(p) +
1 e(p) 2 p∈S1
p∈L1
= I1 + I2 + I3 + I4 + I5 .
(29)
It will be demonstrated in the following section that I1 , I2 , I3 carry the dominating part of the energy and the terms I4 , I5 can be estimated by small surface terms. 2.4. Resummation. This section contains the centerpiece of the proof where the analytical and combinatorial estimates are combined to establish (9). An important step in our analysis consists in creating a link between long range interactions and the volume of big deformed simplices (Lemma 2.10). The difference between the interaction energy of a deformed simplex and the interaction energy of a rigid simplex can be approximated by the difference between the volumes. The error can be controlled by the quadratic distortion of the simplex. The crucial point is that the volume is robust enough to provide information regardless of the structure of possible singularities such as defects or dislocations (Proposition 2.9).
A Proof of Crystallization in Two Dimensions
219
Lemma 2.10. There exists two constants C, α0 > 0 such that for all α ∈ (0, α0 ), 2 2 f1 ∈ C ([0, ∞)), λ ∈ (0, ∞) and all Y = {y0 , y1 , y2 } ⊂ R with the property, |yi − yj | − 1 ≤ α for i = j , the inequality λ √
√ 2 3 f (|yi − yj |) λ f (λ) meas(conv(Y )) − 43 λ2 + 3f (λ) − 21 (30) 1 i=j 2 ≤ C λ |f (λ)| + f L∞ ((1−α)λ,(1+α)λ) (|yi − yj | − λ) i=j
holds. Proof. Let i = j and set δij = λ1 |yi − yj | − 1. Heron’s formula, which gives the area of a triangle in terms of it side lengths, yields meas(conv(Y )) = =
√
3 2 4 λ √ 3 2 4 λ
1+
+
2 3
i=j δij
+O
1 √ λ2 i=j δij 4 3
2 i=j δij
1 2
+ O λ2 i=j δij2 ,
where the O-constant is bounded by a universal number as long as α ∈ (0, 1). By Tay2 lor’s theorem |f (|yi − yj |) − f (λ) − λf (λ)δij | ≤ λ2 f L∞ ((1−α)λ,∞) δij2 . Plugging these two estimates into the left hand side of (30) yields the claim. We extract the leading energy contribution of the long-range interactions. By Lemma 2.10 and decay estimate (5) we obtain for every λ ∈ \ {1} and T ∈ Tλ (y),
√ √ e(p) ≥ 3V (λ) + 2 λ 3 V (λ) meas(conv(y(T ))) − 43 λ2 (31) p⊂T ,#p=2
−Cαλ−7
2
|y(x) − y(x )| − λ
.
{x,x }⊂T
Now we are in the position to carry out the most important step of the proof and localize the long-range interactions. Proposition 2.9 implies that
√ √ 3V (λ) + 2 λ 3 V (λ) meas(conv(y(T ))) − 43 λ2 T ∈Tλ
≥ m(λ)
√ 3V (λ) + 2 3λV (λ) meas(conv(y(S))) −
S∈T1
−Cαλ−3 m(λ)#∂X.
√
3 4
(32)
Another application of Lemma 2.10 transforms the main term in (32) back into a pair sum
√ √ 3V (λ) + 2 λ 3 V (λ) meas(conv(λy(S))) − 43 λ2 2 V λ y(x) − y(x ) − Cαλ−5 |y(x) − y(x )| − 1 , (33) ≥ {x,x }⊂S
{x,x }⊂S
220
F. Theil
where S ∈ T1 (y) is arbitary. For each simplex T ∈ Tλ (y) we choose x ∈ T arbitrary and set = B(z, 20λ) and = B(z, 3λ), where z = 13 x∈T y(x). By Definition 2.6 1 and estimate (25) for α ∈ (0, 2K ) the inclusion y −1 ( ) ⊂ ωT holds and there exists a constant M ∈ N ∩ [λ, 2λ] and a translation vector τ ∈ A2 such that
√2 ∩ A2 ⊂ T (ωT ) T (y(T )) ⊂ τ + conv 00 , M 01 , M 2 3 holds. The L2 -rigidity estimate provided by Proposition 4.3 applied to u(ξ ) := y(−1 (ξ )) gives a bound on the distortion of the big simplex T in terms of the distortion of the smaller simplices S ∈ T1 (y): (|y(x) − y(x )| − λ)2 ≤ C log(λ) (|y(x) − y(x )| − 1)2 . {x,x }∈S {x,x }⊂ωT
{x,x }⊂T
We use this inequality to localize the error term in (31): y(x) − y(x ) − λ 2 λ−7 T ∈Tλ {x,x }⊂T
≤Cλ−7 log(λ)
T ∈Tλ
{x,x }∈S {x,x }⊂ωT
≤ Cλ−5 log(λ)m(λ)
2
|y(x) − y(x )| − 1
2
|y(x) − y(x )| − 1
.
(34)
{x,x }∈S
The final inequality above is a consequence from (28). The error terms which appear in (32), (33) and (34) can be bounded uniformly
(1 + log(λ)) λ
−5
+λ
−3
m(λ) ≤ C
k·
k∈Z2 \{0}
λ∈\{1}
After these considerations we can estimate the big sum
− 3 2 2 1 k < ∞. (35) 1 2
λ∈\{1} T ∈Tλ p⊂T ,#p=2
e(p) by a
much simpler sum which contains only nearest-neighbor terms: 1 e(p) 2 λ∈\{1} T ∈Tλ p⊂T ,#p=2 ≥
1 V (λ|y(x) − y(x )|)m(λ) 2 λ∈\{1} S∈T1 {x,x }⊂S −5 2 −3 −Cα (1 + log(λ))λ (|y(x) − y(x )| − 1) + λ #∂X m(λ) λ∈\{1}
{x,x }∈S
1 V (λ|y(x) − y(x )|)m(λ) 2 λ∈\{1} {x,x }∈S 2 −Cα (|y(x) − y(x )| − 1) + #∂X .
(35)
≥
S∈T1 {x,x }⊂S
(36)
A Proof of Crystallization in Two Dimensions
221
Now we interchange the sum over λ and the sum over S in (36), use formula (22) and obtain the estimate 1 e(p) 2 λ∈\{1} T ∈Tλ p⊂T ,#p=2 1 e∗ ({x, x }) − e({x, x }) − Cα(|y(x) − y(x )| − 1)2 ≥ 2 S∈T1 {x,x }⊂S
−Cα#∂X.
(37)
The lower bound on the minimum distance (13) implies that #(S0 (y)+S1 (y)) ≤ C#∂X. Inequality (12) gives for each p ∈ S(y) the estimates |e∗ (p) − e(p)| ≤ and
1 6
α((1 − α)|ξ |)−5 ≤ Cα
ξ ∈A2 ,|ξ |>1
|e(p) − e∗ (p)| ≤ Cα#∂X.
(38)
p∈S0 ∪S1
Consequentially (37) can be improved to 1 e(p) 2 λ∈\{1} T ∈Tλ p⊂T ,#p=2 e∗ ({x, x }) − e({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα#∂X. ≥ {x,x }∈S
The only difference between (37) and the above inequality is that the right-hand side of the latter inequality contains all nearest-neighbor interactions whereas the former only those which belong to equilateral simplices with side length 1. We end up with the estimate I1 + I 2 + I 3 ≥ e∗ ({x, x }) − Cα(|y(x) − y(x )| − 1)2 − Cα#∂X {x,x }∈S
≥ −3#S +
1 (e∗ (p) + 1) − Cα#∂X. C
(39)
p∈S
The last inequality follows from (7) together with assumptions (1) and (10) if α is sufficiently small. It remains to show that the sum I4 + I5 can be estimated by a small surface term. For every d ∈ [0, ∞) there are at most n(d) := Cd 3 #∂X
(40)
index pairs {x, x } ⊂ X such that ||y(x) − y(x )| − d| ≤ 21 and {x, x } ∈ L0 . To see this we first apply Proposition 2.8.3 and obtain that for each index pair {x, x } ∈ L0 there exist xb ∈ ∂X such that y(x) ∈ B(y(xb ), 28|y(x) − y(x )|). The lower bound on the minimum distance between particles (13) implies that for every xb ∈ ∂X there are at most Cd 2 labels x ∈ X such that y(x) ∈ B(y(xb ), 28|y(x) − y(x )|) and Cd labels x ∈ X such that ||y(x) − y(x )| − d| ≤ 21 . This proves (40).
222
F. Theil
By (4) and (5) this implies that
I4 + I5 =
e(p)
p∈L0 ∪L1
≥ − n( 43 )α + ≥ −Cα#∂X.
∞
n(d + 13 )V
d=2
1 L∞ 3 +d,∞
#∂X (41)
Adding (39) and (41) gives the desired inequality (9). The proof of Theorem 1.1 is finished.
3. Proof of Theorem 1.2 In contrast to the situation of Theorem 1.1 the number of particles is now infinite, with the consequence that there are configurations y(X) such that ∂X = ∅. This is not possible if the number of particles is finite. Infinite energies are avoided by working with special boundary conditions. The usage of the term “boundary conditions” which is typically connected with partial differential equations is slightly non-standard in this context. It simply refers to the fact that we are considering kinematically constrained settings and that the constraints have a natural spatial structure. The idea is to show that a suitable version of the main estimate (9) holds in both cases. We will use this estimate in connection with the competitor y(x) ≡ x to demonstrate that minimizers ymin are rigid, i.e. ∂X = ∅ and |ymin (x) − ymin (x )| = 1 if {x, x } ∈ S which means that all the nearest neighbors have precisely distance 1 from each other. It can be shown with elementary methods that every nonempty set ⊂ R2 with the properties min |y − y | ≥ 1 and
(42)
#{y ∈ | |y − y | ≤ 1} = 7 for all y ∈
(43)
y,y ∈ y=y
is countable and there exists a rotation R ∈ SO(2) and a translation τ ∈ R2 such that R + τ = A2 . Let L ∈ N; a set X ⊂ A2 is L-periodic if X + LA2 = X. We introduce the equivalence relation ∼ on the set of subset ω an L-periodic set X: ω ∼ ω if there is a vector τ ∈ LA2 such that ω = ω + τ . We say that a map y : X → R2 is L-periodic if y(x + τ ) = y(x) + τ for all x ∈ X and all τ ∈ LA2 . For a L-periodic map the defects ∂X, neighbors S and the equilateral simplices Tλ are periodic sets and we can := P/ ∼, := X/ ∼, ∂X := ∂X/ ∼, S := S/ ∼, P define the natural quotient sets X Tλ := Tλ / ∼. The proof of Theorem 1.1 is hinged on the lower bound on the inter-particle distance (13). The proof of (13) is based on a construction where those particles that are too close to each other are moved to infinity and thereby effecively removed from the system without increasing the energy.
A Proof of Crystallization in Two Dimensions
223
per
As the definition of EL doesn’t allow removal of particles we have to relax the minimization problem slightly in order to rescue (13). This is achieved by removing L-periodic subsets from A2 : If X ⊂ A2 is L-periodic we set per EL (X, {y}) := e(p). {p⊂X | #p=2}/∼
per
per
Clearly EL (X, ·) = EL (·) (defined in Theorem 1.2) if X = A2 . Furthermore, since 2 per there are only 2L possible L-periodic sets X, a minimizer (Xmin , ymin ) of EL exists. Due to the more complicated mechanism of particle-removal we have to establish a new version of Lemma 2.2. Lemma 3.1. There exists α0 ∈ (0, 21 ) such that for all α ∈ (0, α0 ), L ∈ N and all minper imizers (Xmin , ymin ) of EL (·, ·) the minimum distance between the particles satisfies estimate (13). Proof. It can be assumed wlog that L ≥ 4 as the cases L ∈ {1, 2, 3} are covered by L = 4L . Let M and A be defined as in the proof of Lemma 2.2. In the periodic situation we have to take the interaction between the particles in A and the mirror images into account. The decay estimate (5) implies that e({x, x }) ≥ −CM 2 α, x∈A x ∈(A+LA2 )\A
where C =
ξ ∈LA2 \{0} (|ξ | − 2)
x∈A x ∈X\(A+LA2 )
−5 . We
replace (14) by the estimate
1 e({x, x }) ≤ − 2α M(M − 1) + CM 2 α,
and (17) by −CM
2
8
2k + α
k=2
∞
−5 ((1 − α) k−1 2 )
1 ≤ − 2α M(M − 1) + CM 2 α.
k=9
Repeating the reasoning at the end of the proof of Lemma 2.2 we obtain that M = 1 if α is sufficiently small, the only difference is that one additional term on the right-hand side is created by periodicity. We claim now that for configurations X, y which satisfy the bound (13) a suitably adapted version of (9) holds: per
EL (X, {y}) ≥
1 C
{x,x }∈S
− 3#X. e∗ ({x, x }) − 1 + 41 #∂X
(44)
The proof of Theorem 1.1 can be adapted to the periodic setting by simply replacing the Tλ and the elements ∂X, S, sets X, ∂X, S, Tλ with the corresponding quotient sets X, of those quotient sets by representatives.
224
F. Theil
As the identity map y(x) ≡ x is L-periodic for every L ∈ N and therefore an per admissible competitor we obtain that EL (Xmin , {ymin }) + 3L2 ≤ 0 and together with (44), 1 min + 3(L2 − #X min ). (e∗ ({x, x }) + 1) + 41 #∂X 0≥ C {x,x }∈S
≤ L2 this estimate shows that #X min = L2 , ∂ X min = ∅, and |ymin (x) − Since #X ymin (x )| = 1 for all {x, x } ∈ S. Hence, the set = {ymin (x) | x ∈ A2 } satisfies (42) and (43) and consequentially R + τ = A2 for a rotation R ∈ SO(2) and a translation τ ∈ R2 . By the periodicity of ymin we can choose R = Id, this is precisely the claim and the proof of Theorem 1.2 is finished. Proof of Corollary 1.3. Again we salvage the lower bound (13) by enlarging the set of competitors slightly. For A ⊂ A we set V (|y(x) − y(x )|), E(A , {y}) = {x,x }⊂X {x,x }∩A =∅
where X = (A2 \ A) ∪ A . As before E(A , ·) = EA (·) if A = A. Let Amin , ymin be the minimizer of E(·, ·). Clearly ymin satisfies the bound (13) since the proof of Lemma 2.2 can be repeated for this case, word by word. We determine the properties of Amin , ymin by comparing the minimization problem with the periodic situation which has been analyzed earlier.
Let Y = A2 ∩ L2 02 √13 Q − L4 ( √33 ), where Q = [0, 1) × [0, 1) ⊂ R2 is the semiopen unit square and let Xper := Y + LA2 and yper (x) := τ + ymin (x − τ ) if x ∈ Xper , τ ∈ LA2 , x + τ ∈ Y . Clearly Xper is L-periodic and yper is a projection of ymin onto the set of L-periodic maps. Due to the compactness of A we can assume that A ⊂ B(0, C) for some number C > 0. Let us define the following sets of pairs: P := {p ⊂ A2 | #p = 2}, Pmin := {p ∈ P | p ∩ (A \ Amin ) = ∅}, Pper := {p ∈ P | p ∩ (A \ Amin + LA2 ) = ∅}, which by construction obey the inclusion Pper ⊂ Pmin ⊂ P. We set e({x, x }) := V (|ymin (x) − ymin (x )|) if {x, x } ∈ Pmin and eper ({x, x }) := V (|yper (x) − yper (x )|) if {x, x } ∈ Pper . By the decay properties of V , for all ε > 0 there exists Lε ∈ N such that eper (p) − e(p) ≤ ε, (45) {p∈Pper | p∩Amin =∅}
|e(p)| ≤ ε
(46)
{p∈Pmin \Pper | p∩Amin =∅}
if L ≥ Lε . Let Sper := {{x, x } ∈ Pper | ||yper (x) − yper (x )| − 1| < α} and define the quantity 1 1 2 ||yper (x) − yper (x )| − 1|2 + #∂X I := per + 3(L − #Y ), C 4 {x,x }⊂Sper /∼
A Proof of Crystallization in Two Dimensions
225
the equivalence relation ∼ has been defined in the previous section. We will demonstrate per now that I = 0 by relating the EA to EL . By Eq. (44) per
I ≤ EL (Y + LA2 , {yper }) + 3L3 1 = eper (p) + 2 {p∈Pper | p∩Amin =∅}
eper (p) + 3L2 .
{p∈Pper | p∩Amin =∅,p⊂Y } per
The last equality is obtained by simply rewriting EL in a more explicit way. By definition of Pper and yper we can replace Amin by A and eper ({x, x }) by er ({x, x }) := V (|x −x |) in the second sum without changing the value of the sum. By (45) and (46) we can replace the periodic configuration yper by the original minimum energy configuration ymin and Pper by Pmin in the first sum without changing its value by more than 2ε, hence 1 I ≤ e(p) + eper (p) + 3L2 + 2ε. 2 {p∈Pmin | p∩Amin =∅}
{p∈Pper | p∩A=∅,p⊂Y }
The first sum is equal to EA (Amin , {ymin }) and can be estimated by EA (A2 ) due to the minimality of ymin . Let Z := Y ∪ A. By definition of Z we can replace the set Y by Z and Pper by P in the second sum without changing it, 1 I ≤ eper (p) + eper (p) + 3L2 + 2ε 2 =
{p∈P | p∩A=∅} per EL (A2 ) + 3L2
{p∈P | p∩A=∅,p⊂Z}
+ 2ε = 2ε, per
where the last two equations follow directly from the definition of EL . As ε is arbitrary and L2 − #Y = #A \ Amin for sufficiently large L, where U := (U + B(0, 1)) ∩ A2 if U ⊂ A2 , we obtain that Amin = A, ∂Xper = ∅ and ||yper (x) − yper (x )| − 1| ≤ Cε for all {x, x } ∈ Sper . Since yper (x) and yper (x ) are independent of L if L is sufficiently big, this proves that |yper (x) − yper (x )| = 1 for all {x, x } ∈ Sper . Hence, the set
= {yper (x) | x ∈ A2 } satisfies (42) and (43) and consequentially R + τ = A2 for a rotation R ∈ SO(2) and a translation τ ∈ R2 . By the periodicity of yper we can choose R = Id. Since yper (x) = ymin (x) for all x ∈ Y we obtain that τ = 0 and ymin (x) = x for all x ∈ A2 . The proof of Corollary 1.3 is finished. Acknowledgement. The author is grateful to Sergio Conti and Benjamin Friedrich for suggesting various improvements of an earlier version of this manuscript.
4. Appendix 4.1. Geometric rigidity. We provide bounds on the global distortion in terms of the local distortion. Both, L∞ and L2 -estimates, are required. Proposition 4.1. Let α ∈ [0, 13 ], ⊂ ⊂ Rd be two domains such that dist(x, ∂ ) ≥ √
2α 1−2α diam( ) for all x
the estimate
holds.
∈ . If u ∈ W 1,∞ ( ) satisfies ∇u − SO(d)L∞ ( ) ≤ α, then |u(x) − u(x )| sup − 1 ≤ α |x − x | {x,x }⊂
(47)
226
F. Theil
Under the additional assumption that the u ∈ C 1 estimate (47) has been established in [7]. Since C 1 is not dense in the set of Lipschitz functions, we have to construct a more modern version of the proof. Proof. The idea of the proof is to demonstrate that for two image points u(x), u(x ) ∈ u( ) the geodesic β ⊂ u( ), which connects those two points, is in fact a straight line. For each pair x = x ∈ Rd we denote by Sα (x, x ) ⊂ Rd = ξ ∈ Rd |ξ − x| + |ξ − x | ≤ 1+2α 1−2α |x − x | the ellipsoid of revolution with foci x and x and eccentricity 1+2α 1−2α . We will now show that the claim is true for all Sα (x, x ) ⊂ . To this end we construct a path γ ∈ C([0, 1], ) which minimizes the arc-length K |u(γ (tk )) − u(γ (tk−1 ))| Iu (γ ) = lim sup →0 k=1 0 = t0 < · · · < tK = 1, max tk − tk−1 < k∈{1...K}
of the image t → u(γ (t)) subject to the boundary condition γ (0) = x, γ (1) = x . Since u is only Lipschitz but not necessarily C 1 , the functional Iu (·) is in general neither continuous in L∞ (0, 1) or similar function spaces nor convex, therefore existence of minimizers can’t be established by a naive application of the direct method. To overcome this difficulty we use the following trick. As the path t → u((1 − t)x + tx ) is an admissible competitor, (1 + α)|x − x | is an upper bound for inf α Iu (γ ). Let C ⊂ W 1,∞ ([0, 1], u ( )) be the set of paths β for which there exists a path γ ∈ W 1,∞ ([0, 1], ) such that β = u ◦ γ and the boundary condition γ (0) = x, γ (1) = x is satisfied. We can assume without loss of generality that β passes through u( ) with 1 ˙ constant speed, i.e. dtd | dβ dt | = 0. Clearly Iu (γ ) = 0 |β| dt =: J (β). We show now that the Lipschitz constant of γ is controlled by the Lipschitz constant of β. First we establish that u is locally injective, i.e. for any t ∈ [0, 1] there exists a pair of open sets O ⊂ , U ⊂ u( ) such that r(t) ∈ O, u(r(t)) ∈ U and u : O → U is injective. Thanks to Kohn’s rigidity estimate [9, (1.5.ii)] we can find a universal constant C > o with the property that for all x, x with B(x, |x − x |) ⊂ there exists a matrix R ∈ SO(d) such that |u(x) − u(x ) − R(x − x )| ≤ C|x − x |α, which implies injectivity of u as long as Cα ≤ 1. Since det(∇u) ≥ 21 the map u is quasiregular, i.e. there exists a constant K > 0 with the property |∇u|d ≤ Kdet(∇u) almost everywhere. By Reshetnyak’s theorem ([6, Theorem I.4.1]) u is discrete and open, and ˙ ≤ by injectivity u : O → U is an imbedding. Now |γ˙ (t)| = (∇u|U (β(t)))−1 β(t) 1+α 1+α 1−α |x − x |. Hence γ is Lipschitz continuous with Lipschitz constant 1−α |x − x |. Let now (βn )n∈N ⊂ C be a minimizing sequence of J (·) which converges in the weak-* topology of W 1,∞ to β. The convexity of J (·) implies that lim inf n→∞ J (βn ) ≥ J (β). The global Lipschitz bound β˙n L∞ ≤ (1+α)|x −x | implies that limn→∞ βn − βC([0,1]) =0, hence β ∈ C([0, 1], u( )). Similarly we obtain that there exists γn , γ ∈ C [0, 1], such that βn = u ◦ γn and limn→∞ γn − γ C([0,1]) = 0. Clearly β = u ◦ γ
A Proof of Crystallization in Two Dimensions
227
and γ satisfies the boundary conditions γ (0) = x, γ (1) = x . This shows that γ is a minimizer of Iu (·). We use the minimality of γ in order to localize the path. Let t ∈ [0, 1], then 1 |x − γ (t)| + |x − γ (t)| ≤ |γ˙ | ds 0
1 1 |∇u(γ )γ˙ | ds ≤ 1−α 0 1 = 1−α Iu (γ ) ≤ 1+α 1−α |x − x |.
(48)
Hence γ (t) ∈ Sα (x, x ) for all t ∈ [0, 1]. The inclusions Sα ⊂ and ∂u( ) ⊂ u(∂ ) (due to the openness of u) imply that the minimal path β is contained in the open set u( ) and consequentially the EulerLagrange equation β = 0 is satisfied. In particular, β is a straight line segment that connects u(x) and u(x ). Inequality (48) shows that |u(x) − u(x )| ≥ (1 − α)|x − x |. The following lemma links gradient estimates for continuous interpolations to the distortion of simplices. Lemma 4.2. There exist two universal constants K, α0 > 0 such that for each pair of triples yi , ηi ∈ R2 , i = 0, 1, 2 with the properties det(y1 − y0 , y2 − y0 ) det(η1 − η0 , η2 − η0 ) ≥ 0, ||yi − yj | − 1| ≤ α0 and |ηi − ηj | = 1 if i = j, the inequalities min |F − R|2 ≤ K max ||yi − yj | − 1|2 ,
(49)
min |F −1 − R|2 ≤ K max ||yi − yj | − 1|2 ,
(50)
i=j
R∈SO(2)
i=j
R∈SO(2)
hold, where |M| := trace(M T M) for a 2 × 2-matrix M. The matrix F ∈ R2×2 is the gradient of the unique affine map u : conv{η0 , η1 , η2 } → R2 which satisfies u(ηi ) = yi for i = 0, 1, 2. Proof. We define the matrix Y0 = 1 ( −1 ).
1 2 √1 2(0 3)
and the vectors v1 = ( 01 ), v2 = ( 01 ),
v3 = As reflections, rotations and translations of ηi and yi leave (49) invariant, we can assume that ηi = Y0 vi and F ηi give the sides of the triangles {ηi } and {yi }, respectively. We choose C > 0, such that for all M ∈ GL(2) the following implication holds: max ||M ηi | − 1| < 1 ⇒
i=1,2,3
min |M − R| ≤ C.
R∈SO(2)
Since |x 2 − 1| ≤ 3|x − 1| for 0 ≤ x ≤ 2, it suffices to prove l :=
min |F − R| ≤ C m
R∈SO(2)
228
F. Theil
√ with m := max ||F ηi |2 − 1|. It is known that l = F T F − Id . For the right-hand i=1,2,3
side we get m ≥ |ηi ·(F T F −Id)ηi | since |ηi | = 1. We set G := F T F −Id and conclude m ≥ |η3 · Gη3 | ≥ 2|η2 · Gη1 | − |η1 · Gη1 | − |η2 · Gη2 |, i.e. 23 m ≥ |η2 · Gη1 |. This shows that |GY0 v1 | = |Gη1 | ≤ Cm. Similarly |GY0 v2 | ≤ Cm and we get √
−1 F T F + Id , we obtain |H | ≤ 2 and |GY0 | ≤ Cm, hence |G| ≤ Cm. For H := thus l = |G H | ≤ 2|G| |H | ≤ C m. This proves (49). To establish (50) let R ∈ SO(2), S ∈ R2×2 sym be the polar decomposition of F , i.e. F = RS and set H := S − Id. The previously derived estimate |G| ≤ Cm implies that |2H + H 2 | ≤ Cm and consequentially |H | ≤ Cm by positivity of H 2 . Since F −1 = (Id − H + H 2 − . . . )R T this implies (50). We apply the previous lemma to bound the distortion of large equilateral simplices by the distortions of the small simplices inside.
√2 } ⊂ R2 Proposition 4.3. Let M ∈ N \ {0, 1} and = conv{ 00 , M 01 , M 2 3 be a dilated unit simplex. There exist a universal constant C > 0 such that for all u : ∩ A2 → R2 the estimate |u(x) − u(x )| − 1 2 (51) min max |u(x) − τ − Rx|2 ≤ C log(M) τ ∈R2 R∈SO(2)
x∈∂ ∩A2
x,x ∈ ∩A2 |x−x |=1
holds. Proof. We extend u to a continuous map in W 1,∞ ( ) by interpolation. By Lemma 4.2 for each triple {x1 , x2 , x3 } ⊂ A2 with the property |xi − xj | = 1 if i = j and each x ∈ int(conv{x1 , x2 , x3 }) we have that dist(∇u(x), SO(2)) ≤ K ||u(xi ) − u(xj )| − 1|2 . i=j
A theorem by M¨uller, Friesecke and James in [8] states that there is a universal constant C which does not depend on u such that for each u we can find a rotation matrix R ∈ SO(2) with the property 2 |∇u − R| ≤ C dist(∇u, SO(d))2 . (52)
Let L = 3M and γ : [0, L] → ∂ be a path which follows ∂ at unit speed, i.e. γ (0) = γ (L) = 0, | dtd γ | = 1 and ∂ = {y(s) | s ∈ [0, L]}. For l ∈ [0, L] ∩ Z let xl = γ (l) ∈ A2 and τ = L1 L l=1 u(xl ). Define for s ∈ [0, L] the function v(s) := u(γ (s)) − τ − Rγ (s). By the standard trace theorem ≤C |∇u − R|2 , (53) v2 1 H 2 ([0,L])
A Proof of Crystallization in Two Dimensions
where v2
H
1 2 ([0,L])
=
v (k)| k∈Z |k||
229
2
1 v (k) = L
and
L
s
v(s)e2πik L ds
(54)
0
is the k th Fourier coefficient of v. We use the convention that [r] = max{s ∈ Z | s ≤ r} L−1 th but only within this proof. For each k ∈ {[ 1−L 2 ], . . . , [ 2 ]} ⊂ Z we define the k discrete Fourier coefficient of v by L−1 l 1 v(l)e2πik L L
v(k) ˇ =
l=0
such that [ L−1 2 ]
v(l) =
−2πik L v(k)e ˇ .
(55)
π2 v (k)| 2 |
(56)
l
k=[ 1−L 2 ]
It will be demonstrated that the inequality |v(k)| ˇ ≤
holds. The case k = 0 is trivial since by definition v (0) = v(0) ˇ = 0. After integrating L−1 the right-hand side in (54) twice by parts one obtains that for k ∈ {[ 1−L 2 ] . . . [ 2 ]}\{0}, L−1 l L 2πik L | v (k)| = (v(l + 1) − 2v(l) − v(l − 1))e 4π 2 k 2 l=0 = 1k 2 2(1 − cos(2π Lk ))v(k) ˇ ≥ π42 |v(k)|, ˇ (2π L )
this implies that
2 |v(k)| ˇ |k| ≤ ( π2 )4 v2
1
H 2 ([0,L])
L−1 k∈{[ 1−L 2 ]...[ 2 ]}
Clearly vL∞ ([0,L]) = maxl∈{0...L} |v(l)| ≤
(57)
.
|v(k)| ˇ by the discrete
L−1 k∈{[ 1−L 2 ]...[ 2 ]}\{0}
reconstruction formula (55). Putting the previous estimate, (56) and (57) together implies that 2 v2L∞ ([0,L]) ≤ |v(k)| ˇ L−1 k∈{[ 1−L 2 ]...[ 2 ]}\{0}
≤
L−1 k∈{[ 1−L 2 ]...[ 2 ]}\{0}
1 |k|
2 |k||v(k)| ˇ ≤ C log(L)u2
L−1 k∈{[ 1−L 2 ]...[ 2 ]}
Together with (53) and (52) this implies (51).
1
H 2 (∂ )
.
230
F. Theil
4.2. Proof of Proposition 2.8. The assertions in Proposition 2.8 are concerned with the question whether certain subsets ω ⊂ X can be imbedded into A2 such that the neighborhood relation which is encoded by the set of edges S is preserved. For the proof we require an existence and uniqueness theorem of such imbeddings (Proposition 4.8). The proof of Proposition 4.8 is constructive: For each path γ which connects two points x1 , x2 ∈ X the imbedding |γ solves a certain ‘ordinary’ second order difference equation (58) which depends on γ . The construction of the difference equation is particularly simple in two dimensions due to the possibility to identify R2 with C. The imbedding is independent of the path γ provided that all paths are equivalent, i.e. they can be homotoped into each other. The equivalence of paths is a nontrivial consequence of the defect-freeness of ω. Definition 4.4. A map γ : [0, K]∩N → X is a discrete path of length K if {γ (k), γ (k + 1)} ∈ S for all k ∈ {0 . . . K − 1}. A path γ is closed if γ (K) = γ (0). A closed path is simple if γ |{0...K−1} is injective. Let γ ⊂ X \ ∂X be a closed, simple path. The set (γ ) ⊂ R2 is the unique bounded domain which has the property ∂ (γ ) = ∪k [y(γ (k)), y(γ (k + 1))] whose existence is guaranteed by Jordan’s curve theorem. We say that γ is positively oriented if γ passes through ∂ (γ ) in the positive sense. The matrix Qγ (k) ∈ SO(2) is the unique rotation matrix which satisfies the equation Qφ(γ (k − 1)) + φ(γ (k + 1)) = 0,
(58)
where φ : N (γ (k)) → A2 is a an arbitrary discrete imbedding such that φ(γ (k)) = 0. The Burgers vector associated to simple closed path γ and a direction v ∈ A2 is defined by b(γ , v) =
k K−1 %
Qγ (i)v,
k=0 i=0
where Qγ (0) = Id and γK = γ0 Since SO(2) is an Abelian group the Burgers vector b is well-defined without further stipulation concerning the order in which the matrices are multiplied. Remark 4.5. It is obvious from the definition that closed paths γ with length smaller or equal than two satisfy vol( (γ )) = b(γ , v) = 0. On the other hand, if γ is an arbitrary closed and simple path with vol( (γ )) = 0, then the length of γ is smaller or equal to two. The Burgers vector measures the defectiveness of a configuration. It depends on the chosen coordinate frame (characterized by v in the two-dimensional situation). If all possible Burgers vectors are 0 then the configuration is a deformed subset of the lattice A2 ; a precise version of this connection is given in the statement and the proof of Proposition 4.8. On the other hand, because of Lemma 4.7 the matrix Q is independent of the choice of φ. The map φ can be reconstructed from (58) by treating γ (k + 1) as unknown. Lemma 4.6. Let γ ⊂ X \ N (∂X) be a simple closed path which is longer than 2 and v ∈ A2 be a direction. There exists a pair of simple closed, positively oriented paths µi ⊂ Xand a pair of directions vi , i = 1, 2 such that
A Proof of Crystallization in Two Dimensions
231
(1) y(µ1 ) ∪ y(µ2 ) ⊂ (γ ), (2) vol( (µ1 )) + vol( (µ2 )) < vol( (γ )) − 41 , & & & (3) k1 ki11=0 Qµ1 (i1 )v1 + k2 ki22=0 Qµ2 (i2 )v2 = k ki=0 Qγ (i)v. We require a lemma which provides elementary geometric assertions. Since there is no multi-scale aspect present in the proof it is omitted. Lemma 4.7. There exists α0 > 0 such that for all α ∈ (0, α0 ) the following assertion is true. Let x ∈ X \ N (∂X) and x ∈ N (x) \ {x}. For each pair ξ, ξ ∈ A2 such that |ξ − ξ | = 1 there exists a unique discrete imbedding φ : N (x) → A2 such that φ(x) = ξ and φ(x ) = ξ . In particular, conv(y(N (x))) ∩ y(X) = y(N (x)). Proof of Lemma 4.6. Let φ : N (γ (0)) → A2 be a discrete imbedding such that φ(γ (0)) = 0. Let x := φ −1 (R π3 φ(γ (1)), where the matrix Rζ ∈ SO(2) is given
ζ − sin ζ by cos sin ζ cos ζ . Case I. x ∈ γ . Define µ1 (k) =
γ (0) if k = 0, x if k = 1, γ (k − 1) if k ∈ {2 . . . K + 1},
µ2 = (γ (0)) and v1 = v2 = R π3 v. Since γ is simple, closed and positively oriented y( x ) ∈ (γ ) and in particular assertion (1) is true. Furthermore, vol(conv{y(γ (0)), y(γ (1)), y( x )}) > 41 if α0 is sufficiently small, therefore assertion (2) holds. The definition of µ1 yields that Qµ1 (1) = R− π3 , Qµ1 (2) = R π3 Qγ (1). Since R π3 +R− π3 = Id a simple calculation shows that also assertion (3) holds. Case II. x ∈ γ . Let K1 := γ −1 ( x ) and define x if k = 0, γ (0) if k = 0, µ1 (k) = µ (k) = γ (k) if k ∈ {1 . . . K1 − 1}, 2 γ (k + K1 ) if x ∈ {1 . . . K −K1 }, v1 = R− π3 v and v2 = R π3 v. Like in the previous case assertions (1)-(3) can be checked by inspection. The following result states that in the absence of singularities it is possible to define a discrete reference configuration, i.e. for a simply connected and defect free subset ω ⊂ X we can find an discrete imbedding : ω → A2 . Proposition 4.8 (Existence of a reference configuration). There exist constants α0 , K > 0 such that for all α ∈ (0, α0 ) and all domains ⊂ ⊂ R2 with the properties dist( , ∂ ) ≥ 3diam( ) + 2, is convex and ∩ y(∂X) = ∅ there exists a discrete imbedding : ω → A2 , where ω = y −1 ( ). Furthermore, the following assertions are true. (1) is unique up to rotation and translation, i.e. for each discrete imbedding : ω → A2 , there exists a rotation matrix R ∈ SO(2) such that (x) − (x ) = R((x) − (x ))
for all x, x ∈ ω.
(59)
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F. Theil
(2) satisfies the rigidity estimate
)| sup |(x)−(x − 1 ≤ Kα. |y(x)−y(x )|
{x,x }⊂ω
(60)
(3) is surjective in the sense that if B(y(x), r) ⊂ for some x ∈ X and r > 0, then (ω) ⊃ B((x), 21 r) ∩ A2 .
(61)
Proof. The proof is a bit lengthy but conceptually very obvious. We define the map by summing along paths and show that the values (x) are independent of the choice of the path. This is achieved by proving that in defect-free patches the closed paths can be deformed to a point. Let 2 = {η ∈ | dist(η, ∂ ) ≥ 2}. We first demonstrate that for almost every η ∈ 2 there exists a small simplex T ∈ T1 such that η ∈ int(conv(y(T ))). It can be assumed wlog that there exists x0 ∈ y −1 ( 2 ), otherwise the assertion is trivial. Let
λ = s ∈ [0, 1] | (1 − s)y(x0 ) + sη ∈ ∪T ∈T1 conv(y(T )) . We will show now that λ is both closed and open relatively to the closed unit interval [0, 1] ⊂ R. λ is closed as it arises as a finite intersection of closed sets. Let s0 ∈ ∂λ ∩ (0, 1). By construction there exists T0 ∈ T1 such that y(T0 ) ⊂ and ξ = (1 − s0 )y(x0 ) + s0 η ∈ ∂conv(y(T0 )). Consequentially we can find σ ∈ [0, 1] and x1 , x2 ∈ T0 such that ξ = σy(x1 ) + (1 − σ )y(x2 ). The measure of set of those points η ∈ R2 with the property that σ = 0 or σ = 1 is zero, therefore we can neglect the extreme cases and assume that σ ∈ (0, 1). By Lemma 4.7 there exists ε > 0 such that for all s ∈ (s0 − ε, s0 + ε) we have that (1 − s)y(x0 ) + sη ∈ ∪T ∈T1 conv(y(T )), which is a contradiction to the assumption that s0 ∈ ∂λ ∩ (0, 1). Since λ is nonempty this implies that λ = [0, 1] and in particular *
2 ⊂ conv(y(T )). (62) T ∈T1
Fix x1 ∈ y −1 ( 2 ). We construct a path which connects x0 and x1 . Let S be the line segment [y(x0 ), y(x1 )]. We can assume that S intersects all edges transversally, i.e. y(x1 )−y(x0 ) is not a multiple of y(x )−y(x ) for any {x , x } ∈ S \{x0 , x1 }, otherwise, we remove the degeneracies by modifying y slightly. By the preceding consideration for almost every point s ∈ S there exists a unique simplex Ts ∈ T1 such that s ∈ conv(Ts ). We enumerate the simplices Ts in the order they intersect S; this yields a finite sequence (Tk )k∈0...K ∈ T1 with the property that x0 ∈ T0 , x1 ∈ TK and #(Tk−1 ∩ Tk ) = 2 for all k = 1 . . . K. The connecting path γ is constructed by walking along the edges of the simplices Tk . It can be assumed wlog that γ is injective, otherwise we have to remove the loops. To construct we first fix a direction v ∈ A2 such that |v| = 1. Let γ be a path of length K which connects x0 and x1 , i.e. γ (0) = x0 and γ (K) = x1 . We define now (x) :=
k K−1 %
Qγ (i)v,
k=0 i=0
where Qγ (i) is defined by (58), and demonstrate that is a discrete imbedding.
(63)
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233
It can be seen without difficulty that continuity (19) and orientation preservation (20) follow directly from the construction of if for each x the value (x1 ) does not depend on the choice of the path γ in (63). Indeed, let x, x ∈ y −1 ( ) be neighbors. We choose an arbitrary path γ which connects x0 with x1 and set γ = (γ , x ). Recall now Remark 4.5, with this particular choice of γ we obtain (x ) − (x) = φ(x ) for all x ∈ N (x),
(64)
where φ is the unique discrete imbedding N (x) → A2 such that φ(x) = 0 and φ(x ) = (x ) − (x). Since φ is a discrete imbedding Eq. (64) proves that is continuous in the discrete sense. Similarly we obtain orientation-preservation. Let {x1 , x2 , x3 } ⊂ y −1 ( 2 ) such that {x1 , x2 }, {x1 , x3 }, {x2 , x3 } ∈ S. Since in particular {x1 , x2 , x3 } ⊂ N (x1 ) and the previously constructed discrete imbedding φ satisfies by construction inequality (20) so does by Eq. (64). In order to show that the value (x) does not depend on the path γ we assume that γ is a different path which connects x to x0 . The path γ is obtained by reversing the order of γ . Let γ = (γ , γ ) be a closed path which starts and ends at x0 . It can be assumed wlog that γ is simple, otherwise we split it up into a collection of simple paths. We want to show that the Burgers vector of γ is zero. To this end we construct a sequence of paths γm,l ⊂ X and directions vl,m ∈ A2 , l ∈ {1 . . . L}, m ∈ {1 . . . l} such that y(γl,m ) ⊂ (γ ), l
b(γl,m , vl,m ) = b(γ , v),
(65)
m=1
vol( (γL,m )) = 0 for all m ∈ {1 . . . L}.
(66)
Equations (65), (66) together with Remark 4.5 show that b(γ , v) = 0. We set γ1,1 = γ , v1,1 = v and determine γl,m , vl,m for l > 1 inductively. If vol( (γl,m )) = 0 for all m ∈ {1 . . . l} we set L = l and are done. Otherwise there exists m∗ ∈ {1 . . . l} such that vol( (γl,m∗ )) > 0 and consequentially the length of γl,m∗ is bigger than 2. We apply Lemma 4.6 to γl,m∗ and set
γl+1,m
γl,m µ1 if = µ2 if γl,m−1
if m ∈ {1 . . . m∗ − 1}, if m = m∗ , if m = m∗ + 1, if m ∈ {m∗ + 2 . . . l},
and vl+1,m analogous. Since both y(µ1 ) and y(µ2 ) are contained in (γ ) all the paths γl,m which are longer than 2 satisfy the assumptions of Lemma 4.6. By construction the total volume lm=1 vol( (γl,m )) decreases by at least 41 in each step, hence this procedure terminates after at most 4vol( (γ )) steps. We show now that is unique in the sense of (59). If is another discrete imbedding it can be seen by exactly the same considerations that the value (x) is completely determined by (x0 ) and (−1 (v)). By Lemma 4.7 φ0 and φ0 differ only by a translation and a rotation. The proof of estimates (60) and (61) is based on geometric rigidity. By Lemma 4.7 we can define for all η ∈ ∪T ∈T1 int(conv(y(T ))) the affine map u by interpolation between
234
F. Theil
the values u(y(x)) = (x) for x ∈ T . By (62) the definition of u can be extended continuously to 2 . Clearly u ∈ W 1,∞ ( 2 ) and by Lemma 4.2 ∇u − SO(2)L∞ ( 2 ) ≤ Kα. Proposition 4.1 gives the bound )| sup |u(η)−u(η − 1 ≤ Kα, |η−η | {η,η }⊂
and in particular (60). Surjectivity (61) follows from the estimate above together with the invariance of domain theorem. Proof of Lemma 2.7. Let z = 13 x∈T y(x), = B(z, 20λ) and = B(η, 3λ). Proposition 4.8 implies that there exists a discrete imbedding : y −1 ( ) → A2 which (1) coincides with T up to rotation and translation and (2) satisfies (60). This proves the claim. Proof of Proposition 2.8. (1) If |y(x1 ) − y(x2 )| ≤ 1 + α, then by Lemma 2.7 T ∈ ∪λ∈\{1} Tλ . Assume from now on that |y(x1 ) − y(x2 )| ≥ 1 + α. Let T1 , T2 ∈ ∪λ∈ Tλ such that T1 = T2 , and let for i ∈ {1, 2} λi , ωi and i denote the associated sidelengths, domains and discrete imbeddings. We will demonstrate first that λ1 = λ2 . Assume wlog that λ1 ≤ λ2 and let z = 21 (y(x1 ) + y(x2 )). Lemma 2.7 implies that |y(x1 ) − y(x2 )| ≤ 43 λ1 , λ2 ≤ 2λ1 and maxi=1,2 |z − 13 x∈Ti y(x)| ≤ λ2 . Let
= B(η, 14λ1 ) and = B(η, 2λ1 ). The assumptions imply that ω := y −1 ( ) ⊂ ω1 ∩ ω2 and {x1 , x2 } ⊂ ω. By Proposition 4.8 there exists a discrete imbedding : y −1 ( ) → A2 and matrices Q1 , Q2 ∈ SO(2) such that (x ) − (x1 ) = Q1 (1 (x ) − 1 (x1 )) = Q2 (2 (x ) − 2 (x1 )) for all x ∈ ω. The choice x = x2 establishes that λ1 = λ2 . Let now T3 ∈ Tλ such that {x, x } ⊂ T3 . It will be demonstrated that T3 necessarily either coincides with T1 or with T2 . By the same argumentation used above we can assume that 3 (x) = (x) for all x ∈ ω. Let {x3 } = T3 \ {x1 , x2 }. The lattice point η = (x3 ) ∈ A2 solves the equation |η − (x1 )| = |η − (x2 )| = λ. (y −1 (B(y(x
(67)
Proposition 4.8 entails that A2 ∩ B((x1 ), λ) ⊂ and con1 ), sequentially η ∈ (ω). In two dimensions (67) has at most two solutions which are exhausted by η ∈ ((T1 ∪ T2 ) \ {x1 , x2 }), therefore T3 necessarily either agrees with T1 or with T2 . (2) For λ = 1 the claim follows from Lemma 4.7. If λ > 1 let = B(z, 28|y(x1 ) − y(x2 )|) and = B(z, 5|y(x1 ) − y(x2 )|), where z = 21 (y(x1 ) + y(x2 )). Proposition 4.8 implies that there exists a discrete imbedding : y −1 ( ) → A2 such that B((x1 ), λ) ∩ A2 ⊂ (ω), where λ = |(x1 ) − (x2 )|. Let η± = (x1 ) + Q± ((x2 ) − (x1 )), where Q± is the rotation by 2π 3 in the clockwise (+) and counter-clockwise (-) direction. Since A2 is invariant under Q± we obtain that η± ∈ A2 . By (61) η± ∈ (y −1 ( )) hence there are two preimages x± = X. Set now T± = {x1 , x2 , x± }, ω± = y −1 ( ). By construction both simplices T± satisfy (23). Estimate (60) implies that | 13 x∈T± y(x) − y(x1 )| ≤ λ, hence the inclusion (24) is satisfied. 4 3 λ)))
A Proof of Crystallization in Two Dimensions
(3) This claim follows directly from Proposition 2.8.2.
235
Proof of Proposition 2.9. We first establish estimate (26). Let for each x ∈ X and λ ∈ , s(x, λ) = #{T ∈ Tλ | x ∈ T } be the number of simplices with side length λ which have x as a corner. We start with the identity s(x, λ) = 3#Tλ , (68) x∈X
which expresses the trivial fact that the sum of the number of books read by each member of a group can also be computed by adding the number of readers of each book. We will demonstrate that for each x ∈ X s(x, λ) ≤ m(λ)s(x, 1).
(69)
This claim is trivial if s(x, λ) = 0. Let T (x, λ) = {T ∈ Tλ | x ∈ T } and set = B(y(x), 28λ), = B(y(x), 4λ) and ω = y −1 ( ). The definition of T entails that
∩ y(∂X) = ∅, hence the prerequisites of Proposition 4.8 are satisfied and we can construct a discrete imbedding : ω → A2 . In particular, we obtain that s(x, 1) = 6. For any other simplex T ∈ T (x, λ) we set {x1 , x2 } = T \ {x}. Lemma 2.7 implies that T ⊂ ω and by uniqueness of we can find a rotation matrix R ∈ SO(2) such that T (x1 ) − T (x2 ) = R((x1 ) − (x2 )). This implies that |(x1 ) − (x2 )| = λ. The number of pairs {η1 , η2 } ⊂ A2 which satisfies the equations |η1 −(x)| = |η2 −(x)| = |η1 − η2 | can be expressed as #({|η − (x)| = λ} ∩ A2 ), together with formula (21) this implies that s(x, λ) ≤ 6m(λ) which yields the bound (69). For the other direction we apply again Proposition 2.8.2 and deduce that for each x ∈ X with the property that s(x, λ) < 6m(λ) there exists xb ∈ ∂X such that y(xb ) ∈ B(y(x), 28λ). Since the minimum distance between particles is bounded from below by 1 2 (Lemma 2.2) we have the bound #y −1 (B(y(xb ), 28λ)) ≤ Cλ2 .
(70)
By (18) and (69) we obtain that the number of simplices in Tλ which have at least one corner lying in B (y(xb ), 28λ) is bounded by Cλ2 m(λ). To prove (27) we associate to each T ∈ Tλ and S ∈ T1 the geometry factor meas (conv(y(S)) ∩ conv(y(T ))) ∈ [0, 1], meas(conv(y(S))) n(S, λ) = µ(S, T ) ∈ [0, λ2 m(λ)].
µ(S, T ) =
T ∈Tλ
If dist(y(S), y(∂X)) > 2λ we can define ω = y −1 (B(y(x), 4λ)) as before and obtain the existence of a discrete imbedding : ω → A2 by applying Proposition 4.8. There are precisely 6m(λ) different sets {η1 , η2 , η3 } ⊂ A2 such that η1 = (x), |ηi − ηj | = λ for i = j . This implies that n(S, λ) ≤ m(λ)λ2
(71)
236
F. Theil
with equality if dist(y(S), y(∂X)) > 20λ. This shows that for each λ ∈ , Lemma 2.7 meas(conv(y(T ))) = meas(conv(y(S)) ∩ conv(y(T ))) T ∈Tλ
T ∈Tλ S∈T1
=
µ(S, T )meas(conv(y(S)))
S∈T1 T ∈Tλ
=
n(S, λ)meas(conv(y(S)))
S∈T1
≤
m(λ)λ2
meas(conv(y(S)))
S∈T1
by (71). For the opposite direction we use that n(S, λ) = m(λ)λ2 if dist(y(S), y(∂X)) > 2λ and obtain that meas(conv(y(S))) − meas(conv(y(S))) ≥ λ2 m(λ) S∈T1
≥ λ2 m(λ)
S∈T1 dist(y(S),y(∂X))≤28λ
meas(conv(y(S))) − Cλ2 #∂X .
S∈T1
The last inequality is due to the fact that for each x ∈ ∂X there are at most Cλ2 simplices with side length 1 within a disk of radius λ2 . Proof of (28). This follows directly from (69) and (70). References 1. Yedder, A.B.H., Blanc, X., Le Bris, C.: A numerical investigation of the 2-dimensional crystal problem. Preprint CERMICS (2003), available at http:// www.ann.jussieu.fr/publications/2003/R03003.html. 2. Gardner, C.S., Radin, C.: The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys. 20(6), 719–724 (1979) 3. Blanc, X., Le Bris, C.: Periodicity of the infinite-volume ground state of a one-dimensional quantum model. Nonlinear Anal. 48(6), Ser. A: Theory Methods, 791–803 (2002) 4. M¨uller, S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Part. Differ. Eqs. 1, no. 2, 169-204 (1993) 5. Radin, C.: The ground state for soft disks. J. Stat. Phys. 26(2), 367–372 (1981) 6. Rickman, S.: Quasiregular mappings. Berlin Heidelberg-New York Springer-Verlag, 1993 7. John, F.: Rotation and strain. Comm. Pure. Appl. Math. 14, 391–413 (1961) 8. Friesecke, G., James, R., M¨uller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55, 1461–1506 (2002) 9. Kohn, R.V. New integral estimates for deformations in terms of their nonlinear strain. Arch. Mech. Anal. 78, (1982), 131–172. Communicated by G. Gallavotti
Commun. Math. Phys. 262, 237–267 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1462-y
Communications in
Mathematical Physics
Nonequilibrium Energy Profiles for a Class of 1-D Models J.-P. Eckmann1,2, , L.-S. Young3, 1 2 3
D´epartement de Physique Th´eorique, Universit´e de Gen`eve, Gen`eve, Switzerland Section de Math´ematiques, Universit´e de Gen`eve, Gen`eve, Switzerland Courant Institute for Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10021, USA
Received: 11 March 2005/ Accepted: 21 May 2005 Published online: 9 December 2005 – © Springer-Verlag 2005
Abstract: As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which contains an energy storage device called a “tank”. Energy exchange among tanks is mediated by tracer particles, which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain. For stochastic and Hamiltonian models of this type, we develop a theory that allows one to derive rigorously – under physically natural assumptions – macroscopic equations for quantities related to heat transport, including mean energy profiles and tracer densities. Concrete examples are treated for illustration, and the validity of the Fourier Law in the present context is discussed.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Main Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 General setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Proposed program: from local rules to global profiles . . . . . . . 3. Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The “random-halves” model . . . . . . . . . . . . . . . . . . . . 3.2 Single-cell analysis . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Single cell in equilibrium with 2 identical heat baths. . . . 3.2.2 Chain of N cells in equilibrium with 2 identical heat baths. 3.3 Derivation of equations of macroscopic profiles . . . . . . . . . . 3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . 3.6 A second example . . . . . . . . . . . . . . . . . . . . . . . . . .
JPE is partially supported by the Fonds National Suisse. LSY is partially supported by NSF Grant #0100538.
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4. Hamiltonian Models . . . . . . . . . . . . . . . . . 4.1 Rotating disks models . . . . . . . . . . . . . . 4.1.1 Dynamics in a closed cell. . . . . . . . 4.1.2 Coupling to neighbors and heat baths. . 4.2 Single-cell analysis . . . . . . . . . . . . . . . 4.3 Derivation of equations of macroscopic profiles 4.3.1 Comparisons of models. . . . . . . . . 4.4 Ergodicity issues . . . . . . . . . . . . . . . . 4.5 Results of simulations . . . . . . . . . . . . . . 4.6 Related models . . . . . . . . . . . . . . . . .
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1. Introduction Heat conduction in solids has been a subject of intensive study ever since Fourier’s pioneering work. An interesting issue is the derivation of macroscopic conduction laws from the microscopic dynamics describing the solid. A genuinely realistic model of the solid would involve considerations of quantum mechanics, radiation and other phenomena. In this paper, we address a simpler set of questions, viewing solids that are effectively 1-dimensional as modeled by chains of classical Hamiltonian systems in which heat transport is mediated by tracer particles. Coupling the two ends of the chain to unequal tracer-heat reservoirs and allowing the system to settle down to a nonequilibrium steady state, we study the distribution of energy, heat flux, and tracer flux in this context. We introduce in this paper a class of models that can be seen as an abstraction of certain types of mechanical models. These models are simple enough to be amenable to analysis, and complex enough to have fairly rich dynamics. They have in common the following basic set of characteristics: Each model is made up of an array of identical cells that are linearly ordered. Energy is carried by two types of agents: storage receptacles (called “tanks”) that are fixed in place, and tracer particles that move about. Direct energy exchange is permitted only between tracers and tanks. The two ends of the chain are coupled to infinite reservoirs that emit tracer particles at characteristic rates and characteristic temperatures; they also absorb those tracers that reach them. To allow for a broad range of examples, we do not specify the rules of interaction between tracers and tanks. All the rules considered in this paper have a Hamiltonian character, involving the kinetic energy of tracers. Formally they may be stochastic or purely dynamical, resulting in what we will refer to as stochastic and Hamiltonian models. Via the models in this class, we seek to clarify the relation among several aspects of conduction, including the role of conservation laws, their relation to the dynamics within individual cells, and the notion of “local temperature”. We propose a simple recipe for deducing various macroscopic profiles from local rules (see Sect. 2.2). Our recipe is generic; it does not depend on specific characteristics of the system. When the local rules are sufficiently simple, it produces explicit formulas that depend on exactly 4 parameters: the temperatures and rates of tracer injection at the left and right ends of the chain. For demonstration purposes, we carry out this proposed program for a few examples. Our main stochastic example, dubbed the “random-halves model”, is particularly sim√ ple: A clock rings with rate proportional to x, where x is the (kinetic) energy of the tracer; at the clock, energy exchange between tracer and tank takes place; and the rule of exchange consists simply of pooling the two energies together and randomly dividing – in an unbiased way – the total energy into two parts. Our main Hamiltonian example
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is a variant of the model studied in [17, 12]. Here the role of the “tank” is played by a rotating disk nailed down at its center, and stored energy is ω2 , where ω is the angular velocity of the disk. Explicit formulas for the profiles in question are correctly predicted in all examples. In terms of methodology, this paper has a theory part and a simulations part. The theory part is rigorous in the sense that all points that are not proven are isolated and stated explicitly as “assumptions” (see the next paragraph). It also serves to elucidate the relation between various concepts regardless of the extent to which the assumed properties hold. Simulations are used to verify these properties for the models considered. Our main assumption is in the direction of local thermodynamic equilibrium. For our stochastic models, a proof of this property seems within reach though technically involved (see e.g. [5, 22, 10] and [11]); no known techniques are available for Hamiltonian systems. Extra assumptions pertaining to ergodicity and mixing issues are needed for our Hamiltonian models. It is easy to “improve ergodicity” via model design, harder to mathematically eliminate the possibility of all (small) invariant regions. In the absence of perfect mixing within cells, actual profiles show small deviations from those predicted for the ideal case. In summary, we introduce in this paper a relatively tractable class of models that can be seen as paradigms for heat conduction, and put forth a program which – under natural assumptions – takes one from the microscopic dynamics of a system to its phenomenological laws of conduction. 2. Main Ideas 2.1. General setup. The models considered in this paper – both stochastic and Hamiltonian – have in common a basic set of characteristics which we now describe. There is a finite, linearly ordered collection of sites or cells labeled 1, 2, . . . , N. In isolation, i.e., when the chain is not in contact with any external heat source, the system is driven by the interaction between two distinct types of energy-carrying objects: • Objects of the first kind are fixed in place, and there is exactly one at each site. These objects play the role of storage facility, and serve at the same time to mark the energy level at fixed locations. For brevity and for lack of a better word, let us call them energy tanks. Each tank holds a finite amount of energy at any one point in time; it is not to be confused with an infinite reservoir. We will refer to the energy in the tank at site i as the stored energy at site i. • The second type of objects are moving particles called tracers. Each tracer carries with it a finite amount of energy, and moves from site to site. For definiteness, we assume that from site i, it can go only to sites i ± 1. With regard to microscopic dynamics, the following is assumed: When a tracer is at site i, it may interact – possibly multiple times – with the tank at that site. In each interaction, the two energies are pooled together and redistributed, so that energy is conserved in each interaction. The times of interaction and manner of redistribution are determined by the microscopic laws of the system, which depend solely on conditions within that site. These laws determine also the exit times of the tracers and their next locations. A priori there is no limit to how many tracers are allowed at each site. We stress that this tracer-tank interaction is the only type of interaction permitted: the tanks at different sites can communicate with each other only via the tracers, and the tracers do not “see” each other directly.
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All stochastic models considered in this paper are Markovian. Typically in stochastic rules of interaction, energy exchanges occur when exponential clocks ring, and energy is redistributed according to probability distributions. In Hamiltonian models, tracers are usually embodied by real-life moving particles, and energy exchanges usually involve some types of collisions. The two ends of the chain above are coupled to two heat baths, which are infinite reservoirs emitting tracers at characteristic temperature (and also absorbing them). It is sometimes convenient to think of them as located at sites 0 and N + 1. The two baths inject tracers into the system according to certain rules (to be described). Tracers at site 1 or N can exit the system; when they do so, they are absorbed by the baths. The actions of the two baths are assumed to be independent of each other and independent of the state of the chain. The left bath is set at temperature TL ; the energies of the tracers it injects into the system are iid with a law depending on the model. These tracers are injected at exponential rates, with mean L . Similarly the bath on the right is set at temperature TR and injects tracers into the system at rate R . To allow for a broad spectrum of possibilities, we have deliberately left unspecified (i) the rules of interaction between tracers and tanks, and (ii) the coupling to heat baths, i.e., the energy distribution of the injected tracers. (Readers who wish to see concrete examples immediately can skip ahead with no difficulty to Sects. 3.1 and 4.1, where two examples are presented.) We stress that once (i) and (ii) are chosen, and the 4 parameters TL , TR , L and R are set, then all is determined: the system will evolve on its own, and there is to be no other intervention of any kind. Remark 2.1. Our approach can be viewed as that of a grand-canonical ensemble, since we fix the rates at which tracers are injected into the system (which indirectly determine the density and energy flux at steady state). An alternate setup would be one in which the density of tracers is given, with particles being replaced upon exit. In this alternate setup, the 4 natural intensive variables would be the temperatures TL and TR , the density of tracers (mean number of tracers per cell) and the mean energy flux. For definiteness, we will adhere to our original formulation. We now introduce the quantities of interest. For fixed N , let µN denote the invariant measure corresponding to the unique steady state of the N -chain (assuming there is a unique steady state). The word “mean” below refers to averages with respect to µN . The main quantities of interest in this paper are • • • •
si = mean stored energy at site i ; ei = mean energy of individual tracers at site i ; ki = mean number of tracers at site i ; Ei = mean total energy at site i, including stored energy and the energies of all tracers present.
For simplicity, we will refer to ei as tracer energy and Ei as total-cell energy. We are primarily interested in the profiles of these quantities, i.e., in the functions i → si , ei , ki and Ei as N → ∞ with the temperatures and injection rates of the baths held fixed. More precisely, we fix TL , TR , L and R . Then spacing {1, 2, . . . , N} evenly along the unit interval [0, 1] and letting N → ∞, the finite-volume profiles i → si , ei , ki , Ei give rise to functions ξ → s(ξ ), e(ξ ), k(ξ ), E(ξ ), ξ ∈ [0, 1]. It is these functions that we seek to predict given the microscopic rules that define a system.
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2.2. Proposed program: from local rules to global profiles. We fix N , TL , TR , L and R , and consider an N -chain with these parameters. To determine the profiles in Sect. 2.1, we distinguish between the following two kinds of information: (a) cell-to-cell traffic, and (b) statistical information pertaining to the dynamics within individual cells. In (a), we regard the cells as black boxes, and observe only what goes in and what comes out. Where left-right exit distributions are known, standard arguments balancing energy and tracer fluxes give easily the mean number of tracers and energy transported from site to site. While these numbers are indicative of the internal states of the cells (for example, high-energy tracers emerging from a cell suggest higher temperatures inside), the profiles we seek depend on more intricate relations than these numbers alone would tell us. We turn therefore to (b). Our very na¨ıve idea is to study a single cell, and to bring to bear on chains of arbitrary length the information so obtained. We propose the following plan of action: (i) Consider a single cell plugged to two heat baths (one on its left, the other on its right), both of which are at temperature T and have injection rate , T and being arbitrary. Finding the invariant measure µT , describing the state of the cell in this equilibrium situation is, in general, relatively simple compared to finding µN . (ii) Suppose the measure µT , has been found. We then look at an N -chain with TL = TR = T and L = R = , and verify that the marginals at site i of the invariant measure µN are equal to µT , . (By the marginal at site i, we refer to the measure obtained by integrating out all variables pertaining to all sites = i.) (iii) Once the family {µT , } is found and (ii) verified, we assume that the structure common to the µT , passes to all marginals of µN as N → ∞ even when (TL , L ) = (TR , R ). More precisely, for all ξ ∈ (0, 1), we assume that all limit points of the marginals of µN at site [ξ N ] (where [x] denotes the integer part of x) inherit, as N → ∞, the structure common to µT , . We observe that (i)–(iii) alone are inadequate for determining the sought-after profiles, for they give no information on which T and are relevant at any given site. The main point of this program is that (a) and (b) together is sufficient for uniquely determining the profiles in question. Remark 2.2. Our rationale for (iii) is as follows: Fix an integer, . As N → ∞, the gradients of temperature and injection rate on the sites centered at [ξ N ] tend to 0, so that the subsystem consisting of these sites resembles more and more the situation in (ii). Though rather natural from the point of view of physics [4], this argument does not constitute a proof. Indeed our program is in the direction of proving the existence of well defined Gibbs measures and then assuming, when the system is taken out of equilibrium, that thermodynamic equilibrium is attained locally; in particular, local temperatures are well defined. The full force of local thermal equilibrium is not needed for our purposes; however. The assumption in (iii) pertains only to marginals at single sites. The rest of this paper is devoted to illustrating the program outlined above in concrete examples.
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3. Stochastic Models 3.1. The “random-halves” model. This is perhaps the simplest stochastic model of the general type described in Sect. 2.1. The microscopic laws that govern the dynamics in each cell are as follows: Let δ > 0 be a fixed number. Each tracer is equipped with two independent exponential clocks. √ Clock 1, which signals the times of energy exchanges with the tanks, rings at rate 1δ x, where x is the (current) energy of the tracer. Clock 2, √ which signals the times of site-to-site movements, rings at rate x. The stored energy at site i is denoted by yi . In the description below, we assume the tracer is at site i. (i) When Clock 1 rings, the energy carried by the tracer and the stored energy at site i are pooled together and split randomly. That is to say, the tracer gets p(x + yi ) units of energy and the tank gets (1 − p)(x + yi ), where p ∈ [0, 1] is uniformly distributed and independent of all other random variables. (ii) When Clock 2 rings, the tracer leaves site i. It jumps with equal probability to sites i ± 1. If i = 1 or N , going to sites 0 or N + 1 means the tracer exits the system. It remains to specify the coupling to the heat baths. Here it is natural to assume that the energies of the emitted tracers are exponentially distributed with means TL and TR . This completes the formal description of the model. Remark 3.1. The rates of the two clocks are to be understood as follows: We assume the √ energy carried by the tracer is purely kinetic, so that its speed is x. We assume also that a tracer travels, on average, a distance δ between successive interactions with the tank, and a distance 1 before exiting each site. Remark 3.2. As we will show, the invariant measure does not depend on the value of δ, which can be large or small. The size of δ does affect the rate of convergence to equilibrium, however. Remark 3.3. While the tracers do not “see” each other in the sense that there is no direct interaction, their evolutions cannot be decoupled. The number of tracers present at a site varies with time. When two or more tracers are present, they interact with the tank whenever their clocks go off, thereby sharing information about their energies. A new tracer may enter at some random moment, bringing its energy to the pool; just as randomly, a tracer leaves, taking with it the energy it happens to be carrying at that time. 3.2. Single-cell analysis. 3.2.1. Single cell in equilibrium with 2 identical heat baths. We consider first the following special case of the model described in Sect. 3.1: N = 1, TL = TR = T , and L = R = . Each state of the cell in this model is represented by a point in =
∞
k
(disjoint union),
k=0
where k = {({x1 , . . . , xk }, y) : x , y ∈ [0, ∞)}. Here {x1 , . . . , xk }, is an unordered k-tuple representing the energies of the k tracers, y denotes the stored energy, and a point in k represents a state of the cell when exactly k tracers are present. Remark 3.4. We motivate our choice of . During a time interval when there are exactly k tracers in the cell – with no tracers entering or exiting – it makes little difference
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whether we think of the tracers as named, and represent the state of the cell by a point in [0, ∞)k+1 , or if we think of them as indistinguishable, and represent the state by a point in k . With tracers entering and exiting, however, thinking of tracers as named will require that all exiting tracers return later, otherwise the system is transient and has no invariant measure. Since any rule that assigns to each departing tracer a new tracer to carry its name is necessarily artificial, and for present purposes exact identities of tracers play no role, we have opted to regard the tracers as indistinguishable. We clarify the relationship between [0, ∞)k+1 and k and set some notation: Let πk : [0, ∞)k+1 → k be the map πk (x1 , . . . , xk , y) = ({x1 , . . . xk }, y), i.e., πk is the (k!)-to-1 map that forgets the order in the ordered k-tuple (x1 , . . . , xk ). For a measure µ˜ on [0, ∞)k+1 that is symmetric with respect to the x coordinates, if µ = (πk )∗ µ, ˜ and σ˜ and σ are the densities of µ˜ and µ respectively, then σ˜ and σ are related by σ ({x1 , . . . , xk }, y) = k! σ˜ (x1 , . . . , xk , y) . We also write d{x1 , . . . , xk }dy = (πk )∗ (dx1 . . . dxk dy), and use I to denote the characteristic function. Proposition 3.5. The model in Sect. 3.1 with N = 1, TL = TR = T , and L = R = has a unique invariant probability measure µ = µT , on . This measure has the following properties: √ • the number of tracers present is a Poisson random variable with mean κ ≡ 2 π/T , i.e., µ(k ) =
κ k −κ , e k!
k = 0, 1, 2, . . . ;
(1)
• the conditional density of µ on k is ck σk d{x1 , . . . , xk }dy, where σk ({x1 , . . . , xk }, y) = I{x1 ,...,xk ,y≥0} √
1 e−β(x1 +···+xk +y) ; x1 · . . . · x k
(2)
here β = 1/T , and ck = β k! (β/π )k/2 is the normalizing constant. Proof. Uniqueness is straightforward, since one can go from a neighborhood of any point in to a neighborhood of any other point via positive measure sets of sample paths. We focus on checking the invariance of µ as defined above. For z, z ∈ , let P h (dz |z) denote the transition probabilities for time h ≥ 0 starting ¯ and seek to prove that from z. We fix a small cube A ⊂ k¯ for some k, d IA (z )P h (dz |z) µ(dz) = 0. dh h=0 On the time interval (0, h), the following three types of events may occur: Event E1 : Entrance of a new tracer Event E2 : Exit of a tracer from the cell Event E3 : Exchange of energy between a tracer and the tank We claim that with initial distribution µ, the probability of more than one of these events occurring before time h is o(h) as h → 0. This assertion applies to events both of the
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same type and of distinct types. It follows primarily from the fact that these events are independent and occur at exponential rates. Of relevance also are the exponential tails of σk and the Poisson distribution of pk := µ(k ) in the definition of µ. To illustrate the arguments involved, we will verify at the end of the proof that the probability of two or more tracers exiting on the time interval (0, h) is o(h), but let us accept the above assertion for now and go on with the main argument. Starting from the initial distribution µ, we let P(Ei , A) denote the probability that Ei occurs before time h resulting in a state in A, and let P(E1c ∩ E2c ∩ E3c , A) denote the probability of starting from a state in A and having none of the Ei occur before time h. We will prove P(E1 , A) + P(E2 , A) + P(E3 , A) + P(E1c ∩ E2c ∩ E3c , A) − µ(A) = o(h)µ(A). (3) Notice that A can be represented as the union of disjoint sets ∪i Ai , where each Ai is of the form ¯ y ∈ [y, Aε (¯z) = {({x1 , . . . , xk¯ }, y) : x ∈ [x¯ , x¯ + ε], = 1, . . . , k, ¯ y¯ + ε]} for some z¯ = ({x¯1 , . . . , x¯k¯ }, y) ¯ ∈ , ε 1, and with the intervals [x¯ , x¯ + ε] pairwise ¯ To prove (3) for A, it suffices to prove it for each Ai provided disjoint for = 1, 2, . . . , k. o(h) in (3) is uniformly small for all i. We consider from here on A = Aε (¯z) with the properties above. Let σ denote the ¯ ¯ density of µ. With ε sufficiently small, we have µ(A) ≈ σ (¯z)ε k+1 = pk¯ ck¯ σk¯ (¯z)ε k+1 , where ck and σk are as in the proposition. The other terms in (3) are estimated as follows: P(E1 , A): E1 is in fact the union of 2k¯ subevents, corresponding to a new tracer coming from the left or right bath and the k¯ approximate values of energy of the new tracer. For definiteness, we assume the new tracer arrives from the left bath, and has energy in [x¯1 , x¯1 + ε]. That is to say, the initial state of the cell is described by B = {({x2 , . . . , xk¯ }, y) : x ∈ [x¯ , x¯ + ε], y ∈ [y, ¯ y¯ + ε]} ⊂ k−1 . ¯ The contribution to P(E1 , A) of this subevent is x¯1 +ε βe−βx dx ≈ h µ(B) e−β x¯1 βε µ(B) h x¯1
≈ h pk−1 σk¯ (¯z) ¯ ck−1 ¯
¯
x¯1 βε k+1 .
Here, h is the probability that a tracer is injected, and the integral above is the probability that the injected tracer lies in the specified range. Summing over all 2k¯ subevents, we obtain k¯ ¯ P(E1 , A) ≈ 2hβ ( x¯ ) pk−1 σk¯ (¯z) ε k+1 . ¯ ck−1 ¯
(4)
=1
P(E2 , A): In order to result in a state in A, the initial state must be in C1 = {({x1 , . . . , xk¯ , x}, y) : x ∈ [x¯ , x¯ + ε], x ∈ [0, ∞), y ∈ [y, ¯ y¯ + ε]} ⊂ k+1 . ¯ We assume here that the tracer with energy x exits between time 0 and time h. This gives ∞ √ 1 ¯ k+1 P(E2 , A) ≈ pk+1 c σ (¯ z ) ε min{h x, 1} √ e−βx dx ¯ ¯ k+1 k¯ x 0 ¯
= pk+1 ck+1 σk¯ (¯z) ε k+1 β −1 (h + o(h)) . ¯ ¯
(5)
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245
P(E3 , A): For definiteness, we assume it is the tracer with energy near x¯1 that is the product of the interaction with the tank. To arrive in a state in Aε , one must start from ¯ x¯1 + y¯ + 2ε]}. D = {({x1 , . . . , xk¯ }, y) : x ∈ [x¯ , x¯ + ε] for ≥ 2, x1 + y ∈ [x¯1 + y, A simple integration using the rule of interaction in Sect. 3.1 gives k¯ √ x¯ ¯ pk¯ ck¯ σk¯ (¯z)ε k+1 . P(E3 , A) ≈ h =1 δ
(6)
P(E1c ∩ E2c ∩ E3c , A): We first note that starting from A, the probability of the tracer with energy ≈ x¯1 exiting is x¯1 +ε √ 1 ¯ ¯ pk¯ ck¯ σk¯ (¯z) x¯1 eβ x¯1 ε k h x √ e−βx dx = h x¯1 pk¯ ck¯ σk¯ (¯z) ε k+1 ; x x¯1 the probability of the tracer with energy ≈ x¯1 interacting with the tank is √ x¯1 ¯ h pk¯ ck¯ σk¯ (¯z) ε k+1 ; δ and the probability of a new tracer entering the cell from the left (resp. right) bath is h µ(Aε ). Thus ¯ P(E1c ∩ E2c ∩ E3c , A) ≈ pk¯ ck¯ σk¯ (¯z) ε k+1 · (1 − h x¯ ) · (1 − h)2 · (1 − h x¯ /δ) 1 k¯ ¯ k+1 k¯ ≈ pk¯ ck¯ σk¯ (¯z) ε . · 1 − h =1 x¯ + 2 + =1 x¯ δ (7) Summing Eqs. (4)–(7), we obtain (3) provided = pk¯ ck¯ 2βpk−1 ¯ ck−1 ¯
and
2pk¯ ck¯ = pk+1 ¯ ck+1 ¯ T .
Note that these two equations represent the same relation for different k. We write this relation as T ck pk = , ck+1 pk+1 2
(8)
and verify that it is compatible with assertion (1): Since 1 1 −βx k ck e−βy dy = 1 , dx =1 √ e x k! √ ∞ and 0 x −1/2 e−βx dx = πT , we have √ ∞ 1 1 −βx 2 ck 2 1 2 π pk+1 = pk = dx pk = pk . √ e √ T ck+1 T k+1 0 k+1 x T To complete the proof, we estimate the probability of two or more tracers exiting before time h. For n = 2, 3, . . . , let E2,n be the event that the initial state is in
¯ y¯ +ε]}, Cn = {({x1 , . . . , xk¯ , x (1) , . . . , x (n) }, y) : x ∈ [x¯ , x¯ +ε], x ( ) ∈ [0, ∞), y ∈ [y,
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and during the time interval (0, h), all n of the tracers carrying energies x ( ) , = 1, 2, . . . , n, exit the system. The probability of ∪n≥2 E2,n is ¯ pk+n z)ε k+1 (O(h))n , ¯ ck+n ¯ σk¯ (¯ n≥2
which, from Eq. (8), is bounded by 2 n ¯ ¯ k+1 pk¯ ck¯ σk¯ (¯z)ε (O(h))n = pk¯ ck¯ σk¯ (¯z)ε k+1 o(h) . T n≥2
Remark 3.6. In the setting of Proposition 3.5, since the cell is in equilibrium with the two heat baths, it is obvious that it ejects, on average, 2 tracers per unit time, and the energies of the tracers ejected have mean T . We observe that the cell in fact reciprocates the action of the bath more strongly than this: the distribution of the energies of the ejected tracers is also exponential. To see this, fix k and consider one tracer at a time. The probability of the tracer exiting with energy > u is ∞ √ 1 ∼ x · √ e−βx dx = β −1 e−βu . x u 3.2.2. Chain of N cells in equilibrium with 2 identical heat baths. We treat next the case of arbitrary N. That is to say, the system is as defined in Sect. 3.1, but with TL = TR = T and L = R = . Let µ be as in Proposition 3.5. Proposition 3.7. The N -fold product µ × · · · × µ is invariant. Remark 3.8. That the invariant measure is a product tells us that at steady state, there are no spatial correlations. We do not find this to be entirely obvious on the intuitive level: one might think that above-average energy levels on the left half of the chain may cause the right half to be below average; that is evidently not the case. This result should not be confused with the absence of space-time correlations. Proof. Proceeding as before, we consider a small time interval (0, h), and treat separately the individual events that may occur during this period. One of the new events (not relevant in the case of a single cell) is the jumping of a tracer from site i ± 1 to site i. We fix a phase point (1)
(1)
(N)
(N)
z¯ = (¯z(1) , . . . , z¯ (N) ) = ({x¯1 , . . . , x¯k1 }, y (1) ; . . . ; {x¯1 , . . . , x¯kN }, y (N) ) , (i) ⊂ N , where A(i) = A (¯ (i) and let A = N ε z ) is as in Proposition 3.5. Let i=1 A (i) (i) σ = σki (¯z ). For definiteness, we fix also an integer n, 1 < n < N , and assume that at time 0, the state of the chain is as follows:
(i) at site n + 1, there are kn+1 + 1 tracers the energies of which lie in disjoint intervals (n+1)
[x¯1
(n+1)
, x¯1
(n+1)
(n+1)
+ ε], . . . , [x¯kn+1 , x¯kn+1 + ε]
and
(ii) at site n, there are kn − 1 tracers whose energies lie in (n)
(n)
(n)
(n)
[x¯2 , x¯2 + ε], . . . , [x¯kn , x¯kn + ε] .
(n)
(n)
[x¯1 , x¯1 + ε] ,
Nonequilibrium Energy Profiles for a Class of 1-D Models
247 (n)
The probability of this event occurring and the tracer with energy ≈ x¯1 jumping from site n + 1 into site n is then given by 21 I · I I · I I I, where I = i=n,n+1 (pki cki σ (i) ε ki +1 ),
(n) (n) I I = pkn −1 ckn −1 σ (n) x¯1 eβ x¯1 ε kn , I I I = pkn+1 +1 ckn+1 +1 σ (n+1) ε kn+1 +1
(n)
x¯1 +ε (n) x¯1
√ 1 h x √ e−βx dx . x
This product can be written as
p h N k −1 ckn −1 pkn+1 +1 ckn+1 +1 (n) · · x¯1 , i=1 pki cki σ (i) ε ki +1 · n 2 pkn ckn pkn+1 ckn+1 which, by Eq. (8), is equal to h 2
(i) ki +1 N i=1 pki cki σ ε
(n) · x¯1 .
(9)
There are many terms of this kind that contribute to IA (z )P h (dz |z)µ(dz), two (n) for x¯ for each pair (n, ). We claim that the system has detailed balance, i.e., the term associated with the scenario above is balanced by the probability of starting from a state (n) in A and having the tracer at site n carrying energy ≈ x¯1 jump to site n + 1. The probability of the latter is 1 N ( pk ck σ (i) ε ki +1 ) · 2 i=1 i i
(n) 1 (n) x¯1 eβ x¯1
ε
·
(n)
x¯1 +ε (n) x¯1
√ 1 h x √ e−βx dx , x
which balances exactly (9) as claimed. An argument combining the one above with that in Proposition 3.5 regarding the injection of new tracers holds at sites 1 and N . Propositions 3.5 and 3.7 are steps (i) and (ii) in the proposed scheme in Sect. 2.2. 3.3. Derivation of equations of macroscopic profiles. Having found a candidate family of equilibrium measures {µT , }, we now complete the rest of the program outlined in Sect. 2.2. The next step, according to this program, is to assume that for N 1, the marginals of the invariant measure µN at site i are approximately equal to µT , for some T = Ti and = i . We identify those parts of our proposed theory that are not proved in this paper and state them precisely as “Assumptions”. Assumption 1. Given TL , TR > 0, L , R ≥ 0, and N ∈ Z+ , the N -chain defined in Sect. 3.1 with these parameters has an invariant probability measure µN . We do not believe this existence result is hard to prove but prefer not to depart from the main line of reasoning in this paper. Once existence is established, uniqueness (or ergodicity) follows easily since any invariant measure clearly has strictly positive density everywhere. A proof of the statement in Assumption 2 below is more challenging. For ξ ∈ (0, 1), let µN,[ξ N] denote the marginal of µN at the site [ξ N ].
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Assumption 2. For every ξ ∈ (0, 1), every limit point as N → ∞ of µN,[ξ N] is a member of the family {µT , , T > 0, ≥ 0}.1 In Sect. 2.1, we introduced four quantities of interest. There is one that was somewhat ambiguously defined, namely ei . Its precise meaning is as follows: ei := ∞ k=1 pi,k ei,k , where pi,k is the probability that the number of tracers at site i is equal to k and ei,k is k1 of the mean total tracer energy conditioned on the number of tracers present being equal to k. Theorem 3.9 is about the profiles of certain observables as N → ∞. We refer the reader to the end of Sect. 2.1 for the precise meaning of the word “profile” in the theorem. Theorem 3.9. The following hold for the “random-halves” model defined in Sect. 3.1 with arbitrary TL , TR , L , R . Under Assumptions 1 and 2 above: • the profile for the mean number of jumps out of a site per unit time is
j (ξ ) = 2 L + (R − L )ξ . • the profile for the mean total energy transported out of a site per unit time is
Q(ξ ) = 2 L TL + (R TR − L TL )ξ ; • the profile for the mean stored energy at a site is s(ξ ) =
Q(ξ ) L TL + (R TR − L TL )ξ = ; L + (R − L )ξ j (ξ )
in the case L = R , this simplifies to s(ξ ) = TL + (TR − TL )ξ ; • the profile for mean tracer energy is e(ξ ) = 21 s(ξ ) ; • the profile for mean number of tracers is π κ(ξ ) = j (ξ ) ; s(ξ ) • the profile for mean total-cell energy is E(ξ ) = s(ξ ) + κ(ξ )e(ξ ) = s(ξ ) +
1 π s(ξ ) j (ξ ) . 2
Proof of Theorem 3.9. We divide the proof into the following three steps: I. Information on single cells. Items (i)–(iv) are strictly in the domain of internal cell dynamics. The setting is that of Proposition 3.5, and the results below are deduced (in straightforward computations) from the invariant density given by that proposition. The parameters are, as usual, T and . (Note that this means the rate at which tracers enter the site is 2.) −βy and mean T ; (i) stored energy has density βe √ β (ii) tracer energy has density √πx e−βx and mean T2 ;
(iii) mean number of tracers, κ = 2 Tπ ;
(iv) mean total-cell energy, E = T · (1 + κ2 ) .
1 As stated in Sect. 2, we assume all marginals µ N,[ξ N ] , N ≥ 1, have uniform tail bounds in energy and in number of tracers of the type suggested in the single-cell analysis.
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Items (v) and (vi) are in preparation for the analysis of cell-to-cell traffic: (v) mean number of jumps out of the cell per unit time, j = 2 ; (vi) mean total energy transported out of the cell per unit time, Q = Tj . II. Global phenomenological equations. Consider now a chain with N cells with settings TL , TR , L and R at the two ends. The following results use only standard conservation laws together with the local rule that when a tracer exits a cell, it has equal probability of going left and right. (A) Balancing tracer fluxes. Let ji denote the number of jumps per unit time out of site i. Then i ji = 2 L + (R − L ) . (10) N +1 Proof. Consider an (imaginary) partition between site i and site i + 1. We let −ji denote the flux across this partition. Then ji = 21 (ji+1 − ji ) for i = 0, 1, . . . , N, where j0 and jN+1 are defined to be 2L and 2R respectively. For i = 0, N , the 21 is there because only half of the tracers out of site i + 1 jump left, and half of those out of site i jump right. The fluxes across partitions between all consecutive sites must be equal, or there would be a pile-up of tracers somewhere. This together with i ji = R − L gives the asserted formula. (B) Balancing energy fluxes. Let Qi denote the mean total energy transported out of site i per unit time. Then
i Qi = 2 L TL + (R TR − L TL ) N +1
.
Proof. The argument is identical to that in (A), with Qi = 21 (Qi+1 − Qi ) and Q0 and QN+1 defined to be 2L TL and 2R TR respectively. III. Combining the results from I and II. Fix ξ ∈ (0, 1). Passing to a subsequence if necessary, we have, by Assumption 2, µN,[ξ N] → µT (ξ ),(ξ ) for some T (ξ ) and (ξ ). We identify the two numbers T (ξ ) and (ξ ) as follows: Let j N (ξ ) denote the mean number of jumps out of site [ξ
N ] per unit timein the N-chain. Then by (A) in Part II, j (ξ ) := limN→∞ j N (ξ ) = 2 L + (R − L )ξ . This is, therefore, the mean number of jumps out of a cell with invariant measure µT (ξ ),(ξ ) . Similarly, we deduce that the mean total energy transported out of a cell with the same
invariant measure is Q(ξ ) := limN→∞ QN (ξ ) = 2 L TL + (R TR − L TL )ξ . Appealing now to the information in Part I, we deduce from (v) and (vi) that (ξ ) = 21 j (ξ ) and T (ξ ) = Q(ξ )/j (ξ ). The rest of the profiles follow readily: we read off s = T from (i), and deduce the relation between s and e by comparing (i) and (ii). The profiles for κ and E follow from (iii) and (iv) together with our knowledge of and T . The proof of Theorem 3.9 is complete. We remark that in the terminology of [7] our random halves model is of gradient type. Remark 3.10. It is instructive to see what Theorem 3.9 says in the special case when L = 0. Since no particles are injected at the left end, clearly, TL cannot matter. But
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since tracers do exit from the left, one expects an energy flux across the system. Upon substituting L = 0 into the formulas above, one gets √ j (ξ ) = 2R ξ , e(ξ ) = 21 s(ξ ) = 21 TR , κ(ξ ) = 2R ξ π/TR , and an energy flux of − 21 Q (ξ ) = −R TR . 3.4. Simulations. Numerical simulations are used to validate Assumptions 1 and 2 in Theorem 3.9. We mention here only those details of the simulations that differ from the theoretical study. Needless to say, we work with a finite number of sites, usually 20. Simulations start in a random initial state, and are first run for a period of time to let the system reach its steady state. All times are in number of events (E1 , E2 , and E3 ). Up to half the simulation time is used to reach stationarity; statistics are then gathered during the remaining simulation time. Since total simulation time is finite, we find it necessary to take measures to deal with tracers of exceptionally low energy: these tracers appear to remain in a cell indefinitely, skewing the statistics on the number of tracers. We opted to terminate events involving a single tracer at a single site after about 0.001 of total simulation time. This was done about 10 times in the course of 109 events. Simulations are performed both to verify directly the properties of the marginals at individual sites and to plot empirically the various profiles of interest. Excellent agreement with predicted values is observed in all runs. A sample of the results is shown in Fig. 1.
3.5. Interpretation of results. We gather here our main observations, discuss their physical implications, and provide explanations for the reasons behind the derived formulas. 1. Linearity of profiles. We distinguish between the following 3 types of profiles: a. Transport of energy and tracers: j (ξ ) and Q(ξ ) are always linear by simple conservation laws and by the imposed left-right symmetry in outflow of tracers and energy from each site. b. Mean stored and (individual) tracer energies: s(ξ ) and e(ξ ) are linear if and only if there is no tracer flux across the system. (See item 4 below.) c. Tracer densities and total-cell energy: κ(ξ ) is never linear (unless TL = TR and L = R ); in addition to the obvious bias brought about by different injection rates, tracers have a tendency to accumulate at the cold end (see item 3 for elaboration). As a result, E(ξ ) is also never linear. 2. Heat flux and the Fourier Law. Heat flux from left to right is given by = TL L − TR R . Thinking of the temperature of the system as given by T (ξ ), Theorem 3.9 says that thermal conductivity is constant and proportional to TR − TL if and only if there is no tracer flux across the system, i.e., if and only if L = R . 3. Distribution of tracers along the chain. In the case L = R , more tracers are congregated at the cold end than at the hot. This is because the only way to balance the tracer equation is to have the number of jumps out of a site be constant along the chain. Inside the cells, however, tracers move more slowly at the cold end, hence they jump less frequently, and the only way to maintain the required number of jumps is to have more tracers. When L = R , the idea above continues to be valid, except that one needs also
Nonequilibrium Energy Profiles for a Class of 1-D Models
251
Fig. 1. Random-halves model with 20 sites, temperatures TL = 10, TR = 100 and injection rates L = 10, R = 5. Top left: Mean tank energies si . Top right: Mean number of tracers κi . Bottom: Mean total energy Ei . Simulations in perfect agreement with predictions from theory
to take into consideration the bias in favor of more tracers at the end where the injection rate is higher. 4. Tracer flux and concavity of stored energy. One of the interesting facts that have emerged is that s(ξ ) is linear if and only if L = R , and when L = R , their relative strength is reflected in the concavity of s(ξ ). This may be a little perplexing at first because no mechanism is built into the microscopic rules for the tanks to recognize the directions of travel of the tracers with which they come into contact. The reason behind this phenomenon is, in fact, quite simple: If there is a tracer flux across the system, say from right to left, then the tank at site i hears from site i + 1 more frequently than it hears from site i − 1 (because ji+1 > ji−1 ). It therefore has a greater tendency to equilibrate with the energy level on the right than on the left, causing si to be > 21 (si+1 + si−1 ). Since this happens at every site, a curvature for the profile of si
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is created. The reader should further note that tracer flux and heat flux go in opposite directions if (L − R ) · (L TL − R TR ) < 0. 5. Individual cells mimicking heat baths. The cells in our models are clearly not infinite heat reservoirs, yet for large N , they acquire some of the characteristics of the heat baths with which they are in contact. More precisely, the i th cell injects each of its two neighbors with i tracers per unit time. These tracers, which have mean energy Ti , are distributed according to a law of the same type as that with which tracers are emitted from the baths (exponential in the case of the random halves model); see Remark 3.6. Unlike the conditions at the two ends, however, the numbers i and Ti are self-selected. 3.6. A second example. We consider here a model of the same type as the “randomhalves” model but with different microscopic rules. The purpose of this exercise is to highlight the role played by these rules and to make transparent which part of our scheme is generic. The rules for energy exchange in this model simulate a Hamiltonian model in which both tracers and tanks have one degree of freedom. Write x = v 2 and y = ω2 , v, ω ∈ (−∞, ∞), and think of energy as uniformly distributed on the circle {(v, ω) : v 2 +ω2 = c}, so that when the tracer interacts with stored energy, the redistribution is such that a random point on this circle is chosen with weight |v| (this is the measure induced on a cross-section by the invariant measure of the flow). That is to say, if (x, y) are the stored energy and tracer energy before an interaction, and (x , y ) afterwards, then for a ∈ [0, x + y], √a
dv 2 √ √ a 0 v 1 + dω dω
P {y > a} = P {|ω | > a} = 1− √ (11) = 1− √
2 x+y x+y dv v 1 + dω 0 dω or, equivalently, the density of y is 1 1 1 √ 2 y x + y
for
y ∈ [0, x + y] .
Assume now that all is as in Sect. 3.1, except that when Clock 1 of a tracer rings, it exchanges energy with the tank according to the rule in (11) and not the random-halves rule. Following the computation in Sect. 3.2 (details of which are left to the reader), we see that Propositions 3.5 and 3.7 hold for the present model provided σk is replaced by σk ({x1 , . . . , xk }, y) = I{x1 ,... ,xk ,y≥0} √
1 e−β(x1 +...+xk +y) . x1 · . . . · xk y
This defines a new family of {µˆ T , } for this model. With µˆ T , in hand, we make the assumption as before that for N 1, the marginals of individual sites have the same form. Proceeding as in Sect. 3.3, we read off the following information on single cells: √ (i) stored energy has density const.e−βy / y and mean T /2 ; √ −βx (ii) tracer energy has density const.e
/ x and mean T /2 ; (iii) mean number of tracers, κ = 2 Tπ ; (iv) mean total-cell energy, E =
T 2 (1 + κ)
.
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The rest of the analysis, including (iv), (v), (A) and (B), do not depend on the local rules (aside from the fact that tracers exiting a cell have equal chance of going left and right). Thus they remain unchanged. Reasoning as in Theorem 3.9, we obtain the following: Proposition 3.11. Under Assumptions 1 and 2, the profiles for the model with energy exchange rule (11) are • j (ξ ) = 2(L + (R − L )ξ ) ; • Q(ξ ) = 2(L TL + (R TR − L TL )ξ ) ; • s(ξ ) = e(ξ ) = 21 Q(ξ )/j (ξ ) ; √ • κ(ξ ) = 2π /s(ξ ) j (ξ )/2 ; √ • E(ξ ) = s(ξ ) + κ(ξ )e(ξ ) = s(ξ ) + 2π s(ξ ) j (ξ )/2 . Numerical simulations give results in excellent agreement with these theoretical predictions. 4. Hamiltonian Models In Sect. 4.1, we introduce a family of Hamiltonian models generalizing those studied numerically in [17, 12]. A single-cell analysis similar to that in Sect. 3 is carried out for this family in Sect. 4.2, and predictions of energy and tracer density profiles are made in Sect. 4.3. We again use the Assumptions in Sect. 3, but the predictions here are made under an additional ergodicity assumption, ergodicity being a property that is easy to arrange in stochastic models but not in Hamiltonian ones. Results of simulations for a specific model are shown in Sect. 4.5. A brief discussion of related models is given in Sect. 4.6. 4.1. Rotating disks models. We describe in this subsection a family of models quite close to those studied numerically in [17, 12]. The rules of interaction (though not the coupling to heat baths) are, in fact, used earlier in [20]. 4.1.1. Dynamics in a closed cell. We treat first the dynamics within individual cells assuming the cell or box is sealed, i.e., it is not connected to its neighbors or to external heat sources. Let 0 ⊂ R2 be a bounded domain with piecewise C 3 boundary. In the interior of 0 lies a (circular) disk D, which we think of as nailed down at its center. This disk rotates freely, carrying with it a finite amount of kinetic energy derived from its angular velocity; it will play the role of the “energy tank” in Sect. 2.1. The system below describes the free motion of k point particles (i.e., tracers) in = 0 \ D. When a tracer runs into ∂0 , the boundary of 0 , the reflection is specular. When it hits the rotating disk, the energy exchange is according to the rules introduced in [20, 17, 12]. A more precise description of the system follows. The phase space of this dynamical system is ¯ k = ( k × ∂D × R2k+1 )/ ∼ , where x = (x1 , . . . , xk ) ∈ k denotes the positions of the k tracers, ϑ ∈ ∂D denotes the angular position of a (marked) point on the boundary of the turning disk, v = (v1 , . . . , vk ) ∈ R2k denotes the velocities of the k tracers, ω ∈ R denotes the angular velocity of the turning disk, and ∼ is a relation that identifies pairs of points in the collision manifold Mk = {(x, ϑ, v, ω) : x ∈ ∂ for some }. The rule of identification is given below.
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¯ k is denoted by ¯ s . As long as no collisions are involved, we have The flow on ¯ s (x, ϑ, v, ω) = (x + sv, ϑ + sω, v, ω) . ¯ s is not We assume at most one tracer collides with ∂ at any one point in time. ( defined at multiple collisions, which occur on a set of measure zero.) At the point of impact, i.e., when x ∈ ∂ for one of the , let v = (vt , vn ) be the tangential and normal components of v . What happens subsequent to impact depends on whether x ∈ ∂0 or ∂D. In the case of a collision with ∂0 , the tracer bounces off ∂0 with angle of reflection equal to angle of incidence, i.e., (vn ) = −vn ,
(vt ) = vt ,
(12)
and the other variables are unchanged. In the case of a collision with the disk, the following energy exchange takes place between the disk and the tracer: (vn ) = −vn ,
(vt ) = ω ,
ω = vt .
(13)
Here we have, for simplicity, taken the radius of the disk, the moment of inertia of the disk, and the mass of tracer in such a way that the coefficients in Eq. (13) are equal to 1. ¯ k is z ∼ z where z, z ∈ Mk are such that all of The identification in the definition of their coordinates are equal except that v and ω in z are replaced by the corresponding quantities with primes in Eqs. (12) and (13) for z . We also write F (z) = z . Note that in both (12) and (13), total energy is conserved, i.e., |v|2 +ω2 = |v |2 +(ω )2 . The energy surfaces in this model are therefore ¯ k,E = ( k × ∂D × SE2k )/ ∼, where SE2k = {(v1 , . . . , vk , ω) ∈ R2k+1 :
|v |2 + ω2 = E} .
We claim that the natural invariant measure, or Liouville measure, of the (discontin¯ s on ¯ k is uous) flow m ¯ k = (λ2 | )k × (ν1 |∂D ) × λ2k+1 , where λd is d-dimensional Lebesgue measure and νd is surface area on the relevant d-sphere. Once the invariance of m ¯ k is checked, it will follow immediately that the ¯ k,E are ¯ s -invariant, as are all induced measures m ¯ k,E = (λ2 | )k × (ν1 |∂D ) × ν2k on ¯ measures on k of the form ψ(E)m ¯ k,E for some ψ : [0, ∞) → [0, ∞). The invariance of m ¯ k is obvious away from collisions and at collisions with ∂0 . Because collisions occur one at a time, it suffices to consider a single collision between a single tracer and the disk. The problem, therefore, is reduced to the following: Consider ¯ s on ¯ 1 , and let M1,D denote the part of the collision manifold involving D. To prove that m ¯ 1 is preserved in a collision with D, it suffices to check that for A ⊂ M1,D and ε > 0 arbitrarily small,
m ¯ 1 ∪−ε<s<0 s (A) = m ¯ 1 ∪0<s<ε s (F (A)) . We leave this as a calculus exercise.
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4.1.2. Coupling to neighbors and heat baths. We now consider a chain of N identical copies of the dynamical system described in Sect. 4.1.1, and define couplings between nearest neighbors and between end cells and heat baths. Let γL and γR be two marked subsegments of ∂0 of equal length; these segments will serve as openings to allow tracers to pass between cells. For now it is best to think of 0 as having a left-right symmetry, and to think of γL and γR as vertical and symmetrically placed (as in Fig. 2), although as we will see, these geometric details are not relevant for the derivation of mean energy and tracer profiles. We call the segments γL and γR (i) (i) (i) (i+1) in the i th cell γL and γR . For i = 1, . . . , N − 1, we identify γR with γL , that is th st to say, we think of the domains of the i cell and the (i + 1) cell as having a wall in (i) (i+1) , and remove this wall, so that tracers that would have common, namely γR and γL collided with it simply continue in a straight line into the adjacent cell. (See Fig. 2.) Tracers are injected into the system as follows. Consider, for example, the bath on the left. We say the injection rate is if at the ring of an exponential clock of rate , (1) a single tracer enters cell 1 via γL . (Note that the rate is not the injection rate per (1) unit length of the opening γL but per unit time.) The points of entry and velocities of entering tracers are iid, the law being the one governing the collisions of tracers with (1) (1) γL . That is to say, the point of entry is uniformly distributed on γL , and the velocity v has density c e−β|v| |v|| sin(ϕ)| dv , 2
(1)
2β 3/2 c= √ , π
(14) (1)
where v ∈ R2 points into γL and ϕ ∈ (0, π) is the angle v makes with γL at the point of entry. (This is the distribution of v at collisions for particles with velocity distribution β 2 π exp(−β|v| )dv.) Here β = 1/T , where T is said to be the temperature of the bath. Observe that the mean energy of the tracers injected into the system by a bath at temperature T is not T but 3T /2. Injection from the right is done similarly, via the opening (1) (N) (N) γR . When a tracer in the chain reaches γL or γR , it vanishes into the baths. This completes the description of our models. We remark that the process above is a Markov process in which the only randomness comes from the action of the baths. Once a tracer is in the system, its motion is governed by rules that are entirely deterministic.
Fig. 2. A row of diamond-shaped boxes with small lateral holes (made by removing vertical walls corre(i) (i+1) sponding to γR = γL ) to allow the tracers to go from one box to the next. The shapes of the “boxes” can be quite general for much of our theory. The configuration shown is the one used in the simulations discussed in Sect. 4.5, but with larger holes for better visibility
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4.2. Single-cell analysis. In analogy with Sect. 3.2, we investigate in this subsection the invariant measure for a single cell coupled to two heat baths with parameters T and . ¯ k and ¯ k,E be as in Sect. 4.1.1. As before, a state of this system is represented Let by a point in ∞ = ∪∞ k=0 k = ∪k=0 ∪E≥0 k,E ,
¯ k and ¯ k,E respectively obtained by identifying where k and k,E are quotients of permutations of the k tracers. With {· · · } representing unordered sets as before, points in are denoted by z = ({x1 , . . . , xk }, ϑ; {v1 , . . . , vk }, ω) or simply ({x }, ϑ; {v }, ω), with v understood to be attached to x . The quotient measures of m ¯ k and m ¯ k,E are respectively mk and mk,E . ¯ s to denote the semi-flow on , ¯ and Abusing notation slightly, we continue to use let s denote the induced semi-flow on . Then s is as in Sect. 4.1.1 except where tracers exit or enter the system. More precisely, if s (z) ∈ k for all 0 ≤ s < s0 , and a tracer exits the system at time s0 , then s0 (z) jumps to k−1 . Similarly, if a tracer is injected from one of the baths at time s0 , then instantaneously s0 (z) jumps to k+1 , the destination being given by a probability distribution. Let |γ | denote the length of the segment γL or γR . Proposition 4.1. There is an invariant probability measure µ with the following properties: (a) the number of tracers present is a Poisson random variable with mean κ where √ λ2 () κ=2 π √ ; |γ | T (b) the conditional density of µ on k is ck σk dmk , where σk ({x }, ϑ; {v }, ω) = e−β(ω
2 +k 2 =1 |v | )
and ck is the normalizing constant. We observe as before that the Poisson parameter κ is proportional to (the √ higher the injection rate, the more tracers in the cell) and inversely proportional to T , i.e., the speed of the tracers (the faster the tracers, the sooner they leave). Unlike the models considered in Sect. 3, where the tracers are assumed to leave the cell at a rate equal to their speed, here the ratio λ2 ()/|γ | appears, as it should: the smaller the passage way, the longer it takes for the tracers to leave. Notice that we have not claimed that µ is unique. We introduce some notation in preparation for the proof. For A ⊂ k and h > 0, we let −h (A) denote the set of all initial states in that in time h evolve into A assuming no new tracers are injected into the system between times 0 and h.2 Then (n) (n) (n) −h (A) = ∪n≥0 −h (A), where −h (A) = −h (A) ∩ k+n , i.e., −h (A) is the set of states where initially k + n tracers are present, and by the end of time h exactly n of these tracers have exited and the remaining k are described by a state in A. Lemma 4.2. Let µ be as in Proposition 4.1, and let Aε be a cube of sides ε in k , ε small enough that µ(Aε ) ≈ pk ck σk (¯z)ε 4k+2 for some z¯ . We assume the following holds for all small h > 0: 2 Notice that (1) { , h ≥ 0} is a semi-flow, and h −h is not defined; (2) −h (A) as defined is = (h )−1 (A).
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(i) no tracers are injected into the system on the time interval (0, h] ; (0) (ii) h (−h (Aε )) = Aε . Then
|γ | µ(−h (Aε )) = σk (¯z)ε 4k+2 pk ck + 2h pk+1 ck+1 + o(h) , c
where pk = µ(k ) and c =
(15)
√2 β 3/2 . π
Proof. The idea is that for a particle to exit in the very short time h, it must be close to the exit γL or γR and move towards it without colliding with the disk or the boundary, or it must have very large speed (and that is improbable). (n) By assumption (i), we have µ(−h (Aε )) = n≥0 µ(−h (Aε )). The n = 0 term is handled easily: By virtue of (i) and (ii), the situation is equivalent to that in Sect. 4.1.1. (0) Since µ|k is invariant for the closed dynamical system with k tracers, we have µ(−h (Aε )) = µ(Aε ). (1) Consider next n = 1. We give the estimate for µ(−h (Aε )) assuming γL and γR are straight-line segments, leaving the general case (where these segments may be curved) to the reader. First some notation: For v ∈ R2 and a > 0, let E(v, a, L) be the parallelogram on the same side of γL as and with the property that one of its sides is γL while the other is parallel to v and has length a; E(v, a, R) is defined similarly. To simplify the discussion, we assume that for a > 0 sufficiently small, E(v, a, L) and E(v, a, R) are contained in , and leave to the reader the verification that “corners” at the end of γL or γR lead to higher order terms (in the variable h used below). (1) Starting from a state in −h (Aε ), we let x and v denote the initial position and velocity of the tracer that exits before time h, and treat separately the cases (1) |v| ≤ ha and (2) |v| > ha . In Case (1), in order for the tracer to exit before time h, we must have x ∈ E(v, h|v|, L) ∪ E(v, h|v|, R), and v must point toward the exits. Since λ2 (E(v, h|v|, L)) = λ2 (E(v, h|v|, R)) = h|v|| sin(ϕ)||γ |, where ϕ is the angle v makes with γL or γR , we obtain (1)
µ(−h (Aε ) ∩ {|v| ≤
a }) h
= σk (¯z)ε 4k+2 pk+1 ck+1 · 2h|γ |
π
dϕ| sin(ϕ)| 0
|v|≤ ha
dv|v|e−β|v|
2
1 = σk (¯z)ε 4k+2 pk+1 ck+1 · 2h|γ | (1 + o(h)) c √ with c = 2β 3/2 / π. For Case (2), we have the trivial estimate σk (¯z)ε 4k+2 pk+1 ck+1 · o(h). To see that the terms corresponding to n > 1 are negligible, we first derive the bound 1 (n) µ(−h (A)) ≤ σk (¯z)ε 4k+2 pk+n ck+n (2h|γ | + o(h))n . c
(16)
Then we compute the growth rate of pk+n ck+n . By the definitions of these numbers, we have −1 √ λ2 () ck+1 pk+1 1 λ2 () 2 · = e−β|v| dv 2 π , √ ck pk k + 1 R2 k+1 |γ | T
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giving pk+n ck+n =
c |γ |
n pk c k ,
n≥1.
(17)
(n)
From (16) and (17) it follows that µ(−h (A)) ≤ σk (¯z)ε 4k+2 (const. · h)n . The asserted bound (15) for µ(−h (A)) is proved. The main difference between the proofs of Propositions 3.5 and 4.1 is that Hamiltonian models have both geometry and memory. In preparation for the proof, we introduce the following language. Let Aε be as in Lemma 4.2. For = 1, . . . , k, we let X denote the projection of Aε onto the plane of its x -coordinate, and V the projection of Aε onto the plane of its v -coordinate (so that X and V are ε-squares in and R2 respectively). We assume for simplicity that for each , either X is a strictly positive distance from γL and γR , in which case we say X is in the interior, or one of its sides is contained in γL or γR . In the latter case, we say X is adjacent to an exit. We further assume that if X is adjacent to an exit, then either all v ∈ V point toward the exit or away from it. Proof of Proposition 4.1. The invariance of µ is already noted in Sect. 4.1.1 except where it pertains to entrances and exits of tracers. We focus therefore on these events, noting that the probability of more than one tracer entering on the time interval (0, h) is o(h), as is the probability of a tracer entering and leaving (immediately) on this time interval. These scenarios will be ignored. d Let Aε be as above. We seek to show as before that dh IAε (z )P h (dz |z)µ(dz)|h=0 = 0. Here it is necessary to treat separately the following configurations for Aε : Case 1. The following holds for all : X can be in the interior or adjacent to an exit, and if it is adjacent to an exit, then all v in V must point toward the exit. Notice that this configuration is relatively inaccessible, meaning the probability of a new tracer entering on (0, h) leading to a state in Aε is o(h)µ(Aε ). Notice also that this configuration has (0) the property h (−h (Aε )) = Aε , so that the contribution of the no-new-tracers event
h
to IAε (z )P (dz |z)µ(dz) is, by Lemma 4.2,
1 (1 − h)2 σk (¯z)ε 4k+2 pk ck + 2h|γ | pk+1 ck+1 + o(h) c 1 = σk (¯z)ε 4k+2 pk ck (1 − 2h) + 2h|γ | pk+1 ck+1 + o(h) c = σk (¯z)ε 4k+2 (pk ck + o(h)) ,
(18)
the last equality being valid on account of Eq. (17). Case 2. X1 is adjacent to an exit and v1 points away from it; X and V for > 1 are as in Case 1. In this configuration, there is a part of X1 that can only be reached in time h if one starts from outside. This region is a parallelogram similar to that in the proof of Lemma 4.2 but with one of its sides equal to X1 ∩ γL or X1 ∩ γR . Following in Case 1, we obtain that the contribution of the no-new-tracers event to the estimates IAε (z )P h (dz |z)µ(dz) in this case is h σk (¯z)ε 4k+2 pk ck 1 − |v¯1 || sin(ϕ¯1 )| + o(h) , (19) ε
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where v¯1 is the v1 coordinate of z¯ and ϕ¯1 is the angle v¯1 makes with γL (or γR ). We now argue that the negative term above is balanced by the contribution of the event in which a new tracer enters on the time interval (0, h). This new tracer must have v1 ∈ V1 and must enter through the ε-segment X1 ∩ γL or X1 ∩ γR . We claim that the probability of this event is pk−1 ck−1 σk (¯z)eβ|v¯1 | ε 4k−2 · h 2
ε 2 · c| sin(ϕ¯1 )||v¯1 |e−β|v¯1 | ε 2 . |γ |
(20)
The first factor in (20) is the µ-measure of the states corresponding to those in Aε but without the tracer with position and velocity (x1 , v1 ); the second factor is the probability of a tracer entering through the designated segment, and the third is the fraction of tracers entering with velocity ∈ V1 (see (14)). That (19) and (20) add up to µ(Aε )(1 + o(h)) again follows from (17). Case 3. X1 and X2 are adjacent to exits, v1 and v2 point away from the exits in question, and X and V are as in Case 1 for > 2. We assume for simplicity that either (X1 × V1 ) ∩ (X2 × V2 ) = ∅ or X1 × V1 = X2 × V2 . In the case (X1 × V1 ) ∩ (X2 × V2 ) = ∅, the contribution of the no-new-tracers event is h h (21) σk (¯z)ε 4k+2 pk ck 1 − |v¯1 || sin(ϕ¯1 )| − |v¯2 || sin(ϕ¯2 )| + o(h) , ε ε and this is cancelled perfectly by the estimate corresponding to (20). ¯ k , where tracer positions and velocities are In the case X1 × V1 = X2 × V2 , on regarded as ordered k-tuples, the set of states where both (x1 , v1 ) and (x2 , v2 ) are not reachable in time h is o(h), and the set where exactly one of these is not reachable is the union of two sets that project to the same set under πk . Thus the estimates for both cases are as in Case 2. The remaining cases are handled similarly. Proposition 4.3. For the N -chain defined in Sect. 4.1.2 with TL = TR = T and L = R = , the N-fold product µ × · · · × µ is invariant. It suffices to check that the transfer of energy from one cell to the next leads to the correct relation between pk ck and pk+1 ck+1 . The proof is left to the reader. 4.3. Derivation of equations of macroscopic profiles. Having found the candidate family of Gibbs measures {µT , }, we now proceed as in Sect. 3.3, seeking to derive the relevant macroscopic profiles under Assumptions 1 and 2; see Sect. 3.3. There are two new problems, leading to two additional assumptions which we now discuss. The first problem is that of uniqueness and ergodicity. Unlike their stochastic counterparts, the Hamiltonian chains defined in Sect. 4.1 may not be ergodic; they are, in fact, easily shown to be nonergodic for certain choices of 0 . Without ergodicity, it is not clear how to make sense of the notion of local temperature, which lies at the heart of Assumption 2. Postponing a discussion to Section 4.4, we bypass this issue by introducing Assumption 1. We assume µN is the unique invariant probability measure for the N -chain defined in Sect. 4.1. It follows that µN is ergodic.
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Another important departure from the stochastic case is that in Hamiltonian models, local rules are purely dynamical: whether a tracer goes to the left or to the right when it exits a cell is determined entirely by local conditions at the time. In the presence of a nonzero temperature gradient, exit distributions are typically asymmetric in the finite chain, and may depend on specific characteristics of the model in question (see below). We first state a general result giving the relation among the various quantities of interest. Let jN,i and QN,i denote respectively the mean number of exits and mean total energy transported out of the i th cell per unit time in the N -chain. Assumption 2. We assume that as N → ∞, the profiles jN,i and QN,i converge in the C 0 sense to functions j (ξ ) and Q(ξ ) on (0, 1). Theorem 4.4. Under Assumptions 1, 1’, 2 and 3, the following hold for the models in Sect. 4.1. • mean stored energy at a site : s(ξ ) = • mean tracer energy : e(ξ ) = 2s(ξ ) ; • mean number of tracers : λ2 () κ(ξ ) = |γ |
1 Q(ξ ) ; 3 j (ξ )
π j (ξ ), 2s(ξ )
where |γ | = |γL | = |γR | is the size of the passage between adjacent cells ; • mean total-cell energy : E(ξ ) = s(ξ ) + κ(ξ )e(ξ ) = s(ξ ) +
λ2 () 2π s(ξ ) j (ξ ) . |γ |
Proof. The proof follows that of Theorem 3.9, except that all quantities here are expressed in terms of the two functions j and Q (which vary from model to model). First we read off the pertinent information from Proposition 4.1 for a single cell connected to two heat baths with parameters T and : (i) stored energy has density
√ √ β e−βy and mean s πy βe−βx and mean T ;3
(ii) tracer energy has density (iii) mean number of tracers, κ =
λ2 () √ √ π 2 |γ | T T (κ + 21 ) ;
=
T 2
;
;
(iv) mean total-cell energy, E = (v) mean number of jumps out of cell per unit time, j = 2 ; (vi) mean total energy transported out of cell per unit time, Q =
3T 2
· j = 3T .
To prove (i), for example, we condition on the event that exactly k tracers are present. 2 Integrating out all other variables, we obtain that the distribution of ω is const.e−βω . Thus the distribution of s = ω2 is as claimed. Items (ii) – (iv) are proved similarly, and (v) and (vi) are deduced from the fact that the cell is in equilibrium with the two baths. To deduce the asserted profiles, fix ξ ∈ (0, 1), and consider the [ξ N ]th cell in the N-chain. By Assumption 2, µN,[ξ N] → µT (ξ ),(ξ ) for some T (ξ ) and (ξ ). Moreover, 3
Note that this is the energy density when the tracers are in the box, to be distinguished from (vi).
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with respect to this limiting distribution, the number of jumps per unit time out of the cell is j (ξ ), and the total energy transported out of the cell is Q(ξ ). We then use the single-cell information above combined with these values of j (ξ ) and Q(ξ ) to identify T (ξ ) and (ξ ). The formula for s is obtained as follows: T = 23 Q/j is from (v) and (vi), and s = 21 T is from (i). Of particular interest to us are models in which there is good mixing within individual cells. In an idealized model in which mixing within individual cells is perfect and instantaneous, exits to the left and the right would be equally likely, as would be the case for mean energy flow. With such a perfect left-right symmetry at each site, j and Q would be linear as explained in the proof of Theorem 3.9. For the class of models described in Sect. 4.1, this idealized state is never attained, but we have found that exit distributions come very close to being symmetric under certain conditions: The most important of these conditions are (i) a geometry of 0 that gives rise to fast mixing for the closed dynamical system (such as concave walls and the absence of “traps”), and (ii) small passageways between adjacent cells (so most tracers stay in the cell for a long time). The presence of large numbers of tracers is also conducive to good mixing. 4 Corollary 4.5. In the setting of Theorem 4.4, if j and Q have approximately linear profiles with j (0) = L , j (1) = R , Q(0) = TL L and Q(1) = TR R , then the profile for mean stored energy is given by s(ξ ) ≈
1 L TL + (R TR − L TL )ξ . 2 L + (R − L )ξ
Other approximate profiles are obtained similarly by substituting j (ξ ) ≈ 2 (L + (R − L )ξ ) into the formulas in Theorem 4.4. Numerical simulations validate these predictions for Hamiltonian chains with small passageways between cells. See Sect. 4.5. Our findings suggest, in fact, C 2 convergences to j and Q. More precisely, let jN,i = jN,i,L + jN,i,R where jN,i,L and jN,i,R are the numbers of exits per unit time that go to the (i − 1)st and (i + 1)st cells respectively. Analogously, let QN,i = QN,i,L + QN,i,R . Then for each compact set of cell configurations (0 , γL , γR ) and parameters TL , TR , L , R > 0, there exists an α ≥ 0 such that for all large N, the following hold for all i: 1 |jN,i,R − jN,i | ≤ Nα ; |QN,i,R − 21 QN,i | ≤ Nα ; 2 1 |(jN,i,R − jN,i ) −(jN,i+1,R − 21 jN,i+1 )| ≤ Nα2 ; 2 1 |(QN,i,R − QN,i ) −(QN,i+1,R − 21 QN,i+1 )| ≤ Nα2 . 2 The situation in Corollary 4.5 corresponds to the case α 1. 4 It is important to distinguish between the following two levels of mixing: mixing within cells, and mixing in the chain. For example, small passageways between cells enhance mixing of the first kind but are obstructions to the latter.
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Remark 4.6. The bounds above are consistent with the following observations: For a cell in the N -chain, the temperature difference between the cell on its left and the one on its right is of order |TL − TR |/N , so one expects the marginal of µN at this site to deviate from the equilibrium measure in Sect 4.2 by the same order of magnitude. This deviation is in turn reflected in the differences |jN,i,R − jN,i,L | and |QN,i,R − QN,i,L |. Similarly, if the second differences are well behaved as we assume, their orders of magnitude as indicated above are dimensionally correct. Detailed dependencies of this asymmetry on the physical parameters are beyond the scope of this paper.5 The discussion of results in Sect. 3.5 (with “linearity” replaced by “approximate linearity”) applies to models satisfying the hypotheses in Corollary 4.5. Statements not involving linearity of j and Q apply to the broader setting of Theorem 4.4. 4.3.1. Comparisons of models. 1. Predicted profiles for Hamiltonian and stochastic models. We observe that the predicted formulas in Theorem 4.4 are of the same type as their counterparts in Theorem 3.9 but the constants are different. The similarity stems from the fact that they are derived from the same general principles. The differences in constants reflect the differences in µT , , which in turn reflect the differences in local rules (see below). 2. Relation between s and e. To highlight the role of the local rules in the profiles studied in this paper, we recall the relation between stored energy s and individual tracer energy in the various models encountered: (a) Random halves (Sects. 3.1–3.3): s = 2e. (At collisions, energy is split evenly on average, but the expected time for the next clock is longer for slower tracers.) (b) Stochastic models simulating Hamiltonian systems where both disk and tracer have a single degree of freedom (Sect. 3.6): s = e. (c) Hamiltonian models in which the disk has one degree of freedom and tracers have two (Sect. 4.1): e = 2s. To this list, we now add one more example, namely (d) Hamiltonian models in which the disk has one degree of freedom and tracers have 3: Consider the model described in Sect. 4.1, but with 0 ⊂ R3 and the disk replaced by a cylinder that rotates along a fixed axis. Here, Liouville measure for a closed system with k tracers is m ¯ k = (λ3 | )k × (ν1 |∂D ) × λ3k+1 (cf. Sect. 4.1.1). From a single-cell analysis similar to that in Sect. 4.2, µT , is easily √ computed. One notes in particular that the distribution of tracer energy is const. xe−βx , while disk energy is as before. A simple computation then gives e = 3s. These examples demonstrate clearly that the relation between s and e is entirely a function of the local structure. In the case of Hamiltonian systems, we see that it is also dimension-dependent.
4.4. Ergodicity issues. Questions of ergodicity for the chains in Theorem 4.4 are beyond the scope of this paper. We include only brief discussions of the following three aspects of the problem: 1. Randomness in the injection process. Among the various features of our models, the one the most responsible for promoting ergodicity is the randomness with which new 5
We thank H. Spohn for interesting correspondence on this point.
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tracers are injected into the system. We observe, however, that this genuinely stochastic behavior occurs only at the two ends of the chain, and even there, the transition probabilities do not have densities with respect to the underlying Lebesgue measure. The problem is thus one of controllability involving the deterministic part of the dynamics. 2. Hyperbolicity of billiard dynamics: a necessary condition. Let N ⊂ R2 denote the playground for the tracers in the N-chain. That is to say, it is the union of N copies of arranged in the configuration shown in Fig. 2 with open passages between adjacent copies of . The presence of one of more tracers being trapped in N without contact (1) (N) with any of the turning disks or the openings at the two ends (i.e., γL and γR ) is clearly an obstruction to ergodicity. This scenario is easily ruled out by choosing 0 to have concave (or scattering) walls. Such a choice of 0 implies that ∂N also has concave boundaries, and the free motion of a particle in a domain with concave boundaries is well known to be hyperbolic and ergodic [21, 16]. We do not know if the absence of trapped tracers in the sense above implies ergodicity. 3. Enhancing ergodicity. Without (formally) guaranteeing ergodicity, various measures can be taken to “enhance” it, meaning to make the system appear for practical purposes as close to being ergodic as one wishes. For example, one can introduce more scattering within each cell by increasing the curvature of the walls of 0 , or alternately, one could add convex bodies inside 0 that play the role of Lorentz scatterers. Another possibility is to add a small amount of noise, and a third is to increase the injection rates: physical intuition says that the larger the number of tracers in the system, the more likely stored energy will behave ergodically. 4.5. Results of simulations. To check the applicability of the theory proposed in Sects. 4.1–4.3 to real and finite systems, we have done extensive simulations some of which we describe in this subsection. The domain 0 used in our simulations is as shown in Fig. 2. Actual specifications of 0 are as follows: We start with a square of sides 2, subtracting from it first 4 disks of radius 1.15 centered at the 4 corners of the square. Two openings corresponding to γL and γR are then created on the left and right; each has length 0.02. This completes the definition of 0 . The disk D is located at the center of the square; it has radius r = 0.0793. Our choice of domain was influenced by the following factors: First, ∂0 is taken to be piecewise concave to promote ergodicity. Second is the size of the disk: A disk that is too small is hit by a tracer only rarely; many tracers may pass through the cell without interacting with the disk (this is analogous to having a large δ in Sect. 3.1). A disk that is too large (relative to the domain in which it can fit) may cause an unduly large fraction of tracers entering the box to exit immediately from the same side. Both scenarios lead to large time-correlations, which are well known to impede the speed of convergence to µN in a finite chain. They may also affect the infinite-volume limit. We have found the geometry and specifications above to work quite well, with a tracer making, on average, about 71 collisions while in a cell. Of these collisions, about 12.5 are with the disk. For the single cell (with the geometry above) plugged to two identical heat baths, we have tested the system extensively for ergodicity. To the degree that one can ascertain from simulations, there is an ergodic component covering nearly 100% of the phase space. The various energy distributions as well as the Poisson distribution of the number of tracers present agree perfectly with those predicted by Proposition 4.1. Simulations for chains of 20 to 60 cells with the choice of r and |γ | above showed very good agreement with the theory. A sample of the fits for Q(ξ ), s(ξ ) and E(ξ ) for
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Fig. 3. Rotating disks model with chain of 30 cells, temperatures TL = 100, TR = 10, and injection rates L = 1, R = 2. Top left: Qi , energy transported out of site i per unit time as a function of i. Top right: Mean disk energy si . Bottom: Mean total energy Ei
9 · 109 events and 30 sites is shown in Fig. 3. Here the ejection rates to the left and right are very close to 50/50. We have also investigated the quantity α toward the end of Sect. 4.3 for various values of r and |γ |, up to r = 0.23 (which is quite close to the maximum-size disk that can be fitted into the domain 0 ) and |γ | = 0.06. Our findings are consistent with the discussion in Sect. 4.3. In addition to these profiles, we have also verified directly Assumption 2, which asserts that the distributions of energy and tracers within each cell are in accordance with those given by µT , for some T , depending on the cell. A sample of these results is shown in Fig. 4. 4.6. Related models. In this subsection we recall from the literature a few models that in their original or slightly modified form can be regarded as approximate realizations
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Fig. 4. Same parameters as in Fig. 3. Top 2 figures show semi-log plots of tracer energy distributions at various sites. Top left: Densities of tracer energies inside boxes (theory predicts βe−βx ). Top right: √ Densities of tracer energies upon exiting the various boxes (theory predicts 2β 3/2 x/π e−βx ). Bottom: Distribution of numbers of tracers at several sites (theory predicts Poisson distribution)
of the class described in Sect. 2.1 of this paper. For more complete accounts, see the review papers [2, 13, 15]. The models which come closest to ours, and which to some degree inspired this work, are those in [17, 12]. In these papers, the authors carried out a numerical study of a system comprised of an array of disks similar to those in Sect. 4.1 but arranged in two rows with periodic boundaries (in the vertical direction). These disks interact via tracers following the rules first used in [20]. We have adopted the same local rules, but have elected to arrange our disks in a single row to simplify the analysis. There is a number of papers dealing with mechanical gadgets that on some level appear similar to ours. For example, in [14, 9], vertical plates are pushed back and forth
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by particles trapped between them. The main difference between these models and ours is that they have exactly one “tracer” in each “cell”. In this respect, these models are closer to our earlier work [6] in which locked-in tracers were considered. Ding-a-ling and ding-dong models belong essentially to the same class [3, 19, 8, 18]. We mention that nonlinearities of profiles are difficult to see when the temperature differences at the two ends are relatively small (in fact, what counts in many cases, including the models studied in this paper, is the ratio of temperatures at the two ends). This may explain why some authors have reported linear profiles when our analysis suggests that may not be the case. We mention also a very well-studied situation, namely that of the Fermi-Pasta-Ulam chain. In this model, and in many others, there is a potential of the form U (xi − xi+1 ) + V (xi ) , with U and V functions that grow to ∞ and xi the coordinates of a chain of anharmonic oscillators. The pinning potential V plays the role of the “tank” in our models, while the interparticle potential is more akin to the role of the tracers. This class of models is difficult to handle because in contrast to the basic setup in our study, there is no clear separation of the pinning and interaction energies. Finally, we mention that Hamiltonian systems with noise have been studied in the context of the Fourier Law. See e.g. [1]. Acknowledgements. The authors thank O. Lanford for helpful discussions. JPE acknowledges the Courant Institute, and LSY the University of Geneva, for their hospitality.
References 1. Bernardin, C., Olla, S.: Fourier’s law for a microscopic model of heat conduction. http://www.ceremade.dauphine.fr/ olla/heatss-2.pdf, 2005, To appear in J. Stat. Phys 2. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Mathematical physics 2000, London: Imp. Coll. Press, 2000, pp. 128–150 3. Casati, G., Ford, J., Vivaldi, F., Visscher W.: One-dimensional classical many-body system having a normal thermal conduction. Phys. Rev. Lett. 52, 1861–1864 (1984) 4. De Groot, S., Mazur P.: Non-Equilibrium Thermodynamics, Amsterdam: North Holland, 1962 5. De Masi, A., Presutti, E.: Mathematical methods for hydrodynamic limits, Vol. 1501 Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1991 6. Eckmann, J.-P., Young, L.-S.: Temperature profiles in Hamiltonian heat conduction. Europhysics Letters 68, 790–796 (2004) 7. Eyink, G., Lebowitz, J.L., Spohn, H.: Hydrodynamics of stationary nonequilibrium states for some stochastic lattice gas models. Commun. Math. Phys. 132, 253–283 (1990) 8. Garrido, P., Hurtado, P., Nadrowski, B.: Simple one-dimensional model of heat conduction which obeys fourier’s law. Phys. Rev. Lett. 86, 5486–5489 (2001) 9. Gruber, C., Lesne, A.: Hamiltonian model of heat conductivity and Fourier law. Physica A, 351, 358–372 (2005) 10. Kipnis, C., Landim, C.: Scaling limits of interacting particle systems, Vol. 320 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Berlin: Springer-Verlag, 1999 11. Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65–74 (1982) 12. Larralde, H., Leyvraz, F., Mej´ıa-Monasterio, C.: Transport properties of a modified Lorentz gas. J. Stat. Phys. 113, 197–231 (2003) 13. Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003) 14. Li, B., Casati, G., Wang, J., Prosen, T.: Fourier law in the alternate mass hard-core potential chain. Phys. Rev. Lett. 92, 254–301 (2004)
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15. Li, B., Casati, G., Wang, J., Prosen, T.: Fourier law in the alternate mass hard-core potential chain. Phys. Rev. Lett. 92, 254–301 (2004) 16. Liverani, C., Wojtkowski, M.P.: Ergodicity in Hamiltonian systems. In: Dynamics reported, Vol. 4 of Dynam. Report. Expositions Dynam. Systems (N.S.), Berlin: Springer, 1995, pp. 130–202 17. Mej´ıa-Monasterio, C., Larralde, H., Leyvraz, F.: Coupled normal heat and matter transport in a simple model system. Phys. Rev. Lett. 86, 5417–5420 (2001) 18. Posch, H.A.: Hoover, W.G.: Heat conduction in one-dimensional chains and nonequilibrium Lyapunov spctrum. Phys. Rev. E 58, 4344–4350 (1998) 19. Prosen, T., Robnik, M.: Energy transport and detailed verification of fourier heat law in a chain of colliding harmonic oscillators. J. Physics. A 25, 3449–3478 (1992) 20. Rateitschak, K., Klages, R., Nicolis, G.: Thermostating by deterministic scattering: the periodic Lorentz gas. J. Stat. Phys. 99, 1339–1364 (2000) 21. Sinaˇı, J.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25, 141–192 (1970) 22. Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics, Heidelberg: Springer-Verlag, 1991 Communicated by G. Gallavotti
Commun. Math. Phys. 262, 269–297 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1489-0
Communications in
Mathematical Physics
Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs J´ozef Dodziuk1 , Varghese Mathai2 , Stuart Yates2,3 1
Ph.D. Program in Mathematics, Graduate Center of CUNY, New York, NY 10016, USA. E-mail: [email protected] 2 Department of Mathematics, University of Adelaide, Adelaide 5005, Australia E-mail: [email protected]., [email protected] 3 Max Planck Institut f¨ ur Mathematik, Bonn, Germany Received: 4 November 2003 / Accepted: 17 October 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: Let Q denote the field of algebraic numbers in C. A discrete group G is said to have the σ -multiplier algebraic eigenvalue property, if for every matrix A ∈ Md (Q(G, σ )), regarded as an operator on l 2 (G)d , the eigenvalues of A are algebraic numbers, where σ∈Z 2 (G, U(Q)) is an algebraic multiplier, and U(Q) denotes the unitary elements of Q. Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ . In the special case when σ is rational (σ n =1 for some positive integer n) this property holds for a larger class of groups K containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators. 1. Introduction This paper is concerned with number theoretic properties of eigenvalues of self adjoint matrix operators that are associated with weight functions on a graph equipped with a free action of a discrete group. These operators form generalizations of the Harper operator and the discrete magnetic Laplacian (DML) on such graphs, as defined by Sunada in [23]. The Harper operator and DML over the Cayley graph of Z2 arise as the Hamiltonian in discrete models of the behaviour of free electrons in the presence of a magnetic field, where the strength of the magnetic field is encoded in the weight function. When the weight function is trivial, the Harper operator and the DML reduce to the Random Walk operator and the discrete Laplacian respectively. The DML is in particular the Hamiltonian in a discrete model of the integer quantum Hall effect (see for example [3]); when the graph is the Cayley graph of a cocompact Fuchsian group, the DML becomes the
The second and third authors acknowledge support from the Australian Research Council.
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Hamiltonian in a discrete model of the fractional quantum Hall effect ([6, 7] and [17]). It has also been studied in the context of noncommutative Bloch theory ([16, 21].) The Harper operator and DML can be thought of as particular examples of weighted sums of twisted right translations by elements of the group; alternatively, they can be regarded as matrices over the group algebra twisted by a 2-cocycle, acting by (twisted) left multiplication. As such, in Sect. 6 we generalize results of [9] to demonstrate that such operators associated with algebraic weight functions have only algebraic eigenvalues whenever the group is in a class of groups containing all free groups, finitely generated amenable groups and fundamental groups of closed Riemann surfaces. When the multiplier associated to the weight function is rational, the algebraicity of eigenvalues extends to groups in a larger class K, defined in Sect. 8. The class K contains all free groups, discrete amenable groups, and groups in the Linnell class C; in particular it includes cocompact Fuchsian groups and many other non-amenable groups. The algebraic eigenvalue properties derived in this paper can be summarized in the following theorem. Theorem 1.1 (Corollary 4.5, Theorems 6.1, 6.2, 6.3). Let σ be an algebraic multiplier for the discrete group G, and let A ∈ Md (Q(G, σ )) be an operator acting on l 2 (G)d by left multiplication twisted by σ , where Q denotes the field of algebraic numbers. Alter natively, consider A to be a finite sum of magnetic translation operators g∈G wg Rgσ , where wg ∈ Md (Q). Then A has only algebraic eigenvalues whenever (1) G is finitely generated amenable, free or a surface group, or (2) σ is rational (σ n =1 for some positive integer n) and G ∈ K, where K is a class of groups containing free groups and amenable groups, and is closed under taking directed unions and extensions with amenable quotients. The case when σ is a rational multiplier is established by relating the spectrum of these operators to the spectrum of untwisted operators on a finite covering graph. The property follows from the class K having the (untwisted) algebraic eigenvalue property (established in Sect. 6, following [9]), and from the fact that K is closed under taking extensions with cyclic kernel, as demonstrated in Sect. 8. We show in Sect. 4 that these operators with rational weight function have no eigenvalues that are Liouville transcendental whenever the group is residually finite or more generally in a certain large class of groups Gˆ containing K. We also show that there is an upper bound for the number of eigenvalues whenever the group satisfies the Atiyah conjecture. However, the case when σ is an algebraic multiplier is established in a significantly different manner: in addition to an approximation argument that parallels that of [9], one also has to use new arguments that rely upon the geometry of closed Riemann surfaces. ˙ [13], We also wish to highlight the remarkable computation of Grigorchuk and Zuk that is recalled in Theorem 5.5. The computation explicitly lists the dense set of eigenvalues of the Random Walk operator on the Cayley graph of the lamplighter group—all of these eigenvalues are algebraic numbers, as predicted by results in this paper and in [18], since the lamplighter group is an amenable group. Section 7 establishes an equality between the von Neumann spectral density function of A ∈ Md (C(G, σ )) for an arbitrary multiplier σ , and the integrated density of states of A with respect to a generalized Følner exhaustion of G, whenever G is a finitely generated amenable group, or a surface group.
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2. Magnetic Translations and the Twisted Group Algebra The Harper operator is an example of an operator that can be described as a sum of magnetic right translations. In this section we will offer a brief description of the magnetic translation operators, and observe that finite weighted sums of these magnetic translations are unitarily equivalent to left multiplication by matrices over a twisted group algebra. Let G be a discrete group and σ be a multiplier, that is σ ∈ Z 2 (; U (1)) is a normalized U (1)-valued cocycle, which is a map from × to U (1) satisfying σ (b, c)σ (a, bc) = σ (ab, c)σ (a, b) ∀a, b, c ∈ G, σ (1, g) = σ (g, 1) = 1 ∀g ∈ G.
(1) (2)
Consider the Hilbert space l 2 (G)d of square-integrable Cd -valued functions on G, where l 2 (G) = {h : G → C : g∈G |h(g)|2 < ∞}. The right magnetic translations are then defined by (Rgσ f )(x) = f (xg)σ (x, g). Obviously, if σ is a multiplier, so is σ . The right magnetic translations commute with left magnetic translations Lσg as follows from (1): Rgσ Lσh = Lσh Rgσ
∀g, h ∈ G,
(3)
where (Lσh f )(x) = f (h−1 x)σ (h, h−1 x). Consider now self-adjoint operators on l 2 (G)d of the form A= A(g)Rgσ ,
(4)
g∈S
where A(g) is a d × d complex matrix for each g, and S is a finite subset of G which is symmetric, i.e., S = S −1 . The self-adjointness condition is equivalent to demanding that the weights A(g) satisfy A(g)∗ = A(g −1 )σ (g, g −1 ). These operators include as a special case the Harper operator and the DML on the Cayley graph of G, where S is the symmetric generating set and A(g) is identically 1 for g in S. For the Harper operator of Sunada [23] on a graph with finite fundamental domain under the free action of the group G, one can construct an operator of the form (4) which is unitarily equivalent to the Harper operator, after identifying scalar valued functions on the graph with Cn -valued functions on the group, where n is the size of the fundamental domain. Note that the operators of the form (4) are given explicitly by the formula (Af )(x) = A(x −1 g)σ (x, x −1 g)f (g). g∈xS
For a given multiplier σ taking values in U(K) = K ∩ U (1) for some subfield K of the complex numbers, one can also construct the twisted group algebra K(G, σ ) and
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2 d examine d × d matrices B ∈ Md (K(G, σ )) acting on l (G) . Elements of K(G, σ ) are finite sums ag g, ag ∈ K with multiplication given by
ag g · bg g = ag bh σ (g, h)k. gh=k
The action of B on a Cd -valued function f is then given by this multiplication, (Bf )(x) = B(xh−1 )σ (xh−1 , h)f (h), h∈Sx
where B(g) denotes the d × d matrix over K whose elements are the coefficients of g in the elements of B. A straightforward computation shows that B= B(g)Lσg , (5) g∈S
where S is a finite subset of G. The left and right twisted translations are unitarily equivalent via the map U σ , (U σ f )(x) = σ (x, x −1 )f (x −1 ), U σ Lσg = Rgσ U σ . As such, operators of the form (4) and (5) will be unitarily equivalent if the coefficient matrices satisfy A(g) = B(g) for all g ∈ G. Hereafter we will therefore concentrate on the latter picture, noting that the results apply equally well to the case of operators described as weighted sums of right magnetic translations. By virtue of (3), the operators B of the form (5) belong to the commutant of the set of magnetic translations {Rgσ | g ∈ G}; the weak closure of the set of operators of the form (5) is actually equal to this commutant as the theorem below shows. The theorem itself is folklore, but we were not able to find the proof in the literature. In the special case of G = Z2 , the details are spelt out in [21]. We will give a self-contained account, adapting the proof for the case of trivial multiplier. Theorem 2.1 (Commutant theorem). The commutant of the right σ -translations on l 2 (G) is the von Neumann algebra generated by left σ¯ -translations on l 2 (G). Similarly, the commutant of the left σ -translations on l 2 (G) is the von Neumann algebra generated by right σ¯ -translations on l 2 (G). Proof. We present a proof of the second statement: the proof of the first statement is analogous. Let WL,σ be the von Neumann algebra generated by the set SL,σ = {Lσg | g ∈ G} of left σ -translations, and WR,σ¯ be the von Neumann algebra generated by the set SR,σ¯ = {Rgσ¯ | g ∈ G} of right σ¯ -translations. We proceed by showing that SL,σ = SR, σ¯ (denoting the commutant by ) and then show that SR,σ¯ = WR,σ¯ . An operator C ∈ B(l 2 (G)) is determined by its components Ca,b = (Cδb , δa ) = . In terms of components, one has that (Cδb )(a) for a, b ∈ G. Suppose C ∈ SR, σ¯ (Rgσ¯ )a,b = δb (ag)σ (a, g) = δa (bg −1 )σ (bg −1 , g), giving (CRgσ¯ )a,b = Ca,bg −1 σ (bg −1 , g)
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and (Rgσ¯ C)a,b = σ (a, g)Cag,b . C commutes with Rgσ¯ for all g, and so substituting bg for b gives C ∈ SR, σ¯ ⇒ Ca,b = σ (a, g)Cag,bg σ (b, g) ∀a, b, g ∈ G. , noting that (Lσ ) −1 −1 Similarly for D ∈ SL,σ g a,b = δb (g a)σ (g, g a) = δa (gb)σ (g, b), σ σ −1 we have (DLg )a,b = Da,gb σ (g, b) and (Lg D)a,b = σ (g, g a)Dg −1 a,b , which after substituting ga for a gives ⇒ Da,b = σ (g, a)Dga,gb σ (g, b) D ∈ SL,σ
∀a, b, g ∈ G.
Consider the product CD for C ∈ SR, σ¯ and D ∈ SL,σ . In terms of components, Ca,g Dg,b = φ(a, g −1 , b)Ca(g −1 b),b Da,(ag −1 )b (CD)a,b = g∈G
=
g∈G
φ(a, a
−1
hb
−1
, b)Da,h Ch,b ,
h∈G
where φ(a, g −1 , b) = σ (a, g −1 b)σ (g, g −1 b)σ (ag −1 , g)σ (ag −1 , b). However we can reduce the expression for φ by applying the cocycle identities to show that it is in fact identically equal to 1: φ(a, k, b) = σ (k −1 , kb) σ (ak, b)σ (a, kb) σ (ak, k −1 ) = σ (k −1 , kb)σ (k, b)σ (a, k)σ (ak, k −1 ) = σ (k −1 , k)σ (k, k −1 ) =1 ∀a, k, b ∈ G. commute So (CD)a,b = (DC)a,b for all a, b ∈ G, demonstrating that operators in SL,σ . with those in SR, σ¯ . The left σ -translations and right σ ⊂ SR, ¯ -translaThis gives the inclusion SL,σ σ¯ . Therefore and thus S ⊂ S tions commute, so we also have that SL,σ ⊂ SR, σ¯ R,σ¯ L,σ . = SR, SL,σ σ¯ A calculation shows that the adjoint of Rgσ¯ is given by (Rgσ¯ )∗ = σ (g, g −1 )Rgσ¯−1 , and so operators that commute with the right σ¯ -translations must commute with their ∗ ∗ adjoints as well. So SR, σ¯ = SR,σ¯ , writing S for the set of adjoints of elements of S. By the von Neumann double commutant theorem, the algebra generated by a set S ∗ ) = S is given by (S ∪ S ∗ ) . So WR,σ¯ = (SR,σ¯ ∪ SR,
σ¯ R,σ¯ = SL,σ . We set the notation WL∗ (G, σ ) = WL,σ = SR, σ¯ for the left twisted group von ∗ Neumann algebra and WR (G, σ ) = WR,σ = SL,σ¯ for the right twisted group von Neumann algebra. The following is a corollary of the theorem.
Corollary 2.2. The commutant of the right σ -translations on l 2 (G)d is the von Neumann algebra WL∗ (G, σ¯ ) ⊗ Md (C). Similarly, the commutant of the left σ -translations on l 2 (G)d is the von Neumann algebra WR∗ (G, σ¯ ) ⊗ Md (C).
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Theorem 2.3 (Existence of trace). There is a canonical faithful, finite and normal trace on the twisted group von Neumann algebras WL∗ (G, σ ) and WR∗ (G, σ¯ ) which is given by tr G,σ (A) = (Aδe , δe ).
(6)
This trace is weakly continuous and can also be written as tr G,σ (A) = (Aδg , δg ),
g ∈ G.
(7)
Proof. It is clear that tr G,σ is linear, finite and weakly continuous (hence normal). Now if A ∈ WL∗ (G, σ ), then (Aδg , δg ) = Ag,g = σ (g, h)Agh,gh σ (g, h) = Agh,gh = (Aδgh , δgh )
(8)
for all h ∈ G. In particular, every diagonal entry of the matrix of A is equal to tr G,σ (A). If A is a self-adjoint operator in WL∗ (G, σ ) such that tr G,σ (A) = 0, then (Aδg , δg ) = 0 for all g ∈ G. But then due to the Cauchy-Schwarz inequality |(Af1 , f2 )|2 ≤ (Af1 , f1 )(Af2 , f2 ), f1 , f2 ∈ l 2 (G), we deduce that (Aδg , δh ) = 0 for all g, h ∈ G, which implies that A = 0. Therefore tr G,σ is faithful. It remains to prove that tr G,σ is a trace. That is, tr G,σ (AB) = tr G,σ (BA),
A, B ∈ WL∗ (G, σ ).
(9)
Since tr G,σ is linear and weakly continuous it is sufficient to consider the case when A = Lσg and B = Lσh for all g, h ∈ G. We compute, tr G,σ (Lσg Lσh ) = (Lσg Lσh δe , δe ) = σ (g, h)(Lσgh δe , δe ) = σ (g, h)(δgh , δe ) σ (g, h) if gh = e, = 0 otherwise. Similarly, tr G,σ (Lσh Lσg )
=
σ (h, g) if hg = e, 0 otherwise.
By the cocycle identity (1) with a = h−1 , b = h and c = h−1 , we see that σ (h, h−1 ) = σ (h−1 , h)
∀h ∈ G.
Therefore tr G,σ (Lσg Lσh ) = tr G,σ (Lσh Lσg ) for all g, h ∈ G as required. The argument for WR∗ (G, σ¯ ) is identical.
Now the matrix algebra Md (C) has the canonical trace Tr given by the sum of the diagonal coefficients of a matrix. Then the tensor product tr G,σ ⊗ Tr is a trace on WL∗ (G, σ )⊗Md (C) and on WR∗ (G, σ¯ )⊗Md (C) which we denote by tr G,σ for simplicity.
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Corollary 2.4. There is a canonical faithful, finite and normal trace on the twisted group von Neumann algebras WL∗ (G, σ )⊗Md (C) and on WR∗ (G, σ¯ )⊗Md (C) which is given by tr G,σ (A) =
d
(Aj,j δe , δe ).
(10)
j =1
This trace is weakly continuous and can also be written as tr G,σ (A) =
d
(Aj,j δg , δg ),
g ∈ G.
(11)
j =1
In the corollary above, we interpret the elements of the tensor product WL∗ (G, σ ) ⊗ Md (C) = Md (WL∗ (G, σ )) as d × d matrices with entries in WL∗ (G, σ ), etc. Suppose that A is a von Neumann algebra of algebras of operators acting on a Hilbert space H. A subspace U of H is termed affiliated if the corresponding orthogonal projection PU onto the closure of U belongs to A. A necessary and sufficient condition for affiliation is that the subspace be invariant under the action of operators in the commutant A of A. We will write U ηA to indicate that the subspace U is affiliated to the algebra A. Given a trace τ on A, the von Neumann dimension dimτ of an affiliated subspace is defined to be the trace of PU . We will use the following properties of the von Neumann dimension (see for example [22], Sect. 2.6 Lemma 2, Sect. 2.26). Lemma 2.5. Let H be a Hilbert space, A a von Neumann algebra of operators on H with (normal, faithful and semi-finite) trace τ and von Neumann dimension dimτ , and let L, N ⊂ H be affiliated subspaces. Then, (1) dimτ L = 0 implies L = {0}, (2) L ⊆ N implies dimτ L ≤ dimτ N, (3) if A ∈ A is an almost isomorphism of L and N , that is, kerA ∩ L = {0} and the set A(L) is dense in N , then dimτ L = dimτ N. The following is an immediate consequence. Lemma 2.6. Let H, A, τ be as in Lemma 2.5. If L is an affiliated subspace of H with corresponding projection PL , then dimτ L = dimτ ker A|L + dimτ im A|L = dimτ (ker A ∩ L) + dimτ im APL . Proof. This follows by noting that A gives an almost isomorphism from the orthogonal complement of its kernel in L to the closure of its image on L.
Hereafter we will use dimG,σ to refer to the von Neumann dimension associated with the trace tr G,σ on the algebra WL∗ (G, σ ) ⊗ Md (C). In the case that σ is trivial, this algebra becomes the von Neumann algebra of G-equivariant operators B(l 2 (G)d )G , and we refer to the trace and dimension by tr G and dimG respectively. Two multipliers σ and σ in Z 2 (G, U(K)) are cohomologous, written σ ∼ σ , if they belong to the same cohomology class in H 2 (G, U(K)). It follows that σ ∼ σ if and only if there exists a map s: G → U(K) such that σ (g, f ) = s(g)s(h)s(gh)σ (g, h)
∀g, h ∈ G.
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The map s gives rise to a unitary equivalence between operators in Md (K(G, σ )) and Md (K(G, σ )).
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Lemma 2.7. Let σ and σ be cohomologous multipliers in Z 2 (G, U(K)). Then for every A in Md (K(G, σ )) acting on l 2 (G)d there is a canonically determined A in Md (K(G, σ )) such that A and A are unitarily equivalent. Proof. Let s : G → U(K) be the map as in (12), such that σ (g, h) = s(g)s(h)s(gh)σ (g, h) for all x, y ∈ G. Writing A(g) ∈ Md (K) for the matrix of coefficients of g in A, as in (5), one has (Af )(g) = A(gh−1 )f (h)σ (gh−1 , h) h∈G
=
A(gh−1 )f (h)s(gh−1 )s(h)s(g)−1 σ (gh−1 , h).
h∈G
Let S be the unitary operator on l 2 (G)d given by multiplication by s: Sf (g) = s(g)f (g). Then letting A (g) = s(g)A(g), one has (SAf )(g) = A (gh−1 )f (h)s(h)σ (gh−1 , h) h∈G
= (A Sf )(g) for all g ∈ G, f ∈ l 2 (G)d . That is, A and A are unitarily equivalent.
It is sometimes convenient to consider only the case when the multiplier satisfies σ (g, g −1 ) = 1 for all g in G. The following lemma shows that there is such a multiplier in every cohomology class when the subfield K is algebraically closed. Lemma 2.8. Suppose K is an algebraically closed subfield of C. Then any multiplier σ ∈ Z 2 (G, U(K)) is cohomologous to a multiplier σ such that σ (g, g −1 ) = 1 for all g ∈ G. Proof. By the cocycle identity, σ (g, g −1 ) = σ (g −1 , g) for all g ∈ G. Choose s : G → U(K) such that s(g) = s(g −1 ) and s(g)2 = σ (g, g −1 ), for example by setting s(g) = eiθ/2 when σ (g, g −1 ) = eiθ , for θ ∈ [0, 2π). The image of s lies in U(K) due to K being algebraically closed. Let σ be the cohomologous multiplier given by s, according to the formula (12). Then σ (g, g −1 ) = s(g)s(g −1 )s(1)σ (g, g −1 ) = σ (g, g −1 )σ (g, g −1 ) = 1, ∀g ∈ G.
3. Algebraic Eigenvalue Property The algebraic eigenvalue property for groups was introduced in [9]. We recall the definition here, and present a class of groups K for which the algebraic eigenvalue property holds. We then define a similar property describing the eigenvalues for matrix operators over the twisted group ring, as described in Sect. 2, termed the σ -multiplier algebraic eigenvalue property. Thus recall the following definition from [9], where Q denotes the set of complex algebraic numbers.
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Definition 3.1 (4.1 of [9]). A discrete group G has the algebraic eigenvalue property, if for every d × d matrix A ∈ Md (QG) the eigenvalues of A, acting on l 2 (G)d , are algebraic numbers. Note that operators without point spectrum satisfy the criterion in the vacuous sense. The trivial group has the algebraic eigenvalue property, since the eigenvalues are the zeros of the characteristic polynomial. The same is true for every finite group. More generally, if G contains a subgroup H of finite index, and H has the algebraic eigenvalue property, then the same is true for G. And if G has the algebraic eigenvalue property and H is a subgroup of G, then H also has the algebraic eigenvalue property. In Sect. 4 of [9] it was shown that the algebraic eigenvalue property holds for all amenable groups and for all groups in Linnell’s class C, which is the smallest class of groups containing all free groups and which is closed under extensions with elementary amenable quotient and under directed unions. This motivates the definition of the class K, a larger class which contains these groups, for which the algebraic eigenvalue property can be shown to hold. Definition 3.2. The class K is the smallest class of groups containing free groups and amenable groups, which is closed under taking extensions with amenable quotient, and under taking directed unions. Remark 3.3. It is clear that the class K contains every discrete amenable group and every group in Linnell’s class C. Recall that the class of elementary amenable groups is the smallest class of groups containing all cyclic and all finite groups and which is closed under taking group extensions and directed unions. As such then K is a strictly larger class of groups than C, as it contains amenable groups which are not elementary amenable, such as the example presented by Grigorchuk in [12]. Remark 3.4. Every subgroup of infinite index in a surface group is a free group. Here is the fundamental group of a compact Riemann surface of genus g > 1. This follows from the fact that such groups are fundamental groups of an infinite cover of the base surface and from the general fact that the fundamental group of a noncompact surface is free (see [1] Chapter 1, § 7.44 and § 8.) Since we have the exact sequence 1 → F → → Z2g → 1, where F is a free group by the argument above and the free abelian group Z2g is an elementary amenable group, we deduce that the surface group belongs to the class C, and hence also to the class K. Remark 3.5. Let be a cocompact Fuchsian group, namely is a discrete subgroup of SL(2, R) such that the quotient space \SL(2, R) is compact. Then there is a torsionfree subgroup G of of finite index such that G is the fundamental group of a compact Riemann surface of genus greater than one. By Remark 3.4 above, G is in K, and since /G is a finite group, it is amenable. Therefore is also in K. Remark 3.6. Consider the modular group SL(2, Z). Then it is well known that there is a congruence subgroup (N) of finite index in SL(2, Z) that is isomorphic to a free group. We conclude by the arguments in Remarks 3.4 and 3.5, that the modular group and all of the congruence subgroups are in K.
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Theorem 3.7. Every group in K has the algebraic eigenvalue property. The proof of this theorem closely follows the argument in [9] for C, and we leave the details to Sect. 8. Remark 3.8. Results of [9] were formulated for operators of the form B=
B(g)Lg
(13)
g∈S
acting on l 2 (G)n , where S ⊂ G is a finite subset, A(g) is an n × n complex matrix and Lg denotes the untwisted left translation on l 2 (G). Since x → x −1 induces a unitary transformation on l 2 (G) that conjugates the right translation Rg with Lg , we see that (13) is unitarily equivalent to
B(h)Rh .
(14)
h∈S
It follows that all results of [9] concerning spectral properties of operators (13) apply to operators of the form (14) equally well. Suppose now we have an operator A ∈ Md (Q(G, σ )) acting on l 2 (G)d by left twisted multiplication, as described in Sect. 2, where Q(G, σ ) is the twisted group algebra over the algebraic numbers Q with multiplier σ . For a fixed σ , one can ask if any such A can have transcendental eigenvalues. Definition 3.9. A discrete group G is said to have the σ -multiplier algebraic eigenvalue property, if for every matrix A ∈ Md (Q(G, σ )), regarded as an operator on l 2 (G)d , the eigenvalues of A are algebraic numbers, where σ ∈ Z 2 (G, U(Q)) is an algebraic multiplier, and U(Q) denotes the unitary elements of the field of algebraic numbers. An immediate consequence of Lemma 2.7 is that for a given group G, the σ -multiplier algebraic eigenvalue property depends only on the cohomology class of σ . Corollary 3.10. Suppose G has the σ -multiplier algebraic eigenvalue property. Then G has the σ -multiplier algebraic eigenvalue property for any σ ∼ σ in Z 2 (G, U(Q)). Proof. Any A ∈ Md (Q(G, σ )) is unitarily equivalent to some A ∈ Md (Q(G, σ )) by Lemma 2.7, and so has only algebraic eigenvalues.
In the following Sect. 4 and 5 we investigate the situation when σ is rational, that is, when σ n = 1 for some n. In particular it is shown that every group in K has the σ -multiplier algebraic eigenvalue property when σ is rational. The case of more general algebraic multipliers is discussed in Sect. 6.
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4. Spectral Properties with Rational σ Suppose the weight function σ is rational with σ r = 1, and let Gσ be the extension of G by Zr as follows, 1 −→ Zr −→ Gσ −→ G −→ 1 (z1 , g1 ) · (z2 , g2 ) = (z1 z2 σ (g1 , g2 ), g1 g2 )
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regarding Zr as a (multiplicative) subgroup of U (1). One can then relate the spectrum of an operator A ∈ Md (K(G, σ )) acting on l 2 (G)d as in (5) to that of an associated operator A˜ on l 2 (Gσ )d . Define a map : Md (K(G, σ )) → Md (KGσ ) as follows. For A ∈ Md (K(G, σ )) with matrices of coefficients A(g) ∈ Md (K), let A˜ = (A) be given by ˜ g) = A(g) if z = 1, A(z, (16) 0 otherwise, acting on l 2 (Gσ )d by left multiplication. Consider the map ξ: l 2 (G)d → l 2 (Gσ )d given by (ξf )(z, g) = zf (g). Then
√1 ξ r
is
an isometry from to the closed subspace R of where R = {f | f (z, g) = zf (1, g) ∀(z, g) ∈ Gσ }. By (15), (1, g)−1 · (z, h) = (z σ (g, g −1 h), g −1 h) and so ˜ )(z, h) = ˜ , g)(ξf )((z , g)−1 · (z, h)) (Aξf A(z l 2 (G)d
l 2 (Gσ )d ,
(z ,g)∈Gσ
=
A(g)(ξf )(zσ (g, g −1 h), g −1 h)
g∈G
=
A(g)f (g −1 h)σ (g, g −1 h)z
g∈G
= (ξ Af )(z, h), for all (z, h) ∈ Gσ , and thus
(A)ξ = ξ A
∀A ∈ Md (K(G, σ )).
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˜ A is therefore unitarily equivalent to the restriction to the subspace R of the operator A. Lemma 4.1. Let A be a bounded linear operator on a Hilbert space H , such that im A|R ⊂ R for a closed subspace R of H . Then regarding A|R as an operator on R, specpoint A|R ⊂ specpoint A and spec A|R ⊂ spec A. Proof. Any eigenvector in R is an eigenvector in H and so the inclusion of point spectrum is immediate. Suppose λ ∈ spec A. Then λ ∈ specpoint A|R and im(A − λ)|R is dense in R. Let B be the inverse of A − λ. For u ∈ R one can find a convergent net uα → u with uα = (A − λ)uα for uα in R. Applying B gives uα → Bu, but R is closed, and so Bu is in R and u is in the image of (A − λ). Therefore (A − λ)|R has inverse B|R and
λ ∈ spec A|R . ˜ We therefore have spectral inclusions for A and A.
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Proposition 4.2. Let A ∈ Md (K(G, σ )) be a bounded linear operator on l 2 (G)d as in (5), and suppose σ is rational. Let A˜ = (A) : l 2 (Gσ )d → l 2 (Gσ )d be the associated Gσ -equivariant operator as described above. Then ˜ spec A ⊆ spec A,
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˜ specpoint A ⊆ specpoint A.
(19)
and
The following is an easy corollary. Corollary 4.3. Let A˜ and A be as described in Proposition 4.2. Then any interval (a, b) that is a gap in the spectrum of A˜ is also contained in a gap of the spectrum of A. In Sect. 8 we prove the following result concerning the class of groups K introduced in Sect. 3. Proposition 4.4. The class K of groups is closed under taking extensions with cyclic kernel. Therefore, by Theorem 3.7, the groups in this class all have the σ -multiplier algebraic eigenvalue property for rational σ . Corollary 4.5 (Absence of eigenvalues that are transcendental numbers). Any A ∈ Md (Q(G, σ )) has only eigenvalues that are algebraic numbers, whenever G ∈ K and σ is a rational multiplier on G. Proof. Let Gσ be the central extension of the group G in the class K, where Gσ is defined in (15), and let A˜ be the operator on l 2 (Gσ )d associated with A as defined in (16). By Proposition 4.4, any central extension of G by a cyclic group Zr is also in the class K, therefore the group Gσ is in K when σ is rational. By Theorem 3.7, we know that every group in the class K has the algebraic eigenvalue property and so A˜ has only algebraic eigenvalues. A therefore has only algebraic eigenvalues by Proposition 4.2.
Recall the definition of the following class of groups from [9]. Definition 4.6. Let G be the smallest class of groups which contains the trivial group and is closed under the following processes: (1) If H ∈ G and G is a generalized amenable extension of H , then G ∈ G. (2) If H ∈ G and U < H , then U ∈ G. (3) If G = limi∈I Gi is the direct or inverse limit of a directed system of groups Gi ∈ G, then G ∈ G. We have the inclusion C ⊂ K ⊂ G, and in particular the class G contains all amenable groups, free groups, residually finite groups, and residually amenable groups. Consider the subclass of groups Gˆ defined as ˆ ∈ G ∀ Zr -extensions G ˆ of G . Gˆ = G ∈ G : G By the results of Sect. 8, we have the inclusion C ⊂ K ⊂ Gˆ ⊂ G.
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Corollary 4.7 (Absence of eigenvalues that are Liouville transcendental numbers). Any self-adjoint A ∈ Md (Q(G, σ )) does not have any eigenvalues that are Liouville transcendental numbers, whenever G ∈ Gˆ and σ is a rational multiplier on G. Proof. Any operator A ∈ Md (Q(G, σ )) is self adjoint if and only if A(g)∗ = A(g −1 ) σ (g, g −1 ) for all g ∈ G. By Lemmas 2.7 and 2.8, there exists an A ∈ Md (Q(G, σ )) such that A and A are unitarily equivalent and where σ (g, g −1 ) = 1 for all g ∈ G. With A being self adjoint, we have that A is self adjoint and thus that A (g)∗ = A (g −1 ) for all g ∈ G. By the construction of Lemma 2.8, if σ is a rational multiplier with σ r = 1, then σ is also rational, with σ 2r = 1. Let A˜ = (A ) ∈ Md (Q(Gσ )), as in (16), where Gσ is the central extension of ˜ g) = δ1 (z)A(g) for G as described in (15). In terms of matrices of coefficients, A(z, σ −1 (z, g) ∈ G . As σ (g, g ) = 1, ˜ ˜ −1 σ (g, g −1 ), g −1 ) = A(z ˜ −1 , g −1 ) A((z, g)−1 ) = A(z and ˜ −1 , g −1 ) = δ1 (z−1 )A (g −1 ) = δ1 (z)A (g)∗ = A(z, ˜ g)∗ , A(z showing that A˜ is self-adjoint. Gˆ is closed under extensions by finite cyclic groups, and so the group Gσ is in the class G. Applying Theorem 4.15 of [9], it follows that A˜ does not have any eigenvalues that are Liouville transcendental numbers. Then by Proposition 4.2, A and thus A do not have any eigenvalues that are Liouville transcendental numbers.
5. On the Finiteness of the Number of Distinct Eigenvalues We deal here with the following situation: G is a discrete group and A ∈ Md (QG). Then A induces a bounded linear operator A : l 2 (G)d → l 2 (G)d by left convolution (using the canonical left G-action on l 2 (G)), which commutes with the right G-action. Let pr ker A : l 2 (G)d → l 2 (G)d denote the orthogonal projection onto ker A. Recall that the von Neumann dimension of ker A is defined as dimG (ker A) := tr G (pr ker A ) :=
d
pr ker A ei , ei l 2 (G)d ,
i=1
where ei ∈ l 2 (G)d is the vector with the trivial element of G ⊂ l 2 (G) at the i th -position and zeros elsewhere. Let G be a discrete group. Let fin(G) denote the additive subgroup of Q generated by the inverses of the orders of the finite subgroups of G. Note that fin(G) = Z if and only if G is torsion free and fin(G) is discrete in R if and only if orders of finite subgroups of G are bounded. Recall the following definition. Definition 5.1. A discrete group G is said to fulfill the strong Atiyah conjecture if the orders of the finite subgroups of G are bounded and dimG (ker A) ∈ fin(G)
∀A ∈ Md (QG);
where ker A is the kernel of the induced map A : l 2 (G)d → l 2 (G)d .
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Linnell proved the strong Atiyah conjecture if G is such that the orders of the finite subgroups of G are bounded and G ∈ C, where C is Linnell’s class of groups that is defined just below Definition 3.1. Theorem 5.2 ([14]). If G ∈ C is such that the orders of the finite subgroups of G are bounded, then the strong Atiyah conjecture is true. In [9], Linnell’s results were generalized to a larger class D of groups, but these groups are all torsion-free and therefore our results do not apply to them. ˜ as given in The following is an easy corollary of the relationship between A and A, (16). Corollary 5.3 (Finite number of distinct eigenvalues). Any self-adjoint A ∈ Md (Q(G, σ)) has only a finite number of distinct eigenvalues whenever G ∈ C is such that the orders of the finite subgroups of G are bounded, and σ is a rational multiplier on G. Proof. Let Gσ be the central extension of the group G in C, where Gσ is defined in (15), and let A˜ be the operator on l 2 (Gσ )d associated with A as defined in (16). By Remark 8.5, the group Gσ as defined in (15) is also in C. Clearly, the orders of finite subgroups of Gσ are bounded as well. Thus Theorem 5.2 above applies to Gσ , and so the dimensions of eigenspaces of A˜ are in the discrete, closed subgroup fin(G) of R and are therefore bounded away from zero. It follows that A˜ can have at most finitely many eigenvalues. The conclusion now follows from (19).
Remark 5.4. Our results do not just apply to operators acting on scalar valued functions but also to vector valued functions. In this case there are many examples where eigenvalues exist. For instance, for the combinatorial Laplacian on L2 , degree zero cochains of a covering space, zero is never an eigenvalue, whereas it is very common for it to be an eigenvalue on L2 cochains of positive degree. This is the case whenever the Euler characteristic of the base is nonzero, which follows for instance from Atiyah’s L2 index theorem for covering spaces [2], and Dodziuk’s theorem on the combinatorial invariance of the L2 Betti numbers, [8]. Let H denote the lamplighter group, namely H is the wreath product of Z2 and Z. ˙ Then there is the following remarkable computation of Grigorchuk and Zuk, Theorem 2 and Corollary 3, [13]. Theorem 5.5. Let A := t + at + t −1 + (at)−1 ∈ ZH be a multiple of the Random Walk operator of H . Then A, considered as an operator on l 2 (H ), has eigenvalues
p 4 cos π | p ∈ Z, q = 2, 3, . . . . (20) q The L2 -dimension of the corresponding eigenspaces is
p 1 dimH ker A−4 cos if p, q ∈ Z, q ≥ 2, with (p, q) = 1. π = q q 2 −1 (21) Note that the number of distinct eigenvalues of A is infinite and dense in some interval! However, the orders of the torsion subgroups of H is unbounded so that Corollary 5.3 is not contradicted. The eigenvalues of A are algebraic numbers as predicted by our Theorem 2.5 in [18]. This can be seen as follows: since (cos(p/qπ )+i sin(p/qπ ))q = (−1)p , this shows that cos(p/qπ )+i sin(p/qπ ) is an algebraic number. Therefore the real part, cos(p/qπ ), is also an algebraic number.
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6. The Case of Algebraic Multipliers The goal in this section is to extend the results that were obtained in the previous sections, from rational multipliers to the more general case of algebraic multipliers. Recall that a generic algebraic multiplier is not necessarily a rational multiplier. We start with an example of algebraic numbers on the unit circle that are not roots of unity. Consider the roots of the polynomial,
√ √ (1+ 4k+1) 2 (1− 4k+1) z −z +(2−k)z −z+1 = z − · z+1 · z − · z+1 (22) 2 2
4
3
2
2
with k a positive integer. This polynomial is irreducible over Z if 4k + 1 is not a square. We look for k such that the first factor has two distinct real roots while the second one has two complex conjugate roots. Thus we seek k so that (1 +
√ 4k + 1)2 −4>0 4
and
(1 −
√
4k + 1)2 − 4 < 0. 4
It is easily seen that the only values of k satisfying these conditions are k = 3, 4, 5. For each of these choices, two of the roots, denoted by eiθ and e−iθ , lie on the unit circle and are roots of the second factor in (22). The two other roots are real, denoted by r and r −1 , where r < 1. The numbers eiθ , e−iθ , r and r −1 are algebraic integers, which are all conjugate to each other. Therefore eiθ is not a root of unity since otherwise all its conjugates would also be roots of unity. However, the numbers eiθ , e−iθ , r and r −1 are units in the corresponding ring of algebraic integers. Since eiθ is not a root of unity, its powers are dense in the unit circle whereas the positive powers of r tend to 0. For fixed α1 , α2 ∈ R such that θ = α2 − α1 , and for all (m, n) ∈ Z2 , let σ ((m , n ), (m, n)) = exp(−i(α1 m n + α2 n m)).
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Then σ is an algebraic multiplier on Z2 whose cohomology class [σ ] ∈ H 2 (Z2 , U (1)) ∼ = U (1) is equal to eiθ , so that σ is not a rational multiplier. It is well known that σ determines the noncommutative torus Aθ , see [4]. The trivial group has the σ -multiplier algebraic eigenvalue property for any σ , since the eigenvalues are the zeros of the characteristic polynomial. The same is true for every finite group. If G has the σ -multiplier algebraic eigenvalue property and H is a subgroup of G, then H also has the σ -multiplier algebraic eigenvalue property. Theorem 6.1. Every free group has the σ -multiplier algebraic eigenvalue property for every σ . Proof. Let G be a free group. Then G has the algebraic eigenvalue property, corresponding to the identity multiplier, by Theorem 4.5 of [9]. However for a free group, every multiplier is cohomologous to the identity, as free groups have no cohomology of degree two or higher. This can be seen by noting that the classifying space of a free group is a bouquet of circles, and so is one dimensional (see for example [5, Chapter II, Sect. 4, Example 1].) By Corollary 3.10 then, G has the σ -multiplier algebraic eigenvalue for every algebraic multiplier σ .
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Theorem 6.2. Suppose that we have a short exact sequence of groups p
1 → H → G → G/H → 1,
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where the quotient group G/H is a finitely generated amenable group. Let σ be an algebraic multiplier on G/H , and let σ = p∗ σ be the pullback of σ . Then if H has the algebraic eigenvalue property, G has the σ -multiplier algebraic eigenvalue property. Proof. We will show that an operator A ∈ Md (Q(G, σ )) has only algebraic eigenvalues by first demonstrating that the point spectrum of A is a subset of the union of the point spectra of a series A(m) of approximations to A, and then showing that each A(m) is equivalent to the untwisted action of a matrix over QH , and thus has only algebraic eigenvalues. In the following let A be the von Neumann algebra WL∗ (G, σ ) ⊗ Md (C), with trace τ = tr G,σ as defined in Corollary 2.4. For finite X ⊂ G/H , let HX be the subspace l 2 (p −1 (X))d of l 2 (G)d and let AX be the commutant B(HX )H of the right H -translations on HX . Picking a right inverse s of p, the isometry ιX : HX = l 2 (p −1 (X))d → l 2 (H )d,#X (ιX f )(h)a,i = f (s(xi )h)a for h ∈ H , xi ∈ X, a = 1, . . . , d induces an isomorphism ψX from AX to WL∗ (H ) ⊗ Md (C) ⊗ M#X (C). Define a trace τX on AX by τX (B) =
1 (tr H ⊗ Tr)(ψX B), #X
where tr H is the usual trace on WL∗ (H ) and Tr is the canonical matrix trace on Md (C) ⊗ M#X (C). In terms of the components (Ba,b )g,k of an operator B ∈ AX (for g, k ∈ G and a, b = 1, . . . , d), the trace is given by τX (B) =
d 1 (Ba,a )s(x),s(x) . #X a=1 x∈X
The von Neumann dimension associated with τX will be denoted by dimX . The multiplier σ (g, h) = 1 for all h ∈ H , so any operator A ∈ A commutes with the right H -translations. For A ∈ A and X ⊂ G/H let A(X) = PX A|HX where PX is the orthogonal projection onto HX . A(X) then belongs to AX and τX (A(X) ) = τ (A). The dimension functions on the algebras AX satisfy the following easily verifiable relations, for finite subsets X ⊆ Y : dimX L ≤ dimτ N for all LηAX , N ηA with L ⊂ N, #Y dimX L = #X dimY L for all LηAX , dimY M ≥ dimY PX (M) for all MηAY .
(25) (26) (27)
As G/H is amenable and finitely generated, it admits a Følner exhaustion by finite subsets {Xm } such that lim
m→∞
#∂Xm = 0, #Xm
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where ∂Xm is the δ-neighbourhood (with regard to the word metric on G/H ) of Xm for any fixed δ. In the following, let A ∈ Md (Q(G, σ )) and let A(m) = A(Xm ) ∈ AXm . Suppose λ is not an eigenvalue of any of the A(m) , and consider the space Eλ of λ-eigenfunctions of A with corresponding orthogonal projection Pλ . For any finite X ⊂ G/H , dimτ Eλ = τ (Pλ ) = τX (PX Pλ |HX ) ≤ dimX im PX Pλ |HX ≤ dimX PX (Eλ ).
( as PX Pλ ≤ 1 )
As A is a matrix over the twisted group algebra, each component is a finite sum of twisted translations, and consequently A has bounded propagation. Explicitly, there are only a finite number of the matrices of coefficients A(g) ∈ Md (C) which are non-zero, and so we can choose a bound κ by κ = max{dG/H (1G/H , gH ) | A(g) = 0} (where dG/H is the word-metric on G/H ) so that for f with support in p −1 (X), Af will have support in p−1 (X ), where X = {x | dG/H (x, X) ≤ κ} is the κ-neighbourhood of Xm . be the κ-neighbourhood of X , and ∂X = X \X be the outer κ-boundary Let Xm m m m m of Xm . Then PXm A = PXm APXm = A(m) PXm + PXm AP∂Xm . For f ∈ Eλ with P∂Xm f = 0 then, PXm Af = λPXm f = A(m) PXm f . By assumption though, λ is not an eigenvalue of A(m) , and so f ∈ Eλ and P∂Xm f = 0 ⇒ PXm f = 0.
(28)
, one has P Picking some superset Y of Xm ∂Xm PY = P∂Xm and PXm PY = PXm , and so (28) implies
ker P∂Xm |PY (Eλ ) ⊆ ker PXm |PY (Eλ ) .
(29)
Applying Lemma 2.6 gives dimY PY (Eλ ) = dimY ker P∂Xm |PY (Eλ ) + dimY P∂Xm (Eλ ) = dimY ker PXm |PY (Eλ ) + dimY PXm (Eλ ), which with the inclusion (29) in turn gives dimY PXm (Eλ ) ≤ dimY P∂Xm (Eλ ) ≤ dimY im P∂Xm =
#∂Xm . #Y
, Then for any m and Y ⊇ Xm
dimτ Eλ ≤ dimXm PXm (Eλ ) =
#Y #∂Xm dimY PXm (Eλ ) ≤ , #Xm #Xm
which goes to zero as m goes to infinity, as the Xm form a Følner exhaustion of G/H . Consequently dimτ Eλ = 0 and λ is not an eigenvalue of A; that is, any eigenvalue of A must be an eigenvalue of A(m) for some m.
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For f ∈ HX , let fˆ = ιX f be the corresponding element in l 2 (H )d,#X defined by fˆ(h)a,i = f (s(xi )h)a , as before. Then for every xi ∈ X, (Af )(s(xi )h) =
#X
A(s(xi )hk −1 s(xj )−1 )f (s(xi )k)σ (s(xi )hk −1 s(xj )−1 , s(xj )k)
j =1 k∈H
=
#X
A(s(xi )hk −1 s(xj )−1 )f (s(xi )k)σ (xi xj−1 , xi )
j =1 k∈H
= (B fˆ)(h)i ∈ Cd , where the matrix of coefficients of h in B ∈ M(d#X) (QH ) is given by B(h)(a,i),(b,j ) = A(s(xi )hs(xj )−1 )a,b · σ (xi xj−1 , xj ) for i, j = 1, . . . , #X and a, b = 1, . . . d. As H has the algebraic eigenvalue property, B and hence A(X) have only algebraic eigenvalues. Consequently, the operators A(m) and A have only algebraic eigenvalues.
