Commun. Math. Phys. 198, 1 – 45 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Renormalization and Localization Expansions. II. Expectation Values of the “Fluctuation” Measures Tadeusz Balaban Rutgers University, Department of Mathematics, 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019, USA Received: 19 December 1997 / Accepted: 12 February 1998
Abstract: This paper is a continuation of [5]. The k+1−st renormalization transformation applied to a density ρk is studied further, the new contributions to the effective action obtained in [5] are renormalized, and their localization expansions are constructed. This completes proofs of Theorems 1 and 4 in [5]. 1. Introduction In the previous paper [5] we have given an inductive description of the space of densities generated by k renormalization transformations, and we have started the analysis of the k + 1 − st transformation S (k) T (k) applied to a density ρk from the space. A result of this analysis is given in the final formula (3.18) of that paper. This formula describes the “new” density S (k) T (k) ρk in terms of the “old” density, the new integral operators T0(k) , and the new contributions to the effective action described in Propositions 3.1–3.4 [5]. The new density has a general form required by the inductive hypotheses (H.1)–(H.7) [5], but it does not yet have all the properties required by them, in particular the new contributions are not renormalized. In this paper we continue the analysis of [5] and we complete the proofs of Theorems 1 and 4. Let us recall the simplified formulations of those theorems. The first describes the renormalization transformation S (k) T (k) as a ¯ They are defined by (1.42) mapping between corresponding spaces Rk (β, a, λ, ν; B, ν). [5] as spaces of densities ρk satisfying the inductive hypotheses (H.1)–(H.7) [5], which include also the hypotheses (H.1)–(H.7) [4]. The theorem can be formulated in the following way. Theorem 1. Under the assumptions of Theorem 1 [4], i.e. under the appropriate as¯ in particular for sumptions on the constants determining the space Rk (β, a, λ, ν; B, ν), ν¯ ≤ βk−1 ≤ 78 L−2 , which implies k ≤ k0 , the transformation S (k) T (k) maps this space ¯ 2 ), but with into a space defined in the same way as the space Rk+1 (β, a, λ, ν; BLd−2 , νL the following changes. For the hypothesis (H.6) the sharp inequality “>” in (1.32) [5]
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T. Balaban
is replaced by the weaker inequality “≥”. For the new contributions in the exponentials, i.e. the contributions with the superscript j = k+1, the hypotheses (H.2)–(H.4), (H.7) are satisfied with an improved constant 23 κ instead of κ. The transformation S (k) T (k) determines uniquely the transformation S (k) T (k) defined on the extended spaces of effective actions (1.6) [4]. It satisfies all the conclusions of Theorem 2 [4], i.e. it establishes the mapping (1.7) [4] between the appropriate extended spaces, which satisfies Eqs. (1.8), (1.9) [4] with ck+1 = 0, dk+1 = bk+1 , and the inequalities (1.10) [4]. Let us recall that we apply the above theorem in the case k < k0 , and all the background configurations in the inductive description are determined by the variational problems without the external field term, that is we put νk = 0 in them. We discuss the cases k ≥ k0 below. The results of the previous paper [5] described in the formula (3.18) [5] go quite a long way towards establishing the above theorem. The density S (k) T (k) ρk has a general form required by the inductive hypotheses, and satisfies most of them. The only missing important properties are the renormalization conditions in (H.2) [5] for the new contributions to the effective action corresponding to j = k + 1. The renormalization of the new contributions has been described in [4], and we use here all the constructions and the results of that paper, with proper modifications and additions. We need also to c , and small field domains determined by the introduce the final small field regions Zk+1 characteristic functions χk+1 . All these issues are discussed in the Sect. 2. This section completes the construction of the small field densities which satisfy the conditions in Theorem 1, in particular it yields the construction of the transformation S (k) T (k) satisfying these conditions. It remains to analyze the resulting large field parts ρ0k of the new effective densities, and this is done in Sect. 3. We obtain there in a simple and natural way inductive definitions of the mult-indices Ak+1 , the integral operators T(k) , Tk+1 , the functions κk+1 , etc., and we prove that the densities ρ0k satisfy all the conditions in the inductive hypotheses, with the possible exception of the inequality (1.32) [5] connected with (H.6) [5]. We obtain instead the weaker inequality, and this completes the proof of Theorem 1. For k < k0 we need to improve the inequality and to obtain the full hypothesis (H.6). This leads to the large field renormalization operation, which will be constructed in the next paper. Once we reach the density ρk0 satisfying all the inductive hypotheses we do not need anymore to improve the weaker inequalities obtained in Theorem 1, and replacing (1.32) [5], the hypothesis (H.6) for ρk0 is enough to control all the remaining steps. We define the remaining densities by (1.44) [5], i.e. by ρk = S (k−1) T (k−1) ρk−1 =
k0 Y
S (j) T (j) ρk0 ,
(1.1)
j=k−1
) where νk0 > 78 L−d βk−1 . Thus we do not starting with ρk0 ∈ Rk0 (β, a, λ, ν; B, βk−1 0 0 perform the large field renormalization operations anymore, and we include the external field term in the variational problems determining the background configurations in the representations of the sequence of the effective densities ρk , starting with k = k0 + 1. Properties of these densities are described in the following theorem. Theorem 4. The densities ρk given by (1.1) satisfy all the inductive hypotheses (H.1)– (H.5), (H.7) [5], and the integral operators Tk (Z, Ak ∩ Z) satisfy the inequality (1.31) [5] with ` = 1, as long as νk ≤ 1.
Renormalization and Localization Expansions
3
A proof of this theorem follows from the proof of Theorem 1, only one additional bound is needed here, and it is given also in Sect. 3. These proofs are based on Propositions 3.1–3.4 [5], and also on Propositions 2.1–2.3 in Sect. 2. These are statements on the existence of localization expansions of various expressions appearing in the course of constructions, expansions satisfying appropriate conditions, and they form an important technical part of any renormalization group method. We construct these expansions in the last section, and the construction is based on the results of the earlier paper [3], where the expansions of local functions of the background configurations were constructed. Here we have to construct expansions of expectation values with respect to the “fluctuation” measures described in Proposition 3.1–3.4 [5], and this is done by a form of a “cluster expansion”. Actually the expansions are constructed in an elementary way, and the cluster expansion of a form discussed in [8, 10] is used only for bounds. Let us remark that each of the renormalization group methods applied until now, for example the ones in [8, 10–12], has its own restrictions on the way the corresponding localizations are constructed. The method applied here has probably the strongest restrictions, because of the non-linear nature of the leading term −βk Ak in the effective action, in particular it demands a complete localization in the sense that a term localized in some domain X depends on other expressions, including covariances, background configurations, etc., all localized in subdomains of X. This is stressed quite explicitly in formulations of the above propositions. The constructions of Sect. 4 cover also expansions which we will need in the next paper on the large field renormalization.
2. The Renormalization of the Effective Actions The renormalization is based on the crucial property stressed several times in the last ˜ ψ) section, namely that localizations of the effective actions E00(k+1) (Zkc , k+1 , 00k+1 ; θ, to subdomains of 00k+1 are equal to localizations to the same subdomains of the effective action E00(k+1) (θ) constructed on the whole torus. Notice that the torus is one of the possible subdomains 00k+1 . Thus we have the functions E00(k+1) (θ), Ek (ψu(k) (θ)), βk+1,u Ak+1,u (θ, φk+1,u (θ)) defined on the whole torus, and we apply the renormalization procedure of Sects. 4,5 [4]. These functions determine the renormalization constants ζk+1 , bk+1 , ck+1 , dk+1 , ek+1 through Eqs. (5.5), (5.14) [4]. Let us recall that in Sect. 5 [4] we have mentioned several simple forms of the supplementary renormalization equations. Now it is convenient to choose the one discussed after the inductive hypotheses in Sect. 1 of [5], see in particular the formulas (1.40), (1.41) [5]. Let us recall also that we have to take the “thermodynamic” limits of the pre-localized functions, the limits as the torus goes to the whole space. For the renormalization constants we have the results formulated in Proposition 5.1 [4], in particular the inequalities (5.31) [4], which are important for future bounds. The constants in turn determine the new renormalized coefficients βk+1 , ak+1 , λk+1 , νk+1 through Eqs. (4.1) [4]. Having the coefficients we can construct renormalized background functions for various generating sets. Let us recall that the first equation in (4.1) [4] is θ = zk+1 ψk+1 , so now ψk+1 is the new spin variable on the lattice T1(k+1) . The renormalization is performed on the actions restricted to a small field domain c , but at first we have to determine this domain. Among the renormalized background Zk+1 functions we have the ones determined by the generating sets Bk+1 (0∼2 ), 0 ∈ πk+1 , and we define the functions χk+1 (0 ) by the formula (1.19) [5] with k replaced by k + 1.
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T. Balaban
∼3 c We denote for simplicity 00∼−3 = ((00c k ) ) , and we introduce the decomposition of k+1 unity X X c (χk+1 (0 ) + χck+1 (0 )) = χck+1 (Sk+1 )χk+1 (Sk+1 ∩ 00∼−3 1= k+1 ) 0 ⊂00∼−3 k+1
Sk+1 ⊂00∼−3 k+1
(2.1) before the exponential in (3.18) [5]. A term in the last decomposition is determined by a new large field region Sk+1 , the last one in the procedure. We have the following simple modification of Lemma 2.1: χ0k+1 (0 )χk+1 (0 ) = χk+1 (0 ).
(2.2)
From this it follows that in the corresponding term in (3.18) [5] the only functions ∼3 ∪ Sk+1 . On the χ0k+1 (0 ) left on the domain 00k+1 are the ones with 0 ⊂ (00c k+1 ) 00∼−3 complement of the last domain restricted to k+1 there are the functions χk+1 (0 ) only. They give the final restrictions on the new variable ψk+1 , in agreement with the c , inductive hypotheses. Thus we can define the last small field domain Zk+1 [ c 0 ∼4 ∼ = {00 ∈ πk+1 : distk+1 (00 , (00c ∪ Sk+1 ) > 2M Rk + M }. (2.3) Zk+1 k+1 ) We can write it again as ∼ L−1 2 c ∼4 ∼ ∼2[Rk ]+2+ L−1 2 )c Zk+1 = (((00c ) ∪ S ) , hence, k+1 k+1 ∼4 ∼ ∪ Sk+1 (00c k+1 )
∼2[Rk ]+2
By the definition the domain c ) (Zk+1
so on the domain denote as before
∼ Wk+1
∼4 ∼ ⊂ Zk+1 ⊂ (00c ∪ Sk+1 k+1 )
(2.4)
∼2[Rk ]+2+ L−1 2
.
(2.5)
c Zk+1
≈2 ∼
has the property that ∼3 ∩ (00c ∪ Sk+1 = ∅, k+1 )
(2.6) 0
there are only the small field functions χk+1 ( ), where we c ≈ )≈2 , Uk+1 = Zk+1 . Wk+1 = (Zk+1
(2.7)
c Having finally the small field domain Zk+1
we can decompose the new contributions c , and boundary in the exponential in (3.18) [5] into expressions localized properly in Zk+1 0(k+1) terms. Let us start with the function E0 . We take the localization expansions discussed in Proposition 3.4 [5], and we define E0(k+1) (z, Y ; ψk+1 ) = E00(k+1) (z, Y, (Zkc , k+1 , 00k+1 ); zk+1 ψk+1 ) for Y ⊂ 00k+1 , X X c E0(k+1) (Zk+1 ; ψk+1 ) = E (k+1) (z, Y ; ψk+1 ).
(2.8) (2.9)
c ∩T (k+1) Y ∈Dk+1 :z∈Y ⊂(Z c )≈ z∈Zk+1 k+1 1
By the crucial property discussed at the beginning of this section the function (2.9) is c of the function E0(k+1) (ψk+1 ) = E00(k+1) (zk+1 ψk+1 ) the localization to the domain Zk+1 defined on the whole torus. The remaining part of the localization expansion of ˜ ψ) can be written in the form (3.3) [5], after resumming over E00(k+1) (Zkc , k+1 , 00k+1 ; θ, points z, with domains Y ∈ Dk+1 (modck+1 ) satisfying the condition Y ∩ Zk+1 6= ∅. Terms of this sum depend on Zk+1 because of the resummation, and we add the sum to the boundary terms B 00(k+1) . Let us change the notations in agreement with the notations
Renormalization and Localization Expansions
5
used in the inductive hypotheses. We denote again by ψ the spin configuration defined on Bk+1 and such that ψ 3j = ψj , j = 0, 1, . . . , k, k + 1. The fluctuation variable on 0 B(3k+1 ) ∩ 00c k+1 , denoted before by ψ, we denote now by ψk . The obtained boundary 00 00(k+1) 0 terms are denoted by B (Ak , k+1 , k+1 , Zk+1 ; ψ, ψk ). Of course they depend also on h, g and the previous fluctuation variables ψj0 . We apply the definitions and decompositions as above to the remaining expressions in (3.18) [5], with some obvious modifications. For F00(k+1) we use the representation (2.52) [4] in terms of M0(k+1) , and we write D E ˜ ψk0 ) = χZ c ∩Z g, M0(k+1) (Zkc , k+1 , 00k+1 ; θ, ˜ ψk0 ) F00(k+1) (Zkc , k+1 , 00k+1 ; θ, k+1 0 k 1 D E 0(k+1) c 00 0 ˜ c g, M + χZk+1 (Zk , k+1 , k+1 ; θ, ψk ) . 0 1 (2.10) The localization expansion of the first term on the right-hand side is added to the boundary terms B 00(k+1) . The localization expansion of the second term is divided into two parts. The first part is the sum over localization domains which are not contained c )≈ , and this part is added to B 00(k+1) also. The second part is the remaining in (Zk+1 c )≈ . For terms of this sum we use sum over localization domains contained in (Zk+1 again the crucial property that they are corresponding localizations of the function F0(k+1) (ψk+1 , g) = F00(k+1) (zk+1 ψk+1 g) defined on the whole torus. This sum is the restricc tion of the function F0(k+1) (ψk+1 , g) to the domain Zk+1 , in agreement with the definition c (1.12) [5] of the hypothesis (H.3) [5], therefore it is denoted by F0(k+1) (Zk+1 ; ψk+1 , g). 0(k+1) Similarly for the function Rn , n = 1, . . . , m, we divide the localization expansion of Rn0(k+1) into two parts. The first part is the sum over localization domains which intersect Zk+1 , and this sum is added to B 00(k+1) . The second part is the sum over domains which c . Terms of this sum are corresponding localizations of the funcare contained in Zk+1 (k+1) tion Rn (ψk+1 ) = Rn0(k+1) (zk+1 ψk+1 ) defined on the whole torus, and we denote the c sum by R(k+1) (Zk+1 ; ψk+1 ), in agreement with the definition (1.17) [5] of the hypothesis n (H.4) [5]. This is the final definition of these functions for n < m, but not for n = m. There will be another important contribution coming from the large filed renormalization operation. For this reason we still keep the “prime” in the notation, i.e. we have 0(k+1) c (Zk+1 ; ψk+1 ). To finish the discussion of these functions let us recall that their Rm localization expansions satisfy the bounds discussed in Propositions 3.2, 3.3 [5], with the constants const. βk−n−1 . According to the inductive hypothesis (H.4) [5] we have to −n . We have replace them by βk+1 n const.βk−n−1 βk+1
= const.
βk+1 βk
n
βk−1 = const.(γk+1 Ld−2 )n βk−1
≤ const.Lm(d−1) βk−1 = const.M d−1 βk−1 ≤
1 for βk large enough, 2
so we obtain certainly the required bounds, with the additional factor 21 we will use yet in the case n = m. After the above decompositions and definitions we obtain the expressions in the exponentials which have the same form as in (3.18) [5], but the functions E00(k+1) , c c c F00(k+1) , Rn0(k+1) there are replaced by E0(k+1) (Zk+1 ), F0(k+1) (Zk+1 ), R(k+1) (Zk+1 ) for n < n
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0(k+1) c m, Rm (Zk+1 ). Of course the boundary terms B 00(k+1) are changed, with all the new contributions added, but this is reflected only in bounds, which we will discuss later on. Consider now the first three terms in the exponential in (3.18) [5]. We would like c and boundary terms. to decompose them into a sum of terms properly localized in Zk+1 We start with the main action, and at first we write the decomposition
A∗k+1,u (Zkc , Bk+1 ; θ, φ0k+1,u (θ˜0 )) = A∗k+1,u (Zkc ∩ Zk+1 , Bk+1 ; θ, φ0k+1,u (θ˜0 )) c + A∗k+1,u (Zk+1 ; θ, φ0k+1,u (θ˜0 )).
(2.11)
We have changed slightly the notation here, we denote φ0k+1,u (θ˜0 ) = φ(B0k+1 ; θ˜0 ), where B0k+1 = (Bk (Wk ) ∩ ck+1 ) ∪ 3k+1 . We consider the second term on the right-hand side above, and we apply the expansion ˜ φ0k+1,u (θ˜0 ) = φ(Bk+1 (Wk+1 ); ψ(Bk+1 (Wk+1 ), B0k+1 ; θ˜0 )) = φ(Bk+1 (Wk+1 ); θ) (2.12) ˜ = φk+1,u (θ) ˜ + δφk+1,u (δψ 0 ). + δφ(Bk+1 (Wk+1 ); ψ(Bk+1 (Wk+1 ), B0k+1 ; θ˜0 ) − θ) The configuration δψ 0 inside the function δφk+1,u is equal to 0 on the domain Wk+1 , and ˜ c (Wk∼ ∩ Bk+1 ; δ, ε) we have |δψ 0 | < K4 δ otherwise small, for example on the space 9 by Proposition 2.3 [3] and the representation (4.21) [2], for suitable δ, ε satisfying the assumptions of Propositions 2.1–2.3 [3]. Then 1 c )∼ , |δφ0k+1,u (δψ 0 )| < K3 K4 δ exp(− γ0 (2M Rk − M − 2M1 )) < exp(−Rk ) on (Zk+1 4 (2.13) and the same bounds for derivative, Laplace operator, etc. of this function. We apply now the formula (2.23) [5] and we obtain c c ˜ ; θ, φ0k+1,u (θ˜0 )) = A∗k+1,u (Zk+1 ; θ, φk+1,u (θ)) A∗k+1,u (Zk+1
∗ c ˜ − h c + δAk+1,u (Zk+1 ; 0, δφk+1,u ) + νk+1,u δφk+1,u , φk+1,u (θ) Z k+1 1 ˜ (δφk+1,u )+ + (δφk+1,u )− , ∂φk+1,u (θ) . + 2 st(Z c )
(2.14)
k+1
Of course the second term on the right-hand side is missing if the external field is included in the variational problem. The bound (2.13) shows that the three terms above are very small, even after multiplying by βk+1,u = βk Ld−2 . We construct their localization expansions using Propositions 2.1–2.3, 4.3 [3], after preliminary pre-localizations in cubes of πk+1 . Terms of these expansions corresponding to localization domains not c are equal to 0, thus we include the expansion into the boundary terms intersecting Wk+1 00(k+1) B . These are small contributions, the bounds hold with constants smaller than any power of βk−1 . We consider similar expressions, but more complicated and containing more terms, later on in more detail, in connection with the renormalization of the main action on the right-hand side of (2.14). We refer the reader to that discussion, if the above statements are too sketchy. We construct a similar expansion for the first term on the right-hand side of (2.11). We define the generating set Bk+1 (∂k+1 ) = (Bk (Wk ) ∩ ck+1 ) ∪ (Bk+1 (Uk+1 ) ∩ k+1 ),
(2.15)
˜ = φk+1,u (∂k+1 ; θ). ˜ We want to and the corresponding function φ(Bk+1 (∂k+1 ); θ) expand the first term around this function. We have
Renormalization and Localization Expansions
7
˜ φ0k+1,u (θ˜0 ) = φ(Bk+1 (∂k+1 ); ψ(Bk+1 (∂k+1 ), B0k+1 ; θ˜0 )) = φ(Bk+1 (∂k+1 ); θ) 0 0 ˜ = φk+1,u (∂k+1 ; θ) ˜ (2.16) + δφ(Bk+1 (∂k+1 ); ψ(Bk+1 (∂k+1 ), Bk+1 ; θ˜ ) − θ) + δφk+1,u (∂k+1 ); δψ 00 ),
and the configuration δψ 00 inside the function δφk+1,u is equal to 0 on the domain Wk ∩ Uk+1 , hence this function satisfies the bounds (2.13) on the domain Zkc ∩ Zk+1 . We apply again the formula (2.23) [5] and we obtain an equality analogous to (2.14), with obvious changes. The last three terms in it have localization expansions with similar properties to the above ones. We include them again into the boundary terms, although this time they contribute to both terms, B 0(k) and B 00(k+1) because some localization domains may be contained in Wk∼ ∩ ck+1 . Thus we have transformed the main action into the following sum ˜ βk+1,u A∗k+1,u (Zkc ∩ Zk+1 , Bk+1 ; θ, φk+1,u (∂k+1 ; θ)) c ˜ + βk+1,u A∗k+1,u (Zk+1 ; θ, φk+1,u (θ)),
(2.17)
with the new contributions to the boundary terms. Now we consider the “old” effective action in (3.18) [5], which is now written as Ek (Zkc ; ψu0(k) (θ˜0 )). It is the sum of terms described in (H.2) [5], and we decompose it into two parts, analogously to (2.11). To the first part we include the terms either with localization points y ∈ Zkc ∩Zk+1 and all possible localization domains, i.e. domains X ⊂ c , but with localization domains intersecting (Zkc )≈ , or with localization points y ∈ Zk+1 c )≈ )c . This includes also the corresponding “counterterms” in (1.8) [5], with the ((Zk+1 condition on the localization domains replaced by the condition on the points x ∈ c )≈ . We denote this part by Ek0 (Zkc ∩ Zk+1 ; ψ 0(k) (θ˜0 )). To the second part (Zkc )≈ \(Zk+1 c ; ψu0(k) (θ˜0 )), we include remaining terms, and it is easy to see that it is equal to Ek (Zk+1 c c th as defined in (H.2) [5], for the domain Zk+1 instead of Zk . j term of this function 0(j) depends on the background configuration ψk+1,u (θ˜0 ), and we would like to replace it by the configuration localized in Bk+1 (Wk+1 ). Using the same notation as in (2.12) we have on the domain Wk+1 , 0(j) (θ˜0 ) = ψ(Bj (Wk ), B0k+1 ; θ˜0 ) ψk+1,u
= ψ(Bj (Wk+1 ), Bk+1 (Wk+1 ); ψ(Bk+1 (Wk+1 ), B0k+1 ; θ˜0 )) ˜ = ψ(Bj (Wk+1 ), Bk+1 (Wk+1 ); θ) + δψ(Bj (Wk+1 ), Bk+1 (Wk+1 ); ψ(Bk+1 (Wk+1 ), B0k+1 ; θ˜0 ) (j) ˜ + δψ (j) (δψ 0 ), (θ) = ψk+1,u k+1,u
(2.18) ˜ − θ)
where the configuration δψ 0 is the same as in (2.12), so it is equal to 0 on Wk+1 ∩ T1(k+1) . (j) A localization expansion of the function δψk+1,u (δψ 0 ) has all terms corresponding to localization domains contained in Wk+1 equal to 0. Using the above decompositions we 0(j) (j) c ˜ up to the expand the actions E (j) (Zk+1 ; ψk+1,u (θ˜0 )) around the configurations ψk+1,u (θ) first order. Combining the previous decomposition with this expansion we obtain c ˜ + Ek0 (Zkc ∩ Zk+1 ; ψu0(k) (θ˜0 )) Ek (Zkc ; ψu0(k) (θ˜0 )) = Ek (Zk+1 ; ψu(k) (θ)) k Z 1 X d (j) c ˜ + δψ (j) (tδψ 0 )). dt E (j) (Zk+1 ; ψk+1,u (θ) + k+1,u dt 0 j=1
(2.19)
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T. Balaban
The second term and the sum on the right-hand side contribute to the boundary terms only, because of their localization properties. We construct localization expansions of these expressions in a more direct way than in the previous section, using the procedure of the proof of Proposition 4.1 [3] only, but there are also some additional problems now. A localization expansion of the second term on the right-hand side above has terms with localization domains contained in Wk∼ ∩ck+1 . We would like to obtain their exponential decay bounds in terms of the function dk . A part of a decay bound comes from the bounds of the functions E (j) (y, X), which are expressed in terms of dj (X), and dj (X) “pseudo-scales” with a factor ≥ L for j < k, but not for j = k. Fortunately for j = k and X ⊂ ck+1 these functions do not depend on ψu0(k) (θ˜0 ), they are already localized in X, so they go unchanged into the localization expansions, and we still obtain the right bounds. Thus we construct the localization expansions using Bk (Wk ) and the η-scale as the basis of the construction for terms with j < k, and then we perform proper resummations. For j = k we treat the whole domain (Zkc )≈ as a part of k+1 , for example we take 0k+1 = (((Zkc )≈ )0 )∼ , where ((Zkc )≈ )0 is a union of all cubes from πk+1 intersecting (Zkc )≈ , and we take the generating set (Bk (Wk ) ∩ 0k+1 ) ∪ 30k+1 , 30k+1 = (0k+1 )(k+1) . We construct the expansions using this generating set and the L−1 η-scale as the basis, and we perform resummations again. In this case we obtain terms of the expansions with localization domains belonging to Dk+1 (modck+1 ) only. These remarks should justify the following proposition. Proposition 2.1. The sum of the second term and the sum over j on the right-hand side of (2.19) has a localization expansion of the form X X∈Dk :X⊂Wk∼ ∩ck+1
+
E 00(k) (X, Zkc ; ψk )
X
E 00(k+1) (Y, Zkc , k+1 , Zk+1 ; θ),
(2.20)
Y ∈Dk+1 (modck+1 ):Y ∩Zk+1 6=∅
with the terms having the properties described in (H.7) [5], and satisfying the bounds there with const. E0 exp(−(κ − 1)dk (X)) on the right-hand sides for the terms of the first sum, and with const. E0 exp(−(2κ − κ0 − 1)dk+1 (Y modck+1 )) on the right-hand sides for the terms of the second sum. Let us remark again that the constant in the bounds can be written explicitly in terms of the previously introduced constants, actually it is almost the same as the constant in the bound (3.131) [1]. We include the expansion (2.20) into the boundary terms, the first sum is included into B 0(k) , and actually into the part of it with localization domains disjoint with ck , which is important for proving the inductive hypotheses, the second sum is included into B 00(k+1) . Consider finally the “old” generating functional Fk (Zkc ; ψu0(k) (θ˜0 ), g) in (3.18) [5]. We decompose it in exactly the same way as in (2.19), using the linearity property in the factor χZkc g as in (2.10), and we construct localization expansions of the corresponding terms in the decomposition. We obtain Proposition 2.2. The terms of the decomposition of the generating functional, the decomposition corresponding to (2.19), have localization expansions of the form (2.20), with E replaced by F, with an additional dependence on g, and the same properties and bounds, but with the constant E0 replaced by 2.
Renormalization and Localization Expansions
9
We include the two sums in the expansion into B 0(k) , B 00(k+1) correspondingly. After all these preparatory operations we have obtained the expression in the exponential in (3.18) [5] transformed into the following form: c c c ˜ + Ek (Zk+1 ˜ + E (k+1) (Zk+1 ; θ, φk+1,u (θ)) ; ψu(k) (θ)) ; ψk+1 ) − βk+1,u A∗k+1,u (Zk+1 0 (k+1) c (k) ˜ c (Zk+1 ; ψk+1 , g) + Fk (Zk+1 ; ψu (θ), g) + F 0
+
m−1 X
c 0(k+1) c R(k+1) (Zk+1 ; ψk+1 , g) + Rm (Zk+1 ; ψk+1 , g) n
n=1
˜ + B 0(k) (Ak , ck+1 ; ψ, ψ 0 , g) − βk+1,u A∗k+1,u (Zkc ∩ Zk+1 , Bk+1 ; θ, φk+1,u (∂k+1 ; θ)) c ; ψ, ψ 0 , g) − Ek Ld |Zkc \k+1 | + B 00(k+1) (Ak , k+1 , 00k+1 , Zk+1
− Ek00 Ld |k+1 \00k+1 | − Ek000 Ld |00k+1 |.
(2.21) Now we are prepared to perform the renormalization of the effective action. It involves the first five terms above, as in Sect. 4 [4]. We perform the same operations as in that c . This causes section, but for the expressions restricted to the small field region Zk+1 some minor changes, so we discuss the main points in these operations, giving more details only if the changes are more substantial. The first operation, and the one which requires a more detailed discussion, is an expansion of the “old” main action around the “new” one determined by the renormalized coefficients. It is a simple generalization of the expansions (4.2), (4.3), (4.13), (4.14) [4]. At first we use Eqs. (4.1) [4] in the special case when ck+1 = 0, dk+1 = bk+1 , and we decompose the first term in (2.21), or rather the corresponding function in the variational problem in (4.2) [4], into a sum of this function with the renormalized coefficients, and a remainder with the renormalization constants. Using the first equality in (4.1) [4] we write θ = ψ + δψ, δψ = χ3k+1 ζk+1 ψk+1 , and we expand the expressions on the right-hand side of the obtained equality around the renormalized configurations. Let us write the corresponding formulas in the case k = k0 , in which we have to expand the configurations φk+1,u , αk+1,u independent of νk+1,u = νk L2 , around the renormalized configurations φk+1 , αk+1 dependent on νk+1 . These configurations are determined by the variational problem for the function Ak+1 (Bk+1 (Wk+1 ); ψ˜ k+1 , φ) with the external field term. In other cases the formulas are basically the same as in Sect. 4 [4], but we will discuss modifications in these cases anyway. Writing φk+1,u = φk+1 + δφk+1 , αk+1,u = αk+1 + δαk+1 and using the formula (2.23) [5] we obtain c c βk+1,u A∗k+1,u (Zk+1 ; θ, φk+1,u ) = βk+1 A∗k+1 (Zk+1 ; ψk+1 , φk+1 ) 1 −1 bk+1 k∂φk+1 k2Z c∗ + βk+1 ak+1 ζk+1 hψk+1 , ψk+1 − Qk+1 φk+1 iZ c ∩3k+1 + βk+1 k+1 k+1 2 β −1 bk+1 1 −1 + k+1 kαk+1 k2Z c + νk+1 βk+1 ek+1 kφk+1 − hk2Z c k+1 k+1 2λk+1 2 c + βk+1 δA∗k+1,u (Zk+1 ; δψ, δφk+1 , δαk+1 ) −1 bk+1 hδφk+1 , −1φk+1 + αk+1 φk+1 iZ c + βk+1 k+1 −1 + νk+1 βk+1 ek+1 hδφk+1 , φk+1 − hiZ c k+1
10
T. Balaban
1 (δφk+1 )+ + (δφk+1 )− , ∂φk+1 + βk+1,u 2 st(Z c
k+1
c βk+1 A∗k+1 (Zk+1 ; ψk+1 , φk+1 )
(k+1)
+ βk+1 C 1 (δφk+1 )+ + (δφk+1 )− , ∂φk+1 + βk+1,u 2 st(Z c =
)
c (Zk+1 ; δψ, δφk+1 , δαk+1 )
(2.22)
,
) k+1
where 1 c ; δψ, δφ, δα) = hδψ − Qk+1 δφ, ak+1 (δψ − Qk+1 δφ)iZ c ∩3k+1 δA∗k+1,u (Zk+1 k+1 2 1 1 −1 (2.23) + (1 + βk+1 bk+1 ) k∂δφk2Z c∗ + hδφ, αk+1 δφiZ c + kδαk2Z c k+1 k+1 k+1 2 λk+1 1 −1 ek+1 )kδφk2Z c . + νk+1 (1 + βk+1 k+1 2 Let us recall that the norms, scalar products and derivatives in all formulas in this section are taken in the L−1 η-scale. In the case considered here the functions δφk+1 δαk+1 satisfy the following variational equations: a a ∗ D Q − 11 + αk+1 + δα δφ = Q∗ δψ − δαφk+1 + f0 , Q −1 −1 1 + βk+1 bk+1 1 + βk+1 bk+1 f0 =
−1 −βk+1 bk+1 νk+1 (−1φk+1 + αk+1 φk+1 ) + (φk+1 − h) −1 −1 1 + βk+1 bk+1 1 + βk+1 bk+1
−1 1 + 2βk+1 bk+1 νk+1 (φk+1 − h), −1 1+ 1 + βk+1 bk+1 1 1 1 (2φk+1 · δφ + |δφ|2 ) = (|φk+1 + δφ|2 − |φk+1 |2 ) = δα, on 1 , 2 2 λk+1 −1 = βk+1 bk+1 Q∗
a
−1 βk+1 bk+1
(Qφk+1 − ψ˜ k+1 ) +
(2.24) where Q denotes the averaging operator determined by the generating set Bk+1 (Wk+1 ), 1 = 1 (Wk+1 ) is the first domain determined by this set, and the last equality in the definition of f0 follows from the variational Eq. (1.10)[2] for the configuration φk+1 . These equations are of the form (3.20) [2], and we can apply to them all the relevant results of [2, 3]. In particular Proposition 3.1 [2] yields the unique solutions δφk+1 δαk+1 . They are analytic functions of φk+1 , αk+1 ; δψ, f0 defined on the spaces (3.21) [2] and satisfying the bounds (3.47)[2]. In variables δψ, f0 they are of first order at least, so if δψ = 0, f0 = 0, then δφk+1 = 0, δαk+1 = 0. We will use this property in future expansions, as in Sect. 4 of [4]. Let us compare Eqs. (2.24) with the corresponding variational Eqs. (4.4), (4.5) [4]. The only essential difference is in the last term of the function f0 in (2.24), which has the coefficient νk+1 , whereas the corresponding term in (4.4) [4] has the coefficient −1 ek+1 . That term has been bounded by 23 εk+1 ε in (4.10) [4], and we would like to νk+1 βk+1 have the same bound here, so we assume that νk+1 < 16 εk+1 ε. In [4] we have taken finally ε = r0 , where r0 has been defined after (5.26)[4], so the assumption is νk+1 < 16 εk+1 r0 . −1 By the remark following (1.9) [5] we have νk+1 = νk0 +1 < 98 Ld βk−1 = 98 Ld βk+1 , so 0 +1 −1 −1 d L β ε ≤ r . We have assumed the the assumption follows from the inequality 27 0 k+1 k+1 4 condition (5.27) [4], which is much stronger than this inequality, so the above assumption is satisfied also. We obtain from it that the configurations δψ, f0 , δφk+1 , δαk+1 satisfy the bounds (4.10), (4.11) [4], hence also the remaining bounds in [4].
Renormalization and Localization Expansions
11
c We consider the function C (k+1) (Zk+1 ) defined by the second equality in (2.23). This definition looks different than the definition (4.14) [4], but in fact one can be obtained from another by using the variational equations. The function is pre-localized c , and we decompose it into a sum of functions pre-localized in cubes 1k+1 (z), z ∈ in Zk+1 c Zk+1 ∩ T1(k+1) , as in (4.15) [4]. They are analytic functions of (ψk+1 , h) defined on a quite large space determined by the assumptions in Propositions 1.1, 3.1 [2], 2.1, 2.2 [3], but we consider them on the space 4ck+1 (Bk+1 (Wk+1 ); εk+1 ). They satisfy the bounds −1 3C4 B6 K5 , |C (k+1) (1k+1 (z); δψ, δφk+1 , δαk+1 )| < βk+1
(2.25)
which are the same as in Proposition 4.1 [4], but we have used the bounds (5.31) [4] for the renormalization constants. The bounds hold on the above space, and for the functions analytically extended in localization parameters. We construct their localization expansions using the procedure of Proposition 4.1 [3], which can be applied without any modifications in this case, and we obtain all the conclusions of this proposition. We can treat in a similar way the surface term, the last term in (2.23). The set c ) can be identified as the set of bonds of the lattice TL−1 η intersectof bonds st(Zk+1 c , with proper directions. We decompose the surface into a union ing the surface ∂Zk+1 of d-1-dimensional M -cubes, which are walls of cubes from πk . This decomposition determines a decomposition of the surface term in (2.23) into a sum over the walls of the terms corresponding to bonds intersecting a wall. A surface term in this sum −1 1 −1 = βk+1 0(1)B6 K3 K42 K5 . We construct can be bounded by 0(1)K3 K42 ε2k+1 B6 K5 βk+1 ε2k+1 localization expansions of these terms as above, and all localization domains in these ∼ and intersect Zk+1 . After proper resummations we expansions are contained in Wk+1 include the obtained expansion into the boundary term B 00(k+1) . Let us make a remark on the basic renormalization group Eqs. (5.5), (5.14), or (5.21) [4] in the considered case k = k0 . For those equations not only bounds are important, we have taken care of that, but also their structure. It differs slightly from the one in [4] in that is now not necessarily of second order in the renormalization variables the function c(k+1) 2 ξ, because of the term with the coefficient νk+1 in f0 . Thus this function contributes also to the terms of zeroth and first order in ξ. Fortunately these contributions are very small, −1 ), and they do not change anything in the analysis they are proportional to νk+1 = O(βk+1 of Sect. 5 [4]. Let us make some comments on the remaining two cases. If k < k0 , then there is no term with νk+1 in the variational equations (2.24), but instead there is the additional c ). The above remark applies to term νk+1 hδφk+1 , φk+1 − hi in the function C (k+1) (Zk+1 this case also. If k > k0 , then we have the same equations and the function as in [4], with obvious changes connected with the generating set and the special renormalization group equations for ck+1 , dk+1 . Finally, let us remark that, for greater clarity, we could divide the operations in the case k = k0 into two steps. In the first step we would include the external field term into the variational problem for the background configurations, without the renormalization. This means that we would write the decomposition (2.22) with ζk+1 = bk+1 = ek+1 = 0, and where δφk+1,u , δαk+1,u are the solutions of the Eqs. (2.24) with f0 = νk+1,u (φk+1,u − −1 h). These solutions are of the order 0(βk+1,u ), and we would include the new term (k+1) c c ) into the function E0 (Zk+1 ). In the second step we would perform −βk+1,u C (k+1) (Zk+1 the usual renormalization operation, as in the case k > k0 , i.e. as in [4]. The next operation is an expansion of the effective action Ek and the generating functional Fk in (2.21) around the renormalized background configurations. It is the
12
T. Balaban
same as in [4]; we use the decompositions (j) ˜ = ψ (j) (ψ˜ k+1 ) + δψ (j) , δψ (j) = δψ (j) (δψ, f0 ), ψk+1,u (θ) k+1 k+1 k+1 k+1
(2.26)
where the last function is given by an obvious generalization of the second formula in (4.16) [4]. We expand the functions Ek , Fk , and we define the new contributions Eu0(k+1) , F 0(k+1) by the formulas (4.17), (4.18) [4], with obvious changes in notations. We define a pre-localized decomposition of the function Eu0(k+1) in the same way as in (3.17) c [5], we take for z ∈ Zk+1 ∩ T1(k+1) , c c Eu0(k+1) (z, Zk+1 ; ψk+1 ) = E (k+1) (z, (Zk+1 )≈ ; ψk+1 ) k Z 1 X d (j) ˜ (j) c dt E (j) (1k+1 (z), (Zk+1 )≈ ; ψk+1 (ψk+1 ) + δψk+1 (tδψ, tf0 )), + dt 0
(2.27)
j=1
c c )≈ ) denotes the sum of terms from E (j) (Zk+1 ) where the symbol E (j) (1k+1 (z), (Zk+1 c with localization points y ∈ 1k+1 (z), and localization domains contained in (Zk+1 )≈ , c )≈ , and the symbol including the counterterms in (1.8) [5] with y ∈ 1k+1 (z), x ∈ (Zk+1 (k+1) c E0 (z, (Zk+1 )≈ ) denotes the sum of terms (2.8) with the given point z and localization c )≈ , i.e. the second sum in (2.9). The function (2.27), more precisely domains Y ⊂ (Zk+1 the sum on the right-hand side, is a non-local function of ψk+1 , and we construct its localization expansion applying the procedure of Proposition 4.1 [3]. We discuss it below together with other expansions. Now we complete the renormalization defining c ; ψk+1 ) E 0(k+1) (z, Zk+1 c = Eu0(k+1) (z, Zk+1 ; ψk+1 ) − βk+1 C (k+1 (1k+1 (z); δψ, δφk+1 , δαk+1 ) − E (k+1) (z), (2.28) where E (k+1) (z) is defined as in (4.21) [4] by the expressions considered on the whole c ) we have to construct lattice. To define a final renormalized contribution E (k+1) (Zk+1 localization expansions of the above functions, and to take terms with proper localization domains. We discuss the localization expansions in the following proposition. c ∩ T1(k+1) has a localization expansion Proposition 2.3. The function (2.28) for z ∈ Zk+1 of the form X c E 0(k+1) (z, Y, Zk+1 ; ψk+1 ) (2.29) ∼ Y ∈Dk+1 :z∈Y ⊂Wk+1
with the terms having the properties described in (H.2) [5], and satisfying the bounds with the constant (N Ld B5 + 1 + const.E0 β −1 L−2γ(k+1) + 3B6 C4 K5 )(1 + K0 ) ≤ (N Ld B5 + 2 + 3B6 C4 K5 )(1 + K0 ), which is determined by the absolute constants introduced before and independent of E0 c for β large enough. If a localization domain Y is contained in (Zk+1 )≈ , then the term of the expansion (2.29) is equal to the corresponding term of the expansion of the function c ; ψk+1 , g) (2.28) defined on the whole lattice. Similarly, the functions M0(k+1) (x, Zk+1 defined by the formula (4.18) [4] have localization expansions of the same general form c (2.29), but with z replaced by x ∈ Zk+1 ∩ TL−1 η , and the constant in the bounds equal to
Renormalization and Localization Expansions
1 −1 β 4 + const.β −1 L−(k+1)α1 2
13
(L−1 η)
d−2 2 +γ−α1
≤ c9 (L−1 η)
d−2 2 +γ−α1
.
c )≈ is equal to the corresponding Again, a term with a localization domain Y ⊂ (Zk+1 term of the localization expansion of the function defined on the whole lattice.
Let us make a remark concerning definition (2.27). It gives a form of the prelocalization, which is an alternative to the definitions (2.57), (4.20) [4], but it requires including the counterterms into the corresponding functions, as in (1.8) [5]. The prelocalized functions defined by (2.27) are generally different from the ones defined by (4.20) [4], but their sums over the points z are of course equal; they are equal to the globally defined new contribution on the whole lattice. Therefore the renormalization equations and the renormalization constants defined by them are also equal. We divide expansion (2.29) into two parts, the first is the sum over localization doc )≈ , the second is the remaining sum over domains intersecting mains contained in (Zk+1 c ≈ c ((Zk+1 ) ) . The second sum is included into the boundary terms B 00(k+1) , after resumc . The first sum is the part of the localization expansion of mations over z ∈ Y ∩ Zk+1 function (2.28) defined on the whole lattice, which is equal to the part contained in c ). The function (2.28) on the whole lattice definition (1.8) [5] of the function E (k+1) (Zk+1 (k+1) (k+1) determines also the functions v1 , v2 , hence the second part, the “counterterm” in definition (1.8) [5]. We subtract and add the counterterm, and the sum with the subtracted c ) to the effective action. It counterterm defines finally the new contribution E (k+1) (Zk+1 satisfies the required properties and bounds, as was discussed at the end of Sect. 5 [4]. c ) to the generating In the same way we construct the final new contribution F (k+1) (Zk+1 functional, but without any counterterms now, and it satisfies also the required properties and bounds. Consider now the added counterterm. It is easy to see, by the antisymmetry of the expression under the double sum, that it is equal to the sum with “minus” sign, the c c )≈ ∩ Zk+1 ∩ T1(k+1) , y ∈ Zk+1 ∩ T1(k+1) . Using the localization summations over z ∈ (Zk+1 expansions of v1(k+1) , v2(k+1) we can write it in the form X
−
X
(k+1) Y ∈Dk+1 :z∈Y c )≈ ∩Z z∈(Zk+1 k+1 ∩T1
D +
v2(k+1) (z, ·, Y
D E 1 (k+1) 2 2 c (ψ v1 (z, ·, Y ), χZk+1 (·) − ψ (z)) k+1 k+1 2
c (νk+1 h · (ψk+1 (·))0 − νk+1 h · (ψk+1 (z))0 ) ), χZk+1
E .
(2.30) The above sum is obviously included into the boundary terms B 00(k+1) , after changing the order of the sums and performing proper resummations. Bounds are given in (3.77), (3.80) [1]. The last operation of this section is the renormalization of the previous main action pre-localized on the domain Zkc ∩ Zk+1 , i.e. of the first term in (2.17). This means simply an expansion of the form (2.22). We need this operation really only in the case k ≤ k0 , but we may do it in all cases as well. A difference in comparison with (2.22)–(2.24) is that now we do not change the dependence of the background configurations on the external field, even in the case k = k0 , so for k ≤ k0 the configurations are independent of the external field before and after the operation. Thus we write
14
T. Balaban
˜ βk+1,u A∗k+1,u (Zkc ∩ Zk+1 , Bk+1 ; θ, φk+1,u (∂k+1 ; θ)) ˜ = βk+1 A∗k+1 (Zkc ∩ Zk+1 , Bk+1 ; ψ, φk+1 (∂k+1 ; ψ)) + βk+1 C (k+1) (Zkc ∩ Zk+1 , Bk+1 ; δψ, δφk+1 (∂k+1 ), δαk+1 (∂k+1 )) 1 ˜ (δφk+1 (∂k+1 ))+ + (δφk+1 (∂k+1 ))− , ∂φk+1 (∂k+1 ; ψ) , + βk+1,u 2 st(Zkc ∩Zk+1 ) (2.31)
˜ −h c where we include the term νk+1 δφk+1 (∂k+1 ), φk+1 (∂k+1 ; ψ) Z ∩Z k
k+1
into the
∩ Zk+1 , Bk+1 ) for k ≤ k0 , and for k > k0 the function is the function C same as in (2.22), with obvious notational changes connected with the different domain and the generating set. For k ≤ k0 the functions δφk+1 (∂k+1 ), δαk+1 (∂k+1 ) satisfy the variational Eqs. (2.24) determined by the generating set Bk+1 (∂k+1 ), where 1 (∂k+1 ) is the first domain determined by the set, and where f0 does not have the term with νk+1 , but has the additional term χ1 (∂k+1 ) 1χc1 (∂k+1 )∩k+1 Q∗k+1 δψ. For k > k0 they satisfy the above described equations, with the additional term νk+1,u added to the operator in the square brackets in the first equation. In both cases we can apply Proposition 3.1 [2] and we obtain the bounds (3.47)[2], from which we conclude the bounds (2.25) for the function C (k+1) (1(y), Bk+1 ), where 1(y) = 1k (y) for y ∈ Zkc ∩ Z k+1 ∩ 3k , and 1(y) = 1k+1 (z) for y = z ∈ Zkc ∩ Zk+1 ∩ 3k+1 . Thus the second term on the right-hand side of (2.31) can be bounded by const. (|Zkc ∩ ck+1 |η + |Zk+1 ∩ k+1 |). The surface term in (2.31) can be estimated in the same way as discussed after (2.25), and we can obtain a bound expressed either in terms of areas of the surfaces ∂Zkc , ∂Zk+1 , or as above. Thus the sum of the last two terms in (2.31) can be estimated by the bound written above. All the terms on the right-hand ∼ , so we do not need to side of (2.31) depend on the variables ψ restricted to Wk∼ ∩ Uk+1 construct localization expansions for the last two terms, and we include them into the boundary terms. We have completed all operations connected with the renormalization of the effective action, and we have transformed the expression in the exponential in (3.18) [5] into an expression in which almost all terms have the form required by the inductive hypotheses, c c c c , Ek+1 (Zk+1 ), Fk+1 (Zk+1 ), R(k+1) (Zk+1 ), all have more precisely the main action on Zk+1 the required form. We divide the sum of all boundary terms into two parts in a way different than the division into B 0(k) , B 00(k+1) . To the first part we include the whole B 0(k) , the sum of terms from B 00(k+1) with localization domains contained in Zk+1 , and the sum of the two last terms in (2.31). The obtained sum is denoted by B (k) (Zkc ∩ c Zk+1 , (Ak , k+1 , 00k+1 , Zk+1 ); ψ, ψ 0 , g), or simply by B (k) (Zkc ∩ Zk+1 ), or B (k) . The sum 00(k+1) c of the remaining terms in B , with localization domains intersecting Zk+1 and Zk+1 , 00 (k+1) c 0 (k+1) (k+1) is denoted by B (Zk+1 , (Ak , k+1 , k+1 , Zk+1 ); ψ, ψ , g), or B (Zk+1 ), or B . We conclude this section with the following proposition. (k+1)
(Zkc
Proposition 2.4. The renormalization operation described in this section has transformed the expression in the exponential in (3.18) [5] into the following one
Renormalization and Localization Expansions
c c Ak+1 (Zk+1 ) + Fk+1 (Zk+1 )+
− −
m−1 X
15
c 0(k+1) c R(k+1) (Zk+1 ) + Rm (Zk+1 ) n
n=1 ∗ c ˜ + B (k) (Zkc ∩ Zk+1 ) βk+1 Ak+1 (Zk ∩ Zk+1 , Bk+1 ; ψ, φk+1 (∂k+1 ; ψ)) Ek Ld |Zkc \k+1 | − Ek00 Ld |k+1 \00k+1 | − Ek000 Ld |00k+1 ∩ Zk+1 |
(2.32)
+ B (k+1) (Zk+1 ). Terms of the above expression satisfy the corresponding inductive hypotheses, and the term B (k) , for which a localization representation is not important anymore, depends on ∼ , and satisfies the bound the variables restricted to the domain Uk+1 |B (k) (Zkc ∩ Zk+1 )| < const.B0 (|Zkc ∩ ck+1 |η + |Zk+1 ∩ k+1 |),
(2.33)
in addition to the analyticity and symmetry properties. We have discussed already almost all statements of the above proposition, except bounds for terms of B (k+1) . We have seen in this section and in the previous paper, that contributions to B (k+1) constructed in various operations have bounds independent of B0 . Summing the bounds for terms with the same localization domain we obtain bounds with the common constant independent of B0 . To finish the analysis of the transformation S (k) T (k) acting on the effective density ρk we have to discuss the large field density obtained as a result of the operations performed in the last three sections. We will do it in the next section. 3. A Definition and Properties of the Large Field Density ρ0k+1 (Zk+1 ) Let us start with a definition of a new multi-index Ak+1 . In this case it is determined by a sequence of domains, so it is defined in purely geometric terms as follows: c }. Ak+1 = Ak ∪ {Pk+1 , Qk+1 , Rk+1 , Sk+1 , k+1 , 00k+1 , Zk+1
(3.1)
Notice that all the new domains are either contained in Zkc ∩ Zk+1 , like the large field regions Pk+1 , Qk+1 , Rk+1 , Sk+1 , or their boundaries are contained in it, like ∂k+1 , ∂00k+1 , ∂Zkc = ∂Zk , so all the new elements of the geometric structure are “localized” in this sense in Zkc ∩ Zk+1 . For this reason we denote by Ak+1 ∩ (Zkc ∩ Zk+1 ) the new sequence of domains added to Ak . If Z is a union of connected components of Zk+1 , then we denote by Ak+1 ∩ (Zkc ∩ Z) the intersections of the new domains, or their boundaries, with the domain Z. This yields in the considered case a precise inductive definition of the symbol Ak ∩ Z used in Sect. 1 [5]. The most important new contribuc . The domain k+1 joins the sequence tions to Ak+1 are the domains k+1 , 00k+1 , Zk+1 determined by the generating set Bk , and defines the new generating set Bk+1 . We have c joins the already made extensive use of this set in the preceding papers. The domain Zk+1 sequence Ck and defines the new sequence Ck+1 . Let us notice also that the expression (2.32) in the exponential depends on the part of the multi-index Ak+1 determined by the c only. domains k+1 , 00k+1 , Zk+1 With the above definition we write now the final form of the new contribution to the large field integral operator. We define
16
T. Balaban
T(k) (Zk+1 , Ak+1 ∩ (Zkc ∩ Zk+1 )) = T0(k) (Zkc , Pk+1 , Qk+1 , Rk+1 , k+1 , 00c k+1 ) · ∼3 c ∩ Sk+1 ))χck+1 (Sk+1 )χk+1 (00∼−3 ∩ Sk+1 ∩ Zk+1 ) · · χ0k+1 (00k+1 ∩ ((00c k+1 ) k+1 ˜ · exp −βk+1 A∗k+1 (Zkc ∩ Zk+1 , Bk+1 ; ψ, φk+1 (Bk+1 (∂k+1 ); ψ)) d c 00 d 00 000 d 00 − Ek L |Zk \k+1 | − Ek L |k+1 \k+1 | − Ek L |k+1 ∩ Zk+1 | .
(3.2)
The definition (2.47) [5] and the above one give a complete and explicit definition of this integral operator. It has obviously all the properties formulated in hypothesis (H.5) ∼ . Integrations in [5], in particular its kernel depends on ψk+1 , ψk , ψk0 restricted to Uk+1 0 this operator are with respect to the variables ψk , ψk restricted to ck+1 , k+1 ∩ 00c k+1 ∼ correspondingly and these domains are disjoint with Wk+1 , so the operator commutes with the small field part of the effective density. Now we define Tk+1 (Zk+1 , Ak+1 ) = T(k) (Zk+1 , Ak+1 ∩ (Zkc ∩ Zk+1 ))Tk (Zk , Ak ) exp B (k) (Zkc ∩ Zk+1 ). (3.3) This is a part of the inductive definition of the integral operators, the part determined by the procedure connected with the transformation S (k) T (k) , and described before in (1.23) [5]. The operator (3.3) has also all the properties formulated in (H.5) [5], together with the ones formulated above. One of the most important properties of the integral operators are the bounds (1.26) [5]. We would like to obtain now a similar, but explicit bound for the operator (3.2), and to obtain this way simultaneously a new contribution to the exponent in (1.26) [5], hence an inductive definition of the function κk . The integral operator (2.2) act on functions ∼ , ψk0 k+1 ∩ 00c F of the variables ψk ck+1 , ψk+1 k+1 ∩ Uk+1 k+1 , and we obtain easily the following preliminary bound, in which we estimate the Gaussian integrations in (2.47) [5]: Z dψk+1 Zk+1 |T(k) (Zk+1 , Ak+1 ∩ (Zkc ∩ Zk+1 ))F | ≤ 1 1 1 |Pk+1 | + (d log M + N log 2) d |Pk+1 |) · Md 2 M 1 1 1 1 · exp(− p21 (βk ) d |Rk+1 | + (d log LM + N log 2) d |Rk+1 |) · 4 M 2 M Z Z ≤ exp(−aLd−2 p20 (βk )
· sup 0 ψk
dψk ck+1
c dψk+1 k+1 ∩Zk+1 χ(Zkc ∩ Zk+1 )χ0c k+1 (Qk+1 )χk+1 (Sk+1 ) ·
(3.4)
˜ · exp −βk+1 A∗k+1 (Zkc ∩ Zk+1 , Bk+1 ; ψ, φk+1 (Bk+1 (∂k+1 ); ψ)) + (O(1) log βk + const.E0 )|Zkc ∩ Zk+1 | |F |. The characteristic function χ(Zkc ∩ Zk+1 ) describes the part of the space (1.24) [5] restricted to the domain Zkc ∩Zk+1 . All other characteristic functions, except the two “large field” functions, have been bounded by 1. To estimate further the above expression we have to use bounds for the main action. We formulate them in the following proposition. Proposition 3.1. Assume that the coefficients βk+1 , ak+1 , λk+1 , νk+1 satisfy the conditions (H.1) [4], and X is a domain such that X ∩ (j \j+1 ) is a union of unit cubes
Renormalization and Localization Expansions
17
in the ξ-scale, determined by points of the lattice T (j) . There exists an absolute positive constant γ0 , i.e. a constant depending on d only, such that βk+1 A∗k+1 (X, Bk+1 ; ψ, φ)
≥ γ0
X k+1
βj
j=0
+
k X j=0
1 βj+1 2
X
1 2
X
|(∂ 1 ψj )(b)|2
b⊂3j ∩X
|ψj+1 (b+ ) − (Qψj )(b− )|
(3.5)
2
,
b∈st(cj+1 ∩T (j+1) )∩X
for arbitrary configurations ψ, φ, ψ defined on Bk+1 ∩X and ψ = ψj on 3j . If they satisfy 2 the conditions ||ψ| − 1| < c5 , νj (1 − h · (ψ)0 ) < c5 on 3j ∩ X, |α| = | λk+1 2 (|φ| − 1)| < j−1 2 c0 (L η) on (j \j+1 ) ∩ X, then we also have βk+1 A∗k+1 (X, Bk+1 ; ψ, φ) ≥ γ0
k+1 X j=0
βj
X 1 (|ψj (y)| − 1)2 2
y∈3j ∩X
(3.6)
+ νj (1 − h · (ψj (y))0 ) . Let us recall that the coefficients βj , λj , νj in the above proposition are connected with βk+1 , λk+1 , νk+1 by the canonical scaling only, i.e. βj = βk+1 (Lj−1 η)d−2 , λj = λk+1 (Lj−1 η)2 , νj = νk+1 (Lj−1 η)2 . The above inequalities are very simple, they have been proved several times before in various places, sometimes as elements of proofs, without formulating them explicitly, for example in Sect. 4 of [6a], in the proof of Lemma 3.1 in [1], in Proposition 2.2 of [2]. In the second inequality (3.6) we could replace the expression in the square brackets by the effective potential Vj (ψj (y), h): in fact it is obvious with the effective potentials, and then we use the bounds of Proposition 2.2 [2] on them. Let us notice that by analyzing the proofs a bit more carefully we could 1 for example, but it is still far from really good bounds. If X is a union take γ0 = 7d+4 of two neighbouring cubes in a proper scale, then we could take γ0 = 13 . This case is usually enough to obtain locally some small factors from large field restrictions, but it is convenient to formulate the inequalities in the general case. They will be used also in the next section. Finally, let us remark that the above mentioned proofs of (3.5), (3.6) are based on trivial and rather universal inequalities, and this is a reason that we have obtained a small constant γ0 . We can obtain an optimal constant γ0 if we take a proper minimal configuration instead of φ and use the results of the renormalization analysis discussed in [1] and Sects. 4,5 of [4]. The analysis yields a precisely defined expansion of the action in terms of the basic “scaling expressions”, which are the expressions on the right-hand sides of the inequalities. Coefficients at leading terms of the expansion, which can be explicitly calculated as in Sect. 5 [4], determine almost uniquely the optimal constant. Now we estimate further the expression on the right-hand side of (3.4). The integrations on Zkc ∩ Zk+1 are restricted by the characteristic function χ(Zkc ∩ Zk+1 ) to a small neighbourhood of the unit sphere in RN , so we bound them by a supremum of the underintegral function with respect to the corresponding ψk , ψk+1 , multiplied by exp O(1)|Zkc ∩ Zk+1 |. Obviously it is a rough bound, we could do much better using the inequalities (3.5), (3.6) to estimate the integral, and then cancel some terms with the factor log βk in the exponential, but it is not important. We use the inequalities to obtain
18
T. Balaban
some small factors for the large field characteristic functions in (3.4). Consider a function 0 0∼3 ˜ ; δ), or the slightly weaker assumption χ0c k+1 ( ). If we assume that ψ ∈ 9(Bk+1 ∩ ˜ satisfies the conditions ˜ k+1 ∩ 0∼2 ; δ), then φk+1,u (Bk+1 (k+1 ∩ 0∼2 , 0∼2 ); θ) ψ˜ ∈ 9(B 9 9 1 (1.15), (1.16) [2] with 8 K1 δ, and if we take 8 K1 δ ≤ 2 δk , then χ0k+1 (0 ) = 1. This means that not all conditions defining the above space are satisfied, and the inequalities yield the factor γ0 1 2 2 1 1 βk δ 2 , βk+1 δ 2 , βk+1 δ 2 , βk+1 δ 2 ) p0 (βk ) . ≤ exp − ( exp −γ0 min 2 2 2 9K1 Taking into account overlap properties of the domains for the above spaces considered for all 0 ⊂ Qk+1 , we obtain the factor 1 8γ0 2 p (βk ) d |Qk+1 | (3.7) exp − d 6 (9K1 )2 0 M for the characteristic function χ0c k+1 (Qk+1 ). By an almost identical, but a slightly simpler reasoning we obtain the factor 1 γ0 2 p (β ) |S | (3.8) exp − k+1 k+1 Md 2 · 6d K12 0 for the function χck+1 (Sk+1 ). Combining the above bounds with (3.4) yields finally Z dψk+1 Zk+1 |T(k) (Zk+1 , Ak+1 ∩ (Zkc ∩ Zk+1 ))F | ≤ 1 ≤ exp −A3 (log βk )2p0 d (|Pk+1 | + |Qk+1 | + |Rk+1 | + |Sk+1 |) M Z c + (O(1) log βk + const.)|Zk ∩ Zk+1 | sup sup dψk Zk χ(Zkc ∩ Zk+1 )|F |, ∼ 0 ψZkc ∩Uk+1 ψk
(3.9) o n 8γ0 2 1 2 , and we have taken p A , A = p . where A3 = min 6d (9K 1 0 2 0 8 1 1) We estimate also the exponential factor in (3.3), which in principle is connected with the operator T(k) , using the bound (2.33), and we include the bound in the above exponential; it changes only the constant. Notice that all the above volumes are measured in the L−1 η-scale. Let us denote the expression in the exponential by −κ(k) (Ak+1 ∩ (Zkc ∩ Zk+1 )). We have written the above bounds on the whole domain Zkc ∩ Zk+1 , but obviously they have the same form if we restrict all the expressions to a union of connected components, in particular the function κ(k) is a sum of its values on the components. From (3.5), (3.3) and (1.26) [5] we obtain the corresponding inequality (1.26) [5] for the operators with the index k + 1, where κk+1 (Z, Ak+1 ∩ Z) = κk (Z ∩ Zk , Ak ∩ (Z ∩ Zk )) + κ(k) (Ak+1 ∩ (Zkc ∩ Z)),
(3.10)
Z is a connected component of Zk+1 . This gives the inductive definition of the function κk+1 . From this definition it is clear that the functions κk are quite universal. They are the same for many classes of models; for example the same functions have been used in the papers [6b,c,d] on lattice gauge field theory, the only difference is in different “running” constants used to express the smallness of the exponential factors (3.7), (3.8),
Renormalization and Localization Expansions
19
etc. We have analyzed these functions extensively in [6d], and we will use the results obtained there. Consider now the second element of the bounds (1.29)–(1.31) [5], the combinatoric factors C(Ak ∩ Z). It is very easy to determine a relation between C(Ak+1 ∩ Z) and C(Ak ∩ Z), Z is a connected component of Zk+1 . There are the new sums over the regions Zk , Pk+1 , Qk+1 , Rk+1 , Sk+1 , k+1 , 00k+1 , or rather their intersections with fixed Z. Forgetting almost everything about the intricate geometric relations between the −d regions we can bound the sum over Zk ∩Z by 2M |Z|η , and the sums over the remaining −d c regions by 26(LM ) |Zk ∩Z|η . The last number can be improved if we use a part of the exponential factor in (3.9). We could replace 6 by a small number, but it is unimportant. Estimating these factors in a simple way we can take C(Ak+1 ∩ Z) = C(Ak ∩ Z) exp(
1 5 1 |Zk ∩ Z|η + |Z c ∩ Z|η ). Md 4 Md k
(3.11)
This inductive definition of the combinatoric factors allows us to write them explicitly, which is unimportant, but implies also easily the inequality (1.28) [5]. Another important definition is that of the function K(Z, Ak ∩Z). We have introduced it sketchily in Sect. 1. Now we would like to give it a more explicit and convenient form. Consider the expression on the left-hand side of the inequality (1.30) [5] multiplied by exp Hk |Z|η . For simplicity of notations let us assume in this paragraph that Z is the only component of Zk , so Z = Zk and Ak ∩ Z = Ak . The inequality (1.30) [5] can be now written in the form C(Ak ) exp(−κk (Zk , Ak ) + Hk |Zk |η ) < exp(−`p2 (βk )).
(3.12)
Consider the expression on the left hand side for the index k + 1, i.e for the multi-index obtained by the above inductive construction. Analyzing the definition of the function κ(k) , and the definitions of all the regions involved, it is easy to see that the expression has the largest possible value if all the large field regions Pk+1 , Qk+1 , Rk+1 , Sk+1 are empty. Thus from the point of view of the bounds (3.12) it is enough to consider the case when no large field regions are created. This leads to the following definitions and notations. For Z ∈ D we define S(Z) = Zk+1 determined by the definitions in Sects. 2,4 under the assumption that Zk = Z and all the large field regions are empty.
(3.13)
In principle the operation S should have an index indicating the scale for which we perform the geometric constructions, but we omit it for simplicity. The domain S(Z) can be determined using the operation “∼”, but we write only a simple inclusion following from (2.4), (2.9), (2.46) [5], and (2.5), S(Z) ⊂ (Z ∼[Rk+1 ]+1 )0∼9+L+3[Rk ] ⊂ Z ∼[Rk+1 ]+10L+L
⊂ Z ∼4L[Rk ] .
(3.14)
diamk+1 (S(Z)) ≤ diamk+1 (Z) + 8M [Rk ] = L−1 diamk (Z) + 8M [Rk ].
(3.15)
2
+3L[Rk ]
From this we obtain the inequality
The operation S can be iterated, and we denote by Sn the composition of n operations on successive scales. The domain Sn (Z) is considered on the scale L−n η, and we have
20
T. Balaban
diamk+n (Sn (Z)) < L−n diamk (Z) +
n X
L−(n−p) 8M [Rk+p−1 ]
p=1
(3.16)
< L−n diamk (Z) + 12M [Rk+n−1 ]. It is clear that for sufficiently large n the domains Sn (Z) shrink to a domain with a diameter smaller than some universal size, for example by (3.15) or (3.16) this size may be taken as equal to 12M [Rk+n−1 ]. When the diameters reach this size, then the next applications of the operation S do not change essentially the sizes of the domains; they keep them almost constant, a bit smaller than the “critical size” 12M [Rk+n−1 ], but greater than 6M [Rk+n−1 ]. Let us denote by A0k+n the multi-indices obtained from Ak under the assumption 0 that there are no new large field regions, then Zk+n = Sn (Zk ). Consider the expression on the left hand side of (3.12) for the above multi-indices. For simplicity of notations in the following formulas we take n = 1. Applying (3.10), (3.11) we obtain 0 0 , A0k+1 ) + Hk+1 |Zk+1 |L−1 η ) C(A0k+1 ) exp(−κk+1 (Zk+1 1 = C(Ak ) exp(−κk (Zk , Ak ) + d |Zk |η + Hk+1 L−d |Zk |η M d 5 L 0 0 0 − κ(k) (A0k+1 ∩ (Zkc ∩ Zk+1 )) + |Zkc ∩ Zk+1 |L−1 η + Hk+1 |Zkc ∩ Zk+1 |L−1 η ). 4 M (3.17) The function κ(k) above is equal to the last term in the exponential (3.9). We combine it L d + H0 log βk+1 with the last two terms above, and we bound O(1) log βk + const. + 45 M by const. log βk+1 . The constant can be written explicitly in terms of the previous basic constants, like L, M, K0 , K1 , etc. For the second and third terms in the last exponential above we use the bound
1 (d − 2) log L 1 1 1 + H0 L−d log βk+1 a = d + + d H0 log βk a < 1 + Hk < Hk , d d M M L L 27 and we obtain the inequality 0 0 , A0k+1 ) + Hk+1 |Zk+1 |L−1 η ) C(A0k+1 ) exp(−κk+1 (Zk+1 0 |L−1 η ). < C(Ak ) exp(−κk (Zk , Ak ) + Hk |Zk |η + const. log βk+1 |Zkc ∩ Zk+1
(3.18)
Obviously it holds for k replaced by k + p − 1, hence we have 0 0 , A0k+1 ) + Hk+n |Zk+n |L−n η ) C(A0k+n ) exp(−κk+n (Zk+n < C(Ak ) exp −κk (Zk , Ak ) + Hk |Z|η
+
n X
(3.19)
const. log βk+p |(Sp−1 (Zk ))c ∩ Sp (Zk )|L−p η = exp(−κk,n (Zk , Ak )),
p=1
where the last equality is a definition of the function κk,n . The sequence κk,n (Zk , Ak ) is decreasing in n, the sequence p2 (βk+n ) is increasing. Hence κk,n (Zk , Ak ) − `p2 (βk+n ) is decreasing and there is an exactly one index K such that the terms of the last sequence are positive for n ≤ K, and non-positive for n > K. Obviously it depends on ` also, but we consider the two values ` = 1, 2 only, and all the statements below are valid
Renormalization and Localization Expansions
21
for both of them. We say then that Ak = Ak ∩ Zk controls K steps, and we define the function K(Zk , Ak ) as equal to the index K. We take ` = 2 for the function used in the formulation of the hypothesis (H.6) [5]. This definition has been formulated in the case when Zk is connected, but in a general case for a connected component Z of Zk we replace Zk , Ak by Z, Ak ∩ Z in all the above formulas and definitions. The function has been considered in [6d], and we formulate below a proposition following from the results obtained there. Let us remark again that this part of the method is completely model-independent, and the results of [6d] hold without changes. Proposition 3.2. (i) If a component Z of Zk has a diameter greater than the critical size in the η-scale, then there is an index K < K(Z, Ak ∩ Z) such that Sk (Z) has a diameter smaller than the critical size in the L−K η-scale, and K + log βk+K a < K(Z, Ak ∩ Z). (ii) If K(Z, Ak ∩ Z) = 0, then there exists an index k1 < k such that for all indices j, k1 ≤ j ≤ k, the domains Zj ∩ Z have diameters smaller than the critical sizes in the Lj η-scales, the corresponding multi-indices Aj ∩ Z = Ak ∩ Zj ∩ Z control at least k − j steps, there are no large field regions in Zkc1 ∩ Z, and n1 = k − k1 > log βk1 a.
(3.20)
(iii) If K(Z, Ak ∩ Z) > 0, Ak+1 is obtained as in (5.1), and Z 0 is a component of Zk+1 containing Z, then K(Z 0 , Ak+1 ∩ Z 0 ) ≥ K(Z, Ak ∩ Z) − 1.
(3.21)
A proof of this proposition is quite simple, although rather lengthy and awkward. Details can be found in [6d], on pp. 383–387 in particular. The proposition will be used in the next paper in a construction of the operation R(k) . Let us remark that this construction creates its own multi-indices, but they have basically the same structure as the above ones, in fact a bit simpler, and all the above considerations and results can be applied to them also. Finally we can conclude the proofs of Theorems 1,4. Proof of Theorem 1. Of course we have been proving this theorem starting with Sect. 2 [5]. We want only to make some final comments. The definitions (3.1)–(3.3) provided the last elements of the inductive description, and combined with (2.47), (3.18) [5], Proposition 2.4, yield the effective density satisfying the inductive hypotheses (H.1)–(H.5), (H.7) [5]. The hypothesis (H.6) [5] may not be satisfied. Instead we have a weaker condition K(Z, Ak+1 ∩ Z) ≥ 0, where Z is a component of Zk+1 , and for some components we may have the equality. This remark ends the proof of the theorem. Proof of Theorem 4. In the same way as above we obtain that the densities ρk given by (1.1) satisfy the inductive hypotheses (H.1)–(H.5), (H.7) [5], so it remains to prove that the integral operators satisfy the inequality (1.31) [5] with ` = 1. The hypothesis (H.6) [5] is satisfied for ρk0 , hence K(Z, Ak0 ∩ Z) > 0 for the admissible multi-indices and components of Zk0 . If we change the condition defining the function K replacing ` = 2 by ` = 1, then we obtain a function K 0 with the same properties as K, except that the corresponding inequalities hold with ` = 1. Using the definition (3.19) of κk,n , the definition of K, Proposition 3.2 and the estimate of the critical size we obtain
22
T. Balaban
κk,n (Z, Ak ∩ Z) > κk,K (Z, Ak ∩ Z) − const.
n X
log βk+p (12M Rk+p )d
p=K+1
> 2p2 (βk+K ) − const.(12M R0 )d
n X
(log βk+p a)2d+1
p=K+1 0
> 2p2 (βk+K ) − const. (log βk+K a)2d+2 > p2 (βk+n ), if n ≤ K + log βk+K a, p2 ≥ 2d + 2 and A2 is sufficiently large. Bounds of the above type will be analyzed in more generality and detail in the next paper, so here we have written only conclusions. From the above inequality we obtain that K 0 (Z, Ak ∩ Z) > K(Z, Ak ∩ Z) + log βk a. Taking k = k0 we conclude that K 0 (Z, Ak0 ∩ Z) is greater than the total number of the remaining steps, so this function is positive for all admissible multi-indices occurring in the densities (1.1). Therefore the integral operators satisfy (1.31) [5] with ` = 1. In the next paper we construct the operation R(k) , and we finally restore the condition (1.32) [5] in the hypothesis (H.6) [5], proving in this way Theorem 2 [5]. To simplify a bit the construction we introduce a modification of the definition of the large field domain Zk+1 . If a component of this domain has a diameter smaller than the critical size, then we replace it by a smallest rectangular parallelepiped containing it. Of course it is a uniquely defined domain in Dk+1 . This way all the large field domains Zj ∩ Z described in Proposition 3.2 are rectangular parallelepipeds also. This condition makes geometric considerations in the next paper much simpler.
4. The Localization Expansions In this section we construct various localization expansions discussed and used in this paper and in [4,5]. There is quite a large number of them, but fortunately they are almost identical, so we discuss more extensively a most important and difficult case, and then we discuss briefly some variations in other cases. This is one of the most technical sections in this sequence of papers. It relies heavily on results and methods of [3], which are combined with some new results, and the results of [7,9], to yield required statements on localization expansions. We refer frequently even to small technical details in [3], and we use many constants and bounds defined and discussed there. We do not repeat even the most important definitions and constructions, because it would increase substantially this already large section. Most important and difficult is a class of the localization expansions connected with fluctuation integrals considered in Sect. 3 of the last paper [5], with their special cases considered in Sect. 2 of [4]. We devote most of this section to a construction of these expansions. Basically we discuss a generic case covering all of them, but we have to consider particularly carefully the most important new contributions (3.14), (3.15) [5] to the effective action and the generating functional. Thus we start with a detailed analysis of the measure (3.16) [5] and the corresponding expectation values (3.14), (3.15) [5], and then we work backwards analyzing the contributions to the measures in (3.11), (3.9), (3.2) [5], and the corresponding expectation values. Our first goal is to construct a localization expansion of the “interaction potential” in the exponential (3.16) [5]. To see clearly properties and bounds of this expression − 21
we expand its terms in βk
up to proper orders, so that we obtain an overall factor
Renormalization and Localization Expansions
23
−1
−1
βk 2 . To study the analyticity properties in βk 2 it is convenient to introduce the rescaled −1
−1
coefficient βk 2 σ in place of βk 2 in the obtained expression. Thus we define the function −1
1
(k) (Zkc ; βk 2 σ, C (k) 2 ψ) Ut,u Z 1 1 ∂3 − 21 1 −1 0 0 2 = −βk σ dt (1 − t ) ( 0 0 0 V (k) )(Zkc ; βk 2 σC (k) 2 t0 tψ), 2 0 ∂ψ ∂ψ ∂ψ (k) 21 (k) 21 (k) 21 C tψ, C tψ, C tψ
Z
−1
+βk 2 σ − 21
dt 0
dt0 (
0 1
Z
+βk σ
1
0
1 ∂ ˜ + β − 2 σC (k) 21 t0 tψ), C (k) 21 tψ Ek )(Zkc ; ψ (k) (θ) k 0 ∂ψ
(4.1)
∂ − 21 c (k) ˜ (k) 21 0 (k) 21 ( 0 Fk )(Zk ; ψ (θ) + βk σC t uψ), C uψ . ∂ψ
For σ = 1 it is equal to the “interaction potential” in the measure (3.16) [5], for σ = 0 it is obviously equal to 0. It is a function of ψ, (θ, h), g, and we want to localize in all these variables. We construct the expansions successively for the three terms in (4.1). Consider the first term and write the function V (k) (Zkc ; ψ 0 ) as the sum (2.30) [5]. A term V (k) (z; ψ 0 ), z ∈ TL(k+1) ∩ Zkc , is given by the formula (2.30) [5] in which the scalar products are restricted to the block Bk+1 (z). It is defined in terms of the functions (1) (1) (1) δφ(1) k , δφk,2 = δφk − δφk , δαk , δαk,2 = δαk − δαk . In Sect. 2 of [3], Proposition 2.1, we have constructed the analytic extensions δφk (s), δαk (s) considered as functions of ψ 0 , φk+1 , αk+1 , s. Here we take the large cube 0 from the partition πk+1 containing the block Bk+1 (z), and X0 = 0∼ , X1 = 0∼L−1 . We construct the extensions for the generating set (Bk (Wk ) ∩ (((Zkc )≈ )0 )c ) ∪ (((Zkc )≈ )0 )(k+1) . The reason for this slight departure from the construction of Sect. 4 [3] is that we want to have X1 contained in the k +1st domain. As it has been noted before the difference is unessential, it means that we take cubes by one scale larger on the domain ((Zkc )≈ )0 ∩ck+1 . Because of the localization in (θ, h) we take also the analytic extensions φk+1 (s), αk+1 (s), see Proposition 2.2 [3], and we consider the functions δφk (ψ 0 , φk+1 (s), αk+1 (s), s), δαk (ψ 0 , φk+1 (s), αk+1 (s), s), which we denote for simplicity again by δφk (s), δαk (s) . They are analytic functions of (θ, h), ψ 0 , s, defined on the spaces ˜ ck+1 (Bk+1 ∩ Wk∼ ; δ, ε) × {ψ 0 : |ψ 0 | < δ1 } × {s : |s| < eκ1 }, 9
(4.2)
where δ ≤ c7 , ε ≤ c0 , νk L2 ε ≤ δ, 4B1 K4 δ ≤ 18 , B3 δ1 ≤ 21 c6 . All these constants have been defined in [3] and they appear in the formulations of Propositions 2.1, 2.2 [3]. The functions satisfy the bounds (2.54)[3], from which we conclude the following ones: |δφk (s)|, |∂ η δφk (s)|, |1η δφk (s)|, |δαk (s)| ≤ K3 |ψ 0 | on Zkc .
(4.3)
(1) 1 0 0 The function δφ(1) k can be defined by the equality δφk (ψ ) = lim t→0 t δφk (tψ ), hence we define
1 lim δφk (tψ 0 , s), δφk,2 (s) = δφk (s) − δφ(1) δφ(1) k (s) = t→0 k (s). t The above functions satisfy the bounds (4.3) also, the second with the factor 2K3 instead of K3 . This is not enough, because δφk,2 (ψ 0 , s) is at least of second order in ψ 0 . We get an improved bound using the analyticity with respect to ψ 0 and writing
24
T. Balaban
δφk,2 (ψ 0 , s) =
2 2πi
Z
Z
1
dt(1 − t) 0
dτ |τ |=r
1 δφk (τ ψ 0 , s), (τ − t)3
with r > 1. The same formula holds for δαk,2 (ψ 0 , s). For simplicity we replace the bound 21 B3−1 c6 on δ1 by C3 c7 , which is ≤ 21 B3−1 c6 . We take r = C|ψ3 c0 |7 above, and we assume that |ψ 0 | < 16 C3 c7 . Simple estimates of the above formula yield the bounds |δφk,2 (s)|, |∂ η δφk,2 (s)|, |1η δφk,2 (s)|, |δαk,2 (s)| ≤ 2K3 (C3 c7 )−1 |ψ 0 |2 on Zkc . (4.4) (1) Substituting the analytically extended functions δφ(1) k (s), δφk,2 (s), δαk (s), δαk,2 (s) into the formula (2.27)[4] for V (k) (z) we obtain an analytic function V (k) (z; s) satisfying the bound (4.5) |V (k) (z; ψ 0 , s)| ≤ 9Ld K32 (C3 c7 )−1 |ψ 0 |3
on the spaces (4.2) with δ1 ≤ 16 C3 c7 . The third order derivative of this function satisfies then the bound 3 d ∂3 (k) 0 0 0 0 (k) 0 0 V , s), δψ , δψ , δψ V (z; ψ + tδψ , s)| (z; ψ = t=0 0 0 0 3 ∂ψ ∂ψ ∂ψ dt Z 3! dτ (k) 0 0 ≤ 6 · 36Ld K32 (C3 c7 )−1 |δψ 0 |3 = V (z; ψ + τ δψ , s) 4 2πi |τ |=r τ (4.6) C3 c 7 1 C3 c7 and r = 9|δψ for arbitrary δψ 0 , if |ψ 0 | < 18 0 | . Finally, we consider the first term on the right-hand side of (4.1). It is the sum of such expressions with V (k) replaced by V (k) (z). These we extend analytically in the way described above, extending also 1 the operators C (k) 2 in s, and the parameter σ to a complex disc. Let us recall that the 1 operator C (k) 2 is extended in the same way as the functions δφk , δαk . The variables φ0 , α0 are replaced by φk+1 (s), αk+1 (s). Thus we obtain the function 3 Z 1 1 ∂ − 21 1 0 0 2 0 −2 (k) (k) 21 3 (k) 21 dt (1 − t ) V (s)ψ, s), ⊗ tC (s)ψ . (z; t tβk σC −βk σ 2 0 ∂ψ 03 (4.7) 1 C3 c7 . The constant C3 is rather large, It is analytic on the spaces (4.2) with δ1 ≤ 18 for simplicity we assume that C3 ≥ 36, and the condition on δ1 can be replaced by the stronger one δ1 ≤ 2c7 . By (4.6) and Proposition 3.1 [3] the function (4.7) can be bounded by −1
βk 2 |σ|K32 Ld
1 3 3 1 −1 B |ψ| < βk 2 p31 (βk )|σ|B53 K32 Ld . c7 5 c7
Now we can construct a localization expansion of the function (4.7) for s = 1 following the proof of Proposition 4.3[3], with a slight modification again. We write the expansion (4.41)[3], or (4.43)[3] with the definition (4.42)[3], but we perform the resummation (4.45)[3] for a different domain Y . Now the part of the generating set contained in the boundary layer Wkc ∩ Wk∼ , i.e. the part Bk (Wk ) ∩ Wkc ∩ Wk∼ , has been introduced for an auxiliary technical reason, as a way of introducing Dirichlet type boundary conditions for the background configurations, so it is completely unimportant for the expansion. We define a generating set B0k+1 as the set obtained from Bk+1 by taking 0k+1 = (Wk∼ )0 instead of k+1 , and we take the set of localization domains Dk+1 (0k+1 ) = {Y ∈ Dk+1 : Y ⊂ 0k+1 }. A domain Y1 in (4.43)[3] determines uniquely a domain Y in Dk+1 (0k+1 ),
Renormalization and Localization Expansions
25
and we perform the resummations (4.45)[3] for these domains Y . It is easy to check that all the remaining considerations of Sect. 4[3] hold for d˜k+1 replaced by dk+1 , and we obtain the corresponding modified Proposition 4.3[3] with the localization expansion X V (k) (z, Y ), (4.8) ((4.7) for s = 1) = Y ∈Dk+1 (0k+1 ):z∈Y
where a term V (k) (z, Y ) is an analytic function of (θ, h) restricted to the domain Y , and satisfying the bound −1
|V (k) (z, Y )| < βk 2 p31 (βk )|σ|B53 K32 Ld −1
1 exp(−2κdk+1 (Y )), c7
(4.9)
−1
if |t0 tβk 2 σC (k) 2 (s)ψ| < βk 2 p1 (βk )|σ|B5 ≤ 2c7 . 1
It is also an analytic function of σ defined on the disc determined by the last condition. In the future we will have to assume that the constant in front of the exponential above is sufficiently small, which yields a much stronger condition than the last one. For a given Y we resum also over all points z ∈ TL(k+1) ∩ Zkc ∩ Y , hence we obtain X X (the first term in the definition (4.1)) = V (k) (z, Y ) . Y ∈Dk+1 (0k+1 )
(k+1) z∈TL ∩Zkc ∩Y
(4.10) This is the desired localization expansion; a term of this expansion is given by the sum in the last parentheses. To estimate this sum we prove at first the following general inequality holding for localization domains X ∈ Dk , for an arbitrary k, |X| ≤ ((2L + 3)M )d (1 + dk (X)) ≤ (3LM )d (1 + dk (X)).
(4.11)
To prove it we take a tree graph 0 in the definition (2.27)[1] and we prove the inequality 1 |0|. Take a vertex v1 ∈ 0 and consider all paths starting (4.11) with dk (X) replaced by M at v1 and ending at some vertex of 0. Take the longest one and denote it by 01 . Assume that |01 | ≥ M , the case |01 | < M is simple and will be discussed at the end of the proof. We choose an orientation of the path 01 considering v1 as an initial point, and we choose a sequence of points x1,0 = v1 , x1,1 , . . . , x1,n(1) ∈ 01 such, that the part of the path contained between two successive points x1,j , x1,j+1 , and denoted by 01 (x1,j , x1,j+1 ) has a length equal to M , i.e. |01 (x1,j , x1,j+1 )| = M, j = 0, 1, . . . , n(1) − 1. The part of the path between the point x1,n(1) and the end-point of 01 has a length < M . For each point x1,j in the sequence, 1 ≤ j ≤ n(1), take a cube 1,j of the size (2L + 1)M , which is a union of large cubes, and which contains the point x1,j in its central large cube. Then the part 01 (x1,j−1 , x1,j ) ∪ 01 (x1,j , x1,j+1 ) of the path is contained in the cube of the size 3M in the center of 1,j and any LM -cube, which is a union of large cubes and which intersects this part of the path, is contained in 1,j . We write this conclusion as follows: n(1) [ [ 1,j , and { : is an LM-cube intersecting 01 } ⊂ j=1
26
T. Balaban
|
[
{ : is an LM-cube intersecting 01 }| ≤
n(1) X
|1,j |
j=1 n(1) X 1 1 = ((2L + 1)M ) |01 (x1,j−1 , x1,j )| ≤ ((2L + 1)M )d |01 |. M M d
j=1
Consider now all vertices of the graph 01 , and all paths of the graph 0 starting at these vertices and disjoint with 01 (more precisely intersecting 01 at the starting point only). Let 02 be a path of maximal length, starting at some vertex v2 . We assume again that |02 | ≥ M , and we repeat the above construction for 02 . Then we consider all vertices of the graph 01 ∪02 and all paths of 0 starting at these vertices and disjoint with 01 ∪02 . We choose again a path of maximal length. The induction should be clear now. After p steps we obtain p paths 01 , 02 , . . . , 0p such that 0i intersects others at most at some vertices, and |0i | ≥ M . On each 0i we have a sequence of points vi = xi,0 , xi,1 , . . . , xi,n(i) , and cubes i,1 , . . . , i,n(i) , with properties described in the case i = 1, such that [
{ : is an LM-cube intersecting
p [
0i } ⊂
i=1
|
[
{ : is an LM-cube intersecting
p [ i=1
= ((2L + 1)M )d
p n(i) [ [
i,j , and
i=1 j=1
0i }| ≤
p X n(i) X
|i,j |
i=1 j=1
p p X n(i) X 1 1 [ |0i (xi,j−1 , xi,j )| ≤ ((2L + 1)M )d | 0i |. M M
Sp
i=1 j=1
i=1
Moreover, the graph i=1 0i is contained in a union of cubes of the size Sp 3M with centers at the centers of the cubes i,j . Take again all vertices of the graph i=1 0i and all paths of 0 starting at the vertices and disjoint with the graph. The construction is finished when there are no paths of a length ≥ SpM . Each point of the graph 0 can be connected by a path with a point in the graph i=1 0i , and a length of this path is < M . Hence each point of 0 belongs to a cube of the size 5M and with a center at a center of one of the cubes i,j . Thus if we take an LM-cube intersecting 0, then it is contained in one of ∼ a union the cubes ∼ i,j . Let us recall that for a localization domain Y we denote by Y of all large cubes contained in Y , or touching Y . The above discussion does not cover the case p = 0, i.e. when at the beginning we have |01 | < M . Then the whole graph 0 is contained in a cube of the size 3M and containing the vertex v1 in the central large cube. By definition the domain X is a union of LM-cubes, and each of these cubes intersect 0. Thus we have X⊂
[
{ : is an LM-cube intersecting 0} ⊂ 0 ∪
p n(i) [ [
∼ i,j ,
i=1 j=1
1 1 [ | 0i | < ((2L + 3)M )d (1 + |0|). M M p
|X| ≤ ((2L + 1)M )d + ((2L + 3)M )d
i=1
Taking the infimum over the tree graphs 0 we obtain the desired inequality (4.11). Notice also that it cannot be essentially improved. We can construct arbitrarily large domains
Renormalization and Localization Expansions
27
for which we have almost the equality in (4.11) modulo an unimportant constant, in fact it can be easily seen that we have the inequality M −d |X| ≥ dk (X) holding quite generally. The inequality (4.11) and the bounds (4.9) yield a bound of a term of the localization expansion. We would like to write it in terms of powers of β and η, using the inductive hypothesis (H.6)[4]. We have the bound pn1 (β) ≤ An1 c2 (np1 , α)β α holding for any α > 0 and β large enough, e.g. β ≥ eA0 . It is easy to find a formula for c2 (np1 , α), 1 np1 1 np1 , or to ( np , but it is unimportant. for example we can take it as equal to ( 2np eα ) α ) Using this bound and the inductive assumption (H.6)[4] with c8 small enough we obtain the bounds X α1 1 − 1 +α1 d−2 −α1 (k) ) β 2 η 2 V (z, Y ) < 2 · 3d A31 B53 K32 L2d M d c2 (3p1 , |σ|. d − 2 c7 c z∈Y ∩Zk
(4.12) · exp(−(2κ − 1)dk+1 (Y )) = const.β
− 21 +α1
η
d−2 2 −α1
|σ| exp(−(2κ − 1)dk+1 (Y )), c
˜ k+1 (Bk+1 ∩ Y ; δ, ε), where α1 can be chosen arbitrarily small. They hold on the spaces 9 where δ ≤ min{c7 , (32B1 K4 )−1 }, ≤ c0 , νk L2 ≤ δ, on the domain {ψ : |ψ| < 1 α1 ) c17 β − 2 +α1 p1 (βk )} of the characteristic function χ(k) and on the disc {σ : B5 c2 (p1 , d−2 d−2 2 −α1
|σ| ≤ 1}. Notice that if we demand that the expression in front of the exponential in (4.12) is ≤ 1, then this is a much stronger condition than the one defining the disc, so σ certainly belongs to the disc. Actually we will have to assume that the expression is sufficiently small. We will discuss later the corresponding restrictions again. Notice also that by the assumption νk ≤ ν ≤ 78 L−2 of Theorem 1 we have νk L2 ≤ 78 < 1. Hence the ˜ ck+1 (Bk+1 ∩ Y ; δ2 , δ2 ), where δ2 = min{c7 , (32B1 K4 )−1 }, satisfy the conditions spaces 9 formulated above. This means that they are the admissible analyticity domains, on which the bounds (4.12) hold. We have the inclusions η
c
c
˜ k+1 (Bk+1 ∩ Y ; 3εk , 3k ) ⊂ 9 ˜ k+1 (Bk+1 ∩ Y ; δ2 , δ2 ), if 3εk ≤ δ2 , 4c (Y, Bk+1 ; {εj }) ⊂ 9 discussed in Sect. 3 [1], so the spaces 4c (Y, Bk+1 {εj }) are also the admissible analyticity domains if εk ≤ 13 δ2 . Consider the second term in (4.1). A most important part of a construction of its localization expansion is covered again by Proposition 4.3 [3]. In order to apply it we have to reconsider Proposition 3.10 and Corollary 3.11 in [1]. The point is that we do not have the action Ek on the whole lattice, but the restricted action Ek (Zkc ) given by the formulas (1.7), (1.8) [5] in (H.2) [5]. In deriving the representations (3.126), (3.127) [1] we have used the fact that some terms are cancelled because of symmetries valid on the whole lattice. Now they are not, and we have to check if we can still interpret them as “irrelevant” terms which can be included into the sum in (3.127)[1]. Lemma 3.8[1] is again a basis of the analysis. We take a term E (j) (y, X; ψj ) in the sum (1.8) [5] and we apply the lemma with σ = C0 Lj η, ε = εk , assuming that j ≤ k − n1 . Let us recall that n1 was defined in [1] as a smallest integer satisfying the inequality (3.62)[1]. We obtain the representation (3.104)[1], and we sum these representations over the domains X ∈ Dj , X ⊂ (Zkc )≈ , y ∈ X, dj (X) ≤ d(ρ) = 2dC0Kα1 Lα M L(k+1)(1+α) . Then we add to both sides the sum over the remaining domains in (1.8) [5]. We obtain the following formula corresponding to (3.109)[1]:
28
T. Balaban
X
E (j) (y, X; ψj ) = V (j) (ψj (y))
X∈Dj :y∈X⊂(Zkc )≈
X
−
E (j) (y, X 0 ; ψj (y))
X∈Dj (ξZd ):dj (X 0 )>d(ρ) or X 0 ∩((Zkc )≈ )c 6=∅
X 1 (j) (j) 2 2 + V (y, x)(ψj (x)−ψj (y))+V2 (y, x)(νj h · (ψj (x))0 −νj h · (ψj (y))0 ) 2 1 x∈(Zkc )≈ D E X 1 V1(j) (y, ·, X), χ(Zkc )≈ (ψj2 (·) − ψj2 (y)) − 2 X∈Dj :y∈X,dj (X)>d(ρ) or X∩((Zkc )≈ )c 6=∅ D E (j) + V2 (y, ·, X), χ(Zkc )≈ (νj h · (ψj (·))0 − νj h · (ψj (y)0 )) 2 d d X X 1 (j,∞) (j,∞) (j,∞) −1 − V1 + ν j V2 ) ψ )(y) (∂ + (V0 µ j 2 2 µ=1 µ,ν=1 D E X V0(j) (y, ·, ·, X 0 ), x1,µ − yµ , x2,ν − yν X 0 ∈Dj (ξZd ):y∈X 0 ,dj (X 0 )>d(ρ) or X 0 ∩((Zkc )≈ )c 6=∅
−
D
V1(j) (y, ·, X 0 )
−
νj V2(j) (y, ·, X 0 ), (xµ
+
E
− yµ )(xν − yν )
X
(∂µ ψj )(y) · (∂ν ψj )(y)
(j) Eirrel (y, X; ψj )
X∈Dj :y∈X⊂(Zkc )≈ ,dj (X)≤d(ρ)
+
X
E (j) (y, X; ψj ).
X∈Dj :y∈X⊂(Zkc )≈ ,dj (X)>d(ρ)
(4.13) By the equality (3.125)[1] the fifth term on the right-hand side is equal to 0, and the third term is cancelled by the corresponding term in the difference (1.8) [5]. Terms of the third sum on the right-hand side with X ⊂ (Zkc )≈ are combined with the corresponding terms of the last sum, as in (3.116)[1], and their sums, or rather differences in this case, satisfy the same bounds as the “irrelevant” terms in the fifth sum, i.e. the bounds (3.105)[1] or (3.128)[1]. We denote them in the same way as the terms of the fifth sum, and we combine the two sums, as in (3.120)[1]. There are still some terms left in the third sum, we obtain the sum over all domains X ∈ Dj such that y ∈ X and X ∩ ((Zkc )≈ )c 6= ∅. Because of this the terms contribute to the boundary terms, and also the exponential factors in the bounds (3.77),(3.80)[1] yield arbitrarily high powers of βk−1 and Lj η, as it follows from the bounds dj (X) ≥ (Lj η)−1 (Rk − 2(L − 1)) > (Lj η)−1 + (log βk a)2 . The first term on the right-hand side of (4.13) is an irrelevant local term, V (j) (ψj (y)) = (j) Virrel (ψj (y)), by Proposition 3.4[1]. The function V (j) , like the renormalized coefficients, is defined independently of any local geometric structure. It is defined globally by the corresponding effective action on the whole lattice, so we combine this term with the L−1 (j) term Eirrel (y, 0 ; ψj ), where 0 = ∼ 2 , ∈ πj , y ∈ . Finally we consider the first and fifth sums. They are sums over domains in the whole lattice, and therefore the condition X 0 ∩ ((Zkc )≈ )c 6= 0 demands an explanation, the domain Zkc is in the torus T . We simply take the domain X 0 and “wrap” it around the torus starting at the point y. We obtain a domain X = projT (X 0 ) in the torus, and we assume that the above condition is
Renormalization and Localization Expansions
29
satisfied for this domain. The considered sums define also local functions of ψj , more precisely functions depending on ψj restricted to st(y), but this time they depend on the geometric structure determined by Ak , more precisely on the large field region Zk , so we (j) cannot combine them with the term Eirrel (y, 0 ; ψj ). Terms of the sums can be obviously bounded by arbitrary powers of Lj η, or βk−1 , or both, so they are irrelevant. We resum them over domains determining a fixed X ∈ Dj , and for X ⊂ (Zkc )≈ we combine the sum with the corresponding term in one of the last two sums in (4.13). For X such that X ∩ ((Zkc )≈ )c 6= ∅ we combine the sum with the corresponding term of the third sum on the right-hand side of (4.13). After all these definitions we obtain the representation X X 1 (j) V1 (y, x)(ψj2 (x) − ψj2 (y)) E (j) (y, X; ψj ) − 2 c X∈Dj :y∈X⊂(Zk )≈ x∈T (j) ∩(Zkc )≈ (4.14) X (j) (j) +V2 (y, x)(νj h · (ψj (x))0 − νj h · (ψj (y))0 ) = Eirrel (y, X; ψj ), X∈Dj :y∈X
where the terms on the right-hand side are analytic functions on the space 9cj (C0 Lj η, εk ) with the additional condition (3.107)[1]. The term corresponding to a domain X depends on ψj , h restricted to X ∩ (Zkc )≈ , and for X ⊂ (Zkc )≈ it satisfies the fundamental bound (3.128)[1]. For X such that X ∩ ((Zkc )≈ )c 6= ∅ it satisfies a much better bound, with a constant in front of the exponential which can be chosen smaller than any power of Lj η and βk−1 for βk large enough. For example we can take it as equal to (Lj η)d+1 βk−1 . The above statements on the representations and the bounds hold for j ≤ k − n1 , but they obviously hold also for k − n1 < j ≤ k, as in the inequality (3.131)[1], using the definition of n1 and the localization expansions of the functions V1(j) , V2(j) . Applying the representations (4.14) we write the second term in (4.1) as a sum of terms Z Z 1 1 dτ (j) −1 ˜ + ψ 0 )), dt0 E (y, X; ψk(j) (ψ (k) (θ) βk 2 σ 2 irrel 2πi τ |τ |=r 0 (4.15) −1
−1
ψ 0 = βk 2 σt0 tC (k) 2 ψ + τ tC (k) 2 ψ = (βk 2 σt0 + τ )tC (k) 2 ψ, 1
1
1
the sum over j = 1, . . . , k, y ∈ T (j) ∩ Zkc , X ∈ Dj such that y ∈ X. The functions (j) Eirrel (y, X; ψj , h) are analytic on the spaces {(ψ, h) : (ψ, h) ∈ 9cj (X ∩ (Zkc )≈ ; C0 Lj η, εk ), |∂ 2 ψ| < K1 Cα1 (Lj η)2−α1 εk } for j ≤ k − n1 , and they satisfy the bounds (3.128)[1]. For j > k − n1 , they are analytic on the spaces 4cj (Bj (X); 1, j ), and they satisfy the bounds in (H.4)[4]. Now we apply Proposition 4.3[3] to the function (4.15), or rather to the underintegral function, with a slight modification. To be in the framework of this proposition we use again the generating set (Bk (Wk ) ∩ (((Zkc )≈ )0 )c ) ∪ (((Zkc )≈ )0 )(k+1) as the basis of the construction of the localization extension. This is much more important here than in the case of the term V (k) (Zkc ), we need the inclusions X ⊂ X1 ⊂ (k + 1st domain) for the exponential decay property (4.52)[3], otherwise we would not have the improved factor 2κ in front 1 of the function dk+1 . This operator C (k) 2 remains unchanged. We use the fact that its construction and bounds are uniform with respect to k+1 . Also the configuration θ is defined on the set Bk+1 . Lemma 4.2[3] holds for the function in (4.15) with a slight modification. The first space in (4.33)[3] is replaced by the space 4c (Wk∼ , Bk+1 ; {εj }).
30
T. Balaban
This is connected again with the fact that X is not contained in k+1 generally. It may intersect ck+1 and we need slightly stronger conditions on θ restricted to a proper neighborhood of ∂k+1 . This modification of the lemma follows from a proof of a more general result of this type given in the next paper, see Lemma 2.5 there. The configuration −1
ψ 0 in (4.15) can be bounded by (βk 2 |σ| + r)B5 p1 (βk ) = δ1 , and the modified lemma holds if this δ1 satisfies the condition (4.38)[3]. Hence if the following two conditions are satisfied: 16B5 K2 K3 Lα
1 − 21 1 βk |σ|p1 (βk ) ≤ η α , 16B5 K2 K3 Lα p1 (βk )r = η α , αα0 αα0
(4.16)
the last equality determines the radius r in (4.15). Using again the bound p1 (β) ≤ c2 (p1 , α)β α and the inductive assumption (H.6)[4] with c8 small enough we obtain that the above inequality is satisfied if 32B5 K2 K3 Lα c2 (p1 ,
1 − 1 +α1 γ−α1 α1 ) β 2 η |σ| ≤ 1. d − 2 αα0
(4.17)
This condition yields restrictions on β and |σ|. We can use a fraction of the negative power of β to make the constant in front smaller than 1, and then the condition takes on a simpler form involving powers of β and η together with |σ|. From this we determine a restriction on |σ|. We will discuss it again later, when we will have all conditions on β and |σ|. Now we assume (4.17) and we apply the proof of Proposition 4.3[3] in the same way as in the construction of the expansion (4.8), with the same resummation (4.45)[3] over the domains Y1 determining a fixed domain Y ∈ Dk+1 (0k+1 ). We obtain a localization expansion of the function (4.15) of the form X E (j) (y, X, Y ; θ, h, ψ), (4.18) (4.15) = Y ∈Dk+1 (0k+1 ):X1 ⊂Y
The terms of this expansion are analytic functions of (θ, h) on the spaces 4c (Y, Bk+1 ; {εj }). They depend on θ, h, ψ restricted to Y , and they satisfy the bounds 1 p1 (βk )η −α E αα0 · exp(−κ0 dj (X) − 2κdk+1 (Y )), −1
|E (j) (y, X, Y ; θ, h, ψ)| < βk 2 |σ|16B5 K2 K3 Lα
(4.19)
where E is a bound of the function inside the integral in (4.15). This bound is equal either to Kα,α1 (Lj η)d+α3 E0 for j ≤ k − n1 , where Kα,α1 is given by (3.129)[1], or to E0 for j > k − n1 . Let us recall also that X1 is a domain in Dk+1 containing the 0 domain X. It is defined as a union of L2 M -cubes from the cover πk+1 , containing an j 0 LM L η-cube from the cover πj contained in X. To get a localization expansion of the second term in (4.1) we sum up all the localization expansions (4.18), and then we do the partial resummation of the terms with a fixed localization domain Y ∈ Dk+1 (0k+1 ). We obtain the sum k X
X
X
j=1 y∈Y
∩T (j) ∩Zkc X∈Dj :y∈X,X⊂(Zkc )≈ ,X1 ⊂Y
E (j) (y, X, Y ; θ, h, ψ),
(4.20)
which is equal to the term of the localization expansion corresponding to the domain Y . It is an analytic function of (θ, h) on the space 4c (Y, Bk+1 (Y ); {εj }), and of ψ on the
Renormalization and Localization Expansions
31
domain {ψ : |ψ| < p1 (βk )} and it depends on θ, h, ψ restricted to Y . We can estimate it in the same way as in (3.131)[1], using (4.19) and the bounds leading to (4.17), and we obtain the bound 1 1 α1 α 2d d ) Kα,α1 32B5 K0 K2 K3 L L (3M ) c2 (p1 , d − 2 αα0 α3 1 1 (4.21) +2(4dC0 K1 L1+α M )d β − 2 +α1 η γ−α1 |σ|E0 exp(−(2κ − 1)dk+1 (Y )) α = const.β − 2 +α1 η γ−α1 |σ|E0 exp(−(2κ − 1)dk+1 (Y )). 1
Later we will have to assume that the constant in front of the exponential is sufficiently small, and this will be the most restrictive condition on β and |σ|, much more restrictive than (4.17). Let us finish at first the discussion of the localization expansion for (4.1). The third term there can be dealt with in exactly the same way as the second term. We represent Fk by the sums in (H.3) [5], and we write each term in the form (4.15). There are two types of terms connected with the functions g(x) · ψ0 (x), g(x) · M(j) (x, X; ψj , h, g),
(4.22)
and the sum is over x ∈ j = 1, . . . , k, X ∈ Dj such that x ∈ Xand X ⊂ (Zkc )≈ . We apply Proposition 4.3[3] as above, assuming (4.17), and we obtain localization expansions of the form (4.18), which we resum as in (4.20), the sum over y in (4.20) replaced by a sum over x ∈ Y ∩ T1 . The resummation yields a localization expansion of the third term in (4.1), and the term of this expansion corresponding to the domain Y is an analytic function of (θ, h), ψ defined on the same space as the expression (4.20), and of g defined on the space {g : g is defined on T1 ∩ Y and has values in Cn , kgk`1 < 1}. This term can be bounded by T1 ∩Zkc ,
32B5 K2 K3 Lα c2 (p1 ,
1 − 1 +α1 γ−α1 α1 3 ) β 2 η |σ|( + 2K0 c9 ) exp(−2κdk+1 (Y )) d − 2 αα0 2
= const.β − 2 +α1 η γ−α1 |σ| exp(−2κdk+1 (Y )), 1
(4.23) and this bound has again the same general form as the bound (4.21). We combine the three localization expansions and we obtain a localization expansion of the function (4.1). We write it as X 1 −1 (k) (k) (Zkc ; βk 2 σ, C (k) 2 ψ) = Ut,u (Y ; ψ), (4.24) Ut,u Y ∈Dk+1 (0k+1 ):Y ∩k+1 6=∅
and the term of this expansion corresponding to a localization domain Y is an analytic function of (θ, h), ψ, g on the space 4c (Y, Bk+1 ; {εj }) × {ψ : |ψ| < p1 (βk )} × {g : kgk`1 < 1},
(4.25)
depending on these variables restricted to the domain Y . It satisfies the bound (k) (Y ; ψ)| < const.β − 2 +α1 η γ−α1 |σ| exp(−(2κ − 1)dk+1 (Y )) |Ut,u 1
(4.26)
with a constant which is a sum of the constants in (4.12),(4.21),(4.23). The positive number α1 above can be chosen as arbitrarily small. The above bound holds under the assumption (4.17), but now we assume that the expression in front of the exponential
32
T. Balaban
above is sufficiently small. A simple way to assure this is to take β and σ satisfying the conditions const.β − 4 +α1 ≤ 1, β − 8 η γ−α1 |σ| ≤ 1. 1
1
(4.27)
Then the expansion can be bounded by β − 8 , which is obviously arbitrarily small for β large enough. The above conditions are stronger than any condition formulated previously in this section, in particular (4.17). Let us remark that there is nothing special about the powers of β above. We could divide the power in (4.26) in many other ways among the three factors. We formulate the obtained localization results in the lemma. 1
−1
1
(k) (Zkc ; βk 2 σ, C (k) 2 ψ) given by the formula (4.1) has the Lemma 4.1. The function Ut,u localization expansion (4.24), whose terms can be extended to analytic functions of 1 (θ, h), ψ, g and σ, defined on the spaces (4.25) and on the disc {σ : β − 8 η γ−α1 |σ| ≤ 1}, depending on (θ, h), ψ, g restricted to the corresponding localization domains Y , and satisfying the bounds
(k) (Y ; ψ)| < ε exp(−(2κ − 1)dk+1 (Y )), |Ut,u
(4.28)
where ε > 0 can be chosen arbitrarily small. For example we can take ε = β − 8 and assume that β is large enough. 1
Let us discuss briefly constructions of localization expansions of the other expression defining the remaining measures in Sect. 3 [5]. In (3.10) [5] we have the sums of terms − 21 (k) 1 (k) ˜ C 2 ψ, h, g) over X ∈ Dk , X ⊂ Z c , and p ≤ n, where n ≤ m. R(k) p (X; ψ (θ) + β k
k
Notice that now we do not have the differences δR(k) p , but it does not matter because these terms are small anyway by (H.4) [5]. Notice also that for X ⊂ ck+1 they are equal to R(k) p (X; ψk ), so they do not depend on the fluctuation variable ψ and can be separated from the expectation value (3.11) [5]. We construct localization expansions for the remaining terms in exactly the same way as for (4.15); we use the same localization extension and the proof of Proposition 4.3[3]. We obtain expansions of the form (4.18) with terms satisfying the bounds (4.19), but with the constant in front of the exponential replaced by βk−p−1 . Actually the construction and the conditions are simpler now. We −1
do not introduce the parameter σ, so we take δ1 = βk 2 B5 p1 (βk ) and we have (4.16), (4.17) with |σ| replaced by 21 . They both hold because we assume the much stronger condition (4.27) on β. We resum the expansions over X and p, with a fixed domain Y , and we obtain a localization expansion of the form (4.24) satisfying Lemma 4.1, where −p− 21
the bounds (4.28) hold with the constant mK0 (3L2 )d βk−p−1 < βk ε=
, so we can take
−p− 1 βk 2 .