One of the main theorems in this section is the following. Theorem 6.3. Let be the fundamental group of a closed Riemann surface of genus g > 1. Then has the σ -multiplier algebraic eigenvalue property, where σ is any algebraic multiplier on . We want to use Theorem 6.2 using the exact sequence of Remark (3.4). To do this, it is necessary to prove that every algebraic multiplier σ on is cohomologous to the pull-back of an algebraic multiplier σ on Z2g . The construction of σ was used in [6, Sect. 7.2]. We follow it closely paying particular attention to algebraicity. Recall that the area cocycle c of the fundamental group of a compact Riemann surface, = g is a canonically defined 2-cocycle on that is defined as follows. Firstly, recall the definition of a well known area 2-cocycle on PSL(2, R). PSL(2, R) acts on H so that H ∼ = PSL(2, R)/SO(2). Then c(γ1 , γ2 ) = AreaH ((o, γ1 · o, γ1 γ2 · o)), where o denotes an origin in H and AreaH ((a, b, c)) is the oriented hyperbolic area of the geodesic triangle in H with vertices at a, b, c ∈ H. The restriction of c to the subgroup is the area cocycle c of . We use the additive notation when discussing area cocycles and remark that (2π)−1 c represents an integral class in H 2 (, R) ∼ = R as follows from the Gauss-Bonnet theorem. Let j denote the (diagonal) operator on l 2 () defined by j f (γ ) = j (γ )f (γ ) ∀f ∈ l 2 () where
γ ·o
j (γ ) =
αj o
∀γ ∈ ,
j = 1, . . . , 2g,
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and where 2g
g
g
{αj }j =1 = {aj }j =1 ∪ {bj }j =1
(30)
is a collection of harmonic 1-forms on the compact Riemann surface g = H/ , generating H 1 (g , R) = R2g . We abuse the notation slightly and do not distinguish between a form on g and its pullback to the hyperbolic plane as well as between an element of and a loop in g representing it. Notice that we can write equivalently j (γ ) = cj (γ ), where the group cocycles cj form a symplectic basis for H 1 (, Z) = Z2g , with generators {αj }j =1,... ,2g , as in (30) and can be defined as the integration on loops on g , αj . cj (γ ) = γ
Define
j (γ1 , γ2 ) = j (γ1 )j +g (γ2 ) − j +g (γ1 )j (γ2 ). Let : H → R2g denote the Abel-Jacobi map x x : x → a1 , b1 , . . . ,
o
o
x
x
ag , o
bg ,
o
x
means integration along the unique geodesic in H connecting o to x. Having chosen an origin o once and for all we make an identification · o ∼ = . Note that acts on R2g in a natural way and the map is -equivariant. In addition, the map is a symplectic map, that is, if ω and ωJ are the respective symplectic 2-forms, then one has ∗ (ωJ ) = kω for a suitable constant k. Henceforth, we renormalize ω (and consequently the area cocycle c) so that ∗ (ωJ ) = ω. One then has the following geometric lemma [6, 17]. where
o
Lemma 6.4.
(γ1 , γ2 ) =
g
j (γ1 , γ2 ) =
ωJ , E (γ1 ,γ2 )
j =1
where E (γ1 , γ2 ) denotes the Euclidean triangle with vertices at (o), (γ1 · o) and (γ1 γ2 · o), andωJ denotes the flat K¨ahler 2-form on the universal cover of the Jacobi g variety. That is, j =1 j (γ1 , γ2 ) is equal to the Euclidean area of the Euclidean triangle E (γ1 , γ2 ). That is, the cocycle = p∗ ( ), where is a 2-cocycle on Z2g and p is defined as the projection, p
1 → F → → Z2g → 1. The following lemma is also implicit in [6, 17].
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Lemma 6.5. The hyperbolic area group 2-cocycle c and the Euclidean area group 2-cocycle on , are cohomologous. Proof. Observe that since ω = ∗ ωJ , one has c(γ1 , γ2 ) = ω= (γ1 ,γ2 )
Therefore the difference
(γ1 , γ2 ) − c(γ1 , γ2 )
ωJ .
((γ1 ,γ2 ))
= E (γ1 ,γ2 ) ωJ − ((γ1 ,γ2 )) ωJ = ∂E (γ1 ,γ2 ) J − ∂((γ1 ,γ2 )) J ,
where J is a 1-form on the universal cover R2g of the Jacobi variety such that dJ = ωJ . Let h(γ ) = ((γ )) J − m(γ ) J , where (γ ) denotes the unique geodesic in H joining o and γ · o and m(γ ) is the straight line in the Jacobi variety joining the points (o) and (γ · o). We can also write h(γ ) = D(γ ) ωJ , where D(γ ) is an arbitrary topological disk in R2g with boundary ((γ )) ∪ m(γ ). Thus the equality above can be rewritten as
(γ1 , γ2 ) − c(γ1 , γ2 ) = h(γ1 ) − h(γ1 γ2 ) + J − J γ1 ·(l(γ2 )) γ1 ·m(γ2 ) = h(γ1 ) − h(γ1 γ2 ) + (γ1 )∗ dJ D(γ2 )
= h(γ1 ) − h(γ1 γ2 ) + h(γ2 ) = δh(γ1 , γ2 ), since ωJ = dJ is invariant under the action of .
Lemma 6.6. Let be the fundamental group of a closed, genus g Riemann surface and p
1 → F → → Z2g → 1, where F is a free group and Z2g the free abelian group as in Remark 3.4. Then every multiplier σ on is cohomologous to a multiplier σ = p∗ (σ ) on , where σ is a multiplier on Z2g . In addition, every algebraic multiplier σ on is cohomologous to an algebraic multiplier σ = p∗ (σ ) on , where σ is an algebraic multiplier on Z2g . More precisely, if σ ∈ Z 2 (, U(Q)) then σ can be chosen from Z 2 (Z2g , U(Q)) so that σ and σ are cohomologous in Z 2 (, U(Q)). Proof. Observe that ( · o) ⊂ Z2g ⊂ R2g . It follows that the Euclidean area cocycle and its pullback represent integral cohomology classes. By the lemma above, the cohomology class of c is integral. Now let σ be an arbitrary multiplier on . Since H 2 (, A) = A for every abelian group A and H 3 (, Z) = 0, we see that σ is cohomologous to a multiplier σ1 = exp(2πiθ c), where θ is a real number. By Lemma 6.5 we see that σ1 is cohomologous to p ∗ (σ ), where σ = exp(2π iθ ) is a multiplier on Z2g . To prove the last claim, we identify the group cohomology with the cohomology of the surface g = H/ . Since the value of the cocycle on the fundamental class
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depends only on the cohomology class and c(g ) = 2g − 2, we see that σ (g ) = exp(2π iθ )2g−2 = σ (g ) is algebraic. It follows that exp(2π iθ) is an algebraic number so that σ is an algebraic cocycle. Now both σ and σ are algebraic cocycles. They are cohomologous in Z 2 (, U (1)). For any coefficients the cohomology class of the cocycle is determined by the value of the cocycle on the fundamental class. Therefore σ and σ represent the same cohomology class in H 2 (, U(Q)), i.e. are cohomologous
in Z 2 (, U(Q)). Proof of Theorem 6.3. Recall that if two multipliers σ , σ on are cohomologous, then has the σ -multiplier algebraic eigenvalue property if and only if has the σ -multiplier algebraic eigenvalue property (Corollary 3.10.) Since the free group F has the algebraic eigenvalue property, and since Z2g is a finitely generated amenable group, by applying Theorem 6.2 and Lemma 6.6, we deduce Theorem 6.3.
7. Generalized Integrated Density of States and Spectral Gaps In this section, we will realize the von Neumann trace on the group von Neumann algebra of a surface group, as a generalized integrated density of states, which is an important step to relating it directly to the physics of the quantum Hall effect. Our first main theorem is the following. Theorem 7.1. Consider the situation of Theorem 6.2, where we have a short exact sequence of groups p
1 → H → G → G/H → 1,
(31)
where the quotient group G/H is finitely generated and amenable. Let σ be a multiplier on G/H , and let σ = p ∗ σ be the pullback of σ . Let A ∈ Md (C(G, σ )) be a self-adjoint operator acting on l 2 (G)d , being a member of the von Neumann algebra A = WL∗ (G, σ ) ⊗ Md (C) with trace τ = tr G,σ . For finite subsets X of G/H , let HX = l 2 (p −1 (X))d be the space of functions with support on p−1 (X), and let AX = B(HX )H be the commutant of the right H -translations on HX . Pick a right inverse s of the projection p and give AX the trace τX as in the proof of Theorem 6.2, which in terms of the components (Ba,b )g,k of an operator B ∈ AX (for g, k ∈ G and a, b = 1, . . . , d) is given by τX (B) =
d 1 (Ba,a )s(x),s(x) . #X a=1 x∈X
Let A(X) = PX A|HX ∈ AX , where PX is the orthogonal projection onto HX . Choose a Følner exhaustion Xm of G/H . Then the spectral density function of A equals the generalised integrated density of states as given by the (normalised) spectral density functions of the operators A(m) = A(Xm ) . That is, with spectral density functions F of A and Fm of the Am , Fm (λ) = τXm (χ(−∞,λ] (A(m) )),
F (λ) = τ (χ(−∞,λ] (A)),
the Fm converge point-wise to F at every λ, lim Fm (λ) = F (λ)
m→∞
∀λ ∈ R.
(32)
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The proof of this theorem in the case of H = 1 was given in [18] and [19] for the discrete magnetic Laplacian. To establish this theorem in our more general situation, we apply the same arguments, slightly generalized as follows, relying upon the notation established in the proof of Theorem 6.2. Lemma 7.2. For any polynomial p, lim τm (p(A(m) )) = τ (p(A)).
m→∞
Proof. The argument is exactly that of [18], Lemma 2.1, and relies upon the amenability of G/H .
Lemma 7.3. Suppose f (λ) and fm (λ) (m = 1, 2, . . . ) are monotonically increasing right continuous functions on R that are zero for λ < a and constant for λ ≥ b, for fixed a and b. Further suppose that lim p dfm = p df (33) m→∞
for all polynomials p, where the integrals are Lebesgue-Stieltjes integrals. Then +
f (λ) = f +(λ) = f (λ) for all λ, where f
+
and f + are defined in terms of the fm by
f (λ) = lim inf fm (λ), f +(λ) = lim f (λ + ), →0+
m
+
f (λ) = lim sup fm (λ),f (λ) = lim f (λ + ).
(34)
→0+
m
In particular f (λ) = limm→∞ fm (λ) at all points of continuity of f , which is at all but a countable number of points. Proof. The proof follows that of part (i) of Theorem 2.6 of [18]. Take a sequence of successively closer polynomial approximations pj to the characteristic function χ(−∞,x] over the interval [a, b) such that χ(−∞,x] (λ) ≤ pj (λ) ≤ χ
1 1 (λ) + j (−∞,x+ j ]
∀λ ∈ [a, b), j ≥ 1.
Then for all j ,
b
fm (x) ≤ a
pj (λ)dfm (λ) ≤ fm (x + j1 ) + j1 (b − a),
f (x) ≤ a
b
pj (λ)df (λ) ≤ f (x + j1 ) + j1 (b − a).
Taking the limit as m goes to infinity, Eqs. (35) and (33) give, b f (x) ≤ pj (λ)df (λ) ≤ f (x + j1 ) + j1 (b − a) ∀j ≥ 1. a
(35) (36)
(37)
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The right continuity of f with Eq. (36) gives b lim pj (λ)df (λ) = f (x), j →∞ a
and so taking the limit of (37) as j goes to infinity gives f (x) ≤ f (x) ≤ f +(x) ∀x.
(38)
Again using the right continuity of f , +
f (x) ≤ f +(x) ≤ f (x) ≤ f +(x) = f (x). f (x) is monotonically increasing in x and bounded, so can have at most a countable number of discontinuities. If f is continuous at x then Eq. (38) implies that f (x) =
f (x) = f (x). Lemma 7.4. Let F and Fm be as in the statement of Theorem 7.1. Then using the notation (34) of Lemma 7.3, +
F (λ) = F (λ) = F +(λ) ∀λ ∈ R, with lim Fm (λ) = F (λ)
m→∞
∀λ ∈ R such that F is continuous at λ.
Proof. This is an immediate consequence of the two preceding lemmas.
(39)
The convergence (39) can be extended to all λ by showing that the jumps of the spectral density functions at points of discontinuity also converge. Lemma 7.5 (Corollary 3.2 of [19]). Let f and fm (for m = 1, 2, . . . ) be monotonically + increasing right continuous functions on R satisfying f (λ) = f +(λ) = f (λ) at all λ, as in Lemma 7.3. Denote the jumps at λ of f and the fm by j and jm respectively, jm (λ) = lim fm (λ) − fm (λ − ), →0+
j (λ) = lim f (λ) − f (λ − ). →0+
Suppose the jm converge to j point-wise at all λ. Then the fm converge to f point-wise at all λ. To obtain point-wise convergence of fm to f at every point, it is in fact sufficient to show that lim inf m jm (λ) ≥ j (λ) at all λ, due to the following lemma. Lemma 7.6. Let f and fm (for m = 1, 2, . . . ) be monotonically increasing right con+ tinuous functions on R satisfying f (λ) = f +(λ) = f (λ) at all λ, as in Lemma 7.3. Denote the jumps at λ of f and fm by j and jm respectively, as in Lemma 7.5. Then lim sup jm (λ) ≤ j (λ) ∀λ ∈ R. m
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Proof. Fix λ. By monotonicity, jm (λ) ≤ fm (λ + ) − fm (λ − )
∀ > 0.
(40)
f is continuous at all but a countable number of points, and at points x of continuity, fm (x) → f (x) as m → ∞. Pick a decreasing sequence k → 0 such that f is continuous at λ + k and λ − k for all k. Then taking the limit in m of (40) gives lim sup jm (λ) ≤ f (λ + k ) − f (λ − k ) m
∀k.
By right continuity of f , f (λ + k ) − f (λ − k ) converges to j (λ) from above as k goes to infinity. Thence on taking the limit in k, lim supm jm (λ) ≤ j (λ).
Now consider the situation of Theorem 7.1. We already have a weak spectral approximation by virtue of Lemma 7.4, so all we require now is to show convergence of the jumps in Fm to those of F . Theorem 7.7. Let D(λ) and Dm (λ) denote the jumps at λ of the spectral density functions F and Fm respectively. Then lim Dm (λ) = D(λ) ∀λ ∈ R.
m→∞
Proof. Let dimX be the von Neumann dimension associated with the trace τX on AX . Note that dim H = dim ker B + dim im B for an operator B in a von Neumann algebra of operators acting on a Hilbert space H, with finite von Neumann dimension dim. So D(λ) = dimτ ker(A − λ) = d − dimτ im(A − λ), Dm (λ) = dimXm ker(A(m) − λ) = d − dimXm im(A(m) − λ). As in the proof of Theorem 6.2, let κ be the propagation of the operator A with respect be the κ-neighbourhood of X so that to the word metric dG/H on G/H and let Xm m f ∈ HXm implies Af ∈ HXm ; equivalently, PXm A|HXm = A|HXm for all m. The space im(A − λ)|HXm is affiliated with AXm . Recall the properties (25), (26) and (27) of dimX as listed in the proof of Theorem 6.2. Then dimXm im(A(m) − λ) = dimXm PXm (im(A − λ)|HXm ) ≤ dimXm im(A − λ)|HXm ≤ dimτ im(A − λ), and dimXm im(A(m) − λ) =
#Xm dimXm im(A(m) − λ). #Xm
The Xm constitute a Følner exhaustion of G/H and so infinity. Taking limits gives
#Xm #Xm
tends to 1 as m goes to
lim inf Dm (λ) = d − lim sup dimXm im(A(m) − λ) m
m
≥ d − dimτ im(A − λ) = D(λ). Finally, applying Lemma 7.6 gives D(λ) ≤ lim inf Dm (λ) ≤ lim sup Dm (λ) ≤ D(λ). m
m
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The proof of Theorem 7.1 now follows from Lemmas 7.4 and 7.5, and Theorem 7.7. The following corollary is an immediate consequence of Theorem 6.6 and Theorem 7.1. Corollary 7.8 (Generalized IDS). Let G = be the fundamental group of a closed Riemann surface of genus g > 1, G/H = Z2g be the abelianisation of G, in which case the commutator subgroup H = F is a free group. Then the equality between the generalized integrated density of states and the von Neumann spectral density function given in Eq. (32) holds for every multiplier σ on . Corollary 7.9 (Criterion for spectral gaps). Consider the situation in Theorem 7.1. The interval (λ1 , λ2 ) is in a gap in the spectrum of A if and only if lim (Fm (λ2 ) − Fm (λ1 )) = 0.
m→∞
(41)
Proof. The interval (λ1 , λ2 ) is in a gap in the spectrum of A if and only if F (λ2 ) = F (λ1 ). By Theorem 7.1, this is true if and only if lim (Fm (λ2 ) − Fm (λ1 )) = F (λ2 ) − F (λ1 ) = 0.
m→∞
8. The Class K and Extensions with Cyclic Kernel In this section we prove the results cited in the earlier sections concerning the class of groups K. Namely we show that the class K is closed under taking extensions with cyclic kernel, and that every group in K has the algebraic eigenvalue property. Recall that the class K is the smallest class of groups which contains the free groups and the amenable groups, and is closed under directed unions and under taking extensions with amenable quotients. It can be seen that every group in K must belong to some Kα defined inductively as follows. Definition 8.1. Define the nested classes Kα , α an ordinal, by • K0 consists of all free groups and all discrete amenable groups, • Kα+1 consists of all extensions of groups in Kα with amenable quotient, and all directed unions of groups in Kα , • Kβ = α<β Kα when β is a limit ordinal. A group is in K if and only if it is in a class Kα for some ordinal α. Recall Theorem 3.7: Theorem. Every group in K has the algebraic eigenvalue property. The proof follows closely that of Corollary 4.8 of [9], and relies upon the same key theorem: Theorem (4.7 of [9]). Let H have the algebraic eigenvalue property. Let G be a generalized amenable extension of H . Then G has the algebraic eigenvalue property. The proof then proceeds by transfinite induction.
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Proof of Theorem 3.7. The algebraic eigenvalue property holds for groups in K0 by Corollary 4.8 in [9]. Proceeding by transfinite induction, suppose Kα has the algebraic eigenvalue property for all α less than some given ordinal β. When β = α + 1 for some ordinal α, a group G is in Kβ if it is an extension of a group in Kα with amenable quotient, or is the directed union of groups Gi ∈ Kα . Note that A ∈ Md (QG) can be regarded as a matrix A in Md (QH ), where H is a finitely generated subgroup of G, generated by the finite support of the Ai,j in G. By Proposition 3.1 of [20], the spectral density functions of A and A coincide. As subgroups of a group with the algebraic eigenvalue property also have the property, it follows that a group has the algebraic eigenvalue property if and only if it holds for all of its finitely generated subgroups. If G ∈ Kα+1 is the directed union of groups Gi ∈ Kα , it follows that every finitely generated subgroup of G is in some Gi and so has the algebraic eigenvalue property. G therefore has the algebraic eigenvalue property. Suppose that G is the extension of a group H in Kα with amenable quotient p R = G/H , i.e. H → G → R, then R = j ∈J Rj is a directed union, where Rj , j ∈ J are finitely generated and amenable (since amenability is subgroup closed). Consider the extensions Gj = p−1 (Rj ) of H , with finitely generated amenable quotient: then Theorem 4.12of [9] applies to show that Gj has the algebraic eigenvalue property. But G = j ∈J Gj is the directed union of groups Gj having the algebraic eigenvalue property, so G also has the algebraic eigenvalue property by the argument given in the previous paragraph. Finally, let β be a limit ordinal, so that Kβ = α<β Kα . If G is in Kβ , then it is in Kα for some α < β, and so has the algebraic eigenvalue property by the induction hypothesis. Therefore groups in Kβ have the algebraic eigenvalue property, and the result follows by induction.
We next address Proposition 4.4. Its proof follows from the following lemmas. Lemma 8.2. The class Kα is subgroup-closed for all ordinals α. Proof. The classes of free groups and amenable groups are both closed under taking subgroups, so K0 is subgroup-closed. Suppose Kα is subgroup closed for all α < β, let G ∈ Kβ and let H be a subgroup of G. If β is a limit ordinal then G ∈ α for some α < β, and so H ∈ Kα , and so in Kβ . Suppose then that β is not a limit ordinal, with β = α + 1 for some ordinal α. If G is a directed union of groups Gj in Kα , then H is a directed union of groups H ∩ Gj which are also in Kα by the induction hypothesis. G must otherwise be an extension of a group N in Kα with amenable quotient B. In this case, H is an extension of N ∩H with quotient H /(N ∩H ) ∼ = N H /N ⊂ G/N ∼ = B, and hence is an extension with amenable quotient of N ∩H which is in Kα by the hypothesis. H is therefore in Kβ .
Lemma 8.3. Let G be a group whose j th derived group G(j ) is in Kα . Then G ∈ Kα+j . Proof. G is an extension with kernel its derived group G by the short exact sequence G −→ G −→ G/G . As the quotient is abelian (and thus amenable), G ∈ Kα implies G ∈ Kα+1 , and the result follows inductively.
Lemma 8.4. Let G be an extension of H ∈ K0 with cyclic kernel. Then G ∈ K2 .
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Proof. Let A be the cyclic group, so that the sequence π
1 −→ A −→ G −→ H −→ 1 is exact. If H is amenable, then G is an amenable group, since A is amenable and the class of amenable groups is closed under taking extensions. Suppose instead that H is free. As free groups have trivial second cohomology, G must be the semi-direct product A φ H with φ : H → Aut A. As before, denote the derived groups of G and H by G and H respectively. Then the following diagram commutes with exact rows, and where the vertical homomorphisms are just inclusions, 1
/ A ∩ G
/ G
1
/A
/G
π|G
π
/ H
/1
/H
/1
G is a semidirect product A ∩ G φ|H H , and φ|H must have image in (Aut A) . As A is cyclic, it has abelian automorphism group, and so (Aut A) is trivial. This in turn implies that φ|H is trivial, and that G ∼ = (A ∩ G ) × H . It follows then that ∼ G = H , which is free and thus in K0 . Therefore G is in K2 .
Proof of Proposition 4.4. Proceeding by transfinite induction, suppose that for a given ordinal β, H ∈ Kα implies that every extension of H with cyclic kernel is in K for all α < β. Let H ∈ Kβ , and G an extension of H with π
1 −→ A −→ G −→ H −→ 1 exact and A cyclic. Choose α < β such that H ∈ Kα+1 ; when β is not a limit ordinal, one can just let α = β − 1. There are two possibilities: either the group H is a directed union of groups Hj in Kα , j ∈ J ; or H is an extension of a group H¯ in Kα . In the first case, G = ∪j ∈J Gj with Gj = π −1 (Hj ). Each Gj is an extension of Hj ∈ Kα with cyclic kernel, and so Gj ∈ K by the induction hypothesis. Therefore G ∈ K. In the second case, let B = H /H¯ be the quotient of the extension, and denote the ¯ = ker ηπ and we get the following ¯ = π −1 (H¯ ). Then G surjection H → B by η. Let G commutative diagram with exact rows. 1
/A
/G ¯
1
/A
/G ηπ
B
π|G¯
π
/ H¯
/1
/H
/1
η
B
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¯ with amenable quotient B, while G ¯ is an extension of G is therefore an extension of G ¯ ∈ K by the induction hypothesis, and so G ∈ K. H¯ ∈ Kα with cyclic kernel A. So G The case for β = 1 holds by virtue of Lemma 8.4, and so by induction, H ∈ K implies that any extension with cyclic kernel of H is in K.
Remark 8.5. Linnell’s class C is also closed under taking extensions with cyclic kernel, by the same argument. C can be written as the union of classes Cα for ordinals α: • C0 is the class of free groups. • Cα+1 is the class of groups which are extensions of groups in Cα with elementary amenable quotient, or are directed unions of groups in Cα . • Cβ = α<β Cα when β is a limit ordinal. As the abelian groups are all elementary amenable, the above procedure for K can be applied directly to C. Acknowledgement. We would like to thank the referee for detailed suggestions on improving the paper.
References 1. Ahlfors, L., Sario, L.: Riemann surfaces, Princeton Mathematical Series, No. 26 Princeton, N.J.: Princeton University Press, 1960 2. Atiyah, M.: Elliptic operators, discrete groups and Von Neumann algebras. Ast´erisque 32–33, 43–72 (1976) 3. Bellissard, J.: Gap Labeling Theorems for Schr¨odinger’s Operators. In: From number theory to physics (Les Houches, 1989), Berlin: Springer, 1992, pp. 538–630 4. Boca, F.: Rotation C ∗ -algebras and almost Mathieu operators. Theta Series in Advanced Mathematics, 1. Bucharest: The Theta Foundation, 2001 5. Brown, K.: Cohomology of groups. Graduate Texts in Mathematics, 87. New York-Berlin: SpringerVerlag, 1982 6. Carey, A., Hannabuss, K., Mathai, V., McCann, P.: Quantum Hall effect on the hyperbolic plane. Commun. Math. Phys. 190 (3), 629–673 (1998) 7. Carey, A., Hannabuss, K., Mathai, V.: Quantum Hall Effect on the hyperbolic plane in the presence of disorder. Lett. Math. Phys. 47, 215–236 (1999) 8. Dodziuk, J.: Rham-Hodge theory for L2 -cohomology of infinite coverings. Topology 16, 157–165 (1977) 9. Dodziuk, J., Linnell, P., Mathai, V., Schick, T., Yates, S.: Approximating L2 -invariants, and the Atiyah conjecture. Commun. in Pure and Appl. Math. 56(7), 839–873 (2003) 10. Dodziuk, J., Mathai, V.: Approximating L2 invariants of amenable covering spaces: a combinatorial approach. J. Funct. Anal. 154(2), 359–378 (1998) 11. Elek, G.: On the analytic zero divisor conjecture of Linnell, Bull. London Math. Soc. 35(2), 236–238 (2003) 12. Grigorchuk, R.: On the Milnor problem of group growth. (Russian) Dokl. Akad. Nauk SSSR 271(1), 30–33 (1983) _ A.: The lamplighter group as a group generated by a 2-state automaton and its 13. Grigorchuk, R., Zuk, spectrum. Geom. Dedicata 87, 209–244 (2001) 14. Linnell, P.A.: Division rings and group von Neumann algebras. Forum Math. 5, 561–576 (1993) 15. L¨uck, W.: Approximating L2 invariants by their finite dimensional analogues. Geom. Func. Anal. 4, 455–481 (1994) 16. Marcolli, M., Mathai, V.: Twisted index theory on good orbifolds, I: noncommutative Bloch theory. Commun. Contemp. Math. 1(4), 553–587 (1999) 17. Marcolli, M., Mathai, V.: Twisted index theory on good orbifolds, II: fractional quantum numbers. Commun. Math. Phys. 217(1), 55–87 (2001) 18. Mathai, V., Yates, S.: Approximating spectral invariants of Harper operators on graphs. J. Funct. Anal. 188(1), 111–136 (2002) 19. Mathai, V., Schick, T., Yates, S.: Approximating spectral invariants of Harper operators on graphs II. Proc. Amer. Math. Soc. 131(6), 1917–1923 (2003)
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20. Thomas Schick,: L2 -determinant class and approximation of L2 -Betti numbers. Trans. Amer. Math. Soc. 353(8), 3247–3265 (2001) 21. Shubin, M.: Discrete Magnetic Schr¨odinger operators. Commun. Math. Phys. 164(2), 259–275 (1994) 22. Shubin, M.: von Neumann algebras and L2 techniques in geometry and topology. Book in preparation 23. Sunada, T.: A discrete analogue of periodic magnetic Schr¨odinger operators. Contemp. Math. 173, 283–299 (1994) Communicated by P. Sarnak
Commun. Math. Phys. 262, 299–315 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1490-7
Communications in
Mathematical Physics
Schr¨odinger Maps and Anti-Ferromagnetic Chains Jalal Shatah1, , Chongchun Zeng2, 1 2
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA. E-mail: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA. E-mail: [email protected]
Received: 28 May 2004 / Accepted: 18 April 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: We study the problem of restricting Schr¨odinger maps onto Lagrangian submanifolds. The restriction is imposed by an infinite constraining potential. We show that, in the limit, solutions of the Schr¨odinger map equations converge to solutions of generalized wave map equations. This result is applied to the anti-ferromagnetic systems where we prove rigorously that the dynamics is governed by wave map equations into S2 . 1. Introduction In this paper we will study the problem of restricting Schr¨odinger maps to Lagrangian submanifolds where we constrain the flow by a singular potential. This problem has its root in the derivation of the equation for the long time dynamics of anti-ferromagnetic chains, a subject which has seen much study in recent years [2, 11, 13, 15]. The basic equations for the magnetic spin {Sn } in a one dimensional chain lattice is described by the Heisenberg model S˙n = Sn ∧ (Sn+1 + Sn−1 ) ,
(H)
where Sn ∈ S2 ⊂ R3 is the unit spin vector at the lattice site n. If the distance between two neighboring points on the lattice is a, then the continuum approximation of (H) is given by the Landau-Lifshitz equations ∂u = a 2 u ∧ uxx , ∂t
(LL)
up to O(a 3 ). These equations have been studied extensively, see [1, 5, 6, 8, 10, 12, 14, 16, 18, 19]. Antiferromagnetic chains are also described by (H), however in this case neighboring spins tend to anti-align, i.e.
The first author is funded in part by NSF DMS 0203485. The second author is funded in part by NSF DMS 0101969 and DMS 0239389.
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S2n + S2n+1 = O(a). Thus the continuum limit describing the long time dynamics of the spins in antiferromagnetic chain is given by two functions u and v taking values in S2 and corresponding to even and odd locations on the lattice. The approximate equations in this case are given by ∂t u = u ∧ −a 2 ∂x2 u − a∂x v + v , ∂t v = v ∧ −a 2 ∂x2 v + a∂x u + u , u + v = O(a) up to O(a 3 ). An accepted fact in the physics community is that the long time dynamics of antiferromagnetic chains, for time of order O( a1 ), is governed by a nonlinear wave map, w ∈ S2 , wtt − wxx + (|wt |2 − |wx |2 )w = 0. By rescaling t →
t a
the antiferromagnetic equations become 1 2 ∂t u = u ∧ −a∂x u − ∂x v + a v ∂t v = v ∧ −a∂x2 v + ∂x u + a1 u u + v = O(a).
(σ )
(AF)
This is a Hamiltonian system of partial differential equations with the Hamiltonian +∞ a 1 H (u, v) = (|ux |2 + |vx |2 ) + v, ux + |u + v|2 dx. 2a −∞ 2 The sigma model (σ ) can be obtained formally as the limit of (AF) as a → 0 [11, 13, 15]. Here we will prove this result rigorously. In fact this result will be a consequence of a more general result that we will prove for a large class of equations which is referred to as Schr¨odinger maps. The setting of our general result is the following: Let M2m , g, ı denote a 2m dimensional K¨ahler manifold M, with a metric g and complex structure ı, and let D denote the covariant derivative on T M. The Schr¨odinger equation into M is defined for maps u : Rd × R → M that satisfy1 ∂t u = −ıDk ∂k u,
(S)
where Dk denotes the covariant derivative on u−1 T M in the direction of x k . Equation (S) is an infinite dimensional Hamiltonian system ∂t u = ıE (u), with energy 1 1 ∂k u, ∂k udx = |∇u|2 dx, E(u) = e(u)dx = 2 2 where , denotes the inner product on T M. An example of such an equation is given by M = S2 ⊂ R3 with the standard metric and complex structure ıv = u ∧ v at u ∈ S2 , 1 Here and throughout the paper we employ the summation convention by summing on repeated indices.
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where ∧ is the cross product in R3 . In this case the equations are referred to as the Landau-Lifshitz equations and are written for u ∈ S2 ⊂ R3 , ∂u (LL) = u ∧ Dk ∂k u = u ∧ u + |∇u|2 u = u ∧ u. ∂t These Schr¨odinger equations are naturally nonlinear, where the nonlinearity arise from the geometric nature of the equation. Other nonlinearities may be present in these equations by considering Hamiltonians with lower order terms such as e(u) = 21 |∇u|2 + A(u)∂u + B(u), which is the case for the anti-ferromagnetic system (AF). Solutions to these type of Hamiltonian equations are referred to as Schr¨odinger maps. The problem that we will study is the limit of solutions for singular Hamiltonians
1 2 E (u) = |∇u| + Wk (u), ∂k u + G(u) dx, (E) 2 as → 0. More precisely, given a K¨ahler manifold M and a compact2 Lagrangian submanifold L ⊂ M, i.e. ıT L = T L⊥ NL, we consider the Cauchy problem for the Hamiltonian (E),
1 ∂t u = ı −Dk ∂k u + Bk ∂k u + G , u(0) = u0 ,
x ∈ Rd , (SM)
where, for each 1 k d, a) Wk : M → T M is a vector field on M, b) Bk (u) = DWk (u)∗ − DWk (u), which is a skew adjoint operator on Tu M, c) G : M → R is a smooth function that satisfies i) G(u) 0 on M and {G(u) = 0} = L, ii) G (p)|Np L I, ∀p ∈ L where Np L = ıTp L the normal space to Tp L.3 The function G can be thought of as a constraining potential that will keep the flow close to L if the data has bounded initial energy E (u0 ) c0 with c0 independent of . Note that for the anti-ferromagnetic equations: i) M = S2 × S2 and L = S2 is the anti-diagonal u + v = 0. ii) W (u, v), ∂x (u, v) = v, ∂x u and G(u, v) = 21 |u + v|2 . We will prove, under some assumptions on B, that the limit of smooth solutions u → p∗ as → 0 for initial data u0 : Rd → L, satisfy a generalized wave map equation. Specifically we consider equation (SM) and assume that on L,
A1) G commutes with ıBk , i.e., [G , ıBk ] = [G , Bk ı] = 0 for each 1 k d. A2) G controls Bk in the following sense:
k {G (Vk ), Vk + G (ıVk ), ıVk } λ|k Bk Vk |2 for some λ > 1 and all V1 · · · Vd ∈ T M. 2 Compactness is not essential, it simplifies the presentation by eliminating the need to bound the solution u pointwise. 3 Throughout the paper, we use G to denote the gradient vector and G the transformation on T M.
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Remark. 1) Assumption A1) implies that, at any p ∈ L, Tp L and Np L are invariant under ıBk and therefore Bk (p)(Tp L) ⊂ Np L and Bk (p)(Np L) ⊂ Tp L. 2) Let DL and WkL be the projections of D and Wk into T L. From the definition of Bk and the previous remark, DL (WkL |L )(p) is self-adjoint. Therefore, WkL is locally a gradient vector field on L. Under these assumptions we have Theorem (Convergence). Given data u0 : Rd → L and ∂u0 ∈ H +1 (Rd ) 4 with > d2 being an integer, then equation (SM) has a unique H +1 solution on [0, T ], where T > 0 for all > 0. Moreover as → 0 the solution u of the Schr¨odinger map equation (SM) converges to p∗ which is a solution of a generalized wave map equation into L, Dt ∂t p∗ − (ıBk + Bk ı) Dt ∂k p∗ − Bk B Dk ∂ p∗ = ıDBk (∂t p∗ )∂k p∗ + Bk DB (∂k p∗ )∂ p∗ − ıG (ıDk ∂k p∗ ) 1 +ıD G ıBk − ıBk G (V )(∂k p∗ ) + D2 ıG (ıG ) (V , V ), 2
(1.1)
where V = (G |Np∗ L )−1 (−ı∂t p∗ − Bk ∂k p∗ ), and the initial data is given by p∗ (0) = u0 , ∂t p∗ (0) = ıBk ∂k u0 . Remark. 1) Local well-posedness for (SM) in Sobolev spaces follows from the methods of [6, 9, 12] and the a priori estimates to be obtained in Sect. 2. 2) Assumption A1) implies that, at p ∈ L, Tp L and Np L are invariant under ıBk , therefore −ı∂t p∗ − Bk ∂k p∗ ∈ Np∗ L and V is well defined. 3) The convergence still holds following the same proof if the data u0 depends on 0) and there exist c1 > 0 and an integer > d2 such that ∂u0 and G (u converge in H as → 0 and |∂u0 |H + |∂u0 |H +1 + |
G (u0 ) |H c1 .
In this case, the limit equation remains the same and
G (u0 ) p∗ (0) = lim u0 , ∂t p∗ (0) = lim ıBk ∂k u0 + ı . →0 →0 4) For the data u0 satisfying the assumptions in 3), since WkL |L is locally a gradient vector field as pointed out in the remarks following Assumptions A1) and A2), we have, | Wk , ∂k udx| c1 , k = 1, . . . , d for some c1 > 0, and thus E (u0 ) c0 . (u0 (−∞) = u0 (+∞) is assumed if d = 1.) 4 Here ∂u ∈ H +1 (Rd ) means that ∂u exists and ∂u belongs to H +1 (Rd ). See more details at 0 0 0 the end of this section.
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Since the derivatives of u and consequently of the limit p∗ are all taken on T M, it may not be immediately clear that, given data p(0) ∈ L and ∂t p(0) ∈ Tp(0) L, the derivatives of p(t) in Eq. (1.1) are on T L and that (1.1) is a generalized wave map Equation on L. Thus we prove Proposition 1.1. Given data p(0, x) ∈ L and ∂t p(0, x) ∈ Tp(0,x) L, such that ∂p(0), ∂t p(0) ∈ H (Rd ), where > min{ d2 , 1} is an integer, the solution p(t, x) ∈ L. The proposition is proved in Sect. 4 by using the normal bundle coordinate system in a neighborhood of L, to be introduced below. Another form of the limit equation, which only depends on the connection on T L, is also given in Sect. 4. In Sect. 3, we show that, for the anti-ferromagnetic equations (AF), Assumptions A1) and A2) are satisfied, and thus the conclusion that the dynamics in this case is governed by σ -model will be a corollary of our result. Corollary (Antiferromagnetic limit). Given smooth data u0 for (AF) so that E (u0 ) ≤ c0 , then as the lattice size a → 0, the solution (u, v) → (p∗ , −p∗ ), where p∗ is a solution to the wave map into S2 , ∂t2 p∗ − ∂x2 p∗ + (|∂t p∗ |2 − |∂x p∗ |2 )p∗ = 0, p∗ (0) = u0 , ∂t p∗ (0) = u0 ∧ ∂x u0 . Notations and function spaces. Most of the notations we employ in this paper are standard. We write x y for x ≤ Cy for some constant C. For any integer , H (Rd ) denotes the usual Sobolev space of functions whose derivatives up to the th order are in L2 . For maps from Rd to M, the space H (Rd ; M) for integers > d2 can be defined by isometrically embedding M into Rk as a Riemannian submanifold. A map u : Rd → M belongs to H (Rd ; M) if φ ◦ u ∈ H (Rd ; Rk ), where φ : M → Rk is the embedding.5 Since the space H (Rd ) = H (Rd ; M) contains only continuous maps for integers > d/2, H (Rd ) can also be defined using local coordinates on M. The Sobolev norm of u is defined in terms of the Sobolev norms of ∂u, viewed as vector fields of the pull-back bundle u−1 T M. Given M ⊂ Rk and a map u ∈ H (Rd , M) for an integer > d2 , the Sobolev norms of vector fields v ∈ u−1 T M can be defined by either using partial derivatives or covariant derivatives. In fact, we have ∂k |v|2 = 2Dk v, v ⇒ |∂k |v(x)|| ≤ |Dk v(x)| . This implies that for u ∈ H (Rd , M), > d2 , and v ∈ u−1 T M, j=0 ∂ j v 2 j=0 D j v 2 j=0 ∂ j v L
L
L2
,
where v is considered as a vector field valued in Rk in ∂v. This can be derived by writing Dk v = ∂k v + v, ∂k na (u) na (u), where {na } is an orthonormal basis to the normal bundle N M. The equivalence now follows by Sobolev embedding and H¨older’s inequality. Therefore for u ∈ H (Rd ), 5
Note that the symplectic structure of M is not involved in the definition of the Sobolev spaces.
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def integers > d2 , the space H (Rd ) = H Rd ; u−1 T M and its norm can be defined by either using partial derivatives or covariant derivatives in the norm. Finally, in this paper we make the convention ∂u = (∂1 u, . . . , ∂d u),
∂ xt u = (∂t u, ∂1 u, . . . , ∂d u).
Meanwhile, D without subscripts will denote only spatial covariant differentiation. 2. Proof of the Main Result To prove convergence, we start by obtaining local in time a priori bounds that are independent of . This will be achieved by using suitable coordinates that distinguish the “tangential” components on L versus the “normal” components on N L = ıT L. These coordinates involve covariant differentiations near submanifolds and exponential maps, see, for example, [17], for more geometric details. The a priori bounds are then obtained in two steps. In the first step, we obtain energy estimates on derivatives of u up to order + 1, > d2 , in terms of derivatives of n , where n is the normal coordinate. In the second step, we use elliptic estimates to bound derivatives of n in terms of + 1 derivatives of u, thus closing the a priori estimates argument. Covariant differentiation on N L. Given a K¨ahler manifold M and a Lagrangian submanifold L ⊂ M, let NL = (p, n) : p ∈ L, n ∈ Np L denote the normal bundle to L. For the manifold N L, we will first introduce a representation of its tangent bundle T (N L) as T L ⊕ N L, and then define a Riemannian on T (N L), which is different from the corresponding metric as well as a connection D Riemannian connection, but compatible with the metric. The submanifold L has a Riemannian structure induced by M and thus an induced covariant derivative DL which is the projection of D onto the tangent subspace T L. The fibers of the bundle N L have an inner product, again induced by M, and a compatible covariant derivative D⊥ , the normal component of D. Given any (p, n) ∈ N L and V ∈ T(p,n) NL, let γ (s) = (p(s), n(s)) ∈ N L be a smooth curve such that γ (0) = (p, n) and γ˙ (0) = V . Let X = ∂s p(0) ∈ Tp L, Y = Ds⊥ n(0) = Ds n(0) − DsL n(0) ∈ Np L. Define L(X, Y ) = V . By using local coordinates, it is straightforward to check that L : T L ⊕ NL → T (N L) is an isomorphism. In terms of the representation L, define an inner product on T (N L) by L(X1 , Y1 ), L(X2 , Y2 ) = X1 , X2 + Y1 , Y2 on T (N L) as and a covariant derivative D DL(X, Y ) = L(DL X, D⊥ Y ).
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is a connection and is compatible with ·, · , i.e. for vector It is easy to check that D fields V1 , V2 on N L, 1 , V2 + V1 , DV 2. ∂ V1 , V2 = DV For later estimates, we need to calculate the tensor created by commuting the covar which iant differential operator D⊥ . This tensor is also a component of the torsion of D is not necessarily the Rieturns out to be generally nonzero except on L and thus D mannian connection corresponding to ·, · . We will need the second fundamental form : N L ⊕ T L → T L. For any p ∈ L, n0 ∈ Np L, and X ∈ Tp L, (n0 , X) is defined as the tangential component of DX n(p) for any normal vector field n of L so that n(p) = n0 . It is standard that (n0 , ·) is self-adjoint for any fixed n0 and is independent of the choice of the vector field n. Given a normal vector n(s1 , s2 ) ∈ Np(s1 ,s2 ) L with parameters s1 , s2 , it is easy to see Ds⊥1 Ds⊥2 n = Ds1 Ds2 n − (n, ps2 ) − (Ds⊥2 n, ps1 ) = Ds1 Ds2 n + (·, ps1 )∗ (n, ps2 ) + tangential terms.
Since the above left side is normal, the tangential terms can be ignored and we obtain Ds⊥1 Ds⊥2 n−Ds⊥2 Ds⊥1 n = R ⊥ (ps2 , ps1 )n+(·, ps1 )∗ (n, ps2 ) − (·, ps2 )∗ (n, ps1 ), (2.1) where R ⊥ is the normal component of the curvature of M and ∗ denotes the adjoint operator. Consequently, given any (p, n) ∈ N L and Vi = L(Xi , Yi ) ∈ T(p,n) N L, i = 1, 2, one can verify T (V1 , V2 ) = L 0, R ⊥ (X2 , X1 )n + (·, X1 )∗ (n, X2 ) − (·, X2 )∗ (n, X1 ) . For simplicity of notations, we make a convention in this paper that, for X ∈ Tp L and Y ∈ Np L, (X, Y ) always mean L(X, Y ) when it appears as a vector in T(p,n) (N L), n ∈ Np L. Geodesic coordinate system near L. Define the exponential map φ : N L → M as φ(p, n) = expp (n),
(p, n) ∈ N L.
It is easy to check that φ is a diffeomorphism from a neighborhood of L ⊂ N L to a neighborhood of L ⊂ M. Moreover, for (X, Y ) ∈ Tp L ⊕ Np L = T(p,0) , N L, φ (p, 0)(X, Y ) = X + Y, which can be easily verified separately for X ∈ Tp L and Y ∈ Np L. We will express equation (SM) in terms of (p, n) by using the exponential map in a neighborhood of L ⊂ NL as a coordinate system for a neighborhood of L ⊂ M. Using (p, n) as a coordinate system and writing u = φ(p, n) property c) of G can be expressed in terms of the Taylor expansion as: 1 a(p)n, n + O(|n|3 ), 2 a(p) : Np L → Np L, a(p) = a(p)∗ and a(p) ≥ I.
G(u) =
(2.2)
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The higher order terms in n will be small, have no effect on estimates on the solution u, and will vanish in the limit6 . Therefore from here on we will assume, without any loss of generality, that G (φ(p, n)) =
1 a(p)n, n , 2
and consequently G is given by: G (u), φ (X, Y ) =
1 a (p)(X)n, n + an, Y 2
or G (φ(p, n)) = (φ ∗ )−1
(2.3)
1 a (p)(.)n, n∗ , an , 2
where X ∈ T L and Y ∈ N L. Energy estimates. To obtain energy estimates we rewrite equation (SM) as
1 ∂t u − ıBk ∂k u = ı −Dk ∂k u + G (u) , and apply the operator Dt − B ıD to obtain ı (Dt − B ıD ) (∂t u − ıBk ∂k u) = −ıDk2 (∂t u − ıB ∂ u) + G (∂t u − ıB ∂ u) ı + G ıBk − ıBk G ∂k u + R1 (D∂u, ∂u) + R2 (∂u, ∂u, ∂ xt u), where multi-linear terms R1 and R2 , smooth in u, come from commuting covariant derivatives and differentiating Bk . Substituting (SM) in the above equation we obtain (Dt − B ıD ) (∂t u − ıBk ∂k u) = − 2 Dk2 D ∂ u + Dk2 G (u) − ıG (ıDk ∂k u) ı ı + 2 G (ıG ) + G , ıBk ∂k u+R1 (D∂u, ∂u)+R2 (∂u, ∂u, ∂ xt u) (2.4) which can be rewritten as Dt ∂t u − (ıBk + Bk ı) Dk ∂t u − B Bk D Dk u = (ıDBk (∂t u) + B DBk (∂ u)) ∂k u ı ı G , ıBk ∂k u + Dk (G (∂k u)) − ıG (ıDk ∂k u) + 2 G (ıG ) + − 2 Dk2 D ∂ u + R1 (D∂u, ∂u) + R2 (∂u, ∂u, ∂ xt u). (2.5) By Assumption A1) G ıBk − ıBk G = 0
The two terms D[G , ıBk ](V )(∂k p∗ ) + 21 D2 ıG (ıG ) (V , V ) in the limit equation seem to depend on G and G(4) on L. However, under the current assumptions, some simple calculation shows they do not. See Sect. 4. 6
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on L and this implies n 1 G ıBk − ıBk G = R3 , where R3 is a smooth function of u acting linearly on n . Moreover since ıG = O(|n|) and ıG → ω ∈ ker(G ) as n → 0, |ıG | then n n 1 , , G (ıG ) = R 4 2 where R4 a bilinear function acting on can be written as
n
which depends smoothly on u. Thus Eq. (2.5)
Dt ∂t u − (ıBk + Bk ı) Dk ∂t u − B Bk D Dk u = Dk (G (∂k u)) − ıG (ıDk ∂k u) + (ıDBk (∂t u) + B DBk (∂ u)) ∂k u − 2 Dk2 D ∂ u n n n ∂k u + R4 , + R1 (D∂u, ∂u) + R2 (∂u, ∂u, ∂ xt u). (2.6) + R3 Lemma 2.1. Under Assumptions A1) and A2) smooth solutions to Eq. (2.6) satisfy the following energy estimate for > d2 2 2 2 |∂u(t)|2H +1 + ∂ xt u(t)H 2 |∂u(0)|2H +1 + ∂ xt u(0)H
t n(s) , |∂ xt u(s)| , |∂u(s)| +1 ds, + g H H H 0 where g is smooth and has only quadratic and higher order terms near zero. Proof. This is a standard energy estimate for wave equations. Differentiate Eq. (2.6) j times with respect to the spatial variables, multiply by D j ∂t u and sum up with respect to all j th order multiple indices to obtain 2 d j 2 D ∂t u + 2 D j +1 ∂u + G (u)D j ∂k u, D j ∂k u dt 2 + G (u)ıD j ∂k u, ıD j ∂k u − Bk D j ∂k u dx
, n , D n , . . . , D j n , ∂ xt u, . . . , D j ∂ xt u, D j +1 ∂u dx, (2.7) = R is a polynomial of the above arguments with coefficients smooth in u. Moreover, where R each of its monomials has a bound in the form n ˜ C(u) j0 · i |D ji −1 | · i |D ji −1 ∂ xt u|, where − j0 + i ji + i j˜i = 2j + 3.
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From the Sobolev inequalities and H¨older inequality, for j ≤ and > d2 , the right side of (2.7) is
n xt gj , |∂ u|H , |∂u|H +1 , H where gj is smooth and has only quadratic and higher order terms near zero. On the other hand, from assumptions c) and (A2) on G, we have, for any vectors ξ1 , . . . , ξd ∈ Tu M, G (u)ξk , ξk + G (u)ıξk , ıξk − |Bk (u)ξk |2 G (u)ξk , ξk + G (u)ıξk , ıξk − c|n| · k |ξk |2 k |ξk |2 for n in a bounded small neighborhood of 0. Therefore 2 G (u)D j ∂k u, D j ∂k u + G (u)ıD j ∂k u, ıD j ∂k u − |Bk D j ∂k u|2 D j ∂u . The conclusion of the lemma follows from integrating (2.7) with respect to t, using the above inequalities and summing up for j = 1, . . . , . Remark. Assumption A2) can be weakened by using better estimates. For example, if we rewrite Eq. (2.6) in terms of V = ∂t − 21 (ıBk + Bk ı)∂k u instead of ∂t u then we can obtain energy estimates of V under a weaker Assumption A2) on B while another qualitative assumption on B would be needed. However, Assumption A2) is sufficient to solve the anti-ferromagnetic problem and helps to keep the presentation simple. Elliptic estimates on n. To estimate derivatives of n we express the Schr¨odinger map equation (SM) in terms of (p, n) by using the geodesic coordinate map φ. With u = φ(p, n) equation (SM) can be written as
2 ⊥ L φ ∂t p, Dt n = ı − φ Dk ∂k p, Dk⊥ n −φ ∂k p, Dk⊥ n , ∂k p, Dk⊥ n
1 1 ∗ ⊥ ∗ −1 + Bk φ ∂k p, Dk n + (2.8) φ a (p)n, n , an . 2
The normal component of Eq. (2.8) is obtained by applying Qφ −1 ı, where Q(p, n) : T(p,n) N L → Np L is the projection Q(p, n)(X, Y ) = Y , 2 −1 1 an Qφ −1 (ıut + Bk ∂k u) = Dk⊥ n − Q φ ∗ φ −1 1 −1 −1 + Qφ −1 φ φ ∂k u, φ ∂k u − Q φ ∗ φ a (p)n, n . 2 Since −1 φ (p, 0)(X, Y ) = X + Y ⇒ Q φ ∗ φ an = an + R5 (p, n)n, where R5 is of order n, we can rewrite the above equation as 2 def − 2 Dk⊥ n + an = R6 (∂ xt u) + O(|n|2 ) = RH S,
(2.9)
where R6 is a smooth function of u, ∂ xt u and and R6 = 0 if ∂ xt u = 0. Equation (2.9) is an elliptic equation for n and from standard L2 theory we obtain
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Lemma 2.2. For sufficiently small, smooth solutions to Eq. (2.9) satisfy n + |n|H +1 ≤ C(|∂ xt u|H ), H where C is smooth and C(0) = 0, for any > d2 . Proof. This is standard L2 theory for semilinear elliptic systems. The only thing that one has to be careful about is the fact that we have a lower bound on a only when restricted to Np L. One notices that Dk n = Dk⊥ n + (n, ∂k p), where is the second fundamental form. From similar identities for higher order derivatives, we have for j ≥ d2 , |n|H j ≤ |(D ⊥ )j n|L2 + C(|∂u|H j −1 )|n|L2 . Therefore we only need to estimate (D ⊥ )j n. j Applying the operator D ⊥ to Eq. (2.9) and multiplying by D ⊥j n we obtain 2 D ⊥j (Dk⊥ )2 n, D ⊥j n + D ⊥j (a(p)n), D ⊥j n dx ≤ D ⊥j RH S, D ⊥j n dx 2 2 1 1 ≤ D ⊥j RH S 2 + D ⊥j n 2 . L L 2 2 By integration by parts and commuting the derivatives with a(p) and using the lower bound on a(p) we obtain for j ≤ , 2 2 2 2 1 1 2 D ⊥j +1 n 2 + D ⊥j n 2 D ⊥j RH S 2 + D ⊥j n 2 L L L L 2 2 ⊥j + C(|∂u|H ) |n|H j −1 |D n|L2
+ 2 |n|2H j + 2 |n|H j −1 |n|H j +1 ,
which implies 2 2 2 2 D ⊥j +1 n 2 + D ⊥j n 2 D ⊥j RH S 2 + |n|2H j −1 . L
L
L
Iterate these equations from 0 to j to obtain 2 2 2 j 2 D ⊥j +1 n 2 + D ⊥j n 2 i=0 D ⊥i RH S 2 , L
L
L
and summing over j = 1, · · · , to get 2 2 ⊥i 2 D ⊥ n + |n|2H i=0 D RH S 2 . H
L
(2.10)
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Finally it is easy to see from (2.9) that for j ≤ , 2 ⊥j D RH S 2 ≤ 2 C1 (|∂ xt u|H ) + C2 (|∂ xt u|H )|n|4H , L
which when combined with (2.10) implies 2 2 D ⊥ n + |n|2H ≤ 2 C1 (|∂ xt u|H ) + C2 (|∂ xt u|H )|n|4H . H
The result now follows for sufficiently small.
Proof of convergence. Based on the above a priori estimates, local smooth solutions to (SM) where the existence time might depend on can be constructed in a variety of methods [6, 9, 12]. To show that the solution will exist on a time interval independent of , first we observe that since the initial data u0 ∈ L then the normal component of u, n ∈ Np L remains small for some time. By Lemmas 2.1 and 2.2 we know that as long as 0 ≤ t ≤ δ, a fixed small constant depending on the initial data but independent of , the following bound holds xt n(t) C(∂ xt u(0) , |∂u(0)| +1 ). (2.11) |∂u(t)|H +1 + ∂ u(t) H + H H H By bootstrapping the solution will exist and satisfy the above bound on the time interval [0, δ]. From these bounds we also have that as → 0, at least for a subsequence, u = φ(p, n) −→ p∗ ∈ L in H , n in H −1 . −→ ζ ∈ Np∗ L The vector field ζ can be computed using the Schr¨odinger map equation (SM), where as → 0 1 G (u) → −ı∂t p∗ − Bk ∂k p∗ , and from Eq. (2.3) where as → 0, 1 G (u) → a(p∗ )ζ = G (p∗ )ζ, which implies that G (p∗ )ζ = −ı∂t p∗ − Bk ∂k p∗ . Moreover since G (p∗ ) = a(p∗ ) is invertible on NL, we have −1 {−ı∂t p∗ − Bk ∂k p∗ } . ζ = G (p∗ )|Np∗ L To derive the equation that p∗ satisfies, we consider Eq. (2.4), (Dt − B ıD ) (∂t u − ıBk ∂k u) = − 2 Dk2 D ∂ u + Dk2 (G (u)) − ıG (ıDk ∂k u) ı ı G , ıBk ∂k u + R1 (D∂u, ∂u) + R2 (∂u, ∂u, ∂ xt u). + 2 G (ıG ) +
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From the estimates (2.11) and |G (u)| = O(|n|) we have (Dt − B ıD ) (∂t u − ıBk ∂k u) → (Dt − B ıD ) (∂t p∗ − ıBk ∂k p∗ ) Dk2 G (u) − 2 Dk2 D ∂ u + DB (∂k u)Dk ∂ u + O(|∂u|3 ) → 0 −ıG (ıDk ∂k u) → −ıG (ıDk ∂k p∗ ),
where the convergence for some higher order derivative terms may be in the distribution sense if d = 1 and l = 1. Expanding the remaining terms in n we have n 1 G , ıBk = D G , ıBk |p∗ + O(), n n 1 1 2 G (ıG ) = {G ıG }| D , + O(). p ∗ 2 2 Therefore as → 0, 1 G , ıBk → D G , ıBk ζ, 1 G (ıG ) → D2 {G (ıG )}(ζ, ζ ). 2 Putting these limits together we obtain ı (Dt − B ıD ) (∂t p∗ − ıBk ∂k p∗ ) = −ıG (ıDk ∂k p∗ ) + D2 {G (ıG )}(ζ, ζ ) 2 + ıD G , ıBk (ζ )∂k p∗ , −1 {−ı∂t p∗ − Bk ∂k p∗ } and this completes the proof. where ζ = G (p∗ )|Np∗ L
3. Anti-Ferromagnetism The anti-ferromagnetic system (AF) can be viewed as a special case of the Schr¨odinger map (SM) with i) M = S2 × S2 and L is the anti-diagonal subset {(u, v) ∈ S2 × S2 | u + v = 0} ∼ S2 . 2 and B(u, v)(ξ1 , ξ2 ) = (−ξ2 + ξ2 , uu, ξ1 − ξ1 , vv) = ii) G(u, v) = |u+v| 2 (−P (u)ξ2 , P (v)ξ1 ) for any ξ1 ∈ Tu S2 , ξ2 ∈ Tv S2 , where P (u) is the orthogonal projection to Tu S2 . It is easy to compute G (u, v) = (v − u, vu, u − u, vv) = (P (u)v, P (v)u), G (u, v)(ξ1 , ξ2 ) = (ξ2 − ξ2 , uu − u, vξ1 , ξ1 − ξ1 , vv − u, vξ2 ), which lead to [G ıB − ıBG ](u, v)(ξ1 , ξ2 ) = ((u + v) ∧ ξ1 − u, v ∧ ξ1 u − ξ1 , vu ∧ v, −(u + v) ∧ ξ2 + v, u ∧ ξ2 v + ξ2 , uv ∧ u).
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Thus it is clear that G ıB − ıBG = 0 on L and Assumption A1) is satisfied, while Assumption A2) turns out to be an equality for λ = 2. In fact, at (u, −u) ∈ L, G (u, −u)(ξ1 , ξ2 ) = (ξ1 + ξ2 , ξ1 + ξ2 ), G (u, −u)(u ∧ ξ1 , −u ∧ ξ2 ) = (−u ∧ ξ2 + u ∧ ξ1 , −u ∧ ξ2 + u ∧ ξ1 ), and the rest follows from direct calculation. Furthermore, we claim D[G ıB − ıBG ] = 0 on L.
(3.1)
In fact, given (u0 , −u0 ) ∈ L and (U, V ), (ξ1 , ξ2 ) ∈ T(u0 ,−u0 ) (S2 × S2 ), let (u(s), v(s)) be a curve on S2 × S2 such that (u(0), v(0)) = (u0 , −u0 ),
(us (0), vs (0)) = (U, V ).
Let (ξ1 (s), ξ2 (s)) be the parallel translation of (ξ1 , ξ2 ) along this curve, i.e. P (u(s))ξ1s (s) = P (v(s))ξ2s (s) = 0. Then D(U,V ) [G ıB − ıBG ](u, v)(ξ1 , ξ2 ) = ∂s {[G ıB − ıBG ](u(s), v(s))(ξ1 (s), ξ2 (s))}|s=0 , where the right side turns out to be 0 through straightforward calculations. On the other hand, ıG (ıG ) = (1 + u, v)(v − u, vu, u − u, vv) =
|u + v|2 (P (u)v, P (v)u), 2
and thus it is clear D(ıG (ıG )) = 0 and D2 (ıG (ıG )) = 0 on L. Moreover, one can verify that, on L, B 2 = −I,
Bı = −ıB,
DB = 0.
Note that G |L is just twice of the projection to N L. Therefore, from Theorem 1, the limit of the anti-ferromagnetic equation (AF) as a → 0 becomes Dt ∂t p∗ + Dx ∂x p∗ = 2DxL ∂x p∗ . Finally, note that L = {(u, v) | u + v = 0} is a geodesic submanifold of M = S2 × S2 , we have DL = D on L and the limit equation turns out to be the wave map equation (σ ).
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4. Proof of Proposition 1.1 In this section, we will show that Eq. (1.1) is intrinsic on L. Equation (1.1) can be rewritten as (Dt − B ıD ) (∂t p∗ − ıBk ∂k p∗ ) = −ıG (ıDk ∂k p∗ ) 1 + ıD G ıBk − ıBk G (V )(∂k p∗ ) + D2 ıG (ıG ) (V , V ), 2
(4.1)
where V = (G |Np∗ L )−1 (−ı∂t p∗ − Bk ∂k p∗ ). Assuming that p∗ ∈ L and ∂t p∗ , ∂k p∗ ∈ Tp∗ L, the differentiation on the left side of Eq. (4.1) generates a normal component due to the imbedding of the submanifold L into M, which we denote by NL. This normal component NL can be expressed in terms of the second fundamental form of L and does not depend on ∂ 2 p∗ , although it comes from differentiation. In order to prove that Eq. (1.1) is intrinsically defined on L, it suffices to show that NL coincides with the normal component of the right side, which we denote by NR. We compute NL and NR by using the normal bundle coordinates φ(p, n) introduced in Sect. 2. The first derivative of φ at (p, 0) is given by φ (p, 0)(X, Y ) = X + Y, for any (X, Y ) ∈ T(p,n) N L in the form given in Sect. 2. To compute the second derivative φ (p, 0), we notice that, even though φ (p, n) is not necessarily symmetric since D˜ is not torsion free, φ (p, 0) is symmetric as the torsion vanishes at (p, 0). Therefore, we only need to compute φ (p, 0)((X, 0), (X, 0)),
φ (p, 0)((X, 0), (0, Y )),
φ (p, 0)((0, Y ), (0, Y )).
The first and the second terms in the above expression can be calculated in terms of the second fundamental form of L in the following manner. Recall that, for any p ∈ L, n0 ∈ Np L, and X ∈ Tp L, (n0 , X) ∈ Tp L is the tangential component of DX n(p) for any normal vector field n on L so that n(p) = n0 . Let p(s) be a geodesic on L so that p(0) = p and ps (0) = X. Let X(s) = ps (s) and Y (s) be extended along p(s) so that Ds⊥ Y (s) = 0, i.e. D˜ s (X(s), 0) = D˜ s (0, Y (s)) = 0. Then φ (p, 0)((X, 0), (X, 0)) = Ds (φ (p(s), 0)(X(s), 0))|s=0 = Ds X(s)|s=0 which is clearly in Np L and for any n ∈ Np L, Ds X(s), n = −X(s), (n, X(s)). Therefore φ (p, 0)((X, 0), (X, 0)) = −(·, X)∗ X. Similarly, φ (p, 0)((X, 0), (0, Y )) = Ds (φ (p(s), 0)(0, Y (s))|s=0 = Ds Y (s)|s=0 = (Y, X). The third term φ (p, 0)((0, Y ), (0, Y )) = 0 since φ(p, sY ) is a geodesic on M. Thus, φ (p, 0)((X1 , Y1 ), (X2 , Y2 )) = (Y1 , X2 ) + (Y2 , X1 ) − (·, X2 )∗ X1 . In the (p, n) coordinates, G has the form G ◦ φ(p, n) =
1 a(p)n, n + G1 (p, n), 2
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where a(p) : Np L → Np L is positive and symmetric and G1 , G1 , and G1 vanish when n = 0, which implies that, as a trilinear form, G 1 (p, 0) vanishes whenever one of the three vectors it acts on belongs to Tp L. Therefore, at any (p, 0) and for any (Xj , Yj ) ∈ T(p,0) N L, j = 1, 2, 3, we have G (X1 + Y1 ) = aY1 , (DX1 +Y1 G )(X2 + Y2 ), X3 + Y3 = (aY1 , X2 ), X3 + (aY2 , X3 ), X1 + (aY3 , X1 ), X2 + a (X1 )Y2 , Y3 + a (X2 )Y3 , Y1 + a (X3 )Y1 , Y2 + (DY1 G1 )(Y2 ), Y3 , where a (p) : Tp L → L(Np L) is linear. Since p∗ ∈ L, we have for any n ∈ Np∗ L, NR, n = ın, G (ıDk ∂k p∗ ) − (DV G )ıBk (∂k p∗ ) − G ı(DBk (V ))(∂k p∗ ) + n, (DBk (V ))G (∂k p∗ ) + Bk (DV G )(∂k p∗ ) 1 − ın, G ı(DV G )(V ) + (DV G )ıG (V ). 2 Since the range of G is Np∗ L and its kernel is Tp∗ L, NR, n = −ın, (DV G )ıBk (∂k p∗ ) + (DV G )ıG (V ) + n, Bk (DV G )(∂k p∗ ). Substituting V and the expression for Dv G , we obtain NR, n = −(aV , ıBk (∂k p∗ )), ın − (aV , ıaV ), ın − (aV , ∂k p∗ ), Bk n = −(aV , ∂t p∗ ), ın − (aV , ∂k p∗ ), Bk n, where we used the fact that V ∈ Np∗ L and Bk is anti-self-adjoint, mapping Np∗ L and Tp∗ L into each other. On the other hand NL, n = −(Dt − ıBk Dk ) (aV ), ın = −(aV , ∂t p∗ ), ın − (aV , ∂k p∗ ), Bk n. Therefore, NL=NR and Eq. (4.1) is intrinsic on L. In fact, the tangential component of (4.1) may also be computed in the same fashion and the limit equation can be rewritten as
DtL − B ıDL (∂t p∗ − ıBk ∂k p∗ ) = −ıaıDkL ∂k p∗ + ıa (ıBk ∂k p∗ )V 1 + Bk a (∂k p∗ )V + ıaı(DBk (V ))∂k p∗ + ıa (ıaV )V + {a (ıaı·)V , V }∗ , 2 (4.2)
where the last term denotes the vector in Tp∗ L given by the duality on Tp∗ L with respect to ·, ·.
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References 1. Anzellotti, G., Baldo, S., Visintin, A.: Asymptotic behavior of the Landau-Lifshitz model of ferromagnetism. Appl. Math. Optim. 23(2), 171–192 (1991) 2. Aharoni,A.: Introduction to the theory of ferromagnetism, Second edition. Oxford: Oxford University Press, 2000, xi+319 pp. 3. Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schr¨odinger equation in H s . Nonlinear Anal. 14(10), 807–836 (1990) 4. Cazenave, T.: Semilinear Schr¨odinger equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, Providence, RI: Amer. Math. Soc., 2003. xiv+323 pp. 5. Chang, N.-H., Shatah, J., Uhlenbeck, K.: Schr¨odinger maps. Commun. Pure Appl. Math. 53(5), 590–602 (2000) 6. Ding, W., Wang, Y.: Local Schr¨odinger flow into K¨ahler manifolds. Sci. China Ser. A 44(11), 1446– 1464 (2001) 7. Grenier, E.: Semiclassical limit of the nonlinear Schr¨odinger equation in small time. Proc. Am. Math. Soc. 126(2), 523–530 (1998) 8. Guo, B., Ding, S.: Initial-boundary value problem for the Landau-Lifshitz system with applied field. J. Part. Differ. Eqs. 13(1), 35–50 (2000) 9. Kenig, C.E., Ponce, G., Vega, L.: Small solutions to nonlinear Schr¨odinger equations. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 10(3), 255–288 (1993) 10. Ko, J.: Ph.D dissertation, Courant Institute, New York University, 2004 11. Komineas, S., Papanicolaou, N.: Vortex dynamics in two-dimensional antiferromagnets. Nonlinearity 11, 265–290 (1998) 12. MacGahagan, H.: Ph.D dissertation, Courant Institute, New York University, 2004 13. Mikeska, H.J., Steiner, M.: Solitary excitations in one-dimensional magnets. Adv. Phys. 40, 191–356 (1991) 14. Nahmod, A., Stefanov, A., Uhlenbeck, K.: On Schr¨odinger maps. Commun. Pure Appl. Math. 56(1), 114–151 (2003) 15. Papanicolaou, N., Tomaras, T.N.: Dynamics of magnetic vortices. Nucl. Phys. B 360(2–3), 425–462 (1991) 16. Prohl, A.: Computational micromagnetism. Advances in Numerical Mathematics. Stuttgart: B. G. Teubner, 2001. xviii+304 pp. 17. Spivak, M.: A comprehensive introduction to differential geometry. Vol. IV, Second edition. Wilmington, DE: Publish or Perish, Inc., 1979. viii+561 pp. 18. Sulem, P.-L., Sulem, C., Bardos, C.: On the continuous limit for a system of classical spins. Commun. Math. Phys. 107(3), 431–454 (1986) 19. Visintin, A.: On Landau-Lifshitz equations for ferromagnetism. Japan J. Appl. Math. 2(1), 69–84 (1985) 20. Zhang, P.: Semiclassical limit of nonlinear Schrdinger equation. II. J. Part. Differ. Eqs. 15(2), 83–96 (2002) Communicated by P. Constantin
Commun. Math. Phys. 262, 317–341 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1493-4
Communications in
Mathematical Physics
Dyson’s Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel Torsten Ehrhardt Department of Mathematics, University of California, Santa Cruz, CA95064, USA. E-mail: [email protected] Received: 22 July 2004 / Accepted: 19 July 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: We prove that the asymptotics of the Fredholm determinant of I − Kα , where Kα is the integral operator with the sine kernel sin(x−y) π(x−y) on the interval [0, α], are given by log det(I − K2α ) = −
α2 log α log 2 − + + 3ζ (−1) + o(1), 2 4 12
α → ∞.
This formula was conjectured by Dyson. The proof for the first and second order asymptotics was given by Widom, and higher order asymptotics have also been determined. In this paper we identify the constant (or third order) term, which has been an outstanding problem for a long time. 1. Introduction Let Kα be the integral operator defined on L2 [0, α] with the kernel k(x, y) =
sin(x − y) . π(x − y)
(1)
Dyson conjectured the following asymptotic formula for the determinant det(I − K2α ), log det(I − K2α ) = −
log α log 2 α2 − + + 3ζ (−1) + o(1), 2 4 12
α → ∞, (2)
(where ζ stands for the Riemann zeta function) and provided heuristic arguments [7]. Later on Jimbo, Miwa, Mˆori and Sato [11] (see also [15]) showed that the function σ (α) = α
d log det(I − Kα ) dα
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satisfies a Painlev´e V equation. Widom [17, 18] determined the highest term in the asymptotics of σ (α) as α → ∞. Knowing these asymptotics one can derive a complete asymptotic expansion for σ (α). From this it follows by integration that the asymptotic expansion of det(I − K2α ) is given by C2n α2 log α + O(α 2N+2 ), − +C+ 2 4 α 2n N
log det(I − K2α ) = −
α → ∞, (3)
n=1
with effectively computable constants C2n . Clearly, the constant C cannot be obtained in this way, and its (rigorous) computation was an outstanding problem for a long time. Only recently, Krasovsky [12] was able to make Dyson’s heuristic derivation of the asymptotic formula (2) rigorous by using the so-called Riemann-Hilbert method. It is the purpose of this paper to give another proof of (2) by using a different approach. The determinant det(I − Kα ) appears in random matrix theory [10]. It is related to the probabilities E2 (n; α) that in the bulk scaling limit of the Gaussian Unitary Ensemble an interval of length α contains precisely n eigenvalues. In particular, we have det(I − Kα ) = E2 (0; α). Since the asymptotics of the ratios E2 (n; α)/E2 (0; α) as α → ∞ can be computed at least for some n (see [4]), the asymptotics of the sine kernel determinant is of relevance also for Eβ (n; α) with general n. Connections between the determinant det(I − Kα ) and corresponding probabilities for the Gaussian Orthogonal and Symplectic Ensembles also exist [4, 10]. A generalization of the determinant det(I −Kα ), where the sine kernel integral operator is considered on a finite union of finite subintervals of R, was also studied, and results were established by Widom [18] and by Deift, Its and Zhou [6]. This generalization has a similar interpretation in random matrix theory [15]. The theory of Wiener-Hopf determinants can explain at least to some extent the reason for the difficulties one faces with the sine kernel determinant. In fact, I − Kα is a truncated Wiener-Hopf operator Wα (φ) with the generating function equal to the characteristic function of the subset (−∞, −1) ∪ (1, ∞) of R. This generating function does not belong to the already difficult class of Fisher-Hartwig symbols [5], i.e., functions which have only a finite number of zeros or discontinuities of a certain type. Notice that even for Fisher-Hartwig symbols the proof of the (conjectured) asymptotics has not yet been achieved completely. It is useful to take a look at the discrete analogue of the determinants, i.e., Toeplitz determinants Tn (χα ), where the generating function χα is equal to the characteristic function of the subarc { eiθ : α < θ < 2π − α } of the complex unit circle. Widom [16] proved that for fixed α ∈ (0, π) the asymptotics are given by α −1/4 1/12 3ζ (−1) α n2 det Tn (χα ) ∼ cos n sin 2 e , 2 2
n → ∞.
(4)
Dyson’s heuristic derivation relies on this formula and on the fact that discretizing the sine kernel operator yields the Toeplitz operator Tn (χα/n ), i.e., limn→∞ det Tn (χα/n ) = det(I − Kα ). He replaces α by α/n in the right-hand side of (4) and takes the limit n → ∞ to arrive at the asymptotic expression given in (2). Krasovsky shows that this (non-rigorous) argumentation can be made rigorous. He determines the asymptotics of the derivative (in α) of log det Tn (χα ) together with an estimate of the error, which holds uniformly on a certain range of the parameter. Upon integration and using Widom’s result to fix the constant, he arrives at the asymptotic formula (2).
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The proof presented in this paper also uses the discretization idea. However, we do not use the asymptotics of a Toeplitz determinant in order to the fix the constant. Instead, we establish an exact identity between the determinant det(I − Kα ) and what could be considered the determinants of Wiener-Hopf-Hankel operators with Fisher-Hartwig symbols. The asymptotics of these Wiener-Hopf-Hankel determinants were determined in the paper [3]. An outline of the main ideas of our proof will be given in Sect. 3 after having introduced some necessary notation. In Sect. 4 we will establish several auxiliary results and the proof will be given in Sect. 5. 2. Basic Notation Let us first introduce some notation. For a Lebesgue measurable subset M of the real axis R or of the unit circle T = {t ∈ C : |t| = 1}, let Lp (M) (1 ≤ p < ∞) stand for the space of all Lebesgue measurable p-integrable complex-valued functions. For p = ∞ we denote by L∞ (M) the space of all essentially bounded Lebesgue measurable functions on M. For a function a ∈ L1 (T) we introduce the n × n Toeplitz and Hankel matrices Tn (a) = (aj −k )n−1 j,k=0 , where ak =
1 2π
2π
Hn (a) = (aj +k+1 )n−1 j,k=0 ,
a(eiθ )e−ikθ dθ,
(5)
k ∈ Z,
0
are the Fourier coefficients of a. We also introduce differently defined n × n Hankel matrices Hn [b] = (bj +k+1 )n−1 j,k=0 ,
(6)
where the numbers bk are the (scaled) moments of a function b ∈ L1 [−1, 1], i.e., 1 1 b(x)(2x)k−1 dx, k ≥ 1. bk = π −1 Given a ∈ L∞ (T) the multiplication operator M(a) acting on L2 (T) is defined by M(a) : f (t) ∈ L2 (T) → a(t)f (t) ∈ L2 (T). We denote by P the Riesz projection P :
∞ k=−∞
fk t ∈ L (T) → k
2
∞
fk t k ∈ L2 (T)
k=0
and by J the flip operator J : f (t) ∈ L2 (T) → t −1 f (t −1 ) ∈ L2 (T). The image of the Riesz projection is equal to the Hardy space H 2 (T) = f ∈ L2 (T) : fk = 0 for all k < 0 .
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For a ∈ L∞ (T) the Toeplitz and Hankel operators are bounded linear operators defined on H 2 (T) by T (a) = P M(a)P |H 2 (T) ,
H (a) = P M(a)J P |H 2 (T) .
(7)
The matrix representation of these operators with respect to the standard basis {t n }∞ n=0 of H 2 (T) is given by infinite Toeplitz and Hankel matrices, T (a) ∼ = (aj −k )∞ j,k=0 ,
H (a) ∼ = (aj +k+1 )∞ j,k=0 .
(8)
The connection to n × n Toeplitz and Hankel matrices is given by Pn T (a)Pn ∼ = Tn (a),
Pn H (a)Pn ∼ = Hn (a),
(9)
where Pn is the finite rank projection operator Pn :
fk t k ∈ H 2 (T) →
k≥0
n−1
fk t k ∈ H 2 (T).
(10)
k=0
An operator A acting on a Hilbert space H is called a trace class operator if it is compact and if the series constituted by the singular values sn (A) (i.e., the eigenvalues of (A∗ A)1/2 taking multiplicities into account) converges. The norm sn (A) (11) A1 = n≥1
makes the set of all trace class operators into a Banach space, which forms also a twosided ideal in the algebra of all bounded linear operators on H . Moreover, the estimates AB1 ≤ A1 B and BA1 ≤ A1 B hold, where A is a trace class operator and B is a bounded operator with the operator norm B. If A is a trace class operator, then the operator trace “trace(A)” and the operator determinant “det(I + A)” are well defined. For more information concerning these concepts we refer to [9]. Given a ∈ L∞ (R) we denote by MR (a) the multiplication operator MR (a) : f (x) ∈ L2 (R) → a(x)f (x) ∈ L2 (R) and by W0 (a) the convolution operator (or, “two-sided” Wiener-Hopf operator) W0 (a) = FMR (a)F −1 , where F stands for the Fourier transform on L2 (R). The usual Wiener-Hopf operator and the “continuous” Hankel operator acting on L2 (R+ ) are given by W (a) = + W0 (a) + |L2 (R+ ) ,
(12)
HR (a) = + W0 (a)Jˆ + |L2 (R+ ) ,
(13)
where (Jˆf )(x) = f (−x) and + = MR (χR+ ) is the projection operator on the positive real half axis. If a ∈ L1 (R), then W (a) and HR (a) are integal operators on L2 (R) with the kernel a(x ˆ − y) and a(x ˆ + y), respectively, where ∞ 1 a(ξ ˆ )= e−ixξ a(x) dx 2π −∞ stands for the Fourier transform of a.
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Finally, let α stand for the projection operator, α : f (t) ∈ L2 (R+ ) → χ[0,α] (x)f (x) ∈ L2 (R+ ).
(14)
The image of α will be identified with the space L2 [0, α]. 3. Main Ideas of the Proof Before explaining the main ideas of the proof, let us introduce the functions, β x−i β x2 uˆ β (x) = , vˆβ (x) = , x ∈ R, β ∈ C. x+i 1 + x2
(15)
These functions are continuous on R \ {0} with limits being equal to 1 as x → ±∞. Moreover, the function uˆ β has a jump discontinuity at x = 0, while the function vˆβ has a zero, a pole, or an oscillating discontinuity at x = 0. The prevailing part of the paper is devoted to prove the identity
α2 det(I − K2α ) = exp − det α (I + HR (uˆ −1/2 ))−1 α 2
× det α (I − HR (uˆ 1/2 ))−1 α . (16) Therein, as we will see, the inverse operators on the right-hand side exist and the determinants can be understood as operator determinants. Such complicated expressions seem to promise little advantage. However, the above determinants are related to determinants of truncated Wiener-Hopf-Hankel operators [3, Prop. 3.14] . Indeed,
Dˆ α+ (β) := det α (W (vˆβ ) + HR (vˆβ )) α
= e−αβ det α (I + HR (uˆ −β ))−1 α (17) if −1/2 < Re β < 3/2, and
Dˆ α− (β) := det α (W (vˆβ ) − HR (vˆβ )) α
= e−αβ det α (I − HR (uˆ −β ))−1 α
(18)
if −1/2 < Re β < 1/2. One minor complication is encountered since the definition of the Wiener-Hopf-Hankel determinants requires vˆβ − 1 ∈ L1 (R), i.e., Re β > −1/2. This complication can be resolved by remarking that the right-hand side of (18) is well defined for −3/2 < Re β < 1/2. Hence Dˆ α− (β) can be continued by analyticity (in β) to the domain where Re β > −3/2. Identity (16) can thus be rewritten as α2 ˆ + det(I − K2α ) = exp − (19) Dα (1/2)Dˆ α− (−1/2). 2 Since the Wiener-Hopf-Hankel determinants (17) and (18) have symbols of Fisher-Hartwig type, their asymptotics might be easier to analyze. Precisely this is done in the paper [3]. Applying these results we obtain the asymptotics of det(I − K2α ).
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In our proof we will not rely on (19), but only on (16). We are mentioning (19) here only because it represents a much simpler interpretation. Let us now proceed with making some remarks on how the proof of (16) is accomplished. We are first going to discretize the sine kernel operator and obtain a Toeplitz determinant, i.e., det(I − Kα ) = lim det Tn (χ αn ). n→∞
Using an exact identity between determinants of symmetric Toeplitz matrices and Hankel matrices, which was established in [2], we reduce the Toeplitz determinant to a Hankel determinant of the form det Hn [b] with a symbol b depending on n. This symbol is supported on a proper subinterval of [−1, 1]. One crucial observation is that since the entries of this type of Hankel matrix are defined as the moments of its symbol, it is possible (by pulling out an appropriate factor) to arrive at a Hankel determinant det Hn [bα,n ] with a certain symbol bα,n which is supported on all of [−1, 1]. It is interesting to observe that the factor which has been pulled out gives precisely (after taking the limit n → ∞) the exponential term in (16). Hence at this point we have shown that α2 det(I − Kα ) = exp − · lim Hn [bα,n ]. n→∞ 8 The next step, which is elaborated on in Sect. 4.2, is to use two other exact determinant identities (which were proved in [2] and [3]) and to establish that
det Hn [b] = det(Tn (a) + Hn (a)) = Gn det Pn (I + H (ψ))−1 Pn for “suitably behaved” functions b, a, and ψ and a constant G (all depending on each other in an explicit way, which will be described in Sect. 4.2). At this point another complication is encountered since the function bα,n is not “suitably behaved”. We can by-pass this complication only by applying an approximation argument, which turns out to be non-trivial and is based on results of [8]. Finally, we will arrive at an identity
det Hn [bα,n ] = det Pn (I + H (ψα,n ))−1 Pn with certain functions ψα,n . The functions ψα,n are of such a form that they have their singularities only at the points t = 1 and t = −1. What one usually tries to do in such cases is to separate the singularities. Indeed, we will prove that the above asymptotics is equal to the asymptotics of
(1) −1 (−1) −1 det Pn (I + H (ψα,n )) Pn · det Pn (I + H (ψα,n )) Pn , (1)
(−1)
where the functions ψα,n and ψα,n have singularities only at t = 1 and t = −1, respectively. Fortunately, the last two expressions are simple enough to analyze, and their limits as n → ∞ equal the constants
det α (I + HR (uˆ −1/2 ))−1 α and det α (I − HR (uˆ 1/2 ))−1 α . (1)
The different sign in front of the Hankel operators comes from the fact that ψα,n has its (−1) singularity at t = 1, while ψα,n has its singularity at t = −1. The proof of the separation of the singularities as well as the last step requires a couple of technical results, which we will establish in Sect. 4.3. The proof of the asymptotic formula as it was outlined here will be given in Sect. 5.
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4. Auxiliary Results 4.1. Invertibility of certain operators I + H (ψ). In this section we are going to prove that for several concrete (piecewise continuous) functions ψ the operator I + H (ψ) is invertible on H 2 (T). Actually, apart from one case, the results that we need have already been established in [3] or can easily be derived from there. For τ ∈ T and β ∈ C, let us introduce the functions uβ,τ (eiθ ) = exp(iβ(θ − θ0 − π )),
0 < θ − θ0 < 2π,
τ = eiθ0 .
(20)
These functions are continuous on T \ {τ } and have a jump discontinuity at t = τ whose size is determined by β. Proposition 4.1. The following operators are invertible on H 2 (T): A1 = I + H (u−1/2,1 ), A3 = I − H (u−1/2,−1 ),
A2 = I − H (u1/2,1 ), A4 = I + H (u1/2,−1 ).
Proof. The invertibility of A1 and A2 follows from Thm. 3.6 in Sect. 3.2 of [3]. The invertibility of A3 and A4 can be obtained by remarking that A3 = W A1 W and A4 = W A2 W , where W is the operator defined by (Wf )(t) = f (−t), t ∈ T.
Next we introduce the function χ (e ) = iθ
i if 0 < θ < π −i if − π < θ < 0.
(21)
For later use, let us observe that χ (t) = u−1/2,1 (t)u1/2,−1 (t) = − u1/2,1 (t)u−1/2,−1 (t).
(22)
We denote by W the Wiener algebra, which consists of all functions in L1 (T) whose Fourier series are absolutely convergent. Moreover, we introduce the Banach subalgebras W± = { a ∈ W : an = 0 for all ± n < 0 } ,
(23)
where an stand for the Fourier coefficients of a. Functions in W± can be continued by continuity to functions which are analytic in {z ∈ C : |z| < 1} and {z ∈ C : |z| > 1} ∪ {∞}, respectively. Notice that a ∈ W+ if and only if a˜ ∈ W− , where a(t) ˜ := a(t −1 ), t ∈ T. Finally, we will denote by GW and GW± the group of invertible elements in the Banach algebras W and W± , respectively. −1 (t)χ (t). Then the operator Proposition 4.2. Let c+ ∈ GW+ and ψ(t) = c˜+ (t)c+ 2 I + H (ψ) is invertible on H (T).
Proof. We first use a result of Power [13] in order to determine the essential spectrum of the Hankel operator H (ψ). Recall that the essential spectrum spess A of a bounded linear operator A acting on a Banach space X is the set of all λ ∈ C for which A − λI is not a Fredholm operator. Also recall that A is a Fredholm operator on X if its image “im A” is a closed subspace of X and if both the kernel “ker A” and the quotient space “X/im A” are finite dimensional. In this case the Fredholm index of A is defined as dim ker A − dim(X/im A).
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The result of Power [13] says that for a piecewise continuous function b the essential spectrum of H (b) is a union of intervals in C, namely,
spess H (b) = [0, ib−1 ] ∪ [0, −ib1 ] ∪ −i bτ bτ¯ , i bτ bτ¯ . τ ∈T+
Here we use the notation bτ = (b(τ +0)−b(τ −0))/2 with b(τ ±0) = limε→±0 b(τ eiε ), and T+ := {τ ∈ T : Im τ > 0}. This result can also be obtained from the more general results contained in [14] and [5, Sects. 4.95-4.102]. For our function ψ we obtain ψ1 = i, ψ−1 = −i, and ψτ = 0 for τ ∈ T \ {1, −1}. Hence spess H (ψ) = [0, 1], which implies that I + H (ψ) is a Fredholm operator. Since H (ψ) − λI is an invertible operator for λ sufficiently large (hence a Fredholm operator with index zero), and since the Fredholm index is invariant with respect to small perturbations, it follows that I + H (ψ) has Fredholm index zero. Thus it remains to prove that the kernel of I + H (ψ) is trivial. In order to prove this we introduce for τ ∈ T and β ∈ C the functions ηβ,τ (t) = (1 − t/τ )β ,
ξβ,τ (t) = (1 − τ/t)β ,
where these functions are analytic in an open neighborhood of { z ∈ C : |z| ≤ 1, z = τ } and { z ∈ C : |z| ≥ 1, z = τ } ∪ {∞}, respectively, and the branch of the power function is chosen in such a way that ηβ,τ (0) = 1 and ξβ,τ (∞) = 1. Notice that uβ,τ (t) = ηβ,τ (t)ξ−β,τ (t),
uβ+n,τ (t) = (−t/τ )n uβ,τ (t).
(24)
Finally, we introduce the Hardy space H 2 (T) = f ∈ L2 (T) : fk = 0 for all k > 0 and notice that f ∈ H 2 (T) if and only if f˜ ∈ H 2 (T). Now suppose that f+ ∈ H 2 (T) belongs to the kernel of I + H (ψ). Then, by (7), f+ (t) + ψ(t)t −1 f˜+ (t) =: f− (t) ∈ t −1 H 2 (T). Using (22) and (24) we can write χ(t) = −t −1 u1/2,1 (t)u1/2,−1 (t) = −t −1 ξ−1/2,1 (t)ξ−1/2,−1 (t)η1/2,1 (t)η1/2,−1 (t), and hence we obtain −1 −1 f0 (t) := t c˜+ (t)ξ1/2,1 (t)ξ1/2,−1 (t)f+ (t) − t −1 c+ (t)η1/2,1 (t)η1/2,−1 (t)f˜+ (t) −1 (t)ξ1/2,1 (t)ξ1/2,−1 (t)f− (t). = t c˜+
It is easy to see that f0 = −f˜0 and that f0 ∈ H 2 (T). Hence f0 = 0, and thus f+ (t) + ψ(t)t −1 f˜+ (t) = 0. Now use again (22) and (24) to write χ (t) = tu−1/2,1 (t)u−1/2,−1 (t) = tξ1/2,1 (t)ξ1/2,−1 (t)η−1/2,1 (t)η−1/2,−1 (t), and it follows c+ (t)η1/2,1 (t)η1/2,−1 (t)f+ (t) = −c˜+ (t)ξ1/2,1 (t)ξ1/2,−1 (t)f˜+ (t). Therein the left-hand side belongs to H 2 (T), whereas the right hand side belongs to H 2 (T). It follows that this expression is zero. Hence f+ = 0. Thus we have proved that the kernel of I + H (ψ) is trivial.
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4.2. A formula for Hankel determinants. The goal of this section is to prove the following formula:
det Hn [b] = Gn det Pn (I + H (ψ))−1 Pn , where b ∈ L1 [−1, 1] is a (sufficiently smooth) continuous and nonvanishing function. Therein G is a constant and ψ is a function (both depending on b). This formula will allow us later to reduce a Hankel determinant to the type of determinant appearing on the right-hand side. To start with let us recall some basic notions. A function a ∈ W is said to admit a canonical Wiener-Hopf factorization in W if it can be represented in the form a(t) = a− (t)a+ (t),
t ∈ T,
(25)
where a± ∈ GW± . It is well known (see, e.g., [5]) that a ∈ W admits a canonical Wiener-Hopf factorization in W if and only if a ∈ GW and if the winding number of a is zero. This, in turn, is equivalent to the condition that a possesses a logarithm log a ∈ W. Clearly, one can then define the geometric mean 1 2π G[a] := exp log a(eiθ ) dθ . (26) 2π 0 Notice that this definition does not depend on the particular choice of the logarithm. Next we are going to cite two results. The first result is from [3, Prop. 3.9]. Recall that a function a defined on T is called even if a = a, ˜ where a(t) ˜ = a(1/t). Proposition 4.3. Let a ∈ GW be an even function which possesses a canonical Wiener−1 Hopf factorization a(t) = a− (t)a+ (t). Define ψ(t) = a˜ + (t)a+ (t). Then I + H (ψ) is 2 invertible on H (T) and
det Tn (a) + Hn (a) = G[a]n det Pn (I + H (ψ))−1 Pn . (27) The second result is from [2, Thm. 2.3]. Proposition 4.4. Let a ∈ L1 (T) be an even function, and let b ∈ L1 [−1, 1] be given by 1 + cos θ iθ b(cos θ ) = a(e ) . (28) 1 − cos θ
Then det Tn (a) + Hn (a) = det Hn [b]. We remark in this connection that under the assumption (28) we have b ∈ L1 [−1, 1] if and only if a(eiθ )(1 + cos θ ) ∈ L1 (T). Combining both results gives immediately the following theorem. Theorem 4.5. Let a ∈ GW be an even function which possesses a Wiener-Hopf factor−1 (t), and define b ∈ L1 (T) by (28). ization a(t) = a− (t)a+ (t). Define ψ(t) = a˜ + (t)a+ Then
det Hn [b] = G[a]n det Pn (I + H (ψ))−1 Pn . (29)
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This result is, however, not yet what we need. As pointed out above we want to derive a formula in which b is a continuous and nonvanishing function on [−1, 1]. But in the previous theorem, the function b(x) has necessarily a zero at x = −1 and a pole at x = 1 (both of order 1/2). The desired result reads as follows. It involves the function χ defined in (21). Notice that the operator I + H (ψ) is invertible by Proposition 4.2. Theorem 4.6. Let c ∈ GW be an even function which possesses a canonical Wiener−1 Hopf factorization c(t) = c− (t)c+ (t). Define ψ(t) = c˜+ (t)c+ (t)χ (t) and b(cos θ) = iθ c(e ). Then
det Hn [b] = G[c]n det Pn (I + H (ψ))−1 Pn . It is interesting to observe that this theorem follows formally from the above theorem with a+ (t) = c+ (t)(1 − t)−1/2 (1 + t)1/2 and a− (t) = c− (t)(1 − t −1 )−1/2 (1 + t −1 )1/2 . Notice in this connection (22) and u−1/2,1 (t) = (1 − t −1 )1/2 (1 − t)−1/2 ,
u1/2,−1 (t) = (1 + t −1 )−1/2 (1 + t)1/2 .
(30)
However, in order to make things precise, we have to use an approximation argument. This approximation leads us to a so-called stability problem, which is somewhat difficult to analyze. In fact, we are going to resort to results of [8] and we apply also Proposition 4.1. In order to carry out the proof we need the following definitions and basic results. In what follows r is a number in [0, 1), which is supposed to tend to 1. A (generalized) sequence of functions ar ∈ L∞ (T) is said to converge to a ∈ L∞ (T) in measure if for each ε > 0 the Lebesgue measure of the set t ∈ T : |ar (t) − a(t)| ≥ ε converges to zero as r → 1. A (generalized) sequence of bounded linear operators Ar on a Banach space X is said to converge strongly on X to an operator A if Ar x → Ax as r → 1 for all x ∈ X. The proof of the following lemma is straightforward (see also [3], Lemma 3.2). Lemma 4.7. Assume that ar ∈ L∞ (T) are uniformly bounded and converge to a ∈ L∞ (T) in measure. Then T (ar ) → T (a)
and
H (ar ) → H (a)
strongly on H 2 (T), and the same holds for the adjoints. A sequence {Ar }r∈[0,1) of operators on X is called stable if there exists an r0 ∈ [0, 1) such that for all r ∈ [r0 , 1) the operators Ar are invertible and sup A−1 r L(X) < ∞. r∈[r0 ,1)
Strong convergence of the inverses and stability are related by the following (basic) result. Lemma 4.8. Suppose that Ar → A strongly on X as r → 1 and that A is invertible. −1 strongly on X as r → 1 if and only if {A } Then A−1 r r∈[0,1) is stable. r →A Proof. One part follows from the Banach-Steinhaus Theorem, while the other part fol−1 −1 −1 lows easily from the identity (A−1
r − A )y = Ar (A − Ar )A y.
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Finally, for r ∈ [0, 1) and τ ∈ T we introduce the following operators Gr,τ acting on L∞ (T): t +r Gr,τ : a(t) → b(t) = a τ . (31) 1 + rt Figuratively speaking, the function a is first stretched at τ and squeezed at −τ , and then rotated on the unit circle such that τ is mapped into 1. It might also be illustrative to remark that for fixed τ, t ∈ T, t = −1, the sequence (Gr,τ a)(t) converges to a(τ ) as r → 1 if a is a continuous function. Now we are ready to give the proof of Theorem 4.6. Proof of Theorem 4.6. For r ∈ [0, 1) introduce the even function (1 − rt)(1 − rt −1 ) ar (t) = c(t) , t ∈ T. (1 + rt)(1 + rt −1 ) The function br corresponding to ar by means of (28) is then given by 2 + 2x 1 + r 2 − 2rx br (x) = b(x) , x ∈ (−1, 1). 2 1 + r + 2rx 2 − 2x It is easy to verify that br → b in the norm of L1 [−1, 1]. Hence (for each fixed n) Hn [b] = lim Hn [br ]. r→1
The canonical Wiener-Hopf factorization of ar is given by ar (t) = ar,− (t)ar,+ (t) with ar,− (t) = c− (t)
(1 − rt −1 )1/2 , (1 + rt −1 )1/2
ar,+ (t) = c+ (t)
(1 − rt)1/2 . (1 + rt)1/2
Upon putting −1 −1 (t) = c˜+ (t)c+ (t) ψr (t) = a˜ r,+ (t)ar,+
1 − rt 1 − rt −1
−1/2
1 + rt 1 + rt −1
1/2 ,
we conclude from Theorem 4.5 and from the fact that G[a] = G[c] that
det Hn [br ] = G[c]n det Pn (I + H (ψr ))−1 Pn . Hence (for n fixed)
det Hn [b] = G[c]n lim det Pn (I + H (ψr ))−1 Pn . r→1
Taking account of (30) it easy to see that fr± :=
1 ∓ rt 1 ∓ rt −1
∓1/2
→ u∓1/2,±1
(32)
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in measure as r → 1. From (22) we thus obtain that ψr → ψ in measure. Moreover, since |fr± (t)| = 1, Lemma 4.7 implies that H (ψr ) → H (ψ) strongly on H 2 (T) as r → 1. We want to conclude that (I + H (ψr ))−1 → (I + H (ψ))−1
(33)
strongly on as r → 1. By Proposition 4.2 and Lemma 4.8 it is necessary and sufficient that the sequence of operators {I + H (ψr )}r∈[0,1) is stable. In order to analyse this stability condition we apply the results of [8, Sects. 4.1–4.2]. These results establish the existence of certain mappings 0 and τ , τ ∈ T, which are defined as H 2 (T)
0 [ψr ] := µ- lim ψr ,
τ [ψr ] := µ- lim Gr,τ ψr ,
r→1
r→1
where µ- lim stands for the limit in measure. Because of (32) we have 0 [fr± ] := µ- lim fr± = u∓1/2,±1 . r→1
Furthermore since
fr±
→ u∓1/2,±1 locally uniformly on T \ {±1}, we have
τ [fr± ] := µ- lim Gr,τ fr± = u∓1/2,±1 (τ ) r→1
for τ = ±1. Finally, ±1 [fr± ]
:= µ- lim
r→1
Gr,±1 fr±
= µ- lim
r→1
1 + rt 1 + rt −1
±1/2
= u±1/2,−1 .
−1 + − Since ψr = c˜+ c+ fr fr we conclude −1 0 [ψr ] = c˜+ c+ u−1/2,1 u1/2,−1 = ψ,
1 [ψr ] = u1/2,−1 , −1 [ψr ] = u−1/2,−1 , τ [ψr ] = constant function,
τ ∈ T \ {−1, 1}.
The stability criterion in [8] (Thm. 4.2 and Thm. 4.3) says that I + H (ψr ) is stable if and only if the operators (i) 0 [I + H (ψr )] = I + H (0 [ψr ]) = I + H (ψ), (ii) 1 [I + H (ψr )] = I + H (1 [ψr ]) = I + H (u1/2,−1 ), (iii) −1 [I + H (ψr )] = I − H (−1 [ψr ]) = I − H (u−1/2,−1 ), (iv) τ [I + H (ψr )] = M(τ [ψr ]) 0 I 0 P 0 0I P 0 I 0 + = 0I 0 Q I 0 0 Q 0I 0 M( τ¯ [ψr ]) (τ ∈ T, Im(τ ) > 0) are invertible. Clearly, by Proposition 4.1 and Proposition 4.2 this is the case. Hence the sequence I + H (ψr ) is stable and (33) follows. We conclude that (for fixed n) the n × n matrices Pn (I + H (ψr ))−1 Pn converge to Pn (I + H (ψ))−1 Pn as r → 1, whence it follows that the corresponding determinants converge, too. This completes the proof.
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4.3. Convergence in trace class norm. In this section we are going to prove a couple of technical results. We are mainly concerned with proving that certain sequences of operators converge in the trace norm. abs stand for the set of all functions on T which are absolutely continuous on Let P C±1 T \ {−1, 1} and which possess one-sided limits at t = 1 and t = −1. abs . Then H (a) is a trace Lemma 4.9. Let a ∈ C(T) be a function such that a ∈ P C±1 2 class operator on H (T) and H (a)1 ≤ C aL∞ (T) + a L∞ (T) + a L1 (T) . (34)
Proof. From partial integration it follows that the Fourier coefficients ak are O(k −2 ) as k → ∞, where the constant involved in this estimate is given in terms of the norms of a, a and a . We write the operator H (a) as a product AB with operators A and B given by its matrix representation with respect to the standard basis by ∞ ∞ , B = diag (1 + k)−1/2−ε . A = aj +k+1 (1 + k)1/2+ε j,k=0
k=0
Both A and B are Hilbert-Schmidt operators if 0 < ε < 1/2 with straightforward estimates for their norms. Hence H (a) is a trace class operator, whose norm can be estimated by (34).
Before proceeding, let us recall that Toeplitz and Hankel operators T (a) and H (a) satisfy the following well-known formulas: ˜ T (ab) = T (a)T (b) + H (a)H (b),
(35)
˜ H (ab) = T (a)H (b) + H (a)T (b),
(36)
˜ := b(t −1 ). where a, b ∈ L∞ (T) and, as before, where b(t) In regard to the following proposition recall the definition (31) of the operators Gr,τ acting on L∞ (T). Their inverse are given by −1 tτ − r G−1 . (37) : b(t) → a(t) = b r,τ 1 − rtτ −1 The proof of the following proposition is very technical, which is due to the fact that we have to prove convergence in the trace norm. It might be helpful to remark the proof of convergence in the operator norm would essentially require only the first two paragraphs of the proof. (The convergence of the derivatives appearing therein would not be necessary.) Proposition 4.10. Let ψµ(1) = G−1 µ,1 (u−1/2,1 − 1),
ψµ(−1) = G−1 µ,−1 (u1/2,1 − 1)
with µ ∈ [0, 1). Then the operators H (ψµ(1) )H (ψµ(−1) ),
H (ψµ(−1) )H (ψµ(1) ),
and
H (ψµ(1) ψµ(−1) )
are trace class operators and converge to zero in the trace norm as µ → 1.
(38)
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T. Ehrhardt
Proof. Let us first notice that (with the proper choice of the square-root), t − µ −1/2 t + µ 1/2 (1) (−1) ψµ (t) = − − 1, ψµ (t) = − 1. 1 − µt 1 + µt (1)
(39) (−1)
In particular, ψµ has a jump discontinuity at t = 1 and vanishes at t = −1 while ψµ has a jump discontinuity at t = −1 and vanishes at t = 1. Moreover, both functions are uniformly bounded and ψµ(1) → 0,
ψµ(−1) → 0,
(40)
uniformly on each compact subset of T \ {1} and T \ {−1}, respectively. (1) (−1) In order to prove the assertion for the operator H (ψµ )H (ψµ ), let f and g be smooth functions on T with f + g = 1 such that f (t) vanishes identically in a neighborhood of 1 (say for | arg t| ≤ π/3) and g(t) vanishes identically in a neighborhood of −1 (say for | arg t| ≥ 2π/3). Then (see (36)) H (ψµ(1) )H (ψµ(−1) ) = H (ψµ(1) )T (f )H (ψµ(−1) ) + H (ψµ(1) )T (g)H (ψµ(−1) ) = H (ψµ(1) f˜)H (ψµ(−1) ) − T (ψµ(1) )H (f˜)H (ψµ(−1) ) (−1) + H (ψµ(1) )H (gψµ(−1) ) − H (ψµ(1) )H (g)T (ψµ ). Clearly, H (f˜) and H (g) are trace class operators. Due to the afore-mentioned fact that (1) (−1) ψµ and ψµ are uniformly bounded and because of the convergence (40), Lemma 4.7 implies that the operators H (ψµ(1) ),
T (ψµ(1) ),
(1) T (ψµ )
H (ψµ(−1) ),
and their adjoints converge strongly to zero as µ → 1. We can conclude that H (ψµ1 ) (−1)
H (ψµ ) is a trace class operator and converges in the trace norm to zero as soon as we have shown that H (ψµ(1) f˜)
and
H (gψµ(−1) )
are trace class operators which converge to zero in the trace norm. On account of Lemma 4.9 this is true if ψµ(1) f˜ ∈ C(T),
abs (ψµ(1) f˜) ∈ P C±1 ,
if ψµ(1) f˜L∞ → 0,
(ψµ(1) f˜) L∞ → 0,
(ψµ(1) f˜) L1 → 0,
(−1)
and if similar statements hold for gψµ . Due to the fact that f vanishes on a neighborhood of 1, these conditions are fulfilled if ψµ(1) |T−1 ∈ C(T−1 ),
(ψµ(1) ) |T−1 ∈ C(T−1 ),
(ψµ(1) ) |T−1 ∈ C(T−1 ),
(41)
and if (1)
ψµ |T−1 L∞ (T−1 ) → 0,
(ψµ ) |T−1 L∞ (T−1 ) → 0, (1)
(ψµ ) |T−1 L1 (T−1 ) → 0. (1)
(42)
Dyson’s Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel
331
(1)
Here we have restricted the function ψµ to the arc T−1 := {t ∈ T : | arg t| ≥ π/4}. (−1) The corresponding (sufficient) conditions for the function ψµ are ψµ(−1) |T1 ∈ C(T1 ),
(ψµ(−1) ) |T1 ∈ C(T1 ),
(ψµ(−1) ) |T1 ∈ C(T1 ),
(43)
and (−1)
ψµ
|T1 L∞ (T1 ) → 0,
(−1) )|
(ψµ
(−1) (ψµ ) |T1 L1 (T1 )
T1 L∞ (T1 )
→ 0,
(44)
→ 0,
where T1 := {t ∈ T : | arg t| ≤ 3π/4}. It is easy to see that conditions (41) and (43) and also the first condition in (42) and in (44) are fulfilled. We will prove the remaining conditions in a few moments, but first we will turn to (−1) (1) (1) (−1) the convergence of the operators H (ψµ )H (ψµ ) and H (ψµ ψµ ). In regard to the (−1) (1) operator H (ψµ )H (ψµ ) we can proceed analogously and it turns out that we arrive at the same sufficient conditions (41)–(44). (1) (−1) As to the operator H (ψµ ψµ ) we have to show (on account of Lemma 4.9) that ψµ(1) ψµ(−1) ∈ C(T),
abs (ψµ(1) ψµ(−1) ) ∈ P C±1
(45)
and that ψµ(1) ψµ(−1) L∞ → 0,
(ψµ(1) ψµ(−1) ) L∞ → 0,
(ψµ(1) ψµ(−1) ) L1 → 0. (46) (1)
(−1)
From the facts stated at the beginning of the proof it follows that ψµ ψµ is contin(1) (−1) uous on T and that ψµ ψµ converges uniformly to zero on T. Moreover, since the (±1) abs , it follows that the derivative of functions ψµ and their derivatives belong to P C±1 (1)
(−1)
abs , too. Thus we are left with the proof of the second and third ψµ ψµ belongs to P C±1 condition in (46). We will prove these assertions by separating the singularities at t = 1 and t = −1:
(ψµ(1) ψµ(−1) ) |T−1 L∞ (T−1 ) → 0,
(ψµ(1) ψµ(−1) ) |T−1 L1 (T−1 ) → 0,
(47)
(ψµ(1) ψµ(−1) ) |T1 L1 (T1 ) → 0.
(48)
and (ψµ(1) ψµ(−1) ) |T1 L∞ (T1 ) → 0,
Now turning back to the proof of the yet outstanding conditions in (42) and (44), we remark that the interval T−1 can be transformed into the interval T1 by a rotation (1) (−1) t → −t. This will not precisely transform the function ψµ into the function ψµ , but into a similar function of the form (39), where only the power 1/2 is replaced by −1/2. Without loss of generality we can thus confine ourselves to the proof of the conditions involving the interval T1 , since the conditions involving the interval T−1 can be reduced to an analogous situation and can be proved in the same way. In order to prove (48) and the last two conditions in (44) we use the linear fractional transformation σ (x) =
1 + ix , 1 − ix
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T. Ehrhardt
which maps the extended real √ √ line onto the unit circle. Clearly, T1 corresponds to σ −1 (T1 ) = [−1 − 2, 1 + 2] =: I0 . We transform the functions into vε (x) = ψµ(1) (σ (x)),
wε (x) = ψµ(−1) (σ (x)),
and we also change the parameter µ ∈ [0, 1) into ε = which we have to prove are then equivalent to (wε ) |I0 L∞ (I0 ) → 0,
1−µ 1+µ
∈ (0, 1]. The conditions
(wε ) |I0 L1 (I0 ) → 0
(49)
(vε wε ) |I0 L1 (I0 ) → 0
(50)
and (vε wε ) |I0 L∞ (I0 ) → 0, as ε → 0. Introduce the functions x − i −1/2 v(x) = − 1, x+i
w(x) =
1 + ix 1 − ix
1/2 − 1,
where v has a jump at x = 0 and the square-root is chosen such that v(±∞) = 0. The function w is continuous on R with w(0) = 0 and limits at x → ±∞. A straightforward computation implies that vε (x) = v(x/ε) and wε (x) = w(xε). The functions v and w and all of their derivatives are bounded on R. Thus the conditions in (49) follow easily. The function w can be written as w(x) = x w(x), ˜ where w˜ is a function which is locally bounded. We write ˜ + v(x/ε)εw (εx), (vε wε ) = v (x/ε)x w(εx) and see immediately that the second term goes uniformly to zero. Moreover, v (x) → 0 as |x| → ∞. Hence xv (x/ε) converges uniformly on I0 to zero, which implies that the first term converges uniformly on I0 to zero. Thus we have proved that (vε wε ) converges uniformly on I0 to zero as ε → 0. Finally, we write the second derivative as ˜ + 2v (x/ε)w (εx) + ε 2 v(x/ε)w (εx). (vε wε ) = ε −1 v (x/ε)x w(εx) The
L1 (I
0 )-norm
(51)
of the first term can be estimated by a constant times |v (x/ε)x/ε|dx ≤ ε |xv (x)| dx, R
I0
which converges to zero. The L1 (I0 )-norm of the second term can be estimated by a constant times |v (x/ε)|dx ≤ ε |v (x)| dx I0
R
and also converges to zero. The last term converges to zero even uniformly. Hence we have proved the conditions (50) and the proof is complete.
In addition to the operators Gµ,τ we introduce operators
1 − µ2 2 Rµ,τ : f ∈ H (T) → g(t) = Gµ,τ (f ) ∈ H 2 (T), 1 + µt where µ ∈ [0, 1) and τ ∈ {−1, 1}.
(52)
Dyson’s Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel
333
Lemma 4.11. For each τ ∈ {−1, 1}, the operator Rµ,τ is unitary on H 2 (T). Moreover, ∗ = τ H (G ∞ Rµ,τ H (a)Rµ,τ µ,τ a) for all a ∈ L (T). Proof. We can define the operators Rµ,τ also on L2 (T). In [8, Sect. 5.1] it is proved that Rµ,τ are unitary on L2 (T) and that ∗ = P, Rµ,τ P Rµ,τ
∗ Rµ,τ M(a)Rµ,τ = M(G−1 µ,τ a),
These statements imply the desired assertions.
∗ Rµ,τ J Rµ,τ = τ J.
In connection with the following proposition recall that the operators I + H (u−1/2,1 ) and I − H (u1/2,1 ) are invertible on H 2 (T) (see Proposition 4.1). Moreover, for α > 0 and n ∈ N define the functions α(1 − t) hα (t) = exp − , (53) 2(1 + t) n t + µα,n hα,n (t) = , (54) 1 + µα,n t where µα,n ∈ [0, 1) is any sequence. (This sequence will be specified later on.) Finally, introduce the functions (1) ψα,n = G−1 µα,n ,1 (u−1/2,1 − 1),
(−1) ψα,n = G−1 µα,n ,−1 (u1/2,1 − 1).
(55)
Proposition 4.12. Let α > 0 be fixed, and consider (53), (54), and (55). Assume that α µα,n = 1 − (56) + O(n−2 ), as n → ∞. 2n Then the following is true: (1)
(−1)
(i) The operators H (ψα,n ) and H (ψα,n ) are unitarily equivalent to the operators H (u−1/2,1 ) and −H (u1/2,1 ), respectively. (ii) The operators (1) −1 Pn (I + H (ψα,n )) Pn − Pn
are unitarily equivalent to the operators An = H (hα,n )(I + H (u−1/2,1 ))−1 H (hα,n ) − H (hα,n )2 , which are trace class operators and converge as n → ∞ in the trace norm to A = H (hα )(I + H (u−1/2,1 ))−1 H (hα ) − H (hα )2 . (iii) The operators (−1) −1 Pn (I + H (ψα,n )) Pn − Pn
are unitarily equivalent to the operators Bn = H (hα,n )(I − H (u1/2,1 ))−1 H (hα,n ) − H (hα,n )2 , which are trace class operators and converge as n → ∞ in the trace norm to B = H (hα )(I − H (u1/2,1 ))−1 H (hα ) − H (hα )2 .
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T. Ehrhardt
Proof. (i) We employ Lemma 4.11 in order to conclude that (1) H (ψα,n ) = Rµ∗ α,n ,1 H (u−1/2,1 )Rµα,n ,1 ,
(−1) H (ψα,n ) = −Rµ∗ α,n ,−1 H (u1/2,1 )Rµα,n ,−1 .
(ii) We first introduce the operator Wn = H (t n ) and remark that Wn2 = Pn and (1) Wn Pn = Pn Wn = Wn . It is easily seen that the operator Pn (I + H (ψα,n ))−1 Pn − Pn (1) −1 is unitarily equivalent to the operator Wn (I + H (ψα,n )) Wn − Wn2 by means of the unitary and selfadjoint operator Wn + (I − Pn ). Now we use the unitary equivalence established in (i) in connection with the fact that Rµα,n ,1 Wn Rµ∗ α,n ,1 = Rµα,n ,1 H (t n )Rµ∗ α,n ,1 = H (hα,n ) (see again Lemma 4.11). Notice that hα,n = Gµα,n ,1 (t n ). This implies the unitary equivalence to An . In order to prove the convergence An → A in the trace norm we write An = H (hα,n )(I + H (u−1/2,1 ))−1 H (u−1/2,1 )H (hα,n ). The function hα,n is uniformly bounded and converges (along with all its derivatives) uniformly on each compact subset of T \ {−1} to the function hα . Hence (by Lemma 4.7) H (hα,n ) → H (hα ),
α ) T (h α,n ) → T (h
strongly on H 2 (T). The same holds for their adjoints. Next we claim that all operators H (u−1/2,1 )H (hα,n ) are trace class operators and converge in the trace norm to H (u−1/2,1 )H (hα ). To see this we choose two smooth functions f and g on T which vanish identically in a neighborhood of −1 and 1, respectively, such that f + g = 1. Then we decompose H (u−1/2,1 )H (hα,n ) = H (u−1/2,1 )T (f )H (hα,n ) + H (u−1/2,1 )T (g)H (hα,n ) = H (u−1/2,1 )H (f hα,n ) − H (u−1/2,1 )H (f )T (h α,n ) +H (u−1/2,1 g)H ˜ (hα,n ) − T (u−1/2,1 )H (g)H ˜ (hα,n ). The Hankel operators H (f ) and H (g) ˜ are both trace class and so are the operators H (f hα,n ) and H (u−1/2,1 g) ˜ since the generating functions are smooth. Moreover, f hα,n → f hα uniformly and the same holds for the derivatives. Hence H (f hα,n ) → H (f hα ) in the trace norm by Lemma 4.9. Along with the strong convergence noted above, it follows that H (u−1/2,1 )H (hα,n ) converges in the trace norm to H (u−1/2,1 )H (f hα ) − H (u−1/2,1 )H (f )T (hα ) + H (u−1/2,1 g)H ˜ (hα ) −T (u−1/2,1 )H (g)H ˜ (hα ), which is trace class and equal to H (u−1/2,1 )H (hα ). (iii) The proof of these assertions is analogous. The only (slight) difference is that Rµα,n ,−1 Wn Rµ∗ α,n ,−1 = Rµα,n ,−1 H (t n )Rµ∗ α,n ,−1 = (−1)n+1 H (hα,n ) as Gµα,n ,−1 (t n ) = (−1)n hα,n . The possibly different sign at this place does not effect the argument.
Dyson’s Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel
335
5. Proof of the Asymptotic Formula In this section we are going to prove the asymptotic formula (2). Our first goal is to discretize the Wiener-Hopf operator I − Kα , which will lead us to a Toeplitz operator. Here and in what follows χα stands for the characteristic function of the subarc {eiθ : α < θ < 2π − α} of T. Proposition 5.1. For each α > 0 we have det(I − Kα ) = lim det Tn (χ αn ).
(57)
n→∞
Proof. Recall that the operator Kα is the integral operator on L2 [0, α] with the kernel K(x − y), where
Introduce the n × n matrices α(j −k) n−1 α K , An = n n j,k=0
K(x) =
sin x . πx
1 1
α Bn = n
K
0
0
α(j −k+ξ −η) n
n−1 dξ dη
. j,k=0
By the mean value theorem the entries of An − Bn can be estimated uniformly by −1 O(n−2 ), whence it follows that the Hilbert-Schmidt norm of √An − Bn is O(n ). Since the Hilbert-Schmidt norm √ of the n × n identity matrix is O( n), we obtain that the trace norm of An − Bn is O(1/ n). The Fourier coefficients of 1 − χ αn are
α if k = 0 πn [1 − χ αn ]k = sin( kα ) n if k = 0. πk Hence it follows that Tn (χ αn ) = In − An . Introduce the isometry Uα,n :
{xk }n−1 k=0
∈ C → n
n−1 n xk χ[ αk , α(k+1) ] ∈ L2 [0, α], n n α k=0
and remark that ∗ Uα,n : f ∈ L2 [0, α] →
n α
It can be verified straightforwardly that
n−1
α 0
f (x)χ[ αk , α(k+1) ] dx
∗ K U Uα,n α α,n
n
n
∈ Cn . k=0
= Bn . Hence
∗ det(I − Kα ) = det(In − Un,α Kα Un,α ) = det(In − Bn ) ∼ det(In − An ) = det Tn (χ αn )
as n → ∞. This completes the proof.
The following result has been established in [2, Cor. 2.5].
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T. Ehrhardt
Proposition 5.2. Let b ∈ L1 [−1, 1] and suppose that b0 (x) = b0 (−x), where 1−x b0 (x) = b(x) . 1+x Then det Hn [b] = det Tn (d) with d(eiθ ) = b0 (cos θ2 ). We use this result in order to reduce our Toeplitz determinant det Tn (χ αn ) to a Hankel determinant. Proposition 5.3. We have 2
det Tn (χ αn ) = (α,n )n det Hn [bα,n ], where
bα,n (x) =
1 + α,n x , 1 − α,n x
α,n = cos
(58)
α . 2n
(59)
Proof. We apply Proposition 5.2 with d(eiθ ) = χ αn (eiθ ), b0 (x) = χ[−α,n ,α,n ] (x), and b(x) =
1+x χ[−α,n ,α,n ] (x). 1−x
It follows that det Tn (χ αn ) = det Hn [b]. The entries of Hn [b] are the moments [b]1+j +k , 0 ≤ j, k ≤ n − 1. A simple computation gives k 1 ) 1 + α,n y 1 1 ( α,n [b]k = b(x)(2x)k−1 dx = (2y)k−1 dy = (α,n )k [bα,n ]k . π −1 π 1 − α,n y −1 Now we can pull out certain diagonal matrices from the left and the right of Hn [b] to obtain the matrix Hn [bα,n ]. The determinants of the diagonal matrices give the factor 2
(α,n )n . In the following result we use the function ψα,n (t) =
1 − µα,n t 1 − µα,n t −1
1/2
1 + µα,n t −1 1 + µα,n t
1/2 χ (t),
(60)
where χ (t) is given by (22) and where µα,n =
1−
2 1 − α,n
α,n
(61)
with α,n given by (59). For later use remark that µα,n ∈ [0, 1) satisfies condition (56). Proposition 5.4. We have lim det Tn (χ αn ) = e−
n→∞
α2 8
lim det Pn (I + H (ψα,n ))−1 Pn .
n→∞
(62)
Dyson’s Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel
337
2
α −4 ) it is readily verified that Proof. We use Proposition 5.3. Since α,n = 1 − 8n 2 + O(n
(α,n )n → e− 2
α2 8
. We obtain lim det Tn (χ αn ) = e−
α2 8
n→∞
lim det Hn [bα,n ].
n→∞
Now we employ Theorem 4.6 with c(e ) = iθ
1 + α,n cos θ . 1 − α,n cos θ
Obviously, (since α,n = 2µα,n /(1 + µ2α,n )) c(t) =
(1 + µα,n t)(1 + µα,n t −1 ) , (1 − µα,n t)(1 − µα,n t −1 )
whence we conclude that G[c] = 1 and that the canonical Wiener-Hopf factorization of c is given by c(t) = c− (t)c+ (t) with c− (t) =
1 + µα,n t −1 1 − µα,n t −1
1/2
,
c+ (t) =
1 + µα,n t 1 − µα,n t
1/2 .
Furthermore, −1 c˜+ (t)c+ (t) =
1 − µα,n t 1 − µα,n t −1
1/2
1 + µα,n t −1 1 + µα,n t
1/2 .
It follows that
det Hn [bα,n ] = det Pn (I + H (ψα,n ))−1 Pn . This implies the desired assertion.
In the following proposition we identify the limit of the determinant
det Pn (I + H (ψα,n ))−1 Pn as n → ∞. Recall the definitions (20), (53), and Proposition 4.1. Proposition 5.5. We have
(63) lim det Pn (I + H (ψα,n ))−1 Pn n→∞
= det H (hα )(I + H (u−1/2,1 ))−1 H (hα ) det H (hα )(I − H (u1/2,1 ))−1 H (hα ) , where all expressions on the right-hand side are well defined.
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T. Ehrhardt
Proof. First of all we remark that the right-hand side is well defined. The inverses exist due to Proposition 4.1. Notice that H (hα )2 is a projection operator. This can be seen as follows. Formulas (35) together with the fact that h˜ α = h−1 α imply that −1 )T (h−1 ) + T (h−1 )H (h ) = I = T (hα )T (h−1 ) + H (h )H (h ) and 0 = H (h α α α α α α α T (h−1 α )H (hα ), whence H (hα )3 = (I − T (hα )T (h−1 α ))H (hα ) = H (hα ). We consider the operators H (hα )(I ± H (u∓1/2,1 ))−1 H (hα ) as being restricted onto the image of H (hα )2 . We can complement these operators with the projection I − H (hα )2 without changing the value of the corresponding determinant,
det H (hα )(I ± H (u∓1/2,1 ))−1 H (hα )
= det I + H (hα )(I ± H (u∓1/2,1 ))−1 H (hα ) − H (hα )2 . By Proposition 4.12(ii)-(iii) we see that this last operator determinant is well-defined. With µ = µα,n given by (61) we obtain from (22), (60) and (37) that ψα,n (t) =
t −µ 1 − µt
−1/2
t +µ 1 + µt
1/2
−1 = G−1 µ,1 (u−1/2,1 )Gµ,−1 (u1/2,1 ).
(±1)
Introduce the functions ψα,n by (55). Then (1) (−1) + 1)(ψα,n + 1). ψα,n = (ψα,n
Proposition 4.10 implies that (1) (−1) (−1) (1) H (ψα,n ) = H (ψα,n ) + H (ψα,n ) + H (ψα,n )H (ψα,n ) + o1 (1),
where o1 (1) stands for a sequence of operators converging in the trace norm to zero as (1) n → ∞. By Proposition 4.12(i) and Proposition 4.1, the operators I + H (ψα,n ) and (−1) I + H (ψα,n ) are invertible and their inverses are uniformly bounded. Hence (1) −1 (−1) −1 )) (I + H (ψα,n )) + o1 (1). (I + H (ψα,n ))−1 = (I + H (ψα,n
Using the formula (I + A)−1 = I − (I + A)−1 A = I − A(I + A)−1 , we can write this as (1) −1 (−1) −1 (I + H (ψα,n ))−1 = −I + (I + H (ψα,n )) +(I + H (ψα,n )) (1) −1 (1) (−1) (−1) −1 + (I +H (ψα,n )) H (ψα,n )H (ψα,n )(I +H (ψα,n )) + o1 (1).
It follows that (1) −1 (−1) −1 Pn (I + H (ψα,n ))−1 Pn = −Pn + Pn (I + H (ψα,n )) Pn + Pn (I + H (ψα,n )) Pn (1) −1 (1) (−1) (−1) −1 +Pn (I +H (ψα,n )) H (ψα,n )H (ψα,n )(I +H (ψα,n )) Pn + o1 (1).
Dyson’s Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel
339
Since I − Pn = I − H (t n )2 = T (t n )T (t −n ) (see (35)), we have (1) (−1) (1) (−1) H (ψα,n )(I − Pn )H (ψα,n ) = H (ψα,n )T (t n )T (t −n )H (ψα,n ) (1) (−1) = T (t −n )H (ψα,n )H (ψα,n )T (t n ) = o1 (1),
where we used also Proposition 4.10 and (36). Hence we obtain Pn (I + H (ψα,n ))−1 Pn (1) −1 (−1) −1 = −Pn + Pn (I + H (ψα,n )) Pn + Pn (I + H (ψα,n )) Pn (1) −1 (1) (−1) (−1) −1 +Pn (I + H (ψα,n )) H (ψα,n )Pn H (ψα,n )(I + H (ψα,n )) Pn + o1 (1) (1) −1 (−1) −1 = Pn (I + H (ψα,n )) Pn (I + H (ψα,n )) Pn + o1 (1).
Therein we employ again (I + A)−1 = I − (I + A)−1 A = I − A(I + A)−1 . Prop(1) osition 4.12(ii)-(iii) implies the uniform boundedness of Pn (I + H (ψα,n ))−1 Pn and (−1) −1 Pn (I + H (ψα,n )) Pn . In connection with the well-known formula | det(I + A) − det(I + B)| ≤ A − B1 exp(max{A1 , B1 }), this proves that
lim det Pn (I + H (ψα,n ))−1 Pn n→∞
(1) −1 (−1) −1 = lim det Pn (I + H (ψα,n )) Pn det Pn (I + H (ψα,n )) Pn . n→∞
These determinants can be written as
(±1) −1 (±1) −1 det Pn (I + H (ψα,n )) Pn = det I + Pn (I + H (ψα,n )) Pn − Pn , and now the convergence in the trace norm stated in Proposition 4.12(ii)-(iii) implies the desired assertion. We remark in this connection that the sequence µα,n defined in (61) satisfies condition (56).
In regard to the next result, recall the definition (15) of the functions uˆ β . Theorem 5.6. We have
α2 det(I − Kα ) = exp − det α2 (I + HR (uˆ −1/2 ))−1 α2 ) 8
× det α2 (I − HR (uˆ 1/2 ))−1 α2 ,
(64)
where all expressions on the right-hand side are well defined. Proof. We combine Proposition 5.1 with Proposition 5.4 and Proposition 5.5 to conclude that
α2 det(I − Kα ) = exp − det H (hα )(I + H (u−1/2,1 ))−1 H (hα ) 8
× det H (hα )(I − H (u1/2,1 ))−1 H (hα ) .
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Let us now introduce a unitary transform S = FU , i.e., U
F
S : H 2 (T) −→ H 2 (R) −→ L2 (R+ ), 1 2 f 1+ix where (Uf ) = √π (1−ix) 1−ix , F is the Fourier transform, and H (R) is the Hardy space on R, i.e., the set of all functions f ∈ L2 (R) for which (Ff )(x) = 0 for x < 0. Using the definitions (7) and (13) it is straightforward to prove that HR (a) = SH (b)S ∗ , where 1 + ix . (65) a(x) = b 1 − ix In particular, HR (uˆ β ) = SH (uβ,1 )S ∗ ,
HR (eixα/2 ) = SH (hα )S ∗ .
It remains to remark that H (eixα/2 )2 = α/2 . The invertibility of I ±HR (uˆ ∓1/2 ) follows from the invertibility of I ± H (u∓1/2 ).
Now we state the following asymptotic formulas for the two operator determinants appearing on the right-hand side of (64). For convenience we make a change in variables α → 2α. These formulas are proved in [3] (see Sect. 3.6). In these formulas, G(z) stands for the Barnes G-function [1]. In regard to the second fomula, the simple computation G(3/2) = G(1/2)(1/2) = G(1/2)π 1/2 has to be done. Theorem 5.7. The following asymptotic formulas hold,
α → ∞, (66) det α (I + HR (uˆ −1/2 ))−1 α ∼ α −1/8 π 1/4 21/4 G(1/2),
det α (I − HR (uˆ 1/2 ))−1 α ∼ α −1/8 π 1/4 2−1/4 G(1/2), α → ∞. (67) Combining the previous results we get the desired asymptotic formula. Theorem 5.8. The asymptotic formula log det(I − K2α ) = −
α2 log α − + C + o(1), 2 4
α → ∞,
(68)
holds with the constant log 2 + 3ζ (−1). 12 Proof. The previous two theorems give the asymptotic formula α2 α −1/4 π 1/2 (G(1/2))2 , det(I − K2α ) ∼ exp − 2 C=
(69)
α → ∞.
(70)
We can express G(1/2) in terms of ζ (−1), where ζ is Riemann’s zeta function. According to [1, p. 290] we have 1 3 log A log 2 log π + − + log G(1/2) = − 4 8 2 24 with A = exp(−ζ (−1) + 1/12) being Glaisher’s constant. Hence log 2 log π + 3ζ (−1) + , 2 log G(1/2) = − 2 12 which implies the desired asymptotic formula (68) with the constant (69).
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References 1. Barnes, E.B.: The theory of the G-function. Quart. J. Pure Appl. Math. XXXI, 264–313 (1900) 2. Basor, E.L., Ehrhardt, T.: Some identities for determinants of structured matrices. Linear Algebra Appl. 343/344, 5–19 (2002) 3. Basor, E.L., Ehrhardt, T.: On the asymptotics of certain Wiener-Hopf-plus-Hankel determinants. New York J. Math. 11, 171–203 (2005) 4. Basor, E.L., Tracy, C.A., Widom, H.: Asymptotics of level-spacing distributions for random matrices. Phys. Rev. Lett. 69(1), 5–8 (1992) 5. B¨ottcher, A., Silbermann, B.: Analysis of Toeplitz operators. Berlin: Springer, 1990 6. Deift, P.A., Its, A.R., Zhou, X.: A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. 146, 149–235 (1997) 7. Dyson, F.J.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, 171–183 (1976) 8. Ehrhardt, T., Silbermann, B.: Approximate identities and stability of discrete convolution operators with flip. Operator Theory: Adv. Appl., Vol. 110, 103–132 (1999) 9. Gohberg, I., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators in Hilbert space. Trans. Math. Monographs 18, Providence, R.I.: Amer. Math. Soc., 1969 10. Metha, M.L.: Random Matrices. 6th ed., San Diego: Academic Press, 1994 11. Jimbo, M., Miwa, T., Mˆori, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlev´e transcendent. Phys. D 1(1), 80–158 (1980) 12. Krasovsky, I. V.: Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. (25), 1249–1272 (2004) 13. Power, S.: The essential spectrum of a Hankel operator with piecewise continuous symbol. Michigan Math. J. 25(1), 117–121 (1978) 14. Power, S.: C*-algebras generated by Hankel operators and Toeplitz operators. J. Funct. Anal. 31(31), 52–68 (1979) 15. Tracy, C.A., Widom, H.: Introduction to random matrices. In: Lecture Notes in Physics, Vol. 424, Berlin: Springer, 1993, pp. 103–130 16. Widom, H.: The strong Szeg¨o limit theorem for circular arcs. Indiana Univ. Math. J. 21, 277–283 (1971/1972) 17. Widom, H.: The asymptotics of a continuous analogue of orthogonal polynomials. J. Appr. Theory 77, 51–64 (1994) 18. Widom, H.: Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals. Commun. Math. Phys. 171, 159–180 (1995) Communicated by J.L. Lebowitz
Commun. Math. Phys. 262, 343–372 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1497-0
Communications in
Mathematical Physics
A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces Jiansheng Geng, Jiangong You Department of Mathematics, Nanjing University, Nanjing 210093, P.R.China. E-mail: [email protected] Received: 12 November 2004 / Accepted: 5 September 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi–periodic solutions and its local Birkhoff normal form. The applications to the higher dimensional beam equations and the higher dimensional Schr¨odinger equations with nonlocal smooth nonlinearity are also given in this paper. 1. Introduction In late 1980’s, motivated by the construction of quasi-periodic solutions for Hamiltonian partial differential equations, the celebrated KAM theory was successfully generalized to infinite dimensional settings by Kuksin [14] and Wayne [20], see also [15–18], which applies to, as typical examples, one-dimensional semi-linear Schr¨odinger equations iut − uxx + mu = f (u), and wave equations utt − uxx + mu = f (u), with Dirichlet boundary conditions. When trying to further generalize the KAM theory so as to apply to the one-dimensional wave equations with periodic boundary conditions and higher dimensional Hamiltonian partial differential equations, the multiplicity of the eigenvalues becomes an obstacle. Especially, the multiplicity goes asymptotically to infinity in the higher dimensional case. On one hand, the multiplicity makes the unperturbed part more complicated at succeeding KAM steps, as a consequence solving the
The work was supported by the National Natural Science Foundation of China (10531050)
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linearized equations becomes very complicated; on the other hand, it makes the measure estimation very difficult since there are so many non-resonance conditions to be satisfied. For those reasons, there is no KAM theorem for higher dimensional Hamiltonian partial differential equations so far. To overcome this difficulty, Craig and Wayne retrieved the origination of the KAM method — Newtonian iteration method together with the Liapunov-Schmidt decomposition which involves the Green’s function analysis and the control of the inverse of infinite matrices with small eigenvalues. They succeeded in constructing periodic solutions of the one-dimensional semi-linear wave equations with periodic boundary conditions. Bourgain further developed the Craig–Wayne’s method and proved the existence of quasi-periodic solutions of partial differential equations in higher dimensional spaces with Dirichlet boundary conditions or periodic boundary conditions. We point out that the Craig-Wayne-Bourgain’s method allows one to avoid explicitly using the Hamiltonian structure of the systems. We will not introduce their approaches in detail. The reader is referred to Craig–Wayne [9], Bourgain [3–7]. Comparing with Craig-Wayne-Bourgain’s approach, the KAM approach has its own advantages. Besides obtaining the existence results it allows one to construct a local normal form in a neighborhood of the obtained solutions, and this is useful for better understanding of the dynamics. For example, one can obtain the linear stability and zero Liapunov exponents. The question is: Is there a KAM theorem which can be applied to Hamiltonian partial differential equations in higher dimensional spaces? This paper is motivated by this question. In this paper, we give a KAM theorem which applies to some Hamiltonian partial differential equations in higher dimensional spaces. We use the theorem to construct the quasi-periodic solutions and prove their linear stability. The KAM theorem can be applied to some Hamiltonian partial differential equations not explicitly containing the space variables and time variable, including the higher dimensional beam equations utt + (− + m)2 u + f (u) = 0,
x ∈ Td
and the higher dimensional Schr¨odinger equations with nonlocal smooth nonlinearities (see Sect. 3 for details) iut + Au + N (u) = 0,
x ∈ Td ,
as well as one-dimensional wave equations under the periodic boundary conditions. Different from the finite dimensional case, the KAM theorem may not be true for infinite dimensional nearly integrable Hamiltonian systems. One has to impose further restrictions both on the unperturbed part and on the perturbation besides smallness. In the existent infinite dimensional KAM theorems, e.g., Kuksin [14], Wayne [20] and P¨oschel [18], some assumptions on the growth of the normal frequencies and the regularity of the perturbation are required (see (A1)–(A3) in the next section). In this paper, we additionally assume that the perturbation has a special form defined in (A4) in the next section. Our proof benefits a lot from such speciality of the perturbation. With the speciality of the form of the perturbation, we can prove that the normal form part of the Hamiltonian remains simple during the iteration. Actually, the normal variables in the normal form part are always uncoupled along the KAM iteration. This makes the measure estimate as easy as the one-dimensional case. Compared with the proof of the existent KAM theorems, an additional job done in this paper is to prove that perturbation always has the special form defined in (A4) along the KAM iteration.
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We remark that although the assumption (A4) looks artificial, the Hamiltonian systems deriving from the Hamiltonian partial differential equations in Td not containing explicitly the space variables and the time variable do have the special form defined in (A4). And our KAM theorem can be applied to some kind of Hamiltonian partial differential equations, such as the beam equations and the Sch¨odinger equations mentioned above. Although the applicable equations of the KAM theorem given in this paper are less general, they already have sufficiently strong physical background. And although the existence results are not new, since Bourgain has obtained the results for more general classes of equations [6], the KAM approach may provide more information about the constructed solutions. We are interested in the establishment of a KAM theorem for higher dimensional Hamiltonian partial differential equations. This paper is a step towards this goal. Moreover our proof is simpler compared with Bourgain’s proof, we think it is of some interest. With this paper we also hope to call more attention to exploit the inherent properties of the considered equations themselves when studying the dynamics of Hamiltonian partial differential equations. Finally, we remark that a result similar to Bourgain’s has been recently announced by Kuksin and Eliasson, but a paper is not yet available. Since the statement of the main result is a bit long, we postpone it to the next section. This paper is organized as follows: In Sect. 2 we give an infinite dimensional KAM theorem; in Sect. 3, we give its applications to higher dimensional beam equations and higher dimensional non-local smooth Schr¨odinger equations. The proof of the KAM theorem is given in Sects. 4, 5, 6. Some technical lemmas are proved in the Appendix.
2. An Infinite Dimensional KAM Theorem for Hamiltonian Partial Differential Equations In this section, we will formulate an infinite dimensional KAM theorem that can be applied to higher dimensional beam equations, higher dimensional nonlocal smooth Schr¨odinger equations and one-dimensional wave equations under periodic boundary conditions. We start by introducing some notations. For given b vectors in Zd , say {i1 , . . . , ib }, we denote Z1d = Zd \ {i1 , . . . , ib }. Let z = (. . . , zn , . . .)n∈Zd , and its complex conjugate 1 z¯ = (. . . , z¯ n , . . .)n∈Zd . We introduce the weighted norm 1
za,ρ =
|zn ||n|a e|n|ρ ,
n∈Z1d
where |n| =
n21 + · · · + n2d , n = (n1 , . . . , nd ) and a ≥ 0, ρ > 0. Denote a neighbor-
hood of Tb × {I = 0} × {z = 0} × {¯z = 0} by D(r, s) = {(θ, I, z, z¯ ) : |Imθ| < r, |I | < s 2 , za,ρ < s, ¯za,ρ < s}, where | · | denotes the sup-norm of complex vectors. Moreover, we denote by O a positive–measure parameter set in Rb . Let α ≡ (· · · , αn , · · ·)n∈Zd , β ≡ (· · · , βn , · · ·)n∈Zd , αn and βn ∈ N with finitely 1 1 β many non-zero components of positive integers. The product zα z¯ β denotes n znαn z¯ nn .
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For any given function F (θ, I, z, z¯ ) =
Fαβ (θ, I )zα z¯ β ,
(2.1)
α,β 1 function in parameter ξ in the sense of Whitney, we define the where Fαβ is a CW weighted norm of F by Fαβ |zα ||¯zβ |, (2.2) F D(r,s),O ≡ sup za,ρ <s ¯za,ρ <s
α,β
where, if Fαβ = k∈Zb ,l∈Nb Fklαβ (ξ )I l eik,θ , (·, · being the standard inner product in Cb ), Fαβ is short for ∂Fklαβ |Fklαβ |O s 2|l| e|k|r , |Fklαβ |O ≡ sup |Fklαβ | + | Fαβ ≡ | (2.3) ∂ξ ξ ∈O k,l
(the derivatives with respect to ξ are in the sense of Whitney). To function F , we associate a Hamiltonian vector field defined by XF = (FI , −Fθ , {iFzn }n∈Zd , {−iFz¯ n }n∈Zd ). 1
Its weighted norm is defined
1
by1
1 XF D(r,s),O ≡ FI D(r,s),O + 2 Fθ D(r,s),O s 1 + Fzn D(r,s),O |n|a¯ e|n|ρ + Fz¯ n D(r,s),O |n|a¯ e|n|ρ . (2.4) s d d n∈Z1
n∈Z1
Remark. In this paper, we require that a¯ > a, i.e., the weight of vector fields is a little heavier than that of z, z¯ . The boundedness of XF Dρ (r,s),O means XF sends a decaying z-sequence to a fastly decaying sequence. The starting point will be a family of integrable Hamiltonians of the form N = ω(ξ ), I + n (ξ )zn z¯ n ,
(2.5)
n∈Z1d
where ξ ∈ O is a parameter, the phase space is endowed with the symplectic structure dI ∧ dθ + i ¯n. d dzn ∧ d z n∈Z1
For each ξ ∈ O, the Hamiltonian equations of motion for N , i.e., dθ = ω, dt
dI = 0, dt
dzn = −i n zn , dt
d z¯ n = i n z¯ n , dt
n ∈ Z1d ,
(2.6)
1 The norm · in (2.2). The vector function G : D(r, s)× O → D(r,s),O for scalar functions is defined Cm , (m < ∞) is similarly defined as GD(r,s),O = m i=1 Gi D(r,s),O .
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admit special solutions (θ, 0, 0, 0) → (θ + ωt, 0, 0, 0) that correspond to an invariant torus in the phase space. Consider now the perturbed Hamiltonian H = N + P = ω(ξ ), I + n (ξ )zn z¯ n + P (θ, I, z, z¯ , ξ ). (2.7) n∈Z1d
Our goal is to prove that, for most values of parameter ξ ∈ O (in Lebesgue measure sense), the Hamiltonians H = N + P still admit invariant tori provided that XP D(r,s),O is sufficiently small. To this end, we need to impose some conditions on ω(ξ ), n (ξ ) and the perturbation P . As we already remarked, the persistence of the lower dimensional torus may not be true if one only assumes the smallness of the perturbation. This is an essential difference between infinite and finite dimensional cases. 1 diffeomorphism between O and its (A1) Nondegeneracy: The map ξ → ω(ξ ) is a CW image. (A2) Asymptotics of normal frequencies: There exists a ι > 0 such that for all n = (n1 , . . . , nd ) ∈ Z1d , n = 0, n ∈ Z1d , (2.8) −ι ¯ ˜ ˜ n = n + n , n = o(|n| ), (2.9) 1 ¯ n ’s are real and independent of ξ while ˜ n ’s are C functions of ξ with where W ) 1 CW -norm bounded by some small positive constant L (depending on det( ∂ω(ξ ∂ξ )); ¯ n is assumed to be as follows: furthermore, the asymptotic behavior of ¯ n = |n|p + o(|n|p ), ¯n− ¯ m = |n|p − |m|p + o(|m|−ι ), |m| ≤ |n|, (2.10) where p ≥ 2 for d > 1 or p ≥ 1 for d = 1. (A3) Regularity of the perturbation: The perturbation P is regular in the sense that XP D(r,s),O < ∞ with a¯ > a. (A4) Special form of the perturbation: The perturbation is taken from a special class of analytic functions, A= P :P = Pklαβ (ξ )I l eik,θ zα z¯ β , b b k∈Z ,l∈N ,α,β
where k, α, β has the following relation b j =1
k j ij +
(αn − βn )n = 0.
(2.11)
n∈Z1d
Remark. Compared with the existent infinite dimensional KAM theorems in literature, we make an additional assumption (A4) on the perturbation. The assumption looks artificial, but it is satisfied by the infinite dimensional Hamiltonian systems derived from Hamiltonian partial differential equations in T d which do not explicitly contain the space variables and the time variable, for example, the Schr¨odinger equations, wave equations and beam equations in Td in the introduction. Now we are ready to state our KAM Theorem.
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Theorem 1. Assume that the unperturbed Hamiltonian N in (2.5) satisfies (A1) and (A2) and P satisfies (A3) and (A4). Let γ > 0 small enough, there is a positive constant ε = ε(b, d, p, ι, a¯ −a, L, γ ) such that if XP D(r,s),O < ε, then the following holds true: There exist a Cantor set Oγ ⊂ O with meas(O \ Oγ ) = O(γ ϑ ) (ϑ is specified in 1 in ξ ) Sect. 6) and two maps (analytic in θ and CW : Tb × Oγ → D(r, s), where is
ε -close γ2
ω˜ : Oγ → Rb ,
to the trivial embedding 0 : Tb × O → Tb × {0, 0, 0} and ω˜
is ε-close to the unperturbed frequency ω, such that for any ξ ∈ Oγ and θ ∈ Tb , the curve t → (θ + ω(ξ ˜ )t, ξ ) is a quasi-periodic solution of the Hamiltonian equations governed by H = N + P . Moreover, the obtained solutions are linearly stable. Remark 1. In the one dimensional case, the growth of n can be sub-linear ( 0 < p < 1). But we can not find any interesting application of it. Remark 2. The regularity a¯ > a is used to control the drifting of the normal frequencies which is crucial in the measure estimation of the survived parameters O \ Oγ for our approach. It seems that the restriction is only of technical reasons. One may expect to have a KAM theorem without the regularity assumption (A3). However this problem remains open so far. Remark 3. The Hamiltonian systems defined by Hamiltonian partial differential equations do have the special form defined in (A4). This fact has been used by many authors when transforming the leading nonlinearity in the perturbation into the partial Birkhoff normal form under Cartesian coordinate systems (see Kuksin–P¨oschel [16], P¨oschel [17], Craig–Worfolk [10], Bourgain [7, 8], Bambusi [1], Bambusi–Berti [2], and Geng– You [12, 13]). Those papers actually use this property for one or two steps. In this paper, we will use this fact at each step of the KAM iteration. For this purpose, we have to prove that the change of action-angle variables and the KAM iteration preserve the special form of the perturbation defined in (A4). Remark 4. In the one dimensional case, assumption (A4) is replaced by a kind of decay property in [13]. Since the decay property is weaker than assumption (A4), the KAM theorem assuming only decay property may have more applications, e.g., when the equation depends on the space variable x. However, the proof for the higher dimensional case would be much more complicated and is not available so far. Remark 5. The parameter γ plays the role of the Diophantine constant for the frequency ω˜ in the sense that there exists τ > 0 (specified in Sect. 6) such that the frequencies of the obtained KAM tori satisfy the following Diophantine conditions: k, ω ˜ ≥
γ , 2|k|τ
∀k ∈ Zb \ {0}.
Notice also that Oγ is claimed to be nonempty only for γ small enough.
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3. Applications 1. Higher dimensional beam equations. In order to avoid the technical complexities, we apply Theorem 1 to the d-dimensional beam equations with a Fourier multiplier Mξ . Actually, Theorem 1 can be applied to the higher dimensional beam equations with 1 constant potentials, i.e., Au ≡ (2 + m) 2 u, x ∈ Rd , m > 0 in (3.1). However, in order to transform the original Hamiltonian into a perturbation of a nonlinear integrable system, the normal form technique is needed. Since the normal form procedure is quite involved and does not fit the main theme of this paper we will handle it in another paper. Consider utt + A2 u + f (u) = 0, Au ≡ (− + Mξ )u, x ∈ Rd , t ∈ R, u(t, x1 + 2π, x2 , . . . , xd ) = · · · = u(t, x1 , x2 , . . . , xd−1 , xd + 2π ) = u(t, x1 , x2 , . . . , xd ),
(3.1)
where f (u) is a real–analytic function near u = 0 with f (0) = f (0) = 0. Here we assume that the operator A = − + Mξ with periodic boundary conditions has eigenvalues {µn } satisfying ωj = µij = |ij |2 + ξj ,
1 ≤ j ≤ b,
n = µn = |n| ,
n = i1 , . . . , ib , (3.2) 1 in,x form a basis in the domain and the corresponding eigenfunctions φn (x) = (2π) de 2
of the operator. Assume that i1 , . . . , ib ∈ Zd are the distinguished sites of Fourier modes (assume 0 ∈ {i1 , . . . , ib } in order to take care of (µn , k) = (0, 0)), and ξ = (ξ1 , . . . , ξb ) is a parameter taking on a closed set O ⊂ Rb of the positive–measure. Introducing v = ut , (3.1) reads ut = v, vt = −A2 u − f (u). Letting q =
1 1 √1 A 2 u − i √1 A− 2 v, 2 2
(3.3)
we obtain
1 1 1 qt = Aq + √ A− 2 f i 2
− 21
A
q + q¯ √ 2
.
(3.4)
Equation (3.4) can be rewritten as the Hamiltonian equations qt = i
∂H ∂ q¯
and the corresponding Hamiltonian is 1 − 21 q + q¯ dx, g A H = (Aq, q) + √ 2 2 Td where (·, ·) denotes the inner product in L2 and g is a primitive of f . Let qn φn (x). q(x) = n∈Zd
(3.5)
(3.6)
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Then system (3.5) is equivalent to the lattice Hamiltonian equations qn φn + q¯n φ¯ n ∂G dx q˙n = i µn qn + , G(q, q) ¯ ≡ g √ ∂ q¯n 2µn Td d
(3.7)
n∈Z
with corresponding Hamiltonian function H = n∈Zd µn qn q¯n + G(q, q). ¯ Sincef (u) is real analytic in u, g(q, q) ¯ is real analytic in q, q. ¯ Making use of q(x) = n∈Zd qn φn (x) again, we may rewrite g as follows g(q, q) ¯ =
gαβ q α q¯ β φ α φ¯ β ,
α,β
hence
qn φn + q¯n φ¯ n dx = G(q, q) ¯ ≡ g Gαβ q α q¯ β , √ 2µ n Td α,β n∈Zd Gαβ = 0, if (αn − βn )n = 0.
(3.8)
n∈Zd
As in [16, 17, 12, 13], the perturbation G in (3.7) has the following regularity property. Lemma 3.1. For any fixed a ≥ 0, ρ > 0, the gradient Gq¯ is real analytic as a map in a neighborhood of the origin with Gq¯ a+1,ρ ≤ cq2a,ρ .
(3.9)
Next we introduce standard action-angle variables (θ, I ) = ((θ1 , . . . , θb ), (I1 , . . . , Ib )) in the (qi1 , . . . , qib , q¯i1 , . . . , q¯ib )-space by letting, Ij = qij q¯ij ,
j = 1, . . . , b,
and qn = zn , q¯n = z¯ n , n = i1 , . . . , ib . So system (3.7) becomes dθj dIj = ωj + PIj , = −Pθj , j = 1, . . . , b, dt dt dzn d z¯ n = −i( n zn + Pz¯ n ), = i( n z¯ n + Pzn ), n ∈ Z1d , dt dt
(3.10)
where P is just G with the (qi1 , . . . , qib , q¯i1 , . . . , q¯ib , qn , q¯n )-variables expressed in terms of the (θ, I, zn , z¯ n ) variables. The Hamiltonian associated to (3.10) (with respect to the symplectic structure dI ∧ dθ + i ¯ n ) is given by d dzn ∧ d z n∈Z1
H = ω(ξ ), I +
n (ξ )zn z¯ n + P (θ, I, z, z¯ , ξ ).
(3.11)
n∈Z1d
Let’s verify that P has the special form defined in (A4) from (3.8), i.e., P (θ, I, z, z¯ , ξ ) ∈ A, which is a key assumption of Theorem 1.
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Denote by en the infinite dimensional vector with the nth component being 1 and the other components being zero, and k = (k1 , . . . , kb ), kj = αij − βij , 1 ≤ j ≤ b, then due to (3.8), Gαβ q α q¯ β G=
n∈Zd (αn −βn )n=0
=
αi
j =1 (αij −βij )ij+ n∈Zd (αn−βn )n=0
b
=
·e
i
b
j =1 (αij −βij )θj
n∈Z1d
α−
j =1 αij eij
b
j =1 αij eij
q¯
b
β−
j =1 βij eij
z¯
b
β−
j =1 βij eij
Pklαβ I l eik,θ zα z¯ β ≡ P .
j =1 kj ij +
b
α−
(αn −βn )n=0
b
z
βi
αi +βi αi +βi Gαβ I1 1 1 · · · Ib b b
b
j =1 (αij −βij )ij +
=
1
αi
βi
Gαβ qi1 1 q¯i1 1 · · · qib b q¯ib b q
n∈Z1d
(αn −βn )n=0
Thus Pklαβ = 0
if
b
kj i j +
j =1
(αn − βn )n = 0,
(3.12)
n∈Z1d
i.e., P ∈ A. Moreover the regularity of P holds true: 1
Lemma 3.2. For any ε > 0 sufficiently small and s = ε 2 , if |I | < s 2 and za,ρ < s, then XP D(r,s),O ≤ ε,
a¯ = a + 1.
(3.13)
To this point, we have verified all the assumptions of Theorem 1 for (3.11) with p = 2, ι = +∞, a¯ − a = 1. Now we are in the position to apply Theorem 1 to get the following result. Theorem 2. For any 0 < γ 1, there is a Cantor subset Oγ ⊂ O with meas(O\Oγ ) = O(γ ϑ ) (ϑ is to be specified in Sect. 6), such that for any ξ ∈ Oγ , Eq. (3.1) parametrized by ξ admits a small-amplitude, quasi-periodic solution of the form un (ω1 t, . . . , ωb t)ein,x , u(t, x) = n∈Zd
where un : Tb → R and ω1 , . . . , ωb are close to the unperturbed frequencies ω1 , . . . , ωb . Moreover, the quasi-periodic solutions we obtained are linearly stable. Remark 1. Theorem 1 also applies to 1D wave equations with periodic boundary conditions utt − uxx + mu + au3 + O(u4 ) = 0, a = 0, u(t, x + 2π) = u(t, x).
(3.14)
Since the proof follows exactly the same steps as that of the beam equations, we omit it.
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2. The higher dimensional nonlocal smooth Schr¨odinger equations. The dD nonlinear Schr¨odinger equations to be considered are iut + Au + N (u) = 0,
Au = −u + Mξ u,
x ∈ Rd , t ∈ R
(3.15)
with periodic boundary conditions u(t, x1 + 2π, x2 , . . . , xd ) = . . . = u(t, x1 , x2 , . . . , xd−1 , xd + 2π ) = u(t, x1 , x2 , . . . , xd ). The operator A is the same as that in the beam equations. The nonlinearity we would like to consider is N (u) = f (|u|2 )u
(3.16)
with the function f real analytic in a neighborhood of 0 ∈ C and vanishing at zero. However for the sake of regularity imposed on the nonlinearity (see (A3)), as those in P¨oschel [19] and Bambusi–Berti [2], we have to assume some nonlocal smoothness for the nonlinearity. Thus we actually consider the nonlinearity N (u) = (f (|u|2 )u),
(3.17)
where : u → ψ ∗u is a convolution operator with a function ψ, which is of smoothness of order δ > 0. More precisely, ua+δ,ρ ≤ cua,ρ .
(3.18)
Equation (3.15) can be rewritten as a Hamiltonian equation ut = i and the corresponding Hamiltonian is 1 H = (Au, u) + 2
∂H ∂ u¯
(3.19)
g(|u|2 ) dx,
(3.20)
Td
where (·, ·) denotes the inner product in L2 and g is a primitive of f . Let u(x) = qn φn (x). n∈Zd
System (3.19) is then equivalent to the lattice Hamiltonian equations ∂G q˙n = i(µn qn + ), G ≡ g(|u|2 )dx , (3.21) ∂ q¯n Td with corresponding Hamiltonian function H = n∈Zd µn qn q¯n + G. φn (x), µn are defined in the last sub-section. Since N (u) is real analytic in u, then making use of u(t, x) = n∈Zd qn (t)φn (x), we may rewrite N (u) as follows N (u) = Nαβ q α q¯ β φ α φ¯ β , α,β
KAM Theorem for Hamiltonian PDEs in Higher Dimensional Spaces
hence
G≡
Td
g(|u|2 )dx =
Gαβ = 0,
if
353
Gαβ q α q¯ β ,
α,β
(αn − βn )n = 0.
(3.22)
n∈Zd
As in [19, 2], the perturbation G in (3.21) has the following regularity property. Lemma 3.3. For any fixed a ≥ 0, ρ > 0, the gradient Gq¯ is real analytic as a map in a neighborhood of the origin with Gq¯ a+δ,ρ ≤ cq3a,ρ .
(3.23)
The same as that of the beam equations, by introducing the standard action-angle variables (θ, I ) = ((θ1 , . . . , θb ), (I1 , . . . , Ib )) in the (qi1 , . . . , qib , q¯i1 , . . . , q¯ib )-space, we arrive at a Hamiltonian system with the Hamiltonian (with respect to the symplectic structure dI ∧ dθ + i ¯n) d dzn ∧ d z n∈Z1
H = ω(ξ ), I +
n (ξ )zn z¯ n + P (θ, I, z, z¯ , ξ ).
(3.24)
n∈Z1d
The same as the beam equations, we can prove that P = Pklαβ (ξ )I l eik,θ zα z¯ β k∈Zb ,l∈Nb ,α,β
satisfies the assumption (A4), i.e., Pklαβ = 0,
if
b
k j ij +
j =1
(αn − βn )n = 0.
(3.25)
n∈Z1d
Moreover the regularity of P holds true: 1
Lemma 3.4. For any ε > 0 sufficiently small and s = ε 2 , if |I | < s 2 and za,ρ < s, then XP D(r,s),O ≤ ε,
a¯ = a + δ.
(3.26)
Remark. When we consider the local smooth nonlinearity N (u) = f (|u|2 )u, we can get XP D(r,s),O ≤ ε with a¯ = a. As a consequence, Theorem 1 can not be applied since (A3) is violated. So we have verified all the assumptions of Theorem 1 for (3.24) with p = 2, ι = +∞, a¯ − a = δ > 0. Then Theorem 1 yields the following result for nonlinear Schr¨odinger equations. Theorem 3. For any 0 < γ 1, there is a Cantor subset Oγ ⊂ O with meas(O\Oγ ) = O(γ ϑ ) (ϑ is to be specified in Sect. 6) such that for any ξ ∈ Oγ , Eq. (3.15) parameterized by ξ ∈ O admits a small-amplitude, quasi-periodic solution of the form u(t, x) = un (ω1 t, . . . , ωb t)ein,x , n∈Zd
where un : Tb → R and ω1 , . . . , ωb are close to the unperturbed frequencies ω1 , . . . , ωb . Moreover, the quasi-periodic solutions obtained here are linearly stable.
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J. Geng, J. You
4. KAM Step Theorem 1 will be proved by a KAM iteration which involves an infinite sequence of change of variables. Each step of KAM iteration makes the perturbation smaller than in the previous step at the cost of excluding a small set of parameters. We have to prove the convergence of the iteration and estimate the measure of the excluded set after infinite KAM steps. At the ν–step of the KAM iteration, we consider a Hamiltonian vector field with Hν = Nν + Pν , where Nν is an "integrable normal form" and Pν ∈ A is defined in D(rν , sν ) × Oν−1 . We then construct a map ν : D(rν+1 , sν+1 ) × Oν → D(rν , sν ) × Oν−1 so that the vector field XHν ◦ν defined on D(rν+1 , sν+1 ) satisfies XPν+1 D(rν+1 ,sν+1 ),Oν = XHν ◦ν − XNν+1 D(rν+1 ,sν+1 ),Oν ≤ ενκ ,
κ>1
with some new normal form Nν+1 . Moreover, the new perturbation Pν+1 still has the special form defined in (A4). To simplify notations, in what follows, the quantities without subscripts refer to quantities at the ν th step, while the quantities with subscripts + denote the corresponding quantities at the (ν + 1)th step. Let us then consider the Hamiltonian H = N + P ≡ e + ω(ξ ), I + n (ξ )zn z¯ n + P (θ, I, z, z¯ , ξ, ε) (4.1) n∈Z1d
defined in D(r, s)×O. We assume that ξ ∈ O satisfies (a suitable τ > 0 will be specified later) |k, ω(ξ )| ≥
γ , |k|τ
k = 0,
|k, ω(ξ ) + n (ξ )| ≥
γ , |k|τ
(4.2)
γ , |k|τ γ |k, ω(ξ ) + n (ξ ) − m (ξ )| ≥ τ , |k|
|k, ω(ξ ) + n (ξ ) + m (ξ )| ≥
|k| + ||n| − |m|| = 0.
Moreover, XP D(r,s),O ≤ ε, and P =
k,l,α,β
(4.3)
Pklαβ I l eik,θ zα z¯ β is in the class A defined in (A4), i.e.,
Pklαβ = 0
if
b j =1
k j ij +
n∈Z1d
(αn − βn )n = 0.
(4.4)
KAM Theorem for Hamiltonian PDEs in Higher Dimensional Spaces
355
Remark 1. According to (4.4), when k = (k1 , · · · , kb ) = 0 and α = en , β = em , we get P0len em = 0
if
b
k j ij +
j =1
(αn − βn )n = n − m = 0.
n∈Z1d
This means that there are not terms of the form n=m P0len em I l zn z¯ m in the perturbation. As a result, the normal variables zn , z¯ m with n = m in the new normal form N+ will not be coupled. Remark 2. Compared with the KAM step in existent KAM theorems in the literature, we make an additional assumption P ∈ A. With this assumption the linearized equations are easy to be solved in Subsect. 4.1, and the new normal form after one step of the iteration still has the form N+ ≡ e+ + ω+ (ξ ), I + n∈Zd + n (ξ )zn z¯ n . This makes the 1 measure estimate available and easier at each KAM step. Subsection 4.5 is an additional work which proves the new perturbation P+ still has the special form defined in (A4), i.e., P+ ∈ A, after one step of the iteration. The proofs in Subsects. 4.2–4.4 are the same as that of the existent KAM theorems. We now let 0 < r+ < r and define s+ =
1 1 sε 3 , 4
4
ε+ = cγ −2 (r − r+ )−c ε 3 .
(4.5)
Here and later, the letter c denotes suitable (possibly different) constants that do not depend on the iteration steps. We now describe how to construct a set O+ ⊂ O and a change of variables : D+ × O+ = D(r+ , s+ ) × O+ → D(r, s) × O such that the transformed Hamiltonian H+ = N+ + P+ ≡ H ◦ satisfies all the above iterative assumptions with new parameters s+ , ε+ , r+ and with ξ ∈ O+ . 4.1. Solving the linearized equations. Expand P into the Fourier-Taylor series P = Pklαβ eik,θ I l zα z¯ β , k,l,α,β
where k ∈ Zb , l ∈ Nb and the multi–indices α and β run over the set of all infinite dimensional vectors α ≡ (· · · , αn , · · ·)n∈Zd with finitely many nonzero components of 1 positive integers. Let R be the truncation of P given by R(θ, I, z, z¯ ) = R0 + R1 + R2 = Pkl00 eik,θ I l k,|l|≤1
+ (Pnk10 zn + Pnk01 z¯ n )eik,θ k,n
+
k20 k11 k02 (Pnm zn zm + Pnm zn z¯ m + Pnm z¯ n z¯ m )eik,θ ,
(4.6)
k,n,m
where Pnk10 = Pklαβ with α = en , β = 0, here en denotes the vector with the nth component being 1 and the other components being zero; Pnk01 = Pklαβ with α = 0, β = en ;
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J. Geng, J. You
k20 = P k11 Pnm klαβ with α = en + em , β = 0; Pnm = Pklαβ with α = en , β = em ; k02 Pnm = Pklαβ with α = 0, β = en + em . Due to assumption (A4), P ∈ A implies that
Pkl00 = 0,
if
b
kj ij = 0,
j =1
Pnk10 = 0, Pnk01 = 0, k20 Pnm
= 0,
k11 = 0, Pnm
k02 = 0, Pnm
if if if if if
b j =1 b j =1 b j =1 b j =1 b
kj ij + n = 0, kj ij − n = 0, (4.7) kj ij + n + m = 0, kj ij + n − m = 0, kj ij − n − m = 0.
j =1
Remark. The special form of P defined in (A4), i.e., P ∈ A, is crucial in this paper. k11 = 0 if k = 0 and n = m, then the With P of such special form, one knows that Pnm 011 terms Pnm zn z¯ m with n = m are absent, i.e., zn , zm with n = m are uncoupled in the new normal form. Rewrite H as H = N + R + (P − R). By the choice of s+ in (4.5) and the definition of the norms, it follows immediately that XR D(r,s),O ≤ XP D(r,s),O ≤ ε.
(4.8)
Moreover, we take s+ s such that in a domain D(r, s+ ), X(P −R) D(r,s+ ) < c ε+ .
(4.9)
In the following, we will look for an F in the class A, defined in a domain D+ = D(r+ , s+ ), such that the time one map φF1 of the Hamiltonian vector field XF defines a map from D+ → D and transforms H into H+ . More precisely, by second order Taylor formula, we have H ◦ φF1 = (N + R) ◦ φF1 + (P − R) ◦ φF1 = N + {N, F } + R 1 1 (1 − t){{N, F }, F } ◦ φFt dt + {R, F } ◦ φFt dt + (P − R) ◦ φF1 + 0 0 011 = N+ + P+ + {N, F } + R − P0000 − ω , I − Pnn zn z¯ n , (4.10) n
KAM Theorem for Hamiltonian PDEs in Higher Dimensional Spaces
where
∂P dθ |z=¯z=0,I =0 , ∂I
ω =
N+ = N + P0000 + ω , I +
011 Pnn zn z¯ n ,
n
P+ =
1 0
357
(1 − t){{N, F }, F } ◦ φFt dt +
0
1
(4.11)
{R, F } ◦ φFt dt +(P − R) ◦ φF1 .
(4.12)
011 Remark. Generally speaking, n Pnn zn z¯ n should be replaced in (4.11) by 011 z z¯ , but in terms of (4.7), P 011 = 0 if n = m. Hence the terms P nm |n|=|m| 011nm n m P z z ¯ are absent. Thus N has the same form as that in the first step. n m + n=m nm We shall find a function F of the form F (θ, I, z, z¯ ) = F0 + F1 + F2 = Fkl00 eik,θ I l + (Fnk10 zn + Fnk01 z¯ n )eik,θ k=0,|l|≤1
+
k,n
k20 k02 (Fnm zn zm + Fnm z¯ n z¯ m )eik,θ +
k11 Fnm zn z¯ m eik,θ
|k|+||n|−|m||=0
k,n,m
(4.13) satisfying the equation {N, F } + R − P0000 − ω , I −
011 Pnn zn z¯ n = 0.
(4.14)
n
Lemma 4.1. F satisfies (4.14) and is in A if the Fourier coefficients of F are defined by the following equations: (k, ω)Fkl00 (k, ω − n )Fnk10 (k, ω + n )Fnk01 k20 (k, ω − n − m )Fnm k11 (k, ω − n + m )Fnm k02 (k, ω + n + m )Fnm
= = = = = =
iPkl00 , iPnk10 , iPnk01 , k20 , iPnm k11 , iPnm k02 . iPnm
k = 0, |l| ≤ 1, (4.15) |k| + ||n| − |m|| = 0,
Proof. Inserting F , defined in (4.13), into (4.14) one sees that (4.14) is equivalent to the following equations {N, F0 } + R0 = P0000 + ω , I , {N, F1 } + R1 = 0, 011 {N, F2 } + R2 = Pnn zn z¯ n .
(4.16)
n
By comparing the coefficients, the first equation in (4.16) is obviously equivalent to the first equation in (4.15). To solve {N, F1 } + R1 = 0, we note that (k, ωFnk10 zn − n Fnk10 zn )eik,θ {N, F1 } = i k,n
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J. Geng, J. You
+i
(k, ωFnk01 z¯ n + n Fnk01 z¯ n )eik,θ k,n
=i (k, ω − n )Fnk10 zn eik,θ k,n
+i
(k, ω + n )Fnk01 z¯ n eik,θ .
(4.17)
k,n
It follows that Fnk10 , Fnk01 are determined by the linear algebraic systems i(k, ω − n )Fnk10 + Rnk10 = 0,
n ∈ Z1d , k ∈ Zb ,
i(k, ω + n )Fnk01 + Rnk01 = 0,
n ∈ Z1d , k ∈ Zb .
Similarly, from {N, F2 } = i
k20 k20 k20 (k, ωFnm zn zm − n Fnm zn zm − m Fnm zn zm )eik,θ
k,n,m
+i
k11 k11 k11 (k, ωFnm zn z¯ m − n Fnm zn z¯ m + m Fnm zn z¯ m )eik,θ
|k|+||n|−|m||=0
+i
k02 k02 k02 (k, ωFnm z¯ n z¯ m + n Fnm z¯ n z¯ m + m Fnm z¯ n z¯ m )eik,θ
k,n,m
=i
k20 (k, ω − n − m )Fnm zn zm eik,θ
k,n,m
+i
k11 (k, ω − n + m )Fnm zn z¯ m eik,θ
|k|+||n|−|m||=0
+i
k02 (k, ω + n + m )Fnm z¯ n z¯ m eik,θ ,
(4.18)
k,n,m k20 , F k11 and F k02 are determined by the following linear algebraic it follows that Fnm nm nm systems k20 k20 (k, ω − n − m )Fnm = iRnm ,
n, m ∈ Z1d , k ∈ Zb ,
k11 k11 (k, ω − n + m )Fnm = iRnm ,
|k| + ||n| − |m|| = 0,
k02 (k, ω + n + m )Fnm
n, m ∈
Moreover, P ∈ A implies F ∈ A.
=
k02 iRnm ,
Z1d ,
(4.19)
k∈Z . b
4.2. Estimation on the coordinate transformation. We proceed to estimate XF and φF1 . We start with the following Lemma 4.2. Let Di = D(r+ + 4i (r − r+ ), 4i s), 0 < i ≤ 4. Then XF D3 ,O ≤ cγ −2 (r − r+ )−c ε.
(4.20)
KAM Theorem for Hamiltonian PDEs in Higher Dimensional Spaces
359
Proof. By (4.2), Lemma 4.1, Assumptions (A1) and (A2), we have |Fkl00 |O ≤ (|k, ω|−1 |Pkl00 |)O < cγ −2 |k|2τ +1 |Pkl00 |O , |Fnk10 |O |Fnk01 |O k20 |Fnm |O k11 |Fnm |O k02 |Fnm |O
≤ ≤ ≤ ≤ ≤
−2
k = 0;
2τ +1
cγ |k| |Pnk10 |; cγ −2 |k|2τ +1 |Pnk01 |; k20 cγ −2 |k|2τ +1 |Pnm |; −2 2τ +1 k11 cγ |k| |Pnm |, −2 2τ +1 k02 cγ |k| |Pnm |.
(4.21) |k| + ||n| − |m|| = 0;
It follows that
1 1 1 Fθ D3 ,O ≤ 2 |Fkl00 | · s 2|l| · |k| · e|k|(r− 4 (r−r+ )) 2 s s k,|l|≤1 1 + |Fnk10 | · |zn | · |k| · e|k|(r− 4 (r−r+ )) k,n
+
1
|Fnk01 | · |¯zn | · |k| · e|k|(r− 4 (r−r+ ))
k,n
+
1
k20 |Fnm | · |zn | · |zm | · |k| · e|k|(r− 4 (r−r+ ))
k,n,m
+
1 k11 |Fnm | · |zn | · |¯zm | · |k| · e|k|(r− 4 (r−r+ )
|k|+||n|−|m||=0
+
1
k02 |Fnm | · |¯zn | · |¯zm | · |k| · e|k|(r− 4 (r−r+ )) )
k,n,m −2
≤ cγ ≤ cγ
−2
(r − r+ )−c XR (r − r+ )−c ε.
(4.22)
Similarly, FI D3 ,O =
1
|Fkl00 |e|k|(r− 4 (r−r+ )) ≤ cγ −2 (r − r+ )−c ε.
|l|=1
Now we estimate XF1 D3 ,O . Since F1zn D3 ,O = Fnk10 eik,θ D3 ,O k
≤
1
|Fnk10 |e|k|(r− 4 (r−r+ ))
k
≤ cγ −2
1
|Pnk10 ||k|2τ +1 e|k|(r− 4 (r−r+ ))
k
and similarly F1z¯n D3 ,O ≤ cγ −2
k
1
|Pnk01 ||k|2τ +1 e|k|(r− 4 (r−r+ )) ,
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J. Geng, J. You
it follows from the definition of the weighted norm2 that c F1zn D3 ,O |n|a¯ e|n|ρ + F1z¯n D3 ,O |n|a¯ e|n|ρ XF1 D3 ,O ≤ s n n ≤ cγ −2 (r − r+ )−c XR ≤ cγ −2 (r − r+ )−c ε.
(4.23)
Moreover, F2zn D3 ,O =
k,m
k20 Fnm zm eik,θ D3 ,O +
≤ cγ −2
k11 Fnm z¯ m eik,θ D3 ,O
k,m
1
k20 |Pnm ||zm ||k|2τ +1 e|k|(r− 4 (r−r+ ) )
k,m
+
k11 |Pnm ||¯zm ||k|2τ +1 e
|k|(r− 41 (r−r+ ))
;
(4.24)
k,m
and similarly F2z¯n D3 ,O ≤ cγ
−2
1
k11 |Pmn ||zm ||k|2τ +1 e|k|(r− 4 (r−r+ ))
k,m
+
1 k02 |Pnm ||¯zm ||k|2τ +1 e|k|(r− 4 (r−r+ ))
.
k,m
Hence we have
c a¯ |n|ρ a¯ |n|ρ XF2 D3 ,O ≤ F2zn D3 ,O |n| e + F2z¯n D3 ,O |n| e s n n ≤ cγ −2 (r − r+ )−c XR
≤ cγ −2 (r − r+ )−c ε.
The conclusion of the lemma follows from the estimates above.
(4.25)
In the next lemma, we give some estimates for φFt . The formula (4.26) will be used to prove our coordinate transformation is well defined. Inequality (4.27) will be used to check the convergence of the iteration. 1
Lemma 4.3. Let η = ε 3 , Diη = D(r+ + 4i (r − r+ ), 4i ηs), 0 < i ≤ 4. If ε ( 21 γ 2 (r − 3
r+ )c ) 2 , we then have φFt : D2η → D3η , −1 ≤ t ≤ 1.
(4.26)
DφFt − I dD1η < cγ −2 (r − r+ )−c ε.
(4.27)
Moreover,
2
Recall (2.4), the definition of the norm.
KAM Theorem for Hamiltonian PDEs in Higher Dimensional Spaces
361
Proof. Let ∂ |i|+|l|+|α|+|β| D m F D,O = max F , |i| + |l| + |α| + |β| = m ≥ 2 . ∂θ i ∂I l ∂zα ∂ z¯ β n n D,O Notice that F is a polynomial of degree 1 in I and degree 2 in z, z¯ . From (2.4), (4.25) and the Cauchy inequality, it follows that D m F D2 ,O < cγ −2 (r − r+ )−c ε,
(4.28)
for any m ≥ 2. To get the estimates for φFt , we start from the integral equation, φFt
t
= id + 0
XF ◦ φFs ds
so that φFt : D2η → D3η , −1 ≤ t ≤ 1, which follows directly from (4.28). Since DφFt = I d +
t 0
(DXF )DφFs ds = I d +
where J denotes the standard symplectic matrix
t 0
J (D 2 F )DφFs ds,
0 −I , it follows that I 0
DφFt − I d ≤ 2D 2 F < cγ −2 (r − r+ )−c ε. Consequently Lemma 4.3 follows.
(4.29)
4.3. Estimation for the new normal form. The map φF1 defined above transforms H into H+ = N + + P+ (see (4.10) and (4.14)). Due to the special form of P defined in (A4), the 011 z z¯ with n = m are absent, i.e., z , z with n = m are uncoupled. terms in n,m Pnm n m n m Hence compared with the normal form in [13], here the normal form N+ is simpler 011 N+ = N + P0000 + ω , I + Pnn zn z¯ n = e+ + ω+ , I +
n
+ n zn z¯ n ,
(4.30)
n
where 011 e+ = e + P0000 , ω+ = ω + P0l00 (|l| = 1), + n = n + Pnn .
(4.31)
Now we prove that N+ has the same properties as N . By regularity of P , set3 δ = min{ι, a¯ − a}, then we have |ω+ − ω|O < ε, 3
Recall (2.9) and (2.4).
011 |Pnn |O < ε|n|−δ .
(4.32)
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It follows that |k, ω + P0l00 | ≥ |k, ω| − |k, P0l00 | ≥
γ γ+ − ε|k| ≥ τ , |k|τ |k|
k = 0, (4.33)
γ+ 011 |k, ω + P0l00 + + , n | ≥ |k, ω + n | − |k, P0l00 + Pnn | ≥ |k|τ
(4.34)
and γ+ , |k|τ γ+ + |k, ω + P0l00 + + , n − m | ≥ |k|τ
+ |k, ω + P0l00 + + n + m | ≥
(4.35) |k| + ||n| − |m|| = 0,
(4.36)
provided that ε|k|τ +1 ≤ c(γ − γ+ ). This means that in the succeeding KAM step, small divisor conditions are automatically satisfied for |k| ≤ K, where εK τ +1 ≤ c(γ − γ+ ). The following bounds will be used for the measure estimates: |ω+ − ω|O < ε,
−δ | + n − n |O < ε|n| .
(4.37)
4.4. Estimation for the new perturbation. Since 1 1 (1 − t){{N, F }, F } ◦ φFt dt + {R, F } ◦ φFt dt + (P − R) ◦ φF1 P+ =
0
0
1
= 0
{R(t), F } ◦ φFt dt + (P − R) ◦ φF1 ,
where R(t) = (1 − t)(N+ − N ) + tR. Hence 1 XP+ = (φFt )∗ X{R(t),F } dt + (φF1 )∗ X(P −R) . 0
According to Lemma 4.3, DφFt − I dD1η < cγ −2 (r − r+ )−c ε,
−1 ≤ t ≤ 1,
thus DφFt D1η ≤ 1 + DφFt − I dD1η ≤ 2,
−1 ≤ t ≤ 1.
Due to Lemma 7.3, X{R(t),F } D2η ≤ cγ −2 (r − r+ )−c η−2 ε 2 , and X(P −R) D2η ≤ cηε, we have XP+ D(r+ ,s+ ) ≤ cηε + cγ −2 (r − r+ )−c η−2 ε 2 ≤ cε+ .
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4.5. Verification of (A4) after one KAM step. The assumption P ∈ A defined in (A4) is used to guarantee that the normal form at each KAM step has the same form as in the first step. To complete one KAM step, we need to prove the new perturbation P+ still has the special form defined in (A4), i.e., P+ ∈ A. Note that 1 1 P+ = P − R + {P , F } + {{N, F }, F } + {{P , F }, F } 2! 2! 1 1 + · · · + {· · · {N, F } · · · , F } + {· · · {P , F } · · · , F } + · · · . n! n! n
n
011 Since P ∈ A we have P −R ∈ A. Moreover, {N, F } = P0000 +ω , I + n Pnn zn z¯ n − R ∈ A. Hence the key point is to prove that A is closed under the Poisson bracket. To this end, we prove the following lemma. Lemma 4.4. If G(θ, I, z, z¯ ), F (θ, I, z, z¯ ) ∈ A, then B(θ, I, z, z¯ ) = {G, F } ∈ A. Proof. Let
G=
Gk1 α1 β1 (I )eik1 ,θ zα1 z¯ β1 ,
k1 ,α1 ,β1
F =
Fk2 α2 β2 (I )eik2 ,θ zα2 z¯ β2 ,
k2 ,α2 ,β2
where the summations are taken over {(k1 , α1 , β1 ),
b
k1j ij +
j =1
(α1n − β1n )n = 0},
(4.38)
(α2n − β2n )n = 0}
(4.39)
n∈Z1d
and {(k2 , α2 , β2 ),
b j =1
k2j ij +
n∈Z1d
respectively. Since ∂Gk α β (I ) 1 1 1 {G, F } = , ik2 Fk2 α2 β2 (I )eik1 ,θ zα1 z¯ β1 eik2 ,θ zα2 z¯ β2 ∂I A1 A2 ∂Fk2 α2 β2 (I ) − ik1 , Gk1 α1 β1 (I )eik1 ,θ zα1 z¯ β1 eik2 ,θ zα2 z¯ β2 ∂I A1 A2 +i Gk1 α1 β1 (I )Fk2 α2 β2 (I )eik1 ,θ eik2 ,θ zα1 −em z¯ β1 zα2 z¯ β2 −em m
−i =
A
3
m
Gk1 α1 β1 (I )Fk2 α2 β2 (I )eik1 ,θ eik2 ,θ zα1 z¯ β1 −em zα2 −em z¯ β2
A4
B(k1 +k2 )(α1 +α2 )(β1 +β2 ) (I )eik1 +k2 ,θ zα1 +α2 z¯ β1 +β2
A5
+
B(k1 +k2 )(α1 +α2 −em )(β1 +β2 −em ) (I )eik1 +k2 ,θ zα1 +α2 −em z¯ β1 +β2 −em ,
A6
(4.40)
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where A1 denotes b
k1j ij +
j =1
(α1n − β1n )n = 0,
n∈Z1d
A2 denotes b
k2j ij +
j =1
(α2n − β2n )n = 0,
n∈Z1d
A3 denotes b j =1 b
k1j ij + (α1m − 1 − β1m )m +
(α1n − β1n )n = −m,
n∈Z1d \{m}
k2j ij + (α2m − (β2m − 1))m +
j =1
(α2n − β2n )n = m,
n∈Z1d \{m}
A4 denotes b
k1j ij + (α1m − (β1m − 1))m +
j =1 b
(α1n − β1n )n = m,
n∈Z1d \{m}
k2j ij + (α2m − 1 − β2m )m +
j =1
(α2n − β2n )n = −m,
n∈Z1d \{m}
A5 denotes b
(k1j + k2j )ij +
j =1
(α1n + α2n − β1n − β2n )n = 0,
n∈Z1d
A6 denotes b
(k1j + k2j )ij + ((α1m + α2m − 1) − (β1m + β2m − 1))m
j =1
+
((α1n + α2n ) − (β1n + β2n ))n = 0.
n∈Z1d \{m}
Thus Lemma 4.4 is obtained.
Corollary 1. The new perturbation P+ ∈ A.
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5. Iteration Lemma and Convergence For any given s, ε, r, γ and for all ν ≥ 1, we define the following sequences rν = r 1 −
ν+1
2−i ,
i=2 4
−2
3 εν = cγ (rν−1 − rν )−c εν−1 , ν+1 γν = γ 1 − 2−i ,
(5.1)
i=2 1 3
ην = εν ,
Lν = Lν−1 + εν−1 , ν−1 13 1 sν = ην−1 sν−1 = 2−2ν ε i s0 , 4 i=0
Kν =
c(εν−1 (γν
− γν+1 ))
1 τ +1
,
where c is a constant, and the parameters r0 , ε0 , γ0 , L0 , s0 and K0 are defined to be r, ε, γ , L, s and 1 respectively. Note that (r) =
∞
3 i
[(ri−1 − ri )−c ]( 4 )
i=1
is a well–defined function of r.
5.1. Iteration lemma. The preceding analysis can be summarized as follows. Lemma 5.1. Let ε is small enough and ν ≥ 0. Suppose that (1) Nν = eν + ων (ξ ), I + n νn (ξ )zn z¯ n is a normal form with parameters ξ satisfying |k, ων | ≥
γν , |k|τ
|k, ων + νn | ≥
k = 0, γν , |k|τ
γν , |k|τ γν |k, ων + νn − νm | ≥ τ , |k|
|k, ων + νn + νm | ≥
|k| + ||n| − |m|| = 0
on a closed set Oν of Rb ; 1 smooth and satisfy (2) ων (ξ ), νn (ξ ) are CW |ων − ων−1 |Oν ≤ εν−1 ,
−δ | νn − ν−1 n |Oν ≤ εν−1 |n| ;
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(3) Pν has the special form defined in (A4) (i.e., Pν ∈ A) and XPν D(rν ,sν ),Oν ≤ εν . Then there is a subset Oν+1 ⊂ Oν ,
Oν+1 = Oν \
Rν+1 k (γν+1 ) ,
|k|>Kν
where Rν+1 k (γν+1 )
= ξ ∈ Oν :
ν+1 ν+1 | < |k, ων+1 | < γ|k| τ |k, ων+1 + n γν+1 ν+1 ν+1 |k, ων+1 ± n ± m | < |k|τ ,
γν+1 |k|τ , or
ν , and a symplectic transformation of variables with ων+1 = ων + P0l00
ν : D(rν+1 , sν+1 ) × Oν → D(rν , sν ), such that on D(rν+1 , sν+1 ) × Oν+1 , Hν+1 = Hν ◦ ν has the form Hν+1 = eν+1 + ων+1 , I + ν+1 n zn z¯ n + Pν+1 ,
(5.2)
(5.3)
n
with |ων+1 − ων |Oν+1 ≤ εν ,
| ν+1 − νn |Oν+1 ≤ εν |n|−δ . n
(5.4)
And also Pν+1 has the special form defined in (A4) (i.e., Pν+1 ∈ A) with XPν+1 D(rν+1 ,sν+1 ),Oν+1 ≤ εν+1 .
(5.5)
5.2. Convergence. Suppose that the assumptions of Theorem 1 are satisfied. Recall that ε0 = ε, r0 = r, γ0 O0 = ξ ∈ O :
= γ , s0 = s, L0 = L, N0 = N, P0 = P , γ0 |k, ω(ξ )| ≥ |k| k = 0, τ , γ0 |k, ω(ξ ) + n | ≥ |k| τ , , γ0 |k, ω(ξ ) + n + m | ≥ |k|τ , γ0 |k, ω(ξ ) + n − m | ≥ |k|τ , |k| + ||n| − |m|| = 0
the assumptions of the iteration lemma are satisfied when ν = 0 if ε0 and γ0 are sufficiently small. Inductively, we obtain the following sequences: Oν+1 ⊂ Oν , ν = 0 ◦ 1 ◦ · · · ◦ ν : D(rν+1 , sν+1 ) × Oν → D(r0 , s0 ), ν ≥ 0, H ◦ ν = Hν+1 = Nν+1 + Pν+1 . ν Let O˜ = ∩∞ ν=0 Oν . As in [17, 18], thanks to Lemma 4.3, it concludes that Nν , , D ν , ων converge uniformly on D( 21 r, 0) × O˜ with ∞ N∞ = e∞ + ω∞ , I + n zn z¯ n . n
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Since 4
4 ν
εν+1 = cγν−2 (rν − rν+1 )−c εν3 ≤ (cγ −2 (r)ε)( 3 ) , it follows that εν+1 → 0 provided that ε is sufficiently small. t be the flow of X . Since H ◦ ν = H Let φH H ν+1 , we have t t φH ◦ ν = ν ◦ φH . ν+1
(5.6)
The uniform convergence of ν , D ν , ων and XHν implies that the limits can be taken on both sides of (5.6). Hence, on D( 21 r, 0) × O˜ we get t t φH ◦ ∞ = ∞ ◦ φH ∞
(5.7)
and 1 ∞ : D( r, 0) × O˜ → D(r, s) × O. 2 It follows from (5.7) that t t ( ∞ (Tb × {ξ })) = ∞ φN (Tb × {ξ }) = ∞ (Tb × {ξ }) φH ∞
˜ This means that ∞ (Tb × {ξ }) is an embedded torus which is invariant for for ξ ∈ O. ˜ We remark here that the frequenthe original perturbed Hamiltonian system at ξ ∈ O. cies ω∞ (ξ ) associated to ∞ (Tb × {ξ }) are slightly different from ω(ξ ). The normal behavior of the invariant torus is governed by normal frequencies ∞ n . 6. Measure Estimates For notational convenience, let O−1 = O, K−1 = 0. Then at ν th step of KAM iteration, we have to exclude the following resonant set Rνkn Rνknm ), (Rνk Rν = |k|>Kν−1 ,n,m
where Rνk = {ξ ∈ Oν−1 : |k, ων (ξ )| <
γν }, |k|τ
Rνkn = {ξ ∈ Oν−1 : |k, ων (ξ ) + νn | <
(6.1) γν }, |k|τ
Rνknm = {ξ ∈ Oν−1 : |k, ων (ξ ) ± νn ± νm | <
(6.2) γν }. |k|τ
(6.3)
Remark. From Sect. 4.3, one has that at the ν th step, small divisor conditions are automatically satisfied for |k| ≤ Kν−1 . Hence, we only need to excise the above resonant set Rν . Note that due to the Special form of the perturbation (see (A4) and (4.7)), there 011 z z¯ in the perturbation P , thus we need not are not the terms of the form n=m Pνnm n m ν ν consider small divisors k, ω + n − νm with k = 0, n = m, which is crucial for this paper.
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Lemma 6.1. For any fixed |k| > Kν−1 , n and m, ! " γν meas Rνk Rνkn Rνknm < c τ +1 . |k| ν−1
j j =0 P0l00 (ξ )
Proof. Recall that ων (ξ ) = ω(ξ ) + |
ν−1
with
j
P0l00 (ξ )|Oν−1 ≤ ε,
(6.4)
j =0
and νn (ξ ) = n (ξ ) +
ν−1
011,j j =0 Pnn
with # # # # #ν−1 011,j # # Pnn ## # # #j =0
≤ ε|n|−δ .
(6.5)
Oν−1
It follows that 4
# # # ∂(k, ων (ξ ) ± νn ± νm ) # # # ≥ c|k|, # # ∂ξ
then the proof of this lemma is evident; we omit it.
Lemma 6.2. The total measure we need to exclude along the KAM iteration is ! " meas Rνk Rνkn Rνknm < cγ ϑ , ϑ > 0. Rν = meas ν≥0 |k|>Kν−1 ,n,m
ν≥0
Proof. We estimate meas
{ξ ∈ Oν−1
|k|>Kν−1 n,m
γ ν : |k, ων (ξ ) + νn − νm | < τ } , |k|
which is the most complicated case. We divide the proof into several cases according to p, d. Case 1. p = 1, d = 1 and p = 2, d > 1. The case p = 1, d = 1 has been proved by P¨oschel in [17]. The case p = 2, d > 1 can be proved similarly. In fact, suppose that |n|2 − |m|2 = l ≥ 0. If l > c|k|, Rν+1 knm = ∅; if l ≤ c|k|, then according to assumption (A2) and (6.5), we have | νn − νm − l| ≤ O(|m|−δ ). It follows that def
Rνknm ⊆ Qνklm = {ξ : |k, ων + l| < 4
Here | · | denotes 1 –norm.
γν + O(|m|−δ )}. |k|τ
(6.6)
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369
Moreover, Qνklm ⊆ Qνklm0 for |m| ≥ |m0 |. Due to Lemma 6.1, one has meas
Rνknm ≤
l≤c|k| |m|<|m0 |
l≤c|k| |n|2 −|m|2 =l
meas(Rνknm ) +
meas(Qklm0 )
l≤c|k|
γ |m0 |C(d) −δ
(6.7)
where C(d) is a constant depending only on space dimension d. By choosing γ |m|k|0 |τ |m0 |−δ , i.e.
C(d)
|m0 | =
|k|τ γ
=
1 δ+C(d)
,
we arrive at meas
Rνknm
δ
l≤c|k| |n|2 −|m|2 =l
γ δ+C(d) δτ
|k| δ+C(d)
(6.8)
.
Case 2. p > 2 for d > 1 and p > 1 for d = 1. Without loss of generality, we assume that |n| ≥ |m|. If |n| = |m|, the proof proceeds in the same way as in Case 1. If |n| > |m|, we have 1 p−2 |n| (|n|2 − |m|2 ), for d > 1, 2 |n|p − |m|p ≥ |n|p−1 (|n| − |m|), for d = 1.
|n|p − |m|p ≥
1
1
1
If |n| > c|k| p−2 for d > 1 or |n| > c|k| p−1 for d = 1, we get Rνknm = ∅; if |n| ≤ c|k| p−2 for d > 1, it follows from Lemma 6.1 that meas(
|n|=|m|
Rνknm ) = meas(
Rνknm ) < c
1 |n|=|m|;|n|,|m|≤c|k| p−2
γ |k|
τ +1− C(d) p−2
.
(6.9)
The case of d = 1 can be proved analogously. δ , τ > (b+1)(δ+C(d)) + C(d) Let ϑ = δ+C(d) δ p−2 (here p > 2), meas(Rν ) ≤ c meas
Rνknm
|k|>Kν−1 n,m
≤c
meas
|k|>Kν−1
< cγ (1 + Kν−1 ) ϑ
Rνknm
n,m −1
.
(6.10)
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The above upper bound (6.10) gives out the measure estimate of excluded parameter set at the ν th step of KAM iteration, then along the KAM iteration, the total measure we need to excise has the upper bound ( ) Rν ≤ meas Rν meas ν≥0
ν≥0
< cγ ϑ
(1 + Kν−1 )−1 < cγ ϑ .
(6.11)
ν≥0
So Lemma 6.2 follows. Remark 1. Note that the regularity is crucial in (6.6)–(6.8). Remark 2. The case p = 1, d ≥ 2 is not considered in this paper since we can not prove Lemma 6.2 in this case. This is the reason why Theorem 1 can not be applied to higher dimensional wave equations.
7. Appendix Lemma 7.1.
Proof. Since (F G)klαβ
F GD(r,s) ≤ F D(r,s) GD(r,s) . = k ,l ,α ,β Fk−k ,l−l ,α−α ,β−β Gk l α β , we have
F GD(r,s) = sup
zρ <s ¯zρ <s
k,l,α,β
k,l,α,β
k ,l ,α ,β
≤ sup
zρ <s ¯zρ <s
|(F G)klαβ |s 2l |zα ||¯zβ |e|k|r |Fk−k ,l−l ,α−α ,β−β Gk l α β |s 2l |zα ||¯zβ |e|k|r
≤ F D(r,s) GD(r,s) , and the proof is finished.
Lemma 7.2 (Cauchy inequalities). c F D(r,s) , σ c ≤ 2 F D(r,s) , s
Fθ D(r−σ,s) ≤ FI D(r, 1 s) 2
and c Fzn D(r, 1 s) ≤ F D(r,s) |n|a e|n|ρ , 2 s c Fz¯ n D(r, 1 s) ≤ F D(r,s) |n|a e|n|ρ . 2 s
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Let {·, ·} denote the Poisson bracket of smooth functions, i.e., * + * + ∂F ∂G ∂F ∂G ∂F ∂G ∂F ∂G {F, G} = , − , − , +i ∂I ∂θ ∂θ ∂I ∂zn ∂ z¯ n ∂ z¯ n ∂zn n then we have the following lemma: Lemma 7.3 If XF D(r,s) < ε ,
XG D(r,s) < ε ,
then X{F,G} D(r−σ,ηs) < cσ −1 η−2 ε ε , 1
η 1. 4
In particular, if η ∼ ε 3 , ε , ε ∼ ε, we have X{F,G} D(r−σ,ηs) ∼ ε 3 . For the proof, see [13].
Acknowledgements. The authors would like to thank the referees for their valuable comments and suggestions which help to improve the presentation of this paper. They also thank R. Cao for his critical reading of the manuscript.
References 1. Bambusi, D.: Birkhoff normal form for some nonlinear PDEs. Commun. Math. Phys. 234, 253–285 (2003) 2. Bambusi, D., Berti, M.: A Birkhoff–Lewis type theorem for some Hamiltonian PDEs. SIAM J. Math. Anal. 37, 83–102 (2005) 3. Bourgain, J.: Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schr¨odinger equations. Ann. Math. 148, 363–439 (1998) 4. Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995) 5. Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE., International Mathematics Research Notices, 1994, pp. 475–497 6. Bourgain, J.: Green’s function estimates for lattice Schr¨odinger operators and applications, Annals of Mathematics Studies, 158. Princeton, NJ: Princeton University Press, 2005 7. Bourgain, J.: Nonlinear Schr¨odinger equations. Park City Series 5. Providence, RI: American Mathematical Society, 1999 8. Bourgain, J.: On diffusion in high–dimensional Hamiltonian systems and PDE. J. Anal. Math. 80, 1–35 (2000) 9. Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations, Commun. Pure. Appl. Math. 46, 1409–1498 (1993) 10. Craig, W., Worfolk, P.: An integrable normal form for water waves in infinite depth. Physica D 84, 513–531 (1995) 11. Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Ann. Sc. Norm. Sup. Pisa 15, 115–147 (1988) 12. Geng, J., You, J.: KAM tori of Hamiltonian perturbations of 1D linear beam equations. J.Math. Anal. Appl. 277, 104–121 (2003) 13. Geng, J.,You, J.: A KAM Theorem for one dimensional Schr¨odinger equation with periodic boundary conditions. J. Diff. Eqs. 209, 1–56 (2005) 14. Kuksin, S.B.: Hamiltonian perturbations of infinite–dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21, 192–205 (1987) 15. Kuksin, S.B.: Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, 1556, Berlin: Springer, 1993 16. Kuksin, S.B., P¨oschel, J.: Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schr¨odinger equation. Ann. Math. 143, 149–179 (1996)
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17. P¨oschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helvetici 71, 269–296 (1996) 18. P¨oschel, J.: A KAM theorem for some nonlinear partial differential equations. Ann. Sc. Norm. Sup. Pisa CI. Sci. 23, 119–148 (1996) 19. P¨oschel, J.: On the construction of almost periodic solutions for a nonlinear Schr¨odinger equations. Erg. Th. Dynam. Syst. 22, 1537–1549 (2002) 20. Wayne, C.E.: Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990) Communicated by G. Gallavotti
Commun. Math. Phys. 262, 373–391 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1498-z
Communications in
Mathematical Physics
KMS States and the Chemical Potential for Disordered Systems Francesco Fidaleo Dipartimento di Matematica, Universit`a di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy. E-mail: [email protected] Received: 17 November 2004 / Accepted: 7 September 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: We extend the theory of the chemical potential associated to a compact separable gauge group, to the case of disordered quantum systems. This is done in the framework of operator algebras. Among the other results, we show that the chemical potential does not depend on the disorder. The situation of the n–torus is treated in some detail. Indeed, the chemical potential is intrinsically defined in terms of the direct integral decomposition of the Connes–Radon–Nikodym cocycle associated to the (normal extension of the) KMS state ω and its trasforms ω ◦ ρ by the (localized) automorphisms ρ of the observable algebra, carrying the Abelian charges of the model under consideration. This description parallels the analogous one relative to the usual (i.e. non disordered) quantum models. 1. Introduction In quantum physics, one often recovers the observable algebra by a principle of global gauge invariance, see [13, 14, 16, 17] and the references cited therein. In order to investigate the thermodynamical behaviour of such physical models, the concept of chemical potential naturally arises. Namely, suppose we have k species of particles, that is the chemical components. Provided that the field algebra F has a natural local structure, one can consider infinite volume limits of states arising from the Gibbs grand–canonical ensemble relative to the inverse temperature β and the k–parametric chemical potential µ = (µ1 , . . . , µk ), one for each species of particles, see e.g. [10], Section 5.2. It is seen that each of these states ϕβ,µ ∈ S(F) satisfies the Kubo–Martin–Schwinger (KMS for short) boundary condition for a one parameter subgroup of the (k + 1)–dimensional Lie group R × G, the gauge group G being isomorphic to the k–dimensional torus in this situation.1 If we denote by H and {N1 , . . . , Nk } the infinitesimal generators of the time 1 Here, in order to simplify matter, we are supposing that the asymmetry subgroup of a KMS state is trivial. For the asymmetry subgroup see Section II.4 of [4], Section 4 and the appendix of the present paper.
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translations and the gauge transformations respectively, the one parameter subgroup mentioned above is generated by −βX, where X has the form X = H − i µi Ni , see e.g. [10], Section 5.2. As the observables are gauge invariant, the restriction ωβ,µ of ϕβ,µ to the observable algebra A = F G , satisfies the KMS boundary condition at the same inverse temperature β for the time evolution. In general, states corresponding to different values of the chemical potential µ, give rise to different KMS states when restricted to the observable algebra. In quantum physics, all the physical content of the model is encoded directly in the observable algebra. On the one hand, the chemical potential has a physical meaning, even if it naturally arises by the use of the field algebra. On the other hand, the KMS boundary condition does not refer to the local structure of the algebra of the observables. In the seminal paper [4], all these questions are studied in detail, see also [5, 22, 23, 27] for strictly connected questions. Namely, if ω is an extremal KMS state on A, it is shown that any weakly clustering extension to all of F is a KMS state relative to a new evolution, modified by a one parameter subgroup of the gauge group. In the situation described above, this one parameter subgroup uniquely determines the values of the chemical potentials {µ1 , . . . , µk }. Unfortunately, the results of [4] are not directly applicable to disordered models for two kinds of reasons. First, the states of interest are typically non factor states. Second, the collection of states obtained by fixing the couplings of the model satisfies no natural invariance condition with respect to the spatial translations. The equilibrium statistical mechanics of models arising from classical spin glasses has been intensively studied, in order to understand the complex behaviour of the set of the temperature states. Furthermore, the phenomenon of the “weak Gibbsianess” naturally appears in the case of disordered systems. Namely, if one considers infinite volume limits of finite volume Gibbs states, he obtains states satisfying the following properties. Their marginal distribution of the couplings is the given probability measure describing the disorder, whereas the conditional distribution of the standard observable variables (the “spin” variables), given the couplings, is some infinite volume Gibbs state almost surely. Such a field of infinite volume Gibbs states gives rise to a Aizenman–Wehr metastate (see [1]) after direct integral decomposition, the last one assuming the meaning of the quantum counterpart of the classical procedure of conditioning with respect to the disorder variables. It can happen that the state so obtained is not necessarily jointly Gibbsian relatively to the standard variables and the couplings, see [19, 25] for some pivotal classical examples. As it is explained in Section 6 of [7], the KMS boundary condition for the algebra generated by spin and disorder variables seems to be the quantum counterpart of the weak Gibbsianess. Moreover, it does not refer to the local structure of the observable algebra. For details on all the mentioned questions, we refer the reader to [9, 18, 28–30] and the literature cited therein. Some attempts to understand the structure of the set of the KMS states of quantum disordered systems are made in [3, 6, 7, 24] by using the standard techniques of operator algebras. Indeed, the systematic investigation of the difference between Gibbsianess and weak Gibbsianess firstly appeared in the paper [24]. In that paper, both equilibrium conditions are connected with some natural variational principles. We extend the algebraic description of the chemical potential to disordered systems without referring to the difference between Gibbsianess and weak Gibbsianess. We use the standard techniques of operator algebras taking advantage twice by the paper [4]. First, we follow its plan. Second, we use the results of this paper in order to describe the occurrence of the chemical potential for disordered models.
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The present paper is organized as follows. After a preliminary section, in Section 3 we investigate some useful ergodic properties, known as self–averaging, of states of interest of disordered systems. Section 4 is devoted to the occurrence of the chemical potential. Starting from a weakly clustering state ϕ on the field algebra F = F ⊗ L∞ (X, ν), normal when restricted to the subalgebra L∞ (X, ν), we show that the stabilizer and the asymmetry subgroup of ϕ coincide almost surely with the corresponding objects relative to the states ϕξ . Here, the measurable equivariant field {ϕξ }ξ ∈X ⊂ S(F) provides the direct integral decomposition of ϕ. Then, we show that for any weakly clustering state ϕ on F whose restriction to A is KMS, there exists a modification of the time evolution by a suitable one parameter group of the gauge group, the same for each ϕξ , such that ϕξ is KMS with respect to this modified evolution almost surely. Section 5 is devoted to an intrinsic description of the chemical potential by objects related to the algebra of observables. The case of the unit circle is treated in some detail, the case of the n–torus being a straightforward generalization. Provided that its zero point is fixed independently on the disorder, the chemical potential is intrinsically defined in terms of the Connes– Radon–Nikodym cocycle associated to the KMS state ω ∈ S(A) under consideration, and its trasforms ω ◦ ρ by the automorphisms ρ of the observable algebra A, carrying the Abelian charges of the model. Indeed, the chemical potential is connected, and is independent almost surely on the disorder, with the Connes–Radon–Nikodym cocycle D(ωξ ◦ ρξ ) : Dωξ of ωξ ◦ ρξ relative to ωξ , see Formula (5.2). Here, {ωξ }ξ ∈X provides the direct integral decomposition of (the normal extension of) ω, and the measurable field of normal automorphisms {ρξ }ξ ∈X gives rise to the normal extensions of ρ to all of πωξ (A) which exist by Proposition 5.1. The present paper is complemented by an appendix where the fiberwise construction of the asymmetry subgroup Nϕξ is outlined in the more general case of non ergodic disordered systems. By the Birkhoff individual ergodic Theorem, our situation will arise as a particular case. 2. Preliminaries We start by recalling the definition of the KMS boundary condition. Let D be the space made of all the infinitely often differentiable compactly supported functions in R. A state φ on the C ∗ –algebra B satisfies the KMS boundary condition at inverse temperature β, which we suppose to be always different from zero, w.r.t. the group of automorphisms {τt }t∈R if function for every A, B ∈ B, (i) t → φ(Aτt (B)) is a continuous where “” (ii) φ(Aτt (B))f (t) dt = φ(τt (B)A)f (t + iβ) dt whenever f ∈ D, stands for Fourier transform. For the equivalent characterizations of the KMS boundary condition, the main results about KMS states, and finally the connections with Tomita theory of von Neumann algebras, see e. g. [10, 32] and the references cited therein. It is well known that the cyclic vector φ of the GNS representation πφ is also separating for πφ (B) . Denote with an abuse of notation, σ φ its modular group. According to this definition, we have φ
σt ◦ πφ = πφ ◦ τ−βt .
(2.1)
C ∗ –algebra
A with an identity I , describing the physical Our set–up is a separable observables.2 We suppose that A is obtained as the fixed point algebra A = F G under 2 In order to avoid technical complications, in quantum field theory the local algebras of observables are enlarged by taking the weak operator closure in the vacuum representation. Namely, the local algebras
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F. Fidaleo
a pointwise–norm continuous action γ : g ∈ G → γg ∈ Aut(F) of a compact second countable group G (the gauge group) on another separable C ∗ –algebra F (the field algebra). This is a typical situation appearing in quantum field theory, when the charges present in the model are described in terms of a principle of (global) gauge invariance, see e.g. [13, 14, 16, 17]. The present description might be applied also to models where the local algebras of observables are full matrix algebras, see Section 5 for some considerations along this line. We suppose that the group {αx }x∈Zd of spatial translations acts on F. We consider also a standard measure space (X, ν) based on a compact separable space X, and a Borel probability measure ν. The group Zd of the spatial translations is supposed to act on the probability space (X, ν) by measure preserving ergodic transformations {Tx }x∈Zd . A one parameter random group of automorphisms ξ
(t, ξ ) ∈ R × X → τt ∈ Aut(F) is acting on F. It is supposed to be strongly continuous in the time variable for each fixed ξ ∈ X, and jointly strongly measurable. ξ Consider, for A ∈ F, the strongly measurable function fA,t (ξ ) := τt (A). We get ξ
fA,t L∞ (X,ν;F ) ≡ esssup τt (A) F = A F , ξ ∈X
ξ
where the last equality follows as τt is isometric. We assume further that τ acts locally. Namely, if A is an element of F, then the function fA,t ∈ L∞ (X, ν; F) belongs to the C ∗ –subalgebra F ⊗ L∞ (X, ν), where the above C ∗ –tensor product is uniquely determined as any commutative C ∗ –algebra is nuclear. We assume the following commutation rules Tx ξ
τt
ξ
α x = α x τt , α x γ g = γ g αx , ξ τt γg
=
(2.2)
ξ γ g τt ,
for each x ∈ Zd , ξ ∈ X, t ∈ R, and g ∈ G. ξ By (2.2), it is immediate to show that αx and τt leave globally stable A. Namely, d Z , R act on A as groups of automorphisms and random automorphisms, respectively. Finally, we address also the situation when Fermion operators are present in F. Namely, there exists an automorphism σ of F commuting with all the gauge transformations, the spatial translations and the random time evolution, such that σ 2 = e.3 We put F+ :=
1 (e + σ )(F), 2
F− :=
1 (e − σ )(F). 2
(2.3)
of observables are typically von Neumann algebras with separable predual, the last being non separable C ∗ –algebras. This does not affect the substance of the theory, see the comments in Section 5. 3 In most of the interesting physical situations, σ ∈ Z (G), G being the gauge group, see e.g. [13, 17]. The situation without Fermion operators corresponds to σ = e.
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In order to achieve the disorder, it is natural to set for the field algebra, F := F ⊗ L∞ (X, ν). Notice that, by identifying F with a closed subspace of L∞ (X, ν; F), each element A ∈ F is uniquely represented by a measurable essentially bounded function ξ → A(ξ ) with values in F. The group Zd of all the space translations is naturally acting on the C ∗ –algebra F as ax (A)(ξ ) := αx (A(T−x ξ )). Furthermore, define on F, ξ
tt (A)(ξ ) := τt (A(ξ )), g := γ ⊗ idL∞ (X,ν) , s := σ ⊗ idL∞ (X,ν) .
(2.4)
It is easy to verify that {ax }x∈Zd , {tt }t∈R and {gg }g∈G define actions of Zd , R and G on F which are mutually commuting, and commute also with the parity automorphism s. The subspaces F+ and F− are defined as in (2.3). Furthermore, by taking into account (2.2) and the definition (2.4) of the action of the gauge group on the disordered field algebra F, ax and tt leave globally stable the disordered observable algebra A. Namely, {ax }x∈Zd and {tt }t∈R define by restriction, mutually commuting actions of Zd and R on A, respectively. In order to study a class of states of interest for disordered systems, we start with ∗–weak measurable fields of states ξ ∈ X → ϕξ ∈ S(F). We suppose that the field {ϕξ }ξ ∈X fulfils almost surely, the equivariance condition ϕξ ◦ αx = ϕT−x ξ
(2.5)
w.r.t. the spatial translations, simultaneously. A state ϕ on F is naturally defined as ϕ(A) =
ϕξ (A(ξ ))ν(dξ ),
A ∈ F.
(2.6)
X
It is immediate to verify that ϕ defined as above is invariant w.r.t. the space translations ax . Moreover, ϕ I ⊗L∞ (X,ν) is a normal state. Equally well, one can start with a a–invariant state ϕ on F, normal when restricted to I ⊗ L∞ (X, ν) (indeed, ϕ I ⊗L∞ (X,ν) = X · ν(dξ )). Then, we can recover a ∗–weak measurable field {ϕξ }ξ ∈X ⊂ S(F) fulfilling (2.5). Such a measurable field provides the direct integral decomposition of ϕ as in (2.6), see [7], Theorem 4.1. Similar considerations can be applied to the observable algebras A as well. In the sequel, we denote by S0 (A), S0 (F) the convex closed subset of states on A, F respectively, fulfilling the properties listed above.
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3. Ergodic Properties of States of Disordered Systems In this section we study some useful ergodic properties of states in S0 (F), as well as in S0 (A). We restrict ourselves to the field algebra, the case relative to the observable algebra being similar. Let C, D ∈ F, and A, B ∈ F+ F− . Put A,B = −1 if A, B ∈ F− and A,B = 1 in the three remaining possibilities. We say that the state ϕ ∈ S(F) is asymptotically Abelian w.r.t. a if lim ϕ C ax (A)B − A,B Bax (A) D = 0. (3.1) |x|→+∞
The state ϕ ∈ S(F) is weakly clustering w.r.t. a if lim N
1 ϕ(Aax (B)) = ϕ(A)ϕ(B), |N |
(3.2)
x∈N
N being the box with a vertex in the origin, containing N d points with positive coordinates.4 Many of interesting states arising from quantum physics are naturally asymptotically Abelian w.r.t. the spatial translations, see e.g. [26], see also [7], Proposition 2.3 for the case of disordered systems. We report the following result for the sake of completeness. Proposition 3.1. Suppose that ϕ ∈ S(F) is a a–invariant asymptotically Abelian state. Then the following assertions are equivalent. (i) ϕ is a–weakly clustering, (ii) ϕ is a–ergodic. Proof. It is a well known fact that (i) always implies (ii). The reverse implication follows as in Proposition 5.4.23 of [10], the last working also under the weaker condition (3.1).
One sees that an asymptotically Abelian invariant state is automatically s–invariant, that is it is an even state, see e.g. [10], Example 5.2.21. The weak clustering property for states in S0 (F) can be translated as a property of the corresponding equivariant fields of states on F. Namely, Let ϕ ∈ S0 (F), and {ϕξ }ξ ∈X ⊂ S(F) the corresponding α–equivariant measurable field of states. Then the results listed below hold true. Proposition 3.2. The following assertions are equivalent. (i) ϕ is a–weakly clustering, 4
By taking into account natural applications to continuous disordered systems, we can consider cases when Rd is acting as the group of spatial translations. We use in (3.2) the natural modification M of the Cesaro mean given on bounded measurable functions, by 1 M(f ) := lim f (x) dd x, D→+∞ vol(D ) D D being a box with edges of length D. Most of the forthcoming analysis can be applied in this situation as well.
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(ii) we have for each A ∈ F, B ∈ F, lim N
1 ϕξ (Aαx (B(T−x ξ ))) = ϕξ (A)ϕ(B), |N | x∈N
in the weak topology of L1 (X, ν). Proof. By a standard density argument, we reduce the situation to elements of the form A ⊗ f, B ∈ F, with f ∈ L∞ (X, ν), A ∈ F. We get f (ξ )(ϕξ (A)ϕ(B))ν(dξ ) ≡ ϕ(A ⊗ f )ϕ(B) X
1 ϕ(A ⊗ f ax (B)) N |N | x∈N
1 = lim f (ξ ) ϕξ (Aαx (B(T−x ξ ))) ν(dξ ). N X |N | = lim
x∈N
The reverse implication follows by reversing the argument.
A sufficient condition for the weak clustering property of ϕ is the pointwise–clustering property for the corresponding equivariant field {ϕξ }ξ ∈X of states, almost surely. Proposition 3.3. Suppose that, for each A, B ∈ F
ϕξ (Aαx (B)) − ϕξ (A)ϕT−x ξ (B) = 0 lim |x|→+∞
(3.3)
almost surely. Then the state ϕ given by (2.6) is weakly clustering. Proof. We can reduce the situation to a measurable set F ⊂ X of full measure such that (3.3) is satisfied simultaneously for each A, B ∈ F. Define, for fixed A, B ∈ F, g ∈ L∞ (X, ν) and ξ ∈ F ,
1 g(T−x ξ ) ϕξ (Aαx (B)) − ϕξ (A)ϕξ (αx (B)) δ1 (N ) := |N | x∈N
1 δ2 (N ) := g(T−x ξ )ϕT−x ξ (B) − ϕ(B ⊗ g). |N | x∈N
We have that δ1 (N ) −→ 0 by hypotesis, and δ2 (N ) −→ 0 almost everywhere on F , by the Birkhoff individual ergodic Theorem. Then we conclude that, for fixed A, B ∈ F, g ∈ L∞ (X, ν), 1 g(T−x ξ )ϕξ (Aαx (B)) − ϕξ (A)ϕ(B ⊗ g) |N | x∈N
≡ δ1 (N ) − ϕξ (A)δ2 (N ) −→ 0 as N −→ +∞, pointwise on a measurable subset of F of full measure.
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F. Fidaleo
Let now f ∈ L∞ (X, ν). We have 1 ϕ(A ⊗ f ax (B ⊗ g)) − ϕ(A ⊗ f )ϕ(B ⊗ g) |N | x∈N
1 = f (ξ ) g(T−x ξ )ϕξ (Aαx (B)) − ϕξ (A)ϕ(B ⊗ g) ν(dξ ) |N | X x∈N
which goes to 0 by the Lebesgue dominated convergence Theorem. The proof follows as we can reduce the situation to the dense subalgebra of F generated by elements of the form A ⊗ f , with A ∈ F and f ∈ L∞ (X, ν). The last results are connected with the well known property of self–averaging for spin glasses, see e.g. [9, 29, 30]. Finally, we point out the fact that there exist examples of weakly clustering states ϕ satisfying the properties listed above without assuming any factoriality condition about ϕ, the last being not natural in the setting of disordered systems, see [7], Section 4 and Section 5. Indeed, it is enough to consider any quasi local algebra F admitting a state ω which is strongly clustering (i.e. mixing) w.r.t. the space translations. Take any probability space (X, ν) on which the space translations act ergodically. Define ϕξ = ω, the constant field. By Proposition 3.3, the state given by ϕ(A) := X ω(A(ξ ))ν(dξ ) satisfies all the required properties. In addition, if the local structure of F consists of full matrix algebras, it follows By Proposition 2.3 of [7], that the states ϕ constructed as above are a–asymptotically Abelian as well. We speak, without any further mention, and if it is not otherwise specified, about asymptotic Abelianess, weak clustering, or ergodicity for states, if they satisfy these properties w.r.t. the spatial translations. 4. Extension of States of Disordered Systems According to the line of [4], the main result of the present section is the following. For each weakly clustering state ϕ ∈ S0 (F) whose restriction to A satisfies the KMS boundary condition, there exists a modification of the time evolution by a suitable one parameter group in the gauge group, the same for each ϕξ , such that ϕξ is KMS w.r.t. this modified evolution almost surely. This is done after showing that the stabilizer Gϕ ⊂ G, as well as the asymmetry subgroup Nϕ ⊂ G coincide almost surely with the corresponding objects relative to the states ϕξ . Let ϕ, ψ ∈ S0 (F), and {ϕξ }ξ ∈X , {ψξ }ξ ∈X be the corresponding α–equivariant measurable fields of states on F. Proposition 4.1. If ϕ, ψ ∈ S0 (F) are weakly clustering states whose restrictions to A are equal, then there exist g ∈ G and a measurable subset F of full measure, such that ξ ∈ F implies ψξ (A) = ϕξ (γg (A)), simultaneously for every A ∈ F. Proof. By Theorem II.1 of [4], there exists g ∈ G such that ψ = ϕ ◦ gg . We compute for each f ∈ L∞ (X, ν) and A ∈ F, f (ξ ) ψξ (A) − ϕξ (γg (A)) ν(dξ ) = 0. X
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This means that for any fixed A ∈ F there exists a measurable set FA of full measure such that ψξ (A) = ϕξ (γg (A)) on FA . Choose a dense countable subset F0 ⊂ F. Put F :=
FA . F is a measurable
A∈F0
subset of X of full measure. For each A ∈ F choose a sequence {An } in F0 converging to A. We have for ξ ∈ F , ψξ (A) = lim ψξ (An ) = lim ϕξ (γg (An )) = ϕξ (γg (A)). n
n
Proposition 4.2. Suppose that each state in S0 (F) is asymptotically Abelian. Then each weakly clustering state ω ∈ S0 (A) extends to a weakly clustering state ϕ ∈ S0 (F). Proof. Let ψ be any extension of ω. It is automatically normal when restricted to L∞ (X, ν). It might be not a–invariant.5 Let m be any invariant mean on Zd which exists as Zd is amenable. Then
A ∈ F → m ψ(ax (A)) ∈ C defines an invariant extension of ω, that is the compact convex subset of S0 (F) consisting of all the a–invariant extensions of ω is nonvoid. Take any extremal element ϕ in such compact convex set. As ω is weakly clustering, it is extremal in the set of all the a–invariant states (i.e. a–ergodic). As ϕ is an extremal extension of ω, we conclude that ϕ is itself extremal in the compact convex set consisting of all the a–invariant states. Namely, ϕ ∈ S0 (F) is a–ergodic. By Proposition 3.1, ϕ is also weakly clustering under our assumptions. Notice that, by the results contained in [7], Proposition 2.3 of [7], there are disordered models satisfying the assumptions of Proposition 4.2. We define in the usual way, the stabilizer of a state ϕ ∈ S(F) as
Gϕ := g ∈ G ϕ ◦ gg = ϕ . The stabilizers Gϕξ of the ϕξ ∈ S(F) are defined analogously as
Gϕξ := g ∈ G ϕξ ◦ γg = ϕξ . The normalizer N (H, G) and the centralizer Z(H, G) of a subgroup H ⊂ G are defined in the usual way as
N (H, G) := g ∈ G gH g −1 = H ,
Z(H, G) := g ∈ G gh = hg , h ∈ H . We show that the stabilizers Gϕξ of the ϕξ are almost surely independent on the disorder, and coincide with the stabilizer Gϕ of ϕ almost everywhere. 5 Notice that, for the measurable field {ψ } ξ ξ ∈X of positive forms giving the direct integral decomposition of ψ, ψξ (I ) = 1 almost everywhere. We have then a decomposition of ψ into a measurable, not necessarily equivariant, field of states.
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F. Fidaleo
Theorem 4.3. Let ϕ ∈ S0 (F). Then there exists a measurable set F ⊂ X of full measure such that ξ ∈ F implies Gϕξ = Gϕ . Proof. Without loss of generality, we can suppose that (2.5) holds true everywhere on X.6 Choose a countable dense subset {h} of G which always exists by separability. Define
Vh,n := g ∈ G dist(g, h) ≤ 1/n ,
Xh,n := ξ ∈ X Gϕ ∩ Vh,n = ∅ , ξ
where “dist” is any metric on G generating its topology. Put fA (ξ, g) := ϕξ (γg (A)) − ϕξ (A),
:= fA−1 ({0}) = (ξ, g) ∈ X × G ϕξ ◦ γg = ϕξ A∈F
≡ (ξ, g) ∈ X × G g ∈ Gϕξ . By separability, we can reduce the last intersection to a countably dense set of F, that is is a measurable subset of X × G. Furthermore, it is immediate to check that
Xh,n = PX (X × Vh,n ) ∩ = ξ ∈ X dist(h, Gϕξ ) < 1/n , PX being the projection w.r.t. the first variable. This means that the sets Xh,n are measurable. By taking into account (2.5), they are also invariant under the action of the spatial translations. So, by ergodicity, we have that ν(Xh,n ) is 0 or 1. Let Sn ⊂ {h} ⊂ G be the set of the h such that ν(Xh,n ) = 1. Define G1 := Vh,n , F1 :=
n>0
h∈Sn
n>0
h∈Sn
Xh,n .
Notice that F1 is a measurable set of full measure. We have that ξ ∈ F1 implies that Gϕξ = G1 . Indeed, if g ∈ G1 , g ∈ Vh,n for h∈Sn each n, which means for each n, dist g, Gϕξ < 2/n whenever ξ ∈ F1 . Hence, ξ ∈ F1 implies G1 ⊂ Gϕξ . Conversely, if g ∈ Gϕξ and ξ ∈ F1 , then g ∈ Vh,n for every n. h∈Sn
Hence, all the Gϕξ are the same for ξ ∈ F1 , and (2.4) implies Gϕ ⊃ Gϕξ for all ξ ∈ F1 . Let now F0 ⊂ F, G0 ⊂ Gϕ be dense countable subsets. Then we obtain for each f ∈ L∞ (X, ν), A ∈ F0 and g ∈ G0 , f (ξ ) ϕξ (γg (A)) − ϕξ (A) ν(dξ ) = 0 X
Let X0 ⊂ X be the measurable set of full measure such that (2.5) is simultaneously satisfied for x ∈ Zd . We reduce the situation to the measurable invariant set Tx X0 of full measure. 6
x∈Zd
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which means that one can choose a measurable set F0 of full measure such that ξ ∈ F0 , A ∈ F0 and g ∈ G0 imply ϕξ (γg (A)) = ϕξ (A), see the proof of Proposition 4.1. Let now g ∈ Gϕ be fixed. Choose a sequence {gn } ⊂ G0 converging to g. We have for each ξ ∈ F0 and A ∈ F0 , ϕξ (γg (A)) = lim ϕξ (γgn (A)) = lim ϕξ (A) ≡ ϕξ (A), n
n
which, by the density of F0 , implies Gϕ ⊂ Gϕξ whenever ξ ∈ F0 . Hence, Gϕ = Gϕξ for all ξ ∈ F := F0 ∩ F1 , which is the assertion. Proposition 4.4. Let ϕ ∈ S0 (F) be a weakly clustering state whose restriction to A is t–invariant. Then there exist a continuous one parameter subgroup t ∈ R → εt ∈ Z(Gϕ , G), and a measurable subset F of full measure such that ξ ∈ F implies that ϕξ is invariant under the modified time translation ξ
ϕξ (A) = ϕξ (τt γεt (A)), simultaneously for every A ∈ F and t ∈ R. Proof. We cannot directly apply Theorem II.2 of [4] as our time translations t enjoy less continuity property than strong continuity, see [7]. However, in order to apply the mentioned result, it is enough to verify that the one parameter group t ∈ R → v¯t ∈ N (Gϕ , G)/Gϕ in pag. 106 of [4] is continuous also in our situation (i.e. when the map t → ϕ(Att (B)C) is continuous for every fixed elements A, B, C ∈ F). Suppose not. There exists an open neighbourhood U ⊃ Gϕ of the stabilizer Gϕ of ϕ such that, for each n, v1/n ∈ U c . Choose a subsequence {v1/nk } converging to some element v0 ∈ U c . We get ϕ(A) = lim ϕ(t1/nk (A)) = lim ϕ(gv1/nk (A)) = ϕ(gv0 (A)) k
k
which is a contradiction as the automorphism gv0 is not in the stabilizer. Hence, the conclusions of Theorem II.2 of [4] hold true also in our situation. Namely, there exists a continuous one parameter group t ∈ R → εt ∈ Z(Gϕ , G) such that v¯t = εt Gϕ , and ϕ(A) = ϕ(tt gεt (A)) for all A ∈ F. We can conclude by reasoning as in the proof of Proposition 4.1, and after choosing countable dense subsets R0 ⊂ R, F0 ⊂ F. Namely, there exists a measurable subset F ⊂ X of full measure such that ξ ∈ F , t ∈ R0 and A ∈ F0 ξ imply ϕξ (A) = ϕξ (τt γεt (A)). Let t ∈ R and A ∈ F. Choose convergent sequences ξ ξ tn → t, An → A. We have on F , by taking into account that τtn γεtn (An ) → τt γεt (A), ξ
ξ
ϕξ (A) = lim ϕξ (An ) = lim ϕξ (τtn γεtn (An )) = ϕξ (τt γεt (A)) n
which is the assertion.7 7
n
Notice that Theorem 4.3 is not directly used in this proof. Namely, Z (Gϕξ , G). {εt }t∈R ⊂
(4.1)
ξ ∈F
After passing to a measurable subset of full measure, we conclude by Theorem 4.3, that the stabilizers Gϕξ appearing in (4.1) coincide with Gϕ .
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Let ϕ ∈ S0 (F), and consider the corresponding equivariant field {ϕξ }ξ ∈X ⊂ S(F). We have shown in Theorem 4.3 that Gϕξ = Gϕ almost surely. This means that the GNS representations πϕξ , as well as πϕ , are equipped with a strongly continuous representation Uϕξ , or Uϕ , of the common subgroup Gϕ ⊂ G implementing the gauge action the set of Gϕ on F or F respectively. Let H ⊂ Gϕ be a closed subgroup. Denote H act of all the irreducible representations of H . It is well known that the elements of H is at most countable on finite dimensional Hilbert spaces (compactness of H ), and H (second countability of H ). The restriction of Uϕξ , or Uϕ to H are denoted as UϕHξ , or UϕH respectively. The of ϕξ , ϕ are the set of the unitary equivalence classes of all H –spectra ϕHξ , ϕH ⊂ H the irreducible representations of H contained in UϕHξ , or UϕH respectively. Following Definition II.3 of [4], we say that ϕHξ (or equivalently ϕH ) is one–sided which enjoys the following properties: if it is contained in a set ⊂ H (i) σ1 , σ2 ∈ implies that every irreducible summand of σ1 ⊗ σ2 is also contained in , (ii) σ, σ¯ ∈ implies that σ = id. Theorem 4.5. Let ϕ ∈ S0 (F). Then (i) the H –spectrum ϕHξ is almost surely independent on ξ ∈ X, (ii) if σ ∈ ϕHξ almost surely, its multiplicity is almost surely independent on ξ ∈ X. Proof. We start by noticing that, by Theorem 4.3, ⊕ Uϕ = Uϕξ ν(dξ ). X
Moreover, we can identify the GNS triplet (πϕT−x ξ , HϕT−x ξ , ϕT−x ξ ) relative to ϕT−x ξ with (πϕξ ◦αx , Hϕξ , ϕξ ) almost surely. Under this identification, UϕHT ξ coincides with −x ∼ U H . In other words, U H = U H almost surely. ϕξ
ϕT−x ξ
ϕξ
Consider the measurable subsets
Fσ,m := ξ ∈ X σ ≺ UϕHξ with multiplicity m of X. By the previous considerations, the Fσ,m are invariant under the action {Tx }x∈Zd , and provide a countable partition of X. The assertion follows as the action of the spatial translations on the sample space (X, ν) is supposed to be ergodic. Corollary 4.6. Let ϕ ∈ S0 (F). (i) If ϕ is weakly clustering and asymptotically Abelian, then UϕHξ satisfies the semigroup property of Theorem II.3 of [4] almost surely, (ii) ϕH is one–sided if and only if ϕHξ is one–sided almost surely. Proof. (i) holds by Theorem 4.5, with the help of Theorem II.3 of [4], (ii) holds by Theorem 4.5. Here, there is the main theorem of the present section concerning the appearance of the chemical potential in the setting of disordered systems.
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Theorem 4.7. Let ϕ ∈ S0 (F) be a weakly clustering asymptotically Abelian state whose restriction to A is (t, β)–KMS state at inverse temperature β = 0. Then there exist a closed normal subgroup N ⊂ Gϕ , a continuous one parameter subgroup t ∈ R → εt ∈ Z(Gϕ , G), a continuous one parameter subgroup t ∈ R → ζt ∈ Gϕ , and a measurable subset F ⊂ X of full measure such that, for each ξ ∈ F , (i) the N –spectrum of ϕξ is one–sided,
(ii) the restriction of ϕξ to F N := A ∈ F γg (A) = A , g ∈ N is a (θ ξ , β)–KMS ξ ξ state for the modified time evolution θt := τt γεt ζt , (iii) the image [ζt ] := ζt N in Gϕ /N is in the centre Z(Gϕ /N ). Proof. We start by noticing that the conclusions of Theorem II.4 of [4] hold true also in our situation. Indeed, the hypothesis of extremality w.r.t. the KMS boundary condition is not used in the proof of that theorem, the last being replaced by the weak clustering property.8 In order to apply those results to our situation, it is enough to replace the dense subset of entire elements used in II.6, pag. 110 with the dense subset of entire elements in a weaker sense, that is those generated by elements of the form ξ Af (ξ ) := f (t)τt (A(ξ )) dt, 9 Hence, we can apply the above mentioned where A runs on elements of F and f ∈ D. theorem to ϕ ∈ S0 (F) as well. Namely, by taking into account Proposition 4.4, there exist a closed normal subgroup N = Nϕ ⊂ Gϕ , a continuous one parameter subgroup t ∈ R → εt ∈ Z(Gϕ , G), a continuous one parameter subgroup t ∈ R → ζt ∈ Gϕ such that the N –spectrum of ϕ is one–sided, and the restriction of ϕ to the N fixed–point algebra FN is a (θ, β)–KMS for θ t := tt gεt ζt , and (iii) holds true. Now, (i) follows by Corollary 4.6, and (ii) follows by Proposition 3.2 of [7]. It is seen in Theorem II.4 of [4], that there exists a closed normal subgroup Nϕ of Gϕ with one–sided ϕ–spectrum, such that the restriction of the weakly clustering asymptotically Abelian state ϕ to FNϕ , is a KMS state for a suitable time evolution. Such a subgroup is explicitely constructed without providing any abstract definition for it, and is called “the asymmetry subgroup of the state ϕ”. Theorem 4.7 has a similar structure. Namely, the same assertions relative to the existence of an “asymmetry subgroup” N hold true almost surely also in our situation with N = Nϕ . Theorem 4.3 and (ii) of Corollary 4.6 tell us that N has a fiber–like interpretation. First, N ⊂ Gϕξ almost surely. Second, the N–spectrum of ϕξ is one–sided almost surely. Thus, Theorem 4.7 is self–contained. However, it is natural to address the following questions. N
(a) Give an abstract definition in terms of the spectral properties of φ φ , for the asymmetry subgroup Nφ ⊂ G of a state φ on a C ∗ –algebra, invariant under the action of a (locally) compact group G. (b) Give a direct fiberwise construction of the asymmetry subgroup Nϕξ in our framework, without assuming the ergodicity of the action of Zd on the sample space (X, ν). 8 See also the analogous results Theorem 4.1 of [22], and Theorem 12 of [23]. In addition, the clustering condition for ϕ is essentially equivalent to the fact that Zπϕ ∼ L∞ (X, ν) in our situation, see Section 5. 9 Another equivalent possibility is to restrict ourselves to the C ∗ –subalgebra F made of elements of 0 F having norm–pointwise continuous orbit w.r.t. the time evolution t.
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It is unclear to the author if (a) has been solved, and/or how to tackle and solve it. Yet, there is some evidence for how one can treat (b). For the reader’s convenience, we report such an analysis in an appendix. We end the present section by noticing that Theorem 4.7 can be applied to the case when the restriction to A of a state in S0 (F) is a KMS state at 0 inverse temperature, that is a trace. Namely, in this situation it is enough to replace (ii) in Theorem 4.7 with (ii’) the restriction of ϕξ to F N is a (γζt , −1)–KMS state.10 5. An Intrinsic Characterization of the Chemical Potential for Disordered Systems In the present section we outline how the chemical potential can arise as an object directly associated to the algebra of observables. We start by proving the following Proposition 5.1. Let ω ∈ S0 (A) be a strongly clustering asymptotically Abelian (t, β)– KMS state at inverse temperature β = 0. Suppose that the automorphism ρ of A is implemented by a unitary in F. If A is simple, then ρ extends to a pointwise–weak measurable field {ρξ }ξ ∈X of normal automorphisms of the weak closure πωξ (A) , almost surely. Proof. Under our assumptions, we can apply Proposition IV. 1 of [4]. Then ω is quasi equivalent to ω ◦ (ρ ⊗ id), where id ≡ idL∞ (X,ν) .11 Then πω is unitarily equivalent to πω◦(ρ⊗id) ∼ = πω ◦ (ρ ⊗ id). This means that there exists a spatial isomorphism of ⊕ πω◦(ρ⊗id) (A) = πωξ (ρ(A)) ν(dξ ) X
onto πω (A) =
⊕ X
πωξ (A) ν(dξ ),
where, under the above ⊕ identification, these von Neumann algebras act on the same Hilbert space Hω = Hωξ ν(dξ ).12 By Theorem IV.8.23 of [33], there exists a meaX
surable field {Uξ }ξ ∈X of unitary operators such that πωξ (ρ(A)) = Uξ πωξ (A) Uξ∗ , where πωξ ◦ ρ = Uξ πωξ ( · )Uξ∗ almost everywhere. As A is supposed to be simple, {πωξ }ξ ∈X is a measurable field of ∗–isomorphism of A onto their images πωξ (A) almost surely. After identifying A with πωξ (A), the searched measurable field of normal automorphisms extending ρ are given, for R ∈ πωξ (A) , by ρξ (R) := Uξ RUξ∗ .
The centralizer Z (Gϕ , G) plays no rˆole in this situation (i.e. εt = e), see [8] for further details. In Proposition IV. 1 of [4], the extremality of the KMS state ω cannot be dropped, as no weak clustering assumption is made there, see Remark II.2 of [4]. 12 See [7], Section 2 for the last equality relative to the direct integral decomposition. 10 11
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We consider the case when the physical spectrum is described in terms of outer automorphisms, see [11, 13–15, 17] for a standard situation arising from quantum field theory. Namely, we suppose that the charges of interest are obtained by considering the unitary equivalence classes of a set of representations {πn }n∈Z of A satisfying πn ∼ = π0 ◦ ηn . Here, π0 is a fixed irreducible representation of A, and “∼ =” stands for unitary equivalence. The ηn , n = 0 are outer transportable automorphisms of A such that, if ρ ∈ [ηk ], there exists a unitary U ∈ F\A with ρ(A) = U AU ∗ , A ∈ A.13 In this situation, the gauge group is the unit circle T. Let ω ∈ S0 (A) be a (t, β)–KMS state such that the centre Zπω := πω (A) πω (A) ∞ is isomorphic to L (X, ν). It is explained in [7], that this situation seems to be the counterpart of the “pure thermodynamical phase” in the case of disordered models. In this situation, we have that ω is weakly clustering (w.r.t. the spatial translation). Namely, as the (τ ξ , β)–KMS state ωξ ∈ S(A) is a factor state almost surely, it satisfies (3.3) almost surely, see [2], Lemma 10.2.14 Then the assertion follows by Proposition 3.3. Take a weakly clustering extension ϕ of ω to all of F which exists by Proposition 4.2. Furthermore, in order to avoid cases when the chemical potential is zero, we suppose 15 that ϕ is gauge invariant (i.e. Gϕ = T). In this situation, we have sp(Uϕ ) = Z ≡ G. Namely, the asymmetry subgroup Nϕ of ϕ is trivial. Let ρ be an automorphism of A carrying the charge n (i.e. ρ ∈ [ηn ]), and consider the unitary U ∈ F implementing ρ on A. Consider the state ϕU := ϕ ◦ adU ⊗I . We have for the Connes–Radon–Nikodym cocycle ([12, 32]), ⊕ ϕ πϕξ (U ∗ )σt ξ (πϕξ (U ))ν(dξ ) DϕU : Dϕ)t = X ⊕ ξ inβµt πϕξ (U ∗ τ−βt (U ))ν(dξ ), (5.1) =e X
for some µ ∈ R. The direct integral decomposition of DϕU : Dϕ) exists by the results in Appendix A of [7], by taking into account the definition of the relative modular operator in terms of the balanced weight (e.g. [32], Section 3). Here, we have used ϕ γθ (U ) = einθ U , and σt ◦ πϕ = πϕ ◦ t−βt ◦ gβµt by Theorem 4.7, by taking into account (2.1). Now, we take advantage from the fact that ω ◦ (ρ ⊗ id) extends to a normal state on all of πω (A) . Denote with the same symbol ωξ the normal extension of ωξ itself to all of πωξ (A) . Under this identification, the equivariant measurable field of states {ωξ ◦ ρξ }ξ ∈X provides the direct integral decomposition of the normal extension of 13 The mentioned automorphisms might be recovered by considering representations π of A which are “strong locally equivalent” to the reference representation π0 , see e.g. [13–15]. Provided that π0 is kept fixed, the last property means
π Ac ∼ = π0 Ac for some region (unbounded regions are also allowed, see e.g. [11]) ⊂ Zd , depending on the represention π. 14 See [31], Proposition 3, when A includes Fermion operators. 15 Let V ∈ F be the unitary implementing the automorphism η on A. This means that γ (V ) = eiθ V . η η θ η The vector n := πϕ (Vηn ⊗ I ) ϕ is an eigenvector of Uϕ (θ ) corresponding to the eigenvalue einθ .
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ω ◦ (ρ ⊗ id). Here, the ρξ are the normal automorphisms of πϕξ (F) appearing in ξ Proposition 5.1. We have by (5.1) and the fact that U ∗ τ−βt (U ) is gauge invariant,
D(ωξ ◦ ρξ ) : Dωξ )t = einβµt πωξ (U ∗ τ−βt (U )) ξ
(5.2)
almost everywhere. Formula (5.2) explains the occurrence of the chemical potential µ ∈ R as an object intrinsically associated to the observable algebra. Furthermore, according with this description, it does not depend on the disorder for states ω on A such that Zπω ∼ L∞ (X, ν). This is in accordance with the standard fact that the physically relevant quantities should not depend on the disorder. Notice that the situation described in the present section extends straightforwardly to the case when the gauge group is the n–dimensional torus or, more generally, to non Abelian finite dimensional Lie groups. As it is explained in [4], Section IV, there could be a freeness in order to define the chemical potential (see Formula (IV.6) together with the comment (i) in pag. 128). By this freeness, the chemical potential would be defined up to a phase factor belonging to the centre of the GNS representation. Such a phase is connected with the zero point of the chemical potential, as the centre is trivial in the situation treated in the above mentioned paper. In our situation, if one makes the choice of the zero point of the chemical potential in a measurable and invariant way, such a zero–point function would lie in L∞ (X, ν).16 In addition, one would conclude by ergodicity that also the freeness in the choice of the zero point of the chemical potential can be avoided. To conclude, we point out the fact that the natural framework for the previous analysis is that of disordered systems consisting of spin models, or equally well disordered models arising from quantum field theory.17 In both situations, some technical and conceptual troubles arise. The Doplicher–Haag–Roberts theory of localized endomorphisms (cf. [11, 13–15]) heavily relies on the Reeh–Schlieder property and the Haag duality of the “vacuum representation” π0 . Unfortunately, the Reeh–Schlieder property cannot hold when the local algebras of observables are finite dimensional. Moreover, it is unclear if one can assume the Haag duality without meeting with trivialities. Thus, the description of the physical spectrum of a spin model in terms of endomorphisms of the observable algebra is a difficult task, even if it cannot be excluded a priori. Relatively to disordered models arising from quantum field theory, the local algebras are usually non separable C ∗ –algebras. However, one could take advantage from the local normality of the states or the endomorphisms, and the split property naturally assumed in quantum field theory. After overcoming the previously mentioned problems, we argue that the theory of the chemical potential will be applied to concrete situations such as spin glasses, or disordered models arising from quantum field theory. 6. Appendix The present appendix is devoted to outline the fiberwise construction of the asymmetry subgroup. We deal with a disordered system satisfying all the assumptions collected in Section 2, except the ergodicity of the action {Tx }x∈Zd on the sample space (X, ν). 16 Our choice in (5.1), (5.2) corresponds to the case when the zero point of the chemical potential is fixed at 0, independently an the disorder. 17 Another possible field for the application of the theory of the chemical potential is some non trivial examples of “continuous” disordered models, see e.g. [20, 21].
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Denote Mx fx the Cesaro mean (w.r.t. the variable x ∈ Zd ) of the sequence fx . In order to avoid technicalities, we make the following slightly stronger assumptions. The field algebra F is norm–asymptotically Abelian, and the equivariant field {ϕξ }ξ ∈X of states on F satisfies
Mx ϕξ (Aαx (B)) − ϕξ (A)ϕξ ◦ αx (B) = 0, (6.1) simultaneously for A, B ∈ F on a measurable set of full measure.18 Let Cϕξ (Gϕξ ) ⊂ ϕ C(Gϕξ ) be the set consisting of the functions fa,bξ on Gϕξ defined for a, b ∈ C(X; F) (i.e. the C ∗ –algebra made of the continuous F–valued functions on the compact space X), as
ϕ fa,bξ (g) := Mx ϕξ ◦ αx (a(T−x ξ )γg (b(T−x ξ ))) . (6.2) The fact that Cϕξ (Gϕξ ) is well–defined on a measurable set of full measure can be viewed in the following way. By the Birkhoff individual ergodic Theorem, the function
ha,b;g (ξ ) := Mx ϕξ ◦ αx (a(T−x ξ )γg (b(T−x ξ ))) exists (and it is invariant under the action of Zd on (X, ν)) almost everywhere, for fixed a, b ∈ C(X; F), g ∈ G. By a standard density argument, there will exist a measurable set F0 ⊂ X of full measure such that all the ha,b;g are simultaneously well–defined. The functions in (6.2) are defined as ϕ
fa,bξ (g) := ha,b;g (ξ )
ξ ∈ F0 , g ∈ Gϕξ .
By reasoning as in pag. 108 of [4], we obtain by asymptotic Abelianess, and the weak clustering property (6.1), ϕ
ϕ ϕ fa,bξ fa ξ,b = Mx faαξx (a ),bαx (b ) ϕξ
ϕ ϕ , fa,bξ + fa ξ,b = Mx fa+α x (a )+λD,b+αx (b )+µD ϕ
where D ∈ C(X; A) is a constant field (depending on ξ ∈ F0 ) with non vanishing fDξ2 ,I , and λ, µ ∈ C are chosen such that ϕξ ϕξ ϕ ϕξ ϕ ϕ ϕξ λµfDξ2 ,I + λ fDb,I + fD,I fb ξ,I + µ faD,I + fa ξ,I fD,I ϕ
ϕ
ϕ
ϕ
+fa ξ,I fb,Iξ + fa,Iξ fb ξ,I = 0. Moreover, Cϕξ (Gϕξ ) is left globally stable by the left and the right action of Gϕξ . This means that ∗ Cϕξ (Gϕξ ) Cϕξ (Gϕξ ) = C(Gϕξ /Nϕξ ) for some closed normal subgroup Nϕξ of Gϕξ , see [4], Lemma A.1. If the action of Zd on (X, ν) is ergodic, then by Theorem 4.1 Gϕξ is almost surely independent on ξ , and by the Birkhoff individual ergodic Theorem, Nϕξ is also almost surely independent on ξ . It remains to show that Nϕξ = Nϕ almost everywhere. 18 The norm–asymptotic Abelianess is automatically satisfied for the quasi–local algebras considered in Section 5. See Proposition 3.2, and Proposition 3.3 for questions connected with the cluster property (6.1).
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By a standard density argument and the Birkhoff individual ergodic Theorem, there ϕ exists a measurable set F ⊂ F0 of full measure such that Gϕ = Gϕξ , and fa,bξ (g) is a constant (depending on a, b; g) on F , simultaneously for a, b ∈ C(X; F) and g ∈ Gϕ . By applying Proposition 3.3, Proposition 3.2 and the Lebesgue dominated convergence Theorem, we conclude that, for ξ ∈ F , a, b ∈ C(X; F) and g ∈ Gϕ , ϕ
fa,bξ (g) = ϕ(agg (b)), that is ξ ∈ F implies Cϕξ (Gϕξ ) ⊂ Cϕ (Gϕ ). Let now a = A ⊗ h, b = B ⊗ k where A, B ∈ F and h, k are essentially bounded measurable functions on X. By the Lusin Theorem, there exists a sequence {gn } ⊂ C(X) such that
gn ∞ ≤ h ∞ k ∞ ,
n ∈ N,
and gn (ξ ) −→ h(ξ )k(ξ ), ν–almost surely. Define for ξ ∈ F and g ∈ Gϕ ,
fnξ (g) := Mx gn (T−x ξ )ϕξ ◦ αx (Aγg (B)) . By the previous considerations and the Lebesgue dominated convergence Theorem, ξ fn (g) −→ ϕ(agg (b)) on F . On the other hand, by passing possibly to subsequences, ξ for each fixed ξ ∈ F , fn converges in the topology of C(Gϕξ ) by the Ascoli–Arzel`a Theorem. This means that Cϕ (Gϕ ) ⊂ Cϕξ (Gϕξ ) whenever ξ ∈ F . Collecting together, we get Cϕξ (Gϕξ ) = Cϕ (Gϕ ), thus Nϕξ = Nϕ if ξ ∈ F . Acknowledgements. The author is grateful to the referee for some useful suggestions.
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13. Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations I. Commun. Math. Phys. 13, 1–23 (1969) 14. Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations II. Commun. Math. Phys. 15, 173–200 (1969) 15. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics I. Commun. Math. Phys. 23, 199–230 (1971) 16. Doplicher, S., Roberts, J.E.: Fields, statistics and non–Abelian gauge groups. Commun. Math. Phys. 28, 331–348 (1972) 17. Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection in particle physics. Commun. Math. Phys. 131, 51–107 (1990) 18. van Enter, A.C.D., van Hemmen, J.L.: Statistical mechanical formalism for spin–glasses. Phys. Rev. A 29, 355–365 (1984) 19. van Enter, A.C.D., Maes, C., Schonmann, R.H., Shlosman, S.: The Griffiths singularity random field. Am. Math. Soc. Trans. 198, 51–58 (2000) 20. Fidaleo, F., Liverani, C.: Ergodic properties of a model related to disordered quantum anharmonic crystals. Commun. Math. Phys. 235, 169–189 (2003) 21. Fidaleo, F., Liverani, C.: Statistical properties of disordered quantum systems. In: Operator Theory: Advances and Applications, Gaspar, D., Gohberg, I., Timotin, D., Vasilescu, F.H., Zsido, L. (eds.), Basel: Birkh¨auser–Verlag, 153, 123–141 (2004) 22. Kastler, D., Takesaki, M.: Group duality and the Kubo–Martin–Schwinger condition. Commun. Math. Phys. 70, 193–212 (1979) 23. Kastler, D., Takesaki, M.: Group duality and the Kubo–Martin–Schwinger condition II. Commun. Math. Phys. 85, 155–176 (1982) 24. Kishimoto, A.: Equilibrium states of a semi–quantum lattice system. Rep. Math. Phys. 12, 341–374 (1977) 25. K¨ulske, C.: (Non–)Gibbsianness and phase transitions in random lattice spin models. Markov Process. Relat. Fields 5, 357–383 (1999) 26. Longo, R.: Algebraic and modular structure of von Neumann algebras of Physics. Proc. Symp. Pure Math. 38, 551–566 (1982) 27. Longo, R.: Notes for a quantum index theorem. Commun. Math. Phys. 222, 45–96 (2001) 28. Mezard, M., Parisi, G., Virasoro, M.A.: Spin–glass theory and beyond. Singapore: World Scientific, 1986 29. Newman, M.N.: Topics in disordered systems. Basel–Boston–Berlin: Birkh¨auser, 1997 30. Newman, M.N., Stein, D.L.: Ordering and broken symmetry in short–ranged spin glasses. J. Phys. Condens. Matter 15, R1319–R1364 (2003) 31. Robinson, D.W.: A characterization of clustering states. Commun. Math. Phys. 41, 79–88 (1975) 32. Strˇatilˇa, S.: Modular theory in operator algebras. Tunbridge-Wells-Kent: Abacus Press, 1981 33. Takesaki, M.: Theory of operator algebras I. Berlin-Heidelberg-New York: Springer, 1979 Communicated by J.L. Lebowitz
Commun. Math. Phys. 262, 393–410 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1499-y
Communications in
Mathematical Physics
Univalent Functions and Integrable Systems Dmitri Prokhorov1 , Alexander Vasil’ev2 1 2
Department of Mathematics and Mechanics, Saratov State University, Saratov 410026, Russia. E-mail: [email protected] Matematisk Institutt, Universitetet i Bergen, Johannes Brunsgate 12, 5008 Bergen, Norway. E-mail: [email protected]
Received: 6 December 2004 / Accepted: 23 August 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to integrable systems. One example of such parameters is complex moments for the Laplacian growth that form a Whitham-Toda integrable hierarchy. Another example we deal with is related to expanding coefficient bodies for conformal maps given by L¨owner subordination chains. The coefficients’ bodies are proved to form a Liouville partially integrable Hamiltonian system for each fixed index and the first integrals are obtained. We also discuss the contact structure of this system. 1. Introduction Complex dynamical systems, iterations and construction of Lie semigroups with respect to the composition operation lead to the study of expanding systems of plane domains. These systems can be thought of as one-parameter families of domains (typically simply connected and univalent) (t) that form subordination chains in the Riemann sphere ˆ which are defined for 0 ≤ t < τ , where τ may be ∞. This means that (t) ⊂ (s), C, (t) = (s), whenever t < s. Kufarev [16] studied a one-parameter family of domains (t), and regular functions g(z, t), g : (t) × [0, τ ) → U , where U = {ζ : |ζ | < 1}. He proved differentiability of g(z, t) with respect to t for z from the Carath´eodory kernel (0) of (t). Sometimes, the evolution of (t) can be characterized by infinite sets of evolution parameters given as a result of integration of certain dynamical systems. Hamiltonian interpretation of such systems yields their integrability. The interest to integrable systems is caused first of all by Hamiltonian systems of mathematical physics where the Hamiltonian is thought of as an energy functional and its critical points correspond to minimal energy of particle systems. However, other areas of mathematics also use Hamiltonian formalism, e.g., optimal control, classical mechanics, etc.
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One of the examples of integrable systems of evolution parameters corresponding to one-parameter families of expanding domains is the problem of Laplacian growth (Hele-Shaw problem). The Laplacian growth means the Dirichlet problem for a harmonic potential where the boundary of the phase domain (t) is unknown a-priori (free boundary), and in fact, is defined by the normality of its motion. The classical (strong) solvability of this problem suggests that phase domains (t) are bounded by analytic curves. It turns out that given an initial domain (0) the set of Richardson’s moments (see the definition in [24]) solves completely the inverse potential problem, thus describing the evolution of phase domains as long as the classical solution exists. As it has been shown in [1, 15, 18, 27], the Laplacian growth problem can be embedded into a larger hierarchy of domain variations (Whitham-Toda hierarchy) for which all the complex moments are treated as independent variables (generalized time variables), and form an integrable system. Another example of evolution parameters is the set of coefficients of the Riemann maps of U onto the family of the phase domains (t) that has attracted attention of the complex analysts for a more than eighty year period of the last century. The Bieberbach conjecture [8] has proved to be the most intriguing problem that forced investigations in geometric function theory during this period. It finally has been solved in 1984 by de Branges [9], who proved that the nth Taylor coefficient bn of a conformal homeomorphism f (ζ ), normalized by f (0) = 0, f (0) = 1, does not exceed its index: |bn | ≤ n. This class of functions we denote by S. The inequality is sharp and the equality is attained only for rotations of the Koebe function k(z) = z(1 − z)−2 . De Branges’ proof has put an end to the numerous attempts to prove or disprove the Bieberbach conjecture. However, a much more general and difficult problem is to describe the range of n first coefficients as a set of values Vn = {(b2 , . . . , bn ), f ∈ S}. In the trivial case V2 is the disk of radius 2. Only the first non-trivial coefficient body V3 = (b2 , b3 ) has been described completely by Schaeffer and Spenser in 1950 in their famous monograph [25]. A qualitative description of Vn , n ≥ 3, has been partially given in [5]. Apart from these two monographs there are only few works where progress in such a problem has been made (see, e.g., [23]). The key stages of our presentation are as follows. Firstly, a boundary point of Vn is given by a unique function whose first coefficients b2 , . . . , bn define the rest of the coefficients. Secondly, this function, unlike the Laplacian growth, presents a full mapping (the complement of the image of U is of measure zero) of the unit disk onto the complex plane minus an analytic graph with the root at infinity. Thirdly, erasing this graph we obtain a family of expanding domains completely described by the initial domain, and in its turn by the point of ∂Vn . Finally, we obtain a (2n − 3) dimensional family of evolution dynamics completely described as a flow generated by a certain vector field, which admits a Hamiltonian formulation. It is proved to be partially integrable in the Liouville sense and the first integrals are obtained. Moreover, it turns out that the first integrals that generate additional directions form a horizontal space so this Hamiltonian system possesses a contact structure. It is important that the evolution exists all the time 0 ≤ t < ∞. Our paper is devoted to a detailed realization of this plan and the description of this system of evolution parameters, its Hamiltonian formulation, and finally, its integration. The central idea of the Hamiltonian formulation is forced by the L¨owner evolution (see, e.g., [2, 17, 20]) that is based on evolution equations in partial and ordinary derivatives mutually linked. The controllable dynamical system for evolution parameters turns out to be Hamiltonian under the assumption that the necessary optimality conditions hold.
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So, we are especially interested in subordination chains of domains generated by Riemann maps corresponding to the critical controls (satisfying the necessary condition of optimality), and therefore, to the trajectories, that in particular, give all boundary points to the coefficient body for univalent maps. From one side our main results describe evolution coefficient systems for a certain type of L¨owner’s dynamics of domains and maps that admit the Hamiltonian formulation and are integrable. The coefficient bodies will form a certain type of hierarchy (dependent on the coefficient indices) and the first integrals will be derived. From the other side, we give all boundary points of the coefficient bodies Vn . We appreciate the referee’s remarks, especially concerning Sect. 5, that allowed us to make the presentation clearer. 2. The L¨owner and L¨owner-Kufarev Evolution There are many papers and monographs devoted to the L¨owner and L¨owner-Kufarev equations. However we revisit this theory from the point of view of correspondence between L¨owner equations in partial and ordinary derivatives and univalence of solutions to the partial derivative L¨owner equation. Let us consider a subordination chain of simply connected hyperbolic domains (t) in the complex plane C, which is defined for 0 ≤ t < τ , where τ may be infinity. We suppose that the origin is an interior point of the kernel of {(t)}τt=0 . Let us normalize the growth of the evolution of this subordination chain by the conformal radius of (t) with respect to the origin to be et . By the Riemann Mapping Theorem we construct a subordination chain of mappings f (ζ, t), ζ ∈ U , where each function f (ζ, t) = et ζ + b2 (t)ζ 2 + . . . is a holomorphic univalent map of U onto (t) for every fixed t. Pommerenke [20, 21] introduced such chains in order to generalize L¨owner’s equation. His result says that given a subordination chain of domains (t) defined for t ∈ [0, τ ), there exists an analytic regular function p(ζ, t) = 1 + p1 (t)ζ + p2 (t)ζ 2 + . . . ,
ζ ∈ U,
such that Re p(ζ, t) > 0 and ∂f (ζ, t) ∂f (ζ, t) =ζ p(ζ, t), ∂t ∂ζ
(1)
for ζ ∈ U and for almost all t ∈ [0, τ ). The class of holomorphic functions p defined as above we denote by C. Equation (1) is called the L¨owner-Kufarev equation due to two seminal papers by L¨owner [17] with p(ζ, t) =
eiu(t) + ζ , eiu(t) − ζ
(2)
where u(t) is a continuous function regarding t ∈ [0, τ ), and by Kufarev [16] in the general case, where this equation appeared for the first time. Equation (1) represents a growing evolution of simply connected domains. Let us consider the reverse process. Given an initial domain (0) ≡ 0 (and therefore, the initial mapping f (ζ, 0) ≡ f0 (ζ )), and a function p(ζ, t) from the class C, we solve Eq. (1) and ask whether the solution f (ζ, t) represents a subordination chain of simply connected domains. The initial condition f (ζ, 0) = f0 (ζ ) is not given on the characteristics
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of the partial derivative equation (1), hence the solution exists and is unique. Assuming s as a parameter along the characteristics we have dt = 1, ds
dζ = −ζp(ζ, t), ds
df = 0, ds
with the initial conditions t (0) = 0, ζ (0) = z, f (ζ, 0) = f0 (ζ ), where z is in U . Obviously, t = s. We still need to give sense to this formalism because the domain of ζ is the entire unit disk, however the solutions to the second equation of the characteristic system range within the unit disk but do not fill it. Therefore, introducing another letter w in order to distinguish the function w(z, t) from the variable ζ , we arrive at the Cauchy problem for the L¨owner-Kufarev equation in ordinary derivatives for a function ζ = w(z, t), dw = −wp(w, t), dt
(3)
with the initial condition w(z, 0) = z. Equation (3) is the nontrivial characteristic equation for (1). Unfortunately, this approach requires the extension of f0 (w −1 (ζ, t)) into U (w−1 means the inverse function) because the solution to (1) is the function f (ζ, t) given as f0 (w −1 (ζ, t)), where ζ = w(z, s) is a solution of the initial value problem for the characteristic equation (3) that maps U into U . Therefore, the solution of the initial value problem for Eq. (1) may be non-univalent. Let S stand for the usual class of all univalent holomorphic functions f (z) = z + a2 z2 + . . . in the unit disk. Solutions to Eq. (3) are regular univalent functions w(z, t) = e−t z + a2 (t)z2 + . . . in the unit disk that map U into itself. Conversely, every function from the class S can be represented by the limit f (z) = lim et w(z, t), t→∞
(4)
where there exists a function p(z, t) from the class C for almost all t ≥ 0, such that w(z, t) is a solution to Eq. (3) (see [21, pp. 159–163]). Each function p(z, t) generates a unique function from the class S. The reciprocal statement is not true. In general, a function f ∈ S can be determined by different functions p ∈ C. From [21, pp. 163] it follows that we can guarantee the univalence of the solutions to the L¨owner-Kufarev equation in partial derivatives (1) assuming the initial condition f0 (ζ ) given by the limit (4) with the function p(·, t) chosen to be the same in Eqs. (1) and (3). In his original paper [17] L¨owner dealt with functions that map U onto domains each of which was obtained by slitting C along a unique Jordan curve. This led him to a subclass S of the class S of one-slit maps which is dense in S. Each function f ∈ S is obtained as the limit (4) where w(z, t) is a solution to the Cauchy problem for the L¨owner equation, dw eiu(t) + w = −w iu(t) , dt e −w
(5)
with the initial condition w(z, 0) = z, where u(t) is a continuous control function. The corresponding L¨owner equation in partial derivatives is just Eq. (1) with the function p(ζ, t) given by (2). The solution to this equation is univalent all the time if and only if the initial condition is given as the limit (4) for the same p given by (2). It maps the unit disk onto a growing family of one-slit domains. The class of such domains
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lies dense in the space of all simply connected domains in the sense of Carath´eodory’s kernel convergence. We proceed with generalized L¨owner chains of domains with several slits. Let (t), 0 ∈ (t), t ∈ [0, τ ), be a subordination chain of simply connected domains each of which is obtained by slitting C along a finite number of Jordan curves with m+1 distinct endpoints. The m + 1th endpoint is at infinity. The one-parameter family f (ζ, t) stands for the corresponding subordination chain of normalized conformal maps f (ζ, t) = et ζ + b2 (t)ζ 2 + . . . , ζ ∈ U , t ∈ [0, τ ). These maps satisfy Eq. (1) with the function p(ζ, t) given by (2), where u(t) is a piecewise continuous control function regarding t ∈ [0, τ ) (see, e.g., Goluzin [12]). However, this representation is not unique and another approach has been suggested originally by Kufarev and described in [2]. It was based on the function p(ζ, t) in (1) given by the following formula: p(ζ, t) =
m
λk (t)
k=1
eiuk (t) + ζ , eiuk (t) − ζ
(6)
where uk (t) are continuous control functions regarding t ∈ [0, τ ) and λk (t) are measur able non-negative control functions, m k=1 λk (t) = 1. Constructing the characteristic equation (3) we come to the L¨owner-Kufarev equation in ordinary derivatives and the limit (4) gives a subclass S of the class S of multi-slit maps which is also dense in S. The function w(z, t) in (4) is a solution to the Cauchy problem for the L¨owner-Kufarev equation eiuk (t) + w dw λk (t) iu (t) = −w , dt e k −w m
(7)
k=1
with the initial condition w(z, 0) = z, where uk (t) and λk (t) are defined as in (6). Similarly to the one-slit evolution, Eq. (1) with the function p given by (6) possesses univalent solutions all the time if and only if the initial condition is given as the limit (4) for the same p in (7) given by (6). Although the representation (1, 6) or (7) is not unique either, it allows to elaborate a unique one. Given a subordination chain (t) of multi-slit domains there exists a unique m set of positive numbers {λk }m , k=1 λk = 1, and a unique set of continuous control k=1 m functions {uk (t)}k=1 , such that (t) = f (U, t), where f is a solution to Eq. (1) with the function p given by (6), and λk (t) ≡ λk . This statement follows from the results of [23]. An analogous uniqueness statement for the representation (7) is true too. 3. Evolution Parameters and Hierarchies As it was stated in the Introduction, one of the examples of integrable systems of evolution parameters corresponding to one-parameter families of domains is the problem of Laplacian growth. In its classical formulation it deals with the smooth family {(t)}, t ∈ [0, τ ) of phase domains, i.e., when ∂(t) are smooth (C ∞ ) interfaces for each t, and the normal velocity vn at the boundary continuously depends on t at any point of ∂(t). Each (t) is supposed to be simply connected in C for any t ∈ [0, τ ) fixed. Starting from the initial moment t = 0 the boundary becomes even analytic smooth for t ∈ (0, τ ). Richardson’s complex moments completely describe this evolution, say given an initial phase domain (0) there exists a unique infinite set of the moments for (t) at each t ∈ (0, τ ) and vice versa. One of the reasons for a slit dynamics to appear
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in this process is extending the solution to exist beyond the blowup time τ . One way of the regularization of the problem is to introduce a surface tension parameter instead of zero boundary condition in the classical Hele-Shaw formulation (see, e.g., [14]). This can produce a crack morphology in which a part of the boundary develops a thin finger penetrating the phase domain. Another proposition (see [13]) is to continue smooth dynamics by slit dynamics beyond the blowup time when a possible cusp is developed at the boundary. Of course, in this case one relaxes smoothness of the boundary permitting the corresponding Riemann map to have singularities at the unit circle. Another reason to consider slit dynamics is the determinate evolution, duality, and univalence of the solutions to the L¨owner and L¨owner-Kufarev equations (in ordinary and partial derivatives) stated in the preceding section. In view of the involved Riemann maps the most natural evolution parameters are the Taylor coefficients of these maps. Of course, the infinite set of coefficients completely describes the evolution given by the maps f : U → (t). However in general, the differential equations for these coefficients obtained from the L¨owner or L¨ownerKufarev equations do not admit any reasonable Hamiltonian formulation that leads to integration of this (infinitely dimensional) system. Instead we propose to construct hierarchies defined by the coefficient bodies as an object to study. Let f ∈ S. We construct the coefficient bodies by setting Vn = {(b2 , . . . , bn ) : f (ζ ) = ζ +
∞
bk ζ k ∈ S},
n ≥ 2,
k=2
in the (2n − 2)-dimensional Euclidean space. Every point of the boundary of Vn is given by a unique function f ∈ S that maps the unit disk onto the plane minus piecewise analytic Jordan arcs forming a tree with a root at infinity having at most n − 1 tips [25, Chap. 6]. This follows from the application of the Goluzin-Schiffer variational method. Indeed, the extremal function is given as a solution of a complex differential equation in quadratic differentials [25, Chap. 2]. By no means is it soluble explicitly for n > 3, but it presents an important qualitative information: the boundary of the image of the unit disk under the extremal function is a part of the trajectory of the quadratic differential that has the above mentioned structure. Each Vn defines a class {0 } of initial domains for multi-slit subordination evolutions {(t)}, and the corresponding systems of differential equations for coefficients form an integrable system Dn for each n, moreover, its dual system Dn∗ obtained by the characteristic equation admits a Hamiltonian formulation. ∗ ∞ The latter systems form hierarchies {Dn }∞ n=2 and {Dn }n=2 . This idea up to some extent will be proved in the next sections. 4. Coefficient Bodies By the coefficient problem for univalent functions we mean the problem of precisely finding the regions Vn defined above. These compact sets have been investigated by a great number of authors, but the most remarkable source is a famous monograph [25] written by Schaeffer and Spencer in 1950. Among other contributions to the coefficient problem we mention a monograph by Babenko [5] that contains a good collection of qualitative results on the coefficient bodies Vn . The results concerning the structure and properties of Vn include (see [5, 25]): (i) Vn is homeomorphic to a (2n−2)-dimensional ball and its boundary ∂Vn is homeomorphic to a (2n − 3)-dimensional sphere;
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(ii) every point x ∈ ∂Vn corresponds to exactly one function f ∈ S which will be called a boundary function for Vn ; (iii) with the exception for a set of smaller dimension, at every point x ∈ ∂Vn there exists a normal vector satisfying the Lipschitz condition; (iv) there exists a connected open set X1 on ∂Vn , such that the boundary ∂Vn is an analytic hypersurface at every point of X1 . The points of ∂Vn corresponding to the functions that give the extremum to a linear functional belong to the closure of X1 . It is worth noting again that all boundary functions have a similar structure. They map the unit disk U onto the complex plane C minus piecewise analytic Jordan arcs forming a tree with a root at infinity and having at most n − 1 tips, as it has been mentioned in the preceding section. This assertion underlines the importance of multi-slit maps in the coefficient problem for univalent functions. The uniqueness of the boundary functions implies that each point of ∂Vn (the set of first coefficients) defines the rest of the coefficients uniquely. L¨owner’s approach is based on the following idea: a function mapping U onto the complement of a single slit admits univalent dynamics for t ∈ [0, ∞) by represented n the L¨owner equation (1–2) forming a subordination chain f (ζ, t) = ∞ n=1 bn (t)ζ . Let us deduce a system of differential equations for the coefficients of f (ζ, t) substituting the expansion of f (ζ, t) into (1). This gives b˙k = kbk + 2
k−1
j bj e−i(k−j )u ,
bk (0) = ak ,
t ≥ 0,
j =1
n k = 1, 2, . . . , where f (ζ, 0) = f0 (ζ ) = ∞ = et . n=1 an ζ . In particular, b1 (t) n Going to the L¨owner equation in characteristics (5) we write w(z, t) = ∞ n=1 an (t)z and substitute this function in (5). To formulate the result we introduce the following notations a1 (t) 0 0 ... 0 0 ... 0 0 · a1 (t) 0 0 . a(t) = · , A(t) = a2 (t) a1 (t) . . . 0 · ... ... ... ... ... an (t) an−1 (t) an−2 (t) . . . a1 (t) 0 Then the differential equation for a(t) is of the form a˙ = −a − 2
n−1
e−isu As a,
s=1
1 0 · a(0) ≡ a 0 = . · · 0 In particular, a1 (t) = e−t and limt→∞ et ak (t) = ak .
(8)
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5. Hamiltonian Formulation for the Coefficient System. Integrability 5.1. Hamiltonian dynamics and integrability. Let us recall briefly the Hamiltonian and symplectic definitions and concepts that will be used in the sequel. There exists a vast amount of modern literature dedicated to different approaches to the definitions of integrable systems (see, e.g., [3, 4, 7, 28]). The classical definition of a completely integrable system in the sense of Liouville applies to a Hamiltonian system. If we can find independent conserved integrals which are pairwise involutory (vanishing Poisson bracket), this system is completely integrable (see e.g., [3, 4, 7]). That is each first integral allows us to reduce the order of the system not just by one, but by two. We formulate this definition in a slightly adopted form as follows. A dynamical system in C2n is called Hamiltonian if it is of the form x˙ = ∇s H (x),
(9)
where ∇s denotes the symplectic gradient given by
∂ ∂ ∂ ∂ . ,..., ,− ,...,− ∇s = ∂ x¯n+1 ∂ x¯2n ∂x1 ∂xn The function H in (9) is called the Hamiltonian function of the system. It is convenient to redefine the coordinates (xn+1 , . . . , x2n ) = (ψ1 , . . . , ψn ), and rewrite the system (9) as ∂H ∂H , ψ˙ k = − , k = 1, 2 . . . , n. (10) x˙k = ∂xk ∂ψ k The system has n degrees of freedom. The two-form ω = nk=1 dx ∧ d ψ¯ admits the standard Poisson bracket {·, ·},
n ∂f ∂g ∂f ∂g , − {f, g} = ∂xk ∂ψ k ∂ψ k ∂xk k=1
associated with ω. The symplectic pair (C2n , ω) defines the Poisson manifold (C2n , {·, ·}). These notations may be generalized for a symplectic manifold and a Hamiltonian dynamical system on it. The system (10) may be rewritten as x˙k = {xk , H },
ψ˙ k = {ψ k , H },
k = 1, 2 . . . , n,
(11)
and the first integrals of the system are characterized by {, H } = 0.
(12)
In particular, {H, H } = 0, and the Hamiltonian function H is an integral of the system (9). If the system (11) has n functionally independent integrals 1 , . . . , n , which are pairwise involutory {k , j } = 0, k, j = 1, . . . , n, then it is called completely integrable in the sense of Liouville. The function H is included in the set of the first integrals. The classical theorem of Liouville and Arnold [3] gives a complete description of the motion generated by the completely integrable system (11). It states that such a system admits action-angle coordinates around a connected regular compact invariant manifold. One can work with real Poisson manifolds instead, making use of the real Hamiltonian function 2Re H keeping all other formulas changeless.
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If the Hamiltonian system admits only 1 ≤ k < n independent involutory integrals, then it is called partially integrable. The case k = 1 is known as the Poincar´e–Lyapunov theorem which states that a periodic orbit of an autonomous Hamiltonian system can be included in a one-parameter family of such orbits under a nondegeneracy assumption. A bridge between these two extremal cases k = 1 and k = n has been proposed by Nekhoroshev [19] and proved later in [6, 10, 11]. The result states the existence of k-parameter families of tori under suitable nondegeneracy conditions. 5.2. Coefficient system. We will show that the system (8) becomes an integrable system when treated as a description of the boundary hypersurface ∂Vn . Solutions a(t) to (8) for different control functions u(t) (piecewise continuous, in general) being multiplied by the factor et represent all points of ∂Vn as t → ∞. The trajectories et a(t), 0 ≤ t < ∞, fill Vn so that every point of Vn belongs to a certain trajectory et a(t). The endpoints of these trajectories can be interior or else boundary points of Vn . In this way, we set Vn as the closure of the reachable set for the control system (8). According to property (ii) of Vn given in the previous section, every point x ∈ ∂Vn is attained by exactly one trajectory et a(t) which is determined by a choice of the piecewise continuous control function u(t). The function f ∈ S corresponding to x is a multi-slit map of U . If the boundary tree of f has only one tip, then there is a unique continuous control function u(t) in t ∈ [0, ∞) that corresponds to f . The case of multi-slit maps we will consider in the next section. To reach a boundary point x ∈ ∂Vn corresponding to a one-slit map, the trajectory et a(t) has to obey extremal properties, i.e., to be an optimal trajectory. The continuous control function u(t) must be optimal, and hence, it satisfies a necessary condition of optimality. The Maximum Principle is a powerful tool to be used that provides a joint interpretation of two classical necessary variational conditions: the Euler equations and the Weierstrass inequalities (see, e.g., [22]). A control that satisfies the Maximum Principle, i.e., maximizes the real Hamiltonian function, is called optimal. Further on, causing no confusion we will use (. . . )T to denote the matrix transposition. To realize the maximum principle we consider an adjoint vector ψ1 (t) · ψ(t) = · , · ψn (t) with complex valued coordinates ψ1 , . . . , ψn , and the real Hamiltonian function T n−1 H (a, ψ, u) = 2Re −a − 2 e−isu(t) As a ψ¯ , s=1
where ψ¯ means the vector with complex conjugate coordinates. To come to the Hamiltonian formulation for the coefficient system we require that ψ¯ satisfies the adjoint system of differential equations d ψ¯ ∂H =− , dt ∂a
0 ≤ t < ∞.
(13)
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Taking into account (8) we rewrite (13) as
n−1 d ψ¯ −isu(t) T s ¯ = E+2 e (s + 1)(A ) ψ, dt
(14)
s=1
where E is the unit matrix. The maximum principle states that any optimal control function u∗ (t) possesses a maximizing property for the Hamiltonian function along the corresponding trajectory, i.e., max H (a ∗ (t), ψ ∗ (t), u) = H (a ∗ (t), ψ¯ ∗ (t), u∗ ), u
t ≥ 0,
where a ∗ and ψ ∗ are solutions to the system (8, 14) with u = u∗ (t). The maximum principle (15) yields that ∂H (a ∗ (t), ψ ∗ (t), u) ∗ = 0. ∂u u=u (t)
(15)
(16)
Evidently, (8), (13), and (16) imply that d H (a ∗ (t), ψ ∗ (t), u∗ (t)) = 0, dt
(17)
for an optimal differentiable control function u∗ (t). The Hamiltonian formalism for system (8, 14) will lead to integrability. First, we show how ψ can be expressed in terms of the phase variable a. Theorem 1. Let a(t) and ψ(t), ψ(τ ) = (v1 , . . . , vn )T obey the system (8, 13), τ ≥ 0. Then ψ¯ k (t) = cn−k+1 , k = 1, . . . , n, where c1 , . . . , cn are the Taylor coefficients of the expansion ∞
(v¯n z + · · · + v¯1 zn )w (z, τ ) ck (t)zk . = w (z, t) k=1
∞
Proof. Let w(z, t) = k=1 ak (t)zk be a solution to the L¨owner differential equation (5). Differentiating (5) with respect to z we immediately have
iu d z z e +w 2eiu w . (18) = + dt w (z, t) w (z, t) eiu − w (eiu − w)2 Considering the expansion ∞
z = qk (t)zk , w (z, t) k=1
we obtain
n−1 d q(t) −isu(t) s = E+2 e (s + 1)A q, dt s=1
(19)
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where q(t) = (q1 (t), . . . , qn (t))T . We observe that system (19) differs from system (14) only by the transposition sign. In order to satisfy the condition q(τ ) = (v¯1 , . . . , v¯n )T we denote by ∞
g(z, t) =
(v¯n z + · · · + v¯1 zn )w (z, τ ) ck (t)zk , = w (z, t) k=1
and observe that g(z, t) obeys the same equation (18) where z/w (z, t) is substituted by g(z, t). Hence,
c1 (t) · c(t) = · · cn (t) obeys the system (19) substituting q(t) by c(t). It is easily seen that v¯n · c(τ ) = · . · v¯1
The difference in the transposition sign implies that ψ¯ k = cn−k+1 , k = 1, . . . , n. This completes the proof. Putting τ = 0 in this theorem we come to the following corollary. ¯ Corollary 1. Let a(t) and ψ(t), ψ(0) = (v1 , . . . , vn )T obey the system (8, 13), t ≥ 0. Then ψ¯ k (t) = cn−k+1 , k = 1, . . . , n, where c1 , . . . , cn are the Taylor coefficients of the expansion ∞
(v¯n z + · · · + v¯1 zn ) = ck (t)zk . w (z, t) k=1
Remark 1. Since Im a1 (t) = 0, the Hamiltonian function H (a, ψ, u) does not depend on Im ψ1 . Therefore, without loss of generality, we can put ψ1 (0) = v1 to be real. This assumption leave a, ψ2 , . . . , ψn changeless. Now we are able to apply the maximum principle. An optimal continuous control function satisfies the maximizing property (15) and obeys Eq. (16). The Hamiltonian H (a, ψ, u) is a trigonometric polynomial with respect to u of degree n − 1 if ψn = 0. Let ψ(0) = (v1 , . . . , vn )T . Note that ψn (t) = et . We assume that vn = 0, and diminish n to n − 1, otherwise. At t = 0, we have H (a 0 , (v1 , . . . , vn )T , u) = −v1 − 2
n−1 s=1
(Re vs+1 cos(su) + Im vs+1 sin(su)).
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For the optimal u∗ , ∗
Hu (a , (v1 , . . . , vn ) , u ) = 2 0
T
n−1
(s Re vs+1 sin(su∗ ) − s Im vs+1 cos(su∗ )) = 0.
s=1
Suppose that v2 , . . . , vn are such that Huu (a 0 , (v1 , . . . , vn )T , u∗ ) = 2
n−1
(s 2 Re vs+1 cos(su∗ ) + s 2 Im vs+1 sin(su∗ )) = 0,
s=1
for the optimal u∗ . As we see, the term with v1 does not influence the process of maximizing, and we will assume it zero later in course. The inequality holds for all (v2 , . . . , vn ) ∈ R2n−2 except for a set A of dimension at most 2n − 4. Indeed, due to the maximizing property of the optimal control function, this condition breaks down if Huu (a 0 , (v1 , . . . , vn )T , u∗ ) = Huuu (a 0 , (v1 , . . . , vn )T , u∗ ) = 0. Thus, the set A is determined by two linearly independent equations n−1
(s 2 Re vs+1 cos(su∗ ) + s 2 Im vs+1 sin(su∗ )) = 0,
s=1 n−1
(s 3 Re vs+1 sin(su∗ ) − s 3 Im vs+1 cos(su∗ )) = 0,
s=1
with the fixed optimal u∗ , varying v2 , . . . , vn . Therefore, A is a linear space of dimension 2n − 4 in R2n−2 . Definition 1. We say that a vector (v2 , . . . , vn ) satisfies the regularity condition on [0, τ ] if Huu (a ∗ (t), ψ ∗ (t), u∗ ) = 0,
t ∈ [0, τ ],
for the optimal u∗ and ψ ∗ (0) = (v1 , . . . , vn )T along the optimal trajectory (et a ∗ (t), ψ ∗ (t)) corresponding to u∗ . Evidently, if (v2 , . . . , vn ) ∈ R2n−2 \ A, then (v2 , . . . , vn ) satisfies the regularity condition on [0, τ ] for τ > 0 and small enough. Let us denote by Y (τ ) the set of all (v2 , . . . , vn ) satisfying the regularity condition on [0, τ ]. Let (v2 , . . . , vn ) ∈ Y (τ ). Then Huu (a, ψ, u) = 0, for (a, ψ, u) from a neighborhood of The equation
t ∈ [0, τ ],
(a ∗ , ψ ∗ , u∗ )
(20)
and this neighborhood depends on t.
Hu (a, ψ, u) = 0, together with the regularity condition on [0, τ ] determines an analytic implicit function u = u(a, ψ) in a neighborhood of (a ∗ , ψ ∗ ), and u(a, ψ) lies in a neighborhood of u∗ ≡ u(a ∗ , ψ ∗ ).
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We substitute u = u(a, ψ) in the phase system (8) and the adjoint system (13) and solve the Cauchy problem with the initial data a(0) = a 0 , ψ(0) = (v1 , . . . , vn )T . Since u = u∗ along (a ∗ , ψ ∗ ), the Hamiltonian H (a, ψ, u) satisfies the maximum principle. Note that the neighborhood of (a ∗ (t), ψ ∗ (t)) can be determined by a neighborhood of the corresponding vector (v2 , . . . , vn ) = (ψ2 (0), . . . , ψn (0)). According to Corollary 1 the function ψ(t) can be represented as a function of the phase variable a and of the initial conditions (v1 , . . . , vn ), say ψ(t) = ϕ(a(t), v1 , . . . , vn ). Theorem 2. Let (v2 , . . . , vn ) satisfy the regularity condition on [0, τ ]. Then the system (8, 14) with u = u(a, ψ) = u(a, ϕ(a, 0, v2 , . . . , vn )) is partially Liouville integrable. Moreover, the first integrals form a contact structure. Proof. We substitute ψ(t) = ϕ(a(t), v1 , . . . , vn ) in the Hamiltonian and obtain H (a(t), ψ(t), u) = H (a(t), ϕ(a(t), v1 , . . . , vn ), u) = H(a(t), v1 , . . . , vn , u). To each point (v1 , . . . , vn ) from a neighborhood of a vector satisfying the regularity condition, there corresponds an optimal continuous control function. The maximum principle implies that Hu (a(t), v1 , . . . , vn , u) = 0, u=u(a,ϕ(a,v1 ,...,vn ))
that, together with (17), gives H(a(t), v1 , . . . , vn , u(a, ϕ(a, v1 , . . . , vn ))) = const.
(21)
In order to prove the partial integrability we will find the first complex integrals 1 , . . . n . We already have one real integral (21). By the corollary from Theorem 1 we deduce that n k=1
v¯n−k+1 zk = w (z, t)
n k=1
ψ¯ n−k+1 zk + w (z, t)
∞
ck z k .
(22)
k=n+1
We denote by (1 , . . . , n )T the vector of the first integrals of the Hamiltonian system (8, 14) given by ¯ ψ1 1 a1 2a2 . . . (n − 1)an−1 nan ¯ ψ 2 0 a1 . . . (n − 2)an−2 (n − 1)an−1 ¯2 3 = 0 0 . . . (n − 3)an−3 (n − 2)an−2 ψ 3 , ... ... ... ... ... ...... n 0 0 ... 0 a1 ψ¯ n with the control u given in the statement of the theorem. Indeed, the equality (22) implies that k = v¯k are constants for all t and k = 1, . . . , n. The first integral (21) allows us to conclude that {k , H} = 0. One checks that {s , k } = −ss+k−1 for all 1 ≤ s < k and s + k − 1 ≤ n. Otherwise, {s , k } = 0 for s + k − 1 > n. This implies that – [n/2] first integrals ([n/2]+1 , . . . , n ) are pairwise involutory; – the integrals (1 , . . . , [n/2] ) are not pairwise involutory but their Poisson brackets give all the rest of the integrals. This structure is said to be contact (which is not uncoupled because, for example, {1 , n } = −n = 0).
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It is clear from the form of the matrix in the above representation of k , k = 1, . . . , n that all these integrals are algebraically (even linearly) independent. However, the last line becomes trivial after the integration of the equations for a1 and ψn .Actually, a1 (t) = e−t . We introduced this new function a1 (t) without loss of sense in order to make the Hamiltonian formulation easier. Other integrals are not trivial because the differential equations for ak and ψn−k+1 , k = 2, 3 . . . , n contain the function ei(k−1)u(t) with the control u(t). This completes the proof. Remark 2. We observe that H (a0 , (v1 , . . . , vn )T , u∗ ) = −v1 − 2
n−1
(Re vs+1 cos(su∗ ) + Im vs+1 sin(su∗ ))
s=1
at t = 0, which is the constant in the right-hand side of (21). It contains an additive term (−v1 ). The optimal control function u∗ does not depend on v1 . Hence, we can put v1 = 0 without loss of generality. This assumption does not influence a, ψ2 , . . . , ψn , and defines uniquely the constant in the right-hand side of (21). Thus, the first integral (21) can be rewritten as H(a(t), v2 , . . . , vn , u(a, ϕ(a, v2 , . . . , vn ))) = const.
(23)
The integral (23) gives a local parametric description of the boundary of the coefficient body for bounded univalent functions from S (such that f ∈ S, |f (z)| < M, 1 < M ≤ eτ ). The points of this part of the boundary correspond to the functions that map U onto the disk |w| < M slit along an analytic curve with exactly one tip at its interior point. The boundary hypersurface is homeomorphic to an open set on the 2n − 3-dimensional sphere in R2n−2 and must be parameterized by 2n − 3 free parameters. We should select 2n − 3 independent parameters among 2n − 2 variables v2 , . . . , vn . Observe that the Hamiltonian H (a, ψ, u) and the solution ψ to the adjoint system (13) are linear with respect to ψ. The implicit function u = u(a, ψ) is invariant upon multiplying ψ by a positive number. Therefore, both H and ψ may be determined up to a positive multiplier without loss of generality. Putting, e.g., |vn | = 1 we reduce the number of free parameters in (23) to 2n − 3 that gives a local parametric representation of the boundary. The integral (21) is continued from [0, τ ] to a bigger interval [0, τ + ε) as long as the regularity conditions are preserved. Let τ > 0 be the minimal positive number such that the regularity conditions break up at τ for v2 . . . , vn , i.e., Huu (a(τ ), v2 , . . . , vn , u(a(τ ), ϕ(a(τ ), v2 , . . . , vn ))) = 0, Huuu (a(τ ), v2 , . . . , vn , u(a(τ ), ϕ(a(τ ), v2 , . . . , vn ))) = 0. Denote by G1 (t, v2 , . . . , vn ) := Huu (a(t), v2 , . . . , vn , u(a(t), ϕ(a(t), v2 , . . . , vn ))), G2 (t, v2 , . . . , vn ) := Huuu (a(t), v2 , . . . , vn , u(a(t), ϕ(a(t), v2 , . . . , vn ))). In these formulas we admit that a(t) is determined by v2 , . . . , vn according to the phase system (8). The system G1 (t, v2 , . . . , vn ) = G2 (t, v2 , . . . , vn ) = 0 of linearly independent equations defines a (2n − 4)-dimensional manifold in the (2n − 3)-dimensional space of free parameters v2 , . . . , vn , |vn | = 1. Thus, all integrals (23) are continued on t ∈ [0, ∞) for all v2 , . . . , vn except for a (2n − 4)-dimensional set. Using continuity of integrals (23) with respect to v2 , . . . , vn we conclude that the relation (23) holds for all (v2 , . . . , vn ) ∈ R2n−3 and for all t ≥ 0.
Univalent Functions and Integrable Systems
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6. Multi-Slit Evolution Now we generalize the results of the preceding section to the multi-slit evolution. We do this for the maps with the boundary tree having exactly two finite tips because other cases can be easily obtained by analogy. The L¨owner-Kufarev equation (7) has the form
eiu1 (t) + w dw eiu2 (t) + w = −w λ iu (t) + (1 − λ) iu (t) , dt e 1 −w e 2 −w where u1 and u2 are continuous control functions and λ ∈ (0, 1). The differential equation for phase variables (8) becomes a˙ = −a − 2
n−1
(λe−isu1 + (1 − λ)e−isu2 )As a.
(24)
s=1
The Hamiltonian is written as H˜ (a, ψ, u1 , u2 , λ) = Re −a − 2
n−1
T (λe−isu1 + (1 − λ)e−isu2 )As a
ψ¯ ,
s=1
and the adjoint system (14) as
n−1 d ψ¯ −isu1 −isu2 T s ¯ (λe + (1 − λ)e )(s + 1)(A ) ψ. = E+2 dt
(25)
s=1
The maximizing condition (15) now splits into two bits max H˜ (a ∗ (t), ψ ∗ (t), u1 , u2 , λ) = H˜ (a ∗ (t), ψ¯ ∗ (t), u∗1 , u∗2 , λ),
u1 ,u2 ,λ
t ≥ 0,
(26)
where a ∗ and ψ ∗ are solutions to the system (24, 25) with u1 = u∗1 (t), u2 = u∗2 (t), and moreover, T n−1 ∗ ∗ (e−isu1 − e−isu2 )As a ψ¯ = 0. (27) Re s=1
Indeed, if condition (27) were not true the value of the optimal λ would be 0 or 1, and the corresponding boundary function would be a one-slit map that contradicts our supposition. The maximum principle (26) implies that ∂ H˜ (a ∗ (t), ψ ∗ (t), u1 , u2 , λ) = 0, j = 1, 2. (28) ∂uj uj =u∗ (t) j
Evidently, (24), (25), and (28) imply that d H˜ (a ∗ (t), ψ ∗ (t), u∗1 , u∗2 , λ) = 0, dt for optimal differentiable control functions u∗1 (t) and u∗2 (t).
(29)
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Obviously, Theorem 1 and the corollary thereafter are still true in this case and ψ can be expressed in terms of the phase variable a. Continuous optimal control functions u∗1 , u∗2 satisfy the maximizing property (26) and obey Eqs. (28). The Hamiltonian H˜ (a, ψ, u1 , u2 , λ) is a trigonometric polynomial with respect to u1 and u2 of degree n−1 if ψn = 0. Let again ψ(0) = (v1 , . . . , vn )T , and we assume that vn = 0, and diminish n to n − 1, otherwise. We have ψn (t) = et . An additional parameter is λ ∈ (0, 1), which is constant. At t = 0, we have H˜ (a 0 , (v1 , . . . , vn )T , u1 , u2 , λ) = λH (a 0 , (v1 , . . . , vn )T , u1 ) +(1 − λ)H (a 0 , (v1 , . . . , vn )T , u2 ). The vector of parameters is (λ, v1 , . . . , vn ), and we put v1 = 0 by the same reason as in the preceding section. Let us consider (λ, v2 , . . . , vn ) ∈ R2n−1 (or better (0, 1)×R2n−2 ). For the optimal u∗1 and u∗2 , H˜ u1 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ) = H˜ u2 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ) = 0. Suppose that λ, v2 , . . . , vn are such that H˜ u1 u1 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ),
or
H˜ u2 u2 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ)
do not vanish. This holds for all (λ, v2 , . . . , vn ) ∈ B ⊂ (0, 1) × R2n−2 except for a set A˜ of dimension at most 2n − 4. The linear space B is obtained by restriction (27) which, being rewritten at t = 0 for the optimal control functions, is equivalent to H (a 0 , (v1 , . . . , vn )T , u∗1 ) − H (a 0 , (v1 , . . . , vn )T , u∗2 ) = 0. So, the dimension of B is 2n − 2. The set A˜ is determined by two linearly independent equations H˜ u1 u1 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ) = H˜ u1 u1 u1 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ) = 0, or H˜ u2 u2 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ) = H˜ u2 u2 u2 (a 0 , (v1 , . . . , vn )T , u∗1 , u∗2 , λ) = 0, with the fixed optimal u∗1 , u∗2 , varying (λ, v2 , . . . , vn ) in B. Therefore, A˜ is a linear space of dimension 2n − 4 in B ⊂ R2n−2 . Definition 2. We say that a vector (λ, v2 , . . . , vn ) satisfies the regularity condition on [0, τ ] for multi-slit evolution if H˜ u1 u1 (a ∗ , ψ ∗ , u∗1 , u∗2 , λ) = 0,
or H˜ u2 u2 (a ∗ , ψ ∗ , u∗1 , u∗2 , λ) = 0,
t ∈ [0, τ ],
for the optimal u∗1 , u∗2 and ψ ∗ (0) = (v1 , . . . , vn )T along the optimal trajectory (et a ∗ (t), ψ ∗ (t)) corresponding to u∗1 , u∗2 .
Univalent Functions and Integrable Systems
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˜ then (λ, v2 , . . . , vn ) satisfies the regularity condition on If (λ, v2 , . . . , vn ) ∈ B \ A, [0, τ ] for τ > 0 and small enough. Let us denote by Y˜ (τ ) the set of all (λ, v2 , . . . , vn ) satisfying the regularity condition on [0, τ ]. Let (v2 , . . . , vn ) ∈ Y˜ (τ ). Then H˜ u1 u1 (a, ψ, u1 , u2 , λ) = 0,
or H˜ u2 u2 (a, ψ, u1 , u2 , λ) = 0,
t ∈ [0, τ ],
for (a, ψ, u1 , u2 , λ) from a neighborhood of (a ∗ , ψ ∗ , u∗1 , u∗2 , λ) and this neighborhood depends on t. The equations Hu1 (a, ψ, u1 , u2 , λ) = Hu2 (a, ψ, u1 , u2 , λ) = 0, together with the regularity condition on [0, τ ] determine analytic implicit functions u1 = u1 (a, ψ, λ) u2 = u2 (a, ψ, λ) in a neighborhood of (a ∗ , ψ ∗ , λ), and each of u1 = u1 (a, ψ, λ) u2 = u2 (a, ψ, λ) lies in a neighborhood of u∗1 ≡ u1 (a ∗ , ψ ∗ , λ), u∗2 ≡ u2 (a, ψ, λ) respectively. We substitute u1 = u1 (a, ψ, λ), u2 = u2 (a, ψ, λ) in the phase system (24) and the adjoint system (25) and solve the Cauchy problem with the initial data a(0) = a 0 , ψ(0) = (v1 , . . . , vn )T , (λ, v2 , . . . , vn ) ∈ B. According to Corollary 1 the function ψ(t) can be represented as a function of the phase variable a and the initial conditions (v1 , . . . , vn ), say ψ(t) = ϕ(a(t), v1 , . . . , vn , λ). As in the preceding section, both H˜ and ψ may be determined up to a positive multiplier without loss of generality. Putting, e.g., |vn | = 1 we reduce the number of free parameters to 2n − 3 that gives a local parametric representation of the boundary. Repeating all further steps of the preceding section we prove the following theorem. Theorem 3. Let (λ, v2 , . . . , vn ) satisfy the regularity condition on [0, τ ]. Then the system (24) with u1 = u1 (a, ψ, λ) = u1 (a, ϕ(a(t), v1 , . . . , vn , λ), λ), u2 = u1 (a, ψ, λ) = u2 (a, ϕ(a(t), v1 , . . . , vn , λ), λ), is partially Liouville integrable. Moreover, this statement is continued for all (λ, v2 , . . . , vn ) ∈ B and for all t ≥ 0. Theorems 2,3 also solve the problem of integrability for the corresponding coefficient systems in Sect. 4 substituting the optimal control functions in the Cauchy problem. One can find exact solutions in explicit form for n = 3 in, e.g., [2] and [26]. References 1. Agam, O., Bettelheim, E., Wiegmann, P., Zabrodin, A.: Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime. Phys. Rev. Lett. 88, 236801 (2002) 2. Aleksandrov, I.A.: Parametric continuations in the theory of univalent functions. Moscow: Nauka, 1976 (in Russian) 3. Arnold, V.I.: Mathematical methods of classical mechanics. New York: Springer-Verlag, 1989 4. Babelon, O., Bernard, D., Talon, M.: Introduction to classical integrable systems. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press, 2003 5. Babenko, K.I.: The theory of extremal problems for univalent functions of class S. Proc. Steklov Inst. Math., No. 101 (1972). Transl. American Mathematical Society, Providence, R.I.: Amer. math. soc., 1975 6. Bambusi, D., Gaeta, G.: On persistence of invariant tori and a theorem by Nekhoroshev. Math. Phys. Electron. J. 8, Paper 1, 13 pp (2002)
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7. Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian systems. Geometry, topology, classification. Boca Raton, FL: Chapman & Hall/CRC, 2004 ¨ 8. Bieberbach, L.: Uber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. S.-B. Preuss. Akad. Wiss. S.940–955 (1916) 9. de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154 , no. 1–2, 137–152 (1985) 10. Fiorani, E., Giachetta, G., Sardanashvily, G.: The Liouville-Arnold-Nekhoroshev theorem for noncompact invariant manifolds. J. Phys. A 36, no. 7, L101–L107 (2003) 11. Gaeta, G.: The Poincaré-Lyapounov-Nekhoroshev theorem. Ann. Physics 297, no. 1, 157–173 (2002) 12. Goluzin, G.M.: Geometric theory of functions of a complex variable. Transl. Math. Monographs, Vol 26, Providence, RI: AMS, 1969 13. Hohlov, Yu.E., Howison, S.D., Huntingford, C., Ockendon, J.R., Lacey, A.A.: A model for nonsmooth free boundaries in Hele-Shaw flows. Quart. J. Mech. Appl. Math. 47, 107–128 (1994) 14. Howison, S.D.: Complex variable methods in Hele-Shaw moving boundary problems. European J. Appl. Math. 3, no. 3, 209–224 (1992) 15. Kostov, I.K., Krichever, I., Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: The τ -function for analytic curves. In: Random matrix models and their applications, Math. Sci. Res. Inst. Publ. 40, Cambridge: Cambridge Univ. Press, 2001, pp. 285–299 16. Kufarev, P.P.: On one-parameter families of analytic functions. Rec. Math. [Mat. Sbornik] N.S. 13(55), 87–118 (1943) 17. L¨owner, K.: Untersuchungen u¨ ber schlichte konforme Abbildungen des Einheitskreises. Math. Ann. 89, 103–121 (1923) 18. Marshakov, A., Wiegmann, P., Zabrodin, A.: Integrable structure of the Dirichlet boundary problem in two dimensions. Commun. Math. Phys. 227, no. 1, 131–153 (2002) 19. Nekhoroshev, N.N.: The Poincar´e-Lyapunov-Liouville-Arnol’d theorem. Funkt. Anal. i Pril. 28, no. 2, 67–69 (1994); translation in Funct. Anal. Appl. 28, no. 2, 128–129 (1994) ¨ 20. Pommerenke, Ch.: Uber die Subordination analytischer Funktionen. J. Reine Angew. Math. 218, 159–173 (1965) 21. Pommerenke, Ch.: Univalent functions, with a chapter on quadratic differentials by G. Jensen. G¨ottingen: Vandenhoeck & Ruprecht, 1975 22. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The mathematical theory of optimal processes. New York-London: Interscience Publishers/John Wiley & Sons, Inc., 1962 23. Prokhorov, D.: Sets of values of systems of functionals in classes of univalent functions. Mat. Sb. 181, no. 12, 1659–1677 (1990); translation in Math. USSR-Sb. 71, no. 2, 499–516 (1992) 24. Richardson, S.: Hele-Shaw flows with a free boundary produced by the injecton of fluid into a narrow channel. J. Fluid Mech., 56, no. 4, 609–618 (1972) 25. Schaeffer, A.C., Spencer, D.C.: Coefficient regions for schlicht functions (with a chapter on the region of the derivative of a schlicht function by Arthur Grad). American Mathematical Society Colloquium Publications, Vol. 35. New York: American Mathematical Society, 1950 26. Vasil’ev, A.: Mutual change of initial coefficients of univalent functions. Matemat. Zametki 38, no. 1, 56–85 (1985); translation in Math. Notes 38, no. 1–2, 543–548 (1985) 27. Wiegmann, P.B., Zabrodin, A.: Conformal maps and integrable hierarchies. Commun. Math. Phys. 213, no. 3, 523–538 (2000) 28. Zakharov, V.E.(ed.): What is integrability? Springer Series in Nonlinear Dynamics. Berlin: SpringerVerlag, 1991 Communicated by L. Takhtajan
Commun. Math. Phys. 262, 411–457 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1475-6
Communications in
Mathematical Physics
The Map Between Conformal Hypercomplex/ Hyper-K¨ahler and Quaternionic(-K¨ahler) Geometry Eric Bergshoeff1 , Sorin Cucu2 , Tim de Wit1 , Jos Gheerardyn2,3 , Stefan Vandoren4 , Antoine Van Proeyen2 1
Center for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands. E-mail: [email protected], [email protected] Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium. E-mail: [email protected], [email protected] 3 Dipartimento di Fisica Teorica, Universit`a di Torino, and I.N.F.N., Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy. E-mail: [email protected] 4 Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3508 TA Utrecht, The Netherlands. E-mail: [email protected]
2
Received: 9 December 2004 / Accepted: 19 July 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by ‘ξ -transformations’, relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-K¨ahler manifolds is mapped to quaternionic-K¨ahler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quaternionic-like Manifolds . . . . . . . . . . . . . . . . . . . . 2.1 General properties of quaternionic-like manifolds . . . . . . 2.2 The ξ -transformations . . . . . . . . . . . . . . . . . . . . . 2.3 Hypercomplex manifolds . . . . . . . . . . . . . . . . . . . 2.4 Hyper-K¨ahler manifolds . . . . . . . . . . . . . . . . . . . 2.5 Quaternionic-K¨ahler manifolds . . . . . . . . . . . . . . . . 2.6 Hermitian Ricci tensor . . . . . . . . . . . . . . . . . . . . 3. The Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hypercomplex manifolds with conformal symmetry . . . . . 3.2 The map from hypercomplex to quaternionic . . . . . . . . . 3.2.1 Suitable coordinates and almost complex structures. 3.2.2 The complex structures and Obata connection. . . . 3.3 The embedded quaternionic space . . . . . . . . . . . . . . 3.3.1 Proof that the small space is quaternionic. . . . . . .
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3.3.2 The local SU(2). . . . . . . . . . . . . . . . . . . . . . . . The map from quaternionic to hypercomplex . . . . . . . . . . . . . 3.4.1 Uplifting a quaternionic manifold. . . . . . . . . . . . . . . 3.4.2 A ξ -transformation for conformal hypercomplex manifolds. 3.5 The map from hyper-K¨ahler to quaternionic-K¨ahler spaces . . . . . 3.5.1 Decomposition of a hyper-K¨ahler metric. . . . . . . . . . . 3.5.2 The inverse map. . . . . . . . . . . . . . . . . . . . . . . . 3.6 The map for the vielbeins and related connections . . . . . . . . . . 3.6.1 Coordinates on the tangent space. . . . . . . . . . . . . . . 3.6.2 Connections on the hypercomplex space. . . . . . . . . . . 3.6.3 Connections on the quaternionic space. . . . . . . . . . . . Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reduction of the Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Symmetries and moment maps . . . . . . . . . . . . . . . . . . . . 5.2 Conformal hypercomplex . . . . . . . . . . . . . . . . . . . . . . . 5.3 The map of the symmetries . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Conformal hypercomplex and quaternionic symmetries. . . 5.3.2 Isometries on conformal hyper-K¨ahler and quaternionic-K¨ahler spaces . . . . . . . . . . . . . . . . . . 5.3.3 The conformal hyper-K¨ahler manifold of quaternionic dimension 1 . . . . . . . . . . . . . . . . . Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The map from hypercomplex manifolds . . . . . . . . . . . . . . . 6.2 The quaternionic manifold . . . . . . . . . . . . . . . . . . . . . . 6.3 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projections Depending on the Complex Structure . . . . . . . . . . . . . 3.4
4. 5.
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1. Introduction Ever since Einstein’s theory of gravity, differential geometry has entered research areas in theoretical physics in various places and contexts. The idea of using geometry to describe and unify the forces of nature continues to play an important role in present day high energy physics. After the invention of supersymmetry, superstrings and their compactifications, connections were found with the holonomy groups and Killing spinors of Riemannian manifolds. It was shown that supersymmetric sigma models were deeply related to geometries with complex structures. K¨ahler manifolds appear already in N = 1 theories in 4 dimensions [1]. Some theories with 8 supersymmetries, i.e. N = 4 in 2 dimensions and N = 2 in 3, 4, 5 or 6 dimensions, exhibit a hypercomplex structure. It was shown that Lagrangians for rigid supersymmetry lead to hyper-K¨ahler [2] manifolds and Lagrangians for supergravity theories lead to quaternionic-K¨ahler manifolds [3]. Many aspects of such geometries are by now well-known and studied by the supersymmetry, supergravity and superstring community, and this has led to new insights and useful applications in these fields. Recently, it has been shown that a generalization of hyperK¨ahler manifolds is possible for rigid N = 2 supersymmetric theories if one does not demand that the field equations are derivable from an action [4,5]. Such a situation arises when in the field equations for the scalar fields, a connection is chosen different from a Levi-Civita connection derivable from a metric on the sigma model target manifold. The
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Table 1. Quaternionic-like manifolds of real dimension 4r. These are the manifolds that have a quaternionic structure satisfying (1.2) and (1.4). The holonomy group is indicated. Generalizations for different signatures are obvious. The table is essentially taken over from [7]. The dot notation means e.g. SU(2) · USp(2r) = SU(2)×USp(2r) Z 2
no SU(2) curvature non-zero SU(2) curvature
no good metric hypercomplex G (r, H) quaternionic SU(2) · G (r, H) field equations
with a good metric hyper-K¨ahler USp(2r) quaternionic-K¨ahler SU(2) · USp(2r) action
rigid supersymmetry supergravity
corresponding geometry is called hypercomplex [6], which differs from a hyper-K¨ahler geometry by the fact that it does not necessarily possess a covariantly constant metric preserved by the connection induced by the hypercomplex structure, as we explain in detail below. We will further refer to a Hermitian metric that is covariantly constant with respect to the latter connection as a ‘good metric’. The manifolds mentioned above are schematically represented in Table 1. There it is indicated that hypercomplex and hyper-K¨ahler manifolds occur in rigid supersymmetric theories, while quaternionic and quaternionic-K¨ahler manifolds occur in supergravity. Manifolds without a good metric occur as long as the field equations are not derived from a conventional action. The conventional actions in supersymmetry and supergravity involve a metric on the manifold of scalars, and the field equations naturally involve the Levi-Civita connection. Hypercomplex and quaternionic manifolds are endowed with a connection that is not necessarily the Levi-Civita connection. For this reason we say they are not derived from an action. The supersymmetry algebra on hypermultiplets requires that we use torsionless connections. However, hyper-K¨ahler torsion (HKT) manifolds, which appear as the moduli space of supersymmetric multi-black holes [8, 9], belong also to the class of hypercomplex manifolds. Indeed, they carry a triplet of complex structures with vanishing Nijenhuis tensor, implying that there exists a torsionless connection. This is similar to the way in which we used two-dimensional non-linear sigma models on group manifolds [10] for the example in [4, Appendix C]. Also there, the connection with torsion corresponding to the three-form field strength is different from the torsionless connection used in the definition of hypercomplex manifolds. Supergravity theories can be constructed by the ‘superconformal tensor calculus’[11– 13], which gives insight in the geometry of the relevant sigma models. These geometries are obtained as projective spaces, related to the dilatation symmetry in the superconformal group, and with possible further projections related to the R-symmetry group, see [14] for a review of these principles. In particular, this has been used in [15–18] in the context of N = 2 theories with hypermultiplets, constructing the link between hyperK¨ahler manifolds and quaternionic-K¨ahler manifolds. This construction starts from a conformal hyper-K¨ahler manifold. We will use the name conformal manifold for a manifold which has a closed homothetic Killing vector. For a manifold with coordinates q X and affine connection XY Z , this is a vector k X , satisfying1 DY k X ≡ ∂Y k X + Y Z X k Z = 23 δY X .
(1.1)
1 The factor 3/2 is a choice of normalization which is convenient for applications of 5-dimensional supergravity theories. The translation between formulations appropriate to supergravities in other dimensions has been considered in [19].
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The presence of such a vector allows the definition of conformal symmetries as a basic step for the superconformal tensor calculus [20]. In this paper, we will formulate and prove the 1-to-1 correspondence2 (locally) between conformal hypercomplex manifolds of quaternionic dimension nH + 1 and quaternionic manifolds of dimension nH . Furthermore, we show that this 1-to-1 correspondence is also applicable between the subset of hypercomplex manifolds that are hyper-K¨ahler and the subset of quaternionic manifolds that are quaternionic-K¨ahler. In the mathematics literature the map between quaternionic-K¨ahler and hyper-K¨ahler manifolds is constructed by Swann [21], and its generalization to quaternionic manifolds is treated in [22]. Here, we give explicit expressions for the complex structures and connections that are needed to apply these results in the context of the conformal tensor calculus in supergravity. To explain the 1-to-1 mapping let us first repeat the basic definitions. A hypercomplex manifold is a manifold with a hypercomplex structure. This means that in any local patch3 there are coordinates q X with X = 1, . . . , 4r and a triplet of complex structures JX Y that satisfy the algebra of the imaginary quaternions, which is that for any vectors A and B, A · J B · J = −
· B 4r A
· J, + (A × B)
Tr J = 0.
(1.2)
This defines an almost hypercomplex structure.4 The closure of the (rigid) supersymmetry algebra on hypermultiplets requires that the complex structures are covariantly constant using a torsionless affine connection XY Z = Y X Z : DX JY Z ≡ ∂X JY Z − XY W JW Z + XW Z JY W = 0.
(1.3)
Stated otherwise, the hypercomplex structures should be integrable.5 A quaternionic manifold is defined by a local span of these three complex structures. This means that the three complex structures can at any point q be rotated as J = R(q)J, where R is a 3 × 3 matrix of SO(3). The covariant constancy condition (i.e. the integrability) should then also be covariant with respect to these rotations. This implies that one needs a connection ω X and the condition (1.3) is replaced by DX JY Z ≡ ∂X JY Z − XY W JW Z + XW Z JY W + 2 ω X × JY Z = 0.
(1.4)
Infinitesimal SO(3) rotations are parametrized by 3 angles (q), δSU(2) JX Y = × JX Y ,
X. δSU(2) ω X = − 21 ∂X + × ω
(1.5)
The connections in (1.4) are not unique. Indeed, they can be changed simultaneously depending on an arbitrary one-form ξ = ξX dq X as [7, 23, 24] Z XY Z → XY Z + 2δ(X ξY ) − 2J(X Z · JY ) W ξW ,
ω X → ω X + JX W ξW . (1.6)
2 As will be explained below, the correspondence is actually 1-to-1 between families (or ‘equivalence classes’) of manifolds. 3 We use the integer r for the quaternionic dimension of any quaternionic-like manifold (the number of hypermultiplets). In the application to the map, this r can be nH or nH + 1 depending on whether we are considering the quaternionic space or the hypercomplex space, respectively. 4 Note that the tracelessness is implied by the first relation. 5 In the context of G-structures, condition (1.3) would rather be called the one-integrability of the hypercomplex structure [7].
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These transformations relate different connections on the same manifold and therefore define equivalence classes. Furthermore, it will turn out that for the conformal hypercomplex manifolds there are related transformations between complex structures and connections. These define equivalence classes between conformal hypercomplex structures. There is however one important difference between the transformations on the quaternionic and on the hypercomplex space. On the quaternionic manifold, the ξ -transformation yields different torsionless connections for a given (integrable) quaternionic structure (i.e. for a given span of three complex structures). Otherwise stated, an integrable quaternionic structure does not determine the connection uniquely. On the conformal hypercomplex space, the transformations relate different (integrable) hypercomplex structures in a continuous fashion. To the best of our knowledge, these transformations between hypercomplex structures on conformal manifolds have not yet been discussed in the literature. The map from conformal hypercomplex to quaternionic manifolds is subject to these transformations, and is 1-to-1 for the equivalence classes. We give an explicit construction of this correspondence by relating the geometric quantities in the corresponding manifolds, i.e. the complex structures, affine connections, curvatures, Killing vectors and moment maps. We prove that the existence of a ‘good metric’ for a hypercomplex manifold is equivalent to the existence of a ‘good metric’ in the corresponding quaternionic space. Also the symmetries of related manifolds are mapped 1-to-1. Our formulation here focuses purely on the geometrical aspects. Our results have been applied already to N = 2 matter coupled D = 5 supergravity [25]. The results of this paper are applicable to the geometry of hypermultiplets independent of whether they are defined in 3, 4, 5 or 6 dimensions. In Sect. 2 we give a summary of the properties of the geometries with a triplet of complex structures (‘quaternionic-like manifolds’). In particular we discuss the curvatures determining the holonomy groups. We will devote special attention to the ξ -transformations and to the properties of the Ricci tensor. At the end of that section we will look at those manifolds that have a Hermitian Ricci tensor, which include all hypercomplex, hyper-K¨ahler and quaternionic-K¨ahler manifolds. Most results in this section are due to [7]. The new work starts in Sect. 3, where we construct the map discussed above. We start by considering conformal hypercomplex manifolds and identify new transformations between connections and the hypercomplex structure that respect the conditions for hypercomplex manifolds. These spaces are then reduced to a submanifold that turns out to be quaternionic. Coordinates are chosen in view of the gauge fixing of dilatation, SU(2) symmetry and special (S) supersymmetry in the superconformal context. We construct the geometric building blocks in this suitable basis. These are the complex structures and affine connections. The freedom of ξ -transformations comes naturally out of this map. Inversely, we associate such a conformal hypercomplex manifold to any quaternionic manifold. This finishes the proof that the mapping between these manifolds is 1-to-1. In a further subsection, we focus on the manifolds with a good metric, i.e. hyper-K¨ahler and quaternionic-K¨ahler manifolds. We show that these are also 1-to1 related and give explicit expressions for the connections. Finally, in this section we also construct the vielbeins and the spin connections (i.e. the connections that transform under local general linear quaternionic transformations). The relation between curvatures of the conformal hypercomplex and the quaternionic manifolds is discussed in Sect. 4. As recalled in Sect. 2, the curvatures of quaternionic-like manifolds are characterized by a symmetric ‘Weyl tensor’ WABC D , where
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A = 1, . . . , 2r are indices in the tangent space. We will therefore connect the Weyl tensors of the hypercomplex and quaternionic spaces. Symmetries of the manifolds that preserve also the hypercomplex structure are called triholomorphic. Such symmetries of conformal hypercomplex manifolds descend to quaternionic symmetries of quaternionic manifolds (which is a statement of preservation of the span of complex structures). We define the moment maps of quaternionic manifolds, and we show in Sect. 5 that these are determined in terms of components of the symmetries of the hypercomplex manifolds in directions that are projected out by the gauge fixing. We summarize and overview all results of this paper in Sect. 6. We have written this in such a way that this section can be read by itself by a reader familiar with quaternionic geometry (that is recapitulated in Sect. 2). We illustrate the results with a schematic picture, Fig. 1. After this summary, we discuss some remaining issues and give some remarks. In the main part of the paper, we choose the signatures in the way that is most relevant for supergravity, i.e. such that the scalars and the graviton have positive kinetic energies. This selects quaternionic-K¨ahler manifolds with a positive definite metric and a negative definite scalar curvature. These are obtained in the map by starting with hyperK¨ahler manifolds of quaternionic signature (− + · · · +). In the discussion section, we indicate how our formulae can be used for other signatures. An appendix gives some useful formulae for calculus with complex structures and Hermitian tensors. 2. Quaternionic-like Manifolds In this section, we review the four different geometries that are used in this paper. They correspond to hypercomplex, hyper-K¨ahler, quaternionic and quaternionic-K¨ahler manifolds, and are distinguished by the properties of their holonomy groups and the presence of a preserved metric, as summarized in Table 1. This section is an extension of Appendix B of [4], a paper where these geometries are discussed in the context of five dimensional conformal hypermultiplets, and which itself makes heavy use of the pioneering paper [7]. We use here the name ‘quaternionic-like manifolds’ for all the manifolds in Table 1. In fact, the formulae for quaternionic manifolds are the most general ones, and in this respect one can argue to just use ‘quaternionic manifolds’ as general terminology. The subtlety is that in principle for hypercomplex manifolds one admissible basis of quaternionic structures is selected, while in quaternionic manifolds only the local span of complex structures is used. For all practical purposes, the formulae of quaternionic manifolds are the most general, and can be applied with ω X = 0 for hypercomplex manifolds. Subsection 2.1 gives these general formulae for an arbitrary quaternionic manifold. The properties of the curvatures are presented. The ξ -transformations mentioned in the introduction are treated in more detail in Subsect. 2.2. Special features for the case of hypercomplex manifolds are given in Subsect. 2.3, for hyper-K¨ahler manifolds in Subsect. 2.4 and for quaternionic-K¨ahler manifolds in Subsect. 2.5. In these 3 cases, the Ricci tensor is Hermitian. We give general properties of quaternionic-like manifolds with Hermitian Ricci tensor in Subsect. 2.6. 2.1. General properties of quaternionic-like manifolds. The common property of all quaternionic-like manifolds, say of dimension 4r, is the existence of a quaternionic
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structure, i.e. a triplet of endomorphisms J, realizing the algebra of the imaginary quaternions (1.2). In general relativity, it is often convenient to introduce (locally) a set of one-forms that define an orthonormal frame: ea = eµ a dx µ . Their components eµ a are the so-called ‘vielbeins’. As such, the so-called flat index a takes values in the orthogonal group, and µ is the one-form index, also known as the curved index. In the present context, the orthogonal group is substituted by the groups mentioned in Table 1. Objects with flat indices will be said to take values in the tangent bundle. Therefore, we locally introduce the coordinates q X with X = 1, . . . , 4r, and we assume the existence of an invertible vielbein fXiA , two matrices ρA B and Ei j (with i = 1, 2 and A = 1, . . . , 2r) that satisfy ρ ρ∗ = −
E E∗ = −
2r ,
2,
(2.1)
together with the reality condition jB
(fXiA )∗ = fX Ej i ρB A .
(2.2)
One can choose a basis such that Ei j = εij , see e.g. [26]. We will further always use this basis. The transformations on variables with an A index are restricted by the reality condiX , satisfies tion to G (r, H) = SU∗ (2r)×U(1). The inverse vielbein, denoted by fiA X fYiA fiA = δYX ,
fXiA fjXB = δji δBA ,
(2.3)
and can be used to define the quaternionic structure as JX Y ≡ −ifXiA σi j fjYA ,
(2.4)
where σ are the three Pauli matrices. These matrices J satisfy (1.2). We use here a slight change of notation with respect to [4] in that we will indicate the triplets by a vector symbol, rather than an α index in [4] (which will be needed below to indicate a coordinate set). All quaternionic-like manifolds admit connections with respect to which the vielbeins are covariantly constant by definition. These are a torsionless affine connection XY Z , a G (r, H) connection ωXA B on the tangent bundle and possibly an SU(2) connection ωXi j , such that jA
DX fYiA ≡ ∂X fYiA − XY Z fZiA + fY ωXj i + fYiB ωXB A = 0,
(2.5)
and similarly for the inverse vielbein. This implies that the quaternionic structure is covariantly constant with respect to the affine connection and the SU(2) connection6 , DX JY Z ≡ ∂X JY Z − XY W JW Z + XW Z JY W + 2 ω X × JY Z = 0.
(2.6)
The G (r, H)-connection ωXA B on the tangent bundle can then be determined by requiring (2.5): Y Z ωXA B = 21 fYiB ∂X fiA (2.7) + XZ Y fiA − ωXi j fjYA . 6 One can make the transition from doublet to vector notation by using the sigma matrices, ω j = Xi X , and similarly ω X = − 21 i σi j ωXj i . This transition between doublet and triplet notation is valid i σi j · ω for any triplet object as e.g. the complex structures.
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A useful concept in describing the integrability of almost quaternionic-like structures is the Nijenhuis tensor. We will use only the ‘diagonal’ Nijenhuis tensor (normalization for later convenience) NXY Z ≡ 16 J[X W · ∂|W | JY ] Z − ∂Y ] JW Z . (2.8) A quaternionic manifold is defined by requiring that the Nijenhuis tensor satisfies (1 − 2 r) NXY Z = −J[X Z NY ]V W · JW V ,
(2.9)
which is equivalent to requiring that NXY Z = −J[X Z · ω Y ] , Op
(2.10)
Op Op Op for some ω X . Since the trace of the Nijenhuis tensor vanishes, ω X obeys JX Y · ω Y = 0. Op Hence, (2.9) is automatically satisfied. The condition (2.10) can be solved for ω X , and used to define an SU(2) connection Op (1 − 2 r) ω X = NXY Z JZ Y .
(2.11)
The condition (2.9) ensures that there exists an appropriate affine connection such that (2.6) is satisfied. This connection is called the Oproiu connection [27], Op Op XY Z ≡ Ob XY Z − J(X Z · ω Y ) , Ob XY Z ≡ − 16 2∂(X JY ) W + J(X U × ∂|U | JY ) W · JW Z .
(2.12) (2.13)
Ob XY Z is the Obata connection, which is the solution if ω X = 0. Conversely, any two connections XY Z and ω X that satisfy (2.6), necessarily imply the condition (2.9) on the Nijenhuis tensor. Moreover, as shown in [4], they must be related to the connections defined by (2.11) and (2.12) by means of the ξ -transformations mentioned in (1.6). We proceed by discussing the curvature tensor of quaternionic manifolds. We first give our conventions: RXY Z W ≡ 2∂[X Y ]Z W + 2V [X W Y ]Z V , RXY B A ≡ 2∂[X ωY ]B A + 2ω[X|C| A ωY ]B C , XY ≡ 2∂[X ω R Y ] + 2ω X × ω Y .
(2.14)
The integrability condition of (2.5) implies that the total curvature on the manifold is the sum of the SU(2) curvature and the G (r, H) curvature. This shows that the (restricted) holonomy splits in these two factors:7 RXY W Z = R SU(2) XY W Z + R G (r,H) XY W Z XY + LW Z A B RXY B A , = −JW Z · R
(2.15)
7 This follows also from the Ambrose-Singer theorem [28], which says that the Lie algebra of the restricted holonomy group of the frame bundle coincides with the algebra generated by the curvature. The direct product structure of the holonomy group is then reflected in these relations.
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where matrices LA B appear. They are defined in Appendix A, where also various useful properties are exhibited. This implies XY = RXY Z W JW Z , 4r R
RXY A B = 21 LW Z A B RXY Z W .
(2.16)
Furthermore, we define the Ricci tensor as RXY = RZXY Z .
(2.17)
For an arbitrary affine connection, it has both a symmetric and antisymmetric part. In general, the antisymmetric part can be traced back to the curvature of the R part in G (r, H) = S (r, H) × R. Indeed, using the cyclicity condition, we find R[XY ] = RZ[XY ] Z = − 21 RXY Z Z = −RR XY ,
A RR XY ≡ RXY A .
(2.18)
Therefore, the antisymmetric part of the Ricci tensor follows completely from the R part. The separate curvature terms in (2.15) do not satisfy the cyclicity condition (unless the SU(2) curvature vanishes), and thus are not bona-fide curvatures. Another splitting of the full curvature can be made where both terms separately satisfy the cyclicity condition, RXY W Z = R Ric XY W Z + R (W) XY W Z .
(2.19)
The first part only depends on the Ricci tensor of the full curvature, and is called the ‘Ricci part’. It is defined by R Ric XY Z W ≡ 2δ[X W BY ]Z − 2δZ W B[XY ] − 4JZ (W · J[X V ) BY ]V ,
(2.20)
where [29] 1 Z W 1 1 BXY ≡ δX δY −XY ZW R(ZW ) + XY ZW R(ZW ) + R[XY ] . 4r 4(r + 2) 4(r + 1) (2.21) Here, we have introduced a projection operator XY ZW ≡ 41 δX Z δY W + JX Z · JY W ,
(2.22)
whose properties are discussed in the Appendix. The symmetric part of BXY can be considered as the candidate for a ‘good metric’ for quaternionic manifolds. Indeed, if there is a good metric, then it is proportional to the symmetric part of this tensor as we will see below in (2.57). The Ricci part does satisfy the cyclicity property, and its Ricci tensor is just RXY . We can further split it as W Ric Ric R Ric XY Z W = Rsymm + Rantis , (2.23) XY Z
where the first term is the construction (2.20) using only the symmetric part of B (the symmetric part of the Ricci tensor), and the second term uses only the antisymmetric part of B (i.e. of the Ricci tensor).
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The second term in (2.19) is defined as the remainder, and its Ricci tensor is zero. For this reason, it is called the ‘Weyl part’ [7]. Following again the discussion in [4], one can rewrite the Weyl part in terms of a symmetric and traceless tensor WABC D , such that8 jB
W RXY Z W = R Ric XY Z W − 21 fXiA εij fY fZkC fkD WABC D ,
(2.24)
Y Z kA (W) WCDB A ≡ 21 ε ij fjXC fiD fkB fW R XY Z W .
(2.25)
with
The G (r, H) curvature can also be decomposed in its Ricci and Weyl part: RXY A B = RRic XY A B + R(W) XY A B , RRic XY A B ≡ 21 LW Z A B R Ric XY Z W = 2δA B B[Y X] + 4L[X V A B BY ]V , jD
R(W) XY A B ≡ 21 LW Z A B R (W) XY Z W = −fXiC εij fY WCDA B .
(2.26)
On the other hand, the SU(2) curvature is determined only by the Ricci tensor, or, equivalently, by the tensor BXY : XY = 2J[X Z BY ]Z . R
(2.27)
We can summarize the different curvature decompositions in the following scheme: Ric Ric RXY Z W = Rsymm + Rantis + R (W) XY Z W (2.28) = R SU(2) +
RR
+ R S (r,H)
XY Z
W.
The terms in the second line depend only on specific terms of the first line as indicated by the arrows. This is the general scheme and thus applicable for quaternionic manifolds, which is the general case, but for specific other quaternionic-like manifolds some parts are absent as can be seen in Table 2. 2.2. The ξ -transformations. The requirement (1.4) for a fixed complex structure does not determine the connections uniquely. Indeed, the affine and SU(2) connections can be changed simultaneously depending on an arbitrary one-form ξ = ξX dq X as ˜ XY Z = XY Z + SXY W Z ξW ,
˜ X = ω ω X + JX W ξW .
(2.29)
Here we have introduced the S-symbols, which can be read of from (1.6). In terms of the projection operator in (2.22), they are Z W Z W SXY ZW ≡ 2δ(X δY ) − 2JX (Z · JY W ) = 4δ(X δY ) − 8(XY ) ZW .
(2.30)
Further properties are given in (A.13)–(A.17). 8 The case of four-dimensional quaternionic manifolds must be treated separately, but they are defined such that (2.24) is still satisfied. See [4] for more details.
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Table 2. The curvatures in quaternionic-like manifolds. The first line gives the decomposition according to Ricci and Weyl curvatures, while the second line gives the decomposition in accordance with the holonomy groups hypercomplex Ric + R (W) Rantis
R R + R S (r,H) quaternionic Ric + R Ric + R (W) Rsymm antis
R SU(2) + R R + R S (r,H)
hyper-K¨ahler R (W) R S (r,H) quaternionic-K¨ahler Ric + R (W) Rsymm R SU(2) + R S (r,H)
Obviously, since the complex structures do not transform, the Nijenhuis tensor is invariant, and so the geometry remains quaternionic. When transforming the affine and the SU(2) connection as in (2.29), also the connection ωXA B transforms, according to its value in (2.7): ω˜ XA B = ωXA B + 21 LY Z A B SXZ Y W ξW .
(2.31)
Note in particular that the R connection transforms as ω˜ XA A = ωXA A + 2(r + 1)ξX .
(2.32)
Clearly, all curvatures will transform under these deformations of the connections, with terms at most quadratic in ξX . A direct computation shows that the full Riemann curvature transforms as ˜ = RXY Z W () + 2SZ[Y W U DX] ξU + 2ST [X W U SY ]Z T V ξU ξV . (2.33) RXY Z W () Defining furthermore9 ηXY ≡ −DX ξY + 21 S(XY ) U V ξU ξV = −D˜ X ξY − 21 SXY U V ξU ξV ,
(2.34)
we find ˜ = RXY () + 4rη(XY ) + 8(XY ) U V ηU V − 4(r + 1)∂[X ξY ] , RXY () XY (ω) XY (ω) R ˜ = R − 2J[Y Z ηX]Z , ˜ = BXY () + ηXY . BXY ()
(2.35)
Using the last expression, one finds that the Ricci part, (2.20), transforms as the full curvature (2.33). Therefore, the Weyl part, R (W) , does not transform, and the W-tensor is invariant. The ξ -transformations can be used to fix the form of the R connection. Indeed, looking at (2.32), we see we can transform away the R connection completely. The R curvature then vanishes. Furthermore, there are residual ξ -transformations depending on a scalar function f (q), i.e. ξX = ∂X f that leave the R curvature invariant. An alternative ξ -choice yields the Oproiu connection that satisfies Op Y = 0, JX Y · ω
(2.36)
which leads to the connection (2.12) [4, 7]. 9 The quadratic terms in these equations can be understood from the consistency of applying these formulae with and ˜ interchanged.
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2.3. Hypercomplex manifolds. Hypercomplex manifolds were introduced in [6]. A very thorough paper on the subject is [7]. Examples of homogeneous hypercomplex manifolds that are not hyper-K¨ahler, can be found in [10,30], and are further discussed in [4]. Non-compact homogeneous manifolds are dealt with in [31]. Various aspects have been treated in two workshops with mathematicians and physicists [32, 33]. A hypercomplex manifold has no SU(2) connection: ω X = 0.
(2.37)
This implies that we do not allow for local SU(2) redefinitions of the hypercomplex structure. The quaternionic structure should thus be covariantly constant with respect to the affine connection only. The unique solution of (2.6) with (2.37) is XY Z = Ob XY Z + NXY Z ,
(2.38)
where the first term is symmetric and the second one (the Nijenhuis tensor) is antisymmetric in XY . The first term is called the Obata connection [34], and is given in terms of the complex structures and their derivatives in (2.13). As mentioned before, and motivated by the supersymmetry algebra, we consider torsionless (symmetric) connections, which requires NXY Z = 0.
(2.39)
By definition, a hypercomplex manifold is a 4r-dimensional manifold M, equipped with a hypercomplex structure with vanishing Nijenhuis tensor. Clearly, hypercomplex manifolds are those quaternionic manifolds with vanishing SU(2) connection. If the quaternionic manifold has a non-vanishing SU(2) connection, it might still be possible to define from it a hypercomplex manifold. For instance, consider the class of quaternionic manifolds with vanishing SU(2) curvature, i.e., XY = 0. R
(2.40)
This only requires the SU(2) connection to be pure gauge. The complex structures are then covariantly constant with respect to this SU(2) connection, and one can still act with ξ -transformations. However, a small calculation shows that one can redefine the complex structures with a local SO(3) matrix R according to J = RJ , such that no SU(2) connection is needed anymore. In such a basis, also the freedom of doing ξ -transformations is fixed. The resulting manifold is then hypercomplex. One can in fact further relax the vanishing condition on the SU(2) curvature by requiring only XY = 2J[X Z ηY ]Z , R
(2.41)
for some ηXY that can be written as (2.34). In particular, the SU(2) connection is nonvanishing. But now, looking at (2.35), this implies that we can do a ξ -transformation such that the SU(2) curvature vanishes, and so we are back in the situation discussed above. In summary, for those quaternionic manifolds that satisfy (2.41) we can associate and define a hypercomplex manifold by making use of the quaternionic, local span of the complex structures, together with the ξ -transformations.
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Further properties of hypercomplex manifolds can be derived. Since they have vanishing SU(2) curvature, the Riemann tensor can be decomposed as jB
Z WABC D . RXY W Z = − 21 fXiA εij fY fWkC fkD
(2.42)
The tensor W is defined as Y Y Z kA RXY B A = 21 ε ij fjXC fiD fkB fW RXY Z W , WCDB A ≡ εij fjXC fiD
(2.43)
and is symmetric in its lower indices. It is however not traceless, and its trace determines the Ricci tensor, jC
RXY = R[XY ] = 21 εij fXiB fY WABC A = −RXY A A .
(2.44)
The tensor B is then antisymmetric, and is just the last term of (2.21). This form of the Ricci tensor implies that it is Hermitian XY U V RU V = RXY .
(2.45)
This follows from jA σj i fX , JX Z fZ iA = −i
σk i · σ j εij = −3εk .
(2.46)
The tensor W in (2.43) should not be confused with the traceless tensor W defined in (2.25). The precise relation is given by WABC D = WABC D −
3 δ D WBC)E E . 2(r + 1) (A
(2.47)
Thus, for hypercomplex manifolds, the Ricci tensor is antisymmetric and Hermitian. Conversely, a quaternionic manifold with antisymmetric and Hermitian Ricci tensor is necessarily hypercomplex. Indeed, using the general result (A.11) for any Hermitian bilinear form, (2.27) then implies that the SU(2) curvature vanishes, and so a basis can be chosen such that it is hypercomplex. We come back to the hermiticity properties of quaternionic-like manifolds at the end of this section. Nowhere in this section have we assumed the existence of a (covariantly constant) metric. When such a tensor exists, hypercomplex manifolds are promoted to hyperK¨ahler manifolds.
2.4. Hyper-K¨ahler manifolds. The crucial difference between hyper-K¨ahler and hypercomplex geometries is that hyper-K¨ahler manifolds admit a Hermitian metric g. This involves 3 conditions: 1. g should be Hermitian. This can be expressed as JX Z gZY = −JY Z gZX . 2. g should be invertible. 3. g should be covariantly constant using the Obata connection.
(2.48)
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If this metric is preserved using the Obata connection, then the hypercomplex manifold is promoted to a hyper-K¨ahler manifold. Equivalently, when the Levi-Civita connection preserves the quaternionic structure, then the manifold is hyper-K¨ahler. It is clear from the discussion of the previous section, that the Levi-Civita and Obata connection on a hyper-K¨ahler manifold must coincide. The Ricci tensor of the Levi-Civita connection is always symmetric. Combined with the fact that the Obata connection has an antisymmetric Ricci tensor, it follows that hyper-K¨ahler manifolds are Ricci flat, RXY = 0.
(2.49)
Using (2.18), this is equivalent to saying that the trace of WABC D vanishes, and so, WABC D = WABC D .
(2.50)
As a consequence, the curvature takes values in USp(2r). The existence of a metric allows us to define the covariantly constant antisymmetric tensors jB
X CAB = 21 fiA gXY ε ij fjYB ,
C AB = 21 fXiA g XY εij fY ,
(2.51)
which satisfy CAC C BC = δA B ,
(2.52)
and which can be used to raise and lower indices according to the NW–SE convention similar to εij : AA = AB CBA ,
AA = C AB AB .
(2.53)
The integrability condition on CAB implies RXY [A C CB]C = 0,
(2.54)
WABCD ≡ WABC E CED ,
(2.55)
and it follows that
is fully symmetric in its four lower indices.
2.5. Quaternionic-K¨ahler manifolds. For some basic references on quaternionic-K¨ahler manifolds, we refer to [35, 36], or the earlier references [34, 37–40]. Similar to hyper-K¨ahler spaces, quaternionic-K¨ahler manifolds admit a Hermitian and invertible metric satisfying (2.48). The connection that preserves this metric, i.e. the Levi-Civita connection, must be related to the Oproiu connection (2.12) by a ξ -transformation. It is a well known fact that quaternionic-K¨ahler spaces are Einstein: RXY =
1 gXY R. 4r
(2.56)
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From the Bianchi identity, one easily shows that the Ricci scalar is constant. The Ricci tensor is obviously symmetric, such that the R part of the G (r, H) curvature is zero. The B-tensor in (2.21) can easily be computed to be BXY = 41 νgXY ,
ν≡
1 R. 4r(r + 2)
(2.57)
Using (2.27), one then finds that the SU(2) curvature is proportional to the quaternionic 2-form: XY = 1 ν JXY . R 2
(2.58)
The Ricci part of the curvature is determined by the curvature of a quaternionic projective space of the same dimension: n R HP ≡ 21 gZ[X gY ]W + 21 JXY · JZW − 21 JZ[X · JY ]W XY W Z
= 21 JXY · JZW + L[ZW ] AB L[XY ]AB .
(2.59)
The full curvature decomposition is then n
RXY W Z = ν(R HP )XY W Z + 21 LZW AB WABCD LXY CD ,
(2.60)
with WABCD completely symmetric. In supergravity, the supersymmetry connects the value of ν to the normalization of the Einstein term in the action. This fixes the value of ν to −κ 2 , where κ is the gravitational coupling constant. The quaternionic-K¨ahler manifolds appearing in supergravity thus have negative scalar curvature, and this implies that all such manifolds that have at least one isometry are non-compact. Properties of the connections of all the quaternionic-like manifolds are summarized in Table 3. 2.6. Hermitian Ricci tensor. In this subsection, we study the properties of quaternioniclike manifolds with a Hermitian Ricci tensor. First of all, it is immediate from the relation between B and the Ricci tensor, see (2.21), that B is Hermitian if and only if R is Hermitian. The important relation we now look at is (2.27). Using (A.8), one can see that, for Hermitian B, the antisymmetric part of B does not contribute. We can therefore conclude that an antisymmetric and Hermitian Ricci tensor is equivalent to the requirement of hypercomplex. Indeed, the SU(2) curvature is then zero, and we use the argument in Sect. 2.3. The other direction was also shown in that section. Furthermore, for the symmetric part of B, the antisymmetrization in (2.27) is automatic in the right-hand side due to (A.11). For a Hermitian Ricci tensor, we thus have, Table 3. The affine connections in quaternionic-like manifolds hypercomplex Obata connection quaternionic Oproiu connection or other related by ξX transformation
hyper-K¨ahler Obata connection = Levi-Civita connection quaternionic-K¨ahler Levi-Civita connection = connection related to Oproiu by a particular choice of ξX
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XY = 2JX Z B(Y Z) , R
R(XY ) = 4(r + 2)B(XY ) .
(2.61)
It is appropriate to define a ‘candidate metric’, gXY =
4 1 B(XY ) = − hXY , ν ν
(2.62)
where ν is an undetermined number. We define hXY such that this number can be avoided in most of our formulae. With the usual normalization in supergravity where κ = 1, this is the metric anyway. We now have XY = − 1 JX Z hY Z . R 2
(2.63)
The identification of the symmetric part of the Ricci tensor as a metric becomes even more appropriate due to the property that it is covariantly constant. To prove this, we start with acting with a DU derivative on (2.63) and antisymmetrizing in [XY U ] using the Bianchi identity for the SU(2) curvature. This leads to 3J[X Z DU hY ]Z = JX Z DU hY Z + JU Z DY hXZ + JY Z DX hU Z = 0.
(2.64)
Multiplying this with (JV U × JW Y ) and taking the symmetric part in (XW ), after using several times the antisymmetry of JX Z hY Z , it leads to DV hXW = 0.
(2.65)
We can summarize this section as follows: 1. If the Ricci tensor is Hermitian and the symmetric part is invertible, then it defines a good metric. Therefore the antisymmetric part of the Ricci tensor is zero in this case (with respect to the Levi-Civita connection of hXY ). On the other hand, if the symmetric part is zero, then a Hermitian Ricci tensor implies zero SU(2) curvature. 2. A quaternionic manifold is quaternionic-K¨ahler if and only if the Ricci tensor is Hermitian and its symmetric part is non-degenerate. Thus, there are 3 cases of Hermitian Ricci tensors on quaternionic-like manifolds: 1. symmetric part is invertible (quaternionic-K¨ahler manifold): there is no antisymmetric part, 2. symmetric part is zero, i.e. antisymmetric Ricci (hypercomplex manifold), 3. symmetric part is non-zero but non-invertible. Then an antisymmetric part is still possible. 3. The Map We now start to discuss the map between the hypercomplex/hyper-K¨ahler and quaternionic(-K¨ahler) manifolds. The first will be called the large space and will be taken to be 4(nH + 1) real dimensional, while the latter will be of real dimension 4nH and be called the small space. Objects on the large space will be denoted by hats, either on their indices or on the objects themselves or on both. The content of this section is as follows. In Subsect. 3.1 we discuss some special properties of conformal hypercomplex manifolds that are important for our discussion. We show that the holonomy of such manifolds is further restricted and we discuss a continuous deformation of the hypercomplex structure. After choosing coordinates
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that are adapted to our setting and rewriting the hypercomplex structure, we prove in Subsects. 3.2 and 3.3 that the large space contains a 4nH dimensional quaternionic subspace. This is the central part of our discussion. Moreover, in Subsect. 3.3.2 we clarify the origin of the quaternionic local SU(2) symmetry. In the following Subsect. 3.4 we show how a quaternionic manifold can be used to construct a hypercomplex one. Subsect. 3.5 considers the map for hypercomplex manifolds that possess a ‘good metric’, and are therefore hyper-K¨ahler manifolds. The conditions for this metric to be a ‘good metric’ are equivalent to the condition that the quaternionic manifold has a ‘good metric’ and is thus promoted to a quaternionic-K¨ahler manifold. Therefore, we prove that the image of the map is a quaternionic-K¨ahler manifold if and only if the original manifold is a conformal hyper-K¨ahler manifold. We show that in order for the affine connection to agree with the Levi-Civita connection, as it should be for quaternionicK¨ahler manifolds, one has to choose a particular ξ -transformation between the allowed connections for quaternionic manifolds. This choice is different from the one that leads to the Oproiu connection. We also prove the inverse map: for all quaternionic manifolds we will define a candidate metric, and if this is a good metric, which is the condition that the Ricci tensor is Hermitian and invertible, then we can construct a good metric for the conformal hypercomplex manifold. After having completed our discussion of the map, we conclude in Subsect. 3.6 by listing the vielbeins and all connection coefficients on the large and small space in the adapted coordinates. 3.1. Hypercomplex manifolds with conformal symmetry. The starting point of our map is given by hypercomplex manifolds that admit a conformal symmetry. By definition, this means there exists a closed homothetic Killing10 vector k X , satisfying (1.1). X Given this homothetic Killing vector k , three more vectors can be constructed naturally: JYX k Y , kX ≡ 13
(3.1)
which generate an SU(2) algebra and satisfy kX JYX . = 21 D Y
(3.2)
It then follows that, under dilatations and SU(2) transformations, the hypercomplex structure is scale invariant and rotated into itself, Y Y − ∂ k Y Z Z Y Z J X JX (3.3) LD k ≡ D k ∂Z JX + D ∂X k JZ = 0, D Z · kZ)∂ · ∂ kY) · ∂ kZ) × L J Y ≡ ( J Y − ( J Z + ( JY = − J Y . ·k
X
Z
X
Z
X
X
Z
X
(local in spacetime but not dependent on Here, we introduced parameters D and the coordinates of the quaternionic space) to generate the infinitesimal dilatations and SU(2) transformations on the coordinates,
δ D q X = D k X ,
· kX · (k Y JYX ) = δSU(2) q X = 13 .
(3.4)
Notice that all this follows from (1.1) and the covariant constancy of the quaternionic structure. 10
Although there is not necessarily a metric defined, we use the same terminology.
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The integrability conditions for (1.1) and (3.2) then read
W kX R Y Z = 0. X
W X kX R Y Z = 0,
(3.5)
Hence, the four vector fields now introduced are zero eigenvectors of the curvature. This implies that the holonomy of a 4(nH + 1) dimensional conformal hypercomplex manifold is contained in SU(2) · G (nH , H), which can easily be understood as follows. On a hypercomplex manifold, we can group all vectors of a given fibre into quaternions. Let us call the fibre F at a certain point p. Given then a vector X ∈ F , we may construct a quaternion as {X, J1 (p)X, J2 (p)X, J3 (p)X} ≡ {X, J(p)X}. In general, the holonomy group yields a G (nH + 1, H) action on F that is generated by the curvature. Since {k(p), 3k(p)} form such a quaternion, the relation (3.5) implies that the holonomy group should leave that quaternion fixed and thus should be included in SU(2) · G (nH , H). Moreover, if one looks at the components of the hypercomplex structures that lie along the other quaternions, it is easy to see that the holonomy group induces an SU(2) · G (nH , H) action on these components. This observation is the heart of our construction, since it strongly suggests that the submanifold along the other 4nH directions is quaternionic.11 As will be motivated in the following sections, the existence of the vector fields k and k implies that we can define a continuous family of hypercomplex structures. Suppose that we start with a conformal hypercomplex manifold with closed homothetic Killing vector k and complex structure J. We can define Y + Y ( Jξ )X = JX
2 3
Y ξ k Y ) − ( ξ k Z ) J , J Z ( X
Z
X
Z
(3.6)
X X ξX ξX for a one-form with components ξX that satisfies k =k = 0. The new complex structures still satisfy the quaternionic algebra and have vanishing Nijenhuis tensor if
∂[X ξY] is Hermitian, and
ξX ξX Lk = Lk = 0.
(3.7)
The last requirement is automatic if the first two are satisfied. In conclusion, we can construct a new hypercomplex structure using (3.6) with a one-form that is constant along the flows of k and k and whose external derivative is Hermitian. One can moreover show that the new hypercomplex manifold is again conformal with the same vector field k. As far as we know this ξ -transformation has not been given before in the mathematical literature.
3.2. The map from hypercomplex to quaternionic. We now start to construct a map between 4(nH + 1)-dimensional hypercomplex/hyper-K¨ahler manifolds, admitting a conformal symmetry, and 4nH -dimensional quaternionic(-K¨ahler) manifolds. 11 More exactly, the four vector fields k and k generate a four dimensional foliation of the hypercomplex space, and we will show that the space of leaves carries a quaternionic structure.
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3.2.1. Suitable coordinates and almost complex structures. We first construct a set of coordinates adapted to our setting. These coordinates should allow us to solve explicitly the constraints imposed by conformal symmetry. The primary object is the homothetic Killing vector (1.1). Therefore, first of all, we will choose one coordinate such that the vector k has a convenient form. One can always find local coordinates q X = {z0 , y p }, where p = 1, ..., 4nH + 3, such that the components of the homothetic Killing vector are
k X = 3z0 δ0X .
(3.8)
This is obtained by choosing at any point the first coordinate in the direction of the vector k X . The factor 3 in (3.8) is a convenient choice for later purposes. Notice that arbitrary coordinate transformations y p (y q ) trivially preserve (3.8). We will make use of this freedom below. Having singled out the ‘dilatation direction’, we now proceed similarly for the SU(2) vector fields (3.1). Frobenius’ theorem tells that the three-dimensional hypersurface spanned by the direction of the three SU(2) vector fields can be parametrized by coordi being one of the indices α. Note nates (zα )α=1,2,3 , such that kX is only non-zero for X that these vectors point in different directions than k X , due to (3.1), and the fact that the complex structures square to − . Therefore, they do not coincide with the direction ‘0’ chosen above. The other 4nH coordinates are indicated by q X . Thus we have at this point q X = z0 , y p = z0 , zα , q X , α = 1, 2, 3, X = 1, . . . , 4nH ,
k X = 3z0 δ0X ,
k0 = kX = 0.
(3.9)
X Now, as kX = 13 k Y J Y and due to our particular choice of coordinates, we find that
1 J0 α = 0 kα , z
J0 0 = 0,
J0 X = 0.
(3.10)
Generically, we allow kα to depend on q X (as it is the case also in group manifold reductions [41]). We assume it to be invertible as a three by three matrix, and define m α as the inverse, in the sense that we have for any vectors A and B, A · m α kα · B = A · B
or
kα · m β = δβα .
(3.11)
It is convenient to introduce 1 0 AX ≡ 0 JX . z
(3.12)
Using (3.10), we can complete the table of complex structures by requiring the quaternionic algebra (1.2). In terms of AX , we find J0 0 = 0, J0 β = z10 kβ , J0 Y = 0,
Jα 0 = −z0 m α, β β Jα = k × m α, Y J α = 0,
JX 0 = z0 AX , JX β = AX × kβ + JX Z (AZ · k β ), (3.13) JX Y = JX Y .
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All dependence on z0 of these complex structures is explicitly shown in this formula. The last equation says that the components of the hypercomplex structures on the large space that lie along the small space satisfy the algebra of the imaginary quaternions. Thus we have decomposed any almost hypercomplex structure on the large space and we find that it is expressed in 3 different quantities: the vectors kα [and their inverses, see (3.11)], vectors AX , which are arbitrary so far, and an almost quaternionic structure on the small space, JX Y . 3.2.2. The complex structures and Obata connection. As was explained in Sect. 2.1, the almost hypercomplex structure J is hypercomplex if the Nijenhuis tensor vanishes, or, equivalently, if there exists a torsionless connection such that the complex structures are covariantly constant. The latter is then the Obata connection, see (2.13). In practice it is easier to first compute that connection. First of all, the relation (1.1) in our coordinate ansatz (3.9) gives rise to
1 1 Y 0 Y Y X0 (3.14) = − δ δ δ 0 . X z0 2 X Further, we can immediately find some more information on the coordinate dependence of the basic quantities. Multiplying the first relation of (3.3) by k X yields that the SU(2) vectors commute with the homothetic Killing vector field. Using (3.9), this yields ∂0 kα = 0,
∂0 m α = 0.
(3.15)
replaced by m One may further use the second line of (3.3) with α and obtain the zα dependence of the SU(2) vector fields that reflect the SU(2) algebra. One can write the corresponding equation in various forms: kγ × ∂γ kα = kα ,
m [α · ∂β] kγ = − 21 (m α × m β ) · kγ . (3.16)
∂[α m β] = − 21 m α × m β,
The connection coefficients can then be written as 00 0 = − 2z10 , 0p 0 = 0,
00 p = 0, p 0q p = 10 δq , 2z
αβ 0 = 21 gαβ , αβ γ = −m (α · ∂β) kγ , X αβ = 0, (3.17) JX 0 · m JX β · m α = 21 α, Xα β = 21 α − m α · ∂X kβ , Xα 0 = 21 z0 AX · m Xα Y = 21 JX Y · m α, 1 0 gXY , XY α =−(∂(X AY ) ) · kα + XYW AW · kα − 21 hZ(X JY ) Z · kα , XY = 2 XY Z = Ob XY Z + 1 J(X δ · m δ δZ + 1 m δ × JY ) Z . 3
Y)
2
Here Ob is the Obata connection defined by (2.13) using the J complex structures, while is the Obata connection using J. We have also introduced the following convenient notation: gαβ ≡ 2 αβ 0 ,
0 gX X Y ≡ 2 Y ,
hXY ≡
1 gXY + AX · AY . z0
(3.18)
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Note that although we have not introduced a metric, we use here suggestive notation since a ‘good metric’ coincides with these definitions, as we will show in Sect. 3.5.1. Hence, we have used g as a shorthand for a complicated function of z0 , kα , AX and Y = 0 and Y = α components of (3.2) in the basis (3.9) and JX . Considering the X using (3.13) leads to kα ≡ gαβ kβ = −z0 m α.
(3.19)
gαβ = − z10 kα · kβ = −z0 m α · m β.
(3.20)
This implies
The requirements of covariantly constant J lead to requirements on the coordinate Y α 0 dependence of the quantities k , AX and JX . From the requirements that D J = 0 and Dα J = 0, we find that ∂0 AX = 0, ∂0 JX Y = 0,
α ×) AX + ∂X m α = 0, (∂α + m α ×) JX Y = 0. (∂α + m
(3.21) (3.22)
Note that the integrability condition for the second relation in (3.21) yields
1 A) V ) α ×) R(− = 0, with R( ≡ 2∂[X VY ] + 2VX × VY . (∂α + m 2
XY
XY
(3.23) X A main non-trivial result comes from D JY 0 = 0. We find
1 A) R(− = 21 hZ[X JY ] Z , 2 XY
(3.24)
with h as in (3.18). We can calculate this expression using the definition of the Obata connection αβ 0 from (2.13) and our particular decomposition (3.13). This leads to
1 A) X U × JY Z ) · R(− 1 A) + ( J . (3.25) hXY = − 13 4J(X Z · R(− 2 2 Y )Z UZ However, this equation can also be obtained from solving h from (3.24). Thus, the equation implies by itself that the matrix h that appears in (3.25) is necessarily the quantity defined in (3.18). Therefore, the integrability condition is equivalent to the requirement that there should be a symmetric matrix h such that (3.24) is satisfied. Finally, the vanishing of the components along the small space of the Nijenhuis tensor of the hypercomplex structure J implies that the Nijenhuis tensor in the small space should be 6NXY Z = − JY ] Z = J[X α · ∂α J[X α · m α × JY ] Z = − 2A[X + AW × J[X W · JY ] Z . (3.26) This equation is the basic equation that determines that the small space is quaternionic. We will further elaborate on this in Sect. 3.3.1. The conclusions of Sect. 3.2.1 can now be completed. There are integrable complex structures on the conformal hypercomplex space for functions kα (q X , zα ),
AX (q X , zα ),
JX Y (q X , zα ).