The measure in the expectation value (3.9) [5] has a most complicated form. The expression in the exponential is the sum of the expression analyzed above, with t = u = 1 and n = m, and the expression inside the expectation value in (3.9) [5] multiplied by the parameter u. This expression can be written in the following form:
Renormalization and Localization Expansions
33
D E 1 1 −1 V (k) (Zkc ; ψ) = βk2 ψ, C (k) 2 (aL−2 Q∗ Q + 1(k) )δ0(k) (βk 2 ψ) E 1 D −1 −1 − βk δ0(k) (βk 2 ψ), (aL−2 Q∗ Q + 1(k) )δ0(k) (βk 2 ψ) 2 Z Z 1 1 dτ (k) c − 21 (k) 1 − 21 2 ψ + (−t + τ β )δ0(k) (β dt V (Z ; β C ψ)) + k k k k 2πi |τ |=r1 τ 2 0 Z Z 1 1 dτ − 21 (k) 1 − 21 ˜ 2 ψ−(t + τ )δ0(k) (β dt (Ek +Fk +R(k) )(Zkc ; ψ (k) (θ)+β + k C k ψ)). 2 2πi |τ |=r τ 0 (4.29) We have used here the formulas (3.5), (3.6) [5]. The above expression differs only in some obvious details from the expression (4.1), and it is clear that its localization expansion can be constructed in exactly the same way. We need to discuss only these 1 small differences, and they are connected mainly with the function δ0(k) (βk2 ψ). Let us recall that it is given by the last equality in (2.36) [5], that is by 1 − 21 (k) − 2 (k) 21 (k) 21 0 (k) ˜ (k) 0 −C ( , 0) βk ψ+ ψ (θ)−ψ ( , 0; θ) on 0 , (4.30) δ0 (βk ψ) = C hence it can be bounded by any power of βk−1 for βk large enough, as it has been noticed in Sect. 2 [5]. We introduce the localization parameters into the expressions in (4.29) as before. The extension of the above function is constructed in the obvious way: we take 1 ˜ and the same extensions but with s = 0 outside the usual extensions for C (k) 2 , ψ (k) (θ), 1 the domain Rk+1 (0 ) for C (k) 2 (0 , 0), ψ (k) (0 , 0; θ). For the extended function (4.30) we have the bound (k) − 1 δ0 (β 2 ψ, s) < β −m−4 , (4.31) k k and we take r1 = βkm+2 , r = βkm+3 in (4.29). We construct the localization expansions of the expressions in (4.29) in the same way as before, even with some simplifications because we do not introduce the parameter σ and we have some bounds smaller. For −1
example, the configurations ψ 0 (s) in (4.29) can be bounded by βk 2 B5 p1 (βk ) + (1 + −1
βkm+3 )βk−m−4 < 2B5 βk 2 p1 (βk ) = δ1 , and the corresponding conditions on δ1 are now weaker than in the case of (4.1). Terms of the localization expansions satisfy stronger −1
bounds now, (4.9) holds with βk 2 |σ| replaced by 8βk−m−2 , and the corresponding change for (4.12), (4.19) holds with the whole constant in front of the exponential replaced by βk−m−3 E, and the corresponding change in the bound (4.21), similarly for (4.23). Localization expansions of the first two expressions on the right-hand side of (4.29) can be constructed in the same way as for V (k) , and they satisfy the same bounds. Thus we obtain Lemma 4.2. The function V (k) (Zkc ; ψ) given the formula (4.29) has the localization expansion of the form (4.24), whose terms V (k) (Y ; ψ) have analytic extensions onto the spaces (4.25), depend on (θ, h), ψ, g restricted to Y , and satisfy the bounds (4.28) with −m− 23
ε = const.βk−m−2 , or with ε = βk
.
Finally we consider the expectation value (3.2) [5]. The measure is determined by the exponential on the left-hand side of (3.1) [5], in which the term B 00(k) is multiplied by the parameter u. The expression in the exponential is equal to the sum of the expressions analyzed above and the expression
34
T. Balaban
1
δψ, 1(k) (1 (Wk ) ∩ Zk )δψ 2 1 η ˜ − βk (δφk (δψ))+ + (δφk (δψ))− , ∂ φk+1 (θ) 2 st(Z c )
βk
k
+ uB
− 1 (k) 1 (k) (Zk , Ak ; ψloc (θ) + βk 2 Cloc 2 ψ), − 1 (k) 1 ˜ δψ (k) (θ) ˜ δψ = βk 2 Cloc 2 ψ − δψ (k) (θ),
00(k)
˜ − ψ (k) (θ) on B(3k+1 ). = ψ (k) (θ) loc (4.32) Let us recall that B 00(k) is the sum of the boundary terms in the expansion (1.34) [5] of the inductive hypothesis (H.7) [5] with localization domains X ∈ Dk (modck ), X∩Zk 6= ∅, X ∩ k+1 6= ∅. It is again obvious how to construct localization expansions for the first two terms in (4.32). We pre-localize the first term in cubes 0 ∈ πk+1 , 0 ⊂ k+1 , for example, by multiplying the first factor δψ by χ0 , the second term in the intersections 0∗ ∩ st(Zkc ), we introduce the same localization extensions as before, we apply Proposition 4.3 [3] and perform proper resummations. We obtain localization expansions of the same form (4.24) but with the domains Y satisfying the conditions Y ∩ k+1 6= ∅, Y ∩ Zk 6= ∅. Therefore terms of the expansions can be bounded by arbitrarily large powers of βk−1 using the exponential decay factors. We need to discuss a localization expansion of the last term in (4.32) in more detail, because it is slightly (k) (θ) + different from the previous expansions. Take a boundary term B (k) (X, Ak ; ψloc where
−1
(k) 1
βk 2 Cloc 2 ψ) in the sum defining B 00(k) in (4.32). We have displayed the dependence on the spin variable restricted to B(3k+1 ), but it depends also on all spin variables corresponding to the multi-index Ak and restricted to X ∩ ck+1 , and on h, g restricted to X. By the definition the domain X satisfies the conditions X ∩ k+1 6= ∅, X ∩ Zk 6= ∅, so the exponential factor exp(−dk (Xmod ck )) can be bounded by exp(−Rk ), hence by a large power of βk−1 , and the boundary term can be bounded, for example, by βk−2 exp(−(κ−1)dk (Xmod ck )). At first we consider this term as localized in X ∩k+1 , we introduce the same localization extension as before, and we apply the proof of Proposition 4.3[3]. The domain X ∩ k+1 may be not connected now, it may have several components, but the proof applies without changes, except that the domains Y1 may have also several components, each of them containing at least one component of X ∩ k+1 . To each Y1 we assign the corresponding domain Y ∈ Dk+1 (modck+1 ) determined as in (4.44)[3]. We perform the same resummation (4.45)[3] controlled by the inequality (4.50)[3]. The inequalities (4.48), (4.49)[3] hold with dj (X), d˜k+1 (Y ) replaced by dk (Xmodck ), dk+1 (Y modck+1 ), and we obtain the localization expansion of the above boundary term over the domains Y containing X. Terms of this expansion are analytic functions on appropriate spaces described in (H.7) [5], and can be bounded by βk−2 exp(−κ0 dk (Xmodck ) − 2κdk+1 (Y modck+1 )). We sum up all these expansions for the boundary terms in B 00(k) , and for a fixed Y we resum over the domains X. This contributes the factor K0 M1 d |Y ∩ k+1 | ≤ K0 (3L2 )d (1 + dk+1 (Y ∩ k+1 )) < K0 (3L2 )d exp dk+1 (Y modck+1 ). Thus we have constructed the localization expansion X B 00(k) (Y, Ak ; ψ), (4.33) (4.32) = Y ∈Dk+1 (modck+1 )
whose terms satisfy the above analyticity properties, and the bounds 3 00(k) B (Y, Ak ; ψ) < β − 2 exp(−(2κ − 1)dk+1 (Y modck+1 )). k
(4.34)
Renormalization and Localization Expansions
35
We have finished the constructions of the localization expansions and we have represented the “interaction potentials” of the measures in (3.15), (3.14), (3.11), (3.9) [5] in the form (4.24), with terms satisfying the bounds (4.28). For the measure in (3.2) [5] this “potential” is a sum of two expressions, the first of the form (4.24), the second of the form (4.33), with terms satisfying the corresponding bounds (4.28), (4.34). This yields the most important parts of constructions of the complete localization expansion discussed in the propositions of Sect.3 [5]. The remaining parts are quite universal and are obtained by applying a form of a “cluster expansion”, see [7,9] for a general discussion. Proof of Proposition 3.4 [5]. We start again with the most important, and most difficult expectation values (3.14), (3.15) [5], which we discuss in detail, and then we discuss briefly the remaining ones. We simplify the notations in Sect.3 [5] omitting the domains k+1 , 00k+1 , the spin variables and the “primes”. For example, we write E1k+1 (z) instead of E10(k+1) (1k+1 (z), k+1 , 00k+1 ; θ, ψ) in (3.17) [5]. The statements of Proposition 3.4 [5] concerning the functions E10(k+1) (Zkc ∩ ck+1 ), E00(k+1) (z) follow from the corresponding statements for E1(k+1) (z), z ∈ Zkc ∩ T (k+1) , and from the results of Sect. 3[3] on the function D(k) , so we consider now E1(k+1) (z). It can be written as the expectation value Z E1(k+1) (z) =
Z
1
ds 0
−1
1
(k) dµs,0 (ψ)Us,0 (z; βk 2 , C (k) 2 ψ),
(4.35)
where −1
1
(k) (z; βk 2 σ, C (k) 2 ψ) Us,0 Z 1 ∂3 − 21 − 21 0 0 (k) 0 (k) 21 = βk σ 1 dt (1 − t ) V (z); β σt sC ψ , k+1 k ∂ψ 0 ∂ψ 0 ∂ψ 0 0 1 1 1 sC (k) 2 ψ, sC (k) 2 ψ, C (k) 2 ψ ∂ − 21 − 21 (k) ˜ (k) 21 (k) 21 +βk σ Ek ψ ,C ψ . 1k+1 (z); ψ (θ) + βk σsC ∂ψ 0
(4.36)
−1
1
(k) (Zkc ; βk 2 σ, C (k) 2 ψ) This function differs only insignificantly from the function Us,0 given by (4.1). The main difference is that it is localized in the cube 1k+1 (z) instead of Zkc . A localization expansion for this function is constructed in the same way as for the function (4.1). Basically it is the same expansion with some obvious minor changes, like that the domains Y must contain the cube 1k+1 (z). Thus we have the expansion of the form X 1 −1 (k) (k) (z; βk 2 σ, C (k) 2 ψ) = Us,0 (z, Y ; ψ), D0 = Dk+1 (0k+1 ), (4.37) Us,0 Y ∈D 0 :z∈Y
whose terms satisfy all the conditions of Lemma 4.1, including the bounds (4.28). Now we are ready to construct a localization expansion for the function E1(k+1) (z). We substitute the expansion (3.37) [5] into the formula (3.35) [5], and we obtain X Z 1 Z (k+1) (k) ds dµs,0 (ψ)Us,0 (z, Y ; ψ). (4.38) E1 (z) = Y ∈D 0 :z∈Y
0
36
T. Balaban
Consider the term of the above sum corresponding to a localization domain Y0 . The expectation value with respect to the measure dµs,0 is not a localized function of (θ, h), because the “interaction potential” (4.1) of the measure depends on (θ, h) on the whole domain Wk∼ . We write a localization expansion of this expectation value by expanding with respect to the terms of the sum in (4.24). We obtain Z X Y Z 1 ∂ (k) (k) (z, Y0 ; ψ) = dtY (Zs,0 (t))−1 . dµs,0 (ψ)Us,0 ∂t Y 0 D⊂D 0 Y ∈D " # (4.39) Z X 1 (k) (k) (k) 2 · dψχ exp − kψk + tY Us,0 (Y ; ψ) Us,0 (z, Y0 ; ψ). 2 Y ∈D
(k) (t) is given by the integral with respect to Let us recall that the normalization factor Zs,0 (k) ψ above, with the function Us,0 (z, Y0 ; ψ) replaced by 1, and the integration is restricted also to the domain 00k+1 , which we do not write explicitly. Let us define a domain Z by the equality [ Y ∪ Y0 . (4.40) Z= Y ∈D
Generally it is not connected, so we may write it as a union of connected components Z = Z0 ∪ Z1 ∪ . . . ∪ Zn , where Z0 is the component containing the domain Y0 . It is easy to see that the integral with respect to ψ factorizes into a product of integrals with integrations restricted to the components of Z, and the integral of the “free” Gaussian factor χ(k) exp[− 21 kψk2 ] over the complement Z c . The same is true for the normalization (k) (t), and the integrals restricted to the domains Z1 , . . . , Zn and Z c are equal factor Zs,0 to the corresponding integrals for the numerator. Thus these integrals cancel, and the integrations in the expectation value on the right-hand side of (4.39) are restricted to the domain Z0 . This means that this expectation value does not depend on parameters tY for Y ⊂ Z1 ∪ . . . ∪ Zn , so if there are components of Z other than Z0 , then the corresponding term in (4.39) vanishes, because of the derivatives ∂t∂Y . Therefore we restrict the sum in (4.39) to subfamilies D such that the domains Z given by (4.40) are connected. Then they are also localization domains from D0 . The terms in (4.39) corresponding to a domain Z depend on (θ, h) restricted to Z, so the sum has the basic property of a localization expansion. It can be written as a sum over domains Z ⊃ Y0 , and for a fixed Z as a sum over subfamilies D determining the domain Z through (4.40). The last sum defines the term of the localization expansion corresponding to the domain Z, thus we have Z 1 Z X (k) ds dµs,0 (ψ)Us,0 (z; Y0 ; ψ) = E1(k+1) (z, Z, Y0 ), (4.41) 0
Z∈D 0 :Y0 ⊂Z
where E1(k+1) (z, Z, Y0 ) Z 1 1 1 1 ds dtY dτY (Z (k) (Z, τ ))−1 = 2πi |τY |=rY (τY − tY )2 s,0 0 0 0 D⊂D Y ∈D " # Z X 1 (k) (k) · dψ Z χ(k) (Z) exp − kψk2Z + τY Us,0 (Y ; ψ) Us,0 (z, Y0 ; ψ), 2 X 0Z
Y Z
Y ∈D
(4.42)
Renormalization and Localization Expansions
37
and the “prime” over the first sum above means that the summation is restricted to the all subfamilies D determining the domain Z by the equality (4.40). We take the radii 1 rY = ε− 2 exp((2κ − 1 − 5κ0 )dk+1 (Y )) and we obtain the bounds 1
(k) (Y ; ψ)| < ε 2 exp(−5κ0 dk+1 (Y )). |τY Us,0
(4.43)
Now we estimate the function (4.42). We have X
Y 3 1 ε 2 exp(−(2κ − 1 − 5κ0 )dk+1 (Y )). 2 D⊂D 0 Y ∈D " # X 1 ·ε exp(−(2κ − 1)dk+1 (Y0 )) exp ε 2 exp(−5κ0 dk+1 (Y ))
|E1(k+1) (z, Z, Y0 )| ≤
Z ·
0
(4.44)
Y ∈D
1 (k) 2 dψ Z χ (Z) exp − kψkZ sup |Zs,0 (Z, τ )|−1 . 2 D,τ (k)
Consider the product over Y ∈ D of the exponential factors above, and take a graph whose vertices are domains of the set D ∪ {Y0 }, and whose edges are pairs of the 00 00 domains {Y 0 , Y } such that their intersections are non-empty, Y 0 ∩ Y 6= ∅. It is a connected graph, because the domain Z is connected. Take a connected subgraph of this graph, which is a tree graph, and which has the same set of vertices D ∪ {Y0 }. Choose one vertex as an initial vertex, for example take Y0 . This determines an order among vertices of each branch of the tree graph beginning at Y0 , so the edges may be replaced 00 by ordered edges. Take such an ordered edge (Y 0 , Y ) and two admissible tree graphs 00 00 1 1 00 , 0 contained in the corresponding domains, and such that M |00 |, M |0 | are close enough to dk+1 (Y 0 ), dk+1 (Y 00 ). Errors may be chosen to be arbitrarily small. The lengths 00 here are taken in the L−1 η-scale. The intersection Y 0 ∩ Y is non-empty, so it contains 00 a cube from the partition πk+1 , and this cube is contained in two cubes 0 , from 00 00 0 0 0 the cover πk+1 , such that ⊂ Y , ⊂ Y . By the definition of admissible graphs, 00 00 00 see (2.27)[1], 00 intersects 0 and 0 intersects . We choose some points x0 , x in the corresponding intersections, and connect them by one of the shortest paths 0x0 ,x00 . For the length in the L−1 η-scale of this path we certainly have |0x,x0 | < 2dLM , hence 00 0 1 00 00 M |0x0 ,x | < 2dL. We connect the graphs 0 , 0 by the path 0x0 ,x , obtaining the 00
00
connected graph 00 ∪ 0x0 ,x00 ∪ 0 , and we assign the number 2dL to the vertex Y . Doing this for all edges of the considered tree graph we obtain a connected graph 0 0 contained in Z. We can choose a subgraph intersecting every cube from the cover πk+1 of 0 which has the same property, and which is a connected tree graph, so we have dk+1 (Z) ≤
X 1 |0| < (dk+1 (Y ) + 2dL) + dk+1 (Y0 ) + (error) M Y ∈D
(the error may be arbitrarily small). From this we can bound the product of the exponential factors in (4.44) by Y 3 1 e4dLκ ε 2 exp(−κ0 dk+1 (Y )) · ε exp(−6κ0 dk+1 (Y0 )) exp(−(2κ − 1 − 6κ0 )dk+1 (Z)). 2
Y ∈D
38
T. Balaban
The third exponential in (4.44) can be bounded by exp K0 ε 2 M −d |Z|, the volume |Z| taken in the L−1 η-scale. This bound is independent of D, so the sum over D in (4.44) is controlled by the product over Y ∈ D of the exponential factors above. We have X0 Y ε0 exp(−κ0 dk+1 (Y )) 1
D ∞ X 1 ≤ n! n=0
X
n Y
Y ∈D
ε0 exp(−κ0 dk+1 (Yi )) = exp
(Y1 ,... ,Yn ):Yi ⊂Z i=1
X
ε0 exp(−κ0 dk+1 (Y ))
Y ⊂Z
(4.45) 1 ≤ exp K0 ε0 d |Z|, M 1
and we take ε0 = 23 e4dLκ ε 2 in the above bound. Using the inequality (4.11) we can 1 bound further the last exponential by exp 23 K0 e4dLκ (3L)d ε 2 (1 + dk+1 (Z)). The previous 1 exponential can be bounded by exp K0 (3L)d ε 2 (1 + dk+1 (Z)), so combining the two 1 bounds we obtain, for example, the bound exp 2K0 e4dLκ (3L)d ε 2 (1 + dk+1 (Z)). We 1 assume that ε is so small that the constant in the exponent is ≤ 1, i.e. 2K0 e4dLκ (3L)d ε 2 ≤ 1. Then the bound is exp(1 + dk+1 (Z)). Combining the above bounds we obtain the inequality (4.44) with the right-hand side replaced by eε exp(−6κ0 dk+1 (Y0 )) exp(−(2κ − 2 − 6κ0 )dk+1 (Z)) multiplied by the last two expressions. It is not so easy to estimate the last expression, (k) because it means that we have to obtain a lower bound for |Zs,0 (Z, τ )|. It seems that the simplest way to obtain such a bound is to use a cluster expansion to represent the (k) (Z, τ ) in an exponential form. We write “partition function” Zs,0 Z
−1 1 (k) dψ Z χ(k) (Z) exp − kψk2Z Zs,0 (Z, τ ) 2 " # Z X 1 −1 (k) 2 = Z0 (Z) dψ Z χ (Z) exp − kψkZ + U0 (Y ; ψ) , 2
(4.46)
Y ⊂Z
(k) (Y ; ψ), and we put τY = 0 for Y ⊂ Z, Y 6∈ D. Thus the where U0 (Y ; ψ) = τY Us,0 functions U0 (Y ; ψ) satisfy the bounds 1
|U0 (Y ; ψ)| < ε 2 exp(−5κ0 dk+1 (Y )).
(4.47)
We use the cluster expansion in a “polymer” form described in many places, for example in [7,9]. It is now completely standard, and we mention only a few main points, and some conclusions. A first step is the expansion (4.39), which is now much simpler, and the derivatives can be calculated explicitly. After a resummation we obtain a “polymer” expansion X U(Z1 ) · . . . · U(Zn ), (4.48) (4.46) = {Z1 ,... ,Zn }
where the sum is over sets of disjoint localization domains {Z1 , . . . , Zn } such that Z1 ∪ . . . ∪ Zn ⊂ Z. The functions U(Z 0 ) are given by simple explicit formulas and satisfy the bounds
Renormalization and Localization Expansions
39
|U(Z 0 )| < 2 · 3d eK0 Ld ε 2 exp(−(4κ0 − 1)dk+1 (Z 0 )). 1
(4.49)
These bounds assure that we can exponentiate the expansion (4.48) ∞ X 1 (4.46) = exp n! n=1
X
ρT (Z1 , . . . , Zn )U (Z1 ) · . . . · U(Zn ),
(4.50)
(Z1 ,... ,Zn )
and the series in the exponent is convergent. See one of the references [7,9] for a defi1 nition of the functions ρT . In our case the series can be bounded by O(1)ε 2 M −d |Z| < 1 O(1)(3L)d ε 2 (1 + dk+1 (Z)), and for ε sufficiently small we conclude that |(4.46)| > e−1 exp(−dk+1 (Z)).
(4.51)
This completes the estimate of the function E1(k+1) (z, Z, Y0 ), and we obtain |E1(k+1) (z, Z, Y0 )| < e2 ε exp(−6κ0 dk+1 (Y0 )) exp(−(2κ − 3 − 6κ0 )dk+1 (Z)).
(4.52)
The last step is to combine the expansions (4.38),(4.41), which gives us the localization expansion X E1(k+1) (z, Z), where E1(k+1) (z) = Z∈Dk+1 (0k+1 ):z∈Z
E1(k+1) (z, Z) =
X
(4.53)
E1(k+1) (z, Z, Y0 ).
Y0 ∈Dk+1 :z∈Y0 ⊂Z
The functions E1(k+1) (z, Z) satisfy all the conditions of Lemma 4.1. In particular they are analytic functions on the spaces (4.25) properly restricted, and with Y replaced by Z, and they satisfy the bounds |E1(k+1) (z, Z)| < e2 K0 ε exp(−(2κ − 3 − 6κ0 )dk+1 (Z)).
(4.54)
Let us notice that the constant e2 K0 ε is already quite small by all the assumptions on ε we have introduced. In fact it can be arbitrarily small for β large enough. Let us recall 1 also that these functions are analytic functions of σ on the disc {σ : |σ| ≤ β 8 η −γ+α1 }, and they are equal to 0 for σ = 0. From this we can obtain the additional small factor 1 2β − 8 η γ−α1 in the bound (4.54). Actually, by the formulas (4.1), (4.36), we should have −1
a constant proportional to βk 2 , or even to βk−1 , in (4.54), but proving this would require −1
“clearing out” the first few orders of a perturbation expansion in βk 2 for the function E1(k+1) . Consider the function E00(k+1) (z) = − 21 D(k) (z)+E1(k+1) (z) for z ∈ 3k+1 , where D(k) (z) is given by the formula (2.60)[4]. We construct the localization extension of D(k) (z) in the same way as all the above extensions, by using Proposition 3.1 [3], and then we use the proof of Proposition 4.3 [3]. This yields a localization expansion of the form (4.53), but with terms depending on (θ, h) only, and satisfying the bounds (4.54) with 2κ instead of 2κ−3−6κ0 , and with the constant in front of the exponential equal to N Ld B5 , where B5 is given by the obvious slight modification of the formula in (3.24)[3]. Combining the two expansions we obtain an expansion of the form (4.53) for E00(k+1) (z), where terms satisfy (4.54) with the constant N Ld B5 + 1. We perform one more resummation in this expansion. For a fixed domain Y ∈ Dk+1 ( modck+1 ) we resum over all domains Z such
40
T. Balaban
that Z “determines” Y , i.e. Z ∩ k+1 = Y ∩ k+1 and Z has a non-empty intersection with each component of ck+1 contained in Y . This resummation yields the following final localization expansion: E00(k+1) (z) =
X
E00(k+1 (z, Y ).
(4.55)
Y ∈Dk+1 (modck+1 ):z∈Y
Let us recall that the terms depend also on the domains Zkc , k+1 , 00k+1 , or rather on their intersections with Y , on the configurations (θ, h) restricted to Y ∩ Wk∼ , ψ = ψk0 restricted to Y ∩ k+1 ∩ 00c k+1 . They are analytic functions on the corresponding spaces in (4.25), and they satisfy the bounds |E00(k+1) (z, Y )| < (N Ld B5 + 1)K0 exp −(2κ − 3 − 7κ0 )dk+1 (Y mod ck+1 ) . (4.56) It is clear from the construction of (4.55) that if Y ⊂ 00k+1 , then the term with this domain is equal to the term with the same domain in the localization expansion constructed on the whole lattice, i.e. for 00k+1 = k+1 = Zkc = T . This ends the proof of the part of Proposition 3.4 [5] concerning the functions E00(k+1) (z), z ∈ 3k+1 . To construct an expansion of E10(k+1) (Zkc ∩ ck+1 ) we sum up the expansions (4.53) over z ∈ Zkc ∩ ck+1 . Then for a fixed Z we resum over z ∈ Z ∩ Zkc ∩ ck+1 , and finally for a fixed Y ∈ Dk+1 (modck+1 ) we resum over Z as above. Notice that the domain Y0 in (4.41), (4.53) satisfies the additional condition Y0 ∩ k+1 6= ∅, otherwise the corresponding term in (4.37) is equal to 0, hence the domains Z, Y satisfy this condition also. After the resummations we obtain an expansion of the form X
E10(k+1) (Zkc ∩ ck+1 ) = Y
∈Dk+1 (modck+1 ):Y
∩k+1 6=∅,Y
E00(k+1) (Y )
(4.57)
∩ck+1 6=∅
whose terms have the same properties as for (4.55), except that the bounds (4.56) hold with 2κ − 4 − 7κ0 in the exponential, and the constant e2 K02 (3LM )d ε in front of it. We can bound this constant by 1 for ε small enough, i.e. β large enough, and we obtain the statements of Proposition 3.4 [5] for this function. It is clear that the above analysis can be applied in an identical way to the function (3.15) [5], or the corresponding functions M00(k+1) , but there is a problem with bounds for the localization expansions. We can obtain bounds of the form (4.23), with 2κ replaced by 2κ − 3 − 6κ0 , which is not enough for the inductive hypothesis (H.7)[4], because it d−2 assumes a higher power of the scaling factor L−1 η. We should have (L−1 η) 2 +γ−α1 −1 γ−α1 instead of (L η) in (4.23). We could weaken the inductive assumption. The exponent γ − α1 is enough to control the sum in (H.7)[4], and the thermodynamic limit, but in the future we will need the exponent d−2 2 + γ − α1 to prove more precise statements on correlation functions. We can obtain better bounds performing one order higher ex−1
pansion in βk 2 , and using more carefully the analyticity assumptions in (H.7)[4]. We start with the formula (3.15) [5] and we expand the underintegral function up to the first −1
order in βk 2 . We simplify the notations again omitting the domains, the “primes”, etc., and we write
Renormalization and Localization Expansions
41
∂ (k) 21 = dt dµ1,t (ψ) δφk (x; 0), C ψ ∂ψ 0 0 + * k X ∂ (j) (j) c (k) ˜ 0 (k) 21 (x, Zk ; ψk (ψ (θ) + ψ ), g) M ,C ψ + ∂ψ 0 ψ 0 =0 j=1 Z 1 1 1 ∂2 −1 − 21 0 (k) 1 2 ψ), C (k) 2 ψ, C (k) 2 ψ [3pt] + βk 2 t dt0 (x; β δφ tt C k k ∂ψ 0 ∂ψ 0 0 k Z 1 X 1 ∂2 −1 (j) ˜ + β − 2 tt0 C (k) 21 ψ), g), dt0 M (x, Zkc ; ψk(j) (ψ (k) (θ) [3pt] + βk 2 t k 0 0 ∂ψ ∂ψ 0 Z
1
Z
M(k+1) (x; g) 0
j=1
1
1
C (k) 2 ψ, C (k) 2 ψ
. (4.58)
Consider the last two terms. The underintegral expression can be represented by the Cauchy formula
−1
Z
βk 2 2t −1
1
dt0
0
+βk 2 2t
1 2πi
k Z X j=1
1
Z |τ |=r0
dt0
0
1 2πi
1 1 dτ −1 δφk (x; βk 2 tt0 C (k) 2 ψ + τ C (k) 2 ψ) 3 τ
Z |τ |=rj
−1
dτ (j) ˜ M (x, Zkc ; ψk(j) (ψ (k) (θ) τ3
(4.59)
+βk 2 tt0 C (k) 2 ψ + τ C (k) 2 ψ), g). 1
1
We construct a localization expansion of this function in the usual way, using the same analytic extensions as before, the localization expansions (1.12) [5] of the functions −1
M(j) , and Proposition 4.3[3]. The function ψ 0 (s) = βk 2 tt0 C (k) 2 (s)ψ + τ C (k) 2 (s)ψ can 1
1
− 21
be bounded as before, |ψ 0 (s)| < βk B5 p1 (βk ) + rj B5 p1 (βk ) = δ1 , but we choose the radii rj in different ways now. For the first term we assume that δ1 ≤ C3 c7 , and we −1
take r0−1 = 2B5 p1 (βk )(C3 c7 )−1 . We assume also that βk 2 B5 p1 (βk ) ≤ 21 C3 c7 , but we have already much stronger conditions on βk , or β. Consider a j th term in the sum in (4.59), or rather a term with the localized function M(j) (x, X), and let us look into the proof of Proposition 4.3[3]. The proof was given with functions E from the effective actions in mind. Now we would like to use the fact that the considered functions are analytic on larger spaces 4cj (Bj (X); 1, εj ). The relevant condition is in (4.37)[3]. We replace the space in it by the same space with εj instead of εk , and then the condition is ε0 + (K2 + 1)δ3 ≤ εj . It is satisfied if the conditions (4.38)[3] hold with ξ α instead of η α , ξ = L−j , and they are satisfied if (4.16) hold with ξ α instead of η α . From the last 1 p1 (βk )ξ −α . With these choices of the radii we can we obtain rj−1 = 16B5 K2 K3 Lα αα 0 bound the term of the localization expansion of (4.59) corresponding to a domain Y by
42
T. Balaban −1
βk 2 2K3 C3 c7 (2B5 p1 (βk )(C3 c7 )−1 )2 exp(−2κdk+1 (Y )) +
k X
−1
βk 2 2K0 (16B5 K2 K3 Lα
j=1
<β
− 21 +α1 γ−α1
η
d−2 1 p1 (βk )ξ −α )2 c9 ξ 2 +γ−α1 exp(−2κdk+1 (Y )) αα0
B52 K3 c2 (2p1 ,
α1 1 1 2 ) + 4K0 (16B5 K2 K3 Lα ) d − 2 c7 αα0
1 α1 1 ) c9 exp(−2κdk+1 (Y )) = const.β − 2 +α1 η γ−α1 exp(−2κdk+1 (Y )). d−2 α (4.60) For the expectation value in (4.58) of this term we have the expansion (4.41), (4.42). Notice that it holds for any values of the parameter t. Terms of this expansion satisfy the 1 bounds (4.52) with ε = const. β − 2 +α1 η γ−α1 , and after the resummation (4.53) we obtain a localization expansion of the last two terms in (4.58). Terms of this expansion satisfy ·Lα c2 (2p1 ,
−1
the bounds (4.54). Let us recall that we have the additional factor βk 2 < 2β − 2 η 2 d−2 multiplying these terms, so altogether we get the right scaling factor η 2 +γ−α1 multiplied by const. β −1+α1 . The last expression is arbitrarily small for β large enough, we 1 1 can even assume that const. β − 2 +α1 ≤ 1 and we are left simply with β − 2 . Consider now the first two terms in (4.58). We write them in the form Hk C
(k) 21
Z
dt 0
(Hk C (k) 2 ψ 0 )(x) = 1
+
1 2πi
d−2
!
Z
1
1
dµ1,t (ψ)ψ (x), where Z |τ |=r0
1 dτ δφk (x; τ C (k) 2 ψ 0 ) 2 τ
(4.61)
Z k X 1 dτ (j) ˜ + τ C (k) 21 ψ 0 ), g). M (x; ψk(j) (ψ (k) (θ) 2πi |τ |=rj τ 2 j=1
1
We construct a localization expansion for the operator Hk C (k) 2 in the usual way, and 1 ξ −α |ψ 0 |. The term of this we take r0−1 = B5 (C3 c7 )−1 |ψ 0 |, rj−1 = 16B5 K2 K3 Lα αα 0 expansion corresponding to a domain Y can be bounded by X 1 0 1 −α |ψ | exp(−2κdk+1 (Y )) + 16B5 K0 K2 K3 Lα ξ . c7 αα0 j=1 d−2 1 ·c9 ξ 2 +γ−α1 |ψ 0 | exp(−2κdk+1 (Y )) ≤ B5 K3 C3−1 + 16B5 K0 c7 1 c9 |ψ 0 | exp(−2κdk+1 (Y )) = const. exp(−2κdk+1 (Y ))|ψ 0 |, ·K2 K3 Lα αα0 γ k
B5 K3 C3−1
(4.62)
and it depends on ψ 0 restricted to Y . We substitute the expectation value in (4.61) instead of ψ 0 in this expansion. Now we use for the first time the fact that the measure dµs,t can be analytically extended in the parameter σ. We display the dependence writing −1
explicitly dµs,t (βk 2 σ, ψ), and we have
Renormalization and Localization Expansions
Z
Z
−1
dµ1,t (βk 2 , ψ)ψ(y) = Z
1
= 0
Z
1 dt0 2πi
43
Z
1
dµ0 (ψ)ψ(y) +
dt0
0
1 dσ (σ − t0 )2 |σ|=r
Z
d dt0
Z
−1
dµ1,t (βk 2 t0 , ψ)ψ(y) (4.63)
−1
dµ1,t (βk 2 σ, ψ)ψ(y),
where we take r−1 = β − 8 η γ−α1 . We construct the corresponding localization expansion (4.41), (4.42) for the above expectation value, and the term of this expansion corresponding to a localization domain Z 3 y can be bounded by 1
2e2 β − 8 η γ−α1 p1 (βk ) exp(−(2κ − 3 − 6κ0 )dk+1 (Z)). 1
(4.64)
1
We combine the two expansions, for the operator Hk C (k) 2 and the expectation value above, and we obtain a localization expansion of (4.61), after proper resummations. Terms of this expansion can be bounded by const.β − 8 η γ−α1 p1 (βk ) exp(−(2κ − 4 − 7κ0 )dk+1 (Z)). 1
(4.65)
This completes the construction of the localization expansion for the function (4.58). We have obtained that X (x; g) = M(k+1) (x, X; g), (4.66) M(k+1) 0 0 X∈Dk+1 (modck+1 ):x∈X
where the terms are analytic functions of (θ, h), g on the spaces (4.25), and they satisfy the bounds (x, X; g)| < const.β − 8 +α1 η γ−2α1 exp(−(2κ − 4 − 7κ0 )dk+1 (Xmodck+1 )). |M(k+1) 0 (4.67) − 21 Multiplying (4.66) by βk we obtain the expansion whose terms satisfy the above bounds d−2 5 with the expression in front of the exponentials replaced by const. β − 8 +α1 . η 2 +γ−2α1 . 3 We can assume that e.g. const. β − 8 +α1 ≤ 1, and we obtain the bound in (H.7)[4] with 1 c9 = β − 4 , and κ replaced by 2κ − 4 − 7κ0 . Finally we assume that κ − 4 − 7κ0 ≥ 21 κ, and this ends the proof of Proposition 3.4 [5]. 1
Proof of Proposition 3.3 [5]. By the considerations in the paragraph after Lemma 4.1 we separate the sum of terms with domains X ⊂ ck+1 from the expectation value (3.11) [5], and the remaining sum has the form (4.24) with terms having the properties described in −n− 1
Lemma 4.1, the bounds (4.28) holding with ε = βk 2 . The measure in (3.11) [5] has exactly the same properties as the measures in (4.35), (4.58), so the proof of Proposition 3.4 [5] can be applied directly here. We obtain a sum, which in the simplified notations has the form X X 0(k+1) Rn−1 (Y0 , Z), (4.68) Y0 ∈D 0 :Y0 ∩k+1 6=∅ Z∈D 0 :Y0 ⊂Z
with the terms satisfying the bounds (4.52). Performing the resummations over Y0 with a fixed Z, and then over Z with a fixed Y ∈ Dk+1 (modck+1 ), we obtain the desired localization expansion over the domains Y : Y ∩ k+1 6= ∅. The constant in the bounds −n− 1
is equal to e2 K02 (3L)2d βk 2 < 21 βk−n , and we have 2κ − 5 − 8κ0 in the exponential, so we assume that κ − 5 − 8κ0 ≥ 21 κ, and we obtain all the statements of Proposition 3.3 [5].
44
T. Balaban
Proof of Proposition 3.2 [5]. It is the same as the above proof, the expectation value (3.9) [5] is the expectation value of the function V (k) (Zkc ; ψ). We apply Lemma 4.2, the proof of Proposition 3.4 [5], and we obtain an expansion of the form (4.68), without the −m− 3
2 . Performing the subscript “n-1”, whose terms satisfy the bounds (4.52) with ε = βk above described resummations, and using the above assumptions, we obtain Proposition 3.2 [5].
Proof of Proposition 3.1 [5]. The only difference here is that terms of the expansion (4.33) satisfy the bounds (4.34) with the function dk+1 (·modck+1 ) instead of the usual dk+1 (·), but this is unessential because this function satisfies the same basic resummation bound (1.31) [5]. Actually we could reverse the resummations in the previous proofs, and do the last ones at the beginning, then we would have in all cases the same situation as above. This does not change at all the proofs, we have to use only the modified concept of connectedness: two domains Y1 , Y2 ∈ Dk+1 (modck+1 ), are connected if Y1 ∩ Y2 ∩ k+1 6= ∅. With this concept all the formulas and bounds in the construction described between (4.37) and (4.54) are valid, with dk+1 (·modck+1 ) instead of dk+1 (·). In the case considered now we obtain the localization expansion (3.3) [5], whose terms −3
satisfy the bounds (3.4) [5] with the constant e2 K0 (3L)d βk 2 in front of the exponential, and 2κ − 4 − 7κ0 inside, hence the bounds (3.4) [5] exactly. We have proved all the propositions in Sect. 3 [5], which means that we have constructed all the localization expansions connected with the fluctuation integrals. There is quite a number of other localization expansions and cluster expansions in Sect. 2, and in the next paper, but they are all covered by the constructions described above, in particular by the ones used in the proof of Lemma 4.1. They are actually simpler because no fluctuation variables and integrals are involved. Almost all of them are straightforward consequences of the results of [3], for example localization expansions of the last three terms in (2.14) can be obtained by a direct application of Propositions 2.1, 4.3 [3]. The second term and the sum on the right-hand side of (2.19) are slightly more complicated, we have to use the representation (4.14) at first, and then apply the same constructions as for the expressions (4.15), using the remarks after (2.19). These constructions yield Proposition 2.1, and in a similar way we obtain Proposition 2.2. The most important localization expansion in Sect. 2 is the one discussed in Proposition 2.3 for the function given by (2.28), with the auxiliary definitions (2.27), (2.22), (2.23). Here the important terms are the function −βk+1 C (k+1) (1k+1 (z)) and the sum over j on the right-hand side of (2.27). These are almost identical with the first two terms in (4.1) pre-localized in 1k+1 (z), and we repeat the proof of Lemma 4.1, which yields the proposition. There will be several other expansions in the next paper, but they are of exactly the same type as the ones discussed above.
References 1. Balaban, T.: A Low Temperature Expansion for Classical N -vector Models. I. A Renormalization Group Flow. Commun. Math. Phys. 167, 103–154 (1995) 2. Balaban, T.: The Variational Problems for Classical N -Vector Models. Commun. Math. Phys. 175, 607–642 (1996) 3. Balaban, T.: Localization Expansions. I. Functions of the “Background” Configurations. Commun. Math. Physics 182, 33–82 (1996) 4. Balaban, T.: A Low Temperature Expansion for Classical N -Vector Models. II. Renormalization Group Equations. Commun. Math. Phys. 182, 675–721 (1997)
Renormalization and Localization Expansions
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5. Balaban, T.: A Low Temperature Expansion for Classical N -Vector Models. III. A Complete Inductive Description, Fluctuation Integrals. Commun. Math. Phys. 182, 675–721 (1997) 6. Balaban, T.: Commun. Math. Phys. a) 89, 571–597 (1983); b) 119, 243–285 (1988); c) 122, 175–202 (1989); d) 122, 355–392 (1989) 7. Brydges, D.: A Short course in Cluster Expansions. In Critical Phenomena, Random Systems, Gauge Theories. Les Houches (1984), London–New York: Elsevier Science Publishers, 1986 8. Brydges, D., Dimock, J., Hurd, T.: Weak Perturbations of Gaussian Measures. In: Mathematical Quantum Theory I. Field Theory and Many Body Theory. CRM Proceedings Lecture Notes 9. Cammarota, C.: Commun. Math. Phys. 85, 517–528 (1982) 10. Feldman, J., Magnen, J., Rivasseau, V., S´en´eor. R.: Commun. Math. Phys. 109, 473 (1987) 11. Abdessalam, A., Rivasseau, V.: An Explicit Large Versus Small Field Multiscale Cluster Expansion. Rev. Math. Phys. To appear 12. Rivasseau, V.: Cluster Expansions with Small/Large Field Conditions In: Mathematical Quantum Theory I: Field Theory and Many Body Theory. CRM Proceedings Lecture Notes 13. Glimm, J., Jaffe, A.: Quantum Physics: Functional Integral Point of View. New York: Springer, 1989 Communicated by D. C. Brydges
Commun. Math. Phys. 198, 47 – 81 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Topological Interpretations of Lattice Gauge Field Theory 1 ´ Doug Bullock1,? , Charles Frohman2 , Joanna Kania-Bartoszynska 1 Department of Mathematics, Boise State University, Boise, ID 83725, USA. E-mail:
[email protected];
[email protected] 2 Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. E-mail:
[email protected]
Received: 6 October 1997 / Accepted: 13 March 1998
Abstract: We construct lattice gauge field theory based on a quantum group on a lattice of dimension one. Innovations include a coalgebra structure on the connections and an investigation of connections that are not distinguishable by observables. We prove that when the quantum group is a deformation of a connected algebraic group G (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of G-characters of the fundamental group of the lattice. Finally, we investigate lattice gauge field theory based on quantum SL2 C, and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice. Introduction Lattice gauge field theory based on an algebraic group G is a finite element approximation of a smooth gauge field theory with G as its structure group. Infinitesimally varying connections and gauge transformations on a principal bundle are discretized via a lattice embedded in the base manifold. To each edge in the lattice a connection imparts an element of G encoding the holonomy along that edge. Gauge fields (i.e., functions on connections) are represented by a copy of the coordinate ring of G associated to each edge. The action of the gauge group is then concentrated at vertices. All computations become merely algebraic with analytic and geometric considerations swept aside. The end result is an algebra of observables (the gauge invariant gauge fields) that can be understood as the character theory for representations of the fundamental group of the lattice into G. Lattice gauge field theory based on a quantum group yields a deformation of this theory. Technically, the result is an algebra of observables which, with respect to the ? The first author is partially supported by an Idaho SBOE Specific Research Grant; the second and third by NSF-DMS-9204489 and NSF-DMS-9626818.
48
D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
standard Poisson structure [2, 11], gives a deformation quantization of the ring of Gcharacters. Here one must think of the fundamental group of the lattice as the fundamental group of a surface with boundary. In this paper we develop, from an elementary and computational viewpoint, the basic objects of lattice gauge field theory based on a quasitriangular ribbon Hopf algebra (e.g., a quantum group). We then use this foundation to begin the study of the structure of algebras of observables (paying particular attention to quantum groups) and to recognize the observables as algebras that have already been studied in a topological framework. The genesis of our approach can be found in the papers [1, 3, 4, 5, 10]. Fock and Rosly were the first to derive the Poisson structure on G-characters from a lattice gauge field theory. Their formula is written in terms of a solution of the modified classical Yang-Baxter equation [9]. Also, recall that the characters of a surface group are only a homotopy invariant while the Poisson structure is a topological invariant. For this reason, Fock and Rosly endow the lattice with extra information, called a ciliation, so that it determines a surface. Passing to quantum groups, Alekseev, Grosse and Schomerus defined an exchange algebra over a ciliated lattice so that basic elements of the algebra of gauge fields commute according to a solution of the quantum Yang-Baxter equation. This algebra is related to a quantization of the characters with respect to the usual Poisson structure. It is based on a full solution of the Clebsch-Gordan problem for the quantum group being used, and has both gauge transformations and gauge fields in the same place. Buffenoir and Roche [4] took this approach farther. First they isolated the gauge fields from the gauge transformations. Their gauge algebra is dual to that of [1], hence they have a coaction of the gauge algebra on the gauge fields. The coinvariant part of the gauge fields is the algebra of observables, which is a deformation of the classical ring of characters. They proceed to define Wilson loops and the Yang-Mills measure and to derive 3-manifold invariants from this setting [3, 5]. We found ourselves unable to compute examples in the exchange algebra formulation. We instead define our gauge fields as functions on the space of connections. This makes the structure of the algebra of observables more clear. Working from the point of view of low-dimensional topology, we assume a familiarity with the basics of knot theory. Otherwise, one can read most of this paper knowing only the definition of a ribbon Hopf algebra and the basics of its representation theory. Kassel [13] and Sweedler [18] are sufficient references. Part 1 is devoted to our translation of the basic objects of a lattice gauge field theory and to our devices for computing in the reformulated version. We do not merely alter the language of [4]. Three significant innovations provide added computing power. The first is to realize that gauge fields come from the restricted dual of the Hopf algebra on which the theory is based. This leads to a coordinate free formulation. Next, we do not multiply gauge fields as abstract variables modulo exchange relations. Rather we comultiply connections in a way that implies the usual exchange relations for fields while preserving their evaluabililty. Finally, we are able to mimic the classical phenomenon of pushing the support of a gauge field around. Our new foundations allow us to compute Wilson loops and many other operators using a simple extension of tangle functors. The second part is devoted to analyzing the structure of the algebra of observables. Our viewpoint is that the observables corresponding to quantum groups generalize the rings studied by Procesi [15]. He arrived at these rings as the invariants of n-tuples of matrices under conjugation. The connection with lattice gauge field theory is that each n-tuple of matrices corresponds to a connection on a lattice with one vertex and n-edges, with the gauge fields based on a classical group.
Topological Interpretations of Lattice Gauge Field Theory
49
In passing from Procesi’s work to ours, we find that the algebra of observables corresponding to a quantum group is a more subtle object. Instead of depending solely on the fundamental group of the lattice, the observables are classified by the topological type of a surface specified by a ciliated lattice. The construction given in this paper leads to an algebra of “characters” of a surface group with respect to any quasitriangular ribbon Hopf algebra. The algebras are interesting from many points of view: They generalize objects studied in invariant theory; they should provide tools for investigating the structure of the mapping class groups of surfaces; and they should give a way of understanding quantum invariants of 3-manifolds. In the case that the data correspond to a connected affine algebraic group G, it is possible to make explicit parallels with the existing theory. The algebra of observables based on U (g) is shown to be the ring of G-characters of the fundamental group of the associated surface. Then, the original motivating problem is solved: Given the ring of G-characters of a surface group, show that the observables based on the corresponding Drinfeld-Jimbo algebra form a quantization with respect to the usual Poisson structure. We also prove for the classical groups that the algebra of observables is generated by Wilson loops. Finally, invoking a quantized Cayley-Hamilton identity, we obtain a new proof, independent of [7], that the Uh (sl2 )-characters of a surface are exactly the Kauffman bracket skein module of a cylinder over that surface. Many further avenues of research present themselves. Working with quantum groups defined over local fields side steps several interesting and subtle structural questions. What happens when one uses a quantum group at a root of unity? How about lattice gauge field theory based on a quasitriangular quasi-Hopf algebra? There is a graphical calculus of the characters of the fundamental group of any manifold with respect to any algebraic group, for example, see [6], [16] or [17]. It derives from the fact that Wilson loops are a pictorial description of characters of the fundamental group of a manifold, after which the tools of classical invariant theory express all functional relations between characters in a diagrammatic fashion. The graphical models have only been worked out for special linear groups. The power of lattice gauge field theory is that it places the representation theory of the underlying manifold and the quantum invariants in the same setting. Ultimately the asymptotic analysis of the quantum invariants of a 3-manifold in terms of the representations of its fundamental group should flow out of this setting. The identification of the representation theory of a quantum group with that of a compact Lie group leads to rigorous integral formulas for quantum invariants of 3-manifolds. This should in turn lead to a simple explication of the relationship between quantum invariants and more classical invariants of 3-manifolds. Finally, there should be a similarly clean treatment of lattice gauge field theory for lattices of higher dimension. Although, in dimension greater than two, the answers will no longer be topological in nature, the constructions and objects should be of interest to geometers, algebraists and analysts. This work was done at the Banach Center, the University of Iowa, Boise State University, The George Washington University, the University of Missouri, MSRI, and the Free University, Berlin. The authors thank all their hosts for their hospitality. We also thank Daniel Altschuler, Jorgen Anderson, Georgia Benkhart, Vic Camillo, Fred Goodman, Joseph Mattes, Michael Polyak, Florin Radulescu, Arun Ram, Justin Roberts, Don Schack and Bob Sulanke for helpful conversations.
50
D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
Part 1. Lattice Gauge Field Theory Herein we develop, from a self contained and axiomatic approach, the machinery of gauge field theory on an abstract, oriented, ciliated graph. For basic background on Hopf algebras we rely on Sweedler [18] and Kassel [13]. The discussion here is restricted to the definitions and basic results confirming that the theory is consistent and computationally viable. For origins of the ideas we refer the reader to [1, 3, 4, 5, 8, 10]. 1. Objects The elementary objects of a lattice gauge field theory are: a ribbon Hopf algebra; an abstract graph which is oriented and ciliated; discretized connections, gauge transformations, and gauge fields. 1.1. Let H be a ribbon Hopf algebra defined over a field k or its power series ring k[[h]]. In the latter case all objects carry the h-adic topology (see [13]), and all morphisms are continuous. In discussions germane to both settings we will refer to the base over which the algebra is defined as b. Following Kassel we let µ, η, 1, and S denote the multiplication, P unit, comultiplication, counit and antipode of H. The universal Rmatrix is R = i si ⊗ ti , which we usually write as s ⊗ t with summation understood. The ribbon element is θ, and we add a charmed element, k = θ−1 S(t)s. The charmed element is grouplike, meaning 1(k) = k ⊗ k. It also satisfies k −1 = S(k) = θtS 2 (s) and S 2 (x) = kxk −1 for all x ∈ H. The Hopf algebra dual is well documented in [18] provided b = k. The topological case, however, has been neglected. If H is a Hopf algebra over k[[h]] then the sets Un = {L ∈ H ∗ | L(H) ⊂ hn k[[h]]} form a neighborhood basis of the origin. An ideal J ≤ H is cofinite if H/J is topologically free and modeled on a finite dimensional vector space; L ∈ H ∗ is cofinite if ker(L) contains a cofinite ideal. The restricted dual, H o , is the completion of the cofinite functionals. It is not hard to check that H o is topologically free. In the case that H is a DrinfeldJimbo deformation of a simple Lie algebra [13], it is clear that H o is modeled on (H/hH)o . To see that it is a Hopf algebra one must check that µ∗ , η ∗ , 1∗ , ∗ and S ∗ restricted to H o or H o ⊗ H o take values in the appropriate spaces. The only point that needs any discussion is why µ∗ (L) ∈ H o ⊗ H o . Suppose that L is the limit of cofinite linear functionals {Li }. It follows that {Li } is a Cauchy sequence in H o . The classical discussion of µ∗ in [18] provides Li ◦µ ∈ H o ⊗H o , which is complete by definition [13]. It is easy to see that Li ◦ µ is also Cauchy, so continuity of µ∗ gives µ∗ (L) ∈ H o ⊗ H o . The Hopf algebra H acts on its restricted dual in two obvious ways: x · φ(y) = φ(xy) and x · φ(y) = φ(yx). A subalgebra of H ◦ is stable if it is invariant under both actions. For the remainder of this section we fix a stable subalgebra B of H ◦ . (Stability implies that B is actually a Hopf subalgebra.) The adjoint action ad : H ⊗ H → H is given, in Sweedler notation, by X x00 yS(x0 ), ad(x ⊗ y) = (x)
which is further compressed to ad(x)y = x00 yS(x0 ). (Readers unfamiliar with this notation for comultiplication are referred to [13].) By taking duals we get the adjoint (right) action of H on B: if x, y ∈ H, and φ ∈ B then
Topological Interpretations of Lattice Gauge Field Theory
51
(φad(x))(y) = φ(ad(x)y) = φ(x00 yS(x0 )). An element φ ∈ B is invariant if for every x ∈ H, φad(x) = (x)φ. Our definition of the adjoint action is chosen so that any function φ with the property that φ(xy) = φ(yx) will be invariant. The invariant elements of B form a subalgebra denoted B H . 1.2. A graph consists of a set E called edges, a fixed point free involution − : E → E, and a partition V of E into subsets called vertices. Let i : E → V be the map sending e to the vertex v containing it. Let t = i ◦ −. We call t(e) the terminal vertex of e and i(e) the initial vertex. An orientation is a choice O of one edge from each orbit of the involution. An oriented graph is denoted by the data (E, −, V, O). There is a one-dimensional CW-complex associated to (E, −, V, O), which is called its geometric realization. The 0-cells are in one to one correspondence with V , the 1cells are in one to one correspondence with O, and the characteristic maps are determined by t and i. A ciliation C of a graph is a linear ordering of each vertex. The additional data is denoted V c , although we will continue to use V for the partition of E. A lattice is an oriented, ciliated graph. The geometric realization of an oriented graph is insufficient to support a ciliation so for a lattice we construct an oriented surface called its envelope. Each vertex becomes an oriented disk and each edge in O becomes an oriented band. The orientation of a fattened vertex induces an orientation on its boundary, one point of which is marked with a cilium. Attach the band corresponding to each e to the disks (or disk) at its initial and terminal ends. The attaching points along the oriented boundary of each disk must be arranged in the order given by the ciliation of the vertex. Further annotate the resulting surface by orienting the core of each band from i(e) to t(e). For example, consider E = {±e1 , ±e2 , ±e3 , ±e4 , ±e5 , ±e6 }, V c = {{−e1 , e2 }, {−e2 , e3 }, {−e3 , e1 , e4 , −e6 }, {−e5 , −e4 }, {e6 , e5 }}, O = {e1 , e2 , e3 , e4 , e5 , e6 }.
and
Ciliation is given by the order in which the elements of each vertex are written above. The envelope of (E, −, V c , O) is shown in Fig. 1, alongside a streamlined schematic version. An envelope determines its lattice. The edge set consists of a pair ±e for each band, with e assigned to the orientation. Label the initial and terminal ends of a band core by e and −e respectively. Each disk forms a ciliated vertex by reading off these labels, beginning at the cilium and traveling along the induced orientation.
Fig. 1. Envelope of an oriented, ciliated graph
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
1.3. For this subsection we fix (E, −, V c , O). A gauge field theory on this lattice is defined by the interactions of three algebraic objects: O He . • A set of connections, A = • A gauge algebra, G =
Oe∈O Hv . v∈V
• And a set of gauge fields, C[A] =
O
Be .
e∈O
The gauge algebra is a Hopf algebra in the natural sense of a tensor power of Hopf algebras, whereas A and C[A] inherit only the vector space structure of H. However, we will shortly endow the connections with a G-action and a comultiplication, which induce dual structures on C[A] via the evaluation pairing. For each v ∈ V c there is a function ordv : v → N that assigns to e ∈ v the ordinal number corresponding to its position in the ciliation. Connections become a left G-module under the action (ord (e)) (ord (e)) . ⊗v∈V yv · ⊗e∈O xe = ⊗e∈O yi(e) i(e) xe S yt(e) t(e) This is a busy formula, even with the summations of Sweedler notation suppressed. For a graphical description of the action (and the usual method of computing it) see [8]. The gauge algebra acts adjointly on gauge fields via (f · y)(x) = f (y · x), so C[A] is a right G-module. A gauge field f is called an observable if for all y ∈ G we have f · y = (y)f . Observables form a submodule O of C[A]. 1.4. There is a construction of gauge fields which uses a direct interpretation of the restricted dual of H. Suppose that W is a finite dimensional left H-module. We use W ∗ to denote the dual with H acting on the left by (x · f )(v) = f (S(x) · v). The action (f ·x)(v) = f (x·v) makes the dual into a right H-module, denoted W 0 . Comultiplication supplies a left action on W ∗ ⊗W , namely x·(f ⊗v) = (x0 ·f )⊗(x00 ·v). Forcing the natural identification of W ∗ ⊗ W with Hom(W, W ) to be an intertwiner makes the later into a left module as well. In Sweedler notation the action is (x · f )(v) = x00 · f (S(x0 ) · v). Now suppose that ρ : H → Hom(W, W ) is the original representation. I.e., x · v = ρ(x)(v). The reader may check that for any x, y ∈ H, we have y · ρ(x) = ρ(ad(y)x). A finite dimensional representation ρ : H → Hom(V, V ) is said to be adapted to B if {h ◦ ρ | h ∈ (Hom(V, V ))0 } ⊂ B. A coloring of a lattice is a labeling of each e ∈ O by a representation adapted to B. Let ∗ We denote the representation associated N to e, and let W−e = We . A coloring naturally associates the left H-module Wv = e∈v We to each vertex. Given a coloring, there is a map of right G-modules, O O O Wv0 → (W−e ⊗ We )0 → (Hom(We , We ))0 → C[A] v∈V
e∈O
e∈O
defined as follows. The first stage is just reordering of the factors with the natural distribution of primes over tensor products. The next is the canonical identification. The last is composition with ⊗e∈O ρe , where the maps ρe : H → Hom(Ve , Ve ) are the actual representations of the coloring.
Topological Interpretations of Lattice Gauge Field Theory
53
Theorem 1. The images of these maps, taken over all colorings, add up to C[A]. Proof. This is evident after establishing the following claim: For each f ∈ B there is a finite dimensional H-module W so that f ∈ {h ◦ ρ | h ∈ (Hom(W, W ))0 } ⊂ B. Fix a nonzero f ∈ B. Choose I to be maximal among ideals of H contained in ker f . Let ρ : H → Hom(W, W ) be the representation induced by left multiplication of H on W = H/I. Define T to be the linear span of the functionals {y 7→ f (xyz) | x, z ∈ H}. Since I is an ideal and it lies in the kernel of f , T may be thought of as a subspace of W ∗ . Choose {x1 , . . . , xn } ⊂ H so that x1 = 1H and so that, in the quotient, this is a basis for WP . By evaluation, each xi is a functional on T . Suppose that, as a functional on T , X = ai xi = 0. For any y, z ∈ H we have f (yXz) = 0. Maximality of I then implies linear independence of {xi } on T , which means T is all of W ∗ . Choose a basis {f1 , . . . , fn } for W ∗ that is dual to {xi }. Let ρ(xj ) be the matrix M j in the basis {xi }. By duality of bases, we know that every element y ∈ H can be written as X fi (y)xi , y=z+ i
where z ∈ I. Hence ρ(y) =
X
fi (y)M i .
i
The j-k entry of the matrix ρ(y) is X
i fi (y)Mjk ,
i
which proves the assertion {h ◦ ρ | h ∈ (Hom(W, W ))0 } ⊂ B. Finally, let hj be the function y 7→
X
i fi (y)Mj1 .
i i Mj1 .
The first column of M i expresses xi x1 in the chosen basis. Note that hj (xi ) = i = δji . Since hj (I) = 0, we have shown that it agrees However, x1 = 1H , so we have Mj1 with fj on all of H. This proves that {h ◦ ρ | h ∈ (Hom(W, W ))0 } contains a spanning set for T . In particular, it contains f . If H is semisimple there is a way of getting an isomorphism out of the construction above. Restrict the colors to lie in an exhaustive list of irreducible representations adapted to B, so that no representation appears twice in the list. Once this has been done then the map in Theorem 1 becomes an isomorphism. This is the definition of gauge fields used in [1, 4]. Let Inv(W ) denote the invariant part of an H-module. Since the maps described above are all intertwiners, we have the following characterization of observables. N Corollary 1. The sum over all colorings of the images of v∈V Inv(Wv0 ) is equal to O.
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
2. Multitangles The goal of this subsection is to develop a functor between two categories: the category of multitangles, M, and the category of connections, A. The objects of M are lattices, and a morphism is a set of tangles in one-to-one correspondence with the vertices of its domain. An object in A is the set of connections on a lattice, viewed as a left module over the gauge algebra of the lattice. The morphisms are pairs of maps, one from the connections in its domain to the connections in its range and the other between the gauge algebras. The second morphism allows us to pull back the connections on the range lattice to a module over the domain gauge algebra. The first map must intertwine this action with the standard one. A multitangle is described by a collection of diagrams in one-to-one correspondence with the vertices of the domain lattice. Each diagram lies in a copy of [0, 1] × [0, 1] with the second factor determining a height function on the entire collection. Each such collection is built by stacking the elementary diagrams described below. A multitangle is an equivalence class of these collections under a relation that will be made explicit shortly. 2.1. Abstractly, a morphism in the category M has a specific domain and range lattice. In practice it will be given by a picture of a multitangle, from which one can easily determine E, O, and V c for both range and domain, but not the involution. However, the specific details of the domain lattice are always clear from context, and the range lattice is then completely determined by the picture. We will now list several elementary morphisms in M by explicitly describing how a domain lattice is altered to produce the range lattice. We also describe an associated picture called an elementary diagram. These are the elementary diagrams. The identity: The domain and range are the same lattice. For each vertex v there is a diagram consisting of arcs with no crossings which are monotonic with respect to the height function. The arcs correspond, from left to right, to the cilial ordering of v. Arcs corresponding to edges in the orientation are directed downwards, others are directed upwards. Fig. 2 shows an envelope and its identity morphism. The convention for ordering and directing arcs used here is standard for all elementary diagrams. It can also be derived from the envelope of a lattice. Allow the band cores to protrude into a fat vertex and unroll it with a M¨obius transformation to the upper half plane, cilium at infinity.
Fig. 2. Identity multitangle
Crossings: The domain and range lattices differ only in the ciliation at a single vertex, v, where a pair of adjacent edges have been transposed. At each vertex other than v there is a trivial diagram as in the identity. The diagram at v has monotonic arcs and
Topological Interpretations of Lattice Gauge Field Theory
55
a single crossing between those arcs corresponding to the transposed edges. Either strand may pass over the other. A geometric example is given in Fig. 3.
Fig. 3. Negative crossing
Triads: Consider an involutory pair of edges, e and −e, in the domain lattice with e ∈ O. In the range lattice, e is removed from its position in a ciliated vertex and replaced with e0 , e00 in that order. Similarly, −e is replaced with −e00 , −e0 . The edges e0 and e00 lie in the orientation of the new lattice, which is otherwise identical to the domain. The diagrams have monotonic arcs without crossings corresponding to all the unchanged edges. The two arcs corresponding to e and −e split at the same height into four arcs corresponding to e0 , e00 , −e00 and −e0 . In an envelope, this is the operation of doubling an edge (Fig. 4).
Fig. 4. Triad
Caps: The edge set of the range will differ from the domain by deleting two adjacent edges from a vertex, exactly one of which lies in the orientation. If the two edges are not an involutory pair, then the two orphaned edges in the range become an involutory pair. The strands corresponding to the deleted edges meet at a local maximum. Otherwise the diagrams are trivial. The effect on envelopes is suggested in Fig. 5. Cups: The range lattice differs from the domain by introducing two new edges, e and −e next to each other at a single vertex. The strands corresponding to the new edges originate in a local minimum and obey the usual directedness rule. Otherwise the diagrams are trivial. This creates a monogon at a vertex as shown in Fig. 6. Stumps: The range lattice is formed from the domain lattice by deleting an involutory pair of edges. The diagrams are trivial except for the two strands corresponding to the deleted edges, which simply terminate. Both stumps must occur at the same height. Figure 7 is an example.
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
Fig. 5. Cap joining a non-involutory pair of edges
Fig. 6. Cup creating an involutory pair at a single vertex
Fig. 7. Stump eliminating an involutory pair
Fig. 8. Switch exchanging an involutory pair
Topological Interpretations of Lattice Gauge Field Theory
57
Fig. 9. Cut splitting a vertex in two
Switches: The range differs from the domain by replacing some e with −e in the orientation. This is indicated in the diagrams by a hash mark on each of the strands involved, with both marks lying at exactly the same height (Fig. 8). Cuts: The domain lattice is altered by dividing the ordered edges at some vertex into two non-empty consecutive sets, which form new ciliated vertices. The diagram for that vertex is trivial, except for a vertical mark at the top which indicates the cilium of the new vertex (Fig. 9). When the range lattice of one collection of diagrams matches the domain of another we may form a new set of diagrams by stacking the first two. It may be necessary to isotop the bases to get arcs to match, and if two diagrams are stacked atop a single diagram with a cut, the cut extends to the top of the new diagram. The height function is then uniformly rescaled. We define a multidiagram to be any such collection formed by stacking elementary diagrams. A multidiagram is a picture of a framed embedding of a 1-dimensional CW-complex into a collection of cubes. One should not think of the hash marks of a switch as vertices; rather they are extra annotation on what is otherwise a smooth 1-cell. The 1-cells of this complex are called the segments of the multidiagram. Alternatively, segments are the components left behind if triads are removed, switches are ignored, and the arcs passing under a crossing are thought of as connected. The critical points, switches, stumps and triads of a multidiagram are collectively referred to as events. A coloring of a multidiagram is an assignment of a finite dimensional H-module to each segment. Let b denote the 1-dimensional module with H acting by its counit, and identify W ⊗ b = b ⊗ W = W . Except for critical points, one may think of a multidiagram as a 1-parameter family of lattices, 0t . A multidiagram is consistently colored if its coloring satisfies: 1. Segments creating an involutory pair of edges in any 0t must have the same color. (I.e., the coloring induces a coloring of 0t in the sense of Sect. 1.4). 2. Triads must obey the condition in Fig. 10. 3. Any segment ending in a stump is colored by b. W1
W2
W1 ⊗ W2
W2
W1
W1 ⊗ W2
Fig. 10. Triad coloring rule
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
2.2. We say that two colored multidiagrams are equivalent if one can be obtained from the other by a sequence of the following local moves. Isotopies: We allow ambient isotopy of the diagrams subject to the following restriction: No two events sharing a segment may occur at the same height. A pair of marks indicating a stump, switch or triad must always remain at the same height. Cuts and stumps cannot become horizontal. Triads must maintain one segment below the horizontal and two above it. Finally, a cap may not occur at the same height as a stump, switch or triad if it shares an involutory pair of edges with the other event. Generalized Reidemeister Moves: These are shown in Fig. 11. The moves are valid regardless of the orientations of the arcs. We also allow the corresponding moves with reversed crossings.
Fig. 11. Generalized Reidemeister moves
Fig. 12. Triad moves
Fig. 13. Cap moves
Interacting Events: These moves describe the interaction of events that share either a common segment or two segments representing an involutory pair of edges. They are divided into triad moves (Fig. 12), cap moves (Fig. 13), stump moves (Fig. 14) and switch moves (Fig. 15). In each picture adjacent pairs of arcs, reading left to right along the base, represent involutory pairs of edges. Subject to that, any orientation of arcs is allowed, as are diagrams with all crossings reversed. The involutory pairs
Topological Interpretations of Lattice Gauge Field Theory
59
Fig. 14. Stump moves
Fig. 15. Switch moves
are shown as adjacent merely to conserve space; the moves are valid even for distant arcs. Some of these moves alter the segments. In each case a consistent coloring of the multidiagram before the move induces an obvious consistent coloring of new multidiagram. Specials: The three moves in Fig. 16 all concern an involutory pair of edges entering a series of triads. In the first two moves three triads have been collapsed into a 5valent vertex using a previously given triad move. In the third move this convention is extended to any number of triads. Note that the colors on the top of each diagram are as generic as possible and unchanged by the move. Consistency conditions then force the indicated colors on the bottom of each diagram. W1 W2 W2 W3
W3
W2
W1
W1 W2 W2 W3
W3
W1
←→ W1 ⊗ W2 ⊗ W2 ⊗ W3
W1 W2 W3 W4
W1 ⊗ W2 ⊗ W2 ⊗ W3
W1 ⊗ W3
W4 W2 W3 W1
W1 W2 W3 W4
W1 ⊗ W3
W4 W2 W3 W1
←→ W4 ⊗ W2 ⊗ W3 ⊗ W1
W1 W2 · · · Wn
W4 ⊗ W2 ⊗ W3 ⊗ W1
W4 ⊗ W3 ⊗ W2 ⊗ W1
Wn · · · W2 W1
W4 ⊗ W3 ⊗ W2 ⊗ W1
W1 W2 · · · Wn
Wn · · · W2 W1
Wn · · · W2 ⊗ W1
Wn · · · W2 ⊗ W1
←→ W1 ⊗ W2 · · · Wn
W1 ⊗ W2 · · · Wn
Fig. 16. Specials
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
The first move is allowed with the switch on the other side of the loop, as well as with opposite orientations. The second move is allowed with a negative crossing. In all cases, coloring is the same across the top and determined by consistency at the bottom. Definition 1. A multitangle is an equivalence class of consistently colored multidiagrams. There is a useful (although somewhat imprecise) topological way of understanding the equivalence of multidiagrams. Think of a multidiagram as a diagram of framed tangles in cubes. For the most part any isotopy relative to the boundary of the cubes is an equivalence, exceptions being the rigidities listed above. Since these involve involutory pairs, and since caps can alter the involution, it is best to be careful when isotoping an event past a cap. A switch is a pair of marks that can slide up or down together unless obstructed by a cup, cap or triad. Any pair of switches that meets will cancel (and thus can be created), and a single switch can be canceled (or created) at a cup. Triads, as long as they are moved about in pairs, can be manipulated as though the performing edge slides on a graph provided the only other events encountered are crossings and triads. Interactions with cups are as expected, and in practice one can usually keep caps at the top of the tangle and switches at the bottom. When this is not possible, it is best to rely on the explicit moves as listed. Stumps can be isotoped almost anywhere (only caps are a problem) so long as they never become horizontal. In particular, one can almost always shrink a segment ending in a stump to be very short, and if the other end is a triad or another stump, the segment can be eliminated. In other words, as long as one avoids caps and keeps triads and stumps upright, any continuous deformation of a multitangle is an equivalence and, with the possible exception of the special moves, the behavior of cups, stumps, triads, and switches is fairly intuitive. Finally, in many applications the coloring of a multidiagram is irrelevant. One may then treat every segment as carrying the color b so that consistency is automatic, even for the special moves. In fact, for trivially colored multidiagrams, the special moves reduce to Fig. 17.
Fig. 17. Special moves with trivial coloring
2.3. We begin building a functor from M to A by sending a lattice to its connections and the elementary diagrams to the morphisms described below.
Topological Interpretations of Lattice Gauge Field Theory
61
The identity: This diagram induces the identity on connections and on the gauge algebra. Crossings: The map on connections is an action of the R-matrix or its inverse. There are 12 cases depending on the sign of the crossing, the directions of the arcs, and the possibility that they are an involutory pair. These are given in Fig. 18, which describes the action in the factors corresponding to the crossing arcs. The map extends linearly on connections, using the identity in all other factors. The map on the gauge algebra is the identity. ty
sx
x
y
S(s)y
tx
x
y
yS(t)
sx
x
yS 2 (s)
x
y
tx
y
ty
xS(s)
x
y
S(s)y
xS(t)
x
y
yS(t)
x
xS(s)
sxS(t)
txS(s)
x
x
y
yS 2 (s)
x
xS(t)
txS 2 (s)
S(s)xS(t)
x
x
y
Fig. 18. Action of R and R−1 at crossings
Triads: Suppose that e and −e are replaced by e0 , e00 , −e00 and −e0 , with e ∈ O originally. The map on connections is comultiplication in the factor corresponding to e and the identity elsewhere, with the requirement that the image of the comultiplication take values in the tensor product of the factors corresponding to e0 and e00 in that order. The distribution of the output, using Sweedler notation with summation suppressed, is illustrated in Figure 19. The diagram acts trivially on G. x0
x00
x
x00
x0
x
Fig. 19. Action of a triad
Caps: There are four cases at a local maximum, depending on orientations and on whether or not the incoming strands represent an involutory pair in the domain lattice. These are listed in Fig. 20, where trW denotes the ordinary trace taken in the H-module W coloring that segment. If the lattice loses a vertex then the map on G is in that factor; otherwise it is the identity. xy
x
y
ykx
x
y
tr W (x)
x
tr W (xk)
x
Fig. 20. Operators corresponding to cups and caps
1
k−1
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
Cups: The action on A is either by the unit of H or by the unit followed by multiplication by k −1 , depending on the orientation of the cup. The two cases are shown in Fig. 20. The map on gauge algebras is the identity. Stumps: A stump acts as the counit in the corresponding factor of the connections. If the range lattice loses one or more vertices because of this, the map on gauge algebras is counit in those factors. Otherwise it is the identity. Switches: For switches the map on connections is x 7→ S(xk) in the factor corresponding to the edge. The map on G is trivial. Cuts: A cut has no effect on connections. It acts trivially on G except in the factor corresponding to the split vertex, where the map is 1 : Hv → Hv0 ⊗ Hv00 . Here v 0 denotes the initial subset of v after the cut, and v 00 the final subset. A general multidiagram is a composition of elementary ones, so it is sent to the corresponding composition of maps. Theorem 2. Equivalent multidiagrams from 0 to 00 induce identical maps on connections and gauge algebras. The map on connections intertwines the action of G0 on A0 with the one on A 00 pulled back via the map on gauge algebras. Proof. To check that equivalent multidiagrams induce the same morphisms one must evaluate both sides of each move under every possible arrangement of orientations and crossings. Number the moves in each of Figs. 11–16, reading left to right and down the page. We will outline the identities and manipulations in H that make each move invariant on connections. That both sides induce the same map on gauge algebras is elementary. Invariance of generalized Reidemeister moves: 1. This follows from the fact that R and R−1 solve the quantum Yang-Baxter equation. 2. Replacing any appearance of S 2 (x) with kxk −1 proves invariance in all cases. 3. This is essentially the identity ( ⊗ 1)(R) = (1 ⊗ )(R) = 1. For some orientations the fact that (S(x)) = (x) is also needed. 4. With the strand directed upwards the left-hand side produces the following morphism, where subscripts indicate successive applications of R and implied sumP mation. Note that R−1 = I S(si ) ⊗ ti : x 7→ xS 2 (s1 )kS(t1 ) = xS(t1 k −1 S(s1 )) = xS(t1 θt2 S 2 (s2 )S(s1 )) = xS(θ)S(t1 t2 S(s1 S(s2 ))) = θx.
5. 6. 7. 8. 9.
Similar computations show that, regardless of orientation, both sides act as multiplication by θ. If the crossings are reversed the action is by θ−1 . This is similar to 2. Those cases that are not immediate follow from an application of S 2 (x) = kxk −1 . RR−1 = 1. Depending on crossings, use one of the identities 1 ⊗ 1(R) = s1 ⊗ s2 ⊗ t1 t2 or 1 ⊗ 1(R) = s1 s2 ⊗ t2 ⊗ t1 . Obvious.
Topological Interpretations of Lattice Gauge Field Theory
63
Triad moves: 1. 1(k −1 ) = k −1 ⊗ k −1 . 2. 1 is coassociative. Cap moves: 1. That S is an anti-algebra morphism suffices. 2. 1 is an algebra morphism. 3. is an algebra morphism and (k) = 1. Stump moves: 1. is the counit for 1. 2. (k −1 ) = 1. 3. From earlier identities, ( ⊗ )(R±1 ) = 1 Switch moves: 1. S(k) = k −1 . 2. ◦ S = and (k) = 1. 3. This is the claim that x 7→ S(xk) is an involution. It follows from S 2 (x) = kxk −1 . 4. (S ⊗ S)(R±1 ) = R±1 . Specials: 1. The definitions of S and . In Sweedler notation the identity is x(1) ⊗ x(2) = x(1) ⊗ S(x(2) )x(3) ⊗ x(4) = x(1) ⊗ x(2) S(x(3) ) ⊗ x(4) . 2. Constrained non-cocommutativity. In Sweedler notation x(1) ⊗ sx(2) ⊗ tx(3) ⊗ x(4) = x(1) ⊗ x(3) s ⊗ x(2) t ⊗ x(4) . 3. S is an anti-coalgebra map and k is grouplike. Checking that every multitangle is a G-module intertwiner is again a matter of checking each elementary diagram under all orientations and crossings. As above, we will indicate the essential identity or manipulation on which the computation rests. Identity: This is obvious. Crossings: Apply the constrained non-cocommutativity identity to prove that a gauge transformation by y at a single vertex is an intertwiner. Any other gauge transformation can be expressed as sums of products of these. Triads: Coassociativity of 1 again. The proof is trivial in Sweedler notation. Caps: Apply the definition of S to check that gauge transformation by y at a single vertex is an intertwiner. Cups: These work for pretty much the same reasons that caps do. Stumps: Follows from the definition of . Switches: S is an anti-algebra map. Cuts: Coassociativity of 1. Remark. We can think of and as single events called positive and negative twists respectively. It is worth remembering that a positive twist acts on a connection as multiplication by θ−1 in that factor. A negative twist acts by θ.
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3. Comultiplication of Connections Fix a lattice 0 = (E, −, V c , O). We define a multitangle whose domain is 0 by repeating the following construction at each vertex: Apply a triad to each arc. Then move the strands corresponding to the x0 ’s to the left of the strands corresponding to the x00 ’s so that the latter segments cross over. Finally, cut the diagrams to separate the x0 ’s from the x00 ’s. An example is given in Fig. 21.
Fig. 21. Multitangle for ∇
The multitangle determines a range lattice denoted 0⊗2 . Its envelope is two disjoint copies of the envelope of 0, but it is important to distinguish them as the prime and double prime copies. The induced morphism on connections is denoted ∇ : A 0 → A0⊗2 . The map on gauge algebras is the standard comultiplication on a tensor power of H. We make the identification A 0 ⊗ A 0 = A0⊗2 , and define 0 = ⊗e∈O e : A0 → b. Theorem 3. The triple (A, ∇, 0 ) is a coalgebra. Proof. Fig. 22 depicts diagrams for (∇ ⊗ 1) ◦ ∇ and (1 ⊗ ∇) ◦ ∇ at one possible trivalent vertex. To see that the left side is equivalent to the right, slide the higher triads down to the lower ones and then back up the other segments. This is possible because 1 is coassociative and because the diagrams separate into three disentangled layers. This phenomenon holds in general, and it is possible to organize this information into an inductive proof that ∇ is coassociative. We leave the details to the reader, with the suggestion that one use a coupon, say n
to denote the diagram for ∇ at a generic n-valent vertex. It is also convenient that
Topological Interpretations of Lattice Gauge Field Theory
65
= n
i
n . −i
The fact that stumps can be retracted and absorbed into triads gives a simple multitangle proof that (0 ⊗ 1) ◦ ∇ = (1 ⊗ 0 ) ◦ ∇ = 1. Thus 0 is a counit for ∇.
Fig. 22. A suggestion of the coassociativity of ∇
The adjoint of ∇, restricted to observables, is denoted by ?: if f, g ∈ O and x ∈ A, then (f ? g)(x) = (f ⊗ g)(∇(x)). Note that 0 is itself an observable. Corollary 2. (O, ?, 0 ) is an algebra. Proof. The intertwining property of morphisms induced by multitangles insures that ? takes values in O. Linearity and associativity follow from linearity and coassociativity of ∇, and 0 is the unit for ? because it is the counit for ∇. 4. Computing in O The algebras of observables for lattices that differ only by orientation or by the cyclic ordering of the ciliated vertices should be isomorphic. Furthermore, mimicking a classical phenomenon, the value of an observable on a connection should be computable from a suitable connection on the complement of a maximal tree in the graph. In the next two sections we will fix a lattice, (E, −, V c , O), and prove that these goals can be met, up to canonical isomorphisms. We will also address the interaction of multitangles with the algebra and coalgebra structures from the previous section. Throughout these two sections, all multitangles are assumed to have trivial coloring. 4.1. Given e ∈ O, let σe denote the map on connections induced by the multitangle which is trivial except for a switch on the strands ±e. In an envelope of 0, σe switches the orientation of the core of the corresponding band. Given v ∈ V , let τv = |v|−1
,
and τv−1 =
|v|−1
.
Here an integer next to an arc indicates that many parallel copies. The orientations are determined by the orientations of the edges at v, and the rest of the multitangle is trivial. The effect on an envelope of τv is to toggle the cilium at v one step counterclockwise, while τv−1 toggles it the other way.
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
Let e ∈ O so that e and −e lie in distinct vertices, e occupies the last position in the ciliation, and −e occupies the first. Define
πe =
n
n
where the coupon denotes n − 1 triads and the two strands entering it represent e and −e. The rest of the multitangle is trivial. The domain and range lattices are identical. The effect of the map on a connection is N described in Fig. 23, which also introduces the convention of writing a simple tensor in e∈O He by labeling the corresponding cores in an envelope. We call this map a push. x(n) yn
yn x
1
x(1) y1
y1
Fig. 23. Effect of a push on a connection
In order to avoid belaboring useless notation, we will assume that the domains of successive applications of switches, toggles and pushes are clear, provided the original domain was specified. We will also suppress the subscripts whenever possible. Any sequence of toggles, switches and pushes defines a G-module map between connection algebras and thus an operator between observables as well. We say that two connections are gauge equivalent if their difference lies in the span of {y · x − (y)x | x ∈ A, y ∈ G}. Observables cannot distinguish gauge equivalent connections. Two morphisms in A (with the same domain and range) are gauge equivalent if the images of every connection are gauge equivalent. The adjoints of a pair of gauge equivalent operators are identical maps on observables. Proposition 1. Let f be any sequence of toggles and switches that begins and ends at the same lattice. The induced operator on connections is gauge equivalent to the identity. Proof. It suffices to check the compositions σ 2 , τ ± τ ∓ , στ ± στ ∓ , and τv±n , where n is the valence of v. Clearly σ is an involution. It follows easily from tangle equivalence that the next two are also the identity map. The multitangle for τ n is trivial away from v. At that vertex it is represented by
n
where the coupon denotes the generator (with positive crossings) of the center of the n-strand braid group. Note that it consists only of crossings and twists. Because such a tangle acts as multiplication in the factors corresponding to each segment, its behavior can be understood by its effect on the connection 1. We can evaluate this as follows:
Topological Interpretations of Lattice Gauge Field Theory
67
n
τ n (1) =
n
=
= 1n−1 (θ−1 ).
=
n
Hence, the effect on an arbitrary connection is gauge action by θ−1 at v. The same proof −|v| with all crossings reversed shows that τv is gauge action by θ. 4.2. The standard tools for manipulating connections and observables are toggles, switches, pushes, triads (in succession), cups, caps (involving non-involutory pairs), stumps and cuts. We will need an understanding of how well they interact with the coalgebra and algebra structures. We already have notation for the first three maps. Let 1ne be the map induced by a succession of triads on the edges e and −e. We extend the notation to include 10 for the identity and 1−1 for a stump. A cup oriented from e1 to e2 (necessarily involutive) induces the map ηee12 . A cap oriented from e1 to e2 , a non-involutive pair, induces µee21 . A cut between e1 and e2 (in that order) induces κee21 . The notation is deliberately parallel with the defining maps of H, and when context permits we will suppress sub- and superscripts. Theorem 4. The maps σ, 1n , η and µ are coalgebra morphisms. Proof. A switch slides up through the multitangle for ∇ ◦ σ and, after a special move, becomes a pair of switches on the appropriate edges. Now apply the special move corresponding to R1 = 1op R to obtain a tangle for (σ ⊗ σ) ◦ ∇. Consider the multitangle for (µ ⊗ µ) ◦ ∇. At the vertex where the cap occurs we can expand this as
i
2
j
Since the caps do not involve involutory pairs, there is a cap move that makes this into a tangle for ∇ ◦ µ. The proof that (η ⊗ η) ◦ ∇ = ∇ ◦ η is similar.
Fig. 24. Triad commuting with ∇ at a valence two vertex
If the lattice has only two edges, the proof that (1 ⊗ 1) ◦ ∇ = ∇ ◦ 1 is just Figure 24. If there are more edges a similar proof works because of the layering phenomenon seen in the proof of Theorem 3. Higher powers of 1 follow immediately; 10 is trivial; and 1−1 comes from retracting stumps into triads.
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
In many applications the output of a multitangle is needed only up to gauge equivalence. There is a particular occurrence which can greatly simplify computations. Let φ be the operator induced by a multitangle that is trivial except for
i
j
.
The difference between the range and domain lattices is just that a subset of consecutive edges in some vertex has been split off to form a new one. The identity map on connections is an operator with the same domain and range as φ. However, it cannot be expressed via the trivial multitangle because the set of edges split off to make the new vertex is not at the end of the original ciliation. Proposition 2. With notation as above, φ is gauge equivalent to the identity. Proof. Given a connection x, we can express φ(x) as an action of φ(1) on x, as in the proof of Proposition 1. In the interest of computing φ(1), replace the i strands of the tangle with a single strand followed by 1i−1 . Tangle equivalence allows the triads to slide over the crossing, after which we find that the action is by 1i−1 (sj · · · s2 s1 ) in the i strands and by t1 ⊗ t2 ⊗ · · · ⊗ tj in the j strands. Now consider the effect of φ on x1 ⊗ · · · ⊗ xj ⊗ y1 ⊗ · · · ⊗ yi in the factors corresponding to the non-trivial part of the tangle. The result is t1 · x1 ⊗ · · · ⊗ tj · xj ⊗ 1i−1 (sj · · · s1 ) · (y1 ⊗ · · · ⊗ yi ), where tk · xk means tk yk or yk S(tk ), depending on orientation of the segment, and similarly for 1i−1 (sj · · · s1 ) · (y1 ⊗ · · · ⊗ yi ). By definition, this is gauge action of 1 ⊗ (sj · · · s1 ) on the connection t1 · x1 ⊗ · · · ⊗ tj · xj ⊗ y1 ⊗ · · · ⊗ yi , so the result is gauge equivalent to (sj · · · s1 )t1 · x1 ⊗ · · · ⊗ tj · xj ⊗ y1 ⊗ · · · ⊗ yi . Since ( ⊗ 1)(R) = 1, this is indistinguishable from the behavior of the identity map. Theorem 5. Up to gauge equivalence, τ ± , κ and π are coalgebra maps. Proof. Draw the tangle for (τv ⊗ τv ) ◦ ∇. By Proposition 2 we can replace the cut with |v|
|v|
The result is equivalent to the tangle for ∇ ◦ τv . Commutation of τ −1 derives from its gauge equivalence with some power of τ . For (κ ⊗ κ) ◦ ∇ the tangle at the cut vertex is
.
Topological Interpretations of Lattice Gauge Field Theory
69
By Proposition 2, this is gauge equivalent to ∇ ◦ κ. Since π is composed of triads, caps, a cut and a cup, it too commutes up to gauge equivalence. Corollary 3. The adjoints of τ ± , σ, π, 1n , η, µ and κ restricted to observables are algebra maps. Since τ and σ are invertible algebra morphisms, we can now see that O is independent of orientation, up to isomorphism, and that it depends only on the cyclic ordering of edges at vertices. 5. Pushes The goal of this section is to prove that a push is invisible to any observable. Let’s begin with a basic fact about invariant tensors. Suppose that U and W are left H-modules. As in Subsect. 1.4, (U ∗ ⊗ W )0 is a right module. This can be identified with Hom(W, U ) as follows: if φ ∈ U ∗ , w ∈ W , then h ∈ Hom(W, U ) becomes the functional sending φ ⊗ w to φ(h(w)). This isomorphism is an intertwiner if we use the following action: for z ∈ H and h ∈ Hom(W, U ), (h · z)(w) = S(z 0 ) · h(z 00 · w). Lemma 1. If h is invariant then, for all z ∈ H and all w ∈ W , h(z · w) = z · h(w). Proof. Let y = S −1 (z). Using Sweedler notation and invariance of h under the action of y 0 , we have h(z · w) = h(S(y) · w) = h((y 0 )S(y 00 ) · w) = (y 0 )h(S(y 00 ) · w) = S(y 0 ) · h(y 00 · S(y 000 ) · w) = S(y 0 ) · h((y 00 )w) = S(y) · h(w) = z · h(w). Now choose a coloring of the lattice with notation ρe , We , and Wv as in Subsect. 1.4. Suppose that our lattice contains the configuration in Fig. 23 and that the vertex on the right is v0 = {−e0 , e1 , e2 , . . . , eN n } (with each ei ∈ O). Choose a gauge field of the form f ⊗ g, where f ∈ Wv0 0 and g ∈ v6=v0 Wv0 , and write the connection depicted in Fig. 23 as z ⊗ x ⊗ y1 ⊗ · · · ⊗ yn . The usual is to convert the gauge field and the connection into N N method of evaluation tensors in e∈E We0 and e∈E We , then tensor them together and contract. (Meaning evaluate the functionals in the We0 ’s on the vectors in corresponding We ’s.) Since these contractions can take place in any order, we can focus on just those taking place between We and We0 for e ∈ v0 . To see the invariance of a push, however, we need to think of these contractions as a composition of morphisms. Let W denote We1 ⊗ · · · ⊗ Wen and let y = y1 ⊗ · · · ⊗ yn . Apply ρe ’s so that x ∈ Hom(We0 , We0 ) and y ∈ Hom(W, W ). We can now view f , and hence x ◦ f ◦ y, as elements of Hom(W, We0 ). Using standard identifications, this becomes a tensor in We0 ⊗ We∗1 ⊗ · · · ⊗ We∗n . The evaluation of the full gauge field is now completed by contracting g with α ⊗ (x ◦ f ◦ y). Lemma 2. With notation as above, if f is invariant then x ◦ f ◦ y = 1 ◦ f ◦ (x(1) y1 ⊗ · · · ⊗ x(n) yn ). Proof. Choose w ∈ W . Reinterpret (x ◦ f ◦ y)(w) as x ∈ H acting on f (y(w)). Then apply Lemma 1.
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In light of Corollary 1, we have established the following result: Theorem 6. The adjoint of a push is the identity on observables. Corollary 4. Every sequence of toggles, switches and pushes from one lattice to another induces the same isomorphism on observables. Proof. Since pushes induce the identity, it suffices to prove this for just toggles and switches. By Proposition 1, any two sequences of toggles and switches are gauge equivalent. We have succeeded in proving that O is independent of orientation and that it depends only on cyclic ordering at vertices. Also, Corollary 4 indicates that we can evaluate observables on whatever configuration is most favorable for the application at hand. There is one more task. We want to construct (as nearly as possible) a universal description of observables. 6. Quantum Holonomy Fix 0 = (E, −, V c , O), with envelope F . We will be quantizing the notion of holonomy along a loop in the lattice. In the classical world, a loop in 0 would be just that, the image of an oriented S 1 with base point. However, in order to quantize, we need the image to be generic and to introduce over- and under-crossings. That is, we need a knot diagram, not just a loop. Also, the base point introduces some technicalities. We will refer to the disks in F representing the elements of V as vertices and the bands as edges. Let α be a proper, generic immersion from a disjoint, finite collection of oriented intervals into F , so that endpoints map to cilia, double points lie in vertices, and the image in each edge consists of arcs parallel to the core. Introduce over- and under-crossings at each interior double point. The resulting object is called a q-net. The special case when the domain is a single interval is called a q-path if the endpoints are distinct, and a q-loop if not. 6.1. A q-net, α, determines an operator, holα , from A to a tensor power of H indexed by the components of α. It is defined as the composition of a pair of multitangles connecting a trio of lattices. The first lattice is 0. The last is determined by the multitangles. The intermediate one comes from α, and it is best defined by constructing its envelope as a subsurface of F . Its fattened vertices are the vertices of F that α meets. Its fattened edges are obtained by thickening the arcs of α in the edges of F . Their cores are oriented according to the orientation of α, and the ciliation is inherited from F . The multitangle connecting the first two lattices is formed as follows: Apply 1m−1 to each pair ±e, where m is the number of times γ meets the corresponding edge of F . The range of this morphism is a lattice identical to the intermediate one, described above, except possibly for orientation. Continue the multitangle by inserting switches whenever the orientations disagree. Color with trivial representations. The second multitangle is formed from the oriented tangles created by α in each vertex of F . Given a vertex, apply a M¨obius transformation mapping it to the upper half plane with the cilium at infinity. Choose a rectangular region that contains all of the image of α except for disjoint arcs running from the top edge to infinity. Rescale this to a unit square and color trivially. The composition of the two induces the operator O Hc . holα : A → c a component of α
Topological Interpretations of Lattice Gauge Field Theory
71
One may think of envelope of the final lattice as the domain of α. Since each vertex is univalent, ciliation is irrelevant. Thus holα is an intertwiner from connections on 0 to connections on the domain of α. 6.2. The notation holα is meant to be read, “holonomy along alpha.” To see it as a quantum analog of holonomy, and to illustrate some computational devices, we will work an example where 0 and α N are as in Fig. 25. Consider the connection x = ei ∈O xi . Since there is no prescribed ordering to the factors, we express x by labeling each edge of 0 with the corresponding xi as in Fig. 26. The evaluation of holα (x) begins with comultiplication applied to each of x4 , x5 and x6 , followed by switches on the output of 1(x5 ). The resulting connection lives over the intermediate lattice described in the definition of holα . The envelope and connection together appear in Fig. 27. e4
e1
e5
e2 e6
e3
Fig. 25. Envelope of 0 and a q-loop, α
x4
x1
x5
x2 x6
x3
Fig. 26. Connection x = ⊗6i=1 xi
x04
x1
x00 4 S(x00 5 k)
S(x05 k)
x2
x00 6 x06
x3
Fig. 27. Intermediate envelope and connection for holα (x)
Next we evaluate the multitangle obtained by expanding the vertex tangles. Let us assume that the multitangle has been isotoped so that evaluation proceeds one vertex at a time. The tangle for the center vertex and the connection in its domain are shown in Figure 28.
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
x3 x1 x04
00 x00 4 x6
x06
Fig. 28. Central vertex tangle
Evaluation leads to the connection and envelope shown in Fig. 29. Note that the two caps give rise to a lattice with (abstractly) four fewer edges and a new involution. This is reflected in the envelope by the concatenation of pairs of edges into single edges. t1 x04
S(x00 5 k)
2 x00 6 S (s2 )ks1 x1
S(x05 k)
x06 kt2 x00 4
x2
x3
Fig. 29. Evaluation of the central vertex tangle
The remaining vertex tangles are caps, none of which involve an involutory pair of strands. Each evaluation glues two edges of the envelope together at a vertex, creating a single edge that bypasses the vertex. The effect on the connection is multiplication in those two factors, with the cap’s orientation determining whether or not k is inserted between them. The final result is holα (x) = t1 x04 kS(x005 k)kx06 kt2 x004 kS(x05 k)kx006 S 2 (s2 )ks1 x1 x2 x3 . There is an alternative, more intuitive, method of computing holonomy. Begin with the connection drawn on the envelope as in Fig. 26. Then draw the intermediate envelope. Label new edges with Sweedler notation, reading right to left with respect to the original orientation. If any edge is not traversed by the loop, remember to counit that factor of the connection. Apply S( · k) whenever the loop’s orientation disagrees with that of the envelope, and the result is again Fig. 27. Next sketch each vertex tangle in which crossings occur. Evaluate only the crossings, writing the result on the intermediate lattice. In our example this gives Fig. 30. Finally, traverse the q-loop from beginning to end, concatenating the symbols on each edge traversed. Additionally, each time the loop passes through a vertex insert k if the cilium lies to the right. In the classical limit k = 1 and R = 1⊗1, so holα is actually accumulating holonomy as the loop is traversed. Quantization occurs when self intersections become over- or under-crossings. For another indication that holα (x) behaves like holonomy, suppose that z(1) ⊗ z(2) denotes the tensor sx1 x2 x3 ⊗ tx4 kS(x5 k)kx6 k with summation on both sides implied. The two factors resemble quantum holonomy around the two lobes of 0, 0 00 z(2) z(1) . and holα (x) = z(2) In a classical setting holonomy is inverted if the direction of the path is reversed. In a cocommutative Hopf algebra inversion is replaced by the antipode. In general, however,
Topological Interpretations of Lattice Gauge Field Theory t1 x04
73
s1 x 1
t2 x00 4 S(x00 5 k)
S(x05 k)
x2
2 x00 6 S (s2 )
x06
x3
Fig. 30. Schematic computation of holα (x)
S is not an involution. So, we have the following quantum analog of the reversing result for holonomy. Proposition 3. Let α be a q-net and α the q-net with a component c reversed. The operator holα is holα followed by a switch in the factor indexed by c. Proof. The map holα is defined by composing two multitangles: T1 , consisting of stumps and triads followed by some switches; and T2 formed from the vertex tangles. If a switch is inserted between T1 and T2 on every edge coming from c, the resulting multitangle defines holα . The new switches pass upwards through T2 , canceling at caps, until only one remains. This will lie on a pair of strands corresponding to the component c in the range of the multitangle T2 ◦ T1 . 7. Wilson Operators Roughly speaking, a Wilson operator is the trace of holonomy around a loop. To make this precise, let us extend the definition of q-net by allowing circle components in the domain. We will call the images of intervals based components and the images of circles unbased components. If there are no based components we call it a q-link. Finally, every unbased component must be colored with a finite dimensional H-module. We assume also that the based components are colored trivially. There is an equivalence relation on q-nets which, like the one on multidiagrams, mimics framed tangle isotopy. Two q-nets are equivalent if they differ by a sequence of isotopies of F , Reidemeister moves of type II and III, and the framing equivalence = . If one thinks of a q-net as a diagram of a framed tangle in F × I, then this equivalence is ambient isotopy fixing the endpoints of the based components and their framing normals. Let L be a q-net. As with quantum holonomy, L determines an operator that is obtained from the composition of two multitangles, T2 ◦ T1 . This time, however, the segments of T2 inherit the colors of the components of L. This coloring extends to T1 via consistency conditions. The resulting color on an edge in the domain of T1 can also be obtained as the right-to-left tensor product of the colors on arcs of L along that edge. The induced operator is denoted WL . It takes values in a tensor power of H indexed by the unbased components. Alternatively, one can think of the domain of L as a set of arcs and circles made into a lattice by ignoring the circles. The target of WL is connections on the domain of L. Proposition 4. If L and L0 are equivalent base point free q-nets with the same coloring then WL = WL0 .
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Proof. Cerf theory implies L and L0 differ by the moves of q-net equivalence, performed within vertices, together with the moves in Fig. 31. The latter alter multitangles by special moves, and the former by generalized Reidemeister moves.
Fig. 31. Slides of a q-net
As is usual in knot theory, we use the L to denote both a q-net (colored) and its equivalence class. The associated operator is called a Wilson operator. If L is a q-link we say WL is a Wilson link, and a Wilson link of one component is called a Wilson loop. In computing the output of a Wilson operator, we can use the same shortcut for caps as in 6.2. The starting point on a closed component does not matter because trace is invariant under cyclic permutation. Wilson operators obey a reversing result: if the reversed component is based the effect is as in Proposition 3, but if not its statement requires more care. Suppose that L and L differ by reversing an unbased component c colored by V . Let AL denote the connections where WL and WL take values. Define a map f : A → Hc ⊗A L as follows: Connect c to a base point to obtain a q-net L• . If the cilium at which c is based lies to its right, then f is WL• followed by right multiplication by k in the factor indexed by c. If it lies to the left, then f = WL• . Finally, define functions tr c and Sc on Hc ⊗ A L as tr V ⊗ 1 and S ⊗ 1 respectively. Proposition 5. With notation as above, WL = tr c ◦ f and WL = tr c ◦ Sc ◦ f . Remark. This is a bit easier to swallow if L = c. In that case the Wilson loop is computed by taking the trace of something, and the reversed loop is computed by taking the trace S of that something. Proof. Repeat the proof of Proposition 3 up to the point where all but one of the new switches have been canceled. This time the switch will be on two involutory strands entering a cap colored by V . The multitangles for WL and WL differ by replacing a cap with a switch followed by a reversed cap. Since the rest of the multitangle induces WL• , it is easy to verify the proposition for the two possible orientations of the cap. Since multitangles intertwine gauge transformations, Wilson links colored by adapted representations are necessarily elements of O. Furthermore, a Wilson link is determined by a geometric object associated to an envelope. While toggles and switches have no effect on the underlying q-net, the alterations to the lattice will effect the evaluation of the Wilson link. Fortunately the variance is natural. Theorem 7. Suppose that 0 and 00 differ by toggles and switches and that f is the induced map on connections. If WL and WL0 are Wilson links built on the same L in the two envelopes, then WL0 ◦ f = WL . Proof. Suppose 0 and 00 differ by a switch σ. In the multitangle for WL0 ◦σ, pass the first switch up through the triads (if any) on that edge and cancel any resulting consecutive switches. The result is a multitangle for WL . Assume now that the lattices differ by a counterclockwise toggle, τ . That part of the multitangle for WL is represented by the
Topological Interpretations of Lattice Gauge Field Theory
75
left side of Fig. 32, where the solid coupons are the switches and powers of 1 and the dashed coupon contains all the crossings of the transformed vertex. The portion of the multitangle for WL0 ◦ τ at that vertex is now the right side of the figure. Since these are equivalent diagrams, WL0 ◦ τ = WL .
i
j
j
n
n
i
Fig. 32. Tangle diagrams for WL and WL ◦ τv
Corollary 5. Let L be a q-link. The unique isomorphism between O0 and O00 identifies the Wilson links determined by L in each algebra. Observables produced by Wilson links can be graphically multiplied. Suppose that L and L0 are q-link. Laying L0 over L and perturbing the result creates a new q-link denoted L ? L0 . This operation is well defined on equivalence classes of q-links. Theorem 8. WL ? WL0 = WL?L0 . Proof. Let F denote the envelope of the lattice. If wt is a tangent vector along a band core, the right hand side of the edge is determined by a normal vector wn so that {wn , wt } is in the orientation of F . By equivalence of q-nets, we may assume that L lies to the right of L0 in every edge of F . At each vertex, the multitangle for WL?L0 begins with an arc for each edge of F , oriented accordingly. Apply a triad to each of these arcs so that it splits into a prime branch and a double prime branch. If L meets a given edge m times, apply 1m−1 to the prime branch for that edge. If L0 meets it n times, apply 1n−1 to the double prime branch. Now insert the necessary switches to make this the first half of the multitangle. Next, focus on the second part of the multitangle at a single vertex v of F . The transformed vertex fits onto the first part of the multitangle so that the arcs of L ∩ v meet the prime branches and the arcs of L0 ∩ v meet the double primes. Since L lies under L0 one can drag L ∩ v to the left and L0 ∩ v to the right until they are disjoint. Finally, move everything between the initial triad on each arc and L ∩ v to just beneath L ∩ v, and similarly for L0 ∩ v. This can be done for every vertex while preserving the simultaneous levels of triads and switches. Inserting a cut in between L ∩ v and L0 ∩ v in each diagram, we have a multitangle that represents (WL ⊗ WL0 ) ◦ ∇. Let bLF denote the linear space over the set of all framed, oriented, colored links in F × I (completed if b = C[[h]]). This is an algebra under ? which – for every 0 whose envelope is homeomorphic to F – is identified with a sub-algebra W0 ⊂ O0 . Corollary 4 makes these algebras into a category. The content of Corollary 5 is that bLF behaves somewhat like a universal object. One of the aims of the next section is to address how much of O is generated by Wilson links.
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Part 2. The Structure of the Algebra of Observables In this part of the paper we investigate the structure of the algebra of observables for various choices of H and B. Suppose first that G is a connected affine algebraic group. We will show that the ring of G-characters of a free group is the algebra of observables for a theory in which H is the universal enveloping algebra of G and B is its coordinate ring. Next we consider the case when H is a Drinfeld-Jimbo deformation of the universal enveloping algebra. Here the observables become a deformation quantization of the classical character ring. Following Fock and Rosly, [10], we show that the Poisson structure with respect to which we are quantizing is the standard Poisson structure, as in the work of [2, 11]. We consider the extent to which one can generate the ring of observables using Wilson loops. Among the groups (and their quantum analogs) for which this is possible is SL2 (C). We conclude with a demonstration that the observables based on (Uh (sl2 ), q SL2 ) are exactly the Kauffman bracket skein algebra of a cylinder over the envelope of the lattice. 8. Classical LGFT 8.1. Let’s recap a few facts from [12]. An affine algebraic group G is a group equipped with a finitely generated algebra of “polynomial” functions that separate points – the coordinate ring, denoted B – so that multiplication and inversion are polynomial maps. Let H be the universal enveloping algebra of G. If the group is connected then its coordinate ring is naturally identified with a stable subalgebra of H o . G acts on itself by conjugation, which adjointly induces an action of G on B. The most important fact for us is that the fixed part of B under this action is exactly the ring B H as defined in Subsect. 1.1. Now suppose that π is any finitely presented group. A representation of π into G is a point in G × · · · × G whose coordinates are the images of the generators. These must satisfy polynomial identities corresponding to the relations of the presentation, so the space of all representations is an affine algebraic set. It has a coordinate ring on which G acts by the adjoint of conjugation in each factor. The fixed subring – which is independent of presentation (up to isomorphism) – is called the affine G-characters of π. We often shorten this to “characters” or “G-characters” when π or G (or both) are understood from context. The notation is XG (π). If π is the fundamental group of a compact manifold M , we write XG (M ) for XG (π1 (M )). 8.2. Fix a connected affine algebraic group G and a lattice 0 = (E, −, V c , O). Let H be the universal enveloping algebra of G and B its coordinate ring, thought of as lying in H o . In order to see what the observables of this theory look like, we rebuild it using groups instead of algebras. Let the connections be the set of functions A : O → G. The gauge group is the set of functions g : V → G, and the gauge fields are the set ⊗e∈O B. We can view the connections as the Cartesian product of copies of G indexed by O. The gauge fields are just the coordinate ring of the Cartesian product. The action of the gauge group on the connections is given by g · A(e) = g(i(e))A(e)g −1 (t(e)), where e ∈ O, g is an element of the gauge group, and A is a connection. This induces a right action of the gauge group on the gauge fields by taking adjoints. Let OG denote the gauge fields fixed by this action.
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Theorem 9. Let G, H, B, 0 = (E, −, O, V c ) and OG be as above, and let O denote the usual observables. 1. O = OG . 2. OG is the G-characters of π1 of the geometric realization of (E, −, V, O). Proof. 1. Note that the gauge group and the gauge algebra act on the same set of gauge fields. A gauge field is invariant under the action of the gauge algebra if and only if it is invariant under elements of the form 1 ⊗ · · · ⊗ y ⊗ · · · ⊗ 1, where y lies in the Lie algebra of G. The proof that these fields are exactly those fixed by the gauge group action is now a simple generalization of [12, Corollary IV.3.2]. 2. If the lattice has one vertex containing all the edges then OG is, by definition, the G-characters of the fundamental group of the geometric realization. For arbitrary 0 consider another lattice 00 that has a single vertex, and so that the two geometric realizations are homotopic. Choose a maximal tree in the geometric realization of 0. There is an obvious map from the connections on 00 into those on 0 that sends them to the edges of O not appearing in the tree. That this is an isomorphism at the level of observables is an elementary exercise. 9. Quantized Characters For this subsection we assume that (H, B) is defined over C[[h]], and H is a DrinfeldJimbo quantization of a simple Lie algebra g [13]. In this case, H/hH is the universal enveloping algebra of g. We will denote such H by Uh (g). Suppose that B/hB is the coordinate ring of a connected affine algebraic group G. The equivalence of the representation theory of Uh (g) and U (g) makes it easy to see that O/hO is the observables associated to the theory based on (U (g), B/hB). We use topological tensor products as in [13], so the gauge fields are topologically free. Since observables form a closed subalgebra, it is also topologically free. Therefore the observables based on (Uh (g), B) are a deformation quantization of the G-characters of the fundamental group of the geometric realization of 0. The only question that remains unanswered is which Poisson structure on the ring of characters is the tangent vector to this deformation. We can assume that Uh (g) has an R-matrix of the form 1 ⊗ 1 + hr + h2 a, where a is some formal power series with coefficients in U (g) ⊗ U (g), and r is a solution of the modified classical Yang-Baxter equation. To be specific, r ∈ g ⊗ g. Such solutions were classified by Belavin and Drinfeld [14]. The standard presentation of U (g) is in terms of generators Xi , Yi , Hi , where i runs over some index set, and each triple generates a copy of U (sl2 ). The Killing form is an element of g∗ ⊗ g∗ , but being nondegenerate, you can contract it with respect to itself to give an element of g ⊗ g. This element can be expressed in the form ai Xi ⊗ Yi + bi Hi ⊗ Hi + ci Yi ⊗ Xi . The element r can be assumed to be of the form r = 2ai Xi ⊗ Yi + bi Hi ⊗ Hi . There are two actions of r on B ⊗B. You can act on the left, by letting r·(f ⊗g)(Z1 ⊗ Z2 ) = f ⊗ g(r Z1 ⊗ Z2 ), or you can act on the right by letting (f ⊗ g) · r(Z1 ⊗ Z2 ) = f (Z1 ⊗ Z2 r). The Poisson bracket on G is given by
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{f, g} = (f ⊗ g) · r − r · (f ⊗ g). In order to write out a formula for the Poisson structure, we need to create a version of r that operates on the gauge fields on a lattice. We also need to distinguish left actions and right actions: if Z acts by right multiplication in the factor corresponding to the edge e, we denote it by Z 0 (e); if it acts by multiplication on the left, we denote it by Z(e). Since the formula we derive involves antisymmetrization, we let Z1 ∧ Z2 = Z1 ⊗ Z2 − Z2 ⊗ Z1 . Now suppose that f and g are gauge fields in the quantum theory. One may compute f ? g − g ? f by writing it as f ⊗ g ◦ ∇ − g ⊗ f ◦ ∇, and expanding the R matrix as a power series in h. The linear term is X 2ai Xi0 (e) ⊗ Yi0 (e) + bi Hi0 (e) ⊗ Hi0 (e) − 2ai Xi (e) ⊗ Yi (e) − bi Hi (e) ⊗ Hi (e)+ e∈O
X
−
v∈V
2ai Xi0 (α) ∧ Yi0 (β) + bi Hi0 (α) ∧ Hi0 (β)
α<β
X
−
X
2ai Xi (−α) ∧ Yi (−β) + bi Hi (−α) ∧ Hi (−β)
−α<−β
+
X
2ai Xi0 (α) ∧ Yi (−β) + bi Hi0 (α) ∧ Hi (−β)
α<−β
+
X
2ai Xi (−α) ∧ Yi0 (β) + bi Hi (−α) ∧ Hi0 (β) .
−α<β
This is the same as the formula derived by Fock and Rosly. Hence the Poisson structure on our algebra of classical observables is the complex linear extension of the standard Poisson structure on the characters of the surface with respect to the compact group.
10. The Kauffman Bracket Skein Module At the end of Sect. 7 we introduced the algebra of links, bLF , which maps naturally into the observables for any lattice with envelope F . For lattice gauge field theory based on one of the groups GLn (C), On (C), Spn (C), the map is onto. This is due to a theorem of Procesi [15], stating that the invariant theory for Cartesian products of these groups is generated by traces. It follows that Wilson links generate observables for theories built on Drinfeld-Jimbo deformations of these groups as well. Furthermore, since each of these groups has a fundamental representation in whose tensor powers one may find all irreducible representations, a single color suffices. The invariant theory of SLn (C) is a quotient of the invariant theory for GLn (C), so here again Wilson links with a single color suffice. If n = 2, we can even specify the kernel of the map from links to observables in terms of skein relations. A space of links divided by skein relations is a skein module. We will show that the observables for a theory built on (Uh (sl2 ),q SL2 ) are the Kauffman bracket skein module. (Actually, there is a sign change involved in the morphism. This could be eliminated by redefining the skein module, but we prefer to keep its original form.)
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To obtain this result we need an explicit formula for the R-matrix in the fundamental representation. The generators of Uh (sl2 ) are X, Y and H. Let 21 be the vector space spanned by e− 1 , e 1 . The standard representation 2
2
1 ρ : Uh (sl2 ) → End( [[h]]) 2 is given by
ρ(X) =
01 00
ρ(Y ) =
00 10
ρ(H) =
1 0 . 0 −1
One may now expand the R-matrix in [13] using these matrices and writing eh/2 for each appearance of q. The trace on Uh (sl2 ) is just the trace under this representation, and takes on values in C[[h]]. The action of S in the fundamental representation is ab d −eh/2 b . 7→ cd a −eh/2 c So tr ◦ S = tr. In SL2 (C), the Cayley-Hamilton identity is: A2 − tr(A)A + I = 0. Putting the term in the trace on the other side of the equation, and multiplying by A−1 we get a linear homogeneous equation.: A + A−1 = tr(A)I. Now multiply by B to get back a bilinear equation: AB + A−1 B = tr(A)B. Finally take the trace: tr(AB) + tr(A−1 B) = tr(A)tr(B). This formula is the Cayley-Hamilton identity as a trace identity for SL2 (C). It is fundamental for SL2 (C) in the sense that every other identity between traces follows from evaluation of this one. The same identity persists in U (sl2 ), except that you need to use the antipode instead of the inverse: tr(ZW ) + tr(S(Z)W ) = tr(Z)tr(W ). Finally the fundamental trace identity for Uh (sl2 ) is: X tr(si Z)tr(ti W ), t tr(ZW ) + t−1 tr(S(Z)W ) = P
i
where i si ⊗ ti is the R-matrix for Uh (sl2 ) and t = eh/4 . Let LM be the set of framed links (including ∅) in a 3-manifold M . Let CLM [[h]] denote formal power series in h with coefficients in the vector space over LM . We define S(M ) to be the topological submodule of CLM [[h]] generated by all expressions of the form
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D. Bullock, C. Frohman, J. Kania-Bartoszy´nska
+t
+ t−1
,
and
2. + t2 + t−2 . These formulas indicate relations that hold among links which can be isotoped in M so that they are identical except in the neighborhood shown. The Kauffman bracket skein module is the quotient K(M ) = CLM [[h]]/S(M ). The Kauffman bracket skein module of F × I is an algebra with multiplication as in the space of links in Sect. 7. We denote it K(F ). From [7] we know that K(F ) is a deformation quantization of the SL2 (C)-characters of the fundamental group of F . Theorem 10. Let F be a compact, connected surface with boundary. If the surface underlying the envelope of 0 is F then the observables of a lattice gauge field theory based on (Uh (sl2 ),q SL2 ) are canonically isomorphic to K(F ). Proof. We define a map ζ : K(F ) → O as follows. Let L be a framed link in F × I. Represent it as a diagram in the envelope of 0 with the blackboard framing. Orient the components arbitrarily and color them with the fundamental representation. Now perturb it so that it becomes a q-link, also denoted L. Finally, map it to the observable (−1)|L| WL . This is well defined at the level of LF ×I by Propositions 4 and 5 and the fact that tr ◦ S = tr. That ζ sends elements of S(F × I) to zero follows from the quantum Cayley-Hamilton identity. By Theorem 8, ζ is an algebra map. To see that it is an isomorphism, consider the map induced between K(F )/hK(F ) and O/hO. This is known to be an isomorphism ([6]), so ζ must be one as well. References 1. Alekseev, A.Y., Grosse, H., Schomerus, V.: Combinatorial quantization of the Hamiltonian ChernSimons theory I–II. hep-th/9403066 and hep-th/9408097 and Commun. Math. Phys. 172, 317–358 (1995) 2. Biswas, I. and Guruprasad, K.: Principle bundles on open surfaces and invariant functions on Lie groups. Int. J. Math. 4, 535–544 (1993) 3. Buffenoir, E.: Chern-Simons theory on a lattice and a new description of 3-manifold invariants. qalg/9509020 4. Buffenoir, E. and Roche, Ph.: Two dimensional lattice gauge theory based on a quantum group. Commun. Math. Phys. 170, 661–698 (1995) 5. Buffenoir, E. and Roche, Ph.: Link invariants and combinatorial quantization of Hamiltonian ChernSimons theory. q-alg/9507001, and Commun. Math. Phys. 181, 331–365 (1996) 6. Bullock, D.: Rings of SL2 (C)-characters and the Kauffman bracket skein module q-alg/9604014, to appear in Comm. Math. Helv. 7. Bullock, D., Frohman, C. and Kania-Bartoszy´nska, J.: Understanding the Kauffman bracket skein module. q-alg/9604013, to appear in J. Knot Theory Ramifications 8. Bullock, D., Frohman, C. and Kania-Bartoszy´nska, J.: Skein quantization and lattice gauge field theory q-alg/9609024, and Chaos, Solitons & Frachels 9, 811–824 (1998) 9. Drinfeld, V.G.: Quantum groups. In: Proc. Int. Congress Math., Berkeley 1986, American Mathematical Society, 1987, pp. 798–820 10. Fock, V.V. and Rosly, A.A.: Flat connections and polyubles. Teoreticheskaya i Matematicheskaya Fizika 95, no. 2, 228–238 11. Goldman, W.: Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85, 263–302 (1986)
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12. Hochschild, G.: Basic Theory of Algebraic Groups and Lie Algebras, Berlin–Heidelberg–New York: Springer Verlag, 1995 13. Kassel, C.: Quantum Groups, Berlin–Heidelberg–New York: Springer-Verlag, 1995 14. Korgodsky, L. and Soibelman, Y.: Algebras of functions on quantum groups. In: Mathematical surreys and monographs; no 56. Providence, Rhode Island: American Mathematical Society, 1998 15. Procesi, C.: The Invariant Theory of n × n Matrices. Adv. Math. 19, 306–381 (1976) 16. Przytycki, J.H. and Sikora, A.: On skein algebras and SL2 (C)-character varieties. q-alg/9705011, to appear in Topology 17. Sikora, A.: A new geometric method for SLn -representations. Preprint 18. Sweedler, M.: Hopf Algebras, New York. W.A. Benjamin, 1969 Communicated by A. Jaffe
Commun. Math. Phys. 198, 83 – 95 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Multidimensional Extension of the Generalized Chowla–Selberg Formula E. Elizalde1,2 1
Consejo Superior de Investigaciones Cient´ıficas (CSIC), Institut d’Estudis Espacials de Catalunya (IEEC), Edifici Nexus 201, Gran Capit`a 2–4, 08034 Barcelona, Spain. E-mail:
[email protected] 2 Departament ECM and IFAE, Facultat de F´ısica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain. E-mail:
[email protected] Received: 20 July 1997 / Accepted: 27 March 1998
Abstract: After recalling the precise existence conditions of the zeta function of a pseudodifferential operator, and the concept of reflection formula, an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous Epstein-type zeta function of the general form X (~nT A~n + ~bT ~n + q)−s , ζA,~b,q (s) = p ~ n∈Z with A the p × p matrix of a quadratic form, ~b a p vector and q a constant, is obtained. It is valid on the whole complex s-plane, is exponentially convergent and provides the residua at the poles explicitly. It reduces to the famous formula of Chowla and Selberg in the particular case p = 2, ~b = ~0, q = 0. Some variations of the formula and physical applications are considered.
1. Introduction: Existence of the Zeta Function of a Pseudodifferential Operator (9DO) A pseudodifferential operator A of order m on a manifold Mn is defined through its symbol a(x, ξ), which is a function belonging to the space S m (Rn ×Rn ) of C∞ functions such that for any pair of multi-indexs α, β there exists a constant Cα,β so that α β ∂ξ ∂x a(x, ξ) ≤ Cα,β (1 + |ξ|)m−|α| . (1) The definition of A is given, in the distribution sense, by Z −n ei<x,ξ> a(x, ξ)fˆ(ξ) dξ, Af (x) = (2π)
(2)
84
E. Elizalde
where f is a smooth function, f ∈ S [remember that S = {f ∈ C ∞ (Rn ); supx |xβ ∂ α f (x)| < ∞, ∀α, β ∈ Rn ], S 0 being the space of tempered distributions and fˆ the Fourier transform of f . When a(x, ξ) is a polynomial in ξ one gets a differential operator. In general, the order m can be complex. The symbol of a 9DO has the form a(x, ξ) = am (x, ξ) + am−1 (x, ξ) + · · · + am−j (x, ξ) + · · · ,
(3)
being ak (x, ξ) = bk (x) ξ k . Pseudodifferential operators are useful tools, both in mathematics and in physics. They were crucial for the proof of the uniqueness of the Cauchy problem [1] and also for the proof of the Atiyah-Singer index formula [2]. In quantum field theory they appear in any analytical continuation process (as complex powers of differential operators, like the Laplacian) [3]. And they constitute nowadays the basic starting point of any rigorous formulation of quantum field theory through microlocalization, a concept that is considered to be the most important step towards the understanding of linear partial differential equations since the invention of distributions [4]. For A a positive-definite elliptic 9DO of positive order m ∈ R, acting on the space of smooth sections of an n-dimensional vector bundle E over a closed, n-dimensional manifold M , the zeta function is defined as X n ≡ s0 . λ−s Re s > (4) ζA (s) = tr A−s = j , m j The quotient s0 = dim M/ord A is called the abscissa of convergence of ζA (s), which is proven to have a meromorphic continuation to the whole complex plane C (regular at s0 ), provided that the principal symbol of A (that is am (x, ξ)) admits a spectral cut: Lθ = {λ ∈ C; Arg λ = θ, θ1 < θ < θ2 }, Spec A ∩ Lθ = ∅ (the Agmon–Nirenberg condition). Strictly speaking, the definition of ζA (s) depends on the position of the cut 0 (0)], which only depends on Lθ , not so that of the determinant [5] det ζ A = exp[−ζA the homotopy class of the cut. The precise structure of the analytical continuation of the zeta function is known in general [6]. The only singularities it can have are simple poles at sk = (n − k)/m,
k = 0, 1, 2, . . . , n − 1, n + 1, . . . .
(5)
The applications of the zeta-function definition of a determinant in physics are very important [7, 8]. A zeta function with the same meromorphic structure in the complex s-plane and extending the ordinary definition to operators of complex order m ∈ C\Z (it is clear that operators of complex order do not admit spectral cuts), has been obtained in Ref. [9]. The construction starts there from the definition of a trace, obtained as the integral over the manifold of the trace density of the difference between the Schwartz kernel of A and the Fourier transformed of a number of first homogeneous terms (in ξ) of the usual decomposition of the symbol (3) of A.
2. Exponentially Convergent Analytic Continuation (with Explicit Poles and Residua) of our Zeta Function A fundamental property shared by zeta functions of any nature is the existence of a reflection formula. For a generic zeta function, Z(s), it has the form:
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85
Z(ω − s) = F (ω, s)Z(s),
(6)
and allows for its analytical continuation in a very easy way – what is the whole story of the zeta function regularization procedure (or at least the main part of it). But the analytically continued expression thus obtained is just another series, which has again a slow convergence behavior, of power series type (actually the same that the original series had, in its own domain of validity). Some years ago, the notorious mathematicians S. Chowla and A. Selberg found a formula, for the case p = 2, ~b = 0 and q = 0 above [10], that yields exponentially quick convergence, and not only in the reflected domain. They were extremely proud of that formula – as one can appreciate by reading the original paper, where actually no hint about its derivation was given. In Ref. [11] we generalized this expression to inhomogeneous zeta functions, but staying always in two dimensions (p = 2), for this was commonly believed to be an unsurmountable restriction of the original formula (see, for instance, Ref. [12]). The generalization we carried out in [11] was already non-trivial and a very detailed account of this step is given in [13] (where a misprint of the original derivation is corrected). Finally, we have realized that an extension to an arbitrary number of dimensions is actually possible. In fact, here we shall obtain formulas for arbitrary p and arbitrary values of ~b and q. The starting point will be Poisson’s resummation formula in p dimensions, which arises from the distribution identity X X ~ x δ(~x − ~n) = ei2πm·~ . (7) p p ~ n∈Z m∈ ~ Z (We shall indistinctly write m ~ · ~x ≡ m ~ T ~x in what follows.) Applying this identity to the function 1 T T ~ f (~x) = exp − ~x A~x + b ~x , (8) 2 with A an invertible p × p matrix, and integrating then over ~x ∈ Rp , one obtains X 1 T T ~ exp − ~n A~n + b ~n 2 p ~ n∈Z T (2π)p/2 X 1 ~ −1 ~ √ = b + 2πim ~ b + 2πim ~ . exp A 2 det A p m∈ ~ Z We are going to consider the following zeta function −s 0 X 1 T (~ ) (~ ) (s) = A + q ζA,~c,q n + ~c n + ~c 2 p ~ n∈Z 0 X −s p ≡ , Re (s) > . Q (~n + ~c) + q 2 p ~ n∈Z
(9)
(10)
The aim is to obtain a formula that gives (the analytical continuation of) this multidimensional zeta function in terms of an exponentially convergent multiseries and which is valid in the whole complex plane, exhibiting the singularities (simple poles) of the meromorphic continuation – with the corresponding residua – explicitly. The prime in
86
E. Elizalde
the summatories of Eq. (10) means that the point ~n = ~0 is to be excluded from the sum. Such restriction is irrelevant as long as q 6= 0 (the contribution of this single point being immediately obtainable), but to define the zeta function in this way is essential in order to be able to reach the limit q → 0 (as we shall do later). The only condition on the matrix A is that it corresponds to a (non-negative) quadratic form, that we call Q. The vector ~c is arbitrary, while q will (for the moment) be a positive constant. 2.1. The main expression (inhomogeneous case). Use of the Poisson resummation formula (9) yields, after some work, the following expression: (2π)p/2 q p/2−s 0(s − p/2) 2s/2+p/4+2 π s q −s/2+p/4 √ √ + 0(s) det A det A 0(s) 0 p X s/2−p/4 × cos(2π m ~ · ~c) m ~ T A−1 m ~ Kp/2−s 2π 2q m ~ T A−1 m ~ , (11) p m∈ ~ Z1/2 ζA,~c,q (s) =
where Kν is the modified Bessel function of the second kind and the subindex 1/2 in ~ ∈ Zp intervene in the sum. That is, if we take Zp1/2 means that only half of the vectors m ~ (as a simple criterion one can, for instance, select an m ~ ∈ Zp we must then exclude −m those vectors in Zp \{~0} whose first non-zero component is positive). Equation (11) fulfills all the requirements demanded before. It is notorious to observe how the only pole of this inhomogeneous Epstein zeta function appears explicitly at s = p/2, where it belongs. Its residue is given by the formula: Ress=p/2 ζA,~c,q (s) = √
(2π)p/2 . det A 0(p/2)
(12)
With a bit of care, it is relatively simple to obtain the limit of expression (11) as q → 0. However, instead of proceeding in this way (what we shall do later), it is advisable to construct first a direct recurrent formula for the case q = 0. This is certainly the natural option in such a case, where no cut-off q exists to safeguard the t-integration (there is no way to use the Poisson formula on all p indices of ~n at once). However, we can still deal with this case by using the Poisson resummation formula on some of the p indices ~n only, say on just one of them, n1 . Poisson’s formula on one index reduces to the celebrated Jacobi identity for the θ3 function ∞ X 2 θ3 z |τ = 1 + 2 q n cos(2nz),
q = eπiτ , |q| < 1, τ ∈ C,
(13)
n=1
the identity being: 2 1 ez /(πiτ ) θ3 θ3 z |τ = √ −iτ
z −1 , τ τ
(14)
or, in other words ∞ X n=−∞
e
n2 πiτ +2niz
=√
∞ X 2 1 e(z−nπ) /(πiτ ) −iτ n=−∞
(15)
Multidimensional Extension of Generalized Chowla–Selberg Formula
87
(for a classical reference see, e.g. Ref. [14]). Here z and τ are arbitrary complex, z, τ ∈ C, with the only restriction that Im τ > 0 (in order that |q| < 1). For the applications, it turns out to be better to recast the Jacobi identity as follows (with πiτ → −t and z → πz): r ∞ ∞ X 2 π X −π2 (n−z)2 /t e−n t+2πinz = e , (16) t n=−∞ n=−∞ equivalently r
∞ X
e
−(n+z)2 t
=
n=−∞
" # ∞ X π −π 2 n2 /t 1+ e cos(2πnz) , t
(17)
n=1
with z, t ∈ C, Re t > 0. 2.2. Recurrent expression (and the homogeneous case). Using this last formula on the first component, n1 , of the summation vector ~n, we obtain (for the sake of simplicity of the final expressions, we shall just consider now the case ~c = ~0, but the result can be generalized to ~c 6= ~0 quite easily): ζA,~0,q (s) = 2
∞ X
an21 + q
−s
n1 =1
+
+
1 0(s) 4π
p−1
~ n2 ∈Z ∞ s X
as/2+1/4
× Ks−1/2 which can be written as
r
0 X
cos
n1 =1
2πn1 √ a
1/2−s π 0(s − 1/2) ~nT2 1p−1~n2 + q a
πn
1/4−s/2 s−1/2 b ~n2 n1 ~nT2 1p−1~n2 + q
1 ~T
a
q T ~n2 1p−1~n2 + q ,
(18)
r
π 0(s − 1/2) ζ1p−1 ,~0,q (s − 1/2) a 0(s) 0 ∞ πn X X 4π s 1 ~T b ~n2 cos + s/2+1/4 a a 0(s) p−1 n1 =1 ~ n2 ∈Z q 1/4−s/2 2πn1 s−1/2 T T Ks−1/2 √ ~n2 1p−1~n2 + q . (19) ~n2 1p−1~n2 + q ×n1 a
ζA,~0,q (s) = ζa,~0,q (s) +
This is clearly a recurrent formula in p, the number of dimensions, the first term of the recurrence being r ∞ X −s π 0(s − 1/2) 1/2−s q = q −s + an2 + q ζa,~0,q (s) = 2 a 0(s) n=1 r ∞ 4π s −1/4−s/2 1/4−s/2 X s−1/2 q a . (20) + q n Ks−1/2 2πn 0(s) a n=1
88
E. Elizalde
To take in these expressions the limit q → 0 is immediate. One obtains: r π 0(s − 1/2) ζ1p−1 ,~0,0 (s − 1/2) ζA,~0,0 (s) = 2a−s ζ(2s) + a 0(s) 0 ∞ πn X X 4π s 1 ~T b ~n2 cos + s/2+1/4 a a 0(s) p−1 n1 =1 ~ n2 ∈ Z q 1/4−s/2 2πn1 s−1/2 T T √ Ks−1/2 ~n2 1p−1~n2 . (21) ~n2 1p−1~n2 ×n1 a In the above formulas, A is a p × p symmetric matrix A = aij i,j=1,2,... ,p = AT , Ap−1 the (p − 1) × (p − 1) reduced matrix Ap−1 = aij i,j=2,... ,p , a the component a = a11 , ~b the p − 1 vector ~b = (a21 , . . . , ap1 )T = (a12 , . . . , a1p )T , and finally, 1p−1 is 1~ the following (p − 1) × (p − 1) matrix 1p−1 = Ap−1 − 4a b ⊗ ~b. Let us now get back to the case q = 0 starting from the beginning (e.g., the zeta function given by Eq. (10), with q = 0). From that expression, the recurrence (21) can be obtained directly in the same way – this is rather obvious, since q plays no role in the derivation. What is not so obvious is to realize that the limit as q → 0 of Eq. (11) is exactly the recurrent formula (21). More precisely, what is obtained in the limit is the reflected formula which one gets after using the well known Epstein zeta function reflection π 2s−p/2 0(p/2 − s)Z(p/2 − s; A−1 ), 0(s)Z(s; A) = √ det A
(22)
being Z(s; A) the Epstein zeta function [15]. After some thinking, such a result is easy to understand. Summing up, we have thus checked that our formula (11) is valid for any q ≥ 0, since it contains in a hidden way, for q = 0, the recurrent expression (21). As announced at the beginning, the formulas derived here can be considered as generalizations (in more than one sense) of the Chowla-Selberg (CS) formula. All share the same properties that are so much appreciated by number-theoretists as pertaining to the CS formula. In a way, these expressions can be viewed as improved reflection formulas for zeta functions; they are in fact much better than those in several aspects. Namely, while a reflection formula connects one region of the complex plane with a complementary region (with some intersection) by analytical continuation, the CS formula and our formulas are valid on the whole complex plane, exhibiting the poles of the zeta function and the corresponding residua explicitly. Even more important, while a reflection formula is intended to replace the initial expression of the zeta function – a power series whose convergence can be extremely slow – by another power series with the same type of convergence, it turns out that the expressions here considered give the meromorphic extension of the zeta function, on the whole complex s-plane, in terms of an exponentially decreasing power series (as was the case with the CS formula, that one being its most precious property). Actually, exponential convergence strictly holds under the condition that q ≥ 0. However, the formulas themselves are valid for q < 0 or even complex. What is not guaranteed for general q ∈C is the exponential convergence of the series, nor its powerlike convergence, for that matter. Those analytical continuations in q must be dealt with specifically, case by case. The physical example of a field theory with a chemical potential falls clearly into this class.
Multidimensional Extension of Generalized Chowla–Selberg Formula
89
2.3. Particular case p = 2. The above statements apply both for the cases q > 0 and q = 0. The last situation is more involved, however. One is led to employ the recurrence relation (21) several times (p − 1, in general), what gives rise each time to an additional series of Bessel functions Kν (exponential convergence). For completeness, let us write down the corresponding series when p = 2 explicitly. They are, with q > 0 [13], ζE (s; a, b, c; q) =
" r ∞ q 1/4 π s X 4 2πq 1−s q s−1/2 √ + n K 2πn −q + √ s−1/2 a qa a (s − 1) 1 0(s) n=1 r r r s X ∞ q a aq 2π + (23) ns−1 Ks−1 4πn a q1 1 n=1 r 1/4−s/2 ∞ X X 4aq 2 (2π)s ns−1/2 cos(πnb/a) d1−2s 1 + 2 + a d n=1 d|n !# r 4aq πn Ks−1/2 1+ 2 , a d −s
where 1 = 4ac − b2 > 0, and, with q = 0, the CS formula [10] −s
√ 22s π as−1
0(s − 1/2)ζ(2s − 1) 0(s)1s−1/2 ∞ πn √ X 2s+5/2 π s s−1/2 n σ (n) cos(πnb/a) K 1 , + √ 1−2s s−1/2 a 0(s) 1s/2−1/4 a
ζE (s; a, b, c; 0) = 2ζ(2s) a
+
n=1
(24) P where σs (n) ≡ d|n ds , sum over the s-powers of the divisors of n. (There is a misprint in the transcription of formula (24) in Ref. [12]). We observe that the rhs’s of (23) and (24) exhibit a simple pole at s = 1, with common residue: 2π Ress=1 ζE (s; a, b, c; q) = √ = Ress=1 ζE (s; a, b, c; 0). 1
(25)
3. The Case of a Truncated Range The most involved case in the family of Epstein-like zeta functions corresponds to having to deal with a truncated range. This comes about when one imposes boundary conditions of the usual Dirichlet or Neumann type [13]. Jacobi’s theta function identity and Poisson’s summation formula are then useless and no expression in terms of a convergent series for the analytical continuation to values of Re s below the abscissa of convergence can be obtained. The best one gets is an asymptotic series expression. However, the issue of extending the CS formula or, better still, the most general expression we have obtained before for inhomogeneous Epstein zeta functions, is not an easy one. This problem has seldom (if ever) been properly addressed in the literature.
90
E. Elizalde
3.1. Example 1. To illustrate the issue, let us consider the following simple example in one dimension: ∞ X
ζG (s; a, c; q) ≡
a(n + c)2 + q
−s
,
Re s > 1/2.
(26)
n=−∞
Associated with this zeta functions, but considerably more difficult to treat, is the truncated series, with indices running from 0 to ∞, ζGt (s; a, c; q) ≡
∞ X
a(n + c)2 + q
−s
,
Re s > 1/2.
(27)
n=0
In this case the Jacobi identity is of no use. How to proceed then? The only way is to employ specific techniques of analytic continuation of zeta functions [13]. The usual method involves three steps [16]. The first step is elementary: to write the initial series as a Mellin transformed one, ∞ ∞ Z X −s 1 X ∞ = dt ts−1 exp −[a(n + c)2 + q]t . (28) a(n + c)2 + q 0(s) 0 n=0
n=0
The second is to expand in power series part of the exponential, while leaving always a converging exponential factor, ∞ ∞ Z ∞ X −s 1 X ∞ X (−a)m 2 (n + c)2m ts+m−1 e−qt . (29) = dt a(n + c) + q 0(s) m! 0 n=0
n=0
m=0
The third and most difficult step is to interchange the order of the two summations – which the aim to obtain a series of zeta functions – which means transforming the second series into an integral along a path on the complex plane, that has to be closed into a circuit (the sum over poles inside reproduces the original series), with a part of it being sent to infinity. Usually, after interchanging the first series and the integral, there is a contribution of this part of the circuit at infinity, which provides in the end an additional contribution to the trivial commutation. More important, what one obtains in general through this process is not a convergent series of zeta functions, but an asymptotic series [13]. That is, in our example, ∞ X n=0
a(n + c)2 + q
−s
∼
∞ X (−a)m 0(m + s) m=0
m! 0(s) q m+s
ζH (−2m, c) + additional terms. (30)
Being more precise, as an outcome of the whole process we obtain the following result for the analytic continuation of the zeta function [17]: ∞ q −s X (−1)m 0(m + s) q −m 1 − c q −s + ζH (−2m, c) ζGt (s; a, c; q) ∼ 2 0(s) m! a m=1 r π 0(s − 1/2) 1/2−s q + a 20(s) ∞ p 2π s −1/4−s/2 1/4−s/2 X s−1/2 a + q n cos(2πnc)Ks−1/2 (2πn q/a). (31) 0(s) n=1
Multidimensional Extension of Generalized Chowla–Selberg Formula
91
(Note that this expression reduces to Eq. (20) in the limit c → 0.) The first series on the rhs is asymptotic [16, 18]. Observe, on the other hand, the singularity structure of this zeta function. Apart from the pole at s = 1/2, there is a whole sequence of poles at the negative real axis, for s = −1/2, −3/2, −5/2, . . . , with residua: Ress=1/2−j ζGt (s; a, c; q) =
(2j − 1)!! q j √ , j = 0, 1, 2, . . . . j! 2j a
(32)
3.2. Example 2. As a second example, in order to obtain the analytic continuation to Re s ≤ 1 of the truncated inhomogeneous Epstein zeta function in two dimensions, ζEt (s; a, b, c; q) ≡
∞ X
(am2 + bmn + cn2 + q)−s ,
(33)
m,n=0
we can proceed in two ways: either by direct calculation following the three steps as explained above or else by using the final formula for the Epstein zeta function in one dimension (example 1) recurrently. In both cases the end result is the same: ζEt (s; a, b, c; q) ≡
∞ X
(am2 + bmn + cn2 + q)−s
m,n=0
∞ (4a) X (−1)m 0(m + s) bn 2m 2 −m−s (2a) (1n + 4aq) ∼ ζH −2m; 0(s) m! 2a m,n=1 −n ∞ X b q 1−s πq 1−s q −s (−1)n 0(n + s − 1)Bn 4aq √ − + + (s − 1)10(s − 1) n! 1 4 2(s − 1) 1 n=0 r r 1 π π 0(s − 1/2) 1/2−s + q + 4 a c 0(s) " r ∞ q 1/4 π s X 1 q s−1/2 2 + n Ks−1/2 2πn √ 0(s) a qa a n=1 r r ∞ s aq 1/4 X a aq + ns−1/2 Ks−1/2 2πn π 1 q1 1 n=1 r r r s X ∞ q a aq 2π + ns−1 Ks−1 4πn a q1 1 n=1 r ∞ X 2 (2π)s ns−1/2 cos(πnb/a) + a n=1 ! r 1/4−s/2 X 4aq 4aq πn d1−2s 1 + 2 Ks−1/2 1 + 2 . (34) × d a d s
d|n
The first series on the rhs is in general asymptotic, although it converges for a wide range of values of the parameters. The second series is always asymptotic and its first term contributes to the pole at s = 1. As in the case of Eq. (23), the pole structure is here explicit, although much more elaborate. Apart from the pole at s = 1, whose residue is
92
E. Elizalde
b π Ress=1 ζEt (s; a, b, c; q) = √ − , 2 1 1
(35)
there is here also a sequence of poles at s = ±1/2, −3/2, −5/2, . . . , with residua: (2j − 1)!! q j 1 1 √ √ Ress=1/2−j ζEt (s; a, b, c; q) = + , j = 0, 1, 2, . . . . (36) j! 2j+2 a c The formula above, Eq. (34), is really imposing and hints already towards the conclusion that the derivation of a general expression in p dimensions for the zeta function considered in Sect. 2 but with a truncated range is not an easy task.
4. Some Uses of the Formulas These formulas are very powerful expressions in order to determine the analytic structure of generalized inhomogeneous Epstein type zeta functions, to obtain specific values of these zeta functions at different points, and from there, in particular, the Casimir effect and heat kernel coefficients, and also in order to calculate derivatives of the zeta function, and from them, in particular, the associated determinant. Notice that obtaining derivatives of the formulas in Sect. 2 presents no problem. Only for truncated zeta functions (Sect. 3) the usual care must be taken when dealing with asymptotical expansions. We shall illustrate these uses with three specific applications. 4.1. Application 1. In a recent paper by R. Bousso and S. Hawking [19], where the trace anomaly of a dilaton coupled scalar in two dimensions is calculated, the zeta function method is employed for obtaining the one-loop effective action, W , which is given by the well known expression W =
1 ζA (0) ln µ2 + ζA 0 (0) , 2
(37)
with ζA (s) = tr A−s . In conformal field theory and in a Euclidean background manifold of toroidal topology, the eigenvalues of A are found perturbatively (see [19]), which leads one to consider the following zeta function: ∞ X
ζA (s) =
(3kl )−s ,
(38)
2 2 + , 2 2(4l2 − 1)
(39)
k,l=−∞
with the eigenvalues 3kl being given by 3kl = k 2 + l2 +
where is a perturbation parameter. It can be shown that the integral of the trace anomaly is given by the value of the zeta function at s = 0. One barely needs to follow the several pages long discussion in [19], leading to the calculation of this value, in order to appreciate the power of the formulae of the preceding section. In fact, to begin with, no mass term needs to be introduced to arrive at the result and no limit mass → 0 needs to be taken later. Using the binomial expansion (the same as in Ref. [19]), one gets
Multidimensional Extension of Generalized Chowla–Selberg Formula
ζ(s) =
∞ X
k 2 + l2 +
k,l=−∞
2 2
−s −
93
−1−s ∞ 2 s X 2 (4l2 − 1)−1 . k 2 + l2 + 2 2 (40) k,l=−∞
From Eq. (23) above, the first zeta function gives, at s = 0, exactly: −π2 /2. And this is the whole result (which does coincide with the one obtained in [19]), since the second term has no pole at s = 0 and provides no contribution. 4.2. Application 2. Another direct application is the calculation of the Casimir energy density corresponding to a massive scalar field on a general, p dimensional toroidal manifold (see [20]). In the spacetime M = R ×6, with 6 = [0, 1]p /∼, which is topologically equivalent to the p torus, the Casimir energy density for a massive scalar field is given directly by Eq. (11) at s = −1/2, with q = m2 (mass of the field), ~b = ~0, and A being the matrix of the metric g on 6, the general p-torus: C = ζg,~0,m2 (s = −1/2). EM,m
(41)
The components of g are, in fact, the coefficients of the different terms of the Laplacian, which is the relevant operator in the Klein-Gordon field equation. The massless case is also obtained, with the same specifications, from the corresponding formula Eq. (21). In both cases no extra calculation needs to be done, and the physical results follows from a mere identification of the components of the matrix A with those of the metric tensor of the manifold in question [20]. Very much related with this application but more involved and ambitious is the calculation of vacuum energy densities corresponding to spherical configurations and the bag model (see [21, 22], and the many references therein). 4.3. Application 3. A third application consists in calculating the determinant of a differential operator, say the Laplacian on a general p-dimensional torus. A very important problem related with this issue is that of the associated anomaly (called the multiplicative or non-commutative anomaly) [23]. To this end the derivative of the zeta function at s = 0 has to be obtained. From Eq. (11), we get 0 p 4(2q)p/4 X cos(2π m ~ · ~c) T A−1 m 2π K 2q m ~ ~ ζ 0 A,~c,q (0) = √ p/4 p/2 det A p ~ m ~ T A−1 m m∈ ~ Z1/2 (2π)p/2 0(−p/2)q p/2 √ , p odd, det A + k k k (−1) (2π) q √ [9(k + 1) + γ − ln q] , p = 2k even, k! det A
(42)
0 and, from here, det A = exp −ζA (0). For p = 2, we have explicitly:
det A(a, b, c; q) = ( r r ∞ √ √ X 1 a aq 2π(q−ln q)/ 1 −2π q/a K1 4πn 1−e e exp −4 n q 1 n=1 ! r X πn 4aq . d exp − 1+ 2 + cos(πnb/a) a d d|n
(43)
94
E. Elizalde
In the homogeneous case (CS formula) we obtain for the determinant: " # √ ∞ √ X π 1 σ1 (n) 1 0 −πn 1/a −4 cos(πnb/a)e , det A(a, b, c) = exp −4ζ (0) − a 6a n (44) n=1
or, in terms of the Teichm¨uller coefficients, τ1 and τ2 , of the metric tensor (for the metric, A, corresponding to the general torus in two dimensions): det A(τ1 , τ2 ) = " # ∞ X (45) 2 πτ τ2 σ (n) 2πnτ 2 1 1 cos exp −4ζ 0 (0) − −4 e−πnτ2 /|τ | . 4π 2 |τ |2 3|τ |2 n |τ |2 n=1
Needless to mention, all the good properties of the expression for the zeta function are just transferred to the associated determinants, which are thus given, on its turn, in terms of very quickly convergent series. Acknowledgement. The author is indebted with Andreas Wipf, Michael Bordag, Klaus Kirsten and Sergio Zerbini for enlightening discussions and with the members of the Institutes of Theoretical Physics of the Universities of Jena and Leipzig, where the main part of this work was done, for very kind hospitality. This investigation has been supported by CIRIT (Generalitat de Catalunya), DGICYT (Spain), project PB96-0925, and by the German-Spanish program Acciones Integradas, project HA1996-0069.
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16. Elizalde, E.: J. Phys. A22, 931 (1989); Elizalde, E. and Romeo, A.: Phys. Rev. D40, 436 (1989); Elizalde, E.: J. Math. Phys. 31, 170 (1990) 17. Elizalde, E.: J. Math. Phys. 35, 6100 (1994) 18. Elizalde, E.: J. Math. Phys. 35, 3308 (1994) 19. Bousso, R. and Hawking, S.W.: Phys. Rev. D56, 7788 (1998) 20. Kirsten, K. and Elizalde, E.: Phys. Lett. B365, 72 (1995) 21. Bordag, M., Elizalde, E., Kirsten, K. and Leseduarte, S.: Phys. Rev. D56, 4896 (1997) 22. Sachs, I. and Wipf, A.: Ann. Phys. (NY) 249, 380 (1996); Wipf, A. and D¨urr, S.: Nucl. Phys. B443, 201 (1995) 23. Elizalde, E., Vanzo, L. and Zerbini, S.: Zeta function regularization, the multiplicative anomaly and the Wodzicki residue. Trento preprint UTF 394, hep-th/9701078, to appear in Commun. Math. Phys.; Elizalde, E., Filippi, A., Vanzo, L. and Zerbini, S.: One-loop effective potential for a fixed charged selfinteracting bosonic model at finite temperature with its related multiplicative anomaly. Trento preprint UTF 405, Imperial/TP/97-98/4, hep-th/9710171, to appear in Phys. Rev. D Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 198, 97 – 110 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Deforming the Lie Algebra of Vector Fields on S 1 Inside the Poisson Algebra on T˙ ∗ S 1 V. Ovsienko1 , C. Roger2 1
C.N.R.S., C.P.T., Luminy-Case 907, F-13288 Marseille Cedex 9, France Institut Girard Desargues, URA CNRS 746, Universit´e Claude Bernard – Lyon I, 43 bd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
2
Received: 13 July 1997 / Accepted: 30 March 1998
Abstract: We study deformations of the standard embedding of the Lie algebra Vect(S 1 ) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle T ∗ S 1 (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of Vect(S 1 ) into the Lie algebra of functions on T˙ ∗ S 1 := T ∗ S 1 \ S 1 which are Laurent polynomials on fibers, and (b) polynomial deformations of the Vect(S 1 ) subalgebra inside the Lie algebra of formal Laurent series on T˙ ∗ S 1 .
1. Introduction 1.1. The standard embedding. The Lie algebra Vect(M ) of vector fields on a manifold M has a natural embedding into the Poisson Lie algebra of functions on T ∗ M . It is defined by the standard action of the Lie algebra of vector fields on the cotangent bundle. Using the local Darboux coordinates (x, ξ) = (x1 , . . . , xn , ξ1 , . . . , ξn ) on T ∗ M , the explicit formula is: π(X) = Xξ,
(1)
Pn Pn where X is a vector field: X = i=1 X i (x)∂/∂xi and Xξ = i=1 X i (x)ξi . The main purpose of this paper is to study deformations of the standard embedding (1). 1.2. Deformations inside C ∞ (T ∗ M ). Consider the Poisson Lie algebra of smooth functions on T ∗ M for an orientable manifold M . In this case, the problem of deformation of the embedding (1) has an elementary solution. The Vect(M ) embedding (1) into C ∞ (T ∗ M ) has the unique (well-known) nontrivial deformation. Indeed, given an arbitrary volume form on M , the expression:
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πλ (X) = Xξ + λdivX, where λ ∈ R, defines an embedding of Vect(M ) into C ∞ (T ∗ M ). The linear map: X 7→ divX is the unique nontrivial 1-cocycle on Vect(M ) with values in C ∞ (M ) ⊂ C ∞ (T ∗ M ) (cf. [2]). 1.3. Two Poisson Lie algebras of formal symbols. Let us consider the following two Lie algebras of Poisson on the cotangent bundle with zero section removed: T˙ ∗ M = T ∗ M \M. (a) The Lie algebra A(M ) of functions on T˙ ∗ M which are Laurent polynomials on fibers; (b) The Lie algebra A(M ) of formal Laurent series on T˙ ∗ M . Lie algebras A(M ) and A(M ) can be interpreted as classical limits of the algebra of formal symbols of pseudo-differential operators on M . We will show that in this case one can expect much more interesting results than those in the case of C ∞ (M ). In both cases, the Poisson bracket is defined by the usual formula: {F, G} =
∂F ∂G ∂F ∂G − . ∂ξ ∂x ∂x ∂ξ
2. Statement of the Problem In this paper we will consider only the one-dimensional case: M = S 1 (analogous results hold for M = R). 2.1. Algebras A(S 1 ) and A(S 1 ) in the one-dimensional case. As vector spaces, Lie algebras A(S 1 ) and A(S 1 ) have the following form: A(S 1 ) := C ∞ (S 1 ) ⊗ C[ξ, ξ −1 ] and A(S 1 ) := C ∞ (S 1 ) ⊗ C[ξ, ξ −1 ]], where C[ξ, ξ −1 ]] is the space of Laurent series in one formal indeterminate. Elements of both algebras: A(S 1 ) and A(S 1 ) can be written in the following form: X ξ k fk (x), F (x, ξ, ξ −1 ) = k∈Z
where the coefficients fk (x) are periodic functions: fk (x + 2π) = fk (x). In the case of algebra A(S 1 ), one supposes that the coefficients fk ≡ 0, if |k| is sufficiently large; for A(S 1 ) the condition is: fk ≡ 0, if k is sufficiently large. 2.2. Formal deformations of Vect(S 1 ) inside A(S 1 ). We will study one-parameter formal deformations of the standard embedding of Vect(S 1 ) into the Lie algebra A(S 1 ). That means we study linear maps π t : Vect(S 1 ) → A(S 1 )[[t]] to the Lie algebra of series in a formal parameter t. Such a map has the following form: π t = π + tπ1 + t2 π2 + · · · ,
(2)
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where πk : Vect(S 1 ) → A(S 1 ) are some linear maps, such that the formal homomorphism condition is satisfied: π t ([X, Y ]) = {π t (X), π t (Y )}. The general Nijenhuis–Richardson theory of formal deformations of homomorphisms of Lie algebras will be discussed in the next section. 2.3. Polynomial deformations of the Vect(S 1 ) inside A(S 1 ). We classify all the polynomial deformations of the standard embedding (1) of Vect(S 1 ) into A(S 1 ). In other words, we consider homomorphisms of the following form: X πk (c)ξ k , (3) π(c) = π + k∈Z
where c = c1 , . . . , cn ∈ R (or C) are parameters of deformations, each linear map πk (c) : Vect(S 1 ) → C ∞ (S 1 ) being polynomial in c, πk (0) = 0 and πk ≡ 0 if k > 0 is sufficiently large. 2.4. Motivations. (a) Lie algebras of functions on a symplectic manifold have nontrivial formal deformations linked with so-called deformation quantization. The problem considered in this paper, is original and has never been discussed in the literature. However, this problem is inspired by deformation quantization. The geometric version of the problem, deformations (up to symplectomorphism) of zero section of the cotangent bundle M ⊂ T ∗ M , has no nontrivial solutions. Existence of nontrivial deformations in the algebraic formulation that we consider here seems to be a manifestation of “quantum anomalies”. Note that interesting examples of deformations of Lie algebra homomorphisms related to deformation quantization can be found in [12]. (b) Lie algebras of vector fields and Lie algebras of functions on a symplectic manifold, have both nice cohomology theories, our idea is to link them together. Lie algebras of vector fields have various nontrivial extensions. The well-known example is the Virasoro algebra defined as a central extension of Vect(S 1 ). A series of nontrivial extensions of Vect(S 1 ) by modules of tensor-densities on S 1 were constructed in [8, 9]. These extensions can be obtained, using a (nonstandard) embedding of Vect(S 1 ) into C ∞ (T˙ ∗ S 1 ), by restriction of the deformation of C ∞ (T˙ ∗ S 1 ) (see [9]). We will show that deformations of the standard embedding relate the Virasoro algebra to extensions of Poisson algebra on T2 defined by A.A. Kirillov (see [4, 11]). (c) The following quantum aspect of the considered problem: deformations of embeddings of Vect(S 1 ) into the algebra of pseudodifferential operators on S 1 , will be treated in a subsequent article. 3. Nijenhuis–Richardson Theory Deformations of homomorphisms of Lie algebras were first considered in [6] (see also [10]). The Nijenhuis–Richardson theory is analogous to the Gerstenhaber theory of formal deformations of associative algebras (and Lie algebras) (see [3]), related cohomological calculations are parallel. Let us outline the main results of this theory.
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3.1. Equivalent deformations. Definition. Two homomorphisms π and π 0 of a Lie algebra g to a Lie algebra h are equivalent (cf. [6]) if there exists an interior automorphism I of h such that π 0 = Iπ. Let us specify this definition for the two problem formulated in Sects. 2.2 and 2.3. (a) Two formal deformations (2) π t and π 0 are equivalent if there exists a linear map It : A(S 1 )[[t]] → A(S 1 )[[t]] of the form: t
It = exp(tadF1 + t2 adF2 + · · · ) = id + tadF1 + t2 (ad2F1 /2 + adF2 ) + · · · , where Fi ∈ A(S 1 ), such that π 0 = It π t . It is natural to consider such an automorphism of A(S 1 )[[t]] as interior. (b) An automorphism I(c) : A(S 1 ) → A(S 1 ) depending on the parameters c = c1 , . . . , cn , which is of the following form: t
I(c) = exp(
n X
ci adFi + ci cj adFij + · · · ),
i=1
where Fi , Fij , · · · ∈ A(S 1 ) is called interior. Two polynomial deformations π(c) and π 0 (c) of the standard embedding Vect(S 1 ) ,→ A(S 1 ) are equivalent if there exists an interior automorphism I(c), such that π 0 (c) = I(c)π(c). 3.2. Infinitesimal deformations. Deformations (2) and (3), modulo second order terms in t and c respectively, are called infinitesimal. Infinitesimal deformations of a Lie algebra homomorphism from g into h are classified by the first cohomology group H 1 (g; h), h being a g-module through π. in (3) are 1-cocycles. Two Namely, the first order terms π1 in (2) and ∂π(c) ∂ci c=0 infinitesimal deformations are equivalent if and only if the corresponding cocycles are cohomologous. Conversely, given a Lie algebra homomorphism π : g → h, an arbitrary 1-cocycle π1 ∈ Z 1 (g; h) defines an infinitesimal deformation of π. 3.3. Obstructions. The integrability conditions are conditions for existence of (formal or polynomial) deformation corresponding to a given infinitesimal deformation. (a) The obstructions for existence of a formal deformation (2) belong to the second cohomology group H 2 (g; h). This follows from the so-called deformation relation (see [6]): dπ t + (1/2)[π t , π t ] = O,
(4)
where [π t , π t ] is a bilinear map from g to h: [π t , π t ](x, y) := {π t (x), π t (y)} − {π t (y), π t (x)}. Note that the deformation relation (4) is nothing but a rewritten formal homomorphism relation (Sect. 1.4).
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Equation (4) is equivalent to a series of nonlinear equations concerning the maps πk : X [πi , πj ]. dπk = −(1/2) i+j=k
The right hand side of each equation is a 2-cocycle and the equations have solutions if and only if the corresponding cohomology classes vanish. (b) Analogous necessary conditions for existence of a polynomial deformation (3) can be easily calculated. 3.4. Remarks. Polynomial deformations. Deformations of algebraic structures (as associative and Lie algebras, their modules and homomorphisms) polynomially depending on parameters are not very well studied. There is no special version of the general theory adopted to this case and the number of known examples is small (see [1]). Theory of polynomial deformation seems to be richer than those of formal ones. The equivalence problem for polynomial deformation has additional interesting aspects related to parameter transformations (cf. Sects. 5.4 and 5.5, formulæ (11)).
4. Polynomial Deformations of the Embedding of Vect(S 1 ) into the Lie Algebra of Formal Laurent Series on T ∗ S 1 Consider the Poisson Lie algebra A(S 1 ). The formula (1) defines an embedding of Vect(S 1 ) into this Lie algebra. The following theorem is the main result of this paper. It gives a classification of polynomial deformations of the subalgebra Vect(S 1 ) ⊂ A(S 1 ). Theorem 1. Every nontrivial polynomial deformation of the standard embedding of Vect(S 1 ) into A(S 1 ) is equivalent to one of a two-parameter family of deformations given by the formula: π
λ,µ
d f (x) dx
λ−µ λ−µ 0 =f x+ ξ + µf x + , ξ ξ
(5)
where λ, µ ∈ R or C are parameters of the deformation; the expression in the right-hand side has to be interpreted as a formal (Laurent) series in ξ. A complete proof of this theorem is given in Sects. 4 and 5. The explicit formula for the deformation π λ,µ is as follows: π λ,µ (f (x)
2 µ2 d λ ) = f (x)ξ + λf 0 (x) + − f 00 (x)ξ −1 + · · · dx 2 2 (λ − µ)k+1 µ(λ − µ)k + f (k+1) (x)ξ −k + · · · + k! (k + 1)!
(50 )
Remark. The formula (5) is a result of complicated calculations which will be omitted. We do not see any a-priori reason for its existence.
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To prove Theorem 1, we apply the Nijenhuis–Richardson theory. The first step is to classify infinitesimal deformations. One has to calculate the first cohomology of Vect(S 1 ) with coefficients in A(S 1 ). Then, one needs the integrability condition under which an infinitesimal deformation corresponds to a polynomial one. 4.1. Algebras A(S 1 ) and A(S 1 ) as Vect(S 1 )-modules. Lie algebra Vect(S 1 ) is a subalgebra of A(S 1 ). Therefore, A(S 1 ) is a Vect(S 1 )-module. Definition. Consider a 1-parameter family of Vect(S 1 )-actions on C ∞ (S 1 ) given by L(λ) f (x)
d dx
(a(x)) = f (x)a0 (x) − λf 0 (x)a(x),
where λ ∈ R. Denote Fλ the Vect(S 1 )-module structure on C ∞ (S 1 ) defined by this action. Remark. Geometrically, L(λ) f d/dx is the operator of Lie derivative on tensor-densities of degree −λ. That means: a = a(x)(dx)−λ . Lemma 4.1. (i) modules:
The Lie algebra A(S 1 ) is decomposed to a direct sum of Vect(S 1 )A(S 1 ) = ⊕m∈Z Fm .
(ii) The Lie algebra A(S 1 ) has the following decomposition as a Vect(S 1 )-module: A(S 1 ) = ⊕m≥0 Fm ⊕ 5m<0 Fm . Proof. Consider the subspace of A(S 1 ) and A(S 1 ) consisting of functions of degree m in ξ: a(x)ξ m . This subspace is a Vect(S 1 )-module isomorphic to Fm . One has: (a)ξ m . {f (x)ξ, a(x)ξ m } = (f a0 − mf 0 a)ξ m = L(m) f d dx
Therefore, algebra of Laurent polynomials is a direct sum of Vect(S 1 )-modules Fm . By definition, an element of algebra A(S 1 ) is a formal series in ξ with a finite number of terms of positive degree. Lemma 4.1 is proven. 4.2. Cohomology groups H 1 ( Vect(S 1 ); A(S 1 )) and H 1 ( Vect(S 1 ); A(S 1 )). It follows that the Vect(S 1 )-cohomology with coefficients in A(S 1 ) is split into a direct sum: H 1 ( Vect(S 1 ); A(S 1 )) = ⊕m∈Z H 1 ( Vect(S 1 ); Fm ). These cohomology groups are well known (see [2]). They are nontrivial if and only if m = 0, −1, −2 and the corresponding group of cohomology are one-dimensional. Therefore, the space of first cohomology H 1 ( Vect(S 1 ); A(S 1 )) is three-dimensional. It is clear that the same result holds for A(S 1 ): H 1 ( Vect(S 1 ); A(S 1 )) = H 1 ( Vect(S 1 ); A(S 1 )) = R3 . 1 1 The cocycles generating nontrivial the following cohomology groups : H Vect(S ); 1 1 1 1 A(S ) and H Vect(S ); A(S ) are
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C0 (f d/dx) = f 0 , C1 (f d/dx) = f 00 (dx), C2 (f d/dx) = f 000 (dx)2 with values in F0 , F−1 , F−2 respectively. 4.3. Infinitesimal deformations. It follows from the Nijenhuis–Richardson theory that the calculated cohomology group classify infinitesimal deformations of the standard embedding of Vect(S 1 ) into the algebras A(S 1 ) and A(S 1 ) (respectively). One obtains the following result: Proposition 4.2. Every infinitesimal deformation of the standard embedding of Vect(S 1 ) into A(S 1 ) and A(S 1 ) is equivalent to: f (x)∂ 7→ f ξ + c0 f 0 + c1 f 00 ξ −1 + c2 f 000 ξ −2 ,
(6)
where c0 , c1 , c2 ∈ R (or C) are parameters. To classify polynomial deformations of the standard embedding of Vect(S 1 ) into A(S 1 ), one needs now the integrability conditions on parameters c0 , c1 , c2 . Remark. We will show (cf. Sect. 6) that in the case of formal deformations of Vect(S 1 ) into A(S 1 ) the corresponding integrability conditions are completely different.
5. Integrability Condition Theorem 5.1. (i) An infinitesimal deformation (6) corresponds to a polynomial deformation of the standard embedding Vect(S 1 ) ,→ A(S 1 ), if and only if it satisfies the following condition: 6c0 3 c2 − 3(c0 c1 )2 − 18c0 c1 c2 + 8c1 3 + 9c2 2 = 0.
(7)
(ii) The polynomial deformation corresponding to a given infinitesimal deformation is unique up to equivalence. The nonlinear relation (7) is the integrability condition for infinitesimal deformations. Given a 1-cocycle C ∈ Z 1 ( Vect(S 1 ); A(S 1 )) which does not satisfy this condition, there is an obstruction for existence of a polynomial deformation. Theorem 5.1 will be proven in the end of this section. Remark. The formula (7) defines a semi-cubic parabola. Indeed, consider the following transformation of the parameters: e c1 = −2c1 + c0 2 , e c2 = 3(c2 − c0 c1 ) + C03 . Then, the relation (7) is equivalent to: −ce1 3 + ce2 2 = 0.
(70 )
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5.1. Homogeneous deformation. Consider an arbitrary polynomial deformation of the standard embedding, corresponding to the infinitesimal deformation (6): X d = f ξ + c0 f 0 + c1 f 00 ξ −1 + c2 f 000 ξ −2 + Pk (c)πk (f )ξ −k , (8) π f (x) dx k∈Z
where c = c1 , c2 , c3 , Pk (c) are polynomials of degree ≥ 2 and πk : Vect(S 1 ) → F−k some differentiable linear maps. Note that since the cocycles C1 , C2 and C3 are defined by differentiable maps, an arbitrary solution of the deformation problem is also defined via differentiable maps. This follows from the Gelfand-Fuks formalism of differentiable (or local) cohomology (see [2]). Definition. Let us introduce a notion of homogeneity for deformations given by differentiable maps. A polynomial deformation (8) is called homogeneous if the sum of the degree in ξ and of the order of differentiation of f in each term of the right-hand side is constant. Since the cocycles C1 , C2 and C3 are homogeneous of order 1, every homogeneous deformation (8) corresponding to a nontrivial infinitesimal deformation, is of homogeneity 1: X Pk (c)f (k+1) ξ −k . (9) π(f (x)∂) = f ξ + c0 f 0 + c1 f 00 ξ −1 + c2 f 000 ξ −2 + k≥3
Lemma 5.2. Every polynomial deformation (8) is equivalent to a homogeneous deformation (9). Proof. It is easy to see that the homomorphism equation: π([f, g]) = {π(f ), π(g)} preserves the homogeneity condition. It means that the first term in (8) (the term of the lowest degree in c) which is not homogeneous of degree 1, must be a 1-cocycle. Such a 1-cocycle is necessarily a coboundary. Indeed, each 1-cocycle is cohomologous to a linear combination of homogeneous of order 1 cocycles: C1 , C2 and C3 (cf. Sect. 4.2). The lemma follows now from the standard Nijenhuis–Richardson technique. One can add (or remove) a coboundary in any term of the polynomial deformation (8) to obtain an equivalent one. Lemma 5.2 is proven. 5.2. Uniqueness of the homogeneous deformation. Proposition 5.3. Given an infinitesimal deformation (6), if there exists a homogeneous polynomial deformation (9) corresponding to the given one, then this homogeneous polynomial deformation is unique. Proof. Substitute the formula (9) to the homomorphism equation. Put P0 = c0 , P1 = c1 , P2 = c2 . Collecting the terms with ξ −k (where k ≥ 3), one readily obtains the following identities for polynomials Pk (c): Pk (c) · (f g 0 − f 0 g)(k) = Pk (c) · (f g (k+1) + (k − 1)f 0 g (k) − f (k+1) g − (k − 1)f (k) g 0 ) X + Pi (c)Pj (c) · (−if (i+1) g (j+2) + jf (i+2) g (j+1) ), i+j=k−1
(10)
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for every f (x), g(x). Each of these identities defines a system of equations of the form: Pk (c) = · · · , where k ≥ 3 and “· · · ” means quadratic expressions of Pi (c) with i < k. Therefore, polynomials Pk (c) with k ≥ 3 are uniquely defined by the constants c0 , c1 and c2 . Proposition 5.3 is proven. 5.3. Integrability condition is necessary. The first three identities (10) give the following system of equations: 2P3 (c) = 2c0 c2 − c1 2 , 5P4 (c) = 3c0 P3 (c) − c1 c2 , ( 9P5 (c) = 4c0 P4 (c) − c1 P3 (c) . 5P5 (c) = 3c1 P3 (c) − 2c2 2 These equations immediately imply the relation (7). This proves that this condition is necessary for integrability of infinitesimal deformations. 5.4. Integrability condition and the universal formula (5). Let us prove that the condition (7) is sufficient for existence of a polynomial deformation. However, it is very difficult to solve the overdetermined system (10) directly. We will use the formula (5). The formula (5) obviously defines a deformation (9) which is polynomial in λ, µ. The first two coefficients c0 and c1 of this deformation are: c0 = λ, c1 =
µ2 λ2 − , 2 2
(11)
and can be taken as independent parameters. Fix the values of c0 and c1 and consider the condition (7) as a quadratic equation with c2 undetermined. The two solutions can be written (using (11)) as expressions from λ and µ: λµ2 µ3 λ3 − + , c2 + = 6 2 3 3 2 3 λµ µ λ c2 − = − − . 6 2 3 The expression c2 + coincides with the third coefficient in the formula (5’). The deformation π λ,µ defined by the formula (5), corresponds to the infinitesimal deformation with c0 , c1 given by (11) and c2 = c2 + . The expression c2 − can be obtained from c2 + using the involution: ∗ : (λ, µ) 7→ (λ, −µ). We have shown that, every infinitesimal deformation satisfying (7) corresponds to a polynomial in c0 , c1 and c2 deformation. Indeed, it follows from existence of the formula (5) that the system (10) has a solution. Theorem 5.1 is proven. Remarks. (a) The parameter µ is a rational parameter on the semi-cubic parabola (7’). c2 = µ3 . Indeed, e c 1 = µ2 , e (b) Suppose that c0 , c1 , c2 ∈ R, then λ and µ in (11) are real if and only if c0 2 ≥ 2c1 .
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5.5. Proof of Theorem 1. We have shown that: (a) Every integrable infinitesimal deformation is equivalent to (6) and obeys the condition (7). (b) Every polynomial deformation is equivalent to a homogeneous one. (c) Given an infinitesimal deformation, there exists a unique homogeneous deformation corresponding to the infinitesimal one. It is given by the universal formula (5). Theorem 1 is proven.
6. Formal Deformations of the Embedding of Vect(S 1 ) into A(S 1 ) We classify formal deformations of the Lie subalgebra Vect(S 1 ) in A(S 1 ). Theorem 2. Every formal deformation of the standard embedding π : Vect(S 1 ) ,→ A(S 1 ) is equivalent to one of the following deformations: (1 − λ)t d (1 − λ)t =f x+ ξ + λtf 0 x + , (500 ) π t f (x) dx ξ ξ where λ ∈ R, the right-hand side is a (Taylor) series in t. In other words, there exists a one-parameter family of formal deformations. The explicit formula for (500 ) is: X ∞ d tk t π f (x) = (1 − (k − 1)λ)(1 − λ)k−1 f (k) (x)ξ −k+1 , dx k!
(5000 )
k=0
note that π (f (x)d/dx) = f ξ + tf 0 + · · · . Classification of infinitesimal deformation was done in Sect. 4.3. In order to prove Theorem 2, let us first classify the infinitesimal deformations which correspond to formal ones. t
6.1. Integrable infinitesimal deformations. Proposition 6.1. An infinitesimal deformation (6) correspond to a formal deformation if and only if c1 = c2 = 0. A 1-cocycle on Vect(S 1 ) corresponding to an integrable infinitesimal deformation is, therefore, proportional to the cocycle C0 from Sect. 4.2. In other words, a nontrivial formal deformation is equivalent to a deformation of the form: d = f ξ + tf 0 + (t2 ) π t f (x) dx (the constant c0 in (6) can be chosen: c0 = 1 up to normalization). Proof of the proposition. Consider an infinitesimal formal deformation: f (x)
d 7→ f ξ + t(c0 f 0 + c1 f 00 ξ −1 + c2 f 000 ξ −2 ) dx
of the standard embedding π : Vect(S 1 ) → A(S 1 ). One must show that it corresponds to a formal deformation π t if and only if c1 = c2 = 0.
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First, note that each term πk , k ≥ 1 of a formal deformation π t (of the expansion (2)) can be chosen in a homogeneous form: πk
d f (x) dx
=
X
αjk f (j+1) ξ j ,
(12)
j
where αjk are some constants. The proof of this fact is analogous to the one of Lemma 5.2. Second, apply the Nijenhuis–Richardson deformation relation (4) (which is equivalent to the homomorphism relation π([f, g]) = {π(f ), π(g)}). In the same way as in Sects. 5.2 and 5.3, collecting the terms with tk , one obtains the following conditions: (a) terms with t2 : 2α32 = 2c0 c2 − c1 2 , 5α42 = −c1 c2 , ( α52 = 0 5α52 = −2c2 2 and therefore, c2 = 0. (b) terms with t3 :
(
,
9α53 = −c1 α32 = (1/2)c1 3 5α53 = 3c1 α32 = −(3/2)c1 3
,
and therefore, c1 = 0. Proposition 6.1 is proven.
Lemma 6.2. The constants αjk in each term πk (given by the formula (12)) of the deformation π t satisfy the condition: αjk = 0 if j ≥ k. Proof. First, one easily shows that αjk = 0 if j > k. In the same way, using the identities (10), one obtains: αkk = 0 for every k ≥ 1. Lemma 6.2 is proven. α53
For example, collecting the terms with t4 one has 9α54 = 4c0 α43 − α12 α32 = 0 and = 4α12 α32 − 2(α22 )2 = −2(α22 )2 , from where α22 = 0.
Lemma 6.3. Every formal deformation π t is equivalent to a formal deformation given by: X d = f ξ + tf 0 + α1 t2 f 00 ξ −1 + α2 t3 f 000 ξ −2 + αk tk+1 f (k+1) ξ −k , π t f (x) dx k≥3
where αi are some constants. This means one can take in (12) αjk = 0 if j ≤ k − 2.
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Proof. Every formal deformation is equivalent to a deformation with α0k = 0 in (12). Indeed, constant α0k is just the coefficient behind tk f 0 . It can be removed (up to equivalence) by choosing a new formal parameter of deformation e t = t + tk α0k . Now, the lemma follows from Proposition 6.1 and homogeneity of the homomorphism condition. Indeed, the terms with j ≤ k − 2 are independent and therefore, the first nonzero term (corresponding to the minimal value of j) must be a 1-cocycle. In the same way as in Lemma 5.2, one shows that such a 1-cocycle is trivial and can be removed up to equivalence. Lemma 6.3 is proven. Now, the expressions Pk = αk tk+1 satisfy the identities (10). Thus, the deformation π t is given by the formula (5) with λ = t. Theorem 2 is proven. 7. Some Properties of the Main Construction Let us study some geometric and algebraic properties of the two-parameter deformation (5). 7.1. Deformation of SL2 (R)-moment map. Consider the standard Lie subalgebra sl2 (R) ⊂ Vect(R) generated by the vector fields: d d d , x , x2 . dx dx dx For every λ and µ, the restriction of the map π λ,µ given by the formula (5) to sl2 (R), defines a Hamiltonian action of sl2 (R) on the half-plane H = {(x, ξ) |ξ > 0} endowed with the standard symplectic structure: ω = dx ∧ dξ. Indeed, the formal series (5’) in this case has only a finite number of nonzero terms and associates to each element of sl2 (R) a well-defined Hamiltonian function on H. Given a Hamiltonian action of a Lie algebra g on a symplectic manifold M , let us recall the notion of so-called moment map from M into the dual space g∗ (see [5]). One associates to a point m ∈ M a linear function m ¯ on g. The definition is as follows: for every x ∈ g, hm, ¯ xi := Fx (m), where Fx is the Hamiltonian function corresponding to x. If the Hamiltonian action of g is homogeneous, then the image of the moment map is a coadjoint orbit of g. In the case of sl2 (R), the coadjoint orbits on sl2 (R)∗ (' R3 ) can be identified with level surfaces of the Killing form. Explicitly, for the coordinates on sl2 (R)∗ , dual to the chosen generators of sl2 (R): y1 y3 − y22 = const. Thus, coadjoint orbits of sl2 (R) are cones (if the constant in the right hand side is zero), one sheet of a two-sheets hyperboloid (if the constant is positive), or a one-sheet hyperboloid (if the constant is negative). Proposition 7.1. The image of the half-plane (ξ > 0) under the SL2 (R)-moment map is one of the following coadjoint orbits of sl2 (R): (i)
λ = 0 or µ = 0, the nilpotent conic orbit;
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(ii) λµ > 0, one sheet of a two-sheets hyperboloid; (iii) λµ < 0, a one-sheet hyperboloid. Proof. The Poisson functions corresponding to the generators of sl2 (R) are: F1 = ξ, F2 = xξ + λ, F3 = x2 ξ + 2λx + λ(λ − µ)ξ −1 , respectively. These functions satisfy the relation: F1 F3 − F22 = λµ.
7.2. The Virasoro algebra and central extension of the Lie algebra C ∞ (T2 ). Consider the Lie algebra C ∞ (T2 ) of smooth functions on the two-torus with the standard Poisson bracket. This Lie algebra has a two-dimensional space of nontrivial central extensions: H 2 (C ∞ (T2 )) = H 2 (T2 ) = R2 . The corresponding 2-cocycles were defined by A.A. Kirillov [4] (see also [11]): Z F dG, c(F, G) = γ
where F = F (x, y), G = G(x, y) are periodic functions: F (x + 2π, y) = F (x, y + 2π) = F (x, y) and γ is a closed path. Recall that the Virasoro algebra is the unique (up to isomorphism) nontrivial central extension of Vect(S 1 ). It is given by the so-called Gelfand-Fuks cocycle: Z 2π w(f (x)d/dx, g(x)d/dx) = f 0 (x)g 00 (x) dx. 0 ∞
Let us show how the central extensions of C (T2 ) are related to the Virasoro algebra via the embedding (5). Let VectPol (S 1 ) be the Lie algebra over C of polynomial vector fields on S 1 . It is generated by: Ln = z n+1 d/dz, where z = eix . The formula (5) with ξ = eiy defines a family of embeddings of VectPol (S 1 ) into C ∞ (T2 )C . It is easy to show that the restriction of two basis Kirillov’s cocycles to the subalgebra VectPol (S 1 ) ,→ C ∞ (T2 )C is proportional to the Gelfand-Fuks cocycle: Z Z 2 F dG) = λ w and ( F dG) = λ2 µ2 w. ( 1 ξ=const x=const VectPol (S ) VectPol (S 1 ) Acknowledgement. We are grateful to F. Ziegler for fruitful discussions. The first author would like to thank Penn. State University for its hospitality.
References 1. Ammar, F.: Syst`emes hamiltoniens compl`etement integrables et d´eformations d’alg`ebres de Lie. Publications Math´ematiques 38, 427–431 (1994) 2. Fuks, D.B.: Cohomology of infinite-dimensional Lie algebras. New York: Consultants Bureau, 1987 3. Gerstenhaber, M.: On deformations of rings and algebras. Annals of Math. 77, 59–103 (1964) 4. Kirillov, A.A.: The orbit method. I. Geometric quantization. Contemp. Math. 145, 1–32 (1994) 5. Kirillov, A.A.: Elements of the theory of representations. Grundlehren der mathematische Wissenschaften, 220, Berlin–Heidelberg–New York: Springer-Verlag, 1976
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6. Nijenhuis, A., Richardson, R.W.: Deformations of homomorphisms of Lie algebras. Bull. AMS 73, 175–179 (1967) 7. Ovsienko, V., Roger, C.: Deformations of Poisson brackets and extensions of Lie algebras of contact vector fields. Russ. Math. Surv. 47:6, 135–191 (1992) 8. Ovsienko, V.Yu., Roger, C.: Extension of the Virasoro Group and the Virasoro Algebra by Modules of Tensor-Densities on S 1 . Funct. Anal. and its Appl. 30 No. 4, 86–88 (1996) 9. Ovsienko, V.Yu., Roger, C.: Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor-densities on S 1 . Indag. Mathem., N.S. 9 (2), 277–288 10. Richardson, R.W.: Deformations of subalgebras of Lie algebras. J. Diff. Geom. 3, 289–308 (1969) 11. Roger. C.: Extensions centrales d’alg`ebres et de groupes de Lie de dimension infinie, alg`ebre de Virasoro et g´en´eralisations. Rep. on Math. Phys. 35, 225–266 (1995) 12. Tabachnikov, S.: Projective connections, group Vey cocycle and deformation quantization. Internat. Math. Res. Notices, 1996, no. 14, 705–722 Communicated by T. Miwa
Commun. Math. Phys. 198, 111 – 156 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
A Non-Gaussian Fixed Point for φ4 in 4 − Dimensions D. Brydges1,? , J. Dimock2,?? , T. R. Hurd3,??? 1 Dept. of Mathematics, University of Virginia, Charlottesville, VA 22903, USA. E-mail:
[email protected] 2 Dept. of Mathematics, SUNY at Buffalo, Buffalo, NY 14214, USA. E-mail:
[email protected] 3 Dept. of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada. E-mail:
[email protected]
Received: 11 September 1997 / Accepted: 30 March 1998
Abstract: We consider the φ4 quantum field theory in four dimensions. The Gaussian part of the measure is modified to simulate 4 − dimensions where is small and positive. We give a renormalization group analysis for the infrared behavior of the resulting model. We find that the Gaussian fixed point is unstable but that there is a hyperbolic non-Gaussian fixed point a distance O() away. In a neighborhood of this fixed point we construct the stable manifold.
1. Introduction We consider a Euclidean φ4 quantum field theory in d dimensions as given by functional integrals of the form R [· · · ]e−V (φ) dµv (φ) R . (1) e−V (φ) dµv (φ) The integral is over some collection of functions φ on Rd and µv is a Gaussian measure with covariance v = (−1)−1 . Up to terms that can be absorbed by adjusting v the potential V has the form Z Z (2) V (φ) = λ : φ4 :v +µ : φ2 :v , where λ is a coupling constant and µ is a mass. It is a basic problem of quantum field theory to establish the existence of such integrals and study their properties. The problem ? ?? ???
Research supported by NSF Grant DMS 9401028. Research supported by NSF Grant PHY9400626. Research supported by the Natural Sciences and Engineering Research Council of Canada.
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is also relevant to the study of critical properties in statistical mechanical systems, in which case φ is interpreted as an order parameter. We are interested in the case when λ and µ are small. Furthermore we focus on the IR (infrared, long distance) behavior of the model and so insert an UV (ultraviolet, short distance) regularization, imposed as a regularization of the covariance v. With this modification the model is well-defined in finite volume and one seeks to take the infinite volume limit. We attack the problem by the renormalization group (RG) method. One successively integrates out short distance modes and then rescales to obtain a new effective potential. The new potential is a lot more complicated. However the leading terms are of the form (2) with new coupling constants (λ0 , µ0 ). One repeats the transformation and studies the flow of the theory. An attracting fixed point is supposed to encapsulate the long distance behavior of all models that flow to it. The massless Gaussian measure (λ = 0, µ = 0) is a fixed point for all d. Suppose d > 4. In a space of general potentials one finds that the linearization around this fixed point has µ growing ( “relevant”) and has λ shrinking ( “irrelevant”). One has a hyperbolic fixed point as in Fig. 1. If one selects µ = µ(λ) to lie on the stable manifold then the fixed point is strictly attracting. These are the critical field theories. If d = 4 then
µ
6
OC C
λ
?
Fig. 1. RG flow for d = 4
the λ term is stable (“marginal”). However this only refers to the linearization, and an analysis of higher order terms shows that λ still flows to zero, so the qualitative picture is the same as for d > 4. A rigorous treatment for d = 4 (in a lattice model) has been given by Gawedzki and Kupiainen [GK85, GK86]. Other treatments can be found in [FMRS87, MP85, MP89]. For d < 4 one finds that λ becomes a second relevant variable and the analysis becomes cloudy. To gain insight into this question one can study the flow equations in 4− dimensions not just for integer , but also for small and positive. Expanding in powers of and keeping the lowest orders one predicts that there is a second fixed point which lies at λ = O() and hence is non-Gaussian. (see Fig. 2.) The stable manifold around this fixed
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point is again supposed to correspond to critical theories. The predictions one gets for these theories (e.g. for critical exponents) turn out to be pretty good not just for small but also for integer. This was one of the early successes of the RG approach (see [WF72, WK74]).
µ
6
6 6
K
* R
?
λ
U
?
Fig. 2. RG flow for d = 4 −
Our goal in this paper is to give a rigorous version of this. We start with the theory in d = 4 but modify the Gaussian measure so that its covariance has scaling behavior appropriate for 4 − dimensions. This is taken as the definition of the theory in 4 − dimensions. Assuming is small we analyze the complete RG flow including all the remainder terms. We prove the existence of the second fixed point and the associated stable manifold corresponding to critical field theories. The analysis in the present paper is carried out at the level of polymer activities, and in infinite volume. In a subsequent paper we hope to make the connection with finite volume. Also in a subsequent paper we hope to analyze the decay of the two-point function. In a preliminary version of this paper we claimed that the two point function has anomalous decay, but we now expect that it is actually canonical, i.e. the same as the free measure. We thank Peter Wittwer for discussions on this point. Canonical decay, if true, would mean that this model does not exhibit all the features one would like to see in low dimensional critical theories. Our analysis uses RG tools created for other models ([BY90, DH91, DH93, BDH94b, BDH94a, BDH95]). We particularly rely on a technical companion paper [BDH96] which contains a number of innovations and has fresh proofs of all the basic theorems. The model is admittedly a bit artificial. However we hope that these techniques can be adapted to study a more realistic model such as N -component φ4 in d = 3 with 1/N as a small parameter. This is a favorite subject for more heuristic treatments. We are also confident that our methods can be adapted rather easily to give a fresh treatment of the d = 4 results. We note that there is previous work on the existence of non-Gaussian fixed points. For hierarchical φ4 models these include a d = 4 − version and a d = 3, N −component
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version [GK83], a computer assisted proof of existence for d=3, N=1 [KW86, KW91], and a proof for the infinitesimal RG [Fel87]. The analysis by Felder [Fel85] of planar graph φ4 in 4 + dimensions is particularly close in spirit to this paper. In the rest of the introduction we define the model, define the RG transformations, and preview our results. 1.1. The model. We shall formulate the model both on R4 and on tori 3N = R4 /(LN Z)4 of side LN . Working for the moment on R4 , the Gaussian measure µv is taken to have mean zero and covariance Z ∞ 2 dα α/2−2 e−|x−y| /4α v(x − y) = Z1 = dp p−2− 0(1 + /2, p2 ) eip·(x−y) , (3) R4
R∞ where the incomplete Gamma function is 0(a, x) = x e−t ta−1 dt. One can see from this that v is the kernel of an ultra–violet regularization of the operator (−1)−1−/2 . This means it scales like |x|−2+ for large values of |x|, and thus can be taken as a definition of a (regularized) inverse Laplacian in 4 − dimensions. The analysis we 2 present actually holds for a wider√ class of covariances, with e−|x−y| /4α replaced by more general functions f (|x − y|/ α). For discrete RG problems such as ours, the covariance is written as a sum v(x − y) =
∞ X
L−(2−)j C(L−j (x − y)),
(4)
j=0
where each term is a rescaling with scale factor L > 1 of the “fluctuation” covariance Z
L2
C(x) =
dα α/2−2 e−x
2
/4α
.
(5)
1
When we work on a torus 3N , we define vN by summing over periods z ∈ (LN Z)4 , and truncating the sum over scales. This yields vN (x − y) =
N X
L−(2−)j Cj (L−j (x − y)),
(6)
j=0
where Cj (x) =
X
C(x + z).
z∈(LN −j Z)4
Note that Cj is almost independent of j when N − j is large. Let µ = µvN be a Gaussian measure with mean zero and covariance vN . Since vN is smooth this measure is realized on a Sobolev space of C 3 functions on 3N . The model is now the study of the measure e−V (φ) dµv (φ) = exp(−V (3N , φ, v; λ, ζ, µ))dµvN (φ), where the potential is
(7)
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Z V (3, φ, v; λ, ζ, µ) = λ 3
Z : φ4 :v +ζ
Z 3
: (∂φ)2 :v +µ
3
: φ 2 :v .
(8)
Here we have allowed for an adjustment in the field strength by including the ζ term. The measure is well defined for N < ∞, but our interest is in the limit(s) as N → ∞. We also need the potential localized in subsets X of 3N . In this case we find it convenient to allow two versions of the field strength and for ζ = (ζ1 , ζ2 ) define V (X, φ, v; λ, ζ, µ) = Z Z 4 : φ :v +ζ1 λ X
Z
Z
2
: φ2 :v . (9)
: φ(−1)φ :v +µ
: (∂φ) :v +ζ2 X
X
X
When X = 3N we can integrate by parts and recover the previous version with field strength ζ1 + ζ2 . −V (φ) . We want to consider integrals of the form R1.2. RG transformations. Let Z(φ) = e Z(φ)dµv (φ) or more generally convolutions Z (µv ∗ Z)(φ) = dµv (ζ) Z(φ + ζ). (10)
Now from (4) or (6) the covariance v = vN has the decomposition vN (x − y) =
N X
Cˆ k (x − y),
(11)
k=0
where Cˆ k (x) = L−(2−)k Ck (L−k x) and Ck = C in infinite volume. The RG idea is that one can then write µv as an iterated convolution µv ∗ Z = µCˆ N ∗ µCˆ N −1 ∗ · · · ∗ µCˆ 1 ∗ µCˆ 0 ∗ Z.
(12)
Zˆj = µCˆ j−1 ∗ · · · ∗ µCˆ 1 ∗ µCˆ 0 ∗ Z,
(13)
Or if we define
then we have a sequence on densities Z˜ 0 = Z, Zˆ1 , Zˆ2 , . . . related by Z˜ j+1 = µCˆ j ∗ Zˆj .
(14)
We want to study ZˆN as N → ∞. We scale the density Zˆj on 3N to Zj on 3N −j by Zj (φ) = Zˆj (φL−j ),
(15)
where for any r > 0 the rescaled field φr (x) is defined to be φr (x) = r1−/2 φ(rx).
(16)
Zj+1 (φ) = (µCj ∗ Zj )(φL−1 )
(17)
Then the densities are related by
This is the basic RG transformation.
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Each fluctuation convolution is well defined and controllable because of the smoothness and exponential decay of the covariances Cj . Each step is almost independent of j for a large torus, and is actually independent of j for infinite volume. The result of all this is that the study of the original integral µv ∗ Z is replaced by the study of an (almost) autonomous discrete flow in the space of densities. The problem is to follow the flow. 1.3. The RG flow. We now sketch our treatment of the RG flow, referring to subsequent sections for precise statements and proofs. To control the flow it is important to keep track of the local structure of the densities Zj . To begin with we have the exponential of a local functional. This form is not preserved but something like it is true. We find that we can always write the densities in the form Zj (φ) = (Exp Aj )(3N −j , φ) ≡
XY
Aj (Xi , φ).
(18)
{Xi } i
Here the Exponential is defined by the sum which is over all partitions {Xi } of the volume 3 into unions of unit blocks called polymers. The functionals Aj (X, φ) are called polymer activities, have tree decay in X, and only depend on the restriction of φ to X. The expansion is called a polymer expansion. Our formalism using these polymer expansions as a tool for controlling the RG was initiated by Brydges and Yau [BY90]. Similar ideas but adapted towards numerical computations were developed by Mack and Pordt [MP89]. We review the results we need in Sect. 2. The RG mapping Zj → Zj+1 of densities is now replaced by a mapping Aj → Aj+1 of polymer activities. At the level of polymer activities the RG transformation also makes sense in infinite volume, and is independent of j. In this paper we only study this infinite volume transformation. The problem of whether this infinite volume transformation correctly represents the infinite volume limit of the model is deferred to another paper. In Sects. 3,4 we will show that the activities Aj can be written in the form Aj (X, φ) = Bj (X, φ, v; λj , ζj , µj , wj ) + Rj (X, φ).
(19)
Here Bj (X, φ) is an explicit functional of certain effective coupling constants (λj , ζj , µj ) and a function wj on 3N −j × 3N −j . The leading contribution comes when X = 1 = a unit block. We have Bj (1, φ, v; λj , ζj , µj , wj ) = exp −V (1, φ, v; λj , ζj , µj ) + · · ·
(20)
with further contributions computed in second order perturbation theory. The term Rj is a remainder term incorporating all higher orders of perturbation theory. The polymer activities Aj are thus parametrized by variables (λj , ζj , µj , Rj , wj ). These variables transform nicely under the RG. Assuming roughly that λj = O(), (ζ1 )j = O(), (ζ2 )j = O(2 ), µj = O(2 ), Rj = O(3 ) and that is sufficiently small we find a flow equation of the form:
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λj+1 = L2 λj − a(wj )λ2j + O(3 ) , (ζi )j+1 = L (ζi )j − bi (wj )λ2j + O(3 ) , µj+1 = L2+ µj − c(wj )λ2j − d(wj )λj (ζ1,j + ζ2,j ) + O(3 ) , Rj+1 = O(3 ), wj+1 (x) = L2− wj (Lx) + C(Lx) ,
(21)
with b2 = 0. The initial conditions should have R0 = 0, w0 = 0. The function wj can be explicitly computed and converges to a limit w∞ as j → ∞. We have nevertheless included it as a dynamical variable to make the flow autonomous in infinite volume. The powers of L in the first three equations are bigger by a factor of L than we would expect if we were really in 4 − dimensions. This is due to the fact that our spatial integrals are in 4 rather than 4 − dimensions. With generic initial data we cannot iterate the RG mapping indefinitely for we have three expanding variables and would soon leave the region of definition of the mapping. Some further analysis is needed. The origin is an unstable fixed point. To find the second fixed point replace a(wj ) by its limiting value a(w∞ ) and ignore the higher order terms in the λ equation. One finds an approximate fixed point at L2 − 1 = O(). λ¯ = 2 L a(w∞ )
(22)
Our further analysis is carried out in a neighborhood of this approximate fixed point. The deviation λ˜ = λ − λ¯ replaces λ and satisfies λ˜ j+1 = (2 − L )λ˜ j + O(3 ).
(23)
Note that (2 − L ) < 1 so that λ˜ j is a contracting variable. ˜ ζ, µ, R, w) we have a mapping on a neighborhood of the In the new variables (λ, origin in a Banach space for which the linearization has two expanding directions (ζ, µ) ˜ R, w), and for which the nonlinear part is very small. By a and the rest contracting (λ, version of the stable manifold theorem there is a hyperbolic fixed point in this neighborhood, and associated with this fixed point is a stable manifold of dimension 2 and an unstable manifold of codimension 2. The stable manifold is given as the graph of a func˜ R, w), µ = µ(λ, ˜ R, w), or specializing to the initial values R = 0, w = 0 tion ζ = ζ(λ, ˜ ˜ Densities corresponding to points on this manifold it has the form ζ = ζ(λ), µ = µ(λ). flow to the density of the fixed point. These are the critical field theories. The details of this argument are given in Sect. 5. 2. Review In order to make the present paper reasonably self–contained, we include here a concise review of the definitions and results for a single RG transformation, adapted to the
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problem at hand. The reader wishing all the details in a more general setting is directed to the original paper [BDH96]. 2.1. Polymer expansions. At the j th RG step, we have a base space 3 = 3N −j which is a 4–dimensional torus R4 /LN −j Z4 of side LN −j . A polymer X is a possibly empty union of blocks of 3 where a block , 1, is an open unit hypercube in 3 centered on a point of the lattice Z4 /LN −j Z4 . Polymer activities are complex valued functions K(X, φ) defined on polymers X and fields φ (in fact, K(X) will depend only on the restriction φ|X ). We include the empty set as a polymer and assume, unless cautioned otherwise, that K(∅) = 0. As we have mentioned, we shall regard the densities Zj as a polymer exponential of a polymer activity Aj (X, φ), Zj = Exp(Aj ) = I + Aj +
1 Aj ◦ A j + · · · . 2!
The polymer exponential is that associated to the following commutative product on the space of polymer functions A(X), B(X), . . . [Rue69, GMLMS71]: X A(Y )B(X \ Y ). (A ◦ B)(X) = Y ⊂X
Here I(∅) = 1 and otherwise I(X) = 0. A local density can be written as the polymer expansion Exp(e−V )(3), where
(X) =
1 if X is a unit block 0 otherwise,
(24)
and V (X, φ) is the local potential (2). In our problem, the densities at every scale are nearly local, which means that they can all be written in the form A = e−V + K, where the activities K(X, φ) are small in a sense we now describe. 2.2. Polymer norms. Our polymer activities K(X, φ) will have certain decay properties depending on the “size” of X, certain growth and decay behaviour depending on the value of φ and its derivatives, and finally analyticity in the variable φ. This will be summarized by finiteness of a certain type of norm for K. ¯ the Banach First we suppose that for any X, K(X, φ) is defined for φ ∈ C 3 (X), space of thrice differentiable functions on X¯ with norm kf k = max sup |∂ β f (x)|. |β|≤3 x
The closure X¯ here just means we assume that the partial derivatives all have continuous boundary values. All φ–derivatives of K are assumed to exist: these are symmetric multi– linear functionals defined by
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X ∂ ∂ ··· K(X, φ + si fi )|s=0 = Kn (X, φ; f1 , · · · , fn ). ∂s1 ∂sn We further impose that K(X, φ) should be Frechet-analytic in φ in a complex strip around the real space C 3 (3). The size of the derivative Kn (X, φ) is measured by the norm ¯ kfj kC 3 (X) ≤ 1} (25) kKn (X, φ)k = sup{|Kn (X, φ; f1 , . . . , fn )| : fj ∈ C 3 (X), for n > 0 and kK0 (X, φ)k = |K0 (X, φ)|. Actually, we need a localized version of this norm, and therefore we consider derivatives restricted to neighborhoods ˜ = {x : dist(x, 1) < 1/4} 1
(26)
of blocks 1. Let 1×n = (11 , . . . , 1n ) be an n-tuple of blocks and define kKn (X, φ)k1˜ ×n ¯ kfj kC 3 (X) ≤ 1, suppfj ⊂ 1 ¯ (27) ˜ j ∩ X}. = sup{|Kn (f1 , . . . , fn )| : fj ∈ C 3 (X), A connection between the natural norm (25) and the localized version is given if we ˜ select a smooth partition of unity χ1 indexed by unit blocks 1 such that suppχ1 ⊂ 1. We assume that each χ1 is a translate of a fixed function χ. We define kχk as the best constant such that kχ1 f k ≤ kχk kf k.
(28)
Then we have kKn (X, φ)k ≤ kχkn
X 1×n
kKn (X, φ)k1˜ ×n .
(29)
The growth of kKn (X, φ)k1˜ ×n in φ will be controlled by a large field regulator which is some variation of the standard choice G = G(κ) = G(κ, X, φ), where G(κ, X, φ) = exp(κkφk2X,2,σ ).
(30)
Here kφk2X,a,b =
X
k∂ β φk2X
(31)
a≤|β|≤b
and kφkX is the L2 (X) norm. We take σ = 6 which is large enough so that this norm can be used in Sobolev inequalities for any low order derivative ∂ α φ. For any such G, we define a norm on derivatives Kn (X, φ) by X kKn (X)kG = sup kKn (X, φ)k1˜ ×n G−1 (X, φ). (32) 3 1×n φ∈C
To control decay in the “size” of X we introduce large set regulators 0p (X) which are defined in dimension d = 4 by
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0p (X) = 2p|X| 0(X), 0(X) = L(d+2)|X| 2(X), Y θ(|b|). 2(X) = inf τ
(33)
b∈τ
The volume |X| of X is the number of blocks in X. The infimum is over trees τ composed of bonds b connecting the centers of the blocks in X, and the length |b| of a bond b = xy is defined to be the `∞ -metric sup1≤j≤d |xj − yj |. θ is a polynomially increasing function chosen so that θ(s) = 1 for s = 0, 1 and θ({s/L}) ≤ L−d−2 θ(s),
s = 2, 3, . . . ,
(34)
where {x} denotes the smallest integer greater than or equal to x. For any polymer function K(X) we define X |K(X)|0(X). (35) kKk0 = sup 1 X⊃1
Now we have the ingredients with which to assemble our norms. For analyticity in a strip of width h > 0, our preferred choice is X hn kKn (X)kG , kK(X)kG,h = n! n kKkG,h,0 = k kK(·)kG,h k0 .
(36)
However, sometimes we change the order and take kKn kG,0 = k kKn (·)kG k0 , X hn kKkG,0,h = kKn kG,0 . n! n
(37)
There is also a limiting case of the norms kKkG,0,h in which G−1 is concentrated at φ = 0. These are called kernel norms and are defined by X kKn (X, 0)k1˜ ×n , |Kn (X, 0)| = 1×n
|K|h,0 = k
X hn n
|K|0,h =
n!
X hn n
n!
|Kn (·, 0)| k0 ,
k |Kn (·, 0)| k0 .
(38)
2.3. Bounds on e−V . We now state a bound on ke−V (X) kG,h for the φ4 potential in 4–dimensions: V (X) = V (X, φ, v; λ, ζ, µ)
(39)
given in Eq. (9) with the regulator G = G(κ)G−1 0 (κ0 ), where G0 (κ0 , X, φ) = exp(κ0 kφk2X ) and G(κ) is defined in (30). The bound is proved under the following hypotheses:
(40)
Non-Gaussian Fixed Point
1. 2. 3. 4.
121
Re(λ)h4 is positive and bounded by a sufficiently small constant. Im(λ)/Re(λ) is bounded by a constant. |µ|h2 ≤ Re(λ)h4 , |ζ|h2 ≤ Re(λ)h4 , κ0 h2 ≤ Re(λ)h4 . h−2 v(0), h−2 ∂ 2 v(0), h−2 κ−1 , h−2 κ−1 0 are all bounded by constants.
In the above, constants are independent of L and |ζ| = max (|ζ1 |, |ζ2 |). Theorem 1. Under the above hypotheses for any polymer X: ke−V (X) kG(κ)G−1 (κ0 ),h ≤ 2|X| ; |e−V (X) |h ≤ 2|X| .
(41)
0
If X is a subset of a unit block 1, then ke−V (X) kG(κ,1)G−1 (κ0 ,1),h ≤ 2; |e−V (X) |h ≤ 2.
(42)
0
Remark. We will need a generalization of this in which the fields : φ4 (x) :, : φ2 (x) :, etc. in V are multiplied by functions of x. The results still hold provided the functions are pointwise bounded above and below by suitable constants. We define P (X, φ) to be a polynomial of degree r if derivatives of higher order than r vanish. The following result will also be needed when we come to control the perturbative part of the analysis 1/2 ), h = Lemma 1. Consider the regulator G(κ)G−1 0 (κ0 ) above with κ = O(λ −1/4 ). For any polynomial P of degree r there is a constant O(1) (depending on r) O(λ such that
kP e−V kG(κ)G−1 (κ0 ),h,0 ≤ O(1)|P |h,01 .
(43)
0
2.4. Results on a single RG map. The RG map is the composition of fluctuation, extraction and scaling. We discuss each of these steps in turn. The j th fluctuation step is the map induced on polymer activities by Gaussian convolution with respect to the measure with covariance C = Cj (x − y) given in Eq. (6). These covariances are smooth functions, invariant under the symmetries of the torus, which decay rapidly in the separation |x−y|. Control of the fluctuation map in general depends on smoothness of the fluctuation covariance C and finiteness of the following norm: X (44) C(11 , 12 )θ (d(11 , 12 )) , kCkθ = 3d sup 11
12
C(11 , 12 ) = kχ11 Cχ12 kC 6 .
(45)
Here θ(s) is the function given by (34), and χ1 (x) is the “bump” function chosen earlier. For Theorem 3 we also require the condition on C: sup
|∂ β C(0)| ≤ O(1).
(46)
0≤|β|≤12
The main fluctuation theorem refers to norms which involve a one–parameter family of regulators t (47) Gt (X, φ) = 2|X| G(2κ, X, φ) G(κ, X, φ)1−t , which has been constructed to satisfy µ(t−s)C ∗ Gs (X, φ) ≤ Gt (X, φ) when κ is small enough depending on a norm of C.
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Theorem 2. For any polymer activity A and any t ∈ [0, 1], there is a unique polymer activity A(t) so that µtC ∗ (ExpA) = Exp A(t) .
(48)
The map F¯t (A) ≡ A(t) is analytic. If h0 < h and kAkG(0),0,h ≤ D ≡
(h − h0 )2 16kCkθ
(49)
then kF¯t (A)kG(t),0,h0 ≤ kAkG(0),0,h .
(50)
It turns out that the family F¯ t (A) = A(t) solves the following flow equation: 1 ∂ E A(t) ≡ A(t) − 1C A(t) − BC A(t), A(t) = 0, ∂t 2
(51)
where the functional Laplacian 1C is the operator Z 1 dµC (ζ) A2 (φ; ζ, ζ) 1C A(φ) = 2 and BC is a bilinear operator on activities: Z dµC (ζ) A1 (φ; ζ) ◦ B1 (φ; ζ). BC (A, B)(φ) = We will improve the estimates in the above theorem by constructing approximate solutions to this flow equation using a perturbative analysis. Given any family t → B(t), we measure how well it matches an exact evolution by the error E B(t) . The following theorem tracks the growth of the remainder R(t) defined by A(t) = B(t) + R(t).