(3.27)
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The kα should satisfy the SU(2) algebra, i.e. (3.16), while the zα dependence of the other quantities is determined by these vector fields using (3.21) and (3.22). Moreover there are the conditions that there should be a symmetric tensor hXY , which is given by (3.25), such that (3.24) is satisfied. The Nijenhuis tensor of the complex structures in the small space should satisfy (3.26), which, as we will show in the next section, has the meaning that it is a quaternionic structure with SU(2) connection determined by AX . We remark that, combining the previous results, the zα -dependence of all quantities can be calculated. E.g the following are useful results: α ×) JX β − JX γ m α · ∂γ kβ = 0. (3.28) ∂α hXY = 0, (∂α + m
3.3. The embedded quaternionic space. 3.3.1. Proof that the small space is quaternionic. As already explained in Sect. 2, contrary to the hypercomplex case where the SU(2) connection is trivial, in the quaternionic case there is a non-trivial SU(2) connection. This means that parallel transport with respect to the affine connection rotates the three complex structures into each other. As a consequence, the integrability condition for the complex structures differs from the hypercomplex case, as the (diagonal) Nijenhuis tensor should now be proportional to the SU(2) connection (2.10). This is exactly what we obtained with (3.26), leading to the SU(2) Oproiu connection satisfying (2.36), Op ω X = − 16 2AX + AY × JX Y . (3.29) Hence, this shows that the small space is quaternionic. The corresponding affine connection is, according to (2.12), Op XY Z = Ob XY Z − J(X Z · ω Y )
Op
(3.30)
= XY Z − 13 AV · J(X V δYZ) + 23 A(X · JY ) Z + 13 AV · J(X V × JY ) Z , where we used the last equation of (3.17). As we have discussed in Sect. 2.2, there is a family of torsionless connections that are compatible with that given structure. Related to that freedom, all SU(2) connections can be written as (3.31) ω X = − 16 2AX + AY × JX Y + JX Y ξY . A particular choice that will be useful below is ξX = 16 JX Y · AY ,
(3.32)
such that we find the connections ω X = − 21 AX ,
XY Z = XY Z + A(X · JY ) Z .
(3.33)
This choice of a quaternionic connection will turn out to be special for two different reasons. First of all, if there is a good metric on the small space (i.e. if the space is quaternionic-K¨ahler) this connection will correspond to the Levi-Civita connection. Secondly, we will show that for this choice the R curvatures for both the large and the
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small space will be equal, R(R) = R(R) . In the hypercomplex space this curvature is proportional to the Ricci tensor12 , which is Hermitian. This implies that the R curvature on the small space is Hermitian. 3.3.2. The local SU(2). One may wonder what the origin is of the local SU(2) invariance on the embedded quaternionic manifold. We will show that this local invariance is already present in conformal hypercomplex manifolds, but the transformations on the complex structures are more complicated than simple vector rotations. Then, we will show how this induces the expected local SU(2) on the quaternionic manifold, with ω X as gauge field. Considering the equations of Sect. 3.2, they are nearly all invariant under a usual vector rotation δ V = × V for all 3-vectors V , especially if only depends on the coordinates of the small space q X . One troublesome equation is (3.21). AX is an arbitrary quantity in the construction, a ‘black box’. It turns out that the local invariance can be obtained by adding a gauge-type transformation for AX . Thus, we consider for the elementary quantities the following SU(2) transformations: δSU(2) AX = ∂X + × AX , δSU(2) kα = × kα ,
δSU(2) JX Y = × JX Y , δSU(2) m α = × m α,
(3.34)
X ) cannot depend on z0 or zα . The main requirement for AX where the parameter (q was (3.24), in which the curvature of A appears. This equation is thus consistent if hXY is invariant under the SU(2) transformations. A long calculation using the definition of 0 the Obata connection shows that X Y is not invariant, but precisely transforms such that hXY defined as in (3.18) is invariant. This proves the local SU(2) symmetry of the conformal hypercomplex manifold. Note that, due to the transformations of AX some components of J do not transform as an ordinary vector. These are JX 0 , JX 0 = z0 ∂X + × δSU(2) JX β = −kβ × ∂X + JX Z kβ · ∂Z + × JX β . δSU(2)
(3.35)
It turns out that the full Nijenhuis tensor is invariant under this SU(2). The complex structures in the small space thus transform as ordinary vectors. We have seen that AX is the gauge field of these SU(2) transformations. In the ξ -gauge where AX is proportional to ω X , see (3.33), this is thus the expected SU(2) gauge field, and we find X. X = − 21 ∂X + × ω δSU(2) ω
(3.36)
Hence, ω X transforms as an true connection since (1.4) now transforms covariantly. Another ξ -choice, as e.g. the Oproiu choice (2.36), is not invariant under SU(2). Hence, if we take this connection, it implies that the remaining SU(2) contains a compensating ξ -transformation, which is ξX = 16 JX Y · ∂Y . 12
(3.37)
Observe that (3.5) implies that the Ricci tensor has only components in the directions of the small space.
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This compensating transformation also contributes to the SU(2) transformation of the affine Oproiu connection Op , such that it cancels other terms that follow from its definition (2.12). Thus the final affine connection is SU(2) invariant, as one should expect for covariance of the covariant derivative of the complex structures (1.4). Hence, the quaternionic local SU(2) is naturally included into the map. 3.4. The map from quaternionic to hypercomplex. 3.4.1. Uplifting a quaternionic manifold. By now, we are in a position to discuss the inverse procedure, namely the construction of a conformal hypercomplex manifold starting from a quaternionic space. We may at this point choose a value for ξ , and we will see below that we have to choose one such that the R curvature (i.e. the antisymmetric part of the Ricci tensor) is Hermitian. This is always possible, as it is clear from (2.32) that we may even choose a ξ such that the R connection vanishes. We now consider a space of 4 real dimensions bigger than the one of the quaternionic space. The extra coordinates are labeled z0 and zα . Let kα (zα , q X ) denote left-invariant vector fields on the SU(2) group manifold, i.e. satisfying (3.16). For now, the dependence on q X is not fixed. One may take the SU(2) vectors independent of q, but an arbitrary dependence is allowed. It will be fixed later. Then construct their inverse m α (z, q) using (3.11). Furthermore, we define AX (q) = −2ω X (q) for ω X (q) the SU(2) connection for the chosen value of ξ . We take the kα , JX Y and AX independent of z0 . The zα dependence of kα is determined by its SU(2) property (3.16). The complex structures are taken to be covariant constant in their zα dependence in the sense of (3.22). This means in fact that when we change the value of zα we go to a different choice of complex structures. These different complex structures are related by an SU(2) rotation. This implies that also the SU(2) connection should change, and indeed this is in agreement with the comparison of (3.34) and (3.21). The latter equation determines the q X -dependence of the SU(2) vector fields kα . One also notices that the curvature of AX is taken to be a covariant vector as shown in (3.23). With these ingredients, we can construct an almost hypercomplex structure as in (3.13). In order for this structure to be integrable, the only remaining condition is (3.24). The condition states that there should exist a symmetric object h such that the SU(2) curvature is related to the hypercomplex structure as indicated. In a quaternionic-K¨ahler manifold this is satisfied with h being the metric. In an arbitrary quaternionic manifold, we have (2.27). We thus just need that the antisymmetric part of B does not contribute to this equation. That is a condition of the form (A.11) for the antisymmetric part of B, which is the antisymmetric part of the Ricci tensor or R curvature. Hence it says that this RR should be Hermitian, which we can obtain by a ξ -transformation as mentioned in the beginning of this section. With these choices, (3.13) defines a hypercomplex structure on a manifold parametrized by the coordinates {z0 , zα , q X }. Moreover, it is easy to see that the vector field 3z0 ∂0 satisfies (1.1), hence the manifold is actually conformal hypercomplex. 3.4.2. A ξ -transformation for conformal hypercomplex manifolds. Suppose we have constructed a hypercomplex manifold from a quaternionic manifold with a Hermitian R curvature RR . We perform a ξ -transformation with Hermitian ∂[X ξY ] . Then the new R curvature RR is also Hermitian. Hence, we can use again the procedure of Sect. 3.4.1 to
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obtain a hypercomplex structure on the large space. As the ξ -gauge modified the SU(2) connection, the vector AX is different from AX , and hence the new hypercomplex structure differs from the original one. This defines a new mapping, dependent on a one-form ξˆ , between hypercomplex structures on conformal hypercomplex manifolds, see (3.38): ξˆ
J
AX
AX
? J, ω
- J 6
ξ
- J, ω
(3.38)
This is the transformation that was announced in (3.6). Remember that the ξ -transformations are defined by one forms ξX dq X that depend only on the quaternionic coordinates and that ξˆ must q X . This says that ξˆX = ξX must be constant along the flows of k and k, R R ˆ ˆ transform a Hermitian R in another Hermitian R , yielding the conditions in (3.7). Hence, we have re-derived the results of Sect. 3.1, and shown how its origin is necessary for the consistency of this picture. The transformation (3.6) is in these coordinates, with complex structures as in (3.13), simply given by δ( ξ )AX = 2JX Z ξZ .
(3.39)
3.5. The map from hyper-K¨ahler to quaternionic-K¨ahler spaces. We will now restrict to the case in which there is a compatible metric. We will show that starting from a hyperK¨ahler space, the small space will be automatically quaternionic-K¨ahler and vice-versa. The existence of a metric will moreover remove the freedom that was implied by the ξ -transformation, since it unambiguously specifies the torsionless connection as being the Levi-Civita connection. Hence, the connection on the small space will be determined uniquely, given the structures that are defined on that manifold. We will show how to construct the quaternionic metric from the one in the hyperK¨ahler space, and show how the affine connection reduces to the Levi-Civita connection. 3.5.1. Decomposition of a hyper-K¨ahler metric. As we already mentioned in Sect.2, a hypercomplex manifold is promoted to a hyper-K¨ahler space if there is a Hermitian metric gX Y that preserves the Obata connection, i.e.
W gX Z( gY)W ∂Z Y − 2 X = 0, Z JX gZY + JYZ gZX = 0.
(3.40) (3.41)
We can now split these equations in various parts in the basis (3.9). The first equation = 0, using (3.14), determines the z0 dependence of the various parts of the metric. for Z = p, X =Y = 0 part leads to Considering furthermore the Z
X Y d s2 ≡ gX Y dq dq = −
1 h00 (dz0 )2 − 2∂p h00 dz0 dy p + z0 hpq dy p dy q , (3.42) z0
=Y = 0 part where the h are independent of z0 . Furthermore we want to invoke the X of (3.41), which using (3.13) and the invertibility of m α implies that ∂α h00 = 0.
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One can further simplify the metric by a redefinition z0 = z0 h00 (q),
zα = zα ,
q X = q X .
(3.43)
This redefinition preserves our previous coordinate choices (3.9). In particular, k has still only components along zα since ∂α h00 = 0. This redefinition accomplishes that (using the new, primed, z0 coordinate, but omitting the primes from now on) g00 = −
1 , z0
g0p = 0.
(3.44)
= p, X = q, Y = 0 part of (3.40) leads to The Z 0 gX X Y = 2 Y .
(3.45)
This coincides with (3.18). At that time we have made a definition for arbitrary hypercomplex manifolds. Here we prove that any good metric on these manifolds is of the form (3.45) after choosing suitable coordinates. Using the table of affine connections, (3.17) and (3.18) we thus obtain (dz0 )2 d s 2 = − 0 + z0 hXY (q)dq X dq Y z + gαβ [dzα − AX (z, q) · kα dq X ][dzβ − AY (z, q) · kβ dq Y ] . (3.46) The metric therefore is a cone13 and since this cone is hyper-K¨ahler, this implies that the base is a tri-Sasakian manifold, see e.g. [42]. We now check the remaining conditions in (3.40) and (3.41). For both it turns out that the only non-trivial components are the ones where the indices are restricted to those in the small space. We then find (3.41) ↔ J(X Z hY )Z = 0,
(3.47) W
(3.40) ↔ ∂Z hXY − 2Z(X hY )W = 0,
(3.48)
where XY Z is given in (3.33). We can thus state that the metric h on the quaternionic space is Hermitian if and only if the metric g on the hypercomplex manifold is Hermitian. The second result, (3.48), states that h is covariantly constant using as connection XY Z . Therefore, it is the Levi-Civita of h. This connection also preserves the quaternionic structure, because we have shown in (3.33) that it is equivalent to the Oproiu connection up to ξ -transformations. We have thus shown that the only ξ -choice that can be taken for quaternionic-K¨ahler manifolds is the one mentioned in Sect. 3.3.1, and in particular that ω X = − 21 AX .
(3.49)
It is seen from (3.46) that the induced metric in the small space is 1 1 (3.50) gXY = z0 hXY = − hXY = 2 hXY . ν κ The overall factors do not play a role in the equations in this section. This metric does not depend on zα , see (3.28). 13 This follows from first extracting the z0 dependence from gˆ , and then defining the radial variable αβ r 2 = z0 .
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3.5.2. The inverse map. The inverse map is of course a special case of the discussion in Sect. 3.4. Therefore, we will not repeat the complete lifting process but merely point out the facts specific to this case. As the small space carries both a quaternionic structure J together with a good metric h, there is a unique torsionless connection compatible with that structure. Hence, we choose as a triplet of vectors AX = −2ω X , where ω X is the SU(2) connection that corresponds to the Levi-Civita connection on the small space, and use it in (3.13). If we introduce z0 and zα dependence as in Sect. 3.4, the resulting almost hypercomplex structure J is automatically integrable. The reason for this is that Eq. (3.24) is now always satisfied since the Ricci tensor, and hence the tensor B (2.21) cannot have an antisymmetric part. Therefore, the large space is already hypercomplex. We can moreover construct a good metric g on the large space, starting from the good metric h on the small space, using the following table: 1 , z0 = z0 AX · m α,
g00 = − gαX
g0α = g0X = 0, gαβ = −z0 m α · m β, gXY = z0 hXY − AX · AY .
(3.51)
Combining this with the remarks of Sect. 3.5.1, we conclude that a conformal hypercomplex manifold is hyper-K¨ahler if and only if the corresponding quaternionic space is quaternionic-K¨ahler.
3.6. The map for the vielbeins and related connections. In the previous part, we did not yet consider the objects that are related to the G (r, H) structure of the manifolds. To discuss these, one needs the vielbeins. These vielbeins are also the starting point for the supersymmetry transformations of hypermultiplets. We have first to choose a suitable basis to express these objects. 3.6.1. Coordinates on the tangent space. Having singled out the z0 and zα coordinates in which the quaternionic structure is given by (3.13), the structure group of the frame bundle reduces from G (nH + 1, H) to SU(2) · G (nH , H), which consists of the set of iA frame reparametrizations on the small space. Similarly, the vielbein fX and its inverse into (i, A) with i = 1, 2 and fiXA can be decomposed. To do so, we split the index A A = 1, . . . , 2nH . To be more precise, we split any G (nH + 1, H)-vector ζ A (q X ) in (ζ i , ζ A ), which i is a vector of SU(2) · G (nH , H), and where ζ is defined by ij 0 A 1 0 i (3.52) ζ . 2 z ζ ≡ −iε fj A
This choice of basis is guided by applications in supergravity, in which ζ A are the super partners of the coordinate scalar fields q X . We will see that the choice of frame in the tangent space is useful for the identification of the quaternionic tangent space within the full hypercomplex tangent space. The formula (3.52) moreover implies 0 fiA = 0. (3.53) fij0 = −iεij 21 z0 ,
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The i factors are needed for the reality conditions (2.2) where the ρi j components of ρAB are ρi j = −Ei j = −εij , and ρ has no off-diagonal elements, i.e. ρiA = ρiA = 0. This is convenient for the formulation of quaternionic manifolds that appear in supergravity, see the discussion in Sect. 6. Writing the first entry of (3.13) in terms of vielbeins, (2.4) yields ij f0 = iεij
1 . 2z0
(3.54)
Furthermore, we redefine ζ A according to ζ A = ζ A − i 2z0 f0iA ζ j εj i .
(3.55)
Using J0 Y = 0 and (3.54), this implies
X A ζ . fiXAζ A = fiA
(3.56)
Therefore, we have in the new basis ζ A = (ζ i , ζ A ), fijX = 0.
(3.57)
After having established this basis, we can drop the primes. Using (2.3) and the form of components of the quaternionic structure in (3.13) combined with (2.4), we find the following table determining the vielbeins of the large space in terms of those of the small space, z0 , kα and AX : fij0 = −iεij 21 z0 , 0 = 0, fiA ij f = iεij 10 , 0
f0iA = 0,
2z
fijα = 2z10 kα · σij , α = fXA α fiA iA X ·k , 0 ij fα = − z2 m α · σ ij , iA fα = 0,
fijX = 0, X = fX, fiA iA (3.58) 0 ij f = z AX · σ ij , X
2
fXiA = fXiA ,
where σi j denote the Pauli matrices, σ ij = εik σk j = σ j i and σij = σi k εkj = σj i . If there is a good metric on the hyper-K¨ahler space, this implies that the scalar product defined by this metric carries over to the tangent space, as was shown in (2.51). In the coordinates (3.9), this symplectic metric decomposes as AB = CAB , C
ij = εij , C
iA = 0. C
(3.59)
Note that the tangent space metric CAB , which can be used to raise and lower flat indices on the small space, corresponds to the metric gXY = z0 hXY .
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3.6.2. Connections on the hypercomplex space. Since we already know the reduction 0 α , A Y X and JX Y , see (3.13), and the form of , ˆ we rules for the JX in terms of z , k can calculate the induced connection ω on the tangent space from (2.7) for the hatted quantities (remember that in the large space there is no SU(2) connection) using the reduction rules for the vielbeins (3.58). The induced connection has the following component expressions: ω0i j = ω0i A = ω0A j = 0, ωαA j = 0, ωαi A = 0 · σ j , ωαi j = i z2 m α i ωXi A = i 2z10 εik fXkA , ωXA B
Y + ω0A B = 21 fYiB ∂0 fiA 0 ωXi j = −i z AX · σi j ,
1 B δ , 2z0 A
2
iB Y ωαA B = 21 f Y ∂α fiA , 0 ωXA i = −i z2 ε ij fjYA hY X , 1 Y + 1 f iB f Z Y Y . = 21 fYiB ∂X fiA iA XZ + 2 AZ · JX 2 Y
(3.60)
3.6.3. Connections on the quaternionic space. By now, we have completely specified our reduction ansatz. This enabled us to give the component expressions for the Obata connection on the hypercomplex manifold in terms of the objects appearing in the ansatz. We now determine an expression for the G (nH , H) connection components on the lower dimensional quaternionic space. Using (3.60), we can dimensionally reduce the G (nH + 1, H) connection as Z iB V ωXA B = ωOp XA B + 16 fiA f(X J Z) · AV +fYiB A(Z · JX) Y −fYiB AV · J (Z V × JX) Y , (3.61) where ωOp XA B is the G (nH , H) connection corresponding to the Oproiu connection (3.30) and the SU(2) connection (3.29). As we have already explained quite extensively, there is a family of possible connections on a quaternionic manifold, related to each other by ξ -transformations, which for the G (nH , H) connection is given by (2.31). The transformation (3.32) gives the drastic simplification ωXA B = ωXA B .
(3.62)
While in quaternionic manifolds this is just one possible choice, we have seen in Sect. 3.5.1 that in quaternionic-K¨ahler manifolds this ξ -choice is imposed. 4. Curvatures In the ‘large space’ many components of the curvatures are zero due to the existence of the homothetic Killing vector field and the SU(2) isometries, (1.1) and (3.2), yielding (3.5):
W W X R X X kX R Y Z =k Y Z = 0.
(4.1)
This (together with the cyclicity properties of the curvatures) shows that the only possible non-zero components of the curvature of the conformal hypercomplex manifolds XY Z W are R . For the Ricci tensor, this implies that the only nonvanishing components are along the quaternionic directions:
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ZXY Z , XY = R R
(4.2)
and this is antisymmetric as in general for hypercomplex manifolds. The other part of the curvature, as shown in Table 2, is the Weyl part. The latter is generally determined by a traceless tensor, see (2.25). But for hypercomplex manifolds there is also a generically ABCD , whose trace part determines the antisymmetric part of the non-traceless tensor W D Ricci tensor, and whose traceless part is W AB C . The vanishing of all the curvature ABCD are only components mentioned above implies that the non-vanishing parts of W ABC D and W ABC i . The latter will not be important for the reduction ABC D , i.e. W W D to the small space. Note, from (2.47), that the traceless part, W AB C , has as non-zero components 3 BC)E E , δD W 2(nH + 2) (A 1 j E ABi j = − =W δ W ABE , 2(nH + 2) i
ABC D = W ABC D − W iAB j = W AiB j W
ABC i = W ABC i . W (4.3)
depends also on the This implies that the Weyl tensor of the hypercomplex space, W, ABE E . trace W We now start the reduction to the small space. We use the ξ -choice (3.33), which gives the easiest formulae for the map, as explained above. The formulae for other ξ choices follow from (2.33)–(2.35). An expression for the SU(2) curvature can be found by considering (3.24). With (3.33), this implies XY = − 1 J[X Z hY ]Z , R 2
(4.4)
which is the equivalence between the SU(2) curvature and the quaternionic two-forms XY Z W of in the quaternionic-K¨ahler case. One can derive the curvature components R the hypercomplex space as a function of the curvature RXY Z W of XY Z by a long but straightforward calculation. We obtain W XY Z W = RXY Z W + 1 δ[X hY ]Z − 21 JZ W · J[X V hY ]V − 21 JZ V · J[X W hY ]V . (4.5) R 2
Taking the trace of this expression gives the Ricci tensor XY = RXY + 2nH + 1 hXY + 1 JX Z · JY W hZW . (4.6) R 2 2 The left hand side is antisymmetric, which determines the antisymmetric part of the quaternionic Ricci tensor RXY , XY = R[XY ] . R
(4.7)
As the R curvature is, up to a sign, equal to this antisymmetric part, see (2.18), we find that the R curvatures are the same for the large and for the small space. As explained in Sect. 3.4.2, they can both be transformed to zero, using a ξ -transformation in the large ξX in the small space. space and a ξ -transformation with ξX = The symmetric part of (4.6) determines the symmetric part of the Ricci tensor for the quaternionic manifold. Using this in (2.21) gives BXY =
1 XY − 1 hXY . R 4(nH + 1) 4
(4.8)
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Then we see that the extra terms in (4.5) constitute the Ricci part of the curvature using the symmetric part of B: Ric W W Rsymm = − 21 δ[X hY ]Z + 21 JZ W · J[X V hY ]V + 21 JZ V · J[X W hY ]V . XY Z
We can then identify
Ric XY Z W = R − Rsymm R
W
XY Z
Ric = R (W) + Rantis
XY Z
W
.
(4.9)
(4.10)
For the full hypercomplex manifold, we have to use r = nH + 1 in (2.21) (with only the antisymmetric part), and thus XY = B
1 nH + 1 Ric Ric W antis = R[XY ] ⇒ R R XY Z W . XY Z 4(nH + 2) nH + 2 antis
(4.11)
and W tensors, The Weyl parts of the curvatures are determined by the traceless W but, as mentioned above in (4.3), in the hypercomplex space this is dependent on the ABC C . We can extract the Weyl part of the curvature of the quaternionic space trace W from (4.10) and find ABC D = WABC D + W
3 BC)E E . δD W 2(nH + 1) (A
(4.12)
ABC D . Hence, WABC D is the traceless part of W Therefore, the main results are: 1. The antisymmetric Ricci tensor of the hypercomplex manifold is the same as the antisymmetric Ricci tensor of the quaternionic manifold. 2. The symmetric Ricci part of the quaternionic manifold is a universal expression in terms of the candidate metric h (the same as for HP nH ). tensor 3. The traceless W tensor of the quaternionic space is the traceless part of the W of the hypercomplex space. As mentioned, we derived everything for a special choice of ξ . However, we have seen in Sect. 2.2 that the Weyl tensor is not changed by a ξ -transformation. Hence, the last conclusion is valid for any ξ . Symbolically, we can represent the dependence of parts of the curvature tensors on basic tensors as follows: = R hXY ↓ Ric R = Rsymm
Ric + R (W) R antis ↑
↑ ABC C WABC D W ↓ ↓ Ric + Rantis + R (W)
(4.13)
For the mapping between a hyper-K¨ahler manifold and a quaternionic-K¨ahler manABC D is ifold, there is no antisymmetric part of the Ricci tensor, and hence also W traceless. Hence, the relation (4.12) reduces to ABC D = WABC D . W
(4.14)
It implies that the hyper-K¨ahler curvature components along the quaternionic directions are the Weyl part of the quaternionic-K¨ahler curvature XY Z W = R (W) XY Z W . R
(4.15)
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The Ricci part of this quaternionic-K¨ahler curvature is the same expression as for HP nH . In fact, we find BXY = − 41 hXY .
(4.16)
The metric of the small space inherited from the large space is gXY = z0 hXY , see (3.46). Comparing (4.16) with the relation (2.57), using this metric implies ν=−
1 . z0
(4.17)
In the context of supergravity, the value of z0 determines the normalization of the Einstein term and is fixed to z0 = κ −2 , where κ is the gravitational coupling constant. Finally, note that for the 1-dimensional case, nH = 1, we had restricted the definition of quaternionic manifolds with special requirements in Appendix B.4 of [4], as was also done in the mathematical literature [43]. Here we find that these relations are automatically fulfilled in the embedded quaternionic manifolds. Hence, they are unavoidable in a supergravity context. 5. Reduction of the Symmetries 5.1. Symmetries and moment maps. The main part of this subsection is a summary of the results on symmetries given in [4]. In the case of manifolds where there is no (good) metric, the question of defining symmetries needs some careful consideration. In general, we consider transformations δq X = kIX (q)I , where the index I runs over the set of possible symmetries. We will define when these are called ‘symmetries’, and then we define ‘quaternionic symmetries’. A set of vector fields kI are symmetry generators if the following condition on the connection and curvature are met: DX DY kIZ = RXW Y Z kIW .
(5.1)
For Riemannian manifolds with a metric gXY , this is just the integrability condition that follows from the Killing equation D(X kY )I = 0, where kXI = gXY kIY . On the other hand, (5.1) does not imply a Killing equation as it is independent of a choice of metric. E.g. the conformal Killing vectors, which do not satisfy the Killing equation, satisfy (5.1). However, it is a sufficient condition to define ‘symmetries’ if there is no metric available. The ‘physical’ origin for this condition is the following. For simplicity, consider the equations of motion for a rigid non-linear sigma model: 2q X = ∂µ ∂ µ q X + ∂µ q Y ∂ µ q Z Y Z X = 0 .
(5.2)
Consider the transformation δq X = kIX I on this field equation: δ2q X = ∂Y kIX I 2q Y + ∂µ q Y ∂ µ q Z DZ DY kIX I − ∂µ q Y ∂ µ q Z kIV I RZV Y X .
(5.3)
Hence, the set of equations of motion is left invariant iff the defining condition (5.1) for a ‘symmetry’ is satisfied .
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Symmetry generators are quaternionic if the vector fields normalize the complex structures, which means LkI JX Y = rI × JX Y ,
(5.4)
for some 3-vectors rI . Using the SU(2) connections, this defines moment maps ν PI (q) by14 ν PI ≡ − 21 rI − kIX ω X.
(5.5)
See [4] for more information, where it was also shown that this leads to a decomposition of the derivatives of the kIX as DX kIY = ν JX Y · PI + LX Y A B tI B A .
(5.6)
The so-called moment maps PI (q) describe the SU(2) content of the symmetry and the tI B A (q) describe the G (r, H) content. The LX Y A B symbols are defined as in (A.2). We can extract ν PI from (5.6) as 4rν PI = −JX Y DY kIX .
(5.7)
If we project the curvature tensors along the symmetry vectors, we get the following relations:15 XY k Y = −νDX PI , R I
RXY B A kIY = DX tI B A .
(5.8)
In supersymmetric models, the condition of quaternionic symmetries is necessary for the invariance of the full field equations including the fermions. The vector fields generate a Lie-algebra: Y X 2k[I ∂Y kJX] = −fI J K kK .
(5.9)
The vector at the left-hand side of this equation for any [I J ] satisfies the two conditions mentioned above, (5.1) and (5.4), and this equation is thus a statement of completeness of the set of quaternionic symmetries. It leads to an important property of the moment maps, which is the ‘equivariance relation’ Y W k Y k W − νfI J K PK = 0. −2ν 2 PI × PJ + R I J
(5.10)
The absence of the SU(2) curvature parts for hypercomplex (and hyper-K¨ahler) manifolds implies that we can take ν = 0 for these manifolds. As also ω X = 0 in that case, we see from (5.4) and (5.5) that the Lie derivative along the symmetry generators of the complex structures vanishes. The symmetries are then called triholomorphic. Though ν = 0 implies that the moment maps do not appear for hyper-K¨ahler manifolds in the above relations, moment maps still appear in the Lagrangian for sigma 14 We define here ν P , where ν is a number. For quaternionic-K¨ahler manifolds, this number is defined I by (2.57), while it is not specified in quaternionic manifolds. This normalization is convenient for comparison with other papers, and, as will be shown below, because the formulae are then applicable to hyper-K¨ahler manifolds upon setting ν = 0. 15 The first of these equations holds for any choice of SU(2)-connection ω X , if a corresponding PI is defined by (5.5).
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models on these manifolds. They are defined by a relation consistent with the equations for quaternionic manifold (5.8) and (2.58) for any ν: JXY kIY = −2∂X PI .
(5.11)
Notice that we have used the existence of a metric here, so this relation only defines moment maps for hyper-K¨ahler manifolds. They should still satisfy the ν = 0 limit of the equivariance relation (5.10): kIX JXY kJY = 2fI J K PK .
(5.12)
We can summarize the various cases as follows: – Hypercomplex : Triholomorphic symmetries must satisfy (5.1) and LkI JX Y = 0.
(5.13)
There exist no moment maps. – Hyper-K¨ahler : conditions for the triholomorphic symmetries as for hypercomplex. The condition for a triholomorphic isometry can be translated to the existence of a triplet of moment maps (5.11). – Quaternionic-K¨ahler : in this case, any isometry normalizes the quaternionic structure [44, 45]. Indeed, if we define the moment maps for any Killing vector kI as in (5.7), and take a covariant derivative, the first equation of (5.8) follows from (5.1), the covariant constancy of J and the decomposition (2.15). The proportionality of the SU(2) curvature and complex structure, (2.58), implies that this can be written as a covariant version of (5.11). Then the Killing equation implies JX Z DZ kIY = −JX Z DY kI Z = 2DY DX PI ,
−DX kIZ JZ Y = −2DX DY PI . (5.14)
It is straightforward to show that (5.4) is satisfied, as this reduces to the sum of these two expressions, see [4, (B.80)]. Thus, all isometries on a quaternionic-K¨ahler manifold are quaternionic. – Quaternionic manifolds: Not all symmetries are quaternionic, i.e. satisfying (5.4). However, when they are, moment maps exist and are given by (5.7). Not all ξ -transformations preserve the symmetries. Indeed, the condition (5.1) is not invariant under general ξ -transformations. Writing (5.1) with ξ -transformed connections modifies it proportional to ZU SXY ξW DU kIW + kIW DW ξU . (5.15) We thus conclude that a symmetry is a symmetry after a ξ -transformation iff LkI ξX = kIY ∂Y ξX + ξY ∂X kIY = 0.
(5.16)
The condition for a symmetry to be quaternionic, (5.4), is invariant if rI is invariant. We can obtain the transformation of the moment map from (5.7). This leads, using (A.16), to = ν P − J Y k X ξ . νP I I X I Y This implies indeed that rI is an invariant for ξ -transformations.
(5.17)
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5.2. Conformal hypercomplex. As explained in Sect. 3.1, hypercomplex and hyperK¨ahler manifolds with a homothetic Killing vector (1.1) have additional properties. First of all, as follows from (3.5), both dilatations and SU(2) transformations define symmetries in the sense of (5.1). Notice that, when there is a metric, these symmetries are not always isometries of the metric. Precisely, the dilatations are of this type, as they satisfy the conformal Killing equation instead of the Killing equation. The dilatation symmetry is triholomorphic. The SU(2) transformations are ‘quaternionic symmetries’, with rI = − 3 as matrix in the 3 components of vectors and three values of I . For the remainder of this section, we concentrate on possible additional symmetries, other than the dilatations and SU(2) transformations. We require such symmetries to commute with the dilatational symmetry. This condition is expressed as ˆ Xˆ 3 Xˆ kY D Yˆ kI = 2 kI .
(5.18)
For triholomorphic symmetries, this implies [4] ˆ Xˆ 1 Yˆ Xˆ kY D Yˆ kI = 2 kI J Yˆ ,
(5.19)
which expresses the fact that they also commute with the SU(2) transformations. For conformal hyper-K¨ahler manifolds, one can deduce more identities. In fact, contracting (5.18) with kXˆ , one finds ˆ
ˆ
k X gXˆ Yˆ kIY = 0.
(5.20)
Moreover, the consistency with the conformal symmetry allows us to integrate (5.11) in terms of the moment maps [17], such that ˆ ˆ ˆ Yˆ = k Xˆ −6P JXˆ Yˆ J Zˆ DYˆ kI Xˆ . kIY = − 23 k X k Z I
(5.21)
(a Fayet-Iliopoulos term) is not allowed in conA possible integration constant in P I formally invariant actions, see [44]. 5.3. The map of the symmetries. In this section, we will discuss the reduction of triholomorphic symmetries of the large space to quaternionic symmetries of the small space. We also illustrate how triholomorphic isometries and moment maps on hyper-K¨ahler spaces descend to quaternionic isometries on quaternionic-K¨ahler manifolds. 5.3.1. Conformal hypercomplex and quaternionic symmetries. We now show that the components of the higher-dimensional symmetry generators lying along the quaternionic directions are bona-fide symmetry generators of the quaternionic space. First of all, for any triholomorphic symmetry on a conformal hypercomplex manifold, the reduction of Eqs. (5.18) and (5.19) leads to the z0 and zα dependence of the higherdimensional symmetry generators. The form of the symmetry vectors is as follows: I , kIα = kα · Q
kI0 = z0 VI (q),
kIX = kIX (q),
(5.22)
where all z0 -dependence is explicitly indicated, and I = 0 , α ×) Q (∂α + m while VI and
kIX
are independent of
zα .
(5.23)
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This can now be used to study the normalization of the complex structures (5.4) on ˆ the quaternionic space. Using the triholomorphicity of kIX , together with (3.13), (3.22) and (5.22), it is easy to show that the vector field with components kIX = kIX normalizes the quaternionic complex structures, I × JX Y . LkI JX Y = Q
(5.24)
I with the vector rI in (5.4). The X 0 component of the normalization This identifies Q condition gives
1 A) I − 2kIY R(− −JX Y ∂Y VI + ∂X − AX × AY kIY − Q = 0. (5.25) 2 YX The relations (5.8), using footnote 15 with ω X = − 21 AX , implies that the last two terms cancel, and we find ∂X VI = 0.
(5.26)
The constant contribution in VI can be set to zero. Indeed, this just reflects that the dilatation vector (3.8) is a triholomorphic symmetry vector. We can indicate this as the symmetry with label I = 0: ˆ ˆ k0X = 3z0 δ0X .
(5.27)
We can thus subtract this from all the other symmetries, redefining them as ˆ ˆ ˆ kIX = kIX − 3z0 δ0X VI .
(5.28)
This redefined symmetry vector is of the form (5.22) with VI = 0. One may now verify explicitly that the condition for a symmetry in the large space reduces to the condition that kIX is a symmetry in the small space. Excluding the dilatation symmetry, the result is thus that any triholomorphic symmetry that commutes with the dilatations is of the form kI0 = 0, kIα = kα · rI , kIX = kIX (q), (5.29) where kIX is a quaternionic symmetry of the small space, satisfying (5.4). With this formula, we can thus also uplift any quaternionic symmetry of the small space to a triholomorphic symmetry of the large space preserving dilatations. Further, we consider the algebra in the large space:
Y X 2kˆ[I ∂YkˆJX] = −fI J K kˆK .
(5.30)
Reducing this equation for the different values of Xˆ leads to Y X X : 2k[I ∂Y kJX] = −fI J K kK ,
Y W k Y k W − νfI J K PK = 0, α : −2ν 2 PI × PJ + R I J 0 0 : fI J K = 0. kK
(5.31)
The first equation says that the algebra in the small space is the same as the algebra in the large space, excluding dilatations. The second equation is the equivariance condition (5.10). The third one says that the dilatations do not appear in the right-hand side of commutator relations. Remark that all equations obtained in this section are invariant under ξ -transformations satisfying (5.16).
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5.3.2. Isometries on conformal hyper-K¨ahler and quaternionic-K¨ahler spaces. When there is a metric, we find that the only non-trivial part of the Killing equation in the large space is (X kˆY )I = z0 D(X kY )I . D
(5.32)
Hence, a triholomorphic symmetry preserving the dilatations is an isometry if and only if it is an isometry of the quaternionic-K¨ahler space. For conformal hyper-K¨ahler manifolds, the moment map is defined in (5.21). Using the decomposition of the symmetry vector (5.29) and (4.17), we find = − 1 k Xˆ Yˆ X 0 1 Xˆ 1 0 ˆ Yˆ P J = − g = z − A k − z + A r = P 2 ω kIX . k k k I I X I X X ˆ ˆ ˆ I 6 2 2 X IY XY I (5.33) As in the quaternionic-K¨ahler spaces we have (3.49), the last term vanishes and the moment map in the hyper-K¨ahler space is equal to the moment map in the quaternionic-K¨ahler space. 5.3.3. The conformal hyper-K¨ahler manifold of quaternionic dimension 1. In quaternionic manifolds, the moment map is completely determined, see (5.7). Fayet-Iliopoulos (FI) terms are in general undetermined constants in the moment maps (see the review [46]). Hence, they are not present in quaternionic-K¨ahler manifolds, except for the ‘trivial situation’ nH = 0. The latter corresponds to a large space of quaternionic dimension 1, which is hyper-K¨ahler as the metric is given in the first line of (3.51): g00 = −
1 , z0
g0α = 0,
gαβ = −z0 m α · m β.
(5.34)
This metric has SU(2) × SU(2) isometries. The first factor is generated by kα , but these are not triholomorphic. However, there is a commuting set of SU(2) Killing vecα , with I = 1, 2, 3, and the minus sign indicating the second SU(2) factor in the tors, k−I holonomy group, different from kα . The generators are thus kI0 = 0,
α kIα = k−I .
(5.35)
These are triholomorphic with respect to the complex structures defined by the first Killing vectors: J0 0 = 0, J0 β = 10 kβ , z
Jα 0 = −z0 m α, Jα β = kβ × m α.
(5.36)
This leads to the moment maps α −2ν PI = rI = m α k−I .
(5.37)
When a gauge fixing is taken, i.e. the zα are fixed to a value, then this leads to some constants. These are the SU(2) FI terms. 6. Summary and Discussion For the convenience of the reader we summarize the main results of this paper. The map is schematically represented in Fig. 1, which will be further explained in this section.
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Fig. 1. The map schematically. The two blocks represent the families of large (upper block) and small spaces (lower block), where the horizontal lines indicate how they are related by ξ , resp. ξ , transformations. They connect parametrizations of the same manifold with different complex structures for hypercomplex manifolds, and different affine and SU(2) connections for the quaternionic manifolds. At the right, the spaces have no R curvature, and part of these are hyper-K¨ahler, resp. quaternionic-K¨ahler. The latter two classes are indicated by the thick lines. The vertical arrows represent the map described in this paper, connecting the manifolds with similar parametrizations. They are a representation of the map between the horizontal lines, which is the map between hypercomplex and quaternionic manifolds. The thick arrow indicates the map between hyper-K¨ahler and quaternionic-K¨ahler spaces
6.1. The map from hypercomplex manifolds. The setup starts from a 4(nH + 1)-dimensional space that is hypercomplex and has a conformal symmetry. The latter is mathematically expressed as the presence of a closed homothetic Killing vector ˆ ˆ ˆ ˆ k Xˆ ≡ ∂ ˆ k Xˆ + D Yˆ Zˆ X k Z = 23 δYˆ X . Y Y
(6.1)
Observe that the coordinates and covariant derivatives in this large space are indicated by hatted quantities. The vector k, which is the generator of dilatations, yields also
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generators of SU(2) transformations: JYX k Y . kX ≡ 13
(6.2)
We use coordinates in this large space that are adapted to these quantities: α = 1, 2, 3, X = 1, . . . , 4nH , q X = z0 , y p = z0 , zα , q X ,
k X = 3z0 δ0X ,
k0 = kX = 0.
(6.3)
This leads to the following form of the complex structures: J0 0 = 0, J β = 1 kβ , 0
z0
J0 Y = 0,
Jα 0 = −z0 m α, β β Jα = k × m α, Y Jα = 0,
JX 0 = z0 AX , JX β = AX × kβ + JX Z (AZ · k β ), (6.4) J Y = J Y . X
X
α is the inverse as a 3 × 3 matrix. The result Here, kα are SU(2) Killing vectors, and m thus depends on z0 , these SU(2) Killing vectors, complex structures on the small space JX Y and a vector AX . The unique torsionless affine connection is the Obata connection ˆ ˆ ˆ ˆ ˆ J(Xˆ U × ∂|Uˆ | (6.5) JWˆ Z . JYˆ ) W + JYˆ ) W · Xˆ Yˆ Z = − 16 2∂(Xˆ The condition that this connection preserves the complex structures implies that α ×) AX + ∂X m α = 0, ∂0 AX = 0, (∂α + m Y ∂0 JX = 0, α ×) JX Y = 0, (∂α + m
1 A) R(− ≡ −∂[X AY ] + 21 AX × AY = 21 hV [X JY ] V , 2 XY
(6.6)
for some symmetric tensor hXY . Furthermore it implies that the JX Y are a quaternionic structure on the small space. Hypercomplex manifolds have no SU(2) curvature, but can have a non-vanishing R curvature, which is Hermitian, i.e. it commutes with the hypercomplex structure, see Appendix A. This R curvature is equal to the Ricci tensor of these manifolds, which is antisymmetric, RR = −RXˆ Yˆ . Xˆ Yˆ
(6.7)
We found that in these conformal hypercomplex manifolds there is a transformation ξXˆ that preserves the hypercomplex structure, Y + 2 Y Z Y Y . Z( ( Jξ )X ξ k ) − ( ξ k ) J J (6.8) = JX X Z Z X 3 The one-form with components ξX satisfies conditions implying, in the parametrization of (6.4), that the vector has only coordinates ξX . Furthermore this vector can only depend on q X and is such that ∂[X ξY ] is Hermitian. The transformation is in this basis generated by ξZ , δ( ξ )AX = 2JX Z
(6.9)
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while kα and JX Y are invariant. These transformations can always be used to obtain R = 0. They are represented by the horizontal lines in the upper part of Fig. 1. R Another invariance that is present in the conformal hypercomplex manifolds is a local SU(2), which acts as δSU(2) AX = ∂X + × AX ,
δSU(2) JX Y = × JX Y .
(6.10)
If the manifold admits a metric, and hence would be hyper-K¨ahler, it must be equal to 0 X gX Y = 2 Y .
(6.11)
In the parametrization that we use, the full form is d s2 = −
(dz0 )2 0 + z hXY (q)dq X dq Y z0 + gαβ [dzα − AX (z, q) · kα dq X ][dzβ − AY (z, q) · kβ dq Y ] , (6.12)
where gαβ = −
1 kα · kβ = −z0 m α · m β, z0
hXY ≡
1 gXY + AX · AY . z0
(6.13)
hXY is the quantity that appears in (6.6). These formulae are very reminiscent of a Kaluza-Klein reduction on an SU(2) group manifold.
6.2. The quaternionic manifold. We have proven that the 4nH -dimensional subspace described by the q X is a quaternionic space. This means that we restrict the 4(nH + 1) dimensional space by 4 gauge choices for the dilatations and SU(2) transformations spanned by kα . In the context of superconformal tensor calculus, these are the transformations that are present in the superconformal group and should be gauge-fixed by the compensating hypermultiplet [26,47]. The gauge fixing of dilatations is done by fixing a value of z0 , which will set the scale of the manifold, see below. On the other hand, SU(2) is gauge-fixed by choosing a value for the zα coordinates. For any value of zα we thus find a quaternionic manifold. These are related by SU(2) transformations. This means that objects that depend on zα are gauge-dependent. Some intrinsic quantities, like the affine connection and the metric on the small space, turn out to be zα -independent. The geometric quantities in the small space are not uniquely defined. In particular, we can take different choices for the affine connection and for the SU(2) connection ω X, which is the gauge field of the transformations (6.10), restricted to the small space X. X = − 21 ∂X + × ω δSU(2) ω
(6.14)
One particular choice is the Oproiu connection [27], Op ω = − 1 2AX + AY × JX Y , X
Op
XY
Z
6
≡
Ob
XY
Z
− J(X Z · ω Y ) , Op
(6.15)
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where Ob XY Z is the Obata connection on the small space. These connections can be changed by ξ -transformations [7, 23, 24], Z ξY ) − 2J(X Z · JY ) W ξW , XY Z → XY Z + 2δ(X
ω X → ω X + JX W ξW .
(6.16)
These ξ -transformations are represented by the horizontal lines in the lower part of Fig. 1. As shown in that picture, they can e.g. be used to remove the R curvature in quaternionic manifolds. They can also be used to obtain a simple form for the connections and curvatures: ω X = − 21 AX ,
XY Z = XY Z + A(X · JY ) Z .
(6.17)
This choice of a quaternionic connection will turn out to be special for two different reasons. With this choice R = −RR = R[XY ] . XY = −R R XY XY
(6.18)
This equality implies that the map can be represented by vertical arrows in Fig. 1. Since in the hypercomplex space this curvature is proportional to the Ricci tensor, which is Hermitian, this implies that the R curvature on the small space is Hermitian. For uplifting a quaternionic manifold to a hypercomplex manifold using (6.4) one thus first has to apply ξ -transformations such that this condition is satisfied, as one can see in Fig. 1. The SU(2) curvature is XY = 2J[X Z BY ]Z , R
where
BXY =
1
XY − 1 hXY . R 4(nH + 1) 4
(6.19)
A quaternionic space is quaternionic-K¨ahler if and only if the Ricci tensor is Hermitian, and its symmetric part is invertible. This implies that the antisymmetric part vanishes, and the symmetric part is proportional to the metric. In the basis that we presented, the induced metric in the small space is gXY = z0 hXY .
(6.20)
This metric does not depend on zα . In supergravity, z0 is fixed to κ −2 , where κ is the gravitational coupling constant. The value of z0 fixes the scale of the manifold in the sense that XY = 1 ν JXY , R 2
R = 4nH (nH + 2)ν,
1 z0 = − , ν
(6.21)
where R is the Ricci scalar. The Levi-Civita connection of the metric is the one in (6.17), and this fixes the ξ -transformations. Therefore, these manifolds can be represented by the vertical thick lines in Fig. 1.
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6.3. Curvatures. The curvatures of conformal hypercomplex manifolds have as only non-vanishing components XY Z W , R
XY Z 0 , R
XY Z α . R
(6.22)
The latter two do not contribute in the map to the quaternionic space. The first one is expressed using vielbeins in a W -tensor, which plays an important role in supersymmetry: ˆ Yˆ Zˆ kA Wˆ CDB A ≡ 1 ε ij fX W j C f iD fkB f ˆ RXˆ Yˆ Zˆ . 2 W
(6.23)
It is symmetric in its lower indices. It is however not traceless in general. Its trace determines the Ricci tensor, and is thus zero for hyper-K¨ahler. The curvatures can be split into a Ricci part and a Weyl part. The former further splits into a ‘symmetric’ and an ‘antisymmetric’ contribution, W Ric XY Z W = R antis (W) XY Z W , R +R XY Z W Ric Ric + Rantis + R (W) XY Z W . (6.24) RXY Z W = Rsymm XY Z
There is no Ricci-symmetric part for the hypercomplex space. With the special ξ -choice as in (6.17), we can relate the different parts as follows: Ric W W Rsymm = − 21 δ[X hY ]Z + 21 JZ W · J[X V hY ]V + 21 JZ V · J[X W hY ]V , XY Z Ric W Rantis = XY Z
nH + 2 Ric R XY Z W , nH + 1 antis jB
W R (W) XY Z W = − 21 fXiA εij fY fZkC fkD WABC D ,
(6.25)
where ABC D − WABC D = W
3 2(nH + 1)
D δ(A WBC)E E .
(6.26)
Observe that the symmetric Ricci part is a universal expression in terms of hXY . The Weyl part of quaternionic manifolds is determined by a tensor WABC D , which is the traceless part of the one mentioned in (6.23). The Weyl tensor of quaternionic manifolds is invariant under the ξ -transformations. For the case of hyper-K¨ahler and quaternionic-K¨ahler manifolds, the antisymmetric parts are absent, and the Weyl parts are identical.
6.4. Symmetries. Triholomorphic symmetries in the hypercomplex space are either the dilatations or they can be decomposed as kI0 = 0,
kIα = kα · rI ,
kIX = kIX (q),
(6.27)
where kIX is a quaternionic symmetry of the small space, such that LkI JX Y = rI × JX Y .
(6.28)
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Using the SU(2) connections, this defines moment maps ν PI by ν PI ≡ − 21 rI − kIX ω X.
(6.29)
For hyper-K¨ahler manifolds there is also a moment map, which is equal to the one in the corresponding quaternionic-K¨ahler space. Only ξ -transformations such that LkI ξX = kIY ∂Y ξX + ξY ∂X kIY = 0
(6.30)
preserve symmetries. The vectors in (6.27) do not change under these ξ -transformations. The moment map transforms as = ν P − J Y k X ξ . νP I I X I Y
(6.31)
A conformal 1-dimensional hypercomplex manifold is always hyper-K¨ahler. In this case there is no quaternionic manifold after the map, but there are possible constant moment maps after a point zα is fixed by the SU(2) gauge. This is the origin of the Fayet-Iliopoulos terms in supergravity when there are no physical hypermultiplets. 6.5. Remarks. In this paper, we have treated the case of negative scalar curvature for the quaternionic-K¨ahler space. This corresponds to ν < 0, see (6.21). If there are continuous isometries, this implies that the space is non-compact. Supergravity restricts us to those manifolds, as is clear from the relation with κ 2 . This is obtained by an indefinite signature in the hyper-K¨ahler space, as can be seen in (6.12). The terms for z0 and zα have negative signature, and we choose hXY to be positive definite. However, we can as well generalize the equations to z0 < 0, which then implies a positive definite metric for the hyper-K¨ahler manifold. The scalar curvature of the quaternionic-K¨ahler manifold is then positive, and this allows compact isometry √ groups. The only place where this is non-trivial is for vielbeins in Sect. 3.6, where z0 appears. However, this can be cured by inserting appropriate i factors. Furthermore, we never used properties of positive definiteness of hXY , so that this choice can easily be relaxed, and the analysis applies to any signature. The signature of h and the sign of ν determine the signature of the large space. We have developed this paper in relation to matter couplings to supergravity in 5 spacetime dimensions. The procedure is, however, independent whether it is applied to 3, 4, 5 or 6 dimensional supersymmetry with 8 supercharges [19]. Therefore the analysis of this paper can be applied to superconformal tensor calculus in general. The generalization of supersymmetric theories with hypercomplex or quaternionic target spaces is related to the idea that physical systems can be defined by field equations without necessity of an action. Indeed, the metric is only necessary for the construction of an action, while supersymmetry transformations and field equations are independent of a metric. Quaternionic-K¨ahler manifolds appear as moduli spaces for type II superstring compactifications on Calabi-Yau 3-folds. It would be very interesting to understand the role of general quaternionic manifolds in the context of such compactifications. In mathematics, the map that is described in this paper was investigated in [21,22]. We have pointed out that the corresponding manifolds in the hypercomplex/hyper-K¨ahler picture are conformal, and discovered some new properties. E.g. the ξ and SU(2) transformations in these manifolds played an important role. We hope that this stimulates
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further investigations in such conformal hypercomplex manifolds. Furthermore, we gave new contributions to the understanding of symmetries, generalizing isometries, in quaternionic manifolds. Acknowledgement. We are grateful to D. Alekseevsky, J. Figueroa-O’Farrill, G. Gibbons, M. Lled´o, G. Papadopoulos, Ph. Spindel and V. Cort´es for interesting and stimulating discussions. We thank the University of Torino for hospitality in the final stage of the preparation of this paper. Work supported in part by the European Community’s Human Potential Programme under contract MRTN-CT-2004-005104 ‘Constituents, fundamental forces and symmetries of the universe’. The work of J.G. is supported by the FWO-Vlaanderen to which he is affiliated as postdoctoral researcher. The work of E.B. and T.d.W. is part of the research program of the Stichting voor Fundamenteel Onderzoek der materie (FOM). The work of S.C., J.G. and A.V.P. is supported in part by the Federal Office for Scientific, Technical and Cultural Affairs through the ‘Interuniversity Attraction Poles Programme – Belgian Science Policy’ P5/27. The work is supported by the Italian M.I.U.R. under the contract P.R.I.N. 2003023852, ‘Physics of fundamental interactions: gauge theories, gravity and strings’.
A. Projections Depending on the Complex Structure The complex structures are defined from the vielbeins as JX Y ≡ −ifXiA σi j fjYA ,
(A.1)
where σ are the three Pauli matrices. Similar matrices are Z iB fW . LW Z A B ≡ fiA
(A.2)
They project e.g. the curvature to the G (r, H) factor as in (2.15). The matrices LA B and J commute and their mutual trace vanishes JX Y LY Z A B = LX Y A B JY Z ,
JZ Y LY Z A B = 0.
(A.3)
Other useful properties are LX Y A B LY Z C D = LX Z C B δA D , LX X A B = 2δA B , L Y B L X D = 2δC B δA D , X A Y C LZ W A B LX Y B A = 21 δX W δZ Y − JX W JZ Y . Bilinear forms are projected to Hermitian ones using XY ZW ≡ 1 δX Z δY W + JX Z · JY W . 4
(A.4)
(A.5)
As a projection operator, it squares to itself: XY ZW ZW U V = XY U V .
(A.6)
Useful relations are JZ U δY V − δZ U JY V − JZ U × JY V J[Z X |X|Y ] (U V ) + J(Z X |X|Y ) [U V ] , δX Z JY U − δY U JX Z − JX Z × JY U = [XY ] (Z|V | JV U ) + (XY ) [Z|V | JV U ] .
4JZ X XY U V = = ZV U 4XY JV =
(A.7) (A.8)
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A Hermitian bilinear form is a tensor FXY such that J α X Z J α Y W FZW = FXY (no sum over α). Any bilinear form can be projected to the space of Hermitian bilinear forms by the projection . It is easy to prove that if XY ZW FZW = FXY ,
(A.9)
this implies the hermiticity of F . Thus, Hermitian bilinear forms are those that satisfy (A.9). Moreover, one can prove that they satisfy FXU JY U = 21 JX U × JY V FU V ,
(A.10)
FXU JY U + JX U FU Y = 0.
(A.11)
and thus also
Inversely, this equation [or (A.10)] is also sufficient for a form to be Hermitian. Finally, another useful matrix between bilinear forms is Z W SXY ZW ≡ 2δ(X δY ) − 2JX (Z · JY W ) = 4δ(XY ) ZW − 8(XY ) ZW .
(A.12)
It satisfies the following identities: V , SW X W V = 4(r + 1)δX ZV U (Z U ) SXY JV = 4δ(X JY ) + 2J(X Z × JY ) U , SZW XV JV Y − JW V SZV XY = 2JZ X × JW Y , SXZ Y W JW Z = 4nH JX Y ,
ST [X (W U SY ]Z V )T = 0,
(A.13) (A.14) (A.15) (A.16) (A.17)
which are used at various places in the main text. References 1. Zumino, B.: Supersymmetry and K¨ahler manifolds. Phys. Lett. B87, 203 (1979) 2. Alvarez-Gaume, L., Freedman, D. Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric sigma model. Commun. Math. Phys. 80, 443 (1981) 3. Bagger, J., Witten, E.: Matter couplings in N = 2 supergravity. Nucl. Phys. B222, 1 (1983) 4. Bergshoeff, E., Cucu, S., de Wit, T., Gheerardyn, J., Halbersma, R., Vandoren, S., Van Proeyen, A.: Superconformal N = 2, D = 5 matter with and without actions JHEP 10, 045, (2002) 5. Gheerardyn, J.: Aspects of on-shell supersymmetry. http://www.arXiv.org/abs/hep-th/0411126, Ph.D. thesis, Leuven University, 2004 6. Salamon, S.: Differential geometry of quaternionic manifolds. Ann. Scient. Ec. Norm. Sup., 4`eme serie 19, 31–55 (1986) 7. Alekseevsky, D. V., Marchiafava, S.: Quaternionic structures on a manifold and subordinated structures. Ann. Matem. pura appl. (IV) 171, 205–273 (1996) 8. Gibbons, G. W., Papadopoulos, G., Stelle, K. S.: HKT and OKT geometries on soliton black hole moduli spaces. Nucl. Phys. B508, 623–658. (1997) 9. Michelson, J., Strominger, A.: The geometry of (super)conformal quantum mechanics. Commun. Math. Phys. 213, 1–17 (2000) 10. Spindel, P., Sevrin, A., Troost, W., Van Proeyen, A.: Extended supersymmetric sigma models on group manifolds. 1. The complex structures. Nucl. Phys. B308, 662 (1988) 11. Ferrara, S., Kaku, M., Townsend, P. K., van Nieuwenhuizen, P.: Gauging the graded conformal group with unitary internal symmetries. Nucl. Phys. B129, 125 (1977)
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12. Kaku, M., Townsend, P. K., van Nieuwenhuizen, P.: Properties of conformal supergravity. Phys. Rev. D17, 3179 (1978) 13. Kaku, M., Townsend, P. K.: Poincar´e supergravity as broken superconformal gravity. Phys. Lett. B76, 54 (1978) 14. Van Proeyen, A.: Special geometries, from real to quaternionic. http://arXiv.org/abs/hep-th/ 0110263, 2001 Proceedings of the ‘Workshop on special geometric structures in string theory’, Bonn, 811/9/2001; Electronic library of Mathematics: http://www.emis.de/proceedings/SGSST2001/, 2001 15. Galicki, K.: Geometry of the scalar couplings in N = 2 supergravity models. Class. Quant. Grav. 9, 27–40 (1992) 16. de Wit, B., Kleijn, B., Vandoren, S.: Rigid N = 2 superconformal hypermultiplets. In: Supersymmetries and Quantum Symmetries, Proc. Int. Sem. Dubna (1997), J. Wess, E.A. Ivanov, (eds.)Lecture Notes in Physics, Vol. 524, Springer, Berlin-Heidelberg, Newyork: 1999, p. 37 17. de Wit, B., Kleijn, B., Vandoren, S.: Superconformal hypermultiplets, Nucl. Phys. B568, 475–502 (2000) 18. de Wit, B., Roˇcek, M., Vandoren, S.: Hypermultiplets, hyperk¨ahler cones and quaternion-K¨ahler geometry. JHEP 02, 039. (2001) 19. Rosseel, J., Van Proeyen, A.: Hypermultiplets and hypercomplex geometry from 6 to 3 dimensions, Class. Quant. Grav. 21, 5503–5518 (2004) 20. Sezgin, E., Tanii, Y.: Superconformal sigma models in higher than two dimensions. Nucl. Phys. B443, 70–84 (1995) 21. Swann, A.: HyperK¨ahler and quaternionic K¨ahler geometry. Math. Ann. 289, 421–450 (1991) 22. Pedersen, H., Poon, Y. S., Swann, A. F.: Hypercomplex structures associated to quaternionic manifolds. Diff. Geom. Appl. 9, 273–292 (1998) 23. Fujimura, S.: Q-connections and their changes on almost quaternion manifolds. Hokkaido Math. J. 5, 239–248 (1976) 24. Oproiu, V.: Integrability of almost quaternal structures. An. st. Univ. Ia¸si 30, 75–84 (1984) 25. Bergshoeff, E., Cucu, S., de Wit, T., Gheerardyn, J., Vandoren, S., Van Proeyen, A.: N = 2 supergravity in five dimensions revisited. Class. Quant. Grav. 21, 3015–3041 (2004) 26. de Wit, B., Lauwers, P. G., Van Proeyen, A.: Lagrangians of N = 2 supergravity - matter systems. Nucl. Phys. B255, 569 (1985) 27. Oproiu, V.: Almost quaternal structures. An. st. Univ. Ia¸si 23, 287–298 (1977) 28. Ambrose, N., Singer, J.: A theorem on holonomy. Trans. AMS 75, 428–443 (1953) 29. Musso, E.: On the transformation group of a quaternion manifold. Boll. U.M.I., (7) 6-B, 67–78 (1992) 30. Joyce, D.: Compact hypercomplex and quaternionic manifolds. J. Diff. Geom 35, 743–761 (1992) 31. Barberis, M. L., Dotti-Miatello, I.: Hypercomplex structures on a class of solvable Lie groups. Quart. J. Math. Oxford (2) 47, 389–404 (1996) 32. Gentili, G., Marchiafava, S., Pontecorvo, M. eds.,: Quaternionic structures in mathematics and physics. 1996, Proceedings of workshop in Trieste, September 1994; ILAS/FM-6/1996, available on http://www.emis.de/proceedings/QSMP94/ 33. Marchiafava, S., Piccinni, P., Pontecorvo, M. eds.,: Quaternionic structures in mathematics and physics. World Scientific, 2001, Proceedings of workshop in Roma, September 1999; available on http://www.univie.ac.at/EMIS/proceedings/QSMP99/ 34. Obata, M.: Affine connections on manifolds with almost complex, quaternionic or Hermitian structure. Jap. J. Math. 26, 43–79 (1956) 35. Ishihara, S.: Quaternion K¨ahlerian manifolds. J. Diff. Geom. 9, 483–500 (1974) 36. Salamon, S.: Quaternionic K¨ahler manifolds. Invent. Math. 67, 143–171 (1982) 37. Wolf, J.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14, 1033–1047 (1963) 38. Bonan, E.: Sur les structures presque quaternioniennes. C.R. Acad. Sc. Paris 258, 792 (1964) 39. Bonan, E.: Connexions presque quaternioniennes. C.R. Acad. Sc. Paris 258, 1696 (1964) 40. Alekseevsky, D. V.: Riemannian spaces with exceptional holonomy group. Funct. Anal. Appl. 2, 97–105 (1968) 41. Pons, J. M., Talavera, P.: Consistent and inconsistent truncations: Some results and the issue of the correct uplifting of solutions. Nucl. Phys. B678, 427–454 (2004) 42. Boyer, C. P., Galicki, K.: 3-Sasakian Manifolds. Surveys Diff. Geom. 7, 123–184 (1999) 43. Alekseevsky, D. V.: Classification of quaternionic spaces with a transitive solvable group of motions. Math. USSR Izvestija 9, 297–339 (1975) 44. de Wit, B., Roˇcek, M., Vandoren, S.: Gauging isometries on hyperk¨ahler cones and quaternion-K¨ahler manifolds, Phys. Lett. B511, 302–310 (2001) 45. Galicki, K.: A generalization of the momentum mapping construction for Quaternionic K¨ahler manifolds. Commun. Math. Phys. 108, 117 (1987)
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46. Van Proeyen, A.: Supergravity with Fayet-Iliopoulos terms and R-symmetry. In: Proceedings of the EC-RTN Workshop ‘The quantum structure of spacetime and the geometric nature of fundamental interactions’, Kolymbari, Crete, 5-10/9/2004, Fortsch. Physik 53, 997–1004 (2005) 47. de Wit, B., van Holten, J. W., Van Proeyen, A.: Central charges and conformal supergravity. Phys. Lett. B95, 51 (1980) Communicated by G.W. Gibbons
Commun. Math. Phys. 262, 459–487 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1476-5
Communications in
Mathematical Physics
Inhomogenous Model of Crossing Loops and Multidegrees of Some Algebraic Varieties P. Di Francesco1 , P. Zinn-Justin2,3 1
Service de Physique Th´eorique de Saclay, CEA/DSM/SPhT, URA 2306 du CNRS, C.E.A.-Saclay, 91191 Gif sur Yvette Cedex, France 2 LIFR–MIIP, Independent University, 119002, Bolshoy Vlasyevskiy Pereulok 11, Moscow, Russia 3 Laboratoire de Physique Th´eorique et Mod`eles Statistiques, UMR 8626 du CNRS, Universit´e Paris-Sud, Bˆatiment 100, 91405 Orsay Cedex, France Received: 12 January 2005 / Accepted: 6 July 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: We consider a quantum integrable inhomogeneous model based on the Brauer algebra B(1) and discuss the properties of its ground state eigenvector. In particular we derive various sum rules, and show how some of its entries are related to multidegrees of algebraic varieties.
1. Introduction Recently, a new connection between quantum integrable models and combinatorics has emerged. This relation can be traced back to the idea, as expressed e.g. in [1], that in stochastic integrable processes, due to the existence of a simple ground state eigenvalue (without any finite-size corrections), the entries of the ground state are integers and must have some combinatorial significance. This idea was based on experience with a particularly successful case: the model of non-crossing loops related to the Temperley–Lieb algebra T L(1), whose special properties [2, 3] led Razumov and Stroganov to conjecture the combinatorial significance of each entry of the ground state [4]. This conjecture has generated a lot of activity (see for example references in [5]) but has not been proved yet in its full generality. The latest model that falls into the framework described above is the model of crossing loops proposed by de Gier and Nienhuis in [6], which is related to the Brauer algebra B(1) and to standard integrable models with symmetry OSp(p|2m) [7, 8], p − 2m = 1. By abuse of language, as in the non-crossing case, we shall call this model the “O(1)” crossing loop model. The novelty in the work [6] is that the entries of the ground state are integers that do not appear to be obviously related to statistical mechanics, but rather belong to the realm of enumerative geometry. Indeed some of them are conjectured to be degrees of algebraic varieties that appear in work of Knutson [9] revolving around the commuting variety. The present article tries to shed some light on the origin of these numbers in the model. In particular we shall see how the algebra of the model naturally
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leads to an action of the symmetric group as divided difference operators, which have well-known meaning in the context of Schubert calculus. Our work is motivated by recent progress in understanding the model of non-crossing loops [5] for the similar Razumov–Stroganov conjecture. The idea of [5] is to make better use of the integrability of the model. It involves in particular the introduction of inhomogeneities (spectral parameters), which give a much more powerful tool to study the ground state, whose coefficients become polynomials in these variables. Here, we shall try to do the same to the O(1) crossing loop model. As in [5], our results include multi-parameter sum rules for the entries of the ground state vector; we find in fact two different sum rules, one for the sum of all entries, and one for the sum in the so-called permutation sector, in which the entries clearly play a special role: these are precisely the coefficients which are conjecturally related to degrees of varieties. In fact we show that this connection is much deeper and that the full polynomial entries are related to so-called multidegrees. We also prove some conjectured properties formulated in [6], involving factorizability of the ground state vector entries. The paper is organized as follows. In Sect. 2 we introduce the model and its ground state eigenvector. Section 3 contains general factorization properties of the entries of the ground state, as well as their construction in terms of divided difference operators in the space of polynomials. Section 4 analyzes in detail the special case of so-called “permutation patterns”, which is also the focus of [6]; we formulate a conjecture that relates some of its entries to (multi)degrees of some algebraic varieties, prove some results including a sum rule, and give a sketch of proof of this generalized de Gier–Nienhuis conjecture. Section 5 concerns recursion relations and the sum rule for all entries. A few concluding remarks are gathered in Sect. 6. The appendices contain some explicit data for n = 2, 3, 4. 2. The Inhomogeneous O(1) Crossing Loop Model: Transfer Matrix and Ground State Vector The O(1) crossing loop model is based on the following solution to the Yang–Baxter equation, expressed as a linear combination of generators of the Brauer algebra B2n (1). These are the identity I , the “crossing” operators fi , and the generators ei of the Temperley–Lieb algebra T Ln (1), i = 1, 2, ..., 2n, with the pictorial representations I=
,
fi =
,
ei =
and acting vertically on the vector space generated by crossing link patterns, that is chord diagrams of 2n labeled points around a circle, connected by pairs via straight lines across the inner disk. We denote by CPn the set of these (crossing) link patterns on 2n points, with cardinality |CPn | = (2n − 1)!!. A simple way of indexing these link patterns is via permutations of S2n with only 2-cycles (fixed-point free involutions), each cycle being made of the labels of the two points connected via a chord. The pictorial representation above makes it straightforward to derive the B2n (1) Brauer algebra relations: ei2 = ei , fi2 = I, ei ei±1 ei = ei , [ei , ej ] = [ei , fj ] = [fi , fj ] = 0 if|i − j | > 1,
fi fi+1 fi = fi+1 fi fi+1 , fi ei = ei fi = ei (2.1)
Inhomogenous Model of Crossing Loops and Some Algebraic Multidegrees
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Looking for a solution for a face transfer matrix operator Xi (u) = a(u)I + b(u)fi + c(u)ei to the Yang–Baxter equation Xi (u)Xi+1 (u + v)Xi (v) = Xi+1 (v)Xi (u + v)Xi+1 (u)
(2.2)
further fixed by the normalization Xi (0) = I , we find the solution Xi (u) = (1 − u)I +
u (1 − u)fi + uei , 2
(2.3)
unique up to scaling of u, as a direct consequence of the relations (2.1). The solution (2.3) also satisfies the unitarity relation Xi (u)Xi (−u) = (1 − u2 )(1 − u2 /4)I.
(2.4)
This solution appeared first in [10], and was further studied in [7], and shown to be related to vertex models based on orthosimplectic groups. We now introduce an inhomogeneous integrable model based on the above solution of the Yang–Baxter equation. It is defined on an infinite cylinder of square lattice of perimeter 2n represented as an infinite strip of width 2n glued along its two borders. A configuration of the model is defined by assigning the plaquettes
,
, or
to each elementary face of the cylinder, with certain weights. In the transfer matrix approach, one considers a semi-infinite cylinder (see Fig. 1). The space of states then represents the pattern of pair connectivity of the 2n labeled midpoints of the boundary edges of the semi-infinite cylinder via plaquette configurations
1
2
3
4
...
... 2n
Fig. 1. A typical configuration of the crossing loop model on a semi-infinite cylinder of square lattice with perimeter 2n
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of the model. Finally, the transfer matrix represents the addition of one row of plaquettes to the semi-infinite cylinder: Tn (t|z1 , . . . , z2n ) 2n = (1 − t + zi ) i=1
(t − zi )(1 − t + zi ) + 2
+ (t − zi )
,
(2.5)
where the weights depend on the label i of the site, and correspond to a tilted version of the operators Xi of Eq. (2.3). The parameter t, which is independent of the row, plays no role in what follows due to the commutativity property [Tn (t), Tn (t )] = 0,
(2.6)
itself a direct consequence of the Yang–Baxter equation. For values of zi and t such that 0 < t −zi < 1, the weights are strictly positive and can be interpreted as unnormalized probabilities, and the transfer matrix as an unnormalized matrix of transition probabilities. Conservation of probability can be expressed in the following way: define the linear form vn with entries in the canonical basis vπ = 1 for all π ∈ CPn . Then summing the weights in Eq. (2.5), we obtain vn Tn (t|z1 , . . . , z2n ) = vn
2n i=1
1 (1 − (t − zi ))(1 + t − zi ). 2
(2.7)
1 This means that 2n i=1 (1 − 2 (t − zi ))(1 + t − zi ) is an eigenvalue of Tn (with left eigenvector vn ), and there must exist a right eigenvector: 2n 1 Tn (t|z1 , . . . , z2n ) − (1 − (t − zi ))(1 + t − zi )I n (z1 , . . . , z2n ) = 0.(2.8) 2 i=1
In the aforementioned range, Eqs. (2.7) and (2.8) are nothing but Perron–Frobenius eigenvector equations for the transpose of Tn and for Tn , and the entries π of n are interpreted, up to normalization, as the equilibrium probabilities, in random configurations of the model on a semi-infinite cylinder, that the boundary vertices be connected according to π . As Tn is polynomial, we may assume that n is also a polynomial of the zi (whose entries are non-identically-zero due to the Perron–Frobenius property). Since we can always factor out the GCD of the entries π , we assume that they are coprime. The main purpose of the present article is the investigation of these entries. A special case, extensively studied in [6], corresponds to choosing the zi to be all equal. In this “homo geneous” case, Tn (t) commutes with the Hamiltonian Hn = 2n i=1 (3−2ei −fi ), and n is the null eigenvector of Hn . It was conjectured in [6] that with proper normalization, the entries of n may be chosen to be all non-negative integers, the smallest of which is 1. Here we use the latter condition to fix the remaining arbitrary numerical factor in the normalization of the entries, so that it coincides in the homogeneous case with that of [6]. Before going into specifics, let us mention a preliminary property satisfied by the entries of n . Our semi-infinite cylinder problem is clearly invariant under rotation by one lattice step. Denoting by ρ = f2n−1 f2n−2 . . . f1 the corresponding rotation operator acting on the crossing link patterns by cyclically shifting the labels i → i + 1, we have
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463
the relation Tn (t|z2 , . . . , z2n , z1 )ρ = ρTn (t|z1 , . . . , z2n ), from which we deduce that ρn (z1 , . . . , z2n ) = λn (z2 , . . . , z2n , z1 ). Noting that n is generically non-zero due to the Perron–Frobenius property, and that λ takes discretes values λ2n = 1 and must therefore be independent of the zi , we immediately get that λ = 1 in the range where π > 0, henceforth the entries of n satisfy the following cyclic covariance relation: ρ·π (z2 , z3 , . . . z2n , z1 ) = π (z1 , z2 , . . . , z2n ).
(2.9)
Similarly, one can prove a reflection relation: if r exchanges i and 2n + 1 − i, r·π (−z2n , −z2n−1 , . . . , −z1 ) = π (z1 , z2 , . . . , z2n ).
(2.10)
3. Factorization and Degree We now establish factorization properties of the transfer matrix Tn and of its eigenvector n . Note that this section (as well as Sect. 5 below) possesses some strong similarities with Sect. 3 of [5], though the model under consideration is different. It is sometimes convenient to use the following pictorial representations for the matrix Xi (t − z) and for the transfer matrix: t−z1 t−z2n t−z2
Xi (u) =
,
u
Tn (t|z1 , . . . , z2n ) =
(3.1) . . .
In this language, the Yang–Baxter and unitarity relations read respectively:
u−v
= u
v
u
u
v
and v−u
111 000 000 000111 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 000111 111 000 111 000 111 000 111 000 111 000 000111 111 000 111
= (1 − u2 )(1 −
u2 ). 4
(3.2)
−u
In all that follows, due to periodic boundary conditions indices are meant modulo 2n (2n + 1 ≡ 1). 3.1. Vanishings and factorizations. Let us show a first intertwining property: Lemma 1. The matrices Tn (t|z1 , . . . , zi , zi+1 , . . . , z2n ) and Tn (t|z1 , . . . , zi+1 , zi , . . . , z2n ) are intertwined by Xi (zi+1 − zi ), namely Tn (t|z1 , . . . , zi , zi+1 , . . . , z2n )Xi (zi+1 − zi ) = Xi (zi+1 −zi )Tn (t|z1 , . . . , zi+1 , zi , . . . , z2n ).
(3.3)
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Proof. This is a direct consequence of the Yang–Baxter relation and reads pictorially: zi+1−zi ... t−zi
...
t−zi+1
=
t−zi+1 t−zi ...
... z i+1 −zi
(3.4) We now remark that at the value 1 of the parameter, the face transfer matrix reduces to Xi (1) = ei . This means that for zi+1 = zi + 1, the above transfer matrices say T ˜ ≡ (. . . zi+1 , zi , . . . ) at zi+1 = zi + 1, and T˜ satisfy T ei = ei T˜ . When acting on 2n n 1 n ˜ ˜ ˜ n is we get: T ei n = ei n , with = i=1 (1 − 2 (t − zi ))(1 − zi + t). Hence ei a non-vanishing vector proportional to n , and there exists a rational function α, such ˜ n . When written in components, this implies that whenever i and i + 1 that n = αei are not connected via a “little arch” in a link pattern π ∈ CPn , the entry π vanishes when zi+1 = zi + 1. We may extend this remark into a: Proposition 1. If the link pattern π ∈ CPn has no arch connecting a pair of points between labels i and j , then the entry π vanishes for zj = zi + 1. The proof is already done in the case j = i + 1. For more distant points, we use a generalized intertwining property T P = P T˜ , where P is a suitable product of X matrices. Using again the fact that X(1) = ei , we see that at zj = zi + 1 the product of X forming P contains a factor ei at the intersection between the lines i and j . We ˜ n has no non-vanishing entry with at least an arch linking two deduce that n = αP points between i and j . Indeed, by expanding the product of Xi that form P as a sum of products of f and e, we see that there is always at least one ek in factor, for i ≤ k < j , which results in the existence of an arch connecting two points in between i and j . This shows that π is divisible by i≤k
1≤i<j ≤2n j −i>n
where is a polynomial yet to be determined. Apart from this, we find a polynomial of total degree n(2n − 1) − n = 2n(n − 1) and partial degree 2n − 2 in each variable. We shall prove in the following that = 1. 3.2. Permutation of variables in n and degree. Let us introduce a normalized version of Xi , which we denote by Rˇ i : Rˇ i (z, w) =
(1 − w + z)I + 21 (w − z)(1 − w + z)fi + (w − z)ei (1 − 21 (w − z))(1 + w − z)
This matrix satisfies the usual unitarity relation Rˇ i (z, w)Rˇ i (w, z) = I .
.
(3.6)
Inhomogenous Model of Crossing Loops and Some Algebraic Multidegrees
465
Theorem 1. The transposition of any two consecutive spectral parameters in n is genˇ erated by the action of R: n (. . . , zi , zi+1 , . . . ) = Rˇ i (zi , zi+1 )n (. . . , zi+1 , zi , . . . ).
(3.7)
Proof. To show this, we apply Lemma 1 to the vector n (z1 , . . . , zi+1 , zi , . . . , z2n ). We find that n (z1 , . . . , z2n ) = αn,i (z1 , . . . , z2n )Xi (zi+1 − zi )n (z1 , . . . , zi+1 , zi , . . . , z2n ), for some rational function αn,i . By the coprimarity assumption for the entries of n , we deduce that αn,i may have no zero, hence it reads αn,i = 1/βn,i , for some polynomial βn,i . Moreover, iterating the above once more, we find that n = Rˇ i (zi+1 , zi )Rˇ i (zi , zi+1 )n βn,i (z1 , . . . , z2n )βn,i (z1 , . . . , zi+1 , zi , . . . , z2n ) = n . (1 − 41 (zi − zi+1 )2 )(1 − (zi − zi+1 )2 )
(3.8)
The only polynomials that satisfy this relation are 1 βn,i (z1 , . . . , z2n ) = (1 + i (zi+1 − zi )) 1 + i (zi − zi+1 ) i 2
(3.9)
for i , i , i = ±1. These signs are further all fixed to be +1 by (i) expressing Eq. (3.9) when all zj = 0 ( i = 1), (ii) expressing it when all zj → ∞ ( i i = 1), and (iii) by applying the Lemma 1 ( i = 1). This yields Eq. (3.7). More explicitly, Eq. (3.7) reads in components: 1 (1 + (zi − zi+1 ))(1 − zi + zi+1 )π (z1 , . . . , z2n ) 2 = (1 + zi − zi+1 )π (z1 , . . . , zi+1 , zi , . . . , z2n ) 1 + (zi+1 − zi )(1 + zi − zi+1 ))fi ·π (z1 , . . . , zi+1 , zi , . . . , z2n ) 2 π (z1 , . . . , zi+1 , zi , . . . , z2n ). (3.10) +(zi+1 − zi ) π , ei ·π =π
This is a very efficient recursion relation, allowing to express all entries of n in terms of the maximally crossing pattern entry. Indeed, two situations may occur for π : (i) π has no little arch joining (i, i + 1). Then Eq. (3.10) translates into fi ·π (z1 , . . . , z2n ) = i π (z1 , . . . , z2n ),
(3.11)
where the linear operator i acts on functions F (z1 , . . . , z2n ) as
i F (z1 , . . . , z2n ) = 2
(1 + zi − zi+1 )F (zi+1 , zi ) − (1 − zi + zi+1 )F (zi , zi+1 ) (zi − zi+1 )(1 − zi + zi+1 ) 1 + zi − zi+1 F (zi+1 , zi ), (3.12) − 1 − zi + zi+1
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where for simplicity we have only represented the arguments i and i + 1 of F (recall that periodic boundary conditions for indices are implied: 2n + 1 ≡ 1). Note here that
i ◦ i = I , in agreement with fi2 = I , a simple consequence of the “gauge formula”
i = (1 + zi − zi+1 ) (2∂i − τi )
1 , 1 + zi − zi+1
(3.13)
where τi and ∂i are respectively the transposition and divided difference operators,1 acting as τi F (zi , zi+1 ) = F (zi+1 , zi ), F (zi+1 , zi ) − F (zi , zi+1 ) , ∂i F (zi , zi+1 ) = zi − zi+1
(3.14)
with the obvious relations τi2 = 1,
∂i2 = 0,
∂i τi = −τi ∂i ,
(3.15)
(ii) π has a little arch joining (i, i + 1). Then Eq. (3.10) translates into
π (z1 , . . . , z2n ) = i π (z1 , . . . , z2n ),
(3.16)
π ∈CPn π =π, ei π =π
where the linear operator i acts as 1 F (zi+1 , zi ) − F (zi , zi+1 ) .
i F (z1 , . . . , z2n ) = (1 + zi − zi+1 )(1 + (zi+1 − zi )) 2 zi − zi+1 (3.17) Note also that i ◦ i = − i , in agreement with (ei − I )2 = −(ei − I ). Some remarks are in order. From the explicit form of i (3.17), one may think that its action on a polynomial increases the degree by one. However this is not the case if the largest total degree piece of the polynomial is symmetric under zi ↔ zi+1 . Such a property will be found below (Lemma 2). Also, it is clear that the set of relations (3.11) and (3.16) is overdetermined. The compatibility between these equations is granted by the Yang–Baxter equation, that translates into relations between the i and i . Finally, we note that the system of equations (3.11) and (3.16) is equivalent to the eigenvector condition (2.8). To see that this system is sufficient to ensure (2.8), we note that in analogy with the Temperley–Lieb case of [5], we may construct yet another transfer matrix Tn that commutes with Tn , and made of a particular product of matrices of the form Rˇ i (zk , z ). More precisely, we have Tn = ni=1 nj=1 Rˇ i+2j −2 (z2j −1 , z2i+2j −2 ), with products taken in the order indicated. As a direct consequence of the Yang–Baxter equation (3.4), we have the commutation relation [Tn , Tn ] = 0, hence if n is an eigenvector of Tn with eigenvalue 1 (that is a Perron–Frobenius eigenvector in appropriate ranges of the zi ), then it is also that of Tn . The set of conditions (3.11) and (3.16) precisely guarantee that n is such an eigenvector of Tn , hence they are sufficient to ensure (2.8). As it turns out, we may generate all the entries π for π ∈ CPn by acting with a number of i on π0 . This is best seen by recalling that the link patterns π ∈ CPn 1 Note the unusual sign convention for ∂ , which will be fixed in Sect. 4 by renumbering variables in i the opposite order.
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can be considered as permutations of S2n with only 2-cycles. As such, fi acts on π as conjugation of π by the elementary transposition i ↔ i + 1, and generates the action of the whole symmetric group S2n . The well-known property that two permutations are conjugate if (and only if) they have the same cycle lengths implies that any π can be obtained from π0 as π = fi1 · · · fik · π0 . We assume that fil+1 · · · fik · π0 does not have a little arch (il , il + 1) for 1 ≤ l ≤ k, i.e. we exclude in such a decomposition any fi that would act on a pattern with a little arch (i, i + 1) since such an action is trivial. We can therefore apply (i) above repeatedly, and express the corresponding entry of n : π = i1 · · · ik π0 .
(3.18)
The procedure is illustrated in Appendix B in the case n = 3. The property (3.18) has an important consequence: by constructing explicitly the entries of , we fix their degree and prove that = 1 in Eq. (3.5): Theorem 2. One has: π0 =
1≤i<j ≤2n j −i
(1 + zi − zj )
(1 + zj − zi )
(3.19)
1≤i<j ≤2n j −i>n
and all the entries of n are polynomials of total degree 2n(n − 1), and partial degree 2(n − 1) in each zi . The proof goes as follows. Starting with the “minimal” candidate (3.19) for π0 subject to the vanishing conditions of Proposition 1 above, we must first show that we can construct all other entries π via Eqs. (3.18) independently of the choice of words fi1 · · · fik , and then check that these moreover satisfy Eqs. (3.16), and are coprime. We first use Eq. (3.18) to express all the π in terms of the minimal choice (3.19) for π0 . It is easy to check that the factorization properties of Proposition 1 are satisfied by all the π thus constructed. As a consequence, all denominators in Eq. (3.12) are cancelled (see the reformulation (3.13)), and the action of i preserves the polynomial character, and the degree. The fact that the values of π are independent of the choice of decomposition π = fi1 · · · fik · π0 which appears in (3.18) is a consequence of the following properties: (i) The affine symmetric group relations satisfied by the ’s, namely 2i = I ,
i i+1 i = i+1 i i+1 , and i j = j i for |i − j | > 1, for all i = 1, 2, . . . , 2n with cyclic boundary conditions, ensure that two words corresponding to the same permutation give rise to the same π . (ii) As the i correspond only to the action of the fi on permutations σ with n cycles of length 2 (the class [2n ] of S2n ), while forbidding the action of fi on permutations σ such that σ (i) = i + 1, we must also enforce the conditions that
i n+i π0 = π0
(3.20)
for i = 1, 2, ..., n. The corresponding elements fi fi+n of S2n are indeed nothing but the remaining generators of the stabilizer subgroup of the class [2n ], once the actions of fi on permutations σ such that σ (i) = i + 1, i = 1, 2, . . . , 2n, have been forbidden.2 Note that the latter condition has the net effect of wiping out any affine symmetric group relation that 2 This action is exactly the “flat first Reidemeister move”. In other words to obtain a true representation of the i without restrictions one should consider link patterns decorated with extra “tadpoles”.
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“winds” around the circle, such as for instance f2 f3 . . . f2n = I , as its action on link patterns would involve at least one of the forbidden moves. The relations (3.20) are readily checked by using the expression (3.19). This defines the vector n unambiguously. To ensure that n satisfies the eigenvector equation (2.8), we must still check that it satisfies the properties (3.16), which have not been used so far. This is done by induction on the number of crossings in π . Assume a link pattern π has a little arch in position (i, i + 1), and that it satisfies Eq. (3.16). As [ i , j ] = 0 for j = i − 1, i, i + 1, we may act upon Eq. (3.16) with any such j , also such that π has no little arch at positions (j, j + 1). This results in a new equation, in which π, π → fi · π, fi · π . This allows to increase the number of crossings of π, hence the induction. We are left with the task of proving Eq. (3.16) for π with maximal number of crossings and with a little arch (i, i + 1). We first note that π0 as given by (3.19) is cyclically invariant (under zi → zi+1 cyclically), henceforth we need only check this property for i = 2n, say. We have to check that
2n
2 1
3
n−1 n
2n 2n−1 n 2n−2 = j =2 j n+2 n+1
2 1 3
2n 2n−1 2n−2
2
+
1
2n
3 j
j+n−1 n−1
n+2 n+1
n
n−1
n
2n−1 2n−2 j+n−1 n+2 n+1
(3.21) or equivalently
2n 1 2 . . . n−1 π0 = (1 + 2n )
n
1 2 . . . j −2 j j +1 . . . n−1 π0 ,
j =2
(3.22) where we have used the commutation of the i for distant enough indices, and the fact that i π0 = n+i π0 for all i, a direct consequence of fi · π0 = fi+n · π0 for all i. Noting also that 1 + i = i × 2/(1 + zi − zi+1 ), and that i is proportional to ∂i , we finally arrive at the condition that the quantity n 2 n = 1 2 . . . n−1 −
1 2 . . . j −2 j j +1 . . . n−1 π0 1 + z2n − z1 j =2
(3.23) must be annihilated by 2n , hence by ∂2n , or equivalently that n must be symmetric under z2n ↔ z1 . We have found an explicit expression for n , displaying this invariance manifestly, namely n = π(n−1) (z2 , z3 , . . . z2n−1 ) 0
n j =2
a1,j a2n,j aj +n−1,1 aj +n−1,2n ,
(3.24)
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where the first term is nothing but the expression for π0 (3.19) but for the size 2n − 2, and with arguments counted cyclically from 2 to 2n−1. We have also used the shorthand notation ai,j = 1 + zi − zj . To prove (3.24), we first note that (3.22) may be equivalently rewritten as n−1 2 n = π0 .
i − (3.25) 1 + z2n − zi i=1
This is easily shown by induction, by noticing that i commutes with 2/a2n,j for j = i + 2, i + 3, . . . , n − 1, and that i (2/a2n,i+1 ) = 4/(a2n,i a2n,i+1 ), from the explicit definition of i (3.12). Finally, acting first with n−1 − 2/a2n,n−1 on π0 , we see that the only terms not symmetric under zn−1 ↔ zn , hence that do not commute with this action, are an−1,n an,2n−1 a2n,n−1 , and from the explicit expression (3.12), we get 2
n−1 − (an−1,n an,2n−1 a2n,n−1 ) = an−1,n a2n,n a2n−1,n−1 . (3.26) a2n,n−1 The same reasoning then applies to the resulting function, whose only term non-symmetric under zn−2 ↔ zn−1 is an−2,n−1 a2n,n−2 an−1,2n−2 ; at the j th step we act with
n−j − 2/a2n,n−j on the product an−j,n−j +1 a2n,n−j an−j +1,2n−j with the result an−j,n−j +1 a2n,n−j +1 a2n−j,n−j , and this goes on until j = n − 1. Collecting all the terms, we finally get Eq. (3.24). We conclude that the n constructed above, normalized by (3.19), satisfies all equations (3.16), and therefore also satisfies (2.8). Moreover, any GCD of the π would divide π0 of (3.19), but all the factors present in (3.19) are required by Proposition 1, hence this GCD must be a constant and the components of n are indeed coprime. Furthermore, due to the structure of i , we find that the entries π are polynomials with total degree and partial degrees in each zi bounded respectively by the total degree 2n(n − 1) and partial degrees 2(n − 1) of π0 . In fact, we have equality of degrees, and an explicit expression for the highest degree terms in π , as the following lemma shows: Lemma 2. The terms of π of maximal degree read πmax (z1 , . . . , z2n ) = (−1)c(π) (zi − zj ) 1≤i<j ≤2n
i<j :π(i)=j
1 , (3.27) zi − z j
where c(π ) is the number of crossings of π. This is proved by induction on the quantity n(n − 1)/2 − c(π ), starting from π = π0 , whose leading degree terms read πmax (z1 , . . . , z2n ) = (−1)n(n−1)/2 (z) 0
n i=1
1 , zi − zi+n
(3.28)
where we used the standard notation (z) = 1≤i<j ≤2n (zi − zj ) for the Vandermonde determinant. This leading term matches (3.27) upon noting that c(π0 ) = n(n − 1)/2. Let us now prove that if π has no little arch linking (i, i + 1), then at leading order in the z, the action of i preserves the form (3.27), with π replaced by fi · π . Assuming that π(i) = k and π(i + 1) = , we see that the product (3.27) may be rewritten as
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πmax = (zi − z )(zi+1 − zk )(zi − zi+1 )π , where the polynomial π is symmetric under the interchange zi ↔ zi+1 . As any polynomial with such a symmetry may be factored out of the action of i (the latter affects only non-symmetric terms), we are left with the task of finding the leading behavior at large zi of
i (zi − z )(zi+1 − zk )(zi − zi+1 ) ∼ −(zi − zk )(zi+1 − z )(zi − zi+1 ),
(3.29)
obtained as an immediate consequence of (3.13), as ∂i decreases the degree strictly, and only the −τi term contributes at leading order. This proves that fmax also has the i ·π form (3.27), the overall minus sign accounting for the decrease by 1 of the number of crossings. This completes the proof of (3.27) for all entries of n , and of Theorem 2. As a side remark, an immediate consequence of Lemma 2 is the property that the maximal degree terms in the ground state vector entries, πmax , are invariant under the interchange zi ↔ zi+1 whenever π has a little arch joining (i, i + 1). Indeed, as an arch joins (i, i + 1), the only term involving zi to be divided out of the Vandermonde determinant in (3.27) is the skew-symmetric term (zi − zi+1 ). This leaves us with a manifestly invariant product, and shows that the action of the operators i on entries of n with a little arch joining (i, i + 1) does not increase the degree, as announced. We conclude this section by stressing that we have explicitly constructed the entries of by repeated action of the i on π0 of (3.19). It is clear that all the entries of thus constructed are polynomials of the zi with integer coefficients. In particular, in the homogeneous limit where all the zi tend to the same value (say t), we see that π0 → 1, and we end up with a vector whose entries are all integers, moreover positive, as then coincides with the Perron–Frobenius eigenvector of the Hamiltonian of [6], with the same normalization condition. This yields a proof that, in the homogeneous case, once is normalized by π0 = 1, all its entries are positive integers. 4. Sector of Permutations 4.1. Definition of the sector Pn and of the associated varieties. A subset Pn of CPn consists of the n! “permutation patterns” that connect the points {1, 2, . . . , n} to the points {n + 1, n + 2, . . . 2n}. Such patterns π ∈ Pn are in one-to-one correspondence with permutations πˆ ∈ Sn of {1, . . . , n} via π(i) = πˆ (n + 1 − i) + n, i = 1, . . . , n. To each pattern π ∈ Pn is naturally associated a homogeneous affine variety Vπ : following Knutson [9], we consider the three conditions on pairs of n × n complex matrices X and Y : (1) XY lower triangular, Y X upper triangular. for i = 1, . . . , n. (2) (XY )ii = (Y X)π(i) ˆ π(i) ˆ (3) The matrix Schubert variety conditions: the rank of any upper left (resp. lower right) rectangular submatrix of X (resp. Y ) is less or equal to the rank of the corresponding submatrix of the permutation matrix πˆ (resp. πˆ −1 ). The Vπ all have the same dimension n(n + 1), or equivalently codimension n(n − 1) in Mn (C)2 . They are conjectured (Sect. 3 of [9]) to be the irreducible components of V which is the set of pairs (X, Y ) that satisfy condition (1) only. Note that we use the “lower-upper scheme” here, as in Sect. 1 of [9], where V is denoted by D 0 ; however, starting with Sect. 2, [9] uses the “upper-upper scheme”, hence slightly differing conventions.
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4.2. Refined de Gier–Nienhuis conjectures. We first provide the following interpretation of the entries π with π ∈ Pn for a special choice of inhomogeneities, which refines a conjecture of Nienhuis and de Gier [6]: Conjecture 1. Set zi = B/(A + B), for i = 1, . . . , n and zi = A/(A + B) for i = n + 1, . . . , 2n. Then we have the following identification for π ∈ Pn :
A+B 2
n(n−1)
π = dπ (A, B),
(4.1)
where dπ (A, B) is the bidegree of Vπ which is related to the separate scaling of the matrices X and Y (such a bidegree is defined in Sect. 2.2 of [9]3 ). The cases n = 3 and n = 4 are given in Appendix C. For A = B = 1, all zi are equal, and our conjecture reduces to that of [6]. More precisely, Eq. (4.1) then expresses identities between on the left hand side the entry of the homogeneous problem (i.e. of the null vector of the Hamiltonian Hn ), and on the right hand side the usual degrees of the algebraic varieties. In order to go further, we note that for any π ∈ Pn , the entry π has no little arch connecting pairs (i, j ) with 1 ≤ i, j ≤ n or n + 1 ≤ i, j ≤ 2n. This motivates the following redefinitions: set pi = zn+1−i , qi = zn+i , i = 1, . . . , n and (1 + zi − zj ) π (z1 , . . . , z2n ) = 1≤i<j ≤n
(1 + zk − zl )
δπ (p1 , . . . , pn , q1 , . . . , qn ),
(4.2)
n+1≤k
where δπ is some polynomial of its 2n variables. To interpret δπ , we also need a more general action of a torus of dimension 2n + 2 that maps (X, Y ) to (aP XQ−1 , bQY P −1 ); it allows to define multidegrees, i.e. torusequivariant cohomology. With respect to this action, the weights of the variables are: [Xij ] = A + pi − qj ,
[Yij ] = B + qi − pj .
(4.3)
We now have the following stronger conjecture: Conjecture 2. δπ (p1 , . . . , pn , q1 , . . . , qn ) is the (2n+2)-multidegree of Vπ associated to the weights (4.3) in which one sets A = B = 1. That Conjecture 2 implies Conjecture 1 is due to the following simple scaling property: if one sets A = B = 1 in Eq. (4.3), and performs the change of variables pi =
A + 2pi , A + B
qi =
B + 2qi , A + B
(4.4)
2 2 one finds [Xij ] = A +B (A + pi − qj )B and [Yij ] = A +B (B + qi − pj ), which are identical to the original weights up to a factor 2/(A + B ). Since the multidegrees are homogeneous polynomials of degree n(n − 1) (the codimension of Vπ ), the 3 Note that we have slightly altered the notation: in [9] the index σ of d (A, B) refers to the permutation σ such that π(i) = n + σ (i), i = 1, 2, . . . n.
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full (2n + 2)-multidegree can be extracted from δπ by simply pulling out the factor (2/(A + B ))n(n−1) . As a special case, if we set all pi = qi = 0, we recover Eq. (4.1). Note also that the terms of highest degree of δπ are now interpreted, using the same scaling argument, as the 2n-multidegree of Vπ associated to the weights (4.3) with A = B = 0. Explicitly, δπmax (p1 , . . . , pn , q1 , . . . , qn ) = (−1)c(π)
n
(pi − qj ).
(4.5)
i,j =1 j =π(i) ˆ
In Sect. 4.5, we shall give a simple and efficient way to compute the δπ for arbitrary n. But first we show some simple properties satisfied by the entries π , π ∈ Pn ; these properties have clear meaning for the multidegrees (see also Sect. 2.2 of [9] for analogous statements on (bi)degrees). As a simple example, the behavior of n under rotation and reflection (Eqs. (2.9)–(2.10)) implies, modulo Conjecture 1, the third and fourth statements in Prop. 4 of [9]; these statements could be easily extended to multidegrees, and indeed, we find for the conjectured multidegrees δπ : δρ n ·π (qn , . . . , q1 , pn , . . . , p1 ) = δπ (p1 , . . . , pn , q1 , . . . , qn ), (4.6a) δr·π (−q1 , . . . , −qn , −p1 , . . . , −pn ) = δπ (p1 , . . . , pn , q1 , . . . , qn ), (4.6b) δr ·π (−pn , . . . , −p1 , −qn , . . . , −q1 ) = δπ (p1 , . . . , pn , q1 , . . . , qn ), (4.6c) n·π =π ·π =π where ρ ˆ 0 πˆ −1 πˆ 0 , r · π = πˆ −1 and r ˆ 0 πˆ πˆ 0 . 4.3. Sum rule for the permutation sector Pn . Theorem 3. The sum of entries of n corresponding to permutation patterns has the following factorized form: Yn (z1 , . . . , z2n ) = π (z1 , z2 , . . . z2n ) =
π ∈Pn
(1 + zi − zj )(2 − zi + zj )
1≤i<j ≤n
(1 + zk − zl )(2 − zk + zl ) .
n+1≤k
(4.7) Proof. Let us introduce the linear form bn that is the characteristic function of Pn , namely with entries in the canonical basis 1 if π ∈ Pn bπ = . (4.8) 0 otherwise bn satisfies bn I = bn ,
bn fi = bn ,
bn ei = 0,
i = n, 2n
(4.9)
as no little arch may connect points among {1, 2, . . . n} or {n + 1, n + 2, . . . , 2n} in π ∈ Pn . We now act with the characteristic function bn of the permutation sector Pn ˇ Using the relations (4.9), we immediately find (4.8) on the matrix R. bn Rˇ i (z, w) =
(1 + 21 (w − z))(1 + z − w) (1 − 21 (w − z))(1 + w − z)
bn ,
i = n, 2n.
(4.10)
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Noting finally that Yn = bn · n , and taking the scalar product of bn with (3.7), we find that the r.h.s. of (4.7) divides the sum on the l.h.s. Moreover, the degree of the r.h.s. is 2n(n − 1), the same as that of the sum Yn by application of Theorem 2. The two terms must therefore be proportional by a constant, further fixed to be 1 by considering the terms of maximal degree (2n(n − 1)) in Yn . Indeed, the maximal degree term in the r.h.s. of (4.7) reads
2 (z1 , . . . zn ) 2 (zn+1 , . . . , z2n ),
(4.11)
where stands for the Vandermonde determinant. On the other hand, resumming all leading behaviors (3.27) over the permutation patterns π ∈ Pn , we get the following quantity Ynmax (z1 , . . . , z2n ) = (z1 , . . . , z2n )
π∈Pn
= (z1 , . . . , z2n ) det
(−1)c(π)
pairs (i,j ) j =π(i), i=1,2,... ,n
1 zi − zn+j
1 zi − z j
1≤i,j ≤n
,
(4.12)
where we have identified the sign (−1)c(π) with the signature of the underlying permutation, eventually leading to the determinant. The identity between (4.11) and (4.12) is just the Cauchy determinant formula:
(z1 , . . . zn ) (zn+1 , . . . , z2n ) 1 n . (4.13) = det zi − zn+j 1≤i,j ≤n i,j =1 (zi − zn+j ) Note that the sum rule (4.7) translates, at the special values of inhomogeneities considered in Sect. 4.1 for Conjecture 1, into the following: π∈Pn
A+B dπ (A, B) = 2
n(n−1) Yn
B A ,... , ,... A+B A+B
= (A + B)n(n−1) , (4.14)
which is in agreement with the first statement in Prop. 4 of [9]. Yet another expression, equivalent to Theorem 3, is: δπ (p1 , . . . , pn , q1 , . . . , qn ) = (2 + pi − pj )(2 − qi + qj ), (4.15) π ∈Pn
1≤i<j ≤n
which is clearly the multidegree of the whole variety V (equations enforcing condition (1)). Remark 1. Rotated versions of the sum rule (4.7) are also available, upon using the gen(i) eral cyclic covariance property of n (2.9), namely involving Yn = Yn (zk → zk+i ), (0) with Yn ≡ Yn . These correspond to sums of entries of n over the rotated permutation (i) pattern sets Pn that connect the points {i+1, i+2, . . . , i+n} to {n+i+1, n+i+2, . . . i} and will be used in Sect. 5.1 (proof of Theorem 4).
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4.4. Factorization in the permutation sector Pn . We may prove a general factorization property for the fully decomposable permutation patterns π ∈ Pn defined as follows. Assume the points 1, 2, . . . n are partitioned into two sets R1 = {1, 2, . . . , r}, R2 = {r + 1, r + 2, . . . , n}; define also S1 = R1 + n, S2 = R2 + n. A permutation pattern π ∈ Pn is called fully decomposable with respect to the partition (R1 , R2 ) if π only connects the points of R1 to those of S1 and the points of R2 to those of S2 . We denote by πi the restriction of π to the set Ri ∪ Si . Proposition 2. For any fully decomposable π , we have the factorization property π (z1 , . . . , z2n ) = (1 + zi − zj ) π1 (zR1 ∪S1 )π2 (zR2 ∪S2 ), (4.16) (i,j )∈X
where X = R1 × R2 ∪ R2 × S1 ∪ S1 × S2 ∪ S2 × R1 , and zI denotes the sequence of zi , i ∈ I . To prove (4.16), we proceed by induction on n(n−1)/2−c(π ). We start from the link pattern π0 , viewed as a fully decomposable pattern with respect to (R1 , R2 ). From the explicit expression (3.19), we immediately get (4.16) upon noting that the restrictions π1 and π2 are themselves the maximally crossing patterns π0 in their respective sets Pr and Pn−r . Assume some fully decomposable π satisfies (4.16). We may reduce by 1 the number of crossings of π by acting on π with some fi , with either i ∈ {1, 2, . . . , r − 1} or i ∈ {r + 1, r + 2, . . . , n − 1}. The corresponding entry fi ·π is obtained by acting with i on π . But this action only affects the corresponding restriction π1 or π2 , within which the uncrossing is performed. Hence the form (4.16) is preserved, and we simply have to substitute π → fi · π and either π1 → fi · π1 or π2 → fi · π2 . This completes the proof of (4.16). As a corollary, when all zi are taken to zero, Eq. (4.16) translates into a factorization property conjectured in [6]. The corresponding property for bidegrees is the second statement of Prop. 4 of [9]. More generally, we can rewrite it in terms of our conjectured multidegrees: δπ (p1 , . . . , pn , q1 , . . . , qn ) =
r n−r
(1 + pi − qj )
i=1 j =1
n
n
(1 + qi − pj )
i=r+1 j =n−r+1
δπ1 (pn−r+1 , . . . , pn , q1 , . . . , qr ) (4.17) δπ2 (p1 , . . . , pn−r , qr+1 , . . . , qn ). n−r r The factors i=1 j =1 (1 + pi − qj ) ni=r+1 nj=n−r+1 (1 + qi − pj ) correspond to equations enforcing the block-triangular shape of the matrices X and Y for a decomposable π. 4.5. Sketch of proof of the multi-parameter de Gier–Nienhuis conjecture. It is not the purpose of the present paper to give a rigorous proof of Conjecture 2. However, we shall indicate the main steps of the proof, leaving aside various technicalities. Note that no results in this paper depend on proving Conjecture 2. The property to be proved is written in short as: deg Vπ = δπ (deg denoting the multidegree as in Conjecture 2, i.e. with A = B = 1). The proof proceeds as usual by induction on the number of crossings of π.
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For the case of the long permutation πˆ 0 , we have the explicit formula (3.19), or with our redefinitions: (1 + pi − qj ) (1 + qi − pj ). (4.18) δπ0 (p1 , . . . , pn , q1 , . . . , qn ) = 1≤i,j ≤n i+j
1≤i,j ≤n i+j >n+1
But the corresponding variety Vπ0 is known explicitly: Vπ0 = {(X, Y ) ∈ Mn (C)2 | X lower right triangular, Y upper left triangular}. (4.19) The equations defining Vπ0 are simply: Xij = 0 for i + j < n + 1 and Yij = 0 for i + j > n + 1; using Eq. (4.3) with A = B = 1, we immediately find that deg Vπ0 equals the r.h.s. of Eq. (4.18). Induction. We want to show the property for a certain permutation pattern, assuming it is true for all permutation patterns with higher number of crossings. We can always write this pattern as fı¯ · π, ı¯ = 1, . . . , n − 1, such that π has one more crossing than fı¯ · π . In terms of permutations, f ˆ i with i = n + 1 − ı¯, that is fı¯ acts (as ı¯ · π = πτ elementary transposition τi ) by multiplication on the right.4 The crucial point is once more to use the formula (3.11) which relates fı¯ ·π to π via the operator j defined by Eq. (3.12). After taking into account our redefinition (4.2), we find the much simpler expression δfı¯ ·π = (2∂i − τi ) δπ ,
(4.20)
following immediately from (3.13). Here τi is the natural action of the symmetric group that exchanges variables pi and pi+1 , and ∂i is the divided difference operator ∂i = 1 pi+1 −pi (τi − 1) (with the usual sign convention). Note that this is a known representation of the symmetric group, studied for example in [11]. It is better to rewrite Eq. (4.20) as: δfı¯ ·π + δπ = (2 + pi − pi+1 )∂i δπ .
(4.21)
Roughly speaking, ∂i corresponds to the action of the elementary transposition τi in the matrix Schubert variety conditions (3), whereas 2 + pi − pi+1 is the additional equation (XY )i i+1 = 0 (part of condition (1)) which was lost in the process. More explicitly, call Xi the row vectors of X and Yi the column vectors of Y . For any variety W inside Mn (C)2 with a torus action, define ∂i W ≡ {(X , Y )|Xj = Xj ∀j = i, = Yi+1 − uYi for (X, Y ) ∈ W, u ∈ C}. Xi = Xi + uXi+1 , Yj = Yj ∀j = i + 1, Yi+1 One can show that W → ∂i W translates into the operator ∂i for multidegrees (indeed it decreases codimension = degree by 1, and clearly does not act on variables other than pi , pi+1 or on symmetric functions of pi , pi+1 ; all these properties characterize ∂i up to normalization, which is easily fixed). Now apply this operation ∂i to the variety Vπ . By direct inspection, ∂i Vπ satisfies condition (3) in which πˆ is replaced with πτ ˆ i (standard reasoning for matrix Schubert varie ties), as well as the set of equations defining condition (1) (noting that Y X = Y X) except for (X Y )i i+1 = Xi |Yi+1 = 0. We therefore compute Xi |Yi+1 = u(Xi+1 |Yi+1 − 4 Note that one could have used a f with ı¯ = n + 1, . . . , 2n − 1; this would correspond to multiplicaı¯ tion of the permutation πˆ on the left. In our multidegree setting (similar to double Schubert polynomials) there is complete symmetry between left and right multiplication, corresponding to operators acting on the pi or on the qi .
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Xi |Yi − u Xi+1 |Yi ), and find that the equation Xi |Yi+1 = 0 nicely factorizes into u = 0, which of course defines Vπ , and a non-trivial linear equation for u: Xi+1 |Yi+1 − Xi |Yi − u Xi+1 |Yi = 0.
(4.22)
Now we want to check condition (2) when this equation is satisfied. We have (Y X )jj (Y X)jj for all j , and (X Y )jj = (XY )jj for j = i, i + 1. Furthermore,
=
(X Y )ii = Xi |Yi = Xi |Yi + u Xi+1 |Yi = Xi+1 |Yi+1 = (XY )i+1 i+1 , (4.23a) (X Y )i+1 i+1 = Xi+1 = Xi+1 |Yi+1 − u Xi+1 |Yi = Xi |Yi = (XY )ii , |Yi+1 (4.23b)
using Eq. (4.22). We conclude that (X , Y ) satisfies condition (2) in which πˆ is replaced with πˆ τi . Since the operation is an involution, all of Vfı¯ ·π is obtained this way. What we have found is that the additional relation (X Y )i i+1 = 0 restricts ∂i Vπ to Vπ ∪ Vfı¯ ·π . This means that it increases codimension by 1, i.e. is independent from other equations and therefore amounts to multiplication by 2 + pi − pi+1 of the multidegree (cf. Eq. (4.3) with A = B = 1). This proves that deg Vfı¯ ·π + deg Vπ = (2 + pi − pi+1 )∂i deg Vπ ; by the induction hypothesis deg Vπ = δπ , and comparing with Eq. (4.21), we conclude that deg Vfı¯ ·π = δfı¯ ·π . As already mentioned at the end of Sect. 3, a corollary of the recursive construction of the π , or equivalently of the δπ in the permutation sector, is that they are sums of products of 1 − zi + zj , and in particular polynomials with integer coefficients; this could also be seen as a consequence, assuming Conjecture 2, of the general Theorem 1.7.1. of [12]. 5. Recursion Relations and Full Sum Rule In this section, we will derive recursion relations relating specialized entries of n to entries of n−1 , that will allow us to prove the full sum rule. 5.1. Recursion relations for the entries of n . Let us examine the case when a little arch connects two neighboring points i, i + 1 in some π ∈ CPn . We have the following Lemma 3. Let ϕi denote the embedding CPn−1 → CPn that inserts a little arch between positions i − 1 and i and relabels the later positions j → j + 2 in any π ∈ CPn−1 ; we also denote by ϕi the induced embedding of vector spaces. We have Tn (t, z1 , . . . , zi , zi+1 = zi + 1, . . . , z2n )ϕi 1 = (t −zi )(1−zi + t)(2 − t +zi )(3−t +zi )ϕi Tn−1 (z1 , .., zi−1 , zi+2 , . . . , z2n ). 4 (5.1)
1−u
=
u
Proof. This is a direct consequence of the unitarity relation (3.2) together with the so-called crossing relation
(5.2)
Inhomogenous Model of Crossing Loops and Some Algebraic Multidegrees
477
in which the second picture has been rotated by +π/2. Indeed, such a rotation exchanges the roles of I and ei while fi is left invariant, and the coefficients in (2.3) are interchanged accordingly under u → 1 − u. However, it is more instructive to prove (5.1) directly, by attaching to the little arch the two plaquettes at sites i and i + 1. We denote this by ui = 1 − t + zi , vi = 21 (t − zi )(1 − t + zi ) and wi = t − zi . Pictorially, this gives rise to 9 situations:
i
i+1
= wi wi+1
+ ui ui+1
+ui vi+1
+ vi wi+1
+wi vi+1
+ vi ui+1
+wi ui+1
+ vi vi+1
+ ui wi+1
(5.3)
We have displayed on the same line the terms contributing to the same picture. We now note that when zi+1 = zi + 1, the first and second line both have a global vanishing factor wi wi+1 + ui ui+1 + ui wi+1 + ui vi+1 + vi wi+1 = wi vi+1 + vi ui+1 = 0. We are only left with the contribution where the little arch has safely gone across the horizontal line, and where in passing the two spaces i and i + 1 have been erased. The prefactor is wi ui+1 + vi vi+1 =
1 (t − zi )(1 − zi + t)(2 − t + zi )(3 − t + zi ), 4
and yields the prefactor in Eq. (5.1).
(5.4)
Together with the results of previous sections, this leads to: Theorem 4. For a given π ∈ CPn , taking zi+1 = zi + 1, we have either of the two following situations: (i) There is no little arch (i, i + 1) in π. Then π (z1 , . . . , zi , zi+1 = zi + 1, . . . , z2n ) = 0.
(5.5)
(ii) There is a little arch connecting i and i + 1. Then we have the recursion relation π (z1 , . . . , zi , zi+1 = zi + 1, . . . , z2n ) =
2n k=1 k =i,i+1
(1 + zi+1 − zk )(1 + zk − zi ) π (z1 , . . . , zi−1 , zi+2 , . . . , z2n ) . (5.6)
between the entry π of n and the entry π of n−1 , where π is the link pattern π with the little arch i, i + 1 removed (π = ϕi π , π ∈ CPn−1 ).
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Proof. The situation (i) is covered by Proposition 1. To prove (5.6), we use Lemma 3, and act on n−1 , with the obvious values of the parameters. We have T ϕi n−1 = / ϕi T n−1 = ϕi n−1 , hence ϕi n−1 is proportional to n , when zi+1 = zi + 1. This yields Eq. (5.6), up to a proportionality factor say γn,i , rational fraction of the parameters. This factor is further fixed by applying (5.5)–(5.6) to the sum over the suitably (i) rotated set of permutation patterns Pn , that allows for little arches in positions (i, i + 1) or (i + n, i + n + 1). Due to the properties (i-ii), the only non-vanishing contributions to this sum of entries when zi+1 = zi + 1 are those for which a little arch connects (i−1) (i, i + 1). These are the images under ϕi of the permutation sector Pn−1 , hence we get (i−1) Yn(i) (z1 , . . . , z2n ) zi+1 =zi +1 = γn,i Yn−1 (z1 , . . . , zi−1 , zi+2 , . . . , z2n ). (5.7) Applying the suitable rotations to the result of Theorem 3 yields the value of γn,i and (5.6) follows. 5.2. Sum rule for the entries. We now compute the sum of all the entries of n and express it in terms of a Pfaffian. We start with the following Lemma 4. The sum of entries of n : Zn (z1 , . . . , z2n ) =
π (z1 , . . . , z2n )
(5.8)
π∈CPn
is a symmetric polynomial. Proof. The linear form vn , with entries vπ = 1, clearly satisfies vn I = vn , vn fi = vn , vn ei = vn , as each of the operators I, fi , ei sends each link pattern to a unique link pattern. Therefore vn Rˇ i (z, w) = vn . The sum over entries is nothing but Zn = vn · n , and taking the scalar product of vn with Eq. (3.7), we deduce that Zn is invariant under the interchange of zi ↔ zi+1 , hence is fully symmetric. By virtue of Theorem 2 and Lemma 4, Zn (z1 , . . . , z2n ), the sum of entries of n , is a symmetric polynomial of total degree 2n(n − 1) and partial degree 2n − 2 in each variable, subject to the recursion relation obtained by summing Eqs. (5.5)–(5.6) over all π ∈ CPn : Zn (z1 , . . . , z2n )|zi+1 =zi +1
2n = (1 + zi+1 − zk )(1 + zk − zi ) Zn−1 (z1 , . . . , zi−1 , zi+2 , . . . , z2n ). k=1 k =i,i+1
(5.9) Together with the initial condition Z1 (z1 , z2 ) = 1, these properties determine it completely and allow us to prove Theorem 5. The sum of all entries Zn has a Pfaffian formulation: 1 − (zi − zj )2 zi − zj × . (5.10) Zn (z1 , . . . , z2n ) = Pf 1 − (zi − zj )2 1≤i,j ≤2n zi − z j 1≤i<j ≤2n
Inhomogenous Model of Crossing Loops and Some Algebraic Multidegrees
479
Proof. The r.h.s. of (5.10), which we denote by Kn , is a symmetric polynomial of the zi , with total degree 2n(n − 1) and partial degree 2(n − 1) in each variable, and such that when say z2 → z1 + 1, the Pfaffian degenerates into (z1 − z2 )/(1 − (z1 − z2 )2 ) times 2 that for z3 , z4 , . . . , z2n , while the other quantity reduces to (1 − (z1 − z2 ) )/(z1 − z2 ) times 3≤j ≤2n (1 + z2 − zj )(1 + zj − z1 ) times that for z3 , z4 , . . . , z2n . As the leading singularity is cancelled, this gives the recursion relation 2n Kn (z1 , . . . , z2n )|z2 =z1 +1 = (1 + z2 − zj )(1 + zj − z1 ) Kn−1 (z3 , . . . , z2n ). j =3
(5.11) This is nothing but (5.9) for i = 1, and we deduce that Zn = αKn for some numerical factor α, fixed to be 1 by the initial condition Z1 (z1 , z2 ) = 1. Note that the Pfaffian is naturally a sum over pairings of 2n objects, indexed equivalently by crossing link patterns, and weighted by a fermionic sign factor. This decomposes naturally Zn into an alternating sum over link patterns. In particular taking all zi to be large, we obtain the leading degree contribution to Zn , Znmax = (z)Pf(1/(zi − zj ))1≤i<j ≤2n , which matches exactly the sum of the leading terms (3.27) obtained in Lemma 2 above. We get an interesting corollary of Theorem 5 by taking all zi to zero. To do this, notice that if Z = Pf(A)/ i<j f (zi − zj ), Aij = f (zi − zj ), f an odd function such that f (0) = 0, then when all zi tend to 0, we have: 1 1 ∂i ∂j Z → n(2n−1) Pf f (z − w)|z=w=0 . (5.12) i!j ! ∂zi ∂w j f (0) 0≤i,j ≤2n−1 With f (x) = x/(1 − x 2 ), this gives i+j Zn (0, . . . , 0) = Pf (−1)j δi+j,1 [2] , i 0≤i,j ≤2n−1
(5.13)
where the Kronecker delta symbol ensures that the matrix element vanishes unless i + j is odd. We also have the following determinantal expression, immediately following from (5.13): 2i + 2j + 1 Zn (0, . . . , 0) = det . (5.14) 2i 0≤i,j ≤n−1 These numbers play for the crossing O(1) loop model the same role as that played for the non-crossing loop model by the numbers An of alternating sign matrices of size n × n. They read 1, 7, 307, 82977, 137460201, 1392263902567 . . .
(5.15)
for n = 1, 2, 3, 4, 5, 6 . . . . A remark is in order. One possible interpretation of the numbers (5.14), via the Lindstr¨om–Gessel–Viennot formula [13], is that they count the total number of n-tuples of non-intersecting lattice paths subject to the following constraints: the paths are drawn on the edges of the square lattice, are non-intersecting, and may only make steps of (−1, 0) (horizontal to the left) or (0, 1) (vertical up), and start at the points (2i, 0), while they end up at the points (0, 2i + 1), i = 0, 1, . . . , n − 1. In (5.14),
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+i the combinatorial number 2i+2j is simply the number of configurations of a single 2i path starting at (2i, 0) and ending at (0, 2j + 1). Finally, it is easy to prove, using standard integrable or matrix model techniques, that 2n2 the numbers (5.14) behave for large n as Zn (0, . . . , 0) ≈ π2 . This is to be compared with the standard asymptotics for the number of n × n alternating sign matrices √ n2 An ≈ 3 4 3 . 6. Conclusion In this paper, we have investigated the ground state vector n of the inhomogeneous O(1) crossing loop model on a cylinder of perimeter 2n. The inhomogeneities translate into a collection of 2n spectral parameters z1 , z2 , . . . z2n , in terms of which we have proved that, when suitably normalized, all entries of n are polynomials of total degree 2n(n − 1). By further characterizing these entries, using in particular an algorithm for generating them recursively, we have been able to find compact formulas for their sum or partial sum over a subset of entries related to permutations of n objects. 6.1. Comparison with the non-crossing loop case. Comparing this case to that of the standard (non-crossing) O(1) loop model studied in [5], the situation is now much simpler. Indeed, the transitivity of the move generated by fi on the crossing link patterns has allowed us to describe all the entries of n simply in terms of successive local actions of the gauged divided difference operators i (3.12) on that corresponding to the maximally crossing link pattern. In particular, as i is degree-preserving, we have had no difficulty in proving that the coefficients of n are polynomials of degree 2n(n − 1), as opposed to the non-crossing case of [5], where no such simple operator is available, and where the degree issue was highly non-trivial, and eventually had to be settled by use of the algebraic Bethe Ansatz. Other properties, such as recursion relations when neighboring spectral parameters become related, turn out to be very similar in the two models. In particular, in light of the formula (5.10) for the sum of all entries of n , we have found an analogous Pfaffian formula for the square of the Izergin–Korepin/Okada determinant ZnI K , equal to the sum of entries in the non-crossing case, reading: zi2 + zi zj + zj2 − z z i j IK 2 × . Zn (z1 , z2 , . . . z2n ) = Pf zi − z j zi2 + zi zj + zj2 1≤i,j ≤2n
1≤i<j ≤2n
(6.1) The latter corresponds to the partition function of the inhomogeneous non-crossing O(1) loop model on a complete (infinite) cylinder of perimeter 2n. The analogies between the crossing and non-crossing loop cases lead us to expect the existence of a much more general treatment of integrable lattice models involving loops, which would display these common features. Actually, the right unifying algebraic setting seems to be the so-called “double affine Hecke algebras” [14] which typically mix loop-like operators with the action of the symmetric group on spectral parameters (see also [15]). 6.2. A positive extension of Schubert polynomials. An intriguing and new feature of the crossing loop model is the emergence of a special subset of entries of n , indexed by
Inhomogenous Model of Crossing Loops and Some Algebraic Multidegrees
481
permutations. Roughly speaking, the latter is obtained by projecting out the generators ei , i = n, 2n, via the relations (4.9). We have shown that the action of the operators i drastically simplifies in this sector so as to resemble the standard divided difference operators commonly used in Schubert calculus. In this respect, the conjecture of [6] relating the ground state of the homogeneous crossing loop model to the degrees of some varieties certainly loses much of its mystery. We have actually refined this conjecture so as to include all spectral parameter dependence, by interpreting a suitably normalized version of the entries of n in the permutation sector as the multidegree of the varieties studied in [9], and gave a sketch of proof of it. Playing around with changes of variables, we have also found an interesting family of polynomials corresponding to the permutation sector, which seems to have only non-negative integer coefficients. These correspond to the specializations zn+1−i = (ti − 1)/(ti + 1), zn+i = 0, for i = 1, 2, . . . , n. More precisely, we define the family of polynomials sπ via: 2−n(n−1) sπ (t1 , t2 , . . . , tn )=
n
i=1 (1 + ti )
1≤i<j ≤n (1 − ti
2(n−1)
+ 3tj + ti tj )
π
tn − 1 t1 − 1 ,... , , 0, . . . tn + 1 t1 + 1
.
(6.2)
The top member of the family reads simply sπ0 = 1≤i≤n tin−i , while all other members are obtained by repeated actions of the operators θi , i = 1, 2, . . . , n − 1, defined by θi sπ = ((1 + ti )(1 + ti+1 )∂i − τi )
(6.3)
(θi implementing as usual multiplication on the right by the elementary transposition of the permutation πˆ such that π(i) = n + π(n ˆ + 1 − i)). The lowest degree term in the sπ , sπmin , is readily identified with the Schubert polynomial indexed by πˆ . The higher degree terms all turn out experimentally to have non-negative integer coefficients. Equation (6.3) gives a very efficient way of computing the bidegrees of conjecture 1 (4.1): indeed, the latter are simply recovered by taking ti = A/B for all i and premultiplying sπ by B n(n−1) . This extension of Schubert polynomials awaits some good algebro-geometric or combinatorial interpretation.
6.3. Other entries of and more combinatorics. This subset of entries of particular interest should not let us forget about the general picture we have obtained. It is actually quite suggestive that for instance the change of variables zi = A/(A + B), zi+n = B/(A + B) for i = 1, 2, . . . , n, together with the global multiplication by (A + B)2n(n−1)−(n−2)−mod(n,2) 2−n(n−1) leads to homogeneous polynomials of A, B for all entries of n (not just those of the permutation sector), all with non-negative integer coefficients. It would be very interesting to find an interpretation for these other entries as well. More combinatorics are undoubtedly hidden in the O(1) crossing loop model. For instance, the specialization z1 = (t − 1)/(t + 1) while all other zi are taken to zero leads after multiplication by a global factor ((1+t)/2)2(n−1) to entries that are all polynomials of t with non-negative integer coefficients. The sum over all these entries produces a “refinement” of the numbers (5.15) into polynomials of t, reading up to n = 4: P1 (t) = 1, P2 (t) = 1 + 5t + t 2 , P3 (t) = 7 + 63t + 167t 2 + 63t 3 + 7t 4 , P4 (t) = 307 + 3991t + 18899t 2 + 36583t 3 + 18899t 4 + 3991t 5 + 307t 6 . (6.4)
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P. Di Francesco, P. Zinn-Justin
The coefficients of these polynomials play for the crossing loop case the same role as that played by refined alternating sign matrix numbers for the non-crossing one [16]. Note that they share with them the property that at t = 0 one recovers the total number of the preceding rank (with t = 1). We have also been able to construct counterparts for the doubly-refined alternating sign matrix numbers [17], by specializing our model to z1 = (t − 1)/(t + 1) and zn+1 = (u − 1)/(u + 1) while all other zi are zero, in which case we still find that all entries of a suitably normalized n are polynomials of t, u with non-negative integer coefficients. All these integers await some combinatorial interpretation. Such an interpretation should be provided by a vertex-type model with fixed boundary conditions on the square grid of size n × n, generalizing the six-vertex model with domain wall boundary conditions of the non-crossing case. The obvious candidates are the OSP (p|2m) vertex models of [7, 8], with p − 2m = 1, but the main problem is that there are many such models, whose row-to-row transfer matrix acts on a Hilbert space of dimension (p + 2m)2n for a system of size n, and that in order to cover at least the (2n − 1)!! dimensions of the Hilbert space of crossing link patterns, both p and m should be at least of the order of the size n. This is the worst scenario that could occur, as the number of spin degrees of freedom on each edge of the lattice must itself grow with the size of the system. The multiplicity of formulations of the vertex model of the non-crossing case as: alternating sign matrices, fully-packed loops, osculating paths, etc leaves much room for imagining a generalization adapted to the crossing loop case, but this is yet to be found. Acknowledgements. P.D.F. thanks B. Nienhuis, V. Pasquier and J.-B. Zuber for discussions. P.Z.-J. thanks F. Hivert and M. Kazarian for informative exchange on Schubert polynomials, and M. Tsfasman for discussions. Special thanks toA. Knutson for his explanations on multidegrees, and other valuable comments.
Appendix A. Entries of Ψ2 The entries of 2 read:
1
4
2
3
1
4
= (1 + z1 − z2 )(1 + z2 − z3 )(1 + z3 − z4 )(1 + z4 − z1 )
= (1 + z2 − z3 )(1 + z4 − z1 ) ×((1 + z3 − z1 )(2 − z4 + z1 ) + (1 + z1 − z2 )(1 + z1 − z3 ))
2
3
1
4
=
(1 + z1 − z2 )(1 + z3 − z4 ) ×((1 + z4 − z2 )(2 − z1 + z2 ) + (1 + z2 − z3 )(1 + z2 − z4 ))
2
3
Appendix B. Entries of Ψ3 from the Action of the Θi We show here how to obtain each entry of 3 by repeated use of Eq. (3.11).
Inhomogenous Model of Crossing Loops and Some Algebraic Multidegrees
483
The coefficient π0 is given by Eq. (3.19). It reads 1
6
2
5 3
= (1 + z1 − z2 )(1 + z1 − z3 )(1 + z2 − z3 )(1 + z2 − z4 ) ×(1 + z3 − z4 )(1 + z3 − z5 )(1 + z4 − z5 )(1 + z4 − z6 ) ×(1 + z5 − z1 )(1 + z6 − z1 )(1 + z6 − z2 )(1 + z5 − z6 )
4
It has total degree 12. The other coefficients are given by 1
6
2
5 3
4
1
6
2 4
1
6
2
5 3
4
1
6
2 4
1
6
2
5 3
6
2
5
6
2
= 2 1 π0
5 3
4
1
6
2
= 2 1 2 π0
= 6 2 1 π0
1
5
2
= 2 3 π0
= 5 3 π0
5
1
1
2
5 3
4
1
6
2
2
5 3
4
1
6
2
5
= 3 2 3 π0
= 1 3 π0
= 4 3 π0
4
1
6
2
5 3
5 3
6
3
6
= 2 π0
4
4
4
6
3
1
= 1 2 π0
= 1 π0
4
3
5 3
1
3
5 3
= 3 π0
= 1 4 3 π0
4
= 2 5 3 π0
4
Unfortunately lack of space does not permit us to produce them explicitly here. Note that we could have restricted ourselves to the computation of the π , taking for π one representative in each orbit under rotation, and extended the result to the remainder of the orbit by use of the cyclic covariance property (2.9). Although we have been able to bypass (3.16) by only using (3.11), we may have applied it to determine the last two entries above 1
6
= ( 6 2 1 2 − 2 1 − 3 − 1)π0
2
5 3
4
1
6
= ( 3 2 1 2 − 1 2 − 3 − 1)π0
2
5 3
4
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P. Di Francesco, P. Zinn-Justin
As already mentioned, both 6 and 3 could in principle increase the degree by one unit, but they both act on the entry 2 1 2 π0 , whose maximal degree (12) contribution reads max
1
6
2
5 3
= (z1 − z2 )(z1 − z3 )(z1 − z4 )(z1 − z5 )(z2 − z3 )(z2 − z4 ) ×(z2 − z6 )(z3 − z5 )(z3 − z6 )(z4 − z5 )(z4 − z6 )(z5 − z6 )
4
according to Lemma 2, Eq. (3.27). This is clearly invariant both under z6 ↔ z1 and z3 ↔ z4 , and therefore the degree is preserved by both operators. Appendix C. Bidegree of the Affine Homogeneous Variety Vπ for n = 3, 4 The specialized entries π , conjectured to be bidegrees of the Vπ , are listed below in decreasing number of crossings of the corresponding link pattern. The 6 bidegrees at n = 3: 1
6
= A3 B 3
d 2
5 3 1
4 6
= A2 B 2 (A2 + AB + B 2 )
d 2
5 3 1
4 6
= A2 B 2 (A2 + AB + B 2 )
d 2
5 3 1
4 6
= AB(A4 + 2A3 B + 4A2 B 2 + 4AB 3 + 2B 4 )
d 2
5 3 1
4 6
= AB(2A4 + 4A3 B + 4A2 B 2 + 2AB 3 + B 4 )
d 2
5 3 1
4 6
= A6 + 3A5 B + 7A4 B 2 + 9A3 B 3 + 7A2 B 4 + 3AB 5 + B 6
d 2
5 3
4
Inhomogenous Model of Crossing Loops and Some Algebraic Multidegrees
485
The 24 bidegrees for n = 4: d
d
d
d
1
8
2
7
3
6 4
5
1
8
2
7
3
6 4
5
1
8 7
3
6 5
1
8
2
7
3
d
d
d
d
=d
5
1
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=d
1 2
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=d
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4
5
8
1
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3
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5
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8 7
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1
8
2
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5
=d
2
7
3
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2
8
2
7
3
6
= A5 B 5 (A2 + AB + B 2 )
5
= A4 B 4 (A4 + 2A3 B + 4A2 B 2 + 4AB 3 + 2B 4 )
6 4
1
7
1
4
8
4 2
=d
= A4 B 4 (A2 + AB + B 2 )2
6 4
1
4
2
4
= A6 B 6
= A4 B 4 (2A4 + 4A3 B + 4A2 B 2 + 2AB 3 + B 4 )
= A3 B 3 (A6 + 3A5 B + 7A4 B 2 + 9A3 B 3 + 7A2 B 4 + 3AB 5 + B 6 )
5
= A3 B 3 (A6 + 3A5 B + 9A4 B 2 +17A3 B 3 + 21A2 B 4 + 15AB 5 + 5B 6 )
= A3 B 3 (5A6 + 15A5 B + 21A4 B 2 + 17A3 B 3 + 9A2 B 4 + 3AB 5 + B 6 )
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8
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d
d
d
d
d
=d
6 4
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1
8
2
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6 4
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1
8 7
3
6 4
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2 3
=d
= A3 B 3 (A2 + AB + B 2 ) (2A4 + 4A3 B + 5A2 B 2 + 4AB 3 + 2B 4 )
= A2 B 2 (A8 + 4A7 B + 13A6 B 2 + 28A5 B 3 + 42A4 B 4 + 42A3 B 5 + 28A2 B 6 + 12AB 7 + 3B 8 ) 6 7
4
5
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= A2 B 2 (3A8 + 12A7 B + 28A6 B 2 + 42A5 B 3 + 42A4 B 4 + 28A3 B 5 + 13A2 B 6 + 4AB 7 + B 8 ) 6 7
3 4
2
3
d
=d
5
= A2 B 2 (3A8 + 12A7 B + 27A6 B 2 + 40A5 B 3 + 45A4 B 4 + 40A3 B 5 + 27A2 B 6 + 12AB 7 + 3B 8 )
= AB(A10 + 5A9 B + 19A8 B 2 + 47A7 B 3 + 81A6 B 4 + 101A5 B 5 + 97A4 B 6 + 73A3 B 7 + 41A2 B 8 + 15AB 9 + 3B 10 )
= AB(3A10 + 15A9 B + 41A8 B 2 + 73A7 B 3 + 97A6 B 4 + 101A5 B 5 + 81A4 B 6 + 47A3 B 7 + 19A2 B 8 + 5AB 9 +B 10 )
= AB(2A10 + 10A9 B+34A8 B 2 + 82A7 B 3 +141A6 B 4 + 169A5 B 5 + 141A4 B 6 + 82A3 B 7 + 34A2 B 8 + 10AB 9 + 2B 10 )
6 4
5
1
8
2
= A12 + 6A11 B + 25A10 B 2 + 70A9 B 3 + 141A8 B 4 + 210A7 B 5 + 239A6 B 6 + 210A5 B 7 + 141A4 B 8 + 70A3 B 9 + 25A2 B 10 + 6AB 11 + B 12 6 7
3 4
5
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References 1. Pearce P.A., Rittenberg V., de Gier J., Nienhuis B.: Temperley–Lieb Stochastic Processes. J. Phys. A35, L661–L668 (2002) 2. Razumov A.V., Stroganov Yu.G.: Spin chains and combinatorics. J. Phys. A34, 3185 (2001) 3. Batchelor M.T., de Gier J., Nienhuis B.: The quantum symmetric XXZ chain at = −1/2, alternating sign matrices and plane partitions. J. Phys. A34, L265–L270 (2001) 4. Razumov A.V., Stroganov Yu.G.: Combinatorial nature of ground state vector of O(1) loop model. Theor. Math. Phys. 138, 333-337 (2004); Teor. Mat. Fiz. 138, 395–400 (2004) 5. Di Francesco P., Zinn-Justin P.: Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule. E. J. Combi. 12(1), R6 (2005) 6. De Gier J., Nienhuis B.: Brauer loops and the commuting variety. JSTAT P01006 (2005) 7. Martins M.J., Ramos P.B.: The Algebraic Bethe Ansatz for rational braid-monoid lattice models. Nucl. Phys. B500, 579–620 (1997) 8. Martins M.J., Nienhuis B., Rietman R.: An Intersecting Loop Model as a Solvable Super Spin Chain. Phys. Rev. Lett. 81 504–507 (1998) 9. Knutson A: Some schemes related to the commuting variety. http://arxiv.org/list/math.AG/0306275, 2003 10. Gaudin M.: La fonction d’onde de Bethe. Paris: Masson 1997 11. Lascoux A., Leclerc B., Thibon J.-Y.: Twisted action of the symmetric group on the cohomology of the flag manifold. Banach Center Publications, Vol. 36, 1996, pp. 111–124 12. Knutson A., Miller E.: Gr¨obner geometry of Schubert polynomials. Ann. Math. 161, no. 3, 1245– 1318 (2003) 13. Lindstr¨om B.: On the vector representations of induced matroids. Bull. London Math. Soc. 5 85–90 (1973); Gessel I. M., Viennot X.: Binomial determinants, paths and hook formulae. Adv. Math. 58 300–321 (1985) 14. Cherednik I.: Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and McDonald’s operators. IMRN (Duke Math. Jour.) 9, 171–180 (1992) 15. Fomin S., Kirillov A.N.: The Yang–Baxter equation, symmetric functions and Schubert polynomials. Discrete Math. 153, 123–143 (1996); The Yang–Baxter equation, symmetric functions and Grothendieck polynomials. http://arxiv.org/list/hep-th/9306005, 1993 16. Di Francesco P.: A refined Razumov–Stroganov conjecture. JSTAT, P08009 (2004) 17. Di Francesco P.: A refined Razumov–Stroganov conjecture II. JSTAT, P11004 (2004) Communicated by L. Takhtajan
Commun. Math. Phys. 262, 489–503 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1460-0
Communications in
Mathematical Physics
Bounds on the Spectral Shift Function and the Density of States Dirk Hundertmark1 , Rowan Killip2 , Shu Nakamura3 , Peter Stollmann4 Ivan Veseli´c4 1
Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois 61801, USA. E-mail: [email protected] Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA. E-mail: [email protected] Graduate School of Mathematical Science, University of Tokyo, 3-8-1 Komaba, Meguro Tokyo, 153-8914 Japan. E-mail: [email protected] 4 Fakult¨at f¨ ur Mathematik, Technische Universit¨at Chemnitz, 09107 Chemnitz, Germany. E-mail: [email protected]; [email protected]
2 3
Received: 25 February 2005 / Accepted: 3 May 2005 c The authors 2005 Published online: 9 December 2005 –
Abstract: We study spectra of Schr¨odinger operators on Rd . First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values µn of the difference of the semigroups as n → ∞ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schr¨odinger operators. The single site potential u is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be H¨older continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies H¨older continuity of the integrated density of states.
1. Introduction and Results In this paper we analyze the spectral properties of multi-dimensional Schr¨odinger operators. First, we consider a pair of operators H1 , H2 which differ by a compactly supported potential u. The singular values µn of the difference of the corresponding exponentials Veff := e−H1 − e−H2 are shown to decay almost exponentially in n. This result allows us to deduce a bound on the Lifshitz-Krein spectral shift function (SSF) of the operator pair H1 , H2 . We give a bound on the SSF when integrated over the energy axis against a bounded, compactly supported function. In turn, the bound on the SSF is used to prove a Wegner estimate for random Schr¨odinger operators of alloy type and H¨older continuity of the integrated density of states (IDS). Our estimates have a better continuity in the energy parameter than previously known bounds. Moreover, we are able to treat random coupling constants, whose distribution does not have c 2005 by the authors. Faithful reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes.
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a density. In particular, for H¨older continuous distributions we prove that the IDS is H¨older continuous, too. We will treat magnetic Schr¨odinger operators H = HA + V = (−i∇ − A)2 + V
(1)
acting on Rd whose potentials (magnetic and electric) obey the following hypotheses: Each component of A is L2loc . The positive part of the electric potential, V+ := max(0, V ), belongs to L1loc and the negative part, V− := max(0, −V ), is in the Kato class. Notice that under our convention, V = V+ − V− . For a general discussion of the Kato-class, see [15]; for its relevance to the Feynman-Kac formula see, e.g., [2, 6, 49]. In particular, V− is in the Kato-class, if 1/p V− Lp = sup |V (y) dy| < ∞, d − (R ) loc,unif
x∈Rd
|x−y|≤1
where p = 1 if d = 1 and p > d/2 if d ≥ 2. Thus the allowed potentials cover all physically relevant cases. Under these hypotheses, one may define H via the corresponding quadratic form (with core Cc∞ ). By the same method, one can define the Dirichlet restriction of H to the cube l = [−l/2, l/2]d , l ≥ 1. This will be denoted H l . Let H1 be a Schr¨odinger operator of the form just described and let H2 = H1 + u, where u = u+ − u− obeys the hypotheses for electric potentials just described. The starting point for our analysis is an estimate on the singular values of Veff := l := e−H1 − e−H2 and on the corresponding object in the finite volume case, namely Veff l l e−H1 − e−H2 . Recall that the singular values of a compact operator A are the squareroots of the eigenvalues of A∗ A. We will enumerate them as µ1 (A) ≥ µ2 (A) ≥ · · · ≥ 0 according to multiplicity. Theorem 1. There are finite positive constants c and C such that the singular values of l obey the operator Veff µn ≤ C e−cn
1/d
.
(2)
In fact, c may be chosen depending only on the dimension, while C depends on the Kato-class norms of u− , V− and on the diameter of the support of u+ . The same estimate holds for the singular values of Veff . p l l Remarks. i) In particular Veff Jp := n µn is finite and thus Veff is an element of the operator ideal Jp := {A compact | AJp < ∞} for any p > 0. Thus our result can be understood as a sharpening and generalization of norm, Hilbert Schmidt, and trace bounds on the difference of semigroups, derived e.g. in [7–9, 16–19, 48, 51, 52]. ii) Note that the estimate (2) depends on the positive part of u only through supp u. Thus, for u = λu, ˜ where u˜ ≥ 0 and λ is a non-negative coupling constant, the estimate is independent of the choice of λ. Moreover (1) u may be taken to +∞ on its support. In this case H2 equals the restriction of H1 to Rd \ supp u with Dirichlet boundary conditions, provided the boundary of supp u obeys some mild regularity conditions; see, e.g., [53].
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(2) Similarly, H1 may be defined on a set strictly smaller than Rd : Let D ⊂ Rd be open, HAD the Dirichlet restriction of HA on D, and H1 = HAD + V , where V satisfies the same conditions as before. In this case Hjl is the Dirichlet restriction of Hj , j = 1, 2 to the set l ∩ D. iii) The proof of Theorem 1 is surprisingly simple. Morally, the result is an immediate consequence of Weyl’s law for the eigenvalue asymptotic of Dirichlet Laplacians on compact domains. This suggests that the decay rate of the singular values of e−H1 − e−H2 is, in fact, given by exp(−cn2/d ). One might ask, however, whether the singular values could not typically decay at a much faster rate. It turns out that a decay rate like exp(−cnα ) for α > 2/d is impossible, see Remark ii) after Theorem 2 below. This leaves open the cases α ∈ (1/d, 2/d]. We conjecture that, in fact, the true bound is of the form µn ≤ C exp(−cn2/d ). Our interest in Theorem 1 comes from the fact that it allows us to derive an integral bound on the the SSF ξ(λ, H2 , H1 ) of the pair of operators H1 , H2 , which shows that the SSF can have only very mild local singularities. (See Sect. 2.1 for a precise definition of the SSF.) The SSF plays a role in different areas of mathematical physics, for instance in scattering theory, cf. e.g. [59], and the study of surface potentials, see [10, 36]. Various of its properties are discussed in the literature: monotonicity and concavity in [20, 22, 35], the asymptotic behaviour in the large coupling constant [43, 46, 44] and semiclassical limit [42, 40]. See [5, 34] for surveys. For t > 0 let Ft : [0, ∞) → [0, ∞) be defined by x Ft (x) = (exp(ty 1/d ) − 1) dy. (3) 0
As the integrand is increasing, Ft is a convex function. Theorem 2. Let ξ be the spectral shift function for the pair H1 , H2 or H1l , H2l . i) Let Ft be defined as above. There exisits a constant K1 , depending on t, such that for small enough t > 0, T Ft (|ξ(λ)|) dλ ≤ K1 eT < ∞ (4) −∞
for all T < ∞. ii) There exist constants K1 , K2 depending only on d, diam supp u+ and the Kato class norms of V− , u− , such that for any bounded compactly supported function f , f (λ) ξ(λ) dλ ≤ K1 eb + K2 {log(1 + f ∞ )}d f 1
(5)
with b = sup supp(f ). Remarks.
i) Note that the function Ft defined in (3) has the asymptotic behavior Ft (x) ∼ d x (d−1)/d exp(tx 1/d )
for large x.
Thus, by part i) of Theorem 2, the spectral shift function can have at most mild logarithmic local singularities. It is tempting to think that, at least for non-negative compactly supported perturbations, the spectral shift function should always be
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bounded. However, this is not the case. For a perturbation of the free Schr¨odinger operator with a constant magnetic field by a compactly supported potential, Raikov and Warzel showed that the spectral shift function diverges at each Landau level Eq like | ln(λ)| d/2 |ξ(Eq + λ)| ∼ as λ ↓ 0, (6) ln | ln λ| see [45] for the cased = 2 and [39] for the generalization to even dimensions. Thus, x setting Ft,α (x) := 0 (exp(ty α ) − 1) dy, the asymptotic (6) implies that Ft,α (|ξ |) is locally integrable if and only if 0 ≤ α ≤ 2/d, whereas Theorem 2 guarantees it only for α ≤ 1/d (and t small enough, if α = 1/d). ii) The proof of Theorem 2 shows that if Theorem 1 holds in the form µn ≤ c1 exp(−c2 nα ), then Ft,α (|ξ |) ∈ L1 (−∞, T ) for small enough t (and all finite T ). Thus the Raikov-Warzel result shows that the estimate in Theorem 1 cannot be improved beyond C exp(−cn2/d ). iii) An example without magnetic fields, where the SSF shows unexpected divergencies, was studied by Kirsch in [28, 29]. It is related to the one with a constant magnetic field in that the high degeneracy of eigenvalues plays a crucial role. Let E > 0, u non-negative, bounded with compact support, and not identically equal to zero, a : [0, ∞) → (0, ∞), and ξl (·) := ξ(·, −l , (− + a(l)u)l ). Then lim supl→∞ ξl (E) = ∞, for any E, a and u as above. This result relies on the degeneracy of eigenvalues of the pure Dirichlet Laplacian on a cube. There is, however, a set of full measure E ⊂ R with dense complement such that limN l→∞ ξl (E) = 0, for all E ∈ E, if a(l) ≤ l −k , k > 3. iv) In contrast to the above unboundedness results, Sobolev, [50] showed that for the pair H1 = − and H2 = − + u with |u(x)| ≤ const. (1 + |x|)−α and α > d, the spectral shift function ξ is, indeed, locally bounded. However, this type of result seems to require very strong hypotheses on H1 , for example, a trace class limiting absorption principle and in particular, that H1 has absolutely continuous spectrum on the positive real axis. Theorem 2.ii) has a nice consequence in the theory of random Schr¨odinger operators. In this case, we take f to be the derivative of a smooth, monotone switch function ρ := ρE,ε : R → [−1, 0]. By a switch function we mean that for a positive ε ≤ 1/2 it has the following properties: ρ ≡ −1 on (−∞, E − ε], ρ ≡ 0 on [E + ε, ∞) and ρ ∞ ≤ 1/ε. Theorem 2.ii) and the Krein trace identity, see §2.1, then imply that there is a constant CE such that Tr [ρ(H2 ) − ρ(H1 )] ≤ CE | log(ε)|d .
(7)
The estimate (7) improves upon the bound derived by Combes, Hislop and Nakamura in [14]. They prove that for any exponent α < 1, there is a constant C˜ E (α) depending only on d, C0 , diam supp u, E + ε and α such that Tr [ρ(H2 ) − ρ(H1 )] ≤ C˜ E (α) ε −α .
(8)
An alloy type model is a random Schr¨odinger operator Hω = H0 + Vω , where H0 = HA + Vper with a periodic potential Vper . The random part of the potential has the form Vω (x) = k∈Zd ωk u(x − k). The coupling constants ωk , k ∈ Zd , are a sequence of bounded random variables, which are independent and identically distributed with
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distribution µ. The expectation of the product measure k∈Zd µ is denoted by E. The single site potential u ≡ 0 is of compact support. Denote for ε > 0, s(µ, ε) = sup{µ([E − ε, E + ε]) | E ∈ R}.
(9)
With this definition, we have Theorem 3. Let Hω be an alloy type model and u ≥ κχ[−1/2,1/2]d for some positive κ. Then for each E0 ∈ R there exists a constant CW such that, for all E ≤ E0 and ε ≤ 1/2, E{Tr[χ[E−ε,E+ε] (Hωl )]} ≤ CW s(µ, ε) (log 1ε )d |l |.
(10)
In particular, if µ is H¨older continuous with exponent α, then the ε-dependence of the RHS of (10) is εα (log 1ε )d . In [54], Stollmann proved a weaker version of (10) with RHS equal to CW s(µ, ε) |l |2 . Bounds like (10) are called Wegner estimates. They were first deduced by physical reasoning by Wegner in [58] for the Anderson model, the finite difference analogue of the alloy type model. Wegner estimates are important a priori estimates, used to derive regularity properties of the integrated density of states (IDS) and to prove localization for random Schr¨odinger operators. In this context, localization means the existence of an energy region where the random operator has almost surely dense pure point spectrum with exponentially decaying eigenfunctions. See, e.g., [55] for a monograph exposition and, e.g., [21] (and the references therein) for more recent developments. The proof of Theorem 3 can be directly applied to Anderson type models, i.e. random Schr¨odinger operators on 2 (Zd ). In this case, a compactly supported potential is a finite rank operator. For such perturbations the supremum-norm of the induced SSF is bounded by the rank of the operator, see e.g. [5, 14]. The uniform bound on the SSF yields a bound like (10) with the RHS side equal to CW s(µ, ε) || and the constant CW independent of the energy E. This gives an easy proof of a Wegner estimate for H¨older continuous single site distributions µ. The IDS N(E) is defined as the limit of the distribution functions, Nωl (E) := |l |−1 #{ eigenvalues of Hωl not greater than E}, as l tends to infinity. For almost all ω ∈ the limit exists and is independent of ω. As a consequence of Theorem 3, the IDS of the above alloy-type model satisfies d 1 , |E1 − E2 | ≤ 1/2, |N(E1 ) − N(E2 )| ≤ CI s(µ, |E1 − E2 |) log |E1 − E2 | where the constant CI may be chosen uniformly if E1 and E2 vary in a compact interval I . Note that this continuity result cannot be obtained from a Wegner estimate with quadratic dependence on the volume of the box l . It also shows that, up to a logarithmic correction, the IDS enjoys the same regularity properties as the distribution of the random potential. Let us return to the discussion of the regularity of the SSF. Denote by H1 = Hω −ω0 u and H2 = H1 + u the alloy type operators where the value of the coupling constant at k = 0 is frozen and equal to 0 and ω0 respectively. The other coupling constants are still random. Despite the examples given above it is still possible that the average of the SSF ξ¯ (λ) := E{ξ(λ, H1 , H2 )} over the random background environment is locally bounded. This then implies no logarithmic loss in the Wegner estimate, and thus the Lipschitz continuity of the integrated density of states.
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Indeed, such a bound on ξ¯ for d ≤ 3 has recently been announced by Combes and Hislop, [23]. They have to assume that the single site distribution has a bounded density with respect to Lebesgue measure. So far, the averaging techniques at our disposal do not seem to be sufficient enough to prove that ξ¯ (λ) is locally bounded for rough single site distributions, even if they are H¨older continuous. In this context we would like to mention bounds on averaged fractional powers of the SSF derived in [1]. As a final remark, we discuss how Theorem 2 can be used to improve Wegner estimates for alloy type models with somewhat different properties than in Theorem 3. First we present an improvement of a recent Wegner estimate by Combes, Hislop and Klopp [12] for single site potentials with small support. Theorem 4. Let Hω be an alloy type model as defined in the paragraph following (8). Assume that Vper has the unique continuation property and is bounded below, ω is distributed according to a bounded density and 0 ≤ u ∈ L∞ is strictly positive on an open set. Then for each E0 ∈ R there exists a constant CW such that E{Trχ[E−ε,E+ε] (Hωl )}
≤ CW ε
1 log ε
d ||
(11)
for all E ≤ E0 and ε ≤ 1/2. This follows directly if one uses Theorem 2 instead of the Lp -estimates on the SSF in the Appendix of [12]. We mention three more disorder regimes where Theorem 2 may be used to simplify proofs of earlier Wegner estimates and to improve the dependence of the estimate in the energy interval length. (1) Single site potentials with small support and singular coupling constants. Using the perturbation technique of Kirsch, Stollmann and Stolz in [32], we can extend the result from Theorem 3 to single site potentials u ≥ κχ[−s,s]d for some κ, s > 0 in the case of zero magnetic field for energies near spectral edges. In this case no assumption on the unique continuation property is needed. (2) Coupling constants whose distribution is continuous merely at the extreme values. In [33], Kirsch and Veseli´c prove a Wegner estimate for alloy type potentials with non-positive single site potentials and coupling constants which have merely in a neighborhood of their maximal value a continuous distribution with bounded density. The estimate applies to energies at the bottom of the spectrum. Using Theorem 2 in the present paper, one can improve the Wegner estimate in [33]. (3) Single site potentials with changing sign. In [24], Hislop and Klopp studied alloy type models with continuous, compactly supported single site potentials, which may take values with both signs, and bounded coupling constants which are distributed according to a bounded, piecewise absolutely continuous density. They prove a Wegner estimate which is H¨older continuous in the energy variable and applies to energies below the spectrum of the non-random, unperturbed operator H0 . The result extends to internal spectral boundaries in the weak disorder regime. An extension of Theorem 2 in the present paper to the case where the perturbation is equal to a potential sandwiched between the square roots of the resolvent would make it possible to improve the Wegner estimate in [24] with regards to the energy interval length.
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Further results on Wegner estimates for alloy type Schr¨odinger operators can be inferred from [4, 13, 27, 30, 31, 36, 56, 57] and the references therein. Let us mention specifically, that if µ has bounded density and u ≥ κχ1 the IDS is actually Lipschitzcontinuous, see [37, 11]. However, the proof of this result is based on quite different methods than ours. It does not use estimates on the SSF; instead the residue theorem is applied to obtain an uniform bound on averaged resolvents. Due to the use of complex analysis it is not clear whether this method can be extended to the case when the single site distribution µ does not have a density. Let us sketch the outline of the paper: The next section contains the definition of the SSF and the proofs of Theorems 1 and 2. In Sect. 3 we prove a lemma which is needed to deal with singular coupling constants and complete the proof of the Wegner estimate, Theorem 3. 2. Bounds on the SSF 2.1. Definition of the SSF. We define the SSF in three steps. Each of them extends the definition to a larger class of operators. For proofs see, e.g., [5 or 59]. Assume first that H1 , H2 are selfadjoint, lower-semibounded with purely discrete spectrum. Then the SSF is defined as the difference of the eigenvalue counting functions, ξ(λ) := #{n | λn (H2 ) ≤ λ} − #{n | λn (H1 ) ≤ λ}, where λn (H ) enumerates the spectrum of H , including multiplicity, in increasing order. Consider now a pair of selfadjoint, lower-semibounded operators such that the difference H2 − H1 is trace class. Then there is a unique function ξ such that Krein’s trace identity (12) Tr [ρ(H2 ) − ρ(H1 )] = ρ (λ) ξ(λ, H2 , H1 ) dλ holds for all ρ ∈ C ∞ with compactly supported derivative (actually, ρ can be taken to lie in a certain Besov space, see [41]). If the operators have discrete spectrum, this definition of ξ coincides with the previous one. It can be recovered choosing a sequence of switch functions ρε which converges to a step function as ε → 0. Finally, we weaken the trace class assumption on the operator difference. Let g : R → [0, ∞) be a monotone, smooth function such that g(H2 ) − g(H1 ) is trace class. Assume that g is bounded on the spectra of H1 and H2 . Then the SSF for the operator pair g(H1 ), g(H2 ) is well defined and we may set
(13) ξ(λ, H2 , H1 ) := sign(g ) ξ g(λ), g(H2 ), g(H1 ) . This definition is independent of the choice of g. Formula (13) is called the invariance principle. This last definition will be sufficiently general to cover the Schr¨odinger operators we are considering. In the sequel we will choose g(x) = e−x . Alternatively, the SSF can be defined via the perturbation determinant from scattering theory by ξ(λ, H1 , H2 ) :=
1 lim arg det 1 + (H1 − H2 )(H2 − λ − iε)−1 ) π ε0
if (H1 − H2 )(H2 + i)−1 is trace class.
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2.2. Decay of singular values. Weyl’s asymptotic law gives the asymptotic behaviour of the nth eigenvalue of the Laplacian on an open ball B for large n. The following simple lemma provides a robust lower bound, very much in the spirit of Weyl’s law, but valid for all n and for general magnetic Schr¨odinger operators. It is the starting point for our proof of Theorem 1. As it costs us nothing in clarity, the lemma will be presented under weaker hypotheses than those described in the introduction. Lemma 5. Let H = HA + V = (−i∇ − A)2 + V as in the introduction (cf. (1)), except that we now require that V− is merely − bounded with relative bound δ < 1. Furthermore, let H U be the Dirichlet restriction of H to an arbitrary open set U with finite volume |U| (also defined via the corresponding quadratic forms). Then, for some constant C, the nth eigenvalue of H U satisfies 2π(1 − δ)d n 2/d En ≥ −C for all n ∈ N. (14) e |U| Proof. Since the Dirichlet Sobolev space H01 (U) is a natural subset of H 1 (Rd ), V− is also relatively form bounded w.r.t. −U , the Dirichlet Laplacian on U with relative bound δ. The diamagnetic inequality, [48], then implies that V− is also relative form bounded w.r.t. to the Dirichlet restriction of HA to U. That is, there exists a C ∈ R such that, as quadratic forms, V− ≤ δHA + C. In particular, since V+ is non-negative, H U ≥ HA − V− ≥ (1 − δ)HA − C, which implies the bound U
Tr(e−2tH ) ≤ e2tC Tr(e−2t (1−δ)HA ) = e2tC e−t (1−δ)HA 2HS 2tC =e |e−t (1−δ)HA (x, y)|2 dx dy, U ×U
where ·HS denotes the Hilbert-Schmidt norm. Again, using the diamagnetic inequality for the Schr¨odinger semigroup, e.g., [48, 26], one has the pointwise bound U |e−t (1−δ)HA (x, y)| ≤ et (1−δ) (x, y). In particular,
−d/2 U U e−t (1−δ)HA 2HS ≤ et (1−δ) 2HS = Tr(e2t (1−δ) ) ≤ |U| 8π t (1 − δ) . U
In the last line we used the fact that the kernel of the Dirichlet semigroup eβ on the diagonal is bounded by the free kernel, i.e., U
eβ (x, x) ≤ eβ (x, x) = (4πβ)−d/2
for all β > 0 and x ∈ U,
which follows immediately from the probabilistic representation of the Dirichlet semigroup [3, 47]. Thus
−d/2 U Tr(e−2tH ) ≤ |U| 8πt (1 − δ) . ˇ sev’s Let N U (E) be the number of eigenvalues of H U smaller or equal to E. By Cebyˇ inequality and the above bound,
Spectral Shift and Wegner Estimates
U
N (E) ≤ e
2tE
E −∞
497 U
e2ts dN U (s) ≤ e2tE Tr(e−2tH )
≤ |U| (8π(1 − δ))−d/2 t −d/2 e2t (E+C) = |U| where, in the last equality, we choose t := implies the lower bound 2π(1 − δ)d En ≥ e on the eigenvalues.
d 4(E+C) .
n |U|
e(E + C) d/2 , 2π(1 − δ)d
Since n ≤ N U (En ), this, in turn,
2/d −C
l requires only minor Proof of Theorem 1. We give the proof for Veff , the adaption to Veff changes. We will use the symbols c and C for constants that vary from line to line; however, their dependence on H1 and H2 will always be as stated in the theorem. Without loss of generality, we can assume that the origin is contained in the support of u. We will estimate the nth singular value by Dirichlet decoupling at an n-dependent radius R. To this end, let R be sufficiently large that supp(u) is contained strictly inside the ball of radius R centered at the origin, which we will denote by BR . Let HjR (j = 1 or 2) be the Dirichlet restriction of Hj to the BR , and let
AR := e−H2 − e−H1 R
R
and DR := Veff − AR .
(15)
As any Kato-class potential is relatively form bounded with respect to the Laplacian n with relative bound zero, we may apply Lemma 5 to deduce that µn (e−Hj ) ≤ C exp(−cn2/d R −2 ) for both j = 1 and j = 2. Since AR is the difference of two nonnegative operators by the min-max theorem its singular values obey the same type of bound: µn (AR ) ≤ C exp(−cn2/d R −2 ).
(16)
If Dn is bounded, then µn (Veff ) ≤ µn (AR ) + Dn . We now proceed to estimate the norm of Dn by using the Feynman-Kac-Itˆo formula for magnetic Schr¨odinger semigroups with Dirichlet boundary conditions, see [6, 48]. Let Ex and Px denote the expectation and probability for a Brownian motion, bt starting at x. If τR = inf{t > 0|bt ∈ BR } denotes the exit time from the ball BR and τn := τRn , then
1 1
(Dn f )(x) = Ex e−iSA (b) e− 0 (V +u)(bs )ds − e− 0 V (bs )ds χ{τn ≤1} (b)f (b1 ) , where SAt is real valued stochastic process corresponding to the purely magnetic part of the Schr¨odinger operator. To be precise, one has to fix a suitable gauge, e.g., Coulomb gauge, i.e., divA = 0, for this and then use gauge invariance for the general case, see [38]. By taking the modulus and using the triangle inequality, one sees that the magnetic vector potential drops out: 1 1 |Dn f |(x) ≤ Ex e− 0 V (bs )ds e− 0 u(bs )ds − 1χ{τn ≤1} (b)|f (b1 )| .
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Moreover, only Brownian paths which both visit supp u and leave BRn within one unit of time contribute to the expectation. Thus if τu is the hitting time for supp(u) and B = {τR ≤ 1, τu ≤ 1}, where we abbreviate R = Rn , then 1 1 |Dn f |(x) ≤ Ex e− 0 V (bs )ds e− 0 u(bs )ds − 1χB (b)|f (b1 )| , so, applying H¨older’s inequality, 1 8 1/8 1/8 − 1 u(b )ds |Dn f |(x) ≤ Ex e−8 0 V (bs )ds Ex e 0 s − 1 1/2 1/4 × Ex χB (b) Ex |f (b1 )|2 . By Kashminskii’s lemma, the Kato-class condition on V− and u− implies that the first two terms are bounded uniformly in x, [2, 49]. Levy’s inequality combined with elementary estimates imply Px=0 {τR ≤ 1} ≤ 2 2Px=0 {|b1 | ≥ R} ≤ Ce−R /4 . As any path in B must cover the distance r between 2 supp u and the complement of the ball BR , we can deduce that Px (B) ≤ Ce−r /4 ≤ √ 2 Ce−R /8 , where we chose without loss of generality r ≥ R/ 2. Thus 1/2 1/2 2 2 |Dn f |(x) ≤ Ce−R /32 Ex |f (b1 )|2 = Ce−R /32 (e |f |2 )(x) , in particular, using the fact that e is an L1 contraction, 1/2 2 2 2 1/2 Dn f 2 ≤ Ce−R /32 (e |f |2 )1 ≤ Ce−R /32 f 2 1 = Ce−R /32 f 2 . To balance the two bounds obtained for µn (AR ) and Dn one chooses Rn := n1/2d , which leads to (2). 2.3. Singular value decay implies SSF estimate. Proof of Theorem 2. Let Ft and the two Schr¨odinger operators H2 = H1 + u be as in the theorem. Using the invariance principle and a change of variables, we have T T F (|ξ(λ, H2 , H1 )|) dλ = F (|ξ(e−λ , e−H2 , e−H1 )|) dλ −∞ −∞ ∞ ≤ eT F (|ξ(s, e−H2 , e−H1 )|) ds. e−T
By an estimate of Hundertmark and Simon [25], the integral on the RHS is bounded by ∞ ∞ F (|ξ(s, e−H2 , e−H1 )|) ds ≤ µn (Veff )(F (n) − F (n − 1)) −∞
n=1
≤
∞
n
µn (Veff )
n=1 ∞
≤C
(ets
1/d
− 1)ds
n−1 1/d
e(t−c)n
n=1
which is finite, if we chose t smaller than the constant c from Theorem 1. This proves (4).
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To prove (5), we dualize the bound (4) with the help of Young’s inequality for an appropriate pair of functions. Note that Ft is non-negative, convex with Ft (0) = 0 and hence its Legendre transform G is well defined and satisfies log(1 + y) d G(y) := sup{xy − F (x)} ≤ y for all y ≥ 0. t x≥0 Thus, by the very definition of G, Young’s inequality holds: yx ≤ F (x) + G(y). So, with b = sup supp(f ), b f (λ)ξ(λ) dλ ≤ F (|ξ(λ)|) dλ + G(|f (λ)|) dλ. (17) −∞
Using the estimate (4), the first integral is bounded by K1 eb . For the second integral in (17), we estimate log(1 + |f (λ)|) d G(|f (λ)|) dλ ≤ |f (λ)| dλ ≤ t −d | log(1 + f ∞ )|d f 1 . t This finishes the proof of Theorem 2.
3. Proof of the Wegner Estimate 3.1. A partial integration formula for singular distributions. The main new idea to deal with single site distributions that are not absolutely continuous, is the following simple Lemma 6. Let µ be a probability measure with support in (a, b)) (or (a, ∞), if its support is unbounded from above) and φ ∈ C 1 (R) be non-decreasing and bounded. Then for any ε > 0, [φ(λ + ε) − φ(λ)] dµ(λ) ≤ s(µ, ε) · [φ(b + ε) − φ(a)], R
where s(µ, ε), the modulus of continuity of µ, is defined in (9). (If b = ∞, φ(b + ) means limx→∞ φ(x), which exists by the properties of φ.) Proof. The proof of this lemma follows immediately from the well-known integrationby-parts formula for Stieltjes integrals. We include it for the convenience of the reader. We write dµ = dM, where M is the distribution function of µ. In the following, all integrals are defined as Stieltjes integrals. Shifting variables and using that M is constant outside of [a, b] gives b+ε [φ(λ + ε) − φ(λ)] dµ(λ) = φ(λ) d[M(λ − ε) − M(λ)]. a
Integrating by parts gives
b+ε [φ(λ + ε) − φ(λ)] dµ(λ) = φ(λ)[M(λ − ε) − M(λ)] a b+ε − φ (λ)[M(λ − ε) − M(λ)] dλ. a
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The first term is zero, since M is constant outside of [a, b] (in case b = ∞, one uses boundedness of φ and limλ→∞ [M(λ−ε)−M(λ)] = 0). The second term is bounded by b+ε b+ε φ (λ)[M(λ) − M(λ − ε)] dλ ≤ sup [M(λ) − M(λ − ε)] · φ (λ) dλ λ
a
a
≤ s(µ, ε) · (φ(b + ε) − φ(a)), since φ ≥ 0.
3.2. Proof of the Wegner estimate. Let ρ be a switch function adapted to the interval [E − ε, E + ε]; see the discussion preceding (7). Then χ[E−ε,E+ε] (x) ≤ ρ(x + 2ε) − ρ(x − 2ε). We may assume without loss of generality k u(· − k) ≥ 1. By the mini-max principle for eigenvalues, we conclude
Tr[ρ(Hωl + ε)] ≤ Tr ρ(Hωl + ε u(· − k)) . k
Assume without loss of generality that l ∈ N. Then l is decomposed in L := l d unit ˜ = ∩ Zd , n → k(n) cubes. We enumerate the lattice sites in l by k : {1, . . . , L} → and set W0 ≡ 0,
Wn =
n
u(· − k(m)),
n = 1, 2, . . . , L.
m=1
Thus E{Tr[χ[E−ε,E+ε] (Hωl )]} ≤ E{Tr[ρ(Hωl + 2ε) − ρ(Hωl − 2ε)]} ≤ E{Tr[ρ(Hωl + 2ε) − ρ(Hωl + 2ε − 4εWL )]} L l l ≤E Tr[ρ(Hω + 2ε − 4εWn−1 )−ρ(Hω +2ε−4εWn )] . n=1
(18) We fix n ∈ {1, . . . , L}, denote k0 = k(n), ω⊥ := {ωk⊥ }k∈˜ , and set
ωk⊥ :=
0 ωk
if k = k0 , if k = k0 ,
φ(ωk0 ) = Tr ρ(Hωl ⊥ − 2ε + 4εWn−1 + ωk0 u(· − k0 )) ,
ωn ∈ R.
The function φ is continuously differentiable, monotone increasing and bounded. By definition of φ, E {Tr[ρ(Hωl + 2ε − 4εWn )) − ρ(Hωl + 2ε − 4εWn+1 )]} ≤ E{ [φ(ωk0 + 2ε) − φ(ωk0 )] dµ(ωk0 )}.
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Let a = inf supp(µ) − 1 and b = sup supp(µ) + 1. Using Lemma 6 and the Krein trace identity (12) together with the second part of Theorem 2, we have [φ(ωk0 + 2ε) − φ(ωk0 )] dµ(ωk0 ) ≤ s(µ, 2ε)[φ(b + 2ε) − φ(a)] ≤ CE s(µ, 2ε) (log(1/ε))d , which implies that (18) is bounded by CE
L
s(µ, 2ε) (log(1/ε))d ≤ CE s(µ, 2ε) (log(1/ε))d l d .
n=1
Note that we apply Theorem 2 successively L times. However, the constant CE depends only on the diameter of u and a local norm of the negative part of the background potential. For this local norm there exist an uniform estimate independent of l and the configuration of the coupling constants ωk , k = k0 . Acknowledgements. Stimulating discussions with W. Kirsch, V. Kostrykin, G. Stolz and S. Warzel are gratefully acknowledged. P.S. acknowledges the kind invitation to the University of Tokyo where part of this work started. D.H and I.V. would like to thank the Department of Mathematics at CalTech, especially B. Simon and C. Galvez, for their warm hospitality. We thank the following institutions for financial support: the National Science Foundation for their support under grants DMS–0400940 (D.H.) and DMS–0401277 (R.K.), the Sloan Foundation (R.K.), the JSPS through Grant-in-Aid for Scientific Research (C) 13640155 (S.N.), the DFG through the SFB 393 (P.S.) and grants Ve 253/1 and /2 within the Emmg-Noether Programme (I.V.). See also www.arxiv.org/math-ph/0412078 or www.ma.utexas.edu/mp arc/c/04/04-423.pdf for an earlier version of this paper.
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Commun. Math. Phys. 262, 505–531 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1439-x
Communications in
Mathematical Physics
Poisson Boundary of the Dual of SUq (n) Masaki Izumi1, , Sergey Neshveyev2, , Lars Tuset2, 1
Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. E-mail: [email protected] 2 Mathematics Institute, University of Oslo, PB 1053 Blindern, Oslo 0316, Norway. E-mail: [email protected]; [email protected] Received: 1 March 2005 / Accepted: 1 March 2005 Published online: 29 October 2005 – © Springer-Verlag 2005
Abstract: We prove that for any non-trivial product-type action α of SUq (n) (0 < q < 1) on an ITPFI factor N, the relative commutant (N α ) ∩ N is isomorphic to the algebra L∞ (SUq (n)/Tn−1 ) of bounded measurable functions on the quantum flag manifold SUq (n)/Tn−1 . This is equivalent to the computation of the Poisson boundary of the dual discrete quantum group SU q (n). The proof relies on a connection between the Poisson integral and the Berezin transform. Our main technical result says that a sequence of Berezin transforms defined by a random walk on the dominant weights of SU (n) converges to the identity on the quantum flag manifold.
0. Introduction The study of random walks on duals of compact groups, by which is meant the study of convolution operators on group von Neumann algebras of compact groups, was initiated by Biane in [B1], who developed a theory which parallels the theory of random walks on discrete abelian groups [B2, B3]. The study of random walks in the more general setting of duals of compact quantum groups [W] was undertaken by the first author [I1], who was motivated by product-type actions of such groups on infinite tensor products of factors of type I (ITPFI). In the classical case such actions are always minimal. For quantum groups this is not so. In fact, the relative commutant of the fixed point algebra can be interpreted as the algebra of bounded measurable functions on the Poisson boundary of the dual discrete quantum group. The general theory of noncommutative Poisson boundaries developed in [I1] was illustrated by the computation of the bound2 ary of SU q (2), which was shown to be isomorphic to the quantum sphere Sq . Later the
Supported by JSPS. Partially supported by the Norwegian Research Council. Supported by the SUP-program of the Norwegian Research Council.
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second and the third author [NT1] computed the Martin boundary of SU q (2), that is, described all (unbounded) harmonic functions, thus generalizing the results in [B3] and establishing a connection between the results in [B3] and [I1]. There are important differences between the classical boundary theory (or even that for duals of compact groups as considered by Biane) and the boundary theory for discrete quantum groups, which we want to stress. In the classical setting, if one forgets about the action of a group on its boundary, the computation of the Poisson boundary reduces to the question whether it is trivial or not. In the quantum case just the description of the (noncommutative) algebra of functions on the Poisson boundary is a highly nontrivial problem. On the other hand, such a central question within the classical theory as the description of minimal harmonic functions, becomes of peripheral interest in the quantum setting. We refer the reader to the survey articles [I2] and [NT2] for further discussions of noncommutative boundaries. The purpose of the present paper is to compute the Poisson boundary of SU q (n).As we already said, this problem can be formulated without any reference to random walks. Suppose one is given a product-type action α of SUq (n) on N . By restriction we get an action of SUq (n) on (N α ) ∩ N . If the action α is faithful, then (N α ) ∩ N is anti-isomorphic to N ∩ (SUq (n) N ). In this case the dual action of SU q (n) on N ∩ (SUq (n) N ) α α induces an action on (N ) ∩ N . The problem is to compute (N ) ∩ N together with the two above actions of SUq (n) and SU q (n). Our main result can be stated in this setting as follows. Theorem A. For any non-trivial product-type action α of SUq (n) (0 < q < 1) on an ITPFI factor N, the relative commutant (N α ) ∩ N is SUq (n)-equivariantly isomorphic to the algebra L∞ (SUq (n)/Tn−1 ) of bounded measurable functions on the quantum α flag manifold SUq (n)/Tn−1 . When the action of SU q (n) on (N ) ∩ N is well-defined, the isomorphism is also SU q (n)-equivariant. As for the boundary interpretation of (N α ) ∩ N , the action of SU q (n) corresponds to the usual action by translations of a group on its boundary, while the action of SUq (n) can be thought of as coming from the symmetry group of the measure. Then an equivalent form of Theorem A is the following. Theorem B. Let G be the q-deformation (0 < q < 1) of the compact group SU (n), or of its quotient by a normal subgroup. Let T ⊂ G be the maximal torus. Then for any ˆ the corresponding Poisson boundary of G-invariant normal generating state on l ∞ (G), ˆ is G- and G-equivariantly ˆ G isomorphic to the quantum flag manifold G/T . There are several heuristic reasons why such a result should be true. For example, by the results of Biane [B2] and Collins [C] the minimal Martin boundary of the dual of SU (n) is the sphere in the dual of the Lie algebra, and the action of SU (n) on the boundary is just the coadjoint action. Thus one can expect that a part of the Martin boundary of SU q (n) carrying the canonical harmonic state is a certain quantization of the sphere. By [H] the action of SUq (n) on the Poisson boundary of SU q (n) is ergodic. This corresponds in the classical case to the fact that the measure is supported on some orbit, and indeed, the typical coadjoint orbit is the flag manifold. Since, however, the Martin boundary seems difficult to compute, we shall not follow the approach suggested above. Our computation of the Poisson boundary is based on a
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careful study of the completely positive map from L∞ (SUq (n)) into the algebra of harmonic functions introduced in [I1]. We call this map the Poisson integral. We show that multiplicativity of restricted to L∞ (SUq (n)/Tn−1 ) is equivalent to convergence to the identity of a certain sequence of operators on L∞ (SUq (n)/Tn−1 ). These operators are analogues of Berezin transforms. In the classical case it is known [D] that Berezin transforms converge to the identity on the flag manifold along any ray in the Weyl chamber. We prove that our operators converge to the identity on the quantum flag manifold along almost every path in the Weyl chamber. It is worth noticing that though we benefited from these analogies, our proof bears no relation to the classical proof. As a matter of fact, our operators are not the most straightforward analogues of Berezin transforms. In particular, in the classical limit they give operators mapping everything to the scalars. Our proof of convergence invokes yet another auxiliary operator, which in the classical limit yields the identity operator, whereas in the quantum case it is uniquely ergodic on C(SUq (n)/Tn−1 ). Multiplicativity of on L∞ (SUq (n)/Tn−1 ) implies injectivity. This can be interpreted as the existence of a surjective map from the Poisson boundary onto the quantum flag manifold. To show that this is an isomorphism we then use a counting argument relying on the already mentioned ergodicity of the action of SUq (n) on the Poisson boundary. Finally note that our approach gives a less computational proof of the result for SUq (2) than in [I1] and [NT1]. For example, we do not need any knowledge of the Clebsch-Gordan coefficients. 1. Preliminaries We will use the same conventions for quantum groups as in [NT1]. We will, however, change the notation slightly to hopefully make it more transparent. So a compact quantum group G is defined by a unital C∗ -algebra C(G) with comultiplication : C(G) → C(G) ⊗ C(G), which is a unital ∗-homomorphism such that ( ⊗ ι) = (ι ⊗ ) and both (C(G))(C(G) ⊗ 1) and (C(G))(1 ⊗ C(G)) are dense in C(G) ⊗ C(G). We will always work in the reduced setting, that is, we assume that the Haar state ϕ on C(G) is faithful. In cases when it is more convenient to deal with von Neumann algebras, we shall use the notation L∞ (G) for πϕ (C(G)) . By a unitary representation of G on a Hilbert space H we mean a unitary corepresentation of (C(G), ), that is, a unitary U ∈ M(C(G) ⊗ B0 (H )) such that ( ⊗ ι)(U ) = U13 U23 . Here B0 (H ) denotes the algebra of compact operators on H . Let I = Irr(G) be the set of equivalence classes of irreducible unitary representations of G. ˆ For each s ∈ I , we fix a representative U s ∈ C(G) ⊗ B(Hs ). Then the algebra c0 (G) ˆ vanishing at infinity is defined as the of functions on the dual discrete quantum group G ˆ is defined as the W∗ -direct sum of C∗ -direct sum of B(Hs ), s ∈ I . The algebra l ∞ (G) B(Hs ), s ∈ I . Let A(G) be the ∗-subalgebra of C(G) generated by the matrix coefficients of finite ˆ ⊂ c0 (G) ˆ be the algebraic direct dimensional unitary representations of G. Let also A(G) ˆ If we fix matrix units sum of B(Hs ), s ∈ I . There is a pairing between A(G) and A(G). {msij }i,j in B(Hs ), and denote by {usij }i,j the corresponding matrix coefficients of U s , then the pairing is given by (usij , mtkl ) = δst δik δj l . For a unitary representation U of G ˆ → B(H ) is given by on H , the corresponding representation πU : l ∞ (G) πU (ω) = (ω ⊗ ι)(U ).