(52)
Theorem 3. 1. Let B(t) be a continuously differentiable function of t ∈ [0, 1]. Suppose h > h0 and kR(0)kG(0),0,h , sup0≤t≤1 kB(t)kG(t),0,h ≤ 41 D, where D is defined in (49). Then kR(t)kG(t),0,h0 ≤ 2(kR(0)kG(0),0,h + t sup kE B(s) kG(s),0,h ). (53) s≤t
2. Suppose further that kR(0)kG(0),0,h , sup0≤t≤1 kB(t)kG(t),0,h ≤ h0 /(2kCkθ ) and h0 ≥ 2. Then for any M ≥ 0, |R(t)|0−1 ,1/2 ≤ O(1)(|R(0)|0−1 ,1 + (h0 )−M kR(0)kG(0),0,h ) + O(1) sup(|E B(s) |0−1 ,1 + (h0 )−M kE B(s) kG(s),0,h ), (54) s≤t
where O(1) depends on M .
Non-Gaussian Fixed Point
123
Now consider the extraction step. Suppose that the polymer activity has the form A = e−V + K. The extraction step consists in removing from K(X) φ–independent terms F0 (X) and φ–dependent terms F1 (X, φ) which are both assumed to satisfy a certain localization property: F (X, φ) has the decomposition F (X, φ) =
X
F (X, 1, φ),
(55)
1⊂X
where 1 is summed over open blocks, and F (X, 1, φ) has the φ dependence localized ¯ The extraction step replaces the potential in 1, i.e. F (X, 1, φ) is a functional on C 3 (1). V by a potential V (F ) defined on a unit block 1 by X (V F ) (1) = V (1) − F (Y, 1).
(56)
Y ⊃1
The following theorem gives the essential properties of the extraction step. The bounds are obtained when we have the following stability of V relative to the perturbation F1 : there are positive numbers f (X) independent of φ and a regulator G such that for all 1 ) ( X z(X)F1 (X, 1) kG,h ≤ 2 (57) k exp −V (1) − X⊃1
for all complex z(X) with |z(X)|f (X) ≤ 2. The variation of Theorem 1 allowing variable coefficients is used to verify this condition. Theorem 4. If K is a polymer activity and F0 (X), F1 (X, φ) satisfy the localization hypothesis (55), then there exists a new polymer activity E(K, F0 , F1 ) so that: Exp(e−V + K)(3) = e
P X
F0 (X)
Exp e−V (F1 ) + E(K, F0 , F1 ) (3),
(58)
where the linearization E1 of E in K, F0 , F1 is E1 (K, F0 , F1 ) = K − (F0 + F1 )e−V .
(59)
If in addition PF1 satisfies stability hypothesis (57), kf k04 , and kKkG,02 ,h are sufficiently small, and Y ⊃1 |F0 (X, 1)| ≤ log 2 then E is jointly analytic in K, F0 , F1 and there is O(1) such that kE(K, F0 , F1 )kG,h,0 ≤ O(1)(kKkG,h,02 + kf k04 ); |E(K, F0 , F1 )|h,0 ≤ O(1)(|K|h,02 + kf k04 ). Finally E≥2 = E − E1 satisfies kE≥2 (K, F0 , F1 )kG,h,0 ≤ O(1)kKkG,h,02 kf k04 , |E≥2 (K, F0 , F1 )|h,0 ≤ O(1)|K|h,02 kf k04 .
(60) (61)
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Note the distinguished role of 3 in Eq. (58): the equation does not generally hold for other polymers X. Now consider the scaling step. The scaled field is φL−1 (x) = L−(1−/2) φ(x/L)
(62)
KL−1 (X, φ) = K(LX, φL−1 ).
(63)
and functionals scale by
The scaling map on polymer activities S(K) = S(K, V ) is defined by the equation (64) Exp(e−V + K)(LX, φL−1 ) = Exp (e−V )L−1 + S(K) (X, φ) from which one derives the explicit formula X Y exp −V (LZ \ X, φL−1 ) K(Xj , φL−1 ) S(K)(Z, φ) = {Xj }→LZ
=
X
j
exp −VL−1 (Z \ L
−1
Y
X, φ)
{Xj }→LZ
KL−1 (L−1 Xj , φ). (65)
j
Here the sum is over disjoint 1–polymers {Xj } with union X such that the L-block closures X¯ jL are “overlap connected” (see [BDH96]) and have union LZ. From Theorem 1, we will verify that in our model, V satisfies the following stability bound: for all L−1 -scale polymers X ⊂ some block 1, k(e−V )L−1 (X)kG,h ≤ 2
(66)
for G = G(κ) with κ small enough. This bound is needed for the main result on scaling. Define dim(φ) = 1 − /2 and hL = L− dim φ h, a = 24 kχk,
(67)
where kχk is the norm (28) of the partition of unity bump function defined earlier. Theorem 5. Let V satisfy the stability assumption (66) and suppose kKkGL ,ahL ,0−5 is sufficiently small. Then kS(K)kG,h,0 ≤ O(1)L4 kKkGL ,ahL ,0−5 , |S(K)|h,0 ≤ O(1)L4 |K|ahL ,0−5 .
(68)
We also need a sharper estimate on the linearization S1 of S, X S1 (K) (Z, φ)) = (e−V )(LZ \ X, φL−1 )K(X, φL−1 ) ¯ L =LZ X:X
=
X
(e−V )L−1 (Z \ L−1 X, φ)KL−1 (L−1 X, φ).
(69)
¯ L =LZ X:X
The new estimate needs the stronger bound: for L−1 scale polymers X contained in any block 1:
Non-Gaussian Fixed Point
125
k(e−V )L−1 (X)kg,h ≤ 2,
(70)
g(X, φ) = G−1 0 (κ0 , X, φ)G(κ/2, X, φ) = exp −κ0 kφk2X + κ/2k∂φk2X,2,σ .
(71)
where
The scaling dimension of a polymer activity K is defined by Definition 1. dim(Kn ) = rn + n(1 − /2); dim(K) = inf dim(Kn ), n
(72)
where the infimum is taken over n such that Kn (X, 0) 6= 0. Here rn is defined to be the largest integer satisfying rn ≤ 3 and Kn (X, φ = 0; p×n ) = 0 whenever p×n is an n–tuple of polynomials of total degree less than rn . Roughly rn gives the number of derivatives in Kn . Theorem 6. Let V satisfy (70). 1. If K(X) is supported on large sets, then kS1 (K)kG,h,0 ≤ O(1)L−1 kKkGL ,ahL ,0−5 , |S1 (K)|h,0 ≤ O(1)L−1 |K|ahL ,0−5 .
(73)
2. If K(X) is supported on small sets, and in addition κ0 h2 ≥ O(1) and κh2 ≥ O(1), then kS1 (K)kG,h,0 ≤ O(1)L4−dim(K) kKkGL ,h/2,0−5 , |S1 (K)|h,0 ≤ O(1)L4−dim(K) |K|h/2,0−5 .
(74)
2.5. Infinite volume. We have explained the RG transformation on finite tori with side LN . We now observe that every polymer formula makes sense when evaluated on finite polymers in the infinite volume R4 . Indeed the only explicit volume dependence is in the fluctuation step where we would replace Cj by C. (There is one exception to this claim which is the extraction formula (58). But this formula only motivates the definition of the extraction map, and is not needed in the definition in the sense that the formula for E in [BDH96] is well defined for infinite volume.) Furthermore, every bound we have stated is uniform for large volumes. Correspondingly there is an infinite volume version of each of these results. In the next two sections we treat finite volume and infinite volume in parallel. In the final section we treat only the infinite volume flow. We leave it to another paper to discuss the relationship between the infinite volume flow and the finite volume flow.
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3. The Specific RG Map A single RG transformation has been defined, and the important estimates recorded. However we have not yet capitalized on the freedom in the extraction step. Our strategy now is to specify the extractions F so that in the rescaling step we have activities K with dim K > 4. Choosing the correct extractions requires second order perturbation theory and control over the higher order remainders. We make the ansatz that after any number of iterations the polymer activity A can be expressed in the following form:
where
A = A(~λ, R, w) = B(~λ, v, w) + R,
(75)
~ B(~λ, v, w) = + Q(~λ, v, w) e−V (λ,v) .
(76)
Here ~λ are effective coupling constants ~λ = (λ, ζ, µ) = (λ, ζ1 , ζ2 , µ),
(77)
w = w(x − y) is a kernel and the remainder R = R(X, φ) is a polymer activity. The potential V = V (~λ, v) is defined in (9). The term Q = Q(~λ, v, w) is a polynomial in the field φ, and is the regular part of the contribution of second order perturbation theory. The exact definition of Q is given below. Once Q is defined our goal is to exhibit a mapping (~λ, R, w) → (~λ0 , R0 , w0 ) such that the transform of the activity A = A(~λ, R, w) is A0 = A(~λ0 , R0 , w0 ), and to demonstrate that the remainder stays small. 3.1. Definition and evolution of Q. The discussion of second order perturbation theory for the φ43 model given in Sects. 3 and 7 of [BDH95] can be largely repeated for our present model. Apart from the change in dimension, and the introduction of the parameter, the other extra ingredient here is the inclusion of the wave function coupling constants ζ1 , ζ2 and corresponding terms in Q involving ∂φ. The present version also has the improvement that leading order extractions are not restricted to small sets. This means that our treatment of the RG flow agrees with the standard perturbative treatment to second order in λ. Let vt = v − tC. Recall that the fluctuation step is an evolution of polymer activities A(t) such that µvt ∗ Exp (A(t)) is constant in t. Then A(t) solves 1 ∂A − 1C A − BC (A, A) = 0. (78) ∂t 2 We consider approximations B(t) to A(t). We will say that B(t) is a solution at order O(~λ) if every term in E(B) is order O(~λ2 ). Lemma 2. The family B(t) = exp −V (~λ, vt ) solves (78) at order O(~λ). E(A) ≡
Proof. The O(~λ) terms in E(B) are (∂/∂t − 1C )V . This vanishes because of the Wick ordering. Indeed for any polynomial P , Wick ordering can be defined by : P (φ) : v = e−1v P (φ) and this implies the ”martingale property” ∂ − 1C : P (φ) : vt = 0. (79) ∂t
Non-Gaussian Fixed Point
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We define Q so that ( + Q) exp (−V ) is an O(~λ2 ) solution. We use the notation Z =
←
→
∂ ∂ w(x, y) dx dy. ∂φ(x) ∂φ(y)
The right functional derivative acts to the right and the left to the left. Multiple lines indicate a product of such factors. such as First we define 1 1 1 (−V ) (−V ) + Q(~λ, v, w) = (−V ) 2 2 2! 1 1 (−V ) (−V ), + 2 4!
(−V ) +
1 1 (−V ) 2 3!
(−V ) (80)
where V = V (~λ, v). This functional depends on a set X through the following localization of the fields in X: Q(~λ, v, w) is a sum of monomials of the form Z Q(m,n) (v, u; X, φ) := : φ(x)m : v u(x − y) : φ(y)n : v dx dy, (81) ˜ X
where X˜ =
1×1 1 × 10 ∪ 10 × 1 ∅
X=1 X = 1 ∪ 10 . otherwise
(82)
Also a primed index indicates the field φ is replaced by ∂φ (and still Wick-ordered if appropriate) and a doubly primed index indicates that φ is replaced by (−1φ). By convention : φ0 : = 1. Lemma 3. Let wt = w + tC, V (t) = V (λ, vt ), and Q(t) = Q(~λ, vt , wt ). Then B(t) = ( + Q(t)) e−V (t) solves (78) at order O(~λ2 ). Proof. The condition for vanishing O(~λ2 ) terms in E(B) is (
∂ − 1C )Q(t) = J(t), ∂t
where J(t) is defined by E(e−V (t) ) = J(t)e−V (t) . Explicitly we have 1R (t) ∂V (t) C(x − y) ∂V |X| ≤ 2 ∂φ(x) ∂φ(y) dxdy . J(t; X, φ) = 2 X˜ 0 otherwise
(83)
(84)
But Q(t) is defined so that (83) holds. This is verified using 1 1 ∂ 1 −1 (−V ) (−V ) = (−V ).....(−V ) − (−V ) ..... (−V ), ∂t 2 2 2 1 1 1 1 1 ∂ −1 (−V ) (−V ) ..... (−V ), (−V ) = (−V ) ..... (−V ) − ∂t 2 2! 2 2 2!
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D. Brydges, J. Dimock, T. R. Hurd
etc., where the dotted line is the t derivative of the solid line Z ..... =
←
→
∂ ∂ C(x, y) dx dy. ∂φ(x) ∂φ(y)
All the terms cancel except 21 (−V ).....(−V ) = J. One can also write out Q explicitly: Q(~λ, v, w) = λ2 8Q(3,3) (v, w) + 36Q(2,2) (v, w2 ) + 48Q(1,1) (w3 ) + 12Q(0,0) (w4 ) h i 0 + λζ1 8Q(3,1 ) (v, ∂w) + 12Q(2,0) (v, (∂w)2 ) h i 00 + λζ2 4Q(3,1) (v, −1w) + 4Q(3,1 ) (v, w) + 12Q(2,0) (v, w(−1w)) + λµ 8Q(3,1) (v, w) + 12Q(2,0) v, w2 h i 0 0 + ζ12 2Q(1 ,1 ) (∂∂w) + Q(0,0) (∂∂w)2 i h 0 0 00 + 2ζ1 ζ2 Q(1 ,1) (−1∂w) + Q(1 ,1 ) (∂w) + Q(0,0) (∂w(−1∂w)) 00 1 h 00 00 + ζ22 Q(1 ,1 ) (w) + 2Q(1 ,1) (−1w) 2 i + Q(1,1) (12 w) + Q(0,0) w12 w + (1w)2 h i 0 + ζ1 µ 4Q(1 ,1) (∂w) + 2Q(0,0) ((∂w)2 ) h i 00 + ζ2 µ 2Q(1,1) (−1w) + 2Q(1 ,1) (w) + 2Q(0,0) (w(−1w)) (85) + µ2 2Q(1,1) (w) + Q(0,0) (w2 ) . Now Q(~λ, v, w) contains terms Q(m,n) (v, u; X, φ) whose kernels u are one of 1w, ∂∂w, 1∂w, 11w, w2 , ∂w∂w, w1w, w3 , w4 . We refer to these kernels as divergent. In fact for any finite number of RG steps they are all smooth functions. However after many renormalization group steps they approximate non-integrable singularities because w(x−y) tends to a fixed point w∗ (x−y) which together with derivatives is exponentially decaying for x − y → ∞, but which has a singularity O(|x − y|−2+ ) as x − y → 0. Each derivative with respect to x, y increases the power of the singularity by one (Lemma 7). We decompose each divergent Q(m,n) (v, u; X, φ) into a local monomial Q(m,n) sing (v, u; X, φ) and a regular part Q(m,n) (v, u; X, φ). For example, set reg Z 1 2 (: φ(x)2 :v − : φ(y)2 :v )2 w(x − y)2 dx dy, Q(2,2) reg (v, w ) = − 2 X˜ Z 2 (v, w ) = (: φ(x)2 :v )2 w(x − y)2 dx dy. (86) Q(2,2) sing ˜ X
Then (2,2) 2 2 Q(2,2) (v, w2 ) = Q(2,2) reg (v, w ) + Qsing (v, w ).
(87)
This is part of the procedure in standard renormalization theory in which nonintegrable kernels are replaced by distributions that coincide with the kernels away
Non-Gaussian Fixed Point
129
from the singularity. The following formulas cover the remaining divergent kernels in (80) and show that there is no conflict with the localization to X: Z (3,1) : φ(x)3 : v (−1w)(x − y) (φ(y) − φ(x)) dx dy, Qreg (v, −1w) = ˜ X Z (3,1) : φ(x)3 : v φ(x)(−1w)(x − y) dx dy, (88) Qsing (v, −1w) = ˜ X
Z 1 =− [(φ(x) − φ(y))2 − [(x − y) · ∂φ(x)]2 ]u(x − y) dx dy, 2 X˜ Z 1 (u) = [φ(x)2 + φ(y)2 − [(x − y) · ∂φ(x)]2 ]u(x − y) dx dy, Q(1,1) sing 2 X˜ Q(1,1) reg (u)
(89)
where u = w3 or 12 w, Z 1 (φ(x) − φ(y))2 1w(x − y) dx dy, Q(1,1) (1w) = − reg 2 X˜ Z (1w) = φ(x)2 1w(x − y) dx dy, Q(1,1) sing
(90)
˜ X
0
0
,1 ) Q(1 reg (∂∂w) = −
0
0
1 2
Z ˜ X
(∂a ∂b w)(x − y) (∂b φ(x) − ∂b φ(y)) dx dy,
Z
,1 ) Q(1 sing (∂∂w) =
(∂a φ(x)) (∂b φ(x)) (∂a ∂b w)(x − y) dx dy,
˜ X
00
(1 ,1) (v, −1w) = − Qreg
(1 ,1) (v, −1w) = Qsing
0
1 2
(91)
Z ˜ X
((−1φ)(x) − (−1φ)(y))
(φ(x) − φ(y)) (−1w)(x − y) dx dy,
Z
00
(10 ,1) (−1∂w) Qreg
(∂a φ(x) − ∂a φ(y))
˜ X
(−1φ)(x)φ(x)(−1w)(x − y) dx dy,
(92)
Z =
(1 ,1) (−1∂w) = Qsing
˜
∂φ(x)[φ(y) − φ(x) − (y − x)∂φ(x)](−1∂w)(x − y) dx dy,
ZX ˜ X
∂φ(x)[φ(x) + (y − x)∂φ(x)](−1∂w)(x − y) dx dy,
(93)
(2,0) (2,0) , Q(2,0) reg = 0; Qsing = Q
(94)
(0,0) (0,0) Q(0,0) . reg = 0; Qsing = Q
(95)
Correspondingly we have a split Q(~λ, v, w) = Qreg (~λ, v, w) + Qsing (~λ, v, w), where all the terms with regular kernels go into Qreg (~λ, v, w).
(96)
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D. Brydges, J. Dimock, T. R. Hurd
Now we define the Q that appears in our ansatz (76) to be the regular part: Q(~λ, v, w) = Qreg (~λ, v, w).
(97)
For the fluctuation step we still want an approximate solution at order O(~λ2 ), and so we still want a solution Q(t) of the flow equation (83). But now the initial condition is Qreg (~λ, v, w) rather than Q(~λ, v, w). Thus we require (
∂ − 1C )Q(t) = J(t), ∂t Q(0) = Qreg (~λ, v, w).
(98)
To find Q(t) we compare it to the old solution Q(t) = Q(~λ, vt , wt ). The difference Q(t) − Q(t) then must satisfy (
∂ − 1C )(Q(t) − Q(t)) = 0, ∂t Q(0) − Q(0) = Qsing (~λ, v, w).
(99)
Because of (79) the solution is to write Qsing (~λ, v, w) as a v-Wick ordered polynomial and then replace v by vt . We calculate Qsing (~λ, v, w) = Q˜ sing (λ, v, w, v(0)) ,
(100)
where for any v(0), Q˜ sing (λ, v, w, v(0))
0 = Q(4,0) v, 36λ2 w2 + Q(2 ,0) v, 48λ2 w3 T
+ Q(2,0) v, 144λ2 v(0)w2 + 48λ2 w3 + 12ζ1 λ(∂w)2 + 12ζ2 λw(−1w) + 12λµw2 1 2 2 (4,0) (20 ,0) 2 (v, 4λζ2 (−1w)) + Q +Q v, 2ζ1 ∂∂w + 2ζ1 ζ2 `(−1∂w) + ζ2 1 wT 2 (100 +1,0) 2 (1+10 ,0) (2ζ1 ζ2 (−1∂w)) v, ζ2 (−1)w + Q +Q 1 2 2 (2,0) (101) v, 12ζ2 λv(0)(−1w) + 2µζ2 (−1)w + ζ2 1 w + Q(0,0) (. . . ). +Q 2 (Actually Q˜ sing ~λ, v, w, v(0) also depends on (∂∂v)(0) but we have suppressed it from the notation.) Here the superscript 1 + 10 indicates the field φ∂φ =: φ∂φ :v and `, T denote the kernels `a (x − y) = −(x − y)a , 1 Tab (x − y) = − (x − y)a (x − y)b . 2
(102)
Then we define Q(t) by Q(t) − Q(t) = Q˜ sing ~λ, vt , w, v(0) , which means that
(103)
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Q(t) = Qreg (~λ, vt , wt ) + FQ (t),
(104)
FQ (t) = Q˜ sing ~λ, vt , wt , vt (0) − Q˜ sing ~λ, vt , w, v(0) .
(105)
where
Although Q˜ sing does not have a useful uniform bound, this will be the case for FQ (t) as we shall see. Looking ahead, an objective is to show that Qreg is RG covariant, F Qreg (~λ, v, w) −→ Qreg (~λ, v # , w# ) + FQ (1) E −→ Qreg (~λ∗ , v # , w# ) S −→ Qreg (~λ0 , v 0 , w0 ),
(106)
where v # = v1 , w# = w1 , where λ∗ is to be specified, and where λ0 , v 0 , w0 are scaled versions of ~λ∗ , v # , w# as defined in the following lemma. The first step will hold since after fluctuation we will have Q(1) given by (104). Extracting FQ (1) and adjusting coupling constants will give the second step. We begin with the third step which follows from the following scaling property: Lemma 4. ~ ~ S1 Qreg (~λ, v, w)e−V (λ,v) = Qreg (~λL , vL , wL )e−V (λL ,vL ) , where kernels scale by vL (x − y) = L2− v (L(x − y)) , wL (x − y) = L2− w (L(x − y))
(107)
and the coupling constants scale by ~λL = (L2 λ, L ζ1 , L ζ2 , L2+ µ). Proof. S1 is linear. By (96) it suffices to prove that ~ ~ S1 Q(~λ, v, w)e−V (λ,v) = Q(~λL , vL , wL )e−V (λL ,vL ) , ~ ~ S1 Q˜ sing ~λ, v, w, v(0) e−V (λ,v) = Q˜ sing ~λL , vL , wL , vL (0) e−V (λL ,vL ) . These are all straightforward computations. For example we have X Q(m,n) (v, w; X, φL−1 ) = Ldm,n Q(m,n) (vL , wL ; Z, φ),
(108)
¯ L =LZ X:X
where dm,n = 8 − (1 − /2)(m + n + 2) − {number of primed indices}.
(109)
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D. Brydges, J. Dimock, T. R. Hurd
3.2. Fluctuation. We summarize what happens when the fluctuation step is applied to the ansatz (75): A = B + R, B = ( + Q)e−V .
(110)
The fluctuation step constructs a new activity A such that µv# ∗Exp(A ) = µv ∗Exp(A), where v # = v − C. #
#
We define A# = A(1), where A(t) is determined by the requirement that for vt = v − tC the quantity µvt ∗ Exp (A(t)) is constant in t and therefore equal to µv ∗ Exp(A). Let B(t) be the approximate solution from the previous section and define R(t) to be the remainder. Thus we have A(t) = B(t) + R(t), B(t) = ( + Q(t)) e−V (t) ,
(111)
where Q(t) is given by (104) and V (t) = V (~λ, vt ). Then defining B # = B(1) and R# = R(1) we have A# = B # + R# , B # = ( + Q# )e−V , #
(112)
where V # = V (1) = V (~λ, v # ), Q# = Q(1) = Qreg (~λ, v # , w# ) + FQ , FQ = FQ (1) = Q˜ sing ~λ, v # , w# , v # (0) − Q˜ sing ~λ, v # , w, v(0) , w# = w1 = w + C.
(113)
We also define K # by A# = e−V + K # #
so that
K # = Q# e−V + R# . #
3.3. Extraction. By Theorem 4, the result of extracting a quantity F = F0 + F1 from K # is a new activity of the form ∗
A∗ = e−V + K ∗ , where V ∗ (1) = V # (1) −
X
(114)
F1 (X, 1),
X⊃1
K ∗ = E(K # , F ) ∗
= K # − F e−V + E≥2 (K # , F ).
(115)
Here F1 is the non-constant part of F , and F1 (X, 1) is related to F1 (X) by the local decomposition:
Non-Gaussian Fixed Point
133
X
F1 (X) =
F1 (X, 1).
1⊂X
We choose F of the form F = FQ + F R ,
(116)
where FQ was defined in (113), and FR is an extraction from R# defined below. Then we have ∗
(117)
∗ # ∗ R∗∗ = (R# − FR e−V ) + Q# e−V − e−V + E≥2 (K # , F ).
(118)
#
where
∗
K ∗ = Q# e−V − FQ e−V + R# − FR e−V + E2 (K # , F ) ∗ = Qreg (~λ, v # , w# )e−V + R∗∗ ,
The local decomposition of FQ,1 is obtained as follows. The functional FQ,1 is a sum of monomials of the form Z : φn (x) : f (x − y) dx dy. (119) F (X) = ˜ X
For each such monomial the local decomposition holds with F (X, 1) defined for X = 11 ∪ 12 by : R R11 ×12 : φn (x) : f (x − y)dxdy if 1 = 11 : φn (x) : f (x − y)dxdy if 1 = 12 . F (X, 1) = (120) 12 ×11 0 otherwise The next lemma is important because it shows that FQ is responsible for the flow of the coupling constants ~λ and is the origin of the explicit formulas for this flow. Lemma 5.
X
FQ,1 (X, 1) = V (1, δ~λQ , v # ),
X⊃1
where
Z 2
δλQ = a(w)λ = 36λ
2
δζ1,Q = b1 (w)λ2 = 48λ2
Z3 3
w#2 − w2 ,
w#3 T − w3 T ,
δζ2,Q = b2 (w)λ2 = 0 δµQ = c(w)λ2 + d(w)λ(ζ1 + ζ2 ) + e(w)λµ Z Z # #3 2 #2 2 2 v (0)w − v(0)w + 48λ w − w3 = 144λ 3 3 Z # 2 2 (∂w ) − (∂w) + 12(ζ1 + ζ2 )λ 3 Z # 2 (w ) − (w)2 . + 12λµ 3
where T was defined in (102).
(121)
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D. Brydges, J. Dimock, T. R. Hurd
Proof. By (119), X
Z F (X, 1) =
Z : φn (x) : dx
1
X⊃1
3
f (x − y)dy ,
(122)
which has the right form to be one of the monomials in V (1, δ~λQ , v # ). The formulas for δ~λQ were obtained from ( 101), (113). Only the first two lines in the definition (101) of Q˜ sing contribute to δ~λQ . The other terms are either constants, or they come from single diagrams in (80). In the latter case we are integrating a derivative and so get zero. For example the O(ζ12 , ζ1 ζ2 ) changes in the field strength vanish since Z 2ζ12
3
(∂a ∂b w# − ∂a ∂b w) = 0
(123)
and since an integration by parts gives Z ζ 1 ζ2
3
(y − x)a (−1∂a w)(x − y)dy Z = 4ζ1 ζ2 (−1w)(x − y)dy 3
= 0.
(124)
We have also used Z Z # w (−1w# ) − w(−1w) = 12λζ2 (∂w# )2 − (∂w)2 12λζ2 3
3
to obtain the λζ2 contribution to δµQ . Construction of FR . FR (X, φ) is zero if X is not a small set; otherwise it is a local polynomial of the form Z 2
FR (X, φ) = α0 (X)|X| + α2,0 (X) +
X
Z α2,2 (X, a, b)
a,b
a
(∂a φ)(∂b φ) + X
Z
φ + X
X
Z α2,1 (X, a)
φ(∂a φ) X
X a,b
0 α2,2 (X, a, b)
Z φ(∂a ∂b φ) X
φ4
+ α4 (X)
(125)
X
with coefficients α determined by the next lemma. Let us introduce the notation Fn = (FR )n (X, 0),
Rn = (R# )n (X, 0),
Vn = Vn# (X, 0),
(126)
but for consistency with extraction, Theorem 4, FR,1 equals FR expanded into Wick powers with the constant (φ0 term) subtracted.
Non-Gaussian Fixed Point
135
Lemma 6. There is a unique choice of coefficients α in FR (X, φ) such that dim(R# − FR e−V ) > 4, X FR,1 (X, 1) = V (1, δ~λR , v # ). #
(127)
X⊃1
Furthermore αm,n (X) and δ~λR are polynomials in eV0 Rn (p×n ) and Vn (p×n ), where n = 0, 2, 4 and p = 1, xa , xb , xa xb . The x-origin can be chosen arbitrarily for each X. The polynomials have O(1) coefficients and are linear in Rn . Proof. Recall that dim was defined in Definition (1). The first four non-vanishing derivatives of R# − FR exp(−V # ) evaluated at φ = 0 are J0 = R0 − F0 e−V0 , J2 = R2 − (F2 − F0 V2 ) e−V0 , J4 = R4 − F4 − 6Sym(F2 V2 ) + 3F0 Sym(V22 ) − F0 V4 e−V0 .
(128)
Referring to the definition of dimension we find that the dim > 4 condition in (127) is equivalent to the vanishing of these derivatives when evaluated on the constant 1 and monomials xa , xa xb as follows: J0 = 0, J2 (1×2 ) = 0, J2 (1, xa ) = 0, J2 (xa , xb ) = 0, J2 (1, xa xb ) = 0, J4 (1×4 ) = 0.
(129)
By using these equations in the order given we determine recursively the coefficients α in (125) in the order 0 (X, a, b), α4 (X) α0 (X), α2,0 (X), α2,1 (X, a), α2,2 (X, a, b), α2,2
and easily see that α is a polynomial (linear) in eV0 Rn and Vn evaluated on 1, xa , xb , xa xb . For example we have 1 V0 e (X)R0 (X), |X| 1 V0 e (X) R2 (X; 1×2 ) + R0 (X)V2 (X; 1×2 ) , α2,0 (X) = 2|X| h α2,1 (X, a) = eV0 (X) R2 (X; 1, xa ) + R0 (X)V2 (X; 1, xa ) Z i 1 ×2 ×2 R2 (X; 1 ) + R0 (X)V2 (X; 1 ) xa . − |X| X α0 (X) =
(130)
Now FR is defined. Define FR,1 (X, 1) by expanding FR into Wick powers, dropping the α0 (X) term and restricting the integrations in (125) to 1. It remains to establish (127). Consider !Z X X α2,1 (X, a) φ(∂a φ). (131) 1
X⊃1
1
136
D. Brydges, J. Dimock, T. R. Hurd
Each iteration of the renormalization group map preserves the symmetry of the initial data under reflections r that map the lattice of unit blocks to itself: K(rX, r† φ) = K(X, φ), where r† φ(x) = φ(rx). By (130) with x-origin chosen at the center of 1 α2,1 (ra X, a) = −α2,1 (X, a),
(132)
where ra is the reflection in the a-direction through 1. Hence the sum above is zero and there is no term of this form in the potential. Analogously using ra , rb we conclude that X α2,2 (X, a, b) = α2,2 δa,b , X⊃1
X
0 0 α2,2 (X, a, b) = α2,2 δa,b ,
(133)
X⊃1
and these contribute to ζ1,R , ζ2,R respectively. From these considerations we find that the changes in the coupling constants are given by (taking into account the Wick ordering) X α4 (X), δλR = X⊃1 0 δζR = (α2,2 , −α2,2 ), X α2,0 (X) + 6v # (0)α4 (X) . δµR =
(134)
X⊃1
We introduce the notation δλR = −r(~λ, w, R), δζi,R = −si (~λ, w, R), δµR = −t(~λ, w, R).
(135)
Note that r, s, t are linear functions of R# , which in turn is a function of (~λ, w, R). The new potential is now V ∗ = V (λ~∗ , v # ), where ~λ∗ = ~λ − δ~λQ − δ~λR . Thus we have λ~∗ given by λ∗ = λ − a(w)λ2 + r(~λ, w, R), ζ ∗ = ζi − bi (w)λ2 + si (~λ, w, R), i
µ∗ = µ − c(w)λ2 − d(w)λ(ζ1 + ζ2 ) − e(w)λµ + t(~λ, w, R).
(136)
Referring to (117) we see that the coupling constants ~λ∗ in the potential do not match the coupling constants ~λ in Q∗∗ = Qreg (~λ, v # , w# ). We make a further adjustment by defining Q∗ = Qreg (~λ∗ , v # , w# ),
(137)
and have ∗
K ∗ = Q∗ e−V + R∗ ,
(138)
where ∗
R∗ = (Q∗∗ − Q∗ )e−V + R∗∗ .
(139)
Non-Gaussian Fixed Point
137
3.4. Scaling. After scaling we have a new activity 0
A0 = e−V + K 0 ,
(140)
where K 0 = S(K ∗ ). By definition of S # V 0 = V (~λ∗ , v # )L−1 = V (~λ∗L , vL ) = V (~λ0 , v 0 ),
(141)
# where v 0 = vL (x − y) and
~λ0 = ~λ∗ = (L2 λ∗ , L ζ ∗ , L2+ µ∗ ). L
(142)
Also 0
K 0 = Q0 e−V + R0 ,
(143)
where 0
∗
Q0 e−V = S1 (Q∗ e−V ), R0 = S1 (R∗ ) + S≥2 (K ∗ ).
(144)
S1 is the linearization of S and S≥2 = S − S1 . From Lemma 4 Q0 = Qreg (λ~0 , v 0 , w0 ) = Q(λ~0 , v 0 , w0 )
(145)
# with w0 = wL which means that the form of our ansatz (75) has been preserved as in (106).
3.5. Summary. Starting with the polymer activity (75) we have found that one RG step transforms the ansatz to ~0 0 (146) A0 = A(~λ0 , R0 , w0 ) = + Q(~λ0 , v 0 , w0 ) e−V (λ ,v ) + R0 . The new coupling constants are obtained from (136),(142) and are given by λ0 = L2 λ − a(w)λ2 + r(~λ, w, R) , ζi0 = L ζi − bi (w)λ2 + si (~λ, w, R) ; b2 = 0, µ0 = L2+ µ − c(w)λ2 − d(w)λ(ζ1 + ζ2 ) − e(w)λµ + t(~λ, w, R) .
(147)
The new remainder R0 is given as a function R0 = U (~λ, R, w)
(148)
by Eqs. (112),(118),(139),(144). Finally the new kernel is w0 (x − y) = L2− (w (L(x − y)) + C (L(x − y))) .
(149)
In infinite volume v 0 = v and everything is reduced to this single mapping in the variables ~λ, R, w.
138
D. Brydges, J. Dimock, T. R. Hurd
4. Estimates We observe the convention that O(1) stands for a constant independent of and L and that C stands for a constant which may depend on L but is independent of . The value of C may vary from line to line. This is consistent with the usage in Sect. 2. 4.1. Estimates on w, a(w), b(w), . . . . We begin with some estimates on w. Recall that w0 = wL + CL , where wL (x) = L2− w(Lx) . Starting with w0 = 0 we get a sequence wj defined by wj+1 = (wj )L + CL or j X
wj (x) =
L(2−)l C(Ll x).
(150)
l=1
This can also be written
Z wj (x) =
1
dα α/2−2 e−x
2
/4α
.
(151)
L−2j
For x 6= 0 this also makes sense for j = ∞. We sometimes write w∞ = w∗ . The basic properties of C and wj are contained in: Lemma 7. For any multi-index β there is a constant O(1) such that for all x and 1≤j≤∞ |(∂ β C)(x)| ≤ O(1)e−2|x|/L , |(∂ β wj )(x)| ≤ O(1)e− 2 |x| |x|−2−|β|+ . 3
(152)
Proof. We have Z
L2
√
dαα/2−2 e−2|x|/
|C(x)| ≤ O(1)
α
≤ O(1)e−2|x|/L
(153)
1
and also
Z wj (x) ≤ O(1)
1
L−2j
≤ O(1)e− 2 |x| 3
√
dα α/2−2 e− 2 |x|− 2 |x|/ α Z ∞ √ 1 dα α/2−2 e− 2 |x|/ α 3
1
0
= O(1)e
− 23 |x|
|x|−2+ .
The bounds for the derivatives follow similarly. We will find it useful to cast this in a Banach space setting. Let W be the Banach space of complex-valued functions on R4 for which the following norm is finite: kwk = sup sup(|x|9/4+|β| |∂ β w(x)|e|x| ).
(154)
|β|≤3 x
By the lemma, the sequence wj starting at w0 = 0 is in W and we can fix a constant k independent of L and such that for 0 ≤ j ≤ ∞, kwj k ≤ k/2 (Definition of k).
(155)
Non-Gaussian Fixed Point
139
The point of taking the power 9/4 here is to make a norm in which the mapping w → w0 is a strong contraction. It is easy to show that for any w, w˜ ∈ W, kw0 − w˜ 0 k ≤ L−1/4 kw − wk. ˜
(156)
Thus, starting from any w0 = w with kwk ≤ k/2 we generate a sequence wj converging to the unique fixed point w∗ and satisfying kwj − w∗ k ≤ L−j/4 kw − w∗ k ≤ L−j/4 k, kwj k ≤ k.
(157)
Hereafter we consider general w ∈ W with kwk ≤ k. Lemma 8. There is a constant C such that for w ∈ W and kwk ≤ k, |a(w)|, |b1 (w)|, |c(w)|, |d(w)|, |e(w)| ≤ C.
(158)
Furthermore there is a constant k 0 independent of L such that |a(w1 ) − a(w2 )| ≤ k 0 kw1 − w2 k
(159)
and similarly for b, c, d, e. Proof. We have, from Lemma 5, Z a(w) = 3e(w) = 36 (2wC + C 2 ), Z b1 (w) = 48 (3w2 C + 3wC 2 + C 3 )T, Z c(w) = 48 6v # (0)wC + 3v # (0)C 2 + 3 (C − C(0)) w2 + 3wC 2 + C 3 , Z (160) d(w) = 12 2∂w∂C + (∂C)2 . The issue is the singularity at x = 0. First consider a(w). For w ∈ W we have |w(x)| ≤ |x|−9/4 e−|x| kwk.
(161)
Therefore w is integrable and since C ≤ O(1) we have that wC is integrable and R wC ≤ O(1). Since |C(x)| ≤ O(1)e−2|x|/L we also have that C 2 is integrable (but the bound depends on L). For the Lipschitz bound only the first term contributes and so we can use the bound (161) for w1 − w2 to get a bound independent of L. For the bound on b1 (w) we use that |T (x)| ≤ O(|x|2 ) and |x|2 |w(x)|2 ≤ |x|−5/2 e−2|x| kwk2
(162)
to bound the first term and get the result. The Lipschitz bound follows similarly. For the bound on c(w) we use |C(x) − C(0)| ≤ O(|x|) to obtain the factor of x needed to integrate w2 . We also use |v # (0)| ≤ O(1). Finally d(w) is bounded since |∂w(x)| ≤ |x|−13/4 e−|x| kwk is integrable.
(163)
140
D. Brydges, J. Dimock, T. R. Hurd
In fact we need a sharper bound on a(w). Lemma 9. Let L be sufficiently large and < (L). Then there exist positive constants c1 , c2 independent of L such that for real w ∈ W and kwk ≤ k , c1 log L ≤ a(w) ≤ c2 log L.
(164)
Proof. As noted the first term in a(w) is bounded independently of L. Thus it suffices to show that Z (165) O(1) log L ≤ C(x)2 dx ≤ O(1) log L. This expression is written as Z
Z
Z
L2
C 2 (x)dx =
Z
L2
dβ (αβ)/2−2
dα 1
e−x
2
/4α −x2 /4β
e
dx.
1
To obtain an upper bound, we write this as twice the integral subject to β ≤ α, and 2 bound e−x /4α by 1. Then provided log L ≤ 1, Z
Z
Z
L2
C 2 (x)dx ≤ 2 1
dβ (αβ)/2−2 1
Z
Z
α
dα
Z
L2
≤ O(1)
2
/4β 4
d x
α
dβ (αβ)/2−2 β 2
dα 1
Z
e−x
1 L2
≤ O(1)
dα α−1 1
≤ O(1) log L. The integral is also bounded from below by the integral over the subregion α ∈ 2 2 (2, L), β ∈ (α/2, α). On this region we can use the lower bound e−x /4α ≥ e−x /4β . Then provided log L ≤ 1 we find Z
Z
Z
L2
C (x)dx ≥ 2 2
α /2−2
dα 2
Z
dβ (αβ)
e−x
2
/4β
2 dx
α/2
Z
Z
L2
≥ O(1)
α
dβ (αβ)/2−2 β 2
dα α/2
2
Z ≥ O(1)
L2
dα α−1 2
≥ O(1) log L. 4.2. The domain D. In the rest of this section we give estimates on the various quantities which enter our flow equations. The control of this flow is relegated to the next section. To get useful estimates we must carefully specify a domain D for the coupling constants ~λ = (λ, ζ, µ), the remainder R, and the kernel w. For the coupling constant we consider a region defined by the inequalities ¯ < α |λ − λ|
|ζ1 | < 1−δ
|ζ2 | < 2−1−δ
|µ| < 2−2δ ,
(166)
Non-Gaussian Fixed Point
141
where λ¯ = O() > 0 given by Eq. (226) is the low order approximation to the fixed point. This domain is supposed to contain the exact fixed point. We pick 1 < α < 2 so that λ = O(). The powers of for ζ, µ are reduced slightly below the naive (integer) values: δ, 1 are small numbers with 1 > 4δ > 0. We also allow α = 1 but then impose the condition ¯ < d, |λ − λ|
(167)
d must be sufficiently small so that Re(λ) > 0. This condition on d is sufficient for Theorem 7 (below). We will impose another smallnesss condition on d in the proof of lemma 16 in the next section. For the remainder R we consider the norms k · k = k · kG,h,0 , | · | = | · |1,0
(168)
as defined in Sect. 2. We take G = G(κ, X, φ) defined in (30) with κ = 1/2 . We also choose h = −1/4 . Now define ||| · ||| = max{2 k · k, | · |}.
(169)
We consider remainders R which satisfy the bound |||R||| ≤ 3−1 .
(170)
kwk ≤ k,
(171)
Finally w should satisfy
where k is the value given by (155) Choose any α, δ, 1 in accordance with these restrictions and let D be the region of complex valued (~λ, R, w) satisfying (166), (170), (171). In the remainder of this chapter we shall prove the following result: Theorem 7. Let L be sufficiently large and sufficiently small (depending on L). Then r, s, t, U are analytic functions of (~λ, R, w) on D and are bounded there by |r|, |s|, |t| ≤ O(1)3−1 , |||U ||| ≤ O(1)L−1+2 3−1 ,
(172)
where O(1) is a constant independent of L, and the parameters defining D. 4.3. Estimates on Q. Our convention is that O(1) is a constant independent of , L and the parameters δ, 1, α, d defining D, whereas a constant C is only independent of . We always assume that the hypotheses of Theorem 7 are satisfied, except we permit any w with kwk < C so that the wt , vt which occurred in the last section are allowed. Lemma 10. For each term in Qreg (~λ, v, w) with X = 1 ∪ 10 , 0
n+m −d(1,1 ) e ; kφk = kφkC 3 (X) |Q(n,m) ¯ . reg (X, φ)| < Ckφk
This bound holds if φ is complex valued.
(173)
142
D. Brydges, J. Dimock, T. R. Hurd
Proof. To show 0
3 2 −d(1,1 ) , |Q(1,1) reg (X, φ)| ≤ Ckwk kφk e
(174)
we refer to (90) and use | [φ(x) − φ(y)]2 − [(x − y) · (∂φ)(x)]2 | = | [φ(x) − φ(y) − (x − y) · (∂φ)(x)] [φ(x) − φ(y) + (x − y) · (∂φ)(x)] | ≤ O(1)|x − y|2 kφk|x − y|kφk
(175)
(by Taylor’s theorem with remainder) followed by Z Z |x − y|3 |w(x − y)|3 dxdy ≤ kwk3 1×10
1×10
|x − y|−15/4 e−3|x−y| dxdy 0
≤ O(1)kwk3 e−d(1,1 ) .
(176)
The other terms, being very similar, are left to the reader. Wick powers such as : φ3 : (x) occurring in Q(3,1) reg are polynomials with O(1) coefficients which offer no obstacles. Lemma 11. Qreg (~λ, v, w) satisfies kQreg e−V k ≤ C1/2 , |Qreg e
−V
| ≤ C
2−2δ
(177) .
(178)
Proof. By Lemma 1 we have kQreg e−V k ≤ O(1)|Qreg |h,01 , |Qreg e−V | ≤ O(1)|Qreg |1,01 .
(179)
Now we bound the right side of these inequalities. P αj fj ) is a polynomial in Let f1 , . . . , fp be C 3 test functions. Then Q(n,m) reg (X, α1 , . . . , αp . We can estimate the p−fold derivative with respect to α1 , . . . , αp at the origin by Cauchy’s theorem and Lemma 10 and thereby estimate the p−fold Gateau derivative 0
−d(1,1 ) . |(Q(n,m) reg )p (X, φ = 0; f1 , . . . , fp )| < Cn,m kf1 k · · · kfp ke
(180)
The derivative vanishes for p > n + m but not necessarily for p < n + m because of the Wick powers of φ. From X 0 0(1 ∪ 10 )e−d(1,1 ) ≤ C (181) 10
it easily follows that n+m . |Q(n,m) reg |h,01 < Cn,m h
(182)
Every term in Qreg is accompanied by O(~λ2 ) coupling constants. Recalling that δ, 1 are small the largest combination λ2 h6 = 1/2 arises from Q(3,3) . If h = 1 the largest term is the O(ζ12 ) = 2−2δ . Thus |Qreg |h,01 < C2 h6 = C1/2 ; |Qreg |1,01 < C2−2δ . Now the lemma follows from (179).
(183)
Non-Gaussian Fixed Point
143
Lemma 12. kFQ e−V k ≤ C1−2δ , #
|FQ e−V | ≤ C2−2δ . #
(184)
Proof. One can prove these estimates directly, but we prefer to give an indirect proof which uses results we have already established. Recall from (104) that FQ = FQ (1), where FQ (t) solves ∂ ∂ ~ − 1C FQ (t) = − ( − 1C )Qreg (λ, vt , wt ) − J(t) . (185) ∂t ∂t {z } | R(t) By construction R is a quartic polynomial in φ. We claim that R is smooth on C 3 (X) (C 0 bounded Gateau derivatives of all orders) and that it decays like e−d(1,1 ) in X = 1∪10 . This is true for Qreg and J(t), see (180). The same is true after ∂/∂t − 1C is applied. This is easily checked for ∂/∂t using the formulas in (90). For 1C we use the definition of 1C , Z 1 1C Q(n,m) dµC (ζ) (Q(n,m) (X, φ) = (186) reg reg )2 (X, φ; ζ, ζ), 2 which reduces the claim to (180). The solution of (185) is Z
t
exp ((t − s)1C ) R(s) ds,
FQ (t) = exp (t1C ) FQ (0) +
(187)
0
where exp (t1C ) is defined as the power series, which terminates after a finite number of terms because R is a polynomial. The first term vanishes because of the boundary condition FQ (0) = 0. Since R is smooth and exponentially decaying the same is true for the second term. Thus FQ is smooth and exponentially decaying. Both FQ (t) and R(t) are quadratic polynomials in the coupling constants ~λ , and the argument we have given so far applies to the coefficient of each term λ2 , λζ1 λ, . . . . As in the proof of the last lemma, smoothness implies C bounds on the kernel norms | |h,01 . The largest coupling constant in the various terms is ζ12 = 2−2δ . The degree of FQ is four, so |FQ |h,01 < C2−2δ h4 = C1−2δ ; |FQ |1,01 < C2−2δ .
(188)
The lemma follows by Lemma 1. (Actually by considering the terms individually the 1−2δ in the first bound can be replaced by .) 4.4. Estimates on r, s, t, U . We continue our C and O(1) convention and we assume that the hypotheses of Theorem 7 are satisfied. We introduce the norms k · k# = k · kG# ,h# ,0# , | · |# = | · |1/2,0# , where G# = G# (2κ, X, φ) and h# = h/2.
(189)
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D. Brydges, J. Dimock, T. R. Hurd
Note that our bound |||R||| < 3−1 implies that kRk ≤ 1−1 , |R| ≤ 3−1 .
(190)
Lemma 13. R# = R# (~λ, R, w) is analytic on D and satisfies there kR# k# ≤ O(1)1−1 , |R# |# ≤ O(1)3−1 .
(191)
Remark. Together with a weaker bound on Q# e−V from Lemma 11 and Lemma 12 we obtain #
kK # k# ≤ C1/2 , |K # |# ≤ C2−2δ .
(192)
Proof. Refer to (111,112) and Theorem 3. With B(t) = ( + Q(t)) e−V (t) and E(t) = E (B(t)) we obtain kR# k# ≤ 2(kRk + sup kE(t)kGt ,h,0 ), t # 3 |R |# ≤ O(1) |R| + sup |E(t)| + O( ) ,
(193)
t
using Gt=1 (X) = 2|X| G# (X) (Gt was defined in (47)), transferring the factor 2|X| to the 0 to get 0−1 = 0# and choosing M = 12. Taking into account (190) and our right to choose after L we see that it suffices to show that kE(t)kGt ,h,0 ≤ C1−3δ , |E(t)|1/2,0 ≤ C3−3δ . Working from the definitions (78) for E(t) and (84) for J, one finds ∂ − 1C + J e−V (t) E(t) = ∂t ∂ + Q(t) · − 1C e−V (t) ∂t ∂Q(t) ∂V (t) −V (t) + C , e ∂φ ∂φ ∂ + − 1C Q(t) − J e−V (t) ∂t 1 −V (t) −V (t) −V (t) + − BC e , e + J(1 − )e 2 1 − BC e−V (t) , Q(t)e−V (t) − BC Q(t)e−V (t) , Q(t)e−V (t) . 2
(194)
(195) (196) (197) (198) (199) (200)
Terms (195),(199) are both zero by the definition of J. The definition of Q is designed so that (198) vanishes.
Non-Gaussian Fixed Point
145
Term (196) is equal to Q(t)J(t)e−V (t) .
(201)
We estimate this and other terms by the same methods as in Lemma 11. In particular using Lemma 1 and Theorem 1 we split off the exp(−V (t)) and are reduced to |QJ|h,0# ≤ |Q|h,0# |J|h,0#
(202)
with either h = −1/4 /2 or h = 1/2. We found in Lemma 11 that Q is C1/2 , C2−2δ (in the two norms). The same estimate holds for J so (196) is C1 , C4−4δ which is consistent with (194). In the proof of Lemma 11 we found, for a monomial of degree p, | · |h,0 ≤ Chp × |coupling constants |
(203)
from which term (197) has the estimate C, C3−3δ and term (200) is C, C3−3δ . Next we introduce norms k · k∗ = k · kG∗ ,h∗ ,0∗ , | · |∗ = | · |1/2,0∗ ,
(204)
where G∗ = G# , 0∗ = (0# )−4 = 0−5 , h∗ = h# . Recall how dimension was defined in Definition 1. Lemma 14. R∗ is analytic on D and can be written in the form R∗ = R1∗ + R2∗ , where dim R1∗ (X) > 4 for X small and kR1∗ k∗ ≤ O(1)1−1 , |R1∗ |∗ ≤ O(1)3−1 , kR2∗ k∗ ≤ C1−4δ , |R2∗ |∗ ≤ C3−4δ .
(205) ∗
Remark. Combining this with a bound from Lemma 11 on Q∗ e−V gives: kK ∗ k∗ ≤ C1/2 , |K ∗ |∗ ≤ C2−2δ .
(206)
Proof. From Sect.3.3, ∗
R1∗ = R# − FR e−V , R2∗ = E≥2 (K # , F ) ∗
∗
+ Q# (e−V − e−V ) + (Q∗∗ − Q∗ )e−V . #
(207)
Then R1∗ has dimension greater than 4 by construction. Our bound |R# |# ≤ O(1)3−1 implies (see Lemma 6) that |FR |# ≤ O(1)3−1 , hence that |(αR )n,m (X)| ≤ O(1)3−1 0# (X)−1
(208)
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D. Brydges, J. Dimock, T. R. Hurd
for (n, m) = (0, 0), (2, 0), . . . and hence that ∗
kFR e−V k∗ ≤ O(1)2−1 , ∗
|FR e−V |∗ ≤ O(1)3−1 .
(209)
(Each field reduces the k · k∗ norm by −1/4 and the maximum number of fields in FR is 4. The sum over X is over the finite number of small sets containing a fixed block.) These are sufficient to give the bounds on R1∗ . By Theorem 4, kE≥2 (K # , F )k∗ ≤ O(1)kf k0# kK # k# , |E≥2 (K # , F )|∗ ≤ O(1)kf k0# |K # |# , where we take for some constant Cf , θ(X)−2 Cf 1−2δ f (X) = 0
|X| ≤ 2 or X small . X large, |X| ≥ 3
(210)
(211)
To see that this f satisfies the Rstability hypothesis (57) of the theorem: The : φ4 : contribution to FQ has the form 1 α(X, 1, x) : φ4 (x) : dx, where |X| ≤ 2. If X = 0 1 ∪ 10 then supx |α(X, 1, x)| ≤ C2−2δ e−d(1,1 ) . Then we have that X X 0 sup |α(X, 1, x)|2/f (X) ≤ C/Cf θ(1, 10 )2 e−d(1,1 ) |X|≤2,X⊃1
x
10
≤ C/Cf .
(212)
By taking Cf sufficiently large this is bounded by a small multiple of so that stability holds by Theorem 1: the λ : φ4 : part of V remains dominant. The same arguments show that the other terms in FQ are also compatible with R stability. The contributions to FR have the form α(X) X p (φ(x)) dx, where α is supported on small sets and |α(X)| ≤ O(3−1 ). Then we have summing over X small only and noting that θ(X) = 1 for X small, X |α(X)|2/f (X) ≤ C2−1+2δ /Cf . (213) X⊃1
Again the sum of these is less than a small multiple of . With this choice of f we have kf k0# ≤ C1−2δ ,
(214)
and combined with the bound (192) on K # this gives kE≥2 (K # , F )k∗ ≤ C1−4δ , |E≥2 (K # , F )|∗ ≤ C3−4δ .
(215)
Using e
−V #
−e
−V ∗
Z
1
=− 0
e−tV
#
−(1−t)V ∗
(V # − V ∗ )dt
(216)
Non-Gaussian Fixed Point
147
one also shows ∗
∗
kQ∗ (e−V − e−V )k∗ + k(Q∗∗ − Q∗ )e−V k∗ ≤ C1−4δ , #
∗
∗
|Q∗ (e−V − e−V )|∗ + |(Q∗∗ − Q∗ )e−V |∗ ≤ C3−4δ #
(217)
to complete the proof. Lemma 15. R0 is analytic on D and satisfies there kR0 k ≤ O(1)L−1+2 1−1 , |R0 | ≤ O(1)L−1+2 3−1 .
(218)
R0 = S1 (R1∗ ) + S1 (R2∗ ) + S≥2 (K ∗ ).
(219)
Proof. From Sect. 3.4:
By Theorem 5 a crude bound is kS(K ∗ )k ≤ O(1)L4 kK ∗ kGL ,ahL ,0−5 ≤ O(1)L4 kK ∗ k∗ ≤ C1/2 |S(K )| ≤ C2−2δ . ∗
(220)
∗ −1 and ahL ≤ h∗ . Here we have used G−1 L ≤ (G ) ∗ For S1 (R1 ) we use Theorem 6, break the estimate into large and small sets and find:
kS1 (R1∗ )k ≤ O(1)L−1+2 kR1∗ k∗ ≤ O(1)L−1+2 1−1 |S1 (R1∗ )| ≤ O(1)L−1+2 3−1 .
(221)
∗
Here we have used dim(R ) ≥ 5 − 2. Indeed from Eq. (128) and the definition of dimension we find that dim(R2∗ ) = 2 dim(φ)+3 = 5−, dim(R4∗ ) = 4 dim(φ)+1 = 5−2 and dim(R6∗ ) = 6 dim(φ) = 6 − 3 . We also have kS1 (R2∗ )k ≤ CkR2∗ k∗ ≤ C1−4δ , |S1 (R2∗ )| ≤ C|R2∗ |∗ ≤ C3−4δ ,
(222)
which is sufficient because 1 > 4δ and is chosen after L. Finally by Cauchy bounds and (220) we find kS≥2 (K ∗ )k ≤ C1 , |S≥2 (K ∗ )| ≤ C4−4δ
(223)
to complete the proof. Proof of Theorem 7. That r(~λ, R, w), s(~λ, R, w), t(~λ, R, w) are analytic in D and satisfy there |r|, |s|, |t| ≤ O(1)3−1
(224)
follows from the same properties for the αn,m (X) in (125). See Lemma 6 and note that the bounds follow from (170). The bound on R0 = U (~λ, R, w) is contained in the previous lemma.
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D. Brydges, J. Dimock, T. R. Hurd
5. Critical Models 5.1. The problem. We now restrict to infinite volume so that the RG flow is the iteration of a fixed mapping. In this section we prove the existence of critical theories by showing that one can pick µ0 = µ0 (λ0 ), ζ0 = ζ0 (λ0 ) so that the RG flow tends to a fixed point. This result is part of the stable manifold theorem, but we have not been able to find a version in the literature which is exactly applicable to our problem, so we have to give an independent proof, at least in part. So far we have proved that the renormalization group transformation can be iterated as long as the parameters (~λj , Rj , wj ) that define the theory remain in the domain D and the flow after j iterations is given by the equations λj+1 = L2 [λj − a(wj )λ2j + r(~λj , Rj , wj )], ζj+1 = L [ζj − b(wj )λ2 + s(~λj , Rj , wj )], j
µj+1 = L2+ [µj − c(wj )λ2j − d(wj )λj (ζ1,j + ζ2,j ) − e(wj )λµ + t(~λj , Rj , wj )], Rj+1 = U (~λj , Rj , wj ) wj+1 = (wj )L + CL .
(225)
The starting point is (~λ0 , 0, 0) with ~λ0 = (λ0 , ζ0 , µ0 ), but we consider the more general case (~λ0 , R0 , w0 ). If we ignore the higher order term r and replace wj by its limiting value w∗ we find the approximate fixed point for λ, L2 − 1 λ¯ = 2 . L a(w∗ )
(226)
In Sect. 4 we required that λ¯ = O(). This is valid since for small we have L2 ∼ 1 + 2 log L + . . . and we have seen that a(w∗ ) ∼ O(log L). (We continue to assume that L is chosen large followed by small, that O(1), O(), etc. are L independent constants, and that C is a possibly L dependent constant which may change from line to line.) ¯ With We now rewrite the equations replacing λj by the deviation λ˜ j = λj − λ. ξj = (λ˜ j , ζj , µj , Rj , wj )
(227)
we have λ˜ j+1 = (2 − L2 )λ˜ j + r(ξ ˜ j ), ˜ j ), ζj+1 = L ζj + s(ξ 2+ µj+1 = L µj + t˜(ξj ), Rj+1 = U (ξj ), wj+1 = (wj )L + CL ,
(228)
where we have defined r(ξ) ˜ = L2 [−a(w∗ )λ˜ 2 + (a(w∗ ) − a(w)) λ2 + r(ξ)], s˜i (ξ) = L [−bi (w)λ2 + si (ξ)], t˜(ξ) = L2+ [−c(w)λ2 − d(w)λ(ζ1 + ζ2 ) − e(w)λµ + t(ξ)].
(229)
Non-Gaussian Fixed Point
149
˜ ζ, µ, R, w), etc. Now ξ0 = (λ˜ 0 , ζ0 , µ0 , R0 , w0 ) is the In this equation r(ξ) = r(λ¯ + λ, initial point. Note that the shift has changed an expanding variable λ into a contracting ˜ Equations (228) are the iteration of a single mapping which we call f so that variable λ. ξj+1 = f (ξj ).
(230)
As a norm on ξ we take ˜ −1+δ |ζ1 |, −2+1+δ |ζ2 |, −2+2δ |µ|, −3+1 |||R|||, k −1 kwk), (231) kξk = sup(−α |λ|, where 1 > 4δ > 0 and 1 < α and k appeared in (157). This norm defines the Banach ¯ space E. The unit ball E(1) is the image of the domain D under the shift λ˜ = λ − λ. −α ˜ −1 ˜ In Sect. 5.3 to come, we will need to replace the factor |λ| by (d) |λ| with d sufficiently small. The parameter d will be chosen independently of L, . This will define a Banach space F . We collect the bounds that we will need. Assuming ξ ∈ E(1) we have from the previous section: |r(ξ)|, |si (ξ)|, |t(ξ)| ≤ O(1)3−1 ; |||U (ξ)||| ≤ O(L−1+2 )3−1 .
(232)
By Lemma 8 and Lemma 9 and noting that L ≤ O(1) because is chosen after L, it follows that (recall b2 = 0) |r(ξ)| ˜ ≤ O(log L)2α + O(1)kw − w∗ k2 + O(1)3−1 , |s˜1 (ξ)| ≤ C2 ; |s˜2 (ξ)| ≤ O(1)3−1 , |t˜(ξ)| ≤ C2−δ .
(233)
Recall that r is analytic on D, sothat r˜ is analytic in E(1). By the Cauchy representation for f (z) = r˜ ξ 0 + z(ξ − ξ 0 ) with contour |z| = O(kξ − ξ 0 k−1 ) we represent f (1) − f (0) to obtain Lipschitz bounds: for ξ, ξ 0 ∈ E(1/2): 2k2 3−1 + , (234) |r(ξ) ˜ − r(ξ ˜ 0 )| ≤ O(1)kξ − ξ 0 k log(L)2α + kw − w∗ k2 where we can use the bound with kw−w∗ k2 when ξ and ξ 0 have the same w component. Furthermore the same bound with α replaced by d holds. In the same way we have |s˜1 (ξ) − s˜1 (ξ 0 )| ≤ C2 kξ − ξ 0 k, |s˜2 (ξ) − s˜2 (ξ 0 )| ≤ O(1)3−1 kξ − ξ 0 k, |t˜(ξ) − t˜(ξ 0 )| ≤ C2−δ kξ − ξ 0 k, |||U (ξ) − U (ξ 0 )||| ≤ O(L−1+2 )3−1 kξ − ξ 0 k.
(235)
5.2. Existence of global solutions. We first want to prove the existence of bounded global solutions, that is sequences ξj ∈ E(1/2) defined for all j = 0, 1, . . . and satisfying ξj+1 = f (ξj ) for j = 0, 1, . . . . We start by looking for finite sequences ξj with 0 ≤ j ≤ k with specified initial conditions λ0 , R0 and specified final conditions ζk = ζf , µk = µf . It is straightforward to check that a sequence ξj is a solution if and only if it satisfies the “integral” equations:
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D. Brydges, J. Dimock, T. R. Hurd
λ˜ j = (2 − L2 )j (λ˜ 0 ) +
j−1 X
(2 − L2 )j−l−1 r(ξ ˜ l ); j = 1, . . . , k,
l=0
ζj = L−(k−j) ζf −
k−1 X
L−(l+1−j) s(ξ ˜ l ); j = 0, . . . , k − 1,
l=j
µj = L−(2+)(k−j) µf −
k−1 X
L−(2+)(l+1−j) t˜(ξl ); j = 0, . . . k − 1,
l=j
Rj = U (ξj−1 ); j = 1, . . . , k.
(236)
The imposition of a mixture of initial and final conditions has given equivalent “integral” equations in which only exponentially decaying factors appear. This is a standard device in the theory of hyperbolic dynamical systems. To obtain a global solution it is natural to consider the formal limit as k → ∞: λ˜ j = (2 − L2 )j (λ˜ 0 ) +
j−1 X
(2 − L2 )j−l−1 r(ξ ˜ l ); j ≥ 1,
l=0
ζj = −
∞ X
L−(l+1−j) s(ξ ˜ l ); j ≥ 0,
l=j
µj = −
∞ X
L−(2+)(l+1−j) t˜(ξl ); j ≥ 0,
l=j
Rj = U (ξj−1 ); j ≥ 1
(237)
When ξj ∈ E(1/2) these sums converge (see below) and one can check that any solution of these equations is a bounded solution of ξj+1 = f (ξj ). Thus we focus on solving this equation. Notice that the equations are now independent of ζf , µf . Let ξ = (ξ0 , ξ1 , ξ2 , . . . ). Then the above equation can be written in the form ξ = F (ξ), where F (ξ) = (F0 (ξ), F1 (ξ), . . . ) and Fj (ξ) = Fjλ (ξ), Fjζ1 (ξ), Fjζ2 (ξ), Fjµ (ξ), FjR (ξ)
(238)
is the right side of the flow equation augmented by the initial data F0λ (ξ) = λ0 , F0R (ξ) = R0 . We regard this as a fixed point equation in the Banach space E of all sequences ξ with norm kξk = supj kξj k . The ball E(1) is all sequences with entries in E(1). Since the wj component of ξj must be the solution of wj+1 = (wj )L + CL starting with w0 , we insert this solution wj into ξ and only consider the first four components ξj = (λ˜ j , ζj , µj , Rj , wj ) to be unknowns in the fixed point equation.
Non-Gaussian Fixed Point
151
Theorem 8. Let 1 < α < 2 − 1 and let the wj components of ξ be fixed as described above. Then 1. If ξ ∈ E(a/2) then F (ξ) ∈ E(a), where 1/8 ≤ a ≤ 1/2. 2. For ξ, ξ 0 in E(1/2) we have kF (ξ) − F (ξ 0 )k ≤ kξ − ξ 0 k/2. 3. For any ξ (0) in E(1/16) the iterates ξ (1) = F (ξ (0) ), ξ (2) = F (ξ (1) ), etc. converge to a limit ξ = limn→∞ ξ (n) in E(1/2) which satisfies ξ = F (ξ) and hence ξj+1 = f (ξj ). Proof. (1) Note that because we have fixed wj , by (157) the bound (233) can be written |r(ξ ˜ j )| ≤ O(1)(log(L)2α + L−j/4 2 + 3−1 ).
(239)
Then we have for j ≥ 1: |Fjλ |
≤ (2 − L ) |λ˜ 0 | + 2 j
j−1 X
(2 − L2 )j−l−1 |r(ξ ˜ l )|
l=0
≤ (2 − L ) (a /2) + 2 j
α
j−1 X
(2 − L2 )j−l−1 O(1)(log(L)2α + L−l/4 2 + 3−1 )
l=0
≤ aα /2 + O(1)(log(L)2α−1 + 2 + 2−1 ) ≤ aα .
(240)
Next we have |Fjζ1 |
≤
∞ X
L−(l+1−j) |s˜1 (ξl )|
l=j
≤
∞ X
L−(l+1−j) C2
l=j
≤ C ≤ a1−δ
(241)
|Fjζ2 | ≤ a2−1−δ .
(242)
and similarly
Next we have |Fjµ | ≤
∞ X
L−(2+)(l+1−j) |t˜(ξl )|
l=j
≤
∞ X
L−(2+)(l+1−j) C2−δ
l=j
≤ C2−δ ≤ a2−2δ .
(243)
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D. Brydges, J. Dimock, T. R. Hurd
Finally we have, for j ≥ 1, |||FjR ||| = |||U (ξj−1 )||| ≤ O(L−1+2 )3−1 ≤ a3−1 .
(244)
(2) We denote δξ = ξ − ξ 0 and δF = F (ξ) − F (ξ 0 ), etc. Since wj is fixed there is no variation in w and −α |δFjλ | ≤
j−1 X
(2 − L2 )j−l−1 −α |δ r(ξ ˜ l )|
l=0
≤ O(1)
j−1 X
(2 − L2 )j−l−1 (log(L)α + L−l/4 2−α + 3−1−α )kδξl k
l=0
≤ O(1)(log(L)α−1 + 2−α + 2−1−α )kδξk ≤ kδξk/2.
(245)
Next we have −1+δ |δFjζ1 | ≤ ≤
∞ X l=j ∞ X
L−(l+1−j) −1+δ |δ s˜1 (ξl )| L−(l+1−j) C1+δ kδξl k
l=j
≤ Cδ kδξk ≤ kδξk/2,
(246)
and similarly −2+1+δ |δFjζ2 | ≤ kδξk/2.
(247)
Next we have −2+2δ |δFjµ | ≤ ≤
∞ X l=j ∞ X
L−(2+)(l+1−j) −2+2δ |δ t˜(ξl )| L−(2+)(l+1−j) Cδ kδξk
l=j
≤ kδξk/2,
(248)
and by (235) −3+1 |||δFjR ||| = −3+1 |||δU (ξj−1 )||| ≤ O(1)L−1+2 kδξk ≤
1 kδξk. 2
(249)
Combining all these gives kδF k ≤ 1/2kδξk (3) Use the previous parts to estimate (the tails of) ξ (0) + standard way.
P
j>0 (ξ
(j)
− ξ (j−1) ) in the
Non-Gaussian Fixed Point
153
Remark. Thus we have shown that for each (λ0 , R0 , w0 ) in E(1/16) there exists (ζ0 , µ0 ) (given by (237)) such that the trajectory ξj starting at that ξ0 = (λ0 , ζ0 , µ0 , R0 , w0 ) stays in E(1/2) for all j. In the next section we will see that (µ0 , ζ0 ) is unique. 5.3. The stable manifold and the fixed point. The existence results of the previous section refer to the Banach space E for which the norm kξk is given by (231) with weight factor ˜ for fixed 1 < α < 2 − 1. For uniqueness we consider the Banach space F with −α |λ| ˜ Since E( 1 ) ⊂ F ( 1 ) norm kξk still given by (231) but now with weight factor (d)−1 |λ|. 2 2 uniqueness in F (1/2) implies the solution in the last section is unique. We now return to regarding w as a dynamical variable so that the problem is again to study the iterations of the mapping f . The Banach space F can be regarded as a product F1 × F2 , where F1 is all triples ˜ R, w) and F2 is all pairs (ζ, µ). Then F1 corresponds to contracting data and F2 (λ, corresponds to expanding data. Let pi be the projection onto Fi , let ξi = pi ξ, and let fi = pi ◦ f . We follow the analysis of Shub, Irwin ([Shu87], p. 51). Lemma 16. kf1 (ξ) − f1 (ξ 0 )k ≤ (1 − )kξ − ξ 0 k. Suppose kξ2 − ξ20 k > kξ1 − ξ10 k so that kξ − ξ 0 k = kξ2 − ξ20 k. Then kf2 (ξ) − f2 (ξ 0 )k ≥ (1 + )kξ − ξ 0 k. Proof. We estimate ˜ U, wL + CL . f1 = (fλ , fR , fw ) = (2 − L2 )λ˜ + r,
(250)
We have, from (234), recalling that k = O(1),
(d)−1 |fλ (ξ) − fλ (ξ 0 )| ≤ (2 − L2 ) + (O(log L)d + O(1)d−1 )) kξ − ξ 0 k ≤ 1 + (−2 log L + O(1)d log L + O(1)d−1 ) kξ − ξ 0 k ≤ (1 − )kξ − ξ 0 k,
(251)
where we first choose d small enough so that −2 + O(1)d < 0 and then choose L large enough for that (−2 + O(1)d) log L + O(1)d−1 < −1. We also have −3+1 |||fR (ξ) − fR (ξ 0 )||| ≤ O(L−1+2 )kξ − ξ 0 k ≤ (1 − )kξ − ξ 0 k
(252)
and 0 k k −1 kfw (ξ) − fw (ξ 0 )k = k −1 kwL − wL
≤ L−1/4 kξ − ξ 0 k ≤ (1 − )kξ − ξ 0 k.
(253)
This completes the proof of the first bound. For the second bound we estimate ˜ L2+ µ + t˜). f2 = (fζ , fµ ) = (L ζ + s,
(254)
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Note that one of −1+δ |ζ1 − ζ10 | and −2+1+δ |ζ2 − ζ20 | and −2+2δ |µ − µ0 | must be equal to kξ2 − ξ20 k = kξ − ξ 0 k. In the first case we have kf2 (ξ) − f2 (ξ 0 )k ≥ −1+δ |fζ1 (ξ) − fζ1 (ξ 0 )| ≥ L −1+δ |ζ1 − ζ10 | − −1+δ ks˜1 (ξ) − s˜1 (ξ 0 )k ≥ (L − C1+δ )kξ − ξ 0 k ≥ (1 + )kξ − ξ 0 k.
(255)
The second case is similar, and in the last case we have kf2 (ξ) − f2 (ξ 0 )k ≥ −2+2δ |fµ (ξ) − fµ (ξ 0 )| ≥ L2+ −2+2δ |µ − µ0 | − −2+2δ kt˜(ξ) − t˜(ξ 0 )k ≥ (L2+ − Cδ )kξ − ξ 0 k ≥ (1 + )kξ − ξ 0 k.
(256)
Now we can state our main result. Consider the RG transformation f on F and we define the stable manifold to be M = {ξ ∈ E(1/16) : ξj = f j (ξ) ∈ F (1/2) for all j = 1, 2, . . . },
(257)
where f j is the j-fold composition of f with itself. Theorem 9. The stable manifold M is the graph of a function ξ2 = h(ξ1 ) which is Lipschitz continuous with Lipschitz constant 1. The RG mapping f is a contraction on M and thus has a unique fixed point ξ ∗ = (ξ1∗ , ξ2∗ ), where ξ2∗ = h(ξ1∗ ). Proof. Suppose ξ, ξ 0 are two points on M. We claim that kξ2 − ξ20 k ≤ kξ1 − ξ10 k.
(258)
If not then kξ2 − ξ20 k > kξ1 − ξ10 k and the previous lemma is applicable. It follows that kf2 (ξ) − f2 (ξ 0 )k ≥ (1 + )kξ − ξ 0 k > (1 − )kξ − ξ 0 k ≥ kf1 (ξ) − f1 (ξ 0 )k,
(259)
whence kf (ξ) − f (ξ 0 )k ≥ (1 + )kξ − ξ 0 k. We can replace ξ and ξ 0 by f (ξ), f (ξ 0 ) in this argument and obtain kf 2 (ξ) − f 2 (ξ 0 )k ≥ (1 + )2 kξ − ξ 0 k. Continuing in this fashion we get kf n (ξ) − f n (ξ 0 )k ≥ (1 + )n kξ − ξ 0 k, which gives a contradiction as n → ∞. Because of the inequality we have that ξ1 = ξ10 implies that ξ2 = ξ20 . Thus there is at most one ξ2 for each ξ1 on M . Recall from the remark at the end of Sect. 5 that we already know that there is at least one ξ2 and hence there is exactly one ξ2 . Thus M is the graph of a function ξ2 = h(ξ1 ) and the inequality says that k(h(ξ1 ) − h(ξ10 )k ≤ kξ1 − ξ10 k. On M we have kξ − ξ 0 k = kξ1 − ξ10 k and hence kf (ξ) − f (ξ 0 )k = kf1 (ξ) − f1 (ξ 0 )k. From Lemma 16 kf1 (ξ) − f1 (ξ 0 )k ≤ (1 − )kξ − ξ 0 k, so f is a contraction on M .
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Remarks. To express M in terms of the original variables, write h as a pair of functions ζ0 = hζ (λ˜ 0 , R0 , w0 ), µ0 = hµ (λ˜ 0 , R0 , w0 ).
(260)
The case of interest is R0 = 0, w0 = 0 which gives ourfunctions µ0 = µ0 (λ0 ), ζ0 = ζ0 (λ0 ). When R0 = 0, w0 = 0 the point ζ0,1 (λ0 ), ζ0,2 (λ0 ) describes the same density as the point (ζ0,1 (λ0 ) + ζ0,2 (λ0 ), 0) so we can get on the stable manifold with a potential of the form (8). Note that the fixed point has the form ξ ∗ = (λ˜ ∗ , ζ ∗ , µ∗ , R∗ , w∗ ), where ζ ∗ = hζ (λ∗ , R∗ , w∗ ), µ∗ = hµ (λ∗ , R∗ , w∗ ). Acknowledgement. David Brydges thanks the Australian National University for its hospitality while part of this work was in progress.
References [BDH94a]
Brydges, D., Dimock, J. and Hurd, T.R.: Applications of the renormalization group. In: J. Feldman, R. Froese, and L. Rosen, eds, Mathematical Quantum Theory I: Field Theory and Many-body Theory. Providence, RI.: AMS, 1994 [BDH94b] Brydges, D., Dimock, J. and Hurd, T.R.: Weak perturbations of Gaussian measures. In: J. Feldman, R. Froese, and L. Rosen, eds, Mathematical Quantum Theory I: Field Theory and Many-body Theory. Providence, RI.: AMS, 1994 [BDH95] Brydges, D., Dimock, J. and Hurd, T.R.: The short distance behavior of φ43 . Commun. Math. Phys. 172, 143–186 (1995) [BDH96] Brydges, D., Dimock, J. and Hurd, T.R.: Estimates on renormalization group transformations. Preprint, 1996 [BY90] Brydges, D. and Yau, H.-T.: Grad ϕ perturbations of massless Gaussian fields. Commun. Math. Phys. 129, 351–392 (1990) [DH91] Dimock, J. and Hurd, T.R.: A renormalization group analysis of the Kosterlitz-Thouless phase. Commun. Math. Phys. 137, 263–287 (1991) [DH93] Dimock, J. and Hurd, T.R.: Construction of the two-dimensional sine-Gordon model for β < 8π. Commun. Math. Phys. 156, 547–580 (1993) [Fel85] Felder, G.: Construction of a nontrivial planar field theory with ultraviolet stable fixed point. Commun. Math. Phys. 102, 139–155 (1985) [Fel87] Felder, G.: Renormalization group in the local potential approximation. Commun. Math. Phys. 111, 101–121 (1987) [FMRS87] Feldman, J., Magnen, J., Rivasseau, V. and S´en´eor, R.: Construction and Borel summability of infrared φ44 by a phase space expansion. Commun. Math. Phys. 109, 437–480 (1987) [GK83] Gawe¸dzki, K. and Kupiainen, A.: Non-gaussian fixed points of the block spin transformation. hierarchical model approximation. Commun. Math. Phys. 89, 191–220 (1983) [GK85] Gawe¸dzki, K. and Kupiainen, A.: Massless lattice φ44 theory: Rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys. 99, 197–252 (1985) [GK86] Gawe¸dzki, K. and Kupiainen, K.: Asymptotic freedom beyond perturbation theory. In: K. Osterwalder and R. Stora, eds, Critical Phenomena, Random Systems, Gauge Theories. Amsterdam: North-Holland, Les Houches, 1984, 1986 [GMLMS71] Gallavotti, G., Martin-L¨of, A. and Miracle-Sol´e, S.: Some problems connected with the description of the coexistence of phases at low temperature in the Ising model. In: A. Lenard, ed, Statistical Mechanics and Mathematical Problems. Lecture Notes in Physics, Vol 20, Batelle Seattle Rencontres, Berlin–Heidelberg–New York: Springer-Verlag, 1971 [KW86] Koch, H. and Wittwer, P.: A non-gaussian renormalization group fixed point for hierarchical scalar lattice field theories. Commun. Math. Phys. 106, 495–532 (1986) [KW91] Koch, H. and Wittwer, P.: On the renormalization group transformation for scalar hierachical models. Commun. Math. Phys. 138, 537 (1991)
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Mack, G. and Pordt, A.: Convergent perturbation expansions for euclidean quantum field theory. Commun. Math. Phys. 97, 267 (1985) Mack, G. and Pordt, A.: Convergent weak coupling expansions for lattice field theories that look like perturbation series. Rev. Math. Phys., 1, 47 (1989) Ruelle, D.: Statistical Mechanics: Rigorous Results. New York: Benjamin, 1969 Shub, M.: Global stability of dynamical systems. New York: Springer-Verlag, 1987 Wilson, K. and Fisher, M.: Critical exponents in 3.99 dimensions. Phys. Rev. Lett. 28, 240–243 (1972) Wilson, K.G. and Kogut, J.: The renormalization group and the ε-expansion. Phys. Rep. 12, 75–200 (1974)
Communicated by A. Jaffe
Commun. Math. Phys. 198, 157 – 186 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Tensor Product Representations of the Quantum Double of a Compact Group T. H. Koornwinder1 , F. A. Bais2 , N. M. Muller2 1 KdV Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. E-mail:
[email protected] 2 Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands. E-mail:
[email protected];
[email protected]
Received: 21 January 1998 / Accepted: 31 March 1998
Abstract: We consider the quantum double D(G) of a compact group G, following an earlier paper. We use the explicit comultiplication on D(G) in order to build tensor products of irreducible ∗-representations. Then we study their behaviour under the action of the R-matrix, and their decomposition into irreducible ∗-representations. The example of D(SU (2)) is treated in detail, with explicit formulas for direct integral decomposition (“Clebsch–Gordan series”) and Clebsch-Gordan coefficients. We point out possible physical applications.
1. Introduction Over the last decade quantum groups have become an important subject of research both in mathematics and physics, see a.o. the monographs [8, 14, 15] and [17]. Of special importance are those quantum groups which are quasi-triangular Hopf algebras, and thus have a universal R-element satisfying the quantum Yang–Baxter equation. Via the QYBE there is a connection with the braid group and thus with the theory of invariants of links and 3-manifolds. In the physical context quantum groups play an important role in the theory of integrable lattice models, conformal field theory (Wess–Zumino–Witten models for example) and topological field theory (Chern–Simons theory). Drinfel’d [10] has introduced the notion of the quantum double D(A) of a Hopf algebra A. His definition (rigorous if A is finite dimensional, and formal otherwise) yields a quasi-triangular Hopf algebra D(A) containing A as a Hopf subalgebra. For A infinite dimensional, various rigorous definitions for the quantum double or its dual have been proposed, see in particular Majid [17] and Podl´es and Woronowicz [20]. An important mathematical application of the Drinfel’d double is a rather simple construction of the “ordinary” quasi-triangular quantum groups (i.e. q-deformations of universal enveloping algebras of semisimple Lie algebras and of algebras of functions on the corresponding groups), see for example [8] and [17].
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In physics the quantum double has shown up in various places: in integrable field theories [6], in algebraic quantum field theory [18], and in lattice quantum field theories. For a short summary of these applications, see [12]. Another interesting application lies in orbifold models of rational conformal field theory, where the physical sectors in the theory correspond to irreducible unitary representations of the quantum double of a finite group. This has been constructed by Dijkgraaf, Pasquier and Roche in [9]. Directly related to the latter are the models of topological interactions between defects in spontaneously broken gauge theories in 2+1 dimensions. In [2] Bais, Van Driel and De Wild Propitius show that the non-trivial fusion and braiding properties of the excited states in broken gauge theories can be fully described by the representation theory of the quantum double of a finite group. For a detailed treatment see [23]. Both from a mathematical and a physical point of view it is interesting to consider the quantum double D(G) of the Hopf ∗-algebra of functions on a (locally) compact group G, and to study its representation theory. For G a finite group, D(G) can be realized as the linear space of all complex-valued functions on G × G. Its Hopf ∗-algebra structure, which rigorously follows from Drinfel’d’s definition, can be given explicitly. In [16] and in the present paper we take the following approach to D(G) for G (locally) compact: We realize D(G) as a linear space in the form Cc (G × G), the space of complex valued, continuous functions of compact support on G × G. Then the Hopf ∗-algebra operations for G finite can be formally carried over to operations on Cc (G × G) for G non-finite (formally because of the occurrence of Dirac delta’s). Finally it can be shown that these operations formally satisfy the axioms of a Hopf ∗-algebra. In [16], we focussed on the ∗-algebra structure of D(G), and we derived a classification of the irreducible ∗-representations (unitary representations). In the present paper, where we restrict ourselves to the case where G is compact, we address questions about “braiding” and “fusion” properties of tensor product representations of D(G), for which the comultiplication and the R-matrix are explicitly needed. We envisage physical applications in nontrivial topological theories such as (2+1)-dimensional quantum gravity, and higher dimensional models containing solitons [4]. In view of these and other applications we present our results on representation theory not just abstractly, but quite explicitly. The outline of the paper is as follows. In Sect. 2 we specify the Hopf ∗-algebra structure of D(G). We then turn to the irreducible unitary representations in Sect. 3, where we first recall a main result of [16], concerning the classification of these representations. We give a definition of their characters, and compare the result to the case of finite G. An outstanding feature of quasi-triangular Hopf algebras is that their non-cocommutativity is controlled by the R-element. Together with the explicit expression for the comultiplication this results in interesting properties of tensor products of irreducible ∗-representations of D(G). In Sect. 4 we define such a tensor product representation, and specify the action of the quantum double. In Sect. 5 we give the action of the universal R-matrix on tensor product states (“braiding”) on a formal level. The rather non-trivial Clebsch–Gordan series of irreducible ∗-representations (“fusion rules”) are discussed in Sect. 6. They are calculated indirectly, namely, via direct projection of states, and the comparison of squared norms. This direct projection results in a very general method to construct the Clebsch–Gordan coefficients of a quantum double in case orthogonal bases can be given for the representation spaces. Finally, Sect. 7 treats the example of G = SU (2) in detail.
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2. The Hopf Algebra Structure of D(G) Drinfel’d [10] has given a definition of the quantum double D(A) of a Hopf algebra A. Write Ao for the dual Hopf algebra to A with opposite comultiplication. Then D(A) is a quasi–triangular Hopf algebra, it is equal to A⊗Ao as a linear space, and it contains A⊗1 and 1 ⊗ Ao as Hopf subalgebras. If A is moreover a Hopf ∗-algebra then D(A) naturally becomes a Hopf ∗-algebra. This definition of the quantum double is only rigorous if A is finite dimensional. If G is a compact group and C(G) the Hopf ∗-algebra of continuous complex valued functions on G, then instead of D(C(G)) we will write D(G) for the quantum double of C(G). For G a finite group we have D(G) ' C(G) ⊗ C[G] ' C(G × G)
(2.1)
as linear spaces. Also in the case of a finite group it is possible to write down the formulas for the Hopf ∗-algebra operations and the universal R-element of D(G), both in the formulation with D(G) = C(G) ⊗ C[G] (see [9]) and with D(G) = C(G × G). In the last picture the formulas may typically involve a summation over the group or a Kronecker delta on G. They suggest analogous formulas for G arbitrarily compact, by simply replacing the summation over G by integration w.r.t. the normalised Haar measure on G, and replacing the Kronecker delta by the Dirac delta. This way we obtain the following definitions, where F, F1 , F2 ∈ C(G × G) and x, y, x1 , y1 , x2 , y2 ∈ G : Multiplication: Z F1 (x, z) F2 (z −1 xz, z −1 y) dz. (2.2) (F1 • F2 )(x, y) := G
∗-operation: F ∗ (x, y) = F (y −1 xy, y −1 ).
(2.3)
1(x, y) = δe (y).
(2.4)
(1F )(x1 , y1 ; x2 , y2 ) = F (x1 x2 , y1 ) δe (y1−1 y2 ).
(2.5)
Unit element
Comultiplication:
Counit:
Z F (e, y) dy.
(F ) =
(2.6)
G
Antipode (S(F ))(x, y) = F (y −1 x−1 y, y −1 ).
(2.7)
R(x1 , y1 ; x2 , y2 ) = δe (x1 y2−1 ) δe (y1 ).
(2.8)
Universal R-element:
Note that due to the occurring Dirac delta’s the unit element in fact does not lie inside D(G). Similarly, the comultiplication doesn’t map into D(G) ⊗ D(G) (not even into the topological completion Cl(C(G × G) ⊗ C(G × G)) ' C(G × G × G × G)),
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and furthermore the R-element doesn’t lie inside D(G) ⊗ D(G). In practice this does not pose a serious problem as we will always formally integrate over these Dirac delta’s, nevertheless we still have to be careful in dealing with the resulting expressions, because it can happen that the Dirac delta is partially fulfilled, giving rise to infinities. With the above operations C(G × G) formally becomes a quasi-triangular Hopf ∗algebra called D(G). For the case of a finite group G this holds rigorously, which is clear just by the quantum double construction. However, for the case of general compact G we have to verify that Eqs. (2.2)–(2.8) do indeed satisfy all axioms of a quasi-triangular Hopf ∗-algebra. In [16] it was observed that C(G × G) with Eqs. (2.2) and (2.3) is a ∗-algebra, and furthermore the irreducible unitary representations of this ∗-algebra were studied and classified. In the present paper we will consider tensor products and braiding properties of these irreducible ∗-representations (from now on mostly referred to as “irreps”) by using the comultiplication and the R-element. 3. Irreducible Representations We recapitulate the contents of Corollary 3.10, one of the main results of [16]. Throughout, when we speak of a compact group (or space), we tacitly assume that it is a separable compact Hausdorff group (or space). Definition 3.1. Let G be a compact group, and Conj(G) the collection of conjugacy classes of G (so the elements of Conj(G) are the sets of the form {xgx−1 }x∈G with g ∈ G). For each A ∈ Conj(G) choose some representative gA ∈ A, and let NA bA (the set of equivalence classes of be the centralizer of gA in G. For each α ∈ N irreducible unitary representations of NA ) choose a representative, also denoted by α, which is an irreducible unitary representation of NA on some finite dimensional Hilbert space Vα . Also, let dz be the normalised Haar measure on G. For measurable functions φ : G → Vα such that for all h ∈ NA it holds that φ(gh) = α(h−1 )φ(g) we put
for almost all g ∈ G,
(3.1)
Z kφk2 := G
kφ(z)k2Vα dz.
(3.2)
Now L2α (G, Vα ), which is the linear space of all such φ for which kφk < ∞ divided out by the functions with norm zero, is a Hilbert space. The elements of L2α (G, Vα ) can also be considered as L2 -sections of a homogeneous vector bundle over G/NA . The space L2α (G, Vα ) is familiar as the representation space of the representation of G which is induced by the representation α of NA . bA we have mutually inequivalent irreTheorem 3.2. For A ∈ Conj(G) and α ∈ N of D(G) = C(G × G) on L2α (G, Vα ) given by ducible ∗-representations 5A α Z (F )φ (x) := F (xgA x−1 , z)φ(z −1 x) dz, F ∈ D(G). (3.3) 5A α G
These representations are moreover k.k1 -bounded (see for this notion formula (33) in [16]). All irreducible k.k1 -bounded ∗-representations of D(G) are equivalent to some 5A α.
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In fact, a much more general theorem holds (see Theorem 3.9 in [16]), namely for the representation theory of so-called transformation group algebras C(X × G), where the compact group G acts continuously on the compact set X, instead of the conjugation action of G on G. We may even assume G and X to be locally compact, under the extra condition of countable separability of the G-action. Then we have to consider Cc (X × G) and use a quasi-invariant measure on G/NA . Also, the rest of the Hopf algebra structure of D(G), in particular the comultiplication, will survive for the case of noncompact G as long as G acts on itself by conjugation. It would be interesting to extend the results of this paper to this case of (special) noncompact G. An interesting issue in representation theory is the character of an irrep. For the case of a finite group G such characters have been derived in [9]. For our case, where irreps are generally infinite dimensional, the operator 5A α (F ) will not be trace class for all F ∈ D(G), so we restrict ourselves to the case of a Lie group G and C ∞ -functions on G × G. In this paper we will only state the formula for the characters of irreps of the quantum double. The proof for it, the orthogonality of the characters, and the related subject of harmonic analysis, will be given in a forthcoming paper. Theorem 3.3. Let χα denote the character of the irreducible ∗-representation α of NA . For an irreducible ∗-representation 5A α of the quantum double D(G) the character is given by Z Z (F ) = F (zgA z −1 , znz −1 ) χα (n) dn dz, F ∈ C ∞ (G × G). (3.4) χA α G
NA
Let us check the connection with the case of a finite group G. As discussed in [16] for a finite group G there is a linear bijection D(G) = C(G) ⊗ C[G] ⇐⇒ C(G × G): X
f ⊗ x 7→ ((y, z) 7→ f (y)δx (z)) , F (. , z) ⊗ z ← F.
(3.5)
Taking f = δg as a function on (finite) G, we obtain Z Z (δ ⊗ x) = δg (zgA z −1 )δx (znz −1 )χα (n) dn dz, χA g α
(3.6)
z∈G
G
NA
which indeed coincides with the definition of the character in [9].
4. Tensor Products In Sect. 3 we have recapitulated the classification of the irreducible ∗-representations of the quantum double D(G). With the coalgebra structure of D(G) that we have derived in Sect. 2 we can now consider tensor products of such representations. B Let 5A α and 5β be irreducible ∗-representations of D(G). For the representation space of the tensor product representation we take the Hilbert space of vector-valued functions on G × G as follows: for measurable functions 8 : G × G → Vα ⊗ Vβ such that for all h1 ∈ NA , h2 ∈ NB it holds that −1 8(xh1 , yh2 ) = α(h−1 1 ) ⊗ β(h2 )8(x, y),
for almost all (x, y) ∈ G × G
(4.1)
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we put Z Z k8k2 := G
G
k8(x, y)k2Vα ⊗Vβ dx dy.
(4.2)
Now the space L2α,β (G × G, Vα ⊗ Vβ ) is defined as the linear space of all such 8 for which k8k < ∞, divided out by the functions of norm zero. Note that this space is the completion of the algebraic tensor product of L2α (G, Vα ) and L2β (G, Vβ ). By Eq. (3.3) B the tensor product representation 5A α ⊗ 5β becomes formally: B B 5A 5A α ⊗ 5β (F ) 8 (x, y) := α ⊗ 5β (1F ) 8 (x, y) Z Z 1F (xgA x−1 , z1 ; ygB y −1 , z2 ) 8(z1−1 x, z2−1 y) dz1 dz2 . = G
G
Then it follows by substitution of Eq. (2.5) and by formally integrating the Dirac delta function that Z A B F (xgA x−1 ygB y −1 , z) 8(z −1 x, z −1 y) dz. (4.3) 5α ⊗ 5β (F ) 8 (x, y) = G
It is easy to see that this is indeed a representation of D(G): there is the covariance property, as given in Eq. (4.1), and the homomorphism property can be readily checked. The functions of the form B 8(x, y) = φA α (x) ⊗ φβ (y) ∈ Vα ⊗ Vβ
(4.4)
A B B (with φA α and φβ basis functions of the representation spaces for 5α and 5β respectively) span a dense subspace of L2α,β (G × G, Vα ⊗ Vβ ). The positive–definite inner product then reads Z Z A B B hφ1 A (4.5) h81 , 82 i := α (x), φ2 α (x)iVα hφ1 β (y), φ2 β (y)iVβ dx dy. G
G
This tensor product representation now enables us to further analyse two important operations which are characteristic for quasi-triangular Hopf algebras, namely “braiding” and “fusion”. They will turn up in several applications of these algebras [3].
5. Braiding of Two Representations Let us investigate the action of the universal R-element in the aforementioned tensor product representation. A simple formal calculation with use of Eqs. (2.8) and (3.3) yields Z Z B ⊗ 5 δe (xgA x−1 z −1 ) δe (w) 8(w−1 x, z −1 y) dw dz (R) 8 (x, y) = 5A α β G
G
−1 −1 x y). = 8(x, xgA
(5.1)
B The braid operator R is an intertwining mapping between 5A α ⊗ 5β on Vα ⊗ Vβ and A 5B β ⊗ 5α on Vβ ⊗ Vα given by
Tensor Product Representations of Quantum Double of Compact Group
A B RAB αβ 8 := σL ◦ 5α ⊗ 5β (R) 8,
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(5.2)
where (σL 8)(x, y) := σ (8(y, x)) ,
σ(v ⊗ w) := w ⊗ v, v ∈ Vα , w ∈ Vβ ,
(5.3)
B so it interchanges the representations 5A α and 5β . Hence −1 −1 B RAB y x) . (5.4) 5A = σ 8(y, ygA αβ 8 (x, y) = σL α ⊗ 5β (R)8(x, y)
To make sure that Eq. (5.4), being derived from a formally defined R-element Eq. (2.8), yields the desired intertwining property for RAB αβ , one can derive this property directly from Eqs. (5.4) and (4.3). Then we must show that A B A AB (x, y). (5.5) 5B RAB αβ 5α ⊗ 5β (F ) 8 (x, y) = β ⊗ 5α (F ) Rαβ 8 The right-hand side of this equation gives Z −1 , F (xgB x−1 ygA y −1 , z) RAB x, z −1 y) dz = αβ 8 (z G Z −1 −1 F (xgB x−1 ygA y −1 , z) σ 8(z −1 y, z −1 ygA y x) dz,
(5.6)
G
which is obviously equal to the left-hand side of Eq. (5.5), using Eq. (5.4) and Eq. (4.3). 6. Tensor Product Decomposition Another general question is the decomposition of the tensor product of two irreducible representations into irreducible representations: M ABγ B NαβC 5C (6.1) 5A α ⊗ 5β ' γ, C,γ
where we suppose that such a tensor product is always reducible. For finite G tensor products of irreps of D(G) indeed decompose into a direct sum over single irreps. For compact G the direct sum over the conjugacy class label C has to be replaced by a direct integral, M Z ⊕ ABγ B ⊗ 5 ' NαβC 5C (6.2) 5A α β γ dµ(C), γ
where µ denotes an equivalence class of measures on the set of conjugacy classes, but the multiplicities must be the same for different measures in the same class, see for instance the last Conclusion in [1] for generalities about direct integrals. Recall that two (Borel) measures µ and ν are equivalent iff they have the same sets of measure zero [1]. By the Radon–Nikodym theorem, µ and ν are equivalent iff µ = f1 ν, ν = f2 µ for certain measurable functions f1 , f2 ≥ 0. If one considers specific states and/or their norms (so elements of specific Hilbert spaces), it is required to make a specific choice for the measure. But if one only compares equivalence classes of irreps, like we do in the Clebsch–Gordan series in Eq. (6.2), the exact measure on Conj(G) is not of importance, only its equivalence class.
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Our aim is to determine the measure µ (up to equivalence) and the multiplicities ABγ of this Clebsch–Gordan series for D(G). In physics these NαβC are often referred to as “fusion rules”, as for example in [9] for the case of G a finite group. In ordinary group theory the multiplicities can be determined using the characters of representations. Recall that for a continuous group H with irreducible representations π a , π b , π c , ... and characters χa , χb , χc the number of times that π c occurs in the π a ⊗π b is given by ABγ NαβC
Z ncab
χc (h)χa (h)χb (h) dh.
=
(6.3)
h∈H
Thus a direct computation of the multiplicities requires an integration over the group. For the quantum double this approach is not very attractive, and we have to take an alternative route. Furthermore, the direct decomposition of the character of a tensor product of irreps into a direct sum/integral over characters of single irreps is problematic, since the tensor product character is not trace class, while the single characters are. The rigorous approach we will take is to look at the decomposition in more detail, B in the sense that we consider the projection of a state in 5A α ⊗ 5β onto states in the irreducible components 5C γ . Subsequently we compare the squared norm of the tensor product state with the direct sum/integral of squared norms of the projected ABγ . The projection states. This will lead to an implicit equation for the multiplicities NαβC involves the construction of intertwining operators from the tensor product Hilbert space to Hilbert spaces of irreducible representations. This construction is described in the next subsection, and the intertwining operators are given in Theorem 6.10. If orthonormal bases are given for the Hilbert spaces of irreducible representations this means we can derive the Clebsch–Gordan coefficients for the quantum double. In Sect. 7 we will work this out explicitly for the case G = SU (2). Since the proof of Theorem 6.10 is quite lengthy, in the following paragraph we first give a brief outline of the procedure we will follow. To prove isometry between the Hilbert space of a tensor product representation and a direct sum of Hilbert spaces of irreducible representations we must construct an intertwining mapping ρ from the first space, whose elements are functions of two variables with a certain covariance property, to the second space (= direct sum of spaces), whose elements are functions of one variable with a similar covariance property. From Eq. (4.3) one can see that the conjugacy class label C of the representation to which 8 must be mapped depends on the “relative difference” ξ between the entries (y1 , y2 ) of 8. This ξ is the variable that remains if (y1 , y2 ) and (zy1 n1 , zy2 n2 ) are identified for all z ∈ G and for all n1 ∈ NA and n2 ∈ NB . So ξ is an element of the double coset GAB = NA \G/NB we have introduced before. Equation (6.7) in Proposition 6.3 shows how C depends on ξ. In Proposition 6.4 we give a map F1 which constructs a function φ on G out of a function 8 on G × G. The action of D(G) on φ depends on the possible “relative differences” ξ between the entries of 8, which is why we say that φ also depends on ξ. Therefore we introduce the function spaces of Eqs. (6.12) and (6.13). Lemma 6.5 shows that the squared norm of 8 equals the direct integral over ξ of the squared norm of φ, and thus that the map F1 is an isometry of Hilbert spaces. One can also think of ξ as a label on φ, which distinguishes its behaviour under the action of D(G), which is in fact shown by Lemma 6.6. These two lemmas together provide the
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R ⊕ C(ξ) B map 5A 5ω dµ(ξ) from the tensor product representation to a direct α ⊗ 5β → integral over “single” (not yet irreducible) representations. into irreducible repSubsequently we must decompose these representations 5C(ξ) ω resentation 5C . Comparing the covariance properties before and after ρ we find the γ restriction on the set γ may be chosen from, which is given in Eq. (6.36). Equation (6.49) gives the isometry of a Hilbert space from the direct integral of Hilbert spaces we constructed before (via the map F1 ) into the direct sum of Hilbert spaces of irreducible representations 5C γ. The combination of these two steps in the tensor product decomposition is summarised in Theorem 6.10. Finally we compare the squared norms before and after the mapping ρ, and arrive at Eq. (6.62), which gives us an implicit formula for the multiplicities. The degeneracy of the irreducible representation 5C γ depends on two things: firstly, the possible non-injectivity of the map ξ 7→ C, which is taken into account by the integration over NA \G/NB with measure dpC (ξ). And secondly by the dimension dγ of Vγ . We now turn to the explicit proof. To start with, fix the conjugacy classes A and B, and also the irreducible unitary representations α ∈ Nˆ A and β ∈ Nˆ B with representation spaces Vα and Vβ of finite dimensions dα = dim Vα and dβ = dim Vβ respectively. The set Conj(G) of conjugacy classes of G forms a partitioning of G. Therefore it can be equipped with the quotient topology, which is again compact Hausdorff and separable. In Definition 3.1 we had already chosen some representative gA ∈ A for each A ∈ Conj(G). We will need the following assumption about this choice: Assumption 6.1. The representatives gA ∈ A can be chosen such that the map A 7→ gA : Conj(G) → G is continuous. In fact, we will make this particular choice. The assumption means that the map from G to G, which assigns to each g ∈ G the representative in its conjugacy class, is continuous. For G a compact connected Lie group we can make a choice of representatives gA in agreement with Assumption 6.1 as follows. Let T be a maximal torus in G, let Tr be the set of regular elements of T (i.e. those elements for which the centraliser equals T ), let K be a connected component of Tr , and let K be the closure of K in T . Take gA to be the unique element in the intersection of the conjugacy class A with K. See for instance reference [7]. Define GAB := NA \G/NB
(6.4)
to be the collection of double cosets of the form NA yNB , y ∈ G. Then GAB also forms a partitioning of G which can be equipped with the quotient topology from the action of G (compact Hausdorff and separable). Now also choose for each ξ ∈ GAB some representative y(ξ) ∈ ξ. We will need the following assumption for this choice of representative: Assumption 6.2. The representatives y(ξ) ∈ ξ can be (and will be) chosen such that the map ξ 7→ y(ξ) : GAB → G is continuous. In other words, the map from G to G which assigns to each g ∈ G the representative in the double coset NA gNB is continuous. For SU (2) a choice of representatives y(ξ) in agreement with Assumption 6.2 will be given in Sect. 7.3. For ξ ∈ GAB define the conjugacy class C(ξ) ∈ Conj(G) by gA y(ξ)gB y(ξ)−1 ∈ C(ξ).
(6.5)
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Then the map λAB : ξ 7→ C(ξ) : GAB → Conj(G)
(6.6)
is continuous. Note that the image of λAB depends on the values of A and B, but that GAB only depends on NA and NB , so not on the precise values of the conjugacy class labels. Proposition 6.3. (a) We can choose a Borel map ξ 7→ w(ξ) : GAB → G such that gA y(ξ)gB y(ξ)−1 = w(ξ)gC(ξ) w(ξ)−1 .
(6.7)
(b) We can choose a Borel map x 7→ (n1 (x), n2 (x)) : G → NA × NB such that x = n1 (x)y(NA xNB )n2 (x)−1 .
(6.8)
(w, C) 7→ wgC w−1 : G × Conj(G) → G
(6.9)
Proof. (a) The map
is continuous (by Assumption 6.1) and surjective. By Corollary A.3 there exists a Borel map x 7→ (wx , Cx ) : G → G × Conj(G) such that x = wx gCx wx−1 . Now take x = gA y(ξ)gB y(ξ)−1 for ξ ∈ GAB , then from Eq. (6.5) it follows that Cx = C(ξ). The map ξ 7→ x is continuous by Assumption 6.2, the map x 7→ wx is Borel. Put w(ξ) := wx , then ξ 7→ w(ξ) is Borel, and Eq. (6.7) is satisfied. (b) The map : NA × NB × GAB → G (n1 , n2 , ξ) 7→ n1 y(ξ)n−1 2
(6.10)
is continuous (by Assumption 6.2) and surjective. By Corollary A.3 there exists a Borel map x 7→ (n1 (x), n2 (x), ξ(x)) : G → NA × NB × GAB such that x = n1 (x)y(ξ(x))n2 (x)−1 . Then ξ(x) = NA xNB , and thus Eq. (6.8) is satisfied. Let Z be the center of G, then Z ⊂ NA and Z ⊂ NB . By Schur’s lemma α(z) and β(z) will be a scalar for z ∈ Z. Define the character ω of Z by α(z) ⊗ β(z) =: ω(z)idVα ⊗Vβ ,
z ∈ Z.