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In particular, if W ∈ B(Hϕ ⊗ Hϕ ) is the multiplicative unitary for G, W ∗ (ξ ⊗ aξϕ ) = (a)(ξ ⊗ ξϕ ), ξ ∈ Hϕ , a ∈ C(G), ˆ → B(Hϕ ). Thus we can (and often then we get a faithful representation πW : l ∞ (G) ∞ ∞ ˆ will) think of L (G) and l (G) as subalgebras of B(Hϕ ). Note that the comultiplicaˆ → l ∞ (G) ˆ ⊗ l ∞ (G) ˆ are then ˆ : l ∞ (G) tions : L∞ (G) → L∞ (G) ⊗ L∞ (G) and given by ˆ = W (x ⊗ 1)W ∗ . (a) = W ∗ (1 ⊗ a)W, (x) Given a unitary representation U of G on H , we define a left and a right action of G on B(H ) by αU,l : B(H ) → L∞ (G) ⊗ B(H ), αU,l (x) = U ∗ (1 ⊗ x)U, ∗ αU,r : B(H ) → B(H ) ⊗ L∞ (G), αU,r (x) = U21 (x ⊗ 1)U21 . ˆ = ⊕s B(Hs ), we get in particular a left and a right action of G on l ∞ (G), ˆ As l ∞ (G) ∞ ˆ which we denote by l and r , respectively. Thinking of l (G) as the group von Neumann algebra of G, they are analogues of the adjoint action. If the representation U is finite dimensional, then the actions of G on B(H ) have canonical invariant states 1 Tr(·πU (ρ −1 )), (ι ⊗ φU )αU,l = φU (·)1, dU 1 ωU = Tr(·πU (ρ)), (ωU ⊗ ι)αU,r = ωU (·)1, dU φU =
where ρ = f1 is the Woronowicz character and dU = Tr(πU (ρ −1 )) = Tr(πU (ρ)) is the quantum dimension of U . We will write φs and ωs instead of φU s and ωU s , respectively. In general, all normal left (resp. right) G-invariant states on B(H ) are given by Tr(·aπU (ρ −1 )) ˆ is a positive element such that Tr(aπU (resp. by Tr(·aπU (ρ))), where a ∈ πU (l ∞ (G)) −1 (ρ )) = 1 (resp. Tr(aπU (ρ)) = 1). ˆ (resp. by Cr (G)) ˆ the space of normal left (resp. right) G-invariant Denote by Cl (G) ∞ ˆ ˆ ˆ is the closed linear span of φs (resp. ωs ), functionals on l (G). Then Cl (G) (resp. Cr (G)) ˆ (as well as Cr (G)) ˆ is an algebra with product φ1 φ2 = (φ1 ⊗φ2 ). ˆ s ∈ I . The space Cl (G) Alternatively, one can define a fusion algebra structure on R(G) = ⊕s∈I Z in the sense of [HI]. Then the quantum dimension function on R(G) defines a convolution algebra ˆ and Cr (G). ˆ The algebra Cl (G) ˆ has an anti-linear l 1 (I ), which is isomorphic to Cl (G) involution φ → φˇ such that φˇ U = φU¯ , where U is the conjugate unitary representation. Let also s → s¯ be the involution on I , so U s¯ ∼ = Us. ˆ ∗ , define the convolution operator Pφ = (φ ⊗ ι) ˆ Given a normal state φ ∈ l ∞ (G) ∞ ˆ on l (G), and set ˆ φ) = {x ∈ l ∞ (G) ˆ | Pφ (x) = x}. H ∞ (G,
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ˆ φ) is (the algebra of bounded measurable functions on) the Poisson boundThen H ∞ (G, ˆ with respect to φ. According to [I1] the algebra structure on H ∞ (G, ˆ φ) is given ary of G by 1 k Pφ (xy). n→∞ n
x · y = s ∗ − lim
n−1 k=0
ˆ Denote by supp φ ⊂ I the set of We will only be interested in the case when φ ∈ Cl (G). ˆ A state φ is called s ∈ I such that φ(Is ) > 0, where Is is the unit in B(Hs ) ⊂ l ∞ (G). generating if ∪n∈N supp φ n = I . If φ is not generating, but supp φ is symmetric, then the norm closure of the linear span of the matrix coefficients usij for s ∈ ∪n∈N supp φ n is the algebra C(H ) for a compact quantum group H . Thus H is a quotient of G, and by the orthogonality relations we have Irr(H ) = ∪n∈N supp φ n . The symmetry assumption for supp φ is needed to ensure that C(H ) is a ∗-algebra. In the case of q-deformations of compact connected semisimple Lie groups [KoS] this assumption is redundant. Indeed, it is well-known that any compact connected semisimple Lie group G has the property that if U is a faithful unitary representation of G, then any irreducible representation of G appears as a subrepresentation of the tensor power U ×n of U for some n ∈ N. It follows that ∪n∈N supp φ n is always symmetric, more precisely, ∪n∈N supp φ n = Irr(H ) with H = G/(∩s∈supp φ Ker U s ). As the fusion algebra is independent of the deformation parameter, we conclude that for the q-deformation G of any compact connected semisimple Lie group, the set ∪n∈N supp φ n is symmetric, so it corresponds to a certain quotient of G. Note that the Poisson boundary in general depends on the generating state. However, in good situations, in particular for duals of q-deformations, it does not. Proposition 1.1. Assume that the fusion algebra R(G) of G is commutative. Then for ˆ we have any generating state φ ∈ Cl (G) ˆ φ) = {x ∈ l ∞ (G) ˆ | Pφs (x) = x for all s ∈ I }. H ∞ (G, Proof. The inclusion ⊃ is obvious as Pφ is a (possibly infinite) convex combination of the operators Pφs , s ∈ I . Conversely, fix s ∈ I . Since φ is generating, there exists n ∈ N such that if we write φ n = t∈I λt φt , we have λs = 0. Then where ψ = (1 − λs )−1
Pφn = λs Pφs + (1 − λs )Pψ ,
λt φt . Since Pφs Pψ = Pψφs and R(G) is commutative, Pφs ˆ Hence by Lemma 1.1 and Pψ are commuting contractions on the Banach space l ∞ (G). in Chapter 5 of [Re], if Pφn (x) = x then Pφs (x) = x. t=s
Recall now the connection between Poisson boundaries and product-type actions [I1]. Let U be a unitary representation of G on H , and φ˜ ∈ B(H )∗ a normal faithful αU,l invariant state. Set −1
˜ (N, ν) = ⊗ (B(H ), φ), −∞
and Nn = · · · ⊗ 1 ⊗ ⊂ N. The actions αU ×n ,l of G on B(H ⊗n ) ∼ = Nn define a left action α of G on N , where U ×n = U12 · · · U1,n+1 . To describe the relˆ Let En : N → Nn be the ˜ U ∈ Cl (G). ative commutant (N α ) ∩ N , consider φ = φπ B(H )⊗n
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ˆ → N the homomorphism defined ν-preserving conditional expectation, and jn : l ∞ (G) by jn (x) = · · · ⊗ 1 ⊗ πU ×n (x). Then En jn+1 = jn Pφ . ∞ ˆ → ⊕ The kernel of jn is ⊕s ∈supp / φ n B(Hs ). So if Fn : l (G) s∈supp φ n B(Hs ) is the ˆ canonical projection, we have jn Fn = jn , and jn is faithful on Fn (l ∞ (G)). As the image of (N α ) ∩ N under En is contained in the relative commutant
⊗n αU ×n ,l B H ∩ B H ⊗n , which coincides with the image of πU ×n , we conclude that the elements of (N α ) ∩ N are in one-to-one correspondence with bounded sequences {xn }∞ n=1 such that xn ∈ ˆ and Fn (l ∞ (G)) xn = Fn Pφ (xn+1 ). Such sequences are called Pφ -harmonic. It turns out that if φ is generating, then any ˆ φ) such that xn = bounded harmonic sequence {xn }n defines an element x ∈ H ∞ (G, Fn (x), and vice versa. Thus we get an isomorphism ∼ α ) ∩ N, j∞ (x) = s ∗ − lim jn (x). ˆ φ)→(N j∞ : H ∞ (G, n→∞
ˆ on l ∞ (G) ˆ to ˆ of G By restricting the left action l of G and the right action ˆ we get actions of G and G on the Poisson boundary. When φ is generating ˆ φ) with (N α ) ∩N , the action of G is just the restriction of α to the and we identify H ∞ (G, relative commutant, since the homomorphisms jn are obviously G-equivariant. As was ˆ is related to the dual action of G ˆ on N ∩ (G N ). conjectured in [I1], the action of G In the rest of this section we want to clarify this point. This will not be used in the subsequent sections. Consider first a more general situation. Suppose we are given a left action of G on a von Neumann algebra N , ˆ φ), H ∞ (G,
α : N → L∞ (G) ⊗ N. The crossed product G N is by definition the W∗ -subalgebra of B(Hϕ ) ⊗ N generated ˆ ⊗ 1 and α(N). Let ν be a normal faithful G-invariant state on N . Then the by l ∞ (G) map v : Hν → Hϕ ⊗ Hν , aξν → α(a)(ξϕ ⊗ ξν ) for a ∈ N, ˆ ˆ is an isometry. As α(N )(l ∞ (G)⊗1) is dense in GN and xξϕ = εˆ (x)ξϕ for x ∈ l ∞ (G), ∞ ˆ we see that the image of v is a (G N )-invariant subwhere εˆ is the counit on l (G), space of Hϕ ⊗ Hν . Let V be the canonical representation of G on Hν implementing the action, V ∗ (ξ ⊗ aξν ) = α(a)(ξ ⊗ ξν ),
ξ ∈ Hϕ , a ∈ N.
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Since (1 ⊗ v)V ∗ (ξ ⊗ aξν ) = (1 ⊗ v)α(a)(ξ ⊗ ξν ) = (ι ⊗ α)α(a)(ξ ⊗ ξϕ ⊗ ξν ) = ( ⊗ ι)α(a)(ξ ⊗ ξϕ ⊗ ξν ) = (W ∗ ⊗ 1)(1 ⊗ α(a))(ξ ⊗ ξϕ ⊗ ξν ) = (W ∗ ⊗ 1)(1 ⊗ v)(ξ ⊗ aξν ), V ∗ = (S ⊗ ι)(V ) and W ∗ = (S ⊗ ι)(W ), where S is the converse for G, we get ˆ vπV (x) = (x ⊗ 1)v for x ∈ l ∞ (G). We also have v ∗ α(a)v = a for a ∈ N . Thus v ∗ (x ⊗ 1)α(a)v = πV (x)a
ˆ a ∈ N. for x ∈ l ∞ (G),
(1.1)
ˆ ˆ where Noting that (Jϕ ⊗ Jν )v = vJν , and recalling that Jϕ x ∗ Jϕ = R(x) for x ∈ l ∞ (G), ˆ we see that Jν πV (l ∞ (G))J ˆ ν = πV (l ∞ (G)). ˆ Thus Rˆ is the unitary antipode on l ∞ (G), ∗ ∞ ˆ the map x → Jν x Jν defines an anti-isomorphism of N ∩ (N ∨ πV (l (G))) onto ˆ ˆ ) = N ∩ (N α ) . N ∩ (N ∨ πV (l ∞ (G))) = N ∩ (N ∩ πV (l ∞ (G)) It follows that the map θ : N ∩ (G N ) → (N α ) ∩ N, x → Jν v ∗ x ∗ vJν , is a normal surjective ∗-anti-homomorphism. ˆ ⊗ (G N ) is defined by The dual action αˆ : G N → l ∞ (G) α(x) ˆ = W˜ (1 ⊗ x)W˜ ∗ , x ∈ G N, where W˜ = ((Jϕ ⊗ Jϕ )W21 (Jϕ ⊗ Jϕ )) ⊗ 1, so that ˆ ˆ α((x ˆ ⊗ 1)α(a)) = ((x) ⊗ 1)(1 ⊗ α(a)) for a ∈ N, x ∈ l ∞ (G). ˆ on G N induces a right action αˆ op of G ˆ on (G N )op , The left action αˆ of G ˆ αˆ op (x) = (ι ⊗ R)(α(x) 21 ). In the simplest case when N = B(H ) all these constructions can be made more explicit. Lemma 1.2. Let U be a unitary representation of G on H , N = B(H ), α = αU,l , ν a normal faithful G-invariant state on B(H ), φ = νπU . Then the map ˆ ⊗ N → G N, θ0 (x) = U ∗ xU, θ0 : l ∞ (G) is an isomorphism with the following properties: ˆ ⊗ N; ˆ ⊗ ι)(x) = αθ (i) (ι ⊗ θ0 )( ˆ 0 (x) for x ∈ l ∞ (G) ˆ we have E0 θ0 (x ⊗ (ii) for the conditional expectation E0 = ι ⊗ ν : G N → l ∞ (G), ˆ ˆ ˆ φ R(x) for x ∈ l ∞ (G); 1) = RP ˆ ˆ (iii) θ θ0 (x ⊗ 1) = πU R(x) for x ∈ l ∞ (G).
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ˆ and 1 ⊗ N generate l ∞ (G) ˆ ⊗ N , the fact that θ0 is an ˆ ∞ (G)) Proof. Since (ι ⊗ πU )(l isomorphism follows from the identities ˆ ˆ U ∗ (1 ⊗ a)U = α(a) for a ∈ N, U ∗ (ι ⊗ πU )(x)U = x ⊗ 1 for x ∈ l ∞ (G), where we have used that U = (ι ⊗ πU )(W ). The equality in (i) obviously holds for x = 1 ⊗ a, a ∈ N . Thus it is enough to ˆ ˆ consider x = (ι ⊗ πU )(y). Then θ0 (x) = y ⊗ 1, so αθ ˆ 0 (x) = (y) ⊗ 1. On the other hand, ˆ ⊗ ι)(x) = (ι ⊗ θ0 )(ι ⊗ ι ⊗ πU )( ˆ ⊗ ι)(y) ˆ (ι ⊗ θ0 )( ˆ ˆ ˆ = (ι ⊗ θ0 )(ι ⊗ ι ⊗ πU )(ι ⊗ )(y) = (y) ⊗ 1. Thus (i) is proved. ˆ we have To prove (ii), note that for any y, z ∈ l ∞ (G) ˆ ⊗ z)) = E0 ((y ⊗ 1)απU (z)) = νπU (z)y = φ(z)y. E0 θ0 (ι ⊗ πU )((y)(1 ˆ ⊗ A(G) ˆ → A(G) ˆ ⊗ A(G) ˆ by r(y ⊗ z) = In other words, if we define a map r : A(G) ˆ (y)(1 ⊗ z), then E0 θ0 (ι ⊗ πU )r = ι ⊗ φ. It is well-known (see e.g. [NT1, Prop. 1.3]) that the map r is bijective with inverse s ˆ we get ˆ ˆ Hence for any y, z ∈ A(G) given by s(y ⊗ z) = (ι ⊗ S)((1 ⊗ Sˆ −1 (z))(y)). ˆ ˆ ⊗ Sˆ −1 (z))(y)). E0 θ0 (y ⊗ πU (z)) = (ι ⊗ φ)s(y ⊗ z) = (ι ⊗ φ S)((1 1 − 21 ˆ Choosing a net ˆ As the state φ is G-invariant and Sˆ = ρ 2 R(·)ρ , we have φ Sˆ = φ R. ˆ {zi }i ⊂ A(G) of central projections converging strongly to the unit, we thus get
ˆ ˆ (y) ˆ ⊗ φ)(Rˆ ⊗ R) ˆ (y) ˆ ⊗ ι) ˆ ˆ φ R(y), ˆ ˆ ˆ R(y) = R(ι = R(φ = RP E0 θ0 (y ⊗ 1) = (ι ⊗ φ R) ˆ This proves (ii). ˆ ˆ = ˆ op R. where we have used that (Rˆ ⊗ R) To prove (iii) we identify Hν with the space H S(H ) of Hilbert-Schmidt operators on 1
H , so that ξν = T02 if T0 ∈ B(H ) is the operator defining ν, ν = Tr(·T0 ). If we further identify H S(H ) with H ⊗ H , we have Jν (ξ¯ ⊗ ζ ) = ζ¯ ⊗ ξ , and a(ξ¯ ⊗ ζ ) = ξ¯ ⊗ aζ for ˆ ∈ B(H ⊗ H ) a ∈ N = B(H ). Thus we have to prove that θ θ0 (x ⊗ 1) = 1 ⊗ πU R(x) ∞ ˆ for x ∈ l (G). To compute θ we will check first that V = U × U . The unitary V ∈ M(C(G) ⊗ B0 (Hν )) is characterized by the properties that it implements the action, V ∗ (1 ⊗ a)V = α(a) for a ∈ N, and that V (ξ ⊗ ξν ) = ξ ⊗ ξν for ξ ∈ Hϕ . The first property is obviously satisfied by U × U . To check the second one, recall that by definition U = (R ⊗ j )(U ) ∈ M(C(G) ⊗ B0 (H )), where R is the unitary antipode on C(G) and j (x)ξ¯ = x ∗ ξ . If we consider C(G) ⊗ H S(H ) as a left module over M(C(G) ⊗ B0 (H ⊗ H )), it is then easy to check that (U × U )(1 ⊗ T ) = (R ⊗ ι)((R ⊗ ι)(U )(1 ⊗ T )U ) for T ∈ H S(H ).
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1
We have to prove that (U × U )(1 ⊗ T02 ) = 1 ⊗ T02 . This follows from the identities 1 1 (R ⊗ ι)(U ) = 1 ⊗ πU ρ − 2 (S ⊗ ι) (U ) 1 ⊗ πU ρ 2 1 1 = 1 ⊗ πU ρ − 2 U ∗ 1 ⊗ πU ρ 2 and
1 1 1 1 −2 2 2 1 ⊗ T0 U = 1 ⊗ πU ρ U 1 ⊗ πU ρ 2 1 ⊗ T0 ,
ˆ . Thus V = U × U . where we have used that πU (ρ)T0 ∈ πU (l ∞ (G)) ∞ ˆ By virtue of (1.1), for any y, z ∈ l (G) we then have ˆ v ∗ θ0 ((ι ⊗ πU )((y)(1 ⊗ z)))v = v ∗ (y ⊗ 1)απU (z)v = πV (y)(1 ⊗ πU (z)) ˆ = (πU¯ ⊗ πU )(y)(1 ⊗ πU (z)) ˆ = (πU¯ ⊗ πU )((y)(1 ⊗ z)). ˆ ˆ is dense in l ∞ (G) ˆ ⊗ l ∞ (G), ˆ for any x ∈ l ∞ (G) ˆ we thus ˆ ∞ (G))(1 Since (l ⊗ l ∞ (G)) get v ∗ θ0 (x ⊗ 1)v = (πU¯ ⊗ πU )(x ⊗ 1) = πU¯ (x) ⊗ 1, whence θ θ0 (x ⊗ 1) = Jν v ∗ θ0 (x ∗ ⊗ 1)vJν = Jν (πU¯ (x ∗ ) ⊗ 1)Jν = 1 ⊗ πU¯ (x ∗ ), where πU¯ (x ∗ )ξ = πU¯ (x ∗ )ξ¯ . By definition of the conjugate representation we have ˆ πU¯ (x ∗ )ξ¯ = πU R(x)ξ . ˆ It follows that θ θ0 (x ⊗ 1) = 1 ⊗ πU R(x).
N
∩ (G N ) for a product-type action defined by a Return now to the study of representation U of G on H and a G-invariant state φ˜ ∈ B(H )∗ . The relative commutant can be described in a way similar to that for (N α ) ∩ N , see [V]. The conditional expectation En : N → Nn extends to the conditional expectation ι ⊗ En : G N → G Nn . Let y ∈ N ∩ (G N ). Then (ι ⊗ En )(y) ∈ Nn ∩ (G Nn ), so by Lemma 1.2, (ι ⊗ En )(y) = (U ×n )∗ (yn ⊗ 1)U ×n ˆ As En En+1 = En , we have for a unique element yn ∈ l ∞ (G). ∗ (U ×n )∗ (yn ⊗ 1)U ×n = (ι ⊗ En ) U ×(n+1) (yn+1 ⊗ 1)U ×(n+1) ∗ ˜ = (U ×n )∗ ((ι ⊗ φ)(U (yn+1 ⊗ 1)U ) ⊗ 1)U ×n ,
whence by Lemma 1.2(ii) ∗ ˆ φ R(y ˆ n+1 ), ˜ yn = (ι ⊗ φ)(U (yn+1 ⊗ 1)U ) = RP
˜ U . Taking into account parts (i) and (iii) of Lemma 1.2 as well as the where φ = φπ description of (N α ) ∩ N in terms of harmonic sequences, we get the following result.
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Proposition 1.3. For the product-type action α of a compact quantum group G defined by a representation U of G on H and a G-invariant normal faithful state φ˜ on B(H ), ˆ there exists a G-equivariant complete order isometry between N ∩ (G N ) and the ∞ ˆ ˆ ˆ space of bounded sequences {yn }∞ n=1 ⊂ l (G) such that R(yn ) = Pφ R(yn+1 ), where ˜ φ = φπU . The surjective anti-homomorphism θ : N ∩ (G N ) → (N α ) ∩ N maps the element defined by a sequence {yn }∞ n=1 to the element defined by the Pφ -harmonic sequence ˆ n )}∞ . {Fn R(y n=1 Note that even if φ is generating, the homomorphism (N ∩(GN ))op → (N α ) ∩N ˆ needs not be injective nor G-equivariant (where we consider (N ∩ (G N ))op with the op α ˆ ˆ φ) to get a right action of G ˆ right action αˆ of G, and identify (N ) ∩ N with H ∞ (G, α on (N ) ∩ N ). Example 1.4. Let G = SU (2) and U be the fundamental representation of G. Identify Irr(G) with 21 Z+ as usual. Set p = ∞ n=1 In− 1 . Then 2
ˆ (p) = p ⊗ (1 − p) + (1 − p) ⊗ p. In particular, Pφ (p) = 1 − p. Consider the sequence {yn }∞ n=1 defined by y2k+1 = 1 − p, y2k = p, and let y be the corresponding element of N ∩ (SU (2) N ). Then y is in the kernel of θ . Moreover, as α(y) ˆ = p ⊗ (1 − y) + (1 − p) ⊗ y, ˆ the G-equivariance also does not hold. It is clear, however, what goes wrong in the previous example. Though the representation U was faithful, U ×U was not, so we had in fact an action of SO(3). More generally, assume G is the q-deformation of a compact connected semisimple Lie group. Let H be the quotient of G defined by U × U , that is, C(H ) is the C∗ -subalgebra of ˇ n. C(G) generated by the matrix coefficients of U × U . Then Irr(H ) = ∪n∈N supp (φφ) k ×k Choose also k ∈ N such that 0 ∈ supp φ , that is, U contains the trivial represenˇ k ⊂ Irr(H ). tation. Since 0 ∈ supp φ k , we have supp φ k ⊂ supp (φˇ k φ k ) = supp (φφ) We have thus shown that by replacing G by its quotient and U by some power U ×k , in the study of product-type actions we can always assume that U × U is faithful, that is, C(G) is generated by the matrix coefficients of U × U . Assume now that U × U is faithful. Then Irr(G) = ∪n∈N supp φ kn for any k ∈ N. Indeed, as we discussed earlier, the set ∪n∈N supp φ kn is always symmetric. Hence it ˇ kn for any n ∈ N. Since 0 ∈ supp (φφ), ˇ ˇ m ⊂ contains supp (φφ) we have supp (φφ) m+1 ˇ supp (φφ) for any m. Thus ˇ n = ∪n∈N supp (φφ) ˇ kn ⊂ ∪n∈N supp φ kn . Irr(G) = ∪n∈N supp (φφ) In particular, we can find k and n such that 0 ∈ supp φ k and supp φ ∩ supp φ kn = ∅. As 0 ∈ supp φ kn , we have supp φ ∩ supp φ kn ⊂ supp φ kn ∩ supp φ kn+1 . By the 0-2 law (see e.g [NT1, Prop. 2.12]) we can conclude that Pφm − Pφm+1 → 0 ˆ such that xn = as m → ∞. It follows that if {xn }∞ is a bounded sequence in l ∞ (G) n=1
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Pφ (xn+1 ), then the sequence is constant, xn = xn+1 . Note also that if y ∈ N ∩ (G N ) ˆ n )}∞ , then by Lemma 1.2(ii), for the conis the element defined by the sequence {R(x n=1 ˆ we have E0 (y) = RP ˆ φ (x1 ). We ditional expectation E0 = ι ⊗ ν : G N → l ∞ (G) have thus proved the following result. Corollary 1.5. Let G be the q-deformation of a compact connected semisimple Lie group, α the product-type action of G on (N, ν) defined by a representation U of G on H and a G-invariant normal faithful state φ˜ on B(H ). Assume that the representation U × U is faithful. Then we have isomorphisms ∼ (N α ) ∩ N ← ∼ H ∞ (G, ˆ φ), (N ∩ (G N ))op → θ
j∞
−1 θ is G-equivariant ˆ ˜ U . Moreover, the homomorphism j∞ where φ = φπ and coincides with the map Rˆ ⊗ ν.
ˆ the homomorphism For a more general group G and a generating state φ ∈ Cl (G), −1 θ is a G-equivariant ∞ ˆ ˆ j∞ isomorphism if any bounded sequence {xn }∞ n=1 ⊂ l (G) such that Pφ (xn+1 ) = xn is constant, that is, xn+1 = xn . For this it is enough to require that supp φ m ∩ supp φ m+1 = ∅ for some m. 2. Poisson Integral and Berezin Transform ˆ be a normal left G-invariant state on Let G be a compact quantum group and φ ∈ Cl (G) ˆ Consider the right action ˆ on L∞ (G), ˆ of G l ∞ (G). ˆ ˆ : L∞ (G) → L∞ (G) ⊗ l ∞ (G), ˆ (a) = W (a ⊗ 1)W ∗ . It was proved in [I1] that ˆ = (ϕ ⊗ ι) ˆ φ). In fact, this is the only normal unital G- and G-equivariant ˆ maps L∞ (G) into H ∞ (G, ∞ ∞ ∞ ∞ ˆ (we consider L (G) and l (G) ˆ with the left actions of map from L (G) into l (G) ˆ given by ˆ and , ˆ G given by and l , respectively, and with the right actions of G ∞ ˆ respectively). Indeed, assume T is such a map. Since the counit εˆ on l (G) is G-invariant, εˆ T is a G-invariant normal linear functional on L∞ (G) such that εˆ T (1) = 1. Hence ˆ εˆ T = ϕ. Then by G-equivariance of T , we get ˆ = (ˆε ⊗ ι)(T ⊗ ι) ˆ = (ϕ ⊗ ι) ˆ = . T = (ˆε ⊗ ι)T ˆ Thus if the Poisson boundary is G- and G-equivariantly isomorphic to a homogeneous ˆ φ). space G/H of G, the map must be an isomorphism of L∞ (G/H ) onto H ∞ (G, We call the map the Poisson integral. To show multiplicativity of we will use the following criterion. Lemma 2.1. Let Ni be a von Neumann algebra, νi a normal faithful state on Ni , i = 1, 2, θ : N1 → N2 a normal ucp map such that ν2 θ = ν1 and σtν2 θ = θσtν1 . Then there exists a normal ucp map θ ∗ : N2 → N1 such that ν2 (θ (x1 )x2 ) = ν1 (x1 θ ∗ (x2 )) for x1 ∈ N1 , x2 ∈ N2 . For any x ∈ N1 , the following conditions are then equivalent:
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(i) x is in the multiplicative domain of θ ; (ii) ||θ (x)||2 = ||x||2 ; (iii) ||θ (x ∗ )||2 = ||x ∗ ||2 ; (iv) θ ∗ θ (x) = x. Proof. Since σtν2 θ = θ σtν1 , the map θ ∗ can equivalently be defined by the condition (θ (x1 )Jν2 x2∗ ξν2 , ξν2 ) = ν2 (θ (x1 )σ ν2i (x2 )) = ν1 (x1 σ ν1i θ ∗ (x2 )) = (x1 Jν1 θ
∗
−2 (x2∗ )ξν1 , ξν1 ).
−2
Hence θ ∗ exists [AC]. A more general result than the equivalence of the conditions (i)-(iv) can be found in [P]. We will give a proof of our particular case for the reader’s convenience. Note that for a contraction T on a Hilbert space and a vector ξ , the equality T ξ = ξ holds if and only if T ∗ T ξ = ξ . This shows that (ii) is equivalent to (iv). Clearly, then (iii) also is equivalent to (iv). For any x ∈ N1 , we have θ (x ∗ )θ (x) ≤ θ (x ∗ x) by Schwarz inequality. Since ν2 is faithful, it follows that the equality ||θ (x)||2 = ||x||2 holds if and only if θ(x ∗ )θ (x) = θ(x ∗ x). It is well-known that (i) is equivalent to the conditions θ(x ∗ )θ (x) = θ(x ∗ x) and θ (x)θ (x ∗ ) = θ (xx ∗ ). Thus (i) implies (ii) and (iii), and (ii) and (iii) together imply (i). Since we already know that (ii) and (iii) are equivalent, we conclude that all four conditions are equivalent. For s ∈ I = Irr(G), consider the map s : L∞ (G) → B(Hs ), s (a) = (ϕ ⊗ ι)(U s (a ⊗ 1)U s ∗ ) = (a)Is . The following lemma shows that there exists a map ∗s : B(Hs ) → L∞ (G) such that φs (s (a)x) = ϕ(a∗s (x)), a ∈ L∞ (G), x ∈ B(Hs ). Lemma 2.2. We have ∗s (x) = (ι ⊗ ωs )l (x) = (ι ⊗ ωs )(U s ∗ (1 ⊗ x)U s ). ϕ
Proof. Recall that for the modular group σt we have ϕ
(σt ⊗ ι)(W ) = (1 ⊗ ρ it )W (1 ⊗ ρ it ).
(2.1)
On the other hand, σt s (x) = ρ −it xρ it for x ∈ B(Hs ). Thus (σt ⊗ σt s )(U s ) = U s (1 ⊗ ρ 2it ). Hence φ
φ
ϕ
φs (s (a)x) = (ϕ ⊗ φs )(U s (a ⊗ 1)U s ∗ (1 ⊗ x)) = (ϕ ⊗ φs )((a ⊗ 1)U s ∗ (1 ⊗ x)(σ−i ⊗ σ−is )(U s )) ϕ
φ
= (ϕ ⊗ φs (·ρ 2 ))((a ⊗ 1)U s ∗ (1 ⊗ x)U s ),
whence ∗s (x) = (ι ⊗ ωs )(U s ∗ (1 ⊗ x)U s ), because ωs = φs (·ρ 2 ).
As a byproduct we get φs s = ϕ. Note also that both maps s and ∗s are G-equivariant. Now we can compute ∗ .
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ˆ be a generating state. Set ν0 = εˆ |H ∞ (G,φ) . Then for the Lemma 2.3. Let φ ∈ Cl (G) ∞ ˆ φ), ν0 ), we have Poisson integral : (L (G), ϕ) → (H ∞ (G, ∗ (x) = s ∗ − lim φ n (Is )∗s (x). n→∞
s∈I
ˆ φ) with the relative commutant Proof. First note that under the identification of H ∞ (G, for a product-type action, the state ν0 coincides with the product-state. It follows that ν0 is faithful, and its modular group is given by the restriction of the modular group ψˆ ˆ to H ∞ (G, ˆ φ). It is then σt = Ad ρ −it of the right-invariant Haar weight ψˆ on l ∞ (G) ϕ easy to see that ν0 = ϕ and using (2.1) that σtν0 = σt . Hence ∗ indeed exists. ˆ φ) is given Recall [I1, Theorem 3.6(2)] that as φ is generating, the product on H ∞ (G, by x · y = s ∗ − lim Pφn (xy). n→∞
ˆ φ) we have Thus for any a ∈ L∞ (G) and x ∈ H ∞ (G, ν0 ((a) · x) = lim εˆ Pφn ((a)x) = lim φ n ((a)x) n→∞ n→∞ = lim φ n (Is )φs (s (a)x) n→∞
= lim
n→∞
s∈I
φ n (Is )ϕ(a∗s (x)),
s∈I
∗ ∗ so s∈I s )s (x) → (x) in weak operator topology. Note, however, that by ∗ ˆ φ) corresponding to an G-equivariance of s if x is in the spectral subspace of H ∞ (G, ∗ irreducible representation of G, then s (x) is in the spectral subspace of L∞ (G), which
φ n (I
is finite dimensional. It follows that the convergence is in norm on a dense ∗-subalgebra ˆ φ). Since of H ∞ (G,
n ∗ φ (Is )s (x) = φ n (Is )φs (x) = φ n (x) = ν0 (x) ϕ s
s
ˆ φ), the operators s φ n (Is )∗s are contractions with respect to the for x ∈ H ∞ (G, 2 ˆ φ). L -norms. Hence we have s ∗ -convergence on the whole space H ∞ (G, From now onwards we assume that the counit ε is bounded on C(G). This is the case for q-deformations of compact connected semisimple Lie groups. Since the map ∗ is G-equivariant, it maps the spectral subspaces of L∞ (G) into The same is true for ∗s s . It follows that for any a ∈ C(G) the sequence themselves. n ∗ ∗ s φ (Is )s s (a) n is in C(G), and it converges in norm to (a). Note now that ∗ G-equivariance of implies that ∗ (a) = (ι ⊗ ε)∗ (a) = (ι ⊗ ε∗ )(a). So to prove that ∗ (a) = a for an element a ∈ C(G), it is enough to show that ε∗ (b) = ε(b) for any element b of the form (ω ⊗ ι)(a), ω ∈ C(G)∗ . As ε∗s = ωs and φ n (Is )ωs = φ n (Is )φs (·ρ 2 ) = φ n (·ρ 2 ) = ωn , s
s
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ˆ we have where ω = φ(·ρ 2 ) ∈ Cr (G), ε∗ (b) = lim ωn (b). n→∞
Thus using equivalence of (i) and (iv) in Lemma 2.1 we get the following criterion for multiplicativity of . ˆ be a generating state. Set ω = φ(·ρ 2 ). Then the Proposition 2.4. Let φ ∈ Cl (G) ∞ n sequence {ω }n=1 of states on C(G) is w ∗ -convergent. For a G-invariant subspace X of C(G) (that is, (X) ⊂ C(G) ⊗ X), the limit state coincides with the counit ε on X if and only if X is in the multiplicative domain of the Poisson integral : L∞ (G) → ˆ φ). H ∞ (G, The operators s and ∗s are analogues of well-known classical constructions [Be, Per]. Let for the moment G be a compact Lie group, U : G → B(H ) a finite dimensional unitary representation. Fix a vector ξ ∈ H , ξ = 1. Let T ⊂ G be the stabilizer of the line Cξ . For an operator S ∈ B(H ), its covariant Berezin symbol σ (S) is defined by σ (S)(g) = (SUg ξ, Ug ξ ). The covariant symbol σ is a G-equivariant map from B(H ) into C(G/T ). Consider the inner products on C(G/T ) and B(H ) given by the G-invariant probability measure and the normalized trace, respectively. Then there exists an adjoint σ˘ : C(G/T ) → B(H ) of σ . Explicitly,
σ˘ (f ) = d f (g)Ug Pξ Ug∗ dg, where d = dim H and Pξ is the projection onto Cξ . A function f is called a contravariant Berezin symbol of σ˘ (f ). The map B = σ σ˘ is called the Berezin transform. If we consider U as a corepresentation of C(G), then the definition of σ can be written as σ (S) = (ι ⊗ ωξ )(U ∗ (1 ⊗ S)U ), where ωξ = (·ξ, ξ ) is the vector-state defined by ξ . Thus we see that our operator ∗s is just σ with ωξ replaced by ωs . Then s = (∗s )∗ is an analogue of σ˘ . Suppose now that G is a semisimple Lie group, U = U λ : G → B(Hλ ) an irreducible representation with highest weight λ, ξ = ξλ a highest weight vector. Let Bλ be the corresponding Berezin transform. Note that ωξnλ = ωξnλ . It is proved in [D] that the sequence {Bnλ }∞ n=1 converges to the identity on C(G/T ) as n → ∞. This is a key step to show that the full matrix algebras B(Hnλ ), n ∈ N, provide a quantization of C(G/T ), see [L, R]. In view of the G-equivariance of the Berezin transform, it is enough to prove the convergence at the unit of G. The proof is based on the observation that the states εBnλ are given by measures which are absolutely continuous with respect to the Haar measure and such that the Radon-Nikodym derivatives, up to normalization, are powers of a single function h such that h(g) = 1 for g ∈ T and h(g) < 1 for g ∈ / T . The proof of our q-analogue will be based on the study of ergodic properties of an auxiliary operator. ˆ define a linear operator Aω : C(G) → For a normal linear functional ω on l ∞ (G) C(G) by ˆ = (ι ⊗ ω)(W (a ⊗ 1)W ∗ ). Aω (a) = (ι ⊗ ω)(a) ˆ so that (ι ⊗ ) ˆ is a right action of G, ˆ ˆ = ( ˆ ⊗ ι), ˆ we have Then ω = ϕAω . Since ˆ ∗ . Thus ωn = ϕAnω , and by Proposition 2.4 Aω1 ω2 = Aω1 Aω2 for any ω1 , ω2 ∈ l ∞ (G)
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to show multiplicativity of on the quantum flag manifold, it is enough to prove the following result. Theorem 2.5. Let G = SUq (n) (0 < q < 1), T ⊂ SUq (n) the maximal torus, and ˆ a normal right G-invariant state, ω = εˆ . Then the counit ε is the only ω ∈ Cr (G) Aω -invariant state on C(G/T ). We will prove the result by induction. For this we first have to establish functorial properties of the operators Aω . Let H be a closed subgroup of G. By this we mean that H is a compact quantum group and that we are given a surjective unital ∗-homomorphism π : C(G) → C(H ) which respects comultiplication. We can define a left and a right action of H on C(G) by the homomorphisms C(G) → C(H ) ⊗ C(G), a → (π ⊗ ι)(a), and C(G) → C(G) ⊗ C(H ), a → (π ⊗ ι)(a), respectively. The corresponding fixed point algebras are denoted by C(H \G) and C(G/H). ˆ as linear functionals on A(G) we can define a By considering the elements of l ∞ (G) ˆ Equivalently, one can consider the unitary dual homomorphism πˆ : l ∞ (Hˆ ) → l ∞ (G). corepresentation U = (π ⊗ ι)(W ) of C(H ), where W is the multiplicative unitary for G, and set πˆ = πU . Lemma 2.6. Let H be a closed subgroup of G defined by π : C(G) → C(H ). Then ˆ ∗; (i) π Aω = Aωπˆ π for any ω ∈ l ∞ (G) ˆ ∗; (ii) Aω (C(G/H )) ⊂ C(G/H ) for any ω ∈ l ∞ (G) ˆ (iii) Aω (C(H \G)) ⊂ C(H \G) for any ω ∈ Cr (G). Proof. Let W and W0 be the multiplicative unitaries for G and H , respectively. Set U = (π ⊗ ι)(W ) as above, so that πˆ = πU . As U = (ι ⊗ πU )(W0 ) = (ι ⊗ πˆ )(W0 ), we get π Aω (a) = (ι ⊗ ω)(U (π(a) ⊗ 1)U ∗ ) = (ι ⊗ ωπˆ )(W0 (π(a) ⊗ 1)W0∗ ) = Aωπˆ π(a), which proves (i). Let a ∈ C(G/H ), so that (ι ⊗ π )(a) = a ⊗ 1. Then using the pentagon equation W12 W13 W23 = W23 W12 we get ∗ ∗ (ι ⊗ π )Aω (a) = (ι ⊗ π ⊗ ω)(W12 W23 (1 ⊗ a ⊗ 1)W23 W12 ) ∗ ∗ ∗ = (ι ⊗ π ⊗ ω)(W13 W23 W12 (1 ⊗ a ⊗ 1)W12 W23 W13 ) ∗ ∗ = (ι ⊗ π ⊗ ω)(W13 W23 ((a) ⊗ 1)W23 W13 ) ∗ ∗ = (ι ⊗ π ⊗ ω)(W13 W23 (a ⊗ 1 ⊗ 1)W23 W13 ) = Aω (a) ⊗ 1,
which shows (ii).
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Suppose now that a ∈ C(H \G). Then ∗ ∗ W23 (1 ⊗ a ⊗ 1)W23 W12 ) (π ⊗ ι)Aω (a) = (π ⊗ ι ⊗ ω)(W12 ∗ ∗ ∗ = (π ⊗ ι ⊗ ω)(W13 W23 W12 (1 ⊗ a ⊗ 1)W12 W23 W13 ) ∗ ∗ = (π ⊗ ι ⊗ ω)(W13 W23 ((a) ⊗ 1)W23 W13 ) ∗ ∗ = (π ⊗ ι ⊗ ω)(W13 W23 (1 ⊗ a ⊗ 1)W23 W13 ) ∗ = (π ⊗ ι ⊗ ω)(W23 (1 ⊗ a ⊗ 1)W23 ) = 1 ⊗ Aω (a),
where in the next to last equality we used right G-invariance of ω. This proves (iii).
ˆ we have ωπˆ ∈ Cr (Hˆ ). Lemma 2.7. For any ω ∈ Cr (G), An equivalent way of saying this is that if ρ0 = f1 is the Woronowicz character for H , then πˆ (ρ0 )ρ −1 commutes with π(l ˆ ∞ (Hˆ )). Yet another equivalent statement is that the homomorphism π intertwines the scaling groups of C(G) and C(H ). Proof of Lemma 2.7. Keeping the notation of the proof of the previous lemma, consider the right actions α = αW,r and α0 = αU,r of G and H , respectively, on B(L2 (G)). Extend ω to a normal G-invariant functional ω˜ on B(L2 (G)). Since (π ⊗ ι)(W ) = U , we have (ι⊗π )α(x) = α0 (x) for any x ∈ B(L2 (G)) (note that the expression (ι⊗π )α(x) makes sense as we have a well-defined homomorphism ι ⊗ π : M(B0 (L2 (G)) ⊗ C(G)) → M(B0 (L2 (G)) ⊗ C(H ))). Hence ω˜ is H -invariant, so ωπˆ = ωπ ˜ U ∈ Cr (Hˆ ). We can now lay the foundation for our induction argument. Lemma 2.8. Suppose η is a state on C(G) such that η = ε on C(H \G/H ) = C(H \G)∩ C(G/H ). Then there exists a state η0 on C(H ) such that η0 π = η. Proof. Suppose a ≥ 0, π(a) = 0. We have to prove that η(a) = 0. Let ϕ0 be the Haar state on C(H ). We have (ϕ0 π ⊗ ι ⊗ ϕ0 π)2 (a) ∈ C(H \G/H ), where 2 = ( ⊗ ι) = (ι ⊗ ). Hence (ϕ0 π ⊗ η ⊗ ϕ0 π)2 (a) = (ϕ0 π ⊗ ε ⊗ ϕ0 π)2 (a) = (ϕ0 π ⊗ ϕ0 π )(a) = (ϕ0 ⊗ ϕ0 )0 π(a) = 0, where 0 is the comultiplication on C(H ). Since the state ϕ0 ⊗ ϕ0 is faithful by our assumptions on quantum groups, we conclude that (π ⊗ η ⊗ π)2 (a) = 0. Applying ε0 ⊗ ι ⊗ ε0 , where ε0 is the counit on C(H ), and using ε0 π = ε we get η(a) = 0. Corollary 2.9. Let T ⊂ H ⊂ G be compact quantum groups, and π : C(G) → C(H ) ˆ Assume the homomorphism defining the inclusion H → G. Let ω be a state in Cr (G). that (i) the counit ε on C(G) is the only Aω -invariant state on C(H \G/H );
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(ii) the counit ε0 on C(H ) is the only Aωπˆ -invariant state on C(H /T ). Then ε is the only Aω -invariant state on C(G/T ). Proof. If η is Aω -invariant, then η = ε on C(H \G/H ), so by the previous lemma η = η0 π for some state η0 on C(H ). By Lemma 2.6(i) and surjectivity of π , the state η0 is Aωπˆ -invariant. Hence η0 = ε0 on C(H /T ). As π(C(G/T )) ⊂ C(H /T ), we have η = η0 π = ε0 π = ε on C(G/T ). Let us now recall the definition of SUq (n), see e.g. [KS]. The algebra C(Uq (n)) of continuous functions on the compact quantum group Uq (n) is generated by n2 + 1 elements uij , 1 ≤ i, j ≤ n, t satisfying the relations uik uj k = quj k uik , uki ukj = qukj uki for i < j, uil uj k = uj k uil for i < j, k < l, uik uj l − uj l uik = (q − q −1 )uj k uil for i < j, k < l, detq (U )t = tdet q (U ) = 1, uij t = tuij for any i, j, where U = (uij )i,j and det q (U ) = w∈Sn (−q)(w) uw(1)1 · · · uw(n)n , with (w) being the number of inversions in w ∈ Sn . The involution on C(Uq (n)) is given by ˆ j
t ∗ = detq (U ), u∗ij = (−q)j −i det q (U iˆ )t, ˆ
where U iˆ is the matrix obtained from U by removing the i th row and the j th column. j Taking the quotient of C(Uq (n)) by the ideal generated by det q (U ) − 1, we obtain the algebra C(SUq (n)). If m < n, then Uq (m) × Tn−m is a subgroup of Uq (n). Namely, if uij , 1 ≤ i, j ≤ m, and t are the generators of C(Uq (m)), and z1 , . . . , zn−m are the canonical generators of C(Tn−m ), then the homomorphism C(Uq (n)) → C(Uq (m) × Tn−m ) is given by if 1 ≤ i, j ≤ m, uij uij → zi−m if i = j > m, 0 otherwise, −1 and thus t → t z1−1 . . . zn−m . Taking the intersection with SUq (n), in other words, tak−1 ing the quotient of C(Uq (m) × Tn−m ) by the ideal generated by 1 − t z1−1 . . . zn−m , n−m n−m−1 ∼ we get a subgroup S(Uq (m) × T ) = Uq (m) × T of SUq (n). The subgroup T = S(Tn ) ∼ = Tn−1 we call the maximal torus of SUq (n). Consider now the filtration T = G1 ⊂ G2 ⊂ · · · ⊂ Gn = SUq (n) of SUq (n), where Gm = S(Uq (m) × Tn−m ), and let πm : C(SUq (n)) → C(Gm ) be the corresponding homomorphisms. We will prove by induction that the counit on C(Gm ) is the only Aωπˆ m invariant state on C(Gm /T ). For m = 1 there is nothing to prove as G1 = T . Thus by Corollary 2.9 we just have to show that the counit is the only Aωπˆ m+1 -invariant state on C(Gm \Gm+1 /Gm ). Let SUq (2) → SUq (n) be the embedding corresponding to the right lower corner of SUq (m + 1). In other words, if uij , 1 ≤ i, j ≤ 2, are the generators of C(SUq (2)), we define a homomorphism θm : C(SUq (n)) → C(SUq (2)) by ui−m+1, j −m+1 if m ≤ i, j ≤ m + 1, uij → otherwise. δij
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Note that θm factorizes through πm+1 : C(SUq (n)) → C(Gm+1 ), so θm = θm πm+1 for some homomorphism θm : C(Gm+1 ) → C(SUq (2)). Lemma 2.10. For each m, 1 ≤ m ≤ n − 1, the homomorphism θm maps C(Gm \Gm+1 / Gm ) isomorphically onto C(T\SUq (2)/T). ∼ (Uq (m) × T)\Uq (m + 1)/(Uq (m) × T). Then the Proof. Note that Gm \Gm+1 /Gm = result can be deduced from Theorem 4.7 in [NYM], which implies that C((Uq (m) × T)\Uq (m + 1)/(Uq (m) × T)) is generated by um+1,m+1 u∗m+1,m+1 . Another possibility is to use the classification of irreducible representations of algebras of functions on homogeneous spaces, see e.g. [PV, DS]. Namely, let either G = Uq (n), H = Uq (n − 1) × T and T = Tn , or G = SUq (n), H = S(Uq (n − 1) × T) and T = S(Tn ) ∼ = Tn−1 . Then the irreducible representations of C(G) can be described as follows [KoS]. Consider the homomorphisms θk : C(G) → C(SUq (2)), 1 ≤ k ≤ n − 1, and π1 : C(G) → C(T ) as above. There exists a canonical irreducible infinite dimensional representation π of C(SUq (2)). Then for a character χ of T and an element w ∈ Sn with a reduced decomposition w = τi1 · · · τik , where τi = (i, i + 1), 1 ≤ i ≤ n − 1, we set πw,χ = (π θi1 ) × · · · × (π θik ) × (χ π1 ) = ((π θi1 ) ⊗ · · · ⊗ (π θik ) ⊗ (χ π1 ))k . Up to equivalence the representation πw,χ is independent of the reduced decomposition of w, and {πw,χ }w∈Sn ,χ∈Tˆ is a complete set of irreducible representations of C(G). If w ∈ Sn−1 , then the representation πw,χ factorizes through C(H ), so its restrictions to C(G/H ) and C(H \G) are given by the counit. If w ∈ Sn \Sn−1 , then w can be written as w1 τn−1 w2 with (w) = (w1 )+(w2 )+1 and w1 , w2 ∈ Sn−1 . Since the representations πw1 ,0 and πw2 ,χ factorize through C(H ), and 2 (C(H \G/H )) ⊂ C(H \G) ⊗ C(G) ⊗ C(G/H ), we see that πw,χ (a) = 1 ⊗ (π θn−1 )(a) ⊗ 1 for any a ∈ C(H \G/H ). Thus the restriction of any irreducible representation of C(G) to C(H \G/H ) factorizes through θn−1 : C(G) → C(SUq (2)). Hence θn−1 : C(H \G/H ) → C(SUq (2)) is injective. As T ⊂ H , the image is obviously contained in C(T\SUq (2)/T). In fact, it coincides with C(T\SUq (2)/T), since e.g. it is easy to check that θn−1 (unn u∗nn ) generates C(T\SUq (2)/T). Now to prove Theorem 2.5 we just have to show that the counit on C(SUq (2)) is the only Aωθˆm -invariant state on C(T\SUq (2)/T) for 1 ≤ m ≤ n − 1. Note that as SU (n) is a simple Lie group, the restriction of a non-trivial representation of SU (n) to a non discrete subgroup is non-trivial. It follows that if ω is non-trivial on l ∞ (SU q (n)), that is, ∞ ω = εˆ , then ωθˆm is non-trivial on l (SU q (2)). Thus we can reduce the proof of Theorem 2.5 to the case of SUq (2), moreover, in this case it suffices to prove that the counit is the only Aω -invariant state on C(T\SUq (2)/T). For this we could use the results of [I1, NT1] saying that the Poisson integral is a homomorphism on C(SUq (2)/T). We will instead give a self-contained probabilistic proof. Let {uij }1≤i,j ≤2 be the generators of C(SUq (2)). Set α = u11 and γ = u21 . Then u11 u12 α −qγ ∗ U= = u21 u22 γ α∗,
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and the relations can be written as α ∗ α + γ ∗ γ = 1, αα ∗ + q 2 γ ∗ γ = 1, γ ∗ γ = γ γ ∗ , αγ = qγ α, αγ ∗ = qγ ∗ α. The comultiplication is determined by the formulas (α) = u11 ⊗ u11 + u12 ⊗ u21 = α ⊗ α − qγ ∗ ⊗ γ , (γ ) = γ ⊗ α + α ∗ ⊗ γ . The homomorphism C(SUq (2)) → C(T) is given by α → z, γ → 0. The monomials α k (γ ∗ )l γ m and (α ∗ )k (γ ∗ )l γ m , k, l, m ≥ 0, span a dense ∗-subalgebra of C(SUq (2)). It is then easy to see that C(T\SUq (2)/T) is generated by γ ∗ γ . The spectrum of γ ∗ γ is the set Iq 2 = {0} ∪ {q 2n }∞ n=0 . Thus we can identify C(T\SUq (2)/T) with the algebra C(Iq 2 ) of continuous functions on Iq 2 . Under this identification the counit is given by the evaluation at 0 ∈ Iq 2 . The Markov operator Aω defines a random walk on Iq 2 \{0}. If this random walk is transient, then νAnω → ε as n → ∞ for any state ν on C(T\SUq (2)/T). In particular, the counit ε is the only Aω -invariant state. As was remarked in [NT1], transience of a random walk on a non-Kac discrete quantum group follows easily from the fact that the Markov operator has a positive eigenvector with eigenvalue strictly smaller than 1. It is natural to expect that the operator Aω acting on the dual side has the same property. Proposition 2.11. Consider the function f on Iq 2 defined by f (0) = 0, f (q 2k ) = ak , where {ak }∞ k=0 is the sequence defined by the recurrence relation 2(1 − q
2k+1
a0 = 1, )ak = q −1 (1 − q 2k+2 )ak+1 + q(1 − q 2k )ak−1 .
Then (i) f ∈ C(Iq 2 ) and f (q 2k ) > 0 for any k ≥ 0;
(ii) for any normal right SUq (2)-invariant state ω = λ ωs ∈ Cr (SU q (2)), the s s 2s+1 element f is an eigenvector for Aω with eigenvalue s λs [2s+1]q . As usual, we identify the set Irr(SUq (2)) with 21 Z+ . Then [2s+1]q = is the quantum dimension of the representation with spin s ∈ 21 Z+ .
q 2s+1 − q −2s−1 q − q −1
Proof of Proposition 2.11. The proof of (i) is analogous to that of [I1, Lemma 5.4]. To see that ak > 0, rewrite the recurrence relation as q −1 (1 − q 2k+2 )(ak+1 − qak ) = (1 − q 2k )(ak − qak−1 ) + q 2k (1 − q)2 ak . It follows by induction that ak+1 − qak ≥ 0, so ak ≥ q k . It remains to show that ak → 0. It is clear that the sequence {ak }∞ k=0 cannot grow faster than a geometric progression. Hence the generating function g(z) =
∞
ak z k
k=0
is analytic in a neighborhood of zero. The recurrence relation can then be written as 2(g(z) − qg(q 2 z)) = q −1 z−1 (g(z) − g(q 2 z)) + qz(g(z) − q 2 g(q 2 z)),
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that is, g(z) =
1 − q 2z 1 − qz
2 g(q 2 z).
We see that g extendsto a meromorphic function with poles at z = q −2k−1 , k ≥ 0. In particular, the series k ak zk converges for |z| < q −1 , whence ak → 0. In fact, since lim (1 − qz) g(z) = 2
z→q −1
2 ∞ 1 − q 2k+1 k=0
1 − q 2k+2
=
(q; q 2 )2∞ , (q 2 ; q 2 )2∞
(q;q 2 )2
we have ak ∼ kq k (q 2 ;q 2 )∞2 . ∞ To prove (ii), first consider the case ω = ω 1 , so that 2
−1 1 q 0 Tr · ω= 0 q [2]q 1
as U 2 = U . Then we have
∗ ∗ −1 1 x0 α γ α −qγ ∗ q 0 Aω (x) = Tr 0x γ α∗ −qγ α 0 q [2]q 1 = (q −1 (αxα ∗ + q 2 γ ∗ xγ ) + q(γ xγ ∗ + α ∗ xα)). [2]q
Identifying C(T\SUq (2)/T) = C ∗ (γ ∗ γ ) with C(Iq 2 ), and using the identities α ∗ (γ ∗ γ )k α = q −2k (γ ∗ γ )k (1 − γ ∗ γ ), α(γ ∗ γ )k α ∗ = q 2k (γ ∗ γ )k (1 − q 2 γ ∗ γ ), we see that the action of Aω on the functions on Iq 2 is given by (Aω h)(t) =
1 −1 q (1 − q 2 t)h(q 2 t) + q 2 th(t) + q th(t) + (1 − t)h(q −2 t . [2]q
Then the definition of {ak }k shows that Aω f = [2]2 q f . To prove that f is an eigenvalue for Aωs for any s, recall from [I1, Sect. 6] that the identity ωs ω 1 = 2
ds− 1 2
ds d 1
ωs− 1 + 2
2
ds+ 1 2
ds d 1
ωs+ 1 , 2
2
where ds = [2s + 1]q , implies that there exists a polynomial p2s of degree 2s such that p2s ω 1 = ωs . Then Aωs = p2s Aω 1 , so f is an eigenvector for Aωs with eigenvalue 2 2 2s+1 p2s [2]2 q . As [2s+1] = ωs (ρ −1 ), and ρ is group-like, we have q p2s
2 [2]q
2s + 1 = p2s ω 1 (ρ −1 ) = p2s ω 1 (ρ −1 ) = ωs (ρ −1 ) = , 2 2 [2s + 1]q
which finishes the proof of the proposition.
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It follows that the random walk defined by Aω on Iq 2 \{0} is transient for any normal right-invariant state ω = εˆ . More precisely, if Aω f = λf , then the probability of visiting a point t2 from a point t1 at the nth step is not larger than f (t1 )f (t2 )−1 λn . Hence νAnω → δ0 = ε as n → ∞ for any state ν on C(Iq 2 ). Note that to see that ε is the only Aω -invariant state is even easier. Indeed, if ν is Aω -invariant, we have ν(f ) = λν(f ), so ν(f ) = 0 and ν = δ0 . This completes the proof of Theorem 2.5. 3. Random Walk on the Center To prove surjectivity of the Poisson integral, we will obtain an estimate on the dimensions ˆ φ). By a result of Hayashi [H], if the fusion algebra of the spectral subspaces of H ∞ (G, of a group G is commutative, then any central harmonic element is a scalar. Equivalently, the action of G on the Poisson boundary is ergodic. This already implies that the ˆ φ) are finite dimensional, more precisely, the dimension spectral subspaces of H ∞ (G, of the spectral subspace corresponding to an irreducible representation U is not larger than the square of the quantum dimension of U [B, HLS]. This estimate is clearly not sufficient for our purposes. We will show that in our situation ergodicity of the action provides a better estimate. The result of Hayashi is in fact more general. It was obtained as a consequence of an analogue of double ergodicity of the Poisson boundary, see e.g. [K]. Since in our situation the proof can be made more concrete, we will present a detailed argument. For Markov operators P and Q on a von Neumann algebra, we say that an element x is (P , Q)-harmonic if P (x) = Q(x) = x. Let now φ and ω be generating normal states ˆ with the same support. Set on l ∞ (G) (N, ν) =
−1
ˆ φ) ⊗ (l ∞ (G),
−∞
+∞ ˆ ω) , ⊗ (l ∞ (G), 0
and let γ be the shift to the right on N . For any finite interval I = [n, m] ⊂ Z we have ˆ → N defined by ˆ (m−n) . Then the space of a normal homomorphism jI : l ∞ (G) ˆ and Qω = (ι ⊗ ω), ˆ can be (Pφ , Qω )-harmonic elements, where Pφ = (φ ⊗ ι) embedded in the space N γ of γ -invariant elements by the homomorphism jZ , jZ (x) = s ∗ − lim j[n,m] (x). n→−∞ m→+∞
Note that if φ = ω, then the automorphism γ is never ergodic [K, Lemma 4]. Nevertheless the following result holds. ˆ be a generating normal left Proposition 3.1. [H, Proposition 3.4]. Let φ ∈ Cl (G) ˆ ω = φ(·ρ 2 ) ∈ Cr (G) ˆ the corresponding right-invariant G-invariant state on l ∞ (G), state. Then the space of central (Pφ , Qω )-harmonic elements consists of the scalars. ˆ Consider the operator Proof. Note that by definition φ n = ωn on the center of l ∞ (G). ˆ Although in general Qφ = Qω , for any left-invariant state φ we have Qφ = (ι ⊗ φ). φ Qnφ = φ n Pφ = ωn Pφ = φ Qnω
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on the center. Similarly, for any right-invariant state ω we have ω Pωn = ω Pφn on the center. It follows that for any central elements z1 and z2 and any intervals I1 ⊂ I2 , I1 = [n, m], I2 = [n − k, m + l], we have ν(jI1 (z1 )jI2 (z2 )) = φ m−n+1 (z1 Qlω Pφk (z2 )). Indeed, e.g. in the case when m + l < 0 we get ν(jI1 (z1 )jI2 (z2 )) = φ m−n+1 (z1 Qlφ Pφk (z2 )) = φ m−n+1 (z1 Qlω Pφk (z2 )). Hence for any central z1 and z2 and any finite intervals I1 ⊂ I2 we have ||jI1 (z1 ) − jI2 (z2 )||2 = ||γjI1 (z1 ) − γjI2 (z2 )||2 . Thus if z is a central (Pφ , Qω )-harmonic element, then the distance ||jZ (z) − γ n jI (z)||2 is independent of n. Since on the one hand this distance goes to zero as I Z, and on the other hand γ n jI (z) converges in weak operator topology to a scalar as n → +∞, we conclude that jZ (z) is a scalar. Corollary 3.2. [H, Corollary 3.5]. Assume that the fusion algebra R(G) of the group ˆ the scalars are the only G is commutative. Then for any generating state φ ∈ Cl (G), central Pφ -harmonic elements. Proof. Commutativity of the fusion algebra means that Pφ = Qω on the center. Thus any central Pφ -harmonic element is (Pφ , Qω )-harmonic, and we can apply the previous proposition. Consider now the random walk on the center in more detail. Identify the center of ˆ with l ∞ (I ), where I = Irr(G). For a fixed generating state φ ∈ Cl (G), ˆ let l ∞ (G) {p(s, t)}s,t∈I be the transition probabilities defined by the restriction of Pφ to l ∞ (I ), so Pφ (It )Is = p(s, t)Is . Let (, P0 ) be the path space of the corresponding random walk, =
∞
I, P0 ({s | s1 = t1 , . . . , sn = tn }) = p(0, t1 )p(t1 , t2 ) · · · p(tn−1 , tn ).
n=1
Denote by πn the projection → I onto the nth factor. Similarly to Sect. 1, set −1
ˆ φ), (N, ν) = ⊗ (l ∞ (G), −∞
ˆ and j∞ (x) = s ∗ − limn jn (x) for x ∈ ˆ n−1 (x) for x ∈ l ∞ (G), jn (x) = · · · ⊗ 1 ⊗ ∞ ∞ ˆ φ). (In Sect. 1 we embedded Fn (l (G)) ˆ into B(H ) for some H and extended H (G, φ to a G-invariant normal faithful state φ˜ on B(H ), which we don’t do now as the relative commutant interpretation of the Poisson boundary will not be important.) As was remarked in [NT1], there is an embedding j ∞ : (L∞ (, P0 ), P0 ) → (N, ν) ˆ If f ∈ l ∞ (I ) is harmonic, then such that f πn → jn (f ) for any f ∈ l ∞ (I ) ⊂ l ∞ (G). the sequence {f πn }∞ is a martingale, so it converges almost everywhere (a.e.) to a n=1 function f∞ ∈ L∞ (, P0 ). Then j∞ (f ) = j ∞ (f∞ ).
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ˆ φ) of central harmonic elements. Denote by H ∞ (I, φ) the space l ∞ (I ) ∩ H ∞ (G, ˆ → l ∞ (I ) be the unique G-equivariant conditional expectation, Let E : l ∞ (G) E(x) = (ϕ ⊗ ι)l (x) = φs (x)Is . s∈I
ˆ φ) we get a conditional expectation H ∞ (G, ˆ φ) → H ∞ (I, φ). By restricting E to H ∞ (G, ˆ φ). Then the sequence {fn }∞ of functions on Proposition 3.3. Let x, y ∈ H ∞ (G, n=1 defined by fn (s) = φsn (xy) converges a.e. to E(x · y)∞ ∈ L∞ (, P0 ). Since fn = E(xy)πn , one can equivalently state that {E(xy)πn }n and {E(x · y)πn }n converge a.e. to the same limit. Proof of Proposition 3.3. Let α be the product-type action of G on N , and E˜ = (ϕ ⊗ ι)α : N → N α the ν-preserving conditional expectation. Since jn (x) → j∞ (x) and jn (y) → j∞ (y) in s ∗ -topology, we have jn (xy) = jn (x)jn (y) → j∞ (x)j∞ (y) = j∞ (x · y) ˜ n (xy) → Ej ˜ ∞ (x · y), and as Ej ˜ n = jn E, we get jn E(xy) → in s ∗ -topology. Hence Ej j∞ E(x · y) in strong∗ operator topology. Using that fn = E(xy)πn , jn E(xy) = j ∞ (E(xy)πn ) and j∞ E(x · y) = j ∞ (E(x · y)∞ ), we conclude that fn → E(x · y)∞ in measure. It remains to show that the sequence {fn }n is a.e. convergent. (n) It is enough to consider the case x = y ∗ . Let L∞ (I n , P0 ) be the subalgebra ∞ of L (, P0 ) consisting of the functions depending only on the first n coordinates, (n) En : L∞ (, P0 ) → L∞ (I n , P0 ) the P0 -preserving conditional expectation. For any ∞ f ∈ l (I ) we have En (f πn+1 ) = Pφ (f )πn . As y ∗ y = Pφ (y)∗ Pφ (y) ≤ Pφ (y ∗ y) by Schwarz inequality, we have E(y ∗ y) ≤ EPφ (y ∗ y) = Pφ E(y ∗ y), whence fn = E(y ∗ y)πn ≤ Pφ (E(y ∗ y))πn = En (E(y ∗ y)πn+1 ) = En (fn+1 ). Thus the sequence {fn }∞ n=1 is a bounded submartingale. By Doob’s theorem, see e.g. [KSK], it must converge a.e. ˆ be a generating state. Assume that the Poisson boundary Corollary 3.4. Let φ ∈ Cl (G) ˆ φ) and almost of the center is trivial, i.e. H ∞ (I, φ) = C1. Then for any x, y ∈ H ∞ (G, every path s ∈ , we have φsn (xy) → εˆ (x · y) as n → ∞. ˆ be a generating state, V an irreducible representation of Corollary 3.5. Let φ ∈ Cl (G) G. Assume that the Poisson boundary of the center is trivial. Then the multiplicity of V ˆ φ) is not larger than the supremum of the multiplicities of V in U × U for in H ∞ (G, all irreducible representations U of G.
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M. Izumi, S. Neshveyev, L. Tuset
ˆ φ) Proof. By the previous corollary, for any finite dimensional subspace X of H ∞ (G, ˆ → and almost every path s ∈ , the restrictions of the irreducible representations l ∞ (G) B(Hsn ) to X are asymptotically isometric in L2 -norm as n → ∞. In particular, these ˆ φ) → B(Hs ) are restrictions are eventually injective. Since the maps H ∞ (G, ∞ ˆ G-equivariant, it follows that the multiplicity of V in H (G, φ) is not larger than the supremum of the multiplicities of V in B(HU ) for all irreducible representations U of G on HU . It remains to note that the G-module B(HU ), or more precisely, the ˆ ˆ ˆ and x ∈ B(HU ), is A(G)-module such that ωx = (S(ω) ⊗ ι)αU,l (x) for ω ∈ A(G) isomorphic to H U ⊗ HU . For the q-deformation G of a compact connected semisimple Lie group the last estimate is optimal. Indeed, let T ⊂ G be the maximal torus. Then for an irreducible representation V of G on H , the multiplicity of V in L∞ (G/T ) is equal to the dimension of the space of zero weight vectors in H , that is, the space of T -invariant vectors. On the other hand, by the Frobenius reciprocity the multiplicity of V in U × U is the U of U in U × V and same as the multiplicity of U in U × V . Both the multiplicity NU,V the dimension m0 (V ) of the space of zero weight vectors are known to be independent U ˇ §131]. It follows of the deformation parameter. Hence NU,V ≤ m0 (V ), see e.g. [Z, ∞ ˆ that the spectral subspaces of H (G, φ) are not larger than the spectral subspaces of L∞ (G/T ). Note also that as εˆ = ϕ and ϕ is faithful, the Poisson integral is injective on its multiplicative domain. Thus we get the following result. Theorem 3.6. Let G be the q-deformation of a compact connected semisimple Lie group, ˆ a generating state. Assume that the Poisson inteT ⊂ G the maximal torus, φ ∈ Cl (G) ∞ ∞ ˆ φ) is a homomorphism. Then it is an isomorphism. gral : L (G/T ) → H (G, Theorems A and B now follow from Proposition 2.4, Theorem 2.5 and Theorem 3.6. Indeed, it follows immediately that if φ ∈ Cl (l ∞ (SU q (n))) is a generating state, then the ∞ n−1 ∞ Poisson integral : L (SUq (n)/T ) → H (SUq (n), φ) is a SUq (n)- and SU q (n)n−1 n equivariant isomorphism, where T = S(T ) is the maximal torus in SUq (n). If φ is not generating, then ∪k supp φ k corresponds to a quotient SU (n)/ of SU (n) and to the quotient G = SUq (n)/ of SUq (n), which we call the q-deformation of SU (n)/ . More explicitly, if is the group of roots of unity of order m, m|n, then C(G) is the subalgebra of C(SUq (n)) generated by the matrix coefficients of U ×m , where U is the fundamental representation of SUq (n). Set T = Tn−1 / , so C(T ) is the image of C(G) under the homomorphism C(SUq (n)) → C(Tn−1 ). Then L∞ (G/T ) = L∞ (SUq (n)/Tn−1 ) ⊂ L∞ (SUq (n)). Since the assumptions of Theorem 2.5 don’t require ω = φ(·ρ 2 ) to be generating on SU q (n), we again conclude that the Poisson integral ∞ ∞ ˆ φ) is a G- and G-equivariant ˆ : L (G/T ) → H (G, isomorphism. This completes the proof of Theorem B. To prove Theorem A, note that the fixed point algebra is independent of whether we consider the action of SUq (n), or the action of its quotient G such that Irr(G) = ∪k supp φ k . Thus (N α ) ∩ N is G-equivariantly isomorphic to L∞ (G/T ) = L∞ (SUq (n)/Tn−1 ). Clearly, the isomorphism is SUq (n)-equivariant. If ˆ φ), so we get an action of G ˆ on (N α ) ∩ N , then we identify (N α ) ∩ N with H ∞ (G, ˆ the isomorphism is also G-equivariant.
Poisson Boundary of the Dual of SU q (n)
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4. Concluding Remarks For any non-trivial product-type action of SUq (n) on N the fixed point algebra N SUq (n) is obviously strictly contained in the fixed point algebra N T for the action of the maximal torus T ⊂ SUq (n) (contrary to what is claimed in [S2]). Moreover, by our results N SUq (n) ∩ N T ∼ = L∞ (T \SUq (n)/T ). Consider the product-type action defined by the fundamental representation of SUq (n) on Cn . Let H∞ (q) be the Hecke algebra, that is, the algebra with generators g1 , g2 , . . . and relations gi2 = (q − q −1 )gi + 1, gi gi+1 gi = gi+1 gi gi+1 , gi gj = gj gi for |i − j | ≥ 2. Then N SUq (n) is the weak operator closure of the image of H∞ (q) under the homomorphism π : H∞ (q) → N defined by π(g1 ) = · · · ⊗ 1 ⊗ q mii ⊗ mii + (q − q −1 ) mii ⊗ mjj + mij ⊗ mj i , i
i<j
i=j
π(gn ) = γ n−1 π(g1 ), where γ : N → N is the shift to the left, see e.g. [KS, Prop. 8.40]. The homomorphism π should not be confused with the homomorphisms π+ and π− defined by π± (g1 ) = · · · ⊗ 1 ⊗ q mii ⊗ mii + (q − q −1 ) mii ⊗ mjj ± mij ⊗ mj i , i
i>j
i=j
π± (gn ) = γ n−1 π± (g1 ). Using that the unique left SUq (n)-invariant state on B(Cn ) is defined by the density matrix 2(n−1) q 0 ... 0 q 2(n−2) . . . 0 1 − q2 0 , . .. . . .. 1 − q 2n .. . . . 0 0 ... 1 we see that if E : N → γ (N) is the ν-preserving conditional expectation, then Eπ+ (g1 ) and Eπ− (g1 ) are scalars, while Eπ(g1 ) is not. According to [PP, S1], one has π+ (H∞ (q)) = π− (H∞ (q)) = N T . As we showed in Sect. 2, if an element a ∈ C(G) is in the multiplicative domain of , then a = lim φ n (Is )∗s s (a). n→∞
s∈I
If in addition the Poisson boundary of the center is trivial, then we have a stronger convergence result: a = lim ∗sn sn (a) n→∞
(4.1)
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ˆ has for almost every path s ∈ . Indeed, first note that as we assume that l ∞ (G) a generating state, the space C(G) is separable. Hence by Corollary 3.4 we have φsn ((a)(b)) → εˆ ((a) · (b)) for almost every path s ∈ and any b ∈ C(G). Since φsn ((a)(b)) = ϕ(∗sn sn (a)b) and εˆ ((a) · (b)) = εˆ ((ab)) = ϕ(ab), we see that convergence (4.1) holds in weak operator topology for almost every path s ∈ . As in Sect. 2, by G-equivariance of ∗sn sn we conclude that the convergence is in norm. Let now G = SUq (n). Identify I = Irr(SUq (n)) with the set of dominant weights of SU (n). Let Vλ be an irreducible representation of SUq (n) with highest weight λ. What we used in Sect. 3, is that the multiplicity of Vλ in B(Hs ) is not larger than the multiplicity of Vλ in C(SUq (n)/T ). In fact, the multiplicities are equal as soon as s is ˇ It is e.g. enough to require s + wλ to be dominant for any sufficiently large, see [Z]. element w of the Weyl group. Recall also that in the classical case Berezin transforms converge to the identity on the flag manifold along any ray in the Weyl chamber. Thus it is natural to conjecture that convergence (4.1) holds for every sequence s = {sn }∞ n=1 such that the distance from sn to the walls of the Weyl chamber goes to infinity. A significant part of our results is valid for q-deformations of arbitrary compact connected semisimple Lie groups. The point where we crucially used that the group was SUq (n), was Lemma 2.10, which allowed us to reduce the proof of Theorem 2.5 to the study of a one-dimensional random walk. Lemma 2.10 is also valid for q = 1, in which case it is an immediate consequence of the fact that S(U (m) × T) ⊂ SU (m + 1) is a Riemannian symmetric pair of rank one. Hence there is hope that similar considerations could work for SO(n), Sp(n) and F4 , see [He, Ch. X, Table V]. This will be discussed in detail elsewhere. For the exceptional groups E6 , E7 , E8 and G2 , however, our reduction procedure leads us to consider random walks of higher dimensions. The ultimate goal would of course be to find a unified proof. For this it could be instructive to understand the origin of the eigenvector constructed in Proposition 2.11, since to prove that ε is the only Aω -invariant state on C(G/T ), it is enough to find a strictly positive eigenvector for Aω in the kernel of ε on C(G/T ) with eigenvalue less than 1. Remark also that by using commutativity of the fusion algebra as in the proof of Proposition 1.1, one can show that if G is the q-deformation of a compact connected simple Lie group and ε is the only Aω -invariant state on C(G/T ) for some ω = εˆ , then ε is the only invariant state for any ω = εˆ . This, however, does not simplify our considerations in Sect. 2. References [AC] [Be] [B1] [B2] [B3] [B] [C]
Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal. 45, 245–273 (1982) Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975) Biane, Ph.: Marches de Bernoulli quantiques. In: S´eminaire de Probabilit´es, XXIV, 1988/89, Lecture Notes in Math. 1426, Berlin: Springer, 1990, pp. 329–344 ´ Biane, Ph.: Equation de Choquet-Deny sur le dual d’un groupe compact. Probab. Theory Related Fields 94, 39–51 (1992) Biane, Ph.: Th´eor`eme de Ney-Spitzer sur le dual de SU(2). Trans. Amer. Math. Soc. 345, 179–194 (1994) Boca, F.P.: Ergodic actions of compact matrix pseudogroups on C ∗ -algebras. In: Recent advances in operator algebras (Orl´eans, 1992). Ast´erisque No. 232, 93–109 (1995) Collins, B.: Martin boundary theory of some quantum random walks. Ann. Inst. H. Poincar´e Probab. Statist. 40, 367–384 (2004)
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Dijkhuizen, M.S., Stokman, J.V.: Quantized flag manifolds and irreducible ∗-representations. Commun. Math. Phys. 203, 297–324 (1999) [D] Duffield, N.G.: Classical and thermodynamic limits for generalized quantum spin systems. Commun. Math. Phys. 127, 27–39 (1990) [H] Hayashi, T.: Harmonic function spaces of probability measures on fusion algebras. Publ. Res. Inst. Math. Sci. 36, 231–252 (2000) [He] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Graduate Studies in Mathematics 34, Providence, RI: Amer. Math. Soc., 2001. xxvi+641 pp. [HI] Hiai, F., Izumi, M.: Amenability and strong amenability for fusion algebras with applications to subfactor theory. Internat. J. Math. 9, 669–722 (1998) [HLS] Høegh-Krohn, R., Landstad, M.B., Størmer, E.: Compact ergodic groups of automorphisms. Ann. of Math. (2) 114, 75–86 (1981) [I1] Izumi, M.: Non-commutative Poisson boundaries and compact quantum group actions. Adv. Math. 169, 1–57 (2002) [I2] Izumi, M.: Non-commutative Poisson boundaries. In: Discrete geometric analysis, Contemp. Math. 347, Providence, RI: Amer. Math. Soc., 2004, pp. 69–81 [KSK] Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov chains. Princeton, NJ-Toronto-London: D. Van Nostrand Co., Inc., 1966. xi+439 pp. [K] Kaimanovich, V.A.: Bi-harmonic functions on groups. C. R. Acad. Sci. Paris S´er. I Math. 314, 259–264 (1992) [KS] Klimyk, A., Schm¨udgen, K.: Quantum groups and their representations. Texts and Monographs in Physics. Berlin: Springer-Verlag, 1997. xx+552 pp. [KoS] Korogodski, L.I., Soibelman, Y.S.: Algebras of functions on quantum groups. Part I. Mathematical Surveys and Monographs 56. Providence, RI: Amer. Math. Soc. 1998. x+150 pp. [L] Landsman, N.P.: Strict quantization of coadjoint orbits. J. Math. Phys. 39, 6372–6383 (1998) [NT1] Neshveyev, S., Tuset, L.: The Martin boundary of a discrete quantum group. J. Reine Angew. Math. 568, 23–70 (2004) [NT2] Neshveyev, S., Tuset, L.: Quantum random walks and their boundaries. In: Analysis of (Quantum) Group Actions on Operator Algebras (Kyoto, 2003), S¯urikaisekikenky¯usho K¯oky¯uroku 1332, 57–70 (2003) [NYM] Noumi, M.,Yamada, H., Mimachi, K.: Finite-dimensional representations of the quantum group GLq (n; C) and the zonal spherical functions on Uq (n − 1)\Uq (n). Japan. J. Math. (N.S.) 19, 31–80 (1993) [Per] Perelomov, A.: Generalized coherent states and their applications. Texts and Monographs in Physics. Berlin: Springer-Verlag, 1986. xii+320 pp. [P] Petz, D.: Sufficiency of channels over von Neumann algebras. Quart. J. Math. Oxford Ser. 39(2), 97–108 (1988) ´ [PP] Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. Ecole Norm. Sup. 19(4), 57–106 (1986) [PV] Podkolzin, G.B., Vainerman, L.I.: Quantum Stiefel manifold and double cosets of quantum unitary group. Pacific J. Math. 188, 179–199 (1999) [Re] Revuz, D.: Markov chains. North-Holland Mathematical Library, Vol. 11. Amsterdam-Oxford: North-Holland Publishing Co. New York: American Elsevier Publishing Co., Inc., 1975. x+336 pp. [R] Rieffel, M.A.: Matrix algebras converge to the sphere for quantum Gromov–Hausdorff distance. Mem. Amer. Math. Soc. 168(796), 67–91 (2004) [S1] Sawin, S.: Relative commutants of Hecke algebra subfactors. Amer. J. Math. 116, 591–604 (1994) [S2] Sawin, S.: Subfactors constructed from quantum groups. Amer. J. Math. 117, 1349–1369 (1995) [V] Vaes, S.: Strictly outer actions of groups and quantum groups. J. Reine Angew. Math. 578, 147–184 (2005) [W] Woronowicz, S.L.: Compact quantum groups. In: Sym´etries quantiques (Les Houches, 1995), Amsterdam: North-Holland, 1998, pp. 845–884 ˇ ˇ [Z] Zelobenko, D.P.: Compact Lie groups and their representations. Translations of Mathematical Monographs, Vol. 40. Providence, R.I.: Amer. Math. Soc. 1973. viii+448 pp. [DS]
Communicated by Y. Kawahigashi
Commun. Math. Phys. 262, 533–553 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1409-3
Communications in
Mathematical Physics
Conservation of Resonant Periodic Solutions for the One-Dimensional Nonlinear Schr¨odinger Equation Guido Gentile1 , Michela Procesi2 1 2
Dipartimento di Matematica, Universit`a di Roma Tre, 00146 Roma, Italy SISSA, 34014 Trieste, Italy
Received: 4 October 2004 / Accepted: 25 March 2005 Published online: 10 October 2005 – © Springer-Verlag 2005
Abstract: We consider the one-dimensional nonlinear Schr¨odinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the mass term). We investigate conservation of small amplitude periodic solutions for a large measure set of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough, say greater than a certain threshold value depending on the number of resonant modes. If the frequencies of the latter are close enough to such a threshold, then they can not be too distant from each other. Hence we can interpret such solutions as perturbations of wave packets with large wave number. 1. Introduction and Set-up We consider the nonlinear Schr¨odinger equation in d = 1 on the interval [0, π ], given by −iut + uxx = ϕ(|u|2 )u, (1.1) u(t, 0) = u(t, π ) = 0, where ϕ(x) is any analytic function ϕ(x) = x + O(x 2 ) with = 0. More generally we can take, instead of ϕ(|u|2 )u, any function f (u, u) which is real analytic in its arguments, provided the two following conditions are fulfilled: the function f is odd in (u, u) and the dominant order is still of the form |u|2 u. The first condition is likely only a technical one, while the second one is a non-degeneracy condition which is essential in the derivation of our results. In principle it could be removed, but then all the forthcoming analysis should be suitably changed. We shall consider the problem of existence of resonant periodic solutions for (1.1), i.e. solutions arising from superpositions of several unperturbed harmonics, and we shall show how suitably adapting the techniques in Ref. [11] we can solve the problem.
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The nonlinear Schr¨odinger equation has many physical applications, in nonlinear optics, plasma physics and fluid dynamics, and generally in any problem of evolution of quasi-monochromatic wave packets with moderate amplitude in strongly dispersive and weakly nonlinear media. The complex amplitude modulation of the packet is found to be described approximately by the cubic nonlinear Schr¨odinger equation. For a discussion of the applicative relevance of the equation we refer to Refs. [1, 5, 16–20], and papers quoted therein. The most studied equation of the form (1.1), with in principle any boundary conditions, is indeed the cubic one, and often equations containing high order terms are called generalised nonlinear Schr¨odinger equations. The particular relevance of the cubic equation is that it is completely solvable (i.e. integrable) on the line, as shown in Ref. [20], and on the interval with periodic boundary conditions, as shown in Ref. [14] by using the finite-gap approach first introduced in Ref. [15]. Other initial boundary values, including the Dirichlet boundary condition on the semi-line and on the interval, have been recently considered in Refs. [8, 7, 12], where the problem is shown to be reducible to the study of a system of ODE with algebraic right-hand side for the spectral data. Existence of periodic (as well as quasi-periodic) solutions for (1.1) is well known; see for instance Refs. [13, 3], and, very recently, Ref. [9], where more general nonlinearities are also considered, including the ones discussed after (1.1). The fact that no linear term as µu (mass term) appears in (1.1) introduces no further difficulties with respect to the case with mass. In this respect the situation is very different from the case of the nonlinear wave equation, where the completely resonant case (µ = 0) can not be studied in the same way as the case with mass: there, when µ = 0 only very special solutions of the linearized equation are found to be continuable in the presence of nonlinearities, for values of the periods which have those of the linearized equation (unperturbed periods) as Lebesgue density points [11] (the last result improves the previous ones where the unperturbed periods were found to be only accumulation points [2]). On the contrary in the case of the nonlinear Schr¨odinger equation, just because the cases µ = 0 and µ = 0 are faced in the same way, all the periodic solutions for µ = 0 are obtained in the quoted papers as continuations of oscillations involving only one single mode. Here we consider directly the case µ = 0, and first we show how to recover the known results with a different technique, based on the Lindstedt series method introduced in Refs. [10, 11]. Hence we discuss how to obtain other more complicated periodic solutions which arise from superposition of several (non-arbitrary) unperturbed modes. Such solutions look like perturbations of wave packets: the larger is the number N of involved harmonics the higher is the minimum allowed wave number of the corresponding wave packet. Moreover the width of the packet can be large with respect to the wave number only if the latter is large too, so that the smaller is the wave number the narrower is the wave packet itself. To give an idea of the phenomenology for instance, already for N = 2, we find that it is possible to continue for ε = 0 a packet involving for instance the two modes 7 and 8, but none with modes, say, 1 and 2. We note that the solutions we find are special, as we impose that all amplitudes are real. We do not think that such a constraint is necessary, but it rather simplifies the analysis of some non-degeneracy conditions of the unperturbed solutions. If ϕ = 0, or f = 0, every solution of (1.1) can be written as
u(t, x) =
∞ n=1
2
Un ein t sin nx =
n∈Z∗
2
an ein t einx ,
a−n = −an ,
(1.2)
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
where we have set Z∗ = Z \ {0}. For ε > 0 we rescale u → obtaining −iut + uxx = ε|u|2 u + O(ε 2 ), u(t, 0) = u(t, π ) = 0,
535
√
ε/u in (1.1), so
(1.3)
where O(ε2 ) denotes an analytic function of u, u and ε of order at least 2 in ε, and we define ωε = 1 + ε. We shall consider ε small and we shall show that for all m0 ∈ N there exists a solution of (1.3), which is 2π/ωε -periodic in t and ε-close to the function u0 (ωε t, x) = a(ωε t, x) − a(ωε t, −x),
(1.4)
m0 A= √ , 3
(1.5)
with 2
a(t, x) = Aeim0 t eim0 x ,
provided ε is in an appropriate Cantor set (depending on m0 and on the nonlinearity f (u, u)). ¯ We shall look for a solution of the form uε (t, x) = einωt+imx uε,n,m , uε,n,m ∈ R, uε,n,m = −uε,n,−m (1.6) (n,m)∈Z2
which is analytic both in x and t, and periodic in t. Then we shall use the norm F (t, x)r = Fn,m er(|n|+|m|)
(1.7)
(n,m)∈Z
2
for analytic functions. Theorem 1. Consider the equation −iut + uxx = f (u, u), u(t, 0) = u(t, π ) = 0,
(1.8)
where f (u, u) is any real analytic function, odd under the transformation (u, u) → (−u, −u), such that f (u, u) = |u|2 u + O(|u|5 ) with = 0. For all m0 ∈ N, define u0 (t, x) = a(t, x) − a(t, −x), with a(t, x) as in (1.5). There is a positive constant ε0 and a set E ∈ [0, ε0 ], both depending on m0 , satisfying lim
ε→0
meas(E ∩ [0, ε]) = 1, ε
(1.9)
such that for all ε ∈ E, by setting ωε = 1 + ε, there exists a 2π/ωε -periodic solution uε (t, x) of (1.1), analytic in (t, x) and of the form (1.6), with √ (1.10) uε (t, x) − ε/ u0 (ωε t, x) ≤ C ε ε, κ
with κ = κ0 log 1/ε0 , for some constants C, κ0 > 0. If f (u, u) = ϕ(|u|2 )u, with ϕ analytic, then one has uε,n,m = 0 for all n = m20 and E = [0, ε0 ], that is no value of ε ∈ [0, ε0 ] has to be excluded.
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As in Ref. [11] we start by considering the case f (u, u) = ϕ(|u|2 )u, with ϕ(x) = x, which contains all the relevant features of the problem. In principle, with respect to the case (1.8), this introduces further symmetry properties which drastically simplify the problem, but if we ignore these simplifications we shall be able immediately to extend the analysis to more general nonlinearities, as in (1.8). This will be done in Sect. 4, where we shall also show how to deal with more general periodic solutions. The result we obtain at the end is the following one. Theorem 2. Consider Eq. (1.1), where ϕ(x) = x +O(x 2 ) is an analytic function, with = 0, or, more generally, Eq. (1.8). For all N ≥ 2 there are sets of N positive integers M+ and sets of real amplitudes {am }m∈M+ , such that the following holds. Define 2 a(t, x) = eim t+imx am , (1.11) m∈M+
and set u0 (t, x) = a(t, x)−a(t, −x). There is a positive constant ε0 and a set E ∈ [0, ε0 ], both depending on the set M+ , satisfying lim
ε→0
meas(E ∩ [0, ε]) = 1, ε
(1.12)
such that for all ε ∈ E, by setting ωε = 1 + ε, there exist a 2π/ωε -periodic solution uε (t, x) of (1.1), analytic in (t, x) and of the form (1.6), with √ (1.13) uε (t, x) − ε/ u0 (ωε t, x) ≤ C ε ε, κ
with κ = κ0 log 1/ε0 , for some constants C, κ0 > 0. In the proof of Theorem 2 a characterization of the sets M+ and of the amplitudes am will be provided. Hence the proof is constructive. What is found is that, by setting 2 m2 , a2 = am , am ∈ R, (1.14) M= m∈M+
m∈M+
2 = 4a − m2 , which fixes the value one has to require a2 = M/(4N − 1) and am of each amplitude, up to the sign (and up to an overall phase in the case (1.1), which, however, can be chosen to be zero under the request for the amplitudes to be real: we shall come back to this in a moment). In words, this means that the integers in M+ have to be large enough and close enough to each other, so that the solutions which can be continued appear as wave packets with large Fourier label (wave number). More precisely we shall find that for fixed N the harmonics have to be large enough – at best proportionally to N 2 -while the width of the packet can not be smaller than O(N ). Then the wave packets with the smallest allowed wave numbers (which are the ones surviving for the largest values of ε) will have a width which is of order of the square root of the wave number. Note that the level of difficulty of Theorem 1 in the case (1.1) is much lower than that of Theorem 2 (or even of Theorem 1 in the case (1.8)). The first is a result on the existence of a nonlinear ground state, and, with the Ansatz u(t, x) = exp(iωε t) v(x), it becomes a bifurcation problem for the function v(x). Hence there is no small divisors problem, and other easier methods could be envisaged to prove that, no matter which linear eigenvalue is considered, there are nearby nonlinear solutions. The advantage of
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
537
the method that we present is that it can be extended, with just a few slight adaptations, to prove existence of the periodic solutions of Theorem 1 in the case (1.8) and, mostly, of Theorem 2, which is on the contrary a substantially more difficult endeavour, as it mixes up different linear modes. We can make a comparison between the nonlinear Schr¨odinger equation and the nonlinear wave equation, in the case of zero mass. For the wave equation only very special periodic solutions can be continued in the presence of nonlinearities, and this is due to the fact that the Q equation involves simultaneously all the harmonics. For the nonlinear Schr¨odinger equation one finds infinitely many periodic solutions, because there are infinitely many (but still not arbitrary) sets of harmonics which can be excited. Note that for a fixed set of harmonics we construct explicitly only a few periodic solutions which continue the unperturbed ones (we have 2N of them because of the arbitrariness of the amplitude signs). For simplicity we looked only for real values of the amplitudes am . Indeed in this way it turns out to be very easy to check some non-degeneracy conditions that we need in order to solve the equations to all orders. The drawback is that we find only special periodic solutions. It can be that, by relaxing the condition that the amplitudes be real, more general solutions can be found. We did not investigate further the problem, but we think that, at least in the case (1.1), one should be able to fix the unperturbed solution up to some arbitrary phases. This should mean that each amplitude am can be written as ρm eiθm , with (ρm , θm ) ∈ R+ × T, and likely there is some arbitrariness in choosing the angles θm (at best there can be N free parameters, because of the symmetry θ−m = θm + π ). For N = 1 this simply yields that the unperturbed solution (1.5) is defined up to an arbitrary phase θ0 . Finally we note that, as the case of non-zero mass µ can be reduced to that of zeromass with the exponential substitution u(t, x) = exp(−iµt)w(t, x), Theorem 2 implies existence of families of quasi-periodic solutions with two-dimensional frequency vectors for the nonlinear Schr¨odinger equation with mass, or even further periodic solutions when the perturbed frequency ωε becomes commensurate with µ. Also such solutions are not known in the literature. From a technical point of view the discussion below heavily relies on [10, 11]. We confine ourselves to explain how the renormalization group analysis developed in those papers applies to the nonlinear Schr¨odinger equation, by outlining the differences everywhere they appear and showing how they can be faced. Hence a full acquaintance with those papers is assumed to follow all the details of the technical parts. The discussion of Theorem 2 requires some new ideas, and involves problems which can be considered as typical of number theory and matrix algebra.
2. Lindstedt Series Expansion 2.1. Strategy of the proof. We proceed in three steps. 1. We perform a Lyapunov-Schmidt type decomposition. Namely we look for a solution of (1.3) of the form u(t, x) =
einωt+imx un,m = v(ωt, x) + w(ωt, x),
(n,m)∈Z2
v(t, x) =
m∈Z
2 t+imx
eim
vm ,
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G. Gentile, M. Procesi
w(t, x) =
eint+imx wn,m ,
(2.1)
(n,m)∈Z n=m2
2
with un,m ∈ R and ω = ωε = 1 + ε. In order to satisfy the Dirichlet boundary conditions, the solutions (if any), must verify un,m = −un,−m ,
(2.2)
for all n, m ∈ Z. Another property that will be used in the following is that the subspace with un,m real is invariant with respect to (1.1) and (1.8). Inserting (2.1) into (1.3) gives two sets of equations, called the Q and P equations [6], which are given, respectively, by Q P
m2 vm = [f (v + w, v + w)]m , ωn − m2 wn,m = ε [f (v + w, v + w)]n,m ,
n = m2 ,
(2.3)
where we denote by [F ]n,m the Fourier component of the function F (t, x) with labels (n, m), so that F (t, x) = einωt+mx [F ]n,m , (2.4) (n,m)∈Z2
and we shorthand [F ]m the Fourier component of the function F (t, x) with label n = m2 ; hence [F ]m = [F ]m2 ,m . We shall consider first the case f (u, u) = |u|2 u. 2. We study the Q equation in (2.3), in the limit ε → 0 (and so wn,m → 0), and prove that it admits non-degenerate solutions u0 (t, x) in appropriate finite dimensional subspaces; in particular in the case of Theorem 1, for ε = 0 we get vm = 0 for all |m| = m0 and v±m0 = ±A so that u0 (t, x) = a(t, x) − a(t, −x). 3. We solve iteratively the P equation for w(t, x) using renormalization techniques; similarly, using the non-degeneracy of u0 (t, x), we solve the Q equation for 2 v(t, x) − u0 (x, t) ≡ V (x, t) = eim t+imx Vm . (2.5) m∈Z
2.2. The Q equation. The Q equation in (2.3), in the limit ε → 0 (and so wn,m → 0) is m2 vm (ε = 0) = [|v(ε = 0)|2 v(ε = 0)]m ,
(2.6)
therefore a non-trivial infinite dimensional equation for the coefficients vm (ε = 0). Following the scheme proposed in the previous section we set: vm (ε = 0) = 0 for all |m| = m0 and v±m0 (ε = 0) = ±A, with A defined as in (1.5), so that u0 (t, x) = a(t, x) − a(t, −x), which is clearly a solution of (2.6). Let us now consider the Q equation at ε = 0. With the notations of (2.4), and recalling that we are considering for the moment being f (u, u) = ϕ(|u|2 )u, with ϕ(x) = x, we have |v + w|2 (v + w) = [|v|2 v]m + [|w|2 w]m + [2|v|2 w + wv 2 ]m + [2|w|2 v + vw 2 ]m m
≡ [|v|2 v]m + [G2 (v, w)]m , where G2 (v, w) is at least linear in w.
(2.7)
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
539
As said in the previous section we write v = a +b+V , with b(ωt, x) = −a(ωt, −x), i.e. 2
b(t, x) = Beim0 t−im0 x ,
B = −A,
(2.8)
so that we obtain [|v|2 v]m = |A|2 Aδm,m0 + |B|2 Bδm,−m0 + 2|A|2 Bδm,−m0 + 2|B|2 Aδm,m0 +2|A|2 Vm + 2|B|2 Vm + 2ABV−m δm,−m0 + 2BAV−m δm,m0 +V m A2 δm,m0 + 2V −m ABδm,±m0 +V m B 2 δm,−m0 + [G1 (v)]m ,
(2.9)
where G1 (v) is at least quadratic in V . Then, by setting G(v, w) = G1 (v) + G2 (v, w), the Q equation in (2.3) can be rewritten for m = m0 as m20 Vm0 = 2|A|2 Vm0 + 2|B|2 Vm0 + V m0 A2 + 2ABV −m0 +2BAV−m0 + [G(v, w)]m0 ,
(2.10)
and for positive m = m0 as m2 Vm = 2|A|2 Vm + 2|B|2 Vm + [G(v, w)]m ,
(2.11)
while the equation for negative values of m can be obtained by using the symmetry properties (2.2), which imply V−m = −Vm . By defining α = |A|2 = AA and using the identities α = AA = BB = −AB = −BA,
β = AA = BB = −AB = −BA, (2.12)
which follow trivially from the definitions of A and B in (1.5) and (2.8) respectively, we can rewrite (2.10) as m20 Vm0 = 4αVm0 + βV m0 − 2βV −m0 − 2αV−m0 + [G(v, w)]m0 ,
(2.13)
where we have used that α = |A|2 = m20 /3. By using once more the identities (2.2) and imposing that the coefficients Vm be real, so that α = β in (2.12), we can write (2.13) and Eq. (2.11), respectively, as m20 Vm0 = 9αVm0 + [G(v, w)]m0 , (2.14) m2 Vm = 4αVm + [G(v, w)]m , so that we find
1 V = − [G(v, w)]m0 , m0 6α Vm =
(2.15) 1 [G(v, w)]m , 2 m − 4α
respectively for positive m0 and m = m0 .
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2.3. The P equation. The P equation in (2.3) involves small divisors; as in Refs. [10, 11] we handle them by appropriately renormalizing the frequencies. Given a sequence {νm (ε)}|m|≥1 , such that νm = ν−m , we define the renormalized frequencies as 2 2 ω˜ m := ωm − νm ,
ωm = |m|,
(2.16)
and the quantities νm will be called the counterterms. By the above definition and the parity properties (2.2) the P equation in (2.3) can be rewritten as 2 ωn − ω˜ m wn,m = νm wn,m + ε[f (v + w, v + w)]n,m (a) (b) = νm wn,m + νm wn,−m + ε[f (v + w, v + w)]n,m , (2.17)
where (a) (b) νm − νm = νm .
(2.18)
Finally we write
(a) (b) wn,m + µνm wn,−m + µε[|v + w|2 (v + w)]n,m , (2.19) wn,m = g(n, m) µνm
where g(n, m) =
1 , 2 ωn − ω˜ m
n = m2 ,
(2.20)
and we look for a solution un,m in the form of a power series expansion in µ, un,m =
∞
µk u(k) n,m ,
(2.21)
k=0
with
(k) un,m
(c)
depending on ε and on the parameters νm , with c = a, b and |m| ≥ 1. (k)
2.4. Recursive equations. So we obtain recursive definitions of the coefficients un,m . (k) The coefficients wn,m verify for k ≥ 1 the equations (k) (a) (k−1) (b) (k−1) wn,m = g(n, m) νm wn,m + νm wn,−m + [|v + w|2 (v + w)](k−1) , (2.22) n,m where [|v + w|2 (v + w)](k) n,m =
k1 +k2 +k3 =k −n1 +n2 +n3 =n −m1 +m2 +m3 =m
with u(0) n,m
u(k) n,m
A, = −A, 0, (k) Vm , = (k) wn,m ,
(k2 ) (k3 ) 1) u(k n1 ,m1 un2 ,m2 un3 ,m3 , (2.23)
if n = m2 and m = m0 , if n = m2 and m = −m0 , otherwise, if n = m2 , if n = m2 ,
k ≥ 1,
(2.24)
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
541
(k)
while the coefficients Vm verify for k ≥ 1 the equations ∗ (k2 ) (k3 ) 1) u(k Vm(k) = g(m2 , m) n1 ,m1 un2 ,m2 un3 ,m3 ,
(2.25)
k1 +k2 +k3 =k −n1 +n2 +n3 =m −m1 +m2 +m3 =m
where
g(m2 , m) =
1 − 2, 2m0
if |m| = m0 , (2.26)
3 , 2 3m − 4m20
if |m| = m0 ,
and the ∗ means that there appear only contributions either with at least one coefficient with n = m2 or with at least two labels ki ≥ 1. (k) It is easy to realize that to any order k one has un,m = 0 whenever n = m20 , and the same remains still true if we replace f (u, u) = |u|2 u with any function of the form f (u, u) = ϕ(|u|2 )u (one can check this obvious property, for instance, by induction on k). For the time being, however, we ignore such a property, and we proceed as if every value of n was possible. The reason to do this is that in such a way the results that we can find for the case f (u, u) = ϕ(|u|)2 u can be extended immediately to the more general case of Eq. (1.8). To prove Theorem 1 we can proceed in two steps as in Ref. [11]. The first step consists in looking for the solution of the recursive equations by considering ω˜ = {ω˜ m }|m|≥1 as a given set of parameters satisfying the Diophantine conditions (called respectively the first and the second Mel nikov conditions) 2 ∀n ∈ Z∗ and ∀m ∈ Z∗ such that n = m2 , ωn ± ω˜ m ≥ C0 |n|−τ 2 2 −τ ± ω˜ m ∀n ∈ Z∗ and ∀m, m ∈ Z∗ such that ωn ± ω˜ m ≥ C0 |n| |n| = |m2 ± (m )2 |,
(2.27)
with positive constants C0 , τ . We can assume without loss of generality C0 ≤ 1/2. We shall show in Sect. 3 how to adapt the discussion in Ref. [11] in order to obtain the following result. Proposition 1. Consider a sequence ω˜ = {ω˜ m }|m|≥1 verifying (2.27), with ω = ωε = 2 − m2 | ≤ C ε for some constant C . For all µ > 0 there 1 + ε and such that |ω˜ m 1 1 0 exists ε0 > 0 such that for |µ| ≤ µ0 and 0 < ε < ε0 there is a sequence ν(ω, ˜ ε; µ) = {νm (ω, ˜ ε; µ)}|m|≥1 , where each νm (ω, ˜ ε; µ) is analytic in µ, such that there are coeffi(k) cients un,m which solve the recursive equations (2.22) and (2.25), with νm = νm (ω, ˜ ε), and define a function u(t, x; ω, ˜ ε; µ) which is analytic in µ, analytic in (t, x) and 2π/ωε -periodic in t. Then in Proposition 1 one can fix µ0 = 1, so that one can choose µ = 1 and set u(t, x; ω, ˜ ε) = u(t, x; ω, ˜ ε; 1) and νm (ω, ˜ ε) = νm (ω, ˜ ε; 1). The second step, also to be proved in Sect. 3, consists in inverting (2.1), with νm = νm (ω, ˜ ε) and ω˜ verifying (2.27). This requires some preliminary conditions on ε, given by the Diophantine conditions |ωn ± m| ≥ 2 C0 |n|−τ0
∀n ∈ Z∗ and ∀m ∈ Z∗ such that n = m,
(2.28)
542
G. Gentile, M. Procesi
with a positive constant τ0 > 1. Then we can solve iteratively (2.1), by imposing further non-resonance conditions besides (2.28). At each iterative step one has to exclude some further values of ε, and at the end the left values fill a Cantor set E with large relative measure in [0, ε0 ] and ω˜ verify (2.27). Of course in the case (1.1), which yields n = m20 , no further condition has to be imposed as one has |m2 − m20 | ≥ 1 > |εm20 | for fixed m0 and ε small enough. The result of this second step can be summarized as follows. Proposition 2. In the case (1.8) here are δ > 0 and a set E ⊂ [0, ε0 ] with complement of relative Lebesgue measure of order ε0δ such that for all ε ∈ E there exists ω˜ = ω(ε) ˜ 2 −m2 | ≤ C ε which solves (2.1) and satisfies the Diophantine conditions (2.27) with |ω˜ m 1 for some positive constant C1 . In the case (1.1) the same result holds for all ε ∈ [0, ε0 ]. The proof follows the same strategy as in Ref. [11]. Slight changes will be discussed in Sect. 3.
3. Renormalization and Proof of Theorem 1 We refer to Sect. 3 in Ref. [11] for the basic definitions of trees (cf. in particular Definition 2). With respect to that paper the diagrammatic rules are changed as follows. (1) We call nodes the vertices such that there is at least one line entering them. We call end-points the vertices which have no entering line. We denote with L(θ ), V (θ ) and E(θ ) the set of lines, nodes and end-points, respectively. For any vertex V (node or end-point) there is one and only one line exiting it, so that we can set = V . (2) There can be two types of lines, w-lines and v-lines, so we associate with each line ∈ L(θ ) a badge label γ ∈ {v, w} and a momentum (n , m ) ∈ Z2 , to be defined in item (6) below. One has γ = v if n = m2 , and γ = w otherwise. One can not have (n , m ) = (0, 0). All the lines coming out from the end-points are v-lines. (3) With each line coming out from a node we associate a propagator g = g(n , m ), with g(n, m) defined in (2.20) and (2.26) if the line comes out from a node, while one has g = 1 if the line comes out from an end-point. (4) If we denote by sV the number of lines entering the node V one can have either sV = 1 or sV = 3. In the latter case we call LV the set of lines entering V: we associate with each line ∈ LV a label s() ∈ {±1} with the constraint ∈LV s() = 1. Also the nodes V can be of w-type and v-type: we say that a node is of v-type if the line coming out from it has label γ = v; analogously the nodes of w-type are defined. We can write V (θ ) = Vv (θ ) ∪ Vw (θ ), with obvious meaning of the symbols; we also call Vws (θ ), s = 1, 3, the set of nodes in Vw (θ ) with s entering lines, and analogously we define Vvs (θ ), s = 1, 3. One has sV = 3 for all V ∈ Vv (so that Vv1 = ∅ unlike Ref. [11]). If V ∈ Vv3 (θ ) and two entering lines come out of end points then the remaining line entering V has to be a w-line. If V ∈ Vw1 (θ ) then the line entering V has to be a w-line. (5) With each end-point V we associate a mode label (nV , mV ), with mV = ±m0 and nV = m20 , and an end-point factor VV =
A, −A,
mV = m0 , mV = −m0 ,
(3.1)
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
543
while with each node V we associate a node factor 1/3, V ∈ Vv (θ ), 3 V ∈ Vw (θ ), ηV = ε, ν (cV ) , V ∈ V 1 (θ ), m V w
(3.2)
where Vv = Vv3 (cf. item (4)), and cV = a if mV = m , where is the line entering V, while cV = b if mV = −m , with the same notations. (6) The momentum (n , m ) of a line = V coming out from a node V is given by n = (−1)S(W,) nW , W∈E(θ ) WV
m =
(−1)S(W,) mW +
W∈E(θ ) WV
(−2mW ),
(3.3)
1 (θ ):c =b W∈Vw W WV
where S(W, ) is the number of lines with s() = −1 between W and . Note that the rules given above look simpler with respect to Ref. [11]. This is due essentially to the fact that the Q equation is much simpler in the present case. Introducing a multiscale decomposition as in Sect. 4 of Ref. [11] we can define for the lines with γ = w the propagator on scale h ≥ −1 as (h)
g
2 = χh (|ωn − ω˜ m |) g =
2 |) χh (|ωn − ω˜ m , 2 ωn − ω˜ m
(3.4)
where χh (x) is a C ∞ function non-vanishing for 2−h−1 C0 < |x| < 2−h+1 C0 if h ≥ 0 and for |x| > C0 if h = −1. This leads to new diagrammatic rules, which differ with respect to the previous ones because item (6) has to be replaced by the following one. (6 ) Each line carries, besides the momentum (n , m ) ∈ Z2 , also a scale label h ≥ −1 for γ = w and a scale label h = −1 for γ = v. The corresponding prop(h ) agator g is given by (3.4) with h = h for γ = w, while is the same as before for γ = v. Then for each tree θ one can define the tree value as (h ) Val(θ ) = (3.5) g ηV VV , ∈L(θ)
so that one has u(k) n,m =
V∈V (θ)
Val(θ ),
V∈E(θ)
(3.6)
(k) θ∈n,m
(k)
where n,m is the set of trees θ of order k, that is with |Vw (θ )| = k, and with momentum (n, m) associated with the root line (we omit the proof, as it proceeds exactly as for Lemma 2 in Ref. [11]). Note that one has |Vv (θ )| ≤ 2|Vw (θ )| = 2k and |E(θ )| ≤ 2(|Vw (θ )| + |Vv (θ )|) + 1 ≤ 6k + 1 (cf. Lemma 3 in Ref. [11]).
544
G. Gentile, M. Procesi
Clusters and self-energy graphs are defined as in Ref. [11] (cf. Definitions 6 and 7). In particular we call 1T and 2T the lines exiting and entering (respectively) the self-energy graph T . Given a self-energy graph T with momentum (n, m) associated to the line 2T the corresponding self-energy value is given by (h ) VTh (ωn, m) = (3.7) g ηV VV , V∈V (T )
∈T
V∈E(T )
(e)
where h = hT is the minimum between the scales of the two external lines of T (they (e) can differ at most by a unit and hT ≥ 0), and, given a self-energy graph, one has 1 n(T ) := (−1)S(V,T ) nV = 0, V∈E(T )
m(T ) :=
1
(−1)S(V,T ) mV +
V∈E(T )
−2mW ∈ {0, 2m},
(3.8)
1 (T ) W∈Vw
cW =b
by definition of the self-energy graph. One says that T is a self-energy graph of type c = a when m(T ) = 0 and a resonance of type c = b when m(T ) = 2m. The following results hold. 2 − m2 | < C ε for all m ≥ 1. Lemma 1. Assume that there is a constant C1 such that |ω˜ m 1 2 If |ωn − ω˜ m | < 1/2 and ε is small enough then min{n , m2 } > 1/4ε. 2 = εn + (n − m2 ) + ν , so that |ωn − ω 2 | > 1/2 for Proof. One has ωn − ω˜ m ˜m m 2 2 n = m and 0 < n < 1/3ε. Moreover if |ωn − ω˜ m | < 1/2 then one has n > 0 and m2 > ωn − |νm | − 1/2 > 1/4ε.
Hence if n < 1/4ε we can bound |g(n , m )| ≤ 2 while if n ≥ 1/4ε in general we can bound |g(n , m )| ≤ 2h+1 C0−1 . To any line with n < 1/4ε we can assign a scale label h = −1. 2 − m2 | < C ε. Define h Lemma 2. Assume that there is a constant C1 such that |ω˜ m 1 0 √ h h +1 0 0 such that 2 ≤ 16C0 / ε < 2 . Then for h ≥ h0 one has
Nh (θ ) ≤ 4k2(2−h)/τ − Ch (θ ) + Sh (θ ) + Mhν (θ ),
(3.9)
where Nh (θ ) is the number of lines in L(θ ) on scale h, Ch (θ ) is the number of clusters (e) in θ on scale h, Sh (θ ) the number of self-energy graphs in θ with hT = h, and Mhν (θ ) is the number of ν-vertices (i.e. nodes V of w-type with sV = 1) in θ ; for more details cf. Definition 8 and Lemma 5 of Ref. [11]. Proof. The proof is as for Lemma 5 of Ref. [11]. Again the only case which deserves attention is when one has a cluster T with two external lines and 1 both on scales ≥ h, so that, with the same notations as in Ref. [11], one has 2 2 2−h+2 C0 ≥ ω(n − n1 ) + η ω˜ m + η1 ω˜ m . (3.10) 1 √ Then |n − n1 | = |m2 ± m21 | would require |n − n1 | ≥ |m | + |m1 | > 1/ ε, while (3.10) would become 2−h+2 C0 > |ε(n − n1 )| − 2C1 ε. Combining the two inequalities √ one would obtain C0 2−h+3 > ε, which contradicts the condition h ≥ h0 . Then one proceeds as in Ref. [11].
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
545
2 − m2 | < C ε. Then one Lemma 3. Assume that there is a constant C1 such that |ω˜ m 1 has h 0 −1
h=0
∈L(θ ) h =h
(h )
|g
| ≤ C2k ε −k/2 ,
(3.11)
for some positive constant C2 .
√ (h ) Proof. If h < h0 one has |g | ≤ C0 2−h0 +1 < ε/4, and the number of lines with scales 0 ≤ h < h0 can be bounded by the total number of lines with label γ = w, which is less than k. The renormalized expansion is defined as in Sect. 5 Ref. [11], with the only difference that now the action of the localization operator L is such that 2 LVTh (ωn, m) = VTh (ω˜ m , m),
(3.12)
so that, in the definition of the set E0 (θ ) (see item (7 ) in Sect. 5 of Ref. [11]), we set 2 /ω. Up to these notational changes no other difference appears with respect to ωm = ω˜ m (k)R the discussion carried out in Ref. [11], hence we introduce the set n,m of renormalized trees by adding the following further rules to the previous ones. (7) With the nodes V of w-type with sV = 1 (ν-vertices) and with h ≥ 0 the minimal (c) scale among the lines entering or exiting V, we associate a factor 2−h νh,m , c = a, b, where (n, m) and (n, ±m), with |m| ≥ 1, are the momenta of the lines, and a corresponds to the sign + and b to the sign − in ±m. (8) The set {h } of the scales associated to the lines ∈ L(θ ) must satisfy the following constraint (which we call compatibility): fixed (n , m ) for any ∈ L(θ ) and replaced R with 11 at each self-energy graph, one must have χh (|ωn | − ω˜ m ) = 0. In terms of the renormalized trees we can write ∞ un,m = u(0) µk Val(θ ), n,m + k=1
(3.13)
(k)R θ∈n,m
(c)
where, for |m| ≥ 1 and h ≥ 0, νh,m is given by 1 kT h (c) (c) + µ VT (σ ω˜ m , m), 2−h µνh,m = µνm 2 σ =± (c)
(3.14)
T ∈T
(c)
with c = a, b, and T
∈L0 (θ)
×
T ∈S0 (θ)
(e)
h RVT T
V∈E0 (θ)
(ωnT , mT ) ,
(3.15)
546
G. Gentile, M. Procesi
where T denotes the line entering T , and we have set (e)
(e)
h
(e)
h
h
RVT T (ωnT , mT ) = VT T (ωnT , mT ) − LVT T (ωnT , mT ),
(3.16)
(e)
h
with VT T (ωnT , mT ) given by (e)
h
VT T (ωnT , mT ) =
(h )
g
V∈V0 (T )
∈L0 (T )
×
(e) hT T
RV
T ∈S0 (T )
ηV
VV
V∈E0 (T )
(ωnT , mT ) ,
(3.17)
if L0 (T ), V0 (T ), E0 (T ) and S0 (T ) are the sets of lines, nodes, end-points and maximal self-energy graphs in T which are not contained in any self-energy graph internal to T . Now we proceed exactly as in Ref. [11]. First (by Lemma 7 of Ref. [11]) we have that the expansion (3.13) is well defined, for νh,m = O(ε); namely we have that the coefficients (3.13) are bounded by |un,m | ≤ 2 D0 ε |m|/4m0 ε |n|/4m0 . 2 The presence of the factor ε |m|/4m0 ε |n|/4m0 , instead of the factor ε e−κ(|n|+|m|)/4 in Eq. (5.8) of Ref. [11], is due to the fact that the end-points carry a mode label (±m0 , m20 ), so that in order to have a momentum (n, m) flowing through the root line one needs at least (k)R k∗ nodes, with k∗ > min{|m|/m0 , |n|/m20 }, and for θ ∈ n,m the tree value Val(θ ) is in general proportional to εk/2 by (3.11) (the discussion is very similar to tat performed in Ref. [10], which we refer to for further details). Then (by Lemma 8 and Lemma 9 of Ref. [11]) we have that under the same conditions also the r.h.s. of (3.14) is well defined. Moreover (by Lemma 10 of Ref. [11]) we find (c) that it is indeed possible to choose νm such that νh,m = O(ε) for any h. We omit the other details which can be easily worked out by looking at the quoted reference. This completes the proof of Proposition 1. We now pass to Proposition 2: we look for the perturbed frequencies ω˜ m (ε) which solve (2.16) and satisfy the Diophantine conditions (2.27) for ε ∈ E, where E is a Cantor set of relative large measure. We proceed as in Sect. 6 of Ref. [11], with some minor differences (which are in fact simplifications) that we outline below. The condition (2.28) on ε can be imposed exactly as in Ref. [11] (Lemma 14), and requires ε ∈ E (0) , with E (0) a suitable subset of [0, ε0 ], with ε0 as in Proposition 1, of large relative measure provided one sets τ0 > 1. (p) We define a sequence ω˜ = {ω˜ m }∞ m=1 by setting (0) 2 2 ) = ωm , (ωm (p)
(p−1)
2 (ω˜ m )2 = ωm − νm (ω˜ m
, ε),
p ≥ 1,
(3.18)
(p)
with νm (ω˜ m , ε) well defined on a Cantor set E (p) where, as in Eq. (6.11) of Ref. [11], (p) the Mel nikov conditions (2.27) are satisfied with ω˜ m = ω˜ m . By reasoning as for Lemma 16 of Ref. [11] the following result is immediately obtained. Lemma 4. For all p ≥ 0 there exists a positive constant C3 such that |νm (ω˜ (p) , ε)| < C3 ε.
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
547 (p+1)
Hence for all fixed p the hypotheses of Proposition 1 are satisfied and νm (ω˜ m , ε) is well defined on some smaller Cantor set E (p+1) ⊂ E (p) . By reasoning as for Lemma 15 of Ref. [11] (Sect. 6) we have that the difference between the frequencies at two subsequent steps decreases exponentially: max |νm (ω˜ (p) , ε) − νm (ω˜ (p−1) , ε)| ≤ Cεp ,
(3.19)
m∈Z
˜ (∞) ≡ ω˜ (∞) (ε) which satisfies Eqs. (2.18) so that the sequence {ω˜ (p) }∞ p=0 has a limit ω (∞) and is well defined on a Cantor set E = E . To conclude the proof of Proposition 2 we still need to impose the Mel nikov con(p) ditions in (2.27), with ω˜ m = ω˜ m (ε) and to verify that the set of ε which satisfies such conditions for all p is of large relative measure and has the origin as a density point. To do this, we evaluate the measure of the complementary set to E (p) in [0, ε0 ] defined by (p)
f1 (ε(t)) := (1 + ε(t))n − (ω˜ m (ε(t)))2 = t ε(t) ∈ (0, ε0 ),
C0 , |n|τ
t ∈ [−1, 1], (3.20)
when dealing with the first Mel nikov conditions (cf. Eq. (6.30) of Ref. [11]), and through (p)
(p)
f2 (ε(t)) := (1 + ε(t))|n |−|(ω˜ m (ε(t)))2 ± (ω˜ m (ε(t)))2 | = t ε(t) ∈ (0, ε0 ),
C0 , |n|τ
t ∈ [−1, 1], (3.21)
when dealing with the second Mel nikov conditions. The functions f1 and f2 depend also on n, m and n, m, m , respectively: we are not making explicit such a dependence in order not to overwhelm the notations. The measure of the complementary set is then bounded by ∗ C0 ∗ C0 −1 max |∂f /∂ε| + max |∂f2 /∂ε|−1 , (3.22) 1 τ ε∈(0,ε ) |n|τ ε∈(0,ε0 ) |n| 0 n,m∈Z
n,m,m ∈Z
where ∗ means that in the sums over n, m, m one only has to consider those values n, m, m such that both f1 (ε) and f2 (ε) can be small, say smaller than 1/4. As in Ref. [11] one has to use that |∂fj /∂ε| ≥ |n|/2 for j = 1, 2. In the case of the first Mel nikov conditions one has to consider only the values of n such that n ≥ N0 = O(ε0−1 ), as |νm | < C1 ε, and for each n the set M0 (n) of m’s such √ that f1 (ε) < 1/4 contains at most 2 + ε0 n values. Therefore the measure of the set of excluded values of ε turns out to be bounded proportionally to ∞ √ C0 2 + ε0 n ≤ const. ε01+δ1 , τ +1 n
(3.23)
n=N0
with δ1 > 0 provided one takes τ > 1. In the case of the second Mel nikov conditions one has to use that if |n| is close to 2 |m − (m )2 | then |n| is of order |m| − |m | (|m| + |m |), with |m| − |m | = 0, so that |m| + |m | ≤ |n|. This means that for each n the number of pairs (m, m ) one has to sum over is at most proportional to 2 + ε0 |n|2 . The same happens (trivially) when |n| is close to m2 + (m )2 . In both cases one has to sum only on the values of n such that |n| ≥ N0 = O(ε0−1 ), so that one has to exclude a set of values of ε whose measure is bounded proportionally to
548
G. Gentile, M. Procesi ∞ C0 2 2 + ε ≤ const. ε01+δ2 , n 0 nτ +1
(3.24)
n=N0
with δ2 > 0 provided one takes τ > 2. The argument above is for fixed p. By taking into account that the centers of the intervals of excluded values of ε get closer and closer at each iterative step p (cf. Lemma 17 in Ref. [11]), we find that we have to apply the construction above only for a finite number of steps p0 (n) (growing proportionally to log |n|), at the price of enlarging the sizes of the first p0 (n) intervals. The conclusion is that if we set τ > 2 we have that E (∞) has large relative measure. Again we refer to Ref. [11] for further details. 4. Extension of the Results and Proof of Theorem 2 The extension of the results of the previous sections to the case in which ϕ(x) is any analytic function with ϕ (0) = 0, can be easily dealt with by reasoning as in Sect. 8 of Ref. [11]. Essentially the diagrammatic rules change as one has to take into account also the contributions of order higher than three arising from the nonlinearity, which means that now sV can be any odd positive integer and for each node V with sV > 1 the node factors ηV depend on the function ϕ. (k) If we use that wn,m = 0 for n = m20 then we see that in fact we have no small 2 | = |ωm2 ± ω 2 | is bounded from below for all divisor problem: the quantity |ωn ± ω˜ m ˜m 0 m ∈ Z \ {±m0 }, provided ε is small enough. Hence in the case (1.1) the discussion can be substantially simplified. The advantage of the proof given in Sect. 3 is that it still applies to any f (u, u) as in (1.8). In such a case there is no longer a symmetry property which imposes n = m20 , so that n can really assume any value. Therefore we need the Diophantine conditions (2.27) and we have to exclude some values of ε. This leads naturally to the set E defined in the statement of Theorem 1. As for the diagrammatic rules, now also the constraints on ∈LV s(), besides the node factors, depend on f . No extra difficulty arises with respect to the analysis of Sect. 3, so we pass directly to discuss the case of more general periodic solutions to be continued. For ε = 0 we call v0 = v(ε = 0) = a + b the solution of the Q equation in (2.3), by writing a(t, x) =
∞
2 t+imx
am eim
,
(4.1)
m=1
with coefficients am ∈ R to be determined, and setting b(t, x) = −a(t, −x). Then the Q equation becomes m2 v0,m = v 0,m1 v0,m2 v0,m3 = 2v0,m v 0,m v0,m + v 0,m v0,m v0,m , −m1 +m2 +m3 =m −m21 +m22 +m23 =m2
m =m
(4.2) so that we obtain
v0,m m2 − 2v0 2 + |v0,m |2 = 0,
(4.3)
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
where we have defined M = m ∈ Z : v0,m = 0 ,
549
M+ = {m ∈ M : m > 0} ,
(4.4)
and set v0 2 :=
2 v0,m =
m∈Z
2 v0,m .
(4.5)
m∈M
Hence (4.3) can be satisfied either if v0,m = 0 or, when v0,m = 0, if v0 2 =
2M , 4N − 1
(4.6)
where we have set 2N = |M| = # {m ∈ M} ,
2M =
m2 .
(4.7)
1 v0 2 , 2
(4.8)
m∈M
By inserting (4.6) into (4.3), setting am = v0,m ,
m > 0,
a2 =
2 am =
m∈M+
and writing M+ = {m1 , m2 , . . . , mN }, with mk < mk+1 , k = 1, . . . , N − 1, we obtain 2 = 4a2 − m2k = am k
k = 1, . . . , N,
4 m21 + m22 + . . . + m2N − m2k , 4N − 1 (4.9)
which makes sense as long as max m2 ≤
m∈M+
4 m2 . 4N − 1
(4.10)
m∈M+
The following result is easily proved. Lemma 5. For all N ≥ 2 there are solutions of (4.9) such that 4a2 is not an integer. Proof. To obtain a solution one can take mk = mN − (N − k) for k = 1, . . . , N, and choose mN ≥ 4N(N − 1). Choose mN = (4N + j )(N − 1), with j ∈ {0, 1}: then 4(m21 + . . . + m2N ) can not be a multiple of 4N − 1 for both j = 0 and j = 1. Here we have confined ourselves only to an existence result. Of course more general solutions can be envisaged, with more spacing between the involved wave numbers mk . The result above can indeed be strengthened as follows. Lemma 6. For all N ≥ 2 and for all increasing lists of positive integers I := {i1 , . . . , iN−1 } there exists mN (I ) (mN for short) such that (4.10) has a solution in the set M+ = {mN − iN−1 , . . . , mN − i1 , mN } with 4a2 = m2 for all m ∈ / M.
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Proof. Fix the set of integers I = {i1 , . . . , iN−1 }, and consider the expression M − (N − 1/4)j 2 for j ∈ N. For j = mN it becomes a polynomial of degree two in mN , with positive leading coefficient 1/4 and positive discriminant. Hence there is an integer K1 such that for all mN > K1 one has f1 (mN ) ≡ M − (N − 1/4)m2N > 0, hence (4.10) is satisfied. The inequality M − (N − 1/4)j 2 > 0 is trivially satisfied for j ≤ mN , so that it is enough to look for an integer mN > K1 such that one has f2 (mN ) ≡ M − (N − 1/4)(mN +1)2 < 0. Again f2 is a polynomial of degree two in mN , with positive leading coefficient 1/4 and positive discriminant, so that there exist two integers K2 < K3 such that f2 (mN ) < 0 for K2 < mN < K3 . Moreover K3 − K2 ≥ 4(N − 1), so that there is mN satisfying (4.10) such that 4a = j 2 for all j ∈ N. The condition 4a2 = m2 for all m ∈ M, required in Lemma 6 and implied by Lemma 5, implies that the solution v0 (t, x) is non-degenerate, namely the linearized operator acting on V is invertible, so that Vm turns out to be defined iteratively to all orders (compare (2.15) in the case of Theorem 1 with Lemma 8 below). The request for the amplitudes am to be real was motivated just with the aim of making straightforward the check of the non-degeneracy condition. To have solutions of (4.9) requires the integers in M+ to be large enough, and not too distant from each other. Indeed, at best, the distance between the harmonics is O(N ), while the harmonics themselves are greater than some threshold value, which is O(N 2 ) – cf. the proof of Lemma 5. Hence the solutions whose existence is stated in Lemma 5 have the form of wave packets centered around some harmonic (wave number) large enough, with a width proportional to the square root of the wave number. In general, if the harmonics are very large with respect to the threshold value, then also the width of the packet can be large. Note however that in order to be sure that the quantity 4a2 is not a squared integer in the proof of Lemma 6 we required mN (hence the wave number of the corresponding packet) to be not too large. Moreover we are more interested in wave packets with wave number not too large, as the larger is the latter the smaller is the corresponding value ε0 appearing in the statement of Theorem 2, as we shall see. Hence we have proved the following result. Lemma 7. For any N there are sets M and functions v0 (t, x) = a(ωt, x) − a(ωt, −x), with 2 a(t, x) = eim ωt+imx am , (4.11) m∈M+
which solve the Q equation with ε = 0. Moreover, by using once more the parity properties V−m = −Vm , one obtains, generalising (2.14) to the case N > 1 and with the same meaning as there for the function G(v, w), for m ∈ M+ , Am,m Vm = [G(v, w)]m , (4.12) m ∈M+
where A is an N × N matrix with entries 2 , m = m , m2 − 4a2 − 5am Am,m = −8am am , m= m .
(4.13)
Conservation of Resonant Periodic Solutions for the 1-D NLS Equation
551
It is also easy to check that, if the amplitudes am are chosen to be real, then also the amplitudes Vm and wn,m can be found to be real (as remarked in Sect. 2 the condition un,m ∈ R is consistent with Eq. (1.8)). Then the following result holds. Lemma 8. For M chosen according to Lemma 6 (or Lemma 5), one has for m ∈ M+ Vm = Dm,m [G(v, w)]m , (4.14) m ∈M+
with D a N × N non-singular matrix. For positive m ∈ / M+ , an analogous, simpler expression is found of the form (4.14) with Vm = (m2 − 4a2 )−1 [G(v, w)]m ,
(4.15)
where the coefficients (m2 − 4a2 ) are not zero by Lemma 6 (or Lemma 5). The amplitudes Vm for negative m are easily obtained by noting that Vm = −V−m . Proof. To obtain (4.14) it is sufficient to prove that the matrix A with entries (4.13) is 2. not singular. By using (4.3) we can write the diagonal entries of A as Am,m = −6am Then one realizes immediately that one has det A = (−1)N det DN (6, 8)
N
2 am ,
(4.16)
m=1
where DN (p, q) is the N × N matrix with diagonal entries p and all off-diagonal entries q. One can easily prove that det DN (p, q) = (p − q)N−1 (p + (N − 1)q). As in our case p = 6 and q = 8 (so that p = q and p < (N − 1)q for all N ≥ 2) the assertion follows. Finally Eq. (4.15) is a direct generalisation of (2.9). This allows us to extend the analysis of the previous section to the case in which the function v0 is of the form considered here. At the end Theorem 2 is obtained, with the set M chosen according to Lemma 6. Following Sect. 2 we insert the series expansion (2.21) in the P equation in (2.19) and in the new Q equation, as given by (4.14) and (4.15). The iterative P Eq. (2.22) is unchanged, while, by Eq. (4.15) and by Lemma 8, the iterative Q equation (2.25) must be substituted by ∗ (k2 ) (k3 ) 1) Vm(k) = g(m2 , m) u(k (4.17) n1 ,m1 un2 ,m2 un3 ,m3 , k1 +k2 +k3 =k −n1 +n2 +n3 =m2 −m1 +m2 +m3 =m
with g(m2 , m) = (m2 − 4a2 )−1 , for m ∈ / M, and ∗ Dm,m Vm(k) = m ∈M+ (k)
k1 +k2 +k3 =k −n1 +n2 +n3 =(m )2 −m1 +m2 +m3 =m (k)
(k2 ) (k3 ) 1) u(k n1 ,m1 un2 ,m2 un3 ,m3
(4.18)
for m ∈ M+ , while Vm = −V−m for m ∈ M \ M+ . Let us consider first the case f (u, u) = |u|2 u. The tree expansion is as in Sect. 3, with the following differences. In item (5) now to each end-point a mode label (nV , mV ), with
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mV ∈ M and nV = m2V , and an end-point factor VV = σV amV , with σV = sgn mV are associated. Moreover each line carries two momentum labels (n , m ) and (n , m ), with m and m both in M or in its complement. If m , m ∈ M they are both either in M+ or in M \ M+ , and the propagator g is not diagonal any more, as it is given by g = Dm,m (see (4.15)), while if m , m ∈ / M then m = m and g = g(m2 , m), with g(m2 , m) given as after (4.17); like in the previous case one has n = m2 and n = (m )2 for such lines. Also for the lines of type w we set n = n and m = m , but one has n = m2 in such a case. The momentum (n , m ) is recursively defined as n V = (−1)s() n , m V = (−1)s() m , (4.19) ∈LV
∈LV
for V ∈ Vw3 (θ ) ∪ Vv3 (θ ), and n V = n ,
mV = δcV ,a m − δcV ,b m ,
(4.20)
for V ∈ Vw1 , if denotes the line entering V. The self-energy graphs T with momentum (n, m) associated to the line 2T are characterized by the relations n(T ) = n1 − n2 = 0 and m(T ) = m1 − m2 ∈ {0, 2m}. T T T T No other differences arise with respect to Sect. 3, so that the analysis can be carried out in the same way. For more general f one reasons as at the beginning of this Section. We do not describe in detail the obvious changes of notations. Of course the value of ε0 depends on the set M, and in particular it goes to zero when N → ∞ (as M diverges in such a case) and, for fixed N , when M → ∞. The conclusion is that infinitely many unperturbed solutions which are trigonometric polynomial with an arbitrary number of harmonics can be continued in the presence of nonlinearities. The case of polynomials of degree 1 (Theorem 1) is the one usually considered in the literature, while the case of polynomials of higher order (Theorem 2) is new. In the latter case the only request on the harmonics is that the corresponding wave numbers have to be close enough to each other and that the larger their number the larger are their values. References 1. Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. SIAM Studies in Applied Mathematics 4, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1981 2. Bambusi, D., Paleari, S.: Families of periodic solutions of resonant PDE’s. J. Nonlinear Sci. 11, no. 1, 69–87 (2001) 3. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schr¨odinger equations. Ann. of Math. (2) 148, no. 2, 363–439 (1998) 4. Bourgain, J.: Global solutions of nonlinear Schr¨odinger equations. American Mathematical Society Colloquium Publications 46, Providence, RI: Amer. Math. Soc., 1999 5. Corones, J.: Solitons as nonlinear magnons. Phys. Rev. B 16, no. 4, 1763–1764 (1977) 6. Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Comm. Pure Appl. Math. 46, 1409–1498 (1993) 7. Degasperis, A., Manakov, S.V., Santini, P.M.: Mixed problems for linear and soliton partial differential equations. Teoret. Mat. Fiz. 133(2), 184–201 (Russian) (2002), translation in Theoret. and Math. Phys. 133, no. 2, 1475–1489 (2002) 8. Fokas, A.S.: Inverse scattering transform on the half-line – the nonlinear analogue of the sine transform. In: Inverse problems: an interdisciplinary study (Montpellier, 1986), Adv. Electron. Electron Phys., Suppl. 19, London: Academic Press, 1987, pp. 409–442
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9. Geng, J.,You, J.: A KAM theorem for one-dimensional Schr¨odinger equation with periodic boundary conditions. J. Differ. Eqs. 209, no. 1, 1–56 (2005) 10. Gentile, G., Mastropietro, V.: Construction of periodic solutions of the nonlinear wave equation with Dirichlet boundary conditions by the Lindstedt series method. J. Math. Pures Appl. (9) 83, no. 8, 1019–1065 (2004) 11. Gentile, G., Mastropietro, V., Procesi, M.: Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions. Commun. Math. Phys. 256, no. 2, 437–490 (2005) 12. Grinevich, P.G., Santini, P.M.: The initial-boundary value problem on the interval for the nonlinear Schr¨odinger equation. The algebro-geometric approach. I. In: Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 212, Providence, RI: Amer. Math. Soc., 2004, pp. 157–178 13. Kuksin, S.B., P¨oschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schr¨odinger equation. Ann. of Math. (2) 143, no. 1, 149–179 (1996) 14. Ma, Y.C., Ablowitz, M.J: The periodic cubic Schr¨odinger equation. Stud. Appl. Math. 65, no. 2, 113–158 (1981) 15. Novikov, S.P.:A periodic problem for the Korteweg-deVries equation. I. Funkcional.Anal. i Priloˇzen. 8 no. 3, 54–66 (Russian) (1974) 16. Schneider, G.: Approximation of the Korteweg-de Vries equation by the nonlinear Schr¨odinger equation. J. Differential Equations 147, no. 2, 333–354 (1998) 17. Schneider, G., Wayne, C.E.: On the validity of 2D-surface water wave models. GAMM Mitt. Ges. Angew. Math. Mech. 25, no. 1–2, 127–151 (2002) 18. Tuszynski, J.A., Dixon, J.M.: Coherent structures in strongly interacting many-body systems. I. Derivation of dynamics. J. Phys. A: Math. Gen. 22, 4877–4894 (1989) 19. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Soviet Phys. J. Appl. Mech. Tech. Phys. 4, 190–194 (1968) 20. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional ˇ Eksper. ` self-modulation of waves in nonlinear media. Z. Teoret. Fiz. 61, no. 1, 118–134 (1971) (Russian), translation in Soviet Physics JETP 34, no. 1, 62–69 (1972) Communicated by G. Gallavotti
Commun. Math. Phys. 262, 555–564 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1437-z
Communications in
Mathematical Physics
Space-Time Localization of a Class of Geometric Criteria for Preventing Blow-up in the 3D NSE Zoran Gruji´c1 , Qi S. Zhang2 1 2
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA. E-mail: [email protected] Department of Mathematics, University of California, Riverside, CA 92521, USA. E-mail: [email protected]
Received: 28 February 2005 / Accepted: 6 April 2005 Published online: 19 October 2005 – © Springer-Verlag 2005
Abstract: A class of local (in the space-time) conditions on the vorticity directions implying local regularity of weak solutions to the 3D Navier-Stokes equations is established. In all the preceding results, the relevant geometric conditions, although being local in nature, have been assumed uniformly throughout the spatial regions of high vorticity magnitude, and uniformly in time. In addition, similar results are obtained assuming a less restrictive integral condition on the vorticity directions. 1. Introduction We aim to study local regularity of solutions to the 3D Navier-Stokes equations ut − u(x, t) + u · ∇u(x, t) + ∇p = 0, ∇ · u = 0, u(x, 0) = u0 (x)
(1.1)
for (x, t) ∈ R3 × (0, ∞), where is the standard Laplacian, a vector field u represents the velocity of the fluid, and a scalar field p the pressure. (The viscosity is normalized, ν = 1.) Since the seminal work of Leray [L], it is known that (1.1) has a global weak solution whenever u0 ∈ L2 (R3 ) is divergence-free. However, the questions of regularity (smoothness) and uniqueness of weak solutions are still open. In the years to follow, an array of conditions implying the regularity has been discovered. Perhaps the most classical ones are Foias-Prodi-Serrin conditions [P, S] on the space-time integrability of u. Some critical cases, as well as the analogous results in the weak Lebesgue spaces, have been treated in, e.g., [ESS, CP, K, KK]. There are also similar results expressed in terms of the vorticity ω = ∇ × u, and in particular Beale-Kato-Majda condition, ω(·)L∞ (R3 ) ∈ L1 (0, T ), originally derived for the 3D Euler equations. (Actually, the time-integrability of the BMO-norm suffices [KoT].)
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A more geometric way of approaching the problem was brought to light by Constantin in [Co]. The vorticity formulation of the 3D NSE reads ωt − ω + u · ∇ω = ω · ∇u. Notice that in the 2D-case, the right-hand side term ω · ∇u–the vortex stretching term–is identically zero. The other part of the nonlinearity is the advective part, i.e., a transport of the vorticity by the velocity. For a vector v, denote by v the unit vectorin the direction of v. The symmetric part of the velocity gradient is denoted by S, S = 21 ∇u + (∇u)T , and a key quantity α is defined by α = S ω· ω. A direct calculation shows that the vortex stretching term equals to Sω, and that Sω · ω = α|ω|2 . Hence, multiplying the vorticity equations by ω in L2 (R3 ) yields 1 d |ω|2 dx + |∇ω|2 dx = Sω · ω dx = α|ω|2 dx. 2 dt Since the boundedness of the enstrophy E, E = |ω|2 dx, is enough to ensure regularity, it is plain that α indeed holds a key to understanding the nonlinear effects in the 3D NSE. Constantin [Co] discovered an integral representation of α which revealed a geometric depletion of the nonlinearity, 3 dy α(x) = P .V . D y, ω(x + y), ω(x) |ω(x + y)| 3 , 4π |y| where the geometric factor D is proportional to the volume spanned by the unit vectors y, ω(x + y) and ω(x). More precisely, D(e1 , e2 , e3 ) = (e1 · e3 ) Det(e1 , e2 , e3 ) for any triplet of unit vectors {e1 , e2 , e3 }. A simple geometric observation shows |D y, ω(x + y), ω(x) | ≤ | sin ϕ ω(x + y), ω(x) | ≤ | ω(x + y) − ω(x)|; hence, local alignment of the vorticity directions will soften up the singularity. This is very interesting since both numerical simulations and laboratory experiments reveal that the regions of high vorticity magnitude exhibit quasi low-dimensional geometry, e.g., vortex sheets and vortex tubes, in which the local alignment of the vorticity directions is evident. The aforementioned ruminations were subsequently exploited by Constantin and C. Fefferman in [CF] where it was shown that Lipschitz regularity of sin ϕ in the regions of high vorticity magnitude, uniformly in time, suffices to control the evolution of the enstrophy. Following their approach, Beirao da Veiga and Berselli [BdVB] scaled the Lipschitz condition down to a 21 -H¨older condition. A more general class of conditions has been obtained in [GR]–essentially, an interpolation between a purely geometric 1 older condition and a purely analytical Beale-Kato-Majda condition. It states it is 2 -H¨ enough to assume that for some q ≥ 2, q 1 1 ω(·)Lq−1 ω(x + y), ω(x) | ≤ |y| q . q (R3 ) ∈ L (0, T ) and | sin ϕ
(1.2)
(As in the previous works, the second condition is assumed throughout the regions of high vorticity magnitude, uniformly in time.) In a recent work [RG], some new cancellation properties in the vortex stretching term based on the integral representation of α have been detected. This led to new geometric criteria for preventing finite time blow-up. In particular, a certain isotropy condition on
Local Regularity
557
the velocity field u has been shown to effectively control the growth of the vorticity magnitude. All of the aforementioned results have in common that a relevant geometric condition preventing finite time blow-up, although being essentially local in nature, is assumed uniformly in the regions of intense vorticity, and uniformly in time. Hence, the natural question arising is whether these results can be localized, both in the space and in the time variable. The localization is expected to hold from the point of view of fluid response–on the other hand, the velocity field is recovered from the vorticity field by a non-local formula and that springs a number of technical difficulties. Another, perhaps more exciting, motivation for the space-time localization is that the existing geometric scenarios for avoiding singularity formation are somewhat complementary–in particular, some are anisotropic and some are isotropic. Thus, the localization is the necessary zero step in an attempt to cover the space-time with different favorable geometric scenarios. In this paper, we show that a full space-time localization of a class of conditions (1.2) preventing the loss of regularity is indeed possible. A limit case q = 2 corresponds to the localization of the purely geometric 21 -H¨older condition. Moreover, we formulate a local integral condition on the vorticity directions guaranteeing local regularity –this condition turns out to be less restrictive than the pointwise 21 -H¨older one. In order to bridge some technical difficulties arising in the localization procedure, a technique developed in a recent work on the parabolic equations with divergence-free singular coefficients [Z] will be utilized. Let us introduce another piece of notation. For a space-time point (x0 , t0 ), and r0 > 0, denote by Qr0 (x0 , t0 ) (or Qr0 ) an open parabolic cylinder B(x0 , r0 ) × (t0 − r02 , t0 ). For clarity of the exposition, we assume that solutions are smooth in an open parabolic cylinder, and prove that the localized enstrophy remains bounded on the closed parabolic cylinder of half the size. An alternative way would be to prove a uniform bound on a sequence of approximate solutions. This approach was used in [CF]. In this case the assumptions need to be imposed on each approximate solution. Theorem 1.1. Let u be a Leray-Hopf solution of (1.1), and suppose ω0 = curlu0 ∈ L1 (R3 ). Given (x0 , t0 ) ∈ R3 × (0, ∞), assume that there exist r0 , d, c > 0, and q ≥ 2, such that (i) ω ∈ Lq, q/(q−1) (Q3r0 (x0 , t0 )), (ii) | sin ϕ( ω(x + y, t), ω(x, t))| ≤ c|y|1/q for (x, t) ∈ {|ω| ≥ d} ∩ Q2r0 (x0 , t0 ) and |y| ≤ r0 . Then, if u is smooth in the open cylinder Q2r0 (x0 , t0 ), the localized enstrophy remains bounded on Qr0 (x0 , t0 ), i.e., sup |ω|2 dx ≤ M < ∞. t∈(t0 −r0 2 ,t0 ) B(x0 ,r0 )
The next theorem is a generalizaton of Theorem 1.1 in case q = 2 where assumption (i) is redundant (a purely geometric case). Theorem 1.2. Let u be a Leray-Hopf solution of (1.1), and suppose ω0 = curlu0 ∈ L1 (R3 ). Given (x0 , t0 ) ∈ R3 × (0, ∞), assume that there exist r0 , d > 0 such that the function 1 λ(y) ≡ sup (1.3) y · ( ω(x + y, t) × ω(x, t)) 3 |y| (x,t)∈{|ω|≥d}∩Q2r (x0 ,t0 ) 0
558
Z. Gruji´c, Q.S. Zhang 6/5
is in Lw (B(0, 3r0 )), the weak L6/5 space. Then, if u is smooth in the open cylinder Q2r0 (x0 , t0 ), the localized enstrophy remains bounded on Qr0 (x0 , t0 ). In other words, if |ω|2 dx = ∞, sup t∈(t0 −r0 2 ,t0 ) B(x0 ,r0 )
then λL6/5 (B(0,3r w
0 ))
= ∞.
Remark 1. If conditions (i) and (ii) with q = 2 in Theorem 1.1 hold, then the function 6/5 λ in (1.3) is bounded above by |y|c5/2 . Hence, it is in Lw (B(0, 3r0 )) and Theorem 1.2 applies. The rest of the paper is organized as follows. We start the proofs of Theorems 1.1 and 1.2 in Sect. 2, and finish in Sect. 3. The proof of Theorem 1.2 is almost the same as that of Theorem 1.1. We present the proof of the latter in detail, and indicate a minor change needed in the proof of Theorem 1.2 in Sect. 3. 2. Local Control of Advection Proof of Theorem 1.1. We will work on the vorticity equations ωt − ω + u · ∇ω = ω · ∇u,
(2.1)
and show that local enstrophy remains bounded. Comparing with the previous works, where the advective term u · ∇ω is simply integrated away, we have to deal with both the advective term and the vortex stretching term ω · ∇u. Let (x0 , t0 ) be a point in the space-time, and let Qr be a parabolic cylinder B(x0 , r) × (t0 − r 2 , t0 ) for some r ≤ r0 . Choose ψ = φ(y)η(s) to be a refined cut-off function satisfying supp φ ⊂ B(x0 , 2r);
φ(y) = 1,
y ∈ B(x0 , r);
|∇φ| C ≤ , δ φ r
0 ≤ φ ≤ 1;
here, δ ∈ (0, 1) will be chosen later (such a function is easily constructed by a scaling argument), and supp η ⊂ (t0 − (2r)2 , t0 ];
η(s) = 1,
|η | ≤ 2/r 2 ;
s ∈ [t0 − r 2 , t0 ];
0 ≤ η ≤ 1.
Denote by Qt2r the parabolic subcylinder of Q2r , Qt2r = B(x0 , 2r) × (t0 − (2r)2 , t) for t in (t0 − (2r)2 , t0 ). Multiplying (2.1) by ωψ 2 and integrating over Qt2r (u is assumed to be smooth on Q2r0 ), one obtains (ω − u · ∇ω + ω · ∇u − ∂s ω) ωψ 2 dyds = 0. Qt2r
Integration by parts yields ∇(ωψ 2 )∇ωdyds = − Qt2r
Qt2r
u · ∇ω(ωψ 2 )dyds +
−
Qt2r
(∂s ω)ωψ 2 dyds.
Qt2r
ω · ∇u ωψ 2 dyds (2.2)
Local Regularity
559
By a direct calculation, ∇(ωψ 2 )∇ωdyds = Qt2r
= =
∇[(ωψ)ψ]∇ωdyds
Qt2r
[∇(ωψ)(∇(ωψ) − (∇ψ)ω) + ωψ∇ψ∇ω]dyds
Qt2r
[|∇(ωψ)|2 − |∇ψ|2 |ω|2 ]dyds.
Qt2r
Substituting this into (2.2), 2 |∇(ωψ)| dyds = Qt2r
u · ∇ω(ωψ )dyds + 2
Qt2r
−
(∂s ω)ωψ dyds +
Qt2r
|∇ψ|2 |ω|2 dyds.
2
Qt2r
ω · ∇u ωψ 2 dyds (2.3)
Q2r
Next, notice that 1 2 (∂s ω)ωψ dyds = (∂s |ω|2 )ψ 2 dyds 2 Qt2r Qt2r 1 |ω|2 φ 2 η∂s ηdyds + |ω|2 (y, t)φ 2 (y)dy. =− 2 B(x0 ,2r) Qt2r Combining this with (2.3), the following bound appears: 1 |∇(ωψ)|2 dyds + |ω|2 (y, t)φ 2 (y)dy 2 B(x0 ,2r) Qt2r ≤ (|∇ψ|2 + |η||∂s η|) |ω|2 dyds Q2r 2 2 + u · ∇ω (ωψ )dyds + ω · ∇u (ωψ )dyds Qt2r
Qt2r
≡ Te + Ta + Ts . Notice that Qt2r
u · (∇ω)(ωψ 2 )dyds
=
1 2
=− =− hence,
Qt2r
1 2
Qt2r
Qt2r
1 div(u · ψ 2 )|ω|2 dyds 2 Qt2r 1 divu |ψω|2 dyds − u · ∇(ψ 2 )|ω|2 dyds 2 Qt2r
u · ψ 2 ∇|ω|2 dyds = −
u · (∇ψ)ψ|ω|2 dyds;
Ta ≤
|u||∇ψ||ψ||ω| dyds ≤ 2
Qt2r
Q2r
|u||∇ψ||ψ||ω|2 dyds ≡ Ta .
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Recall now the pointwise relation ω · ∇u · ω = α|ω|2 , where
3 dy α(x, t) = D y, ω(x + y), ω(x) |ω(x + y, t)| 3 . P .V . 3 4π |y| R The vortex stretching term Ts is then estimated via the splitting into the regions of low and high vorticity, Ts ≤ |ω|2 |∇u|ψ 2 dyds + |α||ω|2 ψ 2 dyds Qt2r ∩{|ω|
≤
Q2r ∩{|ω|
|ω|2 |∇u|ψ 2 dyds +
Collecting the above bounds, (2.4) yields |∇(ωψ)|2 dyds + sup
Q2r ∩{|ω|≥d}
1 2 2 t∈(t0 −(2r) ,t0 )
Q2r
≤
Qt2r ∩{|ω|≥d}
Te + Ta
+ (Ts1
|α||ω|2 ψ 2 dyds ≡ Ts1 + Ts2 .
|ω|2 (y, t)φ 2 (y)dy B(x0 ,2r)
(2.4)
+ Ts2 ).
Notice that the first term, Te , is already in good shape. The vortex stretching terms Ts1 and Ts2 will be estimated in the next section. The following argument is the key step in localizing advection. Notice that, for δ ∈ (0, 1), a ∈ (0, 2) and m ∈ (1, 2], |∇ψ| Ta = |u| ψ 1+δ |ω|2−a δ |ω|a dyds ψ Q2r 1/m m (1+δ)m (2−a)m ≤ |u| ψ |ω| dyds Q2r
×
( Q2r
|∇ψ| m/(m−1) am/(m−1) ) |ω| dyds ψδ
(m−1)/m .
Choose a and δ so that (2 − a)m = 2,
(1 + δ)m = 2.
Then, am/(m − 1) = a
2 2 /( − 1) = 2, 2−a 2−a
δ = (2/m) − 1 < 1,
which paired with the assumptions on the cut-off function ψ yields 1/m (m−1)/m c 2 Ta ≤ |u|m |ψω|2 dyds |ω| dyds . m/(m−1) Q2r Q2r r This implies that, for any > 0,
−m/(m−1) m m 2 |u| |ψω| dyds + C m/(m−1) |ω|2 dyds. Ta ≤ r Q2r Q2r
(2.5)
Local Regularity
561
Since u is a Leray-Hopf solution, |u|4/3 |ψω|2 dyds ≤ k Q2r
indeed,
|∇(ψω)|2 dyds;
Q2r
t0
|u|4/3 h2 dxdt
t0 −4r 2
≤ ≤
B2r t0
2/3 |u|2 dx
t0 −4r 2
B2r
B2r
2/3
|u|2 (x, t)dx
sup t∈[t0 −4r 2 ,t0 ]
≤C
B2r
dt
t0
t0 −4r 2
2/3 |u|2 (x, t)dx
sup t∈[t0 −4r 2 ,t0 ]
1/3 h6 dx
B2r
t0 t0
−4r 2
1/3 h6 dx
dt
B2r
|∇h|2 dxdt B2r
for any h compactly supported in B2r . (The last step is by the Sobolev imbedding.) Substituting the above into (2.5) (m = 4/3), we can find k1 < 1/2 and k2 > 0 such that 1 |∇(ψω)|2 dyds + k2 4 |ω|2 dyds. (2.6) Ta ≤ k1 r Q2r Q2r This establishes a local control on the advective term. In the next section, we provide a local control on the vortex stretching term. 3. Local Control of Vortex Stretching Recall that the vortex stretching term u · ∇ω · ψ 2 ωdyds
(3.1)
Q2r
was estimated via splitting the region of integration by |ω|2 |∇u|ψ 2 dyds + Ts1 + Ts2 = Q2r ∩{|ω|
where α(x, t) =
Q2r ∩{|ω|≥d}
|α||ω|2 ψ 2 dyds, (3.2)
dy 3 D y, ω(x + y), ω(x) |ω(x + y, t)| 3 . P .V . 4π |y| R3
Clearly, Ts1 is bounded above since 1 Ts ≤ cd
(3.3)
|∇u|2 ψ 2 dxdt;
Q2r
hence, we focus on Ts2 =
Q2r ∩{|ω(x,t)|≥d}
α(x, t)|ωψ|2 (x, t)dxdt.
(3.4)
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Z. Gruji´c, Q.S. Zhang
Decompose α as α = α1 + α2
dy 3 P .V . D y, ω(x + y), ω(x) |ω(x + y, t)| 3 . (3.5) ≡ + 4π |y| |y|≤r |y|≥r First, observe that |α2 (x, t)| ≤
3 ω(·, t)L1 (R3 ) , 4πr 3
(3.6)
and this is a priori bounded (cf. [Co2]). Next, we estimate α1 . By our assumption, ω ∈ Lq, q/(q−1) (Q3r ) and sin ϕ( ω(x + y, t), ω(x, t)) ≤ c|y|1/q whenever (x, t) ∈ Q2r ∩ {|ω| ≥ d} and |y| ≤ r. Hence, |ω(x + y, t)| dy. |α1 (x, t)| ≤ c 3−(1/q) |y|≤r |y| After a change of variables, we obtain, for all x ∈ R3 , |ω(y, t)| dy χB(x0 ,2r) (x)|α1 (x, t)| ≤ c |x − y|3−(1/q) |x −y|≤3r 0 |ω(y, t)| χB(x0 ,3r) (y) =c dy. 3 |x − y|3−(1/q) R Since 1 1 1 + =1+ , q 3q/(3q − 1) 3q/2 a version of the weak Young’s inequality implies
2/3q
|α1 (x, t)|
3q/2
dx
B(x0 ,2r)
1/q 1 3q−1 q ≤ c0 |ω(x, t)|q dx , | · | L3w B(x0 ,3r)
where c0 is a constant independent of r, and L3w is the weak L3 -norm. (This inequality can be found, e.g., on p. 107 of [LL].) Taking the q/(q −1)-power in the above inequality, and integrating in time, α1 L3q/2,q/(q−1) (Q2r ) ≤ cωLq,q/(q−1) (Q3r ) .
(3.7)
Using (3.7) and H¨older’s inequality, we arrive at |α1 (x, t)| |ωψ|2 (x, t)dxdt Q2r ∩{|w(x,t)|≥d}
≤ α1 L3q/2,q/(q−1) (Q2r ) |ωψ|2 L3q/(3q−2), q (Q2r ) ≤ cωLq,q/(q−1) (Q3r ) |ωψ|2 L3q/(3q−2), q (Q2r ) = cωLq,q/(q−1) (Q3r ) |ωψ|2L6q/(3q−2), 2q (Q ) . 2r
(3.8)
Local Regularity
563
Write a = 6q/(3q − 2) and b = 2q, and notice that 3q − 2 2 3 3 2 + = + = . a b 2q 2q 2 A version of the Gagliardo-Nirenberg inequality then yields ωψ2L6q/(3q−2), 2q (Q
2r )
≤C
|∇(ωψ)|2 (x, t)dxdt Q2r
+
|ωψ| (x, t)dx . 2
sup t∈(t0 −(2r)2 ,t0 )
B(x0 ,2r)
Inserting this into (3.8), |α1 (x, t)| |ωψ|2 (x, t)dxdt ≤ CωLq,q/(q−1) (Q3r ) |∇(ωψ)|2 (x, t)dxdt Q2r
Q2r ∩{|ω(x,t)|≥d}
+
|ωψ|2 (x, t)dx .
sup t∈(t0 −(2r)2 ,t0 )
(3.9)
B(x0 ,2r)
Collecting the estimates on Ts1 and Ts2 implies a desired bound on the vortex stretching term, Ts1 + Ts2 ≤ C1 ωLq,q/(q−1) (Q3r ) × |∇(ωψ)|2 (x, t)dxdt + Q2r
+C(r, ∇uL2 (Q2r ) ).
|ωψ|2 (x, t)dx
sup t∈(t0 −(2r)2 ,t0 )
B(x0 ,2r)
(3.10)
Here, C1 is independent of r. Now, recall that ω ∈ Lq,q/(q−1) (Q3r0 ). Therefore, there exists r ∗ , 0 < r ∗ ≤ r0 , such that for all r ≤ r ∗ , C1 ωLq,q/(q−1) (Q3r ) < 1/2. Substituting (3.10) and (2.6) into (2.4)’, we deduce 1 2 |∇(ωψ)| dyds + |(ωψ)|2 dyds ≤ c(r) < ∞ (3.11) sup 2 t∈(t0 −(2r)2 ,t0 ) B(x0 ,2r) Q2r for any r ≤ r ∗ . This concludes the proof in the case r ∗ = r0 . If r ∗ < r0 , the conclusion is drawn by shifting the vertex of the cylinder Q2r . Proof of Theorem 1.2. The proof of Theorem 1.2 is almost identical to that of Theorem 1.1. The only modification occurs after (3.6) and before (3.7). By our assumption, whenever (x, t) ∈ Q2r ∩ {|ω| ≥ d} and |y| ≤ r, we have, for the same (x, t) and y, K(y)|ω(x + y, t)|dy, |α1 (x, t)| ≤ c |y|≤r
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where K ∈ Lw . Hence,
χB(x0 ,2r) (x)|α(x, t)| ≤ c =c
|x0 −y|≤3r
R3
K(x − y)|ω(y, t)|dy
K(x − y)|ω(y, t)|χB(x0 ,3r) (y)dy,
and an application of the weak Young’s inequality implies
1/3 5/6 |α1 (x, t)|3 dx ≤ c0 K(·) 6/5 B(x0 ,2r)
Lw
1/2 |ω(x, t)|2 dx
,
B(x0 ,3r)
where c0 is a constant independent of r. The rest of the proof is identical to that of Theorem 1.1 past (3.7), with q = 2. Acknowledgement. Z.G. thanks Department of Mathematics at the University of Chicago for hospitality, and P. Constantin for inspiring discussions.
References [BdVB] [CP] [Co] [Co2] [CF] [GR] [ESS] [K] [KK] [KoT] [L] [LL] [P] [RG] [S] [Z]
Beirao da Veiga, H., Berselli, L.C.: On the regularizing effect of the vorticity direction in incompressible viscous flows. Differ. Integral Eqs. 15 no. 3, 345–356 (2002) Chen, Z-M; Price, W.G.: Blow-up rate estimates for weak solutions of the Navier-Stokes equations. R. Soc. Proc. Ser. A Math. Phys. Eng. Sci. 457, 2625–2642 (2001) Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36 no. 1, 73–98 (1994) Constantin, P.: Navier-Stokes equations and area of interfaces. Commun. Math. Phys. 129 no. 2, 241–266 (1990) Constantin, P., Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42, 775–789 (1993) Gruji´c, Z., Ruzmaikina, A.: Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE. Indiana Univ. Math. J. 53, 1073–1080 (2004) Escauriaza, L., Seregin, G.A., Sverak, V.: L3,∞ -solutions of Navier-Stokes equations and backward uniqueness. Usp. Mat. Nauk 58, 3–44 (2003) Kozono, H.: Removable singularities of weak solutions to the Navier-Stokes equations. Commun. PDE 23, 949–966 (1998) Kim, H., Kozono, H.: Interior regularity criteria in weak spaces for the Navier-Stokes equations. Manus. Math. 115, 85–100 (2004) Kozono, H., Taniuchi, Y.: Bilinear estimates in BMO and the Navier-Stokes equations. Math Z. 235, 173–194 (2000) Leray, J.: Essai sur le mouvements d’ un liquide visqueux emplissant l’ espace. Acta Math. 63, 193–248 (1934) Lieb, E.H., Loss, M.: Analysis. Second edition. Graduate Studies in Mathematics, 14. Providence, RI: Amer. Math. Soc., 2001 Prodi, G.: Un teorema de unicita per le equazioni di Navier-Stokes. Annali di Mat. 48, 173–182 (1959) Ruzmaikina, A., Gruji´c, Z.: On depletion of the vortex-stretching term in the 3D Navier-Stokes equations. Commun. Math. Phys. 247, 601–611 (2004) Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962) Zhang, Q.S.: A strong regularity result for parabolic equations. Commun. Math. Phys. 244, 245–260 (2004)
Communicated by P. Constantin
Commun. Math. Phys. 262, 565–594 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1440-4
Communications in
Mathematical Physics
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis Vincent Rivasseau1 , Fabien Vignes-Tourneret1 , Raimar Wulkenhaar2 1
Laboratoire de Physique Th´eorique, Bˆat. 210, Universit´e Paris XI, 91405 Orsay Cedex, France. E-mail: [email protected]; [email protected] 2 Max-Planck-Institut f¨ ur Mathematik in den Naturwissenschaften, Inselstraße 22–26, 04103 Leipzig, Germany. E-mail: [email protected] Received: 2 March 2005 / Accepted: 21 March 2005 Published online: 29 November 2005 – © Springer-Verlag 2005
Abstract: In this paper we give a much more efficient proof that the real Euclidean φ 4 -model on the four-dimensional Moyal plane is renormalisable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalisation proof based on renormalisation group equations for non-local matrix models. On the other hand, our bounds permit a powerful multi-scale analysis of the resulting ribbon graphs. Here, the dual graphs play a particular rˆole because the angular momentum conservation is conveniently represented in the dual picture. Choosing a spanning tree in the dual graph according to the scale attribution, we prove that the summation over the loop angular momenta can be performed at no cost so that the power-counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph. 1. Introduction Field theories on noncommutative spaces became very popular after the discovery that they arise in compactification of M-theory [15] and in limiting cases of string theory [1, 2]. Although from string theory’s point of view there is no reason that the limit is a well-defined quantum field theory, there has been an enormous activity aiming at renormalisation proofs for noncommutative quantum field theories. Most of the attempts focused at the Moyal plane with the associative and noncommutative product (a b)(x) =
d 4k (2π)4
d 4 y a(x + 21 θ·k) b(x+y) eik·y .
(1.1)
It turned out that the noncommutative analogs of typical field theoretical (in particular four-dimensional) models on the Moyal plane are not renormalisable due to the UV/IRmixing problem [3]. The construction of dangerous non-planar graphs was made precise in [4] where the problem was traced back to divergences in some of the Hepp sectors
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which correspond to disconnected ribbon subgraphs wrapping the same handle of a Riemann surface. Recently, the renormalisation of the noncommutative φ44 -model was achieved [5] within a Wilson-Polchinski renormalisation scheme [6, 7] adapted to non-local matrix models [8]. The renormalisable model is defined by the action functional 1 2 1 λ (x˜µ φ) (x˜ µ φ) + µ20 φ φ + φ φ φ φ (x), S[φ] = d 4 x ∂µ φ ∂ µ φ + 2 2 2 4! (1.2) where x˜µ = 2(θ −1 )µν x ν and the Euclidean metric is used. At first sight, the appearance of the translation invariance breaking harmonic oscillator potential for the φ 4 -action (1.2) might appear strange. However, the renormalisation proof shows that there is a marginal interaction which corresponds to that term and as such requires its inclusion in the initial action. Moreover, thanks to the oscillator potential, the action (1.2) transforms covariantly under the Langmann-Szabo duality [9] which exchanges position space and momentum space. We review the main ideas of the renormalisation proof, in particular the analysis of ribbon graphs, in Sect. 2. However, it must be underlined that the proof given in [5] relies on a numerical determination of the asymptotic scaling dimensions of the propagator. Our paper fills this gap by computing rigorous bounds on the propagator, at least for large enough . This will be done in Sect. 3. On the other hand, our bounds permit another renormalisation strategy which turns out to be much more efficient. See Sect. 4. The strategy is inspired by constructive methods [10]. The key is a scale decomposition of the propagator and an estimation procedure of the ribbon graphs which takes into account the scale attribution. The proof is carried out for the duals of the ribbon graphs, because the set of independent variables is particularly transparent in dual graphs. The methods developed in this paper will be crucial to write a constructive version of [8, 5]. Actually the main obstacle to the construction of the usual φ 4 model is the nonasymptotic freedom of the theory. For the four-dimensional Moyal plane, the parameter controls the UV/IR mixing. When it reaches 1, the entanglement is maximum, and the β function vanishes [11]. In this view, the -region close to 1, for which we prove analytical estimates, is particularly important.
2. Main Ideas of the Previous Renormalisation Proof In order to make this paper self-contained, we review the main ideas of the renormalisation proof given in [5] for the quantum field theory associated with the action (1.2). In order to avoid the oscillating phase factors of the -product in momentum space, the first step is to pass to the matrix base of the Moyal plane, where the action (1.2) becomes1 S[φ] = 4π 2 θ1 θ2
m,n,k,l∈N2
1 2
m,n;k,l φmn φkl +
λ φmn φnk φkl φlm . 4!
(2.1)
1 Note that the symbols and G used in this paper for the (generalised) Laplacian and the Green’s function are transposed in [8, 5].
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
567
As usual, we define the quantum field theory by the partition function, which is expanded into Feynman graphs. As the fields are described by matrices φmn , the resulting Feynman graphs are ribbon graphs built of propagators and vertices,
= Gm,n;k,l ,
= δn1 m2 δn2 m3 δn3 m4 δn4 m1 . (2.2)
The propagator Gm,n;k,l is the inverse of the kinetic matrix m,n;k,l in (2.1). We recall the explicit formula in (3.1) and (3.2). Due to the SO(2) × SO(2)-symmetry of the action, Gm,n;k,l = 0 only if m + k = n + l. Matrix indices which are not determined by this index conservation or as external indices of the graph are summation indices. The corresponding index summation is possibly divergent and requires a regularisation. In [8, 5] the regularisation consists in a smooth cut-off of the propagator indices as a function of a renormalisation scale , i ∂ G m1 n1 k 1 l 1 , (2.3) χ Q m1 n1 k 1 l 1 ( ) = 2 , , ∂ θ ; 2 2; 2 2 1 2 1 2 2 2 2 2 m
n
k
i∈m ,m ,...,l ,l
l
m n
k l
where χ (x) is smooth with χ (x) = 1 for x ≤ 1 and χ (x) = 0 for x ≥ 2. This implies Qm,n;k,l ( ) = 0 only if max(m1 , m2 , . . . , l 1 , l 2 ) ∈ [θ 2 , 2θ 2 ]. The graph is then realised by the differentiated cut-off propagators Q( i ) which regulate the index i summations. At the end, the nested integral over d i is performed within an interval characterised by mixed boundary conditions [7]. Actually, the graphs are built recursively by adding a new propagator. This allows an inductive proof of the power-counting behaviour. On the other hand, one has to carefully discuss the location of the valence of the graph where one attaches a leg of the additional propagator. This discussion alone extends over 20 pages in [8]. It is time for an example. We consider the (planar) one-loop four-point graph
=
R
d 2 2
0 2
d 1 Qm,p+m;p+l,l ( 2 ) Qn,p+m;p+l,k ( 1 ) 1 p
+{ 1 ↔ 2 } + Arm;ls;sk;nr ( R ) .
(2.4)
The choice of the boundary conditions is a preliminary one which ensures convergent integrals at the expense of infinitely many initial data Arm;ls;sk;nr ( R ). This will be corrected later in (2.11). One of the bounds we prove in this paper can be put in the following form: 1 1 2 1 2 1 2 1 2 G m1 n1 k 1 l 1 ≤ K dα e−cα(m +m +n +n +k +k +l +l ) . (2.5) , ; , m2 n2 k 2 l 2
0
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V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar
From the remarks made after (2.3) on the range of the maximal index we conclude
Q 1 1 1 1 ( ) ≤ 32K max χ (x) 1 dα e−cθ 2 α ≤ 32K maxx2χ (x) . (2.6) 0 m ,n k ,l cθ x 2 2; 2 2 m
n
k
l
We thus estimate the summation over p in (2.4) by the maximum of the propagators Q over p and a volume factor (2θ 22 )2 from the support of the cut-off function. This shows that the integral (2.4) is estimated by a constant times ln R . The scaling of (2.6) and the volume of the support of (2.3) with respect to any index seem to suggest that N -point graphs have, as in commutative φ44 -theory, a power-counting degree 4 − N. However, this conclusion is too early: There is a problem in the presence of completely inner vertices, which require additional index summations. The graph
(2.7)
entails four independent summation indices p1 , p2 , p3 and q, whereas for the powercounting degree 4 − N we should only have three of them. It requires a more careful analysis of the scaling behaviour of the propagator to show that the q-summation can actually be performed at no cost, i.e. without a volume factor. The reason is that the propagators show some sort of quasi-locality which implies that the contribution of a propagator Gm,n;k,l to a graph is strongly suppressed if m − l is large. Thus, taking for given m the entire sum over l does not change the power-counting behaviour, K max max Q m1 n1 k 1 l 1 ( ) ≤ θ (2.8) 2 . mi i i , , n ,k 2 2; 2 2 l 1 ,l 2 m
n
k
l
The two bounds (2.6) and (2.8) together ensure the expected power-counting behaviour for all planar ribbon graphs. But (2.8) does even more: it ensures the irrelevance of all non-planar graphs. For instance, in the non-planar graphs
(2.9) the summation over q and q, r, respectively, is controlled by (2.8), i.e. the quasi-locality of the propagator, so that the graphs in (2.9) can be estimated without any volume factor.
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
569
We recall from [8] that the non-planarity of ribbon graphs is classified by the number B of boundary components and the genus g = 1 − 21 (F − I + V ) of the Riemann surface on which the graph is drawn. Here, V and I are the number of vertices and edges (inner double lines) of the graph. To determine the number F of faces we close the external legs, that is, we connect the outgoing arrow labelled mi of an external leg directly with its incoming arrow ni . Then, F is the number of closed single lines and B the number of those closed lines which carry external legs. Then, according to [8, 5], the power-counting degree of a N-leg ribbon graph in four dimensions is ω = (4 − N ) − 4(2g + B − 1) .
(2.10)
The left graph in (2.9) has topology B = 2, g = 0 and the right graph B = 1, g = 1. As a result, there remain only the planar two- and four-leg graphs which can be relevant and marginal. The quasi-locality of the propagator improves the situation in selecting only • the planar four-leg graphs with constant index along the trajectory as marginal, • the planar two-leg graphs with constant index along the trajectory as relevant, • the planar two-leg graphs with an accumulated index jump of 2 along the trajectory as marginal. We refer to [5] for details. The trajectories are the open single lines of the graph (before the closure which identifies the faces). This leaves still an infinite number of divergent graphs. However, there is a discrete Taylor expansion about vanishing external indices which decomposes these divergent graphs into four relevant and marginal base functions and an irrelevant remainder. For instance, the decomposition for the marginal case m = l and n = k of the graph (2.4) reads
d 2 0 d 1 = 2 2 1 × Qm,p;p,m ( 2 ) Qn,p;p,n ( 1 ) − Q0,p;p,0 ( 2 ) Q0,p;p,0 ( 1 ) 0
+
p
R
d 2 2
0 2
d 1 Q0,p;p,0 ( 2 ) Q0,p;p,0 ( 1 ) 1 p
+ { 1 ↔ 2 } + A00;00;00;00;00 ( R ) .
(2.11)
Thus, this definition necessitates a single initial value A00;00;00;00;00 ( R ) which represents the normalisation condition for the coupling constant.
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V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar
Of particular importance are the marginal two-leg graphs with an accumulated index jump by 2, such as
(2.12) The corresponding initial value represents the normalisation condition for the frequency parameter in the initial action (1.2). Therefore, the harmonic oscillator potential must be present from the beginning in order to obtain a renormalisable model.
3. Bounds for the Propagator 3.1. Propagator in the matrix base and cut-offs. The propagator of the noncommutative φ 4 -model in the matrix base of the D-dimensional Moyal plane is given by2 a positive sum [5], analogous to the heat-kernel or parametric α-space representation
∞ 2 2 1 = 0 dα e−α(p +m ) of the ordinary commutative propagator: p2 +m2
Gm,m+h;l+h,l (α) Gm,m+h;l+h,l
θ = 8 √
µ2 θ
1
dα 0
1−α 1 + Cα
= ×
D
D 0 2 (1 − α) 8 +( 4 −1)
(1 + Cα)
D 2
A(m, l, h, u)
u=max(0,−h)
where A(m, l, h, u) =
m m+h l l+h m−u m−u l−u l−u
(3.1)
s=1
m+l+h
min(m,l)
(α)
Gms ,ms +hs ;l s +hs ,l s ,
√
Cα(1 + ) 1 − α (1 − )
m+l−2u , (3.2)
and C is a function of , namely
2 C() = (1−) 4 . ms , l s , hs , us , one
Indices such as m, l, h and u have D2 non-negative components for each symplectic pair of RD . However, due to (3.1) it is enough to prove estimations for a single component. We define the norm of an index by m = D/2 s s s s=1 m . A relation m > n means m > n for all s. We know that cut-offs in the parametric representation for commutative theories are specially convenient both for perturbative and constructive renormalisation. In the same spirit we will divide the integral (3.1) into slices. First we divide it into two different regions, with M > 1: 2 Our representation (3.1) and (3.2) corresponds to (A.17) in [5] with z = 1 − α. The often used index parameter α in [5] is denoted by h.