(6.11)
With this character we now define the linear spaces −1 Funα,β (G × G, Vα ⊗ Vβ ) := {8 : G × G → Vα ⊗ Vβ | 8(un−1 1 , vn2 ) = α(n1 ) ⊗ β(n2 ) 8(u, v) ∀n1 ∈ NA , n2 ∈ NB , u, v ∈ G}, (6.12) Funω (G×GAB , Vα ⊗ Vβ ) := {φ : G×GAB → Vα ⊗ Vβ | φ(xz −1 , ξ) = ω(z)φ(x, ξ), for z ∈ Z}. (6.13)
We will also need the following sets: Go := {x ∈ G | if n1 ∈ NA , n2 ∈ NB and n1 xn−1 2 = x then n1 = n2 ∈ Z}, (6.14) (G × G)o := {(u, v) ∈ G × G | u−1 v ∈ Go }, (GAB )o := {ξ ∈ GAB | y(ξ) ∈ Go }.
(6.15) (6.16)
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They have the following properties, which can be easily verified: (a) If x ∈ Go , n1 ∈ NA , n2 ∈ NB then n1 xn−1 2 ∈ Go . (b) x ∈ Go ⇔ y(NA xNB ) ∈ Go . (c) If x ∈ Go , and m1 , n1 ∈ NA , m2 , n2 ∈ NB then −1 m1 xm−1 2 = n1 xn2 ⇒ ∃z ∈ Z such that m1 = n1 z, m2 = n2 z.
(6.17)
(d) If ξ ∈ (GAB )o then ∃z ∈ Z such that n1 (y(ξ)] = n2 (y(ξ)] = z. (e) If (u, v) ∈ (G × G)o , and m1 ∈ NA , m2 ∈ NB then ∃z ∈ Z such that −1 −1 −1 −1 n1 (m1 u−1 vm−1 2 ) = zm1 n1 (u v), n2 (m1 u vm2 ) = zm2 n2 (u v). (6.18)
The next proposition is the first step in the tensor product decomposition. Roughly speaking, we will consider the functions 8 in Eq. (6.12) as elements of the tensor product representation space. After restriction to (G×G)o these functions 8 can be rewritten in a bijective linear way as functions φ in Eq. (6.13), restricted to G × (GAB )o . The action of D(G) on 8 affects both arguments of 8 (according to Eq. (4.3)), but the corresponding action on φ only affects its first argument, as we will see in Lemma 6.6. The second argument will in fact be directly related to the conjugacy class part of the label (C(ξ), γ) of a “new” irreducible representation of D(G), and thus we will prove that the tensor product representation space is isomorphic to a direct integral of representation spaces C(ξ) is not yet irreducible. of 5C(ξ) ω , where 5ω Proposition 6.4. There is a linear map F1 : 8 7→ φ : Funα,β (G × G, Vα ⊗ Vβ ) → Funω (G × GAB , Vα ⊗ Vβ ) (6.19) given by φ(x, ξ) := 8(xw(ξ)−1 , xw(ξ)−1 y(ξ)),
x ∈ G, ξ ∈ GAB .
(6.20)
This map, when considered as a map 8 7→ φ : Funα,β ((G × G)o , Vα ⊗ Vβ ) → Funω (G × (GAB )o , Vα ⊗ Vβ ), (6.21) is a linear bijection with inversion formula F2 : φ 7→ 8 given by 8(u, v) = α(n1 (u−1 v)) ⊗ β(n2 (u−1 v)) φ(un1 (u−1 v)w(NA u−1 vNB ), NA u−1 vNB ). (6.22) Proof. (i) Let φ be defined in terms of 8 ∈ Funαβ (G × G, Vα ⊗ Vβ ) by Eq. (6.20). The covariance condition of φ w.r.t. Z follows because, for z ∈ Z, φ(xz −1 , ξ) = 8(xz −1 w(ξ)−1 , xz −1 w(ξ)−1 y(ξ)) = 8(xw(ξ)−1 z −1 , xw(ξ)−1 y(ξ)z −1 ) = = α(z) ⊗ β(z) 8(xw(ξ)−1 , xw(ξ)−1 y(ξ)) = ω(z)φ(x, ξ). Moreover, φ restricted to G × (GAB )o only involves 8 restricted to (G × G)o , since for ξ ∈ (GAB )o we have that (xw(ξ)−1 , xw(ξ)−1 y(ξ)) ∈ (G × G)o .
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(ii) F1 is injective because ((F2 ◦ F1 )8) (u, v) = = α(n1 (u−1 v)) ⊗ β(n2 (u−1 v)) 8(un1 (u−1 v), un1 (u−1 v)y(NA u−1 vNB )) = = 8(un1 (u−1 v)n1 (u−1 v)−1 , un1 (u−1 v)y(NA u−1 vNB )n2 (u−1 v)−1 ) = = 8(u, uu−1 v) = 8(u, v) and thus F2 ◦ F1 =id. (Here it is not yet necessary to restrict (u, v) to (G × G)o .) (iii) Let 8 be defined in terms of φ ∈ Funω (G × (GAB )o ; Vα ⊗ Vβ ) by Eq. (6.22). The covariance condition of 8 w.r.t. NA × NB follows because, for m1 ∈ NA , m2 ∈ NB and (u, v) ∈ (G × G)o , −1 −1 −1 −1 −1 8(um−1 1 , vm2 ) = α(n1 (m1 u vm2 )) ⊗ β(n2 (m1 u vm2 )) −1 −1 −1 −1 φ(um−1 1 n1 (m1 u vm2 )w(NA u vNB ), NA u vNB )
= (α(z) ⊗ β(z))(α(m1 ) ⊗ β(m2 ))(α(n1 (u−1 v)) ⊗ β(n2 (u−1 v))) φ(uzn1 (u−1 v)w(NA u−1 vNB ), NA u−1 vNB ) = = (α(m1 ) ⊗ β(m2 ))(α(n1 (u−1 v)) ⊗ β(n2 (u−1 v))) φ(un1 (u−1 v)w(NA u−1 vNB ), NA u−1 vNB ) = α(m1 ) ⊗ β(m2 ) 8(u, v) for some z ∈ Z, where we have used property (e) from above. (iv) F1 is surjective (or: F2 is injective) because for (x, ξ) ∈ G × (GAB )o ((F1 ◦ F2 )φ) (x, ξ) = α(n1 (y(ξ))) ⊗ β(n2 (y(ξ))) φ(xw(ξ)−1 n1 (y(ξ))w(ξ), ξ) = = α(z) ⊗ β(z) φ(xz, ξ) = φ(x, ξ) for some z ∈ Z, where we have used property (d) from above. This concludes the proof. Define a Borel measure µ such that Z Z f (NA yNB ) dy = G
f (ξ) dµ(ξ)
(6.23)
GAB
for all f ∈ C(GAB ). The measure µ has support GAB . We will now specialise the map F1 from Eq. (6.19) to the L2 -case, F1 : 8 7→ φ : L2α,β (G × G : Vα ⊗ Vβ ) → L2ω (G × GAB , Vα ⊗ Vβ ). (6.24) Here the first L2 -space is defined as the representation space of a tensor product representation, see Eqs. (4.1) and (4.2), and the second L2 -space is defined as the set of all measurable φ : G×GAB → Vα ⊗Vβ satisfying, for all z ∈ Z that φ(xz −1 , ξ) = ω(z)φ(x, ξ) almost everywhere, and such that Z Z kφ(x, ξ)k2 dx dµ(ξ) < ∞, (6.25) kφk2 := ξ∈GAB
x∈G
with almost equal φ’s being identified.
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169
Lemma 6.5. Let 8 ∈ Funα,β (G × G, Vα ⊗ Vβ ) and let φ be given by Eq. (6.20). If 8 : G × G → Vα ⊗ Vβ is moreover Borel measurable then φ : G × GAB → Vα ⊗ Vβ is Borel measurable, and Z Z Z Z kφ(x, ξ)k2 dx dµ(ξ) = k8(u, v)k2 du dv. (6.26) ξ∈GAB
x∈G
G
G
In particular, the map F1 : 8 7→ φ is an isometry of the Hilbert space L2α,β (G×G, Vα ⊗ Vβ ) into (not necessarily onto!) the Hilbert space L2ω (G × GAB , Vα ⊗ Vβ ). Proof. It follows from Eq. (6.20) and Proposition 6.3(a) that φ is Borel measurable if 8 is Borel measurable. The left-hand side of Eq. (6.26) equals Z Z k8(xw(ξ)−1 , xw(ξ)−1 y(ξ))k2 dx dµ(ξ) GAB G Z Z = k8(u, uy(ξ))k2 du dµ(ξ) = G GAB Z Z Z Z = k8(u, uy(NA vNB ))k2 du dv = k8(u, un1 (v)−1 vn2 (v))k2 du dv = G G Z Z G ZG 2 2 k8(u, uv)k du dv = k8(u, v)k du dv. = G
G
G×G
Subsequently we can show how the map F1 transfers the action of D(G) on 8 to an action of D(G) on φ: Lemma 6.6. Let 8 ∈ L2α,β (G × G, Vα ⊗ Vβ ), F ∈ D(G) and B 9 := (5A α ⊗ 5β )(F )8.
(6.27)
Let φ be defined in terms of 8 and ψ in terms of 9 via Eq. (6.20). Then Z F (xgC(ξ) x−1 , w) φ(w−1 x, ξ) dw. ψ(x, ξ) =
(6.28)
G
Proof. B −1 −1 ψ(x, ξ) = (5A α ⊗ 5β )(F )8 (xw(ξ) , xw(ξ) y(ξ)) = Z F (xw(ξ)−1 gA y(ξ)gB y(ξ)−1 w(ξ)x−1 , w) = G
Z
8(w−1 xw(ξ)−1 , w−1 xw(ξ)−1 y(ξ)) dw = F (xgC(ξ) x−1 , w) φ(w−1 x, ξ) dw
=
G 2 For C ∈ Conj(G) define a ∗-representation 5C ω of D(G) on Lω (G, Vα ⊗ Vβ ) as follows: Z 5C (F )φ (x) := F (xgC x−1 , w) φ(w−1 x) dw, (6.29) ω G
F ∈ D(G), φ ∈ L2ω (G, Vα ⊗ Vβ ).
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This has the same structure as the defining formula for the representation 5A α as given in Eq. (3.3), but the covariance condition on the functions φ in Eq. (6.29) is weaker, because it only involves right multiplication of the argument with respect to z ∈ Z. Equation (6.28) can also be formulated as: (6.30) ψ(x, ξ) = 5C(ξ) ω (F )φ(., ξ) (x), which clearly shows that Lemmas 6.5 and 6.6 form the first step in a direct integral B decomposition of the representation 5A α ⊗ 5β into irreducible representations. We will also need the following Assumption 6.7. The complement of Go has measure zero in G. This implies that the complement of (G × G)o has measure zero in G × G, and the complement of (GAB )o has measure zero in GAB . For G = SU (n) or U (n) this assumption will be satisfied if A and B are conjugacy classes for which gA and gB are diagonal matrices with all diagonal elements distinct (so they are regular elements of the maximal torus T consisting of diagonal matrices). Then NA = NB = T , and Go certainly contains all g = (gij ) ∈ G which have only nonzero off-diagonal elements, so for which gij 6= 0 if i 6= j. Clearly, Assumption 6.7 is then satisfied. Corollary 6.8. The “isometry into” of Lemma 6.5 can be narrowed down to an “isometry onto”, namely; The map F1 : 8 7→ φ is an isometry of the Hilbert space L2α,β ((G × G)o ; Vα ⊗ Vβ ) onto the Hilbert space L2ω (G × (GAB )o ; Vα ⊗ Vβ ). B The second step in the decomposition of the tensor product representation 5A α ⊗ 5β is the decomposition of the representation 5C(ξ) into irreducible components 5C ω γ . In other words, to decompose the action of D(G) on L2ω (G × GAB , Vα ⊗ Vβ ) as given by Eq. (6.28) or Eq. (6.30). For the moment suppose that ξ can be fixed in Eq. (6.30). Comparison of Eq. (6.30) and Eq. (6.29) with Eq. (3.3) then shows that essentially we have to decompose L2ω (G) 1 as a direct sum of Hilbert spaces L2γ (G, Vγ ) (possibly with multiplicity) on which D(G) acts by the irreducible representation 5C(ξ) , with γ γ ∈ Nˆ C(ξ) . For φ ∈ L2ω (G), C ∈ Conj(G), γ ∈ Nˆ C , dγ := dim Vγ , and i, j = 1, . . . , dγ put Z (x) := γij (n) φ(xn) dn, x ∈ G, (6.31) φC,γ ij NC
where we have chosen an orthonormal basis of Vγ . By construction, for n ∈ NC we have that φC,γ ij (xn) =
dγ X
γik (n−1 )φC,γ kj (x).
(6.32)
k=1
For each j = 1, . . . , dγ the vector φC,γ ij (x) takes values in Vγ , the label i denoting the component. Thus ∈ L2γ (G, Vγ ). (6.33) φC,γ ij i=1,... ,dγ
1
The fact that the elements of L2ω (G) should map to Vα ⊗ Vβ is not important for this argument.
Tensor Product Representations of Quantum Double of Compact Group
Also, for F ∈ D(G) C,γ 5C ω (F )φ ij
i=1,... ,dγ
171
C,γ = 5C γ (F ) φij
, i=1,... ,dγ
(6.34)
which follows from combining Eqs. (3.3), (6.29) and (6.31). However, not all γ ∈ Nˆ C will occur, because φC,γ ij = 0 if γ|Z 6= ω id. This follows from the observation that Z Z Z C,γ γij (n)φ(xn) dn = γij (nz)φ(xnz) dn dz = φij (x) = N Z NC Z Z C Z −1 −1 = γij (nz)ω(z ) dz φ(xn) dn = γ(z)ω(z ) dz φC,γ ij (x). NC
Z
Z
(6.35) Thus we must take γ to be an element of Nˆ C ω = {γ ∈ Nˆ C | γ|Z = ω id}.
(6.36)
From the Peter-Weyl theorem applied to the function n 7→ φ(xn) with x ∈ G we can derive that Z
X
kφ(x)k2 dx = G
γ∈(Nˆ C )ω
dγ
dγ Z X i,j=1
2 kφC,γ ij (x)k dx.
(6.37)
G
Thus as a continuation of the maps in Proposition 6.4 we have an isometry p C,γ dγ φij G1 : φ 7→ i=1,... ,dγ γ∈ Nˆ ( C )ω ,j=1,... ,dγ of the Hilbert space L2ω (G) into the direct sum of (degenerate) Hilbert spaces M d γ L2γ (G, Vγ ) γ∈(Nˆ C )ω
(6.38)
(6.39)
C which is intertwining between the representations 5C ω and ⊕γ∈(Nˆ C ) dγ 5γ of D(G). ω From the existence of an inversion formula we can see that the map G1 is even an d γ isometry onto. To that aim, fix γ ∈ Nˆ C ω and take ψij i,j=1,... ,dγ ∈ L2γ (G, Vγ ) ,
i.e. ψij ∈ L2 (G) for i, j = 1, . . . , dγ and ψij (xn) =
dγ X
γik (n−1 )ψkj (x),
n ∈ NC .
(6.40)
→ L2ω (G)
(6.41)
k=1
The map d γ
G2γ : ψ 7→ φ : L2γ (G, Vγ )
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is defined by φ(x) := dγ
dγ X
ψkk (x).
(6.42)
k=1
Then indeed φ(xz −1 ) = ω(z)φ(x) with z ∈ Z. Furthermore we have that G1 ◦ G2 = id, since for δ ∈ Nˆ C ω and φ given by Eq. (6.42) we have φC,δ ij (x)
= dγ
dγ Z X k=1
= dγ
δij (n)ψkk (xn) dn NC
dγ Z X k,l=1
δij (n)γlk (n) dn NC
ψlk (x) =
ψij (x), δ = γ 0, δ = 6 γ.
(6.43)
We want to apply the above decomposition of L2ω (G) to our case of L2ω (G × GAB , Vα ⊗ Vβ ). A slight problem occurs since in Eq. (6.30) we had fixed ξ, which For varying ξ we will have varying C(ξ) and hence is not allowed in an L2 -space. varying NC(ξ) and Nˆ C(ξ) ω . In order to keep this under control we make the following Assumption 6.9. Conj(G) splits as a disjoint union of finitely many Borel sets Conjp (G), on each of which NC does not vary with C. For G = SU (n) or U (n) this assumption certainly holds, because we can take the representatives gC = diag(eiθ1 , ..., eiθn ) with θ1 ≤ θ2 ≤ ... ≤ θn < θ1 + 2π. Then NC only depends on the partition of the set {1, ..., n} induced by the equalities or inequalities between the θj ’s. We would like to know whether the assumption holds for general compact connected Lie groups G. Let T be a maximal torus in G. For any conjugacy class A in G take the representative gA uniquely as an element t ∈ K ⊂ T (see after Assumption 6.1). Van den Ban [5] has described the centraliser of t in G. From [5] we conclude that the possible centraliser subgroups form a finite collection. This can be seen as follows. Let gC be the complexified Lie algebra of G, let 6 be the root system of T in gC , and let gα be the root space for α ∈ 6. Let W be the Weyl group of the root system 6, which can also be realized as the quotient group W = normaliser G(T ) /T . Let t ∈ T . Then the centraliser of t in G is completely determined by the two sets (each a finite subset of a given finite set): 6(t) := {α ∈ 6 | Ad(t) X = X for X ∈ gα },
W (t) := {w ∈ W | wtw−1 = t}. (6.44)
This also shows that, for t0 ∈ T , the set {t ∈ T | 6(t) = 6(t0 ), W (t) = W (t0 )} is Borel. Thus Assumption 6.9 is satisfied if G is a compact connected Lie group. Note that the Lie algebra of the centralizer of t in G is determined by 6(t) (see for instance Ch. V, Proposition (2.3) in [7]). For determining the centralizer itself, we need also W (t). This can be seen (cf. [5]) by using the so-called Bruhat decomposition for a suitable complexification GC of G. Put NC = Np if C ∈ Conjp (G) and GAB,p := {ξ ∈ GAB | C(ξ) ∈ Conjp (G)}. Similarly to Eq. (6.31) for any φ ∈ L2ω (G × GAB , Vα ⊗ Vβ ) we define
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Z φp,γ ij (x, ξ) :=
γij (n) φ(xn, ξ) dn, Np
x ∈ G, ξ ∈ GAB,p , γ ∈ Nˆ p
(6.45) , i, j = 1, . . . , dγ ω
with of course the same right covariance as Eq. (6.32). Because φ now maps to Vα ⊗ Vβ we can say that 2 (6.46) φp,γ ij i=1,... ,d ∈ Lγ (G × GAB,p , Vα ⊗ Vβ ⊗ Vγ ), γ
where again i denotes the component in Vγ . Equation (6.34) can now be generalised to p,γ (F ) φ(., ξ) ij = 5C(ξ) (F ) φp,γ ξ ∈ GAB,p . 5C(ξ) γ γ ij (., ξ) i=1,... ,dγ , i=1,... ,dγ (6.47) Corresponding to Eq. (6.37) we now have the isometry property Z Z kφ(x, ξ)k2 dx dµ(ξ) = G
GAB
X
X
dγ
p γ∈(Nˆ p ) ω
dγ Z Z X i,j=1
G
GAB,p
2 kφp,γ ij (x, ξ)k dx dµ(ξ) (6.48)
and the isometry from Eq. (6.38) now becomes the isometry p dγ φp,γ G1 : φ 7→ ij i=1,... ,d γ
p;γ∈(Nˆ p )ω ,j=1,... ,dγ
(6.49)
of the Hilbert space L2ω (G × GAB , Vα ⊗ Vβ ) into the direct sum of Hilbert spaces M M d γ . (6.50) L2γ (G × GAB,p , Vα ⊗ Vβ ⊗ Vγ ) p
γ∈(Nˆ p )ω
This isometry is intertwining between the direct integral of representations Z ⊕ M M Z ⊕ 5C(ξ) dµ(ξ) and dα dβ dγ 5C(ξ) dµ(ξ) (6.51) ω γ GAB
p
γ∈(Nˆ p )ω
GAB,p
of D(G). Keep in mind that only the equivalence class of the measure µ matters in a direct integral of representations, as above. Again, to show that G1 is indeed an isometry into, we construct the inverse: for (ψij )i,j=1,... ,dγ ∈ L2γ (G × GAB,p : Vα ⊗ Vβ ⊗ Vγ )dγ define the map G2p,γ : ψ 7→ φ : L2γ (G × GAB,p : Vα ⊗ Vβ ⊗ Vγ )dγ → L2ω (G × GAB , Vα ⊗ Vβ ) (6.52) by φ(x, ξ) := dγ
dγ X
ψkk (x, ξ).
(6.53)
k=1
Then G1 ◦ G2p,γ = id, which can be shown in the same way as under Eq. (6.42). We now combine step one and step two in the procedure described above. The decomposition of the tensor product representation is then given by the intertwining
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T. H. Koornwinder, F. A. Bais, N. M. Muller
isometry ρ := G1 ◦ F1 , and its inverse is given by F2 ◦ G2p,γ . (The latter acting on L2γ (G × GAB,p , Vα ⊗ Vβ ⊗ Vγ )dγ .) Thus we have determinded the Clebsch–Gordan series from Eq. (6.2) Z ⊕ M B ⊗ 5 ' dα dβ dγ 5C(ξ) dµ(ξ), (6.54) 5A α β γ GAB
γ∈Nˆ C
with µ an equivalence class of measures. More precisely, we have to take the variation of NC(ξ) with ξ into account, which splits the direct integral over ξ: M M Z ⊕ A B dα dβ dγ 5C(ξ) dµ(ξ). (6.55) 5α ⊗ 5β ' γ p
γ∈(Nˆ p )ω
GAB,p
Combining F1 from Eq. (6.20) and G1 from Eq. (6.49) we see that a 8 ∈ L2α,β (G×G : Vα ⊗ Vβ ) is taken to an “object” in the direct sum/integral of Hilbert spaces M M Z ⊕ L2γ (G × GAB,p , Vα ⊗ Vβ ⊗ Vγ )dγ dµ(ξ). (6.56) p
γ∈(Nˆ p )ω
GAB,p
This object depends on ξ ∈ GAB , which determines the class label C of the (irreducible) which occurs in the decomposition. It has an index i denoting the representation 5C(ξ) γ component of the vector (with tensor products of vectors in Vα ⊗ Vβ as its entries) in Vγ to which a group element x is mapped, an index p which denotes the Borel set in Conj(G), which in turn determines the set (Nˆ p )ω to which the label γ of the D(G)representation must belong. Finally, the object has an index j indicating the degeneracy . The “vector of tensor products of vectors” means of the irreducible representation 5C(ξ) γ that each component in Vγ of the object in fact depends on the full vector in Vα ⊗ Vβ to which 8 maps a pair (x1 , x2 ) ∈ G × G. We can “dissect” the isometry ρ according to the way it maps the components of 8 to components of the object described above, this results in the following B Theorem 6.10. Let 5A α , 5β be irreducible ∗-representations of D(G), and let p label the finitely many Borel sets in Conj(G), on each of which NC does not vary with C. Take ξ ∈ GAB,p and γ ∈ (Nˆ p )ω . Then, for each k = 1, ..., dα and l = 1, ..., dβ and i, j = 1, ..., dγ a mapping
ρξγ,k,l,j : L2α,β (G × G, Vα ⊗ Vβ ) → L2γ (G, Vγ ) B C(ξ) intertwining the representations 5A is given by α ⊗ 5β and 5γ ρξγ,k,l,j 8 (x) := φp,γ ij (x, ξ) k,l i Z = γij (n) 8kl (xnw(ξ)−1 , xnw(ξ)−1 y(ξ)) dn.
(6.57)
(6.58)
NC(ξ)
An implicit expression for the fusion rules (multiplicities) can now also be obtained by comparing the squared norms before and after the action of ρ on 8. We then would like as a direct integral over to rewrite a direct integral over GAB,p of representations 5C(ξ) γ C Conjp (G) of representations 5γ . However, if the map ξ 7→ C(ξ) : GAB → Conj(G) is
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175
non-injective, which might be the case as we have mentioned before, this rewriting can be difficult. To solve this, we also define a Borel measure ν on Conj(G) such that Z Z F (C(ξ)) dµ(ξ) = F (C) dν(C) (6.59) GAB
Conj(G)
for all F ∈ C(Conj(G)). The measure ν has support λAB (GAB ). By Theorem A.5 there exists for almost each C ∈ Conj(G) a Borel measure pC on GAB such that Z Z Z f (ξ) dµ(ξ) = f (ξ) dpC (ξ) dν(C) (6.60) GAB
C∈Conj(G)
ξ∈GAB
for each f ∈ C(GAB ). If the mapping λAB is injective (like in the case of G = SU (2), as we will discuss in the next section) then the above simplifies to Z Z f (ξ) dµ(ξ) = f (λ−1 (6.61) AB (C)) dν(C), GAB
IAB
where IAB is the image of GAB under λAB . Combining Eqs. (6.26) and (6.48) the isometry property which contains the implicit expression for the multiplicities now reads Z Z k8(u, v)k2 dudv = (6.62) G
G
dα dβ
dβ dα X XX X p
dγ
j=1
k=1 l=1 γ∈(Nˆ p )ω
Z
Z
GAB,p
Conjp (G)
dγ X
dγ Z X ξ k ργ,k,l,j 8 (y)k2 dy dpC (ξ) dν(C). i=1
i
G
Equation (6.62) can be written more compactly as: k8k2 = dα dβ
dβ dα X XX p
Z
k=1 l=1
Conjp (G)
X
dγ
γ∈(Nˆ p )ω
dγ Z X j=1
GAB,p
kρξγ,k,l,j 8k2 dpC (ξ) dν(C).
If λAB is injective then Eq. (6.63) simplifies to dβ Z dγ dα X X XX X −1 λAB (C) dγ kργ,k,l,j 8k2 dν(C) k8k2 = dα dβ p
k=1 l=1
IAB,p
γ∈(Nˆ p )ω
(6.63)
j=1
(6.64)
with IAB,p = λAB (GAB,p ). Note that the measures no longer stand for equivalence classes of measures, but for specific measures, since we are comparing (squared norms of) vectors in Hilbert spaces. The measure ν may involve a nontrivial Jacobian from the mapping λAB . ABγ can now more or less be extracted from Eq. (6.62) or The multiplicities NαβC Eq. (6.63), that is, we can conclude the following:
176
(i) (ii) (iii) (iv)
T. H. Koornwinder, F. A. Bais, N. M. Muller ABγ NαβC = 0 if C 6∈ λAB (GAB ). ABγ NαβC = 0 if γ 6∈ Nˆ ω . ABγ ABγ 6= 0 then NαβC = dα d β d γ . If NαβC The inner product on VγC will depend nontrivially on A and B according to the Jacobian of the mapping λAB and its non-injectivity, which is reflected in the measure pC (ξ).
7. Explicit Results for G = SU (2) To illustrate the above aspects of tensor products of irreducible representations we will now consider the case of G = SU (2). We will only discuss the decomposition of a “generic” tensor product representation and give explicit formulas for the Clebsch– Gordan coefficients in this case. Some applications and the treatment of more special tensor products will be discussed elsewhere [3]. In [16] we have given the classification of the irreducible unitary representations of D(SU (2)). For application of the main result of this paper (the decomposition of the tensor product of such representations into single representations) we first need to establish the notation and parametrisation of elements of SU (2). In this section we use the conventions of Vilenkin [22], because this book contains a complete and explicit list of formulas which are needed in our analysis. For the Wigner functions we use the notation of Varshalovich et al [21] (especially Chapter 4). 7.1. Parametrisation and notation. To specify an SU (2)-element we use both the Euler angles (φ, θ, ψ), and the parametrisation by a single rotation angle r around a given axis n. ˆ In the Euler–angle parametrisation each g ∈ SU (2) can be written as g = gφ aθ gψ with
gφ =
1
e 2 iφ 0 1 0 e− 2 iφ
0 ≤ θ ≤ π,
(7.1)
,
aθ =
0 ≤ φ < 2π,
cos 21 θ − sin 21 θ sin 21 θ cos 21 θ
,
−2π ≤ ψ ≤ 2π.
(7.2)
(7.3)
The diagonal subgroup consists of all elements gφ , and is isomorphic to U (1). The conjugacy classes of SU (2) are denoted by Cr with 0 ≤ r ≤ 2π. The representative of Cr can be taken to be gr , so in the diagonal subgroup. Then Assumption 6.1 which states that the map of the set of conjugacy classes of G to G itself (i.e. the map to representatives) can be chosen to be continuous is satisfied. For r = 0 and 2π the centralizer N0 = N2π = SU (2), for the other conjugacy classes the centralizer Nr = U (1). Let 0 < r < 2π. Then Cr clearly consists of the elements −1 g(r, θ, φ) := gφ aθ gr a−1 θ gφ .
If we take the generators of SU (2) in the fundamental representation to be 1 0 01 0 i τ1 := , τ2 := , τ3 := 0 −1 10 −i 0
(7.4)
(7.5)
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177
and define the unit vector n(θ, ˆ φ) := (cos θ, sin θ cos φ, sin θ sin φ),
(7.6)
then we can also write the element g(r, θ, φ) as r r r g(r, θ, φ) = exp(i n(θ, ˆ φ) · ~τ ) = 11 cos + nˆ · ~τ i sin . (7.7) 2 2 2 This means that there is a 1–1 correspondence between n(θ, ˆ φ) and the cosets ˆ φ) 7→ g(r, θ, φ) : S 2 → Cr is bijective gφ aθ Nr . In other words, the mapping n(θ, from the unit sphere S 2 in R3 onto the conjugacy class Cr . 7.2. Irreducible representations. Next we consider the “generic” irreducible unitary representations of D(SU (2)), i.e. for the case r 6= 0, 2π. The other cases will be treated elsewhere [3]. The centralizer representations will be denoted by n ∈ 21 Z (so not the elements themselves as we did in the sections before, when we discussed the general case). The irreducible unitary representations of Nr are the 1-dimensional representations 1 n : gζ 7→ einζ , (7.8) −2π ≤ ζ ≤ 2π, n ∈ Z. 2 For the generic representations 5rn of D(SU (2)) the representation space is Vnr = {φ ∈ L2 (SU (2), R/2π) | φ(ggζ ) = e−inζ φ(g),
−2π ≤ ζ ≤ 2π}. (7.9)
Vnr
j is given by the Wigner functions Dmn , where the label n An orthogonal basis for is fixed. A thorough treatment of the Wigner functions as a basis of functions on SU (2) can be found in [22]. For g ∈ SU (2) parametrised by the Euler angles as in Eq. (7.1) the j corresponding to the m, nth matrix element in the j th irreducible Wigner function Dmn representation takes the value j j (g) = e−imφ Pmn (cos θ)e−inψ , Dmn
(7.10)
j can be expressed in terms of Jacobi polynomials. For all gζ = eiζ ∈ U (1) where Pmn we have that j j (xgζ ) = e−inζ Dmn (x). Dmn j {Dmn
(7.11)
1 2 N, j
This shows indeed that the set |n fixed, j ∈ ≥ n, −j ≤ m ≤ j} has the right covariance property. The Wigner functions form a complete set on SU (2), so the aforementioned set forms a basis for a Hilbert space corresponding to an irreducible unitary representation of D(SU (2)), with fixed centraliser representation n and arbitrary conjugacy class 0 < r < 2π. In other words, the Hilbert spaces for irreducible unitary representations with the same n and different r are equivalent, and thus can be spanned by identical bases. Recall that the r-dependence of the representation functions φ ∈ Vnr is only reflected in the action of D(SU (2)) on Vnr : Z 5rn (F )φ (y) = F (ygr y −1 , x)φ(x−1 y) dx, φ ∈ Vnr . (7.12) SU (2)
Strictly speaking, we should label the (basis) vectors of Vnr by r as well, then an arbitrary state in a generic representation is written as X X r j φn (x) = cjm rDmn (x), x ∈ G. (7.13) j>n −j≤m≤j
(Note that the sum over j is infinite.) However, since we will always specify which representation 5rn we are dealing with, we will omit the r-label on the functions.
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By Eq. (3.4) the character χrn of a generic representation 5rn is given by Z Z χrn (F ) = F (zgr z −1 , zgζ z −1 ) einζ dζ dz, F ∈ C ∞ (SU (2) × SU (2)). SU (2) U (1) (7.14) 7.3. Clebsch–Gordan series. First we will determine the decomposition of the tensor product of two generic representations 5rn11 and 5rn22 as in Eq. (6.54). It will turn out that p takes only one value, corresponding to generic r3 , and that the map λr1 ,r2 is injective. We have to determine the image Ir1 ,r2 of λr1 ,r2 , the equivalence class of the measure ν, and the set Nˆ r3 . Since the centraliser representations n1 , n2 , n3 are one-dimensional we see that the nonvanishing multiplicities Nnr11 rn22nr33 = 1. We choose y(θ) := aθ as a representative for the double coset Nr1 aθ Nr2 , which is an element of Gr1 r2 = U (1)\SU (2)/U (1). Then Assumption 6.2, stating that the representatives of the double cosets can be chosen in a continuous way, is satisfied. Equation (6.7), which for this case determines r3 (θ) and w(θ), now reads −1 (θ). gr1 aθ gr2 a−1 θ = w(θ)gr3 (θ) w
(7.15)
By computing the trace of the left-hand side of Eq. (7.15) we find for r3 = r3 (θ) that cos
r1 r2 r1 r2 r3 = cos cos − cos θ sin sin , 2 2 2 2 2
(7.16)
which gives us the mapping λr1 ,r2 from Eq. (6.6): 1 1 1 1 λr1 ,r2 (U (1)aθ U (1)) = 2 arccos(cos r1 cos r2 − cos θ sin r1 sin r2 ). (7.17) 2 2 2 2 Thus the mapping λr1 ,r2 : Gr1 r2 → [0, 2π] is injective with image Ir1 r2 = [|r1 − r2 |, min(r1 + r2 , 4π − (r1 + r2 ))].
(7.18)
Now we compute the measures µ and ν from Eqs. (6.59) and (6.60). The measure µ on Gr1 r2 follows from Z Z 1 π f (g) dg = f (aθ ) sin θ dθ (7.19) 2 0 SU (2) for a function f ∈ C(Gr1 r2 ), and thus dµ(θ) =
1 sin θ dθ. 2
(7.20)
The Borel measure ν on the set of conjugacy classes can be derived via Z π Z F (λr1 ,r2 (U (1)aθ U (1))) dµ(θ) = F (r3 ) dν(r3 ) 0
(7.21)
Ir1 ,r2
for an F ∈ C(Conj(SU (2))). With formula (7.17) it follows that ( r sin 23 r1 r2 dr3 , |r1 − r2 | ≤ r3 ≤ min(r1 + r2 , 4π − (r1 + r2 )) 4 sin (7.22) dν(r3 ) = 2 sin 2 0, otherwise.
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179
We conclude that the nongeneric conjugacy classes r3 = 0 and r3 = 2π have ν-measure zero in Ir1 ,r2 . We also see that the measure dν(r3 ) is equivalent with the measure dr3 on Ir1 ,r2 . To determine Nˆ r3 we remark that n1 (z) ⊗ n2 (z) = (z) idVn1 ⊗Vn2 , So Nˆ r3
z = {e, −e} ⊂ SU (2).
(7.23)
= (n1 + n2 ) mod Z. The Clebsch–Gordan series now reads 5rn11 ⊗ 5rn22 '
M n3 ∈(n1 +n2 )modZ
Z
⊕ Ir1 ,r2
5rn33 dr3 .
(7.24)
7.4. Clebsch–Gordan coefficients. We will now explicitly construct the mapping ρ from Eq. (6.58), successively applying the steps of Sect. 6. We can compute w(θ) = gφw aθw by first rewriting Eq. (7.15) as r1 r1 r2 r2 + iτ1 sin )(11 cos + i(cos θ τ1 + sin θ τ2 ) sin ) 2 2 2 2 r3 r3 = 11 cos + i nˆ w · ~τ sin 2 2
(11 cos
(7.25)
(in view of Eqs. (7.4), (7.6), (7.7)), and then comparing coefficients of τ1 , τ2 , τ3 on both sides. This yields sin r21 cos r22 + cos θ cos r21 sin r22 cos θw 1 r r . (7.26) sin θ cos 21 sin 22 nˆ w(θ) = sin θw cos φw = sin r23 sin θ sin φ sin θ sin r1 sin r2 w
w
2
2
It follows from Eqs. (7.15) and (7.26) that θw and φw depend continuously on θ, even for r1 = r2 , in which case the right-hand side of Eq. (7.26) tends to 0 cos r1 (7.27) 2 sin r21 as θ ↑ π, hence θw → π2 , φw → r21 . Thus the Borel map from Proposition 6.3 (a) can be chosen continuously. The first step in the tensor product decomposition is the construction of the map F1 from Corollary 6.8. The isometry F1 : L2n1 ,n2 (SU (2) × SU (2)) → L2 (SU (2) × [0, π]),
= (n1 + n2 ) mod Z (7.28)
is given by φ(x, θ) = 8(xw(θ)−1 , xw(θ)−1 aθ ).
(7.29)
For the inversion formula F2 we need a choice for the Borel map from Proposition 6.3 (b). It follows straightforwardly from the Euler angle parametrisation: write x ∈ SU (2) as x = gφx aθx gψx with 0 ≤ θx ≤ π, 0 ≤ φx < 2π, −2π ≤ ψx < 2π. Put y(U (1)xU (1)) := aθx and n1 (x) := gφx , n2 (x) := gψx . Then F2 : φ 7→ 8 is given by
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T. H. Koornwinder, F. A. Bais, N. M. Muller
8(u, v) = ein1 φu−1 v ein2 ψu−1 v φ(ugφu−1 v w(θu−1 v ), θu−1 v )
(7.30)
with u−1 v ∈ SU (2)o , and SU (2)o =
α −β β α
∈ SU (2) | α, β 6= 0 .
(7.31)
Assumption 6.7, stating that the complement of Go has measure zero in G, is satisfied for this case. The second step in the tensor product decomposition is given by the isometry G1 from Eq. (6.38) M L2n3 (SU (2) × Ir1 ,r2 ). (7.32) G1 : L2 (SU (2) × [0, π]) → n3 ∈(n1 +n2 ) mod Z
Assumption 6.9 about Conj(SU (2)) is satisfied, because there are two sets in Conj(SU (2)) with distinct centralisers: the set p0 = {r = 0, r = 2π} = Z with centraliser SU (2), and the set p1 = {r ∈ (0, 2π)} with centraliser U (1). From Eq. (7.22) we see that the set p0 will give no contribution in the decomposition of the squared norm of the tensor product state, because for r3 = 0, 2π the measure ν(r3 ) on the conjugacy classes is zero. Therefore we only need to compute Eq. (6.45) for p = p1 : Z p1 ,n3 (x, θ) = ein3 ζ φ(xgζ , θ) dζ, n3 ∈ (n1 + n2 ) mod Z (7.33) φ U (1)
with the U (1) over which we integrate embedded in SU (2), so −2π ≤ ζ ≤ 2π, and the Haar measure dζ appropriately normalised. The isometry property of Eq. (6.48) now becomes Z π Z Z Z X 2 |φ(x, θ)| dx dµ(θ) = |φp1 ,n3 (x, r3 )|2 dx dν(r3 ). SU (2) 0 Ir1 ,r2 SU (2) (7.34) n3 ∈(n1 +n2 )modZ The inverse mapping G2p1 reads φ(x, θ) =
X
φp1 ,n3 (x, θ).
(7.35)
n3 ∈(n1 +n2 ) mod Z
This results in the mapping ρ intertwining the representations 5rn11
⊗
5rn22
and
M n3 ∈(n1 +n2 ) mod Z
Z
⊕ Ir1 ,r2
5rn33 dν(r3 ).
(7.36)
We calculate the components of mapping ρ as given in Eq. (6.58). The labels i, j, k, l can be ignored, because Vn1 , Vn2 , Vn3 are one-dimensional. Z θ ein3 ζ 8(xgζ w(θ)−1 , xgζ w(θ)−1 aθ ) dζ. (7.37) ρn3 8 (x) = U (1)
The Clebsch–Gordan series from Eq. (7.24) is contained in
Tensor Product Representations of Quantum Double of Compact Group
Z
181
Z k8(u, v)k2 du dv SU (2) Z Z X
SU (2)
=
Ir1 ,r2
n3 ∈(n1 +n2 ) mod Z
SU (2)
|(ρrn33 8)(x)|2
dx
(7.38) dν(r3 ),
where we have replaced the θ-dependence by r3 -dependence, because the map λr1 ,r2 : Gr1 ,r2 → Conj(SU (2)) is injective, see Eq. (7.17). If we now choose an explicit basis for the representation spaces we can explicitly calculate the Clebsch–Gordan coefficients of D(SU (2)). For the orthogonal bases we j as explained under Eq. (7.9). take the Wigner functions Dmn We will use the notation and definition of the Clebsch–Gordan coefficients of SU (2) as given in [21], chapter 8. Thus jX 1 +j2
j1 j2 (g)Dm (g) = Dm 1 n1 2 n2
j X
j Cjjm Cjjn Dmn (g). 1 m1 j2 m2 1 n1 j 2 n2
(7.39)
j=|j1 −j2 | m,n=−j
are equal to zero if m 6= m1 + m2 . So The Clebsch–Gordan coefficients Cjjm 1 m1 j2 m2 j1 j2 (g)Dm (g) = Dm 1 n1 2 n2
0 X
j (n1 +n2 ) j 1 +m2 ) Cjj1(m m1 j2 m2 Cj1 n1 j2 n2 D(m1 +m2 ) (n1 +n2 ) (g),
(7.40)
j
where the primed summation over j runs from max(|j1 − j2 |, |m1 + m2 |, |n1 + n2 |) to (j1 + j2 ). In the tensor product representation 5rn11 ⊗ 5rn22 we consider the basis function j1 j2 j1 j2 8 = Dm ⊗ Dm : (y1 , y2 ) 7→ Dm (y1 ) Dm (y2 ), 1 n1 2 n2 1 n1 2 n2
ji ≥ ni , −ji ≤ mi ≤ ji , i = 1, 2.
(7.41)
The mapping ρ from Eq. (7.37) takes this basis function to a linear combination of basis functions of a single irreducible unitary representation 5rn33 : Z ρθn3 8 (x) = ein3 ζ 8(xgζ w(θ)−1 , xgζ w(θ)−1 aθ ) dζ U (1)
=
jX 1 +j2
j X
j2 X
Cjjm Cjjp Dpj22 n2 (aθ ) × 1 m1 j2 m2 1 n1 j 2 p 2
j=|j1 −j2 | m,p=−j p2 =−j2
Z
e U (1)
=
( 0 0 X X j
in3 ζ
j X
j j j Dmr (x)Drs (gζ )Dsp (w(θ)−1 ) dζ
r,s=−j
(7.42)
)
j 1 +m2 ) 1 +p2 ) Cjj(m Cjj(n Dpj22 n2 (aθ )D(n (w(θ)) 1 m1 j2 m2 1 n1 j 2 p 2 1 +p2 )n3
j D(m (x), 1 +m2 )n3
p2
where the primed summation over p2 runs from max((−j −n1 ), −j2 ) to min((j −n1 ), j2 ). This shows how 8 ∈ Vnr11 ⊗Vnr22 can be decomposed into single Wigner functions with a fixed label n3 , which form a basis of Vnr33 . The coefficients between the large brackets {} now indeed are the generalised Clebsch–Gordan coefficients for the quantum double group of SU (2). Clearly they depend on the representation labels, so on (r1 , n1 ), (r2 , n2 ) and (r3 , n3 ), where r3 corresponds one–to–one to the double coset θ. They also depend
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T. H. Koornwinder, F. A. Bais, N. M. Muller
on the specific “states” labeled by the j1 , m1 , etc., just as one would expect. Note that aθ and w(θ) are needed to implement the dependence on θ. We can denote these Clebsch– Gordan coefficients by h(r1 , n1 )j1 m1 , (r2 , n2 )j2 m2 | (r3 , n3 )jmi :=
0 X
j 1 +p2 ) Cjjm Cjj(n Dpj22 n2 (aθ )D(n (w(θ)) 1 m1 j2 m2 1 n1 j 2 p 2 1 +p2 )n3
(7.43)
p2
with r3 = λr1 ,r2 (θ). These coefficients are zero if m 6= m1 + m2 . Also, they are zero if n3 6= (n1 + n2 ) mod Z, so n3 must be integer if n1 + n2 integer, and half integer if n1 + n2 half integer. Thus we can write j1 j2 ⊗ Dm (x) ρrn33 Dm 1 n1 2 n2 =
j 0 X X j
j h(r1 , n1 )j1 m1 , (r2 , n2 )j2 m2 | (r3 , n3 )jmi Dmn (x). 3
(7.44)
m=−j
The isometry property of ρ can now be calculated even more explicitly. The left-hand side of Eq. (7.38) gives Z Z j1 j2 j1 j2 Dm (y1 )Dm (y2 )Dm 1 n1 (y1 )Dm2 n2 (y2 ) dy1 dy2 1 n1 2 n2 SU (2) SU (2) (7.45) 1 1 . = 2j1 + 1 2j2 + 1 For the right-hand side of Eq. (7.38) we find Z XZ λ−1 r r (r3 ) j1 j2 2 |ρn3 1 2 ⊗ D dy dν(r3 ), Dm (y)| m 2 n2 1 n1 n3
Ir1 r2
(7.46)
SU (2)
where Ir1 r2 given by Eq. (7.18), and the measure dν(r3 ) by Eq. (7.22). Substituting Eq. (7.42) and Eq. (7.43) yields Z 0 X X XZ j h(r1 , n1 )j1 m1 , (r2 , n2 )j2 m2 | (r3 , n3 )jmiDmn (y) × 3 Ir1 r2
n3
0 X X j0
SU (2)
j
m
h(r1 , n1 )j1 m1 , (r2 , n2 )j2 m2 | (r3 , n3 )j 0 m0 iDm0 n3 (y) dy dν(r3 ). (7.47) j0
m0
The integration over y can be performed, and thus the isometry property of the mapping ρ reads XZ n3
0 X Ir1 r2
j
1 |h(r1 , n1 )j1 m1 , (r2 , n2 )j2 m2 | (r3 , n3 )j(m1 + m2 )i|2 dν(r3 ) 2j + 1
1 1 . = 2j1 + 1 2j2 + 1
(7.48)
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More generally, if we start with the identity of inner products which is immediately implied by Eq. (7.38), we obtain Z 0 XX 1 h(r1 , n1 )j1 m1 , (r2 , n2 )j2 (m − m1 ) | (r3 , n3 )jmi× 2j + 1 Ir1 r2 n j 3
h(r1 , n1 )j10 m01 , (r2 , n2 )j20 (m − m01 ) | (r3 , n3 )jmi dν(r3 ) =
δj1 ,j10 δj2 ,j20 δm1 ,m01 (2j1 + 1)(2j2 + 1)
. (7.49)
This means that the Clebsch–Gordan coefficients (7.43) for D(SU (2)), built from Wigner functions and Clebsch–Gordan coefficients for SU (2), satisfy interesting orthogonality relations, suggesting the existence of a “new” kind of special functions. Remember that the aθ and w(θ) given in Eqs. (7.2) and (7.26) are the choices we made for the Borel mappings y(ξ) and w(ξ) in Assumption 6.2 and Proposition 6.3 which uniquely depend on r3 according to Eq. (7.15). It is now clear that the choice of representatives in the double coset (so the mapping ξ → y(ξ) of Assumption 6.2), and the choice of Borel map ξ → w(ξ) of Proposition 6.3 do not affect the fusion rules: for aθ 7→ gφ aθ gψ and w(θ) 7→ gφ w(θ)gζ the Clebsch–Gordan coefficients from Eq. (7.43) only change by a phase factor ei(n1 φ−n2 ψ+n3 ζ) , and thus the orthonormality relations of Eq. (7.49) do not change. This concludes our discussion of the fusion rules of D(SU (2)). 8. Conclusion In this paper we have focussed on the co-structure of the quantum double D(G) of a compact group G and have used it to study tensor products of irreducible representations. We have explicitly constructed a projection onto irreducible components for tensor product representations, which of course has to take into account the (nontrivial) comultiplication. By subsequently using the Plancherel formula (i.e. by comparing squared norms) we found an implicit formula for the multiplicities, or Clebsch–Gordan series. Also, we have given the action of the universal R-matrix of D(G) on tensor product states. For the example of G = SU (2) we calculated the Clebsch–Gordan series and coefficients explicitly. In a forthcoming article we will expand further on the quantum double of SU (2), in particular the behaviour of its representations under braiding and fusion. These results also will enable us to describe the quantum properties of topologically g interacting point particles, as in ISO(3) Chern–Simons theory, see [3]. A. Some Measure Theoretical Results In this appendix we have collected some measure theoretical results which have been used in Sect. 6. Theorem A.1 (Kuratowski’s theorem, see Parthasarathy, [19], Ch. I, Corollary 3.3). If E is a Borel subset of a complete separable metric space X and λ is a one-one measurable map of E into a separable metric space Y then λ(E) is a Borel subset of Y and λ : E → λ(E) is a Borel isomorphism. Theorem A.2 (Theorem of Federer & Morse [11], see also [19], Ch. I, Thm 4.2). Let X and Y be compact metric spaces and let λ be a continuous map of X onto Y . Then there is a Borel set B ⊂ X such that λ(B) = Y and λ is one-to-one on B.
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The set B is called a Borel section for λ. Since the continuous image of a compact set is compact, we can relax the conditions of Theorem A.2 by not requiring surjectivity of λ. Then λ(B) = λ(X). By Theorem A.1 the mapping λ|B : B → λ(X) is a Borel isomorphism. Let ψ : λ(X) → B be the inverse of λ|B . We will also call the mapping ψ a Borel section for λ. We conclude: Corollary A.3. Let X and Y be compact metric spaces and let λ be a continuous map of X to Y . Then there is a Borel map ψ : λ(X) → X such that λ(ψ(y)) = y for all y ∈ λ(X) and ψ(λ(X)) is a Borel set in X. Theorem A.4 (isomorphism theorem, see for instance [19], Ch. I, Theorem 2.12). Let X1 and X2 be two complete separable metric spaces and let E1 ⊂ X1 and E2 ⊂ X2 be two Borel sets. Then E1 and E2 are Borel isomorphic if and only if they have the same cardinality. In particular, if E1 is uncountable, X2 := R and E2 is an open interval, then E1 and E2 are Borel isomorphic. Next we discuss conditional probability, although we will not deal with probabilistic interpretations. Our reference here is Halmos [13], §48. Let (X, A) and (Y, B) be measurable spaces, i.e. sets X and Y with σ-algebras A and B, respectively. Let λ : X → Y be a measurable map. Let µ be a probability measure on (X, A). Define a probability measure ν on (Y, B) by the rule ν(B) := µ(λ−1 (B)),
B ∈ B.
(A.1)
By the Radon-Nikodym theorem there exists for each A ∈ A a ν-integrable function pA on Y such that Z µ(A ∩ λ−1 (B)) = pA (y) dν(y), B ∈ B. (A.2) B
Then pA (y) is called the conditional probability of A given y. Note that the functions pA are not unique. For fixed A, two choices for pA can differ on a set of ν-measure zero. We will write py (A) := pA (y),
y ∈ Y, A ∈ A.
(A.3)
Then py behaves in certain respects like a measure on (X, A), but it may not be a measure. If f is a µ-integrable function on X then, by the Radon-Nikodym theorem there exists a ν-integrable function ef on Y such that, for every B ∈ B, Z Z f (x) dµ(x) = ef (y) dν(y). (A.4) λ−1 (B)
B
Theorem A.5. If λ is a measurable map from a probability space (X, A, µ) to a measurable space (Y, ν), and if the conditional probabilities pA (y) can be determined such that py is a measure on (X, A) for almost every y ∈ Y , then Z f (x) dpy (x) for y almost everywhere on Y w.r.t. ν. (A.5) ef (y) = X
In particular, if X is an open interval in R, or more generally a complete separable metric space, then pA (y) can be determined such that py is a measure on (X, A) for almost every y ∈ Y , and Eq. (A.4) will hold with ef (y) given by Eq. (A.5).
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This theorem follows from Halmos [13], pp. 210–211, items (5) and (6) together with the above Theorem A.4. Theorem A.5 greatly simplifies if X is a complete separable metric space and, moreover, λ is injective. Then 0, y 6∈ λ(X), py (A) = pA (y) = χλ(A) (y) = δλ−1 (y) (A), y ∈ λ(X) 0, y 6∈ λ(X), ef (y) = f (λ−1 (y)), y ∈ λ(X), Z Z f (x) dµ(x) = f (λ−1 (y)) dν(y). (A.6) λ−1 (B)
B∩λ(X)
Acknowledgement. The third author was supported by the Dutch Science Foundation FOM/NWO. We would like to thank Dr A.A. Balkema for useful discussions on measure theory. Also, we want to thank Dr. E.P. van den Ban for his private communication [5], which we used in discussing our Assumption 6.9.
References 1. Arveson, W.: An Invitation to C ∗ -Algebra. Berlin–Heidelberg–New York: Springer-Verlag, 1976 2. Bais, F.A., de Wild Propitius, M., van Driel, P.: Quantum symmetries in discrete gauge theories. Phys. Lett. B280, 63–70 (1992) 3. Bais, F.A., and Muller, N.M.: Topological field theory and the quantum double of SU (2). hep-th/9804130, to appear in Nucl. Phys. B 4. Bais, F.A., and Schroers, B.J.: Quantisation of monopoles with non-abelian magnetic charge. Nucl. Phys. B512, 250–294 (1998) 5. Ban, E.P. van den: Private communication 6. Bernard, D., and LeClair, A.: The quantum double in integrable quantum field theory. Nucl. Phys. B399, 709 (1993) 7. Br¨ocker, T., and tom Dieck, T.: Representations of Compact Lie Groups. Berlin–Heidelberg–New York: Springer-Verlag, 1985 8. Chari, V., and Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press, 1994 9. Dijkgraaf, R.H., Pasquier, V. and Roche, P.: Quasi Hopf algebras, group cohomology and orbifold models. Nucl. Phys. B (Proc. Suppl.) 18B, 60–72 (1990) 10. Drinfel’d, V.G.: Quantum groups. In: Proceedings of the I.C.M., Berkeley, (1986), Providence, RI: Amer. Math. Soc., 1987, pp. 798–820 11. Federer, H., and Morse, A.P.: Some properties of measurable functions. Bull. Amer. Math. Soc. 49, 270–277 (1943) 12. Hausser, F., and Nill, F.: Doubles of quasi-quantum groups. q-alg/9708023 13. Halmos, P.R.: Measure Theory. Amsterdam: Van Nostrand, 1950 14. Jantzen, J.: Lectures on Quantum Groups. Graduate Studies in Mathematics, Vol. 6, Providence, RI: Am. Math. Soc., 1995 15. Kassel, C.: Quantum Groups. Graduate Texts in Mathematics 155, Berlin–Heidelberg–New York: Springer-Verlag, 1995 16. Koornwinder T.H., and Muller, N.M.: The quantum double of a (locally) compact group. J. Lie Theory, 7, 33–52 (1997); 8, 187 (1998) 17. Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 18. M¨uger, M.: Quantum double actions on operator algebras and orbifold quantum field theories. Commun. Math. Phys. 191, 137–181 (1998) 19. Parthasarathy, K.R.: Probability Measures on Metric Spaces. New York: Academic Press, 1967 20. Podl´es, P., and Woronowicz, S.L.: Quantum deformation of Lorentz group. Commun. Math. Phys. 130, 381–431 (1990) 21. Varshalovich, D.A., Khersonski, V.K. and Moskalev, A.N.: Quantum Theory of Angular Momentum. Singapore: World Scientific, 1988
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22. Vilenkin, N.Ya.: Special Functions and the Theory of Group Representations. Trans. Math. Monographs 22, Providence, RI: Am. Math. Soc., 1968 23. de Wild Propitius, M.D.F. , and Bais, F.A.: Discrete gauge theories. In: Particles and Fields, editor G.W. Semenoff, CRM Series in Math. Phys., Springer Verlag, and M.D.F. de Wild Propitius: Topological Interactions in Broken Gauge Theories. PhD thesis University of Amsterdam, 1995 Communicated by T. Miwa
Commun. Math. Phys. 198, 187 – 197 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
An Analogue of the Kato–Rosenblum Theorem for Commuting Tuples of Self-Adjoint Operators Jingbo Xia? Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14214, USA. E-mail:
[email protected] Received: 22 December 1997 / Accepted: 9 April 1998
Abstract: Suppose that A = (A1 , ..., AN ) and A0 = (A01 , ..., A0N ) are tuples of selfadjoint operators on a Hilbert space H such that [Aj , Ak ] = 0 and [A0j , A0k ] = 0 for all 1 ≤ j, k ≤ N . Suppose that there are z1 , ..., zN ∈ C\R such that (Aj − zj )−1 − (A0j − zj )−1 belongs to the trace class, 1 ≤ j ≤ N . We prove that A|Hnd (A; C1 ) is unitarily equivalent to A0 |Hnd (A0 ; C1 ). Here, H = Hd (A; C1 ) ⊕ Hnd (A; C1 ) and Hd (A; C1 ) is the largest invariant subspace on which A can be simultaneously diagonalized modulo the trace class.
1. Introduction The well-known theorem of Kato and Rosenblum asserts that if T and T 0 are self-adjoint operators on a Hilbert space H such that T − T 0 is of trace class, then T |Hac (T ) and T 0 |Hac (T 0 ) are unitarily equivalent. The purpose of this paper is to prove that such unitary equivalence persists for commuting tuples of self-adjoint operators under traceclass perturbation, solving a problem left open in [11]. Now, for a single self-adjoint operator T , Hac (T ) is defined in terms of absolute continuity with respect to the Lebesgue measure on R. We need to explain what is the proper generalization of Hac (T ) in the case of tuples. Recall, for this purpose, that if we decompose H as H = Hs (T ) ⊕ Hac (T ), then the well-known theorem of Carey and Pincus [1] tells us that T |Hs (T ) can be diagonalized modulo the trace class whereas such diagonalization is not possible on any nonzero invariant subspace of Hac (T ). (The latter assertion is certainly a consequence of the Kato–Rosenblum theorem, but it follows easily from the boundedness of the Hilbert transform on L2 (R) [12,Proposition 2.1].) In other words, the decomposition H = Hs (T ) ⊕ Hac (T ) is equivalent to this go/no-go statement about diagonalization, which is what one needs to replace Hs (T ) and Hac (T ) in ?
Research supported in part by National Science Foundation grant DMS-9703515.
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the case of commuting tuples. The necessity for this is easy to understand: The Lebesgue measure on R serves as a master measure for the problem of trace class perturbation of single operators, but obviously in the case of tuples no single measure can fulfill the same role. Let us denote the trace class by C1 . In this paper all Hilbert spaces are assumed to be separable. We say that a commuting tuple A = (A1 , ..., AN ) of self-adjoint operators can be simultaneously diagonalized modulo the trace class if there exist K1 , ..., Kn ∈ C1 such that (A1 +K1 , ..., AN +KN ) is a commuting tuple of self-adjoint diagonal operators. See [10,Corollary 2.6] for statements equivalent to this. Voiculescu showed that, given a commuting tuple A = (A1 , ..., AN ) of self-adjoint operators on a Hilbert space H, there is a decomposition H = Hd (A; C1 ) ⊕ Hnd (A; C1 ), where both Hd (A; C1 ) and Hnd (A; C1 ) are invariant under A, A|Hd (A; C1 ) can be simultaneously diagonalized modulo C1 , and Hnd (A; C1 ) contains no nonzero invariant subspace on which A can be simultaneously modulo C1 . See [11, p. 80]. Thus A|Hd (A; C1 ) has no invariance whatsoever under trace-class perturbation. We will show that trace-class perturbation preserves A|Hnd (A; C1 ) just as T |Hac (T ) is preserved in the case of single operators. The investigation of perturbations of tuples by arbitrary norm ideals goes back to Voiculescu’s work [10,11]. Indeed given an arbitrary norm ideal C and a commuting tuple A = (A1 , ..., AN ) of self-adjoint operators on H, one has H = Hd (A; C) ⊕ Hnd (A; C), where A|Hd (A; C) is simultaneously diagonalizable modulo C (0) and Hnd (A; C) contains no nonzero invariant subspace on which the same can be done [11]. (C (0) is the k.kC closure of the collection of finite-rank operators, which is not always the same as C [4,9].) One can deduce from Theorem 1.4 of [11] that, if A = (A1 , ..., AN ) and A0 = (A01 , ..., A0N ) are such that Aj − A0j ∈ C (0) for every j, and if C (0) is not the trace class, then A|Hnd (A; C) and A0 |Hnd (A0 ; C) are unitarily equivalent. In this paper we settle the only remaining case, that of trace-class perturbation. Main Theorem. Suppose that A = (A1 , ..., AN ) and A0 = (A01 , ..., A0N ) are tuples of self-adjoint operators on a Hilbert space H such that [Aj , Ak ] = 0 and [A0j , A0k ] = 0 for all j, k ∈ {1, ..., N }. Suppose that there exist z1 , ..., zN ∈ C\R such that (Aj − zj )−1 − (A0j − zj )−1 ∈ C1 , 1 ≤ j ≤ N. Then there exists a unitary operator U from Hnd (A; C1 ) onto Hnd (A0 ; C1 ) such that (A0j − w)−1 |Hnd (A0 ; C1 ) = U {(Aj − w)−1 |Hnd (A; C1 )}U ∗ for all 1 ≤ j ≤ N and w ∈ C\R. Among the norm ideals C with the property C = C (0) , the trace class is the only one whose dual does not consist solely of compact operators. This peculiarity of C1 denies us estimates which were available in case of [11]. C1 is also the only norm ideal which may obstruct the diagonalization of an individual operator in a given tuple. This leads to the realization that, to prove our theorem, the first order of business is to decompose the spectrum of A = (A1 , ..., AN ) on Hnd (A; C1 ) into N + 1 pieces according to (3.3) below. There are the pieces E1 , ..., EN ⊂ RN , where Aj |E(Ej )H is purely absolutely continuous, 1 ≤ j ≤ N . For our purpose these are the trivial pieces; they are taken care of by the Kato–Rosenblum theorem. On the remaining piece, E0 , each Aj is purely
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singular but collectively A still cannot be simultaneously diagonalized modulo C1 . As it happens, what works on E0 , fails on E1 , ..., EN , and vice versa. That is why E1 , ..., EN must be separated from E0 . That AE(E0 ) cannot be simultaneously diagonalized modulo C1 leads to the non-vanishing of the trace of a certain commutator. But the fact that each Aj E(E0 ) is purely singular leads to the vanishing of the traces of other commutators. As it turns out, our desired unitary equivalence can be extracted out of these two facts. The rest of the paper is organized as follows. We collect the necessary preliminaries in Sect. 2. Section 3 contains the spectral decomposition mentioned above. Sect. 4 is the key step of the proof; it deals with E0 . We finish our proof in Sect. 5. In Sect. 6 we discuss some related open problems.
2. Preliminaries The proof of the Main Theorem requires a combination of old tricks and new ones. The purpose of this section is to take care of the old stuff. In order not to leave any loose ends, at the risk of being overcautious, we will sketch a proof whenever a result cannot be found in the literature in the precise form we need. Recall that a commuting tuple A = (A1 , ..., AN ) of bounded self-adjoint operators can be simultaneously diagonalized modulo C1 if and only if there exists a sequence {Tk } of positive contractions of finite rank such that s-limk→∞ Tk = 1 and PN limk→∞ j=1 k[Aj , Tk ]k1 = 0 [10, Cor. 2.6]. Proposition 2.1 ([12, Prop. 2.4]). Suppose that A = (A1 , ..., AN ) is a commuting tuple of bounded self- adjoint operators on a Hilbert space H which cannot be simultaneously diagonalized modulo C1 . Then there exist bounded self- adjoint operators X1 , ..., XN PN on H such that i j=1 [Aj , Xj ] = S+ − S− = S with S+ ≥ 0, S− ≥ 0, S− ∈ C1 and 0 < tr(S) ≤ ∞. Proposition 2.2. Suppose that T is a bounded self-adjoint operator on a Hilbert space H such that Hac (T ) = {0}. If X is a bounded operator on H such that [T, X] ∈ C1 , then tr([T, X]) = 0. There are two distinct proofs for this proposition. The first goes back to the work of Carey and Pincus on the principal function of an almost commuting pair. See, e.g., [2]. The second, which is more pertinent to what we are doing here, is as follows: Since T is purely singular, the well-known theorem of Carey and Pincus [1] asserts that T is diagonalizable modulo C1 . Therefore it follows from Voiculescu’s result [12, Cor. 2.2(i)] that tr([T, X]) = 0. Proposition 2.3. Suppose that A = (A1 , ..., AN ) and A0 = (A01 , ..., A0N ) are commuting tuples of self-adjoint operators (Aj need not commute with A0k ) for which there are z1 , ..., zN ∈ C\R such that (Aj − zj )−1 − (A0j − zj )−1 ∈ C1 , 1 ≤ j ≤ N. Then ϕ(A0 ) − ϕ(A) ∈ C1 for every ϕ ∈ Cc∞ (RN ).
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If the assumption were Aj − A0j ∈ C1 , then this could be found in the literature from the 1970s. But the proof of this version is simple enough to warrant an inclusion QN anyway. Given ϕ ∈ Cc∞ (RN ), define η(t1 , ..., tN ) = ϕ(t1 , ..., tN ) j=1 (tj − zj )2 . Then R η(t) = RN h(λ)eihλ,ti dmN (λ) for some rapidly decreasing function h on RN . Because 0
(A0j − z)−1 (eiλj Aj − eiλj Aj )(Aj − z)−1 Z λj 0 =i eisAj {(Aj − z)−1 − (A0j − z)−1 }ei(λj −s)Aj ds, 0
QN QN C1 contains ( j=1 (A0j − zj )−1 ){η(A0 ) − η(A)}( j=1 (Aj − zj )−1 ). But it is easy to see that the difference between that operator and ϕ(A0 ) − ϕ(A) also belongs to C1 . Let A = (A1 , ..., AN ) be a commuting tuple of self-adjoint operators on a Hilbert space H and let E be its spectral resolution. Then we have the orthogonal decomposition H = Hp (A) ⊕ Hc (A) such that Hp (A) and Hc (A) are both invariant under A, A|Hp (A) is a tuple of diagonal operators, and Ec ({x}) = 0 for every singleton set {x} in RN , where Ec = E|Hc (A). Notation 2.4. Given A and the orthogonal decomposition H = Hp (A)⊕Hc (A) as above, denote the orthogonal projection from H onto Hc (A) by Pc (A). It is obvious that Hnd (A; C1 ) ⊂ Pc (A)H. Definition 2.5. Let A = (A1 , ..., AN ) be a commuting tuple of self-adjoint operators on a Hilbert space H. A sequence {ϕn } of functions on RN is said to be strongly admissible for A if the following hold true: (i) For every n ∈ N, ϕn ∈ C ∞ (RN ), |ϕn | = 1 on RN , and there is an Rn > 0 such that ϕn (x) = 1 whenever |x| ≥ Rn . (ii) w-limn→∞ ϕn (A)Pc (A) = 0. Proposition 2.6. Every commuting tuple A = (A1 , ..., AN ) of self-adjoint operators possesses a strongly admissible sequence. Proof. The proposition obviously follows from the following measure-theoretical result: If µ is a finite, regular Borel measure on RN and if µ has no point masses, then there is a sequence {ϕn } satisfying (i) in Definition 2.5 such that Z f ϕn dµ = 0 whenever f ∈ L1 (RN , µ). (2.1) lim n→∞
To prove this statement, we first note the following: Using an induction on the spatial dimension N and the fact that µ has no point masses, it is easy to show that every Borel set E ⊂ RN is the union of disjoint Borel subsets E + and E − such that µ(E + ) = µ(E − ) = µ(E)/2. Now for each n ∈ N, write RN as the union of pairwise disjoint Borel sets {En,k : k ∈ N} such that diam(En,k ) ≤ 1/n for every k. Let En,k − − + + be the union of disjoint En,k and En,k such that µ(En,k ) = µ(En,k ) = µ(En,k )/2, n, k ∈ N. Define ∞ X + (χEn,k − χE − ), n ∈ N. hn = k=1
n,k
Analogue of Kato–Rosenblum Theorem for Tuples
If g ∈ C0 (RN ), then
R
191
P∞ R
+ − χE − )dµ, where xn,k ∈ (g − g(xn,k ))(χEn,k n,k R En,k whenever En,k 6= ∅. Since diam(En,k ) ≤ 1/n, it follows that limn→∞ ghn dµ = 0 if g ∈ C0 (RN ). By the usual approximation, we have Z f hn dµ = 0 whenever f ∈ L1 (RN , µ). (2.2) lim
ghn dµ =
k=1
n→∞
For every n ∈ N, there is a real-valued C ∞ -function ηn on Rn such that ηn (x) = 0 if |x| ≥ n and such that µ({x ∈ RN : |x| ≤ n − 1, |ηn (x) − π(1 − hn (x))/2| ≥ 1/n}) ≤ 1/n.
(2.3)
N ∞ Let ϕn = exp(iηn ). Then |ϕn | = 1, ϕn ∈ R C (R ), and ϕn (x) = 1 for |x| ≥ n as desired. Equation (2.3) implies limn→∞ |ϕn − exp{iπ(1 − hn )/2}||f |dµ = 0 for every f ∈ L1 (RN , µ). Because exp{iπ(1 − hn )/2} = hn , (2.1) now follows from this and (2.2).
Lemma 2.7. Let A = (A1 , ..., AN ) and A0 = (A01 , ..., A0N ) be commuting tuples of self-adjoint operators on a Hilbert space H (Aj need not commute with A0k ) and let E and E 0 be their respective spectral resolutions. Suppose that there are z1 , ..., zN ∈ C\R such that (Aj − zj )−1 − (A0j − zj )−1 ∈ C1 , 1 ≤ j ≤ N. Let P be an orthogonal projection which commutes with E and let {ϕn } be a sequence of uni-modulus Borel functions such that w-limn→∞ ϕn (A)P = 0. If the weak limit W = w-limn→∞ ϕ∗n (A0 )ϕn (A)P exists, then ξ(A0 )W = W ξ(A) for every ξ ∈ C0 (RN ) and W E(E)H ⊂ E 0 (E)H for every Borel set E ⊂ RN . (ii) Suppose that 1 is a compact set in RN , that X is a bounded operator on H, and that the weak limit (i)
Z = w- lim ϕ∗n (A0 )E(1)XE(1)ϕn (A)P n→∞
exists. If f ∈ C0 (RN ) is such that f = constant on 1, then f (A0 )Z = Zf (A). Proof. (i) Proposition 2.3 implies that ξ(A) − ξ(A0 ) is compact for every ξ ∈ C0 (RN ). Therefore s-limn→∞ (ξ(A) − ξ(A0 ))ϕn (A)P = 0, which yields ξ(A0 )W = W ξ(A). Let a Borel set E ⊂ RN be given and let K be a compact subset of E. For each k ∈ N, let ξk ∈ C0 (RN ) be such that 0 ≤ ξk ≤ 1 and ξk = 1 on K and ξk (x) = 0 whenever d(x, K) ≥ 1/k. Letting k → ∞ in the equality ξk (A0 )W = W ξk (A), we obtain E 0 (K)W = χK (A0 )W = W χK (A) = W E(K). Now (i) follows from the fact that there is a sequence of compact sets {Km } contained in E such that E(E\(∪∞ m=1 Km )) = 0. (ii) Let Zn = ϕ∗n (A0 )E(1)XE(1)ϕn (A)P . Suppose that f = c on 1. Then Zn f (A) = cZn = f (A0 )Zn + ϕ∗n (A0 )(f (A) − f (A0 ))E(1)XE(1)ϕn (A)P. Since f (A) − f (A0 ) is compact by Proposition 2.3, the second term on the right converges to 0 strongly.
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3. Spectral Decomposition Let a commuting tuple A = (A1 , ..., AN ) of self-adjoint operators on a Hilbert space H be given and let E be its spectral resolution. Let End = E|Hnd (A; C1 ) and Ed = E|Hd (A; C1 ). For each w ∈ H, define the measure µw (B) = hE(B)w, wi. We claim that if u ∈ Hnd (A; C1 ) and v ∈ Hd (A; C1 ), then µu ⊥ µv . For, otherwise, there would be a Borel set 6 such that ξ = E(6)u 6= 0 and µξ ≺ µv . This absolute continuity implies that A|Hξ can be simultaneously diagonalized modulo C1 , where Hξ is the closure of {h(A)ξ : h ∈ C0 (RN )} in H. But this is not possible because ξ = E(6)u ∈ Hnd (A; C1 ). This proves our claim. It follows from this claim and the separability of H that there exists a Borel set EA ⊂ RN such that End (B) = E(EA ∩ B)
and
Ed (B) = E((RN \EA ) ∩ B).
(3.1)
Thus if H˜ is a subspace invariant under E and if H˜ contains no nonzero invariant subspace on which A can be simultaneously diagonalized modulo C1 , then H˜ ⊂ Hnd (A; C1 ). Given a set G ⊂ R, define hGij = {(t1 , ..., tN ) : tj ∈ G, tk ∈ R for all k 6= j}. For each 1 ≤ j ≤ N , let Ej be the spectral resolution for Aj . That is, Ej (G) = E(hGij ). Let m denote the Lebesgue measure on R. For each 1 ≤ j ≤ N , there is a Borel set L(j) ⊂ R with m(R\L(j)) = 0 such that the spectral measure G 7→ Ej (G ∩ L(j)) is absolutely continuous with respect to m. Thus we may assume EA ⊃ ∪1≤j≤N hL(j)ij . Define E1 = hL(1)i1 and Ej = hL(j)ij \{∪1≤k<j hL(k)ik } for 2 ≤ j ≤ N.
(3.2)
Set E0 = EA \{∪N j=1 Ej }. The above ensures that, for each 1 ≤ j ≤ N , Aj |E(Ej )H is purely absolutely continuous while Aj E(E0 ) is purely singular. We have the partition EA = E0 ∪ E1 ∪ ... ∪ EN .
(3.3)
4. Commutators and Traces The key to the proof of the Main Theorem is the following: Proposition 4.1. Let A = (A1 , ..., AN ) and A0 = (A01 , ..., A0N ) be tuples of self-adjoint operators on a Hilbert space H, each of which commutes. Let E be the spectral resolution of A. Suppose that there are z1 , ..., zN ∈ C\R such that (Aj − zj )−1 − (A0j − zj )−1 ∈ C1 , 1 ≤ j ≤ N. Let E0 be the Borel set for A given in Sect. 3. Suppose that {ϕn } is a strongly admissible sequence for A such that the weak limit W = w- lim ϕ∗n (A0 )ϕn (A)E(E0 ) n→∞
exists. Then kerW = E(RN \E0 )H. Proof. Set Wn = ϕ∗n (A0 )ϕn (A), n ∈ N. Since ϕn − 1 ∈ Cc∞ (RN ), Proposition 2.3 asserts (4.1) 1 − Wn = ϕ∗n (A0 )((ϕn − 1)(A0 ) − (ϕn − 1)(A)) ∈ C1 , n ∈ N. Obviously kerW ⊃ E(RN \E0 )H. Assuming kerW 6= E(RN \E0 )H, then kerW ∩ E(E0 )H 6= {0}. We will complete the proof by deducing a contradiction from this.
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It follows from Lemma 2.7(i) that kerW is invariant under E. Therefore there is a compact subset ⊂ E0 such that kerW ∩ E()H 6= {0}. Let P : H → kerW ∩ E()H be the orthogonal projection. Then P commutes with E. Pick a real-valued η ∈ Cc∞ (RN ) such that η = 1 on . Define Bj = Aj η(A) and Bj0 = A0j η(A0 ), 1 ≤ j ≤ N . Proposition 2.3 asserts that (4.2) Bj − Bj0 ∈ C1 , 1 ≤ j ≤ N. Since Wn∗ Wn = 1 and w-limn→∞ Wn P = 0, we have limn→∞ k(1 − Wn )f k2 = 2kf k2 if f ∈ P H. Therefore, if K ∈ C1 , K = K ∗ , and range(K) ⊂ P H, then lim tr((1 − Wn )K(1 − Wn )∗ ) = 2tr(K).
n→∞
(4.3)
Since P H ⊂ E(E0 )H, the tuple (B1 |P H, ..., BN |P H) = (A1 |P H, ..., AN |P H) still cannot be simultaneously diagonalized modulo C1 . By Proposition 2.1, there are bounded self-adjoint operators S+ ≥ 0, S− ≥ 0, X1 , ..., XN on P H with S− ∈ C1 such that i
N X
[Bj , Xj ] = i
j=1
N X
[Aj P, Xj ] = S = S+ − S− and 0 < tr(S) ≤ +∞.
(4.4)
j=1
Here and in what follows, we also regard Xj , S± and S as operators on H in the natural way. That is, an operator T on P H is identified with T ⊕ 0 on P H ⊕ (P H)⊥ = H. By Sect. 3, the projection of E0 onto each coordinate axis has Lebesgue measure 0. Hence is totally disconnected. Thus admits partitions {Pk }∞ k=1 such that: (a) For each k ∈ N, Pk = {1k,i : 1 ≤ i ≤ ν(k)}, where each 1k,i is compact, ν(k) 1k,i ∩ 1k,i0 = ∅ if i 6= i0 , and = ∪i=1 1k,i . (b) Pk+1 is a refinement of Pk , k ∈ N. (c) For each k ∈ N, diam(1k,i ) ≤ 1/k, 1 ≤ i ≤ ν(k). Recalling the usual diagonal-selection process and passing to a subsequence of {ϕn } if necessary, we may assume that w-limn→∞ ϕn (A0 )E(1k,i )Xj E(1k,i )ϕ∗n (A) exists for all 1 ≤ j ≤ N , k ∈ N, and 1 ≤ i ≤ ν(k). For 1 ≤ j ≤ N and k ∈ N, we define Xk,j =
ν(k) X
E(1k,i )Xj E(1k,i )
and
i=1
Vk,j = w- lim ϕn (A0 )Xk,j ϕ∗n (A). n→∞
∞ Since kVk,j k ≤ kXj k, there is a subsequence {k(p)}∞ p=1 of {k}k=1 such that the weak limits Vj = w- lim Vk(p),j , 1 ≤ j ≤ N, p→∞
exist. Note that Xk,j = Xk,j P and, consequently, Vj = Vj P , 1 ≤ j ≤ N. Note that the admissibility of {ϕn } implies that w-limn→∞ ϕ∗n (A)Pc (A) = 0. Thus we may invoke Lemma 2.7(ii): If g ∈ C0 (RN ) and if g is a constant on each of the sets 1k0 ,1 , ..., 1k0 ,ν(k0 ) for some k0 = k0 (g) ∈ N, then, because of the condition (b), we have g(A0 )Vk,j = Vk,j g(A) for all k ≥ k0 and 1 ≤ j ≤ N . Thus g(A0 )Vj = Vj g(A), 1 ≤ j ≤ N . Because of (a) and (c), such g’s separate points on the entire RN . (b) tells us that the collection of such g’s is closed under addition and multiplication. Hence we conclude that h(A0 )Vj = Vj h(A) for all h ∈ C0 (RN ) and 1 ≤ j ≤ N . In particular, this means Bj0 Vj = Vj Bj , 1 ≤ j ≤ N . Therefore, by (4.2),
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[Bj , Vj ] = (Bj − Bj0 )Vj ∈ C1 , 1 ≤ j ≤ N. Since Vj = Vj P and Bj = Aj η(A), we have Bj P = Aj P , [Bj , Vj ] = [Bj , Vj ]P , and [Aj P, P Vj ] = P [Bj , Vj ]P ∈ C1 . By the definition of P and (3.3), each Aj P is bounded and purely singular. Thus it follows from Proposition 2.2 that tr((Bj − Bj0 )Vj ) = tr([Bj , Vj ]) = tr([Aj P, P Vj ]) = 0, 1 ≤ j ≤ N.
(4.5)
Let > 0 be given. Since Bj − Bj0 ∈ C1 and w-limp→∞ Vk(p),j = Vj , it follows from (4.5) that there is a κ ∈ N such that N X
|tr((Bj − Bj0 )Vκ,j )| ≤ .
(4.6)
j=1
Let Sκ = i and
PN
j=1 [Bj , Xκ,j ].
tr((1 − Wn )∗ Sκ ) = i
N X
By (4.1), (1 − Wn )∗ ∈ C1 , tr([Bj , (1 − Wn )∗ Xκ,j ]) = 0,
tr((1 − Wn )∗ [Bj , Xκ,j ]) = i
j=1
= −i
N X
tr([1 − Wn∗ , Bj ]Xκ,j )
j=1
N X
tr({Wn∗ (Bj − Bj0 ) − ϕ∗n (A)(Bj − Bj0 )ϕn (A0 )}Xκ,j )
j=1
=i
N X
tr((Wn Xκ,j )∗ (Bj0 − Bj ))
j=1
+i
N X
tr((Bj − Bj0 )ϕn (A0 )Xκ,j ϕ∗n (A)).
j=1
Since w-limn→∞ Wn Xκ,j = W P Xκ,j = 0 and Bj − Bj0 ∈ C1 , we have lim tr((1 − Wn )∗ Sκ ) = i
n→∞
N X
tr((Bj − Bj0 )Vκ,j ).
(4.7)
j=1
Now (1−Wn )∗ (1−Wn ) = (1−Wn )∗ +(1−Wn ) and tr(Sκ (1−Wn )) = tr({(1−Wn )∗ Sκ }∗ ) = tr((1 − Wn )∗ Sκ ). Thus, by (4.6) and (4.7), lim tr((1 − Wn )Sκ (1 − Wn )∗ ) ≤ 2.
(4.8)
n→∞
Let R ∈ C1 be such that R ≤ S and range(R) ⊂ P H. Let Rκ = Note that range(Rκ ) ⊂ P H. Since
Pν(κ) i=1
E(1κ,i )RE(1κ,i ).
ν(κ) N X √ X Sκ = −1 [Bj , Xκ,j ] = E(1κ,i )SE(1κ,i ), j=1
i=1
∗
we have (1 − Wn )Rκ (1 − Wn ) ≤ (1 − Wn )Sκ (1 − Wn )∗ . Thus, by (4.3) and (4.8), 2tr(Rκ ) = lim tr((1 − Wn )Rκ (1 − Wn )∗ ) ≤ lim tr((1 − Wn )Sκ (1 − Wn )∗ ) ≤ 2. n→∞
n→∞
Analogue of Kato–Rosenblum Theorem for Tuples
But tr(Rκ ) = tr(R) because
Pν(κ) i=1
195
E(1κ,i ) = E() and E()R = R. Hence
tr(R) ≤ if R ∈ C1 , range(R) ⊂ P H, and R ≤ S. In view of the arbitrariness of > 0, this is irreconcilable with (4.4).
5. Putting Together the Pieces Given Proposition 4.1, the rest of the proof of the Main Theorem is easy. Let A = (A1 , ..., AN ), A0 = (A01 , ..., A0N ) be given as in the statement of the Main Theorem and let E, E 0 be their respective spectral resolutions. Let EA , E0 , E1 , ..., EN be the Borel sets for A as given in (3.1-3). For 1 ≤ j ≤ N , because Aj |E(Ej )H = Aj |End (Ej )H is purely absolutely continuous and because (Aj − zj )−1 − (A0j − zj )−1 ∈ C1 , by [7, Theorem XI.9] we have the strong limit 0
Uj = s- lim e−itAj eitAj E(Ej ). t→∞
Applying Proposition 2.6 to the tuple A⊕ A0 , there is a sequence {ϕn } which is strongly admissible for both A and A0 . Passing to a subsequence if necessary, we may assume that {ϕ∗n (A0 )ϕn (A)}∞ n=1 is weakly convergent. Define W = w- lim ϕ∗n (A0 )ϕn (A)E(E0 ). n→∞
Proposition 4.1 asserts that kerW = E(RN \E0 )H = H E(E0 )H. Let W = U0 |W | be the polar decomposition of W . That is, U0 is an isometry on E(E0 )H and kerU0 = E(RN \E0 )H. Also, for 1 ≤ j ≤ N , Uj is an isometry on E(Ej )H and kerUj = E(RN \Ej )H. Since E0 , E1 , ..., EN are pairwise disjoint, it follows from Lemma 2.7(i) that Uj H ⊥ Uk H for all 0 ≤ j < k ≤ N . Define U = U0 + U1 + ... +UN . It follows from (3.1-3) that U is an isometry on Hnd (A; C1 ) and kerU = Hd (A; C1 ). By Lemma 2.7(i), W h(A) = h(A0 )W and, therefore, h∗ (A)W ∗ = W ∗ h∗ (A0 ) for all h ∈ C0 (RN ). Therefore |W | commutes with {h(A) : h ∈ C0 (RN )}. Since |W |H is dense in E(E0 )H (Proposition 4.1) and U0 = U0 E(E0 ), we deduce that U0 h(A) = h(A0 )U0 for all h ∈ C0 (RN ). Lemma 2.7 (i) also tells us that Uj h(A) = H(A0 )Uj for all 1 ≤ j ≤ N and h ∈ C0 (RN ). Hence h(A0 )U = U h(A) whenever h ∈ C0 (RN ).
(5.1)
Thus U H = U Hnd (A; C1 ) contains no nonzero invariant subspace of A0 on which A0 can be simultaneously diagonalized modulo C1 . Hence U H ⊂ U Hnd (A0 ; C1 ) by the remark following (3.1). This and (5.1) imply that End is absolutely continuous with respect to 0 0 , where we write End = E|Hnd (A; C1 ) and End = E 0 |Hnd (A0 ; C1 ). Since the roles of A End 0 0 and A are interchangeable, we conclude that End and End are absolutely continuous with respect to each other. Using the notations from Sect. 3, this means that the decomposition (3.3) for EA0 is EA0 = E0 ∪ E1 ∪ ... ∪ EN , the same as that for EA . Thus, repeating the same argument as above, we find that the kernel of V = w-limn→∞ ϕ∗n (A)ϕn (A0 )E 0 (E0 ) is E 0 (RN \E0 )H and V H ⊂ E(E0 )H. Hence V = W ∗ . This implies that U0 H = E 0 (E0 )H. Similarly, Uj H = E 0 (Ej )H, 1 ≤ j ≤ N . In conclusion, U maps Hnd (A; C1 ) isometrically onto Hnd (A0 ; C1 ). Taking (5.1) into account, this completes the proof of the Main Theorem.
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6. Concluding Remarks In the above proof, the main difficulty was the handling of the piece E0 of the spectrum of A = (A1 , ..., AN ), where each individual Aj is purely singular but where collectively A is not diagonalizable modulo C1 . Therefore this proof does not contain a proof of the original Kato–Rosenblum theorem for single operators; indeed we needed the Kato– Rosenblum theorem to complete the proof of our theorem. A variation of the ideas used here, however, does yield a one-page proof of the Kato–Rosenblum theorem which requires no quantitative estimates. We managed to obtain unitary equivalence without the strong convergence of wave operators. But it is still of interest to raise Problem 6.1. Let A = (A1 , ..., AN ) and A0 = (A01 , ..., A0N ) be tuples of self-adjoint operators such that [Aj , Ak ] = 0 = [A0j , A0k ], 1 ≤ j, k ≤ N . Suppose that there are z1 , ..., zN ∈ C\R such that (Aj − z1 )−1 − (A0j − z1 )−1 ∈ C1 , 1 ≤ j ≤ N. Does there necessarily exist a sequence {ϕn } which is strongly admissible for both A and A0 such that the strong limits s- lim ϕ∗n (A0 )ϕn (A)Pnd (A; C1 ) n→∞
and s- lim ϕ∗n (A)ϕn (A0 )Pnd (A0 ; C1 ) n→∞
exist? Here, Pnd (A; C1 ) is the orthogonal projection from H onto Hnd (A; C1 ). Finally, having obtained our unitary equivalence result, it is only natural to explore the analytical description of the spaces Hnd (A; C1 ) and Hd (A; C1 ). In the case where N = 1, we know that this is equivalent to the Lebesgue decomposition of a given measure R with respect to the one-dimensional Lebesgue measure. To motivate the case for arbitrary N , it will be helpful to recall the following: For a regular Borel measure µ on R, = ∞ for µ-a.e. x ∈ R if µ ⊥ m . lim sup r−1 µ((x − r, x + r)) < ∞ for µ-a.e. x ∈ R if µ ≺ m r↓0 Now, given a commuting tuple A = (A1 , ..., AN ) of self-adjoint operators on H with spectral resolution E, for any ξ ∈ H, we define the measure µξ (G) = hE(G)ξ, ξi. 1 1 Using the usual covering lemma, it is easy to show that H = Hsin (A) ⊕ Hinv (A) where 1 Hsin (A) = {ξ ∈ H : lim sup r−1 µξ (B(x, r)) = ∞ for µξ -a.e. x ∈ RN }, r↓0
1 Hinv (A)
= {ξ ∈ H : lim sup r−1 µξ (B(x, r)) < ∞ for µξ -a.e. x ∈ RN }. r↓0
1 (A) [13, Here, B(x, r) = {y ∈ RN : |x − y| < r}. It is known that Hnd (A; C1 ) ⊂ Hinv Theorem 6.2]. There seems to be plenty of evidence supporting 1 1 Conjecture 6.2. Hnd (A; C1 ) = Hinv (A) and, therefore, Hd (A; C1 ) = Hsin (A).
This conjecture is equivalent to the following:
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Conjecture 6.2’. Suppose that µ is a compactly supported regular Borel measure on RN for which there is a constant C > 0 such that µ(B(x, r)) ≤ Cr for all x ∈ RN and 0 < r < 1 and suppose µ(RN ) 6= 0. Then the tuple (M1 , ..., MN ) on L2 (RN , µ) is not simultaneously diagonalizable modulo C1 . Here, (Mj f )(x1 , ..., xN ) = xj f (x1 , ..., xN ), 1 ≤ j ≤ N. References 1. Carey, R., Pincus, J.: Unitary equivalence modulo the trace class for self-adjoint operators. Am. J. Math. 98, 481–514 (1976) 2. Carey, R., Pincus, J.: Mosiacs, principal functions, and mean motion in von Neumann algebra. Acta Math. 138, 153–218 (1977) 3. David, G., Voiculescu, D.: s-Numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions. J. Funct. Anal. 94, 14–26 (1990) 4. Gohberg, I., Krein, M.: Introduction to the theory of linear nonselfadjoint operators. Trans. of Math. Monographs 18, Providence, RI: Am. Math. Soc., 1969 5. Kato, T.: Perturbation of continuous spectra by trace class operators. Proc. Japan Acad. 33, 260–264 (1957) 6. Kato, T.: Perturbation theory for linear operators. New York: Springer-Verlag, 1976 7. Reed, M., Simon, B.: Methods of modern mathematical physics. III Scattering theory. New York: Academic Press, 1979 8. Rosenblum, M.: Perturbations of continuous spectrum and unitary equivalence. Pacific J. Math. 7, 997– 1010 (1957) 9. Schatten, R.: Norm ideals of completely continuous operators. Berlin: Springer-Verlag, 1970 10. Voiculescu, D.: Some results on norm-ideal perturbations of Hilbert space operators. J. Operator Theory 2, 3–37 (1979) 11. Voiculescu, D.: Some results on norm-ideal perturbations of Hilbert space operators. II. J. Operator Theory 5, 77–100 (1981) 12. Voiculescu, D.: On the existence of quasicentral approximate units relative to normed ideals. Part I. J. Funct. Anal. 91, 1–36 (1990) 13. Xia, J.: Diagonalization modulo norm ideals with Lipschitz estimates. J. Funct. Anal. 145, 491–526 (1997) Communicated by B. Simon
Commun. Math. Phys. 198, 199 – 246 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem A. Connes1 , H. Moscovici2 1 2
I.H.E.S., route de Chartres, 91440 Bures-Sur-Yvette, France Department of Mathematics, Ohio State University, Columbus, OH 432400, USA
Received: 3 July 1998/ Accepted: 4 August 1998
Abstract: In this paper we solve a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. We show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific Hopf algebra Hn , associated to each integer codimension. This Hopf algebra reduces transverse geometry, to a universal geometry of affine nature. The structure of this Hopf algebra, its relation with the Lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper, in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the Hochschild cohomology of a Hopf algebra carries a highly nontrivial cyclic structure.
Introduction In this paper we present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. The spaces of leaves of foliations are basic examples of noncommutative spaces and already exhibit most of the features of the general theory. The index problem for longitudinal elliptic operators cf. [Co, M-S] is simple to formulate in the presence of a transverse measure and leads in general to the construction ([C-S]) of a natural map from the geometric group to the K theory of the leaf space, i.e. the K theory of the associated C ∗ algebra. This “assembly map” µ is known in many cases to exhaust the K theory of the C ∗ algebra but property T in the group context and its analogue for foliations give an obstruction to tentative proofs of its surjectivity in general. One way to test the K group, K(C ∗ (V, F )) = K(V /F ) for short, is to use its natural pairing with the K-homology group of C ∗ (V, F ). Cycles in the latter represent “abstract elliptic operators” on V /F and the explicit construction for general foliations of such cycles is
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already a quite elaborate problem. The point is that we do not want to assume any special property of the foliation such as, for instance, the existence of a holonomy invariant transverse metric as in Riemannian foliations. Equivalently, we do not want to restrict in any way the holonomy pseudogroup of the foliation. In [Co1, H-S, C-M], a general solution was given to the construction of transversal elliptic operators for foliations. The first step ([Co1]) consists in passing by a Thom isomorphism to the total space of the bundle of transversal metrics. This first step is a geometric adaptation of the reduction of an arbitrary factor of type III to a crossed product of a factor of type II by a one parameter group of automorphisms. Instead of only taking care of the volume distorsion (as in the factor case) of the involved elements of the pseudogroup, it takes care of their full Jacobian. The second step ([H-S]) consisted in realizing that while the standard theory of elliptic pseudodifferential operators cannot be used to construct the desired K-homology cycle, it suffices to replace it by its refinement to hypoelliptic operators. This was used in [C-M] in order to construct a differential (hypoelliptic) operator D solving the general construction of the K-cycle. One then disposes of a well posed general index problem. The index defines a map: K(V /F ) → Z which is simple to compute for those elements of K(V /F ) in the range of the assembly map. The problem is to provide a general formula for the cyclic cocycle ch∗ (D) which computes the index by the equality, hch∗ (D), ch∗ (E)i = Index DE
∀ E ∈ K(V /F ),
(1)
where the chern character ch∗ (E) belongs to the cyclic homology of V /F . We showed in [C-M] that the spectral triple given by the algebra A of the foliation, together with the operator D in Hilbert space H actually fulfills the hypothesis of a general abstract index theorem, holding at the operator theoretic level. It gives a “local” formula for the cyclic cocycle ch∗ (D) in terms of residues extending the ideas of the Z Wodzicki–Guillemin– Manin residue and the Dixmier trace. Adopting the notation − for such a residue the general formula gives the components ϕn of the cyclic cocycle ϕ = ch∗ (D) as universal finite linear combinations of expressions which have the following general form, Z ∀ aj ∈ A, (2) − a0 [D, a1 ](k1 ) . . . [D, an ](kn ) |D|−n−2|k| where for an operator T in H the symbol T (k) means the k th iterated commutator of D2 with T . It was soon realized that though the general index formula easily reduces to the local form of the Atiyah-Singer index theorem when D is say a Dirac operator on a manifold, the actual explicit computation of all the terms (2) involved in the cocycle ch∗ (D) is a rather formidable task. As an instance of this let us mention that even in the case of codimension one foliations, the printed form of the explicit computation of the cocycle takes around one hundred pages. Each step in the computation is straightforward but the explicit computation for higher values of n is clearly impossible without a new organizing principle which allows one to bypass them. In this paper we shall adapt and develop the theory of cyclic cohomology to the relevant class of Hopf algebras and show that this provides exactly the missing organizing principle, thus allowing to perform the computation for arbitrary values of n. We shall construct for each value of n a specific Hopf algebra H(n), show that it acts on the C ∗ algebra of the transverse frame bundle of any codimension n foliation (V, F ) and that the index computation takes place within the cyclic cohomology of H(n). We compute this
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cyclic cohomology explicitly as Gelfand Fuchs cohomology. The link, between cyclic cohomology and Gelfand Fuchs cohomology is old ([Co1]), what is new is that the entire differentiable transverse structure is captured by the action of the Hopf algebra H(n), thus reconciling our approach to noncommutative geometry to a more group theoretical one, in the spirit of the Klein program. 1. Notations We let M be an n-dimensional smooth manifold (not necessarily connected or compact but assumed to be oriented). Let us first fix the notations for the frame bundle of M , F (M ), in local coordinates xµ
µ = 1, . . . , n
for
x ∈ U ⊂ M.
(1)
We view a frame with coordinates xµ , yjµ as the 1-jet of the map j : Rn → M, j(t) = x + yt
∀ t ∈ Rn ,
(2)
∀ t = (ti ) ∈ Rn . where (yt)µ = yiµ ti Let ϕ be a local (orientation preserving) diffeomorphism of M , it acts on F (M ) by ϕ, j → ϕ ◦ j = ϕ e (j)
(3)
which replaces x by ϕ(x) and y by ϕ0 (x) y where α ϕ0 (x)α β = ∂β ϕ(x)
where ϕ(x) = (ϕ(x)α ).
(4)
We restrict our attention to orientation preserving frames F + (M ), and in the one dimensional case (n = 1) we take the notation y = e−s , s ∈ R.
(5)
In terms of the coordinates x, s one has, ϕ(s, e x) = (s − log ϕ0 (x), ϕ(x))
(6)
and the invariant measure on F is (n = 1) dx dy = es ds dx. y2
(7)
One has a canonical right action of GL+ (n, R) on F + which is given by (g, j) → j ◦ g, g ∈ GL+ (n, R), j ∈ F + .
(8)
It replaces y by yg, (yg)µj = yiµ gji ∀ g ∈ GL+ (n, R) and F + is a GL(n, R)+ principal bundle over M . We let Yij be the vector fields on F + generating the action of GL+ (n, R), Yij = yiµ
∂ = yiµ ∂µj . ∂ yjµ
In the one dimensional case one gets a single vector field,
(9)
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Y = −∂s .
(10)
The action of Diff + on F + preserves the Rn valued 1-form on F + , αj = (y −1 )jβ dxβ .
(11)
One has yjµ αj = dxµ and dyjµ ∧ αj + yjµ dαj = 0. Given an affine torsion free connection 0, the associated one form ω, ωj` = (y −1 )`µ (dyjµ + 0µα,β yjα dxβ )
(12)
is a 1-form on F + with values in GL(n) the Lie algebra of GL+ (n, R). The 0µα,β only depend on x but not on y, moreover one has, dαj = αk ∧ ωkj = −ωkj ∧ αk 0µα,β
(13)
0µβ,α .
= since 0 is torsion free, i.e. The natural horizontal vector fields Xi on F + associated to the connection 0 are, Xi = yiµ (∂µ − 0βα,µ yjα ∂βj ),
(14)
they are characterized by hαj , Xi i = δij
and
hωk` , Xi i = 0.
(15)
For ψ ∈ Diff + , the one form ψe∗ ω is still a connection 1-form for a new affine torsion free connection 00 . The new horizontal vector fields Xi0 are related to the old ones by e ϕ = ψ −1 . e∗ Xi ◦ ψ, Xi0 = ϕ
(16)
When ω is the trivial flat connection 0 = 0 one gets 00 = ψ 0 (x)−1 dψ 0 (x), 0µα,β = (ψ 0 (x)−1 )µρ ∂β ∂α ψ ρ (x).
(17)
2. Crossed Product of F (M ) by 0 and Action of H(n) We let M be gifted with a flat affine connection ∇ and let 0 be a pseudogroup of partial diffeomorphisms, preserving the orientation, ψ : Dom ψ → Range ψ,
(1)
where both the domain, Dom ψ and range, Range ψ are open sets of M . By functoriality of the construction of F (M )+ we let ψe be the corresponding partial diffeomorphisms of F + (M ). We let A = Cc∞ (F + ) >/ 0 be the crossed product of F + by the action of 0 on F + . It can be described directly as Cc∞ (G), where G is the etale smooth groupoid, G = F + >/ 0,
(2)
an element γ of G being given by a pair (x, ϕ), x ∈ Range ϕ, while the composition is, (x, ϕ) ◦ (y, ψ) = (x, ϕ ◦ ψ)
if
y ∈ Dom ϕ and
ϕ(y) = x.
(3)
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In practice we shall generate the crossed product A as the linear span of monomials, f Uψ∗ , f ∈ Cc∞ (Dom ψ),
(4)
where the star indicates a contravariant notation. The multiplication rule is f1 Uψ∗1 f2 Uψ∗2 = f1 (f2 ◦ ψe1 ) Uψ∗2 ψ1 ,
(5)
where by hypothesis the support of f1 (f2 ◦ ψe1 ) is a compact subset of Dom ψ1 ∩ ψ1−1 Dom ψ2 ⊂ Dom ψ2 ψ1 .
(6)
The canonical action of GL+ (n, R) on F (M )+ commutes with the action of 0 and thus extends canonically to the crossed product A. At the Lie algebra level, this yields the following derivations of A, Y`j (f Uψ∗ ) = (Yj` f ) Uψ∗ .
(7)
Now the flat connection ∇ also provides us with associated horizontal vector fields Xi on F + (M ) (cf. Sect. 1) which we extend to the crossed product A by the rule, Xi (f Uψ∗ ) = Xi (f ) Uψ∗ .
(8)
Now, of course, unless the ψ’s are affine, the Xi do not commute with the action of ψ, but using (16) and (17) of Sect. 1 we can compute the corresponding commutator and get, k Ykj , (9) Xi − Uψ Xi Uψ∗ = −γij k are, where the functions γij k γij = yiµ yjα (y −1 )kβ 0βαµ ,
(10)
0µα,β = (ψ 0 (x)−1 )µρ ∂β ∂α ψ ρ (x). It follows that, for any a, b ∈ A one has k (a) Ykj (b), Xi (ab) = Xi (a) b + a Xi (b) + δij
(11)
k where the linear operators δij in A are defined by, k k δij (f Uψ∗ ) = γij f Uψ∗ .
(12)
To prove (11) one takes a = f1 Uψ∗1 , b = f2 Uψ∗2 and one computes Xi (ab) = Xi (f1 Uψ∗1 f2 Uψ∗2 ) = Xi (f1 Uψ∗1 f2 Uψ1 ) Uψ∗2 ψ1 = Xi (f1 ) Uψ∗1 f2 Uψ∗2 + f1 (Xi Uψ∗1 − Uψ∗1 Xi ) f2 Uψ∗2 + f1 Uψ∗1 Xi (f2 Uψ∗2 ). One then uses (9) to get the result. k Next the γij are characterized by the equality ` ψe∗ ω − ω = γjk αk = γ α,
(13)
where α is the canonical Rn -valued one form on F + (M ) (cf. I). ∗ ∗ ∗ ∗ f f f ] The equality ψ 2 ψ1 ω − ω = ψ1 (ψ2 ω − ω) + (ψ1 ω − ω) together with the k k invariance of α thus show that the γij form a 1-cocycle, so that each δij is a derivation of the algebra A,
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A. Connes, H. Moscovici k k k δij (ab) = δij (a) b + a δij (b).
(14)
Since the connection ∇ is flat the commutation relations between the Yj` and the Xi are those of the affine group, (15) Rn >/ GL+ (n, R). c The commutation of the Yj` with δab are easy to compute since they correspond to the c tensorial nature of the δab . The Xi however do not have simple commutation relations c with the δab , and one lets c c = [Xi1 , . . . [Xin , δab δab,i ]...]. 1 ...in
(16)
All these operators acting on A are of the form, T (f Uψ∗ ) = h f Uψ∗ ,
(17)
where h = hψ is a function depending on ψ. In particular they all commute pairwise, 0
c , δac 0 b0 ,i0 ,... ,i0m ] = 0. [δab,i 1 ...in
(18)
1
c It follows that the linear space generated by the Yj` , Xi , δab,i forms a Lie algebra 1 ...in and we let H be the corresponding enveloping algebra. We endow H with a coproduct in such a way that its action on A,
h, a → h(a), h ∈ H, a ∈ A satisfies the following rule, X h(ab) = h(0) (a) h(1) (b)
∀ a, b ∈ A
where
1h =
(19) X
h(0) ⊗ h(1) .
(20)
One gets from the above discussion the equalities 1 Yij = Yij ⊗ 1 + 1 ⊗ Yij ,
(21)
k ⊗ Ykj , 1Xi = Xi ⊗ 1 + 1 ⊗ Xi + δij
(22)
k k k 1δij = δij ⊗ 1 + 1 ⊗ δij .