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
571
• M −1 ≤ α ≤ 1, where we expect an exponential decay in m + l + h of order O(1), • 0 ≤ α ≤ M −1 . This is the UV/IR region which is further sliced according to a geometric progression. For each slice we expect a scaled exponential decay. Then, the decomposition
1
dα =
0
∞ i=1
M −i+1 M −i
(3.3)
dα
leads to the following propagator for the i th slice: Gim,m+h,l+h,l
θ = 8
µ2 θ
M −i+1
M −i
dα
D
D 0 2 (1 − α) 8 +( 4 −1)
(1 + Cα)
D 2
(α)
Gms ,ms +hs ;l s +hs ,l s . (3.4)
s=1
The first slice i = 1 is treated separately. Remark that the factor A in (3.2) is the only one which prevents us from explicitly performing the u-sum. All the bounds in this paper by applying to the bino are obtained n nq mial coefficients in A the simple overestimate q ≤ q! . Of course, this bound is sharp only
n. In the regime n − q n one should rather use the symmetric bound for qn−q n n q ≤ (n−q)! . For α = 0 we see from (3.2) that the propagator vanishes unless u = l = m. This suggests to bound A by √ m−u √ l−u m(h + m) l(h + l) (m + h/2)m−u (l + h/2)l−u ≤ . (3.5) A(m, l, h, u) ≤ (m − u)!(l − u)! (m − u)!(l − u)! Hence, for α ≤ M −1 , √ m+l+h 1−α (α) Gm,m+h;l+h,l ≤ 1 + Cα √ √ m−u l−u α(1−2 ) m(m+h) α(1 − 2 ) l(l+h) √ √ min(m,l) 4 1−α 4 1−α × (m − u)! (l − u)! u=max(0,−h)
≤ e−α(C+1/2)(m+l+h)
min(m,l) u=max(0,−h)
where
X m−u Y l−u , (m − u)! (l − u)!
√ C(1 + )α(1 + α) m(m + h) X= = αD(α) m(m + h) , 1− √ C(1 + )α(1 + α) l(l + h) Y = = αD(α) l(l + h) , 1− 2
(3.6)
(3.7)
a second order Taylor with D(α) = 1− 4 (1 + α). For the inequality (3.6) we performed √ expansion in α and assumed ≥ 1/2 and M ≥ 21 (1 + 5).
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On the other hand, we observe from (3.2) that for α = 1 the propagator vanishes unless u = h = 0. In this situation the bound (3.5) is not suitable anymore because m − u and l − u are of order O(m) and O(l), respectively. Instead, we can use √ u√ h+u ml (m + h)(l + h) ((m + l)/2)u ((m + l + 2h)/2)h+u A(m, l, h, u) ≤ ≤ . u!(h + u)! u!(h + u)! (3.8) Inserting (3.8) into (3.2) we obtain (α)
Gm,m+h;l+h,l ≤
×
m+l+h α(1 − 2 ) 4 + (1 − )2 α u+h √ √ u √ √ 4 1−α ml 4 1−α (m+h)(l+h) min(m,l) α(1−2 ) α(1−2 ) u!
u=max(0,−h)
(u+h)!
.
(3.9) The further procedure will be to use the estimation min(m,l) u=0
X m−u Y l−u ≤ eX+Y . (m − u)! (l − u)!
(3.10)
3.2. Main scaled bounds. The first result is to prove that the propagator shows a scaled exponential decay in any index. This is expressed by Proposition 1. For M large enough there exists a constant K such that for ∈ [0.5, 1], we have the uniform bound Gim,m+h;l+h,l ≤ KM −i e− 3 M
−i m+l+h
(3.11)
.
Proof. We give here the proof for i ≥ 2.√The case of the first slice is √treated in Lemma 7 and Lemma 8 in the appendix3 . Using m(m + h) ≤ m + h/2, l(l + h) ≤ l + h/2 and (3.10), the bound (3.6) becomes Gm,m+h;l+h,l ≤ e−α(C+1/2−D(α))(m+l+h) . (α)
(3.12) 2
Then for α small enough (that is M large enough), C + 21 − D(α) = /2 − 1− 4 α ≥ We can now estimate (3.4) by Gim,m+h;l+h,l ≤
θ 8
M −i+1
M −i
dα e− 3 m+l+hα .
Then, the proposition follows from (3.13) with K =
θ 8 (M
− 1).
3.
(3.13)
We prove in Lemma 7 and Lemma 8 an exponential decay with km + l + h which is possibly −1 smaller than 3 M . We can ignore this discrepancy, because what counts in the renormalisation proof is the sum over matrix indices such as m, see (4.7). Then, the difference is a simple factor which we absorb into K. 3
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
573
Proposition 2. For M large enough there exists two constants4 K and K1 such that for all ∈ [0.5, 1] we have the uniform bound Gim,m+h;l+h,l
D
≤ KM −i e
−i − 4 M m+l+h
2
min 1,
K1
min(ms , l s , ms
+ hs , l s
+ hs )
|ms −l s | 2
Mi
s=1
.
(3.14) Proof. Of course, this bound improves (3.11) only when an index component is smaller than M i /K1 . As K1 ≥ M, there is nothing to prove for the first slice i = 1. Suppose l ≤ m ≤ m + h and δ = m − l. Instead of (3.10) we use the improved estimation l l X m−u Y l−u X m−l+v Y v = (m − u)! (l − u)! (m − l + v)! v! u=0
v=0
l Xm−l 1 2 Xm−l X+Y ≤ (XY )v ≤ . e (m − l)! v! (m − l)!
(3.15)
v=0
Then, the propagator (3.6) takes the form (α) Gm,m+h;l+h,l
≤e
−α(C+1/2−D(α))(m+l+h)
δ √ αD(α) m(m + h) . δ!
(3.16)
By the Stirling formula, e δ 1 ≤ δ! δ ⇒
(α) Gm,m+h;l+h,l
≤e
For x ≥ 0 we estimate
(3.17)
− 3 α(m+l+h)
xδ δ!
6e(D(α))2 αm δ
δ/2
δ
6 α(m + h)
δ!
.
(3.18)
≤ ex and obtain
K2 αm δ
δ/2
e− 4 α(m+l+h) ,
(3.19)
Gm,m+h;l+h,l ≤ e− 4 α(m+l+h) (K2 α)δ/2 ,
(3.20)
(α) Gm,m+h;l+h,l
≤
where K2 = 38 e(1 + M −1 )2 (1 − 2 )2 / 3 . Now we are left with two cases: a) l = 0 ⇔ m = δ: (α)
4 In the following, the K’s will be kinds of “dustbin” constants. It means that their contents changes whereas their names do not.
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V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar
b) l ≥ 1 and δ ≥ 1: using l + δ ≤ 2lδ, Gm,m+h;l+h,l ≤ e− 4 α(m+l+h) (2K2 lα)δ/2 .
(α)
(3.21)
Inserting this into (3.4) and symmetrising with respect to the smallest index we obtain (3.14), with K1 = 2MK2 . This means K1 > M when all ∈ [0.5, 1] are included. Let us now consider a typical graph appearing in the process of renormalisation, that is, with external legs carrying indices lower than the internal ones. The bound (3.14) provides a good factor with respect to power-counting unless the index jump δ = |m − l| is very small, typically δ = 0 or δ = 1. This ensures that if the lower index of a propagator is smaller than the scale we look at, the index is conserved along its trajectory for power-counting relevant and marginal graphs. Unfortunately, that estimation does not carry any information when the lower index is larger than the scale. It leads to a difficulty for graphs which possess completely inner vertices. Therefore, we have to find estimates for propagators with a sum over the index l. The next section is devoted to these bounds.
3.3. Bounds for sums. Now we want to prove that the summation of the propagator (α) Gm,p−l,p,m+l over l, for m and p kept constant, gives the same power-counting as in the previous section. The proof relies on a more accurate estimate of the sum in (3.10). Proposition 3. For M large enough there exists a constant K such that for all ∈ [ 23 , 1], we have the uniform bound p
Gim,p−l,p,m+l ≤ KM −i e− 4 M
−i (p+m)
.
(3.22)
l=−m
Proof. The first slices, say i ≤ 6, are trivial to treat. Using (3.11) we have p
Gim,p−l,p,m+l
2 M −i K s s s ≤ i (m + p s + 1)e− 3 (m +p ) . M
(3.23)
s=1
l=−m
Then, the estimation follows from
(x + 1)e
−i − 3M x
≤
12M i M −i −1 − M −i x 12 e 4 . e
(3.24)
This method fails in the limit i → ∞. Thus, for large i, we have to estimate the propagator (3.6) more carefully, now putting h → p − m − l and l → m + l. Without loss of generality we can assume p ≥ m. We have to divide the range of summation into
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
575
three parts according to the smallest index. Using (3.15) we estimate p
(α)
Gm,p−l;p,m+l
l=−m
≤e
−α(C+1/2)(m+p)
−1 m+l X m−u Y m+l−u (m − u)! (m + l − u)!
l=−m u=0
p−l p Y m+l−u Y p−v X m−u X p−l−v + + (m − u)! (m + l − u)! (p − l − v)! (p − v)! l=0 u=0 l=p−m v=0 −1 p X |l| Yl ≤ + e−α(C+1/2−D(α))(m+p) |l|! l! l=−m l=0 1 p αD(α)(m+p+l) l 2 ≤2 e−α(C+1/2−D(α))(m+p) , (3.25) l! l=0 p−m−1 m
Z
where X = 21 αD(α)(m + p − l) and Y = 21 αD(α)(m + p + l). We can now divide the sum over l into two regions corresponding to l ≤ (2β − 3)(p + m) and l ≥ (2β − 3)(p + m) ≥ (2β − 3)(p + m), where x is the smallest integer which is larger than x and β > 23 will be determined later: l β−1 (2β−3)(p+m) p l αD(α)l 2β−3 ((β − 1)αD(α)(m + p)) Z≤ + . l! l! l=0
l=(2β−3)(p+m)
(3.26) We extend both sums to infinity and use in the second one the identity (3.17): l ∞ β−1 Z ≤ e(β−1)αD(α)(m+p) + αD(α)e . 2β−3
(3.27)
l=0
β−1 For α small enough we have 2β−3 αD(α)e < 1. Then the sum gives a constant which we determine later. Combining (3.27) and (3.25), we have to prove that C + 1/2 − βD(α) ≥ 1 1 4 . Let us define D such that C + 2 − D = 4 . We have D = 4 : 1 ⇔ β(1 + α)(1 − 2 ) ≤ 1 . (3.28) βD(α) ≤ 4 ! 1 35 For α ≤ 10 and β = 22 , we get ≥ 23 ⇒ C + 1/2 − βD(α) ≥ 4 . Under the same β−1 αD(α)e < 35 conditions we have 2β−3 36 so that the second sum in (3.26) is bounded by √ 1 1 . This finishes the 36. For M ≥ 2 ( 5 + 1) we need i ≥ 6 in order to reach α ≤ 10 proof.
The previous estimation for the summed propagator is still not enough for the renormalisation proof, because the index sums are entangled in the graph. We have to prove that the exponential decay is still achieved if for a given summation variable l we maximise the other index p:
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V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar
Proposition 4. For M large enough there exists a constant K such that for all ∈ [ 23 , 1] we have the uniform bound ∞ l=−m
Gim,p−l;p,m+l ≤ KM −i e− 36 M
max
p≥max(l,0)
−i m
(3.29)
.
Proof. Again, the main estimate (3.11) guarantees the desired behavior (3.29) for the first slices, say i ≤ 16. For l ≤ 0, the maximum is attained at p = 0 so that we are in the situation of (3.23) and (3.24). For l > 0, the maximum is attained at p = l so that the l-sum leads to a geometric series. Here, it is important that i is bounded. For i > 16 we have to proceed differently. We divide the domain of summation according to the smallest index at the propagator: ∞
max
l=−m
p≥max(l,0)
−1
=
max
0≤p<m+l
l=−m ∞
+
l=0
−1
+
max
l=−m ∞
max +
l≤p<m+l
l=0
m+l≤p<m
+
−1 l=−m
max p≥m
max .
(3.30)
p≥m+l
We now obtain from (3.6) and (3.15), ∞ l=−m
(α)
max
p≥max(l,0)
≤
−1 l=−m
+
max
0≤p<m+l
−1
−1
+
l=0 ∞
p v=0
m+l≤p<m
e−α(C+1/2)(m+p)
p≥m
m+l u=0
max e
−α(C+1/2)(m+p)
l≤p<m+l
X p−l−v Y p−v (p − l − v)! (p − v)!
m+l u=0
max e−α(C+1/2)(m+p)
l=−m ∞
e
−α(C+1/2)(m+p)
max
l=−m
+
Gm,p−l;p,m+l
X m−u Y m+l−u (m − u)! (m + l − u)!
X m−u Y m+l−u (m − u)! (m + l − u)!
p−l v=0
X p−l−v Y p−v (p − l − v)! (p − v)!
m X m−u Y m+l−u + max e−α(C+1/2)(m+p) p≥m+l (m − u)! (m + l − u)! u=0 l=0 |l| m 1 −α(C+1/2−D(α))(m+p) 2 αD(α)(p + m + |l|) ≤ max e p≥0 |l|! |l|=1
+
∞ l=0
where X =
αD(α) 2 (p
max e
1 −α(C+1/2−D(α))(m+p) 2 αD(α)(p
l!
p≥l
+ m − l) and Y =
αD(α) 2 (p
+ m + l).
+ m + l)
(3.31)
l ,
(3.32)
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis l
The function p → e−γp (p+q) attains its maximum eγ q−l l! need this property for q = m + |l| and
1 l!
577
l l γ
at p =
l γ
γ := α(C + 1/2 − D(α)) . However, we have to take into account the range of p. If e−γp
− q. We
(3.33) l γ
− q < 0 in (3.31), then the
(p+q)l
is monotonously decreasing for all p ≥ 0 so that the maximum is function l! at p = 0. Otherwise we have to insert the specific maximum. This means that we split the sum over l in (3.31) into two pieces: mγ • |l| ≤ 1−γ , where we insert p = 0. Actually, we can safely extend this sum from 0 to m, still keeping p = 0. mγ , where we insert p = 1−γ • |l| ≥ 1−γ γ |l| − m.
This gives
m
max e
|l|=1
p≥0
≤
m
|l|
+ m + |l|) |l|!
1 −α(C+1/2−D(α))(m+p) 2 αD(α)(p
e−γ m
1
l
2 αD(α)(m + l)
+
l!
l=0
m
e(γ −1)l
αD(α)l 2γ
γm l= 1−γ
l
l!
.
(3.34)
For the first sum on the right-hand side we are in the situation of (3.25) with m → 0, p → m so that we can bound that sum according to the steps leading to (3.22) by a constant times e− 4 αm , for ≥ 2/3 and α small enough. In the second sum we use (3.17) so that, for some number > 0, m
e
(γ −1)l
γm l= 1−γ
αD(α)l 2γ
l!
l ≤
m
e
γm l= 1−γ
≤ e−γ m
−l
αD(α)eγ + 2γ
l
l m αD(α)eγ + . 2γ γm
(3.35)
l= 1−γ
In the numerator we can bound γ by 21 , otherwise the sum vanishes. Moreover, we 1 choose = 12 . With (3.33) it thus remains to prove 7
D(α)e 12 <1. 2C + 1 − 2D(α) For α ≤
1 10
this means
7 11−112 12 e 422 −2
< 1, which is indeed satisfied for ≥ 23 .
(3.36)
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V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar
We now pass to (3.32). The condition p ≥ l leads to a splitting of the l-sum at For smaller l we have p = l: ∞ l=0
max e
1 −α(C+1/2−D(α))(m+p) 2 αD(α)(p
.
l
l!
p≥l
≤
+ m + l)
γm 1−2γ
∞
e−γ α(m+l)
l
1
l=0
2 αD(α)(m + 2l) l!
+
∞
e(γ −1)l
αD(α)l 2γ
l!
γm l= 1−2γ
l . (3.37)
The second sum on the right hand side is identical to treat as in the previous case (3.31). The first sum on this right hand side is split as in (3.26) at l = (β − 23 )m. Then it works exactly as in (3.27). This finishes the proof.
3.4. Composite propagators. This section is devoted to the proofs of bounds on the composite propagators introduced in [5]. We define their sliced versions as follows: i(0)
Q
m1 , m2
i(1)
Q
m1 , m2
n1 n1 , m1 ; n2 n2, m2
= Gi
i(0)
n1 n1 , m1 ; n2 n2 m2
=Q
m1 , m2
− Gi
m1 , n1 n1 , m1 ; m2 n2 n2 m2
i(0)
n1 n1 , m1 ; n2 n2 m2
− m1 Q
1, 0
(3.38)
,
0, n1 n1 , 0 ; 0 n2 n2 0
i(0)
n1 n1 , 1 ; n2 n2 0
− m2 Q
0, 1
n1 n1 , 0 ; n2 n2 1
, (3.39)
i( 21 )
Q
m1 + 1, n1 + 1 n1 , m1 ; 2 m2 n2 n m2
= Gi
m1 + 1, n1 + 1 n1 , m1 ; 2 n2 n m2 m2
−
m1 + 1Gi
1, n1 + 1 n1 , 0 ; 2 n 0 0 n2
.
(3.40) Proposition 5. There exist constants Ki such that for ∈ [0.5, 1) and m ≤ have the uniform bounds m1 + m2 − M −i (n1 +n2 ) i(0) e 3 , Q 1 1 1 1 ≤ K0 M −i m , n ;n , m Mi m2 n2 n2 m2 1 2 −i 1 m + m2 2 i(1) −i e− 3 M (n +n ) , Q 1 1 1 1 ≤ K1 M i m n n m M m2 , n2 ; n2 , m2 1 3/2 2 1 −i 1 2 i( 2 ) −i m + m + 1 e− 4 M (n +n ) . Q 1 ≤ K2 M m + 1, n1 + 1 ; n1 , m1 Mi m2 n2 n2 m2
M i , we
(3.41)
(3.42)
(3.43)
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
579
Proof. There is no need to discuss the first slice. From (3.4) we have θ Q 1 1 1 1= m n n m 8 ; i(0)
m2, n2 n2, m2
µ20 θ
(α) (1 − α) 8 (α) (α) Gm1 ,n1 ,n1 ,m1 − G0,n1 ,n1 ,0 Gm2 ,n2 ,n2 ,m2 dα 2 (1 + Cα)
M −i+1
M −i
(α) (α) (α) +G0,n1 ,n1 ,0 Gm2 ,n2 ,n2 ,m2 − G0,n2 ,n2 ,0 .
(3.44) (α)
(α)
Taking (3.11) into account, it remains to obtain estimations for Gm,n,n,m − G0,n,n,0 . From (3.2) we have, expressing m + h by n, (α) (α) G m,n,n,m − G0,n,n,0 √ n √ m min(m,n) 2u 1 − α 1−α m n Cα(1 + ) = − 1 . √ u u 1 + Cα 1 − α (1 − ) 1 + Cα u=0
(3.45) The (u = 0)-part of the sum and the separate term −1 yield √ m m−1 √ j √ 1−α 1−α 1−α 1− = 1− 1 + Cα 1 + Cα 1 + Cα j =0 √ 1−α ≤ m 1− 1 + Cα ≤ mα(C + 1) . Next, using
m v+1
=
m m n u=1
m−v m v+1 v √
and our central estimate
Cα(1 + )
(3.46) m v
≤
mv v! ,
we have
2u
1 − α (1 − ) u m m−1 m − v m Cα(1 + ) v+1 n Cα(1 + ) ≤ √ √ u v+1 v 1 − α (1 − ) 1 − α (1 − ) v=0 u=0 Cα(1 + ) Cα(1 + )(m + n) ≤m √ exp √ 1 − α (1 − ) 1 − α (1 − ) 2 (1 − ) eαD(α)(m+n) . (3.47) ≤ mα √ −1 4 1 − M u
u
Inserting (3.46) and (3.47) back into (3.45) we obtain for α ≤ M −1 the estimation (α) 2 (α) G ≤ αm (1+)2 + (1−) √ e−α(C+1/2−D(α))n . (3.48) − G m,n,n,m 0,n,n,0 −1 4 4 1−M
Comparing (3.48) with (3.12) we obtain in (3.41) the same restrictions to as in Proposition 1.
580
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To approach (3.42) we consider i(1)
Q
m1 , n1 n1 , m1 ; m2 n2 n2 m2 µ20 θ M −i+1 θ (1 − α) 8 = dα 8 M −i (1 + Cα)2 (α) (α) (α) (α) (α) × Gm1 ,n1 ,n1 ,m1 − G0,n1 ,n1 ,0 − m1 G1,n1 ,n1 ,1 − G0,n1 ,n1 ,0 Gm2 ,n2 ,n2 ,m2 (α) (α) (α) (α) +m1 G1,n1 ,n1 ,1 − G0,n1 ,n1 ,0 Gm2 ,n2 ,n2 ,m2 − G0,n2 ,n2 ,0 (α) (α) (α) (α) (α) . (3.49) +G0,n1 ,n1 ,0 Gm2 ,n2 ,n2 ,m2 − G0,n2 ,n2 ,0 −m2 G1,n2 ,n2 ,1 −G0,n2 ,n2 ,0
The fourth line is estimated by (3.48). In the other lines we have for n ≥ 1, (α) (α) (α) (α) G m,n,n,m − G0,n,n,0 − m(G1,n,n,1 − G0,n,n,0 ) n √ m √ √ 1 − α 1−α 1−α − 1 − 1 −m = 1 + Cα 1 + Cα 1 + Cα n+1 m−1 √ 2 √ 1−α 1−α Cα(1 + ) − 1 +mn √ 1 + Cα 1 + Cα 1 − α (1 − ) √ m+n min(m,n) 2u 1−α m n Cα(1 + ) + . (3.50) √ u u 1 + Cα 1 − α (1 − ) u=2 In the third line we use n √Cα(1+) ≤ exp (nαD(α)). The further procedure is as 1−α (1−) before. Finally, we consider i( 21 )
Q
m1 + 1, n1 + 1 n1 , m1 ; 2 m2 n2 n m2 µ20 θ M −i+1 θ (1 − α) 8 = dα (1 + Cα)2 8 M −i (α) (α) (α) × Gm1 +1,n1 +1,n1 ,m1 − m1 + 1G1,n1 +1,n1 ,0 Gm2 ,n2 ,n2 ,m2 (α) (α) (α) + m1 + 1G1,n1 +1,n1 ,0 Gm2 ,n2 ,n2 ,m2 − G0,n2 ,n2 ,0 .
(3.51)
The estimation for the fourth line of (3.51) is immediately obtained from (3.48) and (3.14). In the third line we have (α) √ (α) G m+1,n+1,n,m − m + 1G1,n+1,n,0 √ m min(m,n) √ 2u+1 Cα(1 + ) 1−α (m + 1)(n + 1) m n = √ 1 + Cα u+1 u u 1 − α (1 − ) u=0 n+1 √ 1−α Cα(1 + ) . (3.52) − (m + 1)(n + 1) √ 1 − α (1 − ) 1 + Cα
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
581
We use the estimation n √Cα(1+) ≤ exp 21 nαD(α) and proceed along the same 1−α (1−) lines as before. This finishes the proof. 4. Perturbative Power-Counting 4.1. Ribbon graphs. Consider a given φ 4 -ribbon graph G with N external legs, V vertices, I inner lines and F faces, hence of genus g = 1 − 21 (V − I + F ). There are of the graph and two four indices {m, n; k, l} ∈ N2 associated to each inner line indices for each external line, hence 4I + 2N = 8V such indices. But at every vertex, the left index of a ribbon coincides with the right one of the next ribbon. This creates 4V independent identifications, so we can write the indices of any propagator in terms of a set I of 4V indices, four per vertex, for instance each “left” half-ribbon index. The amplitude of the graph is then the sum AG =
(4.1)
Gmδ (I ),nδ (I );kδ (I ),lδ (I ) δmδ −lδ ,nδ −kδ ,
I δ∈G
where the four basic indices of the propagator G for a line δ are functions of I called {mδ (I), nδ (I); kδ (I), lδ (I)}. The scale decomposition of the propagator being G=
∞
Gi ,
(4.2)
i=0
we have an associated decomposition of any amplitude of the theory as AG =
(4.3)
AG,µ ,
µ
AG,µ =
I δ∈G
Gimδ δ (I ),nδ (I );kδ (I ),lδ (I ) δmδ (I )−lδ (I ),nδ (I )−kδ (I ) ,
(4.4)
where µ = {iδ } runs over all possible attributions of a positive integer iδ for each line δ. Such a µ is therefore called a scale attribution. We recall our two main bounds on the propagator l
Gim,n;k,l ≤ KM −i e−cM max Gim,n;k,l ≤ K M −i e n,k
−i (m+n+k+l)
−c M −i m
,
,
(4.5) (4.6)
for some constants K, K and c, c . A considerable fraction of the 4V indices initially associated to this graph is determined by external indices of the graph and the δ-function in (4.1). The undetermined indices are summation indices. Perturbative power counting for a graph consists in finding the indices for which the summation costs a factor M 2i , and the ones for which it
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V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar
costs only O(1), thanks to (4.6). The factor M 2i follows from (4.5) after summation over some index5 m ∈ N2 , ∞ m1 ,m2 =0
e−cM
−i (m1 +m2 )
=
1 −i
(1 − e−cM )2
=
M 2i (1 + O(M −i )). c2
(4.7)
4.2. Dual graphs. We first want to use as much as possible the δ-functions in (4.1) to reduce the set I to a truly minimal set I of independent indices. It is convenient for this task to consider the dual graph for which the problem becomes analogous to an ordinary problem of momentum routing. The dual graph of a ribbon graph is obtained by associating to each face a vertex and to each vertex a face. Every line bordering two neighbouring faces is replaced by a line joining the two corresponding vertices of the dual graph. Hence, the genus does not change under this duality. We shall write V = F , F = V for the number of vertices and faces of the dual graph (dual quantities are usually distinguished by a prime). If the initial graph is a φ 4 -graph, i.e. has coordination 4 at each vertex, we have 4 = If + Nf for each face f ∈ F , where If and Nf denote the numbers of edges and external valences, respectively, which belong to f . The coordination at the vertices of the dual graph is arbitrary. The construction of the dual of a graph goes as follows: First, for each oriented face of the original ribbon graph, draw an oriented ribbon vertex by assigning: • to a single line of a propagator of the original graph an internal valence of the dual vertex, • to an external valence of the original graph an external valence of the dual vertex, respecting the order according to the arrows on the trajectories. In the second step we connect the valences by the duals of the propagators of the original graph, which is obtained according to
(4.8)
Let us consider the following example of a ribbon graph with a single face:
(4.9)
5 Remember that there are two symplectic pairs, one for spatial dimensions 1 and 2, and the other for spatial dimensions 3 and 4.
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The above rules lead to the following dual vertex:
(4.10)
Now we connect the valences by the duals of the propagators of the original graph, i.e. n1 q with r q , qr with q m2 and n2 m1 with rr :
(4.11)
The dual graph is made of the same propagators as the original graph, only the index assignment is different. Whereas in the original graph we have Gmn;kl = index assignment for propagators in the dual graph is Gmn;kl =
the
(4.12)
The conservation rule δl−m,−(n−k) in (4.1) now states that the difference between outgoing and incoming indices of the half-propagator attached to a dual vertex, namely l − m, is conserved as minus the corresponding difference n − k at the other end of the propagator. Actually, these index differences describe the angular momentum, and the conservation of these differences = l − m and − = n − k is nothing but the angular momentum conservation due to the SO(2) × SO(2) symmetry of the noncommutative φ 4 -action. Thus, taking the incoming indices as the reference, the angular momentum through the dual propagator determines the outgoing indices:
l = m + , n = k + (−) .
(4.13)
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In the same way, there are external angular momenta ℘ which enter the dual graph through the external valences. We shall also use the incoming arrow as the reference so that the angular momentum is the index difference between outgoing and incoming arrows. Furthermore, by cyclicity at any vertex, the sum of all incoming differences at a vertex, i.e. the sum of incoming angular momenta, must be zero. Of course, this implies that the total external angular momentum entering a graph is zero, too. Thus, the angular momentum for the dual graph is exactly like the usual momentum in an ordinary Feynman graph: a momentum that goes out like p at one half-end must go out as −p at the other half-end, and the total momentum is conserved at any vertex. (In Feynman graphs this follows from translation invariance.) It should be stressed that one has to take into account positivity constraints for the angular momenta : they lie in Z, but all indices m, n, k, l must be positive integers. We therefore know that the number of independent index differences is exactly the number of loops L of the dual graph. For a connected graph, this number is L = I − V + 1. Furthermore, each index at a vertex is clearly only a function of the differences at a vertex and of a single reference index for the dual vertex. If the dual vertex is an external one, we take as the reference index the outgoing index at one of the external legs. If the dual vertex is an internal one, we have to make a choice (determined later) for the reference index. These internal vertex reference indices correspond to the loop variable of the original graph. Therefore, after using the conservation rules or δ-functions of each propagator, the number of independent indices to be summed for every graph is simply V − B + L = I + (1 − B). Here, B ≥ 0 is the number of boundary components of the original graph, which coincides with the number of external vertices of the dual graph. Expressing each index in the graph as a function of a set I of such independent indices is therefore identical to the problem called momentum routing in a Feynman graph. It is well-known that the solution is not canonical or unique. A good way to root the momenta is to pick a spanning tree Tµ of the dual graph, with V − 1 lines, and to use the complement set Lµ as the set of fundamental independent differences. The subscript µ refers to the choice of the tree which depends on the scale attribution µ in (4.4).
4.3. Choice of the tree. A given scale attribution µ = {iδ } defines an order of lines in the dual graph. We define δ1 ≤ δ2 ≤ · · · ≤ δI
iδ1 ≤ iδ2 ≤ · · · ≤ iδI .
if
(4.14)
In case of equality we make any choice. Let δ1T be the smallest line with respect to this iδ T
order which is not a tadpole, and Gm1 T ;n T ;k T ,l T be the corresponding propagator. δ1
δ1
δ1
The line δ1T will then connect two vertices v1± We let µ1 := µ \ (δ1 ∪ · · · ∪ δ1T ) and T1 = δ1T
δ1
and forms the first segment of the tree. ∪ v1+ ∪ v1− . In the remaining set µ1 of lines we identify the smallest line δ2T of µ1 which does not form a loop when added to T1 . We define µ2 = µ \ (δ1 ∪ · · · ∪ δ2T ) and
• T2 = T1 ∪ δ2T ∪ v2+ if δ2T connects a vertex v2+ to T1 , / T1 . • T2 = T1 ∪ δ2T ∪ v2+ ∪ v2− if δ2T connects two vertices v2± ∈
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In the nth step, we identify the smallest line δnT of µn−1 which does not form a loop when added to Tn−1 . We define µn = µ \ (δ1 ∪ · · · ∪ δnT ) and • Tn = Tn−1 ∪ δnT ∪ vn+ if δnT connects a vertex vn+ to Tn−1 , / Tn−1 , • Tn = Tn−1 ∪ δnT ∪ vn+ ∪ vn− if δnT connects two vertices vn± ∈ • Tn = Tn−1 ∪ δnT if δnT connects two disjoint subsets of Tn−1 . The (V − 1)th step leads to the desired tree Tµ = TV −1 . The importance of this construction is the fact that any line δjL ∈ Lµ which connects different vertices vj± of the tree has a scale index iδ L which is not smaller than any scale index of the lines in the j
tree which connect vj± . Such a tree optimisation for a given scale attribution is one of the most basic tools of constructive field theory [10], so it is an encouraging sign that it appears also here in a natural way. In the graphical representations we distinguish the tree by triple dashed lines.
4.4. Index assignment for a tree. For the previously constructed tree Tµ we select one of its B ≥ 1 external vertices as the root v0 of the tree. If the graph is a vacuum graph, i.e. with B = 0, we choose any vertex as the root. We relabel the vertices in the tree such that all vertices in the subtree above a vertex vn have a label bigger than n. The order (4.14) of the lines of the graph provides us with a convenient position for the main reference index m at each vertex. If v is an internal vertex, we let δv be the smallest line in (4.14) which is attached to v. By construction of the tree we know that either δv is a tadpole, or it belongs to the tree. We choose the outgoing index (without the arrow when viewed from the vertex) of this particular line δv as the main reference index mv . We let GM be the set of lines at which a main reference index resides. It is possible that both outgoing indices of a line δv = δv attached to v and v are main reference indices. In this case we let δv appear twice in GM . Thus, GM consists of V − B elements. If v is an external vertex, we take as the “main reference index” the outgoing index at any external leg. The following graph shows a typical situation of a tree and its main reference indices, assuming absence of tadpoles and B = 1:
(4.15)
One can now write down every index in a unique way in terms of • • • •
V − B main reference indices mv , B main reference indices at external vertices, L internal angular momenta δ for the set Lµ of “loop lines”, N external angular momenta ℘ for the set N of external lines.
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The rule goes as follows. One writes first the indices for all the “leaves” of the tree, that is the vertices (distinct from the root) with coordination 1 in the tree (i.e. vertices v3 , v5 , v6 , v7 and v8 in (4.15)). For them, starting from the main index mv (at the left of the unique line of Tµ at v that goes towards the root, unless a tadpole at v has the smallest scale), which agrees with the incoming index of the next line at v in the clockwise direction, we compute all other indices by turning clockwise around the vertex and by adding the angular momenta (internal or external ones) associated according to (4.13) with the loop lines δ1 , . . . , δk and possibly external lines 1 , . . . , k . This gives indices mv + 1 , mv + 1 + 2 , . . . until we arrive at mv + 1 + · · · + k+k which is at the right of the unique line at v that goes towards the root. Some of the j could be external angular momenta. Then we can compute the angular momentum associated to that line: it is simply −(1 + · · · + k+k ). The following picture shows these assignments for a particular “leaf”:
(4.16)
Having done this for all leaves, we can prune these leaves and consider the next layer of vertices down towards the root (i.e. vertices v2 and v4 in (4.15)) and reiterate the argument. Any summation index at a vertex v is now clearly equal to mv plus a linear combination of angular momenta δ for the set of lines Lv ∪ Nv hooked to the “subtree above v”, that is the lines hooked at least at one end to a vertex v such that the unique path from v to v0 passes through v. We split therefore the set I of independent indices to be summed into the two sets: • the set Mµ = {mv } of main reference indices at inner vertices, which consists of V − B elements, • the set Jµ = {δ , δ ∈ Lµ } of angular momenta, which consists of L elements. The amplitude of the graph G is now written as: AG =
µ Mµ ,Jµ δ∈G
Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ ) χ (Mµ , Jµ ) , (4.17)
where the sum over Mµ is over positive integers, but the sum over Jµ is over relative integers and the function χ (Mµ , Jµ ) is the characteristic function which states that all the functions {mδ (Mµ , Jµ ), nδ (Mµ , Jµ ); kδ (Mµ , Jµ ), lδ (Mµ , Jµ )} are positive. The dependence of AG on the external indices (B reference indices at external vertices and N external angular momenta) is not made explicit.
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4.5. Power-counting. Our goal is now to prove that sums over difference indices can always be performed through (4.6), hence at no cost, using precisely the propagator Giδ to perform the sum over the angular momentum δ . However, as the angular momenta are entangled, we need appropriate maximisations of the other Giδ over δ . These maximisations require a carefully chosen order. For processing the angular momenta, all main reference indices Mµ and the external indices are kept constant. We introduce in the set of loop lines Lµ another order “>” than the previous order “≤” defined in (4.14), now according to the position in the tree. For this purpose we let vδ be the label of the highest vertex to which δ ∈ Lµ is hooked. We define δ1 > δ2 if • vδ1 > vδ2 , or • when turning clockwise around vδ1 = vδ2 from the main reference index, first δ1 is encountered, then δ2 or the tree line which goes down to the root, or • when turning clockwise around vδ1 = vδ2 from the main reference index, first the tree line down to the root is encountered, then δ2 and finally δ1 . We orient the lines δ ∈ Lµ so that the two indices mδ (Mµ , Jµ ), lδ (Mµ , Jµ ) are hooked to vδ and their difference is precisely δ : lδ (Mµ , Jµ ) − mδ (Mµ , Jµ ) = δ .
(4.18)
For “tadpole” lines δ of Lµ , i.e. hooking vδ to itself, we define the two indices m, l as the ones of the first half-line of δ in the clockwise cyclic ordering between the main reference of vδ and the tree line down to the root. If the tree line is encountered before both half-lines, we have to select instead the indices on the half-line in anti-clockwise order. We define Jµδ+ = {δ ∈ Jµ , δ > δ} and Jµδ− = {δ ∈ Jµ , δ > δ }. Then, we remark that the two distinguished indices mδ (Mµ , Jµ ), lδ (Mµ , Jµ ) at vδ are actually functions mδ (Mµ , Jµδ+ ), mδ (Mµ , Jµδ+ ) + δ which are independent of the indices in +
Jµδ− . We conclude that for δ ∈ Lµ and fixed indices in Mµ and Jµδ , max Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ )
δ ∈Jµδ−
≤ max Giδ nδ ,kδ
mδ (Mµ ,Jµδ+ ),nδ ;kδ ,mδ (Mµ ,Jµδ+ )+δ
,
(4.19)
where on the right-hand side the max is over any indices nδ , kδ in N. It is instructive to look at the example (4.16) where we have δ1 > δ2 > δ3 > δ4 . The indices m2 , l2 depend on the main reference index mv and the “bigger” angular momentum δ1 ∈ Jµ2+ , but not on the “smaller” angular momenta δ3 , δ4 ∈ Jµ2− . For a given scale attribution µ in (4.17) and fixed indices in Mµ we can now evaluate the telescopic sum over the angular momenta in Jµ . With respect to the previously defined order “>” let δL > δL −1 > · · · > δ2 > δ1 be the loop lines. The summation is performed from the smaller to larger labels:
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Jµ δ∈G
Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ ) χ (Mµ , Jµ )
≤
max δ1
δ +
Jµ1
δ∈Lµ , δ>δ1
G
δ1
iδ1
δ +
max
δ1 ,δ2
×
× max δ2
χ (Mµ , Jµ )
δ +
δ∈Tµ
Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ )
δ∈Lµ , δ>δ2
max G δ1
δ2
× max
δ +
δ1 ,δ2
Jµ2
Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ )
mδ1 (Mµ ,Jµ1 ),nδ1 (Mµ ,Jµ );kδ1 (Mµ ,Jµ ),mδ1 (Mµ ,Jµ1 )+δ1
≤
× max ×
Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ )
δ∈Tµ
δ1
Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ )
iδ2
δ +
δ +
mδ (Mµ ,Jµ2 ),nδ2 (Mµ ,Jµ );kδ2 (Mµ ,Jµ ),mδ2 (Mµ ,Jµ2 )+δ2
max G
nδ1 ,kδ1
δ1
iδ1
δ +
δ +
mδ1 (Mµ ,Jµ1 ),nδ1 ;kδ1 ,mδ1 (Mµ ,Jµ1 )+δ1
χ (Mµ , Jµ ) , (4.20)
and so on. At the end we arrive at i Gmδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ ) χ (Mµ , Jµ ) Jµ δ∈G
≤
δ∈Tµ
max Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ ) J µ
× max δ+ δ∈LµJµ
δ ≥−mδ (Mµ ,Jµδ+ )
. δ+ δ+ mδ (Mµ ,Jµ ),nδ ;kδ ,mδ (Mµ ,Jµ )+δ
max Giδ nδ ,kδ
(4.21) The positivity constraints in χ are used to fix the correct range of sums over δ . The result is a bound for the Jµ -summation in (4.17). For tree lines δ ∈ Tµ , where all indices depend on Jµ , the bound, due (4.5), is given by max Gimδ δ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ ) ≤ KM −iδ , Jµ
δ ∈ Tµ , δ ∈ / Gµ . (4.22)
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If one of the indices of δ ∈ Tµ is a main reference index at v, we have max Gimδ v ,nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),lδ (Mµ ,Jµ ) ≤ KM −iδ e−cM
−iδ m v
Jµ
δ ∈ Tµ ∩ Gµ .
,
(4.23) If two indices of δ are main reference indices at v, v , we have max Gimδ v ,nδ (Mµ ,Jµ );m
v ,lδ (Mµ ,Jµ )
Jµ
≤ KM −iδ e−cM
−iδ (m +m ) v v
δ ∈ Tµ , δ ∈ Gµ \ {δ} .
,
(4.24)
Next, each summed propagator which corresponds to a line δ ∈ Li delivers according to (4.21) a factor KM −iδ , max max Giδ δ+ δ+ Jµδ+
δ
mδ (Mµ ,Jµ ),nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),mδ (Mµ ,Jµ )+δ
Jµδ−
≤ K M −iδ ,
δ ∈ Lµ ,
δ∈ / Gµ .
(4.25)
In the case that δ ∈ Li is a tadpole at vi which has the smallest scale index among the set of lines at vi we obtain from (4.6) the bound max max Giδ Jµδ+
mv ,nδ (Mµ ,Jµ );kδ (Mµ ,Jµ ),mv +δ
Jµδ−
δ
≤ K M −iδ e−c M
−iδ m v
δ ∈ Lµ ∩ Gµ .
,
(4.26)
Eventually, there will be indices m , n which are fixed as external ones. Each one −i −i delivers according to (4.5) and (4.6) an additional factor e−cM m and e−cM n , respectively, because these decays cannot be removed by maximising loop momenta. For external indices which are not connected to internal lines we put c ≡ 0. Altogether, the Jµ -summation in (4.17) can be estimated by −iδ e−cM mv(δ ) KM −iδ AG ≤ µ m1 ,...,m ∈N2 V −B
×
N
e−cM
δ ∈Gµ
−i m
=1
N
e−cM
−i n
δ∈G
(4.27)
,
=1
where mv(δ ) is the main reference index at δ ∈ Gµ . After summation over m1 , . . . , mV −B we have AG ≤
µ
×
KI
c2(V N =1
−B)
M−
δ∈G iδ
e
−cM −i m
N
=1
M
2
δ ∈Gµ iδ
e
−cM −i n
.
(4.28)
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The dangerous region of the sum over the scale attribution is at large scale indices. To identify this region, we associate to the order (4.14) of lines a sequence of subgraphs GI ⊂ GI −1 ⊂ · · · ⊂ G1 = G of the original ribbon graph by defining Gγ as the possibly disconnected set of lines δγ ≥ δγ , together with all vertices attached to them. To Gγ we associate the scale attribution µγ which starts from an irrelevant low-scale cut-off iγ −1 ≤ iγ . We conclude from (4.28) that the amplitude AGγ corresponding to the subgraph Gγ converges if ωγ := 2(Vγ − Bγ ) − Iγ = 2Fγ − 2Bγ − Iγ = (2 −
Nγ 2
) − 2(2gγ + Bγ − 1) (4.29)
N
is negative, where Nγ , Vγ , Iγ = 2Vγ − 2γ , Fγ and Bγ are the numbers of external legs, vertices, edges, faces and external faces of Gγ , respectively, and gγ = 1 − 21 (Vγ − Iγ + Fγ ) is its genus. We have thus proven the following Theorem 6. The sum over the scale attribution µ in (4.28) converges if for all subgraphs Gγ ⊂ G we have ωγ < 0. For the total graph γ = G the power-counting degree becomes ω = (2 − B − 1), which reproduces the power-counting degree derived in [8].
N 2 ) − 2(2g
+
4.6. Subtraction procedure for divergent subgraphs. The power-counting Theorem 6 implies that planar subgraphs with two or four external legs are the only ones for which the sum over the scale attribution can be divergent. These graphs require a separate analysis. We first see from Proposition 2 that • only those planar four-leg subgraphs with constant index along the trajectory are marginal, • only those planar two-leg graphs with constant index along the trajectory are relevant, • only those planar two-leg graphs with an accumulated index jump of 2 along the trajectory are marginal. For the other types of graphs there is a sufficient power of M −i through the terms (M −i l)δ in (3.21) which makes the sum over the scale attribution convergent. For the remaining truly divergent graphs one performs similarly as in the BPHZ scheme a Taylor subtraction about vanishing external indices. For instance, a marginal four-leg graph with amplitude Amn;nk;kl;lm is written as Amn;nk;kl;lm = (Amn;nk;kl;lm − A00;00;00;00 ) + A00;00;00;00 .
(4.30)
The difference of graphs Amn;nk;kl;lm − A00;00;00;00 can be expressed as a linear combination involving the composite propagators (3.38). See also [5] for more details. Then, the estimation (3.41) provides an additional factor M −i which makes the sum over the scale attribution for the difference Amn;nk;kl;lm − A00;00;00;00 convergent. Eventually, there remain only the four divergent base functions A 0 0 0 0 0 0 0 0 , ;
;
;
00 00 00 00 A 0 0 0 0 , A 1 0 0 1 − A 0 0 0 0 and A 1 1 0 0 , taking into account the symme; ; ; ; 00 00 00 00 00 00 00 00 try properties of the model. These are normalised to their “experimentally” determined values: the physical coupling constant, the physical mass, the physical field amplitude and the physical frequency of the harmonic oscillator potential, respectively.
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At the end, any graph appearing in the noncommutative φ 4 -model has an amplitude which is uniquely expressed by four normalisation conditions as well as convergent sums over the scale attribution. Thus, the model is renormalisable to all orders. 5. Conclusion For many years, noncommutative quantum field theories were supposed to be ill-behaved due to the UV/IR-mixing problem [3]. Meanwhile, it turned out [8, 5] that at least the Euclidean noncommutative φ44 -model is as good as its commutative version: it is renormalisable to all orders. In fact, the noncommutative φ44 -model is even better than the commutative version with respect to one important issue: the behaviour of the β-function. It is well-known that the main obstacle to a rigorous construction of the commutative φ44 -model is the non-asymptotic freedom of the theory. The noncommutative model is very different: The computation of the β-function [11] shows that the ratio of the bare coupling constant to the square of the bare frequency parameter remains (at the one loop level) constant over all scales, λ2 = const. (This was noticed in [13].) As the bare frequency is bounded by 1, this means that the bare coupling constant is bounded. For appropriate renormalised values, the coupling constant can be kept arbitrarily small throughout the renormalisation flow. We are, therefore, optimistic that a rigorous construction of the noncommutative φ44 -model will be possible. In this paper we have undertaken the first important steps in this direction. We have formulated the perturbative renormalisation proof in a language which admits a direct extension to constructive methods. More details about our program are given in [14]. Moreover, our new renormalisation proof is much more efficient than the previous one (by a factor of 3 when looking at the number of pages). Eventually, we have established analytical bounds for the asymptotic behaviour of the propagator which before were only established numerically. A. The First Slice Let us now have a look at the first slice, that is the region M −1 ≤ α ≤ 1. For technical reasons, we divide it into two subregions, namely M −1 ≤ α ≤ a called the intermediate region and a ≤ α ≤ 1 called the bulk. A.1. The intermediate region. Let us call Ginter (a) the propagator restricted to the region M −1 ≤ α ≤ a. We have then the following lemma: √
1−a 2 Lemma 7. Let min (a) = 1 + 18 25 a ln(1 − a). For each ∈ [ 3 , 1] there exists an a ∈ [0, 1] with ∈ [min (a), 1] and a constant k > 0 such that we have the uniform bound −km+l+h Ginter . m,m+h;l+h,l (a) ≤ Ke
(A.1)
Proof. Here we can’t safely approximate the expressions involving α by their expansions about α = 0. But we have, using (3.5) and (3.10), m+l+h √ 1−α Cα(1 + ) (α) exp √ Gm,m+h;l+h,l ≤ (m + l + h) . (A.2) 1 + Cα 1 − α(1 − )
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To obtain the claimed result, we have to prove that f (α, ) = √
Cα(1 + ) 1 − α(1 − )
+ ln
√
1−α 1 + Cα
<0.
(A.3)
We have
√ 1 − 2 4 α f (α, ) = + ln 1 − α + ln √ 4 4 + (1 − )2 α 1−α 2 √ 1− α ≤ + ln 1 − α . √ 4 1−α In the preceding sections, we bounded from below by 2/3. Thus, 5 α 1 (1 − ) √ + ln(1 − α) . 8 2 1−α
f (α, ) ≤ √
(A.4)
(A.5)
∂1 Thus ∀ ≥ 1 (α) = 1 + 45 1−α α ln(1 − α), f ≤ 0. An expansion of ∂α about α = 0 shows that 1 increases with α. This means that h(α, ) ≤ −k for all α ∈ [M −1 , a] and all √ 18 1 − a ln(1 − a) . (A.6) ≥ min (a) = 1 + 25 a This finishes the proof.
A.2. The bulk. Let us call Gbulk (a) the propagator restricted to the region a ≤ α ≤ 1. We have then the following lemma: √ √ 1−a 2 1−a ln(1 − a) and (a) = 1 − . For each max a a 6 [ √ , 1] with ∈ [min (a), max (a)] and a constant 3+ 10
Lemma 8. Let min (a) = 1 +
18 25
∈ [ 23 , 1) there exists an a ∈ k > 0 such that we have the uniform bound
−km+l+h Gbulk . m,m+h;l+h,l (a) ≤ Ke
(A.7)
Proof. Using (3.9) and (3.10) we obtain √ m+l+h α(1 − 2 ) 4 1 − α (α) Gm,m+h;l+h,l ≤ exp (m + l + h) .(A.8) 4 + (1 − )2 α α(1 − 2 ) Like in the intermediate region, we have now to study √ 4 1−α (1 − 2 )α g(α, ) = + ln 1 − 2 α 4 + (1 − )2 α in order to know in which (α, )-region it is negative, √ 2 1 − 2 1−α g(α, ) ≤ + ln 1− α 4 √ 2 5 1−a ≤ + ln (1 − ) ≡ i(a, ) . 1− a 8 A simple analysis of i considered as a function of allows to draw
(A.9)
(A.10)
Renormalisation of Noncommutative φ 4 -Theory by Multi-Scale Analysis
0
1
2 −
∂i ∂
+ +∞
i(a, 0)
i(a, )
593
i(a, 2 )
√ We have taken into account that M can be as small as 21 ( 5+1) so that
∂i ∂ =0 < 0. We √ 2 1−a . Let us now remember what we are looking for. We want to detera
have 2 = 1 − mine whether there exists a (a, )-region where i(a, ) ≤ 0. Clearly, if i(a, 0) ≤ 0, then i(a, ) < 0 for all ∈ (0, 2 ]. We have i(a, 0) ≤ 0 ⇐⇒ a ≥
2 ln2
8 5
+ −1 +
Actually, the derivation of (A.10) used ≥ 2 ≥
2 3
1 + ln2
8 5
0.9502 .
(A.11)
so that the relevant condition is
2 6 ⇐⇒ a ≥ √ 0.973666 . 3 3 + 10
(A.12)
The study of the behaviour of the propagator in the intermediate region gave us a lower bound (cf. Lemma 7) for which was min . For compatibility of the intermediate and bulk regions, we need min < 2 , which is satisfied under the condition (A.12). 6 This provides a uniform exponential decay of the propagator at least for a ≥ √ and 3+ 10
√
∈ [min (a), max (a)] with max (a) = 2 = 1 − 2 a1−a . The limit a → 1 allows to study the propagator when is close to 1 thanks to lima→1 max (a) = 1. The case = 1 is treated in the next section. B. The Case Ω = 1 The case = 1 can be directly treated.According to (3.2), only the terms with u = m = l survive: G=1 mn;kl
θ = 8 =
µ20
1
dα(1 − α)
µ20 θ D 1 8 +( 4 −1)+ 2 (m+k)
δml δnk
(B.1)
.
(B.2)
0
+
δml δnk 2 D θ (m + n + k + l + 2 )
The exponential decay of the propagator in any index is easily obtained from (B.1) for all slices. Moreover, the l-sum is trivial to perform due to the index conservation δml at each trajectory.
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References 1. Schomerus, V.: D-branes and deformation quantization. JHEP 9906, 030 (1999) 2. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999) 3. Minwalla, S., Van Raamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. JHEP 0002, 020 (2000) 4. Chepelev, I., Roiban, R.: Convergence theorem for non-commutative Feynman graphs and renormalization. JHEP 0103, 001 (2001) 5. Grosse, H., Wulkenhaar, R.: Renormalisation of φ 4 -theory on noncommutative R4 in the matrix base. Commun. Math. Phys. 256, no. 2, 305–374 (2005) 6. Wilson, K.G., Kogut, J.B.: The renormalization group and the epsilon expansion. Phys. Rept. 12, 75 (1974) 7. Polchinski, J.: Renormalization and effective Lagrangians. Nucl. Phys. B 231, 269 (1984) 8. Grosse, H., Wulkenhaar, R.: Power-counting theorem for non-local matrix models and renormalisation. Commun. Math. Phys. 254, 91 (2005) 9. Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168 (2002) 10. Rivasseau, V.: From perturbative to constructive renormalization. Princeton, NJ: Princeton Univ. Press, 1991 11. Grosse, H., Wulkenhaar, R.: The β-function in duality-covariant noncommutative φ 4 -theory. Eur. Phys. J. C 35, 277 (2004) 12. Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of quantum field theory on noncommutative phase spaces. JHEP 0401, 017 (2004) 13. Grosse, H., Wulkenhaar, R.: Renormalisation of φ 4 -theory on noncommutative R4 to all orders. Lett. Math. Phys. 71, no. 1, 13–26 (2005) 14. Rivasseau, V., Vignes-Tourneret, F.: Non-commutative renormalization. http://arxiv.org/abs/ hep-th/0409312, 2004 to appear in Proceedings of conference “Rigorous Quantum Field Theory” in honor of J. Bros (19–21 July 2004, Saclay) 15. Connes, A., Douglas, M. R., Schwarz, A.: “Noncommutative geometry and matrix theory: Compactification on tori,” JHEP 9802, 003 (1998) [arXiv:hep-th/9711162] Communicated by A. Connes
Commun. Math. Phys. 262, 595–609 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1461-z
Communications in
Mathematical Physics
q-Painlev´e VI Equation Arising from q-UC Hierarchy Teruhisa Tsuda, Tetsu Masuda Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan. E-mail: [email protected] Received: 2 March 2005 / Accepted: 14 June 2005 Published online: 24 November 2005 – © Springer-Verlag 2005
Abstract: We study the q-difference analogue of the sixth Painlev´e equation (q-PVI ) by means of tau functions associated with the affine Weyl group of type D5 . We prove that a solution of q-PVI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-PVI in terms of the universal character which is a generalization of the Schur polynomial attached to a pair of partitions. Introduction The sixth q-Painlev´e equation (q-PVI ) is equivalent to the following system of qdifference equations (see [4]): f f = b7 b8
(g + b5 )(g + b6 ) , (g + b7 )(g + b8 )
gg = b3 b4
(f + b1 )(f + b2 ) . (f + b3 )(f + b4 )
(0.1)
Here f = f (a) and g = g(a) are the unknown functions in variables a = (a0 , . . . , a5 ) with a0 a1 a2 2 a3 2 a4 a5 = q; and bi ’s are the parameters given by (1.1) below; the symbols f and g stand for f (. . . , qa2 , q −1 a3 , . . . ) and g(. . . , q −1 a2 , qa3 , . . . ), respectively. Notice that a2 /a3 plays the role of the independent variable and other ai ’s (i = 2, 3) constant parameters of (0.1). This system satisfies the singularity confinement criterion which is a discrete counterpart of the Painlev´e property (see [18]), and actually goes to the sixth Painlev´e differential equation through a certain limiting procedure as q → 1. We have known at least two important aspects of nature of the sixth q-Painlev´e equation. First, q-PVI is closely related to the generalized Riemann–Hilbert problem (see [1]), as analogous to the case of a continuous one; it was shown by Jimbo–Sakai [4] that q-PVI governs the connection preserving deformation of a linear q-difference equation. The second is algebraic geometry of rational surfaces due to Sakai [19]; he presented a class of discrete Painlev´e equations defined by the group of Cremona transformations on certain rational surfaces associated with affine root systems; cf. [2]. Among them,
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q-PVI corresponds to the surface with affine Weyl group symmetry of type D5 , that is the same surface as studied by Looijenga [10]. The aim of the present work is to provide yet another formulation of q-PVI , from the viewpoint of infinite-dimensional integrable systems. An extension of the KP hierarchy, called the UC hierarchy, was proposed in [23]. This hierarchy is considered as an integrable system characterized by the universal character (see [9]) which is a generalization of the Schur polynomial attached to a pair of partitions. Also a q-difference analogue of the hierarchy (q-UC hierarchy) was studied in [25]. In this paper we prove, by using tau functions, that q-PVI coincides with a certain similarity reduction of the q-UC hierarchy. Consequently, we obtain in particular a class of algebraic solutions of q-PVI in terms of the universal character. In Sect. 1, we present the geometric formulation of q-PVI by means of tau functions; then we obtain a birational representation of the affine Weyl group of type D5 (Theorem 1.4). By virtue of this representation, we transform q-PVI equivalently into a system of bilinear equations among tau functions in Sect. 2 (Theorem 2.3). Section 3 is a summary of known results concerning the universal character and the q-UC hierarchy. Finally we see in Sect. 4 that the bilinear equations of q-PVI coincide with a similarity reduction of the q-UC hierarchy under the four-periodic condition; thus we obtain an expression of the solution of q-PVI in terms of that of the hierarchy (Theorem 4.1). Since the q-UC hierarchy admits the universal character as its homogeneous solution, we have immediately a class of algebraic solutions of q-PVI in terms of the universal character (Theorem 4.2). Section 5 is devoted to the verification of Theorem 4.1. Recall that the sixth Painlev´e equation can be deduced from q-PVI as a continuous limit and so are all the other Painlev´e equations. Hence the above relation between q-PVI and the q-UC hierarchy gives a natural explanation why the universal character appears in the solutions of the Painlev´e equations; see [13, 14]. Remark 0.1. We refer to the result [25] where the (higher order) q-Painlev´e equation (1) of type A2g+1 turns out to be a certain similarity reduction of the (2g + 2)-periodic (1)
q-UC hierarchy; cf. [24]. Here we note that the q-Painlev´e equation of type AN−1 can be obtained also from the N-periodic q-KP hierarchy, as previously shown by Kajiwara et al. [7]. It is still an interesting open question why the universal character solves the Garnier system; see [26]. Note. We use the following conventions throughout this paper. q-shifted factorials: (a; q)n =
n−1
(1 − aq i ),
i=0
n−1
(a; p, q)n =
(1 − ap i q j ).
(0.2)
i,j =0
We use also the notation (a1 , . . . , ar ; q)n = (a1 ; q)n · · · (ar ; q)n , and so on. Jacobi’s theta function: θ (a; q) = a, qa −1 ; q . ∞
(0.3)
Elliptic gamma function: (a; p, q) =
pqa −1 ; p, q ∞ (a; p, q)∞
.
(0.4)
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We have (qa; q, q) = θ (a; q), (a; q, q)
(0.5)
θ (qa; q) = −a −1 . θ (a; q)
(0.6)
and
1. The Sixth q-Painlev´e Equation In this section we present, by means of tau functions, the geometric formulation of q-PVI ; cf. [19]. Let (f, g) = (f0 /f1 , g0 /g1 ) denote the inhomogeneous coordinate of P1 × P1 . Consider the eight points, pi (1 ≤ i ≤ 8), given as follows: p1 = (−b1 , 0), p5 = (0, −b5 ),
p2 = (−b2 , 0), p6 = (0, −b6 ),
p3 = (−b3 , ∞), p7 = (∞, −b7 ),
p4 = (−b4 , ∞), p8 = (∞, −b8 ),
where b1 = a3 2 a4 −1 a5 , b2 = a3 2 a4 3 a5 , b3 = a3 −2 a4 −1 a5 , b4 = a3 −2 a4 −1 a5 −3 , (1.1) b5 = a0 −1 a1 a2 −2 , b6 = a0 −1 a1 −3 a2 −2 , b7 = a0 −1 a1 a2 2 , b8 = a0 3 a1 a2 2 , and ai ∈ C× being constant parameters such that a0 a1 a2 2 a3 2 a4 a5 = q. Let ε : X = Xa → P1 × P1 be the blowing-up at eight points pi (1 ≤ i ≤ 8); let ei = ε −1 (pi ) be the exceptional divisor and let h0 = {0} × P1 , h1 = P1 × {0}. We thus have the Picard lattice: Pic(X) = Zh0 + Zh1 + Zei , 1≤i≤8
of rational surface X, equipped with the intersection form (symmetric bilinear form), ( | ), defined by (hi |hj ) = 1 − δi,j ,
(ei |ej ) = −δi,j ,
(hi |ej ) = 0.
(D (1) ) acts on Pic(X) as the First we shall see that the (extended) affine Weyl group W 5 group of Cremona isometries of rational surface X. Here recall that an automorphism σ of Pic(X) is said to be a Cremona isometry (see [10, 19]) iff σ preserves the intersection form ( | ), the canonical divisor KX , and effectiveness of each effective divisor of Pic(X). The anti-canonical divisor −KX is uniquely decomposed into prime divisors: −KX = 2h0 + 2h1 − ei = D 0 + D 1 + D 2 + D 3 , 1≤i≤8
where D0 = h1 − e1 − e2 , D1 = h0 − e5 − e6 , D2 = h1 − e3 − e4 and D3 = h0 − e7 − e8 . Let (−KX )⊥ = {v ∈ Pic(X) | (v|Di ) = 0 for all i}, then we have the Lemma 1.1 (see [19]). (−KX )⊥ Q(D5 ): root lattice of type D5 . (1)
(1)
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T. Tsuda, T. Masuda
We have the canonical root basis B = {α0 , α1 , . . . , α5 } given as follows: α0 = e7 − e8 , α1 = e5 − e6 , α2 = h1 − e5 − e7 , α3 = h0 − e1 − e3 , α4 = e1 − e2 , α5 = e3 − e4 .
(1.2)
The intersection matrix multiplied by −1 actually coincides with the Cartan matrix of (1) D5 : 2 −1 2 −1 −1 −1 2 −1 (Cij ) = −((αi |αj )) = . −1 2 −1 −1 −1 2 −1 2 Now define the action of simple reflection si on Pic(X) corresponding to αi as si (v) = v + (v|αi )αi
for
v ∈ Pic(X),
and also that of the Dynkin diagram automorphism σi as σ1 : h{0,1} → h{1,0} , e{1,2,3,4,5,6,7,8} → e{5,6,7,8,1,2,3,4} , σ2 : e{5,6,7,8} → e{7,8,5,6} . We have in fact the fundamental relations (see e.g. [5]): si2 = 1,
si sj = sj si
(if Cij = 0),
si sj si = sj si sj
(if Cij = −1);
and σ1 ◦ s{0,1,2,3,4,5} = s{5,4,3,2,1,0} ◦ σ1 , σ2 ◦ s{0,1} = s{1,0} ◦ σ2 . One can immediately verify that each action of si and σi is a Cremona isometry. Denote by Cr(X) the group of Cremona isometries of X. (D (1) ). Proposition 1.2 (see [10, 19]). Cr(X) = s0 , . . . , s5 , σ1 , σ2 W 5 (D (1) ) on the multiplicative root variables a = In parallel, we let the action of W 5 (a0 , . . . , a5 ) be as follows: −C
si (aj ) = aj ai ij , σ1 (a{0,1,2,3,4,5} ) = a{5,4,3,2,1,0} −1 ,
σ2 (a{0,1,2,3,4,5} ) = a{1,0,2,3,4,5} −1 .
(1.3)
Secondly we shall realize the action of each element w ∈ Cr(X) as an isomorphism between rational surfaces Xa and Xw(a) . To this end, we now introduce tau functions. Consider the field L = K(τ1 , . . . , τ8 ) of rational functions in indeterminants 1/2 1/2 τi (1 ≤ i ≤ 8) with the coefficient field K = C(a0 , . . . , a5 ). Take a sub-lattice M = i=0,1,2,3 Mi of Pic(X), where Mi = v ∈ Pic(X) (v|v) = −(v|Di ) = −1, (v|Dj ) = 0 (j = i) . Definition 1.3 (cf. [6]). A function τ : M → L is said to be a tau function iff it satisfies the following conditions: (D (1) ); (i) τ (w.v) = w.τ (v) for any v ∈ M and w ∈ Cr(X) W 5 (ii) τ (ei ) = τi (1 ≤ i ≤ 8).
q-Painlev´e VI Equation Arising from q-UC Hierarchy
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We can determine such functions and the action of Cr(X) on them as follows. Suppose f0 = τ (e5 )τ (e6 ),
f1 = τ (e7 )τ (e8 ),
g0 = τ (e1 )τ (e2 ),
g1 = τ (e3 )τ (e4 ),
(1.4)
and 1
1
bi − 2 f0 + bi 2 f1 = τ (h0 − ei )τ (ei ), 1 1 bi − 2 g0 + bi 2 g1 = τ (h1 − ei )τ (ei ),
(i = 1, 2, 3, 4), (i = 5, 6, 7, 8).
(1.5)
Notice that s2 (e5 ) = h1 − e7 and s2 (e7 ) = h1 − e5 ; we then obtain the action of s2 on L. Similarly we obtain the action of s3 by s3 (e1 ) = h0 − e3 and s3 (e3 ) = h0 − e1 . The action of each element si (i = 2, 3) and σj (j = 1, 2) is realized as just a permutation of (D (1) ) τi ’s (1 ≤ i ≤ 8). Summarizing above, we thus have a realization of Cr(X) W 5 on L. Theorem 1.4. Let s0 (τ{7,8} ) = τ{8,7} , s1 (τ{5,6} ) = τ{6,5} , s4 (τ{1,2} )= τ{2,1} , s5 (τ{3,4} ) = τ{4,3} , 1 1 1 1 s2 (τ5 ) = b7 − 2 τ1 τ2 + b7 2 τ3 τ4 τ7 −1 , s2 (τ7 ) = b5 − 2 τ1 τ2 + b5 2 τ3 τ4 τ5 −1 , (1.6) 1 1 1 1 s3 (τ1 ) = b3 − 2 τ5 τ6 + b3 2 τ7 τ8 τ3 −1 , s3 (τ3 ) = b1 − 2 τ5 τ6 + b1 2 τ7 τ8 τ1 −1 , σ1 (τ{1,2,3,4,5,6,7,8} ) = τ{5,6,7,8,1,2,3,4} , σ2 (τ{5,6,7,8} ) = τ{7,8,5,6} , where bi are the parameters defined by (1.1). Then (1.6) with (1.3) gives a representation (D (1) ) = s0 , . . . , s5 , σ1 , σ2 on L. of W 5 (D (1) ) on variables Hence we obtain, by virtue of (1.4), also birational actions of W 5 (f, g) = (f0 /f1 , g0 /g1 ). Theorem 1.5 (see [19]). Let s2 (f ) = f
1
1
1
, 1
b7 − 2 g + b 7 2
b5 − 2 g + b 5 2 σ1 (f ) = g, σ1 (g) = f,
s3 (g) = g
1
1
1
1
b3 − 2 f + b3 2
b1 − 2 f + b 1 2 σ2 (f ) = f −1 ,
,
(1.7)
(D (1) ) on C(a0 , . . . , a5 )(f, g). under (1.1). Then (1.7) with (1.3) realizes the actions of W 5 (1) (D ) gives an isomorphism from Xa to Xw(a) . Moreover, each element w ∈ W 5 (D (1) ) as the sixth qFinally we regard the birational action of a translation in W 5 Painlev´e equation (see [19]): q -PVI = (σ3 ◦ s3 ◦ s5 ◦ s4 ◦ s3 ) ◦ (σ2 ◦ s2 ◦ s0 ◦ s1 ◦ s2 ) : (a0 , a1 , a2 , a3 , a4 , a5 ; f, g) → a0 , a1 , qa2 , q −1 a3 , a4 , a5 ; f , g , (g + b5 )(g + b6 ) , (g + b7 )(g + b8 ) (f + b1 )(f + b2 ) gg = b3 b4 . (f + b3 )(f + b4 ) f f = b7 b8
(1.8a) (1.8b)
Here we let σ3 = σ1 σ2 σ1 and recall that a0 a1 a2 2 a3 2 a4 a5 = q. This system goes to the sixth Painlev´e equation through a certain limiting procedure as q → 1, in fact; see [4].
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Remark 1.6. Theorem 1.5 asserts that q-PVI acts on the family of surfaces X = ∪a Xa as an automorphism; we call X the defining variety of q-PVI . Each fiber Xa of X is called the space of initial conditions of q-PVI . This is a counterpart of Okamoto’s space of initial conditions for the sixth Painlev´e equation; cf. [17]. 2. Bilinear Relations Among Tau Functions This section is devoted to deriving bilinear equations satisfied by tau functions. We will use them later in the following sections to clarify the internal relation between q-PVI and the q-UC hierarchy. (D (1) ) = s0 , . . . , s5 , σ1 , σ2 defined Let ri (0 ≤ i ≤ 3) and π be the elements of W 5 by r0 = s5 s3 s5 ,
r1 = s1 s2 s1 ,
r2 = s4 s3 s4 ,
r3 = s0 s2 s0 ,
π = σ 2 σ1 .
We can easily verify the relations among them: ri 2 = 1, (ri ri±1 )3 = 1, (ri rj )2 = 1 (j = i, i ± 1), and πri = ri+1 π, where we regard the suffix i of ri as an element (A(1) ). Note that the action of the diagram of Z/(4Z). So that r0 , . . . , r3 , π W 3 automorphism π of order four is given as π : a0 , a1 , a2 , a3 , a4 , a5 ; f, g, τ{1,2,3,4,5,6,7,8} 1 → a4 , a5 , a3 , a2 , a1 , a0 ; , f, τ{5,6,7,8,3,4,1,2} . g Now we compute the relations among the chain of four tau functions connected with the action of π: · · · → τ2 → τ6 → τ4 → τ8 → τ2 → · · · .
(2.1)
Let us consider an element (D ), = σ 2 σ 3 s0 s1 s4 s5 s2 s 3 s2 ∈ W 5 (1)
(2.2)
which acts on the parameters as : (a0 , a1 , a2 , a3 , a4 , a5 ) 1 1 1 1 2 2 . , , a0 a1 a2 a3 , a2 a3 a4 a5 , , → a1 a2 a3 a0 a2 a3 a 2 a 3 a5 a 2 a 3 a 4
(2.3)
(A(1) ) and particularly that of q-PVI ; cf. Notice that commutes with each action of W 3 (D (1) ) on tau functions given in Theorem 1.4, [22]. By using the birational actions of W 5 we therefore obtain the bilinear relation between τ2 and τ6 . Lemma 2.1. The following formula holds: a3
−1
(τ2 )
−1
(τ6 ) + (a2 a3 ) − (a2 a3 ) 2
−2
a a 21 0 4 τ2 τ6 − a2 −1 (τ2 ) (τ6 ) = 0. a1 a5 (2.4)
q-Painlev´e VI Equation Arising from q-UC Hierarchy
601
Other relations among the chain, (2.1), can be obtained via the action of π . We can verify also that (τ{1,5,3,7} ) = τ{4,8,2,6} ,
(2.5)
by straightforward computations. Let ξ = a2 a3 and use the notation aˆ i = (ai ) for convenience. Consider the variables τ˜i = τi /ψi , where ψ2 = (q 2 ξ, a2 , aˆ 2 , a3 , q −1 aˆ 3 ; q, q) × −q 2 ξ 2 , −q 2 a2 2 , −q 2 aˆ 2 2 , −q 2 a3 2 , −aˆ 3 2 ; q 2 , q 2 , ∞
(2.6)
and other ψi ’s are also determined by applying π to this formula. We have from Lemma 2.1 the Proposition 2.2. The following formula holds: 1 a 0 a4 2 1 1 a2 (τ˜2 ) −1 (τ˜6 ) + a2 a3 − + τ˜2 τ˜6 a3 a2 a3 a2 a3 a1 a5 a2 −1 (τ˜2 ) (τ˜6 ) = 0. − a2 a3 + a3
(2.7)
Proof. First we notice the formulae (ξ ) = qξ,
(a2 ) = q −1 (a2 ),
(a3 ) = q −1 (a3 ),
and the fact that π and commute with each other. We have (see (0.5)) (q 3 ξ, aˆ 2 , qa2 , aˆ 3 , a3 ; q, q) (ψ2 ) = ψ2 (q 2 ξ, a2 , aˆ 2 , a3 , q −1 aˆ 3 ; q, q) 4 2 −q ξ , −q 2 aˆ 2 2 , −q 4 a2 2 , −q 2 aˆ 3 2 , −q 2 a3 2 ; q 2 , q 2 ∞ × 2 2 −q ξ , −q 2 a2 2 , −q 2 aˆ 2 2 , −q 2 a3 2 , −aˆ 3 2 ; q 2 , q 2 ∞ θ (q 2 ξ, a2 , q −1 aˆ 3 ; q) . −q 2 ξ 2 , −q 2 a2 2 , −aˆ 3 2 ; q 2 ∞
=
Applying the action of π to this, we get θ (q 2 ξ, a3 , q −1 aˆ 2 ; q) (ψ6 ) . = 2 2 ψ6 −q ξ , −q 2 a3 2 , −aˆ 2 2 ; q 2 ∞ Accordingly we have (see (0.6))
2 −ξ , −aˆ 3 2 , −a2 2 ; q 2 ∞ θ (q 2 ξ, a2 , q −1 aˆ 3 ; q) (ψ2 ) −1 (ψ6 ) = θ (qξ, q −1 aˆ 3 , q −1 a2 ; q) −q 2 ξ 2 , −q 2 a2 2 , −aˆ 3 2 ; q 2 ∞ ψ2 ψ6 = (−qξ )−1 (−q −1 a2 )−1 (1 + ξ 2 )(1 + a2 2 ) = (ξ + ξ −1 )(a2 + a2 −1 ).
(2.8)
Applying π , we have also −1 (ψ2 ) (ψ6 ) = (ξ + ξ −1 )(a3 + a3 −1 ). ψ2 ψ6 Substituting τi = ψi τ˜i in (2.4), we arrive at (2.7) via (2.8) and (2.9).
(2.9)
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T. Tsuda, T. Masuda
Let us rename tau functions as follows: ρ{0,1,2,3} = τ˜{2,6,4,8} ,
(2.10)
−1 (ρ{0,1,2,3} ) = τ˜{3,7,1,5} .
(2.11)
so that we have (see (2.5))
Introduce the parameters di ∈ C such that π 2j (A) = q d2j +1 −d2j ,
π 2j −1 (A) = q d2j −1 −d2j ,
A=
a 0 a4 a1 a5
1 2
.
(2.12)
Since π 2 (A) = 1/A, we have d0 − d1 + d2 − d3 = 0. Let ξ = a2 a3
and
η=
a2 . a3
(2.13)
Note that π(ξ ) = ξ and π(η) = η−1 . We then obtain, from Proposition 2.2, the following bilinear equations satisfied by the quartet of tau functions ρi (i = 0, 1, 2, 3): (η + ξ −1 ) (ρ2j ) −1 (ρ2j +1 ) + q d2j +1 −d2j (ξ − ξ −1 )ρ2j ρ2j +1 −(ξ + η) −1 (ρ2j ) (ρ2j +1 ) = 0, (η
−1
+ξ
−1
−(ξ + η
) (ρ2j −1 )
−1
)
−1
−1
(ρ2j ) + q
(2.14a)
d2j −1 −d2j
(ξ − ξ
−1
)ρ2j −1 ρ2j
(ρ2j −1 ) (ρ2j ) = 0.
(2.14b)
Conversely, we can obtain a solution of q-PVI from that of the above system of bilinear equations. In summary, we have the Theorem 2.3. Let {ρi (a)}i=0,1,2,3 be a solution of (2.14), then the pair f (a) =
−1 (ρ3 )ρ1 , −1 (ρ1 )ρ3
g(a) =
−1 (ρ2 )ρ0 , −1 (ρ0 )ρ2
(2.15)
satisfies q-PVI , (1.8). Remark 2.4. If a0 a1 = a4 a5 , then leaves the variable η = a2 /a3 invariant; see (2.3). In this case, (2.14) is equivalent to the bilinear equations of the fifth q-Painlev´e equation (q-PV ); see [7, 12, 25]. We cite the result [22] by Takenawa; he showed that the defining variety of q-PV is actually included in that of q-PVI . 3. Universal Character and q-UC Hierarchy In this section we briefly review the notion of the universal character and the q-UC hierarchy; see [9, 23, 25]. A partition λ = (λ1 , λ2 , . . . ) is a sequence of non-negative integers such that λ1 ≥ λ2 ≥ · · · ≥ 0 and λi = 0 for all sufficiently large i. For a pair of partitions λ = (λ1 , λ2 , . . . , λl ) and µ = (µ1 , µ2 , . . . , µl ), the universal character S[λ,µ] (x, y) is a polynomial in (x, y) = (x1 , x2 , . . . , y1 , y2 , . . . ) defined as follows (see [9, 23]): pµl −i+1 +i−j (y), 1 ≤ i ≤ l . (3.1) S[λ,µ] (x, y) = det pλi−l −i+j (x), l + 1 ≤ i ≤ l + l 1≤i,j ≤l+l
q-Painlev´e VI Equation Arising from q-UC Hierarchy
603
Here pn are the polynomials defined by the generating function: ∞ ∞ n n pn (x)z = exp xn z , n=0
(3.2)
n=1
and pn (x) = 0 for n < 0; note that it can be explicitly written as pn (x) =
k1 +2k2 +···+nkn =n
x1 k1 x2 k2 · · · xn kn . k1 !k2 ! · · · kn !
(3.3)
The Schur polynomial Sλ (see e.g. [11]) is regarded as a special case of the universal character: Sλ (x) = det pλi −i+j (x) = S[λ,∅] (x, y). If we count the degree of each variable xn and yn (n = 1, 2, . . . ) as deg xn = n,
deg yn = −n,
then S[λ,µ] is a (weighted) homogeneous polynomial of degree |λ| − |µ|, where we let |λ| = λ1 + · · · + λl . It is known that the universal character S[λ,µ] describes the irreducible character of a rational representation of the general linear group GL(n; C) corresponding to a pair of partitions [λ, µ], while the Schur polynomial Sλ does that of a polynomial representation corresponding to a partition λ; see [9], for details. The UC hierarchy, introduced in [23], is an extension of the KP hierarchy and is an infinite-dimensional integrable system characterized by the universal character in a similar sense that the KP hierarchy is characterized by the Schur polynomial; see [15, 20, 21]. The q-UC hierarchy is a q-difference analogue of the UC hierarchy defined as follows. Consider finite subsets I ⊂ Z>0 and J ⊂ Z<0 . Let ti (i ∈ I ∪ J ) be the independent variable and Ti;q its q-shift operator defined by (i ∈ I ), qti (3.4) Ti;q (ti ) = q −1 ti (i ∈ J ), and Ti;q (tj ) = tj (i = j ). We use also the notation Ti1 i2 ...in ;q = Ti1 ;q Ti2 ;q · · · Tin ;q for the sake of simplicity. Definition 3.1 (see [25]). The following system of q-difference equations is called the q-UC hierarchy: (ti − tj )Tij ;q (ω0 )Tk;q (ω1 ) + (tj − tk )Tj k;q (ω0 )Ti;q (ω1 ) +(tk − ti )Tki;q (ω0 )Tj ;q (ω1 ) = 0,
(3.5)
where i, j, k ∈ I ∪ J . Recall that the q-UC hierarchy includes the q-KP hierarchy (see [7]) as a special case. In what follows we extend, for convenience, the universal character S[λ,µ] , (3.1), to be defined for a pair of arbitrary sequences of integers [λ, µ]. Note that for any pair ˜ µ] of sequences of integers [λ, µ], we have a unique pair of partitions [λ, ˜ such that
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S[λ,µ] = ±S[λ˜ ,µ] ˜ . Now let us consider the function s[λ,µ] = s[λ,µ] (t) in ti (i ∈ I ∪ J ) defined by s[λ,µ] (t) = S[λ,µ] (x, y), with
xn =
i∈I ti
yn =
i∈I
n
− qn
j ∈J tj
(3.6)
n
, n(1 − q n ) ti −n − q −n j ∈J tj −n n(1 − q −n )
(3.7a) .
(3.7b)
Note that we have an expression of Hn (t) = pn (x) by the following generating function: ∞
Hn (t)zn =
i∈I,j ∈J
n=0
(qtj z; q)∞ . (ti z; q)∞
(3.8)
The universal characters solve the q-UC hierarchy in the sense of Proposition 3.2 (see [25]). For every integer m and pair of sequences of integers [λ, µ], ω0 = s[λ,µ] (t),
ω1 = s[(m,λ),µ] (t),
(3.9)
satisfy the q-UC hierarchy (3.5). 4. The Solutions of the Sixth q-Painlev´e Equation In this section we see that a solution of q-PVI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular a class of algebraic solutions of q-PVI in terms of the universal character. Let I = {1, 2}, J = {−1, −2} and replace the base q with q 2 . Consider the following chain of the q-UC hierarchy: (t1 − t−2 )T1,−2;q 2 (ω2j )T−1;q 2 (ω2j +1 ) + (t−2 − t−1 )T−2,−1;q 2 (ω2j )T1;q 2 (ω2j +1 ) +(t−1 − t1 )T−1,1;q 2 (ω2j )T−2;q 2 (ω2j +1 ) = 0, (4.1a) 1 1 1 1 T−1;q 2 (ω2j −1 )T1,−2;q 2 (ω2j )+ T1;q 2 (ω2j −1 )T−2,−1;q 2 (ω2j ) − − t1 t−2 t−2 t−1 1 1 + T−2;q 2 (ω2j −1 )T−1,1;q 2 (ω2j ) = 0, − (4.1b) t−1 t1 with the four-periodic condition: ωi+4 = ωi .
(4.2)
Let {ωi = ωi (t−2 , t−1 , t1 , t2 )}i=0,1,2,3 be a solution of (4.1) satisfying the similarity condition: ωi (qt−2 , qt−1 , qt1 , qt2 ) = q di ωi (t−2 , t−1 , t1 , t2 ),
(4.3)
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where di ∈ C being the parameters given by (2.12). Let a2 a0 a1 a2 ξ = a2 a3 , η = , ζ = q. a3 a3 a4 a5
(4.4)
Define the functions Fi (ξ, η, ζ ) by Fi (ξ, η, ζ ) = ωi (t−2 , t−1 , t1 , t2 ),
(4.5)
t−1 = −q −2 ξ,
(4.6)
under the substitution t1 = η,
t2 = ζ,
t−2 = −q −2 ξ −1 .
We have the following expression of the solution of q-PVI in terms of that of the q-UC hierarchy. Theorem 4.1. The quartet (ρ0 , ρ1 , ρ2 , ρ3 ) = F0 (ξ, η, ζ ), F1 (ξ, η, q −2 ζ ), F2 (ξ, η, ζ ), F3 (ξ, η, q −2 ζ ) , (4.7) solves the system of bilinear equations of q-PVI , (2.14). Consequently the pair f =
−1 (ρ3 )ρ1 , −1 (ρ1 )ρ3
g=
−1 (ρ2 )ρ0 , −1 (ρ0 )ρ2
(4.8)
satisfies q-PVI , (1.8). The proof of the theorem is given in the next section. Finally, we shall present algebraic solutions of q-PVI . Introduce a function R[λ,µ] (ξ, η, ζ ), for each pair of partitions [λ, µ], defined by R[λ,µ] (ξ, η, ζ ) = s[λ,µ] (t) = S[λ,µ] (x, y),
(4.9)
under the substitution (4.6), or ηn + ζ n − (−ξ )n − (−ξ )−n , n(1 − q 2n ) η−n + ζ −n − (−ξ )n − (−ξ )−n yn = . n(1 − q −2n ) xn =
(4.10a) (4.10b)
Recall that the universal character S[λ,µ] is a homogeneous solution of the q-UC hierarchy whose degree equals |λ| − |µ| (see [25], or Prop. 3.2). Hence we can immediately obtain from Theorem 4.1 a class of algebraic solutions of q-PVI in terms of the universal character; cf. [24, 25]. Theorem 4.2. For every m, n ∈ Z, the quartet (ρ0 , ρ1 , ρ2 , ρ3 ) = R[(m−1)!,(n−1)!] (ξ, η, ζ ), R[m!,(n−1)!] (ξ, η, q −2 ζ ), R[m!,n!] (ξ, η, ζ ), R[(m−1)!,n!] (ξ, η, q −2 ζ ) ,
(4.11)
satisfies (2.14) with d1 − d0 = d2 − d3 = m and d2 − d1 = d3 − d0 = −n. Consequently the pair f =
−1 (ρ3 )ρ1 , −1 (ρ1 )ρ3
g=
−1 (ρ2 )ρ0 , −1 (ρ0 )ρ2
(4.12)
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gives an algebraic solution of q-PVI , (1.8), when a0 a4 = q m−n and = q m+n . (4.13) a1 a5 Here n! stands for the two-core partition (n, n−1, . . . , 1); also we let n! = (−1−n)! for n < 0. Remark 4.3. The algebraic solutions obtained in Theorem 4.2 are reduced to those of the sixth Painlev´e equation, in parallel with the continuous limit from q-PVI to the equation; cf. [13]. Remark 4.4. Let us consider the function R[(n),∅] (ξ, η, ζ ) = Rn (ξ, η, ζ ) = pn (x) under the substitution (4.10), where pn (x) is the polynomial given by (3.2), or (3.3). We then have the following expression by means of the generating function (see (3.8)): ∞ −ξ z, −ξ −1 z; q 2 ∞ n Rn (ξ, η, ζ )z = . (4.14) ηz, ζ z; q 2 ∞ n=0 Hence we obtain Rn = ζ n
(−ξ −1 ζ −1 ; q 2 )n 2 φ1 (q 2 ; q 2 )n
q −2n , −ξ η−1 2 2 ; −q ξ η , q −q 2−2n ξ ζ
(4.15)
where 2 φ1 denotes the basic hypergeometric series (see e.g. [3]): ∞ (a; q)n (b; q)n n a, b q; x = x . 2 φ1 c (c; q)n (q; q)n n=0
We remark also that Rn (ξ, η, ζ ) is equivalent to a kind of q-orthogonal polynomial called the Al-Salam–Chihara polynomial; see [8]. Example 4.5. Let us consider the polynomial P[λ,µ] (ξ, η, ζ ) = ξ |λ|+|µ| η|µ| ζ |µ| q −2|ν|
1 − q 2h(i,j )
(i,j )∈λ
q 2h(k,l) − 1 R[λ,µ] (ξ, η, ζ ).
(k,l)∈µ
Here we denote by h(i, j ) the hook-length, that is, h(i, j ) = λi + λj − i − j + 1 (see [11]); let ν = (ν1 , ν2 , . . . ) be a sequence of integers defined by νi = max{0, µi − λi }. It is interesting that P[λ,µ] seems to be a polynomial whose coefficients are all positive integers. Some examples of the polynomial are given as follows: λ
µ P[λ,µ] (ξ, η, ζ )
∅
∅ 1
(1)
∅ 1 + ξ 2 + (η + ζ )ξ
(2)
∅ q 2 (1 + ξ 4 ) + (1 + q 2 )(η + ζ )(ξ + ξ 3 ) + ((1 + q 2 )(1 + ηζ ) + η2 + ζ 2 )ξ 2
(1, 1) ∅ 1 + ξ 4 + (1 + q 2 )(η + ζ )(ξ + ξ 3 ) + ((1 + q 2 )(1 + ηζ ) + q 2 (η2 + ζ 2 ))ξ 2 ∅
(1) ηζ (1 + ξ 2 ) + (η + ζ )ξ
∅
(2) η2 ζ 2 (1 + ξ 4 ) + (1 + q 2 )ηζ (η + ζ )(ξ + ξ 3 ) + (q 2 (η2 + ζ 2 ) + (1 + q 2 )ηζ (1 + ηζ ))ξ 2
(1) (1) q 2 ηζ (1 + ξ 4 ) + q 2 (η + ζ )(1 + ηζ )(ξ + ξ 3 ) + ((1 + q 2 )2 ηζ + q 2 (η2 + ζ 2 ))ξ 2 (1) (2) q 2 η2 ζ 2 (1 + ξ 6 ) + q 2 ηζ (η + ζ )(1 + q 2 + ηζ )(ξ + ξ 5 ) +((1 + 3q 2 + 2q 4 + q 6 )η2 ζ 2 + q 2 (q 2 (η2 + ζ 2 ) + (1 + q 2 )ηζ (1 + η2 + ζ 2 )))(ξ 2 + ξ 4 ) +(η + ζ )((1 + q 2 )(1 + q 2 + q 4 )ηζ + q 4 (η2 + ζ 2 ) + q 2 (1 + q 2 )η2 ζ 2 )ξ 3
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We regard this polynomial as a q-analogue of the Umemura polynomial associated with algebraic solutions of the sixth Painlev´e equation; see [13, 16]. 5. Verification of Theorem 4.1 In order to prove the theorem, we shall see that the system of bilinear equations of q-PVI , (2.14), arises naturally from that of the q-UC hierarchy, (4.1), through a certain similarity reduction. Proof of Theorem 4.1. We have from (2.3) and (4.4) that ±1 (ξ ) = q ±1 ξ,
±1 (η) = q −1 ζ,
±1 (ζ ) = qη.
(5.1)
Notice that function ωi = ωi (t−2 , t−1 , t1 , t2 ) can be regarded as a function in variables (x, y) = (x1 , x2 , . . . , y1 , y2 , . . . ) via t1 n + t2 n − q 2n (t−1 n + t−2 n ) , n(1 − q 2n ) t1 −n + t2 −n − q −2n (t−1 −n + t−2 −n ) yn = . n(1 − q −2n )
xn =
(5.2a) (5.2b)
Let us consider the substitution of variables (see (4.6)): t1 = η,
t2 = ζ,
t−1 = −q −2 ξ,
t−2 = −q −2 ξ −1 .
We verify by using (5.1) that q 2n t1 n + t2 n − q 2n (t−1 n + q −2n t−2 n ) n(1 − q 2n ) 2n n n q η + ζ − (−ξ )n − q −2n (−ξ )−n = n(1 − q 2n ) (q 3 η)n + (qζ )n − (−qξ )n − (−qξ )−n = q −n n(1 − q 2n ) 2 n (q ζ ) + (q 2 η)n − (−ξ )n − (−ξ )−n = q −n ; n(1 − q 2n )
T1,−2;q 2 (xn ) =
and similarly
T1,−2;q 2 (yn ) = q n
(q 2 ζ )−n + (q 2 η)−n − (−ξ )n − (−ξ )−n n(1 − q −2n )
.
Combine this with the similarity condition (4.3); we obtain T1,−2;q 2 (ω2j ) = q −d2j (F2j (ξ, q 2 η, q 2 ζ )). One can verify in the same way that T−1;q 2 (ω2j +1 ) = q −d2j +1 −1 (F2j +1 (ξ, q 2 η, ζ )); and also T−2,−1;q 2 (ω2j ) = q −2d2j F2j (ξ, q 2 η, q 2 ζ ), T1;q 2 (ω2j +1 ) = F2j +1 (ξ, q 2 η, ζ ), T−1,1;q 2 (ω2j ) = q −d2j −1 (F2j (ξ, q 2 η, q 2 ζ )), T−2;q 2 (ω2j +1 ) = q −d2j +1 (F2j +1 (ξ, q 2 η, ζ )).
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Substitute (4.6) and the above formulae into the bilinear equation of q-UC hierarchy, (4.1a); replacing η and ζ with q −2 η and q −2 ζ , respectively, hence we obtain η + ξ −1 (F2j (ξ, η, ζ )) −1 (F2j +1 (ξ, η, q −2 ζ )) +q d2j +1 −d2j ξ − ξ −1 F2j (ξ, η, ζ )F2j +1 (ξ, η, q −2 ζ ) − (ξ + η) −1 (F2j (ξ, η, ζ )) (F2j +1 (ξ, η, q −2 ζ )) = 0. In parallel, we have also from (4.1b) that η−1 + ξ −1 (F2j −1 (ξ, η, q −2 ζ )) −1 (F2j (ξ, η, ζ )) +q d2j −1 −d2j ξ − ξ −1 F2j −1 (ξ, η, q −2 ζ )F2j (ξ, η, ζ ) − ξ + η−1 −1 (F2j −1 (ξ, η, q −2 ζ )) (F2j (ξ, η, ζ )) = 0.
(5.3)
(5.4)
These formulae, (5.3) and (5.4), coincide with the system of bilinear equations of q-PVI , (2.14). The proof is now complete. Acknowledgement. The authors wish to thankYasushi Kajihara, Masatoshi Noumi,Yasuhiro Ohta, Kazuo Okamoto, Hidetaka Sakai, Tomoyuki Takenawa, andYasuhikoYamada for valuable discussions and comments. This work is partially supported by a fellowship of the Japan Society for the Promotion of Science (JSPS).
References 1. Birkhoff, G. D.: The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations. Amer. Acad. Proc. 49, 521–568 (1913) 2. Dolgachev, I., Ortland, D.: Point sets in projective spaces and theta functions. Ast´erisque 165 (1988) 3. Gasper, G., Rahman, M.: Basic hypergeometric series. Encyclopedia of Mathematics and its Applications 35, Cambridge: Cambridge University Press, 1990 4. Jimbo, M., Sakai, H.: A q-analog of the sixth Painlev´e equation. Lett. Math. Phys. 38, 145–154 (1996) 5. Kac, V. G.: Infinite dimensional Lie algebras. 3rd ed., Cambridge: Cambridge University Press, 1990 6. Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y.: 10 E9 solution to the elliptic Painlev´e equation. J. Phys. A: Math. Gen. 36, L263–L272 (2003) 7. Kajiwara, K., Noumi, M., Yamada, Y.: q-Painlev´e systems arising from q-KP hierarchy. Lett. Math. Phys. 62, 259–268 (2002) 8. Koekoek, R., Swarttouw, R. F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Tech. Report 98-17, Department of Technical Mathematics and Informatics, Faculty of Information Technology and Systems, Delft: Delft University of Technology, 1998 9. Koike, K.: On the decomposition of tensor products of the representations of the classical groups: By means of the universal characters. Adv. Math. 74, 57–86 (1989) 10. Looijenga, E.: Rational surfaces with an anticanonical cycle. Ann. of Math. 114, 267–322 (1981) 11. Macdonald, I. G.: Symmetric Functions and Hall Polynomials. 2nd ed., Oxford Mathematical Monographs, New York: Oxford University Press Inc., 1995 12. Masuda, T.: On the rational solutions of q-Painlev´e V equation. Nagoya Math. J. 169, 119–143 (2003) 13. Masuda, T.: On a class of algebraic solutions to the Painlev´e VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46, 121–171 (2003) 14. Masuda, T., Ohta,Y., Kajiwara, K.: A determinant formula for a class of rational solutions of Painlev´e V equation. Nagoya Math. J. 168, 1–25 (2002) 15. Miwa, T., Jimbo, M., Date, E.: Solitons: Differential equations, symmetries and infinite dimensional algebras. Cambridge Tracts in Mathematics 135, Cambridge: Cambridge University Press, 2000
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16. Noumi, M., Okada, S., Okamoto, K., Umemura, H.: Special polynomials associated with the Painlev´e equations II. In: Integrable Systems and Algebraic Geometry, Saito, M.-H., Shimizu, Y., Ueno, K. (eds.), Singapore: World Scientific, 1998, pp. 349–372 17. Okamoto, K.: Sur les feuilletages associ´es aux e´ quation du second ordre a` points crtiques fixes de P. Painlev´e. Japan J. Math. 5, 1–79 (1979) 18. Ramani, A., Grammaticos, B., Hietarinta, J.: Discrete versions of the Painlev´e equations. Phys. Rev. Lett. 67, 1829–1832 (1991) 19. Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlev´e equations. Commun. Math. Phys. 220, 165–229 (2001) 20. Sato, M.: Soliton equations as dynamical systems on an infinite dimensional Grassmann manifold. RIMS Koukyuroku 439, 30–46 (1981) 21. Sato, M.: Soliton equations and universal Grassmann varieties. Lecture notes taken by M. Noumi, Saint Sophia Univ. Lecture Notes 18, 1984 (in Japanese) (1) 22. Takenawa, T.: Weyl group symmetry of type D5 in the q-Painlev´e V equation. Funkcial. Ekvac. 46, 173–186 (2003) 23. Tsuda, T.: Universal characters and an extension of the KP hierarchy. Commun. Math. Phys. 248, 501–526 (2004) 24. Tsuda, T.: Universal characters, integrable chains and the Painlev´e equations. Adv. Math. 197, 587–606 (2005) 25. Tsuda, T.: Universal characters and q-Painlev´e systems. Commun. Math. Phys. 260, 59–73 (2005) 26. Tsuda, T.: Toda equation and special polynomials associated with the Garnier system. Adv. Math., in press Communicated by Y. Kawahigashi
Commun. Math. Phys. 262, 611–644 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1443-1
Communications in
Mathematical Physics
Notes on Certain (0, 2) Correlation Functions Sheldon Katz, Eric Sharpe Departments of Mathematics and Physics, MC-382, University of Illinois, Urbana, IL 61801, USA. E-mail: [email protected]; [email protected] Received: 9 March 2005 / Accepted: 16 March 2005 Published online: 19 October 2005 – © Springer-Verlag 2005
Abstract: In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way. Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Half-Twisted (0, 2) Theory . . . . . . . . . . . . . . . . . . . . . . Relevant Massless States in (0, 2) Theories . . . . . . . . . . . . . Classical Correlation Functions . . . . . . . . . . . . . . . . . . . . Quantum Correlation Functions – Formal Discussion . . . . . . . . 5.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Integration over moduli space . . . . . . . . . . . . . . . . . . 5.3 Evaluation map . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Stable maps and compactifications . . . . . . . . . . . . . . . 5.5 Generalization of obstruction sheaves . . . . . . . . . . . . . 6. Linear-Sigma-Model-Based Compactifications . . . . . . . . . . . . 6.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bundles presented as cokernels, and a check of the (2, 2) locus 6.3 A technical aside . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In this paper we shall describe some correlation function computations in (large radius) heterotic nonlinear sigma models with K¨ahler (but not necessarily1 Calabi-Yau) targets, as part of a program to generalize techniques developed to understand ordinary mirror symmetry, to (0, 2) mirrors. We shall describe how to generalize the rational-curve3 counting of type II theories to generalizations of the 27 coupling in heterotic strings. We shall also check some of the predictions made recently in [1]. Recall that a perturbative heterotic string compactification is defined by not only a manifold, but also a bundle over that manifold. We will only consider holomorphic vector bundles over complex manifolds. To fix notation, we shall denote the target space by X, and we shall denote the holomorphic vector bundle2 on the target space X by E. We shall assume throughout this paper that E possesses the following two properties: 1. top E ∨ ∼ = KX , 2. ch2 (E) = ch2 (T X). The second of these properties is the well-known anomaly cancellation condition in heterotic strings, and we shall see that it plays an important role in defining the quantum product structure. The first of these properties plays a different3 and equally important role. We shall see that it is required to make sense of product structures (at both the classical and quantum levels), as well as for the additive structure of the heterotic chiral ring to be well-behaved. Also, when the target space is Calabi-Yau, this condition guarantees that a left-moving U (1) symmetry (precisely the U (1)R on the (2, 2) locus) is nonanomalous. The left-moving U (1) in question is used to build representations of the full low-energy gauge group. Hence for Calabi-Yau compactifications we preferentially consider holomorphic vector bundles with vanishing first Chern class. In any event, both of the properties above – the anomaly cancellation condition as well as the condition on first Chern classes – will play an important role in our analyses. Technically, although we will sometimes speak of heterotic chiral ‘rings,’ following [1], in general one will not always have a well-behaved ring structure. It would be more accurate to say that we will study correlation functions between certain operators in (0, 2) models, which in certain special cases (e.g. the gauge bundle is a deformation of the tangent bundle) can be interpreted as corresponding to a ring structure. If the gauge bundle is a deformation of the tangent bundle, then this ring is a deformation of the usual (2, 2) chiral ring. The ring structure encodes the correlation functions in the same 1 As a result, these sigma models have (0, 2) worldsheet supersymmetry, but beta functions do not necessarily vanish. 2 Occasionally one can use more general sheaves than just locally-free sheaves [2]; however, the circumstances under which this is allowed are not well-understood at present, so for simplicity throughout this paper we shall assume the gauge sheaf is locally-free on X. 3 In [3] a condition analogous to this one appeared in making sense of the path integral measure in certain theories used to examine singlet correlation functions in heterotic theories. There, however, the analogue of E turned out to be the tangent bundle in heterotic computations, and not the gauge bundle.
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manner that the (2, 2) correlation functions are encoded in the quantum cohomology ring. A related point concerns the possibility of loop corrections. When the target is Calabi-Yau, then our three-point correlation functions correspond to spacetime superpotential terms, and there is an old nonrenormalization theorem based on Peccei-Quinn symmetries that spacetime superpotentials do not receive loop corrections in α , only worldsheet instanton corrections (see e.g. [4]). However, in this paper we are also interested in massive two-dimensional theories in which the target space is not Calabi-Yau, and we do not have a suitable generalization of the nonrenormalization result to nonCalabi-Yau spaces. Thus, in this paper we compute worldsheet instanton corrections to certain correlation functions, designed to generalize mathematical computations, but sometimes, some of these correlation functions might also conceivably get loop corrections in addition. As our purpose in this note is to generalize some old mathematical computations, we will not speak of this possibility further. Another minor technical issue revolves around the spacetime interpretation of our correlation functions. When the target is Calabi-Yau, our three-point functions compute superpotential terms. However, the spacetime superpotential is not a function, but rather a section of a line bundle over the CFT moduli space [5]. This fact is realized on the worldsheet via U (1)R rotations as one moves about on CFT moduli space [6]. The ‘gauge choice’ needed to uniquely define a superpotential term at any point in CFT moduli space therefore amounts to a choice of normalization of the vertex operators, which cannot be extended globally over CFT moduli space unless the line bundle to which the superpotential couples is trivial. We will not speak further of this issue in this paper. We begin in Sect. 2 with a review of the half-twisted (0, 2) model in which we shall be computing correlation functions, which reduces to the A model twist on the (2, 2) locus. In Sect. 3 we describe the states that will enter into our correlation functions – essentially, generalizations of 27’s. In Sect. 4 we describe classical (no worldsheet instantons) computations of the correlation functions, and in Sect. 5 we give a formal discussion of how to take into account worldsheet instantons. Our calculations generalize the rational curve counting of type II theories, and in particular, the obstruction sheaf story seems to generalize in an intriguing fashion. To make sense of our formal calculations, we must compactify the moduli space of worldsheet instantons and describe how to extend certain sheaves (induced by the gauge bundle) over the compactification. In Sect. 6 we review how linear sigma models can be used to provide a natural compactification, and furthermore how they naturally define and extend the relevant sheaves, in a fashion compatible with symmetries. Finally in Sect. 7 we apply this technology to check some predictions of [1]. We should mention some related work. For example, considerations of (2, 2) sigma models on supermanifolds [7] lead one to study holomorphic vector bundles on Calabi-Yau manifolds, and their extensions over moduli spaces of rational curves. Some related work in heterotic compactifications is [3, 8–10], where correlation functions of gauge singlets were studied (and shown to vanish). In this paper, we are concerned with heterotic correlation functions between vertex operators for charged states – no gauge singlets appear in our correlation functions, and so the vanishing results of [3, 8–10] are not relevant. Finally, it should be noted throughout this paper that we will not couple our models to worldsheet gravity – our computations are performed for a fixed complex structure on the worldsheet – and furthermore, we will only consider genus zero worldsheets.
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2. Half-Twisted (0, 2) Theory In typical quantum cohomology calculations, the worldsheet theory is the A-model topological field theory. A (0, 2) theory cannot be twisted to give a completely topological theory, but we can do most of the twisting, and will recover most of the corresponding results. √ In a standard (0, 2) theory, one has right-moving fermions ψ+ coupling to K ⊗ φ ∗ T X and left-moving fermions λ− coupling to K ⊗ φ ∗ E, where denotes the worldsheet. We define the “half-twisted theory” by coupling fermions to bundles as follows:4 i ψ+ ∈ C ∞ φ ∗ T 1,0 X , ∨ ı ∗ 1,0 ψ + ∈ C ∞ K ⊗ φ T X , ∨ λa− ∈ C ∞ K ⊗ φ ∗ E , λa− ∈ C ∞ φ ∗ E , and it is this half-twisted theory that we shall be studying throughout this paper. In other words, just as in a standard topological twist, we make the worldsheet spinors into worldsheet scalars and vectors. Note that although we perform this operation on both left- and right-movers, the resulting theory will not be a topological field theory in general, since it only has (0, 2) worldsheet supersymmetry, and not (2, 2). Also note that on the (2, 2) locus, where E = T X, the twisting above is equivalent to the A model twist. See [8] for another discussion of half-twisted theories in heterotic compactifications. One of the reasons two-dimensional topological field theories are useful is that they simplify the computation of correlation functions (see e.g. [11]). Many three-point functions in physical (type II) theories are the same as certain three-point functions in corresponding topological field theories. Schematically, < ψψφ >phys = < ψψψ >tf t In a nutshell, the reason for this equivalence is that the spectral flow insertion used to generate the vertex operator for the spacetime boson φ above from the vertex operator for the corresponding spacetime spinor can also be interpreted as generating the topological field theory. Although the mechanism for relating physical and topological correlation functions is well-known, let us pause for a moment to very briefly and schematically review it. In a nutshell, to twist the worldsheet theory means adding a term proportional to 21 ωψψ to the worldsheet action, where ω is the worldsheet spin connection. Now, ψψ is proportional to the U (1)R current J . If we bosonize J ∼ ∂φ, then the term added looks like 2 Rφ. By concentrating the worldsheet curvature at points, so that R ∼ δ (z − z0 ), we see that topological twisting is essentially the same as inserting factors of the form exp(φ), which is spectral flow. Analogous statements can also be made relating correlation functions in physical (0, 2) theories to corresponding correlation functions in the half-twisted theory described above. Recall that a three-point correlation function in a physical (0, 2) theory 4 We have used Hermitian metrics on X and E to write φ ∗ T 0,1 (X) as (φ ∗ T 1,0 (X))∨ and φ ∗ E as (φ ∗ E )∨ . We do this to make a Riemann-Roch calculation in Sect. 4 more transparent.
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has two aspects, coming from the left- and right-movers. In a three-point function the right-movers behave just as in a type II theory, as described above. The left-movers encode gauge information, and require an analogue of spectral flow, which can also be alternately interpreted as twisting. For example, suppose we have an irreducible rank 3 three holomorphic vector bundle, and we wish to compute a 27 coupling5 . The bundle breaks E8 to E6 , and the E6 is built on the worldsheet from SO(10) × U (1), the SO(10) from the left-moving fermions in the E8 but not coupled to the bundle, and the U (1) from a left-moving symmetry6 on the fermions coupled to the bundle, under which λa and λa have equal and opposite charges. (On the (2, 2) locus, this U (1) is the U (1)R of the left-moving N = 2 algebra.) Just as a physical three-point correlation function involves both spacetime spinors and a spacetime boson, one analogously uses both the 16 and 10 representations of SO(10) that are part of the 27 of E6 : 27 = 10−1 ⊕ 161/2 ⊕ 12 3
under SO(10) × U (1). In particular, the physical 27 correlation function can be computed in the form < ψ16 ψ16 φ10 >phys . Just as in the right-movers, the vertex operator for φ could be obtained from that for ψ via spectral flow, similarly the left-moving 10 can be obtained from the left-moving 16 through the action of a left-moving analogue. Just as in (2, 2) theories, where these spectral flow insertions could be interpreted as giving topological twists, in (0, 2) theories these insertions of (analogues of) spectral flow can also be interpreted as giving the half-twisted theory, hence < ψ16 ψ16 φ10 >phys = < ψψψ >half −twist . Because the half-twisted theory simplifies computations of correlation functions as described above, in this paper we shall use it exclusively to compute correlation functions. In computations of rational curve corrections in type II theories, the A model is often coupled to worldsheet topological gravity in the literature. In this paper, we will not couple to topological gravity. Furthermore, when computing n-point correlation functions on P1 for n > 3, we shall not use descendants of n − 3 of the vertex operators; we shall compute products in the purely half-twisted (0, 2) theory. 3. Relevant Massless States in (0, 2) Theories In a type II string theory, the chiral ring consists of massless RR sector states, which (at large radius) are counted by Dolbeault cohomology H p,q (X) of the target space X, or, equivalently, elements of the sheaf cohomology groups H q X, p T ∗ X . 3
On the (2, 2) locus, the 27 gets worldsheet instanton corrections, whereas the 273 does not. On the 3 (2, 2) locus, the 27 corresponds to an A model calculation, whereas the 273 corresponds to a B model computation. 6 It is straightforward to compute that in a physical (0, 2) theory on a Calabi-Yau, this left-moving U (1) will be nonanomalous precisely when c1 (E ) = 0. 5
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By Hodge theory, these Dolbeault cohomology groups generate the full de Rham cohomology. In a heterotic string compactification on a space X with gauge bundle E, recall from [12, Sect. 3] that the charged massless RR states are counted (at large radius) by H q X, p E ∨ which is clearly analogous to the type II string result. In the special case that E = T X, we duplicate the type II result. We shall sometimes, when sensible, speak of a heterotic chiral ring associated to certain (0, 2) heterotic theories. This ring will be described additively by the sum of sheaf cohomology groups of the form above, i.e. ∗,∗ ≡ Hhet
H q X, p E ∨
p,q
Note that this ring is naturally bigraded. Physically, that bigrading corresponds to the distinction between left- and right-movers. The heterotic chiral ring is associated to part of the gauge sector of the heterotic theory. The multiplication in the ring is described by the correlation functions. In principle, there are additional massless states in a large radius heterotic compactification – there are gauge singlets that correspond to complex, K¨ahler, and bundle moduli, as well as additional charged matter fields, such as elements of H n (X, End E) come from the NS-R sector. However, we shall not consider such fields here. Clearly we have states of the form H 1 (X, E ∨ ) in our state space, the (0, 2) generalization of H 1 (T ∨ ), corresponding to K¨ahler moduli on the (2, 2) locus. Note that the (0, 2) theory does still contain K¨ahler moduli, but in the (0, 2) theory those states are strictly gauge neutral. A (2, 2) theory contains both gauge neutral as well as gauge charged states corresponding to complex structure moduli – i.e. in addition to singlet moduli fields in the target space theory corresponding to complex moduli, there are also charged 27’s corresponding to complex moduli. A (0, 2) theory contains singlets corresponding to complex moduli, but even for a rank three E, the 27’s are counted by H 1 (X, E) and not H 1 (X, T ). In the introduction we mentioned that, in addition to the anomaly cancellation condition, throughout this paper we shall impose the constraint that top E ∨ ∼ = KX .
(1)
We can see one motivation for this constraint already in the additive structure of our chiral states. Recall that Serre duality in (0, 2) theories maps spectra back into themselves [12]. Serre duality acts as ∗ H i X, j E ∨ ∼ = H n−i X, j E ⊗ KX ∗ ∼ = H n−i X, r−j E ∨ ⊗ r E ⊗ KX , where r is the rank of E. Notice that when the line bundle r E ⊗ KX
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on X is trivial, i.e. r E ∨ ∼ = KX , Serre duality still closes our states back into themselves, but if this line bundle is nontrivial, Serre duality does not close the states back into themselves. Thus, just to make the additive structure well-behaved under Serre duality, we must require the constraint above. The condition (1) also impacts correlation functions. For definiteness, let E have rank r and let X have dimension n. Note that the condition r E ∨ ∼ = KX implies that there is an isomorphism φ : H n (X, r E ∨ ) → C. This isomorphism already allows classical products of operators in the chiral ring to be evaluated, and we see that it will similarly be crucial in the evaluation of the quantum products. For Calabi-Yau compactifications, another physical motivation for this constraint comes from the fact that our bundles must be embeddable inside E8 . Although SU (n) has natural E8 embeddings for small n which cleanly correspond to worldsheet physics, realizing U (n) embeddings on the worldsheet is not as well understood. In particular, for a U (n) embedding there is a worldsheet anomaly in a left-moving U (1) symmetry used to build vertex operators. Thus, although one can work with (0, 2) CFT’s with c1 = 0, to make them useful for a compactification one typically adds extra left-movers to cancel out the c1 , hence as a practical matter we only consider holomorphic vector bundles with c1 = 0. 4. Classical Correlation Functions In a (2, 2) chiral ring, the classical product is obtained by wedging together enough differential forms to get a top form on the target space X, which can then be integrated over X to produce a number. In other words, for the product of k operators we have the maps H q1 (X, p1 T ∗ X) ⊗ · · · ⊗ H qk (X, pk T ∗ X) −→ H n (X, n T ∗ X) ∼ = C, (2)
where qi = pi = n. The first map is given by cup and wedge products, and the second map is given by integration of a top form on X. This mathematical analysis is backed-up physically by e.g. ghost number conservation, which implies that the only nonzero correlation functions can come from top-degree forms. The classical correlation functions in our heterotic theory involve a product of states such that the sums of the degrees of the sheaf cohomology groups is the dimension n of the target-space X, and that the number of E’s in the coefficients equals the rank r of E. Indeed, wedging together representative differential forms, from such a product we get an element of H n X, r E ∨ . (3) This is the analogue of a top-degree form in the present case. However, we can only get a number from this top-degree form in special circumstances. In the special case that r E ∨ ∼ = KX , then we can identify an element of (3) with a top form on X, which can then be integrated to get a number. Next, let us take a moment to consider the physical situation more carefully.
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i couple to φ ∗ T 1,0 X and In the half-twisted theory, as described in Sect. 2, the ψ+ ∨ ı couple to K ⊗ φ ∗ T 1,0 X , hence the number of ψ i zero modes, minus the the ψ+ + ı zero modes, is given by Hirzebruch-Riemann-Roch as number of ψ+ χ φ ∗ T 1,0 X = c1 (φ ∗ T X) + n(1 − g).
This is the anomaly in the (right-moving) U (1)R symmetry. (Compare e.g. [1] above Eq. (154).) This anomaly gives us a selection rule that is responsible for the top-degree-form constraint when there are no worldsheet instantons. Consider the case that our vacuum consists of constant maps φ, i.e. no worldsheet instantons. Then φ ∗ T X is a trivial bundle over the worldsheet, so c1 (φ ∗ T X) = 0, hence χ = n(1 − g). Thus at string tree level i zero modes, hence the degrees of the sheaf cohomology groups must add there are nψ+ up to n (= dim X) in order to hope to get a nonzero result for the correlation function. As outlined in Sect. 2, to most efficiently see the product structure in the chiral ring, we also want to consider the case that the left-moving fermions are also twisted, so that ∨ λa− couples to K ⊗ φ ∗ E and λa− couples to φ ∗ E. Here we compute an anomaly χ φ ∗ E = c1 φ ∗ E + r(1 − g). When we are expanding about constant φ, we have c1 (φ ∗ E) = 0. Thus at string tree level we recover the rule that for a correlation function to be nonvanishing, the number of λa− ’s appearing must equal the rank of E. In any event, from counting zero modes of worldsheet fermions, we see that when there are no worldsheet instantons, for a product to be nonvanishing the sum of the degrees of the sheaf cohomology groups must match the dimension of the target, and the sum of the exterior powers of the coefficients must match the rank of E. In other words, for a product to be non-vanishing, the wedge product of differential forms must give us the analog of a top form, as we outlined earlier. As noted above, in the special case that r E ∨ ∼ = KX , this wedge product living in (3) can be identified with a top form and subsequently integrated to get a number. A precise choice of trivialization of r E ⊗KX∨ is needed to normalize correlation functions. This is analogous to the (2, 2) chiral ring, where a precise choice of nowhere-zero holomorphic top form is needed to normalize the correlation functions. Thus, the constraint r E ∨ ∼ = KX not only makes the path integral well-defined, but is also used in order to be able to define the classical product. We will see the same constraint in defining quantum products. 5. Quantum Correlation Functions – Formal Discussion 5.1. Generalities. The selection rules derived above from anomalies no longer apply in the presence of worldsheet instantons, but rather are corrected. Previously if the sum of the degrees of the sheaf cohomology groups was greater than the dimension of the manifold, the resulting correlation function must vanish, reflecting a basic property of the (‘classical’)Yoneda product, really just a property of wedge products. In an instanton background, however, it is possible to have a nonzero correlation function even when that sum of degrees is strictly greater than the dimension of the target manifold. This is what gives rise to “quantum” products – products that are nonzero, but become zero when α → 0.
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Suppose that we want to compute a correlation function associated to elements of the heterotic chiral ring. Specifically, recall that the selection rule on right-movers is that the sum of the degrees of the sheaf cohomology groups must equal c1 (φ ∗ T X) + n(1 − g),
(4)
where n is the dimension of the target and g is the genus of the worldsheet, and the sum of the powers of the bundle E ∨ appearing must be c1 (φ ∗ E) + r(1 − g),
(5)
where r is the rank of E. In a one-instanton background, neither c1 (φ ∗ T X) nor c1 (φ ∗ E) need vanish. There is a significant complication, not present in the (2, 2) case, coming from operator determinants. In a (2, 2) topologically-twisted theory, the operator determinants for the left- and right-moving worldsheet fermions precisely cancel against the operator determinant for the worldsheet bosons. In a physical theory, no such cancellation occurs, and certainly in a (0, 2) theory, twisted or not, no such cancellation is possible. (See also [9] for a more extensive discussion of this issue.) Thus, all string tree level correlation functions necessarily contain numerical factors amounting to the partition function of left-moving fermions on S 2 . If these numerical factors were the same for all worldsheet instanton contributions, then we could ignore them, or absorb them into redefinitions. However, these numerical factors for any one worldsheet instanton depend upon the restriction of E to that rational curve, as after all they represent the partition function for left-moving fermions coupling to E. If two different rational curves have non-isomorphic restrictions of E, then the numerical factors will be different. Far worse now is the case of a family of curves, in which the cohomology of the restriction of E can jump in the family [13]. If the cohomology of the restriction of E jumps, then surely the corresponding numerical factor jumps, and so the required integral over the moduli space of zero modes can not have nearly so simple a form as appeared in [14]. We shall ignore this possibility, and work only in g = 0 (at fixed complex structure), so that this determinant is just a number that can be reabsorbed into other definitions. We will also ignore group theory factors associated to the representations of the unbroken gauge group needed to specify the gauge sector contributions to the operators. In the rest of this section we shall only work formally, on smooth not necessarily compact spaces of maps of fixed degree from the worldsheet into X of fixed degree. Our goal in this section is to outline the general ideas, not to fill in all details. In Sect. 6 we shall describe how to compactify the moduli spaces and extend these constructions over those compactifications. 5.2. Integration over moduli space. In this subsection, we shall formally outline how to compute rational curve corrections in the special case that there are no excess zero modes, that there is no analogue of the obstruction sheaf. (We shall make this restriction precise momentarily.) The general case will be discussed in Sect. 5.5. Readers unfamiliar with these computations for (2, 2) theories might wish to consult [15, Sect. 3.3], [16, Sect. 3.3] for a very readable review. In this section we shall only work formally. We consider an idealized situation which is almost never realized: we suppose that we have a smooth and compact instanton moduli space M of maps from the fixed worldsheet into X of fixed degree. On the other
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hand, it is quite common to find noncompact but smooth moduli spaces. In Sect. 6 we shall describe how to compactify the moduli spaces and extend these constructions over those compactifications. Recall in the (2, 2) case, the chiral ring is composed of (p, q) forms on the target space X. There is a selection rule that correlation functions are nonzero when the sum of the p’s and q’s separately equals (dim X) (1 − g) + c1 φ ∗ T X (the same as the selection rule (4) on the degrees of sheaf cohomology groups). Each of those (p, q) forms on X is associated to a (p, q) form on the moduli space M of worldsheet instantons of given degree, and the selection rule above becomes the statement that the wedge product of those differential forms on M is formally a top form, since formally the dimension of M is given by (dim X) (1 − g) + c1 φ ∗ T X . ∼ KX as well as the In the heterotic case, with a rank r bundle E obeying r E ∨ = anomaly cancellation condition, elements of the sheaf cohomology groups H p X, q E ∨ are mapped to sheaf cohomology groups H p M, q F ∨ on the moduli space M, where F is a sheaf on M of rank (X, φ ∗ E). We will discuss the precise map in the next subsection, but for the purpose of outlining the general ideas, for now we merely assert such a map exists. Let us assume that we have a universal instanton α : × M → X, where is the worldsheet.7 We have the natural projection π : × M → M. Now we can define a sheaf F on M by F = π∗ α ∗ E. In this section, we will assume that R 1 π∗ α ∗ E = 0 = R 1 π∗ α ∗ T X.
(6)
In the case where E = T X, we have that F = T M, as the fiber of F at a point of M corresponding to an instanton φ : → X is H 0 (, φ ∗ T X), the space of first-order deformations of the map φ.8 The first equality in (6) tells us that the map φ is unobstructed. In particular M is smooth. The second equality is the natural extension. By Riemann-Roch, this condition implies that the fibers of F have the same dimension, so that F is a vector bundle of rank given in Eq. (5). Physically the assumption (6) means that there are no excess zero modes. We will discuss the more general case in Sect. 5.5. 7 Here is where we will find it hard to enforce the compactness of M in practice. In a typical situation maps from the worldsheet can degenerate to maps from the worldsheet with some 2-spheres attached. 8 Note that we are not considering stable maps here as we are not identifying maps differing by an automorphism of . As we have not coupled to worldsheet gravity, worldsheet automorphisms are irrelevant. Unlike topological string theories, in a topological field theory, even at genus zero one can have nonzero two-point couplings.
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Under the assumption (6), we can compute c1 (F) by Grothendieck-Riemann-Roch. Letting a subscript of k on a cohomology class denote its complex codimension k component, and letting η be the pullback to × M of the cohomology class of a point of , we have c1 (F) = ch(F)1 = π∗ ch(α ∗ E)Td(T ) 2 = π∗ α ∗ c1 (E)2 /2 − c2 (E) + (1 − g) ηα ∗ c1 (E) .
(7)
In (7) we have used ch2 (E) = c12 (E)/2 − c2 (E). If we now apply (7) to E = T X we obtain c1 (T M) = π∗ ch(α ∗ T X)Td(T ) 2 = π∗ α ∗ c1 (T X)2 /2 − c2 (T X) + (1 − g) ηα ∗ c1 (T X) . By the anomaly condition together with (1), we conclude that c1 (F) = c1 (T M), implying9 that top F ∨ ∼ = KM , as desired. The selection rules (4) and (5) then imply that we can get nonvanishing correlation functions by a straightforward extension of the procedure (2), replacing X by M and T ∗ X by F ∨ : H q1 (M, p1 F ∨ ) ⊗ · · · ⊗ H qk (M, pk F ∨ ) −→ H N M, R F ∨ ∼ = H N (M, KM ) ∼ = C,
(8)
given by cup and wedge product of cohomology classes followed by integration of a top form. Technically, as mentioned in the previous section, the correlation function should be more properly described as
det ∂ φ ∗ E H top (M, KM ) M det ∂ φ ∗ T X instead of merely
M
H top (M, KM ) .
But at genus zero, the ratio of operator determinants is just a number, which we shall suppress. We shall also studiously ignore the possibility that the splitting behavior of φ ∗ E changes over M in such a way that this ratio changes on a set of nonzero measure on M. Understanding this ratio of operator determinants would be much more important for calculations at higher genus, and for properly understanding coupling to worldsheet gravity. In (8), N is the dimension of M as given by the selection rule (4) and R is the rank of F as given by the selection rule (5). Also, note that the anomaly cancellation condition played a crucial role in defining this product, as it is only because of the anomaly cancellation condition that we have R F ∨ ∼ = KM . 9 At least when M is smooth, K¨ahler, and simply-connected, assumptions we will freely make for the purposes of the formal discussion of this section.
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In the next section, we will describe maps ψk : H q X, p E ∨ −→ H q M, p F ∨ , the subscript k corresponding to the k th insertion point. This will complete our description of the correlation functions in our ideal situation. Given
cohomology classes ηi ∈
H qi X, pi E ∨ satisfying the selection rules qi = N and pi = R, we apply (8) to the ψi (ηi ) to get the value of the desired correlation function, using algebraic geometry. 5.3. Evaluation map. In order to make sense of the calculations just described, we need to describe the maps ψi : H q (X, p E ∨ ) → H q (M, p F ∨ ). More precisely, by restricting the universal map α : × M → X to pi ∈ (where pi is the i th insertion point), we have an evaluation map evi : M −→ X, evi (φ) = φ(pi ). Note that upon identifying M with {pi } × M, we see that evi is just the restriction of α to M. We need to define a map ψi : H q X, p E ∨ → H q M, p F ∨ . (9) First, note that pulling back by evi , we get a map ∨ H q X, p E ∨ −→ H q M, p α ∗ E |{pi }×M .
(10)
So to derive the map (9) we just need to find a sheaf map ∨ p α ∗ E |{pi }×M −→ p F ∨ yielding a corresponding map on cohomology ∨ H q M, p α ∗ E |{pi }×M −→ H q M, p F ∨ to compose with the pullback map (10) to arrive at ψi . For simplicity of notation, we complete the argument in the case p = 1; the generalization to arbitrary p is straightforward. We shall demonstrate the existence of the dual map F → (α ∗ E) |{pi }×M , and will work in terms of local sections of the corresponding sheaves. Let U be an open subset of M, then F(U ) = π∗ α ∗ E (U ) (by definition) = α ∗ E (π ∗ U ) = H 0 × U, α ∗ E so we have now expressed local sections of F in terms of local sections of α ∗ E. Next, we can restrict those sections, getting a map H 0 × U, α ∗ E −→ H 0 U, α ∗ E |{pi }×M . These are, however, precisely the local sections of the sheaf (α ∗ E) |{pi }×M over the open set U ⊆ M. Thus, we have a map of local sections F(U ) −→ α ∗ E |{pi }×M (U ) as desired, completing the description of the maps ψi .
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Note that if E = T X, then the composition T M = F → α ∗ E|{pi }×M = evi∗ E = evi∗ T X is easily checked to be thedifferential map of the evaluation evi : M → X. Thus, in this case, the map ψi : H q X, p E ∨ → H q M, p F ∨ coincides with the pullback by evi on ordinary cohomology H p,q (X) → H p,q (M). It follows that on the (2, 2) locus, the correlation functions coincide with those of ordinary Gromov-Witten theory, with minor modifications due to the turning off of topological gravity. In summary, the heterotic chiral ring is a generalization of the chiral ring of GromovWitten theory and quantum cohomology, and as such, deserves to be better understood mathematically. In the next section, we relate to standard Gromov-Witten theory a little more closely.
5.4. Stable maps and compactifications. In this section, we describe how to modify our basic construction of the preceding sections to more realistic situations. Suppose we want to compute a correlation function with k insertions. Let M g,k (X, d) be the moduli space of genus g stable maps to X of degree d 10 as introduced in [17]. These are maps f : C → X of degree d, where C is a connected algebraic curve of genus g with marked points p1 , . . . , pn in the smooth locus of C modulo automorphisms of C preserving f and the pi . The stability condition is that each genus 0 component of C on which f is constant has at most 3 special (nodal or marked) points. There are evaluation maps ei : M g,k (X, d) → X defined by ei (f ) = f (pi ). There is a forgetful map ρ : M g,k (X, d) → M g,k which forgets the map and contracts any components of C which have become unstable after forgetting the map. Let , p1 , . . . , pk ∈ M g,n be our worldsheet with its fixed complex structure and insertion points. For ease of notation we denote this simply by ∈ M g,n . Then our more realistic model for the instanton moduli space is M = ρ −1 (). Note that M is compact, and its elements correspond to degree d maps φ : → X, as well as maps from the union of with trees of 2 spheres to X of total degree d. Now there is still a universal map. It is well known that the universal family of curves over M g,k (X, d) is given by M g,k+1 (X, d). There is a map πk : M g,k+1 (X, d) → M g,k (X, d) which is given by forgetting the last marked point and contracting unstable components. The fiber of πk corresponds to the location of an extra point on C, which explains why M g,k+1 (X, d) is the universal curve over M g,k (X, d). The map ek+1 : M g,k+1 (X, d) → X is then clearly the universal map. There are k natural sections si : M g,k (X, d) → M g,k+1 (X, d) of πk given by sending a stable map f : C → X with marked points pi to the point pi , identified with the corresponding point on the universal curve. 10 More properly, we should replace d with a homology class β ∈ H (X, Z), but we just denote this 2 by d and call it a “degree” for ease of locution.
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We now let C = πk−1 (M), and let α be the restriction of ek+1 to C. Then let π be the restriction of πk to C. This gives the diagram C π
−→α
X .
M We now see that this is the proper generalization of the universal map α : × M → X and projection π : × M → M considered previously. In particular, we can now define F = π∗ α ∗ E as before. To get the generalized maps ψi : H q X, p E ∨ → H q M, p F ∨ , we just use the sections si : M g,k (X, d) → M g,k+1 (X, d) in place of the embeddings M = {pi } × M ⊂ × M and proceed as before. The price to be paid for this general construction using well-known mathematics is that these M are almost never smooth except in simple cases like projective spaces and g = 0. So to evaluate correlation functions, we would need to develop a virtual fundamental class in this context. This is very similar to Gromov-Witten theory, except that topological gravity is turned off (by restricting to the fiber of ρ and the virtual fundamental class must be modified accordingly). Furthermore, the techniques of Gromov-Witten theory do not apply in general. However, the techniques of localization can be applied if X admits a torus action and the bundle E is equivariant for that torus action. Such bundles are studied in [18, 19] if X is a toric variety. We do not develop these techniques here since one of our goals in the present paper is to verify the claims of [1] and the gauge bundles appearing there are not of this type. It would nevertheless be interesting to develop computational techniques for this equivariant situation. In Sect. 6 we will describe another compactification, the linear sigma model compactification. We expect this to coincide with the nonlinear heterotic theory for simple spaces such as products of projective spaces, and, by analogy with [20], to be related to the heterotic theory by a change of variables.
5.5. Generalization of obstruction sheaves. Next, let us turn to the case in which R 1 π∗ α ∗ T X = 0, R 1 π∗ α ∗ E = 0, i.e. the case that would call for obstruction sheaves on the (2, 2) locus. Our analysis in Sect. 5.2 only makes sense physically in the case that there are ψ i zero modes, but no ψ ı zero modes, and λa zero modes, but no λb zero modes. The index theorems quoted in Sect. 5.2 only specify the difference between the number of such zero modes. We shall describe a proposal for how this case should be described mathematically, that will generalize the obstruction bundle story of the (2, 2) locus. We shall check that our proposal satisfies basic consistency tests, and reduces to the ordinary obstruction bundle story on the (2, 2) locus. However, there is a crucial mathematical statement we have not yet been able to prove.
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Write the number of zero modes of the four types of worldsheet fermions as follows: ψ i : h0 (, φ ∗ T X) = m + p, ψ ı : h1 (, φ ∗ T X) = p, λa : h1 (, φ ∗ E) = q, λa : h0 (, φ ∗ E) = r + q. The numbers m, r are calculated by index theory, and the numbers p, q count ‘excess’ zero modes. We still need to make a simplifying assumption. We assume that the number of excess modes p and q are constant on the instanton moduli space M, although possibly nonzero. As a consequence, M will still be smooth, of dimension m + p, and F will still be a bundle, of rank r + q. But now the selection rule on the correlators gives bundle-valued forms of degree m, r – so the selection rules no longer define top forms on the moduli space in the presence of excess zero modes. We will fix this problem by wedging with more bundle-valued differential forms, but before we discuss the mathematics, let us review the physics. Just as in [14], we can soak up these excess zero modes by using the four-fermi term in the worldsheet action. Recall the four-fermi term has the form Fi ab ψ i ψ λa λb , (11)
where F is the curvature of the bundle E. (Note, in particular, that the structure of the four-fermi term in a sigma model is asymmetric between the left- and right-movers: the curvature of the left-moving bundle E appears in the action, but there is no term representing the curvature of the right-moving bundle T X.) Each time we bring down a factor of this four-fermi term, we soak up one of each type of worldsheet fermion. So long as p = q, which is the case for (2, 2) theories, we can use this four-fermi term to soak up all of the excess zero modes. In general, however, p need not equal q. Suppose, without loss of generality, that p > q. Then we could use q factors of the four-fermi terms to soak up all of the excess λ zero modes, and all but p − q of the excess ψ zero modes. We could bring down an additional p − q factor of the four-fermi term, and then use λ propagators to contract away the excess λ fermions. The resulting correlation function would exhibit a dependence on the positions on the worldsheet in which the correlators were inserted. The result is not topological, but since this theory is not a topological field theory, there is no good reason to believe that all correlators under consideration can be expressed purely topologically. As we are only interested in those correlators that can be expressed purely topologically, henceforth we shall only consider theories for which H 1 , φ ∗ T X = H 1 , φ ∗ E
(12)
for all φ, i.e. p = q everywhere on M. Phrased more formally, we shall assume that rank R 1 π2∗ α ∗ E = rank R 1 π2∗ α ∗ T X everywhere on M, i.e. that p = q in the notation introduced above.
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Let us now return to the mathematical description of our correlators. As outlined above, the physical selection rule says that images of the vertex operators as sheaf cohomology on M wedge together to form an element of H m M, r F ∨ but the dimension of M is m + p, and the rank of F is r + q, so we need an additional factor if we wish the correlation functions to be expressible as integrals of top forms. This problem has a well-known solution on the (2, 2) locus. In [14], bringing down factors of the four-fermi term was interpreted in terms of wedging the differential forms representing correlators (whose total degree was too small to be a top form) with a differential form representing the Euler class (coming from the four-fermi term factors), which had the effect of making the integrand a top form, so that correlation functions again naturally generate numbers. There is an analogous phenomenon here. If we try to interpret each (0, 2) four-fermi term (11) as a bundle-valued differential form on the moduli space, then each such four-fermi term should be identified with an element of H 1 M, F ∨ ⊗ F1 ⊗ G1∨ (13) on symmetry grounds, where F1 ≡ R 1 π∗ α ∗ E, Gi ≡ R i π∗ α ∗ T X are the sheaves over the moduli space defined by the zero modes of the fermions. (The ψ i is responsible for having degree one sheaf cohomology; the other three fermions are responsible for the coefficients.) We should stress at this point, however, that this is an ansatz. One would like to be able to prove mathematically that the curvature F of E defines an element of the sheaf cohomology group above, but we have not yet been able to do this. Nevertheless, we can check that this ansatz is consistent. First, we shall show that with this ansatz, correlation functions can be expressed as integrals of top forms, i.e. naturally generate numbers. Previously in Sect. 5.2, when R 1 π∗ α ∗ E and R 1 π∗ α ∗ T X both vanished, we saw that the heterotic anomaly cancellation condition and GrothendieckRiemann-Roch implied that ∨ top F = KM
exactly as needed for our correlation functions to generate a number. However, more generally we have a slightly different statement. If we define F1 ≡ R 1 π∗ α ∗ E, Gi ≡ R i π∗ α ∗ T X, as above, then the anomaly cancellation condition and Grothendieck-Riemann-Roch imply top F ⊗ top F1∨ ∼ = top G0 ⊗ top G1∨ . In particular, since G0 ∼ = T M, this means that top F ∨ ⊗ top F1 ⊗ top G1∨ ∼ = KM
Notes on Certain (0, 2) Correlation Functions
so that we can fix up correlation functions by wedging with a representative of H q M, q F ∨ ⊗ q F1 ⊗ q G1∨
627
(14)
(recall F1 has rank q, matching the rank of G1 ). Note, however, that if we bring down enough copies of the (0, 2) four-fermi term to absorb the ‘excess’ zero modes, then from our ansatz (13), we generate precisely the factor (14) above. In other words, our interpretation of the (0, 2) four-fermi terms is precisely what we need to describe correlators as integrals of top forms. Let us now check that the description above gives correct results on the (2, 2) locus. In this case, F1 G1 so that the F1 and G1 factors in (14) cancel out. Nevertheless, the particular class of (14) used depends on G1 , as we will see. Recall that in the (2, 2) case, correlators are described by differential forms on M, not sheaf cohomology groups, and the factor (14) is replaced by the top Chern class of G1 , i.e. cq (G1 ). Chern classes and sheaf cohomology can be related via the Atiyah class of the bundle, which we shall briefly review. Consider expressing Chern classes in terms of the curvature of the connection on a holomorphic vector bundle E on a K¨ahler manifold X: cr ∝ Tr F ∧ F ∧ · · · ∧ F. Taking advantage of hermitian fiber metrics on vector bundles, we can write F = Fi ab dzi ∧ dz ∧ λa ∧ λb . Furthermore, because of the Bianchi identity, if F is a holomorphic connection (i.e. Fij = Fı = 0), then F is ∂-closed. Thus, we can think of F as a holomorphic (0, 1)form valued in T X∨ ⊗ E ⊗ E ∨ , and so in particular the curvature F defines an element of H 1 X, T X∨ ⊗ E ⊗ E ∨ . The sheaf cohomology class above is known as the Atiyah class and is independent of the choice of connection [21]. Taking the trace defines a map H 1 X, T X ∨ ⊗ E ⊗ E ∨ → H 1 X, T X ∨ ∼ = H 1,1 (X) whose image is the first Chern class of the bundle. Furthermore, note that by wedging r copies of F together we create an element of H r X, r T X ∨ ⊗ r E ⊗ r E ∨ and the trace defines a map from the sheaf cohomology group above to H r X, r T X ∨ ∼ = H r,r (X) whose image is the r th Chern class. In particular, on the (2, 2) locus, the factor (14) is the sheaf cohomology description of the q th Chern class of G1 , as claimed. Thus, even though we do not yet understand mathematically how the Atiyah class of E induces an element of the sheaf cohomology group (13), our proposal satisfies reasonable consistency tests. This gives us some confidence in our interpretation.
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6. Linear-Sigma-Model-Based Compactifications 6.1. Generalities. Linear sigma models were used in [20] to provide natural compactifications of moduli spaces of worldsheet instantons. Let us take a moment to review their construction. Recall that given a toric variety expressed as a GIT quotient of a set of homogeneous polynomials xa by a set of C× actions, the linear sigma model moduli space is described as a GIT quotient of the space of zero modes of the xa . Let na denote the weights of homogeneous coordinate xa with respect to the C× actions, and let d denote the (fixed) degree of the worldsheet instantons (expressed as a vector of weights with respect to the same C× actions). Then we can expand each xa as xa ∈ H 0 P1 , O( na · d) = xa0 uda + xa1 uda −1 v + · · · where u, v are homogeneous coordinates on the worldsheet P1 . To for da = na · d, construct the linear sigma model moduli space, we take the space of xai , and quotient by the same set of C× ’s as defined for the original toric variety, such that each xai has weight vector na (same as for the original xa ), after removing an exceptional set. The original (2,2) gauged linear sigma models of [22] were generalized in [23] to describe (0,2) theories, that is, Calabi-Yau’s together with bundles. The bundles are presented physically in a very special form, as the cohomology of a short complex E
F
na ) −→ ⊕O(m i) ⊕O −→ ⊕O( sometimes known as a ‘monad’. The bundles in question are bundles over11 the ambient toric variety. If a superpotential is present to specify a Calabi-Yau subvariety, then the bundle on the subvariety is obtained by restricting the bundle on the ambient space. The methods of [20] can be extended in a very straightforward fashion to define extensions of the sheaves F, F1 over the (compact) linear sigma model moduli spaces, as we will discuss in numerous examples in the rest of this section. The basic idea is to expand each worldsheet field corresponding to a line bundle, in a basis of zero modes, and associate the coefficients to line bundles on M. For example, if we had a reducible bundle E = ⊕a O( na ), described by a set of free left-moving fermions of charges na , then our ansatz yields the induced bundles ⊗C O( F = ⊕a H 0 P1 , O( na · d) na ), ⊗C O( F1 = ⊕a H 1 P1 , O( na · d) na ). It can be shown12 that this ansatz matches the R i π∗ α ∗ E on the open subset of M corresponding to honest maps, and furthermore that this ansatz has the desired rank for F F1 11 In this paper, we will work with monads describing bundles over the ambient space. However, historically monads have been used to describe bundles only over a Calabi-Yau complete intersection. The distinction involves whether the composition of maps F ◦ E vanishes identically (as we will assume throughout this paper) or only up to hypersurface equations (as is more typical in the older literature [2, 23]). 12 Clearly
na ) R i π∗ α ∗ E = ⊕a R i π∗ α ∗ O( for i = 0, 1, and these sheaves have the same ranks as the F , F1 listed. To show that the R i π∗ α ∗ E completely decompose into a direct sum of line bundles one uses the T -equivariant nature of the line bundles O( na ), as we will describe in detail in the section on bundles presented as cokernels.
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as well as the correct determinant line bundle top F ∨ ⊗ top F1 . These verifications are special cases of arguments we will present later, so we omit details. In the rest of this section we will study this for the case of bundles presented as cokernels, which will allow us to check that this ansatz correctly reproduces known results on the (2, 2) locus. The analysis for more general presentations is very closely related, and so for reasons of brevity, is omitted from this paper. (See instead [26] for a more detailed discussion.) We will also check that this ansatz for induced sheaves is compatible with needed results, e.g. that top F ∨ ⊗ top F1 ∼ = KM ⊗ top G1 continues to hold after extending the sheaves over the compactification of the moduli space, and that in all cases, our ansatz gives sheaves that agree with the R i π∗ α ∗ E on the open subset of M corresponding to honest maps. An important technical issue is that the anomaly cancellation condition in gauged linear sigma models is slightly stronger than the mathematical statement ch2 (E) = ch2 (T X), as shown in [2], and can even distinguish different presentations of the same gauge bundle. We specifically require the stronger form, as described in [2], in order for our constructions to be consistent. In particular, in Sect. 6.3 we shall see some examples which satisfy ch2 (E) = ch2 (T X) but fail the stronger linear sigma model anomaly cancellation condition, despite merely being alternative presentations of well-behaved linear sigma models. In these examples, our construction fails. We will discuss the linear sigma model anomaly cancellation condition in more detail later. A related issue, also discussed in Sect. 6.3, that the extension of the sheaves R i π∗ α ∗ E over the compactification divisor depends upon the presentation of E, and different presentations can give very different extensions.
6.2. Bundles presented as cokernels, and a check of the (2, 2) locus. 6.2.1. General analysis of bundles presented as cokernels. Given a bundle E presented in the linear sigma model as a cokernel of the form na ) −→ E −→ 0 0 −→ O⊕k −→ ⊕a O( for worldsheet instantons of degree d we have an induced long exact sequence ⊗C O −→ ⊕a H 0 P1 , O( ⊗C O( na · d) na ) −→ F 0 −→ ⊕k H 0 P1 , O(0 · d) 1 1 1 1 ⊗C O −→ ⊕a H P , O( ⊗C O( −→ ⊕k H P , O(0 · d) na · d) na ) −→ F1 −→ 0. The long exact sequence above is obtained by expanding worldsheet fields in their zero modes, and interpreting the coefficients in the expansion as defining line bundles on the linear sigma model moduli space. Each map of the original cokernel sequence induces a map between the zero modes of the worldsheet fields, hence induces a map between bundles on the linear sigma model moduli space. This long exact sequence simplifies to give the short exact sequence, ⊗C O( 0 −→ O⊕k −→ ⊕a H 0 P1 , O( na · d) na ) −→ F −→ 0 (15)
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and the relation
⊗C O( F1 ∼ na · d) na ). = ⊕a H 1 P1 , O(
(16)
To verify that this method is giving reasonable results, we shall next check the following things: • First, we shall check that this gives the correct results on the (2, 2) locus. • We shall argue that more generally, F and F1 agree with R 0 π∗ α ∗ E, R 1 π∗ α ∗ E, respectively, on the open subset of M corresponding to honest maps. • Finally, we shall check that rank (F F1 ) and top F ∨ ⊗top F1 satisfy the correct relations for our formal analysis of correlation functions to proceed. First, let us check that our results are correct on the (2, 2) locus. Note that when E is the tangent bundle of a toric variety, F as described above is the tangent bundle to the linear sigma moduli space of [20]. When E = T X, the sheaf F1 is known as the obstruction bundle13 , as we shall now check. First, a couple of easy tests of this statement. Loosely, the obstruction sheaf is ı zero modes, and that is precisely the the sheaf over the moduli space defined by the ψ+ physical meaning of F1 . More concretely, since − na · d − 1 na · d ≤ −1 1 1 h P , O( na · d) = 0 otherwise we see that the top Chern class of the obstruction sheaf F1 is given by c1 (O( na ))−na ·d−1 , na ·d≤−1
a result that precisely matches [20][Eq. (3.62)]. Next, let us check more systematically that on the open subset of the linear sigma model moduli space M corresponding to honest maps, F1 really is the obstruction sheaf. Recall that on the (2,2) locus, the obstruction sheaf is R 1 π∗ α ∗ T X, at least over the locus where there is a universal instanton α : P1 × M → X. Now if X is toric, present the tangent bundle as 0 −→ O⊕m −→ ⊕a O( na ) −→ T X −→ 0. Applying π∗ α ∗ gives rise to an exact sequence including terms na ) −→ R 1 π∗ α ∗ T X −→ 0. R 1 π∗ α ∗ O⊕m −→ ⊕a R 1 π∗ α ∗ O( The first term is zero since H 1 (P1 , O) = 0, so we get an isomorphism na ). R 1 π∗ α ∗ T X ∼ = ⊕a R 1 π∗ α ∗ O( To reconcile with the computation of F1 in Eq. (16), we need to compare R 1 π∗ α ∗O( na) with ⊗C O( H 1 P1 , O( na · d) na ). 13 Technically, we shall show that F agrees with the obstruction sheaf over the open subset of the 1 moduli space corresponding to honest maps. Strictly speaking, we are not aware of a previous general definition of obstruction sheaf over (compact) linear sigma model moduli spaces.
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is the fiber of These sheaves both have the same ranks, since H 1 P1 , O( na · d) R 1 π∗ α ∗ O( na ). In addition, note that the induced torus action decomposes R 1 π∗ α ∗ O( na ) into eigenbundles, each of which have precisely the same torus weight as the bundle O( na ). Since there is a single weight and M is itself a toric variety, we simply get a sum of line bundles, as expected. Thus, our sheaf F1 really does match the obstruction sheaf on the (2,2) locus, as advertised. We shall outline an example of an obstruction sheaf in Sect. 6.2.2. Note that from Eq. (16), we see that the obstruction sheaf over any linear sigma model moduli space is always a direct sum of line bundles. Now that we have checked that on the (2, 2) locus the sheaves F, F1 are precisely the tangent bundle and obstruction sheaf, as expected, next we shall check that F and F1 match R 0 π∗ α ∗ E and R 1 π∗ α ∗ E on the open subset of M corresponding to honest maps. The analysis is very similar to our earlier comparison of F1 on the (2, 2) locus to the obstruction sheaf. To do this, we return to the definition of E as a cokernel 0 −→ O⊕k −→ ⊕a O( na ) −→ E −→ 0 which induces, on the open subset of M corresponding to honest maps, the long exact sequence 0 −→ ⊕k R 0 π∗ α ∗ O −→ ⊕a R 0 π∗ α ∗ O( na ) −→ R 0 π∗ α ∗ E 1 ∗ 1 ∗ −→ ⊕k R π∗ α O −→ ⊕a R π∗ α O( na ) −→ R 1 π∗ α ∗ E −→ 0. Since R 1 π∗ α ∗ O = 0, this long exact sequence simplifies to become the short exact sequence, 0 −→ O⊕k −→ ⊕a R 0 π∗ α ∗ O( na ) −→ R 0 π∗ α ∗ E −→ 0 and the relation R 1 π∗ α ∗ E ∼ na ). = ⊕a R 1 π∗ α ∗ O( For the same reasons as in our discussion of the obstruction sheaf, the sheaves R i π∗ α ∗ O( na ) all split into a direct sum of line bundles, and so we now see explicitly that when restricted to the open subset U of M corresponding to honest maps, F|U ∼ = R 0 π∗ α ∗ E 1 ∗ ∼ and F1 |U = R π∗ α E. Let us next check that the sheaves F, F1 derived above do indeed possess the properties discussed in Sect. 5.5, namely that top F ∨ ⊗ top F1 ∼ = KM ⊗ top G1 and that rank F − rank F1 = χ (φ ∗ E) = c1 (φ ∗ E) + rank E. To perform this verification, let us present the tangent bundle to the toric variety in the form 0 −→ O⊕m −→ ⊕n O( qn ) −→ T X −→ 0.
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Writing n = (ni ), and using E as described above, the anomaly cancellation condition in the (0,2) gauged linear sigma model takes the form j j nia na = qni qn a
n
for each i, j , a condition which is very slightly stronger than the statement of matching second Chern characters [2]. Similarly, we shall also impose the requirement that for each i, nia = qni , a
n
which is a slightly stronger form of the constraint ∼ KX . top E ∨ = First, consider the ranks of F and F1 , rank F = na · d + 1 − k na ·d≥0
− na · d − 1
rank F1 =
na ·d<0
from which we see that rank F − rank F1 = rank E +
na · d
a
= rank E + c1 (φ ∗ E) precisely as desired. Next, consider the product of top exterior powers of F and F1 . From their expressions above, we see that c1 (F) = na · d + 1 nia Ji , na ·d≥0
c1 (F1 ) =
− na · d − 1 nia Ji ,
na ·d<0
where the Ji generate degree two cohomology of M (and may be constrained by relations). Thus,
i i c1 (F F1 ) = na · d na Ji , na Ji + a
a
where denotes the K-theoretic difference. For the tangent bundle,
i i c1 (T M G1 ) = qn J i + qn · d qn Ji . n
n
However, using the anomaly cancellation condition above plus the statement that top E ∨ ∼ = KX , we see immediately that c1 (F F1 ) = c1 (T M G1 ) precisely as desired.
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6.2.2. Example: Obstruction sheaves and Hirzebruch surfaces. So far we have proven general properties of this class of (0, 2) gauged linear sigma models, and described details of the special case of the tangent bundle of PN−1 . Next, we shall consider a special case on the (2, 2) locus in which the obstruction sheaf is nontrivial, to further illustrate these ideas. Specifically, we will consider moduli spaces of maps into the curve of self-intersection −n on a Hirzebruch surface Fn . We can describe the Hirzebruch surface in terms of four homogeneous coordinates s, t, u, v, with weights under two C× actions as follows: s 1 0
λ µ
t 1 0
u n 1
v 0 1
The homogeneous coordinates on the P1 fibers are u, v. Maps into the P1 fiber have weight (0, 1) under (λ, µ), and similarly maps into the base have weight (1, 0). Following [24], Chap. IV], maps into the curve of self-intersection −n should have weight (1, 0) − n(0, 1) = (1, −n) under (λ, µ). For such maps, when we expand in zero modes, we find that s and t are sections of OP1 (1), u is a section of OP1 , and v is a section of OP1 (−n). Let x and y be homogeneous coordinates on the worldsheet P1 , then we can expand s t u v
= ax + by, = cx + dy, = e, = 0,
for a, b, c, d, e complex numbers which we can be identified with homogeneous coordinates on the linear sigma model moduli space. The moduli space is determined by a pair of C× actions, whose actions on the homogeneous coordinates a, b, c, d, e are determined by the actions on the homogeneous coordinates s, t, u, v on Fn . If we denote the C× actions on the moduli space by (λ, µ), then the homogeneous coordinates a, b, c, d, e have weights given by λ µ
a 1 0
b 1 0
c 1 0
d 1 0
e n 1
The obstruction sheaf on the moduli space, computed by the methods given earlier, is given by n−1 H 1 P1 , O(−n) ⊗C O(0, 1) = O(0, 1). 1
But O(0, 1) is the trivial bundle on M since the apparent freedom of e has been removed as noted earlier. Thus the obstruction bundle is On−1 . Since the euler class of On−1 is trivial for n ≥ 2, all correlation functions will vanish. This was to be expected, as the virtual dimension is E · c1 (Fn ) + 2 = 4 − n, so that all three point functions vanish for n ≥ 2 by simple dimension considerations.
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6.3. A technical aside. A linear sigma model describes a physically natural compactification of moduli spaces of maps, and (0, 2) linear sigma models describe physically natural extensions of the sheaves R i π∗ α ∗ E over the compactifications of the moduli spaces. However, we shall see in this section that the precise extensions depend upon the presentation of the gauge bundle E, and not all presentations yield consistent linear sigma models. In particular, it is important for our construction that the linear sigma model anomaly cancellation condition be met, which is somewhat stronger than the mathematical statement ch2 (E) = ch2 (T X). We shall see that the linear sigma model anomaly cancellation condition can even distinguish different presentations of the same gauge bundle. Presentations that fail the linear sigma model anomaly cancellation condition have different extensions over the compactification divisor which do not have desirable properties. If we schematically let charges of left-moving fermions be denoted na and charges of right-moving fermions be denoted qs , then we have imposed two conditions, namely a
nia =
a j nia na
qsi for each i,
s
=
j
qsi qs for each i, j.
s
The first of these statements is the linear-sigma-model version of the requirement top E ∨ ∼ = KX , and the second [2] is the linear-sigma-model version of the anomaly cancellation condition ch2 (E) = ch2 (T X). The tangent bundle of P1 × P1 gives an example of this phenomenon. Consider the following three presentations of this bundle: 1. The standard (2,2) presentation, as a cokernel 0 −→ O⊕2 −→ O(1, 0)2 ⊕ O(0, 1)2 −→ T P1 × P1 −→ 0 corresponding physically to four charged fermi superfields (part of the (2, 2) chiral superfields describing the homogeneous coordinates) and two neutral chiral superfields (part of the (2,2) gauge superfields corresponding to the two U (1)’s). 2. A cokernel presentation for one T P1 , and a free fermion presentation for the other: 0 −→ O −→ O(1, 0)2 ⊕ O(0, 2) −→ T P1 × P1 −→ 0. 3. A free fermion presentation for both sides: T P1 × P1 ∼ = O(2, 0) ⊕ O(0, 2). Only the standard (2, 2) presentation satisfies the strong linear sigma model anomaly cancellation condition for a gauge bundle on P1 × P1 ; the second two presentations do not satisfy this condition. As a result, each of these physical presentations of the tangent bundle gives a different induced sheaf F over the compactified moduli space, and only for the (2, 2) presentation does the resulting induced sheaf have desirable properties. The first presentation yields F isomorphic to the tangent bundle of the moduli space P2d1 +1 × P2d2 +1 .
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The second presentation yields 0 −→ O −→ ⊕21 H 0 P1 , O(d1 ) ⊗C O(1, 0) ⊕ H 0 P1 , O(2d2 ) ⊗C O(0, 2) −→ F −→ 0 or, equivalently, 2d +1 F ∼ = π1∗ T P2d1 +1 ⊕ ⊕1 2 O(0, 2).
The third presentation yields F = H 0 P1 , O(2d1 ) ⊗C O(2, 0) ⊕ H 0 P1 , O(2d2 ) ⊗C O(0, 2) 1 +1 2 +1 = ⊕2d O(2, 0) ⊕ ⊕2d O(0, 2). 1 1
So long as d1 and d2 are both positive, F1 ≡ 0 in each case. Note that for each presentation, rank F = (2d1 + 1) + (2d2 + 1) = rank T P1 × P1 + c1 φ ∗ T P1 × P1 . However, although the ranks match, it is straightforward to check that, just as for T P1 , the first Chern classes do not match, and (except for the first presentation) do not have the desired symmetry property top F ∨ ∼ = KM . The reason for this is the same as for P1 : the linear sigma model anomaly cancellation condition is only met for the first presentation. Thus, we see in these examples several important technicalities of linear sigma models and our constructions: • First, different presentations of a single, fixed gauge bundle E can define different extensions of R i π∗ α ∗ E across the compactified moduli space. • Second, the linear sigma model anomaly cancellation condition can distinguish different presentations of the same bundle. • Third, if a given presentation of the gauge bundle fails the linear sigma model anomaly cancellation condition, then even if there exists an alternate presentation that satisfies that condition, the induced sheaves F, F1 determined by the failing presentation have the wrong Chern classes to be useful in our calculations of correlation functions.
7. Verification of Results of Adams-Basu-Sethi In this section we shall give first-principles verifications of some conjectures made in [1].
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7.1. Physical prediction for quantum cohomology. One of the first examples of a heterotic quantum cohomology ring14 considered in [1] [Sect. 6.2] concerns a heterotic theory on P1 × P1 , with gauge bundle given by a deformation of the tangent bundle. Recall that we can describe the tangent bundle of P1 × P1 as the cokernel of the map
O⊕O
x1 0 x2 0 0 x 1 0 x2
−→
O(1, 0)2 ⊕ O(0, 1)2 ,
where x1 , x2 are homogeneous coordinates on the first P1 , and x1 , x2 are homogeneous coordinates on the second P1 . In [1] [Sect. 6.2] a deformation of the tangent bundle of P1 × P1 is described as the cokernel of the map x1 α1 x1 + α2 x2 x2 α1 x1 + α2 x2 β x + β x x1 1 1 2 2 β1 x1 + β2 x2 x2
O⊕O
−→
O(1, 0)2 ⊕ O(0, 1)2 ,
where αi , αi , βi , βi are constants. In the special case that α1 = 1 and α2 = 2 , and all other constants vanish, the authors of [1] obtain what can be described as a deformation of the chiral ring relations for a (2, 2) model on P1 × P1 , namely X˜ 2 = exp(it2 ), X − (1 − 2 ) X X˜ = exp(it1 ), 2
where t1 , t2 are K¨ahler parameters describing the sizes of the P1 ’s, and X, X˜ are degree two generators. (The sign ambiguity is meaningless, because 1 , 2 act as homogeneous coordinates on the deformation space.) Note in the special case of the (2, 2) locus, where 1 = 2 , the relations above reduce precisely to the ordinary quantum chiral ring relations for P1 × P1 . In terms of correlation functions, the quantum cohomology claim above allows us to compute quantum corrections in terms of classical results. We shall describe the classical computation in Sect. 7.2. Without even doing the classical computation, it is not hard to see that consistency15 of the ring structure forces a nonzero value for X 2 when 1 = 2 (i.e. off the (2, 2) locus) while the other classical correlation functions are unchanged. In any event, the classical correlation functions can be shown to be given by < X˜ 2 > = < 1 > = 0, < X X˜ > = 1, < X 2 > = 1 − 2 .
(17)
We can formally compute a higher-order correlator by using the quantum cohomology relations, e.g. < X X˜ 3 > = < X X˜ X˜ 2 >, (18) 14 As emphasized earlier, in a heterotic theory, one will not be able to make sense of a ring structure in general, but in special cases – such as the deformation of the tangent bundle considered here – such a structure can still be meaningful. 15 Write < X 2 >= g for some function g of , , with the other classical correlation functions as 1 2 2 ˜2 ˜3 listed. Onecan then compute < X X >= exp(it2 ) and < X X >= g exp(it2 ). Then, plug those values 2 2 into < X˜ X − (1 − 2 )XX˜ >= 0 to find that g = 1 − 2 .
Notes on Certain (0, 2) Correlation Functions
= < X X˜ > exp(it2 ) = exp(it2 ), 2 < X X˜ 5 > = < X X˜ X˜ 2 > = exp(2it2 ), < X X˜ 7 > = exp(3it2 ), < X˜ 4 > = < 1 > exp(2it2 ) = 0, < X2 X˜ 2 > = < X2 > exp(it2 ) = (1 − 2 ) exp(it2 ), < X X˜ X2 − (1 − 2 )X X˜ > = < X X˜ > exp(it1 ) = exp(it1 ), < X3 X˜ > = exp(it1 ) + (1 − 2 )2 exp(it2 ), < X4 > = < X2 exp(it1 ) + (1 − 2 ) X X˜ >
637
(19) (20) (21) (22) (23) (24) (25) (26)
= 2 (1 − 2 ) exp(it1 ) + (1 − 2 )3 exp(it2 ) (27) and so forth. In Sect. 7.2, we shall directly compute the four-point functions listed above, and we will directly confirm that they do have the form listed. In general, one would only expect to obtain the answer up to a coordinate change, of course, but in this simple example, we shall recover the advertised four-point functions. One quick physical way to derive the quantum cohomology statement above (following [20], but slightly different from [1]) is to compute the one-loop effective superpotential 1 1 σ1 + 1 σ2 σ1 + 2 σ2 W˜ eff = ϒ1 i τˆ1 − log − log 2π 2π σ σ 1 1 2 2 + ϒ2 i τˆ2 − log − log 2π 2π (where the ϒ’s are (0,2) gauge multiplets), from which setting ∂ W˜ eff = 0 ∂ϒa gives the relations (σ1 + 1 σ2 ) (σ1 + 2 σ2 ) = q1 , σ22 = q2 which, after a change of variables, are equivalent to the relations given in [1]. In the case of interest, the gauge bundle E is given by
0 −→ O ⊕ O
x1 1 x1 x2 2 x2 0 x 1 0 x2
−→
O(1, 0)2 ⊕ O(0, 1)2 −→ E −→ 0.
(28)
For degree (1, 0) maps, the linear sigma model moduli space is P3 × P1 . If we let α0 , α1 , α0 , α1 denote homogeneous coordinates on the P3 (obtained from expanding out the homogeneous coordinates x1 , x2 in terms of their zero modes), and let β0 , β1 denote the
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homogeneous coordinates on the P1 , then the sheaf F on the moduli space is described by
0 −→ O ⊕ O
α0 α1 α0 α 1 0 0
1 α0 1 α1 2 α0 2 α1 β0 β1
−→
O(1, 0)4 ⊕ O(0, 1)2 −→ F −→ 0.
(29)
Next, we shall find polynomial representatives for the relevant sheaf cohomology groups. From (28), we derive the long exact sequence 0 −→ H 0 X, E ∨ −→ H 0 O(−1, 0)2 ⊕ O(0, −1)2 −→ H 0 O2 (30) −→ H 1 X, E ∨ −→ H 1 O(−1, 0)2 ⊕ O(0, −1)2 −→ ··· from which we read off that H 1 X, E ∨ = H 0 X, O2 = C2 .
(31)
In particular, the two elements of H 1 (X, E ∨ ) can both be represented by constants. Essentially the same calculation using (29) in place of (28) reveals that H 1 M, F ∨ = H 0 M, O2 = C2 . We therefore have a natural map H 1 (X, E ∨ ) C2 H 1 (M, F ∨ ) which is the linear sigma model version of the maps ψi (9). 7.2. Computation of the correlation functions. Let us now explicitly compute the classical correlation functions listed in Eq. (17) and outline the computation of the four-point functions listed in Eqs. (27), (25), (23), (19), and (22). The classical contributions to these four-point functions all vanish, and the only worldsheet instanton contributions can come from the (1, 0) and (0, 1) sectors. Before we begin, we need to observe that the natural basis (31) for H 2 (P1 × P1 , E ∨ ) does not specialize to the usual basis for H 2 (P1 × P1 ) generated by the hyperplane classes of the P1 factors on the (2, 2) locus. Let’s denote the basis determined by (31) as Y, Y˜ . The 4 × 2 matrix in (28), specialized to the (2, 2) locus 1 = 2 , shows that Y, Y˜ is related to the standard quantum cohomology basis X, X˜ by X = Y + Y˜ ,
X˜ = Y˜ ,
(32)
as can be inferred from the first two rows (resp. the last 2 rows). Here we have put = 1 = 2 . To explain the method, we compute the classical correlation functions of Y and Y˜ in detail. We start with the dual of (28)
0 → E ∨ → O(−1, 0)2 ⊕ O(0, −1)2
x1 x2 0 0 1 x1 2 x2 x˜1 x˜2
−→
O2 → 0.
(33)
Notes on Certain (0, 2) Correlation Functions
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Let us denote by e1 , e2 , f1 , f2 the natural basis for O4 , corresponding to the columns of the matrix in (33). We want to compute the cohomology classes Y, Y˜ ∈ H 1 (E ∨ ). We do this by computing images of (1, 0) (0, 1) ∈ H 0 (O2 ) = C2 of the coboundary mapping in (30). We compute using the Cech cover Uij = {xi = 0, x˜j = 0} of P1 × P1 , where 1 ≤ i, j ≤ 2. Let us first compute Y˜ as the coboundary of (0, 1). First we must lift (0, 1) to a section of O(−1, 0)2 ⊕ O(0, −1)2 over Uij , and the simplest lift is x˜j−1 fj . Then Y˜ has a Cech representative ˜j−1 fj . Y˜ij,i j = x˜j−1 fj − x
(34)
To make sense of (34) we fix an ordering of the sets in the open cover, say U11 , U12 , U21 , U22 . Similarly, we lift (1, 0) to xi−1 ei − i x˜j−1 fj on Uij , yielding the Cech representative −1 ˜j−1 fj − i x˜j−1 Yij,i j = xi−1 ei − x i ei + i x fj .
(35)
To compute the classical correlator Y˜ 2 , we take the cup and wedge product of Y˜ with itself to get a Cech representative of H 2 (2 E ∨ ). From our constraint (1) this is identified with an element of H 2 (K) C so we will get a number. In the present situation of P1 × P1 , we have K O(−2, −2). To compute explicitly, note the inclusion 2 E ∨ O(−2, −2) → 2 O(−1, 0)2 ⊕ O(0, −1)2 O(−2, 0) ⊕ O(−1, −1)4 ⊕ O(0, −2).
(36)
Our strategy is to first compute the cup product as a representative of H 2 (2 (O(−1, 0)2 ⊕ O(0, −1)2 ) = H 2 (O(−2, 0)⊕O(−1, −1)4 ⊕O(0, −2)) using the inclusion (36). Then we interpret this cocycle as a representative of H 2 (2 E ∨ ) = H 2 (O(−2, −2)). We do this explicitly by computing the inclusion (36). So the computation proceeds from (35) and (34) by the explicit computation of (36) and the explicit computation of the cup product. For any sheaves S, T , the cup product H 1 (S) ⊗ H 1 (T ) → H 2 (S ⊗ T ) is given in terms of Cech representatives ω and η of elements of H 1 (S) and H 1 (T ) by (ω ∪ η)abc = ωab ⊗ ηbc , where restrictions to appropriate smaller open sets are understood and suppressed from the notation. Note that ω ∪ η is indeed a cocyle if ω and η are: δ (ω ∪ η)abcd = ωbc ⊗ ηcd − ωac ⊗ ηcd + ωab ⊗ ηbd − ωab ⊗ ηbc = (ωbc − ωac ) ⊗ ηcd + ωab ⊗ (ηbd − ηbc ) = −ωab ⊗ ηcd + ωab ⊗ ηcd = 0, and similarly ω ∪ η is a coboundary if either ω or η is a coboundary while the other is a cocycle.
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To describe the map (36) we note that any map O(−2, −2) → O(−2, 0) ⊕ O(−1, −1)4 ⊕ O(0, −2) is given by multiplication with an element of O(0, 2) ⊕ O(1, 1)4 ⊕ O(2, 0)
(37)
which we now compute. From (33) we compute that on the open set where x˜1 = 0, the sheaf E ∨ is the row space of x˜1 x2 −x˜1 x1 (2 − 1 )x1 x2 0 0 0 x˜2 −x˜1 with a similar expression when x˜2 = 0. The maximal minors of this matrix are 0 x2 x˜1 x˜2 −x˜12 x2 −x1 x˜1 x˜2 x1 x˜12 (2 − 1 )x1 x2 x˜1 . Note that this is a multiple of 0 x2 x˜2 −x˜1 x2 −x1 x˜2 x1 x˜1 (2 − 1 )x1 x2 .
(38)
We get precisely the same factor over the open set where x˜2 = 0, therefore this must be the desired element of (37). We can rewrite this as x2 x˜2 e1 ∧ f1 − x˜1 x2 e1 ∧ f2 − x1 x˜2 e2 ∧ f1 + x1 x˜1 e2 ∧ f2 + (2 − 1 )x1 x2 f1 ∧ f2 . (39) This gives a computational simplification. When we compute a cup and wedge product of elements of H 1 (E), then the resulting Cech representative written as a representative of H 2 (O(−2, 0) ⊕ O(−1, −1)4 ⊕ O(0, −2)) must be a multiple of (39) on each open set. To find the multiple, hence the class in H 2 (O(−2, −2)), we need only compute the coefficient of one of the ei ∧ fj , say, e1 ∧ f1 . Let’s compute Y˜ 2 . Since there are no ei in the Cech representatives (34) for Y˜ij,i j , we cannot obtain an e1 ∧ f1 term in the cup product. Hence Y˜ 2 = 0. For Y Y˜ we compute using (34) and (35), omitting terms not involving e1 or f1 , = 1 x˜1−1 f1 ∧ −x˜1−1 f1 = 0, Y ∪ Y˜ 11,12,21 1 Y ∪ Y˜ = −x1−1 e1 + 1 x˜1−1 − 2 x˜2−1 f1 ∧ −x˜1−1 f1 = e1 ∧ f 1 , 11,21,22 x1 x˜1 Y ∪ Y˜ = 1 x˜1−1 f1 ∧ 0 = 0, 11,12,22 1 Y ∪ Y˜ = −x1−1 e1 − 2 x˜1 −1 f1 ∧ −x˜1−1 f1 = e1 ∧ f 1 . 12,21,22 x1 x˜1 After dividing by the coefficient x2 x˜2 of e1 ∧ f1 from (39), we learn that the Cech representative φ of Y Y˜ as an element of O(−2, −2) is given by φ11,12,21 = 0,
φ11,21,22 = (x1 x2 x˜1 x˜2 )−1 ,
φ11,12,22 = 0,
φ12,21,22 = (x1 x2 x˜1 x˜2 )−1 .
(40)
Notes on Certain (0, 2) Correlation Functions
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We now make an explicit choice of isomorphism H 2 (O(−2, −2)) C. First of all, it is not hard to see that Cech representatives of H 2 (O(−2, −2)) C that do not have a term involving (x1 x2 x˜1 x˜2 )−1 are coboundaries. For example, a section of the form (x12 x˜1 x˜2 )−1 can be extended to U11 ∩U12 and then used to construct a coboundary. So we only need to look at coefficients of 1/(x1 x2 x˜1 x˜2 ). Let Aij,i j ,i j be these coefficients. Note that the cocycle condition implies A12,21,22 − A11,21,22 + A11,12,22 − A11,12,21 = 0.
(41)
Now take a section ρ on U11 ∩ U22 and compute δρ11,12,22 = δρ11,21,22 = ρ. Similarly, if ρ is a section on U12 ∩ U21 we compute δρ11,12,21 = δρ12,21,22 = ρ. Since these generate all possible ways of getting terms (x1 x2 x˜1−1 x˜2 )−1 in coboundaries we conclude that the coboundaries satisfy A11,12,22 = A11,21,22 ,
A11,12,21 = A12,21,22 .
(42)
We therefore can define tr : H 2 (O(−2, −2)) → C,
tr(ω) = Aω11,12,22 − Aω11,21,22 ,
(43)
where Aω is the coefficient of (x1 x2 x˜1−1 x˜2 )−1 in the Cech cocycle ω. Note that this is the unique (up to a multiple) linear functional on the A’s subject to the cocycle condition (41) which vanishes on coboundaries. Finally, applying (43) to (40) we get tr(φ) = 1. Thus Y Y˜ = 1. We similarly compute the cup product of Y with itself using (35) and omitting terms not involving e1 ∧ f1 , 1 e1 ∧ f 1 , x1 x˜1 2 =− e1 ∧ f 1 , x1 x˜1 1 = e1 ∧ f 1 , x1 x˜1 2 =− e1 ∧ f 1 . x1 x˜1
(Y ∪ Y )11,12,21 = (Y ∪ Y )11,21,22 (Y ∪ Y )11,12,22 (Y ∪ Y )12,21,22
Thus the Cech representative ρ in H 2 (O(−2, −2) satisfies ρ
ρ
ρ
ρ
A11,12,21 = 1 , A11,21,22 = −2 , A11,12,22 = 1 , A12,21,22 = −2 , leading to tr(ρ) = −(1 + 2 ). Hence Y 2 = −(1 + 2 ). Summarizing the classical computation, we have computed Y˜ 2 = 0,
Y Y˜ = 1,
Y 2 = −(1 + 2 ).
We now make the substitution X = Y + 1 Y˜ ,
X˜ = Y˜ ,
(44)
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which extends (32) which held on the (2, 2) locus. With these new variables, we once again get the usual classical correlators (17). Computations in the instanton sectors are very similar to the classical computation. We shall merely outline those calculations for the (1, 0) and (0, 1) sectors, starting with the (1, 0) sector. As we have seen, there are several parts to giving a mathematical formulation of the correlation functions. First, we need to give a map ψ : H p (X, q E ∨ ) → H p (M, q F ∨ ). To compute a correlation function φ1 , . . . , φn , we start by multiplying the ψ(φi ) (via cup and wedge products) to get an element of H 4 (M, 4 F ∨ ), so we next need to be able to compute cup and wedge products. Finally, we need to evaluate this class numerically. Since 4 F ∨ KM , we have H 4 (M, 4 F ∨ ) C. As in the classical calculation, we will use Cech cohomology and a trace map tr : H 4 (M, 4 F ∨ ) → C. Then the correlation functions are φ1 , . . . , φn 1,0 = tr (ψ(φ1 ) ∪ · · · ∪ ψ(φn )) ,
(45)
where the subscript emphasizes that we are looking only at instanton contributions of a fixed degree. As we have seen before, from the dual of (28) we have H 1 (X, E ∨ ) = C2 while from the dual of (29) we have H 1 (M, F ∨ ) = C2 . Explicitly the exact sequence 0 → F ∨ → O(−1, 0)4 ⊕ O(0, −1)2 → O2 → 0
(46)
whose coboundary map H 0 (M, O2 ) → H 1 (M, F ∨ ) is an isomorphism by the vanishing of the cohomologies of O(−1, 0)4 ⊕ O(0, −1)2 . Thus the identification of H 1 (F ∨ ) with C2 is canonical. Similarly, the isomorphism of H 1 (X, E ∨ ) with C2 is canonical. Composing these canonical isomorphisms gives the desired isomorphism ψ : H 0 (X, E ∨ ) H 0 (M, F ∨ ). Proceeding as in the classical case, one finds that ˜ 1,0 = 1, X4 1,0 ∝ 1 − 2 . (47) X˜ 4 1,0 = XX˜ 3 1,0 = X 2 X˜ 2 1,0 = 0, X3 X (See [26] for more details of these calculations.) Similarly, in the (0, 1) sector one finds that X˜ 4 0,1 = 0, X X˜ 3 0,1 = 1, X2 X˜ 2 0,1 ∝ 1 − 2 , ˜ 0,1 = (1 − 2 )f1 (), X 4 0,1 = (1 − 2 )f2 (), X3 X
(48)
where fi () is a homogeneous polynomial of degree i in 1 , 2 . Combining (47) with (48), we find agreement with (19), (22), (23), (25), and (27).16 16 The exact form of f and f in (48) has recently been worked out with the assistance of D. Christie, 1 2 completing the verification of (19), (22), (23), (25) and therefore of the predictions of [1].
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8. Conclusions 3
In this paper we have described the computation of generalizations of the 27 coupling in perturbative heterotic string compactifications. These calculations amount to a heterotic version of curve-counting, generalizing the standard A model calculations. We spent the first third of this paper describing formally how one can calculate these correlation functions. We saw how old ideas from A model calculations generalize in the heterotic context – for example, the obstruction sheaf story seems to generalize in an interesting way. Our formal methods in the first third of this paper suffered from needing not only a compactification of the moduli space of worldsheet instantons, but also extensions of induced sheaves over that compactification divisor, in a fashion that preserves certain necessary properties of the Chern classes. The second third of this paper was devoted to solving this problem using linear sigma models, which not only compactify moduli spaces, but as we described, also provide the needed sheaf extensions. In the final part of this paper, we applied this technology to check some predictions of [1] for heterotic curve counting. There are several open problems that need to be solved: 1. We have described how to map H p (X, q E ∨ ) → H p (M, q F ∨ ) on open subsets of the moduli space, and in special cases involving linear sigma models, have used calculational tricks to extend the map over the compactification of the moduli space. However, a more general prescription for extending the map over the compactification is lacking. 2. In Sect. 5.5, we described a proposal for generalizing obstruction sheaf constructions. From physics, we conjecture that the Atiyah class of E determines an element of H 1 M, F ∨ ⊗ F1 ⊗ G1∨ with the property that when E = T X, the element of the sheaf cohomology group above is the Atiyah class of the obstruction sheaf. With those assumptions, an easy Grothendieck-Riemann-Roch argument showed how the resulting product of sheaf cohomology groups generated a top form which can be integrated over the moduli space, and we also checked that we reproduce the usual obstruction sheaf story on the (2, 2) locus. However, although the underlying physics seems clear, these mathematical conjectures need to be checked. There are other extensions of this work that would be interesting to pursue. For example, it would be interesting to understand how these correlation functions change when the K¨ahler class passes through a stability subcone wall [25], which would be the heterotic analogue of a flop. It would also be interesting to better understand the effect of the ratio of operator determinants that we outlined earlier. For the calculations in this paper, at genus zero, that ratio is just a constant, which we have ignored. However, at higher genus, it is a nontrivial function of both the moduli of the Riemann surface and of the bundle. Acknowledgements. We would like to thank A. Adams, J. Bryan, D. Christie, R. Plesser, and S. Sethi for useful conversations. The work of SK has been partially supported by NSF grant DMS 02-96154 and NSA grant MDA904-03-1-0050. The work of ES has been partially supported by NSF grant DMS 02-96154.