(23)
These rules, together with the equality 1(h1 h2 ) = 1h1 1h2
∀ hj ∈ H
(24)
suffice to determine completely the coproduct in H. As we shall see H as an antipode S, we thus get an Hopf algebra H(n) which only depends upon the integer n and which acts on any crossed product, (25) A = Cc∞ (F ) >/ 0 of the frame bundle of a flat manifold M by a pseudogroup 0 of diffeomorphisms. We shall spend most of this paper understanding the structure of the Hopf algebra H(n) as well as its cyclic cohomology. We shall concentrate on the case n = 1 for notational simplicity but all the results are proved in such a way as to extend in a straightforward manner to the general case. Let us show now that provided we replace A by a Morita equivalent algebra we can bypass the flatness condition of the manifold M . To do this we start with an arbitrary manifold M (oriented) and we consider a locally finite open cover (Uα ) of M by domains
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205
` of local coordinates. On N = Uα , the disjoint union of the open sets Uα , one has a natural pseudogroup 00 of diffeomorphisms which satisfy π ψ(x) = π(x)
∀ x ∈ Dom ψ,
(26)
where π : N → M is the natural projection. Equivalently one can consider the smooth etale groupoid which is the graph of the equivalence relation π(x) = π(y) in N , G0 = {(x, y) ∈ N × N ; π(x) = π(y)}.
(27)
One has a natural Morita equivalence, Cc∞ (M ) ' Cc∞ (G0 ) = Cc∞ (N ) >/ 00
(28)
which can be concretely realized as the reduction of Cc∞ (G0 ) by the idempotent, e ∈ Cc∞ (G0 ) = Cc∞ (N ) >/ 00 , e2 = e,
(29)
associated to a partition of unity in M subordinate to the cover (Uα ), X
ϕα (x)2 = 1, ϕα ∈ Cc∞ (Uα )
(30)
by the formula, e(u, α, β) = ϕα (u) ϕβ (u).
(31)
We have labelled the pair (x, y) ∈ G0 by u = π(x) = π(y) and the indices α, β so that x ∈ U α , y ∈ Uβ . This construction also works in the presence of a pseudogroup 0 of diffeomorphisms of M since there is a corresponding pseudogroup 00 on N containing 00 and such that, with the above projection e, (Cc∞ (N ) >/ 00 )e ' Cc∞ (M ) >/ 0.
(32)
Now the manifold N is obviously flat and the above construction of the action of the Hopf algebra H(n) gives an action on A0 = Cc∞ (F + (N )) >/ 00 , (A0 )e = A = Cc∞ (F + (M )) >/ 0. We can thus summarize the above discussion as follows, Proposition 1. a) For each n, let H(n) be the bialgebra which, as an algebra, is the c , enveloping algebra of the Lie algebra linearly generated by the Yj` , Xi , δab,i 1 ...in and is endowed with the coproduct given by (21), (22), (23). Then H(n) is a Hopf algebra. b) Let M be a smooth oriented manifold, 0 be a pseudogroup of orientation preserving diffeomorphisms of M , and A = Cc∞ (F + (M )) >/ 0. For each open cover (Uα ) of M by affine patches, one has ` a canonical action of H(n) on 0 the algebra Uα , and the algebras A and A are Morita A0 = Cc∞ (F + (N )) >/ 00 , with N = equivalent.
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Proof. a) One checks that the presentation of the Lie algebra linearly generated by the c , is compatible with the coproduct. The only point is to check the Yj` , Xi , δab,i 1 ...in existence of the antipode. The antipode S is characterized abstractly as the inverse of the element L(a) = a in the algebra of linear maps L from H to H with the product X X L1 (a(1) ) L2 (a(2) ), 1a = a(1) ⊗ a(2) , a ∈ H. (33) (L1 ∗ L2 )(a) = A simple computation shows that S is the unique antiautomorphism of H(n) such that, S(Yj` ) = −Yj` ,
c c S(δab ) = −δab ,
c S(Xa ) = −Xa + δab Ycb .
(34)
Note that the square of S is not the identity. b)This follows from the above discussion, the desired Morita equivalence is given by the equality (A0 )e = A = Cc∞ (F + (M )) >/ 0. Since cyclic cohomology is invariant under Morita equivalence we shall not need for this paper to refine the statement b) of the proposition, but we shall simply mention that the obstruction to lift the action of H(n) on A0 to an action on A, is exactly the curvature of the manifold M , and provides in general the correct generalization of the notion of Riemannian curvature in the framework of Noncommutative Geometry. 3. One Dimensional Case, the Hopf Algebras Hn We first define a bialgebra by generators and relations. As an algebra we view H as the enveloping algebra of the Lie algebra which is the linear span of Y , X, δn , n ≥ 1 with the relations, [Y, X] = X, [Y, δn ] = n δn , [δn , δm ] = 0
∀ n, m ≥ 1, [X, δn ] = δn+1
∀ n ≥ 1. (1)
We define the coproduct 1 by 1 Y = Y ⊗ 1 + 1 ⊗ Y , 1 X = X ⊗ 1 + 1 ⊗ X + δ1 ⊗ Y , 1 δ1 = δ1 ⊗ 1 + 1 ⊗ δ1 (2) and with 1 δn defined by induction using (1). One checks that the presentation (1) is preserved by 1, so that 1 extends to an algebra homomorphism, 1:H→H⊗H (3) and one also checks the coassociativity. For each n we let Hn be the algebra generated by δ1 , . . . , δn , Hn = {P (δ1 , . . . , δn ) ; P polynomial in n variables}. We let Hn,0 be the ideal,
Hn,0 = {P ; P (0) = 0}.
(4) (5)
By induction on n one proves the following Lemma 1. For each n there exists Rn−1 ∈ Hn−1,0 ⊗ Hn−1,0 such that 1 δn = δn ⊗ 1 + 1 ⊗ δn + Rn−1 .
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Proof. It holds for n = 1, n = 2. Assuming that it holds for n one has 1 δn+1 = 1 [X, δn ] = [1 X, 1 δn ] = [X ⊗ 1 + 1 ⊗ X + δ1 ⊗ Y, δn ⊗ 1 + 1 ⊗ δn + Rn−1 ] = δn+1 ⊗ 1+1⊗δn+1 +[X ⊗1+1⊗X, Rn−1 ]+δ1 ⊗[Y, δn ]+[δ1 ⊗Y, Rn−1 ] = δn+1 ⊗1+1⊗δn+1 +Rn , where (6) Rn = [X ⊗ 1 + 1 ⊗ X, Rn−1 ] + n δ1 ⊗ δn + [δ1 ⊗ Y, Rn−1 ]. Since [X, Hn−1,0 ] ⊂ Hn,0 and [Y, Hn−1,0 ] ⊂ Hn−1,0 ⊂ Hn,0 , one gets that Rn ∈ Hn,0 ⊗ Hn,0 . For each k ≤ n we introduce a linear form Zk,n on Hn ∂ hZk,n , P i = P (0). ∂ δk
(7)
One has by construction, hZk,n , P Qi = hZk,n , P i Q(0) + P (0) hZk,n , Qi
(8)
and moreover ε, hε, P i = P (0) is the counit in Hn , hL ⊗ ε, 1 P i = hε ⊗ L, 1 P i = hL, P i
∀ P ∈ Hn .
(9)
(Check both sides on a monomial P = δ1a1 . . . δnan .) Thus in the dual algebra Hn∗ one can write (8) as 1 Zk,n = Zk,n ⊗ 1 + 1 ⊗ Zk,n .
(10)
Moreover the Zk,n form a basis of the linear space of solutions of (10) and we need to determine the Lie algebra structure determined by the bracket. We let for a better normalization, 0 = (k + 1)! Zk,n . Zk,n
(11)
0 0 0 , Z`,n ] = (` − k) Zk+`,n if k + ` ≤ n and 0 if k + ` > n. Lemma 2. One has [Zk,n
Proof. Let P = δ1a1 . . . δnan be a monomial. We need to compute h1 P, Zk,n ⊗ Z`,n − Z`,n ⊗ Zk,n i. One has 1 P = (δ1 ⊗ 1 + 1 ⊗ δ1 )a1 (δ2 ⊗ 1 + 1 ⊗ δ2 + R1 )a2 . . . (δn ⊗ 1 + 1 ⊗ δn + Rn−1 )an . We look for the terms in δk ⊗ δ` or δ` ⊗ δk and take the difference. The latter is non-zero only if all aj = 0 except aq = 1. Moreover since Rm is homogeneous of degree m + 1 0 0 , Z`,n ] = 0 if k + ` > n. One then computes by one gets q = k + ` and in particular [Zk,n induction using (6) the bilinear part of Rm . One has R1(1) = δ1 ⊗ δ1 , and from (6) (1) Rn(1) = [(X ⊗ 1 + 1 ⊗ X), Rn−1 ] + n δ 1 ⊗ δn .
(12)
(1) Rn−1 = δn−1 ⊗ δ1 + Cn1 δn−2 ⊗ δ2 + . . . + Cnn−2 δ1 ⊗ δn−1 .
(13)
This gives
`−1 and we get Thus the coefficient of δk ⊗ δ` is Ck+` `−1 k−1 [Zk,n , Z`,n ] = (Ck+` − Ck+` ) Zk+`,n .
One has result.
(k+1)! (`+1)! (k+`+1)!
`−1 k−1 (Ck+` − Ck+` )=
`(`+1)−k(k+1) k+`+1
(14)
= ` − k thus using (11) one gets the
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For each n we let A1n be the Lie algebra of vector fields f (x) ∂/∂x
,
f (0) = f 0 (0) = 0
(15)
modulo xn+2 ∂/∂x. xk+1 0 ∂/∂x are related by (11) to Zk,n = xk+1 ∂/∂x which The elements Zk,n = (k+1)! satisfy the Lie algebra of Lemma 2. Thus A1n is the Lie algebra of jets of order (n + 1) of vector fields which vanish at order 2 at 0. Proposition 3. The Hopf algebra Hn is the dual of the enveloping algebra U (A1n ), Hn = U(A1n )∗ . Proof. This follows from the Milnor–Moore theorem.
Since the A1n form a projective system of Lie algebras, with limit the Lie algebra A1 of formal vector fields which vanish at order 2 at 0, the inductive limit H1 of the Hopf algebras Hn is, (16) H1 = U(A1 )∗ . The Lie algebra A1 is a graded Lie algebra, with one parameter group of automorphisms, αt (Zn ) = ent Zn which extends to U(A1 ) and transposes to U(A1 )∗ as ∂ h[Y, P ], ai = P, αt (a)t=0 ∀ P ∈ H1 , a ∈ U (A1 ). ∂t
(17)
(18)
Indeed (αt )t is a one parameter group of automorphisms of H1 such that αtt (δn ) = ent δn .
(19)
One checks directly that αtt is compatible with the coproduct on H1 and that the corresponding Lie algebra automorphism is (17). Now (cf. [Dx] 2.1.11) we take the basis of U(A1 ) given by the monomials, a
n−1 . . . Z2a2 Z1a1 , aj ≥ 0. Znan Zn−1
(20)
To each L ∈ U (A1 )∗ one associates (cf. [Dx] 2.7.5) the formal power series X L(Z an . . . Z a1 ) n 1 xa1 1 . . . xann , an ! . . . a 1 !
(21)
in the commuting variables xj , j ∈ N. It follows from [Dx] 2.7.5 that we obtain in this way an isomorphism of the algebra of polynomials P (δ1 , . . . , δn ) on the algebra of polynomials in the xj ’s. To determine the formula for δn in terms of the xj ’s, we just need to compute P
hδn , Znan . . . Z1a1 i.
(22)
Note that (22) vanishes unless j aj = n. In particular, for n = 1, we get ρ (δ1 ) = x1 ,
(23)
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209
where ρ is the above isomorphism. We determine ρ (δn ) by induction, using the derivation D(P ) =
X
δn+1
∂ (P ) ∂ δn
(24)
(which corresponds to P → [X, P ]). One has by construction, hδn , ai = hδn−1 , Dt (a)i
∀ a ∈ U (A1 ),
(25)
where Dt is the transpose of D. By definition of Zn as a linear form (7) one has, Dt Zn = Zn−1 , n ≥ 2 , Dt Z1 = 0.
(26)
Moreover the compatibility of Dt with the coproduct of H1 is ∀ a, b ∈ U (A1 ),
Dt (ab) = Dt (a) b + a Dt (b) + (δ1 a) ∂t b
(27)
where a → δ1 a is the natural action of the algebra H1 on its dual hP, δ1 ai = hP δ1 , ai
∀ P ∈ H1 , a ∈ U (A1 ).
(28)
To prove (27) one pairs both sides with P ∈ H1 . The `.h.s. gives hP, Dt (ab)i = h1 [X, P ], a ⊗ bi = h[X ⊗ 1 + 1 ⊗ X + δ1 ⊗ Y, 1 P ], a ⊗ bi. The terms in [X ⊗ 1, 1 P ] yield h1 P, Dt a ⊗ bi and similarly for [1 ⊗ X, 1 P ]. The term [δ1 ⊗ Y, 1 P ] yields h1 P, δ1 a ⊗ ∂t bi, thus one gets (27). Lemma 4. When restricted to U (A2 ), Dt is the unique derivation, with values in U (A1 ) satisfying (26), moreover D t (Znan . . . Z2a2 Z1a1 ) = Dt (Znan . . . Z2a2 ) Z1a1 + Znan . . . Z2a2
a1 (a1 − 1) a1 −1 Z1 . 2
Proof. The equality 1 δ1 = δ1 ⊗ 1 + 1 ⊗ δ1 shows that a → δ1 a is a derivation of U(A1 ). One has δ1 Zn = 0 for n 6= 1 so that δ1 = 0 on U(A2 ) and the first statement follows from (27) and (26). The second statement follows from, Dt (Z1m ) =
m(m − 1) m−1 Z1 2
which one proves by induction on m using (27).
(29)
Motivated by the first part of the lemma, we enlarge the Lie algebra A1 by adjoining an element Z−1 such that, [Z−1 , Zn ] = Zn−1
∀ n ≥ 2,
(30)
we then define Z0 by [Z−1 , Z1 ] = Z0 , [Z0 , Zk ] = k Zk .
(31)
The obtained Lie algebra A, is the Lie algebra of formal vector fields with Z0 = x ∂∂x , xn+1 ∂ Z−1 = ∂∂x and as above Zn = (n+1)! ∂ x. We can now compare Dt with the bracket with Z−1 . They agree on U(A2 ) and we need to compute [Z−1 , Z1m ]. One has
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m(m − 1) m−1 Z1 + m Z1m−1 Z0 (32) 2 Z1m−1 + m Z1m−1 Z0 Z1 + Z1m Z0 = m(m−1) + m Z1m + ([Z−1 , Z1m Z1 ] = m(m−1) 2 2 (m + 1) Z1m Z0 ). Thus, if one lets L be the left ideal in U(A) generated by Z−1 , Z0 we get, [Z−1 , Z1m ] =
Proposition 5. The linear map Dt : U(A1 ) → U(A1 ) is uniquely determined by the equality Dt (a) = [Z−1 , a] mod L. Proof. For each monomial Znan . . . Z1a1 one has Dt (a) − [Z−1 , a] ∈ L, so that this holds a for any a ∈ U (A1 ). Moreover, using the basis of U (A) given by the Znan . . . Z1a1 Z0a0 Z−1−1 we see that U(A) is the direct sum L ⊕ U(A1 ). a
(The linear span of the Znan . . . Z0a0 Z−1−1 with a0 + a−1 > 0 is a left ideal in U (A) b−1 a bm since the product (Zm . . . Z−1 ) (Znan . . . Z0a0 Z−1−1 ) can be expressed by decomposing c b−1 an c−1 bm (Zm . . . Z−1 Zn . . . Z1a1 ) as a sum of monomials Zq q . . . Z1c1 Z0c0 Z−1 which are then a0 a−1 multiplied by Z0 Z−1 which belongs to the augmentation ideal of U (Lie algebra of Z0 , Z1 ).) We now define a linear form L0 on U(A) by a
L0 (Znan . . . Z1a1 Z0a0 Z−1−1 ) = 0 unless a0 = 1, aj = 0
∀ j,
(33)
and L0 (Z0 ) = 1. Proposition 6. For any n ≥ 1 one has hδn , ai = L0 ([ |{z} . . . [Z−1 , a] . . . ])
∀ a ∈ U (A1 ).
n times
Proof. Let us first check it for n = 1. P We let a = Znan . . . Z1a1 . Then the degree of a is P j aj and L0 ([Z−1 , a]) 6= 0 requires j aj = 1 so that the only possibility is a1 = 1, aj = 0 ∀ j. In this case one gets L0 ([Z−1 , Z1 ]) = L0 (Z0 ) = 1. Thus by (23) we get the equality of Proposition 6 for n = 1. For the general case note first that L is stable under right multiplication by Z−1 and hence by the derivation [Z−1 , ·]. Thus one has (Dt )n (a) = [Z−1 , . . . [Z−1 , a] . . . ] mod L
∀ a ∈ U (A1 ).
(34) a
Now for a ∈ L one has L0 ([Z−1 , a]) = 0. Indeed writing a = (Znan . . . Z1a1 ) (Z0a0 Z−1−1 ) = a bc with b ∈ U(A1 ), c = Z0a0 Z−1−1 , one has [Z−1 , a] = [Z−1 , b] c + b [Z−1 , c]. Since b ∈ U(A1 ) and [Z−1 , c] has strictly negative degree one has L0 (b [Z−1 , c]) = 0. Let Znbn . . . Z1b1 Z0b0 be a non zero component of [Z−1 , b], then unless all bi are 0 it contributes by 0 to L0 ([Z−1 , b] c). But [Z−1 , b] ∈ U (A0 )0 has no constant term. Thus one has L0 ([Z−1 , a]) = 0
a
∀ a = Znan . . . Z1a1 Z0a0 Z−1−1
(35)
except if all aj = 0, j 6= 1 and a1 = 1. L0 ([Z−1 , Z1 ]) = 1. Using (25) one has hδn , ai = hδ1 , (Dt )n−1 (a)i and the proposition follows.
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One can now easily compute the first values of ρ (δn ), ρ (δ1 ) = x1 , ρ (δ2 ) = x2 + x3
x21 2 ,
ρ (δ3 ) = x3 + x2 x1 + 21 , ρ (δ4 ) = x4 + x3 x1 + 2 x22 + 2 x2 x21 + 43 x41 . The affine structure provided by the δn has the following compatibility with left multiplication in U(A1 ), P k k Proposition 7. a) One has Rn−1 = Rn−1 ⊗ δk , Rn−1 ∈ Hn−1,0 . 1 k b) For fixed a0 ∈ U (A ) there are λn ∈ C such that X λkn hδk , ai. hδn , (a0 a)i = hδn , a0 i ε(a) + k , a0 i. Proof. a) By induction using (6). b) Follows, using λkn = hRn−1
The antipode S in U (A1 ) is the unique antiautomorphism such that S Zn = −Zn
∀ n.
(36)
It is non-trivial to express in terms of the coordinates δn . In fact if we use the basis Zj of A1 but in reverse order to construct the map ρ z2
we obtain a map ρe whose first values are ρe (δ1 ) = z1 , ρe (δ2 ) = z2 + 21 , ρe (δ3 ) = am z3 +3P z1 z2 + 21 z13 , ρe (δ4 ) = z4 +2 z22 +6 z1 z3 +9 z12 z2 + 43 z14 . One has hδn , S (Zm . . . Z1a1 )i = P aj am am (−1) P hδn , Z1a1 . . . Zm i so that ρ (S t δn ) = hδn , S (Zm . . . Z1a1 )i xa1 1 . . . xamm = P am i xa1 1 . . . xamm = ρe (δn ) with zj = −xj in the latter expres(−1) aj hδn , Z1a1 . . . Zm sion. x3 x2 Thus ρ (S t δ1 ) = −x1 , ρ (S t δ2 ) = −x2 + 21 , ρ (S t δ3 ) = −x3 +3 x1 x2 − 21 , ρ (S t δ4 ) = −x4 + 2 x22 + 6 x1 x3 − 9 x21 x2 + 43 x41 . We thus get S t δ1 = −δ1 , S t δ2 = −δ2 + δ12 , S t δ3 = −δ3 + 4 δ1 δ2 − 2 δ13 , . . . .
(37)
The antipode S is characterized abstractly as the inverse of the element L(a) = a in the algebra of linear maps L from U(A1 ) to U (A1 ) with the product X X L1 (a(1) ) L2 (a(2) ), 1a = a(1) ⊗ a(2) , a ∈ U . (38) (L1 ∗ L2 )(a) = Thus one has
X
(S t δn,(1) ) δn,(2) = 0
∀ n , 1 δn =
X
δn,(1) ⊗ δn,(2)
(39)
writing S t δn = −δn + Pn , where Pn (δ1 , . . . , δn−1 ) is homogeneous of degree n, this allows one to compute S t δn by induction on n. Note also that the expression σ = δ2 − 21 δ12 is uniquely characterized by ρ (σ) = x2
(40)
which suggests to define higher analogues of the Schwartzian as ρ−1 (xn ). Let us now describe in a conceptual manner the action of the Hopf algebra H∞ on the crossed product, (41) A = Cc∞ (F ) >/ 0 of the frame bundle of a 1-manifold M by the group 0 → Diff + M . We are given a flat connection γ on M , which we view as a GL equivariant section,
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γ : F → J(M )
(42)
j
from F to the space of jets Rn → M . For α ∈ F , γ (α) is the jet γ (α) = (exp ∇) ◦ α,
(43)
where exp ∇ is the exponential map associated to the connection ∇. We let G be the groupoid F >/ 0 and let (ϕ, α) ∈ G with s (ϕ, α) = α ∈ F , r (ϕ, α) = ϕ e α ∈ F , ϕ ∈ 0.
(44)
We let G(A1 ) ⊂ U(A1 ) (completed I-adically, I = augmentation ideal) be the group like elements, ψ , 1 ψ = ψ ⊗ ψ. (45) We then have a canonical homomorphism γ from G to G(A1 ) given by γ (ϕ, α) = γ (ϕ e α)−1 ◦ ϕ ◦ γ (α),
(46)
where we identify G(A1 ) with the group of germs of diffeomorphisms by the equality A f = f ◦ ϕ−1
∀ A ∈ G(A1 ) , f function on R.
Theorem 8. For any f ∈ Cc∞ (G) and n ∈ N one has, (δn f )(g) = δn (γ (g)−1 ) f (g)
∀ g ∈ G.
Proof. We first define a representation π of A1 in the Lie algebra of vector fields on F (R) preserving the differential form es ds dx, y = e−s , π (Zn ) = − n
xn xn+1 ∂s + ∂x . n! (n + 1)!
(47)
n+1
x es ds which is closed.) Let then H be the (One has iZn (es ds dx) = − xn! es dx − (n+1)! function of F (R) given by H(s, x) = s. (48)
By construction the representation π is in fact representing A, and moreover for any a ∈ U (A) one has, (49) L0 (a) = −(π(a) H)(0). a
Indeed, for a = Z0 the r.h.s. is 1 and given a monomial Znan . . . Z1a1 Z0a0 Z−1−1 , it vanishes if a−1 > 0 or if a0 > 1 and if a−1 = 0, a0 = 0. If a−1 = 0, a0 = 1 the only case in which it does not vanish is aj = 0 ∀ j > 0. One has π (Z−1 ) = ∂x and it follows from Proposition 6 that, hδn , ai = −(∂xn π(a) H)(0).
(50)
Now if a = A ∈ G(A1 ) we have, with ψ = ϕ−1 , that (π(a) f )(s, x) = f (s − log ψ 0 (x), ψ(x)) and we thus have,
∀ f function on F (R)
hδn , Ai = (∂xn log ψ 0 (x))x=0 .
(51) (52)
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
213
We now consider F (M ) with the same notations in local coordinates, i.e. (y, x) with y = e−s . In crossed product terms we have, δn (f Uϕ ) = f γn Uϕ , γn (y, x) = y n ∂xn (log ψ 0 (x)) , ψ = ϕ−1 .
(53)
Now Uϕ , as a function on G is the characteristic function of the set {(ϕ, α); α ∈ F } and one has γ(ϕ, α), for α = (y, x), given by t → (ϕ (x + y t) − ϕ (x))/y ϕ0 (x) = γ (ϕ, α)(t).
(54)
4. The Dual Algebra H∗ To understand the dual algebra H∗ we associate to L ∈ H∗ , viewed as a linear form on H, assumed to be continuous in the I-adic topology, the function with values in U(A1 ), f (s, t) , hf (s, t), P i = hL, P etX esY i
∀ P ∈ H1 .
(1)
We shall now write the product in H∗ in terms of the functions f (s, t). We first recall the expansional formula, ∞ Z X A+B = eu0 A B u1 A B . . . eun A π duj . (2) e P uj =1,uj ≥0
n=0
We use this formula to compute 1 etX , say with t > 0, ∞ Z X 1 etX = 5 dsi δ1 (s1 ) . . . δ1 (sn ) etX ⊗ Y (s1 ) . . . Y (sn ) etX , (3) n=0
0≤s1 ≤...≤sn ≤t
where δ1 (s) = esX δ1 e−sX , Y (s) = esX Y e−sX = Y − sX. One has, (Y − s1 X) etX esY = (∂s + (t − s1 ) ∂t ) etX esY .
(4)
(Since etX Y esY = (Y − tX) etX esY .) We thus get the following formula for the product, (t > 0), ∞ Z X (f1 f2 )(s, t) = 5 dsi δ1 (s1 ) . . . n=0
0≤s1 ≤...≤sn ≤t
(5)
δn (sn ) f1 (s, t) (∂s + (t − sn ) ∂t ) . . . (∂s + (t − s1 ) ∂t ) f2 (s, t). We apply this by taking for f1 the constant function, f1 (s, t) = ϕ ∈ G(A1 ) ⊂ U(A1 ),
(6)
while we take the function f2 to be scalar valued. ∞ X n One has δ1 (s) = esX δ1 e−sX = δn+1 sn! , and its left action on U (A1 ) is given, n=0
on group like elements ϕ by δ1 (s) ϕ = hδ1 (s), ϕi ϕ.
(7)
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(Using hδ1 (s) ϕ, P i = hϕ, P δ1 (s)i = h1 ϕ, P ⊗ δ1 (s)i = hϕ, P i hϕ, δ1 (s)i.) Moreover by (50) Sect. 3, one has hδ1 (s), ϕi = −
X sn ∂ n+1 (π(ϕ) H)0 n! x
(8)
while, with ψ = ϕ−1 , one has (π(ϕ) H)(s, x) = s − log ψ 0 (x), so that (8) gives 00 00 X sn ψ ψ n ∂ (x)x=0 = (s), ψ = ϕ−1 . hδ1 (s), ϕi = n! x ψ 0 ψ0 Thus we can rewrite (5) as (t > 0) ∞ Z X (ϕ f )(s, t) = n=0
0≤s1 ≤...≤sn ≤t
5 dsi
(9)
n 00 Y ψ
ψ0
1
(si ) (10)
(∂s + (t − sn ) ∂t ) . . . (∂s + (t − s1 ) ∂t ) f (s, t). We first apply this formula to f (s, t) = f (s), independent of t, we get Z t 00 n ∞ X 1 ψ ∂sn f (s) = f (s + log ψ 0 (t)). (s) ds n! ψ0 0 n=0
We then apply it to f (s, t) = t. The term (∂s + (t − sn ) ∂t ) . . . (∂s + (t − s1 ) ∂t ) f gives (t − sn ), thus we get, Z sn 00 n−1 ∞ Z X ψ 00 (sn ) 1 ψ (t − sn ) dsn + t (u) du 0 ψ0 0≤sn ≤t ψ (sn ) (n − 1)! 0 n=0
Z
t
=t+ 0
Z
t
=t+
ψ 00 (s) (exp log ψ 0 (s))(t − s) ds ψ 0 (s)
ψ 00 (s)(t − s) ds = t + ψ(t) − t ψ 0 (0) − ψ(0) . k 1
0
k 0
Thus in general we get, (ϕ f )(s, t) = f (s + log ψ 0 (t), ψ(t)) , ψ = ϕ−1 .
(11)
5. Hopf Algebra H(G) Associated to a Pair of Subgroups In this section we recall a basic construction of Hopf algebras ([K, B-S, M]). We let G be a finite group, G1 , G2 be subgroups of G such that, G = G1 G2 ,
(1)
i.e. we assume that any g ∈ G admits a unique decomposition as g = ka
,
k ∈ G1 , a ∈ G2 .
(2)
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
215
Since G1 = G/G2 one has a natural left action of G on G1 which for g ∈ G1 coincides with the left action of G1 on itself, it is given by g(k) = π1 (gk)
∀ g ∈ G, k ∈ G1 ,
(3)
where πj : G → Gj are the two projections. For g ∈ G1 one has g(k) = gk while for a ∈ G2 one has, a(1) = 1
∀ a ∈ G2 .
(4)
(Since π1 (a) = 1 ∀ a ∈ G2 .) Since G2 = G1 /G, one has a right action of G on G2 which restricted to G2 ⊂ G is the right action of G2 on itself, a · g = π2 (ag)
∀ a ∈ G2 , g ∈ G.
(5)
As above one has 1·k =1
∀ k ∈ G1 .
(6)
Lemma 1. a) For a ∈ G2 , k1 , k2 ∈ G1 one has a(k1 k2 ) = a(k1 )((a · k1 )(k2 )). m b)] For k ∈ G1 , a1 , a2 ∈ G2 one has (a1 a2 ) · k = (a1 · a2 (k))(a2 · k). Proof. a) One has a k1 = k10 a0 with k10 = a(k1 ), a0 = a · k1 . Then (a k1 ) k2 = k10 a0 k2 = k10 k20 a00 with k20 = a0 (k2 ). Thus k10 k20 = a(k1 k2 ) which is the required equality. b) One has a2 k = k 0 a02 with k 0 = a2 (k), a02 = a2 · k. Then a1 a2 k = a1 (k 0 a02 ) = (a1 k 0 ) a02 = k 00 a01 a02 , where a01 = a1 · k 0 , thus (a1 a2 ) · k = a01 a02 as required. One defines a Hopf algebra H as follows. As an algebra H is the crossed product of theP algebra of functions h on G2 by the action of G1 . Thus elements of H are of the form hk Xk with the rule, Xk h Xk−1 = k(h), k(h) (a) = h(a · k)
∀ a ∈ G2 , k ∈ G1 .
The coproduct 1 is defined as follows, X εa ⊗ εb , εc (g) = 1 if g = c and 0 otherwise. 1 εc =
(7)
(8)
ba=c
1 Xk =
X
hkk0 Xk ⊗ Xk0 , hkk0 (a) = 1 if k 0 = a(k) and 0 otherwise.
(9)
k0
One first checks that 1 defines a covariant representation. The equality (8) defines a representation of the algebra of functions on G2 . Let us check that (9) defines a (4) that 1 X1 = X1 ⊗ X1 . One has representation ofX G1 . First, for k = 1 one gets byX k1 k2 0 0 h k 0 X k 1 h k 0 X k 2 ⊗ X k1 k2 = hkk10 k1 (hkk20 ) Xk1 k2 ⊗ Xk10 k20 . 1 X k 1 1 Xk 2 = k10 ,k20
1
2
k10 ,k20
1
2
For a ∈ G2 one has (hkk10 k1 (hkk20 ))(a) 6= 0 only if k10 = a(k1 ), k20 = (a · k1 )(k2 ). Thus 1 2 given a there is only one term in the sum to contribute, and by Lemma 1 , one then has k10 k20 = a(k1 k2 ), thus, X 1 X k 1 1 Xk 2 = hkk100k2 Xk1 k2 ⊗ Xk00 = 1 Xk1 k2 . (10)
216
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Next, one has 1 Xk 1 εc =
XX k0
that 1 Xk 1 εc = One has 1 εc·k−1 1 Xk =
X k0
hkk0 Xk εa ⊗ Xk0 εb , and Xk εa = εa·k−1 Xk , so
ba=c
XX k0
hkk0 εa·k−1 Xk ⊗ εb·k0 −1 Xk0 .
(11)
ba=c
X
hkk0 εa0 Xk ⊗ εb0 Xk0 . In (11), given a, b with
b0 a0 =c·k−1
ba = c the only k 0 that appears is k 0 = (a · k −1 )(k). But (a · k −1 )(k) = a(k −1 )−1 and −1 b · k 0 = b · a(k −1 ) so that by Lemma 1, (b · a(k −1 ))(a · k −1 ) = (ba) · k −1 = c · k −1 . Thus one gets 1 Xk 1 εc = 1 εc·k−1 1 Xk , (12) which shows that 1 defines an algebra homomorphism. To show that the coproduct 1 is coassociative let us identify the dual algebra H∗ with the crossed product, (13) (G1 )space >/ G2 . For f Ua∗ ∈ H∗ we define the pairing with H by hh Xk , f Ua∗ i = f (k) h(a)
(14)
while the crossed product rules are Ua∗ f = f a Ua∗ , f a (k) = f (a(k)) ∗ = Ub∗ Ua∗ Uab
∀ k ∈ G1 ,
∀ a, b ∈ G2 .
(15)
What we need to check is (16) h1 h Xk , f Ua∗ ⊗ g Ub∗ i = hh Xk , f Ua∗ g Ub∗ i. X We can assume that h = εc so that 1 h Xk = εa Xk ⊗ εb Xa(k) . The left-hand side ba=c
of (16) is then f (k) g(a(k)) or 0 according to ba = c or ba 6= c which is the same as the right-hand side. Let us now describe the antipode S. The counit is given by ε (h Xk ) = h (1)
1 unit of G2 .
(17)
We can consider the Hopf subalgebra H1 of H given by the h Xk , for k = 1. The antipode S 1 of H1 is given by (group case) (S 1 h)(a) = h (a−1 ) = e h (a).
(18)
−1 εa−1 . S (εa Xk ) = Xa(k)
(19)
Thus it is natural to expect,
One needs to check that given c ∈ G2 one has X X S (εa Xk ) εb Xa(k) = εa Xk S (εb Xa(k) ) = ε (εc Xk ). ba=c
(20)
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
The first term is
X
217
−1 Xa(k) εa−1 εb Xa(k) which is 0 unless c = 1. When c = 1 it is equal
ba=c
to 1 since the (a−1 ) · a(k) = (a · k)−1 label G2 when a varies in G2 . −1 εb−1 which is non-zero only if a · Similarly the second term gives εa Xk Xb(a(k)) −1 −1 −1 k c(k) = b , i.e. a · k = b · c(k), i.e. if c · k = 1 (since by Lemma 1, one has (b−1 c) · k = ((b−1 ) · c(k))(c · k)). Thus c = 1 and the sum gives 1. Let us now compute the H-bimodule structure of H∗ . Lemma 2. a) The left action of h Xk ∈ H on f Ua∗ ∈ H∗ is given by (h Xk ) · (f Ua∗ ) = ha Xk (f ) Ua∗ , where for k1 ∈ G1 , ha (k1 ) = h(a · k1 ) and Xk (f )(k1 ) = f (k1 k). b) The right action of h Xk ∈ H on f Ua∗ ∈ H∗ is given by (f Ua∗ ) · (h Xk ) = −1 ∗ h(a) L−1 k (f ) Ua·k , where for k1 ∈ G1 , (Lk f )(k1 ) = f (k k1 ). Proof. a) By definition h(h Xk ) · f Ua∗ , h0 Xk0 i = hf Ua∗ , h0 Xk0 h Xk i. Thus one has to check that f (k0 k) h0 (a) h(a · k0 ) = h0 (a) (ha Xk (f ))(k0 ) which is clear. b) One has h(f Ua∗ )·h Xk , h0 Xk0 i = hf Ua∗ , h Xk h0 Xk0 i = f (k k0 ) h(a) h0 (a·k) while ∗ hh(a) L−1 k (f ) Ua·k , h0 Xk0 i = h(a) f (k k0 ) h0 (a · k). 6. Duality Between H and Cc∞ (G1 ) >/ G2 , G = G1 G2 = Diff R Let, as above, H be the Hopf algebra generated by X, Y , δn . While there is a formal group G(A1 ) associated to the subalgebra A1 of the Lie algebra of formal vector fields, there is no such group associated to A. As a substitute for this let us take, G = Diff (R).
(1)
(We take smooth ones but restrict to real analytic if necessary.) We let G1 ⊂ G be the subgroup of affine diffeomorphisms, ∀ x ∈ R,
(2)
ϕ ∈ G, ϕ(0) = 0, ϕ0 (0) = 1.
(3)
k(x) = ax + b and we let G2 ⊂ G be the subgroup,
Given ϕ ∈ G it has a unique decomposition ϕ = k ψ, where k ∈ G1 , ψ ∈ G2 and one has, ϕ(x) − ϕ(0) . (4) a = ϕ0 (0), b = ϕ(0), ψ(x) = ϕ0 (0) The left action of G2 on G1 is given by applying (4) to x → ϕ(ax + b), for ϕ ∈ G2 . This gives (5) b0 = ϕ(b), a0 = a ϕ0 (b) which is the natural action of G2 on the frame bundle F (R). Thus, Lemma 1. The left action of G2 on G1 coincides with the action of G2 on F (R). Let us then consider the right action of G1 on G2 . In fact we consider the right action of ϕ1 ∈ G on ϕ ∈ G2 , it is given by (ϕ · ϕ1 )(x) =
ϕ(ϕ1 (x)) − ϕ(ϕ1 (0)) . ϕ0 (ϕ1 (0)) ϕ01 (0)
(6)
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A. Connes, H. Moscovici
Lemma 2. a) The right action of G on G2 is affine in the coordinates δn on G2 . b) When restricted to G1 it coincides with the action of the Lie algebra X, Y . Proof. a) By definition one lets δn (ψ) = (log ψ 0 )(n) (0).
(7)
With ψ = ϕ · ϕ1 , the first derivative ψ 0 (x) is ϕ0 (ϕ1 (x)) ϕ01 (x)/ϕ0 (ϕ1 (0)) ϕ01 (0) so that up to a constant one has, log ψ 0 (x) = (log ϕ0 ) (ϕ1 (x)) + log ϕ01 (x).
(8)
Differentiating n times the equality (8) proves a). To prove b) let ϕ1 (x) = ax + b while, up to a constant, ∞ X δn0 n x (log ϕ )(x) = n! 0
,
δn0 = δn (ϕ).
(9)
1
Then the coordinates δn = δn (ϕ · ϕ1 ) are obtained by replacing x by ax + b in (9), which gives ∂ 0 δn = δn+1 at b = 0, a = 1. (10) δn = an δn0 if b = 0, ∂b We now consider the discrete crossed product of Cc∞ (G1 ) by G2 , i.e. the algebra of finite linear combinations of terms f Uψ∗ , f ∈ Cc∞ (G1 ) , ψ ∈ G2 ,
(11)
where the algebraic rules are Uψ∗ f = (f ◦ ψ) Uψ∗ .
(12) Cc∞ (G1 ) >/ G2
We define a pairing between the (enveloping) algebra H and equality, hh Xk , f Uψ∗ i = h(ψ) f (k) ∀ k ∈ G1 , ψ ∈ G 2 .
by the (13)
In order to make sense of (13) we need to explain how we write an element of H in the form h Xk . Given a polynomial P (δ1 , . . . , δn ), we want to view it as a function on G2 in such a way that the left action of that function h given by lemma 2 of Sect. 5 coincides with the multiplication of Uψ∗ by P (γ1 , . . . , γn ), γj =
∂ ∂x
j
log ψ 0 (x) e−js , k = (e−s , x) ∈ G1 .
(14)
The formula of Lemma 2 of Sect. 5 gives the multiplication by h(ψ · k)
(15)
which shows that with δn defined by (7) one has, h = P (δ1 , . . . , δn ).
(16)
We then need to identify the Lie algebra generated by X, Y with the Lie algebra G1 of G1 (generated by Z−1 , Z0 ) in such a way that the left action of the latter coincides with
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
Xf
Uψ∗
=
e
−s
∂ f ∂x
Uψ∗
, Yf
Uψ∗
=−
∂ f ∂s
219
Uψ∗
(k = (e−s , x)).
(17)
The formula of Lemma 2 of Sect. 5 gives Xk f Uψ∗ = (Xk f ) Uψ∗ with (Xk f ) (k1 ) = f (k1 k). One has f (k) = f (s, x) for k = (e−s , x), i.e. k(t) = e−s t+x. With k1 = (e−s1 , x1 ) and k(ε) = (eε , 0) one gets ∂ ∂ f (k1 k)ε=0 = − f (k1 ) = (Y f )(k1 ) ∂ε ∂s1
(18)
so that Y corresponds to the one parameter subgroup (eε , 0) of G1 . With k(ε) = (1, ε) one has ∂ f (k1 kε )ε=0 = (e−s ∂x1 f )(k1 ) = (Xf )(k1 ) (19) ∂ε so that X corresponds to the one parameter subgroup (1, ε) of G1 . Now the element etX esY of G1 considered in Sect. 4 is given by k = etX esY = (es , t)
(20)
which has the effect of changing s to −s in our formulas and thus explains the equality (11) of Sect. 4. This gives a good meaning to (13) as a pairing between H and the crossed product Cc∞ (G1 ) >/ G2 . 7. Hopf Algebras and Cyclic Cohomology In this section we shall associate a cyclic complex (in fact a 3 module, where 3 is the cyclic category), to any Hopf algebra satisfying a suitable unimodularity condition. This unimodularity condition is fulfilled by the Hopf algebra H(n). The resulting cyclic cohomology seems the natural candidate for the analogue of Lie algebra cohomology in the context of Hopf algebras, where both the Hochschild cohomology (also called Sweedler cohomology) or the transposed (also called Harrison cohomology) give too naive results. In short, what we find is that the natural cosimplicial space, δ0 (h1 ⊗ . . . ⊗ hn−1 ) = 1 ⊗ h1 ⊗ . . . ⊗ hn−1 , δj (h1 ⊗ . . . ⊗ hn−1 ) = h1 ⊗ . . . ⊗ 1 hj ⊗ . . . ⊗ hn−1 , δn (h1 ⊗ . . . ⊗ hn−1 ) = h1 ⊗ . . . ⊗ hn−1 ⊗ 1, σi (h1 ⊗ . . . ⊗ hn+1 ) = h1 ⊗ . . . ε(hi+1 ) ⊗ . . . ⊗ hn+1 , 0 ≤ i ≤ n, attached to the coalgebra H together with its counit 1, 11 = 1 ⊗ 1, possesses a hidden cyclic structure. The latter is determined by giving the action on the n fold tensor power H⊗n of the operator τn which is, e 1 )) h2 ⊗ . . . ⊗ hn ⊗ 1, τn (h1 ⊗ . . . ⊗ hn ) = (1n−1 S(h e It is nontrivial that the where one uses the product in H⊗n and a twisted antipode S. n + 1 power (τn )n+1 = 1, and that the compatibility relations (cf. (9) below) with the cosimplicial structure are actually verified. We first return to our specific example and make sense of the right action of H on Cc∞ (G1 ) >/ G2 . We use the formula of Lemma 2.b) Sect. 5,
220
A. Connes, H. Moscovici ∗ (f Uψ∗ ) · (h Xk ) = h(ψ) L−1 k (f ) Uψ·k ,
(1)
where (L−1 k (f ))(k1 ) = f (k k1 ) for k1 ∈ G1 . For the action of functions h = P (δ1 , . . . , δn ) we see that the difference with the left action is that we multiply Uψ∗ by a constant, namely h(ψ). Next, since we took a discrete crossed product to get Cc∞ (G1 ) >/ G2 , we can only act by the same type of elements on the right, i.e. (2) Finite linear combinations of h Xk . e has little in common with H, but both are multipliers of the smooth This algebra H e acts on both sides on C ∞ (G1 ) >/ G2 but only the left crossed product by G1 . In fact, H c action makes sense at the Lie algebra level, i.e. as an action of H. e ⊗ H. e e since 1(etX ) cannot be written in H The coproduct 1 is not defined for H Thus there is a problem to make sense of the right invariance property of an n-cochain, ϕ(x0 , . . . , xn ) ; xj ∈ Cc∞ (G1 ) >/ G2
(3)
which we would usually write as X
ϕ(x0 y(0) , . . . , xn y(n) ) = ε(y) ϕ(x0 , . . . , xn )
(4)
P e for 1n y = y(0) ⊗ . . . ⊗ y(n) , y ∈ H. In fact it is natural to require as part of the right invariance property of the cochain, that it possesses the right continuity property in the variables ψj ∈ G2 so that the integration required in the coproduct formula (3) Sect. 4, does make sense. This problem does not arise for n = 0, in which case we define the functional, Z ∗ τ0 (f Uψ ) = 0 if ψ 6= 1, τ0 (f ) = f (s, x) es ds dx, (5) where we used k = (e−s , x) ∈ G1 . One has (f ◦ ψ)(s, x) = f (s − log ψ 0 (x), ψ(x)) by (5) Sect. 6, so that τ0 is a trace on the algebra Cc∞ (G1 ) >/ G2 = H∗ . Let us compute τ0 ((f Uψ∗ )(h Xk )) and compare it with ε(h Xk ) τ0 (f Uψ∗ ). First ∗ f Uψ∗ h Xk = h(ψ)(L−1 k f ) Uψ·k so that τ0 vanishes unless ψ · k = 1, i.e. unless ψ = 1. We can thus assume that ψ = 1. Then we just need to compare τ0 (L−1 k f ) with τ0 (f ). For k = (e−s1 , x1 ) one has k ◦ (e−s , x)(t) = e−s1 (e−s t + x) + x1 = e−(s+s1 ) t + (e−s1 x + x1 ). This corresponds to ψ(x) = e−s1 x + x1 and preserves τ0 . Thus Lemma 1. τ0 is a right invariant trace on H∗ = Cc∞ (G1 ) >/ G2 . Let us now introduce a bilinear pairing between H⊗(n+1) and H∗⊗(n+1) by the formula, hy0 ⊗ . . . ⊗ yn , x0 ⊗ x1 ⊗ . . . ⊗ xn i = τ0 (y0 (x0 ) . . . yn (xn ))
(6)
∀ yj ∈ H , xk ∈ H∗ . This pairing defines a corresponding weak topology and we let Definition 2. An n-cochain ϕ ∈ C n on the algebra H∗ is right invariant iff it is in the range of the above pairing.
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
221
We have a natural linear map λ from H⊗(n+1) to right invariant cochains on H∗ , given by λ(y0 ⊗ . . . ⊗ yn )(x0 , . . . , xn ) = hy0 ⊗ . . . ⊗ yn , x0 ⊗ . . . ⊗ xn i
(7)
and we investigate the subcomplex of the cyclic complex of H∗ given by the range of λ. It is worthwhile to lift the cyclic operations at the level of ∞ M
H⊗(n+1)
n=0
and consider λ as a morphism of 3-modules. Thus let us recall that the basic operations in the cyclic complex of an algebra are given on cochains ϕ(x0 , . . . , xn ) by, (δi ϕ)(x0 , . . . , xn ) = ϕ(x0 , . . . , xi xi+1 , . . . , xn )
i = 0, 1, . . . , n − 1,
(δn ϕ)(x0 , . . . , xn ) = ϕ(xn x0 , x1 , . . . , xn−1 ), (8) (σj ϕ)(x0 , . . . , xn ) = ϕ(x0 , . . . , xj , 1, xj+1 , . . . , xn ) j = 0, 1, . . . , n, (τn ϕ)(x0 , . . . , xn ) = ϕ(xn , x0 , . . . , xn−1 ). These operations satisfy the following relations: τn δi = δi−1 τn−1 1 ≤ i ≤ n, τn δ0 = δn , 2 τn σi = σi−1 τn+1 1 ≤ i ≤ n, τn σ0 = σn τn+1 ,
(9)
τnn+1 = 1n . In the first line δi : C n−1 → C n . In the second line σi : C n+1 → C n . Note that (σn ϕ)(x0 , . . . , xn ) = ϕ(x0 , . . . , xn , 1), (σ0 ϕ)(x0 , . . . , xn ) = ϕ(x0 , 1, x1 , . . . , xn ). The map λ maps H⊗(n+1) to C n thus there is a shift by 1 in the natural index n. We let δi (h0 ⊗ h1 ⊗ . . . ⊗ hi ⊗ . . . ⊗ hn−1 ) = h0 ⊗ . . . ⊗ hi−1 ⊗ 1 hi ⊗ hi+1 ⊗ . . . ⊗ hn−1 (10) and this makes sense for i = 0, 1, . . . , n − 1. One has τ0 (h0 (x0 ) . . . hi (xi xi+1 ) hi+1 (xi+2 ) . . . hn−1 (xn )) i h(0) (xi ) hi(1) (xi+1 ) . . . hn−1 (xn )), δn (h0 ⊗ h1 ⊗ . . . ⊗ hn−1 ) = which is compatible with h0 (xn x0 ) = property of τ0 . With ε : H → C the counit, we let
X P
=
P
τ0 (h0 (x0 ) . . .
h0(1) ⊗ h1 ⊗ . . . ⊗ hn−1 ⊗ h0(0)
(11)
h0(0) (xn ) h0(1) (x0 ), together with the trace
σj (h0 ⊗ . . . ⊗ hn+1 ) = h0 ⊗ . . . ⊗ ε(hj+1 ) hj+2 . . . ⊗ hn+1
j = 0, 1, . . . , n
(12)
which corresponds to hj (1) = ε(hj ) 1. Finally we let τn act on H⊗(n+1) by τn (h0 ⊗ . . . ⊗ hn ) = h1 ⊗ h2 ⊗ . . . ⊗ hn−1 ⊗ hn ⊗ h0
(13)
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which corresponds to τ0 (h0 (xn ) h1 (x0 ) . . . hn (xn−1 )) = τ0 (h1 (x0 ) . . . h0 (xn )). One checks that with these operations H\ is a 3-module where 3 is the cyclic category. To the relations (9) one has to add the relations of the simplicial 1, namely, i ≤ j, δj δi = δi δj−1 for i < j, σj σi = σi σj+1 δi σj−1 i < j if i = j or i = j + 1 σ j δi = 1 δ σ i > j + 1. i−1 j
(14)
The small category 3 is best defined as a quotient of the following category E 3. The latter has one object (Z, n) for each n and the morphisms f : (Z, n) → (Z, m) are non-decreasing maps, (n, m ≥ 1) f : Z → Z , f (x + n) = f (x) + m
∀ x ∈ Z.
(15)
In defining 3 (cf. [Co]) one uses homotopy classes of non decreasing maps from S 1 to S 1 of degree 1, mapping Z/n to Z/m. Given such a map we can lift it to a map satisfying (15). Such an f defines uniquely a homotopy class downstairs and, if we replace f by f + km, k ∈ Z the result downstairs is the same. When f (x) = a (m) ∀ x, one can restrict f to {0, 1, . . . , n − 1}, then f (j) is either a or a + m which labels the various choices. One has 3 = (E 3)/Z. We recall that δi is the injection that misses i, while σj is the surjection which identifies j with j + 1. Proposition 3. H\ is a 3-module and λ is a 3-module morphism to the 3-module C ∗ (H∗ ) of cochains on H∗ = Cc∞ (G1 ) >/ G2 . This is clear by construction. Now the definition of H\ only involves ((10) . . . (13)) the coalgebra structure of H, it is thus natural to compare it with the more obvious duality which pairs H⊗(n+1) with H∗⊗(n+1) namely, hh0 ⊗ h1 ⊗ . . . ⊗ hn , x0 ⊗ . . . ⊗ xn i =
n Y
hhj , xj i.
(16)
0
One has hhi , xi xi+1 i = h1 hi , xi ⊗ xi+1 i, so that the rules (10) and (11) are the correct ones. One has hhj+1 , 1i = ε(hj+1 ) so that (12) is right. Finally (13) is also right. This means that C ∗ (H∗ ) = H\∗∗ as 3-modules. Thus, λ : H\ → H\∗∗
(17)
is a cyclic morphism. To understand the algebraic nature of λ, let us compute it in the simplest cases first. We first take H = C G1 , where G1 is a finite group, and usePthe Hopf algebra H(G) for λk Xk , k ∈ G1 . As a right G = G1 , G2 = {e}. Thus as an algebra it is the group ring invariant trace on H∗ we take X f (k) , ∀ f ∈ H∗ . (18) τ0 (f ) = P
G1
The pairing hh, f i, for h = λk Xk , f ∈ H∗ is given by h(f ) is given by Lemma 2 Sect. 5, i.e.
P
λk f (k). The left action
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
h(f )(x) = Thus the two pairings are, for (16): X k,ki
X
X
λk f (xk).
223
(19)
h0 (k0 ) f0 (k0 ) . . . hn (kn ) fn (kn ) and for (6),
ki
h0 (k0 ) f0 (k k0 ) . . . hn (kn ) fn (k kn ). Thus at the level of the h0 ⊗ . . . ⊗ hn the map λ is just the sum of the left translates, X (Lg ⊗ Lg ⊗ . . . ⊗ Lg ). (20) G1
Next, we take the dual case, G1 = {e}, G2 = G, with G finite as above. Then H is the algebra of functions h on G2 , and the dual H∗ is the group ring of Gop 2 ' G2 , with generators Ug∗ , g ∈ G2 . For a trace τ0 on this group ring, the right invariance under H means that τ0 is the regular trace, X τ0 fg Ug∗ = fe . (21) This has a natural normalization, τ0 (1) = 1, for which we should expect λ to be an idempotent. The pairing between h and f is, X (22) hh, f i = h(g) fg . P Thus the two pairings (16) and (6) give respectively, for (16) h0 (g0 ) f0P (g0 ) h1 (g1 ) ∗ ∗ fg Ug∗ = fP 1 (g1 ) . . . hn (gn ) fn (gn ) and for (6), knowing that h(Ug ) = h(g) Ug , i.e. h ∗ h(g) fg Ug one gets, X h0 (g0 ) f0 (g0 ) h1 (g1 ) f1 (g1 ) . . . hn (gn ) fn (gn ). (23) gn ...g1 g0 =1
Thus, at the level of H⊗n+1 the map λ is exactly the localisation on the conjugacy class of e. These examples clearly show that in general Ker λ 6= {0}. Let us compute in our case how τ0 is modified by the left action of H on H∗ . By Lemma 2, Sect. 5 one has (h Xk )(f Uψ∗ ) = hψ Xk (f ) Uψ∗ and τ0 vanishes unless ψ = 1. In this case hψ is the constant h(1) = ε(h), while (24) Xk (f )(k1 ) = f (k1 k). R Thus we need to compare f ((e−s , x)(a, b)) es ds dx with its value for a = 1, b = 0. With k −1 = (a−1 , −b/a) the right multiplication by k −1 transforms (y, x) to (y 0 , x0 ) 0 ∧ dx0 = a dy with y 0 = y a−1 , x0 = x − y b/a, so that dy y 2 ∧ dx, y0 2 τ0 ((h Xk ) · f Uψ∗ ) = ε(h) δ(k) τ0 (f Uψ∗ ),
(25)
where the module δ of the group G1 is, δ(a, b) = a.
(26)
In fact we view δ as a character of H, with δ(h Xk ) = ε(h) δ(k).
(27)
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(Note that 1 · k = 1 for all k ∈ G1 so that (27) defines a character of H.) Thus in our case we have a (non trivial) character of H such that τ0 (y(x)) = δ(y) τ0 (x)
∀ x ∈ A, y ∈ H.
(28)
In fact we need to write the invariance property of τ0 as a formula for integrating by parts. To do this we introduce the twisted antipode, X X e = y(0) ⊗ y(1) . (29) S(y) δ(y(0) ) S(y(1) ) , y ∈ H, 1 y = e = S(σ(y)), where σ is the automorphism obtained by composing (δ ⊗ 1) ◦ One has S(y) 1 : H → H. One can view Se as δ ∗ S in the natural product (cf.(38) Sect. 5) on the algebra of linear maps from the coalgebra H to the algebra H. Since S is the inverse of the identity map, i.e. I ∗ S = S ∗ I = ε, one has (δ ∗ S) ∗ I = δ, i.e. X e (0) ) y(1) = δ(y) ∀ y ∈ H. (30) S(y The formula that we need as a working hypothesis on τ0 is, e (b)) τ0 (y (a) b) = τ0 (a S(y)
∀ a, b ∈ A, y ∈ H.
(31)
Using we shall now determine Ker λ purely algebraically. We let h = P i this formula h0 ⊗ . . . ⊗ hin ∈ H⊗(n+1) , we associate to h the following element of H⊗(n) : X e i0 )) hi1 ⊗ . . . ⊗ hin , t(h) = (1n−1 S(h (32) i
where we used both the coproduct of H and the product of H⊗(n) to perform the operations. Lemma 4. h ∈ Ker λ iff t(h) = 0. Proof. Let us first show that h − 1 ⊗ t(h) ∈ Ker λ for any h. One can assume that h = h0 ⊗ . . . ⊗ hn . Using (31) one has τ0 (h0 (x0 ) h1 (x1 ) . . . hn (xn )) = τ0 (x0 (Se h0 )(h1 (x1 ) . . . hn (xn ))) = τ0 (x0 (Se h0 )(0) h1 (x1 )(Se h0 )(1) h2 (x2 ) . . . (Se h0 )(n−1) hn (xn )). It follows that if t(h) = 0 then h ∈ Ker λ. Conversely, let us show that if 1 ⊗ t(h) ∈ Ker λ then t(h) = 0. We assume that the Haar measure τ0 is faithful, i.e. that ∀ a ∈ A implies b = 0. τ0 (ab) = 0 P e hin , one has Thus, with e h = t(h), e h= hi1 ⊗ . . . ⊗ e X e hi1 (x1 ) . . . e hin (xn ) = 0 ∀ xj ∈ A.
(34)
Applying the unit 1 ∈ H to both sides we get, X hin , xn i = 0 he hi1 , x1 i . . . he
(35)
which implies that e h = 0 in H⊗n .
∀ xj ∈ A,
(33)
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225
Definition 5. The cyclic module C ∗ (H) of a Hopf algebra H is the quotient of H\ by the kernel of t. We let HC ∗ (H) be the corresponding periodic cyclic cohomology. We shall give shortly an equivalent formulation of the above cyclic module as an additional cyclic structure on the cosimplicial space associated to the augmented coalgebra H. Note that to define t we needed the module δ : H → C, but that any reference to analysis has now disappeared in the definition of C ∗ (H). Note also that the construction of C ∗ (H) uses in an essential way both the coproduct and the product of H. As we shall see it provides a working definition of the analogue of Lie algebra cohomology in general. (Though we assumed τ0 was a trace. This is an unwanted unimodularity restriction which one should remove using the modular theory.) When H = U(G) is the enveloping algebra of a Lie algebra, there is a natural interpretation of the Lie algebra cohomology, H ∗ (G, C) = H ∗ (U(G), C),
(36)
where the right-hand side is the Hochschild cohomology with coefficients in the U(G)bimodule C obtained using the augmentation. In general, given a Hopf algebra H one can dualise (this is the construction of the Harrison complex), the construction of the Hochschild complex C n (H∗ , C), where C is viewed as a bimodule on H∗ using the augmentation, i.e. the counit of H∗ . This gives the following operations: H⊗(n−1) → H⊗n , defining a cosimplicial space δ0 (h1 ⊗ . . . ⊗ hn−1 ) = 1 ⊗ h1 ⊗ . . . ⊗ hn−1 , δj (h1 ⊗ . . . ⊗ hn−1 ) = h1 ⊗ . . . ⊗ 1 hj ⊗ . . . ⊗ hn−1 ,
(37)
δn (h ⊗ . . . ⊗ h ) = h ⊗ ... ⊗ h ⊗ 1, 1 n+1 1 i+1 σi (h ⊗ . . . ⊗ h ) = h ⊗ . . . ε(h ) ⊗ . . . ⊗ hn+1 ,
(38) (39)
1
n−1
1
n−1
0 ≤ i ≤ n.
Proposition 6. The map t is an isomorphism of cosimplicial spaces. to an element of the Proof.PModulo Ker t = Ker λ any element of H⊗(n+1) P 1 is equivalent form 1 ⊗ h1i ⊗ . . . ⊗ hni = ξ. One has t(ξ) = hi ⊗ . . . ⊗ hni . It is enough to show that the subspace 1 ⊗ H⊗∗ is a cosimplicial subspace isomorphic to (37) through t. Thus we let h0 = 1 in the definition (10) of δ1 and (11) of δn and check that they give (37). Similarly for σi . Thus, the underlying cosimplicial space of the cyclic module C ∗ (H) is a standard object of homological algebra attached to the coalgebra H together with its counit,1, 11 = 1 ⊗ 1. The essential new feature, due to the Hopf algebra structure is that this cosimplicial space carries a cyclic structure. The latter is determined by giving the action of τn which is, e 1 )) h2 ⊗ . . . ⊗ hn ⊗ 1, τn (h1 ⊗ . . . ⊗ hn ) = (1n−1 S(h
(40)
e It is nontrivial to check where one uses the product in H⊗n and the twisted antipode S. n+1 = 1, for instance for n = 1 this means that Se is an involution, i.e. directly that (τn ) 2 e S = 1. Note that the antipode S of the Hopf algebra of Sect. 3 is not an involution, while Se is one. The second assertion of the following proposition shows that, when applied to the Hopf algebra H of functions on an algebraic group, the cyclic cohomology HC ∗ (H),
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gives a highly nontrivial answer. The dual-Hochschild or Harrison cohomology gives simply the vector space of invariant twisted forms, and the operator B of cyclic cohomology captures their differential. Proposition 7. 1) The periodic cyclic cohomology HC ∗ (H), for H = U (G) the enveloping algebra of a Lie algebra G is isomorphic to the Lie algebra homology H∗ (G, C) where C is a G-module using the module δ of G. 2) The periodic cyclic cohomology HC ∗ (H), for H = U(G)∗ , is isomorphic to the Lie algebra cohomology of G with trivial coefficients, provided G is an affine space in the coordinates of H. This holds in the nilpotent case. Proof. One has a natural inclusion G ⊂ U (G). Let us consider the corresponding inclusion of 3n G in H⊗n , given by X (−1)σ Xσ(1) ⊗ . . . ⊗ Xσ(n) . (41) X1 ∧ . . . ∧ Xn → Let b : H⊗n → H⊗(n+1) be the Hochschild coboundary, one has Im b ⊕ 3n G = Ker b
∀ n.
(42)
For n = 1 one has b(h) = h ⊗ 1 − 1h + 1 ⊗ h so that b(h) = 0 iff h ∈ G. For n = 0, b(λ) = λ − λ = 0 so that b = 0. In general the statement (42) only uses the cosimplicial structure, i.e. only the coalgebra structure of H together with the element 1 ∈ H. This structure is unaffected if we replace the Lie algebra structure of G by the trivial commutative one. More precisely let us define the linear isomorphism, π : S(G) → U(G) , π(X n ) = X n
∀ X ∈ G.
(43)
Then 1 ◦ π = (π ⊗ π) ◦ 1S , where 1S is the coproductP of S(G). Indeed it is enough to check this equality on X n , X ∈ G and both sides give Cnk X k ⊗ X n−k . The result then follows by dualising the homotopy between the standard resolution and the Koszul resolution S(E) ⊗ 3(E) of the module C over S(E) for a vector space E. Let us then compute B(X1 ∧ . . . ∧ Xn ). Note that B0 (X1 ∧ . . . ∧ Xn ) corresponds to the functional (−1)σ τ0 (Xσ(1) (x0 ) . . . Xσ(n) (xn−1 )) which is already cyclic. Thus it is enough to compute B0 . One has P
e S(X) = −X + δ(X)
∀ X ∈ G,
(44)
e thus 1n−2 1 ⊗ . . . ⊗ X ⊗ . . . ⊗ 1 + δ(X) XS X = − X1 ⊗ 1 . . . ⊗ 1. We get B0 (X1 ∧ . . . ∧ σ (−1) δ(Xσ(1) ) Xσ(2) ⊗ . . . ⊗ Xσ(n) − (−1)σ Xσ(2) ⊗ . . . ⊗ Xσ(1) Xσ(j) ⊗ Xn ) = σ
. . .⊗Xσ(n) =
P
k+1
(−1)
∨
σ
δ(Xk ) X1 ∧. . . ∧ Xk ∧ . . .∧Xn +
X
(−1)i+j [Xi , Xj ]∧X1 ∧
i<j ∨
∨
. . . ∧ Xi ∧ . . . ∧ Xj ∧ . . . ∧ Xn . This shows that B leaves 3∗ G invariant and coincides there with the boundary map of Lie algebra homology. The situation is identical to what happens in computing cyclic cohomology of the algebra of smooth functions on a manifold. 2) The Hochschild complex of H is by construction the dual of the standard chain complex which computes the Hochschild homology of U (G) with coefficients in C (viewed as a bimodule using ε). Recall that in the latter complex the boundary dn is
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
227
dn (λ1 ⊗ . . . ⊗ λn ) = ε(λ1 ) λ2 ⊗ . . . ⊗ λn − λ1 λ2 ⊗ . . . ⊗ λn + . . . +(−1)i λ1 ⊗ . . . ⊗ λi λi+1 ⊗ . . . ⊗ λn + . . . + (−1)n λ1 ⊗ . . . ⊗ λn−1 ε(λn ).
(45)
One has an homotopy between the complex (45) and the subcomplex V (G) obtained by the following map from 3∗ G, X (−1)σ Xσ(1) ⊗ . . . ⊗ Xσ(n) ∈ U (G)⊗n . (46) X1 ∧ . . . ∧ Xn → This gives a subcomplex on which dn coincides with the boundary in Lie algebra homology with trivial coefficients, dt (X1 ∧. . .∧Xn ) =
X
∨
∨
(−1)i+j [Xi , Xj ]∧X1 ∧. . . ∧ Xi ∧ . . . ∧ Xj ∧ . . .∧Xn . (47)
i<j
It is thus natural to try and dualise the above homotopy to the Hochschild complex of H. Now a Hochschild cocycle X (48) h= hi1 ⊗ . . . ⊗ hin ∈ H⊗n gives an n-dimensional group cocycle on G = {g ∈ U (G)completed ; 1g = g ⊗ g}, where, X c(g1 , . . . , gn ) = hhi1 , g1 i . . . hhin , gn i. (49) These cocycles are quite special in that they depend polynomially on the gi ’s. Thus we need to construct a map of cochain complexes from the complex of Lie algebra cohomology to the complex of polynomial cocycles and prove that it gives an isomorphism in cohomology, (50) s : 3n (G)∗ → H⊗n . If we let j be the restriction of a polynomial cochain to 3(G)∗ we expect to have j ◦s = id and to have a homotopy, s ◦ j − 1 = dk + kd. (51) Using the affine coordinates on G we get for g 0 , . . . , g n ∈ G, an affine simplex ( n ) X X 0 n i λi g ; λi ∈ [0, 1], λi = 1 , 1(g , . . . , g ) =
(52)
0
moreover the right multiplication by g ∈ G being affine, we have, 1(g 0 g, . . . , g n g) = 1(g 0 , . . . , g n ) g.
(53)
The map s is then obtained by the following formula, (s ω)(g 1 , . . . , g n ) = γ(1, g 1 , g 1 g 2 , . . . , g 1 . . . g n ),
(54)
Z
where γ(g 0 , . . . , g n ) =
w 1(g 0 ,... ,g n )
where w is the right invariant form on G associated to ω ∈ 3n (G)∗ . To prove the existence of the homotopy (51) we introduce the bicomplex of the proof of the Van Est
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theorem but we restrict to forms with polynomial coefficients: An (G) and to group cochains which are polynomial. Thus an element of C n,m is an n-group cochain c(g 1 , . . . , g n ) ∈ Am
(55)
and one uses the right action of G on itself to act on forms, (g ω)(x) = ω (xg).
(56)
The first coboundary d1 is given by (d1 c)(g 1 , . . . , g n+1 ) = g 1 c(g 2 , . . . , g n+1 ) − c(g 1 g 2 , . . . , g n+1 ) + . . . +(−1)n c(g 1 , . . . , g n g n+1 ) + (−1)n+1 c(g 1 , . . . , g n ).
(57)
The second coboundary is simply (d2 c)(g 1 , . . . , g n ) = dc(g 1 , . . . , g n ).
(58)
(One should put a sign so that d1 d2 = −d2 d1 .) We need to write down explicitly the homotopies of lines and columns in order to check that they preserve the polynomial property of the cochains. In the affine coordinates δµ on G we let X ∂ (59) X= δµ ∂ δµ be the vector field which contracts G to a point. Then the homotopy k2 for d2 comes from, Z 1 dt (60) (iX ω)(t δ) kω = t 0 which preserves the space of forms with polynomial coefficients1 . The homotopy k1 for d1 comes from the structure of induced module, i.e. from viewing a cochain c(g1 , . . . , gn ) as a function of x ∈ G with values in 3m G∗ , (k1 c)(g1 , . . . , gn−1 )(x) = c(x, g1 , . . . , gn−1 )(e) .
(61)
One has (d1 k1 c)(g1 , . . . , gn )(x) = (k1 c)(g2 , . . . , gn )(x g1 ) − (k1 c)(g1 g2 , . . . , gn )(x) + . . . + (−1)n−1 (k1 c)(g1 , . . . , gn−1 gn )(x) + (−1)n (k1 c)(g1 , . . . , gn−1 )(x) = {c(x g1 , g2 , . . . , gn ) − c(x, g1 g2 , . . . , gn ) + . . .
1
+ (−1)n−1 c(x, g1 , . . . , gn−1 gn ) + (−1)n c(x, g1 , . . . , gn−1 )}(e) = c(g1 , . . . , gn )(x) − (d1 c)(x, g1 , . . . , gn )(e) = c(g1 , . . . , gn )(x) − (k1 d1 c)(g1 , . . . , gn )(x). R1
For instance for G = R, ω = f (x) dx one gets kω = α, α(x) = x
0
f (tx) dt. In general take
ω = P (δ) d δ1 ∧ . . . ∧ d δk , then iX ω is of the same form and one just needs to know that is still a polynomial in δ, which is clear.
R1 0
P (t δ) ta dt
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
229
This homotopy k1 clearly preserves the polynomial cochains. This is enough to show that the Hochschild cohomology of H is isomorphic to the Lie algebra cohomology H ∗ (G, C). But it follows from the construction of the cocycle s ω (54) that (s ω)(g1 , . . . , gn ) = 0
g1 . . . g n = 1 ,
if
(62)
and this implies (as in the case of discrete groups) that the corresponding cocycle is also cyclic. Our goal now is to compute the cyclic cohomology of our original Hopf algebra H. This should combine the two parts of Proposition 7. In the first part the Hochschild cohomology was easy to compute and the operator B was non trivial. In the second part b was non trivial. At the level of G1 , one needs to transform the Lie algebra cohomology into the Lie algebra homology with coefficients in Cδ . The latter corresponds to invariant currents on G and the natural isomorphism is a Poincar´e duality. For G1 with Lie algebra {X, Y }, [Y, X] = X, δ(X) = 0, δ(Y ) = 1, one gets that X ∧ Y is a 2-dimensional cycle, while since b Y = 1, there is no zero dimensional cycle. For the Lie algebra cohomology one checks that there is no 2-dimensional cocycle. We shall start by constructing an explicit map from the Lie algebra cohomology of G = A, the Lie algebra of formal vector fields, to the cyclic cohomology of H. As an intermediate step in the construction of the map θ we shall use the following double complex (C n,m , d1 , d2 ). For 0 ≤ k ≤ dim G1 and let k (G1 ) = k be the space / {0, . . . , dim G1 }. We let of de Rham currents on G1 , and we let k = {0} for k ∈ C n,m = {0} unless n ≥ 0 and − dim G1 ≤ m ≤ 0, and let C n,m be the space of totally → −m such that, antisymmetric polynomial maps γ : Gn+1 2 γ(g0 g, . . . , gn g) = g −1 γ(g0 , . . . , gn )
∀ g i ∈ G2 , g ∈ G
(63)
where we use the right action of G on G2 = G1 \G to make sense of gi g and the left action of G on G1 = G/G2 to make sense of g −1 γ. The coboundary d1 : C n,m → C n+1,m is given by (d1 γ)(g0 , . . . , gn+1 ) = (−1)m
n+1 X
∨
(−1)j γ(g0 , . . . , gj , . . . , gn+1 ).
(64)
j=0
The coboundary d2 : C n,m → C n,m+1 is the de Rham boundary, (d2 γ)(g0 , . . . , gn ) = dt γ(g0 , . . . , gn ).
(65)
For g0 , . . . , gn ∈ G2 , we let 1(g0 , . . . , gn ) be the affine simplex with vertices the gi in the affine coordinates δk on G2 . Since the right action of G on G2 is affine in these coordinates, we have 1(g0 g, . . . , gn g) = 1(g0 , . . . , gn ) g
∀ gi ∈ G2 , g ∈ G.
(66)
Let ω be a left invariant differential form on G associated to a cochain of degree k in the complex defining the Lie algebra cohomology of the Lie algebra A (cf. [G0]). For each pair of integers n ≥ 0, − dim G1 ≤ m ≤ 0 such that n + m = k − dim G1 , let Z m(m+1) hCn,m (g 0 , . . . , g n ), αi = (−1) 2 π1∗ (α) ∧ ω (67) (G1 ×1(g 0 ,... ,g n ))−1
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for any smooth diferential form α, with compact support on G1 and of degree −m. In this formula we use G1 × 1(g0 , . . . , gn ) as a cycle in G and we need to show that if Ki ⊂ Gi are compact subsets, the subset of G, K = {g ∈ G ; π1 (g) ∈ K1 , π2 (g −1 ) ∈ K2 }
(68)
is compact. For g ∈ K one has g = ka with k ∈ K1 and π2 (g −1 ) ∈ K2 so that π2 (a−1 k −1 ) ∈ K2 . But π2 (a−1 k −1 ) = a−1 · k −1 and one has a−1 ∈ K2 · K1−1 and a ∈ (K2 · K1−1 )−1 , thus, the required compactness follows from K ⊂ K1 (K2 · K1−1 )−1 .
(69)
∗
We let C (A) be the cochain complex defining the Lie algebra cohomology of A and let C be the map defined by (67). Lemma 8. The map C is a morphism to the total complex of (C n,m , d1 , d2 ). Proof. Let us first check the invariance condition (63). One has (66) so that for Cn,m (g 0 g, . . . , g n g) the integration takes place on {h ∈ G ; π2 (h−1 ) ∈ 1(g 0 , . . . , g n ) g} =
0 X
.
P0 P P = g −1 in G, with = {h ∈ Since π2 (h−1 ) g −1 = π2 (h−1 g −1 ) one has G ; π2 (h−1 ) ∈ 1(g 0 , . . . , g n )}. One has, with β = π1∗ (α) ∧ ω, the equality Z Z (70) P β = P Lg β, g −1
where g → Lg is the natural action of G on forms on G by left translation. One has π1 (gk) = g π1 (k) so that Lg π1∗ (α) = π1∗ (Lg α). Moreover ω is left invariant by hypothesis, so one gets, hCn,m (g 0 g, . . . , g n g), αi = hCn,m (g 0 , . . . , g n ), Lg αi,
(71)
which since Ltg = Lg−1 , is the invariance condition (63). Before we check that Cn,m is polynomial in the g i ’s, let us check that C(dω) = (d1 + d2 ) C(ω). One has
d(π1∗ (α) Z
∧ ω) =
π1∗ (dα)
∧ ω + (−1)
dβ =
m
X
π2 (g −1 )∈1(g 0 ,... ,g n+1 )
π1∗ (α)
Z
(72)
∧ dω and since
(−1)i
∨
β.
(73)
π2 (g −1 )∈1(g 0 ,... ,g i ,... ,g n ,g n+1 )
With β = π1∗ (α) ∧ ω, the r.h.s. gives (−1)m (d1 C) (g 0 , . . . , g n+1 ), while the l.h.s. gives hC(g 0 , . . . , g n+1 ), dαi + (−1)m hC 0 (g 0 , . . . , g n+1 ), αi, with C 0 = C(dω). Thus we get, 0 d1 Cn,m + d2 Cn+1,m−1 = Cn+1,m m(m+1)
(74)
provided we use the sign: (−1) 2 in the definition (67) of C. We shall now be more specific on the polynomial expression of Cn,m (g 0 , . . . , g n ) and write this de Rham current on G1 in the form,
Hopf Algebras, Cyclic Cohomology and Transverse Index Theorem
X
c j ρj ,
231
(75)
where ρ0 = 1, ρ1 = ds, ρ2 = es dx, ρ = ρ1 ∧ ρ2 form a basis of left G1 invariant forms on G1 , while the cj are functions on G1 which are finite linear combinations of finite products of the following functions, k ∈ G1 → δp (gj · k) , j ∈ {0, . . . , n}.
(76)
The equality (67) defines Cn,m as the integration over the fibers for the map π1 : G → G1 e of integration on (G1 × 1)−1 = of the product of the smooth form ω by the current 1 {g ∈ G ; π2 (g −1 ) ∈ 1}, e (77) Cn,m = (π1 )∗ (ω ∧ 1). e along the fibers, and its Thus c0 which is a function is obtained as the integral of ω ∧ 1 value at 1 ∈ G1 is Z c0 (1) =
g −1 ∈1
ω/G2 .
(78)
To obtain the value of cj (1) by a similar formula, one can contract by a vector field Z on G1 given by left translation, k ∈ G1 → ∂ε (k(ε)k)ε=0 , k(ε) ∈ G1 . Let Ze be the vector field on G given by the same left translation, g ∈ G1 → ∂ε (k(ε)g)ε=0 . The equality π1 (k(ε) g) = k(ε) π1 (g) shows that Ze is a left of Z for the fibration π1 . It follows that for any current β on G one has iZ π1∗ (β) = π1∗ (iZe β).
(79)
R ∗ π1 (iZ α) ∧ β = hiZ π1∗ R(β), αi = hπ1∗ (β), iZ αi = hβ, π1∗ (iZ α)i = R(One has iZe π1∗ (α) ∧ β = π1∗ (α) ∧ iZe β.) e e = (i ω) × 1 e + (−1)∂ω ω ∧ i 1. Next one has iZe (ω ∧ 1) e e The contribution of the Z Z first term is simple, Z g −1 ∈1
(iZe ω)/G2 .
(80)
To compute the contribution of the second term, one needs to understand the current e on G. In general, if β is the current of integration on a manifold M ,→ G (possibly iZe 1 with boundary), the contraction iZe β is obtained as a limit for ε → 0 from the manifold e e one gets the M × [−ε, ε] which maps to G by (x, t) → (exp t Z)(x). Applying this to 1 map from G1 × 1 × [−ε, ε] to G given by (k, a, t) → k(t) a−1 k −1 .
(81)
One has k(t) a−1 k −1 = (ka k(t)−1 )−1 = k 1 (a · k(t)−1 ))−1 where the a · k −1 (t) belong to the simplex 1 · k(t)−1 while k 1 ∈ G1 is arbitrary. Thus, if we let Z 0 be the vector field on G2 given by, a → ∂ε (a · k −1 (ε))ε=0 we see that the contribution of the second term is Z iZ 0 ω e /G2 , ω e (g) = ω(g −1 ). g∈1
(82)
(83)
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Thus there exists a differential form µj on G2 obtained from ω e by contraction by suitable vector fields and restriction to G2 , and such that, Z µj . (84) cj (1) = 1(g 0 ,... ,g n )
The value of cj at k ∈ G1 can now be computed using (71) for g = k ∈ G1 . The forms ρj are left invariant under the action of G1 and by (71) one has L−1 g C(g0 , . . . , gn ) = C(g0 g, . . . , gn g), g = k. Thus cj (k) corresponds to the current L−1 k C(g0 , . . . , gn ) evaluated at 1 and one has, Z Z µj = µj . (85) cj (k) = 1(g 0 ,... ,g n )·k
1(g 0 ·k,... ,g n ·k)
Let us now apply (71) for g ∈ G2 . The point 1 ∈ G1 , is fixed by the action of G2 while the forms ρj vary as follows under the action of G2 , −1 L−1 g ρj = ρj , j 6= 1 , Lg ρ1 = ρ1 − δ1 (g · k) ρ2
(at k ∈ G1 ).
(86)
Thus we see that while µj , j 6= 1 are right invariant forms on G2 , the form µ1 satisfies, µ1 (ag) = µ1 (a) − δ1 (g) µ2 (a).
(87)
Now in G2 the product rule as well as g → g −1 are polynomial in the coordinates δn . It follows that the forms µj are polynomial forms in these coordinates and that the formula (84) is a polynomial function of the δj (g k ). Using (85) we obtain the desired form (76) for ci . We shall now use the canonical map 8 of [Co] Theorem 14, p. 220, from the bicomplex (C n,m , d1 , d2 ) to the (b, B) bicomplex of the algebra H∗ = Cc∞ (G1 ) >/ G2 . What we need to prove is that the obtained cochains on H∗ are right invariant in the sense of Definition 2 above. Let us first rewrite the construction of 8 using the notation f Uψ∗ for the generators of H∗ . As in [Co], we let B be the tensor product, B = A∗ (G1 ) ⊗ 3 C (G02 )
(88)
where A∗ (G1 ) is the algebra of smooth forms with compact support on G1 , while we label the generators of the exterior algebra 3 C (G02 ) as δψ , ψ ∈ G2 , with δ0 = 0. We take the crossed product, (89) C = B >/ G2 of B by the product action of G2 , so that Uψ∗ ω Uψ = ψ ∗ ω = ω ◦ ψ
∀ ω ∈ A∗ (G1 ),
Uψ∗1 δψ2 Uψ1 = δψ2 ◦ψ1 − δψ1
∀ ψj ∈ G2 .
(90)
The differential d in C is given by d(b Uψ∗ ) = db Uψ∗ − (−1)∂b b δψ Uψ∗ ,
(91)
where the first term comes from the exterior differential in A∗ (G1 ). Thus the δψ play the role of
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233
−δψ = (d Uψ∗ ) Uψ = −Uψ∗ d Uψ .
(92)
A cochain γ ∈ C n,m in the above bicomplex determines a linear form γ e on C, by, e(b Uψ∗ ) = 0 γ e(ω ⊗ δg1 . . . δgn ) = hω, γ(1, g1 , . . . , gn )i , γ
if ψ 6= 1.
(93)
What we shall show is that the following cochains on H∗ satisfy Definition 2, e(dxj+1 . . . dx` x0 dx1 . . . dxj ) , xj ∈ H∗ . ϕ(x0 , . . . , x` ) = γ We can assume that γ(1, g1 , . . . , gn ) =
n Y
(94)
Pj (δ(gj · k) ρi (k), where each Pj is a poly-
j=1
nomial (in fact monomial) in the δn . We take the ρj , j = 0, 1, 2, 3 as a basis of A∗ (G1 ) viewed as a module over Cc∞ (G1 ) and for f ∈ Cc∞ (G1 ) we write df as df = −(Y f ) ρ1 + (Xf ) ρ2 , Y = −
∂ ∂ , X = e−s , ∂s ∂x
(95)
which is thus expressed in terms of the left action of H on H∗ = Cc∞ (G1 ) >/ G2 . Moreover, using (86), one has Uψ∗ ρj Uψ = ρj , j 6= 1 and Uψ∗ ρ1 Uψ = ρ1 − δ1 (Uψ∗ ) Uψ ρ2
(96)
or in other terms ρ1 Uψ∗ = δ1 (Uψ∗ )ρ2 + Uψ∗ ρ1 . This shows that provided we replace some of the xj ’s in (94) by the hj (xj ), hj ∈ H, we can get rid of all the exterior differentials df and move all the ρj ’s to the end of the expression which becomes, γ e (f 0 δψ0 Uψ∗0 f 1 δψ1 Uψ∗1 . . . f ` δψ` Uψ∗` ρi ),
(97)
provided we relabel the xj ’s in a cyclic way (which is allowed by definition 2) and we omit several δψj . To write (97) in the form (6) we can assume that ψ` . . . ψ1 ψ0 = 1 since otherwise one gets 0. We first simplify the parenthesis using the crossed product rule in C and get, f 0 (f 1 ◦ ψ 0 )(f 2 ◦ ψ 1 ◦ ψ 0 ) . . . (f ` ◦ ψ `−1 . . . ψ0 ) δψ0 (δψ1 ψ0 − δψ0 ) . . . (δψ` ...ψ0 − δψ`−1 ...ψ0 ) ρi .
(98)
Let f = f 0 (f 1 ◦ ψ 0 ) . . . (f ` ◦ ψ `−1 . . . ψ0 ). When we apply γ e to (98) we get Z f (k) G1
` Y
Pj (δ(ψ j−1 . . . ψ 0 · k)) ρ(k),
(99)
j=1
where we used the equality δg2 = 0 in 3 C G02 . The same result holds if we omit several δψj , one just takes Pj = 1 in the expression (99). We now rewrite (99) in the form, τ0 (f (P0 (δ) Uψ∗0 ) Uψ0 (P1 (δ) Uψ∗1 ψ0 ) Uψ1 ψ0 . . . (P` (δ) Uψ∗`−1 ...ψ0 ) Uψ`−1 ...ψ0 ).
(100)
Let us replace f by f 0 (f 1 ◦ψ 0 ) . . . (f ` ◦ψ `−1 . . . ψ0 ) and move the f j so that they appear without composition, we thus get
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τ0 (f 0 (P0 (δ)Uψ∗0 ) f 1 (Uψ0 P1 (δ)Uψ∗1 ψ0 ) f 2 (Uψ1 ψ0 P2 (δ) Uψ∗2 ψ1 ψ0 ) . . . ). We now use the coproduct rule to rearrange the terms, thus X (1) P1 (δ) Uψ∗0 P1(2) (δ) Uψ∗1 , P1 (δ) Uψ∗1 ψ0 =
(101)
(102)
and we can permute f 1 with Uψ0 P1(1) (δ) Uψ∗0 and use the equality (P (δ) Uψ∗0 )(Uψ0 Q (δ) Uψ∗0 ) = (P Q)(δ) Uψ∗0 .
(103)
Proceeding like this we can rewrite (101) in the form, τ0 (f 0 Q0 (δ) Uψ∗0 f 1 Q1 (δ) Uψ∗1 . . . f ` Q` (δ) Uψ∗` ),
(104)
which shows that the functional (94) satisfies Definition 2. Thus the map 8 of [Co] p. 220 together with Lemma 7 gives us a morphism θ of complexes from the complex C ∗ (A) of the Lie algebra cohomology of the Lie algebra A of formal vector fields, to the (b, B) bicomplex of the Hopf algebra H. Since the current c(g 0 , . . . , g n ) is determined by its value at 1 ∈ G1 , i.e. by X cj (1) ρj (105) we can view the map C as a map from C ∗ (A) to the cochains of the group cohomology of G2 with coefficients in the module E, E = 3 G1∗
(106)
which is the exterior algebra on the cotangent space T1∗ (G1 ). Since the action of G2 on G1 fixes 1 it acts on T1∗ (G1 ) and in the basis ρi the action is given by (86). Since A is the direct sum A = G1 ⊕ G2 of the Lie algebras of G1 and G2 viewed as Lie subalgebras of A (it is a direct sum as vector spaces, not as Lie algebras), one has a natural isomorphism 3 A∗ ' 3 G1∗ ⊗ 3 G2∗ (107) of the cochains in C ∗ (A) with cochains in C ∗ (G2 , E), the Lie algebra cohomology of G2 with coefficients in E. Lemma 9. Under the above identifications, the map C is the cochain implementation: C ∗ (G2 , E) → C ∗ (G2 , E) of the Van Est isomorphism, which associates to a right invariant form µ on G2 with values in E, the totally antisymmetric homogeneous cochain, Z C(µ)(g 0 , . . . , g n ) = µ. 1(g 0 ,... ,g n )
Proof. By (84) we know that there exists a right invariant form, µ = with values in E, such that Z µj . cj (1) =
P
µj ρj on G2 (108)
1(g 0 ,... ,g n )
The value of µj at 1 ∈ G2 is obtained by contraction of ω evaluated at 1 ∈ G, by a suitable element of 3 G1 . Indeed this follows from (80) and the vanishing of the vector field Z 0 of (82) at a = 1 ∈ G2 . Thus the map ω → µ is the isomorphism (107).
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Of course the coboundary d1 in the cochain complex C ∗ (G2 , E) is not equal to the coboundary d of C ∗ (A), but it corresponds by the map C to the coboundary d1 of the bicomplex (C n,m , d1 , d2 ). We should thus check directly that d1 and d anticommute in C ∗ (A). To see this, we introduce a bigrading in C ∗ (A) associated to the decomposition A = G1 ⊕ G2 . What we need to check is that the Lie algebra cohomology coboundary d transforms an element of bidegree (n, m) into a sum of two elements of bidegree (n + 1, m) and (n, m + 1) respectively. It is enough to do that for 1-forms. Let ω be of bidegree (1, 0), then dω(X1 , X2 ) = −ω([X1 , X2 ])
∀ X1 , X2 ∈ A.
(109)
This vanishes if X1 , X2 ∈ G2 thus showing that dω has no component of bidegree (0, 2). We can then decompose d as d = d1 + d2 , where d1 is of bidegree (1, 0) and d2 of bidegree (0, 1). Let us check that d1 is the same as the coboundary of Lie algebra cohomology of G2 with coefficients in 3 G1∗ . Let α ∈ 3m G1∗ and ω ∈ 3n G2∗ . The component of bidegree (n + 1, m) of d(α ∧ ω) = (dα) ∧ ω + (−1)m α ∧ dω is d1 α ∧ ω + (−1)m α ∧ d1 ω,
(110)
where d1 ω takes care of the second term in the formula for the coboundary in Lie algebra cohomology, P P
∨
(−1)i+1 Xi ω(X1 , . . . , Xi , . . . , Xn+1 )+
i+j i<j (−1)
∨
∨
(111)
ω([Xi , Xj ], X1 , . . . Xi , . . . , Xj , . . . , Xn+1 ).
Thus, it remains to check that d1 α ∧ ω corresponds to the first term in (111) for the natural action of G2 on E = 3∗ G1 , which can be done directly for α a 1-form, since d1 α is the transpose of the action of G2 on G1 . We can now state the main lemma allowing to prove the surjectivity of the map θ, Lemma 10. The map θ from (C ∗ (A), d1 ) to the Hochschild complex of H is an isomorphism in cohomology. Proof. Let us first observe that by Lemma 8 and the above proof of the Van Est theorem, the map C gives an isomorphism in cohomology from (C ∗ (A), d1 ) to the complex (C ∗ , d1 ) of polynomial G2 -group cochains with coefficients in E. Thus it is enough to show that the map 8 gives an isomorphism at the level of Hochschild cohomology. This of course requires to understand the Hochschild cohomology of the algebra H∗ with coefficients in the modulo C given by the augmentation on H∗ . In order to do this we shall use the abstract version ([C-H] Theorem 6.1 p. 349) of the Hochschild-Serre spectral sequence. A subalgebra A1 ⊂ A0 of an augmented algebra A0 is called normal iff the right ideal J generated in A0 by the Ker ε (of the augmentation ε of A1 ) is also a left ideal. When this is so, one lets (112) A2 = A0 /J be the quotient of A0 by the ideal J. One has then a spectral sequence converging to ∗ (C) of A0 with coefficients in the module C (using the the Hochschild cohomology HA 0 augmentation e) and with E2 term given by p q (HA (C)). HA 2 1
(113)
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To prove it one uses the equivalence, for any right A2 -module A and right A0 -module B, HomA2 (A, HomA0 (A2 , B)) ' HomA0 (A, B),
(114)
where in the left term one views A2 as an A2 − A0 bimodule using the quotient map ϕ : A0 → A2 to get the right action of A0 . Also in the right term, one uses ϕ to turn A into a right A0 -module. Replacing A by a projective resolution of A2 -right modules and B by an injective resolution of A0 -right modules, and using (since A2 = C ⊗A1 A0 ) ) the equivalence, HomA0 (A2 , B) = HomA1 (C, B) ,
(115)
one obtains the desired spectral sequence. In our case we let A0 = H∗ and A1 = Cc∞ (G1 ). These algebras are non unital but the results still apply. We first need to prove that A1 is a normal subalgebra of A0 . The augmentation ε on H∗ is given by, ε (f Uψ∗ ) = f (1)
∀ f ∈ Cc∞ (G1 ), ψ ∈ G2 .
(116)
Its restriction to A1 is thus f → f (1). Thus the ideal J is linearly generated by elements g Uψ∗ where g ∈ Cc∞ (G1 ) , g(1) = 0.
(117)
We need to show that it is a left ideal in A0 = H∗ . For this it is enough to show that Uψ∗1 g Uψ∗2 is of the same form, but this follows because, ψ(1) = 1
∀ ψ ∈ G2 .
(118)
Moreover, with the above notations, the algebra A2 is the group ring of G2 . We thus obtain by [C-H] loc. cit., a spectral sequence which converges to the Hochschild cohomology of H and whose E2 term is given by the polynomial group cohomology of G2 with coefficients in the Hochschild cohomology of the coalgebra U(G1 ), which according to Proposition 6.1 is given by 3 G1∗ . It thus follows that, combining the Van Est theorem with the above spectral sequence, the map θ is an isomorphism in Hochschild cohomology. We can now conclude this section by the following result, Theorem 11. The map θ defines an isomorphism from the Lie algebra cohomology of A to the periodic cyclic cohomology of H. This theorem extends to the higher dimensional case, i.e. where the Lie algebra A is replaced by the Lie algebra of formal vector fields in n-dimensions, while G = Diff Rn , with G1 the subgroup of affine diffeomorphisms, and G2 = {ψ; ψ(0) = 0, ψ 0 (0) = id}. It also admits a relative version in which one considers the Lie algebra cohomology of A relative to the subalgebra of SO(n) ⊂ G0 = GL(n) ⊂ G1 .
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237
8. Characteristic Classes for Actions of Hopf Algebras In the above section we have defined and computed the cyclic cohomology of the Hopf algebra H as the Lie algebra cohomology of A, the Lie algebra of formal vector fields. The theory of characteristic classes for actions of H extends the construction ([Co2]) of cyclic cocycles from a Lie algebra of derivations of a C ∗ algebra A, together with an invariant trace τ on A. At the purely algebraic level, given an algebra A and an action of the Hopf algebra H on A, H ⊗ A → A , h ⊗ a → h(a) satisfying h1 (h2 a) = (h1 h2 )(a) ∀ hi ∈ H and X h(ab) = h(0) (a) h(1) (b)
(1)
∀a, b ∈ A,
we shall say that a trace τ on A is invariant iff the following holds, e τ (h(a)b) = τ (a S(h)(b))
∀ a, b ∈ A, h ∈ H.
(2)
One has the following straightforward, Proposition 1. Let τ be an H-invariant trace on A, then the following defines a canonical map from HC ∗ (H) to HC ∗ (A), γ(h1 ⊗ . . . ⊗ hn ) ∈ C n (A), γ(h1 ⊗ . . . ⊗ hn )(x0 , . . . , xn ) = τ (x0 h1 (x1 ) . . . hn (xn )). In the interesting examples the algebra A is a C ∗ algebra and the action of H on A is only densely defined. It is then crucial to know that the common domain, A∞ = {a ∈ A ; h(a) ∈ A,
∀ h ∈ H}
(3)
is a subalgebra stable by the holomorphic functional calculus. It is clear from the coproduct rule that A∞ is a subalgebra of A, the question is to show the stability under holomorphic functional calculus. Our aim is to show that for any action of our Hopf algebra H (of Sect. 2) on a C ∗ algebra A the common domain (3) is stable under holomorphic functional calculus. For short we shall say that an Hopf algebra H is differential iff this holds. Lemma 2. Let H be a Hopf algebra, H1 ⊂ H a Hopf subalgebra. We assume that H1 is differential and that as an algebra H is generated by H1 and an element δ ∈ H, (ε(δ) = 0) such that the following holds, a) For any h ∈ H1 , there exist h1 , h2 , h01 , h02 ∈ H1 such that δh = h1 δ + h2 , hδ = δh01 + h02 . b) There exists R ∈ H1 ⊗ H1 such that 1 δ = δ ⊗ 1 + 1 ⊗ δ + R. Then H is a differential Hopf algebra.
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Proof. Let A1,∞ = {a ∈ A ; h(a) ∈ A ∀ h ∈ H1 }. By hypothesis if u is invertible in A and u ∈ A1,∞ one has u−1 ∈ A1,∞ . It is enough, modulo some simple properties of the resolvent ([Co]) to show that the same holds in A∞ . One has using a) that A∞ = ∩ k
1 ≤ j ≤ k} = ∩ A∞,k . Let u ∈ A∞ be invertible in A, k P hi ⊗ki then u−1 ∈ A1,∞ . Let us show by induction on k that u−1 ∈ A∞,k . With R = it follows from b) that X hi (u) ki (u−1 ) (4) δ(u u−1 ) = δ(u) u−1 + u δ(u−1 ) + {a ∈ A1,∞ ; δ j (a) ∈ A1,∞
so that,
δ(u−1 ) = u−1 δ(1) − u−1 δ(u) u−1 −
X
u−1 hi (u) ki (u−1 ).
(5)
Since ε (δ) = 0 one has δ(1) = 0 and the r.h.s. of (5) belongs to A1,∞ . This shows that u−1 ∈ A∞,1 . Now by a), A∞,1 is stable by the action of H1 and by b) it is a subalgebra of A1,∞ . Thus (5) shows that δ(u−1 ) ∈ A∞,1 , i.e. that u−1 ∈ A∞,2 . Now similarly since, (6) A∞,2 = {a ∈ A∞,1 ; δ(a) ∈ A∞,1 } we see that A∞,2 is a subalgebra of A∞,1 stable under the action of H1 , so that by (5) we get δ(u−1 ) ∈ A∞,2 and u−1 ∈ A∞,3 . The conclusion follows by induction. Proposition 3. The Hopf algebra H of Sect. 2 is differential. Proof. Let H1 ⊂ H be the inductive limit of the Hopf subalgebras Hn . For each n the inclusion of Hn in Hn+1 fulfills the hypothesis of Lemma 2 so that H1 = U Hn is differential. Let then H2 ⊂ H be generated by H1 and Y , again the inclusion H1 ⊂ H2 fulfills the hypothesis of the Lemma 2 since 1Y = Y ⊗ 1 + 1 ⊗ Y while h → Y h − hY is a derivation of H1 . Finally H is generated over H2 by X and one checks again the hypothesis of Lemma 2, since in particular XY = (Y + 1) X. It is not difficult to give examples of Hopf algebras which are not differential such as the Hopf algebra generated by θ with 1 θ = θ ⊗ θ.
(7)
9. The Index Formula We use the notations of Sect. 2, so that A = Cc∞ (F + ) >/ 0 is the crossed product of the positive frame bundle of a flat manifold M by a pseudogroup 0. We let v be the canonical Diff + invariant volume form on F + and, L2 = L2 (F + , v),
(1)
be the corresponding Hilbert space. The canonical representation of A in L2 is given by e (π(f Uψ∗ ) ξ)(j) = f (j) ξ(ψ(j))
∀ j ∈ F + , ξ ∈ L2 , f Uψ∗ ∈ A.
(2)
The invariance of v shows that π is a unitary representation for the natural involution of A. When no confusion can arise we shall write simply aξ instead of π(a) ξ for a ∈ A, ξ ∈ L2 .
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The flat connection on M allows to extend the canonical action of the group G0 = GL+ (n, R) on F + to an action of the affine group, G1 = Rn >/ G0
(3)
generated by the vector fields Xi and Y`k of Sect. 1. This representation of G1 admits the following compatibility with the representation π of A, Lemma 1. Let a ∈ A one has, 1) Y`k π(a) = π(a) Y`k + π(Y`k (a)) k (a)) Ykj . 2) Xi π(a) = π(a) Xi + π(Xi (a)) + π(δij Proof. With a = f Uψ∗ ∈ A and ξ ∈ Cc∞ (F + ) one has Y`k π(a) ξ = Y`k (f ξ ◦ ψ) = Y`k (f ) ξ ◦ ψ + f Y`k (ξ ◦ ψ). Since Y`k commutes with diffeomorphisms the last term is (Y`k ξ) ◦ ψ which gives 1). The operators Xi acting on A satisfy (11) of Sect. 2 which one can specialize to b ∈ Cc∞ (F + ), to get k Xi (f Uψ∗ b) = Xi (f Uψ∗ ) b + f Uψ∗ Xi (b) + δij (f Uψ∗ ) Ykj (b).
(4)
One has Xi (f Uψ∗ b) = Xi (f (b ◦ ψ)) Uψ∗ = Xi (π(a)b) Uψ∗ , Xi (f Uψ∗ ) b = π(Xi (a)) b Uψ∗ , k k f Uψ∗ Xi (b) = π(a) Xi (b) Uψ∗ and δij (f Uψ∗ ) Ykj (b) = π(δij (a)) Ykj (b) Uψ∗ , thus one gets 2). Corollary 2. For any element Q of U(G1 ) there exists finitely many elements Qi ∈ U (G1 ) and hi ∈ H such that X Q π(a) = π(hi (a)) Qi ∀ a ∈ A. Proof. This condition defines a subalgebra of U(G1 ) and we just checked it for the generators. Let us now be more specific and take for Q the hypoelliptic signature operator on F + . It is not a scalar operator but it acts in the tensor product h = L2 (F + , v) ⊗ E,
(5)
where E is a finite dimensional representation of SO(n) specifically given by E = 3 Pn ⊗ 3 Rn , Pn = S 2 Rn .
(6)
The operator Q is the graded sum, Q = (d∗V dV − dV d∗V ) ⊕ (dH + d∗H ),
(7)
where the horizontal (resp. vertical) differentiation dH (resp. dV ) is a matrix in the Xi resp. Y`k as well as their adjoints (which also involve scalars). When n is equal to 1 or 2 modulo 4 one has to replace F + by its product by S 1 so that the dimension of the vertical fiber is even (it is then 1 + n(n+1) ) and the vertical signature operator makes sense. The 2 longitudinal part is not elliptic but only transversally elliptic with respect to the action of SO(n). Thus to get an hypoelliptic operator one restricts Q to the Hilbert space,
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H1 = (L2 (F + , v) ⊗ E)SO(n)
(8)
and one takes the following subalgebra of A, A1 = (A)SO(n) = Cc∞ (P ) >/ 0 , P = F + /SO(n).
(9)
Let us note that the operator Q is in fact the image under the right regular representation of the affine group G1 of a (matrix valued) hypoelliptic symmetric element in U (G1 ). By an easy adaptation of a theorem of Nelson and Stinespring, it then follows that Q is essentially selfadjoint (with core any dense, G1 -invariant subspace of the space of C ∞ vectors of the right regular representation of G1 ). We let τ be the trace on A1 which is dual to the invariant volume from v1 on P , that is, Z ∗ f dv1 . (10) τ (f Uψ ) = 0 if ψ 6= 1, τ (f ) = P
Also we adapt the result of the previous sections to the relative case, and use the action of H on A to get a characteristic map, HC ∗ (H, SO(n)) → HC ∗ (A1 )
(11)
associated to the trace τ . Proposition 3. Let us assume that the action of 0 on M has no degenerate fixed point. Then any cochain on A1 of the form, Z 1 ϕ(a0 , . . . , an ) = − a0 [Q, a1 ](k1 ) . . . [Q, an ](kn ) (Q2 )− 2 (n+|k|) ∀ aj ∈ A1 (with T (k) = [Q2 , . . . [Q2 , T ] . . . ]), is in the range of the characteristic map. Proof. By Lemma 2 one can write the operator T = a0 [Q, a1 ](k1 ) . . . [Q, an ](kn ) in the form of a matrix of operators of the form, X 1 α n a 0 hα hj ∈ H , Qα ∈ U (G1 ), (12) 1 (a ) . . . hn (a ) Qα α
thus we just need to understand the cochains of the form, Z Z − a0 h1 (a1 ) . . . hn (an ) R = − a R,
(13)
where R is pseudodifferential in the hypoelliptic calculus and commutes with the affine subgroup of Diff, ∀ ψ ∈ G01 ⊂ Diff. (14) R Uψ = Uψ R Since R is given by a smoothing kernel outside the diagonal and the action of 0 on F + = F + (M ) is free by hypothesis, one gets that Z − f Uψ∗ R = 0 if ψ 6= 1. (15) Z Also by (14) the functional − f R is G01 invariant and is hence proportional to
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Z f dv1 .
(16)
We have thus proved that ϕ can be written as a finite linear combination, X 1 α n τ (a0 hα ϕ(a0 , . . . , an ) = 1 (a ) . . . hn (a )).
(17)
α
Since we restrict the subalgebra A1 ⊂ A, the cochain X α c= hα 1 ⊗ . . . ⊗ hn
(18)
should be viewed as a basic cochain in the cyclic complex of H, relative to the subalgebra H0 = U(SO(n)). Now by Theorem 11 of Sect. 5 one has an isomorphism, H∗ (An , SO(n)) → HC ∗ (H, SO(n)), θ
(19)
where the left hand side is the relative Lie algebra cohomology of the Lie algebra of formal vector fields. Let us recall the result of Gelfand-Fuchs (cf. [G]) which allows to compute the left hand side of (19). One lets G0 = GL+ (n, R) and G0 its Lie algebra viewed as a subalgebra of G1 ⊂ An . One then views the natural projection, π : An → G0
(20)
as a connection 1-form. Its restriction to G0 ⊂ An is the identity map and, π ([X0 , X]) = [X0 , π(X)]
∀ X 0 ∈ G0 , X ∈ A n .