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References 1. Adams, A., Basu, A., Sethi, S.: (0, 2) duality. Adv. Theor. Math. Phyp. 7, 865–950 (2004) 2. Distler, J., Greene, B., Morrison, D.: Resolving singularities in (0, 2) models. Nucl. Phys. B481, 289–312 (1996) 3. Beasley, C., Witten, E.: Residues and worldsheet instantons. JHEP 0310, 065 (2003) 4. Dine, M., Seiberg, N., Wen, X.-G., Witten, E.: Nonperturbative effects on the string worldsheet. Nucl. Phys. B278, 769–789 (1986) 5. Witten, E., Bagger, J.: Quantization of Newton’s constant in certain supergravity theories. Phys. Lett. B115, 202–206 (1982) 6. Distler, J.: Notes on N = 2 sigma models. In Trieste 1992, Proceedings, String theory and quantum gravity ’92, Singapore: world scientific, 1992, pp. 234–256 7. Schwarz, A.: Sigma models having supermanifolds as target spaces. Lett. Math. Phys. 38, 91–96 (1996) 8. Silverstein, E., Witten, E.: Criteria for conformal invariance of (0, 2) models. Nucl. Phys. B444, 161–190 (1995) 9. Berglund, P., Candelas, P., de la Ossa, X., Derrick, E., Distler, J., Hubsch, T.: On the instanton contributions to the masses and couplings of E6 singlets. Nucl. Phys. B454, 127–163 (1995) 10. Basu, A., Sethi, S.: Worldsheet stability of (0, 2) linear sigma models. Phys. Rev. D68, 025003 (2003) 11. Antoniadis, I., Gava, E., Narain, K., Taylor, T.: Topological amplitudes in string theory. Nucl. Phys. B413, 162–184 (1994) 12. Distler, J., Greene, B.: Aspects of (2, 0) string compactifications. Nucl. Phys. B304, 1–62 (1988) 13. Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces. Boston: Birkh¨auser, 1980 14. Aspinwall, P., Morrison, D.: Topological field theory and rational curves. Common. Math. Phys. 151, 245–262 (1993) 15. Witten, E.: Topological sigma models. Common. Math. Phys. 118, 411–449 (1988) 16. Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nucl. Phys. B340, 281–332 (1990) 17. Kontsevich, M.: Enumeration of rational curves via torus actions. In The Moduli Space of Curves (Texel Island, 1994), Dijkgraaf, R., Faber C., van der Geer, G. eds., Progress in Math. 129, Boston-Basel-Berlin: Birkh¨auser, 1995, pp. 335–368 18. Klyachko, A. A.: Equivariant bundles on toral varieties. Math. USSR Izvestiya 35, 337–375 (1990) 19. Knutson, A., Sharpe, E.: Sheaves on toric varieties for physics. Adv. Theor. Math. Phys. 2, 865–948 (1998) 20. Morrison, D., Plesser, R.: Summing the instantons: quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B440, 279–354 (1995) 21. Atiyah, M.: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85, 181–207 (1957) 22. Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B403, 159–222 (1993) 23. Distler, J., Kachru, S.: (0, 2) Landau-Ginzburg theory. Nucl. Phys. B413, 213–243 (1994) 24. Beauville, A.: Complex Algebraic Surfaces. Second edition, Cambridge: Cambridge University Press, 1996 25. Sharpe, E.: K¨ahler cone substructure. Adv. Theor. Math. Phys. 2 1441–1462 (1999) 26. Katz, S., Sharpe, E.: Notes on certain (0, 2) correlation functions, http://arxiv.org/list/hepth/0406226, 2004, the preprint version of this paper Communicated by N.A. Nekrasov
Commun. Math. Phys. 262, 645–661 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1481-8
Communications in
Mathematical Physics
A Genus-3 Topological Recursion Relation Takashi Kimura1, , Xiaobo Liu2, 1 2
Department of Mathematics and Statistics, 111 Cummington St., Boston University, Boston, MA 02215, USA. E-mail: [email protected] Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail: [email protected]
Received: 14 March 2005 / Accepted: 13 June 2005 Published online: 20 December 2005 – © Springer-Verlag 2005
Abstract: In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves. Let Mg,n be the moduli space of genus-g stable curves with n marked points. It is well known that relations in the tautological rings on Mg,n produce universal equations for the Gromov-Witten invariants of compact symplectic manifolds. Examples of genus-1 and genus-2 universal equations were given in [Ge1, Ge2], and [BP]. Relations among known universal equations were discussed in [L2]. It is expected that for manifolds with semisimple quantum cohomology, such universal equations completely determine all higher genus Gromov-Witten invariants in terms of its genus-0 invariants. This has been proven for the genus-1 case in [DZ] and for the genus-2 case in [L1]. However for genus bigger than 2, no explicit universal equations had been found except for those which follow from an obvious dimension count. The main purpose of this paper is to introduce a new genus-3 universal equation, namely, a genus-3 topological recursion relation. Associated to the Gromov-Witten invariants of a compact symplectic manifold M is its big phase space, a product of infinitely many copies of H ∗ (M; C). We will choose a basis {γα | α = 1, . . . , N} of H ∗ (M; C). The quantum product W1 ◦ W2 of two vector fields W1 and W2 on the big phase space was introduced in [L1]. This is an associative product without an identity element. An operator T on the space of vector fields on the big phase space was also introduced in [L1] to measure the failure of the string vector field to be an identity element with respect to this product. This operator turns out to be a very useful device to translate relations in the tautological rings of Mg,n into universal equations for Gromov-Witten invariants. We will write universal equations of
Research of the first author was partially supported by NSF grant DMS-0204824 Research of the second author was partially supported by NSF grant DMS-0505835
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Gromov-Witten invariants as equations among tensors W1 · · · Wk g which are defined to be the k th covariant derivatives of the generating functions of genus-g Gromov-Witten invariants with respect to the trivial connection on the big phase space. We will briefly review these definitions in Sect. 1 for completeness. The main result of this paper is the following theorem. Theorem 0.1. For Gromov-Witten invariants of any compact symplectic manifold, the following topological recursion relation holds for any vector field W on the big phase space: T 3 (W) 3
1 5 =− W T (γα ◦ γ α ) 2 + T (γα ) {W ◦ γ α } 2 252 42 13 41 α β + T (γ ) 2 γα W γ γβ 0 + T (γ α ) 2 {γα ◦ W} 1 168 21 13 1 α − + W ◦ γα ◦ γ Wγ α 1 γα γ β ◦ γβ 1 2 168 280 β 23 α 47 α − γ 1 γα W γ ◦ γβ 1 − γ 1 γα γ β 1 γβ W γ µ γµ 0 5040 5040 5 23 α α β µ − W γ 1 γα γ γβ γ 0 γµ 1 + γ 1 γα Wγ β γβ γ µ 0 γµ 1 1008 504 11 α β 4 α γ γ 1 γα γβ ◦ W 1 − γ 1 γα γ β 1 γβ ◦ W 1 + 140 35 2 89 α W γ α 1 γα ◦ γβ 1 γ β 1 + γ 1 γα W γ β γ µ 0 γβ 1 γµ 1 + 105 210 1 α 1 β − γ 1 γα γ γβ ◦ W 1 + W γ α γ β 1 γα ◦ γβ 1 210 140 23 α β 3 α β µ + γ γ 1 γα γβ Wγ 0 γµ 1 − γ γ 1 γα ◦ γβ W 1 140 140 13 α 1 α β µ Wγ 1 γα γβ γ γµ γ 0 + γ 1 γα Wγ β γβ γ µ γµ 0 − 4480 8064 1 41 α β α β µ − Wγ γ 1 γα γβ γ γµ 0 + γ γ 1 γα γβ Wγ µ γµ 0 2240 6720 1 1 + Wγ α γα γ β γβ γ µ γµ 0 − W ◦ γ α 1 γα γ β γβ 1 53760 210 1 1 α Wγ α γα γ β ◦ γβ 1 − γ γα γ β 1 γβ Wγ µ γµ 0 − 5760 2688 1 α 1 β γ γα γ γ β ◦ W 1 + Wγα γβ γµ 1 γ α γ β γ µ 0 − 5040 3780 1 α β µ γα γβ γµ 1 Wγ γ γ 0 . + (1) 252 A by-product of the proof of this theorem is a new relation in the tautological ring of M3,1 which will be given in Sect. 3. This relation is equivalent to the genus-3 topological recursion relation. The main idea behind the proof of Theorem 0.1 is that universal equations for Gromov-Witten invariants can be written as linear combinations of finitely many terms
Genus-3 Topological Recursion Relation
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of a given dimension each corresponding to a boundary stratum, suitably decorated, in Mg,n . The goal is to determine the coefficients in these linear combinations. Suppose that all of the Gromov-Witten invariants of a particular manifold are known, then since these invariants must satisfy the universal relations, it will imply relations between these coefficients. In the case of our genus-3 topological recursion relation, the Gromov-Witten invariants of a point and of CP 1 , both of which are completely known, completely determine the coefficients and, hence, this universal relation. This method may be adapted to obtain more universal equations of any genus and will be explored in a future paper. Finally, we observe that these universal equations not only apply to Gromov-Witten theory but also to other cohomological field theories, in the sense of Kontsevich-Manin, such as r-spin theory [JKV], where r ≥ 2 is an integer. The correlators in r-spin theory are intersection numbers on the moduli space of r-spin curves, (cf. [JKV]). r-spin theory is interesting because of the generalized Witten conjecture which states that its big phase space potential function solves the r th KdV integrable hierarchy. When r = 2, this reduces to the ordinary Witten conjecture [W] proven by Kontsevich [K]. While this conjecture has been proven in special cases for general r in low genus, it is still open in general. Since Eq. (1) also applies to r-spin theory, we use it to calculate some correlators in this theory. After the completion of this paper, the preprint [AL] appeared in which a topological recursion relation equivalent to ours on M3,1 was obtained assuming that the so-called invariance conjectures hold. However, since these conjectures have not yet been established, their work does not yet prove that our topological recursion relation holds on M3,1 . 1. Preliminaries Let M be a compact symplectic manifold. The big phase space is by definition the infinite product P :=
∞
H ∗ (M; C).
n=0
} of H ∗ (M; C), where γ
Fix a basis {γ0 , . . . , γN 0 is the identity element, of the ordinary cohomology ring of M. Then we denote the corresponding basis for the nth copy of H ∗ (M; C) in P by {τn (γ0 ), . . . , τn (γN )}. We call τn (γα ) a descendant of γα with descendant level n. We can think of P as an infinite dimensional vector space with a basis {τn (γα ) | 0 ≤ α ≤ N, n ∈ Z≥0 }, where Z≥0 = {n ∈ Z | n ≥ 0}. Let (tnα | 0 ≤ α ≤ N, n ∈ Z≥0 ) be the corresponding coordinate system on P . For convenience, we identify τn (γα ) with the coordinate vector field ∂t∂α on P for n ≥ 0. If n < 0, τn (γα ) is n understood to be the 0 vector field. We also abbreviate τ0 (γα ) by γα . We use τ+ and τ− to denote the operators which shift the level of descendants by 1, i.e.
fn,α τn (γα ) = fn,α τn±1 (γα ), τ± n,α
n,α
where fn,α are functions on the big phase space. We will adopt the following notational conventions: Lower case Greek letters, e.g. α, β, µ, ν, σ, . . . , etc., will be used to index the cohomology classes on M. These indices
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run from 0 to N . Lower case English letters, e.g. i, j , k, m, n, . . . , etc., will be used to index the level of descendants. These indices run over the set of all non-negative integers, i.e. Z≥0 . All summations are over the entire ranges of the corresponding indices unless otherwise indicated. Let ηαβ = γα ∪ γ β M
be the intersection form on H ∗ (M, C). We will use η = (ηαβ ) and η−1 = (ηαβ ) to lower and raise indices. For example, γ α := ηαβ γβ . Here we are using the summation convention that repeated indices (in this formula, β) should be summed over their entire ranges. Let k τn1 (γα1 ) τn2 (γα2 ) . . . τnk (γαk ) g,d := ( ini ∪ evi ∗ γαi ) [Mg,n (M;d)]virt i=1
be the genus-g, degree d, descendant Gromov-Witten invariant associated to γα1 , . . . , γαk and nonnegative integers n1 , . . . , nk (cf. [W, RT, LiT]). Here, Mg,k (M; d) is the moduli space of stable maps from genus-g, k-pointed curves to M of degree d ∈ H2 (M; Z). i is the first Chern class of the tautological line bundle over Mg,k (M; d) whose geometric fiber over a stable map is the cotangent space of the domain curve at the i th marked point while evi : Mg,n (M; d) → M is the i th evaluation map for all i = 1, . . . , k. Finally, [Mg,n (M; d)]virt is the virtual fundamental class. The genus-g generating function is defined to be 1 Fg = tnα11 · · · tnαkk q d τn1 (γα1 ) τn2 (γα2 ) . . . τnk (γαk ) g,d , k! k≥0
d∈H2 (V ,Z)
α1 ,... ,αk n1 ,... ,nk
where q d belongs to the Novikov ring. This function is understood as a formal power series in the variables { tnα } with coefficients in the Novikov ring. Introduce a k-tensor · · · ·· defined by k
W1 W2 · · · Wk g :=
fm1 1 ,α1 · · · fmk k ,αk
m1 ,α1 ,... ,mk ,αk
∂k αk Fg , · · · ∂tm k
α1 α2 ∂tm 1 ∂tmk
∂ i i for vector fields Wi = m,α fm,α α , where fm,α are functions on the big phase space. ∂tm This tensor is called the k-point (correlation) function. For any vector fields W1 and W2 on the big phase space, the quantum product of W1 and W2 is defined by W1 ◦ W2 := W1 W2 γ α 0 γα . Define the vector field
T (W) := τ+ (W) − W γ α 0 γα
Genus-3 Topological Recursion Relation
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for any vector field W. The operator T was introduced in [L1] as a convenient tool in the study of universal equations for Gromov-Witten invariants. Let ψi be the first Chern class of the tautological line bundle over Mg,k whose geometric fiber over a stable curve is the cotangent space of the curve at the i th marked point. When we translate a relation in the tautological ring of Mg,k into differential equations for generating functions of Gromov-Witten invariants, each ψ class corresponds to the insertion of the operator T . Let ∇ be the trivial flat connection on the big phase space with respect to which τn (γα ) are parallel vector fields for all α and n. Then the covariant derivative of the quantum product satisfies ∇W3 (W1 ◦ W2 ) = (∇W3 W1 ) ◦ W2 + W1 ◦ (∇W3 W2 ) + W1 W2 W3 γ α 0 γα and the covariant derivative of the operator T is given by ∇W2 T (W1 ) = T (∇W2 W1 ) − W2 ◦ W1 for any vector fields W1 , W2 and W3 (cf. [L1, Eq. (8) and Lemma 1.5]). We need to use these formulas in order to compute derivatives of universal equations. 2. Proof of the Genus-3 Topological Recursion Relation g
The cohomology class ψ1 vanishes on Mg,1 due to a result of Ionel (cf. [Io]). It was g proven in [GV] and [Io] that ψ1 on Mg,1 is supported on the locus of curves which has at least one genus-0 component. Furthermore, by a result of Faber and Pandharipande g [FP], ψ1 is equal to a class from the boundary strata which is tautological, and therefore is a linear combination of products of ψ and κ classes and fundamental classes of some boundary strata. For g = 3, κ classes do not occur in this linear combination since components of curves in the boundary strata have genus at most 2 and κ1 can be represented as linear combinations of ψ classes and fundamental classes of boundary strata on the moduli spaces of stable curves of genus less than or equal to 2 (cf. [AC]). Therefore, it follows that ψ13 on M3,1 can be written as a linear combination of products of the ψ classes and the fundamental classes of some boundary strata. By taking into consideration the genus-0 and genus-1 topological recursion relations as well as Mumford’s genus-2 relation, we can translate these results into the following universal equations with unknown constants a1 , . . . , a30 : 0 = (W) := − T 3 (W) 3 + a1 T (W) γα ◦ γ α 2 + a2 W T (γα ◦ γ α ) 2 + a3 T (γ α ) 2 γα W γ β γβ 0 + a4 T (γ α ) 2 {γα ◦ W} 1 + a5 W ◦ γα ◦ γ α 2 + a6 Wγ α 1 γα γ β ◦ γβ 1 + a7 γ α 1 γα W γ β ◦ γβ 1 + a8 γ α 1 γα γ β 1 γβ W γ µ γµ 0 + a9 W γ α 1 γα γ β γβ γ µ 0 γµ 1 + a10 γ α 1 γα Wγ β γβ γ µ 0 γµ 1 + a11 γ α γ β 1 γα γβ ◦ W 1 + a12 γ α 1 γα γ β 1 γβ ◦ W 1 + a13 W γ α 1 γα ◦ γβ 1 γ β 1 + a14 γ α 1 γα W γ β γ µ 0 γβ 1 γµ 1
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+ a15 γ α 1 γα γ β γβ ◦ W 1 + a16 W γ α γ β 1 γα ◦ γβ 1 + a17 γ α γ β 1 γα γβ Wγ µ 0 γµ 1 + a18 γ α γ β 1 γα ◦ γβ W 1 + a19 Wγ α 1 γα γβ γ β γµ γ µ 0 + a20 γ α 1 γα Wγ β γβ γ µ γµ 0 + a21 Wγ α γ β 1 γα γβ γ µ γµ 0 + a22 γ α γ β 1 γα γβ Wγ µ γµ 0 + a23 Wγ α γα γ β γβ γ µ γµ 0 + a24 W ◦ γ α 1 γα γ β γβ 1 + a25 Wγ α γα γ β ◦ γβ 1 + a26 γ α γα γ β 1 γβ Wγ µ γµ 0 + a27 γ α γα γ β γβ ◦ W 1 + a28 T (γα ) {W ◦ γ α } 2 + a29 Wγα γβ γµ 1 γ α γ β γ µ 0 + a30 γα γβ γµ 1 Wγ α γ β γ µ 0 , (2) where W is any vector field on the big phase space. Using the genus-2 equation discovered by Belurousski-Pandharipande [BP], we can write T (W) γα ◦ γ α 2 as a linear combination of other terms on the right hand side of Eq. (2) (cf. [L1, Eq. (21)]). Therefore, we can set a1 = 0.
(3)
Note that Eq. (2) holds for any compact symplectic manifold. However, we shall see that the Gromov-Witten invariants of a point and of CP 1 already completely determine the coefficients a2 , . . . , a30 . 2.1. Relations obtained from the Gromov-Witten invariants of a point. When the target manifold is a point, all stable maps must necessarily have degree 0. Hence, we will omit any reference to the degrees of these Gromov-Witten invariants. In fact, the moduli space of stable maps into a point is isomorphic to the moduli space of stable curves. Since the cohomology space of a point is one dimensional, coordinates on the big phase space are simply denoted by t0 , t1 , t2 , · · · . We also identify vector fields ∂t∂m with τm on the big phase space. Gromov-Witten invariants of a point obey the string equation
τ0 τn1 · · · τnk
g
=
k
τn1 · · · τnj −1 · · · τnk
j =1
g,d
+ δg,0 δk,2 δn1 ,0 δn2 ,0
and the dilaton equation
1 δg,1 δk,0 . = (2g − 2 + k) τn1 · · · τnk g + 24 We can use these two equations to compute Gromov-Witten invariants involving only τ0 and τ1 . More complicated Gromov-Witten invariants of a point can be computed using the Virasoro constraints, or equivalently the KdV hierarchy which were conjectured by Witten and proven by Kontsevich (cf. [W] and [K]). To determine the coefficients a2 , . . . , a30 in Eq. (1), we will only need the genus-2 invariants τ1 τn1 · · · τnk
τ4 2 =
g
1 , 1152
τ2 τ3 2 =
29 , 5760
τ2 τ2 τ2 2 =
7 , 240
Genus-3 Topological Recursion Relation
651
and the genus-3 invariants τ7 3 =
1 82944 , 607 τ4 τ4 3 = 1451520 , 583 τ3 τ3 τ3 3 = 96768 , 193 τ2 τ2 τ2 τ2 τ3 3 = 288 .
τ2 τ6 3 =
77 414720 , 17 τ2 τ2 τ5 3 = 5760 , 53 τ2 τ2 τ2 τ4 3 = 1152 ,
τ3 τ5 3 =
503 1451520 , 1121 τ2 τ3 τ4 3 = 241920 , 205 τ2 τ2 τ3 τ3 3 = 3456 ,
All other invariants needed to determine the coefficients a2 , . . . , a30 can be computed from the string and dilaton equations. We will compute derivatives of (τm ) restricted to the origin t = 0 of the big phase space. By Eq. (2), these values must all be equal to zero. We thus obtain some linear relations between the coefficients a2 , . . . , a30 in Eq. (1). From (τ4 ) |t=0 = 0, we obtain 0=−
1 a2 a29 a25 + + a23 + + . 82944 384 24 24
(4)
From τ5 (τ0 ) |t=0 = 0, we obtain 0=−
503 a2 a25 a5 a27 a28 a29 + + + a23 + + + + . 1451520 288 1152 24 24 288 24
(5)
From τ4 (τ1 ) |t=0 = 0, we obtain 0=−
607 a25 a2 a3 a26 a29 a30 + + + 5a23 + + + + . 1451520 96 384 6 24 6 24
(6)
From τ3 (τ2 ) |t=0 = 0, we obtain 0=−
503 7a25 29a2 a6 a18 a19 a22 7a29 + + + + + + 10a23 + + . 1451520 1440 576 576 24 24 24 24
(7)
From τ2 (τ3 ) |t=0 = 0, we obtain 0=−
77 7a25 29a2 a7 a16 a20 a21 7a29 + + + + + + 10a23 + + . 414720 1440 576 576 24 24 24 24
(8)
From τ3 τ3 (τ0 ) |t=0 = 0, we obtain 583 29a2 29a5 a6 a11 a18 a19 a22 + + + + + + + 96768 576 2880 288 288 288 12 12 7a27 29a28 7a29 7a25 + 20a23 + + + + . 12 12 576 12
0=−
(9)
From τ2 τ4 (τ0 ) |t=0 = 0, we obtain 1121 11a2 a3 a4 11a5 a7 a15 a16 a20 + + + + + + + + 241920 288 384 9216 1440 576 576 576 24 a21 11a25 a26 11a27 11a28 11a29 a30 a24 + +15a23 + + + + + + + . (10) 24 576 24 24 24 288 24 24
0=−
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From τ2 τ3 (τ1 ) |t=0 = 0, we obtain 29a2 29a3 a6 a7 a8 a16 a17 a18 1121 + + + + + + + + 241920 288 1440 288 192 576 192 576 288 35a25 a20 a21 a22 7a26 35a29 7a30 a19 + + + + 60a23 + + + + . + 12 6 8 8 24 24 24 24
0=−
(11)
From τ2 τ2 (τ2 ) |t=0 = 0, we obtain 7a2 a6 a7 a9 a10 a13 a16 17 + + + + + + + 5760 48 144 72 288 288 6912 72 a18 a19 a20 a21 a22 + + + + + + 90a23 + 2a25 + 2a29 . 144 4 2 3 6
0=−
(12)
From τ2 τ2 τ3 (τ0 ) |t=0 = 0, we obtain 5a2 29a3 29a4 5a5 a6 7a7 a8 a9 a10 205 + + + + + + + + + 3456 12 720 17280 72 72 288 288 288 288 a11 a12 a13 7a15 7a16 a17 a18 5a19 5a20 7a21 5a22 + + + + + + + + + + + 72 6912 6912 288 288 288 72 12 6 12 12 7a24 59a25 7a26 59a27 5a28 59a29 7a30 + 210a23 + + + + + + + . (13) 288 12 12 12 12 12 12
0=−
From τ2 τ2 τ2 (τ1 ) |t=0 = 0, we obtain 7a2 7a3 a6 a7 a8 a9 a10 a13 a14 a16 53 + + + + + + + + + + 1152 8 48 24 12 48 48 32 1152 2304 12 a18 3a19 15a20 a17 3a22 + + + + 2a21 + + 630a23 + 48 24 2 4 2 + 12a25 + 2a26 + 12a29 + 2a30 . (14)
0=−
From τ2 τ2 τ2 τ2 (τ0 ) |t=0 = 0, we obtain 7a3 7a4 7a5 a6 a7 a8 a9 a10 a11 193 49a2 + + + + + + + + + + 288 12 12 288 12 6 3 12 12 8 6 a12 a13 a14 a15 a16 a17 a18 + + + + + + + + 6a19 + 15a20 + 8a21 + 6a22 288 288 576 3 3 12 6 a24 49a28 + 2520a23 + (15) + 48a25 + 8a26 + 48a27 + + 48a29 + 8a30 . 3 12
0=−
2.2. Relations obtained from the Gromov-Witten invariants of CP 1 . When the target manifold is CP 1 , the degrees of the stable maps are indexed by H2 (CP 1 ; Z) ∼ = Z. The degree d part of any equation for generating functions of Gromov-Witten invariants is the coefficient of q d in the Novikov ring. We choose the basis {γ0 , γ1 } for H ∗ (CP 1 ; C) with γ0 ∈ H 0 (CP 1 ; C) being the identity of the ordinary cohomology ring and γ1 ∈ H 2 (CP 1 ; C) the Poincar´e dual to a point. Coordinates on the big phase space are denoted by {tn0 , tn1 | n ∈ Z+ }. We identify vector fields ∂t∂0 and ∂t∂1 with τn,0 n n and τn,1 respectively. We also define τn,α = 0 if n < 0.
Genus-3 Topological Recursion Relation
653
The Gromov-Witten invariants of CP 1 obey three basic equations: the string equation
τ0,0 τn1 ,α1 · · · τnk ,αk
g,d
=
k
τn1 ,α1 · · · τnj −1,αj · · · τnk ,αk
j =1
g,d
+ δg,0 δd,0 δk,2 δn1 ,0 δn2 ,0 δα1 +α2 ,1 , the dilaton equation
τ1,0 τn1 ,α1 · · · τnk ,αk
g,d
1 = (2g − 2 + k) τn1 ,α1 · · · τnk ,αk g,d + δg,1 δk,0 δd,0 , 12
and the divisor equation τ0,1 τn1 ,α1 · · · τnk ,αk g,d = d τn1 ,α1 · · · τnk ,αk g,d +
k j =1
δαj ,0 τn1 ,α1 · · · τnj −1,1 · · · τnk ,αk g,d
+ δg,0 δd,0 δk,2 δn1 ,0 δn2 ,0 δα1 ,0 δα2 ,0 −
1 δg,1 δd,0 δk,0 . 24
We can use these three equations to compute Gromov-Witten invariants involving only τ0,0 , τ1,0 , and τ0,1 . More complicated Gromov-Witten invariants for CP 1 can be computed using the Virasoro constraints which was conjectured in [EHX] and proven in [Gi]. A computer program for computing such invariants based on the Virasoro constraints was written by Andreas Gathmann (cf. [Ga]). To determine a2 , · · · a30 in Eq. (1), we only need a small number of such invariants. In Appendix A we will list all of the necessary invariants which are obtained from Gathmann’s program. To obtain more relations on a2 , · · · a30 in Eq. (1), we will compute derivatives of (τm,α ) at the origin of the big phase space t = 0. From the degree 0 part of
(τ1,1 ) |t=0 = 0, we obtain 0=
31 a13 a14 − − . 967680 13824 13824
(16)
From the degree 0 part of (τ2,0 ) |t=0 = 0, we obtain 0=−
41 7a2 a7 a10 a13 a16 + + + + + . 290304 960 288 288 6912 288
(17)
From the degree 1 part of (τ3,1 ) |t=0 = 0, we obtain 1 7a2 a7 a10 a13 − − + + 322560 2880 288 288 13824 a14 a16 a20 a29 a25 − − − + 8a23 + + . 13824 288 6 6 12
0=−
(18)
From the degree 0 part of τ2,1 (τ0,0 ) |t=0 = 0, we obtain 0=
31 7a4 a12 a13 a14 − − − − . 96768 46080 13824 13824 13824
(19)
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From the degree 0 part of τ3,0 (τ0,0 ) |t=0 = 0, we obtain 1501 7a2 a4 7a5 a7 a10 a12 + + + + + + 725760 720 1920 2880 288 288 6912 a13 a15 a16 a24 7a28 + + + + + . 6912 288 288 288 720
0=−
(20)
From the degree 0 part of τ2,0 (τ1,0 ) |t=0 = 0, we obtain 2329 7a2 7a3 a6 a7 a8 a9 a10 a13 + + + + + + + + 1451520 240 960 288 96 288 288 96 1728 a14 a16 a17 a18 + + + + . (21) 2304 96 288 288
0=−
From the degree 1 part of τ3,1 (τ0,1 ) |t=0 = 0, we obtain 31 7a4 7a5 a7 a10 a12 a13 a14 + + − + + + − 96768 138240 2880 288 288 13824 13824 13824 a20 a25 a27 7a28 a29 a30 a16 a24 − + 8a23 − + + + + − . (22) − 288 6 288 6 12 1440 8 24
0=−
From the degree 1 part of τ4,0 (τ0,1 ) |t=0 = 0, we obtain 0=
277 a2 13a4 a5 a7 a12 a13 a14 a15 + − − − + + + − 207360 360 46080 120 144 13824 13824 13824 144 a16 a24 a20 a25 a27 11a28 a29 a30 − − + 16a23 − + + − + + . (23) 144 6 144 2 6 720 3 12
From the degree 1 part of τ3,1 (τ1,0 ) |t=0 = 0, we obtain 0=
41 7a2 7a3 a6 a7 a8 a9 a10 a13 a14 − − − − − + + + + 193536 720 2880 288 96 288 288 96 4608 13824 2a25 a17 a18 a19 2a20 a26 a29 a30 a16 − − − − + 40a23 + + + + . (24) − 96 288 288 6 3 3 6 3 12
From the degree 1 part of τ2,1 (τ1,1 ) |t=0 = 0, we obtain 1 7a3 7a4 a6 a7 a8 a9 a10 a12 + − − − + + + − 46080 960 46080 288 288 288 288 96 13824 a25 a13 a16 a19 a20 a22 a29 a30 + − − − − + 24a23 + + + . (25) 6912 288 6 2 12 6 8 24
0=−
From the degree 1 part of τ3,0 (τ1,1 ) |t=0 = 0, we obtain 83 7a2 a3 a4 7a5 a6 5a7 a8 a10 a12 − − + + − − − + + 193536 720 40 1920 2880 144 288 144 288 6912 5a14 a15 5a16 a17 a18 a19 2a20 a22 7a13 + + − − − − − + + 56a23 + 13824 13824 288 288 144 144 6 3 6 a24 a26 7a28 7a29 + + a25 − + + . (26) 288 6 720 12
0=−
Genus-3 Topological Recursion Relation
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From the degree 1 part of τ2,1 (τ2,0 ) |t=0 = 0, we obtain 19 a2 7a4 a6 5a7 a9 a10 a12 + + + − − + + 138240 320 46080 144 288 144 288 13824 5a16 a19 a13 a21 a29 − + − a20 − + 72a23 + a25 + . + 6912 288 3 6 2
0=−
(27)
From the degree 1 part of τ1,1 (τ2,1 ) |t=0 = 0, we obtain 0=
7a2 a7 5a10 a13 a14 a16 − + + − − 960 288 288 3456 2304 288 2a20 a25 a21 a29 − − + 24a23 + + . 3 12 6 6
(28)
From the degree 1 part of τ2,0 (τ2,1 ) |t=0 = 0, we obtain 1 a2 7a3 7a4 a6 a7 a8 a9 a10 a12 − + − − − + − − − 15360 240 960 46080 288 96 288 288 288 13824 a14 a29 5a16 a17 a18 2a20 a22 + − + − − − + 72a23 + a25 + . (29) 2304 288 288 288 3 6 2
0=−
From the degree 1 part of τ1,1 (τ3,0 ) |t=0 = 0, we obtain 1 107a2 a7 a10 7a13 5a14 a16 − − + + + − 46080 2880 32 288 13824 13824 32 5a20 5a25 a21 7a29 − + + 56a23 + + . 6 6 6 12
0=−
(30)
From the degree 1 part of τ1,1 τ2,1 (τ1,0 ) |t=0 = 0, we obtain 1 7a2 7a3 7a4 a6 a7 a9 a10 a12 + + − − − + + − 9216 240 480 46080 144 72 48 16 13824 a13 a16 a17 a18 5a19 19a20 a21 a22 + − − − − − − − + 144a23 1152 72 288 288 6 6 4 4 a26 19a29 5a30 5a25 + + + . (31) + 6 6 24 24
0=−
From the degree 1 part of τ2,0 τ2,0 (τ1,1 ) |t=0 = 0, we obtain 661 a2 179a3 263a4 7a5 7a6 7a7 5a8 5a9 − − + + − − − − 161280 96 1440 69120 480 144 144 144 144 5a10 5a20 a11 a12 a13 a15 13a16 5a17 7a18 − + + − + − − − − a19 − 144 144 768 1152 48 144 144 144 3 a22 19a25 7a29 a30 a24 7a28 − + 528a23 + + − a26 + + − . (32) 3 48 3 96 2 6
0=−
Remark. We have also checked more than 18000 other combinations of derivatives of for the CP 1 case using a Mathematica program, but the relations obtained are just linear combinations of the relations (4) to (32). This explicitly verifies that these relations are indeed consistent.
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Proof of Theorem 0.1. It is straightforward to solve a2 , . . . , a30 from Eqs. (4) to (32). The answers are 1 , a2 = − 252
a3 =
13 168 , 1 23 , a7 = − 5040 , a6 = 280 23 11 a10 = 504 , a11 = 140 , 89 1 a14 = 210 , a15 = − 210 , 3 1 a18 = − 140 , a19 = − 4480 , 41 1 a22 = 6720 , a23 = 53760 , 1 1 , a27 = − 5040 , a26 = − 2688 1 a30 = 252 .
a4 =
41 21 , 47 a8 = − 5040 , 4 a12 = − 35 , 1 a16 = 140 , 13 a20 = 8064 , 1 a24 = − 210 , 5 a28 = 42 ,
13 a5 = − 168 , 5 a9 = − 1008 ,
a13 = a17 = a21 = a25 = a29 =
2 105 , 23 140 , 1 − 2240 , 1 − 5760 , 1 3780 ,
(33)
Together with Eqs. (3) and (2), this proves Eq. (1).
We make the following observations about topological recursion relations of genus less than or equal to 3: 1. T (W) does not appear in the lower genus terms of the genus-g topological recursion relations for T g (W) when g = 1, 2, 3. 2. With the exception of a25 , all of the denominators in the coefficients of the genus-3 topological recursion relation (1) have a factor 7. We also note that the genus-2 topological recursion relations (i.e. Mumford’s equation), have denominators in its coefficients of lower genus terms containing a factor of 10, while for the genus-1 topological recursion relation, the corresponding factor is 24. 3. For g = 1, 2, 3, the coefficients of the terms consisting of purely genus-0 functions in the genus-g topological recursion relations are 1 . (2g + 1)!! 8g We conjecture that this should also hold for all genera. 3. A New Relation in Tautological Ring of M3,1 Note that Eq. (2) is a direct translation of a relation in the tautological ring of M3,1 with undetermined coefficients a1 , . . . a30 . Those coefficients were determined during the proof of topological recursion relation (1). Therefore we have also obtained a proof to the following theorem: Theorem 3.1. In the tautological ring of M3,1 , the following relation holds:
ψ13 = −
+
1 126 41 21
f
2
f
2
f
f
13 84
2
f −
13 84
+
1
f
f
f
2
+
f
5 42 f
f
2
f
Genus-3 Topological Recursion Relation
657
+
1 140
−
47 2520
+
11 70
−
1 105
+
23 70
−
5 504
1
f
f
1
f +
23 126
+
4 105
1
f
f
1
f +
89 35
f
f 1
1
f
f 1
1
f
1
f
f
f
f
f
f −
4 35
f
1
f
f 1
f
1 105
1
f
1
f
f
f
1
f
f −
1
f
1
3 70
1
1 70
+
1
f
1
23 2520
−
1
f
1
−
f
f
f
f
f
−
1 560
−
1 1440
−
1 560
f
f +
1
f
f
f
1
f
f
1
f
1
f
1
1 42 f 1
1
f
1
f
1
f
1 630
f
1
f
1
1
+
f
1
1
f
f
+
f
+
f −
1
1
f
f 1
41 1680 −
13 1008
1 1260
1 672
f
1
f
f
f
1
f
+
f
1 1120
@f
(34) In this formula, each stratum in Mg,n is represented by its dual graph. We adopt the conventions of [Ge2] for dual graphs with a slight modification. We denote vertices of genus 0 by a hollow circle f, and vertices of genus g ≥ 1 by gf. A vertex with an incident arrowhead denotes the ψ class associated to the marked point (which is at a node) on the irreducible component associated to that vertex.
658
T. Kimura, X. Liu
Note that when translating relations in the tautological ring of Mg,n to GromovWitten invariants, we need to divide the coefficient of each stratum by the number of elements in the automorphism group of the corresponding dual graph. This explains the difference between the coefficients in this formula and those in Eq. (1). 4. Application to Higher Spin Curves We briefly review the moduli space of r-spin curves and r-spin theory. For details, we refer the reader to [JKV]. Let r ≥ 2 be an integer. For each r, r-spin theory is a cohomological field theory, in the sense of Kontsevich-Manin, just as is the Gromov-Witten theory of a compact, symplectic manifold M. Consequently, the correlation functions in this theory satisfy the same universal equations as does Gromov-Witten theory. In particular, they satisfy the topological recursion relations, the string, and dilaton equations. However, there is no analog of the divisor equation but we will not need this. The role of the moduli space of stable maps is replaced by the moduli space of 1/r r-spin curves Mg,n . Let [n] := { 0, . . . , n} for any nonnegative integer n then we 1/r 1/r have the disjoint union Mg,n := α∈[r−2]n Mg,n (α). While the definition of an r-spin curve is rather involved, over a smooth stable curve (C; p1 , . . . , pn ) of genus g with n 1/r marked points when r is prime, a point in Mg,n (α) consists of a line bundle L → C together with an isomorphism L⊗r → ω(− ni=1 αi pi ), where α = (α1 , . . . , αn ) and ω is the canonical bundle of C. For degree reasons, such a line bundle exists only if (2 − 2g + ni=1 αi )/r is an integer. When r is not prime then one should introduce all d th roots for all d which divides r. This moduli space admits a compactification which is a smooth Deligne-Mumford stack by allowing the bundles to degenerate over the nodes. There are also the tautological classes i associated to the cotangent line at the marked 1/r points pi . When nonempty, the dimension of Mg,n (α) is the same as that of Mg,n , namely 3g − 3 + n, since there are only a finite number of r-spin structures over a given stable curve. There is no analog in r-spin theory of the degree d of a stable map into a manifold so we will ignore this parameter. The virtual fundamental class in Gromov-Witten theory is replaced by the virtual class 1/r c1/r , a cohomology class in H 2D (Mg,n (α)), where D = 1r ((r − 2)(g − 1) + ni=1 αi ). The analog of the cohomology ring of M is the vector space H with a basis { γ0 . . . γr−2 } with the pairing ηαβ = δα+β,r−2 . The correlation functions are defined by
τn1 (γα1 ) τn2 (γα2 ) . . . τnk (γαk ) 1/r
g
:= r
1−g 1/r
[Mg,k (α)]
c1/r ∪
k
ni ,
i=1 1/r
where [Mg,k (α)] denotes the fundamental class of the stack Mg,k (α). As far as dimensional considerations are concerned, r-spin theory behaves as if it were a Gromov-Witten theory of a manifold with real dimension 2(r − 2)/r and γα behaves as if it were a cohomology class of dimension 2α/r for all α = 0, . . . , r − 2. Applying Theorem 0.1 and dimensional arguments, we obtain the following. Proposition 4.1. Let r ≥ 2. In r-spin theory, the only possible nontrivial 1-point genus-3 correlators are as follows: 1 , r = 2 : τ7,0 3 = 82944
Genus-3 Topological Recursion Relation
r = 3 : τ6,1 3 = r = 4 : τ6,0 3 =
659
1 31104 , 3 20480 ,
r = 5 : All 1-point genus 3-correlators vanish for dimensional reasons, 2561 r = 6 : τ5,4 3 = 20901888 , r ≥ 7 : τ5,4 3 . Can be reduced to 5-point genus-0 correlators which can be calculated using the WDVV equation. Furthermore, when r = 3, the only possible nontrivial 2-point correlators are
τ7,0 τ0,1 τ4,0 τ3,1 τ1,0 τ6,1
3
3 3
= 1/15552,
τ6,0 τ1,1
3
= 19/77760,
τ5,0 τ2,1
= 67/77760, τ3,0 τ4,1 3 = 443/77760, τ2,0 τ5,1 = 5/31104, τ0,0 τ7,1 3 = 1/31104.
3 3
= 47/77760, = 103/217728,
Remark 4.2. When r = 3, our results for τ7,0 τ0,1 3 and τ6,1 3 both agree with the 1/3
results of Shadrin [Sh] who calculated them by studying the geometry of M3,n . He also showed that both of these numbers were consistent with the 3-rd KdV hierarchy as predicted by the generalized Witten conjecture. Appendix A. Gromov-Witten Invariants of CP 1 Used to Determine the Genus-3 Topological Recursion Relation We need the following Gromov-Witten invariants of CP 1 which are obtained using Gathmann’s program based on the Virasoro constraints. Genus-1 invariants:
2 τ1,1
1,1
2 = τ3,0 1,1 = 0, τ1,1 τ2,0 1,1 = 1/8, τ2,0
1,1
= −1/6, τ2,1 1,1 = 1/24.
Genus-2 invariants:
2 τ1,1 τ2,0 2,0 = 7/1920, τ2,0 = −5/288, 2,0 2,0 4 τ2,1 2,0 = 7/5760, τ3,0 2,0 = −1/240, τ1,1 = 0, 2,1 3 τ 2 τ2 3 τ1,1 = 0, τ1,1 τ1,1 τ2,0 = 17/64, 2,0 2,0 2,1 = 1/32, 2,1 2,1 2 τ 4 τ2,0 = 5/6, τ1,1 = 0, τ1,1 τ2,0 τ2,1 2,1 = 1/192, 2,1 2,1 2,1 2 τ 2 2 τ τ2,0 = 23/576, τ2,1 = 1/576, τ1,1 = 0, 2,1 3,0 2,1 2,1 2,1 2 τ τ1,1 τ2,0 τ3,0 2,1 = 1/32, τ2,0 = 11/96, τ2,1 τ3,0 2,1 = 1/192, 3,0 2,1 2 τ3,0 = 29/1440, τ1,1 τ3,1 2,1 = 0, τ2,0 τ3,1 2,1 = 1/192, 2,1 τ τ = 1/384, τ2,0 τ4,0 2,1 = 41/2880, τ4,1 2,1 = 1/1920, 1,1 4,0 2,1 τ5,0 2,1 = 1/576. 2 τ1,1
= 0,
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T. Kimura, X. Liu
Genus-3 invariants 2 τ 2 τ τ1,1 = 0, τ1,1 τ2,0 τ2,1 3,0 = 0, τ2,0 = −31/10752, 2,1 2,1 3,0 3,0 2 2 τ2,1 = 0, τ2,1 τ3,0 3,0 = −31/96768, τ3,0 = 1501/725760, 3,0 3,0 τ1,1 τ3,1 3,0 = 0, τ2,0 τ3,1 3,0 = −31/96768, τ1,1 τ4,0 3,0 = −31/193536, 2 τ4,1 3,0 = −31/967680, τ5,0 3,0 = 41/290304, τ3,1 = 0, 3,1 2 τ τ1,1 τ2,1 τ4,0 3,1 = 1/9216, τ3,1 τ4,0 3,1 = 1/9216, τ2,0 = 7/5760, 4,1 3,1 τ τ = 1/46080, τ τ = 1/9216, τ τ = 19/138240, 2,1 4,1 3,1 3,0 4,1 3,1 2,1 5,0 3,1 τ1,1 τ5,1 3,1 = 0, τ2,0 τ5,1 3,1 = 1/15360, τ1,1 τ6,0 3,1 = 1/46080, τ6,1 3,1 = 1/322560.
All other invariants needed to determine the genus-3 topological recursion relation can be computed from the string, dilaton, divisor equations, and the 0-point invariants 0,1 = 1,
1,0 = 0.
Acknowledgement. The authors would like to thank Andreas Gathmann for allowing them to use his computer program for computing Gromov-Witten invariants of CP 1 based on the Virasoro constraints. ´ We would also like to thank the Institut des Hautes Etudes Scientifiques, where this paper was initiated, for their hospitality and financial support and the Focus on Mathematics Program at Boston University for the usage of their main server whether they are aware of it or not.
References [AC]
Arbarello, E., Cornalba, M.: Calculating cohomology groups of moduli spaces of curves via ´ algebraic geometry. Inst. Hautes Etudes Sci. Publ. Math. No. 88, 97–127 (1998) [AL] Arcara, D., Lee, Y.-P.: Tautological equation in M¯ 3,1 via invariance conjectures. http://arxiv.org/ list/math.AG/0503184, 2005 [BP] Belorousski, P., Pandharipande, R.: A descendent relation in genus 2. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29, 171–191 (2000) [DZ] Dubrovin, B., Zhang, Y.: Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation. Commum. Math. Phys. 198, 311–361 (1998) [EHX] Eguchi, T., Hori, K., Xiong, C.: Quantum Cohomology and Virasoro Algebra. Phys. Lett. B402, 71–80 (1997) [FP] Faber, C., Pandharipande, R.: Relative maps and tautological classes. J. Eur. Math. Soc. (JEMS) 7(1), 13–49 (2005) [Ga] Gathmann, A.: Topological recursion relations and Gromov-Witten invariants in higher genus. http://arxiv.org/list/math.AG/0305361, 2005 [Ge1] Getzler, E.: Intersection theory on M¯ 1,4 and elliptic Gromov-Witten Invariants. J. Am. Math. Soc. 10, 973–998 (1997) [Ge2] Getzler, E.: Topological recursion relations in genus 2. In: Integrable systems and algebraic geometry, (Kobe, Kyoto, 1997), RiverEdge, NJ: World Scientific Publishing, 1998, pp. 73–106 [Gi] Givental, A.: Gromov-Witten invariants and quantization of quadratic hamiltonians. Moscow Math. J. 1(4), 551–568 (2001) [GV] Graber, T., Vakil, R.: Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. http://arxiv.org/list/math.AG/0309227, 2003 [Io] Ionel, E.: Topological recursive relations in H 2g (Mg,n ). Invent. Math. 148(3), 627–658 (2002) [JKV] Jarvis, T.: Kimura, T., Vaintrob, A.: Moduli spaces of higher spiin curves and integrable hierarchies. Compositio Math. 126(2), 157–212 (2001) [K] Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix airy function. Commun. Math. Phys. 147(1–23), (1992)
Genus-3 Topological Recursion Relation [LiT] [L1] [L2] [Sh] [RT] [W]
661
Li, J., Tian, G.: Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds. In: Topics in symplectic 4-manifolds (Irvine, CA, 1996), R. Stem, (ed.), Cambridge, MA: International press, 1998, pp. 47–83 Liu, X.: Quantum product on the big phase space and Virasoro conjecture. Adv. Math. 169, 313–375 (2002) Liu, X.: Relations among universal equations for Gromov-Witten invariants. C. Hertling, M. Marcolli (eds.), Aspects of Mathematics, Bonn: Max-Planck-Institute for Mathematics, 2004, pp. 169 –180 Shadrin, S.: Geometry of meromorphic functions and intersections on moduli spaces of curves. Int. Math. Res. Not. no. 38, 2051–2094 (2003) Ruan, Y., Tian, G.: Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130, 455–516 (1997) Witten, E.: Two dimensional gravity and intersection theory on Moduli space. Surveys in Diff. Geom. 1, 243–310 (1991)
Communicated by L. Takhtajan
Commun. Math. Phys. 262, 663–675 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1480-9
Communications in
Mathematical Physics
Universal Bounds for Eigenvalues of a Buckling Problem Qing-Ming Cheng1, , Hongcang Yang2, 1 2
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan. E-mail: [email protected] Academy of Mathematics and Systematical Sciences, CAS, Beijing 100080, P.R. China. E-mail: [email protected]
Received: 14 March 2005 / Accepted: 13 July 2005 Published online: 9 December 2005 – © Springer-Verlag 2005
Abstract: In this paper, we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n-dimensional Euclidean space, which is also called a buckling problem. We introduce a new method to construct “nice” trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions. Thus, we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem, which was proposed by Payne, P´olya and Weinberger in [14] and this problem has been mentioned again by Ashbaugh in [1]. 1. Introduction Let ⊂ Rn be a bounded domain in an n-dimensional Euclidean space Rn . An eigenvalue problem of Dirichlet Laplacian is given by u = −λu, in , (1.1) u = 0, on ∂, which is also called a fixed membrane problem. The spectrum of this eigenvalue problem is real and purely discrete: 0 < λ1 < λ2 ≤ · · · ≤ λk ≤ · · · → ∞. Let λk+1 be the (k + 1)th eigenvalue of the eigenvalue problem (1.1). The universal bounds for the eigenvalue λk+1 are studied by many mathematicians. The main contributions were obtained by Payne, P´olya and Weinberger [14] and Thompson [15], Research partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science. Research partially supported by SF of CAS.
664
Q.-M. Cheng, H.C. Yang
Hile and Protter [9] and Yang [16] (cf. Cheng and Yang [7]), that is, Payne, P´olya and Weinberger [14] and Thompson [15] proved, for k = 1, 2, · · · , k
λk+1 − λk ≤
4 λi . kn
(1.2)
i=1
Hile and Protter [9] improved the above result of Payne, P´olya and Weinberger and Thompson to k i=1
λi kn ≥ . λk+1 − λi 4
(1.3)
Further, Yang [16] has obtained a very sharp inequality:
λk+1
k 2 1 ≤ 1+ λi n k i=1 2 2 1/2 k k k 4 1 1 1 2 λj − λk − 1 + λi . + nk n k k i=1
j =1
(1.4)
i=1
Yang’s inequality (1.4) is sharper than the inequalities (1.2) and (1.3) ( see [1] and [2] for details). In order to obtain the universal bounds for the eigenvalues of the fixed membrane problem (1.1), it is important to find appropriate trial functions. For the fixed membrane problem (1.1), one chooses ϕi = x p ui −
k
aij uj ,
aij =
j =1
x p u i uj ,
as trial functions by making use of the basic method of Payne, P´olya and Weinberger, where uj is an orthonormal eigenfunction corresponding to the j th eigenvalue λj , x p denotes the pth standard coordinate function of Rn . On the other hand, in order to describe vibrations of a clamped plate, we need to consider an eigenvalue problem for Dirichlet biharmonic operator, so-called a clamped plate problem: 2 u = λu in ,
(1.5)
u|∂ = ∂u ∂ n ∂ = 0, where 2 is the biharmonic operator in Rn and n denotes the unit outward normal vector on the boundary ∂ of . For this clamped plate problem (1.5), Payne, P´olya and Weinberger [14] also established the following inequality for the eigenvalues: k
λk+1 − λk ≤
8(n + 2) 1 λi . n2 k i=1
(1.6)
Universal Bounds for Eigenvalues of a Buckling Problem
665
Hile and Yeh [10] obtained, as a generalization of their result, −1/2 k 1/2 k λi n2 k 3/2 ≥ λi . λk+1 − λi 8(n + 2) i=1
(1.7)
i=1
They used an improved method of Hile and Protter [9]. Further, Hook [11], Chen and Qian [4], independently, proved the following inequality: k 1/2 λ1/2 n2 k 2 i λi . (1.8) ≤ 8(n + 2) λk+1 − λi i=1
i=1
The following Problem 1 has been proposed by Ashbaugh in [1]: Problem 1. Whether one can obtain a universal inequality for the eigenvalues of the clamped plate problem (1.5), which is similar to Yang’s universal inequality (1.4) for the eigenvalues of the fixed membrane problem (1.1)? This Problem 1 has been solved by Cheng and Yang in [6], as stated: λk+1
k 4(n + 2) 1 ≤ 1+ λi n2 k i=1 2 1/2 2 k k k 4(n + 2) 1 8(n + 2) 1 1 λ + λ − − λ . i i j n2 k n2 k k i=1
i=1
(1.9)
j =1
In order to prove the universal inequalities for the eigenvalues of the clamped plate problem (1.5), we see that ϕi = x p ui −
k
aij uj ,
aij =
j =1
x p ui uj ,
are also very appropriate trial functions. In this paper, we consider an eigenvalue problem: 2 in , u = −u
∂u
u|∂ = = 0, ∂ n ∂
(1.10)
which is called a buckling problem. This eigenvalue problem (1.9) is used to describe the critical buckling load of a clamped plate subjected to a uniform compressive force around its boundary. It is the purpose of this paper to derive a universal inequality for the (k + 1)th eigenvalue of the buckling problem (1.9). It is very hard to discuss this problem. The reason is that we can not findappropriate trial functions because we must use the Dirichlet inner product (f, h)D = ∇f · ∇h for functions f and h. As in the fixed membrane problem and the clamped plate problem, one wants to choose functions p
ϕ i = x ui −
k j =1
aij uj ,
aij =
∇(x p ui ) · ∇uj
666
Q.-M. Cheng, H.C. Yang
as trial functions. However, they are not very appropriate trial functions for the buckling problem (1.9) because the aij ’s are not symmetric (cf. Ashbaugh [1] and Payne, P´olya and Weinberger [14]). Thus, the crucial point for the solution of finding a universal bound for the eigenvalues of the buckling problem (1.9) is how to find “nice” trial functions. In [14], Payne, P´olya and Weinberger proposed the following: Problem 2. Whether one can obtain a universal inequality for the eigenvalues of the buckling problem (1.9), which is similar to the universal inequalities for the eigenvalues of the fixed membrane problem (1.1) or of the clamped plate problem (1.5)? Remark 1. In [1], Ashbaugh has mentioned Problem 2 of Payne, P´olya and Weinberger again. In this paper, first of all, we introduce, in Sect. 2, a new method to construct trial functions for the buckling problem (1.9). Secondly, we derive a universal inequality for the (k + 1)th eigenvalue of the buckling problem (1.9) by making use of the trial functions. Thus, we give an affirmative answer for Problem 2 of Payne, P´olya and Weinberger [14]. We shall review known results about estimates on the eigenvalues of the buckling problem (1.9). From now, we shall use i and ui to denote eigenvalues and eigenfunctions of the buckling problem. In [14], Payne, P´olya and Weinberger proved, for n = 2, 2 ≤ 31 . For general n, one obtained 2 ≤ (1 +
4 )1 . n
Subsequently, Hile and Yeh [10] improved the above results. They proved 2 ≤
n2 + 8n + 20 1 . (n + 2)2
Ashbaugh [1] obtained n
i+1 ≤ (n + 4)1 .
(1.11)
i=1
From our knowledge, these are the only known results about the universal bounds for the eigenvalues of the buckling problem (1.9). In this paper, we prove the following: Theorem. Let i be the i th eigenvalue of the buckling problem (1.9). Then, we have k i=1
k
(k+1 − i )2 ≤
4(n + 2) (k+1 − i )i . n2 i=1
(1.12)
Universal Bounds for Eigenvalues of a Buckling Problem
667
Corollary 1. Under the assumption of the theorem, we have k 2(n + 2) 1 i k+1 ≤ 1 + n2 k i=1
k
2(n + 2) 1 + i n2 k i=1
2
2 21 k k 4(n + 2) 1 1 j − − 1+ i . n2 k k j =1
i=1
Corollary 2. Under the assumption of the theorem, we have 2 2 21 k k k 1 1 2(n + 2) 4(n + 2) 1 j − i − 1 + i . k+1 −k ≤ 2 n2 k n2 k k i=1
j =1
i=1
Remark 2. From the corollary 1, we have k+1 − k ≤ k+1 −
k
k
i=1
i=1
1 4(n + 2) 1 i ≤ i . k n2 k
(1.13)
2. The Construction of Trial Functions For functions f and h, we define the Dirichlet inner product (f, h)D of f and h by ∇f · ∇h. (f, h)D =
The Dirichlet norm of a function f is defined by f D = {(f, f )D }
1/2
=
n
α=1
1/2 |∇α f |
2
.
Let ui be the i th orthonormal eigenfunction of the buckling problem (1.9) corresponding to the eigenvalue i , namely, ui satisfies 2 ui = −i ui in , u | = ∂ui
= 0, i ∂ (2.1) ∂ n ∂ (ui , uj )D = ∇ui · ∇uj = δij , u 2 = (u , u ) = |∇u |2 = 1. i D i i D i Define H22 () by H22 () = {f : f, ∇α f, ∇α ∇β f ∈ L2 (),
α, β = 1, . . . , n}.
Then, H22 () is a Hilbert space with norm · 2 : 1/2 n f 2 = |f |2 + |∇f |2 + (∇α ∇β f )2 .
β,α=1
668
Q.-M. Cheng, H.C. Yang
2 () be a subspace of H 2 () defined as Let H2,D 2
∂
2 f = 0 . () = f ∈ H22 () : f |∂M = H2,D ∂ n ∂ 2 () with The biharmonic operator 2 defines a self-adjoint operator acting on H2,D discrete eigenvalues {0 < 1 < 2 ≤ · · · ≤ k ≤ · · · } for the buckling problem (1.9) and the eigenfunctions defined in (2.1),
{ui }∞ i=1 = {u1 , u2 , . . . , uk , . . . } 2 (). form a complete orthonormal basis for Hilbert space H2,D 2 If ϕ ∈ H2,D (), (ϕ, uj )D = 0, for j = 1, 2, . . . , k, then, from the Rayleigh-Ritz inequality, we have k+1 ϕ 2D ≤ ϕ2 ϕ = − ∇ϕ · ∇(ϕ). (2.2)
We define an inner product (f, h) for vector-valued functions f = (f 1 , f 2 , . . . , f n ) ∈ Rn and h = (h1 , h2 , . . . , hn ) ∈ Rn by n f ·h= f α hα . (f, h) ≡
α=1
The norm of f is defined by f = (f, f)
1/2
=
n
α=1
1/2 α 2
(f )
.
Denote a Hilbert space H12 () of the vector-valued functions as H12 () = {f : f α , ∇β f α ∈ L2 (), for α, β = 1, . . . , n} with norm · 1 : f 1 = f 2 +
n α,β=1
1/2 |∇α f β |2
.
2 () ⊂ H2 () be a subspace of H2 () spanned by the vector-valued functions Let H1,D 1 1 2 (). It is easy to see that {∇ui }∞ , which form a complete orthonormal basis of H1,D i=1 2 (), ∇f ∈ H2 () and for any h ∈ H2 (), there exists a function for any f ∈ H2,D 1,D 1,D 2 f ∈ H2,D () such that h = ∇f . Let g = x p for a fixed p with p = 1, 2, . . . , n. For a vector-valued function g∇ui , i = 1, . . . , k, we decompose it into
g∇ui = fi + wi ,
(2.3)
2 () and w ⊥ H 2 () , namely, where fi is the projection of g∇ui onto H1,D i 1,D 2 2 (), fi = ∇hi , for some hi ∈ H2,D () fi ∈ H1,D
(2.4)
Universal Bounds for Eigenvalues of a Buckling Problem
669
and (wi , ∇u) =
n α=1
2 wiα ∇α u = 0, for any u ∈ H2,D ().
(2.5)
2 () is dense in L2 () and C 1 () is dense in L2 (), we have, for Therefore, since H2,D any function h ∈ C 1 () ∩ L2 (),
(wi , ∇h) = 0.
(2.6)
Hence, from the definition of wi and (2.6), we have wi |∂ = 0, divwi 2 = 0, ( divwi ≡ nα=1 ∇α wiα ).
(2.7)
From (2.3) and (2.4), we can write g∇α ui = fiα + wiα , α = 1, . . . , n, 2 (). Since with fiα = ∇α hi for some function hi ∈ H2,D β
∇β (g∇α ui ) − ∇α (g∇β ui ) = ∇β wiα − ∇α wi
(2.8)
holds, we infer, from (2.7) and (2.8), ∇wi 2 = = =
n
α,β=1 n
1 2 1 2
β
∇α wi 2
α,β=1 n
β
∇β wiα − ∇α wi 2 + divwi 2 ∇β (g∇α ui ) − ∇α (g∇β ui ) 2
α,β=1
= 1 − ∇p ui 2 .
(2.9)
Now we can construct the trial functions ϕi by ϕi = hi −
k
bij uj ,
(2.10)
j =1
where
∇hi = fi and bij =
g∇ui · ∇uj = bj i .
It is easy to check, from the definition (2.3) of fi , that ϕi satisfies ϕi |∂ =
∂ϕi |∂ = 0 and (ϕi , uj )D = (∇ϕi , ∇uj ) = 0, ∂ n
(2.11)
for any j = 1, 2, . . . , k. Hence, we know that ϕi is a trial function. In Sect. 3, we shall make use of the trial function ϕi to prove our theorem.
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3. Proof of Theorem In this section, we shall use the notation in Sect. 2. Proof of Theorem. Defining a vector-valued function vi = ∇ϕi , we have vi = fi −
k
bij ∇uj ,
(vi , ∇uj ) =
j =1
∇uj · vi = 0,
i, j = 1, · · · , k.
(3.1)
From (2.10), (2.11), (3.1) and the Rayleigh-Ritz inequality (2.2), we have k+1 vi 2 ≤ −(vi , vi ).
(3.2)
From the definition of fi , we have g∇ui 2 = fi 2 + wi 2
(3.3)
and, from (2.6), we infer −(fi , fi ) = −
(g∇ui − wi )((g∇ui ) − wi )
= −(g∇ui , (g∇ui )) + 2(wi , (g∇ui )) − (wi , wi ) = −(g∇ui , (g∇ui )) + 2(wi , (fi + wi )) − (wi , wi ) = −(g∇ui , (g∇ui )) − ∇wi 2 ,
(3.4)
−(g∇ui , (g∇ui )) = − g∇ui · g∇ui + 2∇(∇g · ∇ui ) = gui g2 ui + 2gui ∇g · ∇ui −2 g∇ui · ∇(∇g · ∇ui ),
(3.5)
gui g2 ui = −i ui g 2 ui = 2i gui ∇g · ∇ui + i g∇ui 2 2 2 2 = −i ui + i fi + wi .
Because of −2
g∇ui · ∇(∇g · ∇ui ) = 2 ∇ui + 2 2
(3.6)
we have
g∇ui · ∇(∇g · ∇ui ),
−2
g∇ui · ∇(∇g · ∇ui ) = ∇ui 2 .
(3.7)
Universal Bounds for Eigenvalues of a Buckling Problem
671
According to 2 2 gui ∇g · ∇ui = 2 ∇ui + 4 ∇p ui − 2 gui ∇g · ∇ui , 2
we derive
2
gui ∇g · ∇ui = ∇ui 2 + 2 ∇p ui 2 .
(3.8)
From (3.5), (3.6), (3.7) and (3.8), we obtain −(g∇ui , (g∇ui ))
= 2 ∇ui 2 + 2 ∇p ui 2 − i ui 2 + i fi 2 + wi 2 .
(3.9)
From (3.1), we have −(vi , vi ) = −(fi −
k
bij ∇uj , fi −
j =1
= −(fi , fi ) + 2
k
bij ∇uj )
j =1 k
bij (fi , ∇uj ) +
j =1
k
2 j bij .
(3.10)
j =1
Here we used (∇uj , ∇u ) = δj . From (2.3), (2.4) and (2.5), we derive 2
k
bij (fi , ∇uj ) = 2
j =1
= −2
k
bij (∇hi , ∇uj )
j =1 k
j bij (fi , ∇uj ) = −2
j =1
k
2 j bij .
j =1
From (3.1), we infer fi 2 = vi 2 +
k
2 bij .
(3.11)
j =1
Therefore, from (3.2), (3.4), (3.9), (3.10) and (3.11), we infer k+1 vi 2 ≤ 2+2 ∇p ui 2 − ∇wi 2 +i ( vi 2 − ui 2 + wi 2 )+
k
2 (i −j )bij .
j=1
Thus, from (2.9), we obtain (k+1 − i ) vi 2 ≤ 1 + 3 ∇p ui 2 − i ( ui 2 − wi 2 ) +
k
2 (i − j )bij .
j =1
(3.12)
672
Q.-M. Cheng, H.C. Yang
2 (), for h = ∇(gu ) − f ∈ H2 (), we have Since ∇(gui ) = ui ∇g + g∇ui ∈ H1,D i i i 1,D
ui ∇g = hi − wi . Hence, we obtain ui 2 = ui ∇g 2 = wi 2 + hi 2 . Because of (∇(∇g · ∇ui ), wi ) = 0, we infer 2 ∇p ui 2 =−2 ui ∇g · ∇(∇g · ∇ui ) =−2 hi · ∇(∇g · ∇ui ) ≤ i hi 2 +
(3.13)
(3.14)
1 ∇p ∇ui 2 . i
Then, from (3.12), (3.13) and (3.14), we have k
(k+1 − i ) vi 2 ≤ 1 + ∇p ui 2 +
1 2 ∇p ∇ui 2 + (i − j )bij . i j =1
According to
−2
g∇ui · ∇(∇g · ∇ui ) = 2 ∇p ui 2 + 2 gui ∇p ui = 2 ∇p ui 2 − 2 ui ui − 2 gui ∇p ui =2+2 g∇ui · ∇(∇g · ∇ui ),
we have
1 = −2
On the other hand,
g∇ui · ∇(∇g · ∇ui ).
1 = −2
= −2
g∇ui · ∇(∇g · ∇ui ) (fi + wi ) · ∇(∇g · ∇ui )
= −2
(vi +
= −2
k
bij ∇uj ) · ∇(∇g · ∇ui )
j =1
vi · ∇(∇g · ∇ui ) − 2
k j =1
bij cij ,
(3.15)
Universal Bounds for Eigenvalues of a Buckling Problem
where
673
cij =
∇(∇g · ∇ui ) · ∇uj = −cj i .
Hence, we have k
(k+1 − i )2 (1 + 2
bij cij )
j =1
= (k+1 − i )2 −2vi , ∇(∇g · ∇ui ) −
k
cij ∇uj
j =1
1 (k+1 − i ) ∇p ∇ui 2 − A
≤ A(k+1 − i )3 vi 2 +
k
2 , cij
j =1
where A is a positive constant. From (3.14), we infer k
(k+1 − i )2 (1 + 2
bij cij )
j =1
1 2 ≤ A(k+1 − i ) 1 + ∇p ui + ∇p ∇ui 2 + (i − j )bij i k 1 2 + (k+1 − i ) ∇p ∇ui 2 − . cij A 2
2
j =1
By taking the sum for i from 1 to k and noticing that bij is symmetric and cij is antisymmetric, we have k
k
(k+1 − i )2 − 2
i=1
≤ A (k+1 − i ) −
k
(k+1 − i )(i − j )bij cij
i,j =1 2
1 1 + ∇p ui + ∇p ∇ui 2 i
2
2 (k+1 − i )(i − j )2 bij
i,j =1
j k 1 2 + , (k+1 − i ) ∇p ∇ui 2 − (k+1 − i )cij A i=1
i,j =1
namely, k
(k+1 − i )2 ≤ A
i=1
k i=1
1 (k+1 − i )2 1 + ∇p ui 2 + ∇p ∇ui 2 i k
+
1 (k+1 − i ) ∇p ∇ui 2 . A i=1
674
Q.-M. Cheng, H.C. Yang n 2 p=1 ∇p ∇ui
Taking sum for p from 1 to n and noticing n
k
(k+1 − i )2 ≤ A
k
i=1
k
1 (k+1 − i )i . A
(k+1 − i )2 (n + 2) +
i=1
Putting A =
n 2(n+2) , k
= i , we have
i=1
we have k
(k+1 − i )2 ≤
i=1
4(n + 2) (k+1 − i )i . n2 i=1
This finishes the proof of the theorem.
Proof of Corollary 1. From the theorem, we know that (1.11) is a quadratic inequality of k+1 . Hence, we infer k 2(n + 2) 1 i k+1 ≤ 1 + n2 k +
i=1
k 2(n + 2) 1
n2
k
2 i
− 1+
i=1
This completes the proof of Corollary 1.
4(n + 2) n2
k 1
k
j =1
j −
k 1
k
2 21 i .
i=1
From Corollary 1, it is obvious that we have k+1
k 4(n + 2) 1 ≤ 1+ i . n2 k i=1
Proof of Corollary 2. From the theorem, we know that (1.11) is a quadratic inequality of k+1 . On the other hand, since k is any positive integer, we know that k also satisfies the same quadratic inequality. Hence, we obtain
2(n + 2) k ≥ 1 + n2 −
k
1 i k i=1
k 2(n + 2) 1
n2
k
2 i
i=1
2 21 k k 4(n + 2) 1 1 j − − 1+ i . n2 k k j =1
i=1
Thus, we derive 2 2 21 k k k 2(n + 2) 1 4(n + 2) 1 1 k+1 −k ≤ 2 j − i − 1+ i . n2 k n2 k k i=1
Hence, Corollary 2 is true.
j =1
i=1
Universal Bounds for Eigenvalues of a Buckling Problem
675
Remark 3. We conjecture that under the assumption of the theorem, the following inequality is satisfied k
4 (k+1 − i ) ≤ (k+1 − i )i . n 2
i=1
Acknowledgement. We would like to express our gratitude to the referee for his/her valuable comments and suggestions.
References 1. Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In: Spectral theory and geometry (Edinburgh, 1998), Davies, E.B., Yu Safalov (eds.), London Math. Soc. Lecture Notes, Vol. 273, Cambridge: Cambridge Univ. Press, 1999, pp. 95–139 2. Ashbaugh, M.S.: Universal eigenvalue bounds of Payne-P´olya-Weinberger, Hile-Protter and H.C. Yang. Proc. Indian Acad. Sci. Math. Sci. 112, 3–30 (2002) 3. Ashbaugh, M.S., Hermi, L.: A unified approach to universal inequalities for eigenvalues of elliptic operators, Pacific J. Math. 217, 3–30 (2004) 4. Chen, Z.-C., Qian, C.-L.: Estimates for discrete spectrum of Laplacian operator with any order. J. China Univ. Sci. Tech. 20 , 259–266 (1990) 5. Cheng, Q.-M., Yang, H.C.: Estimates on Eigenvalues of Laplacian, Math. Ann. 331, 445–460 (2005) 6. Cheng, Q.-M., Yang, H.C.: Inequalities for eigenvalues of a clamped plate problem. To appear in Trans. Amer. Math. Soc. 7. Cheng, Q.-M., Yang, H.C.: Bounds on eigenvalues of Dirichlet Laplacian. Preprint 8. Harrell, E.M., Stubbe, J.: On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Amer. Math. Soc. 349, 1797–1809 (1997) 9. Hile, G.N., Protter, M.H.: Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29, 523–538 (1980) 10. Hile, G.N., Yeh, R.Z.: Inequalities for eigenvalues of the biharmonic operator. Pacific J. Math. 112 , 115–133 (1984) 11. Hook, S.M.: Domain independent upper bounds for eigenvalues of elliptic operator. Trans. Amer. Math. Soc. 318, 615–642 (1990) 12. Payne, L.E.: Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal. 4, 517– 529 (1955) 13. Payne, L.E.: A note on inequalities for plate eigenvalues, J. Math. and Phys. 39, 155–159 (1960/61) 14. Payne, L.E., P´olya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. and Phys. 35, 289–298 (1956) 15. Thompson, C.J.: On the ratio of consecutive eigenvalues in n-dimensions. Stud. Appl. Math., 48, 281–283 (1969) 16. Yang, H.C.: An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of ICTP, Trieste, 1991 Communicated by B. Simon
Commun. Math. Phys. 262, 677–702 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1463-x
Communications in
Mathematical Physics
Abrikosov Lattices in Finite Domains Y. Almog∗ Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel Received: 15 March 2005 / Accepted: 12 May 2005 Published online: 29 November 2005 – © Springer-Verlag 2005
Abstract: In 1957 Abrikosov published his work on periodic solutions to the linearized Ginzburg-Landau equations. Abrikosov’s analysis assumes periodic boundary conditions, which are very different from the natural boundary conditions the minimizer of the Ginzburg-Landau energy functional should satisfy. In the present work we prove that the global minimizer of the fully non-linear functional can be approximated, in every rectangular subset of the domain, by one of the periodic solution to the linearized Ginzburg-Landau equations in the plane. Furthermore, we prove that the energy of this solution is close to the minimum of the energy over all Abrikosov’s solutions in that rectangle. 1. Introduction Consider a planar superconducting body which is placed at a sufficiently low temperature (below the critical one) under the action of an applied magnetic field . Its energy is given by the Ginzburg-Landau energy functional which can be represented in the following dimensionless form [11] 2 i ||4 2 2 E= −|| + + |h − hex | + ∇ + A dx1 dx2 , (1.1) 2 κ in which ∈ H 1 (, C) is the superconducting order parameter, such that || varies from || = 0 (when the material is at a normal state) to || = 1 (for the purely superconducting state). The magnetic vector potential is denoted by A ∈ H 1 (, R2 ) (the magnetic field is, then, given by h = ∇ × A), hex is the constant applied magnetic field, and κ is the Ginzburg-Landau parameter which is a material property. The ∗
USA
Present address: Department of Mathematics, Louisiana State University, Baton Rolge, LA 70803,
678
Y. Almog
superconductor lies in , which is a smooth connected domain. Its Gibbs free energy is given by E. Note that E is invariant to the gauge transformation → eiκζ
;
A → A + ∇ζ,
(1.2)
where ζ is any smooth function. Thus, we confine ourselves in the sequel to competitors from the space (1.3a) H = (ψ, A) ∈ H 1 (, C) × H 1 (, R2 ) | A satisfies (1.3b) , ˆ =0 ∇ · (A − A) ˆ (1.3b) (A − A) · nˆ ∂ = 0 , Aˆ = h x iˆ ex 1 2 where iˆ2 is a unit vector in the x2 direction. For sufficiently large magnetic fields it is well known, both from experimental observations [21] and from theoretical predictions [15], that superconductivity is destroyed and the material must be in the normal state. If the applied magnetic field is then decreased there is a critical field where the material enters the superconducting phase once again. This field is called “the onset field” and is denoted by HC3 . At the bifurcation from the normal state, superconductivity remains concentrated near the boundary, which is why this phenomenon has been termed “surface superconductivity” [24, 8, 9, 19, 12, 17]. In the absence of boundaries the critical field at which superconductivity nucleates is denoted by HC2 and is significantly weaker than HC3 (HC3 ≈ 1.7κ whereas HC2 = κ). Furthermore, the bifurcating modes are periodic lattices, named after Abrikosov [2, 10, 4] that have been experimentally observed [14]. It has been conjectured, therefore, by Rubinstein [23] that superconductivity remains concentrated near the boundary for HC2 < hex < HC3 . When hex ≈ HC2 (either for κ large or for large domains) a bifurcation of Abrikosov’s lattices far away from the wall was conjectured [23]. Recently, it has been proved both in the large κ limit [22, 7], and in the large domain limit [5] that as long as HC2 < hex < HC3 superconductivity remains concentrated near the boundaries. However, the second part of the conjecture in [23] is still open. In [25] it is shown for the global minimizer of (1.1) (ψκ , Aκ ), that ψκ diminishes as hex ↑ HC2 away from the boundaries. However, the exact structure of ψκ is that limit has never been found. In [6] the bifurcation of periodic solutions from the one-dimensional surface superconductivity solution introduced in [22] was studied. Nevertheless, the analysis in [6] was performed in a half-plane and under the assumption that the solutions are periodic in the direction parallel to the boundary. In the present contribution we focus on the emergence of Abrikosov’s lattices in finite domains in R2 . We prove that when hex is slightly smaller than κ the global minimizer of (1.1) in H can be approximated in an appropriately chosen rectangle in by one of Abrikosov’s solutions. Furthermore, we prove that the energy of this solution is close to the minimum of the energy over all Abrikosov’s solutions in that rectangle. Thus, we prove the following result. Theorem 1.1. Let 1 . κ −1/5 = [1 − hex /κ]1/2 log κ Let further (ψκ , Aκ ) denote the global minimizer of (1.1) in H. Then, denote by R a rectangle in whose side lengths are given by
Abrikosov Lattices in Finite Domains
679
ωN L1 = √ hex κ
L2 =
;
2π , √ ω hex κ
where N(κ, ) ∈ N and ω(κ, ) ∈ R are such that 1 (L1 L2 )1/2 ; κ 5
r1 <
L2 < r2 , L1
where r1 and r2 are independent of κ and . Finally, let u(x1 + L1 , x2 ) = eihex κL1 x2 u(x1 , x2 ) a.e. , u(x1 , x2 + L2 ) = u(x1 , x2 ) a.e. i i hex ∞ ˆ ˆ ¯ ¯ uφ ∀φ ∈ Cc (R) , ∇ +A u· − ∇ +A φ = κ κ R R κ
1 PR = u ∈ Hloc (R2 ) UR = u ∈ PR
where Aˆ is given in (1.3). Then, there exists u0 (κ, ) ∈ UR such that
|ψ − u0 | ≤ C 2
|ψ|2 ,
2
R
(1.4a)
R
JR (u0 ) ≤ inf JR (v)[1 − δ] < 0 ∀v ∈ UR , v∈UR
(1.4b)
where δ ≤ α/4 ∀α < 1
(1.4c)
and JR (u) =
|u|4 − 2 |u|2 .
(1.4d)
R
Furthermore, ˆ H 1 () ≤ C 3 ; Aκ − A
ˆ H 2 () ≤ C 2 . Aκ − A
(1.4e)
This result proves that we can approximate ψκ in every rectangular subset R of , and possibly even in diminishingly small rectangles (as κ → ∞), by some function u0 in UR , which is the space of Abrikosov’s periodic solutions in R. Furthermore, the theorem shows that u0 can be found by studying the minimization problem of JR in UR , which is a finite dimensional subspace. The rest of this contribution is arranged as follows: in the next section we review some of the results obtained for the linear periodic problem, analyzed first by Abrikosov [2]. In § 3 we obtain some a-priori estimates that are valid for any critical point of (1.1). In § 4 we obtain upper and lower bounds for (1.1) in R which enable the proof of Theorem 1.1. Finally, in § 5 we briefly summarize the main results of this work and emphasize some additional key points.
680
Y. Almog
2. The Periodic Problem Consider the problem
i ∇ + Aˆ κ
2 u=
hex u κ
in R2 ,
(2.1)
where Aˆ = hex x1 iˆ2 .
(2.2)
Let ω ∈ R, and let ωN L1 = √ hex κ
L2 =
;
2π , √ ω hex κ
where N ∈ N. The periodic boundary conditions u should satisfy are given by v(x1 + L1 , x2 ) = eiκhex L1 x2 v(x1 , x2 ) . v(x1 , x2 + L2 ) = v(x1 , x2 ) We can now apply the transformation x→
κhex x,
(2.3)
to obtain
i∇ + x1 iˆ2
2
v = v,
(2.4a)
v(x1 + L 1 , x2 ) = eiωNx2 v(x1 , x2 ), v(x1 , x2 + L 2 ) = v(x1 , x2 ),
(2.4b) (2.4c)
where in the new coordinates L 1 = ωN
L 2 =
;
2π . ω
It can be easily verified that the phase change around ∂R, where R = [0, L 1 ] × [0, L 2 ] is 2π N. Thus, the number of vortices in R (including multiplicities) is N [13]. The general solution of (2.4a) and (2.4c) is given in the form ∞
v=
gn (x1 )eiωnx2 ,
n=−∞
where gn satisfies gn
− [(x1 − nω)2 − 1]gn = 0. The general solution of the above ordinary differential equation is 1
gn = Cn e− 2 (x1 −nω) + Dn G(x − nω), 2
where G can be expressed in terms of Parabolic Cylinder functions [1].
(2.5)
Abrikosov Lattices in Finite Domains
681
From (2.4b) we conclude that for all x1 ∈ R we must have gn (x1 + ωN ) = gn−N (x1 ), or equivalently that Cn = Cn−N
;
Dn = Dn−N .
By the lemma of Riemann-Lebesgue we must have gn (x1 ) −−−−→ 0 |n|→∞
∀x1 ∈ R,
and since G(x1 ) is unbounded in R, we must have Dn = 0 for all n ∈ Z. Consequently, the general solution of (2.4) is v=
∞
1
Cn e− 2 (x1 −nω) eiωnx2 , 2
n=−∞
where Cn+N = Cn for all n. Thus, we can write that v=
N−1
(2.6a)
Cn f n ,
n=0
where fn =
∞
1
ei(n+rN)ωx2 e− 2 [x1 −(n+rN)ω] . 2
(2.6b)
r=−∞
Note that fn+1 (x1 , x2 ) = eiωx2 fn (x1 − ω), and hence fn 2L2 (R) =
2π 3/2 ω
∀0 ≤ n ≤ N − 1.
(2.7)
We now define the spaces u(x1 + L , x2 ) = eiωNx2 u(x1 , x2 ) a.e. 1 1 (R2 ) , (2.8a) P = u ∈ Hloc u(x1 , x2 + L 2 ) = u(x1 , x2 ) a.e. ∞ ˆ ˆ ¯ ¯ i∇ + x1 i2 u · −i∇ + x1 i2 φ = U = u∈P uφ ∀φ ∈ Cc (R) , R R (2.8b) and state the following result: Lemma 2.1. Let w ∈ P. Then, ˜ w = w0 + w,
(2.9)
U0⊥
(the orthogonal complement with respect to the where w0 ∈ U0 and w˜ ∈ inner product). Furthermore, 2 ˆ ˜ 2. (2.10) −i∇ + x1 i2 w˜ ≥ 3 |w| R
L2 (R)
R
Proof. Since the restriction of P to H 1 (R) is a closed subspace of H 1 (R), and since U0 is a closed subspace of P (2.9) follows immediately.
682
Y. Almog
To prove (2.10) we recall first that w˜ must satisfy (2.4c), and hence it can be represented by the Fourier series ∞
w˜ =
w˜ n (x1 )eiωnx2 ,
(2.11)
n=−∞
where by (2.4b) we have w˜ n (x1 + N ω) = w˜ n−N (x1 ).
(2.12)
Consequently, R
|w| ˜ 2 = L 2
∞
L 1
n=−∞ 0
|w˜ n |2 dx1 = L 2
N−1 ∞ n=0
−∞
|w˜ n |2 dx1 ,
(2.13)
from which we also obtain that w˜ n ∈ L2 (R) for all 0 ≤ n ≤ N − 1. Since w˜ ∈ U0⊥ we must have w, ˜ fk = 0
∀0 ≤ k ≤ N − 1,
where fk is given by (2.6b) . Substituting (2.6b) into the above and making use of (2.11) and (2.12) yields ∞ 1 2 w˜ n exp − (x1 − nω) dx1 = 0. (2.14) 2 −∞ We now make use of (2.11) to obtain ∞ 2
ˆ −i∇ + x1 i2 w˜ = L2
L 1
n=−∞ 0
R
=
L 2
N−1 ∞ n=0
−∞
|w˜ n − (x1 − nω)w˜ n |2 dx1
|w˜ n − (x1 − nω)w˜ n |2 dx1 ,
1 (R), where from which we obtain that w˜ n ∈ Hmag 1 Hmag (R) = w |w|2 + |w − xw|2 dx < ∞ .
R
It is well-known [16] that
inf
1 (R) u∈Hmag
u⊥e−x
2 /2
R |u
− xu|2 dx = 3. 2 R |u| dx
Consequently, by (2.15) and (2.13) N−1 2 −i∇ + x1 iˆ2 w˜ ≥ 3L 2 R
n=0
∞ −∞
|w˜ n |2 dx1 = 3
|w| ˜ 2. R
(2.15)
Abrikosov Lattices in Finite Domains
683
3. A priori Estimates In this section we obtain some a priori estimates which should be satisfied by any solution of the Euler-Lagrange equations and the natural boundary conditions associated with (1.1). Thus, (ψ, A) must satisfy the equations
2 i (3.1a) ∇ + A ψ = ψ 1 − |ψ|2 , κ i ∗ −∇ × ∇ × A = (3.1b) ψ ∇ψ − ψ∇ψ ∗ + |ψ|2 A, 2κ together with the boundary conditions
i ∇ + A ψ · nˆ = 0 κ
;
h = hex .
(3.1c,d)
Since (3.1) are invariant to the gauge transformation (1.2) we fix the gauge (1.3b) for A. The first a priori estimates include the following well-known results: Lemma 3.1. Let hex ≥ κ − o(κ). Then, any solution of (3.1) must satisfy ρ L∞ () < 1, h − hex C 1 () ¯ ≤ C, i ∇ + A ψ ∞ ¯ ≤ C. κ
(3.2a) (3.2b) (3.2c)
L ()
Here and in the sequel, C is independent of κ and . Proof. See [18, 25]. Let now = (1 − hex /κ)1/2 be positive and satisfy κ −1/5
1 , log κ
(3.3)
as κ → ∞. In this case, according to the following result, every solution must be close ˆ to the normal state (ψ, A) = (0, A): Lemma 3.2. Let |ψ| = ρ. Any solution of (3.1) must satisfy ρ 4 ≤ C 4 , |h − hex |2 ≤ C 6 .
(3.4a) (3.4b)
Furthermore, let Aˆ : → R2 be given by (2.2), Then, ˆ Lp () ≤ C(p) 3 , A − A ˆ H 2 () ≤ C 2 , A − A for any p ≥ 1.
(3.4c) (3.4d)
684
Y. Almog
Proof. Let f =h−κ +
1 2 ρ . 2κ
(3.5)
In [5] it was shown that 2 1 ˆ ρ4, ∇ f − ρ f = κ J + κ − 2κ 2
2
(3.6)
where 2 ˆ 2 J ρ = |∇f |2 .
(3.7)
We note that Jˆ is continous at points where ρ = 0 [5]. Integrating (3.6) over yields ∇f 2 ∂f 1 1 4 2 2 2 + ρ + ρ (h − h ) − ρ ≤ . ex ρ κ κ ∂ ∂n Since by (3.2), C ∂f 1 1 2 ρ (h − h ) + , ex ≤ κ ∂ ∂n κ κ we obtain 1/2 ∇f 2 C C 4 2 2 2 4 + + + C ρ ≤ ρ ≤ ρ . κ κ ρ
(3.8)
Using (3.3), (3.4a) easily follows. Furthermore, 1/2 1/2 ∇f 2 2 |∇f | ≤ ρ ≤ C 3 . ρ Hence,
|∇f | +
|f − (hex − κ)| ≤ C 3 . ∂
Poincar´e inequality [20] then yields, |f − (hex − κ)|W 1,1 () ≤ C 3 . Consequently, by the Gagliardo-Nirenberg inequality we obtain that |f − (hex − κ)|2 ≤ C 6 ,
from which (3.4b) readily follows. To prove (3.4c) we utilize the well-known inequality ˆ H 1 () ≤ ˆ 2= A − A |∇ × (A − A)| |h − hex |2 (3.9)
Abrikosov Lattices in Finite Domains
685
and Sobolev embedding. Finally, in order to prove (3.4d) we observe that by (3.1b), 2 2 2 i |∇h| ≤ ρ ∇ + A ψ . κ Multiplying (3.1a) by ρ 2 ψ¯ and integrating by parts we obtain 2 i ρ 2 ∇ + A ψ ≤ ρ4. κ Hence, by (3.4a)
|∇h|2 ≤ C 4 .
Combining the above with (3.9) we obtain ˆ H 2 () ≤ C 2 . A − A We next proceed to obtain some L∞ estimates. Lemma 3.3. Let δ denote the domain δ = {x ∈ | d(x, ∂) ≥ δ}, and let α denote, here and in the sequel, any real number smaller than 1. Then, ψ L∞ (1/κ ) ≤ Cα α/2 , i ≤ Cα α/2 , κ + A ψ ∞ L (1/κ )
(3.10a) (3.10b)
h − hex L∞ (1/κ ) ≤ Cα α .
(3.10c)
Proof. Let χr denote a smooth cutoff function satisfying C C 0 |x| > r |∇χr | ≤ |∇ 2 χr | ≤ 2 . χr (x) = 1 |x| < 2r r r
(3.11)
Let x0 ∈ 2/(κ) . We multiply (3.6) by χ1/(κ) (x − x0 ) and integrate by parts to obtain ∇f 2 +κ ∇χ · ∇f − χρ 2 (h − κ) = κ χ χρ 4 , (3.12) − ρ B1 B1 B1 B1 κ
κ
κ
κ
where Br = B(x0 , r). For the first integral on the right-hand side of (3.12) we have 1/2 2 1/2 ∇f ∇χ · ∇f ≤ |∇χ |2 ρ 2 ρ B B B1 1 1 κ κ κ 1/4 2 1/2 ∇f . ≤ C 1/2 κ 1/2 ρ4 ρ B
1 κ
B
1 κ
686
Y. Almog
For the second integral we have, utilizing (3.2b), 2 2 ≤ χρ (h − κ) ρ |h − h | + ρ 2 |hex − κ| ex B1 B1 B1 κ κ κ 1/2 1/2 1 ρ4 + C ρ4 . ≤ C κ B1 B1 κ
Let
κ
1/2
ρ4
Xn =
;
Yn =
B 2n−1
B 2n−1
κ
1/2 ∇f 2 . ρ
κ
Then, by (3.12) we have X02 + Y02 ≤ C
1/2 1/2 2 X Y + X 1 1 . κ 1/2 1 κ
(3.13)
Since, 2 2 ∇f 2 ≤ 2 ∇h + 2 |∇ρ|2 = 2 i ∇ + A ψ , ρ ρ 2 κ κ we have by (3.2), X02 + Y02 ≤
C . κ 2
Since x0 was arbitrarily chosen, we can cover B1/κ by a finite number of discs of radius 1/(2κ). Consequently, X12 + Y12 ≤
C . κ 2
Substituting in (3.13) yields X02 + Y02 ≤
C 1/4 κ 2
.
The above procedure can recursively be applied to obtain X02 + Y02 ≤
Cn 2αn , κ2
where αn can be determined by the recurrence relation αn =
1 3 + αn−1 4 4
;
1 α0 = − . 2
Clearly, αn → 1, and hence, we can conclude that 2α ρ 4 ≤ Cα 2 ∀α < 1. κ B1 κ
(3.14)
Abrikosov Lattices in Finite Domains
687
We now apply the transformation x → κ(x − x0 ) to (3.1a) in B(x0 , 1/κ) to obtain 2
i ˜ 2 in B(0, 1), ∇ + A˜ ψ˜ = ψ˜ 1 − |ψ| κ ˜ where ψ(x) = ψ(κ(x −x0 )), A˜ = A(k(x −x0 )). Standard elliptic estimates [3], together with (1.2), then show that ˜ H 2 [B(0,1)] ≤ C ψ ˜ L2 [B(0,1)] = Cκ ψ L2 [B(x ,1/κ)] ≤ Cκ 1/2 ψ L4 [B(x ,1/κ)] , ψ 0 0 mag wherein
u 2H 2 (U ) mag
=
˜ 2 + |(i∇ + A) ˜ 2 u|2 . |u|2 + |(i∇ + A)u|
U
Since ψ 4L4 [B(x ,1/κ)] 0
≤
ρ 4 ≤ Cα B
1 κ
2α κ2
∀α < 1,
(3.15)
(which is a rather crude estimate, but appears to be difficult to improve) (3.10a) follows immediately from Sobolev embedding. To prove (3.10b) we use bootstrapping and Sobolev embedding. Finally, to prove (3.10c) we notice that by (3.1b), i |∇h| ≤ ρ + A) ψ . κ Consequently, by (3.10a), (3.10b), (3.2b), and (3.1d) we have |h − hex | ≤ Cα α +
C . κ
We note, once again, that (3.10) is not the optimal estimate. Ideally, one should obtain i ψ L∞ (1/κ ) + + A ψ ≤ C, ∞ κ L (1/κ ) however, in view of the crudeness of (3.15), (3.10) is the best we can obtain here. Let R(, κ) ⊂ denote a rectangle whose sides length are given by ωN L1 = √ hex κ
;
L2 =
2π , √ ω hex κ
where ω ∈ R, and N ∈ N are chosen such that r1 ≤
L1 ≤ r2 L2
;
1 (L1 L2 )1/2 , κ 5
(3.16)
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Y. Almog
where r1 and r2 are constant as κ → ∞ and → 0 according to (3.3). Denote by x1 and x2 the coordinates in the respective directions of L1 and L2 . Let R denote the image of R under the transformation (2.3). Let PR denote the restriction of (2.8a) to H 1 (R) and PR denote its image under the inverse of (2.3), i.e, the restriction to H 1 (R) of ! u(x1 + L1 , x2 ) = exp iωN √hex κx2 u(x1 , x2 ) a.e. 1 . (R2 ) u ∈ Hloc u(x1 , x2 + L2 ) = u(x1 , x2 ) a.e. Let η denote a smooth cutoff function satisfying 1 1 x ∈ R : d(x, ∂R) ≥ κ η= |∇η| ≤ Cκ 0 x ∈ R2 \ R
|∇ 2 η| ≤ Cκ 2 2 .
(3.17)
Let (ψκ , Aκ ) denote the global minimizer (which depends on as well) of (1.1) in H. (To keep the notation consistent with the one in the next section, we state the rest of the results in this section for the global minimizer, although they could have been stated for any solution of (3.1).) Clearly, ηψκ ∈ PR . Let UR denote the restriction of (2.8b) to H 1 (R). Let UR denote its image under the inverse of (2.3), i.e, hex ∞ ˆ ˆ ¯ ¯ uφ ∀φ ∈ Cc (R) . i∇ + A u · −i∇ + A φ = u ∈ PR κ R
R
From the results of § 2 it follows that UR is a finite dimensional subspace of PR . Furthermore, we can now write ˜ ηψκ = u0 + u, where u0 ∈ UR and u˜ ∈
UR⊥ .
The next lemma estimates the
(3.18) L2 (R)
norm of u. ˜
Lemma 3.4. Let u˜ be defined by (3.18). Then, 2 2 |u| ˜ ≤ C |ψκ |2 + Cα (L1 L2 )α 5 . R
(3.19)
R
Proof. We first multiply (3.1a) by η2 ψ¯ κ and integrate over R to obtain 2 i 2 2 ∇ + Aκ (ηψκ ) = 1 |∇η| |ψ | + η2 |ψκ |2 (1 − |ψκ |2 ). κ κ κ2 R R R We now write 2 2 i i ∇ + Aκ (ηψκ ) = ∇ + Aˆ (ηψκ ) κ κ R R i 2 2 ˆ ¯ ¯ + η (Aκ − A) · ψκ ∇ψκ − ψκ ∇ ψκ + |ψκ | Aκ κ R ˆ 2. − η2 |ψκ |2 |Aκ − A| R
For the second integral on the right-hand-side of (3.21) we have ˆ · i ψ¯ κ ∇ψκ − ψκ ∇ ψ¯ κ + |ψκ |2 Aκ I2 = η2 (Aκ − A) κ R 2 1/2 1/2 i ∇ + Aκ (ηψκ ) ˆ2 η2 |ψκ |2 |Aκ − A| . ≤ κ R R
(3.20)
(3.21)
Abrikosov Lattices in Finite Domains
689
By (3.20) and H¨older inequality we thus have, ˆ L4 (R) ψκ L4 (R) ψκ L2 (R) I2 ≤ C Aκ − A 1
≤ Cp (L1 L2 ) 2
1 − 4p
ˆ L4p (R) ψκ 2 4 Aκ − A ∀p > 1. L (R)
Since by (3.4c), ˆ Lp (R) ≤ Aκ − A ˆ Lp () ≤ C 3 , Aκ − A we obtain I2 ≤ Cα (L1 L2 )α 5
∀α < 1.
For the last integral on the right-hand-side of (3.21) we have η2 |ψκ |2 |Aκ − A| ˆ 2 ≤ Aκ − A ˆ 2 4 ψκ 2 4 ≤ Cα (L1 L2 )α 8 . L (R) L (R)
(3.22)
(3.23)
R
Consequently, by (3.21), (3.22), and (3.23), 2 2 i i ∇ + Aκ (ηψκ ) = ∇ + Aˆ (ηψκ ) + O((L1 L2 )α 5 ). (3.24) κ κ R R Combining the above with (3.20) and (3.18) we obtain 2 i 2 ∇ + Aˆ u˜ − |u| ˜ + η2 |ψκ |4 κ R R R 1 2 2 2 ∼ |∇η| |ψκ | + |u0 |2 + O((L1 L2 )α 5 ). = 2 κ R R
(3.25)
By (2.10), however, 2 i ∇ + Aˆ u˜ ≥ 3 hex |u| ˜ 2. κ κ R R Consequently, 1 |u| ˜ 2+ η2 |ψκ |4 ≤ 2 |∇η|2 |ψκ |2 + 2 |u0 |2 + Cα (L1 L2 )α 5 . κ R R R R In view of (3.17) we obtain 1 κ2
(3.26)
|∇η|2 |ψκ |2 ≤ C 2 R
|ψκ |2 , R
and since u0 L2 (R) ≤ ψκ L2 (R) , we can combine the above with (3.25) to obtain (3.19). We note that the error term in (3.19) is indeed small compared with the first term on the right-hand-side of (3.19). We shall demonstrate this point later when we obtain a lower bound for ψκ L2 (R) at the end of § 4. Lemma (3.4) basically shows that ψκ is indeed close, in the L2 (R) sense to u0 . However, in order to obtain a minimization problem for u0 we need an estimate of (1.1). To this end, we need the following L∞ estimate
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Y. Almog
Lemma 3.5. Let u˜ be defined in (3.18). Then, u ˜ L∞ (R) ≤ Cα α , i ∇ + Aˆ u˜ ≤ Cα α . ∞ κ L (R)
(3.27a) (3.27b)
Proof. Since the proof is rather lengthy we divide it into several steps. Step 1. Let χr be given by (3.11), and let x0 ∈ R1/(κ) (see Lemma 3.3 for the definition (0) of Rδ ). Clearly, χ1/2κ (x − x0 )ψκ ∈ PR . Denote by u0 and u˜ (0) the respective projection of χ1/2κ (x − x0 )ψκ on UR and UR⊥ . Step1.1. Prove that 2α |u˜ (0) |2 ≤ Cα 2 . (3.28) κ R To prove (3.28) we repeat the same steps leading to (3.26) to obtain 1 Cα (0) (0) 2 2 4 2 2 2 |u˜ | + χ |ψκ | ≤ 2 |∇χ | |ψκ | + |u0 |2 + 2α 2α 5 κ κ R R R R (where χ stands for χ1/2κ (x − x0 )). Consequently, in view of (3.11) and (3.14) we have for 5/6 < α < 1, 2α Cα 2 5 |u˜ (0) |2 ≤ C 2 |ψκ | + 2α 2α ≤ Cα 2 . 1 κ κ R B x0 , κ Step 1.2. Prove that u˜ (0) L∞ (R) ≤ Cα α , i (0) ˆ ≤ Cα α . κ + A u˜ ∞ L (R)
(3.29a) (3.29b) (0)
We use standard elliptic estimates to prove (3.29). Since u˜ (0) = χ ψκ − u0 we have in view of (3.1a),
2 i hex (0) 1 ˆ 2 χ ψκ ∇ + Aˆ u˜ (0) − u˜ = 2 χ ψκ − χ |ψκ |2 ψκ − 2 ψκ ∇ 2 χ −|Aκ − A| κ κ κ
ˆ · i ∇ + Aκ (χ ψκ ) + 2 i ∇χ · i ∇ + Aκ ψκ . +2(Aκ − A) (3.30) κ κ κ ˜ ∈ P , we can extend it periodically to R2 , i.e., Furthermore since, u(0) R N ˜ (x + L , x ) = u(0) ˜ (x , x + L ) = u(0) ˜ (x , x ). (3.31) exp −iω √ x2 u(0) 1 1 2 1 2 2 1 2 hex κ ˜ satisfies (3.30) for every x ∈ R2 if the right-hand side The periodic extension of u(0) of it is extended in exactly the same manner.
Abrikosov Lattices in Finite Domains
Applying (2.3) to (3.30) we obtain 2 i∇ + x1 iˆ2 u˜ (0) − u˜ (0) =
691
1 [ 2 χ ψ˜ κ − χ |ψ˜ κ |2 ψ˜ κ ] 1 − 2 −ψ˜ κ ∇ 2 χ − |(1 − 2 )−1/2 A˜ κ − x1 iˆ2 |2 χ ψ˜ κ +2((1 − 2 )−1/2 A˜ κ − x1 iˆ2 ) · i∇ + (1 − 2 )−1/2 A˜ κ (χ ψ˜ κ ) (3.32) +2i∇χ · i∇ + (1 − 2 )−1/2 A˜ κ ψ˜ κ ,
where (ψ˜ κ , A˜ κ ) denotes (ψκ , Aκ ) in the stretched coordinates (2.3). To apply standard elliptic estimates we need an L2 estimate of the right-hand-side of (3.32) in B(x, 1) for every x ∈ R2 . By (3.10a) and (3.14) we have that χ 2 |ψκ |6 ≤ Cα 5α ∀α < 1, B(x,1)
and by (3.11) and (3.10a), |ψκ ∇ 2 χ |2 ≤ C 4 B(x,1)
|ψκ |2 ≤ Cα 5α .
B(x,1)
In view of (3.4d) and (3.10) we also have 2 |(1 − 2 )−1/2 Aκ − x1 iˆ2 |2 · i∇ + (1 − 2 )−1/2 Aκ ψκ ≤ C 5α . B(x,1)
Furthermore,
B(x,1)
2 |∇χ |2 i∇ + (1 − 2 )−1/2 Aκ ψκ ≤ C 3α .
Finally,
|(1 − 2 )−1/2 Aκ − x1 iˆ2 |2 χ 2 |ψκ |2 ≤ Cα 5α .
B(x,1)
Combining the above and (3.28), we may rely on the framework in [3] to obtain u˜ (0) H 2 (B(x,1) ≤ Cα α
∀α < 1.
Sobolev embedding then yields (3.29a). Bootstrapping and Sobolev embedding prove (3.29b). Step 2. Prove (3.27). We first note that (3.27) and (3.29) are different: while u˜ is the projection of ηψκ on UR⊥ , u˜ (0) is the projection of χ ψκ on the same space. To obtain (3.27) we thus need to relate χ and η. Let then {xi }M i=1 denote a set of points in R satisfying M " 1 1. R1/(κ) ⊆ B xi , κ . i=1 # 1 1 2. B xi , 4κ B xj , 4κ = ∅ if i = j .
692
Y. Almog
Let {χi }M a set of C ∞ functions satisfying i=1 denote 1 1. supp χi ⊆ B xi , κ . $M 2. i=1 χi = 1, ∀x ∈ R1/(κ) . C 3. |∇χi | ≤ κ , |∇ 2 χi | ≤ κ 2C 2 . Let η=
M
(3.33)
χi .
i=1
Clearly η satisfies (3.17) and hence, we may use it in (3.18) to define u0 and u. ˜ Furthermore, let u˜ (i) denote the projection of χi ψκ on UR⊥ . Then, u˜ =
M
u˜ (i) .
i=1
Furthermore, |u| ˜ ≤
M
|u˜ (i) |,
(3.34a)
i=1
M i i ˆ u˜ ≤ ˆ u˜ (i) . + A + A κ κ
(3.34b)
i=1
Since M is a large number, we seek an estimate for u˜ (i) when |x − xi | > 2/(κ). Step 2.1. Prove that 1 2 |u˜ (i) (x)| ≤ Cα α/2 N exp − hex κ dp2 (x, xi ) − 2 2 , (3.35a) 4 κ i ˆ u˜ (i) (x) ≤ CN exp − 1 κ 2 dp2 (x, xi ) − 2 , (3.35b) + A κ 4 κ 22 where dp (x, xi ) =
min
j,k=−1,0,1
|x − xi − (kL1 , j L2 )|.
(3.35c)
(i)
(i)
Since u0 = −u˜ (i) for every x ∈ R \ B(xi , 1/(κ)) we prove (3.35) for u0 . Recall from (2.6) that (i)
u0 =
N−1
Cn(i) fn (x),
n=0
where fn is given by fn =
∞ r=−∞
ei(n+rN)ω
√
√ hex κx2 − 21 [ hex κx1 −(n+rN)ω]2
e
.
(3.36)
Abrikosov Lattices in Finite Domains
693
(i)
Since u0 is the projection of χi ψκ on UR we have hex κω (i) Cn = χi ψfn . 2π 3/2 R Let ψˆ ik =
√ hex κω L2 −iωk √hex κx2 e χi ψdx2 . 2π 0
Let xi = (xi1 , xi2 ). Then, since ψˆ ir (x1 ) is supported in (xi1 − 1/(κ), xi1 + 1/(κ)) we have % ∞ xi1 + 1 κ 1 √ hex κ 2 (i) Cn = ψˆ in+rN e− 2 [ hex κx1 −(n+rN)ω] dx1 . π r=−∞ xi1 − κ1 By (3.10) and (3.11) we have that |ψˆ in+rN | ≤ Cα α/2
1 κL2
∀α < 1,
(recall that L2 1/(κ 5 )). Thus ∞ Cn(i) ≤ Cα α/2 hex κ
1 κ
1 r=−∞ − κ
1
√ hex κ(x1 −xi1 )−(n+rN)ω]2
e− 2 [
dx1 .
Let dp1 (x1 , y) =
min
k=−1,0,1
|x1 − y − kL1 |.
Then,
1 1 2 . exp − hex κ dp1 (xi1 , nω/ hex κ) − 2 2 2 κ
Cn(i)
≤ Cα
α/2
(3.37)
By (3.36) we have the estimate
1 2 |fn (x)| ≤ C exp − hex κdp1 x1 , nω/ hex κ . 2 Consequently, (i)
Since
N−1
& 1 2 − hex κ dp1 x1 , nω/ hex κ 2 n=0 1 2 +dp1 . xi1 , nω/ hex κ − 2 2 κ
|u0 (x)| ≤ Cα α/2
& ' 2 2 2 dp1 x1 , nω/ hex κ + dp1 xi1 , nω/ hex κ , (x1 , xi1 ) ≤ 2 dp1
(3.38)
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Y. Almog
we have that 2 1 2 . N − hex κ dp1 x1 , xi1 − 2 2 4 κ
(i) |u0 (x)|
≤ Cα
α/2
(3.39)
(i)
To prove that u0 decays in the x2 direction we use the fact that w0 = e−ihex κx1 x2 u0 (i)
(i)
to satisfy the problem 2 (i) hex (i) i κ ∇ − hex x2 iˆ1 w0 = κ w0 (i) (i) w0 (x1 + L1 , x2 ) = w0 (x1 , x2 ) (i) (i) w0 (x1 , x2 + L2 ) = w0 (x1 , x2 )eiκhex L2 x1
.
(i)
Consequently, w0 must decay in the x2 direction according to (3.39). We thus obtain 2 1 2 . N − hex κ dp1 x2 , xi2 − 2 2 4 κ
(i) |u0 (x)|
≤ Cα
α/2
(3.40)
Combining (3.40) and (3.39) yields (3.35a), from which one can easily prove (3.35b) using standard elliptic estimates. Step 2.2. Prove (3.27). Substituting (3.29) and (3.35) into (3.34) we obtain − 12 α |u(x)| ˜ ≤ Cα + N e . Since L1 L2 ≤ || we have N ≤ C 2 κ 2 . Consequently, in view of (3.3), (3.27) is proved. 4. Upper and Lower Bounds All the results of the previous section could have been formulated for any solution of the Euler-Lagrange equations. In this section we concentrate, however, on the energy functional 2 4 |ψ| i ER (ψ, A) = + |h − hex |2 + ∇ψ + Aψ . −|ψ|2 + (4.1) 2 κ R We obtain estimates for it in terms of the reduced functional JR which appears in Theorem 1.1. We start by proving the following upper bound
Abrikosov Lattices in Finite Domains
695
Lemma 4.1. Let η be defined by (3.33), and let u0 and u˜ be defined by (3.18). Then, for all v ∈ UR ER (ηψκ , Aκ ) ≤ JR (v) + Cα α−1 where
JR (v) = − 2
|v|2 + R
1 2
(L1 L2 )1/2 , κ
|v|4 .
Proof. Following [25], let η˜ denote a smooth cutoff function satisfying 1 x ∈ R η˜ = 0 x ∈ A2 |∇ η| ˜ ≤ Cκ |∇ 2 η| ˜ ≤ Cκ 2 2 , 1 x ∈ \ R d(x, ∂R) ≥ 3 κ
Let further
(4.2b)
R
Furthermore, (L1 L2 )1/2 + (L1 L2 )α 7α . |u| ˜ 4 ≤ Cα 2α |u0 |4 + Cα 2α κ R R
where
(4.2a)
(4.3)
(4.4)
k−1 k ≤ d(x, ∂R) ≤ . Ak = x ∈ \ R | κ κ
ηv ˜ p ψ˜ = ηψ ˜ κ
d(x, ∂R) ≤ d(x, ∂R) ≥
3 2κ 3 2κ
or x ∈ R , and x ∈ \ R
(4.5a)
where v p denotes a periodic extension, according to (3.31), of some v ∈ UR , and A˜ satisfies Aˆ x ∈ R ∪ A1 (4.5b) A˜ = 2 Aκ d(x, ∂R) ≥ κ and x ∈ \ R and ∇ × A˜ − hex L∞ () ≤ C 2 ,
(4.5c)
(cf. [25]). In view of the above, since (ψκ , Aκ ) is the minimizer of E in H we have ˜ ≥ E(ψκ , Aκ ). ˜ A) E(ψ, Consequently, ˆ − ER (ψκ , Aκ ) ˜ − E(ψκ , Aκ ) = ER (v, A) ˜ A) 0 ≤ E(ψ, 2 2 i i ˜ ˜ + κ ∇ + A ψ − κ ∇ + Aκ ψκ A + |∇ × A˜ − hex |2 − |∇ × Aκ − hex |2 A & ' & ' 1 ˜ 4 − |ψκ |4 − ˜ 2 − |ψκ |2 , |ψ| |ψ| + 2 A A
(4.6)
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Y. Almog
where A = A1 ∪ A2 ∪ A3 . By (4.5) and (3.10) we have (L1 L2 )1/2 |∇ × A˜ − hex |2 ≤ Cα 4α . κ A Moreover, by (4.4) and (3.10) we have 2 2 1/2 i ∇ + A˜ ψ˜ − i ∇ + Aκ ψκ ≤ Cα α−1 (L1 L2 ) . κ κ κ A3
(4.7)
(4.8)
To estimate 2 2 i ∇ + A˜ ψ˜ − i ∇ + Aκ ψκ , κ κ A1
we choose first v ∈ UR such that
|v|4 ≤ 4 L1 L2 ,
(4.9)
R
which yields |v|2 ≤ 2 L1 L2 . R
In this case
2 2 i i p p ∇ + Aˆ ηv ∇ + Aˆ ηv = ˜ ˜ κ κ A1 R∪A1 i 2 1 − ∇ + Aˆ v p = η˜ 2 |v p |2 + 2 |∇ η| ˜ 2 |v p |2 κ κ A1 R A1 ≤ (1 + C 2 ) |v p |2 . A1
Let p
vl = v p (x1 + l1 , x2 + l2 ), where 0 ≤ l1 ≤ L1 and 0 ≤ l2 ≤ L2 . Clearly, p 2 |vl | dl1 dl2 = |A| |v|2 . A
R
R
Consequently, inf
(l1 ,l2 )∈R A
p
|vl |2 ≤
|A| |R|
|v|2 ≤ C R
L 1 L2 . κ
Abrikosov Lattices in Finite Domains
697 p
Denote by (l1m , l2m ) the values of l1 and l2 which minimize the L2 (A) norm of vl and let vm = v p (x1 + l1m , x2 + l2m ). If we choose v p = vm in (4.5a) we obtain 2 i L1 L2 ∇ + Aˆ ηv ˜ m ≤ C . κ κ A1 Combining the above with (4.8) we obtain 2 1/2 i ∇ + A˜ ψ˜ ≤ Cα α−1 (L1 L2 ) . κ κ A
(4.10)
Combining (4.6), (4.7), (4.9), and (4.10) we obtain ˆ + Cα α−1 ER (ψκ , Aκ ) ≤ ER (vm , A)
(L1 L2 )1/2 . κ
Since ˆ = JR (vm ) = JR (v), ER (vm , A) (4.2) is proved as long as (4.9) holds. To prove (4.2) for any v ∈ UR , we write 1 JR (γ v) = − 2 γ 2 |v|2 + γ 4 |v|4 . 2 R R It is easy to show that for a given v ∈ UR , JR (γ v) is minimal for 2 2 2 R |v| . γ0 = 4 R |v| Let w = γ0 v. Then,
|w| = 4
R
and hence
|w|2 ,
2
(4.11)
R
|w|2 ≤ 2 L1 L2 . R
Consequently, ER (ψκ , Aκ ) ≤ JR (w) + Cα α−1
(L1 L2 )1/2 (L1 L2 )1/2 ≤ JR (v) + Cα α−1 , κ κ
which proves (4.2). Let η be given by (3.33). It is easy to show that ER (ηψκ , Aκ ) ≤ ER (ψκ , Aκ ) + Cα α−1
(L1 L2 )1/2 . κ
(4.12)
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By (3.24), (3.18), and (2.10) we obtain 1 2 2 4 4 ER (ηψκ , Aκ ) ≥ − |u0 | + η |ψκ | + |u| ˜ 2 − Cα (L1 L2 )α 5 . 2 R R R
(4.13)
Combining the above and (4.2) with v = u0 we obtain that 1/2 1 2 4 α−1 (L1 L2 ) α 5 + (L1 L2 ) . |u| ˜ ≤ |u0 | + Cα 2 R κ R By (3.27) we then have
|u| ˜ 4 ≤ Cα 2α
R
(L1 L2 )1/2 + (L1 L2 )α 7α . |u0 |4 + Cα 2α κ R
Using (4.3) we can obtain a lower bound for ER (ψκ , Aκ ) in terms of the reduced functional JR . Lemma 4.2. Let η be given by (3.33) and u0 be given by (3.18). Then, 1/2 α−1 (L1 L2 ) α 5 ER (ψκ , Aκ ) ≥ JR (u0 ) − Cα + (L1 L2 ) κ
(4.14)
for all α < 1. Proof. By (4.12) and (4.13) we have that ER (ψκ , Aκ ) ≥ −
2
1 |u0 | + 2 R
η4 |ψκ |4 − Cα (L1 L2 )α 5 − Cα α−1
2
R
(L1 L2 )1/2 κ (4.15)
for all α < 1. Since η4 |ψκ |4 ≥ |u0 |4 − 4|u0 |3 |u|, ˜ we have by (4.3) and H¨older inequality that η4 |ψκ |4 ≥ |u0 |4 [1 − Cα α/2 ] R
R
−Cα (L1 L2 ) + α 5
α−1 (L1 L2 )
κ
1/2 1/4
3/4 |u0 |
4
. (4.16)
R
By (4.3) we also have that u0 L4 (R) ≤ ηψκ L4 (R) + u ˜ L4 (R) ≤ ηψκ L4 (R) 1/4 (L1 L2 )1/2 +Cα α/2 u0 4L4 (R) + (L1 L2 )α 5 + α−1 . κ
(4.17)
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To estimate u0 L4 (R) , we thus need an estimate for ηψκ L4 (R) . By (3.26), (3.33), and (2.10) we have that 1/2 4 4 2 2 2 α 5 α−1 (L1 L2 ) , η |ψκ | ≤ η |ψκ | + Cα (L1 L2 ) + κ R R (recall that u0 L2 (R) ≤ ηψκ L2 (R) ). Consequently, (L1 L2 )1/2 η4 |ψκ |4 ≤ 4 L1 L2 + Cα (L1 L2 )α 5 + α−1 . κ R
We note that by (3.16) the first term on the right-hand-side of the above inequality is much greater than the second one if α is sufficiently close to 1, such that (L1 L2 )1−α . Consequently |ηψκ |4 ≤ 2 4 L1 L2 , R
and hence, by (4.17) we obtain |u0 |4 ≤ 2 4 L1 L2 . R
Substituting in (4.16) and then in (4.15) we obtain (4.14).
Proof of Theorem 1.1. Combining (4.2) and (4.14) we obtain 1/2 α−1 (L1 L2 ) α 5 JR (u0 ) ≤ JR (v) + Cα + (L1 L2 ) . κ To prove (1.4a) we thus need an estimate for inf v∈UR JR (v). Let |v|4 β = inf B(v) := L1 L2 inf R . 2 2 v∈UR v∈UR R |v|
(4.18)
(4.19)
Since UR is finite-dimensional, it is easy to show that there exists w ∈ UR satisfying
2 |w|4 = βL1 L2 |w|2 . (4.20) R
R
Furthermore, since β is invariant to the transformation w → γ w for every γ ∈ R, we can choose w such that (4.11) is satisfied. Combining (4.11) and (4.20) we obtain L1 L2 |w|2 = 2 . β R Thus, in view of (4.11) 1 JR (w) = − 2 2
|w|2 = − 4 R
L 1 L2 . 2β
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In [2, 10, 4] B(v) was calculated in various cases. In particular, it was found that when v is the well-known square lattice, i.e, when v=C
N−1
fn ,
n=0
where fn is given by (2.6b), that B(v) ≈ 1.18 independently of N and the scale of R. It follows therefore, in view of (4.19) that inf JR (v) ≤ −C 4 L1 L2 ,
(4.21)
v∈UR
where C > 0. Substituting (4.21) into (4.18) we obtain (1.4b). To prove (1.4a) we write
|ψκ − u0 | ≤ 2 R
|η − 1| |ψκ | +
2
2
|u| ˜
2
R
2
.
R
The right-hand-side of the above inequality can be bounded utilizing (3.19) and (3.17) to obtain (L1 L2 )1/2 2 2 . (4.22) |ψκ − u0 | ≤ C |u0 |2 + Cα (L1 L2 )α 5 + Cα α−1 κ R R Since u0 L2 (R) ≤ ψκ L2 (R) we need a lower bound for u0 L2 (R) to complete the proof of (1.4a). To this end we use (1.4b) and (4.21) to obtain β u0 4L2 (R) − 2 u0 2L2 (R) ≤ JR (u0 ) ≤ −C 4 L1 L2 , 2L1 L2 from which we obtain u0 2L2 (R) ≥ C 2 L1 L2 . Substituting in (4.22) proves (1.4a).
5. Conclusion Let R(κ, ) = [0, L1 ] × [0, L2 ], where L1 and L2 are given by (3.16). In the previous sections the following main results were proved: 1. We proved that the L2 (R) distance of ψκ from the space of Abrikosov solutions in R, UR is much smaller than the L2 (R) norm of ψκ . 2. We proved that the energy, which is given by (1.4d), of the projection of ηψκ on UR , where η is given by (3.33), is approximately the minimum over all UR of (1.4d).
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We note that the above results do not show that ψκ is nearly periodic, inasmuch as every function in L2 (R) can be approximated by a periodic function. Nevertheless, since the energy of the above projection is close to the minimum of JR over UR , we can obtain an approximation of ψκ by studying a much simpler minimization problem than the minimization of (1.1) in H. It is widely believed that the minimizer of JR in UR (which is a finite dimensional space), is the well-known triangular lattice [2, 10, 4], as long as N , in (3.16), is even. If in addition to that, any u ∈ UR whose energy is close to the minimum must be close, in some sense to the triangular lattice, then ψκ is indeed nearly periodic. It seems worthwhile to note here that the direction of the lattice cannot be determined by the energy considerations applied in the previous section. Thus if 0
ˆ u0 (x) = u0 (Qx) ; A = hex Q , x1 where Q is a 2 × 2 rotation matrix, then, since the cells affected by the rotation are only those near the boundary, we have that 1/2 ˆ ≤ C (L1 L2 ) . ER (u 0 , Aˆ ) − ER (u0 , A) κ Clearly, the above error is indistinguishable by the lower and upper bounds, (4.14) and (4.2), obtained in § 4. Finally, we note that the limitations (3.3) could have been replaced by the weaker assumptions 1 4 κ
;
(L1 L2 )1/2
1 , κ 4
if only we could overcome the crudeness of the estimate (3.15). However, to extend the analysis to the case 4 ∼ O(1/κ), a completely different approach is necessary, since in that case the surface energy, which is of O(1/κ) is at least equally important to ER which is of O( 4 L1 L2 ). References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. New York: Dover, 1972 2. Abrikosov, A.A.: On the magnetic properties of superconductors of the second group. Soviet Phys. J.E.T.P., 5, 1175–1204 (1957) 3. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of partial differential. Commum. Pure Appl. Math. 17, 35–92 (1964) 4. Almog, Y.: On the bifurcation and stability of periodic solutions of the Ginzburg-Landau equations in the plane. Siam J. Appl. Math. 61, 149–171 (2000) 5. Almog, Y.: Non-linear surface superconductivity for type II superconductors in the large domain limit. Arch. Rat. Mech. Anal. 165, 271–293 (2002) 6. Almog, Y.: The loss of stability of surface superconductivity. J. Math. Phys. 45, 2815–2832 (2004) 7. Almog, Y.: Non-linear surface superconductivity in the large κ limit. Rev. Math. Phys. 16, 961–976 (2004) 8. Bauman, P., Philips, D., Tang, Q.: Stable nucleation for the Ginzburg-Landau system with an applied magnetic field. Arch. Rat. Mech. Anal. 142, 1–43 (1998) 9. Bernoff, A., Sternberg, P.: Onset of superconductivity in decreasing fields for general domains. J. Math. Phys. 39, 1272–1284 (1998) 10. Chapman, S.J.: Nucleation of superconductivity in decreasing fields I. European J. Appl. Math. 5, 449–468 (1994)
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11. Chapman, S.J.: Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model. Quart. Appl. Math. 53, 601–627 (1995) 12. del Pino, M., Felmer, P.L., Sternberg, P.: Boundary concentration for eigenvalue problems related to the onset of the superconductivity. Commun. Math. Phys. 210, 413–446 (2000) 13. Eliot, C.M., Matano, H., Qi, T.: Zeros of complex Ginzburg-Landau order parameters with applications to superconductivity. European J. Appl. Math. 5, 431–448 (1994) 14. Essmann, U., Tr¨auble, H.: The direct observation of individual flux lines in type II superconductors. Phys. Lett. A24, 526–527 (1967) 15. Giorgi, T., Philips, D.: The breakdown of superconductivity due to strong fields for the GinzburgLandau model. SIAM J. Math. Anal. 30, 341–359 (1999) 16. Helffer, B.: Semi-Classical Analysis for the Schr¨odinger Operator and Applications, No. 1336 in Lecture Notes in Mathematics, Berlin-Heidelberg New York: Springer-Verlag, 1988 17. Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185, 604–680 (2001) 18. Helffer, B., Pan, X.B.: Upper critical field and location of nucleation of surface superconductivity. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 20, 145–181 (2003) 19. Lu, K., Pan, X.B.: Gauge invariant eigenvalue problems in R2 and R2+ . Trans. Am. Math. Soc. 352, 1247–1276 (2000) 20. Maz’ja, V.G.: Sobolev spaces. Springer Series in Soviet Mathematics, Berlin: Springer-Verlag, 1985. Translated from the Russian by T. O. Shaposhnikova 21. Meissner, W., Ochsenfeld, R.: Naturwissenschaffen 21, 787 (1933) 22. Pan, X.B.: Surface superconductivity in applied magnetic fields above hC2 . Commum. Math. Phys. 228, 327–370 (2002) 23. Rubinstein, J.: Six lectures on superconductivity. In: Boundaries interfaces and transitions, M. Delfour, (ed.), Vol. 13 of CRM proceedings and lecture notes, Providence, RI: Am. Math. Soc., 1998, pp. 163–184 24. Saint-James, D., de Gennes, P.: Onset of superconductivity in decreasing fields. Phys. Let. 7, 306–308 (1963) 25. Sandier, E., Serfaty, S.: The decrease of bulk-superconductivity near the second critical field in the Ginzburg-Landau model. SIAM J. Math. Anal. 34, 939–956 (2003) Communicated by P. Constantin
Commun. Math. Phys. 262, 703–728 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1442-2
Communications in
Mathematical Physics
Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups Julien Bichon1 , An De Rijdt2 , Stefaan Vaes2,3 1
Laboratoire de Math´ematiques Appliqu´ees, Universit´e de Pau et des Pays de l’Adour, IPRA, Avenue de l’Universit´e, 64000 Pau, France. E-mail: [email protected] 2 Department of Mathematics, K.U.Leuven, Celestijnenlaan 200B, 3001 Leuven Belgium. E-mail: [email protected] 3 Institut de Math´ematiques de Jussieu, Alg`ebres d’Op´erateurs, 175 rue du Chevaleret, 75013 Paris, France. E-mail: [email protected] Received: 17 March 2005 / Accepted: 18 March 2005 Published online: 10 October 2005 – © Springer-Verlag 2005
Abstract: We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C∗ -algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C∗ -algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood. Introduction By a well known theorem of Høegh-Krohn, Landstad and Størmer [16], compact groups only admit ergodic actions on tracial C∗ -algebras. Indeed, the (unique) invariant state is necessarily a trace. Moreover, given a compact group G acting ergodically on a C∗ -algebra B, one studies the so-called spectral subspaces: the action of G on B yields a unitary representation of G which can be decomposed into a direct sum of irreducible representations. One proves that the multiplicity of an irreducible representation is necessarily bounded by the dimension of this irreducible representation. A deeper analysis of the spectral structure of ergodic actions of compact groups has been made by A. Wassermann [27–29]. In the culmination of his work, Wassermann shows that the compact group SU(2) only admits ergodic actions on von Neumann algebras of finite type I. In the 1980’s, Woronowicz introduced the notion of a compact quantum group and generalized the classical Peter-Weyl representation theory. Many fascinating examples of compact quantum groups are available by now: Drinfel’d and Jimbo [14, 17] introduced the q-deformations of compact semi-simple Lie groups and Rosso [21] showed that they fit into the theory of Woronowicz. The universal orthogonal and unitary quantum groups were introduced by Van Daele and Wang [25] and studied in detail by Banica [2, 3]. Other
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examples of compact quantum groups, related with graphs and metric spaces have been constructed by the first author [9] and by Banica [4, 5]. The abstract theory of ergodic coactions of compact quantum groups on C∗ -algebras has been initiated by Boca [10] and Landstad [18]. The major difference with the compact group case, is the following: the (unique) invariant state is no longer a trace. Indeed, Wang [26] gave examples of ergodic coactions of universal unitary quantum groups on type III factors. Moreover, in the work of Boca, the multiplicity of an irreducible corepresentation is proved to be bounded by the quantum dimension rather than the ordinary dimension of the corepresentation. Nevertheless, in all the examples known up to now, the multiplicity is actually bounded by the ordinary dimension. In this paper, we provide examples of ergodic coactions where the multiplicity of an irreducible corepresentation is strictly larger than the ordinary dimension of the corepresentation. In [29], A. Wassermann gives a complete classification of the ergodic actions of SU(2), essentially labeling them by the finite subgroups of SU(2). It would, of course, be great to give a complete classification of ergodic coactions of the deformed SUq (2). In [23], Tomatsu provides a first step in this direction: he computes all ergodic coactions of SUq (2) on ‘virtual’ quotient spaces SUq (2)/ . (More precisely, he describes all the coideals of the quantum group SUq (2).) By construction, the ergodic coactions of SUq (2) on its virtual quotient spaces are such that the multiplicity of an irreducible corepresentation is bounded by its dimension. The results of this paper imply in particular that there are much more ergodic coactions of SUq (2) than the ones studied by Tomatsu. The major tool to produce our new examples of ergodic coactions of compact quantum groups, is the notion of monoidal equivalence of quantum groups. One can look at a compact quantum group with several degrees of precision. At first, we study only the fusion rules in the representation theory: we label the irreducible corepresentations and describe how a tensor product of irreducibles breaks up into irreducibles. Taking into account only these fusion rules, we loose a lot of information: for example, the qdeformed compact Lie groups have the same fusion rules as their classical counterparts. In a next approximation, one studies the corepresentation theory of a compact quantum group as a monoidal category, but without its concrete realization (the so-called forgetful functor to the category of Hilbert spaces). This is crucial: by the Tannaka-Krein reconstruction theorem [32], the concrete monoidal category of (finite-dimensional) corepresentations essentially determines the compact quantum group. Note that knowing the representation theory of a compact quantum group as a monoidal category, comes down to knowing the fusion rules together with the 6j -symbols, see Remark 3.3. Closely related to the notion of monoidally equivalent quantum groups, is the notion of a unitary fiber functor on a compact quantum group. Essentially, a unitary fiber functor gives another concrete realization, different from the tautological realization, of the representation theory of a compact quantum group. In this paper, we choose not to use the abstract language of categories. We give ‘down-to-earth definitions’ of monoidally equivalent quantum groups and unitary fiber functors, see 3.1 and 3.7. This makes the construction of associated C∗ -algebras and coactions straightforward. This concrete approach is well adapted to the language of corepresentations of compact quantum groups. In this way, using previous results of Banica [2, 3], we can show very easily as well the monoidal equivalence of the universal orthogonal and universal unitary quantum groups. The results in this paper can be summarized as follows.
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• In Sect. 2, we recall the theory of spectral subspaces [10, 18] and provide a simple proof for the multiplicity bound. We also introduce the notion of quantum multiplicity of an irreducible corepresentation in an ergodic coaction and this can be strictly larger than the ordinary multiplicity. • In Theorem 3.9, we show that there is a natural bijective correspondence between certain ergodic coactions of compact quantum groups and unitary fiber functors. These coactions are called of full quantum multiplicity. These are precisely the ergodic coactions for which the crossed product is isomorphic with the compact operators, see [18]. They can also be described as the ergodic coactions for which the inequality between quantum multiplicity and quantum dimension, becomes an equality. • In Sect. 4, we study the special case of unitary fiber functors preserving the dimension. This leads to a bijective correspondence with unitary 2-cocycles on the dual, discrete, quantum group. The ideas for this section come from the work of Wassermann [28]. • In Sect. 5, we establish the monoidal equivalence between the universal orthogonal quantum groups Ao (F ). Recall that, for any F ∈ GL(n, C) satisfying F F = ±1, one defines the compact quantum group Ao (F ) as the universal quantum group generated by the coefficients of a unitary n by n matrix U with relations U = F U F −1 . The comultiplication on Ao (F ) is (uniquely) defined in such a way that U becomes a corepresentation. We show that Ao (F1 ) is monoidally equivalent with Ao (F2 ) if and only if the signs of the Fi Fi agree and Tr(F1∗ F1 ) = Tr(F2∗ F2 ). In particular, if √ 0 < q ≤ 2 − 3, there is a continuous family of non-isomorphic Ao (F ) monoidally equivalent with SUq (2). • In Sect. 6, we prove similar results for the universal unitary quantum groups Au (F ), defined as the universal quantum group generated by the coefficients of a unitary n by n matrix U with the relation that F U F −1 is unitary. Again, the comultiplication is defined such that U becomes a corepresentation. We show that the quantum dimension of U , i.e. Tr(F ∗ F ) Tr((F ∗ F )−1 ), is a complete invariant for the Au (F ) up to monoidal equivalence. • Using the previous results, we obtain a complete classification of the ergodic coactions of full quantum multiplicity of Ao (F ) and Au (F ), as well as a computation of the 2-cohomology of their duals (Corollary 5.9). In particular, we construct ergodic coactions of SUq (2) such that the multiplicity of the fundamental corepresentations is arbitrarily large (Corollary 5.8). In the theory of Hopf algebras, ergodic coactions of full quantum multiplicity correspond to Hopf-Galois extensions. In this algebraic setting, several results related to ours have been obtained. The relation between Hopf-Galois extensions and fiber functors is due to Ulbrich [24] and the relation between monoidal equivalence of Hopf algebras and Hopf-bi-Galois extensions has been established by Schauenburg [22]. Fiber functors preserving the dimension and 2-cocycles have been studied by Etingof and Gelaki [15]. The main difference between these Hopf algebraic results and our work, lies in dealing with the ∗ -structure and positivity. In a sense, we are dealing with the real forms of (certain) Hopf algebras. This allows us to construct Hilbert space representations and C∗ -algebras. The compatibility of fiber functors with ∗ -structures is a severe restriction. Indeed, there exist many fiber functors on the representation category of SU(2) (see [11]), but the forgetful functor is the only one compatible with the ∗ -structure. After completion of a first version of this paper – signed by the last two authors and circulated as a preprint – the first author joined the project and his preprint [6] was taken into account, yielding the current paper as a final result.
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1. Preliminaries Notation 1.1. Consider a subset S of a C∗ -algebra. We denote by S the linear span of S and by [S] the closed linear span of S. The symbol ⊗ denotes tensor products of Hilbert spaces, minimal tensor products of C∗ -algebras as well as algebraic tensor products of ∗ -algebras. If A is a ∗ -algebra and U ∈ Mn1 ,n2 (C) ⊗ A, we denote by U the matrix U ij = Uij∗ . We make use of the leg numbering notation. For instance, if v ∈ A ⊗ B, then v12 denotes the element in A ⊗ B ⊗ C defined by v12 = v ⊗ 1. We analogously use the notations v13 , v23 , etc. Definition 1.2. A compact quantum group G = (A, ) consists of a unital C∗ -algebra A together with a unital *-homomorphism : A → A⊗A satisfying the coassociativity relation (ι ⊗ ) = ( ⊗ ι) and the cancellation properties [(A)(A ⊗ 1)] = A ⊗ A = [(A)(1 ⊗ A)] . If G = (A, ) is a compact quantum group, there exists a unique state h on A which is invariant under the comultiplication: (ι ⊗ h)(a) = (h ⊗ ι)(a) = h(a)1 for all a ∈ A. We call h the Haar state of G. Definition 1.3. Let H be a Hilbert space. A unitary corepresentation v of G on H is a unitary element of M(K(H ) ⊗ A) satisfying (ι ⊗ )(v) = v12 v13 . The dimension of the underlying Hilbert space H is called the dimension of v and denoted by dim v. The tensor product of the unitary corepresentations v and w is defined by T w := v13 w23 . v T w and w T v are in general not unitarily equivalent. Note that the corepresentations v This is a crucial feature of quantum groups.
Notation 1.4. Let G = (A, ) be a compact quantum group. Given two unitary corepresentations v ∈ M(K(H ) ⊗ A) and w ∈ M(K(K) ⊗ A), we denote by Mor(v, w) the intertwiners between v and w: Mor(v, w) = {S ∈ B(H, K) | (S ⊗ 1)v = w(S ⊗ 1)} . Terminology 1.5. A unitary corepresentation v ∈ M(K(H ) ⊗ A) is called irreducible if Mor(v, v) = C1. A unitary corepresentation w ∈ M(K(L) ⊗ A) is called unitarily equivalent to v if Mor(v, w) contains a unitary operator. In this paper, all corepresentations are assumed to be unitary. Recall the following well known facts (see [30]). Every irreducible corepresentation of a compact quantum group is finite dimensional and every corepresentation decomposes as a direct sum of irreducible corepresentations.
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the set of Notation 1.6. Let G = (A, ) be a compact quantum group. We denote by G equivalence classes of irreducible corepresentations of G and we choose unitary rep We denote by the (class of the) trivial resentatives U x ∈ B(Hx ) ⊗ A for all x ∈ G. corepresentation 1 ∈ A. there is a unique x ∈ G such that Mor(, x T x) = 0. The irreducFor every x ∈ G, T x), t = 0 ible corepresentation x is called the adjoint of x. Take now t ∈ Mor(, x and define the antilinear map St : Hx → Hx : ξ → (ξ ∗ ⊗ 1)t . Define Qx := St∗ St and normalize t in such a way that Tr(Qx ) = Tr(Q−1 x ). This uniquely determines Qx and fixes t up to a number of modulus 1. If we take the unique −1 ∗ T x) such that (t ⊗ 1)(1 ⊗ t˜) = 1, then St˜ = St and Qx = (St St∗ )−1 . t˜ ∈ Mor(, x is defined as Tr(Qx ) and denoted by Notation 1.7. The quantum dimension of x ∈ G dimq (x). Observe that t ∗ t = dimq (x)1 and that dimq (x) = dimq (x) ≥ dim(x). The orthogonality relations can then be written as follows (see [30]): for ξ ∈ Hx , η ∈ Hy , (ι ⊗ h)(U x (ξ η∗ ⊗ 1)(U y )∗ ) =
δx,y 1 Qx ξ, η . dimq (x)
(1.1)
Notation 1.8. Let G = (A, ) be a compact quantum group. We denote by A the set of coefficients of finite dimensional corepresentations of G. Hence, ξ, η ∈ Hx . A = (ωξ,η ⊗ ι)(U x ) | x ∈ G, Then, A is a unital dense ∗ -subalgebra of A. Restricting to A, A becomes a Hopf ∗ -algebra. Terminology 1.9. Let B be a unital ∗ -algebra. A linear functional ω : B → C is said to be a faithful state on B if ω(1) = 1 and if ω(aa ∗ ) ≥ 0 for all a ∈ B, with equality holding if and only if a = 0. Observe that the Haar state of a compact quantum group (A, ) is a faithful state on the underlying Hopf ∗ -algebra A. The dual (discrete) quantum group of (A, ), can be defined as Aˆ =
B(Hx )
ˆ with (a)S = Sa
ˆ S ∈ Mor(x, y T z) . for all a ∈ A,
x∈G
ˆ ), ˆ using the left regular corepresenThere is, of course, another way to define (A, tation of (A, ) and the theory of multiplicative unitaries. We only need this in Sect. 4, see Notation 4.6.
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2. Spectral Subspaces and Quantum Multiplicity In this section, we give a brief overview of the general theory of ergodic coactions of compact quantum groups. We study spectral subspaces and prove in particular that they are finite dimensional. The results in this section are well known (see [16] for the classical case of compact groups and [10, 18, 23] for compact quantum groups). We give a short presentation for the convenience of the reader. denotes the set of Let G = (A, ) be a compact quantum group. Recall that G equivalence classes of irreducible corepresentations of G and that we chose unitary We have (ι ⊗ )(U x ) = U x U x . representatives U x ∈ B(Hx ) ⊗ A for all x ∈ G. 12 13 Definition 2.1. Let B be a unital C*-algebra. A (right) coaction of (A, ) on B is a unital ∗ -homomorphism δ : B → B ⊗ A satisfying (δ ⊗ ι)δ = (ι ⊗ )δ and [δ(B)(1 ⊗ A)] = B ⊗ A . The coaction δ is said to be ergodic if the fixed point algebra B δ := {x ∈ A | δ(x) = x ⊗ 1} equals C1. Remark 2.2. If δ : B → B ⊗ A is an ergodic coaction of (A, ) on B there is a unique state ω on B which is invariant under δ, given by ω(b)1 = (ι ⊗ h)δ(b). In what follows, we fix a coaction δ : B → B ⊗ A of a compact quantum group G = (A, ). We define the spectral subspace associated with x by Definition 2.3. Let x ∈ G. x }. Kx = {X ∈ H x ⊗ B | (ι ⊗ δ)(X) = X12 U13
Defining Hom(Hx , B) = {S : Hx → B | S linear and δ(Sξ ) = (S ⊗ ι)(U x (ξ ⊗ 1))} , we have Kx ∼ = Hom(Hx , B), associating to every X ∈ Kx the operator SX : Hx → B : ξ → X(ξ ⊗ 1). Definition 2.4. We define B as the subspace of B generated by the spectral subspaces, i.e. X ∈ Kx , ξ ∈ Hx . B := X(ξ ⊗ 1) | x ∈ G, Observe that B is a dense unital *-subalgebra of B and that the restriction δ : B → B ⊗ A defines a coaction of the Hopf ∗ -algebra (A, ) on B. Terminology 2.5. A coaction δ : B → B ⊗ A of (A, ) on B is said to be universal if B is the universal enveloping C∗ -algebra of B. It is said to be reduced if the conditional expectation (ι ⊗ h)δ of B on B δ is faithful. Remark 2.6. Observe that an ergodic coaction is reduced if and only if the unique invariant state is faithful. In the special case where B = A and δ = , the ∗ -algebra B coincides with the underlying Hopf ∗ -algebra A ⊂ A consisting of coefficients of finite-dimensional corepresentations. So, we obtain the usual notions: a compact quantum group (A, ) is said to be universal if A is the universal enveloping C∗ -algebra of A and reduced if
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the Haar state is faithful on A. Clearly, any compact quantum group has its universal and reduced companion. In the case where (A, ) is the dual of a discrete group, these notions coincide with the full, resp. reduced group C∗ -algebra. A compact quantum group is said to be co-amenable if its universal and reduced companion coincide. Equivalently, a compact quantum group is co-amenable if we have on the same C∗ -algebra a bounded co-unit and a faithful Haar state. It is then clear that a coaction of a co-amenable compact quantum group is always both universal and reduced. Examples of co-amenable compact quantum groups include SUq (n) and other q-deformations of compact Lie groups. Fix now an ergodic coaction δ : B → B ⊗ A of a compact quantum group (A, ) on a unital C∗ -algebra B. Denote by ω the unique invariant state on B. the spectral subspace Kx is For every X, Y ∈ Kx , XY ∗ ∈ B δ = C1. So, for x ∈ G, a Hilbert space with scalar product X, Y 1 = XY ∗ . Terminology 2.7. Let δ be an ergodic coaction of (A, ) on B. The dimension of the Hilbert space Kx is called the multiplicity of x in δ and denoted by mult(x). Xx ∈ B(Hx , K x ) ⊗ B such that (X x )∗ (Y ⊗ 1) = Y ∗ for all Define, for every x ∈ G, Y ∈ Kx . Observe that X x (X x )∗ = 1. Therefore, (Xx )∗ X x ∈ B(Hx ) ⊗ B is a projection. ∗ Take t ∈ Mor(, x T x), normalized in such a way that t t = dim q (x). Let x ∈ G. Define the antilinear map Tt : Kx → Kx : Tt (Y ) = (t ∗ ⊗ 1)(1 ⊗ Y ∗ ) . Since t is fixed up to a number of modulus one, Lx := Tt∗ Tt is a well defined positive element of B(Kx ). √ Definition 2.8. We put mult q (x) := Tr(Lx ) Tr(Lx ) and call mult q (x) the quantum multiplicity of x in δ. We prove in Theorem 2.9 that multq (x) ≤ dimq (x) for all x ∈ G. If equality holds for all x ∈ G, we say that δ is of full quantum multiplicity. Theorem 2.9. Let δ : B → B ⊗ A be an ergodic coaction of a compact quantum group G = (A, ) on a unital C∗ -algebra B. For every irreducible corepresentation x ∈ G, mult(x) ≤ mult q (x) ≤ dimq (x) . With Xx defined as above, the ergodic coaction is of full quantum multiplicity if and only if X x is unitary for all x ∈ G. ∗ and take t ∈ Mor(, x T x), normalized in such a way that t t = Proof. Let x ∈ G ∗ T x) satisfying (1 ⊗ t )(t˜ ⊗ 1) = 1, we dimq (x). If we take the unique t˜ ∈ Mor(, x have t˜∗ t˜ = dimq (x) and hence, it is clear that Tt˜ = Tt−1 and Lx = (Tt Tt∗ )−1 . So, multq (x) = Tr(Tt Tt∗ ) Tr((Tt Tt∗ )−1 ) ≥ dim(kx¯ ) = mult(x) ,
By definition for all x ∈ G. Lx X, Y 1 = (t ∗ ⊗ 1)(1 ⊗ Y ∗ ), (t ∗ ⊗ 1)(1 ⊗ X ∗ )1 = (t ∗ ⊗ 1)(1 ⊗ Y ∗ X)(t ⊗ 1) . (2.1)
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Since t ∗ (1 ⊗ a)t = Tr(Q−1 x a) for all a ∈ B(Hx ), we conclude that x ∗ x Tr(Lx )1 = (Tr(Q−1 x ·) ⊗ ι)((X ) X ) .
Since (Xx )∗ X ∗ is a projection, it follows that Tr(Lx ) ≤ Tr(Q−1 x ) = dim q (x) for all Applying this inequality to x and x, we conclude that mult q (x) ≤ dimq (x) for x ∈ G. all x ∈ G. if and only if Moreover, multq (x) = dimq (x) for all x ∈ G x ∗ x −1 (Tr(Q−1 x ·) ⊗ ι)((X ) X ) = Tr(Qx )1
(2.2)
This last statement holds if and only if (Xx )∗ X x = 1 for all x ∈ G, i.e. for all x ∈ G. when Xx is unitary for all x ∈ G. It is also straightforward to show that the invariant state ω is a KMS state, at least for universal and for reduced ergodic coactions. Proposition 2.10. If the ergodic coaction δ of (A, ) on B is either universal or reduced, the invariant state ω is a KMS state. The elements of the dense ∗ -subalgebra B ⊂ B (Definition 2.4) are analytic with respect to the modular group, given by it σtω Y (ξ ⊗ 1) = (L−it x Y )(Qx ξ ⊗ 1) for all x ∈ G, Y ∈ Kx , ξ ∈ Hx . and Y ∈ Ky , Z ∈ Kz . One verifies that Proof. Let y, z ∈ G (U z )∗ ((ι ⊗ ω)(Z ∗ Y ) ⊗ 1)U y = (ι ⊗ ω)(Z ∗ Y ) ⊗ 1 . ω)(Z ∗ Y )
Hence, (ι ⊗ conclude that
(2.3)
equals 0 if y = z and is scalar if y = z. Applying ω to (2.1), we (ι ⊗ ω)(Z ∗ Y ) =
δy,z 1 Ly Y, Z . dimq (y)
If moreover ξ ∈ Hy , η ∈ Hz and if we put a = Y (ξ ⊗ 1) and b = Z(η ⊗ 1), we get ω(b∗ a) =
δy,z ξ, ηLy Y, Z . dimq (y)
(2.4)
Using (1.1), one checks that ω(ab∗ ) = (ι ⊗ ω)(Y (ξ η∗ ⊗ 1)Z ∗ ) =
δy,z 1 Qy ξ, η Y, Z . dimq (y)
ω As a linear space, B ∼ = x∈G (Kx ⊗ Hx ). So, we can define linear maps σt : B → B by the formula it σtω Y (ξ ⊗ 1) := L−it y (Y )(Qy ξ ⊗ 1) . It is clear that (σtω ) is a one-parameter group of linear isomorphisms of B. Observe that all elements of B are analytic with respect to (σtω ) and that ω(σiω (a)b∗ ) = ω(b∗ a) = ω(aσiω (b)∗ ) for all a, b ∈ B. Since (2.4) implies that ω is faithful on B, it follows that ω (a ∗ ) for all a ∈ B. Standard σiω : B → B is multiplicative and that σiω (a)∗ = σ−i ω ∗ complex analysis allows to conclude that the σt are -automorphisms of B. It is also clear that ω is invariant under σtω . If δ is a universal coaction, the one-parameter group (σtω ) extends to B by universality. If δ is a reduced coaction, we can extend σtω to B because ω is invariant under σtω and ω is faithful on B. In both cases, it follows that (σtω ) satisfies the KMS condition with respect to ω and so, ω is a KMS state.
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Finally observe that ω is a trace if and only if Lx = 1 and Qx = 1 for all x ∈ G with Kx = 0. 3. Commuting Coactions and Monoidal Equivalence of Quantum Groups Our main goal in this section is to show the relation between ergodic coactions of full quantum multiplicity and monoidal equivalence of compact quantum groups. In Sects. 5 and 6, we shall give examples of monoidally equivalent compact quantum groups, giving rise to new examples of ergodic coactions. The relation between ergodic coactions of full quantum multiplicity and monoidal equivalence allows us to classify completely such coactions for the unitary and orthogonal quantum groups Au (F ) and Ao (F ), in particular for SUq (2). Definition 3.1. Two compact quantum groups G = (A, ) and G2 = (A2 , 2 ) are said →G 2 satisfying ϕ() = , to be monoidally equivalent if there exists a bijection ϕ : G together with linear isomorphisms T ··· T xr , y1 T ··· T yk ) ϕ : Mor(x1 T ··· T ϕ(xr ), ϕ(y1 ) T ··· T ϕ(yk )) → Mor(ϕ(x1 )
satisfying the following conditions: ϕ(1) = 1, ϕ(S ⊗ T ) = ϕ(S) ⊗ ϕ(T ), ∗ ∗ ϕ(S ) = ϕ(S) , ϕ(ST ) = ϕ(S)ϕ(T ),
(3.1)
whenever the formulas make sense. In the first formula, we consider 1 ∈ Mor(x, x) = T ) = Mor(x, T x). Such a collection of maps ϕ is called a monoidal Mor(x, x equivalence between G and G2 . Remark 3.2. To define a monoidal equivalence between G and G2 , it suffices to define → G 2 satisfying ϕ() = , together with linear isomorphisms a bijection ϕ : G T · · · T yk ) → Mor(ϕ(x), ϕ(y1 ) T · · · T ϕ(yk )) for k = 1, 2, 3, satisfying ϕ : Mor(x, y1 ϕ(1) = 1, (3.2) T y), T ∈ Mor(b, x T y), S ∈ Mor(a, x (3.3) ϕ(S)∗ ϕ(T ) = ϕ(S ∗ T ), T z), S ∈ Mor(b, x T y),(3.4) ϕ((S ⊗ 1)T ) = (ϕ(S) ⊗ 1)ϕ(T ), T ∈ Mor(a, b T b), S ∈ Mor(b, y T z). (3.5) ϕ((1 ⊗ S)T ) = (1 ⊗ ϕ(S))ϕ(T ), T ∈ Mor(a, x Indeed, such a ϕ admits a unique extension to a monoidal equivalence. Again, (3.2) T ) = Mor(x, T x). should be valid for 1 ∈ Mor(x, x) = Mor(x, x T c) → Remark 3.3. Observe that the existence of the linear isomorphisms ϕ : Mor(a, b T ϕ(c)) only says that G and G2 have the same fusion rules. Adding Mor(ϕ(a), ϕ(b) (3.2)–(3.5) means that G and G2 moreover have the same 6j -symbols (see [12]). Indeed, T c), we can write two natural orthonormal taking orthonormal bases for all Mor(a, b T y T z), one given by elements (S ⊗1)T , the other given by elements bases for Mor(a, x (1 ⊗ S)T . The coefficients of the transition unitary between both orthonormal bases are called the 6j -symbols of G.
Remark 3.4. As we shall see in Sects. 5 and 6, there are natural examples of monoidal equivalences where dim(ϕ(x)) = dim(x). On the other hand, it is clear that, for all x ∈ G and all monoidal equivalences ϕ, dimq (ϕ(x)) = dimq (x). We shall see in Sect.
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6 that for a certain class of compact quantum groups (the universal unitary ones), this equality of quantum dimension is the only constraint for monoidal equivalence. Remark 3.5. It is clear that a monoidal equivalence ϕ in the sense of Definition 3.1, defines a monoidal equivalence in the usual sense (preserving the ∗ -operation), between the monoidal categories of finite dimensional corepresentations of G and G2 ([20]). Moreover, this monoidal equivalence is uniquely determined up to isomorphism. We prefer to work with the ‘concrete’data of Definition 3.1, avoiding all kinds of identifications. Notation 3.6. If two compact quantum groups G = (A, ) and G2 = (A2 , 2 ) are monoidally equivalent, we write G ∼ G2 . mon
Closely related to the notion of monoidal equivalence, is the following notion of unitary fiber functor (see Proposition 3.12 for the relation between both notions). Definition 3.7. Let G = (A, ) be a compact quantum group. A unitary fiber functor a finite dimensional Hilbert space Hϕ(x) and consists on G associates to every x ∈ G further of linear maps T ··· T xr , y1 T ··· T yk ) ϕ : Mor(x1 → B(Hϕ(x1 ) ⊗ · · · ⊗ Hϕ(xr ) , Hϕ(y1 ) ⊗ · · · ⊗ Hϕ(yk ) )
satisfying Eqs. (3.1) in Definition 3.1. Remark 3.8. We make a remark analogous to 3.2. To define a unitary fiber functor on a finite-dimensional Hilbert space Hϕ(x) , with G, it suffices to associate to every x ∈ G Hϕ() = C and to define linear maps T ··· T yk ) → B(Hϕ(x) , Hϕ(y1 ) ⊗ · · · ⊗ Hϕ(y ) ) ϕ : Mor(x, y1 k
for k = 1, 2, 3, satisfying (3.2) – (3.5) as well as the non-degenerateness assumption S ∈ Mor(a, b T c), ξ ∈ Hϕ(a) } {ϕ(S)ξ | a ∈ G,
is total in
Hϕ(b) ⊗ Hϕ(c)
for all b, c ∈ G. Moreover, it follows from Proposition 3.12 below, that a unitary fiber functor ϕ on G naturally defines a compact quantum group G2 such that ϕ becomes a monoidal equivalence between G and G2 . Theorem 3.9. Consider a compact quantum group G = (A, ) and let ϕ be a unitary fiber functor on G. – There exists a unique unital ∗ -algebra B equipped with a faithful state ω and unitary satisfying elements X x ∈ B(Hx , Hϕ(x) ) ⊗ B for all x ∈ G, y z T z), (S ⊗ 1) = (ϕ(S) ⊗ 1)X x for all S ∈ Mor(x, y 1. X13 X23 2. the matrix coefficients of the Xx form a linear basis of B, 3. (ι ⊗ ω)(X x ) = 0 if x = . – There exists a unique coaction δ : B → B ⊗ A satisfying x x (ι ⊗ δ)(X x ) = X12 U13
for all x ∈ G.
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– The state ω is invariant under δ. Denoting by Br the C∗ -algebra generated by B in the GNS-representation associated with ω and denoting by Bu the universal enveloping C∗ -algebra of B, the coaction δ admits a unique extension to a coaction on Br , resp. Bu . These coactions are reduced, resp. universal and they are ergodic and of full quantum multiplicity. – Every reduced, resp. universal, ergodic coaction of full quantum multiplicity, arises in this way from a unitary fiber functor. Proof. Let ϕ be a unitary fiber functor on G = (A, ). Define the vector space B = ∗ ∗ ⊕x∈G B(Hx , Hϕ(x) ) . We shall turn this vector space into a -algebra. x Define natural elements X ∈ B(Hx , Hϕ(x) ) ⊗ B by (ωx ⊗ ι)(X x ) = (δx,y ωx )y∈G for all ωx ∈ B(Hx , Hϕ(x) )∗ . By definition, the coefficients of the X x form a linear basis of B. Hence, it suffices to define a product and an involution on the level of the X x . It is clear that there exists a unique bilinear multiplication map B × B → B such that y
z X13 X23 (S ⊗ 1) = (ϕ(S) ⊗ 1)X x
for all
T z) . S ∈ Mor(x, y
But then y
a b c c (X14 X24 )X34 ((S ⊗ 1)T ⊗ 1) = (ϕ(S) ⊗ 1 ⊗ 1)X13 X23 (T ⊗ 1) = ((ϕ(S) ⊗ 1)ϕ(T ) ⊗ 1)X x = (ϕ((S ⊗ 1)T ) ⊗ 1) X x T b), T ∈ Mor(x, y T c). Since intertwiners of the form (S ⊗ 1)T for all S ∈ Mor(y, a T b T c), we conclude that linearly span Mor(x, a
a b c (X14 X24 )X34 (S ⊗ 1) = (ϕ(S) ⊗ 1)X x T b T c). Analogously, for all S ∈ Mor(x, a
a b c X14 (X24 X34 ) (S ⊗ 1) = (ϕ(S) ⊗ 1)X x T b T c). This proves the associativity of the product on B. It is for all S ∈ Mor(x, a clear that X provides the unit element of B. Observe also that
y
x (ϕ(S)∗ ⊗ 1)X13 X23 = Xz (S ∗ ⊗ 1)
(3.6)
T y). for all S ∈ Mor(z, x We define an antilinear map b → b∗ on B such that
x (X x )∗13 (ϕ(t) ⊗ 1) = X23 (t ⊗ 1)
(3.7)
t ∈ Mor(, x T x). This antilinear map is well defined: taking t ∈ for all x ∈ G, ∗ T x) and t˜ ∈ Mor(, x T x), normalized in such a way that (t ⊗ 1)(1 ⊗ t˜) = 1, Mor(, x we define ∗ (ωξ,η ⊗ ι)(X x ) := (ω(ξ ∗ ⊗1)t,(1⊗η∗ )ϕ(t˜) ⊗ ι)(X x ) for all ξ ∈ Hx and η ∈ Hϕ(x) .
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For ξ ∈ Hx , η ∈ Hϕ(x) , we compute
∗ ∗
(ωξ,η ⊗ ι)(X x )
∗ = (ω(ξ ∗ ⊗1)t,(1⊗η∗ )ϕ(t˜) ⊗ ι)(X x )
= (ω(((ξ ∗ ⊗1)t)∗ ⊗1)t˜,(1⊗((1⊗η∗ )ϕ(t˜))∗ ϕ(t)) ⊗ ι)(X x ) = (ωξ,η ⊗ ι)(X x ) ,
in the last step using our particular choice of t and t˜. We also get x ∗ x ) = (ϕ(t)∗ ⊗ 1)X13 . (t ∗ ⊗ 1)(X23
Because x x X23 (t ⊗ 1) = ϕ(t) ⊗ 1 (X x (X x )∗ )13 (ϕ(t) ⊗ 1) = X13
and because, by (3.6), (t ∗ ⊗ 1)((X x )∗ X x )23 = t ∗ ⊗ 1 , the elements Xx are unitaries. and S ∈ Mor(z, x T y), Since for all x, y, z ∈ G y
y
x x ∗ X23 (S ⊗ 1))∗ = ((ϕ(S) ⊗ 1)X z )∗ = (X z )∗ (ϕ(S)∗ ⊗ 1) = (S ∗ ⊗ 1)(X23 )∗ (X13 ) (X13
by (3.6) and the fact that the X x are unitary, our involution is anti-multiplicative. We conclude that B a *-algebra. Denote by ω the linear functional ω : B → C given by ω(1) = 1 and (ι⊗ω)(X x ) = 0 for all x = . We show that ω is a faithful state on B. ∗ Take t ∈ Mor(, x T x) such that t t = dim q (x). Take the unique Let x, y ∈ G. ∗ T x) such that (t ⊗ 1)(1 ⊗ t˜) = 1. Then, t˜ ∈ Mor(, x y
x ). (ωµ,ρ ⊗ ι)(X y )(ωξ,η ⊗ ι)(X x )∗ = (ωµ⊗(ξ ∗ ⊗1)t,ρ⊗(1⊗η∗ )ϕ(t˜) ⊗ ι)(X13 X23
We conclude that ω (ωµ,ρ ⊗ ι)(X y )(ωξ,η ⊗ ι)(X x )∗ 1 = δx,y µ ⊗ (ξ ∗ ⊗ 1)t, t ϕ(t), ρ ⊗ (1 ⊗ η∗ )ϕ(t˜) dimq (x) 1 = δx,y (ξ ∗ ⊗ 1)t, (µ∗ ⊗ 1)tρ, η dimq (x) 1 = δx,y Qx µ, ξ ρ, η . dimq (x) Choose orthonormal bases (fix ) for every space Hϕ(x) . Any element a ∈ B admits a unique decomposition (ωξix ,fix ⊗ ι)(X x ) a= x,i
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in terms of vectors ξix ∈ Hx . Then, ω(aa ∗ ) =
x,i
1 Qx ξix , ξix . dimq (x)
It follows that ≥ 0 for all a and that ω(aa ∗ ) = 0 if and only if ξix = 0 for all x and i, i.e. if and only if a = 0. The definition of the coaction δ : B → B ⊗ A is obvious and it is clear that ω is invariant under δ. It follows that we can extend δ to coactions δr , resp. δu of (A, ) on Br , resp. Bu . Moreover, ω(x)1 = (ι ⊗ h)δr (x) for all x ∈ Br and analogously for x ∈ Bu . It follows that δr and δu are ergodic coactions. Given the unitary elements X x and Theorem 2.9, it follows that δr and δu are of full quantum multiplicity. By definition, the coaction δr on Br is reduced and the coaction δu on Bu is universal. Indeed, the canonical ∗ -subalgebra of Bu generated by the spectral subspaces for δu , is exactly B. It remains to show that any reduced, resp. universal ergodic coaction of full quantum multiplicity arises as above from a unitary fiber functor. Let δ : B → B ⊗ A the unibe an ergodic coaction of full quantum multiplicity. Construct for x ∈ G, x tary elements X ∈ B(Hx , K x ) ⊗ B as in Sect. 2. Define Hϕ(x) := K x . Let S ∈ T ··· T xr , y1 T ··· T yk ). The element Mor(x1 ω(aa ∗ )
y
y
x1 xr 1 k X1,k+1 · · · Xk,k+1 (S ⊗ 1)(X1,r+1 · · · Xr,r+1 )∗
is invariant under ι ⊗ δ. So, we can define ϕ(S) by the formula y
y
x1 xr 1 k X1,k+1 · · · Xk,k+1 (S ⊗ 1) = (ϕ(S) ⊗ 1)X1,r+1 · · · Xr,r+1 T ··· T xr , y1 T ··· T yk ). It is clear that ϕ is a unitary fiber functor for all S ∈ Mor(x1 on G. Denote by B the ∗ -subalgebra of B generated by the spectral subspaces of δ as defined in Sect. 2. By definition, B is generated by the coefficients of the Xx . In order to show that B is isomorphic to the ∗ -algebra defined by the unitary fiber functor ϕ, it suffices to show that the coefficients of the Xx form a linear basis of B. But this follows immediately from (2.4).
Definition 3.10. Two unitary fiber functors ϕ and ψ on a compact quantum group G are said to be isomorphic if there exist unitaries ux ∈ B(Hϕ(x) , Hψ(x) ) satisfying ψ(S) = (uy1 ⊗ · · · ⊗ uyk )ϕ(S)(u∗x1 ⊗ · · · ⊗ u∗xr ) T ··· T xr , y1 T ··· T yk ). for all S ∈ Mor(x1
Proposition 3.11. Let ϕ and ψ be unitary fiber functors on G and denote by δϕ , resp. δψ the associated coactions on Bϕ , resp. Bψ . Then the following statements are equivalent. 1. The fiber functors ϕ and ψ are isomorphic. 2. There exists a ∗ -isomorphism π : Bϕ → Bψ satisfying (π ⊗ ι)δϕ = δψ π . Proof. Straightforward.
The following proposition follows immediately from the Tannaka-Krein reconstruction theorem, but we give a detailed statement for clarity. Its proof is completely analogous to the proof of Theorem 3.9.
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Proposition 3.12. Let ϕ be a unitary fiber functor on a compact quantum group G = (A, ). – There exists a unique universal compact quantum group (A2 , 2 ) with underlying Hopf ∗ -algebra (A2 , 2 ) and unitary corepresentations U ϕ(x) ∈ B(Hϕ(x) ) ⊗ A2 satisfying ϕ(y) ϕ(z) T z), 1. U13 U23 (ϕ(S) ⊗ 1) = (ϕ(S) ⊗ 1)U ϕ(x) for all S ∈ Mor(x, y 2. the matrix coefficients of the U ϕ(x) form a linear basis of A2 . – {U ϕ(x) | x ∈ G} is a complete set of irreducible corepresentations of (A2 , 2 ) and ϕ is a monoidal equivalence of compact quantum groups. The following result is then a corollary of Theorem 3.9. Proposition 3.13. Consider two compact quantum groups G = (A, ) and G2 = (A2 , 2 ). Let ϕ : G → G2 be a monoidal equivalence. In particular, ϕ is a unitary fiber functor on G. Denote by Br , resp. Bu the C∗ -algebras associated to ϕ as in Theorem 3.9, with dense ∗ -subalgebra B. Denote by δ the corresponding coaction of (A, ) on B. Denote by X x ∈ B(Hx , Hϕ(x) ) ⊗ B the unitaries generating B. ϕ(x)
x for – There is a unique coaction δ2 : B → A2 ⊗ B satisfying (ι ⊗ δ2 )(X x ) = U13 X23 all x ∈ G. The coaction δ2 commutes with δ and extends to Br , resp. Bu , yielding a reduced, resp. universal, ergodic coaction of full quantum multiplicity. – Every pair of commuting reduced, resp. universal, ergodic coactions of full quantum multiplicity arises in this way from a monoidal equivalence.
Proof. Given the monoidal equivalence ϕ, it is obvious to construct the coaction δ2 . It remains to show the second statement. Let δ : B → B ⊗ A and δ2 : B → A2 ⊗ B be commuting ergodic coactions of full quantum multiplicity. Denote by B the unital ∗ -subalgebra of B generated by the spectral subspaces of δ. Using Theorem 3.9, we get a unitary fiber functor ϕ on G and we may assume that B and δ are constructed from ϕ as in Theorem 3.9. In particular, B is generated by the coefficients of X x ∈ B(Hx , Hϕ(x) )⊗B. Because δ and δ2 commute, the element (ι⊗δ2 )(X x )(X x )∗13 is invariant under (ι⊗ι⊗ δ). Since δ is ergodic and X x unitary, we get a unitary element U ϕ(x) ∈ B(Hϕ(x) ) ⊗ A2 ϕ(x) x such that (ι ⊗ δ2 )(X x ) = U12 X13 . Because δ2 is a coaction, we easily compute that ϕ(x) U is a unitary corepresentation of G2 . is a complete set of irreducible unitary It remains to show that {U ϕ(x) | x ∈ G} corepresentations of G2 and that ϕ is a monoidal equivalence. Assume that S ∈ Mor(ϕ(x), ϕ(y)). The element (X y )∗ (S ⊗ 1)X x belonging to B(Hx , Hy ) ⊗ B is invariant under ι ⊗ δ2 , so it has the form T ⊗ 1, with T ∈ B(Hx , Hy ). It follows that T ∈ Mor(x, y) = δx,y C and hence, S ∈ δx,y C. So, the U ϕ(x) are mutually inequivalent irreducible corepresentations of G2 . exhausts all irreducible corepresentaIn order to show that the set {U ϕ(x) | x ∈ G} tions of G2 , it suffices to show, for all a ∈ A2 , that (ι ⊗ h2 )((1 ⊗ a)U ϕ(x) ) = 0 for all implies a = 0. But, given the formula for δ2 , we get (h2 ⊗ ι)((a ⊗ 1)δ2 (x)) = 0 x ∈ G, for all x ∈ B. Since δ2 is of full quantum multiplicity, this implies that a = 0. It remains to show that ϕ is a monoidal equivalence. For this, it suffices to show T ϕ(z)) = ϕ(Mor(x, y T z)). If S ∈ Mor(x, y T z), we use the that Mor(ϕ(x), ϕ(y)
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multiplicativity of δ2 to obtain ϕ(x)
x = (ι ⊗ ι ⊗ δ2 )((ϕ(S) ⊗ 1)X x ) (ϕ(S) ⊗ 1 ⊗ 1)U12 X13 y
z = (ι ⊗ ι ⊗ δ2 )(X13 X23 (S ⊗ 1)) ϕ(y)
y
ϕ(y)
ϕ(z)
ϕ(z)
z (S ⊗ 1 ⊗ 1) = U13 X14 U23 X24 x . = U13 U23 (ϕ(S) ⊗ 1 ⊗ 1)X13 T ϕ(z)). The converse inclusion is shown It follows that S ∈ Mor(ϕ(x), ϕ(y) analogously.
4. Unitary Fiber Functors Preserving the Dimension We study in this section unitary fiber functors ϕ on a compact quantum group G pre Taking into serving the dimension, i.e. satisfying dim Hϕ(x) = dim Hx for all x ∈ G. account Theorem 3.9, this comes down to the study of ergodic coactions of full quantum multiplicity satisfying mult(x) = dim(x) for all x ∈ G. We establish a relation between unitary fiber functors preserving the dimension (up ˆ ). ˆ The to isomorphism) and the 2-cohomology of the dual, discrete quantum group (A, following definition is due to Landstad [19] and Wassermann [28], who consider it for the dual of a compact group. ˆ is said to be a 2-cocycle if it satisfies Definition 4.1. A unitary element ∈ M(Aˆ ⊗ A) ˆ ⊗ ι)()( ⊗ 1) = (ι ⊗ )()(1 ˆ ( ⊗ ) .
(4.1)
Two 2-cocycles 1 and 2 are said to differ by a coboundary if there exists a unitary ∗ ⊗ u∗ ). We denote this relation by ∼ and ˆ such that 2 = (u)(u ˆ u ∈ M(A) 1 2 observe that ∼ is an equivalence relation on the set of 2-cocycles. Remark 4.2. In the quantum setting, there is no reason that a product of two 2-cocycles ˆ ) ˆ as the is again a 2-cocycle. So, although we could define the 2-cohomology of (A, set of equivalence classes of 2-cocycles, this set has no natural group structure. the minimal central projections of Aˆ = ⊕ B(Hx ). Notation 4.3. Denote by px , x ∈ G, x
Remark 4.4. Up to coboundary, we can and will assume that a unitary 2-cocycle is normalized, i.e. (p ⊗ 1) = p ⊗ 1
and (1 ⊗ p ) = 1 ⊗ p .
ˆ ), ˆ the dual of G = (A, ). Denote Let be a normalized unitary 2-cocycle on (A, ˆ ⊗ ι)()( ⊗ 1) = (ι ⊗ )()(1 ˆ ⊗ ) . (2) := ( It follows from Remark 3.2 that there is a unique unitary fiber functor ϕ on G satisfying Hϕ (x) = Hx ,
ϕ (S) = ∗ S ,
ϕ (T ) = ∗(2) T ,
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T z) and T ∈ Mor(a, x T y T z). Observe that we implicitly used for all S ∈ Mor(x, y ˆ and hence, ∗ S is a well defined element of that B(Hy ⊗ Hz ) is an ideal in M(Aˆ ⊗ A) T z). B(Hx , Hy ⊗ Hz ) whenever S ∈ Mor(x, y From Proposition 3.12, we get a compact quantum group (A , ), whose dual ˆ ) is given by (Aˆ ,
Aˆ = ⊕x B(Hx ) = Aˆ
ˆ (a)ϕ (S) = ϕ (S)a and
T z). Also, ϕ becomes a monoidal equivalence for all a ∈ B(Hx ), S ∈ Mor(x, y between (A, ) and (A , ). Observe that
ˆ (a) = ∗ (a) ˆ for all a ∈ Aˆ = Aˆ . Proposition 4.5. Let ϕ be a unitary fiber functor on a compact quantum group G = Then there exists a normalized (A, ) such that dim Hϕ(x) = dim Hx for all x ∈ G. ˆ ˆ unitary 2-cocycle on (A, ), uniquely determined up to coboundary, such that ϕ is isomorphic with ϕ . Proof. Denote by δ : B → B ⊗ A the reduced ergodic coaction associated with ϕ by Theorem 3.9. Consider the generating unitaries X x ∈ B(Hx , Hϕ(x) ) ⊗ B satisfying x U x for all x ∈ G. (ι ⊗ δ)(Xx ) = X12 13 Since dim Hϕ(x) = dim Hx , we can take unitary elements ux : Hϕ(x) → Hx . Take u = 1. Define Y x = (ux ⊗ 1)X x and consider Y := ⊕x Y x ∈ M(Aˆ ⊗ B). Because ˆ ⊗ ι)(Y )Y ∗ Y ∗ is invariant under (ι ⊗ ι ⊗ δ), we find a unitary element the element ( 23 13 ˆ such that ∈ M(Aˆ ⊗ A) ˆ ⊗ ι)(Y ) = ( ⊗ 1)Y13 Y23 . ( ˆ ⊗ ι ⊗ ι and ι ⊗ ˆ ⊗ ι to this equality, we obtain that is a unitary 2-cocycle Applying ˆ ). ˆ on (A, T z). Then, It remains to show that ϕ and ϕ are isomorphic. Let S ∈ Mor(x, y ˆ ⊗ ι)(Y )(S ⊗ 1) = ( ⊗ 1)Y13 Y23 (S ⊗ 1) (S ⊗ 1)Y x = ( y z (S ⊗ 1) = ((uy ⊗ uz ) ⊗ 1)X13 X23 = ((uy ⊗ uz ) ⊗ 1)(ϕ(S) ⊗ 1)X x = ((uy ⊗ uz )ϕ(S)u∗x ⊗ 1)Y x . T z). Hence, ϕ (S) = ∗ S = (uy ⊗ uz )ϕ(S)u∗x for all S ∈ Mor(x, y It is obvious that ϕ1 is isomorphic with ϕ2 if and only if the 2-cocycles 1 and 2 differ by a coboundary.
ˆ ) ˆ and consider the unitary fiber functor Fix a normalized unitary 2-cocycle on (A, ϕ . Theorem 3.9 yields C∗ -algebras Br and Bu with ergodic coactions δr and δu of full quantum multiplicity. It is, of course, possible to describe these C∗ -algebras directly in ˆ ). ˆ Before we terms of : they correspond to the -twisted group C∗ -algebras of (A, can prove such a statement, we have to introduce a few notations and a bit of terminology. Such -twisted group C∗ -algebras have been studied by Landstad [19] and Wassermann [28] when is a unitary 2-cocycle on the dual of a compact group.
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Notation 4.6. Denote by H the L2 -space of the Haar state h on (A, ). We consider A and Aˆ as being represented on H . Denote by V the right regular corepresentation of (A, ). The operator V is a multiplicative unitary in the sense of [1] and belongs to M(Aˆ ⊗ A). There exists a canonical unitary u ∈ B(H ) with u2 = 1 such that Vˆ := (u ⊗ 1)V (u ⊗ 1) is a multiplicative unitary. We know that uAu ⊂ A , ˆ ⊂ Aˆ and uAu ˆ (x) = V ∗ (1 ⊗ x)V = Vˆ (x ⊗ 1)Vˆ ∗ , ˆ Vˆ ) = Vˆ12 Vˆ13 . (ι ⊗ )(V ) = V12 V13 , (ι ⊗ )( The unitaries V and Vˆ satisfy the pentagonal equation V12 V13 V23 = V23 V12 and Vˆ12 Vˆ13 Vˆ23 = Vˆ23 Vˆ12 . ˆ ) ˆ on a Hilbert space K is a unitary Definition 4.7. An -corepresentation of (A, ˆ X ∈ M(A ⊗ K(K)) satisfying ˆ ⊗ ι)(X) = ( ⊗ 1)X13 X23 . ( The following lemma can be checked immediately using the formulas in Notation 4.6 and in particular, the commutation relation V12 Vˆ23 = Vˆ23 V12 . := (1 ⊗ u)∗ (1 ⊗ u), the unitary V ∈ M(Aˆ ⊗ K(H )) Lemma 4.8. Denoting is an -corepresentation. It is called the right regular -corepresentation. The next lemma is crucial to define the twisted quantum group C∗ -algebras. ˆ ) ˆ on K. Then, the closed linear Lemma 4.9. Let X be an -corepresentation of (A, ∗ ∗ ˆ space [(µ ⊗ ι)(X) | µ ∈ A ] is a unital C -algebra. Proof. Write B := [(µ ⊗ ι)(X) | µ ∈ Aˆ ∗ ]. From the defining relation for an corepresentation, it follows that B is an algebra acting non-degenerately on K. Since ˆ ⊗ ι)(X)X ∗ = ( ⊗ 1)X13 , we have ( 23 ∗ ˆ ⊗ ι)(X)X23 | µ, η ∈ Aˆ ∗ ] B = [(µ ⊗ η ⊗ ι) ( ∗ ∗ = [(µ ⊗ η ⊗ ι) Vˆ12 X13 Vˆ12 X23 | µ, η ∈ Aˆ ∗ ] ∗ ∗ = [(µ ⊗ η ⊗ ι) X13 Vˆ12 X23 | µ, η ∈ Aˆ ∗ ] ∗ = [(µ ⊗ η ⊗ ι) X13 (1 ⊗ K(H ))Vˆ ∗ (K(H ) ⊗ 1) 12 X23 | µ, η ∈ Aˆ ∗ ] ∗ = [(µ ⊗ η ⊗ ι) X13 (K(H ) ⊗ K(H ) ⊗ 1)X23 | µ, η ∈ Aˆ ∗ ] = [(µ ⊗ ι)(X)(η ⊗ ι)(X)∗ | µ, η ∈ Aˆ ∗ ] = [BB ∗ ] . Here we used the regularity of the multiplicative unitary Vˆ . It follows that B is a C∗ algebra. Since is normalized, we have (p ⊗ 1)X = p ⊗ 1 and hence, B is unital. Definition 4.10. We define the unital C∗ -algebras V ) | µ ∈ Aˆ ∗ ] and Cu∗ (G, ) := [(µ ⊗ ι)(X) | µ ∈ Aˆ ∗ ] , Cr∗ (G, ) := [(µ ⊗ ι)( where X denotes a universal -corepresentation.