(21)
The curvature of this connection, = dπ +
1 [π, π] 2
(22)
is easy to compute and is given by, (cf. [G]) (X, Y ) = [Y1 , X−1 ] − [X1 , Y−1 ],
∀ X, Y ∈ An ,
(23)
in terms of the projections X → (X)j associated to the grading of the Lie algebra An . It follows from the Chern–Weil theory that one has a canonical map from the Weil complex, S ∗ (G0 ) ∧ 3∗ (G0 ) = W (G0 ), ϕ
W (G0 ) → C ∗ (An ).
(24)
For ξ ∈ G0∗ viewed as an odd element ξ − ∈ 3∗ G0 , one has ϕ(ξ − ) ∈ A∗n ,
ϕ(ξ − ) = ξ ◦ π.
(25)
For ξ ∈ G0∗ viewed as an even element ξ + (∈ S ∗ G0 ), one has ϕ(ξ + ) ∈ 32 A∗n ,
ϕ(ξ + ) = ξ ◦ ,
(26)
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moreover the map ϕ is an algebra morphism, which fixes X it uniquely. By construction ϕ S r (G0∗ ). By definition one lets vanishes on the ideal J of W = W (G0 ) generated by r>n
Wn = W/J
(27)
be the corresponding differential algebra and ϕ e the quotient map, ϕ e : H ∗ (Wn ) → H ∗ (An ).
(28)
It is an isomorphism by [G-F]. The discussion extends to the relative situation and yields a subcomplex W SO(n) of the elements of Wn which are basic relative to the action of SO(n). Again by [G-S], the morphism ϕ e gives an isomorphism, ϕ e : H ∗ (W SO(n)) → H ∗ (An , SO(n)).
(29)
A concrete description of H ∗ (W SO(n)) is obtained (cf. [G]) as a small variant of H ∗ (W O(n)), i.e. the orthogonal case. The latter is the cohomology of the complex E(h1 , h3 , . . . , hm ) ⊗ P (c1 , . . . , cn ) , d,
(30)
where E(h1 , h3 , . . . , hm ) is the exterior algebra in the generators hi of dimension 2i−1, (m is the largest odd integer less than n) and i odd ≤ n, while P (c1 , . . . , cn ) is the polynomial algebra in the generators ci of degree 2i truncated by the ideal of elements of weight > 2n. The coboundary d is defined by, dhi = ci , i odd , dci = 0 for all i.
(31)
One lets pi = c2i be the Pontrjagin classes, they are non trivial cohomology classes for 2i ≤ n. One has H ∗ (W SO(n)) = H ∗ (W O(n)) for n odd, while for n even one has H ∗ (W SO(n)) = H ∗ (W O(n)) [χ]/χ2 − cn .
(32)
Let us now recall the index theorem of [C-M] for spectral triples (A, H, D) whose dimension spectrum is discrete and simple, which is the case (cf. [C-M]) for the transverse fundamental class, (we treat the odd case) Z Theorem 4. a) The equality − P = Resz=0 Trace (P |D|−z ) defines a trace on the algebra generated by A, [D, A] and |D|z , z ∈ C. b) The following formula only has a finite number of non zero terms and defines the components (ϕn )n=1,3... of a cocycle in the (b, B) bicomplex of A, Z X cn,k − a0 [D, a1 ](k1 ) . . . [D, an ](kn ) |D|−n−2|k| ϕn (a0 , . . . , an ) = k (k) ∀ aj ∈ A where one lets T√ = ∇k (T ), ∇(T ) = D2 T − T D2 and where k is a |k| multiindex, cn,k = (−1) 2i (k1 ! . . . kn !)−1 (k1 + 1)−1 . . . (k1 + k2 + . . . + kn + n −1 n) 0 |k| + 2 , |k| = k1 + . . . + kn . c) The pairing of the cyclic cohomology class (ϕn ) ∈ HC ∗ (A) with K1 (A) gives the Fredholm index of D with coefficients in K1 (A).
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Let us remark that the theorem is unchanged if we replace everywhere the operator D by Q = D |D|. (33) This follows directly from the proof (cf. [C-M]). In our case the operator Q is differential, given by (7), so that by Proposition 3 we know that the components ϕn belong to the range of the characteristic map. Since the computation is local we thus get, collecting together the results of this paper, Theorem 5. There exists for each n a universal polynomial Ln ∈ H ∗ (W SOn ) such that, Ch∗ (Q) = θ (Ln ), where θ is the isomorphism of Theorem 11 Sect. 7 and we let θ be the composition of θ with the characteristic map of the action of the Hopf algebra H on A. One can end the computation of Ln by evaluating the index on the range of the assembly map, (34) µ : K∗,τ (P >/0 E 0) → K(A1 ), provided one makes use of the conjectured (but so far only partially verified, cf. [He, K-T]) injectivity of the natural map, Hd∗ (0n , R) → H ∗ (B0n , R)
(35)
from the smooth cohomology of the Haefliger groupoid 0n to its real cohomology. One then obtains that Ln is the product of the usual L-class by another universal expression in the Pontrjagin classes pi , accounting for the cohomological counterpart of the K-theory Thom isomorphism, β : K∗ (C0 (M )>/0) → K∗ (C0 (P )>/0)
(36)
of [Co1,V]. This can be checked directly in small dimension. It is noteworthy also that the first Pontrjagin class p1 already appears with a non zero coefficient for n = 2. Appendix, the One Dimensional Case In the one dimensional case the operator Q is readily reduced to the following operator on the product Fe of the frame bundle F + by an auxiliary S 1 whose corresponding periodic coordinate is called α (and whose role is as mentioned above to make the vertical fiber even dimensional) (1) Q = QV + Q H . We work with 2 copies of L2 (Fe , es dαdsdx) and the following gives the vertical operator QV , 2 −∂α + ∂s (∂s + 1) −2∂α ∂s − ∂α . (2) QV = −2∂α ∂s − ∂α ∂α2 − ∂s (∂s + 1) The horizontal operator QH is given by QH =
1 −s e ∂x γ2 , i
(3)
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0 −i where γ2 = anticommutes with QV . i 0 We use the following notation for 2 × 2 matrices 01 0 −i 1 0 γ1 = , γ2 = , γ3 = . 10 i 0 0 −1
(4)
We can thus write the full operator Q acting in 2 copies of L2 (Fe , es dαdsdx) as Q = (−2∂α ∂s − ∂α )γ1 +
1 −s e ∂x γ2 + −∂α2 + ∂s (∂s + 1) γ3 . i
(5)
Theorem 6. Up to a coboundary ch∗ Q is equal to twice the transverse fundamental class [Fe ]. The factor 2 is easy to understand since it is the local index of the signature operator along the fibers of Fe . The formulas of Theorem 4 for the components ϕ1 and ϕ3 of the character are, Z √ 1 0 1 −(a0 [Q, a1 ](Q2 )−1/2 ) (6) ϕ1 (a , a ) = 2i0 2 Z √ 1 3 − 2i 0 −(a0 ∇[Q, a1 ](Q2 )−3/2 ) 2 2 Z √ 1 1 5 + 2i 0 −(a0 ∇2 [Q, a1 ](Q2 )−5/2 ) 2 3 2 Z √ 1 7 − 2i 0 −(a0 ∇3 [Q, a1 ](Q2 )−7/2 ), 2·3·4 2 Z √ 1 3 −(a0 [Q, a1 ][Q, a2 ][Q, a3 ](Q2 )−3/2 ) ϕ3 (a , a , a , a ) = 2i 0 3i 2 Z √ 1 5 − 2i 0 −(a0 ∇[Q, a1 ][Q, a2 ][Q, a3 ](Q2 )−5/2 ) (7) 2·3·4 2 Z √ 1 5 − 2i 0 −(a0 [Q, a1 ]∇([Q, a2 ][Q, a3 ](Q2 )−5/2 ) 3·4 2 Z √ 1 5 − 2i 0 −(a0 [Q, a1 ][Q, a2 ]∇[Q, a3 ](Q2 )−5/2 ). 2·4 2 0
1
2
3
The computation gives the following result, ϕ1 (a0 , a1 ) = 0
∀ a0 , a1 ∈ A,
(8)
in fact, each of the 4 terms of (6) turns out to be 0, g] + bψ), ϕ3 = 2([F
(9)
g] is the tranverse fundamental class ([Co]), i.e. the extension to the crossed where [F product of the following invariant cyclic 3-cocycle on the algebra Cc∞ (Fe ),
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Z µ(f 0 , f 1 , f 2 , f 3 ) =
e F
f 0 df 1 ∧ df 2 ∧ df 3 .
We shall now give the explicit form of both bψ and ψ with Bψ = 0. We let τ be the trace on A given by the measure Z f) → f es dαdsdx. f ∈ Cc∞ (F e F
(10)
(11)
This measure is invariant under the action of Diff + and thus gives a dual trace on the crossed product. The two derivations ∂α and ∂s of Cc∞ (Fe ) are invariant under the action of Diff + and we denote by the same letter their canonical extension to A, ∂α (f Uψ∗ ) = (∂α f )Uψ∗ , ∂s (f Uψ∗ ) = (∂s f )Uψ∗ .
(12)
We let δ1 be the derivation of A defined in Sect. 2. By construction both δ1 and τ are invariant under ∂α but neither of them is invariant under ∂s . But the following derivation ∂u : A → A∗ commutes with both ∂α and ∂s , h∂u (a), bi = τ (δ1 (a)b).
(13)
We then view ∂α , ∂s and ∂u as three commuting derivations, where ∂u cannot be iterated and use the notation (us, α, s) for the cochain a0 , a1 , a2 , a3 → ha0 , (∂u ∂s a1 )(∂α a2 )(∂s a3 )i. The formula for bψ is then the following, 1 (− (u, α, s) + (u, s, α) + (α, s, u) − (s, α, u)), 8 1 (− (us, α, s) + (α, us, s) + (α, s, us) 2 + (us, s, α) + (s, us, α) − (s, α, us) + (α, uα, α)) − (u, αs, s) − (s, uα, s) − (s, αs, u), 1 (− (u, α, ss) + (ss, u, α) − (ss, α, u) 4 + (u, ss, α) + (α, u, ss) + (α, ss, u) − (u, ααα) + (αα, u, α) − (αα, α, u) − (u, αα, α) + (α, u, αα) − (α, αα, u)). This formula is canonical and a possible choice of ψ is given by, 1 ((α, su) − (αu, s) − (s, αu) + (su, α)), 8 1 1 (−(ααu, α) − (ααα, u) + αu, αα)), 4 3 1 1 1 1 ((uαs, s) − (α, ssu) + (αss, u) + (αs, us) + (αu, ss) − (ssu, α)). 4 2 2 2 The natural domain of ψ is the algebra C 3 of 3 times differentiable elements of A where the derivation ∂u is only used once.
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References [B-S] Baaj, S. and Skandalis, G.: Unitaires multiplicatifs et dualit´e pour les produits crois´es de C ∗ -alg`ebres. Ann. Sci. Ec. Norm. Sup. 4 s´erie, t. 26, 425–488 (1993) [C-E] Cartan, H. and Eilenberg, S.: Homological algebra. Princeton, NJ: Princeton University Press, 1956 [C-M] Connes, A. and Moscovici, H.: The local index formula in noncommutative geometry. GAFA 5, 174–243 (1995) [Co] Connes, A.: Noncommutative geometry. London–New York: Academic Press, 1994 [Co1] Connes, A.: Cyclic cohomology and the transverse fundamental class of a foliation. In: Geometric methods in operator algebras, Kyoto, 1983 Pitman Res. Notes in Math. 123, Harlow: Longman, 1986. pp. 52–144 [Co2] Connes, A.: C ∗ alg`ebres et g´eom´etrie differentielle. C.R. Acad. Sci. Paris, Ser. A-B 290, 1980 [Dx] Dixmier, J.: Alg`ebres enveloppantes. Paris: Gauthier-Villars, 1974 [G-F] Gelfand, M. and Fuks, D.B.: Cohomology of the Lie algebra of formal vector fields. Izv. Akad. Nauk SSSR 34, 322–337 (1970); Cohomology of Lie algebra of vector fields with nontrivial coefficients. Funct. Anal. 4, 10–45 (1970); Cohomology of Lie algebra of tangential vector fields. Funct. Anal. 4, 23–31 (1970) [G] Godbillon, G.: Cohomologies d’alg`ebres de Lie de champs de vecteurs formels. Seminaire bourbaki (1971/1972), expos´e No.421, Lecture Notes in Math. Vol 383, Berlin: Springer, 1974, pp. 69–87 [H] Haefliger, A.: Sur les classes caract´eristiques des feuilletages. Seminaire bourbaki (1971/1972), expos´e No.412, Lecture Notes in Math. Vol 317, Berlin: Springer, 1973, pp. 239–360 [He] Heitsch, J.L.: Independent variations of secondary classes. Ann. of Math. Vol 108, 421–460 (1978) [H-S] Hilsum, M. and Skandalis, G.: Morphismes K-orient´es d’espaces de feuilles et fonctorialit´e en th´eorie de Kasparov. Ann. Sci. Ecole Norm. sup. (4) Vol. 20, 325–390 (1987) [K] Kac, G.I.: Extensions of Groups to Ring Groups. Math. USSR Sbornik Vol. 5 No 3, (1968) [K-T] Kamber, F. and Tondeur, Ph.: On the linear independence of certain cohomlogy classes of B 0q . Studies in algebraic topology, Adv. in Math. Suppl. Studies, 5, 213–263 (1979) [M] Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 [M-S] Moore, C.C. and Schochet, C.: Global analysis on foliated spaces. it Math. Sci. Res. Inst. Publ. 9, New York: Springer, 1988 Communicated by A.Jaffe
Commun. Math. Phys. 198, 247 – 281 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Bosonization of Massive Fermions J. Dimock? Dept. of Mathematics, SUNY at Buffalo, Buffalo, NY 14214, USA. E-mail:
[email protected] Received: 19 May 1997 / Accepted: 30 March 1998
Abstract: We study the Euclidean sine-Gordon field theory on the plane with β < 16π/3 and an interaction density confined to a finite square. For β = 4π we construct correlation functions for the field : sin φ : and show that they are equal to the pseudoscalar ψ0ψ correlation functions for a free fermion theory with mass term confined to the finite square.
1. Introduction On a two dimensional space time certain boson and fermion quantum field theories are equivalent, a phenomenon discovered by Coleman [10]. This equivalence has found applications to statistical mechanics and string theory. The simplest case is the equivalence of massless free fermions and massless free bosons on the plane with a Euclidean metric. The massless free fermion theory is given formally by a (fermionic) functional measure Z ¯ (1) exp (ψi∂/ψ)(x) dx dψ dψ. The massless free boson theory is given formally by the functional measure Z 1 |∂φ(x)|2 dx dφ. exp − 2β
(2)
The equivalence is the statement that all correlation functions (moments) are equal if β = 4π and we make the correspondence ?
Research supported by NSF Grant PHY9400626
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c : cos(φ(x)) : ic : sin(φ(x)) :
⇐⇒ ⇐⇒
(ψψ)(x), (ψ0ψ)(x),
(3)
where 0 = γ5 = iγ0 γ1 and c is a positive constant. This massless equivalence is now well-understood. There is a large literature, but for our purposes the best references are [15, 13]. Equivalence has been found on other two dimensional manifolds as well, see for example [1]. The correspondence becomes increasingly complicated with the topology of the manifold. Already on the torus it seems that one must take a circle valued field to make the correspondence. In any case the massless equivalence suggests the equivalence of the massive free fermions with mass µ and measure Z ¯ ¯ (4) exp ((ψi∂/ψ)(x) + µ(ψψ)(x)) dx dψ dψ and sine-Gordon bosons at β = 4π with coupling constant ζ and measure Z 1 2 exp (− |∂φ(x)| + ζ : cos(φ(x)) :) dx dφ 2β
(5)
provided ζ = µc. Indeed the two theories are formally equivalent order by order in perturbation theory. The result is remarkable because it is an equivalence between a linear and a non-linear field theory. The sine-Gordon theory is of particular interest because it describes the classical statistical mechanics of a Coulomb gas at temperature β −1 and activity ζ/2. The purpose of this paper is to give a rigorous proof of this widely conjectured equivalence. What has been missing until now is a proof that perturbation theory for the sine-Gordon model converges to the actual model at β = 4π. This involves showing that the model exists and is analytic for ζ in a neighborhood of the origin and explicitly computing the derivatives at ζ = 0. Both steps are non-trivial due to the fact that at this value of β the model requires an infinite energy renormalization. We work in the plane, at first with an ultraviolet cutoff N . For β < 16π/3 we prove that correlation functions are analytic in ζ small uniformly in N . Then for β = 4π we take the N → ∞ limit and compute the derivatives at ζ = 0. A parallel and much more elementary analysis is given for fermions. Comparing derivatives at ζ = µ = 0 we conclude that correlation functions at non-coinciding points are the same in the two models. The following qualifying remarks need to be made. First, although the free action is taken on the whole plane, the interaction/mass term is confined to a finite volume. Taking the infinite volume limit becomes a separate issue which is not addressed here. Secondly we only show equality between correlation functions for : sin φ : and ψ0ψ. The ordinary correlation functions for : cos φ : and ψψ do not exist. ( Truncated functions are a different story.) An earlier result on boson–fermion equivalence is due to Fr¨ohlich and Seiler [15] who consider the sine-Gordon model for β < 4π. In this case no energy renormalization is required and the convergence of perturbation theory is simpler. They prove equivalence with a more complicated Thirring-Schwinger model. Our control of the ultraviolet limit uses a renormalization group technique, and owes much to an earlier paper on sine-Gordon by Dimock and Hurd [12]. This work considered the model on the torus and allowed β < 8π. Unfortunately there is a mistake
Bosonization of Massive Fermions
249
in the proof, and up until now it has not been repaired.1 Here we use a variation of the proof, which is simpler and avoids the error, but only works for β < 16π/3. Our use of the free massless action for the whole plane introduces some special infrared divergences not present or not important for compact manifolds like the torus, and also not present for the massive sine-Gordon model (Yukawa gas). The massive sine-Gordon model is studied in [14, 3, 2, 18, 9]. We particularly note the paper of Brydges and Kennedy [9], where the analyticity of the renormalized partition function is established for β < 16π/3 by a precursor of the method used in the present paper. The organization of the paper is as follows. In Sect. 2.1 we review some material about exponential functions of the free boson field, and prove a new result about truncated correlation functions. In Sect. 2.2 we prove several forms of the equivalence with the free fermi field. In Sect. 2.3 we define the sine-Gordon theory in a box, quote basic existence and analyticity results for β < 16π/3 and then prove the equivalence at β = 4π with massive free fermions in a box. In Sect. 3 we introduce the renormalization group and use it to prove the N uniform bounds needed in Sect. 2.
2. Boson–Fermion Equivalence 2.1. Exponentials of the free boson field. Our treatment follows Fr¨ohlich-Seiler [15] , and Fr¨ohlich- Marchetti [13], but with some innovations (Lemmas 2, 5). The massless boson field theory on R2 is defined as the family of Gaussian random variables φ(x) with covariance which is the inverse Laplacian. We start with a regularized version of this adding a mass r to remove the infrared difficulties and an ultraviolet cutoff at LN (for fixed large L) to get a more regular theory. Thus consider the operator vr,N whose kernel is Z ip(x−y) 2 −2N e e−p L dp (6) vr,N (x − y) = (2π)−2 (p2 + r2 ) and consider the Gaussian random variables φ(x) with covariance βvr,N . We can realize it concretely by taking the Gaussian measure µβ,r,N on S 0 (R2 ) with covariance βvr,N . Because of the smoothness of the kernel the measure is actually supported on smooth functions and we take φ(x) to be evaluation at x. We have the expectation: + * Z X X si φ(xi )) = exp(i si φ(xi )) dµβ,r,N (φ) exp(i i
β,r,N
= exp −
i
1X 2
si sj βvr,N (xi − xj ) .
(7)
i,j
We are interested in taking the limits N → ∞ and r → 0. To get something nontrivial we need to make some adjustments. First define Wick-ordered exponentials by : eiφ(x) :r,N = exp(βvr,N (0)/2) eiφ(x)
(8)
1 The problem is that for the homotopy property one needs κ = O(L−2 ), not κ = O(1). Then the Sobolev inequalities require κ(h∗1 )2 ≥ O(1) and hence h∗1 = O(L). This spoils the estimate above line (49) in [12]. This error also spoils the results in [11].
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(β dependence is suppressed from the notation here). These normalize the exponential so that
iφ(x) :r,N β,r,N = 1. (9) :e We consider the case where there is a mismatch between the Wick ordering, taken to be : eiφ(x) :1,N , and the measure µβ,r,N . Lemma 1. For non-coinciding points x1 , ..., xn , y1 , ..., ym the limit *n + m Y Y iφ(xi ) −iφ(yj ) :e : :e : i=1
* = lim lim
r→0 N →∞
j=1
n Y
:e
iφ(xi )
β m Y
:1,N
i=1
+ :e
iφ(yj )
(10)
:1,N
j=1
β,r,N
exists. The limit is zero if n 6= m, and if n = m is given by Q c−n−m 0
1≤i
|xi − xi0 | Qn Qm i=1
Q 1≤j<j 0 ≤m
j=1
|yj − yj 0 |
!β/2π (11)
|xi − yj |
for some positive constant c0 . Proof. The expectation with r > 0, N < ∞ is evaluated as X β v1,N (0)(n + m) − exp vr,N (xi − x0i ) 2 0 i,i
−
X
vr,N (yj − yj0 ) + 2
j,j 0
X
vr,N (xi − yj )
(12) .
i,j
The terms i = i0 and j = j 0 contribute −vr,N (0)(n + m) to the bracket. Combining this with the first term and using lim (v1,N (0) − vr,N (0)) = (2π)−1 log r,
(13)
N →∞
we get a contribution (rβ/4π )(n+m) as N → ∞. We also have for x 6= 0 the existence of Z ∞ −√p2 +r2 |x| e p vr (x) = lim vr,N (x) = (2π)−1 dp. (14) N →∞ −∞ 2 p2 + r 2 Thus we find that the N → ∞ limit of the expectation is X vr (xi − x0i ) (rβ/4π )(n+m) exp β − i
−
X j<j 0
vr (yj − yj0 ) +
X i,j
vr (xi − yj )
(15) .
Bosonization of Massive Fermions
251
Now as r → 0 we have vr (x) = v1 (rx) = −(2π)−1 log(r|x|) + γ + h(rx),
(16)
where h(x) is bounded on compact subsets of R2 and goes to 0 as x → 0. With c0 = e−γ we find that our expression can be written (rβ/4π )(n−m) c−n−m+(n−m) 0 !β/2π Q Q 0 0 1≤i
2
(17)
where H(x, y) is bounded on compact subsets of R2(n+m) and goes to one as (x, y) → 0. The r → 0 limit gives 0 if n 6= m and the stated result if n = m. Remark. It is useful to introduce complex variables zi = x1i + ix0i and wj = yj1 + iyj0 . Then |xi − yj | = |zi − wj | and so forth. Then use the Cauchy lemma (see for example [16]) to obtain Q Q (wi − wj ) 0 1≤i
j=1
(19)
β
This is useful for estimates and essential for the identification with fermions. More generally we have the following result which is valid for n 6= m. Note that when n = 0 it is the standard expansion of the Vandermode determinant. Lemma 2. For n ≤ m Q
Q − zi0 ) 1≤j<j 0 ≤m (wj − wj 0 ) Qn Qm = det{aij }, i=1 j=1 (zi − wj )
1≤i
where {aij } is the m × m matrix with entries (zi − wj )−1 1 ≤ i ≤ n . aij = (wj )m−i n < i ≤ m
(20)
(21)
Proof. The proof is by induction on n. If n = m it is the Cauchy formula. We assume it is true for n and prove it for n − 1. On the left side of the identity for n the factor zn occurs n − 1 times in the numerator and m times in the denominator. If we let k = m − n + 1 multiply by znk and then let zn → ∞ we get the left side of the identity for n − 1. We have to check that the same operation applied to the right side for n gives the right side for n − 1. The first n − 1 rows of the determinant are independent of zn and expanding out these rows we see that the problem is reduced to an evaluation of the limit of certain determinants of k × k matrices. Relabeling the wj ’s the required limit is
252
J. Dimock
(z − w1 )−1 w1k−2 k lim z det ... z→∞ w 1 1
... (z − wk )−1 k−1 w1 ... wkk−1 k−2 ... wk ... ... ... = det ... ... . ... w w 1 k ... wk 1 ... 1 ... 1
(22)
But the first row on the left can be expanded as z −1 (1, ..., 1) + z −2 (w1 , ..., wk ) + ... + z −k (w1k−1 , ..., wkk−1 ) + O(z −k−1 ).
(23)
The first k − 1 terms do not contribute to the determinant, the k th term gives the answer, and the remainder goes to zero. Next we investigate the integrability of these correlation functions. Lemma 3. For β < 4π the expectation + *n m Y Y iφ(xi ) −iφ(yj ) :e : :e : i=1
j=1
β
is locally integrable. Proof. First consider the case n = m and 2π ≤ β ≤ 4π. Then −1
| det{(zi − wj )
}|
β/2π
≤
n XY π
≤ const
!β/2π |zi − wπ(i) |
−1
i=1 n XY π
|zi − wπ(i) |−β/2π ,
(24)
i=1
where the sum is over permutations π of (1, ..., n) and in the last step we have used the inequality (a + b)p ≤ 2p−1 (ap + bp ) valid for p ≥ 1. This shows the local integrability. If β < 2π take out the region where the determinant is less than one and then increase β to 2π to get an integrable function. If n 6= m we use (20) to get the result by a similar argument. Now define
Z : eiφ :1,N (f ) =
: eiφ(x) :1,N f (x) dx,
(25)
where the test functions f are taken to be bounded measurable functions of compact support. Lemma 4. For β < 4π and test functions fi , gj the limit + *n m Y Y iφ −iφ : e : (fi ) :e : (gj ) i=1
j=1
β
Bosonization of Massive Fermions
* = lim lim
r→0 N →∞
253 n Y
:e
iφ
m Y
:1,N (fi )
i=1
+ :e
−iφ
:1,N (gj )
(26)
j=1
β,r,N
exists and is equal to + Z *Y n m Y Y Y : eiφ(xi ) : : e−iφ(yj ) : fi (xi ) gj (yj ) dx dy. = i=1
j=1
i
β
(27)
j
Proof. With the cutoffs in place it is easy to change the order of intergration and obtain + *n m Y Y iφ −iφ : e :1,N (fi ) :e :1,N (gj ) i=1
=
Z *Y n
:e
iφ(xi )
j=1
:1,N
i=1
m Y
β,r,N
+ :e
−iφ(yj )
Y
:1,N
j=1
β,r,N
fi (xi )
i
Y
gj (yj ) dx dy. (28)
j
Thus to prove the result one has to show that the right side of this equation converges to the same thing without cutoffs. However it is difficult to find a dominating function, since one does not exactly have a determinant representation for N < ∞. The actual proof for the N → ∞ limit requires some trickery, see Fr¨ohlich [14], Theorem 3.7, for the details. The proof is given for n = m and does use the determinant representation (18) at N = ∞. It also works for n 6= m if we use the representation (20) at N = ∞. The r → 0 limit is easier. One uses (17), (18) and dominated convergence. We also need to discuss truncated correlation functions . For these we will be able to integrate over coinciding points even at β = 4π. Truncated correlations are defined for any random variables {χi } and any expectation h...i (including our singular limits) inductively by *
n Y
+T
*
χi
=
i=1
n Y
+ χi
−
XY 5
i=1
j
*
Y
+T χi
,
(29)
i∈5j
where the sum is over the proper partitions 5 = {5j } of (1, ..., n). The truncated functions are sums of products of ordinary correlation functions. For β < 4π we have by the previous lemma that *
n Y
:e
i=1
=
iφ
: (fi )
m Y
+T :e
−iφ
: (gj )
j=1
Z *Y n i=1
: eiφ(xi ) :
m Y j=1
β
+T : e−iφ(yj ) : β
Y i
fi (xi )
Y
gj (yj ) dx dy.
(30)
j
Note that these expectations are zero unless n = m. We want to take the limit β % 4π.
254
J. Dimock
Lemma 5. Except for n = m = 1, *n Y
:e
i=1
iφ
*
= lim
β%4π
: (fi )
m Y
+T :e
−iφ
: (gj )
j=1 n Y
4π
: eiφ : (fi )
i=1
m Y
+T
(31)
: e−iφ : (gj )
j=1
β
exists and is equal to Z *Y n
:e
iφ(xi )
:
i=1
m Y
+T :e
−iφ(yj )
j=1
Y
: 4π
fi (xi )
i
Y
gj (yj ) dx dy.
(32)
j
Proof. It suffices to take n = m ≥ 2. Let x = (x1 , ..., xn ) and y = (y1 , ..., yn ) and define *n + n Y Y iφ(xi ) −iφ(yj ) Gβ (x, y) = :e : :e : (33) i=1
j=1
β
and let GβT (x, y) be the truncated function. We claim that for each n there is a locally integrable function h(x, y) such that |GβT (x, y)| ≤ h(x, y)
(34)
for all β ≤ 4π. Then the integral in (32) exists. Furthermore we can take the limit β % 4π in (30) by the dominated convergence theorem and obtain (31). The proof of (34) is by induction on n. We suppose the result is true for all smaller n (and make no assumption in case n=2). It suffices to find a dominating function in the sector |x1 − y1 | ≤ |xi − yj |.
(35)
˜ and have We write x = (x1 , x) ˜ y) ˜ − GβT (x, y) = Gβ (x, y) − Gβ (x1 , y1 )Gβ (x,
0 X
Y
GβT (x5j , y5j0 ),
(36)
{(5j ,5j 0 )} i
where the sum is over partitions {(5j , 5j 0 )} of two copies of (1, 2, ...n) except that partitions containing (1, 1) are excluded. The first two terms in (36) can be written as a constant times | det(zi − wj )−1 |β/2π − |(z1 − w1 )−1 det(z˜i − w˜ j )−1 |β/2π .
(37)
We use the identity for α ≥ 1, ||a|α − |b|α | ≤ |a − b|α(|a|α−1 + |b|α−1 ) to bound this by a constant times:
(38)
Bosonization of Massive Fermions
255
| det(zi − wj )−1 − (z1 − w1 )−1 det(z˜i − w˜ j )−1 | × (| det(zi − wj )−1 |β/2π−1 + |(z1 − w1 )−1 det(z˜i − w˜ j )−1 |β/2π−1 ),
(39)
which in turn is bounded by X Y |zi − wπ(i) |−1 2 + | det(zi − wj )−1 | + |(z1 − w1 )−1 det(z˜i − w˜ j )−1 | . (40) π:π(1)6=1 i This function is locally integrable since the second factor is locally in L2− for any > 0 as in (24) and the first factor is locally L2+ for sufficiently small. To see this last 0 be the intersection with the sector (35). statement let B2n be a ball in R2n and let B2n For a permutation π with π(1) 6= 1 the corresponding term in the first factor has an L2+ 0 norm on B2n which is Z Y | |xi − yπ(i) |−2− dx dy 0 B2n
Z
i
|x1 − y1 |−n dx1 dy1 . . . (dyπ(1) )∧ . . . dyn ,
≤ const
(41)
Bn
and this is finite if n < 2. Now consider the last term in (36). Each term in the sum will be a product of some two-point truncated functions (n=m=1), and some truncated functions which are dominated by integrable functions by the inductive hypothesis. The truncated two point functions equal the ordinary two point functions which are a constant times |x−y|−β/2π . Thus we have to dominate terms of the form !−β/2π Y |xi − yπ(i) | , (42) i∈I
where I is a subset of (1, ..., n) and π is a permutation with π(1) 6= 1. This is dominated by one on the set where the product is greater than one and by its value at β = 4π on the complement. The latter is locally integrable as in (41). 2.2. Massless fermions and equivalence. Now we turn to massless fermions, mostly following [13]. The massless free fermion field theory on the plane is defined by the two point function which is a fundamental solution S(x − y) for the Dirac operator i∂/. Here ∂/ = γµ ∂µ , where γµ are the gamma matrices satisfying {γµ , γν } = −2δµν . We choose a skew adjoint representation in which γ0 = −iσ2 , γ1 = −iσ1 and 0 = iγ0 γ1 = σ3 , where the σi are the Pauli matrices. One can write down S(x−y) directly, but we prefer to obtain it as a limit of regularized quantities. We define Z M Z N p/ −2 dp0 dp1 eipx 2 . (43) SM N (x) = (2π) p −M −N In the limit M → ∞ one can do the p0 integral by closing the contour. For x0 > 0 one finds Z N (iγ0 |p1 | + γ1 p1 ) dp1 e(−|p1 |x0 +ip1 x1 ) (44) SN (x) = (2π)−1 2|p1 | −N which is a bounded function.
256
J. Dimock
Lemma 6. 1. For x 6= 0, S(x) = limM,N →∞ SM N (x) exists, is continuous, and is a fundamental solution for the Dirac operator. 2. If z = ix0 + x1 then for x 6= 0 0 z¯ −1 −1 S(x) = lim SN (x) = (2π) . (45) z −1 0 N →∞ 3. Uniformly in N : |SN (x)| ≤ O(1)|x|−1 .
(46)
Proof. The first part follows by integrating by parts twice. This argument also shows that one can take the limit in either order. Thus for x0 > 0, Z (−|p1 |x0 +ip1 x1 ) e (iγ0 |p1 | + γ1 p1 ) dp1 . (47) S(x) = lim SN (x) = (2π)−1 N →∞ 2|p1 | Then one can compute matrix elements Z (−|p1 |x0 +ip1 x1 ) e (|p1 | + p1 ) dp1 S21 (x) = −i(2π)−1 2|p1 | Z ∞ = −i(2π)−1 eip1 z dp1 0 −1
= (2πz)
.
(48)
Similarly S12 (x) = (2π z) ¯ −1 , S11 (x) = 0, S22 (x) = 0. The same formulas hold for x0 < 0, and by continuity for all x 6= 0. With a cutoff we have instead Z N −1 eip1 z dp1 = O(|z|−1 ). (49) (SN )21 (x) = −i(2π) 0
This completes the proof.
The regularized theory is defined by a fermionic functional integral with respect to the Gaussian measure with covariance SN . We have for correlation functions + *n Z Y n Y ψαi (xi )ψ βi (yi ) = ψαi (xi )ψ βi (yi ) dµSN i=1
N
i=1
= det{(SN )αi βi (xi − yj )}
(50)
with all other correlation functions equal to zero. For non-coinciding points we can take the limit N → ∞ and obtain + *n Y ψαi (xi )ψ βi (yi ) = det{Sαi βi (xi − yj )}. (51) i=1
We introduce chiral fermions b, c by b = (1 + 0)ψ/2 = ψ1 , c¯ = (1 − 0)ψ/2 = ψ2 .
(52)
Bosonization of Massive Fermions
257
Then our results so far can be characterized by the statement that we have a free fermion field theory with two point functions
¯ c(y) = (2π z¯ − w) b(x)¯ ¯ −1 , hc(x)b(y)i = (2πz − w)−1 .
(53)
and all others equal to zero. Lemma 7. For non-coinciding points, + *n m Y Y ¯ i) bb(x c¯c(yj ) = i=1
j=1
0 n 6= m 1 2 }| n=m (2π)−2n | det{ zi −w j
.
(54)
¯ b to be taken at the same point and still ¯ = 0 we can allow b, Remark. Since b(x)b(x) N have an N → ∞ limit. No Wick ordering is required. Proof. The expectation can be explicitly computed by Wick’s theorem. We have two ¯ c¯)) and they give complex conjugate independent field theories (namely (b, c) and (b, expectations whence + *n + *n m m Y Y Y Y ¯ i) bb(x c¯c(yj ) = | b(xi ) c(yj ) |2 . (55) i=1
j=1
i=1
If n 6= m the expectation is zero, while if n = m we have + *n n Y Y b(xi ) c(yj ) = (2π)−n det{ i=1
j=1
j=1
1 }. zi − w j
(56)
This completes the proof.
Now by comparing (19) and (54) we have: Theorem 1 (Massless equivalence). For non coinciding points and β = 4π, + *n m Y Y n+m : exp(iφ(xi )) : : exp(−iφ(yj )) : c0 i=1
*
= (2π)n+m
j=1 n Y
¯ : (xi ) : bb
i=1
m Y
+
: c¯c : (yj ) .
β
(57)
j=1
This shows that the massless free theories are equivalent if we take c = c0 /π and make the identification c ¯ : exp(iφ) : ⇐⇒ bb, 2 c : exp(−iφ) : ⇐⇒ c¯c. (58) 2 We can also write this as c : cos(φ) : ic : sin(φ) :
⇐⇒ ⇐⇒
¯ + c¯c, ψψ = bb ¯ − c¯c. ψ0ψ = bb
(59)
258
J. Dimock
Next we smear with test functions, still taken to be bounded measurable functions with compact support. We study scalars formally given by Z (60) (ψψ)(f ) = (ψψ)(x)f (x) dx. These are not defined directly but by their expectations given by Z
(ψψ)(f1 )...(ψψ)(fn ) N = (ψψ)(x1 )...(ψψ)(xn ) N f (x1 ) . . . f (xn ) dx. (61) Similarly we define expectations with pseudo-scalars ψ0ψ. We cannot take the limit N → ∞ here unless the test functions have non-overlapping supports. However for truncated correlation functions like Z T
T
(ψψ)(x1 ) . . . (ψψ)(xn ) N f (x1 ) . . . f (xn ) dx(62) (ψψ)(f1 ) . . . (ψψ)(fn ) N = (with ψψ treated as a unit) we can take the limit. Lemma 8. For n + m ≥ 3, *
n Y
(ψψ)(xi )
i=1
m Y
+T (ψ0ψ)(yj )
(63)
j=1
is locally integrable and *n +T *n +T m m Y Y Y Y (ψψ)(fi ) (ψ0ψ)(gj ) = lim (ψψ)(fi ) (ψ0ψ)(gj ) i=1
exists and equals Z *
N →∞
j=1
n Y i=1
(ψψ)(xi )
m Y
+T (ψ0ψ)(yj )
j=1
i=1
n Y i=1
fi (xi )
j=1
m Y
gj (yj ) dx dy.
(64)
N
(65)
j=1
Proof. Suppose that m = 0 and n ≥ 3. The general case would just involve inserting 00 s T
at appropriate places. It suffices to bound ψψ(x1 )...ψψ(xn ) N by a locally integrable function independent of N . Then the integrability follows and one can use dominated convergence to take the limit in (62). Truncated correlations for free field theories are given as sums over connected Feymann diagrams. In this case the only connected diagram is a circle and we have X T
tr(SN (x1 − xπ(1) ) . . . SN (xπn−1 (1) − x1 )), (ψψ)(x1 ) . . . (ψψ)(xn ) N = (−1)n+1 (66) π where the sum is over the cyclic permutations π of (1, 2, . . . , n). Each term has the form tr(SN (x1 − x2 )SN (x2 − x3 ) . . . SN (xn − x1 )). On any ball B use Lemma 6 to bound this by
(67)
Bosonization of Massive Fermions
259
h(x1 , . . . xn ) = O(1)g(x1 − x2 )g(x2 − x3 ) . . . g(xn − x1 ),
(68)
g(x) = χ2B (x)|x|−1 .
(69)
where
Then by Young’s inequality Z h ≤ |B|kg ∗ g ∗ . . . ∗ gk∞ B
≤ |B|kgk3/2 kg ∗ . . . ∗ gk3 ≤ |B|kgk23/2 kg ∗ . . . ∗ gk3/2 ≤ |B|kgk33/2 kg ∗ . . . ∗ gk1 ≤ |B|kgk33/2 kgkn−3 1 and this is finite since g ∈ Lp for p < 2.
(70)
Remark. Note that since the number of permutations in bounded by n!, it follows from our proof that for any f there are constants A, B such that T
(71) (ψψ)(f )n N ≤ AB n n! for all N . Theorem 2. For n + m ≥ 3, +T +T *n *n m m Y Y Y Y (ψψ)(fi ) (ψ0ψ)(gj ) = (c)n (ic)m : cos φ : (fi ) : sin φ : (gj ) . i=1
j=1
i=1
j=1
4π
(72) Proof. The pointwise identity follows from Theorem 1. The correlations smeared with test functions are given as integrals over the pointwise correlation functions by Lemma 5 and Lemma 8. Hence the result. 2.3. The main result. 2.3.1. The sine-Gordon model. We consider the the sine-Gordon model on the plane with an interaction density supported on a square 3. The model is defined by the interaction Z : cos φ(x) :1,N dx. (73) ζ : cos φ :1,N (χ3 ) = ζ 3
By scaling we can always arrange that 3 = 1 = a unit square, and we make this choice for the rest of the paper. Define normalized expectations by R [. . . ] exp(ζ : cos φ :1,N (χ1 ))dµβ,r,N (φ) . (74) h. . . iζ,β,r,N = R exp(ζ : cos φ :1,N (χ1 ))dµβ,r,N (φ) We study expectations of products of the fields sin φ. The main technical result is:
260
J. Dimock
Theorem 3. Let f1 , . . . , fn be test functions in C0∞ (R2 ) with disjoint supports in 1. Then for all r, N and for β in a compact subset of (0, 16π/3) the functions *n + Y : sin φ :1,N (fi ) (75) Wβ,r,N (ζ; f1 , . . . , fn ) = ζ n i=1
ζ,β,r,N
are analytic in ζ and uniformly bounded on a fixed neighborhood of the origin. The same T (ζ; f1 , . . . , fn ). is true for the truncated functions Wβ,r,N The proof is given in subsequent sections and involves taking some shortcuts. This accounts for the presence of the ζ n which is not really necessary. Assuming the theorem we can prove: Lemma 9. Under the hypotheses of the theorem, + *n Y n : sin φ : (fi ) : W(ζ; f1 , . . . , fn ) = ζ i=1
(76) ζ,4π
defined by W(ζ; f1 , . . . , fn ) = lim lim lim Wβ,r,N (ζ; f1 , . . . , fn ) β%4π r→0 N →∞
(77)
exists and is analytic in a neighborhood of the origin. The same is true for truncated functions +T *n Y T n : sin φ : (fi ) : (78) W (ζ; f1 , . . . , fn ) = ζ i=1
ζ,4π
and for these we have the derivatives at zero (W T )(k) (0; f1 , . . . , fn ) = ( 0 k
Qn . k! k−n T : sin φ : (f )(: cos φ : (χ )) k ≥n i 1 i=1 (k−n)! 4π
(79)
Remark. We are only asserting that the N → ∞, r → 0 limits exist for β < 4π and then defining the theory for β = 4π as the limit from below. In fact the N → ∞, r → 0 limits no doubt exist for all β < 16π/3 and give the same answer at β = 4π. This should follow from our subsequent analysis as in [12]. Proof. It suffices to prove that the limit exists for the truncated functions. By a standard computation we have the derivatives (W T )(k) β,r,N (0; f1 , . . . , fn ) = ( 0 k
Qn . k! k−n T : sin φ : (f )(: cos φ : (χ k ≥n )) 1,N i 1,N 1 i=1 (k−n)! β,r,N
(80)
k By the theorem we have Cauchy bounds |(W T )(k) β,r,N (0)| ≤ AB k! uniformly in β, r, N .
Bosonization of Massive Fermions
261
The derivatives also have limits as N → ∞ given by (79). For n + k ≥ 3 this follows by Lemma 4 and Lemma 5. For n = 1, k − n = 0 and n = 1, k − n = 1 the limit vanishes, in the latter case by φ → −φ symmetry. For n = 2, k − n = 0 the limit exists by the pointwise limit and the assumption of non-overlapping supports. Thus in the Taylor series expansion (W T )β,r,N (ζ; f1 , . . . , fn ) =
∞ X
k (W T )(k) β,r,N (0; f1 , . . . , fn )ζ /k!
(81)
k=0
we can take the various limits and all the claimed results follow.
2.3.2. Massive fermions. Now consider massive fermions. We again start with a regularized theory defining D E [. . . ]eµ(ψψ)(χ3 ) E N. (82) h. . . iµ,N = D eµ(ψψ)(χ3 ) N
Again by scaling it suffices to consider 3 = 1 = a unit square. We study particularly expectations of products of ψψ and ψ0ψ. With the cutoff in place an expression like + *n Y µ(ψψ)(χ1 ) (ψψ)(fi )e i=1
N
can be defined by a power series in µ. The coefficient of µk /k! is an (n + k) × (n + k) determinant with bounded entries. For fixed n it can be estimated by O(1)k k! and hence the series converges for |µ| small. (Actually it converges for all µ.) These considerations
Q n (ψψ)(f show that i ) µ,N is well-defined and analytic for |µ| small. The same is i=1 true with the fields ψ0ψ. Lemma 10. For test functions with disjoint supports the limit *n *n + + Y Y (ψ0ψ)(fi ) = lim (ψ0ψ)(fi ) i=1
µ
N →∞
i=1
(83) µ,N
exists and is analytic in µ in a neighborhood of the origin. The same is true for the truncated functions and these satisfy: dk dµk
*
n Y
+T
* |µ=0 =
(ψ0ψ)(fi )
i=1
n Y
+T k
(ψ0ψ)(fi )((ψψ)(χ1 ))
.
(84)
i=1
µ
Proof. It suffices to consider the truncated functions. The derivatives at zero can be computed as dk dµk
*
n Y i=1
+T
* |µ=0 =
(ψ0ψ)(fi ) µ,N
n Y i=1
+T k
.
(ψ0ψ)(fi )((ψψ)(χ1 ))
N
(85)
262
J. Dimock
The derivatives have bounds of the form AB k k! uniformly in N by estimates like (71). The derivatives have limits as N → ∞ given by (84). For n + k ≥ 3 this follows by Lemma 8. For n = 1, k = 0 and n = 1, k = 1 the expression vanishes, in the latter case by reflection invariance. For n = 2, k = 0 the limit exists by the assumption of non-overlapping supports.
Qn Now in the Taylor series expansion for i=1 (ψ0ψ)(fi ) µ we can take the limit N → ∞ and all the results follow. 2.3.3. Massive equivalence. Our main result is the following Theorem 4. Let f1 , . . . fn be test functions with disjoint support in 1. If ζ = µc is sufficiently small, + *n + *n Y Y n (ic) : sin φ : (fi ) : = (ψ0ψ)(fi ) . (86) i=1
ζ,4π
i=1
µ
Proof. It suffices to show that the truncated functions are equal. If ζ = µc = 0 the result follows from Theorem 2. If ζ = µc 6= 0 it suffices to establish: *n *n +T +T Y Y n n : sin φ : (fi ) : =µ (ψ0ψ)(fi ) . (87) (iζ) i=1
ζ,4π
i=1
µ
Both sides are now analytic functions of µ on a neighborhood of the origin. To establish the identity it suffices to show that the derivatives at zero agree. The derivatives are computed in Lemma 9 and Lemma 10. To prove the identity we need that *n +T *n +T Y Y n k k k : sin φ : (fi )(: cos φ : (χ1 )) = (ψ0ψ)(fi )((ψψ)(χ1 )) . (ic) c (88) i=1 i=1 4π But this follows from Theorem 2. This completes the proof of equivalence.
Remark. If we try to include terms like : cos φ : and ψψ in the statement of the theorem we run into the difficulty that divergent terms like (ψψ)(f )(ψψ)(χ1 ) N occur in the perturbation expansion. 3. The sine-Gordon Model 3.1. Overview. We now study the sine-Gordon model on the plane with an interaction density supported on a square. The goal is to obtain estimates uniform in the ultraviolet cutoff N and prove Theorem 3. The analysis uses a renormalization group technique, especially the tools developed in an earlier treatment of the sine-Gordon model by Dimock and Hurd [11, 12] (see the remarks in the introduction). The structure of the proof is now somewhat different in that the leading terms in perturbation theory play a prominent role. This makes it possible to treat remainder terms more crudely. In particular we do not make any extractions from the remainders, which is problematic for dimensionless fields. The closer one is to 8π the more orders of perturbation theory
Bosonization of Massive Fermions
263
one has to take to render the remainder irrelevant. For this paper we only go to second order perturbation theory, and the proof works for β < 16π/3 which is sufficient for our purposes. Possibly something similar will work all the way up to β = 8π. The present model also differs from [12] in that the base manifold is the plane and not the torus, and because there are now genuine infrared divergences in the problem (the r → 0 limit). Our version of the renormalization group follows the general framework of Brydges and Yau [8]. We also adopt some recent innovations in this technique due to Brydges, Dimock, and Hurd [5, 6, 7]. In particular we use open polymers and background potentials. 3.2. The renormalization group. The model is defined in Sect. 2. Consider expressions of the form Z 0 ˜ ˜ 4r,N (ζ, φ) = µβ,r,N ∗ eVN (ζ,3) (φ) = eVN (ζ,3,φ+φ ) dµβ,r,N (φ0 ), Z ˜ VN (ζ, 3, φ) = ζ : cos(φ(x)) :1,N dx (89) 3
and µβ,r,N has covariance βvr,N defined in (6). Then 4r,N (ζ, 0) is the partition function, and derivatives of 4r,N (ζ, φ) yield correlation functions. We want to obtain control over 4r,N uniformly in r, N as N → ∞ and r → 0. Scaling up by a factor LN we find 4r,N (ζ, φ) = (µβ,rN ,0 ∗ eV (ζN ,3N ) )(φ(·/LN )), Z cos φ(x) dx, V (ζ, X, φ) = ζ
(90)
X
where 3N = LN 3, ζN = L−2N eβv1,N (0)/2 ζ, rN = L−N r.
(91)
Note that the Wick ordering constant exp(βv1,N (0)/2) has been absorbed into the coupling constant. The measure µβ,rN ,0 has the covariance βvrN ,0 and it is convenient to write this as βvrN ,0 = β 0 vˆ N , where β 0 = erN β, 2
vˆ N (x) = e−rN vrN ,0 (x), Z 2 eipx −(p2 +rN −2 ) e = (2π) dp. 2 p2 + r N 2
(92)
We seek a sequence of densities Zir,N (φ) for i = N, N − 1, . . . , 1, 0 localized in 3i such that 4r,N (φ) = (µβ 0 vˆ i ∗ Zir,N )(φ(·/Li )).
(93)
264
J. Dimock
r,N Initially this holds with ZN = eV (ζN ,3N ) . Given such a density for i we define the density for i − 1 as follows. First define Ci by
Ci (x) = vˆ i (x) − vˆ i−1 (x/L),
(94)
µβ 0 vˆ i = µβ 0 vˆ i−1 (·/L) ∗ µβ 0 Ci .
(95)
whence
Then scaling down by L changes the first factor to µβ 0 vˆ i−1 and gives the expression for i − 1 with r,N )(φ) = (µβ 0 Ci ∗ Zir,N )(φ(·/L)). (Zi−1
(96)
This is the basic renormalization group (RG) transformation. Since Ci is only weakly dependent on i the study of Zir,N is essentially the study of iterations of this mapping. Following the flow down to i = 0 gives Z0r,N , and for this functional it will be possible to analyze the r, N dependence. 3.3. Polymer expansions. Control over the densities Zir,N depends on keeping careful track of the localization of the functional. This is accomplished by polymer expansions. We only sketch the formalism here, referring to [8, 4, 5] for more details A (open) polymer X is a union of open unit squares centered on points of the integer lattice in R2 . A polymer activity is a function K(X, φ) depending on polymers X and fields φ with the property that the dependence on φ is localized in X. One can define a product on polymer activities and also an exponential function Exp. If is the characteristic function of unit blocks, then we have for an additive polymer activity V and a general polymer activity K Exp(eV + K)(X, φ) =
XY
K(Xi , φ)eV (X\∪i Xi ,φ) ,
(97)
{Xi } i
where the sum is over collections of disjoint polymers {Xi } in X. The densities will all be represented in this form. In particular the initial density has this form with K = 0: r,N (φ) = eV (ζN ,3N ,φ) = Exp(eV (ζN ) )(3N , φ). ZN
(98)
It is useful to keep track of the dependence on fields and their derivatives separately. Thus we consider modified polymer activities K(X, ψ) defined on pairs ψ = (ψ0 , ψ1 ) consisting of a continuous function ψ0 and a continuous vector field ψ1 . A basic polymer activity K 0 (X, φ) is represented as a modified polymer activity K(X, ψ) if K(X, ψφ ) = K 0 (X, φ), where ψφ = (φ, ∂φ). Modified polymer activities K, K 0 are equivalent, written K ≈ K 0 , if K(X, ψφ ) = K 0 (X, ψφ ). Norms are defined on the modified polymer activities. Let Kn (X, ψ) be the nth derivative of K(X, ψ) with respect to ψ. This is a multilinear functional on pairs f = (f0 , f1 ) of continuous functions, and we assume that restricted to components it is given by a measure of bounded variation supported on X. We let kKn (X, ψ)k denote total variation norm (with a sum over components) and let kKn (X, ψ)k1 denote this norm restricted to a product of unit blocks 1 = 11 × · · · × 1n .
Bosonization of Massive Fermions
265
Next we define a large field regulator
X Z
G(κ, X, φ) = exp κ
1≤|α|≤3
|∂ α φ|2
(99)
X
with constant 0 ≤ κ ≤ 1. Then define X sup kKn (X, ψφ )k1 G(X, φ)−1 . kKn (X)kG =
(100)
3 1 φ∈C
Dependence on the set X is controlled by a large set regulator 0(X) which is such that 0(X)−1 has polynomial tree decay in X. We define X 0(X)kKn (X)kG . (101) kKn kG,0 = sup 1 X⊃1
Finally, for h > 0 we define kKkG,0,h =
X
hn /n! kKn kG,0 .
(102)
n
Our polymer expansions will always involve modified polymer activities K(X, ψ) for which this norm is finite. Such activities are analytic in ψ in a strip of width h. Sometimes we do that last two sums in the opposite order and call the norm kKkG,h,0 . If K is translation invariant then one can drop the supremum over 1 in (101) and so kKkG,h,0 = kKkG,0,h . Our K ’s are generally translation invariant, although they appear in expressions like Exp(eV + K)(3, ψφ ) which are not. Another variation is that sometimes we replace 0 by 0n (X) = 2n|X| 0(X).
(103)
3.4. The RG flow to second order. As a preliminary we study some aspects of the RG flow to second order in ζ. Suppose that after some number of steps we have basic polymer activities of the form K = QeV + O(ζ 3 ), where V = O(ζ), Q = O(ζ 2 ). Then our density has the form Exp(( + Q)eV )(3, φ) + O(ζ 3 ).
(104)
We want to find new densities Q1 , V1 and more generally continuous functions Qt , Vt , t ∈ [0, 1] such that Q0 = Q, V0 = V and such that for β = β 0 , C = Ci , µtβC ∗ Exp(( + Q)eV ) = Exp(( + Qt )eVt ) + O(ζ 3 ).
(105)
In general if one defines At by µtβC ∗ Exp(A) = Exp(At ),
(106)
then one finds the At satisfies the differential equation E(At ) ≡ ( where
∂ − 1βC )At − BβC (At , At ) = 0, ∂t
(107)
266
J. Dimock
Z 1 dµC (ζ) A2 (φ; ζ, ζ), 2 Z 1 dµC (ζ) A1 (φ; ζ) ◦ B1 (φ; ζ), BC (A, B)(φ) = 2 1C A(φ) =
(108)
and where An (φ) is the nth derivative of A(φ). Furthermore approximate solutions to (107) give approximate solutions to (106). This is the approach we take. Starting with A = ( + Q)eV we ask for At = ( + Qt )eVt so that E(At ) = O(ζ 3 ). Associated with V (ζ) = V (ζ, X, φ) defined in (90), we define polymer activities Q0 (ζ, W ) for W = (W1 , W2 ) and |X| ≤ 2 by Z ζ2 cos(φ(x) + φ(y))W1 (x − y) dx dy Q0 (ζ, W ; X, φ) = 4 X˜ Z ζ2 cos(φ(x) − φ(y))W2 (x − y) dx dy. (109) + 4 X˜ Here for X = 1 ∪ 10 we define X˜ = 1 × 10 ∪ 10 × 1.
(110)
If |X| ≥ 3 then Q0 (ζ, W ; X, φ) = 0. Lemma 11. Let Vt = V (ζt ) and Q0t = Q0 (ζt , Wt ). Then At = ( + Q0t )eVt satisfies E(At ) = O(ζ 3 ) provided ζt = e−tβC(0)/2 ζ, W1,t (x) + 1 = e−tβC(x) (W1,0 (x) + 1), W2,t (x) + 1 = e+tβC(x) (W2,0 (x) + 1).
(111)
Proof. For the first order terms in E to vanish we need ∂Vt − 1βC Vt = 0. ∂t
(112)
This follows by our choice of ζt . For the second order terms in E to vanish it turns out we need: ( where for |X| ≤ 2,
∂ − 1βC )Q0t − Jt = 0, ∂t
(113)
Z
1 Jt (X, φ) = 2 =−
˜ X ζt2
8
βC(x − y)(Vt )1 (φ, x)(Vt )1 (φ, y) dx dy X Z eipφ(x) pqβC(x − y)eiqφ(y) dx dy.
p,q=±1
(114)
˜ X
If Q0t has the general form Q0t (X, φ) =
Z ζt2 X eipφ(x) Wpq,t (x − y)eiqφ(y) dx dy. 8 ˜ X p,q=±1
(115)
Bosonization of Massive Fermions
267
Then for (113) to vanish the kernel must satisfy
∂ + pqβC(x) Wpq,t (x) = −pqβC(x) ∂t
which can also be written
∂ + pqβC(x) (Wpq,t (x) + 1) = 0. ∂t
(116)
(117)
This has the solution Wpq,t (x) + 1 = e−tpqβC(x) Wpq,0 (x) + 1 .
(118)
The theorem is the special case W1,1 = W−1,−1 and W1,−1 = W−1,1 . If it is true at t = 0 then it is true for all t. Denoting the first pair by W1 and the second pair by W2 we have Q0t = Q0 (ζt , Wt ) and (118) becomes (111). Now we modify our definition of Q by defining Q(ζ, W ; X, ψ). We say that a polymer with |X| ≤ 2 is a small set if X¯ is connected. Otherwise it is a large set. We define Q(ζ, W ; X, ψ) = Q1 (ζ, W1 ; X, ψ) + Q2 (ζ, W2 ; X, ψ),
(119)
where Z ζ2 Q1 (ζ, W ; X, ψ) = cos (ψ0 (x) + ψ0 (y)) W (x − y) dx dy, 4 X˜ ( ζ2 R ˜ cos (ψ R0 (x) − ψ0 (y)) W (x − y) dx dy X large (120) Q2 (ζ, W ; X, ψ) = ζ42 RX x cos X small ψ 1 W (x − y) dx dy ˜ 4 X y and the integral we restrict:
Rx y
¯ We recover our earlier definition when is over the shortest path in X. Q(ζ, W ; X, ψφ ) = Q0 (ζ, W ; X, φ).
(121)
Now define Qt (X, ψ) = Q(ζt , Wt ; X, ψ). We do not have (113) for all ψ, but (113) does imply
∂ − 1βC ∂t
Qt − Jt (X, ψφ ) = 0.
(122)
The result follows from the identity (1βC Qt )(X, ψφ ) = (1βC Q0t )(X, φ), where 1βC is the Laplacian in ψ; see the Appendix. 3.5. Estimates. In estimates we make the convention that constants denoted O(1) are independent of L, while constants denoted C may depend on L. The value of C may vary from line to line.
268
J. Dimock
We start with some estimates on Ci (x) which is given by Z eipx −(p2 +ri2 ) −2 −L2 (p2 +ri2 ) e dp − e Ci (x) = (2π) p2 + ri2 Z Z L2 2 2 −2 ipx e e−α(p +ri ) dα dp = (2π) 1
Z
= (4π)−1
L2
α−1 e−x
2
/4α −αri2
e
dα.
(123)
1
Then we have the following uniform bounds: Lemma 12. For any multi-index β 6= 0, 0 < Ci (x) ≤ O(log L)e−|x|/L , |∂ β Ci (x)| ≤ O(1)e−|x|/2L .
(124)
Proof. For α ≤ L2 we have |x| |x| |x|2 ≥ √ −1≥ − 1. 4α L α
(125)
This gives the first bound. The second bound is similar. We always get at least one extra power of α−1 so the α integral has a bound independent of L. Next we study the evolution of the kernel W = (W1 , W2 ) under the transformation (111) at t = 1, with β = β 0 , followed by a scaling operation. Starting with WN = (W1,N , W2,N ) define inductively W1,i−1 (x) + 1 = e−β W2,i−1 (x) + 1 = e
0
Ci (Lx)
0
+β Ci (Lx)
(W1,i (Lx) + 1),
(W2,i (Lx) + 1).
(126)
This is equivalent to W1,i (x) + 1 = e−β W2,i (x) + 1 = e+β
0
0
wi (x)
wi (x)
,
,
(127)
where wi (x) =
N X
Ck (Lk−i x)
k=i+1 −2
Z
= (2π)
= (4π)−1
Z
2 2 −2(N −i) 2 2 eipx (e−(p +ri )L − e−(p +ri ) ) dp p2 + ri2
1
α−1 e−x
2
/4α −αri2
e
dα.
(128)
L−2(N −i)
Lemma 13. 1. For |x| ≤ 1 1 log |x|−1 + O(1), 2π 0 |W1,i (x)|, |x|β /2π |W2,i (x)| ≤ O(1). 0 < wi (x) ≤
(129)
Bosonization of Massive Fermions
269
2. For |x| ≥ 1 0 < wi (x) ≤ O(1)e−|x|/2 , |W1,i (x)|, |W2,i (x)| ≤ O(1)e−|x|/2 .
(130)
Proof. For |x| ≤ 1, wi (x) ≤ (4π)−1 = (4π)
−1
Z Z
1
α−1 e−x
2
/4α
dα
0 |x|−2 0
≤ O(1) + (4π)−1
α−1 e−1/4α dα Z
|x|−2
α−1 dα
1
1 log |x|−1 . = O(1) + 2π
(131)
This gives the first bound on wi and the bounds on W1,i , W2,i follow. For |x| ≥ 1 and α ≤ 1 we have |x| 1 |x| |x|2 ≥ √ −1≥ + √ − 1. 4α 2 α 2 α This gives the second bound on wi and the bounds on W1,i , W2,i follow.
(132)
Now we turn to some bounds on polymer activities. We start with bounds on V (X) = V (ζ, X, φ) defined in (90). The following hold for any G = G(κ, X, φ). Lemma 14. For a unit block 1: kV (1)kG,h ≤ |ζ|eh .
(133)
Proof. The nth derivative of V (1, φ) with respect to φ is the multilinear function on continuous fj = (fj , 0) given by Z Vn (1, φ; f1 , . . . fn ) = ζ cos(n) (φ(x))f1 (x) . . . fn (x) dx. (134) 1
Then Vn (1, φ) is given by a measure satisfying kVn (1, φ)k ≤ |ζ|.
(135)
Similarly kVn (ζ, 1)kG ≤ |ζ|. Multiplying by hn /n! and summing over n gives the result. Lemma 15. For |ζ|e4h sufficiently small keV (1) kG,h ≤ 2, keV (1) − 1kG,h ≤ 2|ζ|eh , keV (X) kG,h ≤ 2|X| .
(136)
270
J. Dimock
Proof. First suppose that X = 1. Then after some combinatorics (see [5]) one finds that ∞
X hn hn V (1,φ) k(e kVn (1, φ)k) ≤ exp(|ζ|eh ). )n k ≤ exp( n! n!
(137)
hn V (1) k(e )n kG ≤ exp(|ζ|eh ). n!
(138)
n=0
Similarly
Now replace h by 4h, multiply by 4−n , and sum over n to obtain the first bound for |ζ|e4h small. The second bound follows from the first and the representation Z 1 V (1) −1= V (1)esV (1) ds. (139) e 0
The last bound follows by keV (X) kG,h ≤
Y
keV (1) kG,h .
(140)
1⊂X
Lemma 16. Let |ζ|e4h be sufficiently small, and let Q be a translation invariant polymer activity. Then kQeV kG,0,h ≤ kQkG,01 ,h .
(141)
Proof. Using Lemma 15 we have kQ(X)eV (X) kG,h ≤ kQ(X)kG,h keV (X) k1,h ≤ 2|X| kQ(X)kG,h .
(142)
Then multiplying by 0(X) and summing gives the result with norm k·kG,h,0 = k·kG,0,h . Next we give some bounds on Q(ζ, W ). For good bounds one would like W1 , W2 to be locally integrable and exponentially decaying. Referring to Lemma 13 we see that this is true for W1 , but not for W2 if β 0 ≥ 4π. Thus we only assume the weaker bounds Z 0 |W1 (x − y)|dxdy ≤ Ce−d(1,1 )/2L , 1×10 Z 0 |x − y|2 |W2 (x − y)|dxdy ≤ Ce−d(1,1 )/2L . (143) 1×10
0
Note that W1,i , W2,i satisfy this, even with stronger decay e−d(1,1 )/2 . For W2,i we need 0 that |x|2−β /2π is locally integrable and this is true for β 0 < 8π. For a good uniform bound on Q2 (ζ, W2 ) we subtract the most singular part which is the constant Qsing (ζ, W2 ; X, ψ) = Q2 (ζ, W2 ; X, 0) Z ζ2 W2 (x − y) dx dy. = 4 X˜ For the difference the bound (143) will be adequate.
(144)
Bosonization of Massive Fermions
271
Lemma 17. Let W1 , W2 satisfy (143). Then for some constant C: kQ1 (ζ, W1 )kG,0,h ≤ C|ζ|2 eO(1)h , kQ2 (ζ, W2 ) − Qsing (ζ, W2 )kG,0,h ≤ C|ζ|2 eO(1)h κ−1 , kQ(ζ, W ) − Qsing (ζ, W2 )kG,0,h ≤ C|ζ|2 eO(1)h κ−1 . (145) Proof. The last follows from the first two. For the first bound we compute for fj = (fj , 0), (Q1 )n (X, ψφ ; f1 , . . . fn )
=
ζ2 4
Z ˜ X
cos(n) (φ(x) + φ(y))W1 (x − y)
n Y
(fj (x) + fj (y)) dx dy.
(146)
j=1
This leads to
Z
0
k(Q1 )n (1 ∪ 1 , ψφ )k ≤ O(1) |ζ| n
2 1×10
|W1 (x − y)| dx dy 0
≤ CO(1)n |ζ|2 e−d(1,1 )/2L .
(147)
The norm restricted to 1 has the same bound, and since there are 2n terms in the sum over 1 it follows that 0
k(Q1 )n (1 ∪ 10 )kG ≤ CO(1)n |ζ|2 e−d(1,1 )/2L .
(148)
Now multiply by 0(X) and sum over X, then multiply by hn /n! and sum over n, and get the result. The second bound proceeds in just the same way for large sets. We avoid the singularity at x = y by the large set hypothesis and the subtraction has the same bound as the function. For small sets X we compute for continuous vector-valued fj = (0, fj ), (Q2 − Qsing )n (X, ψφ ; f1 , . . . fn ) Z x Z n Z x Y ζ2 (n) cos ∂φ − δn,0 W2 (x − y) fj dx dy. (149) = 4 X˜ y y j=1
It follows that k(Q2 − Qsing )n (X, ψφ )k Z Z 2 (n) ≤ O(1)|ζ| cos ˜ X
x y
∂φ − δn,0 |W2 (x − y)||x − y|n dx dy. (150)
For n = 0, 1 Sobolev inequalities give Z x cos ∂φ − 1 ≤ |x − y|2 ( sup |∂φ(x)|)2 ≤ |x − y|2 κ−1 G(κ, X, φ) x∈X y Z x sin ∂φ ≤ |x − y| sup |∂φ(x)| ≤ |x − y|κ−1/2 G(κ, X, φ). (151) y
x∈X
272
J. Dimock
Thus we always have a factor |x − y|2 , and so k(Q2 − Qsing )n (1 ∪ 10 , ψφ )k Z n 2 −1 ≤ O(1) |ζ| κ G(κ, X, φ)
1×10
|x − y|2 |W2 (x − y)| dx dy
≤ CO(1)n |ζ|2 κ−1 G(κ, X, φ).
(152)
The restriction to 1 has the same bound. Now multiply by G−1 , taking the supremum over φ, and sum over 2n terms 1 to obtain k(Q2 − Qsing )n (1 ∪ 10 )kG ≤ CO(1)n |ζ|2 κ−1 .
(153)
The result now follows as before.
3.6. The full RG flow. We now study the full RG flow. The building blocks are the polymer activities Vir,N = V (ζir,N ), = Q(ζir,N , Wi ) − Qsing (ζir,N , W2,i ), Qr,N i where ζir,N = L−2i exp
β0 β (v1,N (0) − vr,N (0)) + vˆ i (0) ζ. 2 2
(154)
(155)
r,N = ζN . Also note using β 0 vˆ i (0) ≤ βvri ,0 (0) and vr,N (0) = Note that initially ζN vrN ,0 (0) and
vr,0 (0) = − log r/2π + O(1)
(156)
we have that |ζir,N | ≤ O(1)|ζ|i , |ζ|i = L−(2−β/4π)i |ζ|.
(157)
We define hi = 3 −
∞ X
2−j
(158)
j=i
so that 1 ≤ hi ≤ 3. We also define Gi (X, φ) = G(κi , X, φ), where κi = κ0 2−i
(159)
for some constant κ0 . For the main theorem we have the following restrictions on the parameter values. First fix β0 < 16π/3 and allow β ≤ β0 . We assume rN is small enough so β 0 = 2 erN β ≤ β1 < 16π/3. Then choose sufficiently small (depending on β0 ), L sufficiently large (depending on β0 , ), κ0 sufficiently small (depending on β0 , , L), and finally |ζ| sufficiently small (depending on β0 , , L, κ0 ).
Bosonization of Massive Fermions
273
By the previous lemmas we have r,N
keVi Vir,N
k(e
kGi ,0,hi ≤ C,
− 1)kGi ,0,hi ≤ C|ζ|i ≤ |ζ|1− , i r,N
kQr,N eV i i
2− kGi ,0,hi ≤ C|ζ|2i κ−1 . i ≤ |ζ|i
(160)
Here we have used that L−(2−β/4π)i 2i ≤ 1 for L sufficiently large, and that C|ζ| κ−1 0 ≤ 1 for |ζ| sufficiently small. Theorem 5. For β < 16π/3 and |ζ| sufficiently small as above, there exist polymer such that activities Kir,N (X, ψ) and constants r,N i r,N
r,N
Zir,N (φ) = ei Exp(eVi
+ Kir,N )(3i , ψφ ).
(161)
The Kir,N are analytic in ζ and translation invariant. They have the form r,N
eVi Kir,N = Qr,N i
+ Rir,N ,
(162)
where for all β, r, N, i: , kRir,N kGi ,0,hi ≤ |ζ|3− i
(163)
kKir,N kGi ,0,hi ≤ 2|ζ|2− . i
(164)
and hence
corresponds to the vacuum energy in second order perturbation Remark. The term r,N i theory. It does not have a uniform bound for β ≥ 4π. Proof. We suppress r, N from the notation. The proof that i , Ri exist and that Ri satisfies the bounds is by induction on i, starting with i = N and working down to i = 0. We assume the result is true for i and prove it for i − 1. For i=N the result holds with WN = 0, QN = 0, RN = 0, N = 0. Each step preserves the analyticity in ζ. Step 1. For the fluctuation integral we have in general that for any polymer activity M , µtC ∗ Exp( + M ) = Exp( + Ft (M )),
(165)
where Ft (M ) is a certain analytic function of M . (See the Appendix for more details.) For a family Gt (X, φ), 0 ≤ t ≤ 1, of large field regulators satisfying for t > s µ(t−s)C ∗ Gs ≤ Gt
(166)
kFt (M )kGt ,0,h0 ≤ kM kG0 ,0,h
(167)
kM kG,0,h ≤ (h − h0 )2 /16kCkθ ,
(168)
we have the general bound [8]
provided h0 < h and
where kCkθ is a certain norm on C.
274
J. Dimock
We study Zi−1 (φ) = ei µβ 0 Ci ∗ Exp(eVi + Ki )(3i , ψφ,L−1 ),
(169)
where ψL−1 (x) = (ψ0 (x/L), L−1 ψ1 (x/L)). Before the scaling the polymer exponential is rewritten µtβ 0 Ci ∗ Exp(eVi + Ki )(3i , ψφ ) = µtβ 0 Ci ∗ Exp( + Mi )(3i , ψφ ) = Exp( + Ft (Mi ))(3i , ψφ ),
(170)
Mi = (eVi − 1) + Ki .
(171)
Gt (X, φ) = (2|X| Gi−1 (X, φ))t Gi (X, φ)1−t .
(172)
where
Now define
This satisfies (166) as in [6]. We use that κi−1 = 2κi and that κ0 is sufficiently small in order that κi kβ 0 Ci k be sufficiently small for various Sobolev norms. We conclude from (167) that kFt (Mi )kGt ,0,hi−1 ≤ kMi kGi ,0,hi ≤ 3|ζ|1− . i
(173)
The condition (168) is satisfied since for L sufficiently large and |ζ| sufficiently small 16kMi kGi ,0,hi kβ 0 Ci kθ ≤ C|ζ|1− ≤ 2−2i+2 = (hi − hi−1 )2 . i
(174)
Next we isolate the second order contribution. Write Mi = Ni + Ri , where Ni = (eVi − 1) + Qi eVi .
(175)
As the leading term in Ft (Mi ) = Ft (Ni + Ri ) we take Nt = (eVt − 1) + Qt eVt , where with ζt = e−tβ
0
Ci (0)/2
(176)
ζi ,
Vt = V (ζt ), Qt = Q(ζt , Wt ) − Qsing (ζi , W2,i ),
(177)
and Wt is defined by (111) with β = β 0 , C = Ci , and W0 = Wi . This is consistent with the discussion in Sect. 3.4; the subtraction of the constant Qsing will have no effect to second order. Now provided that Rt ≈ Ft (Ni + Ri ) − Nt we can rewrite (170) as Exp( + Rt + Nt )(3i , ψφ ) = Exp(eVt + Kt )(3i , ψφ ),
(178)
where Kt = Qt eVt + Rt To define Rt we write Ft (Ni + Ri ) − Nt = (Ft (Ni + Ri ) − Ft (Ni )) + (Ft (Ni ) − Nt ) Z t Z 1 (Ft )1 (Ni + sRi ; Ri ) ds − Ft−s 1 (Ns ; Es ) ds, = 0
0
(179)
Bosonization of Massive Fermions
275
where (Ft )1 is the derivative of Ft and Et = E(Nt ) = E(+Nt ). An explicit computation [5] shows that ∂ − 1β 0 Ci Qt − Jt eVt + β 0 Ci ((Vt )1 , (Vt )1 ) Qt eVt Et = ∂t − β 0 Ci ((Qt )1 , (Vt )1 ) eVt − 2Bβ 0 Ci eVt , Qt eVt (180) − B β 0 Ci Q t e V t , Q t e V t . In the first term Qsing does not contribute and so the term is equivalent to zero by (122). Define Et0 to be equal to Et but with the first term removed so Et0 ≈ Et . Now define Z 1 Z t Rt = (Ft )1 (Ni + sRi ; Ri ) ds − (181) Ft−s 1 (Ns ; Es0 ) ds, 0
0
and we have Rt ≈ Ft (Ni + Ri ) − Nt as required (see the Appendix). Now we give some bounds. First write Qt = Q(ζt , Wt ) − Qsing (ζt , W2,t ) + Qsing (ζt , W2,t ) − Qsing (ζi , W2,i ).
(182)
The kernels Wt still satisfy the bounds (143) and we apply Lemma 17 with κ = tκi−1 + (1 − t)κi . Just as in the bound on Qi we find that . kQ(ζt , Wt ) − Qsing (ζt , W2,t )kGt ,0,hi ≤ |ζ|2− i
(183)
For the second term we write Qsing (ζt , W2,t , X) − Qsing (ζi , W2,i , X) Z 0 ζ2 (etβ (Ci (x−y)−Ci (0)) − 1)W2,i (x − y) dx dy = i 4 X˜ Z 0 0 ζi2 e−tβ Ci (0) (etβ Ci (x−y) − 1) dx dy. + 4 X˜
(184)
This also satisfies . kQsing (ζt , W2,t ) − Qsing (ζi , W2,i )kGt ,0,hi ≤ |ζ|2− i
(185)
Indeed in the first term the (Ci (x − y) − Ci (0)) supplies the factor |x − y| needed for a good bound on W2,i (x − y), and in the second term the Ci (x − y) gives the needed exponential decay. We conclude that kQt kGt ,0,hi−1 ≤ O(1)|ζ|2− . The same bound i . A similar first order bound holds for holds with 01 so kQt eVt kGt ,0,hi−1 ≤ O(1)|ζ|2− i Nt . Estimating (Ft )1 and (Ft−s )1 by Cauchy bounds we have from (181) ! kRt kGt ,0,hi−1 ≤ 2 kRi kG,0,hi + sup kEs0 kGs ,0,hi
.
(186)
s≤t
All the terms in Es0 are O(ζ 3 ) and by the techniques we have been using one can show . It follows that kRt kGt ,0,hi−1 ≤ O(1)|ζ|3− . that kEs0 kGs ,0,hi ≤ |ζ|3− i i
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Now let ζ # , W # , V # , Q# , K # , R# be the values of ζt , Wt , Vt , Qt , Kt , Rt at t = 1. Then we have Zi−1 (φ) = ei Exp(eV + K # )(3i , ψφ,L−1 ), #
(187)
where K # = Q# eV + R# . After switching a factor 2−|X| from G−1 1 to 0 we have #
#
, kQ# eV kGi−1 ,0−1 ,hi−1 ≤ O(1)|ζ|2− i kR# kGi−1 ,0−1 ,hi−1 ≤ O(1)|ζ|3− , i
(188)
and hence a bound on K # . Step 2. We extract a constant Fi (X) which is the last two terms in Q# : Fi (X) = Qsing (ζ # , W # , X) − Qsing (ζi , W2,i , X).
(189)
. The extraction is a rearrangement [5]2 From (185) we have kFi kGi−1 ,0−1 ,hi−1 ≤ |ζ|2− i ∗
Exp(eV + K # )(3i , ψ) = eδi Exp(eV + K ∗ )(3i , ψ), #
(190)
where V ∗ = V # and K ∗ = E(K # , Fi ) #
= (K # − Fi eV ) + E≥2 (K # , Fi ), X δi = Fi (X).
(191)
X⊂3i ∗
Now we can write K ∗ = Q∗ eV + R∗ , where Q∗ = Q(ζ # , W # ) − Qsing (ζ # , W2# ), R∗ = R# + E≥2 (K # , Fi ).
(192)
A general bound on E≥2 is kE≥2 (K, F )kG,0,h ≤ O(1)kKkG,01 ,h kF kG,0,h ).
(193)
Using this and (188) to bound R∗ , and (183) to bound Q∗ we have: ∗
kQ∗ eV kGi−1 ,0−2 ,hi−1 ≤ O(1)|ζ|2− , i kR∗ kGi−1 ,0−2 ,hi−1 ≤ O(1)|ζ|3− , i
(194)
and hence a bound on K ∗ . Now we have ∗
Zi−1 (φ) = ei−1 Exp(eV + K ∗ )(3i , ψφ,L−1 ), i−1 = i + δi .
(195)
2 In [5], 3 was a torus, not a rectangle in the plane. Nevertheless this formula holds with the same E and i the same δ .
Bosonization of Massive Fermions
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Step 3. Finally we scale down. The scaling transformation is [5]: ∗
Exp(eV + K ∗ ))(3i , ψφ,L−1 ) = Exp(eVi−1 + K 0 )(3i−1 , ψφ ),
(196)
where K 0 = S(K ∗ ) is a certain function of K ∗ . We have used L2 ζ # = L2 e−β
0
/2πCi (0)
ζi = ζi−1
(197)
to identify V ∗ (LX, φ(·/L)) = Vi−1 (X, φ). Now write S(K ∗ ) = S1 (K ∗ ) + S≥2 (K ∗ ), ∗ where S1 is the linearization. Then we have K 0 = S1 (Q∗ eV ) + Ri−1 , where Ri−1 = S1 (R∗ ) + S≥2 (K ∗ ).
(198)
However an explicit calculation using (121), (197), and W # (Lx) = Wi−1 (x) shows that ∗
S1 (Q∗ eV ) ≈ Qi−1 eVi−1 .
(199)
Thus if we define Ki−1 = Qi−1 eVi−1 + Ri−1 , then Ki−1 ≈ K 0 , we can replace K 0 by Ki−1 in (196), and obtain the required Zi−1 (φ) = ei−1 Exp(eVi−1 + Ki−1 )(3i−1 , ψφ ).
(200)
A basic S bound is kS(K)kG,0,h ≤ O(1)L2 kKkG,0−2 ,h .
(201)
The same bound holds for S1 and S≥2 has a bound proportional to kKk2 . We conclude for β ≤ β0 < 16π/3 , sufficiently small, and L sufficiently large, kRi−1 kGi−1 ,0,hi−1 ≤ O(1)L2 |ζ|3− i = O(1)L2 L−(2−β/4π)(3−) |ζ|3− i−1 ≤ |ζ|3− i−1 . This completes the proof.
(202)
3.7. Correlation functions. We now study the correlation functions, see also Hurd [17]. As in (75) we consider: Wβ,r,N (ζ; f1 , . . . , fn ) = ζ n h: sin φ :1,N (f1 ) . . . : sin φ :1,N (fn )iζ,β,r,N (203) with test functions fi ∈ C0∞ (R2 ) with non-overlapping supports in 1. These have a nice representation as functional derivatives of 4r,N (ζ, φ) as defined in (89). Indeed we have (−1)n Wβ,r,N (ζ, f1 , . . . , fn ) =
(4r,N )n (ζ, 0; f1 . . . ., fn ) . 4r,N (ζ, 0)
(204)
Equations (93) and (161) for i = 0 give the representation for the 4r,N . We have r,N
4r,N (ζ, φ) = e0 µβ 0 vˆ 0 ∗ Exp(Ar,N 0 )(1, ψφ ) r,N
= e0 µβ 0 vˆ 0 ∗ Ar,N 0 (1, ψφ ),
(205)
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where r,N
= eV0 Ar,N 0
+ K0r,N .
(206)
The divergent constants exp(r,N 0 ) cancel in the expression (204) and the functional derivatives can be taken inside the integral to obtain R r,N (A0 )n (1, ψφ ; ψf1 , . . . , ψfn ) dµβ 0 vˆ 0 (φ) n . (−1) Wβ,r,N (ζ, f1 , . . . ., fn ) = R r,N A0 (1, ψφ ) dµβ 0 vˆ 0 (φ) (207) We need uniform bounds on this function. In preparation we have R Lemma 18. For κ0 sufficiently small G(κ0 , 1, φ)dµβ 0 vˆ 0 (φ) is bounded by a constant independent of r. Proof. Let B be the quadratic form X Z
(φ, Bφ) =
1≤|α|≤3
Then we have
Z
1
|∂ α φ|2 .
(208)
Z G(κ0 , 1, φ) dµβ 0 vˆ 0 (φ) =
eκ0 (φ,Bφ) dµβ 0 vˆ 0 (φ)
= (det(1 − κ0 β 0 vˆ 0 B vˆ 0 ))−1/2 1/2
1/2
1/2
(209)
1/2
provided vˆ 0 B vˆ 0 is trace class and κ0 is sufficiently small. We give a bound on the trace norm which is uniform in r and hence get a uniform bound on the integral for κ0 P 1/2 1/2 1/2 sufficiently small. We have vˆ 0 B vˆ 0 = α A∗α Aα , where Aα = χ1 ∂ α vˆ 0 . We show that Aα is Hilbert-Schmidt with a uniformly bounded Hilbert–Schmidt norm to obtain the result. We have Z Z 2 1/2 2 dx dy (∂ α vˆ 0 )(x − y) kAα kHS = =
1 1/2 k∂ α vˆ 0 k22
= O(1)
Z
|pα |2 −(p2 +r2 ) e dp, p2 + r 2
and this is bounded uniformly in r since |α| ≥ 1. This completes the proof.
(210)
We paraphrase Theorem 3 as: Theorem 6. Under the hypotheses of Theorem 5, Wβ,r,N (ζ, f1 , . . . ., fn ) is analytic in ζ in a neighborhood of the origin and is bounded there uniformly in β, r, N . Proof. If suffices to assume kψfi k ≤ 1. By the definition of the norm r,N r,N −1 (A0 )n (1, ψφ ; ψf1 , . . . , ψfn ) ≤ n!h−n 0 G(κ0 , 1, φ)0(1) kA0 kG0 ,0,h0 . (211)
Bosonization of Massive Fermions
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Using this and the previous lemma we get for each n, Z (Ar,N )n (1, ψφ ; ψf , . . . , ψf ) dµβ 0 vˆ (φ) ≤ O(1)kAr,N kG ,0,h . 1 n 0 0 0 0 0
(212)
By Theorem 5, kAr,N 0 kG0 ,0,h0 ≤ C. Thus the numerator in (207) has a uniform bound and the analyticity follows from the analyticity of Ar,N 0 . The same holds for the denominator, but we also have using (160) Z 1 − Ar,N (1, ψφ ) dµβ 0 vˆ 0 (φ) ≤ O(1)k − Ar,N kG0 ,0,h0 ≤ O(1)|ζ|1− , (213) 0 0 so the denominator is bounded away from zero. This completes the proof of Theorem 6 and Theorem 3. 4. Appendix We collect some notes about the fluctuation step. First suppose we have basic polymer activities K = K(X, φ). We have the definitions Z (µC ∗ K)(φ) = K(φ + ζ) dµC (ζ), Z 1 K2 (φ; ζ, ζ) dµC (ζ), (1C K)(φ) = 2 Z 1 K1 (φ; ζ) ◦ K1 (φ; ζ) dµC (ζ). (214) BC (K, K)(φ) = 2 We define Kt (X, φ) by µtC ∗ Exp( + K) = Exp( + Kt ).
(215)
This determines Kt since the exponential function Exp is invertible. Then one finds the Kt satisfy the the differential equation E(Kt ) = E( + Kt ) = 0 or ∂ − 1C Kt − BC (Kt , Kt ) = 0 (216) ∂t with the initial condition K0 = K. This can be integrated to give an integral equation Z t µ(t−s)C ∗ BC (Ks , Ks ) ds. (217) Kt = µtC ∗ K + 0
A finite iteration gives an explicit formula for Kt (X, φ), also written (Ft K)(X, φ). Now suppose we have modified activities K = K(X, ψ). Now define Z (µC ∗ K)(ψ) = K(ψ + ψζ ) dµC (ζ), Z 1 K2 (ψ; ψζ , ψζ ) dµC (ζ), (1C K)(ψ) = 2 Z 1 K1 (ψ; ψζ ) ◦ K1 (ψ; ψζ ) dµC (ζ), (218) BC (K, K)(ψ) = 2
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where we recall ψζ = (ζ, ∂ζ). Again define Kt (X, ψ) = (Ft K)(X, ψ) by (215), and again Eqs. (216), (217) hold. Now consider the relation between the two formulations. Suppose that K(ψφ ) = K 0 (φ). Directly from the definition we have (µC ∗K)(ψφ ) = (µC ∗K 0 )(φ) and it follows that (Ft K)(ψφ ) = (Ft K 0 )(φ). A corollary is that if K ≈ K 0 then Ft (K) ≈ Ft (K 0 ). If also A ≈ A0 then (Ft )1 (K; A) ≈ (Ft )1 (K 0 ; A0 ). We also note explicitly that if K(ψφ ) = K 0 (φ) then (1C K)(ψφ ) = (1C K 0 )(φ), BC (K, K)(ψφ ) = BC (K 0 , K 0 )(φ).
(219)
d K(ψφ + sψf )|s=0 ds d 0 K (φ + sf )|s=0 = ds = (K 0 )1 (φ; f )
(220)
This follows from K1 (ψφ ; ψf ) =
and similarly for the second derivative. Acknowledgement. I would like to thank T. Hurd and D. Brydges for helpful remarks, and A. Wightman for insisting that somebody should establish results of this type.
References 1. Alvarez-Gaume, L., Bost, J.B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. 112, 503–552 (1987) 2. Benfatto, G.: An iterated expansion for the Yukawa gas. J. Stat. Phys. 41, 671 (1985) 3. Benfatto, G., Gallavotti, G., Nicolo, F.: On the massive sine-Gordon equation in the first few regions of collapse. Commun. Math. Phys. 83, 387–410 (1982) 4. Brydges, D.C.,Dimock, J. and Hurd, T.R.: Weak perturbations of Gaussian measures. In: J.Feldman, R.Froese, and L.Rosen, eds, Mathematical Quantum Theory I: Field Theory and Many-body Theory. Providence, RI.: AMS, 1994 5. Brydges, D.C.,Dimock, J. and Hurd, T.R.: The short distance behavior of φ43 . Commun. Math. Phys. 172, 143–186 (1995) 6. Brydges, D.C.,Dimock, J. and Hurd, T.R.: Estimates on renormalization group transformations. Preprint 7. Brydges, D.C., Dimock, J. and Hurd, T.R.: A non-Gaussian fixed point for φ4 in 4 − dimensions. Commun. Math. Phys. 8. Brydges, D.C. and Yau, H.T.: Grad φ perturbations of massless Gaussian fields. Commun. Math. Phys. 129, 351–392 (1990) 9. Brydges, D.C. and Kennedy, T.: Mayer expansions and the Hamilton-Jacobi equation. J. Stat. Phys. 48, 19–49 (1987) 10. Coleman, S.: Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D 11, 2088 (1975) 11. Dimock, J. and Hurd, T.R.: A renormalization group analysis of the Kosterlitz–Thouless phase. Commun. Math. Phys. 137, 263–287 (1991) 12. Dimock, J. and Hurd, T.R.: Construction of the two-dimensional sine-Gordon model for β < 8π. Commun. Math. Phys. 156, 547–580 (1993) 13. Fr¨ohlich, J. and Marchetti, P.: Bosonization, topological solitons, and fractional charges in two dimensional quantum field theory. Commun. Math. Phys. 116, 127–173 (1988) 14. Fr¨ohlich, J.: Classical and quantum statistical mechanics in one and two dimensions: Two component Yukawa and Coulomb systems. Commun. Math. Phys. 47, 233–268 (1976)
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15. Fr¨ohlich, J. and Seiler, E.: The massive Thirring-Schwinger model (QED2 ): Convergence of perturbation theory and particle structure. Helv. Phys. Acta 49, 889 (1976) 16. Hua, L.: Harmonic analysis of functions of several complex variables in the classical domains. Providence, RI: AMS, 1963 17. Hurd, T.R.: Charge correlations for the two-dimensional Coulomb gas. In: V. Rivasseau, ed, Constructive Physics, Berlin: Springer, 1995, pp. 311–326 18. Nicolo, F., Renn, J. and Steinmann, A.: On the massive sine-Gordon equation in all regions of collapse. Commun. Math. Phys.105, 291–326 (1986) Communicated by A. Jaffe
Commun. Math. Phys. 198, 283 – 309 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Quantum Screening Operators and Canonical q -de Rham Cocycles Abdellah Sebbar1,2 1 Centre de Recherches Mathematiques (CRM) Universite de Montreal, C. P. 6128, Succursale Centre Ville, Montreal, Quebec H3C 3J7, Canada 2 Centre Interuniversitaire de Calcul Mathematique Algebrique (CICMA), Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada. E-mail:
[email protected]
Received: 24 December 1997 / Accepted: 2 April 1998
Abstract: In this paper a close connection is established between certain cohomology c2 ), and a twisted q-de Rham spaces of representations of the quantum affine algebra Uq (sl (Jackson–Aomoto) cohomology of configuration spaces using the quantum screening operators.
1. Introduction The representation theory of Virasoro algebras, established by Feigin and Fuchs [7], gives a way to obtain intertwining operators between the Fock space representations, out of the top homology classes of certain one dimensional local systems over configuration spaces. A similar construction exists for affine Kac-Moody algebras. These intertwining operators are built up from the so-called screening operators. In [10], Ginzburg and Schechtman made the remark that in fact these screening operators contain more information. Namely, they provide canonical cocycles of the Virasoro and affine Kac-Moody Lie algebras, with coefficients in the de Rham complex of an operator-valued local system on the configuration space. This makes it possible to obtain canonical morphisms from higher homology groups of the above local system to appropriate Ext-groups between the Fock modules. The purpose of this paper is to investigate this connection between the geometry of configuration spaces and representation theory in the case of the quantum affine algebras c2 ). The quantized universal enveloping algebras of semisimple Lie algebras were Uq (sl studied in this context in [17], and the representations in question were the Verma modules. In this paper we consider the q-analog of the Wakimoto modules [19] via bosonization of the quantum affine algebras, using three scalar boson fields. From the same boson fields, we construct the screening operator S(z). We introduce a certain c2 ) and operators between bracket h·, ·i which is a pairing between elements of Uq (sl
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the q-Wakimoto modules. This bracket is defined using the Hopf algebra structure of c2 ). Then we solve the difference equation Uq (sl c2 ). hx, S(z)i = ∂q S(x, z) x ∈ Uq (sl It turns out that the solution S(z) and S(x, z) have very nice homological properties, in particular, S(x, z) satisfies a cocycle condition of the form: S(xy, z) = x · S(y, z) + ε(y)S(x, z), where the dot means the action of x on the Hom space between the modules, and ε is c2 ). The above difference equation is built by the counit map of the Hopf algebra Uq (sl means of q-difference operators of Jackson type, which generate a flat connection in a one dimensional vector bundle over the n-dimensional torus. We then consider a “q-de Rham” complex of the space of formal algebraic q-differential forms over the n-torus, whose “differentials” are the above q-difference operators. The homology groups of this complex can be regarded as the homology groups of the n-torus with coefficients in a local system with stalk C. From these data, we construct the canonical cocycles, which in fact live in the total space of a double complex produced by mixing the above q-de Rham complex and a Hochshild complex which will be introduced below. These cocycles yield canonical maps between the homology groups of the local systems and the Ext-spaces between the representations in question. c2 ), in fact, I should mention the emphasis made on the Hopf algebra structure of Uq (sl we make sense of the above notions only by the use of this structure. Also, techniques such as the vertex operators calculus and the normally ordered products were very helpful. 2. Hopf Algebras and Their Actions In this section we follow closely the treatment in [17] with less details. We present some constructions on Hopf algebras and their representations, and establish some results related to them. All the algebraic structures will be over the field of complex numbers. 2.1. Composition of maps. Let H be a Hopf algebra, and let 1, A, and ε denote respectively the comultiplication, the antipode and the counit maps. If M and N are two H-modules, one can define a structure of left module on both M ⊗ N and Hom(M, N ) by: X (x0 m) ⊗ (x00 n) (m ∈ M, n ∈ N , x ∈ H), x · (m ⊗ n) = (x)
(x · f )(m) =
X
x0 f (A(x00 )m)
(m ∈ M, f ∈ Hom(M, N ), x ∈ H),
(x)
where we have used the Sweedler notation for the comultiplication: X 1(x) = x0 ⊗ x00 (x ∈ H). (x)
(If more than one superscript
0
or 00 is involved, we will use numerical subscripts.)
Quantum Screening Operators and Canonical q-de Rham Cocycles
285
Each vector space carries a structure of an H-module through the map ε. Therefore, the dual space M∗ = Hom(M, C) is an H-module with the action of H given by: (x · φ)(m) = φ(A(x)m)
x ∈ H, φ ∈ M∗ , m ∈ M.
Lemma 2.1. For every x ∈ H, the following relation holds in H ⊗ H ⊗ H: X X x1 ⊗ 1 ⊗ x2 = x1,1 ⊗ A(x1,2 )x2,1 ⊗ x2,2 . (x)
(2.1)
(x)
Proof. We prove the identity by applying the coassociativity of the map 1 several times. Let x ∈ H, by coassociativity we have: x1 ⊗ x2,1 ⊗ x2,2 = x1,1 ⊗ x1,2 ⊗ x2 . Now, applying 1 ⊗ 1 ⊗ 1 to the left side, and 1 ⊗ 1 ⊗ 1 to the right side, we obtain by coassociativity: x1 ⊗ x2,1,1 ⊗ x2,1,2 ⊗ x2,2 = x1,1 ⊗ x1,2 ⊗ x2,1 ⊗ x2,2 . Therefore, x1,1 ⊗ A(x1,2 )x2,1 ⊗ x2,2 = x1 ⊗ A(x2,1,1 )x2,1,2 ⊗ x2,2 = x1 ⊗ ε(x2,1 ) ⊗ x2,2 = x1 ⊗ 1 ⊗ x 2 , using the axioms of the counit. This proves the lemma.
Proposition 2.2 (Composition lemma). If M, N and P are three H-modules, then for every f ∈ Hom(N , P) and for every g ∈ Hom(M, N ) and x ∈ H, we have: X (x0 · f ) ◦ (x00 · g). x · (f ◦ g) = (x)
Proof. The relation can be written as: x1 f (g(A(x2 )m)) = x1,1 f (A(x1,2 )x2,1 g(A(x2,2 )m)))
(m ∈ M),
which follows from the above lemma after applying 1 ⊗ 1 ⊗ A to its two sides.
The composition lemma seems to be just a consequence of the axiomatic definition of the Hopf algebra, especially from the coassociativity. Let us consider the composition map (f, g)−→f ◦ g. It is a bilinear map and therefore induces a linear map ◦
Hom(N , P) ⊗ Hom(M, N )−→Hom(M, P). Using the action of H on the tensor product and on the Hom space, we can restate the composition lemma as the following result Corollary 2.3. The composition map is H-linear. Remark 2.1. The linearity of the composition map is known and proved in the literature only when N (or both M and P) is finite dimensional, in which case Hom(M, N ) is isomorphic to M∗ ⊗ N , see [13]. Here, we established it for the general case, for all the representations we will be considering are infinite dimensional.
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2.2. A bracket and a cochain complex. Let H be a Hopf algebra over C with the associated maps as above. We define a bilinear map h·, ·i on H ⊗ H by X x0 yA(x00 ) − ε(x)y (x, y ∈ H). (2.2) hx, yi = (x)
This bracket satisfies the following relations: Proposition 2.4. For all x, y, z in H, we have : (1) hxy, zi = hx, hy, zii + ε(x) hy, zi + ε(y) hx, zi, (2) ε(hx, yi) = 0,
(3) A2 (hx, yi) = A2 (x), A2 (y) . Proof. The first relation follows from that fact that 1 is an algebra homomorphism and that A is an algebra anti-homomorphism. The second relation follows from the antipode axiom: X X x0 A(x00 ) = A(x0 )x00 = ε(x) · 1, (x)
(x)
and from the fact that ε is an algebra homomorphism. We will prove the third relation: From the identity X A(x00 ) ⊗ A(x0 ) 1(A(x)) = (x)
we obtain
1(A2 (x)) =
X
A2 (x0 ) ⊗ A2 (x00 ).
Using this relation and the fact that ε(A(x)) = ε(x) we obtain:
2 A (x), A2 (y) = A2 (hx, yi). Note that h1, xi = hx, 1i = 0 for every x ∈ H. And if H is commutative then hx, yi = 0 for every x, y ∈ H, while if H is cocommutative, then hx, A(y)i = A hx, yi . If M is a left H-module, then m ∈ M is called an H-invariant if xm = ε(x)m for every x ∈ H. If N is another H-module, we set hx, φi = x · φ − ε(x)φ
x ∈ H, φ ∈ Hom(M, N ).
(2.3)
Notice the analogy with the definition of hx, yi. We say that φ is an invariant if hx, φi = 0 for every x ∈ H. Proposition 2.5. For all x, y in H and for all φ, ψ in Hom(M, N ) we have: hxy, φi = x · hy, φi + ε(y) hx, φi , X hx0 , φi hx00 , ψi + hx, φi ψ + φ hx, ψi . hx, φψi =
(2.4) (2.5)
(x)
Proof. The first relation is the same as the relation (1) in Proposition 2.4 when we substitute z by φ. The P second relation is a consequence of the composition lemma and the fact that ε(x) = (x) ε(x0 )ε(x00 ).
Quantum Screening Operators and Canonical q-de Rham Cocycles
287
The importance of the bracket h·, ·i will appear throughout this work. The expression for hx, φi is simply the difference of two actions of x on φ. In the case of the Universal enveloping algebras of Lie algebras, this bracket coincides with the Lie bracket, and for the quantized version of these algebras the appearance of the trivial action in this bracket will emphasize the role of the group-like elements as will be seen in the next sections. Another importance of this bracket will appear below in defining the differential for the cochain complex. Let M be an H-module, and let us consider the following sequence: C • = C(H ⊗• , M) : 0−→M−→Hom(H, M)−→ · · · −→Hom(H ⊗n , M)−→ · · · , and the linear map d : Hom(H ⊗n−1 , M)−→Hom(H ⊗n , M) defined as follows: If φ ∈ Hom(H ⊗n−1 , M) and x1 ⊗ x2 ⊗ . . . ⊗ xn ∈ H ⊗n , then dφ(x1 , x2 , . . . , xn ) = x1 · φ(x2 , . . . , xn ) n−1 X + (−1)i φ(x1 , . . . , xi−1 , xi xi+1 , xi+2 , . . . , xn ) i=1
+ (−1)n φ(x1 , x2 , . . . , xn−1 )ε(xn ). One can look at this sequence as a Hochshild complex of the associative algebra H with a left and a right action which commute on the space M, the right action is given by the trivial action, i.e. by ε, [3]. It follows that (C • , d) is a cochain complex, i.e. d2 = 0. One can choose the coefficients of the cochains in Hom(M, N ), where M and N are two H-modules. Thus we obtain a complex C • (H, M, N ) = Hom(H ⊗• , Hom(M, N )). If φ ∈ Hom(M, N ), then dφ(x) = hx, φi. Hence, dφ = 0 implies that hx, φi = 0 for all x ∈ H. It follows that the 0th cohomology space is the space of H-invariants. We will see that in the case of quantum groups, the space of invariants coincides with the space of intertwiners. More generally, the cohomology spaces are the Ext-spaces Ext•H (M, N ). c2 ) 3. The Quantum Affine Algebra Uq (sl c2 . We recall the definition of the affine algebra sl c2 and we 3.1. The affine algebra sl fix some notations. Let E, F , H be the standard generators of the Lie algebra sl2 . For X, Y in sl2 , we set (X, Y ) = tr(XY ), this defines an invariant bilinear form on sl2 . Thus (E, F ) = (F, E) = 1, (H, H) = 2. We fix a complex number k and we set c2 is defined by the generators B(X, Y ) = k(X, Y ). The corresponding affine algebra sl Xn (X ∈ sl2 , n ∈ Z) and 1, and the relations (a)
[Xn , Ym ] = [X, Y ]m+n + nB(X, Y )δm+n,0 .1
(X, Y ∈ sl2 , m, n ∈ Z).
This algebra is realized as a central extension of the loop algebra sl2 ⊗ C[z, z −1 ] by C.1, c2 is isomorphic to the Kac-Moody and we identify Xn with X ⊗ z n . The affine algebra sl
288
A. Sebbar
algebra sl2 (1) corresponding to the Cartan matrix of affine type A(1) 1 . To be more precise, c we have to add a derivation to sl2 , that is an element D satisfying: [D, Xn ] = nXn and [D, 1] = 0. And we have the identification sl2 (1) = sl2 ⊗ C[z, z −1 ] ⊕ C.1 ⊕ C.D. Let P = Z30 ⊕ Z3 ⊕ Zδ be the weight lattice and let Q = Zα0 ⊕ Zα1 be the root lattice endowed with a symmetric bilinear form (., .) defined by (30 , λ0 ) = 0, (30 , α1 ) = 0, (30 , δ) = 1, (α0 , α1 ) = 2, (α1 , δ) = 0, (δ, δ) = 0, where 31 = 30 + α21 , δ = α 0 + α1 . We define P ∗ = ZH0 ⊕ ZH1 ⊕ ZD as the dual space of P . The dual pairing is defined by (i = 0, 1) for λ ∈ P. hHi , λi = (αi , λ) If k is a nonnegative integer we denote by Pk = {(k − i)30 + i31 , i = 0, 1, . . . , k} the set of dominant integral weights of level k, and we set λi = (k − i)30 + i31 . 3.2. Deformation of the affine algebra. The quantum affine algebra Uq (sl2 (1) ) is an associative algebra over Q(q) with 1, where q is a transcendental complex number, generated by ei , fi , i = 0, 1, and q h (h ∈ P ∗ ). The defining relations are as follows: 0
0
q h q h = q h+h , q h ei q −h = q hh,αi i ei ,
q0 = 1 ,
q h fi q −h = q −hh,αi i fi ,
[ei , fj ] = δi,j
q hi − q −hi , q − q −1
e3i ej − [3]e2i ej ei + [3]ei ej e2i − ej e3i = 0 fi3 fj Here [n] =
−
[3]fi2 fj fi
+
[3]fi fj fi2
−
fj fi3
(i 6= j),
=0
(i 6= j).
q n −q −n q−q −1 .
c2 ) to be the subalgebra of Uq (sl2 (1) ) generated by We consider the algebra Uq (sl ei , fi , ti = q hi (i = 0, 1). The algebra Uq (sl2 (1) ) has a Hopf algebra structure. The comultiplication is given by 1(ei ) = ei ⊗ 1 + ti ⊗ ei , 1(fi ) = fi ⊗ t−1 i + 1 ⊗ fi 1(q h ) = q h ⊗ q h
(i = 0, 1) ,
(h ∈ P ∗ ).