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Remark that a -corepresentation X on K is said to be universal if for any -corepresentation Y on K1 , there exists a non-degenerate ∗ -homomorphism π : [(µ ⊗ i) (X)|µ ∈ Aˆ ∗ ] → B(K1 ) satisfying (i ⊗ π )(X) = Y . It is clear that a universal -corepresentation exists and that the C∗ -algebra Cu∗ (G, ) is well defined up to an isomorphism. Proposition 4.11. Denote by B the unital ∗ -algebra associated by Theorem 3.9 with the unitary fiber functor ϕ . Consider the unitaries X x ∈ B(Hx ) ⊗ B generating B. Denote by Br and Bu the associated reduced and universal C∗ -algebra. ˆ ) ˆ on a Hilbert space K, we obtain a represenFor any -corepresentation X of (A, Taking X = V , we tation π of B on K given by (ι⊗π )(X x ) = (px ⊗1)X for all x ∈ G. ∗ ∼ get an isomorphism Br = Cr (G, ). Taking X to be a universal -corepresentation, we get an isomorphism Bu ∼ = Cu∗ (G, ). Proof. It is immediate that the formula (ι ⊗ π )(X x ) = (px ⊗ 1)X defines a one-to-one ˆ ). ˆ This correspondence between representations of B and -corepresentations of (A, already shows the isomorphism Bu ∼ = Cu∗ (G, ). V . Denote by ω the Consider π : B → B(H ) given by (ι ⊗ π )(X x ) = (px ⊗ 1) unique invariant state on B. To prove the isomorphism Br ∼ = Cr∗ (G, ), it suffices to define a faithful state ω1 on Cr∗ (G, ) such that ω1 π = ω. Define α : Cr∗ (G, ) → V ) = ( V )12 V13 . M(K(H ) ⊗ A) : α(a) = V (a ⊗ 1)V ∗ . One verifies that (ι ⊗ α)( ∗ ∗ Hence, α : Cr (G, ) → Cr (G, ) ⊗ A is a coaction satisfying απ = (π ⊗ ι)δ. It follows that (ι ⊗ h)α(a) ∈ C1 for all a ∈ Cr∗ (G, ). So, we can define ω1 by the formula ω1 (a)1 = (ι ⊗ h)α(a). Clearly, ω1 π = ω. 5. Monoidal Equivalence for Ao (F ) Recall the following definition of the compact quantum group Ao (F ) [25]. Definition 5.1. For all n ∈ N and F ∈ GL(n, C) with F F = c1 ∈ R1, A0 (F ) is defined as the universal compact quantum group generated by the coefficients of the corepresentation U ∈ Mn (C) ⊗ Ao (F ) with relations U is unitary and U = (F ⊗ 1)U (F −1 ⊗ 1) . Then, (Ao (F ), U ) is a compact (matrix) quantum group. When the matrix F has dimension 2, we precisely obtain the quantized versions of the classical Lie group SU(2), as considered by Woronowicz. Definition 5.2 ([31]). Let q ∈ [−1, 1] \ {0}. We define SUq (2)to be the universal ∗ C∗ -algebra generated by 2 elements α, γ such that U = γα −qγ is a unitary corepα∗ resentation. Then (SUq (2), U ) is a compact (matrix) quantum group.
0 1 Observe that SUq (2) ∼ = Ao −q −1 0 and that dimq (U ) = q + q1 . Banica [3] has shown that the irreducible corepresentations of Ao (F ) can be labelled by N, in such a way that the fusion rules are identical to the fusion rules of the compact T U ) is one-dimensional and generated by Lie group SU(2). In particular, Mor(, U tF := ei ⊗ F e i , i
where (ei ) is the standard basis of
Cn .
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Theorem 5.3. Let F ∈ GL(n, C) with F F = c1 and c ∈ R. Consider G = Ao (F ). – Take F1 ∈ GL(n1 , C) satisfying F1 F 1 = c1 1 and Tr(Fc∗ F ) = Tr(Fc1∗ F1 ) . There exists a 1 unitary fiber functor ϕF1 on G, uniquely determined up to isomorphism, such that ϕ √
1 1 tF1 . tF = ∗ Tr(F F ) Tr(F1∗ F1 )
(5.1)
– Every unitary fiber functor ϕ on G is isomorphic with one of the form ϕF1 . Moreover, ϕF1 is isomorphic with ϕF2 if and only if n1 = n2 and there exists a unitary v ∈ U(n1 ) and a λ = 0 such that F2 = λvF1 v t . Proof. Take β ∈ R \ {0}. Consider the universal graded C∗ -algebra (A(n, m))n,m∈N that satisfies An,m = {0} if n − m is odd and that is generated by elements v(r, s) ∈ A(r +s, r +s+2) with relations (denoting 1n the unit of the C∗ -algebra A(n) := A(n, n)) v(r, s)∗ v(r, s) = 1r+s , v(r, s + 1)∗ v(r + 1, s) = β 1r+s+1 , v(r, k + l + 2)v(r + k, l) = v(r + k + 2, l)v(r, k + l), v(r, k + l + 2)∗ v(r + k + 2, l) = v(r + k, l)v(r, k + l)∗ . Take F ∈ GL(n, C) with F F = c1 and c ∈ R. Put β = Tr(Fc∗ F ) . Let G = (Ao (F ), ) and denote by U n the n-fold tensor product of the fundamental corepresentation, with the convention that U 0 = . Take the isometric t ∈ Mor(, U 2 ) given 1 ∗ by t = √Tr(F ∗ F ) tF . Then, (t ⊗ 1)(1 ⊗ t) = β 1. We get a natural ∗ -homomorphism π : (A(n, m))n,m∈N → (Mor(U n , U m ))n,m∈N given by π(v(r, s)) = 1r ⊗ t ⊗ 1s . Because of Proposition 1 in [2], π is surjective. It follows from the comments after Th´eor`eme 4 in [3], that π is faithful on A(n, n) for all n. But then, π is faithful on A(n, m) because π(T ) = 0 ⇔ π(T ∗ T ) = 0 ⇔ T ∗ T = 0 ⇔ T = 0 . We conclude that π is a ∗ -isomorphism. Take F1 ∈ GL(n1 , C) satisfying F1 F 1 = c1 1 and Cn1
and denote by K n
c Tr(F ∗ F )
=
c1 Tr(F1∗ F1 ) .
Write K =
the n-fold tensor product of K, with the convention that K 0 = C. From the preceding discussion, we obtain a faithful ∗ -homomorphism π : (Mor(U n , U m ))n,m∈N → (B(K n , K m ))n,m∈N satisfying π(t) = √
1 tF , π(1) Tr(F1∗ F1 ) 1
= 1 and π(1 ⊗ T ⊗ 1) = 1 ⊗ π(T ) ⊗ 1 for all T .
= N as follows. We define uniquely a projecWe choose a concrete identification G n n tion Pn ∈ Mor(U , U ) satisfying Pn T = 0 for all r < n and all T ∈ Mor(U r , U n ). We define Un as the restriction of U n to the image of Pn . We then identify T ··· T nr , m1 T ··· T mk ) Mor(n1 = (Pm1 ⊗ · · · ⊗ Pmk ) Mor(U n1 +···+nr , U m1 +···+mk )(Pn1 ⊗ · · · ⊗ Pnr ).
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T ··· T nr , Define Hϕ(n) := π(Pn )K n and define ϕ(S) by restricting π to Mor(n1 T ··· T mk ). It is now obvious that ϕ is a unitary fiber functor on G. m1 Suppose conversely that ϕ is a unitary fiber functor on Ao (F ). We continue to use = N introduced above. Up to isomorphism, we may asthe concrete identification G n 1 sume that Hϕ(1) = C and we denote K = Cn1 . We define the ∗ -homomorphism π : (Mor(U n , U m ))n,m → (B(K n , K m ))n,m by restricting ϕ. Define the matrix F1 such that ϕ(tF ) = tF1 . Then, F1 F 1 = c1 1 with c1 = c and Tr(F1∗ F1 ) = Tr(F ∗ F ). Since t generates the graded C∗ -algebra (Mor(U n , U m ))n,m , π coincides with the ∗ -homomorphism constructed in the first part of the proof starting with F1 . Denoting by Tn ∈ n T · · · T 1) the embedding, we get unitary operators ϕ(Tn ) : Hϕ(n) → π(Pn )K Mor(n, 1 that implement the isomorphism between ϕ and ϕF1 .
Corollary 5.4. Let F ∈ GL(n, C) with F F = c1 and c ∈ R. Consider G = Ao (F ). A compact quantum group G1 is monoidally equivalent with G if and only if there exists F1 ∈ GL(n1 , C) satisfying F1 F 1 = c1 1 and Tr(Fc∗ F ) = Tr(Fc1∗ F1 ) such that G1 ∼ = Ao (F1 ). 1
Proof. The unitary fiber functor ϕF1 constructed in Theorem 5.3 yields a monoidal equivalence Ao (F ) ∼ Ao (F1 ). Since these fiber functors ϕF1 are, up to isomorphism, mon the only unitary fiber functors on Ao (F ), we are done. So, we exactly know when the compact quantum groups Ao (F1 ) and Ao (F2 ) are monoidally equivalent. If this is the case, Proposition 3.13 provides us with a universal C∗ -algebra Bu and a pair of ergodic coactions of full quantum multiplicity. It is possible to give an intrinsic description of this C∗ -algebra Bu . Theorem 5.5. Let Fi ∈ GL(ni , C) be such that Fi F i = ci 1 for ci ∈ R (i = 1, 2). Assume that c1 = c2 and Tr(F1∗ F1 ) = Tr(F2∗ F2 ). – Denote by Ao (F1 , F2 ) the universal unital C∗ -algebra generated by the coefficients of Y ∈ Mn2 ,n1 (C) ⊗ Ao (F1 , F2 ) with relations Y unitary and Y = (F2 ⊗ 1)Y (F1−1 ⊗ 1) . Then, Ao (F1 , F2 ) = 0 and there exists a unique pair of commuting universal ergodic coactions of full quantum multiplicity, δ1 of Ao (F1 ) and δ2 of Ao (F2 ), such that (ι ⊗ δ1 )(Y ) = Y12 (U1 )13 and (ι ⊗ δ2 )(Y ) = (U2 )12 Y13 . Here, Ui denotes the fundamental corepresentation of Ao (Fi ). – (Ao (F1 , F2 ), δ1 , δ2 ) is isomorphic with the C∗ -algebra Bu and the coactions on it given by Proposition 3.13 and the monoidal equivalence Ao (F1 ) ∼ Ao (F2 ) of Corollary mon 5.4. – The multiplicity of the fundamental corepresentation U1 in the coaction δ1 equals n2 . Remark that the condition on the matrices F1 and F2 is not really less general than the condition in Theorem 5.3, but just a normalization: if Tr(Fc1∗ F1 ) = Tr(Fc2∗ F2 ) , we multiply 1 2 F2 by a scalar and obtain c1 = c2 and Tr(F1∗ F1 ) = Tr(F2∗ F2 ).
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Proof. Take Fi as in the statement of the theorem and denote by ϕ the unitary fiber functor on G1 := Ao (F1 ) given by Theorem 5.3 and (5.1). We continue to use the iden1 with N. Theorem 3.9 provides us with a ∗ -algebra B generated by the tification of G coefficients of unitary operators Xn ∈ B(Hn , Hϕ(n) )⊗B, n ∈ N. Define X := X 1 . Since 1 appears in a tensor power of the fundamental corepresentation and every element of G since 1 = 1, it follows that the coefficients of X generate B as an algebra. Moreover (3.7) precisely says that X = (F2 ⊗ 1)X(F1−1 ⊗ 1). It follows that Ao (F1 , F2 ) = 0. Denoting by C the unital ∗ -subalgebra of Ao (F1 , F2 ) generated by the coefficients of Y , we get a surjective ∗ -homomorphism ρ : C → B satisfying (ι ⊗ ρ)(Y ) = X. It remains to show that ρ is a ∗ -isomorphism. Denote by U the fundamental corepresentation of Ao (F1 ) on H = Cn1 and by n U its nth tensor power, on H n . As in the proof of Theorem 5.3, we denote by Pn ∈ Mor(U n , U n ) the projection onto the irreducible corepresentation Un . Denote K = Cn2 and denote by K n the nth tensor power of K. Recall that we constructed a faithful ∗ homomorphism π : (Mor(U n , U m ))n,m → (B(K n , K m ))n,m and that ϕ is defined by restricting π to the relevant subspaces. The graded C∗ -algebra (Mor(U n , U m ))n,m is generated by the elements t ∈ Mor(, U 2 ) and 1r ⊗ t ⊗ 1s . Put Y n = Y1,n+1 · · · Yn,n+1 ∈ B(H n , K n )⊗Ao (F1 , F2 ). Since Y 2 (t ⊗1) = π(t)⊗1, it follows that Y m (S ⊗ 1) = (π(S) ⊗ 1)Y n for all S ∈ Mor(U n , U m ). Define a linear map γ : B → C by the formula (ι ⊗ γ )(Xn ) = Y n (Pn ⊗ 1) = (π(Pn ) ⊗ 1)Y n . We claim T m). Then, that γ is multiplicative. Take a, n, m ∈ N and S ∈ Mor(a, n n m (ι ⊗ ι ⊗ γ )(X13 X23 )(S ⊗ 1) = (ϕ(S) ⊗ 1)(ι ⊗ γ )(X a ) = (ϕ(S) ⊗ 1)(Pa ⊗ 1)Y a = (π(S) ⊗ 1)Y a = Y n+m (S ⊗ 1) = (Y n (Pn ⊗ 1))13 (Y m (Pm ⊗ 1))23 (S ⊗ 1) = (ι ⊗ γ )(X n )13 (ι ⊗ γ )(X m )23 (S ⊗ 1).
Because (ι ⊗ γρ)(Y ) = Y , because γ is multiplicative and because the coefficients of Y generate C as an algebra, it follows that γ is the inverse of ρ. So, ρ is indeed a ∗ -isomorphism. Remark 5.6. A combination of Proposition 6.2.6 in [6] and the results in [7] yields an alternative proof for the fact that Ao (F1 , F2 ) = 0. Remark 5.7. Combining the co-amenability of SUq (2) with Remark 2.6, we obtain the following. If either F1 or F2 is a two by two matrix (but not necessarily both), the (unique) invariant state on the universal C∗ -algebra Ao (F1 , F2 ) is faithful. So, the ergodic coactions of Ao (F1 ) and Ao (F2 ) are both universal and reduced. This is somehow remarkable, because Ao (F ) is not co-amenable when F has dimension strictly bigger than 2. Moreover, still supposing that either F1 or F2 is a two by two matrix, we also get that Ao (F1 , F2 ) is a nuclear C∗ -algebra. Indeed, supposing that F2 ∈ GL(2, C), Ao (F1 , F2 ) is Morita equivalent with its double crossed product, i.e. the crossed product of the compact operators with the co-amenable quantum group Ao (F2 ), yielding a nuclear C∗ -algebra. See [13] for details. A precise parameterisation of the unitary fiber functors (and hence, the ergodic coactions of full quantum multiplicity) on the quantum groups Ao (F ), amounts to the study
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of matrices F ∈ GL(n, C) satisfying F F = ±1, up to the equivalence relation F1 ∼ F
if and only if there exists a unitary v ∈ U(n) such that F1 = vF v t . (5.2)
Let F ∈ GL(n, C) with F F = ±1. Denote H = Cn and J : H → H the complex conjugation. We rather look at the anti-linear operator F = J F , satisfying F 2 = ±1. In the case where F 2 = 1, our data come down to giving a real vector space together with a Hilbert space structure on the complexification. In the case where F 2 = −1, H becomes a right module on the quaternions such that, restricting the quaternions to C, we get a Hilbert space. In particular H is even-dimensional. It is then straightforward to provide a fundamental domain for the equivalence relation (5.2) (see e.g. [33]). Take F ∈ GL(n, C) with F F = ±1. Let F = J |F | be the polar decomposition of F. Then, J is an anti-unitary, J 2 = ±1 and J |F |J ∗ = |F |−1 . Define H< as the subspace of H spanned by the eigenvectors of |F | with eigenvalue λ < 1. Define H> = J H< , which is as well the subspace of H spanned by the eigenvectors of |F | with eigenvalue λ > 1. Finally, let H1 be the subset of eigenvectors of |F | with eigenvalue 1. Take an orthonormal basis ξ1 , . . . , ξk of H< of eigenvectors of |F | with eigenvalues 0 < λ1 ≤ · · · ≤ λk < 1. If F F = 1, we have J 2 = 1 and we take an orthonormal basis µ1 , . . . , µn−2k for H1 of real vectors: J µi = µi . If (ei ) denotes the standard basis for Cn and w : (ei ) → (ξi , J ξi , µi ) denotes the transition unitary, we find that 0 D(λ1 , . . . , λk ) 0 w t F w = D(λ1 , . . . , λk )−1 (5.3) 0 0 . 0 0 1n−2k Here, D(λ1 , . . . , λk ) denotes the diagonal matrix with the λi along the diagonal. If F F = −1, we have J 2 = −1, H1 has even dimension and we take an orthonormal basis µ1 , . . . , µr , J µ1 , . . . , J µr for H1 . If w : (ei ) → (ξi , µi , J ξi , J µi ) denotes the transition unitary, we find that 0 D(λ1 , . . . , λn/2 ) wt F w = , (5.4) −D(λ1 , . . . , λn/2 )−1 0 where 0 < λ1 ≤ · · · ≤ λn/2 ≤ 1. Since the spectrum of F ∗ F is invariant under the equivalence relation (5.2), a fundamental domain is given by the matrices in (5.3) with 2k ≤ n and 0 < λ1 ≤ · · · ≤ λk < 1 and the matrices in (5.4) with 0 < λ1 ≤ · · · ≤ λn/2 ≤ 1. Corollary 5.8. Let 0 < q ≤ 1. For every even natural number n with 2 ≤ n ≤ q + q1 , the quantum group SUq (2) admits an ergodic coaction of full quantum multiplicity such that the multiplicity of the fundamental corepresentation is n. If −1 ≤ q < 0, the same statement holds for every natural number n with 2 ≤ n ≤ q + q1 . Corollary 5.9. Let F ∈ GL(n, C) with F F = c1 and c ∈ R. Denote G = Ao (F ). For all F1 ∈ GL(n1 , C) satisfying F1 F 1 = c1 and Tr(F1∗ F1 ) = Tr(F ∗ F ), we denote by δF1 the coaction of G on Ao (F, F1 ) defined in Theorem 5.5. – Up to isomorphism, the δF1 yield all universal ergodic coactions of full quantum multiplicity of G. Moreover, δF1 is isomorphic with δF2 if and only if n1 = n2 and there exists a unitary v ∈ U(n1 ) such that F2 = vF1 v t .
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– For all F1 as above with n1 = n, we denote by (F1 ) the unitary 2-cocycle on the dual of G associated with the unitary fiber functor ϕF1 . The (F1 ) describe, up to coboundary, all unitary 2-cocycles on the dual of G. Moreover (F1 ) and (F2 ) differ by a coboundary if and only if there exists a unitary v ∈ U(n) such that F2 = vF1 v t . Corollary 5.10. Every unitary 2-cocycle on the dual of SUq (2) is a coboundary. 6. Monoidal Equivalence for Au (F ) In this section, we prove, for the unitary quantum groups Au (F ) studied by Banica [3], analogous results as for Ao (F ). As in the previous section, we give a complete classification of unitary fiber functors, monoidally equivalent quantum groups, ergodic coactions of full quantum multiplicity and 2-cohomology. Recall the following definition. Definition 6.1. For all n ∈ N and F ∈ GL(n, C), we define Au (F ) as the universal compact quantum group generated by the coefficients of the corepresentation U ∈ Mn (C) ⊗ Au (F ) with relations U and (F ⊗ 1)U (F −1 ⊗ 1) are unitary . Then, (Au (F ), U ) is a compact (matrix) quantum group. Banica [3] has shown that the irreducible corepresentations of Au (F ) can be labelled by the free monoid N N generated by α and β. He also computed the corresponding fusion rules. β T U ) Defining U α := U and U β := (F ⊗ 1)U (F −1 ⊗ 1), the spaces Mor(, U α β α T and Mor(, U U ) are one-dimensional and generated by −1 ei ⊗ F ei resp. sF := ei ⊗ F ei . tF := i
i
Theorem 6.2. Let F ∈ GL(n, C) be normalized such that Tr(F ∗ F ) = Tr((F ∗ F )−1 ). Let G = Au (F ). – If F1 ∈ GL(n1 , C) satisfies Tr(F1∗ F1 ) = Tr((F1∗ F1 )−1 ) = Tr(F ∗ F ), there exists a unitary fiber functor ϕF1 on G, uniquely determined up to isomorphism, such that ϕ(tF ) = tF1 and ϕ(sF ) = sF1 . – Every unitary fiber functor ϕ on G is isomorphic with one of the form ϕF1 . Moreover, ϕF1 is isomorphic with ϕF2 if and only if n1 = n2 and there exist unitary elements v, w ∈ U(n1 ) such that F2 = vF1 w. Proof. Let N N be the free monoid generated by α and β. Denote by e the empty word. Elements of N N are words in α and β. Take a parameter c > 0. Let (A(p, q))p,q∈NN be the universal graded C∗ -algebra generated by elements Vx (p, q) ∈ A(pq, pxq) for p, q ∈ N N, x ∈ {αβ, βα}
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J. Bichon, A. De Rijdt, S. Vaes
with relations (denoting by 1p the unit of the C∗ -algebra A(p) := A(p, p)) Vx (p, q)∗ Vx (p, q) = 1pq , Vαβ (p, αq)∗ Vβα (pα, q) = c 1pαq , Vβα (p, βq)∗ Vαβ (pβ, q) = c 1pβq , Vy (p, qxr)Vx (pq, r) = Vx (pyq, r)Vy (p, qr), Vx (p, qyr)∗ Vy (pxq, r) = Vy (pq, r)Vx (p, qr)∗ . Take F ∈ GL(n, C) normalized in such a way that Tr(F ∗ F ) = Tr((F ∗ F )−1 ). Put c = Tr(F ∗ F ). Consider the quantum group G = (Au (F ), ). Define for every p ∈ NN q T U . the unitary corepresentation U p of G inductively by U pq := U p 1 Defining t ∈ Mor(, U αβ ) and s ∈ Mor(, U βα ) by the formulas t = √Tr(F ∗ F ) tF and s =
√
1 s , Tr(F ∗ F ) F
we get a natural ∗ -homomorphism
π : (A(p, q))p,q∈NN → (Mor(U p , U q ))p,q∈NN given by π(Vαβ (p, q)) = 1p ⊗ t ⊗ 1q and π(Vβα (p, q)) = 1p ⊗ s ⊗ 1q . It follows from Proposition 4, the proof of Th´eor`eme 1 and Proposition 3 in [3] that π is an isomorphism of C∗ -algebras. Take F1 ∈ GL(n1 , C) satisfying Tr(F1∗ F1 ) = Tr((F1∗ F1 )−1 ) = Tr(F ∗ F ). Write α K = K β = Cn1 and define inductively K p , for all p ∈ NN such that K pq = K p ⊗K q . We take K e = C. From the preceding discussion, we obtain a faithful ∗ -homomorphism π : (Mor(U p , U q ))p,q∈NN → (B(K p , K q ))p,q∈NN satisfying π(t) = √
1 tF , π(s) Tr(F1∗ F1 ) 1
=√
1 sF Tr(F1∗ F1 ) 1
and π(1⊗S ⊗1) = 1⊗π(S)⊗1
for all S. with N N as follows. We define, for We choose a concrete identification of G p ∈ N N, Pp ∈ Mor(U p , U p ) as the unique projection satisfying Pp T = 0 for all r ∈ N N with length r < length p and all T ∈ Mor(U r , U p ). We define Up as the restriction of U p to the image of Pp . We then identify T ··· T pr , q1 T ··· T qk ) Mor(p1 = (Pq1 ⊗ · · · ⊗ Pqk ) Mor(U p1 ···pr , U q1 ···qk )(Pp1 ⊗ · · · ⊗ Ppr ) . T ··· T pr , Define Hϕ(p) := π(Pp )K p and define ϕ(S) by restricting π to Mor(p1 T ··· T qk ). It is obvious that ϕ is a unitary fiber functor. q1 The converse statement is proven in exactly the same way as in the proof of Theorem 5.3.
The next two results are proven in exactly the same way as the corresponding results for Ao (F ). Corollary 6.3. Let F ∈ GL(n, C) with Tr(F ∗ F ) = Tr((F ∗ F )−1 ) and consider G = Ao (F ). A compact quantum group G1 is monoidally equivalent with G if and only if there exists F1 ∈ GL(n1 , C) satisfying Tr(F1∗ F1 ) = Tr((F1∗ F1 )−1 ) = Tr(F ∗ F ) such that G1 ∼ = Ao (F1 ).
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So, we exactly know when the compact quantum groups Au (F1 ) and Au (F2 ) are monoidally equivalent. If this is the case, Proposition 3.13 provides us with a universal C∗ -algebra Bu and a pair of ergodic coactions of full quantum multiplicity. It is again possible to give an intrinsic description of this C∗ -algebra Bu . The proof is analogous to the proof of Theorem 5.5. Again, the fact that Au (F1 , F2 ) = 0 can be deduced from Proposition 6.2.6 in [6] and the results in [8]. Theorem 6.4. Let Fi ∈ GL(ni , C) be such that Tr(F1∗ F1 ) = Tr((F1∗ F1 )−1 ) = Tr(F2∗ F2 ) = Tr((F2∗ F2 )−1 ). – Denote by Au (F1 , F2 ) the universal unital C∗ -algebra generated by the coefficients of X ∈ Mn2 ,n1 (C) ⊗ Au (F1 , F2 ) with relations X and (F2 ⊗ 1)X(F1−1 ⊗ 1) are unitary . Then, Au (F1 , F2 ) = 0 and there exists a unique pair of commuting universal ergodic coactions of full quantum multiplicity, δ1 of Au (F1 ) and δ2 of Au (F2 ), such that (ι ⊗ δ1 )(X) = X12 (U1 )13 and (ι ⊗ δ2 )(X) = (U2 )12 X13 . Here, Ui denotes the fundamental corepresentation of Au (Fi ). – (Au (F1 , F2 ), δ1 , δ2 ) is isomorphic with the C∗ -algebra Bu and the coactions on it given by Proposition 3.13 and the monoidal equivalence Au (F1 ) ∼ Au (F2 ) of Corollary mon 6.3. Remark 6.5. Exactly as in Corollary 5.9, a combination of Theorems 6.2 and 6.4 gives a complete classification of the ergodic coactions of full quantum multiplicity of Au (F ) and of the 2-cohomology of the dual of Au (F ). A precise parameterisation of the unitary fiber functors on the quantum groups Au (F ) is easy. If F1 , F ∈ GL(n, C), we write F1 ∼ F
if and only if there exist unitary v, w ∈ U(n) such that F1 = vF w.
We study matrices F ∈ GL(n, C) satisfying Tr(F ∗ F ) = Tr((F ∗ F )−1 ) up to the equivalence relation ∼. It is obvious that for any such F , there exist unique 0 < λ1 ≤ · · · ≤ λn satisfying i λ2i = i λ−2 i such that F ∼ D(λ1 , . . . , λn ). Here D(λ1 , . . . , λn ) denotes again the diagonal matrix with the λi along the diagonal. References 1. Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualit´e pour les produits crois´es de C∗ -alg`ebres. Ann. Scient. Ec. Norm. Sup. 26, 425–488 (1993) 2. Banica, T.: Th´eorie des repr´esentations du groupe quantique compact libre O(n). C. R. Acad. Sci. Paris S´er. I Math. 322, 241–244 (1996) 3. Banica, T.: Le groupe quantique compact libre U(n). Commun. Math. Phys. 190, 143–172 (1997) 4. Banica, T.: Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224, 243–280 (2005) 5. Banica, T.: Quantum automorphism groups of small metric spaces. Pac. J. Math. 219, 27–52 (2005) 6. Bichon, J.: Galois extension for a compact quantum group. http://arxiv.org/list/math.QA/9902031, 1999 7. Bichon, J.: The representation category of the quantum group of a non-degenerate bilinear form. Comm. Algebra 31, 4831–4851 (2003)
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8. Bichon, J.: Corepresentation theory of universal cosovereign Hopf algebras. http://arxiv.org/ list/math.QA/0211059, 2002 9. Bichon, J.: Quantum automorphism groups of finite graphs. Proc. Amer. Math Soc. 131, 665-673 (2003) 10. Boca, F.: Ergodic actions of compact matrix pseudogroups on C∗ -algebras. In: Recent advances in operator algebras (Orl´eans, 1992). Ast´erisque 232, 93–109 (1995) 11. Brugui`eres, A.: Dualit´e tannakienne pour les quasi-groupo¨ıdes quantiques. Comm. Algebra 25, 737–767 (1997) 12. Carter, J.S., Flath, D.E., Saito, M.: The classical and quantum 6j -symbols. Mathematical Notes 43, Princeton, NJ: Princeton University Press, 1995 13. Doplicher, S., Longo, R., Roberts, J.E., Zsid´o, L.: A remark on quantum group actions and nuclearity. Rev. Math. Phys. 14, 787–796 (2002) 14. Drinfel’d, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, Providence, RI: Amer. Math. Soc., 1987, pp. 798–820 15. Etingof, P., Gelaki, S.: On cotriangular Hopf algebras. Amer. J. Math. 123, 699–713 (2001) 16. Høegh-Krohn, R., Landstad, M.B., Størmer, E.: Compact ergodic groups of automorphisms. Ann. of Math. (2) 114, 75–86 (1981) 17. Jimbo, M.: A q-difference analogue of U (g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985) 18. Landstad, M.: Simplicity of crossed products from ergodic actions of compact matrix pseudogroups. In: Recent advances in operator algebras (Orl´eans, 1992). Ast´erisque 232, 111–114 (1995) 19. Landstad, M.: Ergodic actions of nonabelian compact groups. In: Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), Cambridge: Cambridge Univ. Press, 1992, pp. 365–388 20. MacLane, S.: Categories for the working mathematician. Graduate Texts in Mathematics 5, Berlin–New York: Springer-Verlag, 1971 21. Rosso, M.: Alg`ebres enveloppantes quantifi´ees, groupes quantiques compacts de matrices et calcul diff´erentiel non commutatif. Duke Math. J. 61, 11–40 (1990) 22. Schauenburg, P.: Hopf bi-Galois extensions. Comm. Algebra 24, 3797–3825 (1996) 23. Tomatsu, R.: Compact quantum ergodic systems. http://arxiv.org/list/math.OA/0412012, 2004 24. Ulbrich, K.-H.: Galois extensions as functors of comodules. Manuscripta Math. 59, 391–397 (1987) 25. Van Daele, A., Wang, S.: Universal quantum groups. Internat. J. Math. 7, 255–263 (1996) 26. Wang, S.: Ergodic actions of universal quantum groups on operator algebras. Commun. Math. Phys. 203, 481–498 (1999) 27. Wassermann, A.: Ergodic actions of compact groups on operator algebras. I. General theory. Ann. of Math. (2) 130, 273–319 (1989) 28. Wassermann, A.: Ergodic actions of compact groups on operator algebras. II. Classification of full multiplicity ergodic actions. Canad. J. Math. 40, 1482–1527 (1988) 29. Wassermann, A.: Ergodic actions of compact groups on operator algebras. III. Classification for SU(2). Invent. Math. 93, 309–354 (1988) 30. Woronowicz, S.L.: Compact quantum groups. In: Sym´etries quantiques (Les Houches, 1995), Amsterdam: North-Holland, 1998, pp. 845–884 31. Woronowicz, S.L.: Twisted SU(2) group. An example of a non-commutative differential calculus. Publ. Res. Inst. Math. Sci. 23, 117–181 (1987) 32. Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups. Invent. Math. 93, 35–76 (1988) 33. Yamagami, S.: Fiber functors on Temperley-Lieb categories. http://arxiv.org/list/math. QA/ 0405517, 2004 Communicated by A. Connes
Commun. Math. Phys. 262, 729–755 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1464-9
Communications in
Mathematical Physics
Blowup of Smooth Solutions for Relativistic Euler Equations Ronghua Pan1 , Joel A. Smoller2 1 2
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. E-mail: [email protected] Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA. E-mail: [email protected]
Received: 18 March 2005 / Accepted: 13 June 2005 Published online: 18 November 2005 – © Springer-Verlag 2005
Abstract: We study the singularity formation of smooth solutions of the relativistic Euler equations in (3 + 1)-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a nonvacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial “generalized” momentum is sufficiently large. 1. Introduction In this paper, we study the singularity formation of solutions of the Einstein equations for an isentropic perfect fluid. Due to the hyperbolic nature of these nonlinear equations, one expects singularity formation in the solutions. Indeed, one even expects black holes to form. However, singularity formation in relativistic flow is not yet well-understood; the theory is most lacking in the multi-dimensional case, (3+1)-dimensional spacetime. As a first step in this direction, we consider here the relativistic Euler equations for a perfect fluid in 4-dimensional Minkowski spacetime, Div T = 0,
(1.1)
T ij = (p + ρc2 )ui uj + pg ij ,
(1.2)
where
is the stress-energy tensor for a perfect fluid, and g ij denotes the flat Minkowski metric, g ij = diag(−1, 1, 1, 1), x = (x 0 , x 1 , x 2 , x 3 )T with x 0 = ct. ρ is the mass-energy
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density, p is the pressure, c is the speed of light, and u is the 4-velocity of the fluid. dx Recall that since u = 1c dτ (τ is the proper time, u is a unit 4-vector in Minkowski space), it follows that (u0 )2 −
3
(uα )2 = 1,
α=1
and thus only three of the quantities u0 , u1 , u2 , u3 are independent. We now fix our space-time coordinates as (t, x 1 , x 2 , x 3 )T , set x = (x 1 , x 2 , x 3 )T , u = (u1 , u2 , u3 )T , and let cu . v= (1 + |u|2 ) One easily derives from Eq. (1.1) the relativistic Euler equations: 2 +p 2 +p − cp2 ) + ∇x • ( ρc v) = 0, ∂t ( ρc c2 −v 2 c2 −v 2
+p +p ∂t ( ρc v) + ∇x • ( ρc v ⊗ v) + ∇x p = 0, c2 −v 2 c2 −v 2 2
2
(1.3)
in the unknowns ρ, v and p. Here ∇x denotes the spatial gradient operator. Given a scalar k and 3-vectors a and b, by the notion a ⊗ b we mean the matrix abT , while ∇x • (kabT ) = (∇x • (ka1 b), ∇x • (ka2 b), ∇x • (ka3 b))T . We consider the Cauchy problem for (1.3) with initial data ρ(x, 0) = ρ0 (x), v(x, 0) = v0 (x).
(1.4)
Equations (1.3) close if we assume an equation of state, p = p(ρ), p(0) = 0 with p(ρ) ≥ 0, 0 < p (ρ) < c2 , p (ρ) ≥ 0, f or ρ ∈ (ρ∗ , ρ ∗ ),
(1.5)
where 0 ≤ ρ∗ < ρ ∗ ≤ ∞. For a γ -law, p(ρ) = σ 2 ρ γ with γ ≥ 1, the constant ρ ∗ is chosen as follows: if γ = 1, then ρ ∗ = ∞; and if γ > 1, then p (ρ ∗ ) = c2 . Thus the unknowns for the Cauchy problem (1.3)–(1.4) are ρ and v. For more details on the derivation of Eqs. (1.3) and a discussion of (1.5), see [14]. We are interested in the life span of smooth solutions for the Cauchy problem (1.3)– (1.4). For this purpose, we shall discuss two different cases: the case of finite initial energy, and the case of infinite initial energy. For the first case, we shall prove that if the initial data has compact support, then the life span of any non-trivial smooth solution for the Cauchy problem (1.3) and (1.4) is finite. For the second case, we show that if the initial data is a compactly supported perturbation around a non-vacuum background, then the life span of smooth solutions is finite provided that the radial component of the initial “generalized” momentum is sufficiently large; cf. Theorem 3.2. We start with the infinite energy case. The local existence of classical solutions of the Cauchy problem (1.3)–(1.4) has been established by Makino and Ukai ([6, 7]) provided that the initial data is in the subluminal region away from the vacuum. A sharper result is proved here in Theorem 2.1 in Sect. 2, where the stability of the solution with respect to the initial data (cf. Corollary 2.2) and the properties of finite propagation speed (cf. Lemma 2.3) are presented. In Sect. 3, we first derive some interesting structural properties of (1.3) in Lemma 3.1, then we prove a blowup result for smooth solutions (cf. Theorem
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3.2). Our proof is in the spirit of the work of Sideris for classical Euler equations [11] and is based on the largeness of the initial radial component of “generalized” momentum, which of course implies the largeness of the initial velocity. However, in our case, the velocity is still subluminal. In Sect. 4, we prove our blowup result for smooth solutions of (1.3)–(1.4) with non-trivial initial data that has compact support. In Sect. 5, we make some remarks concerning our results. A discussion on the type of singularity is also given. The existence of initial data satisfying our blowup conditions is also shown there. All the results in Sect. 2 are based on the existence of a strictly convex entropy function for (1.3), which was constructed by Makino and Ukai in [6, 7]. For the reader’s convenience, we present the construction in the Appendix, correcting a few errors in the original papers. Before proceeding, we now briefly review the methods and results of singularity formation for nonlinear hyperbolic systems. In one space dimension, the theory is fairly complete. It was proved that a singularity develops in finite time no matter how small and smooth the initial data is; cf. [4, 5, 13]. These results were established by the characteristic method, which is quite powerful in one space dimension. In more than one space dimension, there are no general theorems available mainly because the characteristics become intractable. However, the approach via certain averaged quantities was introduced by Sideris [11] to prove the formation of singularities in three-dimensional compressible fluids. This idea avoids the local analysis of solutions. A similar technique was used to prove other formation of singularity theorems. We refer to [8, 9, 16] for classical fluids, and [2, 10] for relativistic fluids. Blowup results for relativistic Euler equations are announced in [2] and [10]. However, as remarked on p. 154 of [2], “ the unpublished proof in [10] contained an error which invalidated the argument ”. Furthermore, we note that the coefficient matrices in (2.15) of [2] constructed through (2.16) of [2] are not symmetric away from the equilibrium. But the symmetry of (2.15) in [2] is crucial to prove the finite propagation speed property needed in their proof. Thus the argument in [2] is not complete. Furthermore, we note that the equation of state used in [2] and [10] is different from ours. In addition, the approach of [2] is also different from ours. Our approach is closer to the method of Sideris [11]. Finally, we remark that the equation of state (1.5) in this paper is interesting for cosmology. It includes many physical cases, e.g. γ -laws, p(ρ) = σ 2 ρ γ , γ ≥ 1. For instance, the case p(ρ) =
1 2 c ρ 3
is very important in cosmology; it is the equation of state for the Universe in earliest times after the Big-Bang; see [15]. Some cases discussed in [2] (e.g. when s = const.) satisfy (1.5) as well. Another important example (see [15, p.319]) is the equation of state for neutron stars, where p = Ac5 a(y), ρ = Ac3 b(y), y y q4 a(y) = dq, b(y) = 3 q 2 1 + q 2 dq. 1 + q2 0 0
(1.6)
Here A is a positive constant. This equation of state implies the following asymptotics: p → 13 c2 ρ as ρ → ∞ and p → 15 A2/3 ρ 5/3 as ρ → 0. It is easy to see that p (ρ) =
c2 y 2 2 (1 + y 2 )−5/2 > 0, > 0, p (ρ) = 2 3(1 + y ) 9Acy
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R. Pan, J.A. Smoller
whenever, y > 0. We also note that y = 0 is equivalent to ρ = 0. Thus the equations (1.6) also satisfy (1.5). 2. Existence of Solutions: Infinite Energy Case In this section, we consider the local existence of smooth solutions for the Cauchy problem (1.3)–(1.4) when the initial data is away from the vacuum. For this purpose, we introduce some convenient notation: ρc2 + p , c2 − v 2 2 ρc + p p ρˆ = . − c2 − v 2 c2
ρ˜ =
The Cauchy problem (1.3)–(1.4) becomes ˜ = 0, ρˆt + ∇x • (ρv) (ρv) ˜ t + ∇x • (ρv ˜ ⊗ v) + ∇x p(ρ) = 0, ρ(x, 0) = ρ (x), v(x, 0) = v (x). 0 0
(2.1)
(2.2)
Let ρ∗ < ρ ∗ be non-negative constants in (1.5) subject to the subluminal condition p (ρ ∗ ) ≤ c2 . We set z = (ρ, v1 , v2 , v3 )T and define the region z by z = {z : ρ∗ < ρ < ρ ∗ , v 2 < c2 }.
(2.3)
Theorem 2.1. Assume an equation of state is given as in (1.5). Suppose the initial data z0 (x) = (ρ0 (x), v0 (x))T is continuously differentiable on R3 , taking values in any compact subset D of z and that ∇x z0 (x) ∈ H l (R3 ) for some l > 3/2. Then there exists T∞ , 0 < T∞ ≤ ∞, and a unique differentiable function z(x, t) = (ρ(x, t), v(x, t))T on R3 × [0, T∞ ), taking values in z , which is a classical solution of the Cauchy problem (1.3)–(1.4) on R3 × [0, T∞ ). Furthermore, ∇x z(·, t) ∈ C 0 ([0, T∞ ); H l ).
(2.4)
The interval [0, T∞ ) is maximal, in the sense that whenever T∞ < ∞, lim sup ∇x z(·, t) L∞ = ∞
t→T∞
(2.5)
and/or the range of z(·, t) escapes from every compact subset of z as t → T∞ . This theorem will be proved by applying Theorem 5.1.1 in Dafermos [1] for hyperbolic conservation laws endowed with a strictly convex entropy. We state this theorem here for readers convenience.
Blowup of Smooth Solutions for Relativistic Euler Equations
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Theorem A. Assume that the system of conservation laws
Ut +
m
∂xα Gα (U ) = 0, x ∈ Rm , U ∈ O ⊂ Rn ,
(*)
α=1
is endowed with an entropy η with ∇ 2 η(U ) positive definite, uniformly on a compact subset of O. Suppose the initial data U (x, 0) = U0 (x) is continuously differentiable on Rm , takes values in some compact subset of O and ∇U0 ∈ H l for some l > m/2. Then there exists T∞ , 0 < T∞ ≤ ∞, and a unique continuously differentiable function U on Rm × [0, T∞ ), taking values in O, which is a classical solution of the initial-value problem (∗) with initial data U0 on [0, T∞ ). Furthermore, ∇U (·, t) ∈ C 0 ([0, T∞ ); H l ). The interval [0, T∞ ) is maximal, in the sense that whenever T∞ < ∞, lim sup ∇U (·, t) L∞ = ∞,
t→T∞
and/or the range of U (·, t) escapes from every compact subset of O as t → T∞ . Proof of Theorem 2.1. We first rewrite (1.3) or (2.2) in the form of conservation laws, θt +
3
(f k (θ ))xk = 0,
(2.6)
k=1
where θ = (θ0 , θ1 , θ2 , θ3 )T and f k (θ ) = (θk , f1k , f2k , f3k )T are defined by θ0 = ρ, ˆ θj = ρv ˜ j, ˜ j vk + pδj k , j = 1, 2, 3. fjk = ρv
(2.7)
By Theorem A, it is sufficient to show that (2.6) has an entropy η(θ ) with ∇ 2 η(θ ) positive definite in z . Such an entropy, due to Makino and Ukai [7], is constructed in the Appendix of this paper. Define ρ c2 φ(ρ) = (2.8) dr, K = ρm c2 + p(ρm ), 2 ρm rc + p(r) ρm being any fixed number in (ρ∗ , ρ ∗ ). The entropy given in (6.25) below is cKeφ(ρ) η = c2 ρˆ − √ . c2 − v 2
(2.9)
We now verify that ∇ 2 η(θ ) is positive definite in z . To this end, we first compute ∇θ η(θ). By the chain rule, we have w T = (∇θ η) = (∇z η)(∇z θ)−1 ,
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R. Pan, J.A. Smoller
where (∇z θ )−1 is defined in (6.10), and w T = (w0 , w1 , w2 , w3 ) is given by 3 (ρ) 2 w0 = − (cc2 −v 2 )1/2 + c , wj =
c (ρ) v , (c2 −v 2 )1/2 j
j = 1, 2, 3,
(2.10)
with (ρ) =
Keφ(ρ) . (ρc2 + p)
(2.11)
We remark that w can serve as a symmetric variable which reduces (1.3) to a symmetric hyperbolic system [1, 3]. For the Hessian matrix H of η, we compute H = ∇ 2 η(θ ) = ∇θ w T = (∇z w T )(∇z θ)−1 c (ρ)E1 = (ρc2 +p)(c 2 −v 2 )1/2 H1 A1 A2 v T c (ρ)E1 . ≡ (ρc2 +p)(c2 −v 2 )1/2 A2 v A3 vv T + A4 I3 Here, E1 =
1 c4 −p v 2
(2.12)
is given in (6.11) below, and the Ai are given by
A1 = c4 (p c2 + 2p v 2 + c2 v 2 ), A2 = −c2 (c4 + 2c2 p + p v 2 ), (2.13) A3 = (c4 + 2c2 p + p v 2 + 2p (c2 − v 2 )), A4 = (c2 − v 2 )(c4 − p v 2 ). We now show that H is positive definite. From (2.12), we see that it is sufficient to show H1 is positive definite. Let r = (r0 , r T )T be any 4-vector with r ∈ R3 . We calculate: A1 A2 v T T r H1 r = (r0 , r) (r0 , r)T A2 v A3 vv T + A4 I3 = (A1 r02 + 2A2 r0 v T r + A3 (v T r)2 + A4 r 2 ). Letting A˜ 1 = (1 − δ)A1 with
1 2
> δ > 0 to be determined in (2.14) below, we have
(A1 r02 + 2A2 r0 v T r + A3 (v T r)2 + A4 r 2 )
A2 T 2 1 δ A22 T 2 2 ˜ = A1 r0 + v r + δA1 r02 + A4 r 2 v r − (A − A1 A3 ) + A1 2 1 − δ A1 A˜ 1
1 δ A22 2 2 2 2 ≥ A4 − (A − A1 A3 )v − v r + δA1 r02 A1 2 1 − δ A1
A22 2 2 p (c2 − v 2 )2 (c4 − p v 2 ) ≥ − 2δ v r + δA1 r02 (p c2 + 2p v 2 + c2 v 2 ) A1 ≥ δA1 r02 + δr 2 . Here, we determine δ by 0 < δ + 2δ
A22 2 p (c2 − v 2 )2 (c4 − p v 2 ) v < . A1 (p c2 + 2p v 2 + c2 v 2 )
(2.14)
Blowup of Smooth Solutions for Relativistic Euler Equations
735
We thus conclude that rT H1 r ≥ (δA1 r02 + δr 2 ). This proves H1 is positive definite in z . Hence, H is positive definite, and η is strictly convex on z . This completes the proof of Theorem 2.1. The existence of a strictly convex entropy guarantees that classical solutions of the initial-value problem depend continuously on the initial data, even within the broader class of admissible bounded weak solutions; see [1]. Here, by admissible bounded weak solution, we mean bounded functions satisfying the initial value problem and entropy inequality in the sense of distributions. The following Theorem B is Theorem 5.2.1 in Dafermos [1]: Theorem B. Assume that the system of conservation laws (∗) is endowed with an entropy η with ∇ 2 η(U ) positive definite, uniformly on compact subset of O. Suppose U is a classical solution of (∗) on [0, T ), taking values in a convex compact subset N of O, with initial data U0 . Let U¯ be any admissible weak solution of (∗) on [0, T ), taking values in N, with initial data U¯ 0 . Then |U (x, t) − U¯ (x, t)|2 dx ≤ aebt |U0 (x) − U¯ 0 (x)|2 dx |x|
|x|
holds for any R > 0 and t ∈ [0, T ), with positive constants s, a, depending only on N , and a constant b that also depends on the Lipschitz constant of U . In particular, U¯ is the unique admissible weak solution of (∗) with initial data U¯ 0 (x) and values in N . The following corollary is a consequence of the convexity of η and Theorem B. Corollary 2.2. Let θ be a classical solution of (2.6) obtained in Theorem 2.1 with initial data θ0 (x) taking values in a compact subset D of z , and let θ˜ be any admissible weak solution of (2.6) on [0, T∞ ), taking values in D , with initial value θ˜0 (x) ∈ D. Then |θ (x, t) − θ˜ (x, t)|2 dx ≤ aebt |θ0 (x) − θ˜0 (x)|2 dx |x|
|x|
holds for any R > 0 and t ∈ [0, T∞ ), with positive constants s, a, depending only on D, and a constant b that also depends on the Lipschitz constant of θ . In particular, θ˜ is the unique admissible weak solution of (2.6) with initial data θ˜0 (x) and values in D. In the next lemma, we will show that a compactly supported perturbation around a non-vacuum background propagates with finite speed. For this purpose, we consider the following Cauchy problem ˜ = 0, ρˆt + ∇x • (ρv) (2.15) (ρv) ˜ t + ∇x • (ρv ˜ ⊗ v) + ∇x p(ρ) = 0, (ρ (x) − ρ, ¯ v (x)) = 0, f or |x| ≥ R, 0 0 where, R > 0, and 0 < ρ¯ ∈ (ρ∗ , ρ ∗ ) satisfies the subluminal condition, p (ρ ∗ ) ≤ c2 .
(2.16)
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Lemma 2.3. Let (ρ, v)(x, t) be a C 1 solution of the Cauchy problem (2.15) (equivalent to (1.3) with the same initial data), where the initial data (ρ0 , v0 )(x) takes values in a compact subset of z (cf. (2.3)). Then the support of (ρ − ρ, ¯ v)(x, t) is contained in the ball B(t) = {x : |x| ≤ R + st} where s = p (ρ) ¯ is the sound speed in the far field. Proof. This lemma is a consequence of the local energy estimates. It will be proved using the method of [11] for symmetric hyperbolic systems. For this purpose, we first observe that w = (∇θ η)T given in (2.10) (where η is as in (2.9)) renders the system (2.15) symmetric hyperbolic [3]: ∂w i ∂w A (w) = 0, + ∂t ∂xi 3
A0 (w)
(2.17)
i=1
where the coefficient matrices Aα (w) = (Aαmn ), α, m, n = 0, 1, 2, 3 are given by (6.5); that is A0 = (∇θ2 η)−1 , Ak = (∇θ f k )(∇θ2 η)−1 . We now compute the explicit form of these matrices. First of all, we have a1 a2 v T , A0 = (∇θ2 η)−1 = (ρ) a2 v a3 vv T + a4 I3
(2.18)
where I3 is the 3 × 3 identity matrix, and
(ρ) =
1 (ρc2 + p)2 e−φ(ρ) , K
(2.19)
and a1 = a3 =
4 c2 +p v 2 c4 +3p v 2 , a2 = cc3+2p , c3 p (c2 −v 2 )3/2 p (c2 −v 2 )3/2 2 c +3p 1 , a4 = c(c2 −v 2 )1/2 . cp (c2 −v 2 )3/2
(2.20)
For Ak , k = 1, 2, 3, we first compute (∇θ f k ) = (∇z f k )(∇z θ )−1 =
0
ekT
[c2 (c2 + v 2 )p E1 ek − c2 (c2 + p )E1 vk v] [−C4 vekT + vk I3 ]
with ek = (δ1k , δ2k , δ3k )T , where (∇z f k ) is given by
B 3 vk B2 vk v T + B4 ekT k (∇z f ) = ; B3 vk v + p ek B2 vk vv T + B4 vekT + B4 vk I3
(2.21) ,
Blowup of Smooth Solutions for Relativistic Euler Equations
737
cf. (6.13) below. Then we have Ak = (∇θ f k )(∇θ2 η)−1 = (∇θ f k )A0
a2 vk [a3 vk v T + a4 ekT ] (2.22) = (ρ) . T T T [a3 vk v + a4 ek ] [a3 vk vv + a4 vk I3 + a4 (ek v + vek )] It is clear that the matrices Ak (w) are all real symmetric and smooth in z . Furthermore A0 (w) = (∇ 2 η)−1 is positive definite in z . Now, we choose ρm = ρ¯ subject to (2.16) for convenience. In this setting, the background state in the w-variable becomes w¯ = w(ρ, ¯ 0) = 0. Set A˜ i (w) = (A0 (w))−1 Ai (w), and define Q(λ, ξ ) = λI4 −
3
ξα A˜ α (0),
(2.23)
α=1
where (λ, ξ ) ∈ R × S 2 (S 2 is the unit 2-sphere). Using the real symmetry of Aα (0), for each ξ ∈ S 2 , we see that the characteristic equation detQ(λ, ξ ) = 0, has real roots λi (ξ ), i = 0, 1, 2, 3, called the characteristic speeds. Let λ¯ be the largest absolute value of these characteristic speeds. For any fixed (x0 , t0 ) ∈ R3 × (0, T∞ ), we define the family of cones ¯ 0 − s), 0 ≤ s ≤ τ }, Cτ = {(s, x) : |x0 − x| ≤ λ(t
(2.24)
parametrized by τ ∈ [0, t0 ) and the associated cross sections Eσ = {(µ, x) ∈ Cτ : µ = σ }.
(2.25)
We introduce the linear partial differential operator ∂ ∂ Aα (0) , + ∂t ∂xα 3
P = A0 (0)
(2.26)
α=1
where again 0 is the background state in the w-variable. Equation (2.17) reads ∂w α ∂w + (A (0) − Aα (w)) . ∂t ∂xα 3
P w = (A0 (0) − A0 (w))
(2.27)
α=1
Now we multiply both sides of (2.27) by 2wT , to get 3 ∂ T α ∂t [w T A0 (0)w] + ∂xα [w A (0)w] α=1 3 ∂w ∂w T 0 0 α α = 2w (A (0) − A (w)) ∂t + (A (0) − A (w)) ∂xα . α=1
(2.28)
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R. Pan, J.A. Smoller
We integrate (2.28) over Cτ to obtain
3
∂α [w T Aα (0)w](x, σ ) dxdσ
Cτ α=0
=
3
(Aα (0) − Aα (w))∂α w Cτ α=0 τ ≤ C max |∇w| |w|2 (x, σ ) dxdσ, 2w T
Cτ
0
(2.29)
Eσ
where ∂0 = ∂t , and we have used the mean value theorem in the estimation of the last step. For the left hand side of (2.29), we want to apply the divergence theorem since it is in divergence form. For this purpose, we need to determine the boundary of Cτ and the associated unit outer normal vector. The boundary of Cτ consists of three parts: the cap Eτ with unit outer normal (1, 0, 0, 0)T , the base E0 with unit outer normal (−1, 0, 0, 0)T , and along the surface ¯ 0 − σ ), 0 ≤ σ ≤ τ } Rτ = {(σ, x) : |x0 − x| = λ(t
(2.30)
the unit outer normal vector is n=
1 1 + λ¯ 2
¯ −ν T )T , ν = (λ,
(x0 − x) . |x − x0 |
(2.31)
To see (2.31), we note that on Rτ one has (λ¯ , −ν T )T • ((t0 − σ ), (x0 − x)T )T = 0. We now apply the divergence theorem to the left hand side of (2.29),
3
∂α [w T Aα (0)w](x, τ ) dxdτ
Cτ α=0
=
(w T A0 (0)w)(x, τ ) dx −
Eτ
1
+ λ¯ 2 + 1
τ 0
(w T A0 (0)w)(x, 0) dx
(2.32)
E0
¯ T A0 (0)w − w T (λw ∂Eσ
3
να Aα (0)w)(x, σ ) dSx dσ,
α=1
where dSx denotes the surface element on ∂Eσ . The third term of (2.32) on the right hand side can be simplified as follows: ¯ T A0 (0)w − w T (λw
3
να Aα (0)w)
α=1 3
να A˜ α (0))w
¯ 4− = wT A0 (0)(λI
(2.33)
α=1
¯ ν)w. = wT A0 (0)Q(λ, We recall that A0 (0) > 0 and A0 (0)Q(λ, ν) = λA0 (0) −
3 α=1
να Aα (0)
(2.34)
Blowup of Smooth Solutions for Relativistic Euler Equations
739
is real symmetric. We claim that for any ν ∈ S 2 , ¯ ν) ≥ 0, A0 (0)Q(λ,
(2.35)
which will be verified at the end of this proof. Therefore, we conclude from (2.29), (2.32), (2.33) and (2.35) that (w T A0 (0)w)(x, τ ) dx Eτ τ T 0 ≤ (w A (0)w)(x, 0) dx + C1 max |∇w| |w|2 (x, σ ) dxdσ. Cτ
E0
0
Eσ
(2.36) Since A0 (0) > 0, there are positive constants C2 and C3 such that τ 2 2 |w| (x, τ ) dx ≤ C2 |w| (x, 0) dx + C3 |w|2 (x, σ ) dxdσ, Eτ
0
E0
which, by Gronwall’s inequality implies that |w|2 (x, τ ) dx ≤ C2 eC3 τ ( Eτ
Eσ
|w|2 (x, 0) dx).
(2.37)
E0
¯ 0 , then w(x, τ ) = 0 for any τ ∈ [0, t0 ) and Therefore, if w(x, 0) = 0 for |x − x0 | ≤ λt ¯ 0 − τ ). This implies that, if w(x, 0) = 0 for |x| > R, then w(x, t) = 0 |x − x0 | ≤ λ(t ¯ for |x| > R + λt. The next step is to verify that λ¯ = p (ρ). ¯ For this purpose, we compute the largest possible characteristic speed at a constant background state. Now we compute the eigenvalues of 3
ξα A˜ α (0),
ξ ∈ S2.
(2.38)
α=1
Since A˜ α = (∇θ2 η)(∇θ f α )(∇θ2 η)−1 , the matrix in (2.38) is similar to the matrix M(ξ ) =
3
ξα ∇θ f α (θ¯ ), ξ ∈ S 2 ,
(2.39)
α=1
where θ¯ = (ρ, ¯ 0, 0, 0)T is the background state in the θ -variable. It is easy to compute: ∇z f α (∇z θ )−1|ρ=ρ,v=0 ∇θ f α (θ¯ ) = ¯ 0 eαT = , p (ρ)e ¯ α 0 where ei = (δ1i , δ2i , δ3i )T . Thus, one has 0 ξT . M(ξ ) = p (ρ)ξ ¯ 0
(2.40)
(2.41)
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R. Pan, J.A. Smoller
Now we claim that M(ξ ) has rank 2. This can be seen by 1 0 , MM T = 0 p (ρ) ¯ 2ξ ξ T since rank(ξ ξ T ) = 1. Thus, M(ξ ) has two non-zero eigenvalues and two zero eigenvalues. We need to find all non-zero eigenvalues of M(ξ ). We compute −r ξT 0 = det(M(ξ ) − rI4 ) = det ¯ −rI3 p (ρ)ξ 0 ξT = det (2.42) (p (ρ) ¯ − r 2 )ξ −rI3 0 ξT = (p (ρ) , ¯ − r 2 ) det ξ −rI3 ¯ are the two distinct non-zero eigenvalues of M(ξ ). Therefore, and this implies ± p (ρ) we have ¯ (2.43) λ¯ = p (ρ). Notice that we did not use (2.35) to obtain (2.43). The last step is to verify (2.35). Since λ¯ = s = p (ρ), ¯ we have A0 (0) =
K c2
1
K 0 eαT 0 , Aα (0) = 2 , α = 1, 2, 3, 0 I3 c eα 0
s2
and thus ¯ ν)) = A (0)Q(λ, 0
K c2
−ν T −ν sI3 1 s
(2.44)
.
Let r = (r0 , r T )T be any 4-vector with r ∈ R3 , we compute:
1 T −ν ¯ ν))r = K2 (r0 , r T ) s (r0 , r T )T rT A0 (0)Q(λ, c −ν sI3 K 1 = 2 ( r02 − 2r0 ν T r + s|r|2 ) c s K 1 ≥ 2 ( r02 − 2r0 |r| + s|r|2 ) c s K = 2 (r02 − 2sr0 |r| + s 2 |r|2 ) sc K = 2 (r0 − s|r|)2 ≥ 0. sc
(2.45)
Here, we have used (ν T r) ≤
(ν 2 )(r 2 ) = |r|.
Thus, we have proved (2.35). The proof of this lemma is complete.
Blowup of Smooth Solutions for Relativistic Euler Equations
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3. Singularity Formation: Infinite Energy Case In this section, we prove the singularity formation of smooth solutions of (2.15) when the initial radial “generalized” momentum is large. To begin, we prove the following two easy but useful identities. Lemma 3.1. ρˆ and ρ˜ satisfy the following identities: ρˆ = ρ˜ =
1 ρv ˜ 2 c2 1 ρv ˜ 2 c2
+ ρ, + (ρ +
p ). c2
(3.1)
Proof. From (2.1), it is easy to see 2 ρc + p p ρˆ = − 2 c2 − v 2 c = = = = = Hence, ρ˜ = ρˆ +
p c2
ρc4 + pc2 − pc2 + pv 2 c2 (c2 − v 2 ) ρc2 c2 + pv 2 c2 (c2 − v 2 ) ρc2 v 2 + ρc2 (c2 − v 2 ) + pv 2 c2 (c2 − v 2 ) ρc2 + p 2 ρc2 (c2 − v 2 ) v + 2 2 2 c (c2 − v 2 ) c (c − v 2 ) 1 2 ρv ˜ + ρ. c2
implies ρ˜ =
p 1 2 ρv ˜ +ρ + 2. c2 c
We again denote the sound speed in the far field by s = p (ρ), ¯ and define the following quantities: M(t) = [ρ(ρ, ˆ v) − ρ( ˆ ρ, ¯ 0)](x, t) dx, F (t) = ρv ˜ • x dx.
(3.2)
By Lemma 2.3, both M(t) and F (t) are well-defined as long as the smooth solution exists. Using these two quantities, we shall show that the smooth solution of (2.15) obtained in Theorem 2.1 blows up in finite time if the initial data is subject to some restrictions. Roughly speaking, if M(0) > 0, and F (0) > 0 is sufficiently large, then the solution will blow up in finite time.
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Theorem 3.2. Assume that the initial data of (2.15) and ρ¯ are chosen such that M(0) > 0, F (0) > 0 and s 2 < 13 c2 . If F (0) > =
32πs 3(1 −
3s 2 ) c2
R 4 max ρˆ0 (x),
then the smooth solution of the Cauchy problem (2.15) obtained in Theorem 2.1 blows up in finite time. Proof. Using Lemma 3.1, we know that ρ(ρ, ˆ v) =
1 2 ρv ˜ + ρ, c2
thus, ρ( ˆ ρ, ¯ 0) = ρ. ¯
(3.3)
This implies that ρ − ρ¯ = (ρ(ρ, ˆ v) − ρ( ˆ ρ, ¯ 0)) −
1 2 ρv ˜ . c2
(3.4)
From the first equation of (3.2), it is easy to see that d M(t) = (ρ(ρ, ˆ v) − ρ( ˆ ρ, ¯ 0))t dx dt = ρˆt dx = − ∇x • (ρv) ˜ dx = 0, where we have used the first equation in (2.2) or (2.15). Hence, M(t) = M(0) = (ρˆ0 (x) − ρ) ¯ dx > 0. Using the second equation in (2.2) and (3.2), we compute F (t) = (ρv) ˜ t • x dx = − [∇ • (ρv ˜ ⊗ v) + ∇p(ρ)] • x dx.
(3.5)
(3.6)
But if p¯ = p(ρ), ¯ we have ∇p • x = ∇(p − p) ¯ •x = ∇ • [x(p − p)] ¯ − (p − p)∇ ¯ •x = ∇ • [x(p − p)] ¯ − 3(p − p). ¯
(3.7)
Blowup of Smooth Solutions for Relativistic Euler Equations
743
We also note that ∇ • (ρv ˜ ⊗ v) • x = = =
3 i,j =1 3
∂xi (ρv ˜ i vj )xj
[∂xi (ρv ˜ i vj xj ) − ρv ˜ i vj ∂xi xj ]
i,j =1 3
(3.8)
[∂xi (ρv ˜ i vj xj ) − ρv ˜ i vj δij ]
i,j =1
= −ρv ˜ 2+
3 j =1
∇ • (ρv ˜ ⊗ vxj ),
where v 2 = v T v. Inserting (3.7) and (3.8) into (3.6), and using the divergence theorem, we obtain F (t) = ρv ˜ 2 dx + 3 (p − p) ¯ dx. (3.9) Since p (ρ) ≥ 0, p (ρ) is a non-decreasing function of ρ. It is clear that ρ p (ξ ) dξ ≥ p (ρ)(ρ ¯ − ρ). ¯ p(ρ) − p(ρ) ¯ = ρ¯
Thus, using (3.4) and (3.5), one has 2 2 F (t) ≥ ρv (ρ − ρ) ¯ dx ˜ dx + 3s 3s 2 2 2 = ρv ˜ dx + 3s M(t) − 2 ρv ˜ 2 dx c 3s 2 = 1− 2 ρv ˜ 2 dx + 3s 2 M(0) c 3s 2 ≥ 1− 2 ρv ˜ 2 dx. c On the other hand, we have the following estimate: 2 F (t) = ( ρv ˜ • x dx)2 |x|2 ρ˜ dx)( ρv ˜ 2 dx) ≤( B(t) B(t) ≤ 2( |x|2 ρˆ dx)( ρv ˜ 2 dx), B(t)
(3.10)
(3.11)
(3.12)
B(t)
where we have used the following fact: ρ˜ ≤ 2ρ. ˆ
(3.13)
To see this, we note that from the subluminal condition p (ρ) < c2 , together with p(0) = 0, we get p(ρ) ≤ c2 ρ. Thus ρ˜ =
1 2 p 1 2 ρv ˜ + ρ + 2 ≤ 2 ρv ˜ + 2ρ = ρˆ + ρ ≤ 2ρ. ˆ c2 c c
744
R. Pan, J.A. Smoller
Due to (3.3) and (3.5), we have the following estimate: ρ|x| ˆ 2 dx ≤ (R + st)2 ρˆ dx B(t) B(t) 2 = (R + st) (M(t) + ρ¯ dx) B(t) = (R + st)2 (M(0) + ρ¯ dx) B(t) = (R + st)2 ρˆ0 (x) dx
(3.14)
B(t)
4π ≤ (R + st)5 (max ρˆ0 (x)). 3 Hence, (3.12) gives F 2 (t)
8π 5 ≤ ρv ˜ 2 dx) (R + st) (max ρˆ0 (x))( 3 B(t) ≡ K0 (R + st)5 (
ρv ˜ 2 dx),
(3.15)
B(t)
where K0 =
8π (max ρˆ0 (x)). 3
Thus (3.11) and (3.15) imply that 3s 2 F (t) ≥ 1 − 2 K0−1 (R + st)−5 F 2 (t), c so F ≥ K1 (R + st)−5 , F2 where K1 = (1 −
3s 2 )K0−1 . c2
1 F (0)
(3.17)
(3.18)
Integrating (3.18) with respect to t, one has
1 1 K1 −4 ≤ − [R − (R + st)−4 ] ≡ ψ(t). F (t) F (0) 4s Now ψ(0) =
(3.16)
(3.19)
> 0 by assumption, and ψ(+∞) =
1 K1 −4 − R < 0, F (0) 4s
if F (0) >
4sR 4 ≡ . K1
(3.20)
Therefore, 1 = 0, f or some t0 > 0. F (t0 )
(3.21)
Thus the life-span T of smooth solutions satisfies T < t0 . This completes the proof of Theorem 3.2.
Blowup of Smooth Solutions for Relativistic Euler Equations
745
4. Singularity Formation: Finite Energy Case Due to the hyperbolic nature of Einstein equations, one expects the finite propagation speed of waves in the solutions. We will prove in the following lemma that for any smooth solution with compactly supported initial data, the support of the solution is invariant in time. Lemma 4.1. Let (ρ, v)(x, t) be a smooth solution of the Cauchy problem (1.3)–(1.4) up to some time T > 0. If the support of initial data is contained in the ball BR (0) centered at the origin with radius R, then the support of (ρ, v)(x, t) is contained in the same ball BR (0) for any t ∈ [0, T ). Proof. Assume that the initial support of the solution is contained in a ball BR (0), the support of the smooth solution will remain compact by the hyperbolic nature of the system (1.3). We denote by x(t; x0 ) the particle path starting at x0 when t = 0, i.e., d x(t; x0 ) = v(x(t; x0 ), t), dt
x(t = 0; x0 ) = x0 ,
(4.1)
and by Sp (t) the closed region that is the image of BR (0) under the flow map (4.1). Hence, the support of the smooth solution of (1.3)–(1.4) will remain inside Sp (t). Thus, fixing any x0 on the boundary of BR (0), we have ρ0 (x0 ) = 0 and v0 (x0 ) = 0, and x(t; x0 ) is on the boundary of Sp (t). Furthermore, d x(t; x0 ) = v(x(t; x0 ), t) = 0, dt
(4.2)
due to continuity of v(x, t) and the fact that x(t; x0 ) sits at the boundary of the support of the solution. Therefore, x(t; x0 ) = x0 for any t ∈ [0, T ) whenever |x0 | = R. Hence, Sp (t) = BR (0). This proves this lemma. Based on Lemma 4.1, we shall prove the following blowup result. Theorem 4.2. Suppose the support of the smooth functions (ρ0 (x), v0 (x)) is non-empty and contained in a ball BR (0) centered at the origin with radius R. Then the smooth solution of (1.3)-(1.4) with the initial data (ρ0 (x), v0 (x)) blows up in finite time. Proof. We first introduce the following functions: 1 H (t) = ρ|x| ˆ 2 dx, F (t) = ρv ˜ • x dx, E(t) = ρˆ dx. 2
(4.3)
Here, H (t) is the second moment of ρ, ˆ F (t) is the total radial “generalized” momentum, and E(t) is the total “generalized” energy. These functions are well defined in the domain where the smooth solutions exist. Interesting relations between them can be obtained by the following calculations. E(t) is conserved, because using (2.2) one has E (t) = ρˆt dx = − ∇x • (ρv) ˜ dx = 0. We thus have
E(t) = E(0) =
ρˆ0 (x) dx > 0,
(4.4)
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R. Pan, J.A. Smoller
for non-trivial initial data. For H (t), we have H (t)
1 = 2
ρˆt |x|2 dx 1 2 [∇x • (ρv)]|x| ˜ dx =− 2 = ρv ˜ • x dx
(4.5)
= F (t), where we have used the relation 2 2 ˜ = ∇x • (ρv|x| ˜ ) − ρv ˜ • 2x. [∇x • (ρv)]|x|
From the second equation in (2.2) and integrating by parts, we have ˜ t • x dx H (t) = F (t) = (ρv) ˜ ⊗ v)] • x + (∇x p • x)) dx = − ([∇x • (ρv = ρv ˜ 2 dx + 3p dx 1 2 3p ( 2 ρv ˜ + 2 ) dx. = c2 c BR (0) c
(4.6)
By Jensen’s inequality, we have
p(ρ) dx = BR (0)
4π 3 R 3
4π ≥ ( R 3 )p 3
BR (0) p(ρ) 4π 3 3 R
dx
BR (0) ρ dx 3 ( 4π 3 R )
(4.7) ,
so (4.6) and (4.7) imply
H (t) ≥ c
2 BR (0)
1 2 3 4π ρv ˜ dx + 2 ( R 3 )p(ρB ) c2 c 3
≡ c2 N (t), where
ρB =
BR (0) ρ dx 3 ( 4π 3 R )
is the mean density over BR (0). Since E(t) = E(0) =
BR (0)
1 2 ρv ˜ +ρ c2
dx,
(4.8)
Blowup of Smooth Solutions for Relativistic Euler Equations
747
it is possible to bound N (t) from below using E(0). We consider two cases. First if 1 2 1 ρv ˜ dx ≥ E(0), 2 2 BR (0) c we get N (t) ≥ On the other hand, if
then as
(4.9)
BR (0)
1 2 1 ρv ˜ dx ≤ E(0), c2 2
E(t) = E(0) = BR (0)
we have
1 E(0). 2
1 2 ρv ˜ +ρ c2
ρ dx ≥ BR (0)
dx ≤
1 E(0) + 2
ρ dx, BR (0)
1 E(0). 2
Thus 3 4π 3 ( R )p(ρB ) c2 3 1 3 4π ≥ 2 ( R 3 )p( EB (0)) c 3 2 ≡ B1 E(0) > 0,
N (t) ≥
where B1 =
4π R 3 p( 21 EB (0)), c2 E(0)
(4.10)
E(0) . 3 ( 4π 3 R )
Define B = c2 min{ 21 , B1 }; then
H (t) ≥ BE(0) > 0.
(4.11)
and EB (0) =
(4.8)–(4.10) imply that
This gives a lower bound on H (t): H (t) ≥
1 BE(0)t 2 + F (0)t + H (0). 2
(4.12)
In order to refine (4.12), we estimate F (t) in terms of H (t) and E(t). Using (3.10), we have |F (t)| = | (ρv ˜ • x) dx| 1 1 2 2 dx) ( ρv ˜ 2 dx) 2 ≤ ( ρ|x| ˜ √ 1 1 (4.13) ≤ 2H (t) 2 (c2 E(t)) 2 √ 1 = c 2[H (t)E(t)] 2 1
≡ D[H (t)E(t)] 2 .
748
R. Pan, J.A. Smoller
We derive from (4.12) and (4.13) that H (t) ≥
1 1 BE(0)t 2 − D[H (0)E(0)] 2 t + H (0). 2
(4.14)
We note that (4.14) implies that H (t) tends to infinity as t goes to infinity. However, we have the following uniform upper bound for H (t): 1 ρ|x| ˆ 2 dx 2 1 ρ|x| ˆ 2 dx = 2 BR(0) 1 ≤ R 2 ρˆ dx 2 1 = E(0)R 2 . 2
H (t) =
(4.15)
Thus (4.12) or (4.14) together with (4.15) imply that the life-span of the smooth solutions must be finite if E(0) > 0. This completes the proof of Theorem 4.2. Notice that, for non-trivial initial data, we have 1 H (0) − E(0)R 2 2 1 1 ρˆ0 |x|2 dx − R 2 ρˆ0 dx = 2 2 1 2 ρˆ0 (|x| − R 2 ) dx = 2 BR (0) < 0.
(4.16)
This enables us to estimate the life-span as follows: from (4.14) and (4.15) we have for smooth solutions 1 1 BE(0)t 2 − D H (0)E(0)t + H (0) ≤ E(0)R 2 . 2 2 This is equivalent to φ(t) = Bt 2 − 2Ddt + 2d 2 − R 2 ≤ 0, where d 2 = satisfies
H (0) E(0) , and φ(0)
(4.17)
< 0 by (4.16). Hence, the life-span T of the smooth solution
T ≤
Dd +
√ D 2 d 2 − 2Bd 2 + R 2 B . B
(4.18)
Blowup of Smooth Solutions for Relativistic Euler Equations
749
5. Concluding Remarks We have proved the blowup of smooth solutions of relativistic Euler equations in both cases: finite initial energy (Theorem 4.2) and infinite initial energy (Theorem 3.2). In contrast to the characteristic method, we adapted the approach via some functions: total “generalized” energy, total radial “generalized” momentum, and the second moment. Our approach depends on the beautiful structure of the equations and several quantities constructed from the natural variables; cf. Lemma 3.1. Although the relativistic Euler equations are much more complicated than the classical Euler equations, these structures make our proofs possible. We will now make some remarks on our results and discuss some related issues. Remark 1. In our blowup theorems, the velocity in the far field is assumed to be zero initially. For the more general case, say v0 (x) = v¯ off a bounded set, the change of variables (Sideris [11]) v → v − v, ¯ x → x + t v¯ will reduce this problem to the case we considered. Remark 2. The condition p (ρ) ¯ <
c2 3
in Theorem 3.2 arises naturally in the proof. Here, 3 is the spatial dimension. In d dimensions, 3 is replaced by d. In particular, for d = 1, this condition is that the sound speed is subluminal. For p(ρ) = σ 2 ρ, d = 1, this condition guarantees the genuinely nonlinearity of the relativistic Euler equations and allows the existence of global solutions in BV; see Smoller and Temple [14]. This condition is not required in Theorem 4.2. Remark 3. Our blowup results crucially depend on the compact support of the perturbations. Singularity formation for more general initial data remains open. Remark 4. The type of singularity which occurs is another open problem. The possibilities are: a) shock formation, b) violation of the subluminal conditions; e.g. |v| tends to c, or p (ρ) → c2 , c) concentration of the mass. For p(ρ) = σ 2 ρ and d = 1, the singularity must be a shock if the initial data is away from the vacuum. It was shown in Smoller and Temple [14] that weak solutions exist globally in time with bounded total variation, subluminal velocity and positive density uniformly bounded from above and below. Furthermore, Smoller and Temple proved in [14] that the subluminal condition guarantees the genuine nonlinearity of the equations, so one concludes that the singularities in the solutions must be shocks by Lax’ theory [4]. It would be interesting to clarify the types of singularities for relativistic Euler equations in multi-dimensions. However, black hole formation is impossible for our problem, since our spacetime is fixed to be flat Minkowski spacetime. Remark 5. The singularity in our Theorem 3.2 looks like shock formation. The largeness condition in radial “generalized” momentum, (3.20), implies that the particle velocity must be supersonic in some region relative to the sound speed at infinity. One can guess that the singularity formation is detected as the disturbance overtaking the wave front
750
R. Pan, J.A. Smoller
thereby forcing the front to propagate with supersonic speed. To see these things, we argue as follows. Using the fact ρ˜ ≤ 2ρ(cf. ˆ (3.13)), one has F (0) ≤
8π 4 R (max ρˆ0 (x)) max |v0 (x)|, 3
(5.1)
while = ≥
4s 8π 4 R max ρˆ0 (x) 2 1− 3s2 3 c 4 4s 8π 3 R (max ρˆ0 (x)).
(5.2)
max |v0 (x)| ≥ 4s.
(5.3)
Hence, F (0) > implies that
This insures the initial particle velocity is supersonic in some region. However, the rigorous proof of shock formation is still open. Remark 6. The lower bound of the initial radial “generalized” momentum in (3.20) depends on the initial velocity through ρ. ˆ This is different from the Newtonian case, where ρˆ is replaced by ρ and so in the Newtonian case it does not depend on the velocity. On the other hand, the velocity has to be subluminal. Therefore, we must show that the set of initial data required in Theorem 3.2 is non-empty. From (5.1)–(5.3), we find the set is non-empty if c > max |v0 (x)| ≥
4s 1−
3s 2 c2
.
A simple calculation shows that the necessary condition for s to satisfy is √
7 2 s< − c. 3 3
(5.4)
(5.5)
Since ρ˜ ≥ ρ, ˆ F (0) is of the same order as the upper bound in (5.1). Thus, if s is chosen to be small (this can be done by choosing ρ¯ small), one can easily find initial data satisfying the conditions required in Theorem 3.2. Remark 7. The equation of state p(ρ) satisfying (1.5) is quite general for isentropic fluids. It can be weakened by replacing p (ρ) ≥ 0 with p (ρ) non-decreasing. This includes the well-known γ -law, p(ρ) = σ 2 ρ γ , γ ≥ 1 as a particular case. In fact, in the case of a γ -law, (3.13) can be refined, and thus (3.20) can be replaced by a weaker condition, as we now show. 2 2 ˜ 2 by When γ = 1, s = σ . Equation (3.13) is refined as ρ˜ < (1 + σc2 )ρˆ = ρ˜ + σc4 ρv Lemma 3.1. Thus, (3.20) can be weakened to: σ 2 4π 4 4σ 1+ 2 F (0) > 1 = R max ρˆ0 (x). (5.6) 2 c 3 1 − 3σ2 c
When γ > 1, we observe that the subluminal condition p (ρ) = γ σ 2 ρ γ −1 ≤ c2
Blowup of Smooth Solutions for Relativistic Euler Equations
implies that
p c2
≤
1 γ ρ.
751
Thus,
1 1+ γ
1 1 ρ + 2 ρv ˜ 2 γ γc 1 ≥ ρˆ + ρ γ p ˜ ≥ ρˆ + 2 = ρ. c
ρˆ = ρˆ +
ˆ and then (3.20) is replaced by the following We can thus refine (3.13) to ρ˜ < (1 + γ1 )ρ, weaker condition: 1 4π 4 4s 1 + R max ρˆ0 (x). (5.7) F (0) > 2 = 2 γ 3 1 − 3sc2 6. Appendix For the reader’s convenience, we justify the construction of a strictly convex entropy function for (1.3) due to Makino and Ukai in [7], and we will also correct several errors. To this end, we first record (2.6)–(2.7) here , θt +
3
(f k (θ ))xk = 0,
(6.1)
k=1
where θ = (θ0 , θ1 , θ2 , θ3 )T and f k (θ ) = (θk , f1k , f2k , f3k ) are defined by ˆ θj = ρv ˜ j, θ0 = ρ, ˜ j vk + pδj k , j = 1, 2, 3. fjk = ρv
(6.2)
The scalar function η = η(θ ) is called an entropy function and scalar functions q k (θ ), k = 1, 2, 3 are called entropy flux functions, if they satisfy: ∇θ η(θ )∇θ f k (θ ) = ∇θ q k (θ ).
(6.3)
Since the the right-hand side of (6.3) is a gradient of the function q k , the relevant integrability condition (cf. [1], p. 39) is (∇θ2 η)(∇θ f k ) = (∇θ f k )T (∇θ2 η).
(6.4)
If we find such an η that is strictly convex, the change of variables θ → w = (∇θ η)T will render (6.1) into the symmetric form (2.17); see [1], where A0 = (∇θ2 η)−1 , Ak = (∇θ f k )(∇θ2 η)−1 .
(6.5)
To see this, we apply chain rule: ∂α θ = (∇θ w)−1 ∂α w = (∇θ2 η)−1 ∂α w.
(6.6)
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R. Pan, J.A. Smoller
Substituting (6.6) into (6.1), we obtain (∇θ2 η)−1 wt +
3
(∇θ f k )(∇θ2 η)−1 wxk = 0.
k=1
A0 is positive definite if and only if η is strictly convex. To verify the real symmetry of Ak , we use (6.4). Multiplying both sides of (6.4) by (∇θ2 η)−1 on the left and right, we see Ak = (Ak )T . We will solve (6.3) keeping the mechanical energy of classical Euler equations in mind. Thus, instead of θ, we will use z = (ρ, v1 , v2 , v3 )T as independent variables. We compute: B1 B2 v T , (6.7) ∇z θ = B3 v B2 vv T + B4 I3 where B1 = B3 =
2 +p c2 +p − cp2 , B2 = 2 (cρc 2 −v 2 )2 , c2 −v 2 2 +p c2 +p , B4 = ρc . c2 −v 2 c2 −v 2
(6.8)
Moreover, det (∇z θ ) =
(ρc2 + p)3 (c4 − v 2 p ) > 0, c2 (c2 − v 2 )4
in z . We can thus compute the inverse of ∇z θ : c2 (c2 + v 2 )E1 −2c2 E1 v T −1 , (∇z θ ) = −c2 (c2 + p )E1 E2 v 2p E1 E2 vv T + E2 I3
(6.9)
(6.10)
with E1 =
1 c2 − v 2 , E = . 2 c4 − p v 2 ρc2 + p
(6.11)
Based on (6.10), we will solve (6.3) using z as independent variables for convenience. In the z-variables, (6.3) becomes ∇z ηC k = Dz q k , k = 1, 2, 3, where
B 3 vk B2 vk v T + B4 ekT , B3 vk v + p ek B2 vk vv T + B4 vekT + B4 vk I3
(6.12)
(∇z f k ) =
(6.13)
and C k = (∇z θ )−1 (∇z f k )
C3 ekT c 2 C 1 vk , = −C1 C2 vk v + C2 ek −C4 vekT + vk I3
(6.14)
Blowup of Smooth Solutions for Relativistic Euler Equations
753
with C1 = C3 =
(c2 −v 2 ) c2 −p , C2 = p E2 = p ρc 2 +p , c4 −p v 2 c2 (ρc2 +p) p (c2 −v 2 ) , C4 = c4 −p v 2 . c4 −v 2 p
(6.15)
Formally, (6.12) is an over-determined system, consisting of 12 equations for 4 unknowns. We seek solutions with the special form: η = η(ρ, y), q k = Q(ρ, y)vk , y = v 2 = v12 + v22 + v32
(6.16)
to reduce the number of equations in (6.12). Substituting this ansatz into (6.12), we obtain the following first order linear system: ηy = Qy , (6.17) c2 C1 ηρ + 2C2 (1 − C1 y)ηy = Qρ , C η − 2C yη = Q. 3 ρ 4 y This seems still an over-determined system. However, it is possible to derive a decoupled equation for Q from (6.17). We first multiply the second equation of (6.17) by (ρ 2 c2 + p), and using (c2 − p )C3 = c2 C1 (ρ 2 c2 + p), we have (c2 − p )C3 ηρ + 2C2 (ρc2 + p)(1 − C1 y)ηy = (ρc2 + p)Qρ .
(6.18)
Then, we compute (c2 − p ) × (6.17)3 : (c2 − p )C3 ηρ − 2(c2 − p )C4 yηy = (c2 − p )Q.
(6.19)
We subtract (6.19) from (6.18) and substitute ηy with Qy , using (6.15); this reduces (6.17) into the following decoupled system: ηy = Qy , (6.20) 2(c2 − y)p Qy = (ρc2 + p)Qρ − (c2 − p )Q. We now proceed to solve (6.20) with the help of (6.17). First, (6.20)1 gives η = Q(ρ, y) + G(ρ).
(6.21)
Substitute this into (6.17)3 to get c2 − y 2 c2 − y 2 Q− Qρ , 2 ρc + p c2
Gρ = or equivalently Gρ =
1 1 q − 2 qρ , q = (c2 − y)Q. ρc2 + p c
We observe that G depends on ρ only, so we have a linear first order ODE for q, 1 f (ρ) 1 q − 2 qρ = 2 , +p c c
ρc2
(6.22)
754
R. Pan, J.A. Smoller
which has the solution q(ρ) = eφ(ρ) (g(ρ) + h(y)).
(6.23)
Here φ(ρ) is defined in (2.8). Substituting (6.23) into (6.20)2 , and separating variables, one has ρc2 + p dg dh − g = 2(c2 − y) + h = m(ρ, y), dρ dy p
(6.24)
where the first term in (6.24) is independent of y, while the second term is independent of ρ. Thus, m is independent of both ρ and y. We conclude that m = const. Thus, by integrating (6.24), we have √ q = D1 (ρc2 + p) + D2 eφ(ρ) c2 − v 2 , G = − Dc21 p + D3 , 2 φ(ρ) (6.25) Q = D ρc +p + D √e , η =
1 c2 −v 2 2 +p D1 ρc c2 −v 2
2
c2 −v 2 φ(ρ)
+ D2 √e 2
c −v 2
−
D1 p c2
+ D3 ,
where D1 , D2 and D3 are integration constants. With K as in (2.8), one choice is D1 = c2 , D2 = −cK, D3 = 0, thus cKeφ(ρ) . η = c2 ρˆ − √ c2 − v 2
(6.26)
The associated entropy-flux is (q 1 , q 2 , q 3 )T defined by qk =
c2 (ρc2 + p) cKeφ(ρ) v − vk . √ k c2 − v 2 c2 − v 2
(6.27)
Moreover η is strictly convex as was verified in Sect. 2. References 1. Dafermos, C.: Hyperbolic conservation laws in continuum physics. Berlin-Heidelberg-New York: Springer-Verlag, 2000 2. Guo, Y., Tahvildar-Zadeh, S.: Formation of singularities in relativistic Fluid dynamics and in spherically symmetric plasma dynamics. Contemp. Math. 238, 151–161 (1999) 3. Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975) 4. Lax, P.: Development of singularity of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys. 5, 611–613 (1964) 5. Liu, T.: The development of singularity in the nonlinear waves for quasi-linear hyperbolic partial differential equations. J. Differ. Eqs. 33, 92–111 (1979) 6. Makino, T., Ukai, S.: Local smooth solutions of the relativistic Euler equation. J. Math. Kyoto Univ. 35-1, 105–114 (1995) 7. Makino, T., Ukai, S.: Local smooth solutions of the relativistic Euler equation. II. Kodai Math. J. 18, 365–375 (1995) 8. Makino, T., Ukai, S., Kawashima, S.: Sur la solutions a´ support compact de l’equation d’Euler compressible. Japan J. Appl. Math. 3, 249–257 (1986)
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9. Perthame, B.: Non-existence of global solutions to Euler-Poisson equations for repulsive forces. Japan J. Appl. Math. 7, 363–367 (1990) 10. Rendall, A.: The initial value problem for self-gravitating fluid bodies. Mathematical Physics X (Leipzig, 1991), Berlin: Springer, 1992 pp. 470–474 11. Sideris, T.: Formation of singularity in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–485 (1985) 12. Sideris, T.: Formation of singularities of solutions to nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 86, 369–381 (1984) 13. Smoller, J.: Shock waves and reaction diffusion equations Berlin-Heidelberg-New York: SpringerVerlag, 2nd Edition, 1993 14. Smoller, J., Temple, B.: Global solutions of the relativistic Euler equations. Commun. Math. Phys 156, 67–99 (1993) 15. Weinberg, S.: Gravitation and Cosmology: principles and applications of the general theory of relativity. New York: Wiley, 1972 16. Xin, Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51, 229–240 (1998) Communicated by G.W. Gibbons
Commun. Math. Phys. 262, 757–791 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1484-5
Communications in
Mathematical Physics
Orbital Stability of Double Solitons for the Benjamin-Ono Equation Aloisio Neves, Orlando Lopes Departamento de Matematica, IMECC-UNICAMP, C.P. 6065, Campinas, SP-Brasil, CEP 13083-970, Brasil. E-mail: [email protected]; [email protected] Received: 5 April 2005 / Accepted: 5 July 2005 Published online: 9 December 2005 – © Springer-Verlag 2005
Abstract: In this paper we prove the orbital stability of double solitons for the Benjamin-Ono equation. In the case of the KdV equation, this stability has been proved in [17]. Parts of the proof given there rely on the fact that the Euler-Lagrange equations for the conserved quantities of the KdV equation are ordinary differential equations. Since this is not the case for the Benjamin-Ono equation, new methods are required. Our approach consists in using a new invariant for multi-solitons, and certain new identities motivated by the Sylvester Law of Inertia.
1. Introduction A common feature of integrable evolution equations describing nonlinear wave motion is that they have an infinite sequence of conservation quantities (first integrals) V1 (u), V2 (u), V3 (u), . . . In concrete examples, the solutions of the equation V2 (u) + cV1 (u) = 0 are one-solitons (traveling waves, standing waves). These solutions evolve without changing their shape and their stability has been proved in [2, 4 and 10] for the KdV equation and in [1] for the Benjamin-Ono (BO) equation. On the other hand, the higher order Euler-Lagrange equation V3 (u) + αV2 (u) + βV1 (u) = 0
(1.1)
gives rise to more complicated solutions called double solitons (or two-solitons). In some sense, these solutions represent the superposition of two one-solitons and their speeds are related to the multipliers α and β through an algebraic equation. As in the case of the one-soliton, to prove that the double soliton is stable we have to show that they locally minimize V3 subject to given values of V1 and V2 . This proof involves the spectral analysis of the one-parameter family of self-adjoint operators L(t), which are the linearization of (1.1) at the double soliton.
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An alternative way to construct double solitons is to use the self-adjoint operator M that appears in the Lax pair associated to the integrable equation (see [7] for the construction of the Lax pair for the Benjamin-Ono equation). The double solitons are the potentials for which the self-adjoint operator M has two eigenvalues. Although the discreteness of the spectrum of M combined with Inverse Scattering Theory might suggest some stability property of the double solitons, the first rigorous proof of the Liapunov stability of double solitons for the KdV equation has been given in [17] using the variational approach introduced above. In this case, the linearized operator L(t) is a fourth order self-adjoint linear ordinary differential operator. In [17] the authors first give spectral conditions for critical points of abstract constrained variational problems to be local minimizers. As a consequence of this abstract result, they show that the stability of the double soliton follows from the following spectral conditions: C.1 for any t, L(t) has exactly one negative eigenvalue; C.2 for any t, zero is a double eigenvalue of L(t). Their method to show that these two conditions are indeed satisfied relies very heavily on the ODE structure of L(t). More specifically, they use the concept of stable and unstable subspaces for linear equations to show that the multiplicity of zero as an eigenvalue of L(t) is exactly two for any t. They also use some results from the Calculus of Variations that are related to the concept of conjugate points (for unbounded intervals) to count the number of negative eigenvalues of L(t). In this paper we prove the stability of double solitons for the BO equation. As in the case of the KdV equation, to prove this stability we have to show that the family of self-adjoint operators L(t) satisfies Conditions C.1 and C.2 above. However, in the case of BO equation, L(t) is not an ordinary differential operator anymore because the Hilbert transform appears in it. This makes the spectral analysis more complicated and then a new approach is required. The first step of our method (which is presented in Sects. 2 and 3) consists in making a simplification in the spectral problem to reduce the spectral analysis of the one-parameter family L(t) to the analysis of the spectra of two stationary operators L1 and L2 . We now describe this simplification in more detail. We begin by describing how the operators L1 and L2 are constructed. The double soliton u(t) appears in the coefficient of the self-adjoint operator L(t). For instance, in the case of the KdV equation, L(t) is given by L(t)h = h(4) + 10uhxx + 10uxx h + 10ux hx + 30u2 h + α(−hxx − 6uh) + βh, where u = u(t) is the double soliton. Let u1 and u2 be the one-solitons associated to that double soliton u(t) and let L1 and L2 be the stationary “limit operators” that we get when we replace u(t) by u1 and u2 in L(t), respectively. The notation is such that u1 is the soliton with lower speed. Suppose we can show the following properties of the spectra of L1 and L2 : C.3 L1 has one negative eigenvalue and L2 has no negative eigenvalue; C.4 zero is a simple eigenvalue of L1 and of L2 . Our reduction procedure consists in proving that conditions C.3 − C.4 imply C.1 − C.2. In other words, what we do is to reduce the proof of the stability of the double soliton to the study of the stationary “limit” uncoupled operators L1 and L2 , whose coefficients depend only on the one-solitons u1 and u2 , respectively. This fact is implicit in [17], but that is so because they use techniques which are very specific to variational problems whose Euler-Lagrange equations are ODE’s.
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To carry out this simplification we use a new invariant for multi-solitons (part two of Theorem 3) which has been introduced by one of the authors in [16]. We also use a theorem about the asymptotic behavior of the spectrum of a sequence of self-adjoint operators (Theorem 4). This simplification is done in an abstract framework. Hence, in principle, it works for any integrable equation, independently of the structure of the Euler-Lagrange equations for the conserved quantities. However, the procedure to verify that the spectral conditions C.3 − C.4 are satisfied is specific to the integrable equation under study. In the case of BO equation our method to verify those conditions is based on certain new identities (and some variants of them) of the type MQM t = N Q0 N t .
(1.2)
In (1.2) Q is a certain self-adjoint operator, M and N are auxiliary operators and Q0 is another self-adjoint operator which is “simpler” than Q. Identities of type MQ = Q0 M
(1.3)
are well-known and they are related to the Darboux factorization ([5] and [6]). If the spectrum of Q0 is known, the kernel of M has dimension one and (1.3) holds, then the spectrum of Q can be calculated. We will come back to this point later in this paper. However, for some linear operators that we encounter in this paper, identities of type (1.3) do not seem to exist, while identities of type (1.2) can still be found. Identities of type (1.2) are motivated by the Sylvester Law of Inertia (Theorems 1 and 2). As we will see, if (1.2) holds and we have some information about the spectrum of Q0 and about the kernel and image of M and of N, then certain qualitative properties of the spectrum of Q can be obtained. In other words, identities of type (1.3) are stronger in the sense that they allow us to calculate spectra while identities of type (1.2) give qualitative information only. Fortunately, this qualitative information is enough for our needs. To illustrate how our method works, we first use it to provide an alternative proof of the stability of double solitons for the KdV equation. Then, we extend the method to the new case of the BO equation. As in [17], our approach is purely variational. In particular, Inverse Scattering Theory is not used here. Inverse scattering has been used in [21] to prove stability results for the KdV equation but, in that case, the distances between the initial conditions and the corresponding solutions at later times are measured by different norms. In [17] and in the present paper, the phase space where stability is proved is the largest Sobolev space where the first integrals V1 , V2 and V3 make sense (H 2 (IR) for the KdV equation and H 1 (IR) for the BO equation.) It is likely that our method can be extended to multi-solitons of the BO equation and of its hierarchy but the algebra may become prohibitive. The hierarchy for the BO equation has been constructed in [8]. The possibility of extending our method to show stability of double solitons (and one-solitons also) for other integrable equations depends only on finding identities of type (1.2) for these equations. Perhaps we should expect that, for general integrable equations, the stability of multisolitons should be a consequence of stability of one-solitons. Unfortunately, such a result has not been proved yet. Using the method we develop in this paper it can conceivably be shown that if the one-solitons are stable for all equations in its hierarchy, then multisolitons are also stable. In [18] some results of stability and asymptotic stability are proved for the generalized KdV equations for subcritical powers of the nonlinearity. To be more specific,
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it is shown in [18] that if the profile of an initial condition for the generalized KdV equation is close to the profile of N traveling waves which are far apart, then this profile is preserved for positive time. This type of stability (which holds also for non-integrable nonlinearities) does not include the results of [17] (neither does ours.) The reason is that when we minimize conserved quantities, we get the stability of the whole orbit of the double soliton (and not just of its tail) for both positive and negative times. This paper is organized as follows: in Sect. 2 we recall a new invariant for multisolitons which has been introduced in [16]. In Sect. 3 we prove a result about the asymptotic behavior of the spectrum of a certain sequence of self-adjoint operators. With these two abstract results we will be able to make the simplification of the spectral problem. In Sect. 4 we recall a result that gives the spectral condition for critical points of constrained variational problems to be local minimizers. In Sect. 5 we introduce identities of type (1.3) for the KdV equation and we prove the stability of double solitons for that equation. Finally, in Sect. 6 we extend the method to the BO equation. We also include an appendix where we recall some properties of the Hilbert transform; the proofs of some lemmas which depend on long calculations involving Hilbert transform are also left to the appendix.
2. A New Invariant for Multi-Solitons We begin by recalling the definition of inertia of a real symmetric matrix. Definition 1. If A is a real N × N symmetric matrix then the inertia in(A) is a triplet (n, z, p) of nonnegative integers where n, z and p are the number of negative, zero and positive elements (counted according to their multiplicity) of the spectrum of A . The next result is known as the Sylvester Law of Inertia (see [9]). Theorem 1. If A is a real symmetric N × N matrix and M is a real nonsingular (not necessarily orthogonal) N × N matrix, then in(MAM ∗ ) = in(A). The unbounded self-adjoint operators L we will be dealing with in this paper satisfy the following property: there is a δ > 0 such that the spectrum of L to the left of δ consists of a finite number of eigenvalues and the corresponding spectral projections have finite dimensional range. For that class of self-adjoint operators we give the following: Definition 2. The inertia in(L) of a self-adjoint operator as above is the pair (n, z) of nonnegative integers where, as in the case of matrices, n is the dimension of the negative subspace of L and z is the dimension of the null space of L. Now we can state the Generalized Sylvester Law of Inertia: Theorem 2. If L with domain D(L) is a self-adjoint operator as above and M is an invertible bounded operator, then in (MLM ∗ ) = in (L), where MLM ∗ is the selfadjoint operator with domain (M ∗ )−1 (D(L)). The proof of Theorem 2 follows exactly as in the matrix case [9] because the only thing it only uses is the variational characterization of the eigenvalues of self-adjoint operators. We now introduce certain one-parameter families of self-adjoint operators L(t) that are isoinertial, that is, have a constant inertia. The families of self-adjoint operators that
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arise in the stability theory of multi-solitons fit in that class. Let us consider an abstract evolution equation u˙ = f (u)
(2.1)
in a Hilbert space X. We suppose that it has a first integral V (u). In order to be precise, we consider three Hilbert spaces X, X1 and X2 and we make the following hypotheses: H.1 X2 ⊂ X1 ⊂ X with continuous embedding; the embedding from X2 into X1 will denoted by i; H.2 V : X1 → R is a C 3 functional; H.3 f : X2 → X1 is a C 2 function; H.4 for any u ∈ X2 we have V (i(u))f (u) = 0.
(2.2)
If u(t) is a strong solution of (2.1) and , is the scalar product of X, we suppose that there is a self-adjoint operator L(t) : D(L) ⊂ X → X with constant domain D(L) such that L(t)h, k = V (u(t))(h, k) for h and k in a subspace Z ⊂ D(L) ∩ X2 which is dense in X. We consider also another operator B(t) : D(B) ⊂ X → X such that B(t)h = −f (u(t))h for h ∈ Z and we make the final assumption H.5 The closed operators B(t) and B ∗ (t) have a common domain independent of t, and the Cauchy problems u˙ = B(t)u
u˙ = B ∗ (t)u
are well posed for both positive and negatives times in the space X. Theorem 3. Let u(t) be a strong solution of (2.1) and suppose H1 to H5 are satisfied. – (P.Lax [15]) If for some t0 we have V (u(t0 )) = 0 then V (u(t)) = 0 for any t; in other words, the set of the critical points of the first integral V (u) is invariant under (2.1). – If u(t) is as in the first part (that is, u(t) is a solution of (2.1) satisfying V (u(t)) = 0 for all t ∈ R), then the inertia in(L(t)) of the self-adjoint operator L(t) that represents V (u(t)) as above is independent of t. The proof of Theorem 3 is given in [16] but we recall what is the main idea for the second part. A very famous device for finding isospectral families of self-adjoint operators (families with constant spectrum) is the Lax pair ([14]). The motivation is the following: let L(t) be a family of self-adjoint operators and let us impose that L(t) = M(t)L(0)M ∗ (t)
(2.3)
for any t, where M(t) is orthogonal and satisfies ˙ M(t) = B(t)M(t)
M(0) = I
(2.4)
with B(t) skew-adjoint. Differentiating (2.3) with respect to t and using (2.4) we get ˙ L(t) = B(t)L(t) − L(t)B(t).
(2.5)
Conversely, if (2.5) holds with B(t) skew-adjoint then L(t) = M(t)L(0)M ∗ (t), where M(t) is orthogonal and this implies that the spectrum of L(t) is constant.
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Suppose that instead of constructing isospectral families, we want to construct isoinertial families. Then in view of Theorem 2 we impose L(t) = M(t)L(0)M ∗ (t)
(2.6)
and we assume that M(t) evolves in time satisfying a linear equation ˙ M(t) = B(t)M(t)
M(0) = I.
(2.7)
Differentiating (2.6) with respect to t and using (2.7) we get ˙ L(t) = B(t)L(t) + L(t)B ∗ (t).
(2.8)
Conversely, if (2.8) is satisfied then (2.6) holds and, if M(t) is invertible (which is guaranteed by assumption H.5), then according to Theorem 2 the inertia in(L(t)) of L(t) is constant. Therefore, without imposing that B(t) is skew-adjoint, Eq. (2.8) governs isoinertial families of operators. The proof of part two of Theorem 3 consists in showing that the self-adjoint operator L(t) satisfies Eq. (2.8) and this follows easily differentiating (2.2) one and two times with respect to u (details are given in [16].) As we have pointed out in the introduction, to prove the stability of double solitons we have to verify the spectral conditions C.1 and C.2 for a certain one-parameter family of self-adjoint operators L(t) whose coefficients depend on the double soliton u(t). These two conditions mean precisely that the inertia of L(t) is the pair (1,2). Since multi-solitons fit in the framework of Theorem 3, we conclude that the inertia in(L(t)) of L(t) is independent of t. Therefore, we can choose a convenient t to calculate the inertia and the best thing we can do is to calculate the inertia in(L(t)) as t goes to ∞. In that case the double soliton splits into two one-solitons u1 and u2 far apart. If L1 and L2 are the operators that we get when we replace u(t) by u1 and u2 in L(t), respectively, then in the next section we show that, as t goes to ∞, the spectrum σ (L(t)) of L(t) converges to the union of the spectra σ (L1 ) and σ (L2 ) of L1 and of L2 . 3. Asymptotic Behavior of the Spectrum of Certain Sequences of Self-Adjoint Operators As a model for the main result of this section we take X = L2 (R) and we denote by τn the family of isometries (τn h)(x) = h(x −n). We consider also the following symmetric operators: (Ah)(x) = h(4) (x) + αh (x) + βh(x), where α and β are such that A is invertible; (Bh)(x) = b(x)h (x) + b (x)h (x) + b0 (x)h(x); (Ch)(x) = c(x)h (x) + c (x)h (x) + c0 (x)h(x); (Cn h)(x) = (τn−1 Cτn h)(x) = c(x + n)h (x) + c (x + n)h (x) + c0 (x + n)h(x); (Dn h)(x) = dn (x)h (x) + dn (x)h (x) + d0,n (x)h(x). The functions b(x), b0 (x), c(x) and c0 (x) are smooth and, together with some derivatives, tend to zero at infinite; dn (x) and d0,n (x) are sequences of smooth functions whose L∞ norm and of some of their derivatives tend to zero as n tends to ∞.
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In the model above, the operator Ln = A + B + Cn + Dn is the sum of an operator A which is translation invariant, an operator Dn whose coefficients tend to zero as n goes to ∞ and two other operators B and Cn whose coefficients have “support” far apart when n tends to infinity. We will show that as n gets large, the spectrum of Ln tends to the union of the spectra of A + B and of A + C. The abstract framework is the following: let X be a real Hilbert space and τn be a sequence of isometries of X. Suppose also that A, B, C are linear operators, Cn , Dn are two sequences of linear operators with Cn = τn−1 Cτn and that the following assumptions are satisfied: A.1 A, A + B, A + C, A + Cn , A + B + Cn + Dn are self-adjoint with the same domain D(A); A.2 A is invertible and A commutes with τn , that is, τn−1 Aτn = A; A.3 there is a number δ > 0 such that the spectra of all self-adjoint operators A, A + B, A + C, A + B + Cn + Dn to the left of δ consists of a finite number of eigenvalues and the spectral projection corresponding to any such eigenvalue has finite dimensional range; A.4 for any λ ∈ ρ(A + B) ∩ ρ(A + C), the operators A(A + C − λI )−1 and A(A + B − λI )−1 are bounded ; A.5 |Dn A−1 | tends to zero as n tends to ∞; A.6 for any element u ∈ D(A), |Cn u| → 0 as n tends to ∞; A.7 for any element u ∈ X, τn u tends to zero weakly in X as n tends to infinity; A.8 for any λ in the resolvent set ρ(A + C) of A + C, the operator B(λI − A − C)−1 is compact. Theorem 4. Under assumptions A.1 to A.8, if λ < δ and λ ∈ ρ(A + B) ∩ ρ(A + C), then there is a number N0 such that for n ≥ N0 , λ belongs to the resolvent set ρ(Ln ) of Ln = A+B +Cn +Dn . Moreover, if λ0 ∈ σ (A+B)∪σ (A+C) and > 0 is given, then there is an integer N1 such that for n ≥ N1 , the dimension of the range of the spectral projection of Ln corresponding to the circle centered at λ0 and radius is equal to the sum of the dimensions of the ranges of the spectral projections of A + B and A + C associated to λ0 . In particular, if the dimension of the null space of Ln is constant and ≥ 2 and the sum of the dimensions of the null spaces of A + B and A + C is equal to 2, then the dimension of the null space of Ln is equal to 2; moreover, as n tends to infinity, a nonzero eigenvalue of Ln cannot accumulate at zero. Proof. First of all we notice that λ ∈ ρ(A + C) if and only if λ ∈ ρ(A + Cn ) and (A + Cn − λI )−1 = τn−1 (A + C − λI )−1 τn .
(3.1)
If λ ∈ ρ(A + B) ∩ ρ(A + C), λ < δ and n is large, we have to show that λ ∈ ρ(Ln ). From Assumption A.3, all we have to do is to show u = 0 is the unique solution of Au + Bu + Cn u + Dn u − λu = 0.
(3.2)
Therefore, assuming that u solves (3.2) we get u = (A + B − λI )−1 (−Cn u − Dn u)
(3.3)
u = (A + Cn − λI )−1 (−Bu − Dn u).
(3.4)
and
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If we replace the u following B in (3.4) by the value of u given by (3.3) we get: u = Wn u = (A + Cn − λI )−1 B(A + B − λI )−1 (Cn u + Dn u) −(A + Cn − λI )−1 Dn u.
(3.5)
Now we claim that |(A + C − λI )(A + Cn − λI )−1 | = (A + C − λI )τn−1 (A + C − λI )−1 τn | is uniformly bounded. In fact, we have Aτn−1 (A + C − λI )−1 = τn−1 A(A + C − λI )−1 and Cτn−1 (A + C − λI )−1 = CA−1 Aτn−1 (A + C − λI )−1 and then the claim follows from Assumption A.4. Using the fact that the norm of a bounded operator is equal to the norm of its adjoint we can write the following: |(A + Cn − λI )−1 B(A + B − λI )−1 Cn | = |Cn (A + B − λI )−1 B(A + Cn − λI )−1 | = |Cn (A + B − λI )−1 B(A + C − λI )−1 (A + C − λI )(A + Cn − λI )−1 |. Since Cn (A + B − λI )−1 = Cn A−1 A(A + B − λI )−1 converges to zero in the strong operator topology in view of Assumptions A.4 and A.6, and B(A+C −λI )−1 is compact in view of Assumption A.8, we see that |(A + Cn − λI )−1 B(A + B − λI )−1 Cn | tends to zero. Putting this information together with Assumption A.5, we conclude that the norm |Wn | of the operator Wn given by (3.5) tends to zero as n gets large. Then u = 0 for n large and this proves the first part of the theorem. To prove the second part, suppose that λ0 < δ, λ0 ∈ σ (A + B) ∪ σ (A + C) and > 0 is given. From the first part we know that there is N0 such that if n ≥ N0 and λ ∈C I is such that |λ − λ0 | = then λ ∈ ρ(Ln ). If we set (λI − Ln )−1 f = −u then u is the unique solution of Au + Bu + Cn u + Dn u − λu = f.
(3.6)
To find a more convenient form for the resolvent operator (λI − Ln )−1 f , we argue as in the first part in the following way: if u is the unique solution of (3.6) then u also solves u = (A + B − λI )−1 (−Cn u − Dn u + f )
(3.7)
u = (A + Cn − λI )−1 (−Bu − Dn u + f ).
(3.8)
and
If we replace the u following B in (3.8) by the value of u given by (3.7) we get: u = Wn u − (A + Cn − λI )−1 B(A + B − λI )−1 f + (A + Cn − λI )−1 f, (3.9) where Wn u is given by (3.5). As we have proved in the first part, for |λ − λ0 | = the norm |Wn | of Wn goes to zero as n goes to infinity and this implies that the unique solution of (3.9) is u = Yn (λ)f with Yn (λ)f = (I − Wn )−1 [−(A + Cn − λI )−1 B(A + B − λI )−1 f +(A + Cn − λI )−1 f ]
(3.10)
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and then (λI − Ln )−1 f = −Yn (λ)f . We denote by the circle centered at λ0 with radius and oriented counterclockwise. To calculate the spectral projection 1 1 −1 Tn = ˆ (λI − Ln ) dλ = − Yn (λ) dλ 2πi 2π i of Ln corresponding to the disk |λ − λ0 | = , we start by calculating the operator 1 Sn = [(A + Cn − λI )−1 − (A + Cn − λI )−1 B(A + B − λI )−1 ] dλ. (3.11) 2π i The first term of Sn is simply −Qn , Qn being the spectral projection of A+Cn associated to the eigenvalue λ0 . If we denote by Q the spectral projection of A + C corresponding to the eigenvalue λ0 , from (3.1) we have Qn = τn−1 Qτn ; in particular, the ranges of Qn and of Q have the same dimension. To calculate the second term in (3.11) we use the following result (see [12]): if U is a self-adjoint operator and λ0 is an isolated point of the spectrum σ (U ), then the function (λI − U )−1 has a simple pole at λ0 and (λI − U )−1 =
P + P ⊥ (λ0 I − U )−1 P ⊥ + V (λ), (λ − λ0 )
(3.12)
where P is the orthogonal spectral projection corresponding to λ0 , P ⊥ = I − P and V (λ) is analytic at λ = λ0 with V (λ0 ) = 0. Therefore, if we denote by P the spectral projection associated to the operator A + B at λ0 and Qn is as before we can write: Rn (λ) = ˆ (λI − A − Cn )−1 B(λI − A − B)−1 Qn =[ λ − λ0 −1 ⊥ +Q⊥ n (λ0 I − A − Cn ) Qn P +Vn (λ)]B + P ⊥ (λ0 I − A − B)−1 P ⊥ + V (λ) , λ − λ0 where Vn (λ) and V (λ) are analytic at λ = λ0 and vanish there. Then the residue of Rn (λ) at λ0 is equal to −1 ⊥ Qn BP ⊥ (λ0 I − A − B)−1 P ⊥ + Q⊥ n (λ0 I − A − Cn ) Qn BP .
(3.13)
Since P projects on the null space of A + B − λ0 I , we have BP = (A − λ0 I )P and then the second term of (3.13) can be written as −1 ⊥ ⊥ −1 ⊥ Q⊥ n (λ0 I − A − Cn ) Qn BP = Qn (λ0 I − A − Cn ) Qn (A − λ0 I )P −1 ⊥ = Q⊥ n (λ0 I − A − Cn ) Qn [(A + Cn − λ0 I ) − Cn ]P ⊥ −1 = Q⊥ n P − Qn (λ0 I − A − Cn ) Qn Cn P −1 = P − Qn P − Q ⊥ n (λ0 I − A − Cn ) Qn Cn P .
Moreover, if P and Q are expressed as Pu =
N i=1
u, φi φi ,
Qu =
M j =1
u, ψj ψj ,
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where φi and ψj are unity vectors, from Assumption A.7 we see that the norm of the operator Qτn P u = u, φi τn φi , ψj ψj i,j
goes to zero as n goes to ∞ and then the norm of Qn P = τn−1 Qτn P also goes to zero. According to Assumption A.6 the norm of Cn P goes to zero. The norm of Wn , as we have seen in the proof of the first part, also goes to zero. Therefore, we see that the norm |Tn − (−Sn )| goes to zero and putting all these things together we conclude that the norm of Tn − Qn − P tends to zero and this proves the second part of the theorem. The final assertions follow immediately from the first and second parts and the theorem is proved. 4. Local Minimizers for Constrained Variational Problems In this section we recall a result proved in [17] and [10] which gives the spectral condition for a critical point of a constrained variational problem to be a local minimizer. For simplicity we consider the case of two constraints only. Therefore, we consider three smooth functionals V1 (u), V2 (u) and V3 (u) in a Hilbert space X and we assume that for real numbers α and β in a certain range, u = u(α, β) is a smooth family of solution of the Euler-Lagrange equation V3 (u) + αV2 (u) + βV1 (u) = 0.
(4.1)
We consider also the second derivative V3 (u) + αV2 (u) + βV1 (u) of the augmented lagrangian V3 (u) + αV2 (u) + βV1 (u) and we calculate it at u = u(α, β). That second derivative is a continuous bilinear form in the Hilbert space X that can be represented by an (in general) unbounded self-adjoint operator L = L(α, β) with a certain domain. We assume that there is a δ > 0 such that the essential spectrum of L is contained in [δ, +∞) and that the family u(α, β) is nondegenerate in the sense that zero is not an eigenvalue of L. Considering also the real valued function V (α, β) = V3 (u(α, β)) + αV2 (u(α, β)) + βV1 (u(α, β)) we can state the following result: Theorem 5. The family of critical points u(α, β) is a local minimizer for V3 subject to V2 (u) = k2 and V1 (u) = k1 iff the number of negative eigenvalues of L = L(α, β) is equal to the number of positive eigenvalues of the hessian matrix V (α, β). To prove the orbital stability of double solitons, we have to show that a certain two dimensional manifold u(t, τ ) is a local minimizer for a certain constrained variational problem. In the notation u(t, τ ), t will be time and τ denotes translation in the space variable x. Then, instead of having a single operator L we have a family L(t, τ ) of operators. As it has been remarked in [17], the manifold u(t, τ ) is made of degenerate critical ∂u ∂u points because and are eigenfunctions of L associated to the zero eigenvalue. ∂t ∂x Since the definition of orbital stability takes those degeneracies in account, the condition for the critical points u(t, τ ) to be nondegenerate is that the kernel of L(t, τ ) has dimension two; in other words, the kernel of L(t, τ ) is spanned by the tangent vectors ∂u ∂u and to the orbit u(t, τ ). ∂t ∂τ
Orbital Stability of Double Solitons for the Benjamin-Ono Equation
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Due to the noncompactness of the orbit u(t, τ ), another question we have to worry about is to verify that a nonzero eigenvalue of L(t, τ ) cannot accumulate at zero as t goes to infinity. The fact that this cannot happen in our case will become clear from our arguments and it has already been considered in Theorem 4. 5. Stability of Double Solitons for the KdV Equation The stability of double solitons of the KdV equation ut + 6uux + uxxx = 0
(5.1)
has been proved in [17]. In this section, using the results we have proved in Sects. 2 and 3 and further arguments, we give an alternative proof for that result. In Sect. 6 we extend that method to the BO equation. The first three conserved quantities for (5.1) are 1 +∞ 2 V1 (u) = u dx, 2 −∞ +∞ 1 V2 (u) = (u2 − 2u3 ) dx, 2 −∞ x 1 +∞ 2 V3 (u) = (u − 10uu2x + 5u4 ) dx. 2 −∞ xx Defining ηi = −bi x + 4bi3 t for i = 1, 2 and A = (b2 − b1 )2 /(b1 + b2 )2 , the double soliton is given by u(t, x) = 2
d2 log(1 + e2η1 (t,x) + e2η2 (t,x) + Ae(2η1 (t,x)+2η2 (t,x)) ) dx 2
(5.2)
(see [11]) and the speeds are ci = 4bi2 . The traveling wave with speed c = 4b2 is √ c 1 uc = u(t, x) = 2b2 sech2 (b(x − 4b2 t)) = sech2 ( (x − ct)). (5.3) 2 2 As t tends to +∞, the double soliton splits in two separate one-solitons that are far apart in the following sense: if we define w(t, x) = u(t, x) − 2b12 sech2 b1 (x − b12 t) − 2b22 sech2 b2 (x − b22 t + 2 ), where 2 = log A is the phase shift, then lim w(t)k,p = 0
t→+∞
1≤p≤∞
k∈N
(5.4)
and lim un1 (t)um 2 (t)k,p = 0
t→+∞
1 ≤ p ≤ +∞ m, n ∈ N, m, n ≥ 1,
(5.5)
where |.|k,p denotes the norm in the Sobolev space Wk,p (R). The Euler-Lagrange equation V3 (u) + αV2 (u) + βV1 (u) = 0 is u(4) + 10uuxx + 5u2x + 10u3 + α(−uxx − 3u2 ) + βu = 0
(5.6)
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and the soliton (5.2) solves (5.6) if c 1 + c2 = α
and
c1 c2 = β;
(5.7)
(see [14]). In other words, the constants c1 and c2 are related to the multipliers through the quadratic equation c2 − αc + β = 0.
(5.8)
The linearized operator of (5.6) at the double soliton u(t, ·) is L(t)h = h(4) +10uhxx +10uxx h + 10ux hx + 30u2 h + α(−hxx − 6uh) + βh. (5.9) Definition 3. We say that the double soliton (5.2) is orbitally stable if defining the two dimensional set S = {u(t, · + τ ), t, τ ∈ R}, where u(t, ·) is the double soliton, then for any > 0, there is δ > 0 such that if u0 ∈ H 2 (IR) and d(u0 , S) < δ, then d(u(t, u0 ), S) < , where d is the distance taken in the H 2 (IR) norm and u(t, u0 ) is the solution of (5.1) such that u(0, u0 ) = u0 . This section is devoted to prove the following: Theorem 6. The double soliton (5.2) is orbitally stable. The proof of Theorem 6 will be done as follows: we write a chain of statements S.i, i = 0, . . . , 4 where S.0 is Theorem 6 and S.4 is Theorem 7 below. Using the theory that we have developed in the previous sections together with simple arguments, we show that each statement implies the previous one and at the end we prove Theorem 7. Since the Cauchy problem for (5.1) is well posed in the space H 2 (IR) and V1 , V2 and V3 are conserved quantities (see [13]), then, as in [17], Theorem 6 follows from: Statement S.1: The set S is made of local minimizers of V3 for given values of V2 and of V1 . Defining V (α, β) = V3 (u(t)) + αV2 (u(t)) + βV1 (u(t)), where u(t) is the double soliton, it has been proved in [17] that det (V (α, β)) < 0. Therefore the hessian matrix V (α, β) has one positive eigenvalue and one negative one. In view of Theorem 5 and the comments following it, we see that Statement S.1 is a consequence of Statement S.2: For every t ∈ IR, the operator L(t) defined by (5.9) has exactly one negative eigenvalue, it has zero as a double eigenvalue and there is no accumulation of the spectrum of L(t) at zero as t tends to infinity. As we have pointed out in the introduction, Statement 2 has been proved in [17] using ODE methods. We now present a different proof that can be extended to BO equation. The stationary limit operators obtained by replacing in (5.9) the double soliton u(t, x) by the one-solitons ui are: Li h = h(4) + 10ui hxx + 10ui,xx h +10ui,x hx + 30u2i h + α(−hxx − 6ui h) + βh.
(5.10)
The notation is such that c1 < c2 . According to Theorem 3, part ii, the inertia of L(t) is constant; moreover, from (5.4) and (5.5) we see that the assumptions of Theorem 4 are satisfied for any sequence tn that goes to infinity; therefore Statement S.2 follows from the following:
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Statement S.3: – L1 has zero as a simple eigenvalue and no negative eigenvalues; – L2 has zero as a simple eigenvalue and exactly one negative eigenvalue. In order to simplify the calculations, we normalize one speed taking c = 4; in this case, from (5.8) we get α = 4 + β4 . The profile of the traveling wave with speed c = 4 is 2sech2 x; from now on, this profile will be denoted by u = u(x) and we use capitals to denote the multiplication operator by the corresponding function. The operators L1 and L2 given by (5.10) calculated at this particular wave is L = D 4 + 10U D 2 + 10Uxx + 10Ux D + 30U 2 + α(−D 2 − 6U ) + β, (5.11) where D =
β d . If in this last equation we replace α by 4 + as above, we get dx 4 L = Lβ = Q +
β K, 4
(5.12)
where Qh = D 4 + (10U − 4)D 2 + 10Uxx + 10Ux D + (30U 2 − 24U )
(5.13)
K = −D 2 − 6U + 4I.
(5.14)
and
Notice that c = 4 is the lower speed if β < 16 and c = 4 is the higher speed if β > 16. Finally, using a simple scaling argument it is easy to show that Statement S.3 follows from Theorem 7. – For 0 < β = 16, zero is a simple eigenvalue eigenvalue of Lβ ; – for 0 < β < 16, Lβ has one negative eigenvalue and for β > 16, Lβ has no negative eigenvalue. We use the subscript odd to denote space of odd functions and the subscript ev to denote space of even functions. Notice that K and Q map even functions in even functions and odd functions in odd functions. We now state three lemmas from which the proof of Theorem 7 follows. Lemma 1. 1 (IR) we have Kh, h ≥ 0 and Kh, h = 0 if and only if h is a multiple 1. For h ∈ Hodd of v = u ; 1 the operator K has exactly one negative eigenvalue and zero is not an eigen2. in Hev value.
Lemma 2. For any h ∈ H 2 (IR) we have Qh, h ≥ 0 and Qh, h = 0 iff h is a multiple of u = v.
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Lemma 3. For β = 16 the function duc (x) 1 2 , (x tanh x − 1)sech x = 2 dc c=4 where uc (x) is the one-soliton (5.3), is an (even) eigenfunction of L16 associated to the zero eigenvalue. Remark 1. The spectral conditions given by Lemma 1 are precisely the conditions that are needed to prove the orbital stability of one-solitons. Admitting Lemmas 1, 2 and 3 we prove Theorem 7. 2 (IR). As we have Proof of Theorem 7. First we analyse the operator Lβ in the space Hodd pointed out, Qu = 0 = Ku . Moreover, from Lemmas 1 and 2 we see that, in the space 2 (IR), the operators Q and K are positive in the subspace of functions orthogonal Hodd to u ; hence, zero is an eigenvalue of Lβ associated to the eigenfunction v = u and Lβ 2 (IR), for any β > 0, is also positive in the subspace orthogonal to u . Therefore, in Hodd Lβ has zero as a simple eigenvalue with eigenfunction u and all other eigenvalues are positive. 2 (IR). If we divide L by β/4 the sign of the eigenvalues are Now we turn to Hev β preserved. Then we define γ = 4/β and the operator
L˜ γ = γ Q + K
(5.15)
2 (IR). For γ = 0, from and we denote by λ1 (γ ) the lowest eigenvalue of L˜ γ in Hev Lemma 1 we conclude that λ1 (γ ) is negative and all the other eigenvalues of L˜ 0 are 2 (IR), h = 0, from the variational characterpositive. Since Qh, h > 0 for h ∈ Hev ization of the eigenvalues for self-adjoint operators, we conclude that all eigenvalues of L˜ γ move strictly to the right as γ increases. Moreover, from Lemma 3 we see that λ1 (1/4) = 0; we conclude that λ1 (γ ) < 0 for γ < 1/4 and λ1 (γ ) > 0 for γ > 1/4 and this proves the theorem. Then, all is left is to prove Lemmas 1, 2 and 3. The proof of Lemma 3 follows by elementary calculation and the rest of this section will be devoted to prove Lemmas 1 and 2. We start defining two auxiliary linear operators that will play a crucial role in the proof of Lemma 1 and Lemma 2 and later in our method:
Mh = h (x) + 2 tanh xh(x),
M t h = −h (x) + 2 tanh xh(x).
(5.16)
We also define K0 = −D 2 + 4I.
(5.17)
Lemma 4. The following identity holds: MKM t = M t K0 M.
(5.18)
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The proof of (5.18) follows by expansion of both sides. Before using that identity to prove Lemma 1, we make a few comments. First let us recall that if Kh = −hxx + p(x)h and
K0 h = −hxx + p0 (x)h
(5.19)
are second order ordinary differential operators and the spectrum of K0 is known, then the spectrum of K can be calculated if an identity of type MK = K0 M
(5.20)
holds. In (5.20) M is an auxiliary operator whose kernel and range are known. This method has been used in [5] (see also [6], Sect. 2.4 for a presentation of it). Starting with an operator with constant coefficients and imposing (5.20) recursively, it can be shown that for the reflectionless potentials p(x) = −m(m + 1)sech2 x, where m is an integer, the spectrum of K given by (5.19) can be calculated explicitly. The auxiliary operator M in (5.20) is given by Mh = h (x) + n tanh x, where n is an integer related to m. Since in the definition (5.14) of the operator K u is given by u = 2sech2 (x), we see that the potential −6u in (5.14) is equal to −12sech2 (x). Then it is the reflectionless potential with m = 3 and we conclude that the spectrum of K can be calculated explicitly. In other words, (5.20) gives information which is much better than Lemma 1. However, as we have pointed out in the introduction, in the study of the stability of double solitons, we will have to make the spectral analysis of some higher order operators for which, at least apparently, there is no identity of type (5.20) but there is an identity of type (5.18). Therefore, we give the proof of Lemma 1 using identity (5.18) to illustrate how our method works. The first thing is to find the kernel and the range of the auxiliary operators M and M t . Lemma 5. If M, M t : D(M) = D(M t ) = H 1 (R) ⊂ L2 (R) → L2 (R) are given by (5.16) then i) the null space of M is spanned by u and M t is injective; ii) M is onto and the image of M t is the subspace orthogonal to u. Proof. The general solution of h (x) = −2 tanh(x)h(x) and
h (x) = 2 tanh(x)h(x)
are h(x) = Csech2 (x) and
h(x) = C cosh2 (x)
2 respectively, and this x proves part one. If h ∈ L (R) is given, it is easy to see that g(x) = sech2 (x) cosh2 (s)h(s) ds solves g (x) + 2 tanh(x)g(x) = h(x) and it 0
2 remains to show x that the operator (T h)(x) = g(x) maps L (R) boundedly into itself. Since | cosh2 (s) ds ≤ cosh(x) sinh(|x|) we see that T maps L∞ (R) boundedly 0
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into itself. Moreover, for b > 0 we have b x 2 2 sech (x) cosh (s)|h(s)| ds dx = 0
0
b
x
2
cosh (s)|h(s)| ds)(tanh(x) 0
= tanh(b)
= 0
b
cosh2 (x)|h(x) dx
0 b
−
dx
0
tanh(x) cosh2 (x)|h(x)| dx
0 b
(tanh(b) − tanh(x)) cosh2 (x)|h(x)| dx
b
≤
(1 − tanh(x)) cosh2 (x)|h(x)| dx.
0
Furthermore, from (1 − tanh(x)) cosh2 (x) = (1 + exp−x )/2 ≤ 1 for x ≥ 0, we get b x b sech2 (x) cosh2 (s)|h(s)| ds dx ≤ |h(x) dx. 0
0
0
The case b < 0 is treated in a similar way. This shows that T maps L1 (R) boundedly into itself and then, by interpolation, T maps L2 (R) boundedly into itself. Finally, if h belongs to the range of M t then it has to be orthogonal to the null space of +∞
Q, that is, h has to be orthogonal to u. Conversely, suppose that 0 and let us define g(x) = −
x
−∞
cosh2 x h(s) ds = cosh2 s
x
+∞
−∞
sech2 (x)h(x) dx =
cosh2 x h(s) ds. cosh2 s
An easy calculation shows that g(x) solve the equation −g (x) + 2 tanh(x)g(x) = h(x) and then, it remains to show that the operator S(h) = g maps L2 (R) boundedly into itself. x cosh2 x h(s) ds for x ≤ 0. Arguing exactly as above First we define (S1 h)(x) = − 2 −∞ cosh s we can show that S1 maps L∞ ((−∞, 0]) boundedly into itself and also L1 ((−∞, 0]) boundedly itself. For x ≥ 0 we get the same estimates for the operator (S2 h)(x) = +∞ cosh2 x h(s) ds and this proves the lemma. cosh2 s x Proof of Lemma 1. First we notice that M and M t map odd functions in even functions and even functions in odd functions. According to Lemma 5, any h ∈ H 2 (IR) can be 2 (IR). In this case, α = 0, written as h = αu + M t k. We consider first the case h ∈ Hodd k is even and using (5.18) we get: Kh, h = KM t k, M t k = MKM t k, k = M t K0 Mk, k = K0 M t k, M t k. Since K0 s, s ≥ 0 for any s ∈ H 2 (IR) and K0 s, s = 0 iff s = 0, we conclude that if 2 (IR), we have Kh, h ≥ 0 and Kh, h = 0 iff M t k = 0. Moreover, accordh ∈ Hodd ing to Lemma 5, M t k = 0 implies that k has to be a multiple of u and then h = M t k is also a multiple of u .
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Suppose now that h is even; then h = αu + M t k, with k odd. In the hyperplane α = 0 we have Kh, h = K0 Mk, Mk ≥ 0 and Kh, h = 0 iff Mk = 0 and, according to Lemma 5, this implies k = 0 because k is even. Therefore Kh, h > 0 in a hyperplane and this implies that K can have at most one nonpositive eigenvalue. Since Ku = −3u2 we see that Ku, u < 0 and this implies that K has exactly one negative eigenvalue in 2 (IR) and the lemma is proved. Hev It remains to deal with the more complicated higher order operator Q defined by (5.13). First we need a lemma whose proof follows expanding both sides of the identity. Lemma 6. Defining Q0 = D 4 − 4D 2
(5.21)
MQM t = M t Q0 M,
(5.22)
we have
where M and
Mt
are given (5.16)
Remark 2. From (5.18) and (5.22), we see that the auxiliary operator M that conjugates K with K0 is the same that conjugates Q with Q0 . Proof of Lemma 2. According to Lemma 5, any h ∈ H 2 (IR) we can be written as h = αu + M t k and then Qh, h = α 2 Qu, u + 2αMQu, k + MQM t k, k = α 2 Qu, u + 2αMQu, k + Q0 Mk, Mk. If we define s by k = αq + s and we denote by T = M t Q0 M the right-hand side of (5.22) we get Qh, h = α 2 (Qu, u − T q, q − 2MQu − T q, q) +2α(MQu − T q), s + T s, s. To eliminate the cross term of this quadratic form in α and s we have to choose q in such way that MQu = T q
(5.23)
and, in this case, Qh, h = α 2 (Qu, u − T q, q) + T s, s. 1 We can verify that q = xsech2 x satisfies (5.23) and then performing some calculation 3 we get: 36608 2 36608 2 α + T s, s = α + Q0 Ms, Ms. 315 315 Since Q0 y, y ≥ 0 and Q0 y, y = 0 iff y = 0, we conclude that Qh, h ≥ 0 and Qh, h = 0 iff α = 0 and Ms = 0. Moreover, according to Lemma 5, Ms = 0 implies that s has to be a multiple of u and then k = s is also a multiple of u. Hence, h = M t s is a multiple of u and this proves the lemma. Qh, h =
Remark 3. For the operator K, zero is an isolated eigenvalue; however, for the operator Q, zero belongs to its essential spectrum.
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6. Stability of Double Solitons for the BO Equation In this section we extend the method we used in Sect. 5 to prove the stability of double solitons of the Benjamin-Ono equation ∂u(t, x) + 4u(t, x)ux (t, x) + H (uxx (t, x)) = 0, ∂t
(6.1)
where H is the Hilbert transform. In the appendix we give the definition and some properties of the Hilbert transform that will be used. We also develop a calculus involving 1 . The proofs of some identities are the Hilbert transform and the function u = 1 + x2 also left to the appendix. The first three first integrals of (6.1) are: 1 +∞ 2 V1 (u) = u dx, 2 −∞ 1 +∞ 4 V2 (u) = (−uH ux − u3 ) dx, 2 −∞ 3 +∞ 1 V3 (u) = (u2 + 3u2 H ux + 2u4 ) dx. 2 −∞ x These three first integrals will be considered in the space H 1 (R) and they can be found in [20] (a change of scale has to be made because the coefficients of BO equation in [20] are different from ours). Formulae (6.2)–(6.7) can be found in [19]. The one-soliton with speed c > 0 for (6.1) is: u(t, x) =
c2 (x
c − ct)2 + 1
and its profile uc (x) =
c c2 x 2
(6.2)
+1
satisfies −H uc − 2u2c + cuc = 0.
(6.3)
The double soliton with speeds c1 > 0 and c2 > 0, c1 < c2 is u(t, x) =
c2 θ12 + c1 θ22 + (c1 + c2 )c12 , (θ1 θ2 − c12 )2 + (θ1 + θ2 )2
where θn = cn (x − cn t), n = 1, 2 and c12 = If we define
c1 + c 2 c1 − c 2
(6.4)
2 .
f = −θ1 θ2 + i(θ1 + θ2 ) + c12 ,
(6.5)
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then the double soliton (6.4) is also given by u(t, x) = and H u(t, x) = −
i ∂ f ∗ (t, x) ln 2 ∂x f (t, x)
(6.6)
1 ∂f (t, x) 1 ∂f ∗ (t, x) 1 + ∗ . 2 f (t, x) ∂x f (t, x) ∂x
(6.7)
The double soliton is a superposition of two one-solitons in the following sense: – if we define w(t, x) = u(t, x) − u1 (t, x) − u2 (t, x), where u1 and u2 are the onesolitons with speeds c1 and c2 , respectively, then lim w(t)k,p = 0
t→+∞
1 ≤ p ≤ +∞
k ∈ N,
(6.8)
where |.|k,p denotes the norm in the Sobolev space Wk,p (R) (as it has been remarked in [19], unlike the KdV equation, no phase shift appears as the result of collisons between solitons of BO equation); – lim un1 (t)um 2 (t)k,p = 0
t→+∞
1 ≤ p ≤ +∞
m, n ∈ N, m, n ≥ 1.
(6.9)
If α and β are constants the Euler-Lagrange equation V3 (u) + αV2 (u) + βV1 (u) = 0 is −uxx + 3uH ux + 3H (uux ) + 4u3 + α(−H ux − 2u2 ) + βu = 0.
(6.10)
As in the case of the KdV, we have to find the relationship between the speeds c1 and c2 and the multipliers α and β in such way that the double soliton (6.4) satisfies (6.10). Applying the operator H on both sides of Eq. (6.1) and taking in account that H 2 = −I we see that 1 ∂H u(t, x) uxx (t, x) − . H (uux ) = 4 ∂t Therefore, all terms in (6.10) can be calculated in terms of f given by (6.5) and performing the calculation we find that (6.10) is satisfied if c1 + c 2 =
4α 3
and
c 1 c2 =
4β . 3
(6.11)
In other words, for given multipliers α and β in a certain range, the speeds of the double soliton are the positive solutions of the quadratic equation c2 −
4β 4α c+ = 0. 3 3
(6.12)
The profiles of the corresponding one-solitons are: u1 (x) =
c1 c12 x 2 + 1
u2 (x) =
c2 . c22 x 2 + 1
(6.13)
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Definition 4. We say that the double soliton (6.4) is orbitally stable if defining the two dimensional set S = {u(t, · + τ ), t, τ ∈ R}, where u(t, ·) is the double soliton, then for any > 0 there is δ > 0 such that if u0 ∈ H 1 (IR) and d(u0 , S) < δ then d(u(t, u0 ), S) < , where d is the distance taken in the H 1 (IR) norm and u(t, u0 ) is the solution of (6.1) such that u(0, u0 ) = u0 . This section is devoted to prove the following: Theorem 8. The double soliton (6.4) is orbitally stable. The proof follows the same steps as in the case of the KdV equation. However, the proof of Theorem 9 below is much more diffficult than the proof of Theorem 7. The linearized operator for (6.10) at the double soliton is L(t)h = −hxx + 3hH ux (t, x) + 3u(t, x)H hx + 3H (ux (t, x)h) + 3u(t, x)hx + 12u(t, x)2 h + α(−H hx − 4u(t, x)h) + βh. (6.14) We denote by L1 and L2 the reduced operators obtained by replacing the double soliton u(t, .) in (6.14) by the one-solitons u1 and u2 given by (6.13). If we normalize one of the speeds to be equal to one then, in view of (6.12), we have 3 α=β+ . 4 Therefore, the reduced operator calculated at the one-soliton u = one is
(6.15) 1 with speed 1 + x2
Lβ = Q + βK,
(6.16)
3 Q = −D 2 + 6U 2 + 3U H D + 3H V + 3H U D − H D 4
(6.17)
K = −H D − 4U + I.
(6.18)
where
and
In (6.17) and (6.18), v = u = −2xu2 , the capitals U and V denote multiplication d . Moreover, c = 1 is the lower speed if β < 3/4 and it is the operators and D = dx higher speed if β > 3/4. To formulate the spectral condition for stability of the double soliton, according to Sect. 4, first we have to calculate V (α, β) = V (u(t, x, α, β)) = V3 (u(t, x, α, β)) + αV2 (u(t, x, α, β)) + βV1 (u(t, x, α, β)), where u(t, x, α, β) is the double soliton. Since V3 , V2 and V1 are first integrals, to calculate V (α, β) we can pass to the limit as t → +∞. Using estimates (6.8) and (6.9) we see that V (α, β) = V3 (u1 ) + V3 (u2 ) + α(V2 (u1 ) + V2 (u2 )) + β(V1 (u1 ) + V1 (u2 )),
Orbital Stability of Double Solitons for the Benjamin-Ono Equation
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c is the profile of where u1 and u2 are given by (6.13). If, as before, uc (x) = 2 2 c x +1 +∞ 1 , we have I1 = π , a simple wave with speed c and we define In = (1 + x 2 )n −∞ I2 = π/2, I3 = 3π/8, I4 = 15π/48 and +∞ +∞ +∞ u2c (x) dx = cI2 , u3c (x) dx = c2 I3 , u4c (x) dx = c3 I4 . −∞
−∞
−∞
Morever, since uc satisfies (6.3) we have +∞ +∞ 3 uc (x)H uc (x) dx = −2 uc (x) dx + c −∞
and
+∞
−∞
−∞
u2c (x)H uc (x) dx = −2
Furthermore +∞ uc (x)2 (x) dx = 4c6 −∞
+∞ −∞
= 4c3
+∞
−∞
u4c (x) dx + c
−∞
u2c (x) dx
−∞
+∞ −∞
x2 dx = 4c3 2 2 (c x + 1)4
+∞
+∞
u3c (x) dx.
+∞ −∞
(y 2
y2 dy + 1)4
+ 1) − 1 dy = 4c3 (I3 − I4 ). (y 2 + 1)4
(y 2
Collecting all calculations we get V3 (uc ) = c3 (−2I4 + 3I3 ), V2 (uc ) = c2 (−2I3 + 2I2 ), V1 (uc ) = cI2 /2.
(6.19)
From (6.11) we also get 16 2 8 α − β, 9 3 64 16 c13 + c23 = (c1 + c2 )(c12 − c1 c2 + c22 ) = α 3 − αβ, 27 3
c12 + c22 = (c1 + c2 )2 − 2c1 c2 =
(6.20)
and, finally, using (6.19), we find that V (α, β) = (c13 + c23 )(−2I4 + 3I3 ) + α(c12 + c22 )(2I3 + 2I2 ) + β(c1 + c2 )I2 /2 π = (116α 3 − 189αβ). 27 The determinant of the hessian matrix V (α, β) is independent of α and β and it is equal to −49π 2 < 0; this implies that V (α, β) has exactly one positive and one negative eigenvalue. The well-posedness of the Cauchy problem for (6.1) in the space H 1 (IR) has been proved in [23]. From the comments above we see that the proof of Theorem 8 goes exactly as in the case of the KdV equation and the only thing that we have to prove is the counterpart of Theorem 7 which is the following:
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A. Neves, O. Lopes
Theorem 9. 1. For 0 < β = 3/4 the operator Lβ defined by (6.16) has zero as a simple eigenvalue; 2. for 0 < β < 3/4, Lβ has one negative eigenvalue and for 3/4 < β the operator Lβ has no negative eigenvalue. We use the subscript odd to denote space of odd functions and the subscript ev to denote space of even functions. Notice that K and Q map even functions in even functions and odd functions in odd functions. As in the case of the KdV equation, the proof of Theorem 9 is a consequence of the following three lemmas: Lemma 7. 1 (IR) we have Kh, h ≥ 0 and Kh, h = 0 if and only if h is a multiple 1. For h ∈ Hodd of v = u ; 1 K has exactly one negative eigenvalue and zero is not an eigenvalue. 2. in Hev
Lemma 8. For any h ∈ H 1 (IR) we have Qh, h ≥ 0 and Qh, h = 0 iff h is a multiple of u = v. Lemma 9. For β = 3/4 the function −u + 2u2 =
duc (x) , dc c=1
where uc is given by (6.2), is an (even) eigenfunction of L3/4 associated to the zero eigenvalue. Remark 4. As in the case of the KdV equation, the spectral conditions given by Lemma 7 are precisely the conditions that are needed to prove the orbital stability of one-solitons for the BO equation. The proof of Theorem 9 admitting Lemmas 7, 8 and 9, follows exactly as the proof of Theorem 7 for the KdV equation and we will not repeat it. To prove Lemma 9 we simply replace −u + 2u2 in L3/4 and we make the calculation according to formulae given in the appendix. The rest of this section will be dedicated to the proofs of Lemmas 7 and 8. As we will see, the proofs of them are much more difficult than the proofs of Lemmas 1 and 2. We start analysing the operator K given by (6.18) which is precisely the linearized operator for the one-soliton. The one-soliton with speed c = 1 is u=
1 1 + x2
(6.21)
and it solves −HDu − 2u2 + u = 0.
(6.22)
The linearized operator for (6.22) is the operator K given by (6.18). Remark 5. In [3] a resolution of the identity for the operator K has been found; in particular, the spectrum of K has been calculated explicitly and Lemma 7 actually follows from that. However, as in the case of the KdV equation, we give a proof for Lemma 7 because the method will be used to prove the much more complicated Lemma 8.
Orbital Stability of Double Solitons for the Benjamin-Ono Equation
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To extend the method we have used for the KdV equation, we have to find identities similar to (5.18). We start defining the following functions: a = a(x) = x 2 − 1;
b = b(x) = −2x;
v = v(x) = u (x) = −2xu2 ;
u = u(x) =
1 ; 1 + x2
w = w(x) = u − 2u2 ;
and we denote by A, B, U, V and W the operators defined by multiplication by a, b, u, v and w, respectively. Notice that au = 1 − 2u. We also define the following bounded operators from L2 (IR) into itself: M = AUH + BU
N = −AUH + BU,
(6.23)
and their adjoint M t = −HAU + BU
N t = HAU + BU.
(6.24)
Lemma 10. Setting K0 = −H D + I,
(6.25)
MKM t = N K0 N t .
(6.26)
then the following identity holds
The proof of Lemma 10 will be given in the appendix. Remark 6. If we look at identity (5.18), instead of (6.26) it would be more natural to look for an identity of the type: MKM t = M t K0 M
(6.27)
2x . To eliminate 1 − x2 the singularity we multiply both sides of (6.27) on the right and on the left by 1 − x 2 . We also compose the resulting equation with U to get bounded auxiliary operators and the result is identity (6.26). (that is the way we started). Formally, (6.27) holds with M = H +
To use identity (6.26) to prove Lemma 7, we need to calculate the image and the null space of the auxiliary operators appearing there. Notice that if h is an even (odd) function, then M(h), N (h), M t (h), N t (h) are odd (even). Denoting by Ker(T ) and I m(T ) the null space and the image of a bounded operator T then the following is true: Lemma 11. If M and N are considered from L2 (IR) into itself then 1. Ker(M) = [w, xw]; 2. Ker(N t ) = [u, xu]; 3. I m(M t ) = [w, xw]⊥ ; 4. I m(N ) = [u, xu]⊥ ; 5. MN = I ; 6. N t M t = I. In particular, M t and N are one-to-one and M and N t are onto. 1 (IR) the set The proof of Lemma 11 will be given in the appendix. We denote by Hev 1 1 of the elements of H (IR) which are even and by Hodd (IR) the odd ones.
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Proof of Lemma 7. First we notice that K0 s, s ≥ 0 and K0 s, s = 0 iff s = 0 because −H s (ξ ) = |ξ |ˆs (ξ ) (see the appendix.) If h is odd then, according to Lemma 11, Part 3, h can be decomposed as h = αxw + M t k, where k is even and then, using (6.26) we get Kh, h = α 2 K(xw), xw + 2αMK(xw), k + MKM t k, k πα 2 = + N K0 N t k, k 4 because MK(xw) = M(xw) = 0 and K(xw), xw = π/4. Therefore, Kh, h ≥ 0 and Kh, h = 0 iff α = 0 and N t k = 0 and this implies k = cu (Lemma 11, Part 2). Hence, h = cM t u = −4cxu2 = 2cu and this proves the first part of Lemma 7. To prove the second, we first notice that Ku, u < 0 because K(u) = −2u2 . This implies that K has at least one negative eigenvalue in the space of the even functions. Moreover, if h is even then, according to Lemma 11, Part 3, we can make the decomposition h = αw + M t k, k odd. Taking α = 0 and using identity (6.26) we see that Kh, h = N K0 N t k, k ≥ 0 and Kh, h = 0 only if N t k = 0 and this implies k = cxu (Lemma 11, Part 2). We conclude that Kh, h ≥ 0 for h belonging to the codimension one subspace I m(M t )ev and then K cannot have two negative eigenvalues. 1 (IR). By contradiction, It remains to prove that zero is not an eigenvalue of K in Hev suppose K has one negative eigenvalue with eigenfunction φ1 and one zero eigenvalue with eigenfunction φ2 . In this case, the codimension one subspace I m(M t )ev has to intercept the subspace spanned by φ1 and φ2 ; since Kh, h ≤ 0 in S = [φ1 , φ2 ] and Kh, h = 0 for h ∈ S only if h is a multiple of φ2 we see that I m(M t )ev has to intercept S in [φ2 ]. However, as before, Kh, h = 0 for h ∈ I m(M t )ev only if h is a multiple of M t (xu) = 4(u2 − u); since the function u2 − u is not an eigenfunction of K associated to the zero eigenvalue we conclude that K cannot have a zero eigenvalue in the space of the even functions and Lemma 7 is proved. Now we concentrate on the proof of Lemma 8 which will be broken in several lemmas. 1 (IR) and As in the case of the KdV equation, the spectrum of Q considered in both Hodd 1 (IR) accumulate at zero (from the right.) Arguing as in the case of the KdV equaHev tion, we start trying to find an identity of type (5.22) involving the operator Q under consideration and a “better” operator Q0 . In view of identities (5.18), (5.22) and (6.26), perhaps we should expect that the following identity holds 3 MQM t = N (−D 2 − H D)N t , 4 because −D 2 − 43 H D is the part of Q that has constant coefficients. Regretfully, this identity is false! The next attempt is to find coefficients k1 , k2 and k3 in such way that 3 Q0 = −D 2 + k1 U + k2 U 2 + k3 U H D + k3 H V + k3 H U D − H D 4 is “better” than Q and MQM t = N Q0 N t .
(6.28)
We show how to accomplish that. According to formulae presented in the appendix, for any integer m the operators H (um h) − um H h
and
H (xum h) − xum H h
Orbital Stability of Double Solitons for the Benjamin-Ono Equation
781
have finite dimensional range. Therefore, modulo a finite dimensional range operator, H commutes with U m and xU m . With this fact in mind and expanding both sides of (6.28), we see that the infinite dimensional part of (6.28) is satisfied if k1 = 3, k2 = −2 and k3 = −1. Using these coefficients to define the operator 3 Q0 = −D 2 + 3U − 2U 2 − U H D − H V − H U D − H D 4 we start with
(6.29)
Lemma 12. The following identity holds: 3 Q0 = (H D − U )2 + 3U (1 − U ) − H D. 4
(6.30)
In particular Q0 h, h ≥ 0 and Q0 h, h = 0 iff h = 0. Proof. The verification of (6.30) is trivial. The final statement follows from 0 < u(x) ≤ 1 and u(x) = 1 only for x = 0 and the lemma is proved. Since Q0 is positive and we want to prove that Q is positive, some progress has been made. However, using Q0 defined by (6.29) to calculate MQM t − N Q0 N t we see that this difference is a nonzero operator with finite dimensional range. In other words, the infinite dimensional part of both sides of (6.28) are equal but the difference of them contains a residue (an operator with finite dimensional range). We state this as Lemma 13. If M, N, Q and Q0 are as above then MQM t = N Q0 N t + R,
(6.31)
where Rk =
1 9u − 12u2 , ku + −12u + 16u2 , ku2 π
+xu + 8xu2 , kxu + 8xu − 16xu2 , kxu2 .
(6.32) (6.33)
The proof of Lemma 13 will be given in the appendix. For reasons that will be given later, identity (6.31) still is not convenient. To prove Lemma 8 we will use variants of (6.31) which are obtained perturbing M by an operator with finite dimensional range; in other words, we will use identities like M1 QM1t = N Q0 N t + R1 .
(6.34)
Suppose we have proved an identity of type (6.34) in such way that for any h ∈ H 1 (IR), there is a decomposition h = αφ + M1t k where φ spans ker(M1 ). Then Qh, h = α 2 Qφ, φ + 2αQφ, M1t k + QM1t k, M1t k = α 2 Qφ, φ + 2αM1 Qφ, k + M1 QM1t k, k. Defining s = k + αq, where q will be found, and denoting by T the right-hand side of (6.34) we have Qh, h = α 2 (Qφ, φ − T q, q − 2M1 Qφ − T q, q) +2α(M1 Qφ − T q), s + T s, s.
(6.35)
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A. Neves, O. Lopes
We look at this last equality as a quadratic form in α and s and to eliminate the cross term we have to choose φ and q in such way that M1 Qφ = T q.
(6.36)
Qh, h = α 2 (Qφ, φ − T q, q) + T s, s.
(6.37)
In this case,
Since we want to show that Q is positive, the solution q of (6.36) has to be known explicitly because we need to know that the coefficient of α 2 in (6.37) is positive. Also, we have to choose certain parameters appearing in the definition of M1 in such way that we can conclude that the operator T given by the right side of (6.34) is positive. We analyse the case of even and odd functions separately. As we will see, the case of even functions is much more difficult. We start with the case of odd functions. The reason for (6.31) not to be convenient is that, apparently, Eq. (6.36) cannot be solved explicitly in q for φ in the Kernel of M. Our next attempt is to perturb M in order to incorporate the residue R in the perturbed operator. To be more specific we have: Lemma 14. Defining M1 (h) = M(h) + m5 xu, hu + m6 xu, hu2
(6.38)
M1t (k) = M t (k) + m5 u, kxu + m6 u2 , kxu
(6.39)
for h odd and for k even, with m5 = 12/π and m6 = −16/π, and M and M t given by (6.23) and 2 (IR) we have (6.24), then for k ∈ Hev M1 QM1t (k) = N Q0 N t (k).
(6.40)
The proof of Lemma 14 will be given in the appendix. Notice that M1 (as M and N ) maps even functions in odd functions and odd functions in even functions. To carry out the procedure above we need the following proposition whose proof is also left to the appendix: Lemma 15. If φ = −5xu2 + 8xu3 then Ker(M1 ) = [φ] and I m(M1t ) = [φ]⊥ , where ker(M1 ) is taken in the space of odd functions and I m(M1t ) is calculated for M1t as a map from even functions into odd functions. 1 (IR) then, according to Lemma 15, we have Proof of Lemma 8 for h odd. If h ∈ Hodd t the decomposition h = αφ + M1 k with k even. If we define s = k + αq then Qh, h is given by (6.35), where T = N Q0 N t (the right-hand side of (6.40)). Moreover, taking q = −2u2 we have M1 Qφ = T q and then, according to (6.37) and performing some calculation, we get
9πα 2 + N Q0 N t s, s. 64 Since Q0 is positive definite by Lemma 12, we conclude that Qh, h ≥ 0 and Qh, h = 0 iff α = 0 and N t s = 0 = N t k. In this case, according to Lemma 11, Part 2, k has to be a multiple of u and this implies that h has to be a multiple of M1t (u) = 2u and Lemma 8 is proved for odd functions. Now we turn to the more complicated case of even functions. We start with the following identity whose proof is left to the appendix: Qh, h =
Orbital Stability of Double Solitons for the Benjamin-Ono Equation
783
Lemma 16. Defining M2 h = Mh +
32 2 u , hxu2 π
(6.41)
M2t k = M t k +
32 xu2 , ku2 π
(6.42)
for h even and
for k odd, where M and M t are given by (6.23) and (6.24), then for k odd we have M2 QM2t (k) = N Q0 N t (k) + R2 (k),
(6.43)
where R2 (k) = c11 xu, kxu + c12 xu2 , kxu + c12 xu, kxu2 + c22 xu2 , kxu2
(6.44)
with c11 =
1 π
c12 = −
28 π
c22 =
400 . π
(6.45)
Remark 7. If instead of (6.41) we define M2 (h) = M(h) + m1 u, hxu + m2 u, hxu2 + m3 u2 , hxu + m4 u2 , hxu2 , where the parameters m4 and m2 are given by m4 = −4m3 −
32 π
m2 = −4m1 +
32 π
and m1 and m3 belong to the (nonempty) ellipse 13π 2 m21 + 19π 2 m1 m3 + 7π 2 m23 − 48πm1 − 36π m3 + 16 = 0, then (6.43) holds with no residue (R2 = 0). However, in that case, KerM2 has dimension two and, apparently, the equation M2 Qφ = T q = N Q0 N t q cannot be explicitly solved in q for two linearly independent functions φ belonging to 1 (IR) Ker(M2 ). Probably this is related to the fact that the infimum of Qh, h in Hev (which is zero) is not achieved (at a nonzero element). Besides the choice given by 32 (6.41), which corresponds to m1 = m2 = m3 = 0, m4 = , there may be others that π may work (with different nonzero residue R2 ). Lemma 17. If T = N Q0 N t + R2 is the right-hand side of (6.43) and s is odd then T s, s ≥ 0 and T s, s = 0 iff s = 0.
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Proof. We define N t s = p and then s = αxu + M t p (Lemma 11, Parts 4 and 6) and xu, s = xu, αxu + M t p = αxu, xu + M(xu), p =
απ − u, p 2
and xu2 , s = xu2 , αxu + M t p = αxu2 , xu2 + M(xu2 ), p =
απ 1 − u, p. 8 2
Therefore, R2 s, s =
π2 π (16c11 + 8c12 + c22 )α 2 + (−8c11 − 6c12 − c22 )u, pα 64 8 1 + (4c11 + 4c12 + c22 )u, p2 . (6.46) 4
Since 16c11 + 8c12 + c22 = 192/π , the coefficient of α 2 in (6.46) is positive and then the 5 minimum of (6.46) considered as a quadratic function of α is achieved at α = 2 u, p π 2 2 and the value of this minimum is − u, p2 . Then R2 s, s ≥ − u, p2 and this π π implies that T s, s ≥ Q0 p, p −
2 u, p2 π
(6.47)
5 and equality holds only for α = 2 u, p. Hence, Lemma 17 is a consequence of the π following: Lemma 18. Consider the operator T0 (p) = Q0 p −
2 u, pu π
acting on even functions. Then T0 p, p ≥ 0 and T0 p, p = 0 if and only if p = 0. Remark 8. According to Lemma 12, the operator Q0 is positive but it is easy to see that zero belongs to its essential spectrum. Therefore, the positivity of Q0 is very sensitive 2 to negative perturbation in any direction. The constant is optimal for Lemma 18 to be π true. Proof of Lemma 18. First let us notice that for any two functions p, z in the space H 1 (IR) we have Q0 z, p2 ≤ Q0 p, pQ0 z, z
(6.48)
and equality holds if and only if p and z are linearly dependent. This follows from Schwarz inequality because according to Lemma 12, the bilinear form [p, z] = Q0 p, z is a scalar product. Formally, Lemma 18 follows from (6.48) taking z ≡ 1 because Q0 (1) = 2u and u, 1 = π. Since the function z ≡ 1 does not belong to the space H 1 (IR), that procedure has to be justified. This will be done next.
Orbital Stability of Double Solitons for the Benjamin-Ono Equation
785
Let φ : IR → IR be an even function belonging to H 2 (IR) such that φ(0) = 1 and x let us define φλ (x) = φ( ). Then H φ ∈ H 2 (IR) and (H φλ )(x) = (H φ)(x/λ); in λ particular H φ ∈ L∞ . Taking z = φλ in (6.48) we get Q0 φλ , p2 ≤ Q0 , pQ0 φλ , φλ .
(6.49)
We interrupt the proof of Lemma 18 to analyse the limit of (6.49) as λ tends to infinity. Lemma 19. As λ tends to infinity 1. Q0 φλ , p tends to 2u, p;
3 3 2. Q0 φλ , φλ tends to 2u, 1 − H Dφ, φ = 2π − H Dφ, φ. 4 4 Proof. The first limit is easier because it is linear in φλ and so we will prove only the second. A change of variables shows that: 1
|φλ |L2 = λ− 2 |φ|L2 , |φλ |L2
=λ
− 23
(6.50)
|φ|L2 .
(6.51)
We expand Q0 φλ , φλ and first we collect the terms that tend to zero: 1 |φλ , φλ | = |φλ , φλ | |φ|2L2 ; λ |uH φλ , φλ | = |φλ , H (uφλ )| ≤ |φλ |L2 |φλ |L∞ |u|L2 = |φλ |L2 |φ|L∞ |u|L2 ; |H (uφλ ), φλ | = |uφλ , H φλ | ≤ |uφλ |L1 |H φλ |L∞ ≤ |u|L2 |φλ |L2 ||H φλ |L∞ = λ−1/2 |u|L2 |φ|L2 |H φ|L∞ . The next terms:
+∞
−∞
and
and
+∞
−∞
−∞
(3u − 2u
2
)φλ2 dx
H (vφλ )φλ dx =
+∞
−∞
+∞
+∞ −∞
→
+∞ −∞
(3u − 2u2 ) dx
H (vφλ )(φλ − 1) dx +
(Lebesgue)
+∞ −∞
H (v)φλ dx
H (vφλ )(φλ − 1) dx ≤ |(v(φλ − 1)|L1 |H φλ |L∞ = |(v(φλ − 1)|L1 |H φ|L∞ → 0(Lebesgue)
H (v)φλ dx →
Furthermore, −H φλ , φλ =
+∞
−∞
and the lemma is proved.
+∞ −∞
H (v) dx =
φλ (H φλ ) dx =
+∞
−∞
+∞
−∞
(u − 2u2 ) dx
(Lebesgue).
φ (H φ) dx = −H φ , φ dx
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We see that the term −H Dφλ , φλ is self similar with respect to dilations and then a further limit process will be required. End of proof of Lemma 18. According to Lemma 19, for any even function φ ∈ H 2 (IR) such that φ(0) = 1 and any h ∈ H 1 (IR) the following inequality holds: 4u, p2 ≤ Q0 p, p(2π − H Dφ, φ).
(6.52)
To get rid of the term H Dφ, φ we claim the following: there is a sequence of even functions φn ∈ H 2 (IR) such that φn (0) = 1 and H Dφn , φn tends to zero. In fact, denoting by fn (ξ ) = φˆn (ξ ) the Fourier transform of φ and using the inverse Fourier transform we have +∞ −1 φn (x) = (2π) eixξ fn (ξ ) dξ −∞
and then φn (0) = (2π)−1
+∞ −∞
fn (ξ ) dξ.
Therefore, it is sufficient to show that there is a sequence fn satisfying the following conditions: 1. fn is real, even, bounded and has support in the interval [−1, 1]; +∞ fn (ξ ) dξ = 2π ; 2. −∞ +∞ 3. |ξ |fn2 (ξ ) dξ tends to zero as n tends to infinity. 0
Let fn be the sequence of even functions defined in the following way for ξ ≥ 0 : cn 1 for ≤ ξ ≤ 1, ξ n fn (ξ ) = 0 for ξ > 1.
fn (ξ ) = Imposing
+∞
−∞
fn (ξ ) dξ = 2π we get cn =
π and then log n
+∞
|ξ |f 2 (ξ ) dξ =
0
π2 and this proves the claim. 4 log n Replacing φ by φn in (6.52) and passing to the limit in n we get T0 p, p ≥ 0 and this proves the first part of Lemma 18. 1 (IR) we have T p , p = 0. In this case Suppose now that for some 0 = p0 ∈ Hev 0 0 0 2 we must have T0 p0 = Q0 p0 − u, p0 u = 0 because T0 p, p has a minimum at π p = p0 . If we had u, p0 = 0 then Q0 p0 = 0 and this would imply p0 = 0 (Lemma 12). Therefore, multiplying p0 by some constant, we may assume that Q0 p0 = 2u. Since Q0 is injective and formally Q0 (1) = 2u, we try to show that Q0 p0 = 2u implies p0 = 1 a.e. In order to justify this conclusion we proceed in the following way: from Q0 p0 = 2u we get p0 , Q0 φλ = 2u, φλ and from Lemma 19 we also get 2p0 , u = 2u, 1. Moreover, as λ tends to infinity Q0 (p0 − φλ ), (p0 − φλ ) = Q0 p0 , s0 − 2Q0 p0 , φλ + Q0 φλ , φλ
Orbital Stability of Double Solitons for the Benjamin-Ono Equation
787
3 tends to − H Dφ, φ. Furthermore, from (6.30) we conclude that for any finite real 4 number b we have b 3u(1 − u)(p0 − φλ )2 ≤ Q0 (p0 − φλ ), (p0 − φλ ), −b
and then, passing to the limit in λ, we get b 3 3u(1 − u)(p0 − 1)2 ≤ − H Dφ, φ. 4 −b Replacing φ by the sequence φn as above and passing to the limit in n, we conclude that p0 = 1 a.e. and this is a contradiction because p0 ∈ H 1 (IR) and Lemma 18 is proved. The next lemma, whose proof will be given in the appendix, provides a useful decomposition of even functions. 9 3 u − u2 + u3 , then Ker(M2 ) = [φ] and I m(M2t ) = [φ]⊥ , where 16 8 Ker(M2 ) is calculated on even functions and I m(M2t ) is calculated for M2t as a map from odd functions into even functions.
Lemma 20. If φ =
1 (IR), according to Lemma 20, we have Proof of Lemma 8 for h even. For any h ∈ Hev the decomposition h = αφ + M2t k. If we set s = k + αq, where q will be chosen, we get
Qh, h = α 2 (Qφ, φ − T q, q − 2M2 Qφ − T q, q) +2α(M2 Qφ − T q), s + T s, s, 11 1 where T is the right-hand side of (6.43). If we take q = − xu + xu2 we have 64 4 M2 Qφ = T q and then Qh, h = α 2 (Qφ, φ − T q, q) + T s, s =
9 2 α + T s, s, 4096
and the conclusion follows from Lemma 17 and Lemma 8 is proved.
7. Appendix In this section we recall some properties of the Hilbert transform and we prove some lemmas involving it. Following [3], we define the Hilbert transform by: 1 ∞ f (x − y) 1 ∞ f (x + y) (Hf )(x) = P V dy = P V dy, (7.1) π −∞ y π −∞ y where PV stands for principal value. In some classical books on Harmonic Analysis ([22], for instance) the definition of Hilbert transform is (7.1) with a minus sign in front of it. Some properties of Hilbert transform. (ξ ) = isign(ξ )fˆ(ξ ); ([22]); 1. Hf 2. H is a bounded operator from L2 (IR) into itself (follows from 1); 3. H 2 = −I, H ∗ = −H and −H Df (ξ ) = |ξ |fˆ(ξ ) (follow from 1);
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4. H commutes with translation; in particular, H D = DH. Some formulae. Let us recall the definition and some identities involving functions previously defined: a = x 2 − 1; x 2 u = 1 − 2u;
1 ; au = 1 − 2u; 1 + x2 v = ux = −2xu2 ; w = u − 2u2 = au2 .
b = −2x;
u=
Theorem 10. For a dense set of functions in L2 (IR) the following identities hold: 1 1, h, π 1 H (x 2 h) = x 2 H (h) + [x, h + 1, hx], π 1 H (uh) = uH (h) + [−xu, hu − u, hxu], π 1 H (ah) = aH (h) + [x, h + 1, hx], π 1 H (auh) = auH (h) + [2xu, hu + 2u, hxu], π 1 2 2 H (u h) = u H (h) + [−xu2 , hu − xu, hu2 π −u2 , hxu − u, hxu2 ], 1 H (xuh) = xuH (h) + [u, hu − xu, hxu], π 1 2 2 H (xu h) = xu H (h) + [−u + u2 , hu + u, hu2 π −xu2 , hxu − xu, hxu2 ], H (xhx ) = xH (hx ), 1 H (x 2 hx ) = x 2 H (hx ) − 1, h, π 1 H (uhx ) = uH (hx ) + [−u + 2u2 , hu − 2xu2 , hxu], π 1 H (xuhx ) = xuH (hx ) + [2xu2 , hu + −u + 2u2 , hxu], π H (xhxx ) = xH (hxx ), H (x 2 hxx ) = x 2 H (hxx ). H (xh) = xH (h) +
(7.2) (7.3) (7.4) (7.5) (7.6)
(7.7) (7.8)
(7.9) (7.10) (7.11) (7.12) (7.13) (7.14) (7.15)
Proof. Equation (7.2) follows from (7.1) and (7.3) follows by iteration of (7.2). From (7.3) we get H ((1 + x 2 )h) = (1 + x 2 )H (h) +
1 [x, h + 1, hx]; π
if in this last identity we replace h by uh and divide the result by 1 + x 2 we get (7.4). All the other formulae follow from the first three and the theorem is proved.
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The Hilbert transform of special functions. 1 H (u2 ) = −xu2 − xu, 2 1 1 2 3 2 2 3 3 H (xu ) = u − u H (u ) = −xu − xu − xu. 2 2 8 The first formula is proved in [3] and the rest follows from formulae given in Theorem 10. Now we prove some lemmas stated in section 6. H (u) = −xu
H (xu) = u
Proof of Lemma 11. Proof. Using the Hilbert transform of the special functions given above it is easy to see that [w, xw] ⊂ Ker(M). On the other hand, if h ∈ Ker(M), we have auH (h) − 2xuh = 0,
(7.16)
and then applying H to both sides of (7.16) and using the relations (7.6) and (7.8) we get −auh − 2xuH (h) = αu + βxu,
(7.17)
where α and β are constants. Since −2xu au det = 1, −au −2xu if we look at (7.16) and (7.17) as a system in h and H (h) and solve it in h, we get h = au(αu + βxu) = αw + βxw and this proves Part 1. From the definitions of the operators M and N t it follows that M(auh) = auN t (h). Therefore, Part 2 follows from Part 1. We always have I m(M t ) ⊂ Ker(M)⊥ and then we have only to prove that [w, xw]⊥ = [au2 , xau2 ]⊥ ⊂ I m(M t ). If h ∈ [au2 , xau2 ]⊥ then, using the Hilbert transform of the special functions u, xu and the formula (7.6) with h replaced by H (auh), it is easy to see that H (auH (auh)) = −a 2 u2 h. From the formula (7.8) with auh instead h we obtain H (xau2 h) = H (xu auh) = xuH (auh). Hence, taking f = N t (h) = H (auh) − 2xuh, we have M t (f ) = −H (auH (auh) − 2xau2 h) − 2xu(H (auh) − 2xuh) = (a 2 u2 + 4x 2 u2 )h = h, and this proves Part 3. If h ∈ [u, xu]⊥ , one can verify that N (M(h)) = h and this proves Part 4. Part 5 follows from the relations (7.6) and (7.8) and Lemma 11 is proved. The proofs of Lemmas 15 and 20 follow in a similar way.
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Proof of Lemma 10. Using that H 2 = −I and H D = DH , we first expand both sides of (6.26): (AU H + BU )(−H D − 4U + I )(−H AU + BU )(h) = −auH (ax uh) − auH (auhx ) + abx u2 h + abu2 hx + 4auH (uH (auh)) −4auH (bu2 h) + a 2 u2 h + auH (buh) − bax u2 h − bau2 hx − buH (bx uh) −buH (buhx ) + 4bu2 H (auh) − 4b2 u3 h − buH (auh) + b2 u2 h (7.18) and (−AU H + BU )(−H D + I )(H AU + BU )(h) = −auH (ax uh) − auH (auhx ) − abx u2 h − abu2 hx + a 2 u2 h − auH (buh) +bax u2 h + bau2 hx − buH (bx uh) − buH (buhx ) + buH (auh) + b2 u2 h. (7.19) Making the difference (7.18) - (7.19) we get 2abx u2 h − 2ax bu2 h + 4auH (uH (auh)) − 4auH (bu2 h) + 2auH (buh) +4bu2 H (auh) − 4b2 u3 h − 2buH (auh). (7.20) Using the relations (7.2)-(7.15), the expression (7.20) can be rewritten in the form (2abx u2 − 2ax bu2 − 4a 2 u3 − 4b2 u3 )h +(−4au3 b + 2abu2 + 4au3 b − 2abu2 )H (h) + R, where R contains all the finite dimensional terms. If we replace the values of a, b and u in this last expression we see that the infinite dimensional part (the coefficients of h and H h) are zero as well as the finite dimensional part R and the lemma is proved. The proofs of the identities given by Lemmas 13, 14 and 16 involve much longer calculations but, similarly to the proof of Lemma 10, they follow by expansion of both sides and using formulae given by Theorem 10. References 1. Albert, J. Bona, J., Henry, D.: Sufficient conditions for stability of solitary-wave solutions of model equation for long waves. Physica D 24, 343–366 (1987) 2. Benjamin, T.: The stability of solitary waves. Proc. R. Soc. London A 328, 153–183 (1972) 3. Bennett, D., Brown, R., Stansfield, S., Stroughair, J., Bona, J.: The stability of internal waves. Math. Proc. Camb. Phil. Soc. 94, 351–379 (1983) 4. Bona, J.: On the stability of solitary waves. Proc. R. Soc. London A 344, 363–374 (1975) 5. Deift, P., Trubowitz, E.: Inverse scattering on the line. Commum. Pure Appl. Math. 32, 121–151 (1979) 6. Dodd, R. et al.: Solitons and Nonlinear Wave Equations. London-New york: Academic Press, 1982 7. Fokas, A., Ablowitz, M.: The inverse scattering transform for the Benjamin-Ono equation: A pivot to multidimensional problems. Stud. Appl. Math. 68, 1–10 (1983) 8. Fokas, A., Fuchssteiner, B.: The hierarchy of the Benjamin-Ono equation. Phys. Lett. 86A, 341–345 (1981) 9. Golub, G., van Loan, C.: Matrix Computations. Baltimore, MD-London: The John Hopkins University Press, Second Edition, 1989 10. Grillakis, M., Shatah, J., Strauss, W.: Stability Theory of Solitary Waves in the Presence of Symmetry, Part I. J. Funct Anal 74(1), 160–197, (1987) part II, 94(2), 308–348 (1990) 11. Hirota, R.: Direct methods in soliton theory. In: Solitons, R.K., Bullough, P.J. Caudrey, (eds), New-York: Springer-Verlag, 1980, pp. 157–176
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12. Kato, T.: Perturbation theory for linear operators. Berlin: Springer-Verlag, 1995 (reprint of the 1980 edition) 13. Kenig, C., Ponce, G., Vega, L.: Well-posedness and scattering results for generalized Korteweg-deVries equations via contraction principle. Commum. Pure Apll. Math. 46, 527–620 (1993) 14. Lax, P.: Integrals of nonlinear equations of evolution and solitary waves. Commum. Pure Appl. Math. 21, 467–490 (1968) 15. Lax, P.: Periodic solutions of the KdV equation. Commum. Pure Appl. Math. 28, 141–188 (1975) 16. Lopes, O.: A class of isoinertial one parameter families of self-adjoint operators. In: Nonlinear Equations: Methods, Models and Applications, Progress in Nonlinear Differential Equations and Their Applications, Lupo, D., et al., Basel-Boston: Birkhauser, 2003 17. Maddocks, J., Sachs, R.: On the stability of KdV multi-solitons. Commum. Pure. Appl. Math. 46, 867–902 (1993) 18. Martel, Y., Merle, F.: Stability and Asymptotic stability in the Energy space of the sum of N solitons for subcritical gKdV Equations. Commun. Math. Phys. 231, 347–373 (2002) 19. Matsuno,Y.: Interaction of the Benjamin-Ono solitons. J. Phys. A, Math. Gen. 13, 1519–1536 (1980) 20. Satsuma, J., Ablowitz, M., Kodama, Y.: On an internal wave equation describing a stratified fluid with finite depth. Phys Lett 73A(4), 283–286 (1979) 21. Scharf, G., Wreszinski, W.: Stability for the Korteweg-de Vries Equation by inverse scattering theory. Ann. Phys. 134, 56–75 (1981) 22. Stein, E., Weiss, G.: Introduction to Fourier analysis on Euclidian spaces. Princeton, NJ: Princeton University Press, 1971 23. Tao, T.: Global well-posedness of the Benjamin-Ono equation in H 1 (IR). J. Hyperbolic Diff. Eq 1(1), 27–49 (2004) Communicated by P. Constantin