The antipode A is given by A(q h ) = q −h ,
A(ei ) = −t−1 i ei ,
A(fi ) = −fi ti
(i = 0, 1) .
The counit ε is given by ε(ei ) = ε(fi ) = 0
(i = 0, 1) ,
ε(q h ) = 1.
This definition of the quantum affine algebra by Chevalley generators is due to Drinfeld c2 ) [6], which is the and Jimbo [5, 11]. Later, Drinfeld gave another realization for Uq (sl c loop algebra version of the above algebra. In this realization Uq (sl2 ) is an associative
Quantum Screening Operators and Canonical q-de Rham Cocycles
289
algebra generated by {En , n ∈ Z}, {Fn , n ∈ Z}, {Hn , n ∈ Z − {0}} and invertible k K and q 2 satisfying the following relations: 1 [Hn , Hm ] = δn+m,0 [2n][k], n
[Hn , K] = 0 ,
KEn K −1 = q 2 En , [Hn , Em ] =
[2n] −|n|k q 2 En+m , n
KFn K −1 = q −2 Fn , [Hn , Fm ] = −
[2n] |n|k q 2 Fn+m , n
En+1 Em − q 2 Em En+1 = q 2 En Em+1 − Em+1 En , Fn+1 Fm − q −2 Fm Fn+1 = q −2 Fn Fm+1 − Fm+1 Fn , k(n−m) k(m−n) 1 2 2 q , ψ − q φ [En , Fm ] = n+m n+m q − q −1 where ψn and φn are related to Hl by X
ψn z −n = K exp (q − q −1 )
∞ X
n∈Z
X
φn z
! Hl z −l ,
(3.1)
l=1
−n
=K
−1
exp −(q − q
−1
)
∞ X
n∈Z
! H−l z
l
.
(3.2)
l=1
Here, φn = ψ−n = 0 for n ≥ 0. We define H0 by the formula −1 H0 . K = exp (q − q ) 2 The standard Chevalley generators {ei , fi , ti } are given by the identification t0 = q k K −1 , t1 = K , e1 = E0 , f1 = F0 , e0 t1 = F1 , t−1 1 f0 = E−1 .
(3.3)
This identification leads to an algebra isomorphism between the above realizations. Equivalently, the Drinfeld realization can be obtained using the generators En , Fn k (n ∈ Z), φ−n , ψn (n ∈ N) and q ± 2 . And if we consider the currents E(z) =
X
En z −n , F (z) =
n∈Z
X
Fn z −n , φ(z) =
n∈Z
∞ X
φ−n z n , ψ(z) =
n=0
∞ X
ψn z −n ,
n=0
then the defining relations can be written as φ0 ψ0 = ψ0 φ0 = 1, φ(z)φ(w) = φ(w)φ(z) ,
φ(z)ψ(w) =
(3.4)
ψ(z)ψ(w) = ψ(w)ψ(z),
(3.5)
(zq k−2 − w)(zq −k+2 − w) ψ(w)φ(z), (zq k+2 − w)(zq −k−2 − w)
(3.6)
290
A. Sebbar k
φ(z)E(w) =
zq − 2 +2 − w k
zq − 2 − wq 2
E(w)φ(z),
(3.7)
F (w)φ(z),
(3.8)
E(w)ψ(z),
(3.9)
F (w)ψ(z),
(3.10)
k
φ(z)F (w) =
zq 2 −2 − w k
zq 2 − wq −2 k
ψ(z)E(w) =
wq − 2 − zq 2 k
wq − 2 +2 − z k
ψ(z)F (w) =
[E(z), F (w)] =
wq 2 − q −2 z k
wq 2 −2 − z
w −k k w −k 1 k 2 ) − δ( 2 ) q q )ψ(wq δ( )φ(wq , q − q −1 z z
(3.11)
(z − q 2 w)E(z)E(w) = (q 2 z − w)E(w)E(z),
(3.12)
(z − q −2 w)F (z)F (w) = (q −2 z − w)F (w)F (z),
(3.13)
where δ(z) =
X
zn.
n∈Z
These relations are understood to be between formal power series. Since the Drinfeld currents are suitable for bosonization, we will use this realization for the rest of this chapter. Drinfeld also gave the Hopf algebra structure for this current realization. c2 ) has a Hopf algebra structure given by Proposition 3.1. [4] The algebra Uq (sl • Comultiplication 1
1(q k ) = q k ⊗ q k , k
1(E(z)) = E(z) ⊗ 1 + φ(zq 2 ⊗1 ) ⊗ E(zq k⊗1 ), k
1(F (z)) = 1 ⊗ F (z) + F (zq 1⊗k ) ⊗ ψ(q 1⊗ 2 ), k
k
1(φ(z)) = φ(zq −1⊗ 2 ) ⊗ φ(zq 2 ⊗1 ), k
k
1(ψ(z)) = ψ(zq 1⊗ 2 ) ⊗ ψ(zq − 2 ⊗1 ). • Counit ε
ε(q k ) = ε(ψ(z)) = ε(φ(z)) = 1, ε(E(z)) = ε(F (z)) = 0.
Quantum Screening Operators and Canonical q-de Rham Cocycles
• Antipode A
291
A(q k ) = q −k , k
A(E(z)) = −φ(zq − 2 )−1 E(zq −k ), k
A(F (z)) = −F (zq −k )ψ(zq − 2 )−1 , A(φ(z)) = φ(z)−1 ,
If φ(z) =
P∞
A(ψ(z)) = ψ(z)−1 .
n=0
φ−n z n then the formula for 1(φ(z)) means that X mk nk 1(φ(z)) = (q 2 φ−n z n ) ⊗ (q − 2 φ−m z m ). m,n
4. Bosonization and q-Analog of Wakimoto Modules c2 ). Here we follow [1, 9], with some modifications, 4.1. Free boson realization of Uq (sl we introduce the Heisenberg algebra generated by three free boson fields a, b and c. We construct a homomorphism from the quantum affine algebra to the Heisenberg algebra which will enable us to express the Drinfeld generators in terms of the Heisenberg generators. The generators of the quantum Heisenberg algebra Hq (sl2 ) are an , bn , cn , n ∈ Z, pa , pb , and pc . The relations are [an , am ] =
1 [(k + 2)n][2n]δn+m,0 , n
[a0 , pa ] =
1 [bn , bm ] = − [n]2 δm+n,0 , n [cn , cm ] =
1 2 [n] δm+n,0 , n
[b0 , pb ] =
[c0 , pc ] =
4h (k + 2), q − q −1
−2h , q − q −1 2h , q − q −1
(4.1)
(4.2)
(4.3)
where q = eh . The remaining commutators vanish. eq (sl2 ) of Hq (sl2 ) as follows: We define the completion H eq (sl2 ) = lim Hq /In H ←−
(n > 0),
where In is the left ideal of Hq (sl2 ) generated by all the polynomials in am , bm , cm , m > 0, of degrees greater than or equal to n (we set degam = degbm = degcm = m). We form the generating functions: ! ∞ X a 0 + a±n z ∓n , a± (z) = ±(q − q −1 ) 2 n=1
b± (z) = ±(q − q
−1
∞
)
b0 X + b±n z ∓n 2 n=1
! ,
292
A. Sebbar
b(z) = −
X bn q − q −1 z −n + b0 log z + pb , [n] 2h n6=0
c± (z) = ±(q − q
−1
∞
c0 X + c±n z ∓n 2
)
! ,
n=1
c(z) = −
X cn q − q −1 z −n + c0 log z + pc . [n] 2h n6=0
For a real number α we define: a(z; α) = −
X n6=0
1 an q −α|n| z −n + [(k + 2)n] k+2
q − q −1 a0 log z + pa . 2h
Let : : denote the normal ordering of a product of operators defined by moving the creation operators to the left and moving the annihilation operators to the right. In our case the annihilation operators are {an , bn , cn n ≥ 0} and the creation operators are {an , bn , cn , pa , pb , pc , n < 0}. For example X bn z −n : exp(b(z)) := exp − [n] n<0
!
X bn z −n exp − [n]
! e pb z
q−q −1 2h
b0
.
n>0
c2 ) to H eq (sl2 ) which is Proposition 4.1 ([1, 9]). There is a homomorphism ω from Uq (sl defined on generators as follows: ω[E 0 (z)] = − : eb+ (z)−(b+c)(zq) : + : eb− (z)−(b+c)(zq ω[F 0 (z)] = e −e ω[ψ(z)] = e
k a+ (zq 2 +1 )
: eb+ (zq
k a− (zq − 2 −1 )
a+ (zq) b+ (zq
ω[φ(z)] = ea− (zq
e
−1
)
e
k+2
)+(b+c)(zq
: eb− (zq k 2
)+b+ (zq
k b− (zq − 2
−k−2
k +2 2 )
k+1
)
−1
)
:,
:
)+(b+c)(zq −k−1 )
:,
, k
)+b− (zq − 2 −2 )
,
where E 0 (z) = (q − q −1 )E(z) and F 0 (z) = (q − q −1 )F (z). In order to prove this proposition, we need the following two lemmas, which will be used later on too. Lemma 4.2 ([8]). Let X and Y be two operators such that [X, Y ] commutes with X and with Y , then [X, eY ] = [X, Y ]eY
and
eX eY = eY eX e[X,Y ] .
The following lemma can be proved by direct computation of the operator product expansions. It will be used later on without any mention of a specific relation.
Quantum Screening Operators and Canonical q-de Rham Cocycles
293
Lemma 4.3. We have the following commutation relations: ea+ (z) ea− (w) =
(w − zq k+4 )(w − zq −k−4 ) a− (w) a+ (z) e e , (w − zq k )(w − zq −k )
eb+ (z) eb− (w) =
(z − w)2 eb− (w) eb+ (z) , (z − wq 2 )(z − wq −2 )
(z − wq 2 )(z − wq −2 ) c− (w) c+ (z) e e , (z − w)2 z − wq b(w) b+ (z) zq − w c(w) c+ (z) :e :e eb+ (z) : eb(w) := :e , ec+ (z) : ec(w) := :e , zq − w z − wq wq − z b(w) b− (z) w − zq c(w) c− (z) :e :e :e , ec− (z) : ec(w) := :e , eb− (z) : eb(w) := w − zq wq − z ec+ (z) ec− (w) =
: e(b+c)(z) :: e(b+c)(w) :=: e(b+c)(w) :: e(b+c)(z) : . c2 ) and their For simplicity we will use the same notation for the elements of Uq (sl e images in H(sl2 ). Using the above lemma and the fact that φ 0 = e−
q−q −1 2
(a0 +b0 +c0 )
and
ψ0 = e
q−q −1 2
(a0 +b0 +c0 )
,
one can easily prove that E(z), F (z), φ(z) and ψ(z) satisfy the defining relations of the c2 ), except for the relation involving [E(z), F (w)] which needs an explaalgebra Uq (sl 0 (z) nation. We look at z and w as complex variables and we set E 0 (z) = −E+0 (z) + E− and F 0 (z) = F+0 (z) − F−0 (z) in the obvious way. Then we have E+0 (z)F+0 (w) = 0 E− (z)F−0 (w) =
F+0 (w)E+0 (z) = 0 F−0 (w)E− (z) =
zq −1 − wq k+1 : E+0 (z)F+0 (w) : z − wq k
(|z| > |wq k |),
zq − wk −k−1 0 : E− (z)F−0 (w) : z − wq −k
(|z| > |wq −k |),
wq k+1 − zq −1 : F+0 (w)E+0 (z) : wq k − z
(|z| < |wq k |),
wq −k−1 − zq 0 : F−0 (w)E− (z) : wq −k − z
(|z| < |wq −k |).
The other products have no poles, more precisely: E+0 (z)F−0 (w) = F−0 (w)E+0 (z) =: E+0 (z)F−0 (w) : , 0 0 0 (z)F+0 (w) = F+0 (w)E− (z) =: E− (z)F+0 (w) : . E−
For |z| |w|, it follows that: zq −1 − wq k+1 zq − wq −k−1 0 0 0 : E (z)F (w) : − : E− (z)F−0 (w) : + + z − wq k z − wq −k 0 (z)F+0 (w) :, + : E+0 (z)F−0 (w) : + : E−
E 0 (z)F 0 (w) = −
294
A. Sebbar
and for |z| |w| we have: zq −1 − wq k+1 zq − wq −k−1 0 : F+0 (w)E+0 (z) : − : F−0 (w)E− (z) : k z − wq z − wq −k 0 (z) : + : F−0 (w)E+0 (z) : . + : F+0 (w)E−
F 0 (w)F 0 (z) = −
Since the normally ordered product does not depend on the order of the factors, we conclude that E 0 (z)F 0 (w) and F 0 (w)E 0 (z) have the same analytic continuation. The coefficient of z −n−1 in the Laurent expansion of E 0 (z)F 0 (w) − F 0 (w)E 0 (z) is Z Z 1 1 0 0 n E (z)F (w)z dz − F 0 (w)E 0 (z)z n dz, 2πi CR 2πi Cr where CR and Cr are circles on the z-plane of radii R |w| and r |w| respectively, which is equal to the sum of the residues of the common analytic continuation. The latter is equal to 0 (wq −k )F−0 (w) : . (wq k )n+1 (q − q −1 ) : E+0 (wq k )F+0 (w) : −(wq −k )n+1 (q − q −1 ) : E−
Moreover,
k
: E+0 (wq k )F+0 (w) := ψ(wq 2 ) and Hence
k
0 : E− (wq −k )F−0 (w) := φ(wq − 2 ).
w k k w [E 0 (z), F 0 (w)] = (q − q −1 ) δ( q k )ψ(wq 2 ) − δ( q −k )φ(wq − 2 ) , z z
which provides the right formula for [E(z), F (w)]. c2 ) are created 4.2. Representations. The infinite dimensional representations of Uq (sl from the Fock module of the Heisenberg algebra via the homomorphism ω. We start by considering the vacuum state of the boson Fock space which satisfies (n ≥ 0).
an . = bn . = cn . = 0 We define the vector r,s by: r,s = exp r
pa + s(pb + pc ) 2(k + 2)
(r, s ∈ Z).
Let F be the free Q(q)-algebra generated by {an , bn , cn , n < 0} and let Fr,s be the Fock module defined by Fr,s := F.r,s . It is clear that φ(z) and ψ(z) map Fr,s to Fr,s ⊗ C((z)) and from the simple observation that pa e±(pb +pc ) r,s = er 2(k+2) +(s±1)(pa +pb ) , we deduce that E(z) maps Fr,s to Fr,s−1 ⊗ C((z −1 )) and F (z) maps Fr,s to Fr,s+1 ⊗ C((z −1 )).
Quantum Screening Operators and Canonical q-de Rham Cocycles
295
c2 ) of highest weight λ and generated We denote by V (λ) the Verma module over Uq (sl by the highest weight vector vλ . Thus e 0 v λ = e1 v λ = 0 ,
t0 v λ = q α v λ ,
t1 v λ = q β v λ ,
(4.4)
where α30 + β31 is the classical part of λ; 30 and 31 are the fundamental weights of sl2 . We refer to (4.4) as the highest weight condition. Proposition 4.4. The vector r,0 satisfies: En r,0 = Fn r,0 = Hn r,0 = 0 E0 r,0 = 0, Kr,0 = q r r,0 . Proof. We have E(z)r,0
X bn + c n (zq)−n = exp [n]
if n > 0,
! r,−1
n<0
+ exp (q − q
−1
)
X
bn z
−n
n<0
X bn + c n (zq −1 )−n + [n]
! r,−1 .
n<0
Hence, for n ≥ 0, En r,0 = 0. We have a similar formula for F (z)r,0 except that, due to the presence of a± (z), the first term comes with the factor q r and the second term comes with the factor q −r , which implies that Fn r,0 = 0 for n < 0 only. Since ψ(z)r,0 = q r r,0 by (3.1), we deduce that Hn r,0 = 0 for n ≥ 0 and Kr,0 = q r r,0 . Corollary 4.5. The vector r,0 satisfies the highest weight condition (4.4) and can be identified with the highest vector vλr , where λr = (k − r)30 + r31 . Proof. This follows from the proposition and the identification (3.3).
The following proposition follows from the identification of vλr with r,0 and the c2 ) on r,0 : action of the generators of Uq (sl Proposition 4.6. There is an embedding of the highest weight module in a direct sum of Fock modules: M Fr,s . (4.5) V (λr ),→ s∈Z
Remark 4.1. The module V (λr ) is reducible as is known in Conformal Field Theory. To obtain irreducible modules, one has to use the q-analog of the Felder resolution, see [14]. We set Wr :=
M
Fr,s .
(4.6)
s∈Z
c2 )-module structure defined by the bosonization formulae This Fock space carries a Uq (sl of Proposition 4.1. These are the q-analogs of the Wakimoto modules [19].
296
A. Sebbar
5. Screening Operators 5.1. The operator S(z). The so-called screening operator S(z) is used to investigate the irc2 ), and to compute correlation functions in Conformal reducible representations of Uq (sl e 2 ) which acts then on Fock space representations. Field Theory. It is an element of H(sl Definition 5.1 ([1]). The screening operator is given by S(z) =
−1 k+2 1 : e−a(z, 2 ) : : e−b− (z)−(b+c)(qz) : − : e−b+ (z)−(b+c)(q z) : . −1 (q − q )z
For simplicity we write z(q − q −1 )S(z) = S1 (z) − S2 (z). Proposition 5.1. The operator S(z) sends Fr,s to Fr−1,s−1 , and therefore sends the r module Wr to the module Wr−2 ⊗ z − k+2 C((z −1 )). pb +pc a Proof. exp −p sends r,s to r,s−1 , and k+2 sends r,s to r−1,s , exp − 2 e
−pa k+2
q−q −1
e− 2h(k+2) a0 log z r,s = e
−pa k+2
q−q −1
rpa
e[− 2h(k+2) a0 log z, 2(k+2) ] r,s
r
= z − k+2 r−2,s .
Now we compute the pairing between E(z), F (z), φ(z), ψ(z) and S(w). Proposition 5.2. The operator S(w) satisfies: k
k
hE(z), S(w)i = E(z)S(w) − φ(zq 2 )S(w)φ(zq 2 )−1 E(z), k
(5.1) k
hF (z), S(w)i = F (zq −k )S(w)ψ(zq − 2 )−1 − S(w)F (zq −k )ψ(zq − 2 )−1 , k
k
hφ(z), S(w)i = φ(zq 2 )S(w)φ(zq 2 )−1 − S(w), k
k
hψ(z), S(w)i = ψ(zq − 2 )S(w)ψ(zq − 2 )−1 − S(w), where these relations are understood to be between formal power series. Proof. We have k
1(E(z)) = E(z) ⊗ 1 + φ(zq 2 ⊗1 ) ⊗ E(zq k⊗1 ) X k = E(z) ⊗ 1 + φ−m q 2 m−km z m ⊗ En z −n , m≥0,n
hence (id ⊗ A)1(E(z)) = E(z) ⊗ 1 +
X m,n
therefore
k
φ−m q 2 m−kn z m ⊗ A(En )z −n ,
(5.2)
(5.3)
(5.4)
Quantum Screening Operators and Canonical q-de Rham Cocycles
hE(z), S(w)i = E(z)S(w) +
X
297
k
φ−m q 2 m−kn z m S(w)A(En )z −n
m,n
= E(z)S(w) +
X
k
φ−m q 2 z m S(w)A(En )(q k z)−n
m,n k
k
= E(z)S(w) − φ(zq 2 )S(w)φ(zq 2 )−1 E(z). This proves (5.1). The proof of the other formulas is similar, i.e. by expanding the currents and applying A to the second component of the tensor product, and using the k fact that q 2 is central. Proposition 5.3. We have the following relations between formal power series: φ(z)S(w) = S(w)φ(z) =: φ(z)S(w) :, ψ(z)S(w) = S(w)ψ(z) =: ψ(z)S(w) :, E(z)S(w) = S(w)E(z) and the products have no poles.
(5.5) (5.6) (5.7)
Proof. The relations (5.5) and (5.6) follow from direct computation. Equation (5.7) needs some explanation. Set E1 (z) =: exp(b+ (z) − (b + c)(zq)) :
and
E2 (z) =: exp(b− (z) − (b + c)(zq −1 )) : .
Then we have E1 (z)S1 (w) = S1 (w)E1 (z) = q : E1 (z)S1 (w) :, and E2 (z)S2 (w) = S2 (w)E2 (z) = q −1 : E2 (z)S2 (w) : . The other products have a pole at z = w: E1 (z)S2 (w) = S2 (w)E1 (z) = q = qw
z − wq −2 : E1 (z)S2 (w) : z−w
1 − q −2 : E1 (w)S2 (w) : + regular part at z = w z−w
and E2 (z)S1 (w) = S1 (w)E2 (z) = q −1 = q −1 w
z − wq 2 : E2 (z)S1 (w) : z−w
1 − q2 : E2 (w)S1 (w) : + regular part at z = w. z−w
It follows that E1 (z)S2 (w) + E2 (z)S1 (w) = S2 (w)E1 (z) + S1 (w)E2 (z) =w
q − q −1 (: E1 (w)S2 (w) : − : E2 (w)S1 (w) :) + · · · . z−w
It is easy to see that : E1 (w)S2 (w) :=: E2 (w)S1 (w) : and (5.7) follows.
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A. Sebbar
Before investigating how the Fourier coefficients behave with S(z), we introduce the notion of q-difference operator: For a function f (z) and for α ∈ C, we define f (zq α ) − f (zq −α ) Dα f (z) := . dq z z(q − q −1 ) We can also write Dα f (z) = In particular
f (zq α ) − f (zq −α ) dq z. z(q − q −1 )
(5.8)
Dα z n = [αn]z n−1 dq z.
If f and g are two functions, the q-difference of the products is given by: Dα (f (z)g(z)) = f (zq α )Dα (g(z)) + Dα (g(z))f (zq −α ) = f (zq −α )Dα (g(z)) + Dα (f (z))g(zq α ). For p, s ∈ C − {0} with |p| < 1 and for a function f (u), we define Z
s∞
f (u)dp u = s(1 − p)
0
∞ X
f (spn )pn
n=−∞
whenever it is convergent. This is a q-difference analog of the ordinary integration, and it is called the Jackson integral along a q-interval [0, s∞]. This integral has the following property: Z s∞ Dα f (u)dp u = 0. dp u 0 All these notions extend to multi-variable functions. Theorem 5.4. For every integer n, we have: hEn , S(w)i = 0, hφ−n , S(w)i = hψn , S(w)i = 0 (n ≥ 0), k+2 k Dk+2 hFn , S(w)i = (wq)n : e−a(w,− 2 ) ψ(wq 2 )−1 : . dq w
(5.9) (5.10) (5.11)
Proof. The relations (5.9) and (5.10) follow immediately from Proposition 5.2 and Proposition 5.3. Set (q − q −1 )F (zq −k ) = F1 (zq −k ) − F2 (z −k ), where F1 (zq −k ) = ea+ (zq and F2 (zq −k ) = ea− (zq
− 3k −1 2 )
− k +1 2 )
: exp(b+ (zq 2 ) + (b + c)(zq)) :
: exp(b− (zq −2k−2 ) + (b + c)(zq −2k−1 )) : .
We look at z and w as complex variables. To normally order F1 (zq −k )S1 (w), we need the following brackets:
Quantum Screening Operators and Canonical q-de Rham Cocycles
a+ (zq
− k2 +1
=
k+2 ), −a w, 2
299
X ∞ w n −1 q −2n q − q −1 a0 , pa − [an , a−n ] (q − q −1 ) 2 k+2 [(k + 2)n] z n=1
∞ w n X 1 −4n (q = −2h + − 1) . n z n=1
And [b+ (zq 2 ), −b− (w) − b(qw)] + [b(zq) − pb , −b− (w)] gives h+
∞ w n X 1 (1 − q −4n ) . n z n=1
Therefore F1 (zq −k )S1 (w) = q −1 : F1 (zq −k )S1 (w) : .
(5.12)
Meanwhile S1 (w)F1 (zq −k ) = e[
q−q −1 2
b0 ,pb ]
: S1 (w)F1 (zq −k ) := q −1 : F1 (zq −k )S1 (w) : . (5.13)
Similarly, we have: F2 (zq −k )S2 (w) = S2 (w)F2 (zq −k ) = q : F2 (zq −k )S2 (w) : .
(5.14)
The other products have poles: F1 (zq −k )S2 (w) = q −1
S2 (w)F1 (zq −k ) = q
z−w : F1 (zq −k )S2 (w) : z − wq −2
w−z : S2 (w)F1 (zq −k ) : z − wq 2
(|w| |z|),
(|z| |w|).
(5.15)
(5.16)
Notice that the expressions at the right are the same. And F2 (zq −k )S1 (w) = q
S1 (w)F2 (zq −k ) = q −1
z − wq 2k : F2 (zq −k )S1 (w) : z − wq 2k+2 w − zq −2 : S1 (w)F2 (zq −k ) : w − zq −2k−2
(|w| |z|),
(5.17)
(|z| |w|). (5.18)
Since the normally ordered products do not depend on the order of the factors, we deduce k k from (5.12)−(5.18) that F (zq −k )S(w)ψ(zq − 2 )−1 and S(w)F (zq −k )ψ(zq − 2 )−1 have k the same analytic continuation (ψ(zq − 2 )−1 contains only positive modes, so it stays to the right when the products are normally ordered).
300
A. Sebbar
Therefore, in view of Proposition 5.2, we obtain: Z k 1 F (zq −k )S(w)ψ(zq − 2 )−1 z n−1 dz hFn , S(w)i = 2πi CR Z k 1 − S(w)F (zq −k )ψ(zq − 2 )−1 z n−1 dz, 2πi Cr where CR and Cr are circles of radii R |w| and r |w| respectively. It follows that hFn , S(w)i is equal the sum of the residues of −1 z n−1 q (z − w) : F1 (zq −k )S2 (w) : −1 2 (q − q ) w z − wq −2 k q(z − wq 2k ) −k : F (zq )S (w) : ψ(zq − 2 )−1 , + 2 1 2k+2 z − wq therefore k 1 −(wq −2 )n : F1 (wq −k−2 )S2 (w)ψ(wq − 2 −2 )−1 : −1 (q − q )w 3k + (wq 2k+2 )n : F2 (wq k+2 )S1 (w)ψ(wq 2 +2 )−1 : .
hFn , S(w)i =
Meanwhile : F1 (wq −k−2 )S2 (w) : =: ea+ (wq =: e
− k −1 2 )−a(w, k+2 2 )
−a(wq −k−2 ,− k+2 2 )
:
:.
Notice the change of sign in the argument of a. Similarly : F2 (wq k+2 )S1 (w) :=: e−a(wq
k+2
,− k+2 2 )
:.
It follows that hFn , S(w)i = which proves (5.11).
k+2 k Dk+2 (wq k )n : e−a(w,− 2 ) ψ(wq 2 )−1 : , dq w
As a consequence, the bracket of the Jackson integral of S(w) for p = q k+2 with the c2 ) vanishes. Using(5.11) we have generators of Uq (sl R s∞ Corollary 5.5. The Jackson integral, 0 S(w)dp w (p = q k+2 ), of the screening opc2 ). erator is an invariant for Uq (sl 5.2. Difference equations. We will state and prove two results concerning the screening operators, which deal with the following difference equation: hx, S(w)i = c2 ). for every x ∈ Uq (sl
Dk+2 S(x, w), dq w
(5.19)
Quantum Screening Operators and Canonical q-de Rham Cocycles
301
c2 ) such Theorem 5.6. There exists a well defined operator S(x, w), for each x ∈ Uq (sl that: k
(1) S(x, w) vanishes on the Fourier coefficients of E(z), φ(z), ψ(z), and on q ± 2 and S(Fm , w) = (wq)m : e−a(w,−
k+2 2 )
k
ψ(wq 2 )−1 :
(m ∈ Z).
c2 ) we have: (2) For every x, y ∈ Uq (sl S(xy, w) = x · S(y, w) + ε(y)S(x, w).
(5.20)
Proof. We consider S(x, w) to be linearly dependent on x and defined on the generators as in (1), and we extend it to the free algebra generated by Em , Fm , φ−n , ψn (m ∈ Z, k c2 ), we need n ∈ N) and q ± 2 . In order to have S(x, w) well defined for every x ∈ Uq (sl c2 ). By to show that the relation (5.20) is compatible with the defining relations of Uq (sl linearity of S(x, w), we have: S(E(z), w) = S(φ(z), w) = S(ψ(z), w) = 0, and
S(F (z), w) = δ
wq k z
: e−a(w,−
k+2 2 )
(5.21)
k
ψ(zq 2 )−1 : .
(5.22)
It is clear that the relation (5.20) is compatible with the relations (3.4)−(3.7), (3.9) and (3.12) since (5.20) vanishes on both sides of these relations (they do not involve F (z)). We will prove the compatibility with (3.8) and (3.11). From (3.8) we have k
φ(z1 )F (z2 ) = gk (z)F (z2 )φ(z1 ),
with gk (z) =
z1 q 2 −2 − z2 k
z1 q 2 − z2 q −2
.
And S(φ(z1 )F (z2 ), w) = φ(z1 ) · S(F (z2 ), w) + ε(F2 (z2 ))S(φ(z1 ), w) k k k+2 k k wq 2 : e−a(w,− 2 ) ψ(wq 2 )−1 : φ(z1 q 2 )−1 . = φ(z1 q )δ z2 Meanwhile, k
k
ψ(wq 2 )−1 φ(z1 q 2 )−1 =
k k (z1k+2 − w)(z1 q −k−2 − w) φ(z1 q 2 )−1 ψ(wq 2 )−1 . k−2 −k+2 (z1 q − w)(z1 q − w) (5.23)
And since ∞ X z 1 n k k+2 −a w, − , −a(z1 q 2 −1 ) = 2h + , q (k+2)n − q (k−2)n 2 w n=1
we obtain e−a(w,−
k+2 2 )
k
φ(z1 q 2 )−1 =
k k+2 wq 2 − z1 q k φ(zq 2 )−1 e−a(w,− 2 ) . w − z1 q k+2
(5.24)
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A. Sebbar
From (5.23) and (5.24) we obtain: wq 2 − z1 q −k δ S(φ(z1 )F (z2 ), w) = w − z1 q −k+2
wq k z2
: e−a(w,−
k+2 2 )
k
ψ(wq 2 )−1 : . (5.25)
We need to compare this expression with S(gk (z)F (z2 )φ(z1 ), w). Since S(gk (z)φ(z1 ), w) = 0, we have: S(gk (z)F (z2 )φ(z1 ), w) = ε(gk (z)φ(z1 ))S(F (z2 ), w) = g0 (z)S(F (z2 ), w). Now we use techniques from Vertex Operator Calculus [8], and we have: g0 (z)δ
wq k z2
z2 − q −2 δ = −2 q z2 − z1
wq k z2
wq k − z1 q −2 δ = −2 q z2 − z1
wq k z2
.
This follows from the identity: z −n δ
w z
= w−n δ
w z
.
Therefore S(gk (z)F (z2 )φ(z1 ), w) =
q 2 w − q −k z1 S(F (z2 ), w), w − q −k+2 z1
which is equal to (5.25). The case of (3.10) is similar. We will use a different approach to prove the compatibility with
[E(z), F (w)] =
w −k k 1 w −k k 2 ) − δ( 2 ) q q )ψ(wq )φ(wq . δ( q − q −1 z z
Since S(x, w) vanishes if x is the right side, we need to show that S(E(z)Fm , w) and S(Fm E(z)) (m ∈ Z), given by (5.20), are equal. On one hand S(Fm E(z), w) = Fm · S(E(z), w) + ε(E(z))S(Fm , w) = 0.
Quantum Screening Operators and Canonical q-de Rham Cocycles
303
On the other hand S(E(z)Fm , w) = E(z) · S(Fm , w) + 0 k
= E(z)S(Fm , w) − (wq k )n φ(zq 2 ) : e−a(w,−
k+2 2 )
k
k
ψ(wq 2 )−1 : φ(zq 2 )−1 E(z).
Using (3.9), we have k
ψ(wq 2 )−1 E(z) =
k w − zq −k+2 E(z)ψ(wq 2 )−1 . 2 −k wq − zq
Keeping in mind the relations (5.23) and(5.24), we see that the second term in the expression of S(E(z)Fm , w) is E(z)S(Fm , w) times (w − q k+2 z)(w − q −k−2 ) wq 2 − zq k wq −2 − q −k z . . . (w − q −k+2 )(w − q k−2 z) w − q k+2 w − q −k−2 z This product is easily seen to be equal to 1. Therefore S(E(z)Fm , w) = 0. The case of (3.13) is similar. Theorem 5.7. The screening operator S(z) satisfies: hx, S(z)i =
Dk+2 S(x, z), dq z
(5.26)
c2 ). for every x ∈ Uq (sl c2 ) by Theorem 5.4. Since both Proof. This is already satisfied by the generators of Uq (sl c sides are well defined for every x ∈ Uq (sl2 ), assume that the equality holds for x and y, then, by Theorem 5.6, we have hxy, S(z)i = x · hy, S(z)i + ε(y) hx, S(z)i Dk+2 (x · S(y, z) + ε(y)S(x, z)) = dq z Dk+2 = S(xy, z), dq z which proves the theorem.
6. Canonical q-de Rham Cocycles 6.1. The simple case. From now on, we set α = k + 2 and we suppose α 6= r0 (the level is noncritical), and let r be a complex number. Recall that the operators z α S(z) r c2 )) send Wr to Wr−2 ⊗ C((z −1 )). We consider the following and z α S(x, z) (x ∈ Uq (sl complex c2 )⊗• , Hom(Wr , Wr−2 ) . (6.1) Hom Uq (sl The differential d0 of this complex was introduced in Sect. 2 and is given by
304
A. Sebbar
0
d φ(x1 , . . . , xn ) = x1 · φ(x2 , . . . , xn ) +
n−1 X
(−1)i φ(x1 , . . . , xi−1 , xi xi+1 , xi+2 . . . , xn )
i=1
+ (−1)n ε(xn )φ(x1 , . . . , xn−1 ). The cohomology groups are Ext • c (Wr , Wr−2 ). And the 0-cohomolgy group is exUq (sl2 ) c2 )-invariants. actly the space of Uq (sl Let 0 = C((z −1 )) and 1 = C((z −1 ))dq z (the space of formal algebraic qdifferentials 1-forms). We consider the following complex: • : 0−→0 −→1 −→0.
(6.2)
The differential of this complex is given by d00 (f (z)) = Dα0 (f (z)) − [r]
q r−α f (z)dq z , z
where Dα0 (f (z)) = Dα f (zq α ) (the last derivative should be read as f 0 (zq α ) and not as (f (zq α ))0 ). We consider the double complex c2 )⊗• , Hom(Wr , Wr−2 ⊗ • ) C •• = Hom Uq (sl c2 )⊗• ⊗ Wr , Wr−2 ⊗ • . = Hom Uq (sl Let C • be the simple complex associated with this double complex. Set r
S 01 = q 2r z α S(zq α )dq z, r
S 10 (x) = z α S(x, z)
c2 )). (x ∈ Uq (sl
Then (S 01 , S 10 ) is a one-cochain in C 1 , and we have: Theorem 6.1. The cochain (S 01 , S 10 ) is a one-cocycle in C 1 . Proof. We need to prove that d00 S 01 = 0, d0 S 10 (z) = 0, d0 S 01 = d00 S 10 .
(6.3) (6.4) (6.5)
Equation (6.3) derives from the fact that the complex • has length 1, and (6.4) is a consequence of Theorem 5.6. The relation (6.5) also follows from Theorem 5.6 in the following way: r
r
r
Dα (z α S(x, z)) = Dα (z α )S(x, zq −α ) + (zq α ) α Dα S(x, z) r
r
= [r]z α −1 S(x, zq −α ) + q r z α Dα S(x, z), therefore
Quantum Screening Operators and Canonical q-de Rham Cocycles r
305
r
r
r
r
Dα0 (z α S(x, z)) = [r]q r−α z α −1 S(x, z) + q 2r z α Dα S(x.zq α ) = [r]q r−α z α −1 S(x, z) + q 2r z α hx, S(zq α )i , by Theorem 5.7. We deduce that
r r d00 (z α S(x, z)) = x, q 2r z α S(zq α ) = d0 S 01 .
6.2. Compositions. Let p be a positive integer, and let us consider the ring p Y Ap = C[[z1 , . . . , zp ]][ zi−1 ]; i=1 th
we look at Ap as the ring of functions on the p power of the formal punctured disk Xp . Let a (1 ≤ a ≤ p) denote the space of algebraic q-differential a-forms on the formal variety Xp . Thus, 0 is just Ap , and elements of a have the form X f (z1 , . . . , zp )dq zi1 ∧ . . . ∧ dq zia (f (z1 , . . . , zp ) ∈ Ap ). We consider the following complex • : 0−→0 −→ . . . −→p −→0,
(6.6)
its differential is given by 00
d =
Dα0
−
p X
q r−α−2(i−1) [r − 2(i − 1)]
i=1
where, in several variables,
Dα0
d q zi , zi
is defined by:
Dα0 (f ) =
p X Dα0 (f ) ∧ d q zi . d q zi i=1
For i = 1, . . . , p, we set: r−2(i−1) α
S 0 (zi ) = q 2(r−i) zi
r−2(i−1) α
S 0 (x, zi ) = zi 0
S(zi q α ),
S(x, zi ).
0
The operators S (zi ) and S (x, zi ) are elements of
Hom Wr−2(i−1) , Wr−2i ⊗ Ap .
We consider the following double complex: c2 )⊗• , Hom(Wr , Wr−2p ⊗ • ) C •• = Hom Uq (sl c2 )⊗• ⊗ Wr , Wr−2p ⊗ • . = Hom Uq (sl And for each m = 0, . . . , p, we define the operators
306
A. Sebbar
c2 )⊗m , Hom(Wr , Wr−2p ⊗ p−m ) S m,p−m ∈ Hom Uq (sl as follows: S m,p−m (x1 , . . . , xm ) X m(m+1) = (−1) 2
(−1)i1 +···+im S(x1 , . . . , xm ; i1 , . . . , im ),
1≤i1 ≤...≤im ≤p
and S is given by X
S 0 (z1 ) · . . . · S 0 (x01 , zi1 )x001
(x1 ,... ,xn )
· S 0 (zi1 +1 ) · . . . · S 0 (x0m , zim )x00m · (S 0 (zim +1 ) · . . . · S 0 (zp )) dq z1 ∧ . . . ∧ dbq zi1 ∧ . . . ∧ dbq zim ∧ . . . ∧ dq zp .
In this expression, we consider initially the composition S 0 (z1 ) · . . . · S 0 (zp ), and we substitute each S 0 (zik ) by S 0 (x0k , zik )x00k · . . . and x00k is acting on all the remaining factors on the right if there are any; if not, it is just S 0 (xk , zik ) (which happens only when ik = p) . And the sum is taken over all the terms involved in the Sweedler notation for the comultiplication of x1 , . . . , xm . The element S = (S 0,p , . . . , S p,0 ) is p-cochain in the simple complex associated with the double complex C •• . Theorem 6.2. The cochain S is a p-cocycle. We need to prove that
d00 S 0,p = d0 S p,0 = 0,
and for k = 0, . . . , p − 1, d0 S k,p−k = (−1)k d00 S k+1,p−k−1 . These relations follow from Theorem 5.6 and Theorem 5.7 and some lemmas on Hopf algebras. We give the proof in the case p = 2 in order to illustrate the techniques used, the general case is proved exactly in the same way but with lengthy formulas. Let us assume p = 2, we have: = S 0 (z1 )S 0 (z2 )dq z1 ∧ dq z2 , X S 0 (x0 , z1 )x00 · S 0 (z2 )dq z2 − S 0 (z1 )S 0 (x, z2 )dq z1 , S 1,1 (x) = S 0,2
(x)
S
2,0
(x, y) =
X
S 0 (x0 , z1 )x00 · S 0 (y, z2 ).
(x)
We will use the counit axiom and the composition lemmas several times without mentioning them. We also will drop the σ in the Sweedler notation to simplify the expressions. Since • is of length 1, it is clear that d00 S 0,2 = 0. Let us prove that d0 S 2,0 = 0, we have: d0 S 2,0 (x, y, z) = x · S 0 (y 0 , z1 )y 00 · S 0 (z, z2 ) − S 0 (x0 y 0 , z1 )(x00 y 00 ) · S 0 (z, z2 ) + S 0 (x0 , z1 )x00 · S 0 (yz, z2 ) − ε(z)S 0 (x0 , z1 )x00 · S 0 (y, z2 ),
Quantum Screening Operators and Canonical q-de Rham Cocycles
307
using Theorem 5.6, we get: x · S 0 (y 0 , z1 )y 00 · S 0 (z, z2 ) = x0 · S 0 (y 0 , z1 )(x00 y 00 ) · S 0 (z, z2 ), S 0 (x0 y 0 , z1 )(x00 y 00 ) · S 0 (z, z2 ) = x0 · S 0 (y 0 , z1 )(x00 y 00 ) · S 0 (z, z2 ) + S 0 (x0 , z1 )(x00 y) · S 0 (z, z2 ), S 0 (x0 , z1 )x00 · S 0 (yz, z2 ) = S 0 (x0 , z1 )(x00 y) · S 0 (z, z2 ) + ε(z)S 0 (x0 , z1 )x00 · S 0 (y, z2 ). Adding up, we get d0 S 2,0 = 0. Now we prove that d0 S 1,1 = −d00 S 2,0 : d0 S 1,1 (x, y) = x · (S 0 (y 0 , z1 )y 00 · S 0 (z2 ))dq z2 − x · (S 0 (z1 )S 0 (y, z2 ))dq z1 − S 0 (x0 y 0 , z1 )(x00 y 00 ) · S 0 (z2 )dq z2 + S 0 (z1 )S 0 (xy, z2 )dq z1 + ε(y)S 0 (x0 , z1 )x00 · S 0 (z2 )dq z2 − ε(y)S 0 (z1 )S 0 (x0 z2 )dq z1 . Using Theorem 5.6, we get: d0 S 1,1 (x, y) = − hx0 , S(z1 )i x00 · S 0 (y, z2 )dq z1 − S 0 (x0 , z1 )x00 · hy, S 0 (z2 )i dq z2 . On the other hand: r
Dα0 S 2,0 (x, y) =[r]q r−α z1α r
−1
r−2
z2 α S(x0 , z1 )x00 · S(y, z2 )dq z1
r−2
+ q 2r z1α z2 α hx0 , S(zq α )i x00 · S(y, z2 )dq z1 r
r−2
+ [r − 2]q r−2−α z1α z2 α r
r−2 α
+ q 2(r−2) z1α z2
−1
S(x0 , z1 )x00 · S(y, z2 )dq z2
S(x0 , z1 )x00 · hx, S(z2 q α )i dq z2 ,
where we have used Theorem 5.7. Therefore r
r−2
d00 S 2,0 (x, y) =q 2r z1α z2 α hx0 , S(zq α )i x00 · S(y, z2 )dq z1 r
r−2
+ q 2(r−2) z1α z2 α S(x0 , z1 )x00 · hx, S(z2 q α )i dq z2 = hx0 , S(z1 )i x00 · S 0 (y, z2 )dq z1 + S 0 (x0 , z1 )x00 · hy, S 0 (z2 )i dq z2 , which show that d0 S 1,1 = −d00 S 2,0 . Finally, we need to prove that d0 S 0,2 = d00 S 1,1 : d0 S 0,2 (x) =x · (S 0 (z1 )S 0 (z2 ))dq z1 ∧ dq z2 − ε(x)S 0 (z1 )S 0 (z2 )dq z1 ∧ dq z2 =x0 · S 0 (z1 )x00 · S 0 (z2 )dq z1 ∧ dq z2 − ε(x)S 0 (z1 )S 0 (z2 )dq z1 ∧ dq z2 = hx0 , S 0 (z1 )i x00 · S 0 (z2 )dq z1 ∧ dq z2 + ε(x0 )S 0 (z1 )x00 · S 0 (z2 )dq z1 ∧ dq z2 − ε(x)S 0 (z1 )S 0 (z2 )dq z1 ∧ dq z2 = hx0 , S 0 (z1 )i x00 · S 0 (z2 )dq z1 ∧ dq z2 + S 0 (z1 ) hx, S 0 (z2 )i dq z1 ∧ dq z2 .
308
A. Sebbar
Meanwhile, r
Dα0 (S 1,1 (x)) =q r−α [r]q 2(r−2) z1α r
−1
r−2
z2 α S(x0 , z1 )x00 · S(z2 )dq z1 ∧ dq z2
r−2
+ q 2r q 2(r−2) z1α z2 α hx0 , S(z1 q α )i x00 · S(z2 )dq z1 ∧ dq z2 r
r−2
+ q 2r [r − 2]q r−2−α z1α z2 α r α
r−2 α
+ q 2r q 2(r−2) z1 z2
−1
S(z1 )S(x, z2 )dq z1 ∧ dq z2
S(z1 ) hx, S(z2 q α )i dq z1 ∧ dq z2 .
Therefore r
r−2
d00 S 1,1 (x) =q 2r q 2(r−2) z1α z2 α hx0 , S(z1 q α )i x00 · S(z2 )dq z1 ∧ dq z2 r
r−2
+ q 2r q 2(r−2) z1α z2 α S(z1 ) hx, S(z2 q α )i dq z1 ∧ dq z2 = hx0 , S 0 (z1 )i x00 · S 0 (z2 )dq z1 ∧ dq z2 + S 0 (z1 ) hx, S 0 (z2 )i dq z1 ∧ dq z2 , which proves that d0 S 0,2 = d00 S 1,1 . Finally, by homological considerations we deduce: Corollary 6.3. The cocycle S induces canonical linear maps sm : Hm (• )∗ −→Ext p−m Wr , Wr−2p 0 ≤ m ≤ p. Uq (sc l2 ) Acknowledgement. I would like to thank V. Schechtman for his suggestions and help, A. Knapp for his advices, and L. Takhtajan and H. Bougourzi for stimulating discussions.
References c 1. Awata, H., Odake, S., Shiraishi, J.: Free Boson realization of Uq (sl N ). Commun. Math. Phys. 162, 61–83 (1994) 2. Bougourzi, A.H.: Uniqueness of the bosonization of the Uq (su(2)k ) quantum current algebra. Nuc. Phys. B 404, 457–482 (1993) 3. Cartan, H., Eilenberg, S.: Homological Algebra. Princeton Math. Ser. 19, Princeton, NJ: Princeton University Press, 1956 [ Commun. 4. Ding, J., Frenkel, I.: Isomorphism of two realizations of quantum affine algebra Uq (gl(n)). Math. Phys. 156, 277–300 (1993) 5. Drinfeld, V.G.: Hopf algebras and the quantum Yang–Baxter equation. Sov. Math. Dokl. 32, 254–258 (1985) 6. Drinfeld, V.G.: New realizations of Yangian and quantum affine algebra. Sov. Math. Dokl. 36, 212–216 (1987) 7. Feigin, B., Fuchs, D.: Representations of the Virasoro algebra. In: Representations of Lie groups and related topics. A.M. Vershik, D.P. Zhelobenko (eds.), New York: Gordon and Breach, 1990, pp. 465–554 8. Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator calculus. In: Mathematical Aspects of String Theory. ed. by S.-T. Yau, Singapore: World Scientific, 1987, pp. 150–188 9. Frenkel, E., Reshetekhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebra. Commun. Math. Phys. 178, 237–264 (1996) 10. Ginzburg, V., Schechtman, V.: Screenings and a universal Lie-de Rham cocycle. Preprint q-alg/9704014 11. Jimbo, M.: A q-difference analog of U( g) and the Yang–Baxter equation. Lett. Math. Phys. 10, 63–69 (1985) 12. Kac, V.: Infinite dimensional Lie algebras. 3rd Edition, Cambridge: Cambridge U. Press, 1990 13. Kassel, C.: Quantum groups. New York: Springer-Verlag, 1995 14. Konno, H.: BRST cohomology in quantum affine algebra Uq (sl2 ). Mod. Phys. Lett. A 9 1253–1265 (1994)
Quantum Screening Operators and Canonical q-de Rham Cocycles
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15. Lusztig, G.: Introduction to quantum groups. Basel–Boston: Birkhauser, Progress in Mathematics, 110 16. Matsuo, A.: Quantum algebra structure of certain Jackson integrals. Commun. Math. Phys. 157, 479–498 (1993) 17. Sebbar, A.: Quantum groups and canonical q-de Rham cocycles. To appear in J. Alg. 18. Sweedler, M.E.: Hopf Algebras. New York: Benjamin, 1969 19. Wakimoto, M.: Fock representations of the affine Lie algebra A(1) 1 . Commun. Math. Phys. 104, 605–609 (1986) Communicated by T. Miwa
Commun. Math. Phys. 198, 311 – 361 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation Boris Dubrovin1 , Youjin Zhang2 1
SISSA, Via Beirut 2–4, 34014 Trieste, Italy. E-mail:
[email protected] Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. E-mail:
[email protected]
2
Received: 16 January 1998 / Accepted: 2 April 1998
Abstract: We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W -algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov–Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity. 1. Introduction According to [5, 8, 31], the primary free energy of the matter sector of a 2D topological field theory (TFT) with n primaries as a function F (t) of the coupling constants t = (t1 , . . . , tn ) must satisfy WDVV equations of associativity. The problem of selection of physical solutions among all the solutions to WDVV equations is still open. Reformulating, the problem is to understand which part of the building of a 2D TFT can be constructed taking an arbitrary solution of WDVV equations as the basement. The first problem to be settled is coupling of a given matter sector to topological gravity. In the full theory, besides the primaries φ1 = 1, φ2 , . . . , φn that we now redenote φ1,0 , . . . , φn,0 , there are infinite number of their gravitational descendents φ1,p , . . . , φn,p , p = 1, 2, . . . . The generating function of their correlators is the full free energy of the theory D P α,p E F(T ) = e T φα,p . (1.1) Here T α,p are the coupling constants correspondent to the fields φα,p , XZ ...e−S[ψ] [dψ] h...i := g≥0
6g
(1.2)
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(the sum over the fields ψ living on the surface of genus g, S is the classical action). According to the idea of Witten [32] this procedure of coupling to topological gravity must be described by an integrable hierarchy of PDEs. The unknown functions of the hierarchy are the particular two-point correlators D E P β,q ∂ 2 F (T ) , (1.3) vα := φα,0 φ1,0 e T φβ,q = ∂T α,0 ∂T 1,0 T 1,p , . . . , T n,p are the times of the pth flow of the hierarchy, and the cosmological constant X := T 1,0 plays the role of the spatial variable of the hierarchy. The partition function of the full theory is the τ -function of a particular symmetric solution of the hierarchy. This idea works perfectly well for the case of pure gravity (the matter sector is trivial, n = 1, F (t) = 16 t3 ). According to the theory of Witten - Kontsevich [26, 33] the partition function of 2D gravity is a particular τ -function of the KdV hierarchy. For a 2D TFT with a nontrivial matter sector the correspondent integrable hierarchies are not known, although there are interesting conjectures about their structure for topological minimal models [7], for CP 1 topological sigma-model [11, 14, 15, 16, 17]. However, the properties of genus expansion of a 2D TFT provide us with certain nontrivial assumptions about the structure of the hypothetical hierarchy. Denote Fg the genus g part of the free energy D P α,p E (1.4) Fg := e T φα,p , g
F=
X
Fg .
(1.5)
g≥0
Particularly, the primary free energy is obtained restricting F0 onto the small phase space T α,p>0 = 0, F (t) = F0 |T α,0 =tα ,
T α,p>0 =0 .
(1.6)
The procedure of genus expansion consists of the following two parts. 1) We introduce slow spatial and time variables rescaling X 7→ εX,
T α,p 7→ εT α,p .
(1.7)
2) We change F 7→
∞ X
ε2 g−2 Fg .
(1.8)
g=0
The indeterminate ε is called string coupling constant. As ε → 0 one has a singular limit of the tau-function (i.e., of the partition function) of the theory (1.9) τ (T, ε) := exp ε−2 F0 + F1 + ε2 F2 + ... . Also all of the correlators become series in ε. Particularly, the series of the two-point correlators (1.2) have the form vα =
∞ X g=0
ε2g vαg ,
(1.10)
Bihamiltonian Hierarchies in 2D Topological Field Theory
313
where E D P β,q vαg := φα,0 φ1,0 e T φβ,q = g
∂ 2 Fg . ∂T α,0 ∂X
(1.11)
The genus zero (i.e., the tree-level) approximation of the theory corresponds to the 0 0 , ..., v to the hierarchy dispersionless approximation of the hierarchy. The solution v 1 D E Pn β,q
is given by the genus zero two-point correlators vα0 = φα,0 φ1,0 e T φβ,q as the 0 functions of the couplings. The solution is specified by the initial data on the small phase space vα0 |T α,p>0 =0 = ηαβ T β,0 ,
(1.12)
where the constant “metric” ηαβ is specified by the primary correlators of the form ηαβ := hφ1,0 φα,0 φβ,0 i0 |T α,p>0 =0 .
(1.13)
The construction of the would-be dispersionless approximation of the unknown integrable hierarchy for an arbitrary solution to equations of associativity and of the needed τ -function of it was given in [9] in terms of the geometry of WDVV equations (see also [11]). The bihamiltonian structure of the hierarchy was found in [10]. We briefly recollect this construction in Sect. 2 below. (We also describe more accurately the quasihomogeneity property of the hierarchy and of the τ -function formulated in [9] only for a generic solution of WDVV equations.) One can try to go beyond the tree-level approximation expanding the unknown hierarchy in a series w.r.t. ε2 . The string coupling constant ε plays the role of the dispersion parameter. The reader can keep in mind the dispersion expansion ut = u u x +
1 2 ε uxxx 12
(1.14)
of the KdV equation as an example of such a series. Particularly, for the one-loop (i.e., genus= 1) approximation of the theory, it is sufficient to retain the terms of the hierarchy up to the ε2 order. Particular solutions of the one-loop approximation must have the form vα = vα0 (T ) + ε2 vα1 (T ) + O(ε4 ) E D E D P β,q P β,q = φα,0 φ1,0 e T φβ,q + ε2 φα,0 φ1,0 e T φβ,q + O(ε4 ) 0
∂2 = F0 (T ) + ε2 F1 (T ) + O(ε4 ). α,0 1,0 ∂T ∂T
1
(1.15)
So for ε = 0 the one-loop approximation becomes the already known tree-level approximation of the hierarchy. We will call the genus one approximation of the integrable hierarchy the one-loop deformation of the genus zero hierarchy. Our result is that, under the assumption of semisimplicity (see below) the one-loop deformation of the hierarchy exists for any solution of WDVV equations and it is uniquely determined by the general properties of the genus one correlators proved by Dijkgraaf and Witten [6] and by Getzler [21]. (For the solution of WDVV equations with one and two primaries the one-loop approximation of the hierarchy was constructed in [6, 16]). Recall that the genus one part of the free energy has the form 1 log det Mαβ (t, ∂X t) + G(t) , (1.16) F1 (T ) = 24 t=v 0 (T )
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where the matrix Mαβ has the form Mαβ (t, ∂X t) = cαβγ (t) ∂X tγ , cαβγ (t) = ∂α ∂β ∂γ F (t),
(1.17) (1.18)
and G(t) is a certain function specified by Getzler’s equation [21] (see also Sect. 6 below). The first part of the formula becomes trivial on the small phase space T α,p = 0 for p > 0. The second part describes, in the topological sigma-models, the genus one Gromov–Witten invariants of the target space. For this function we derive the following formula τI (1.19) G = log 1/24 J (as above, semisimplicity of the solution of WDVV is assumed). Here J is the Jacobian of the transform between canonical and flat coordinates (see Sect. 2 below). To explain who is τI we recall that, in the semisimple case, WDVV can be reduced to equations of isomonodromy deformations of a certain linear differential operator with rational coefficients [9]. Our τI is the tau-function of the solution of these equations of isomonodromy deformations in the sense of [24]. According to [24, 30] the tau-function appears as the Fredholm determinant of an appropriate Riemann–Hilbert boundary value problem (see [12] for reduction of WDVV equations to a boundary value problem). Remarkably, the formula makes sense for an arbitrary semisimple solution of WDVV equations. Using explicit expressions (2.17), (2.19) for τI one can derive from (1.19) the proof of main conjectures of the recent paper of Givental [22]. As a byproduct of our computations, we obtained a nice formula for the generating function of elliptic Gromov–Witten invariants of complex projective plane. Namely, the function ψ :=
φ000 − 27 , 8(27 + 2 φ0 − 3 φ00 )
(1.20)
where φ(z) =
X k≥1
Nk(0) ek z , (3 k − 1)!
(1.21)
Nk(0) = the number of rational curves of degree k on CP 2 passing through generic 3 k − 1 points is the generating function for the numbers Nk(1) of the elliptic curves of degree k on CP 2 passing through generic 3 k points: 1 X (1) k ψ(z) = − + ek z . Nk 8 (3 k)!
(1.22)
k≥1
We prove also that the compatible pair of Poisson brackets describing the treelevel hierarchy admits a unique deformation to give a bihamiltonian structure, modulo O(ε4 ), of the one-loop hierarchy. The deformed bihamiltonian structure turns out to be a nonlinear extension of the Virasoro algebra (i.e., a classical W -algebra) with the central charge
Bihamiltonian Hierarchies in 2D Topological Field Theory
12ε2 c= (1 − d)2
"
# n X 1 2 1 n−2 (qα − d) . 2 2
315
(1.23)
α=1
Here ε is the string coupling constant, d and qα are the “dimension” and the “charges” of the theory. In the case of quantum cohomology of X (i.e., the topological sigma-model with the target space X) d coincides with the complex dimension of the target space X and qα are the halfs of the degrees of the basic elements in H ∗ (X). Remarkably, this formula works not only in quantum cohomologies. It gives the correct value for the central charge [20] of the classical W -algebras for the topological minimal models of A - D - E type (see below Sect.8)! We can continue this procedure trying to construct a higher genera approximation of the unknown integrable hierarchy. Of course, it would be too optimistic to expect that our procedure will go smoothly for any genus g for an arbitrary solution of equations of associativity. Moreover, from [16] it follows that, constructing the integrable hierarchy, probably for a generic solution of WDVV one cannot go beyond the genus one. However, our results suggest that in an arbitrary physical 2D TFT coupling to gravity is given by an integrable bihamiltonian hierarchy of 1 + 1 PDEs. Bihamiltonian structure of the hierarchy is to be described by a classical W -algebra with the prescribed central charge and the conformal dimensions of the primaries. So, we embed the problem of coupling to topological gravity into the problem of classification of a certain class of classical W -algebras. We briefly discuss this project in the final section, postponing the study of the higher genera corrections for a subsequent work. The paper is organized as follows. In Sect. 2 we recall some important points of the theory of WDVV equations of associativity (equivalently, the theory of Frobenius manifolds) and the construction of coupling to gravity at tree-level. The main results of the paper are formulated in Sect. 3. In Sect. 4 we derive some useful identities of the theory of semisimple Frobenius manifolds used in the proof of the main results. The derivation of the bihamiltonian structure of the hierarchy in the genus one approximation is given in Sect. 5. In Sect. 6 we solve Getzler’s equations for elliptic Gromov–Witten invariants for any semisimple Frobenius manifold. The examples of the deformed bihamiltonian hierarchies are given in Sect. 7. In the last Sect. 8 we discuss the programme of study of higher genera corrections in the setting of classical W -algebras.
2. WDVV Equations of Associativity and the Structure of a 2D TFT at Genus Zero WDVV equations of associativity is the problem of finding a function F (t) = F (t1 , . . . , tn ), a constant symmetric nondegenerate matrix (η αβ ), numbers q1 , . . . , qn , r1 , . . . , rn , d such that ∂α ∂β ∂λ F (t) η λµ ∂µ ∂γ ∂δ F (t) = ∂δ ∂β ∂λ F (t) η λµ ∂µ ∂γ ∂α F (t)
(2.1)
for any α, β, γ, δ = 1, . . . , n, (2.2) ∂1 ∂α ∂β F (t) ≡ ηαβ , where (ηαβ ) = (η αβ )−1 , X 1 (1 − qα ) tα + rα ∂α F (t) = (3 − d) F (t) + Aαβ tα tβ + Bα tα + C (2.3) 2
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B. Dubrovin, Y. Zhang
for some constants Aαβ , Bα , C. The numbers qα , rα , d and Aαβ , Bα , C must satisfy the following normalization conditions (see [12]): q1 = 0, rα 6= 0 only if qα = 1, Aαβ 6= 0 only if qα + qβ = d − 1, Bα 6= 0 only if qα = d − 2, C 6= 0 only if d = 3, X A1α = ηαε rε , B1 = 0.
(2.4)
ε
We will usually normalize the coordinates tα reducing ηαβ to the antidiagonal form ηαβ = δα+β,n+1 .
(2.5)
This can always be done for d 6= 0. Then qα + qn−α+1 = d,
qn = d.
(2.6)
Any solution of WDVV equations provides the space of parameters M n 3 (t1 , . . . , tn ) with a structure of Frobenius manifold. That means that there exists a unique structure of a Frobenius algebra (At , < , >) on the tangent planes Tt M n such that h∂α · ∂β , ∂γ i = ∂α ∂β ∂γ F (t),
h∂α , ∂β i = ηαβ .
(2.7)
Explicitly ∂α · ∂β = cγαβ (t) ∂γ
where cγαβ (t) = η γε ∂ε ∂α ∂β F (t).
(2.8)
The vector field e = ∂1
(2.9)
is the unity of the algebra. We introduce also the Euler vector field on M n E(t) = E ε (t) ∂ε :=
n X
(1 − qε ) tε + rε ∂ε .
(2.10)
ε=1
This is the generator of the scaling transformations (2.3). All the equations (2.1)–(2.3) can be easily reformulated in a covariant way (see [11]). One of the main geometrical objects on a Frobenius manifold is a deformation of the Levi–Civita connection ∇ for < , >: e u v = ∇u v + z u · v. ∇
(2.11)
Here u, v are two vector fields on M n , z is the parameter of the deformation. The connection (2.11) is flat for any z. It can be extended to a flat connection on M n × C∗ e ∇ where
d dz
v = ∂z v + E · v −
1 µ v, z
(2.12)
Bihamiltonian Hierarchies in 2D Topological Field Theory
317
1 (2 − d) = diag(µ1 , . . . , µn ), 2 hµ a, bi = − ha, µ bi .
µ := −∇ E +
d µ α = qα − , 2
(2.13) (2.14)
e d doing an (Comparing with [11] we change the normalization of the component ∇ dz n ∗ elementary gauge transform.) The connection on M × C is still flat. The Frobenius manifold is said to satisfy the semisimplicity condition (or, briefly, it is semisimple) if the algebras At are semisimple for generic t. On the open domain of the points of semisimplicity one can introduce canonical coordinates u1 , . . . , un such that ∂ ∂ ∂ · = δij , i, j = 1, . . . , n. ∂ui ∂uj ∂ui
(2.15)
(We will use all lower indices working with the canonical coordinates. No summation over the repeated indices will be assumed in this case.) In these coordinates WDVV can be reduced to a commuting family of nonstationary Hamiltonian flows on the Lie algebra so(n) with the standard Poisson bracket ∂V = {V, Hi (V ; u)}, i = 1, . . . , n ∂ui
(2.16)
(the definition of the matrix V = (Vij ), V T = −V ∈ so(n) see below in Sect.4), the canonical coordinates u1 , . . . , un play the role of the times and the quadratic Hamiltonian has the form Hi =
1 X Vij2 . 2 ui − uj
(2.17)
j6=i
These are the equations of isomonodromy deformations of the operator 1 d − U − V, U = diag(u1 , . . . , un ) dz z
(2.18)
with rational coefficients [9]. The tau-function τI of a solution in the theory of isomonodromy deformations is defined [24] by the quadrature d log τI =
n X
Hi dui .
(2.19)
i=1
(We denote this function τI to avoid confusions with the tau-function (1.9) of the integrable hierarchy.) Another geometric object is a deformation of the flat metric < , > on M n [9, 11]. We introduce the intersection form (ω1 , ω2 )t := iE (ω1 · ω2 ),
ω1 , ω2 ∈ Tt∗ M n .
(2.20)
The metric ( , )t − λ < , >t
(2.21)
on Tt∗ M n does not degenerate for almost all (λ, t). It is flat for these (λ, t). In the coordinates tα
318
B. Dubrovin, Y. Zhang αβ g αβ (t) := (d tα , d tβ ) = E ε cαβ (t) + Aαβ , ε = (d + 1 − qα − qβ ) F
(2.22)
where 0
F αβ (t) := η αα η ββ
0
∂ 2 F (t) , ∂tα0 ∂tβ 0
0
0
Aαβ := η αα η ββ Aα0 β 0 .
(2.23)
We give also the formula for the Levi–Civita connection for the flat (but not constant in the coordinates tα !) metric ( , ) 1+d αβ αε β − qβ cαβ (2.24) 0γ (t) := −g (t) 0εγ (t) = γ (t), 2 where 0
0
αα ββ cαβ η ∂α0 ∂β 0 ∂γ F (t). γ (t) = η
(2.25)
The flat metric (2.22) is responsible not only for the second Poisson bracket of the integrable hierarchy (see below), but also for the relation between Frobenius manifolds and reflection groups [11]. The genus zero approximation of the needed integrable hierarchy will be an infinite family of dynamical systems on the loop space L(M n ). We supply the loop space with a Poisson bracket αβ 0 δ (X − Y ), {v α (X), v β (Y )}(0) 1 =η
(2.26)
(to avoid confusions we redenote tα → v α the coordinates on M n when dealing with the hierarchy; comparing with the above notations of the Introduction we omit the label 0, i.e., v α = η αε vε0 ). The second Poisson bracket on the same loop space has the form γ αβ {v α (X), v β (Y )}(0) (v(X)) δ 0 (X − Y ) + 0αβ γ (v(X)) vX δ(X − Y ). (2.27) 2 =g
Particularly, for d 6= 1 the Poisson bracket of T (X) :=
2 v n (X) 1−d
(2.28)
has the form 0 {T (X), T (Y )}(0) 2 = [T (X) + T (Y )] δ (X − Y ).
(2.29)
This coincides with the Poisson bracket on the dual space to the Lie algebra of onedimensional vector fields. Therefore the full Poisson bracket (2.27) can be considered as a nonlinear extension of this algebra (the classical W-algebra with zero central charge). Observe that 2 rα 2 (1 − qα ) α α v = δ 0 (X − Y ) + vX (X) + δ(X − Y ). {v α (X), T (Y )}(0) 2 1−d 1−d (2.30) So T (X) plays the role of the stress-energy tensor, and the conformal dimensions of the fields v α having qα 6= 1 are 1α =
2(1 − qα ) . 1−d
(2.31)
Bihamiltonian Hierarchies in 2D Topological Field Theory
319
When qα = 1 the variable sα := exp v α has the Poisson bracket with the stress-energy tensor of the form {sα (X), T (Y )}(0) 2 =
2rα α s (X)δ 0 (X − Y ) + sα X δ(X − Y ). 1−d
(2.32)
So it is a primary field with the conformal dimension 1α =
2rα . 1−d
(2.33)
The two Poisson brackets are compatible, i.e., any linear combination (0) a1 { , }(0) 1 + a 2 { , }2
(2.34)
with arbitrary constant coefficients a1 , a2 gives a Poisson bracket on L(M n ) [11]. This gives a possibility to construct a hierarchy of commuting flows on L(M n ) starting from the Casimirs of the first Poisson bracket Z (2.35) H α,−1 = v α (X) dX, α = 1, . . . , n using the standard bihamiltonian recursion procedure [29] α,p−1 (0) }2 {·, H α,p }(0) 1 = kα,p {·, H
(2.36)
for appropriate constants kα,p . These constants are to be chosen in a clever way to make the hierarchy compatible with the genus zero recursion relations for the topological correlators. For the genus zero approximation the needed normalization of the Hamiltonians is given by an alternative procedure [9] using the flat coordinates of the deformed e connection ∇. e are functions θ(t, z) such that The flat coordinates of ∇ e dθ = 0. ∇
(2.37)
e Then Let us forget for the moment about the last component (2.12) of the connection ∇. the flat coordinates θ are specified by the equation ∂α ∂β θ = z cγαβ ∂γ θ. A basis of the solutions θ1 (t, z), . . . , θn (t, z) can be obtained as power series X θγ,p (t) z p , θγ (t, z) = tγ +
(2.38)
(2.39)
p≥1
where the coefficients θγ,p (t) are determined recursively from the equations ∂α ∂β θγ,p+1 (t) = cραβ (t)∂ρ θγ,p ,
θγ,0 (t) = tγ = ηγε tε .
(2.40)
One can normalize the deformed flat coordinates requiring h∇ θα (t, z), ∇ θβ (t, −z)i ≡ ηαβ . There still remains some freedom in the choice of the deformed flat coordinates
(2.41)
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θα (t, z) 7→ θε (t, z) Gεα (z)
(2.42)
with an arbitrary matrix-valued series G(z) = (Gβα (z)), G(z) = 1 + z G1 + z 2 G2 + . . . , G(z) η G(−z)T ≡ η.
(2.43) (2.44)
Later we put also the Eq. (2.12) into the game. This will fix the deformed flat coordinates almost uniquely. The Hamiltonians of the genus zero hierarchy have the form Z (2.45) Hβ,p = θβ,p+1 (v(X)) dX, p = 0, 1, . . . . The hierarchy itself reads ∂v (0) = Kβ,p (v, vX ) = {v, Hβ,p }(0) 1 = ∂X ∇ θβ,p+1 (v) = ∇ θβ,p (v) · ∂X v, (2.46) ∂T β,p (we treat ∂X v and ∂T β,p v as tangent vectors to the Frobenius manifold). Observe that the coefficients in front of ∂X v are functions well-defined everywhere on the Frobenius manifold. The genus zero two-point functions E D P β,q (2.47) vα0 (T ) = φα,0 φ1,0 e T φβ,q 0
give a particular solution of the commutative hierarchy (2.46) specified by the following symmetry reduction ! X T α,p ∂T α,p v 0 = 0. (2.48) ∂T 1,1 − α,p
(We identify T 1,0 and X. So the variable X is supressed in the formulae). The solution can be found in the form X X T β,q ∇ θβ,q (T0 ) + T β,q T γ,p ∇ θβ,q−1 (T0 ) · ∇ θγ,p (T0 ) + . . . . v 0 (T ) = T0 + (2.49) q>0 p,q>0 This is a power series in T α,p>0 with the coefficients depending on T0 := (T1,0 , . . . , Tn,0 ), Tα,0 := ηαβ T β,0 . The series can be found as the fixed point t = v 0 of the gradient map Mn → Mn t = ∇ 8T (t), where 8T (t) =
X
(2.50)
T α,p θα,p (t).
(2.51)
Defining the functions α,p;β,q (t) on the Frobenius manifold by the following generating function ∞ X α,p;β,q (t) z p wq , (z + w)−1 h∇ θα (t, z), ∇ θβ (t, w)i − ηαβ = p,q=0
(2.52)
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we complete the construction of the genus zero free energy of the TFT coupled to gravity by setting log τ = F0 (T ) =
1 X α,p;β,q (v 0 (T ))Teα,p Teβ,q , 2
(2.53)
where Teα,p = T α,p if (α, p) 6= (1, 1), Te1,1 = T 1,1 − 1.
(2.54)
The resulting function F0 (T ) satisfies the string equation 1 ∂F0 (T ) X α,p = T ∂T α,p−1 F0 (T ) + ηαβ T α,0 T β,0 . 1,0 ∂T 2
(2.55)
On the small phase space T α,p>0 = 0 one has F0 (T )| T α,p>0 =0 = F (t).
(2.56)
T α,0 =tα
Also the derivatives of the function F0 (T ) satisfy the genus zero recursion relations of Dijkgraaf and Witten. Observe that ∂ 2 F0 (T ) = α,p;β,q (v 0 (T )). ∂T α,p ∂T β,q
(2.57)
The proofs of all these results can be found in [9]. e d of the deformed connection to fix the densities of We now use the last component ∇ dz the commuting Hamiltonians Hα,p . Let us consider first the non-resonant case µα −µβ 6∈ e can be constructed Z6=0 for α 6= β. Then the system of deformed coordinates t˜α (t, z) of ∇ in the form t˜α (t, z) = θα (t, z) z µα =
∞ X
θα,p (t) z p+µα ,
(2.58)
p=0
e ∇
d dz
dt˜α (t, z) = 0.
(2.59)
The coefficients θα,p (t) are now defined uniquely by (2.40) and by the quasihomogeneity equation following from (2.59), 2−d + µα θα,p (t). (2.60) LE θα,p (t) = p + 2 The functions α,p;β,q (t) are also quasihomogeneous of the degree p + q + 1 + µα + µβ . From this one easily derives the quasihomogeneity constraint for F0 (see [9]). Let us now consider the non-generic case. We describe first the fundamental matrix solution of the linear system e ∇
d dz
dt˜ = 0.
(2.61)
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We rewrite this system for the gradient (∇ t˜)α = η αβ ∂β t˜
(2.62)
of the deformed flat coordinates. So the columns of the fundamental matrix are the gradients of the deformed flat coordinates t˜1 (t, z), . . . , t˜n (t, z). The fundamental matrix has the form Y (t, z) = ∇t˜1 (t, z), . . . , ∇t˜n (t, z) = (∇θ1 (t, z), . . . , ∇θn (t, z)) z µ z R , (2.63) where the constant1 matrix R = (Rβα ) satisfies the following requirements: 1. Rβα 6= 0 only if µα − µβ is a positive integer, α Rβ if µα − µβ = k . = 2. Let Rk α β 0 otherwise We have R = R1 + R2 + . . .
(2.64)
(finite number of terms). Then we must have hRk a, bi + (−1)k ha, Rk bi = 0,
k = 1, 2, . . .
(2.65)
for any two vectors a, b. The matrix R is determined uniquely up to the transformations R 7→ G−1 R G,
(2.66)
where the matrix G = (Gα β ) must satisfy the following conditions: 1. Gα β 6= 0 only if µα − µβ is a non-negative integer. 2. Define the decomposition G = G0 + G1 + . . .
(2.67)
G0 = 1
(2.68)
similar to (2.64). We must have
and the matrix G must satisfy the following orthogonality condition hG+ a, G bi =< a, b >,
(2.69)
G+ = G0 − GT1 + GT2 − GT3 + . . . .
(2.70)
for any a, b, where
1 Constancy of the matrix R is a manifestation of the general isomonodromicity property proved in the theory of Frobenius manifolds [9, 11, 12].
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Proof can be found in [12]. The class of equivalence of the matrix R modulo the transformations (2.66) together with the matrix µ completely specifies the class of gauge e d modulo gauge transformations of the form (2.42)– equivalence of the operator ∇ dz (2.44) near the singularity at z = 0. Particularly, the coefficients Aαβ Bα , C in (2.3) have the form Aαβ = ηα (R1 )β ,
(2.71)
Bα = η1ε (R2 )εα ,
(2.72)
1 C = − η1ε (R3 )ε1 , 2
(2.73)
e d = 0 we obtain the following Plugging the formula (2.63) into the equation ∇ dz quasihomogeneity constraint for the function θα,p (t) p X 2−d LE θα,p (t) = p + + µα θα,p (t) + θε,p−k (t) (Rk )εα + const. (2.74) 2 k=1
(Observe that the functions θα,p (t) are defined up to an additive constant.) A more involved computation shows that LE α,p;β,q (t) = (p + q + 1 + µα + µβ )α,p;β,q (t) +
p X
(Rr )εα ε,p−r;β,q (t)
r=1
+
q X
(Rr )εβ α,p;ε,q−r (t) + (−1)q Rp+q+1
ε α
ηεβ .
(2.75)
r=1
Using this and the explicit formula (2.53) we arrive at Proposition 1. The genus zero partition function τ satisfies the following constraint L0 τ = 0, where L0 =
X 1
(2.76)
X + k + µλ Teλ,k ∂T λ,k + Teλ,k (Rr )εα ∂T ε,k−r
2 ε 1 X (−1)q Teα,p Teβ,q Rp+q+1 α ηεβ . + 2
(2.77)
Here Teα,p are defined by (2.54). Example. For topological sigma-models R coincides with the matrix of multiplication by the first Chern class c1 (X) in the classical cohomologies of the target space X [11]. Since deg c1 (X) = 1 we have R = R1 .
(2.78)
The recursion relation (2.74) in this case coincide with the recursion relation of Hori [23], and the particular case of (2.75) was obtained in [17]. We infer that the coefficients
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θα,p (t) of Dthe expansion flat coordinates coincide2 with the two-point P of the deformed E n
functions φα,0 φ1,0 e
α=1
tα φα,0
defined in terms of intersection theory on the moduli 0
spaces of instantons S 2 → X. The general identity (2.76) in this particular case coincides with the L0 Virasoro constraint derived in [23]. Remark. Applying an appropriate recursion procedure to the operator L0 we can derive a half-infinite sequence of the Virasoro constraints Lk τ = 0,
k ≥ −1
(2.79)
generalizing the constraints of [18]. All the operators Lk , k ≥ −1 are given in terms of the monodromy data (µ, R) at z = 0. We will present these results in a separate publication. We conclude this section with an explicit formula for the bihamiltonian structure of genus zero hierarchy (2.46). Proposition 2. Let (α, p) be a pair of indices such that p + µα +
1 6= 0. 2
(2.80)
Then the equation ∂v = {v, Hα,p }(0) (2.81) 1 ∂T α,p of the hierarchy (2.46) is also a Hamiltonian flow w.r.t. the second Poisson bracket (2.27) (0) ˆ {v, Hα,p }(0) 1 = {v, Hα,p }2 .
The Hamiltonian Hˆ α,p has the form X ε Hˆ α,p = (−1)k Rp−l,k α k,l
where the matrices Rk,l are defined as follows R0,0 = 1,
Rk,0 = 0
for
k > 0, Rk,l =
(2.82)
Hε,l−1 , (p + µα + 21 )k+1 X
Ri1 ...Ril
(2.83)
for
i1 +...+il =k
Proof. We use the identity Z Z 1 (0) ˜ { · , t(v(X), z)dX}2 = {·, t˜(v(X), z)dX}(0) ∂z − 1 2z
l > 0. (2.84)
(2.85)
e (see Lemma H.3 in [11]). Inverting, we obtain valid for an arbitrary flat coordinate t˜ of ∇ Z Z z Z 1 1 {·, z2 w− 2 t˜α (v(X), w)dw dX}(0) = { · , t˜α (v(X), z)dX}(0) 2 1 . (2.86) Integrating the expansions in both sides of the equation and using X ε t˜α (t, z) = θε,p z p+µε z R α we obtain the formula (2.82). Proposition is proved. 2
Our normalization of the correlators differs from that of Hori
(2.87)
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3. Formulation of the Main Results We formulate now the main requirement to uniquely specify the one-loop correction to the hierarchy (2.46). We want to find a hierarchy of equations of the form ∂t (0) (1) = Kβ,p (t, tX ) + ε2 Kβ,p (t, tX , . . . ) ∂T β,p
(3.1)
such that the following property holds true (cf. [6, 16]): Main assumption. For any solution v = v(T ) of the hierarchy (2.46) the function t(T ) = (t1 (T ), . . . , tn (T )) t(T ) := v(T ) + ε2 w(T ), where ∂2 wα (T ) = ∂T α,0 ∂T 1,0
(
(3.2) )
1 log det Mαβ (t, tX ) + G(t) 24
(3.3) t=v(T )
satisfies (3.1) modulo terms of the order ε4 . Here the matrix Mαβ (t, tX ) is defined by (1.17), and G(t) is the G-function of the Frobenius manifold (see below). We denote t = (t1 , . . . , tn ) the dependent variables of the hierarchy to emphasize that they live on the Frobenius manifold M n . So (3.1) is still a dynamical system on the loop space L(M n ). (1) It is clear that the corrections Kβ,p are determined uniquely. Indeed, the deformed hierarchy (3.1) is obtained from the tree-level hierarchy (2.46) by the infinitesimal B¨acklund transform vα 7→ vα + ε2 wα (v, vX , vXX , vXXX ) = tα ,
(3.4)
where the functions wα are defined by the formula (3.3). The functions wα are polynomials in vXX , vXXX but they are rational functions in vX . Remarkably, all the denominators will disappear from the deformed hierarchy. We will prove that the corrections are polynomials in tX , tXX , tXXX for the case of semisimple Frobenius manifold (see the definition in Sect. 2 above). Observe that Mαβ is the matrix of multiplication by the vector ∂X t. So the determinant det Mαβ vanishes identically on the nilpotent part of the algebra At . (1) First we observe that the correction Kβ,p can be subdivided into two parts (1) 0 00 Kβ,p = Kβ,p + Kβ,p ,
(3.5)
0 00 where Kβ,p is the contribution of the first term in the r.h.s. of (3.3), and Kβ,p is the contribution of the second term respectively. The main difficulty is in the computation 0 . of Kβ,p
Theorem 1. There exists a unique hierarchy of the form 0 ∂t (0) 0 = Kβ,p (t, tX ) + ε2 Kβ,p;λ (t) tλXXX + Kβ,p;λµ (t) tλXX tµX β,p ∂T 0 + Kβ,p;λµν (t) tλX tµX tνX
(3.6)
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such that the function t(T ) = (t1 (T ), . . . , tn (T )) satisfies (3.6) up to terms of order ε4 for an arbitrary solution v(T ) of (2.46) tα (T ) = vα (T ) +
ε2 ∂2 log det Mαβ (t, tX ) t=v(T ) . 24 ∂T α,0 ∂T 1,0
(3.7)
0 0 0 (t), Kβ,p;λµ (t), Kβ,p;λ (t) of the hierarchy are analytic functions The coefficients Kβ,p;λµν on the Frobenius manifold. The hierarchy (3.6) admits a representation
0 ∂t 0 = t(X), Hβ,p + ε2 δHβ,p + O(ε4 ), β,p 1 ∂T
(3.8)
where the perturbation of the first Poisson bracket has the form {tα (X), tβ (Y )}01 = ε2 µν αβ η µν cαβ cµν (t(Y )) δ 000 (X − Y ) µν (t(X)) + η 24 0 ε2 µν 2 αβ 4 − η ∂X (cµν (t(X))) + η µν ∂Y2 (cαβ µν (t(Y ))) δ (X − Y ) + O(ε ). (3.9) 24
{tα (X), tβ (Y )}(0) 1 +
The operation { , }01 is skew-symmetric and it satisfies the Jacobi identity modulo O(ε4 ). The perturbations of the Hamiltonians have the form Z 0 (3.10) δHβ,p = χβ,p+1;µν (t(X)) tµX tνX dX, where χβ,p;µν = χβ,p;νµ are given by χα,0;µν = 0, χα,p+1;µν =
1 γ ∂θα,p 1 γ ξσ ζ ∂θα,p−1 w c c c − , 24 µν ∂tγ 24 ξζ ν σµ ∂tγ
p ≥ 0.
(3.11)
α are defined by Here θα,−1 = 0 and wµν µν α α α µν α µν α ν αµ = cµν wβγ µβ cνγ − cγ cµνβ = cµβ cνγ + cµγ cνβ − ∂µ (cβγ cν ).
(3.12)
0 The Hamiltonians Hβ,p + ε2 δHβ,p commute pairwise modulo O(ε4 ) w.r.t. the bracket (3.9). α α αβ αβ αβ Here and below cα γµ , cγµν , cγµνσ , cγµ , cγµν , cγµνσ are obtained by taking derivatives 1 N of the function F (t , . . . , t ) with respect to the coordinates t1 , . . . , tN and by using η αβ to raise the indices, for example, 0
αα ββ cαβ η γµ = η
0
∂tα0
∂ 4 F (t) . ∂tβ 0 ∂tγ ∂tµ
(3.13)
Remark. The first theorem does not use the quasihomogeneity condition (2.3). The next theorem does use it.
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Theorem 2. The following formulae give the perturbation of the second Poisson bracket: {tα (X), tβ (Y )}02 = {tα (X), tβ (Y )}(0) 2 αβ 2 000 αβ + ε h (t(X)) δ (X − Y ) + rγ (t(X)) tγX δ 00 (X − Y ) αβ (t(X)) tγX tµX δ 0 (X − Y ) + fγαβ (t(X)) tγXX + qγµ
γ µ γ µ ν γ αβ αβ + bαβ γµ (t(X)) tX tXX + aγµν (t(X)) tX tX tX + pγ (t(X)) tXXX δ(X − Y )
+ O(ε4 ), where
(3.14) 1 1 µν αβ ∂ν (g µν cαβ c ) + , c µ 12 2 ν µ 1 1 µν ( − µβ ) cαβ = µν cγ , 12 2 1 1 β β ( − µβ ) η ασ (∂σ ∂ν wγµ = + ∂σ ∂γ wµν 72 2 β β β β + ∂σ ∂µ wγν − 2 ∂µ ∂ν wγσ − 2 ∂µ ∂γ wνσ − 2 ∂ν ∂γ wµσ )
hαβ =
(3.15)
pαβ γ
(3.16)
aαβ γµν
β ασ β ασ β + η ξζ (6 cασ ξζ cσγµν + 3 cξζγ cσµν + 3 cξζµ cσγν
bαβ γµ
rγαβ
β ασ β ασ β ασ β +3 cασ ξζν cσγµ + cξζγµ cσν + cξζγν cσµ + cξζµν cσγ ) , 1 1 β β β ( − µβ ) η ασ (−2 ∂γ wµσ = + ∂σ wγµ − ∂µ wγσ ) 12 2 3 ασ β 1 β +η ξζ (3 cασ cξζγ cσµ + cασ cβ ) , ξζ cσγµ + 2 2 ξζµ σγ 3 1 3 1 3 αβ αν βµ αµ − µβ cγ cνµ − − µα cβν = ∂γ h + γ cνµ , 2 24 2 24 2
βα αβ fγαβ = rγαβ + pαβ , γ + pγ − ∂ γ h 1 1 1 αβ βα qγµ ∂γ rµαβ + ∂µ rγαβ − ∂µ ∂γ hαβ − = (bαβ µγ + bµγ ) + 2 2 2 1 αβ βα − ∂µ (pγ + pγ ). 2 The Jacobi identity for a linear combination
a1 { , }01 + a2 { , }02
(3.17)
(3.18) (3.19) (3.20)
(3.21)
(3.22)
with arbitrary constant coefficients a1 , a2 holds true modulo O(ε ). The equations of the perturbed hierarchy for those (β, p) for which 4
1 6= 0 (3.23) 2 are Hamiltonian flows also w.r.t. the second Poisson bracket (3.14) with the Hamiltonian p + µβ +
0 Hˆ β,p =
X
(−1)k Rp−l,k
k,l
Here Rl,k are defined in (2.84).
0 H,l−1 + ε2 δH,l−1 β
(p + µα + 21 )k+1
.
(3.24)
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The proofs will be given in Sect. 5. The deformations (3.9) and (3.14) of the Poisson brackets are obtained by applying the same infinitesimal B¨acklund transform (3.7) to the Poisson brackets (2.26) and (2.27) resp. We prove that after this transform each of the deformed Poisson brackets is a combination of δ(X − Y ), . . . , δ 000 (X − Y ) with the coefficients being polynomial in tX , tXX , tXXX . Coefficients of these polynomials are functions analytic on the Frobenius manifold (assuming the Frobenius manifold to be analytic itself). Applying similar procedure to the Hamiltonians (2.45) we obtained the deformed Hamiltonians (3.10). The structure (3.6) of the deformed hierarchy follows from the formulae (3.8)–(3.11). Finally, the same infinitesimal B¨acklund transform gives the deformation the linear pencil (2.34) of the Poisson brackets. We describe now the effect of adding the second term in the formula (3.3). At the moment we consider G(t) as an arbitrary function on some domain in the Frobenius manifold. We will compute this function in Theorem 3 below. Proposition 3. Inserting an arbitrary function G(t) in (3.3) we preserve the structure of the hierarchy, of the Hamiltonian, and of the Poisson brackets. The Hamiltonians get 00 with a correction ε2 δHα,p 00 = δHα,p
Z cγξν cσξ µ
∂θα,p ∂G µ γ t t dx. ∂tγ ∂tσ x x
(3.25)
The deformations of the first and of the second Poisson brackets get the correction ε2 { , }001 and ε2 { , }001 with {tα (X), tβ (Y )}001 = a˜ αβ (t(X)) δ 000 (X − Y ) + b˜ αβ (t(X)) δ 00 (X − Y ) + e˜αβ (t(X)) δ 0 (X − Y ),
(3.26)
{tα (X), tβ (Y )}002 = aαβ (t(X)) δ 000 (X − Y ) + bαβ (t(X)) δ 00 (X − Y ) + eαβ (t(X)) δ 0 (X − Y ) + ∂X (q αβ (t(X))) δ(X − Y ),
(3.27)
where ∂G(t) , ∂tµ µ 3 ∂2G βρ αρ ∂X a˜ αβ + σ ρ cασ − cβσ tX , µ η µ η 2 ∂t ∂t 2 αβ ∂X b˜ αβ − ∂X a˜ , ∂G(t) γβ 2 cαµ , (3.28) γ g ∂tµ ∂ 2 G(t) µ 3 γβ γα γβ ∂G(t) µ ∂X aαβ + cαρ − cβρ t + (µα − µβ ) cαρ t , µ g µ g γ cµ 2 ∂tγ ∂tρ X ∂tρ X 2 ∂ G ασ βρ ∂ 2 G βσ ρα ∂G αβ ρσ µ ν 1 − µβ c c − c c + σ cρµ cν tX tX , µ ν 2 ∂tσ ∂tρ ∂tσ ∂tν ρ µ ∂t
a˜ αβ = 2 cαβµ b˜ αβ = e˜αβ = aαβ = bαβ = q αβ =
2 αβ eαβ = q αβ + q βα + ∂X bαβ − ∂X a .
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The full Poisson brackets of the one-loop deformed hierarchy are { , }1 = { , }01 + { , }001 and { , }2 = { , }02 + { , }002 . For the case of quantum cohomology the function G(t) must be the generating function of the elliptic Gromov–Witten invariants of the target space. The recursion relations for the elliptic Gromov–Witten invariants were found by Getzler [21]. He proved that the generating function G(t) must satisfy a complicated system of differential equations (see (6.1) below). This system makes sense on an arbitrary Frobenius manifold. Our next result is the solution of this system on an arbitrary semisimple Frobenius manifold. Theorem 3. For an arbitrary semisimple Frobenius manifold the system (6.1) has a unique, up to an additive constant, solution G = G(t2 , . . . , tn ) satisfying the quasihomogeneity condition LE G = γ
(3.29)
with a constant γ. This solution is given by the formula G = log
τI , J 1/24
(3.30)
where τI is the isomonodromic tau-function (2.19) and α ∂t J = det ∂ui
(3.31)
is the Jacobian of the transform from the canonical coordinates to the flat ones. The scaling anomaly γ in (3.29) is given by the formula 1 X 2 nd , µα + 4 48 n
γ=−
(3.32)
α=1
where d µα = qα − , α = 1, . . . , n. 2 Corollary 1. For d 6= 1 the variable T (X) = bracket
2 n 1−d t (X)
{T (X), T (Y )}2 = [T (X) + T (Y )] δ 0 (X − Y ) + ε2
(3.33) has the Virasoro Poisson c 000 δ (X − Y ) 12
(3.34)
with the central charge 12ε2 cε = (1 − d)2 2
"
# n X 1 2 n−2 µα . 2
(3.35)
α=1
So, the bihamiltonian structure of the conjectured integrable hierarchy at the oneloop approximation looks like a classical W -algebra with the central charge (3.35) and the conformal weights (2.31).
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4. Some Formulas Related to the Canonical Coordinates of Frobenius Manifold The canonical coordinates on a semisimple Frobenius manifold M n are denoted by (u1 , . . . , un ). They satisfy the multiplication table ∂ ∂ ∂ · = δij . ∂ui ∂uj ∂ui
(4.1)
The Pn invariant 2metric becomes diagonal in the canonical coordinates, i.e. < , >= ∂ i=1 ηii (u)dui . We assume that the unit vector field of the Frobenius manifold is e = ∂t1 , then the rotation coefficients γij (u) are defined by √ ∂j ηii (u) 1 ∂i ∂j t1 (u) = p γij (u) = p , for i 6= j, (4.2) 2 ηjj (u) ∂i t1 (u) ∂j t1 (u) ∂ where ∂i = ∂u . They are symmetric with respect to their indices and satisfy the followi ing equations:
∂γij = γik γkj , ∂uk N X ∂γij = 0. ∂uk
i, j, k distinct,
(4.3) (4.4)
k=1
Define ∂i tα (u) . ψiα (u) = √ ηii (u)
(4.5)
The matrix (ψiα ) satisfies the following identities: n X
ψlα ψlβ = ηαβ ,
l=1
n X
ψlα ψlβ = η αβ ,
(4.6)
l=1
where ψjα = ψjγ η γβ . We list here the following useful identities: (see [11]) cαβγ =
n X ψiα ψiβ ψiγ i=1
ψi1
,
(4.7)
∂ui ψiα ∂tα = ψi1 ψiα , = , ∂ui ∂tα ψi1 X ∂ψiα ∂ψiα = γik ψkα , i 6= k, =− γik ψkα , ∂uk ∂ui
(4.8) (4.9)
k
ψlβ ψiβ ∂ψiα X = γil ψlα ( − ), ∂tβ ψl1 ψi1 N
(4.10)
l=1
Denote σi = ψiα tα X.
(4.11)
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331
Then we have ∂σi X ψlα ψiα = γil σl ( − ). ∂tα ψl1 ψi1
(4.12)
A = (ψiα ψjβ cαβγ tγX ),
(4.13)
n
l=1
Let’s consider the matrix
by using the identity (4.6) we have det A = det(η αβ ) det(cαβγ tγX ).
(4.14)
We see that the matrix cαβγ tγX diagonalizes in the canonical coordinates. From (4.7) we see that the following expression for F (1) (t, tX ) holds true: 1 log det(cαβγ tγX ) + G(t) 24 1 log det(η αβ ) + log det(A) + G(t) 24 X ψkα ψkβ ψkγ γ 1 log det(η αβ ) + log det(ψiα ψjβ tX ) + G(t) 24 ψk1
F (1) (t, tX ) = 1 24 1 =− 24 =−
k
Y Y 1 1 1 log( ψlγ tγX ) − log( ψl1 ) + G(t) − log det(η αβ ), = 24 24 24 n
n
l=1
=
1 log 24
n Y
l=1
σl −
l=1
1 log( 24
n Y
ψl1 ) + G(t) −
l=1
1 log det(η αβ ). 24
(4.15)
This expression of the function F (1) is crucial in the proof of Theorems 1, 2. Let’s denote 1 log det(cαβγ tγX ) 24 n n Y Y 1 1 1 log log( ψl1 ) − log det(η αβ ). σl − = 24 24 24
F (1) :=
l=1
l=1
By a direct calculation we get also the following formulas:
(4.16)
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B. Dubrovin, Y. Zhang
Ft(1) α = X
n 1 X ψiα , 24 σi
(4.17)
i=1
n 1 X ψiα ψiβ , 24 σi2 i=1 n 1 X γjk ψkα ψkβ ψjβ = − 24 σj ψk1 ψj1
F (1) =− α β tX t X
Ft(1) α tβ X
(4.18)
j,k=1
N ψjβ 1 X γjk σk ψjα ψkβ , − − 24 ψk1 ψj1 σj2
(4.19)
N 1 X ψiα ψjα ψiα ψj1 σj ψjα γij − + − , 2 24 σi ψj1 ψi1 ψi1 ψi1 i,j=1
(4.20)
j,k,γ=1
Ft(1) α = cαγ βµ
=
N X
γij
i,j=1
×
ψjµ ψj1
! ψiα ψiβ ψjγ ψiα ψiγ ψjβ ψiγ ψiβ ψjα ψiγ ψiα ψiβ ψj1 + + − 2 ψi1 ψi1 ψi1 ψi1 ψiµ − . (4.21) ψi1
All these formulae do not use the quasihomogeneity (2.3). In the quasihomogeneous case the canonical coordinates u1 (t), . . . , un (t) are the roots of the characteristic equation det(g αβ (t) − u η αβ ) = 0. Here g
αβ
(4.22)
is the intersection form. The matrix γij in this case has the form γij = −(ui − uj )−1 Vij ,
(4.23)
where Vij =
n X
µα ψiα ψjα .
(4.24)
α=1
The columns of the matrix 9 = (ψiα ) are the eigenvectors of the matrix V with the eigenvalues µα . Particularly, ψi1 is the eigenvector of V with the eigenvalue µ1 = −d/2. It follows that P (uj − uk )γik γkj − γij , (4.25) ∂k γij = γik γkj , k 6= i, j, ∂i γij = k ui − uj ψkα uj − uk ∂γij X ψiα ui − uk ψjα = (γik γkj + γik γkj + γik γkj α ∂t ψk1 ui − uj ψi1 uj − ui ψj1 N
k=1
−
γij ψiα γij ψjα − ). ui − uj ψi1 uj − ui ψj1
(4.26)
We also write down the following useful formulae: ψiγ g γβ = ui ψiβ , X d (qα − ) ψiα ψjα . (uj − ui )γij = 2 α
(4.27) (4.28)
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333
5. Proofs of Theorem 1 and 2 We begin with the proof of Theorem 1. So we assume here that G = 0 in the formulae (3.2), (3.3). Doing the infinitesimal B¨acklund tranform (3.2) (with G(t) = 0) we obtain 0
{tα (X), tβ (Y )}1 = η αβ δ 0 (X − Y ) β ∂wβ (t(Y )) 2 ∂w (t(Y )) ∂wβ (t(Y )) 2 + ∂ + ∂X +ε Y γ ∂tγ ∂tY ∂tγXX α ∂wβ (t(Y )) 3 ∂w (t(X)) ∂wα (t(X)) αγ 0 + ∂ δ (X − Y ) + + ∂X η γ Y ∂tY Y Y ∂tγ ∂tγX ∂wα (t(X)) 2 ∂wα (t(X)) 3 γβ 0 + ∂ + ∂ δ (X − Y ) + O(ε4 ), η X X ∂tγXX ∂tγXXX
(5.1)
where wα (t) = wα (t, tX ) is the function obtained from wα (v, vX ) by replacing v µ and their X-derivatives by tµ and by the correspondent X-derivatives of tµ . Recall that wα (t) α α depends not only on tα , but also on tα X , tXX , tXXX . More explicitly, we have αβ γ αβ wα (t) = F (1) tXXX + (Ft(1) + F (1) γ β cµ β cγ β t tX
+
cαβ tX t µ γ
X
F (1) tµ tβ X X
γ µ cαβ γ tXX tXX
+
F (1) tν tβ X X
µ αβ γ + 3 F (1) β cγµ ) tXX tX
γ µ ν cαβ γµ tX tX tXX
tX
αβ γ + Ft(1) tXX β cγ
γ µ ν (1) αβ (1) αβ αβ αβ γ µ + (F (1) + Ft(1) β ν cγµ + F β cγµν ) tX tX tX + (Ftβ tµ cγ β cγµ ) tX tX , (5.2) tX t
tX
where F (1) = F (1) (t, tX ) is defined in (4.16). Whenever there is no risk of confusion we will omit the arguments of a function henceforth. 0 In the Poisson bracket {tα (X), tβ (Y )}1 , the coefficient of ε2 δ (4) (X − Y ) is equal to zero, so it can be written as 0 {tα (X), tβ (Y )}1 = η αβ δ 0 (X − Y ) + ε2 hˆ αβ δ (3) (X − Y ) + rˆαβ δ 00 (X − Y ) + fˆαβ δ 0 (X − Y ) + pˆαβ δ(X − Y ) , (5.3) where hˆ αβ , rˆαβ , fˆαβ , pˆαβ are functions of tµ (X) and their X-derivatives. We have the following two lemmas on the coefficients hˆ αβ and rˆαβ : Lemma 1. The coefficients hˆ αβ have the expression 1 µν αβ hˆ αβ = η cµν . 12
(5.4)
Lemma 2. The coefficients rˆαβ are symmetric w.r.t. α and β, i.e., rˆαβ = rˆβα .
(5.5)
The proofs of Lemma 1 and Lemma 2 are similar to those of Lemma 5 and Lemma 6 which will be given below, however, the computation is much more simple, so we omit the proofs here.
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Lemma 3. For the Hamiltonians Hα,p defined in (2.45) the following identity holds true: Z Hα,p =
Z θα,p+1 (v(X))dX =
0 θα,p+1 (t(X))dX + ε2 δHα,p + O(ε4 ),
(5.6)
0 where δHα,p are defined by (3.10) and (3.11).
Proof. By using (2.40) we have Z Hα,p Z = Z = Z =
θα,p+1 (v(X)) dX
=
Z
∂θα,p+1 (t(X)) µ w dX + O(ε4 ) ∂tµ Z 2 ∂ θα,p+1 (t(X)) ν ∂F (1) (t(X), tX (X)) 2 tX dX + O(ε4 ) θα,p+1 (t(X)) dX + ε ∂tµ ∂tν ∂Tµ,0 Z ∂θα,p ν (1) σµ ρ σµ ρ ξ θα,p+1 (t(X)) dX + ε2 cγµν t tX t X Ftσ cρ tX + Ft(1) σ c X ρξ ∂tγ X σµ ρ + Ft(1) tXX dX + O(ε4 ). (5.7) σ cρ θα,p+1 (t(X)) dX − ε
2
X
Now formulas (4.7), (4.17), (4.20) and (4.21) amount to ∂θα,p ν (1) σµ ρ (1) σµ ρ ξ t t + F t t F σ c σ cρ t X t X X X X ρξ ∂tγ γ ψjγ 1 ∂θα,p ψiγ 2 ψi ψj1 2 2 = γij + σ + σ 2 σ i j j 4 2 2 ψ 24 ∂tγ ψi1 ψi1 ψj1 ψi1 j1 ! γ γ ψj ψ − 3 σi σj 3i − σi σj 2 , ψi1 ψj1 ψi1
cγµν
and Z
∂θα,p ν (1) σµ ρ t Fσ c t dX ∂tγ X tX ρ XX Z γ 1 ∂θα,p ψj ψjµ ψjν ν ψiσ ψkσ ψkµ ψkρ ρ = tX tXX 24 ∂tγ ψj1 σi ψk1 Z γ 1 ∂θα,p ψj ψjµ σj ψiσ ψkσ ψkµ ψkρ ρ = tXX 24 ∂tγ ψj1 σi ψk1 ! Z Z γ γ 1 1 ∂θα,p ψj ψjρ ρ ∂ ∂θα,p ψj ψjρ = tXX dX = − tρX tνX dX. 2 2 24 ∂tγ 24 ∂tν ∂tγ ψj1 ψj1 cγµν
So the Hamiltonians H α,p can be expressed in the form (5.6), (3.10) with
Bihamiltonian Hierarchies in 2D Topological Field Theory
χα,p+1;µν =
γij ∂θα,p 24 ∂tγ
335
γ 2 ψiµ ψiν ψiγ ψj1 ψjµ ψjν ψj ψjµ ψjν ψiγ + + 4 2 2 ψ ψi1 ψi1 ψj1 ψi1 j1
ψiµ ψjν ψjγ 3 ψiµ ψjν ψiγ 3 ψiν ψjµ ψiγ − − 3 3 2 ψ 2 ψi1 2 ψi1 2 ψi1 j1 ! ! γ ψiν ψjµ ψjγ ψ ψ 1 ∂ 1 ∂ ∂θα,p j jµ − − − 2 ψ 2 48 ∂tν ∂tγ 48 ∂tµ 2 ψi1 ψj1 j1
−
=
1 γ ∂θα,p 1 γ ξσ ζ ∂θα,p−1 wµν c c c − , γ 24 ∂t 24 ξζ ν σµ ∂tγ
γ ∂θα,p ψj ψjν 2 ∂tγ ψj1
!
p ≥ −1,
α are defined in (3.12). Lemma is proved. where θα,−2 = θα,−1 = 0 and wµν
Proof of Theorem 1. We first prove the formula (3.9) for the first Poisson bracket. From Lemma 1 and Lemma 2 we already know the expression of the coefficients hˆ αβ and the anti-symmetric part of the coefficients rˆαβ R in the formula (5.3). Now from the fact that the Casimirs of the first Poisson bracket v γ (X) dX have the expression Z
Z v γ (X) dX =
tγ (X) dX + O(ε4 )
(5.8)
we see that pˆαβ = 0
(5.9)
in the formula (5.3). So the anti-symmetry condition of the first Poisson bracket gives us the following relations: rˆαβ + rˆβα = 3 ∂X hˆ αβ , 2 ˆ αβ h , fˆβα − fˆαβ + 2 ∂X rˆαβ = 3 ∂X αβ 3 ˆ αβ 2 αβ ˆ ∂X f + ∂X h − ∂X rˆ = 0.
(5.10) (5.11) (5.12)
Identity (5.10) together with (5.5) gives us the expression for rˆαβ , while from the identity (5.12) it follows that 2 ˆ αβ h , fˆαβ = ∂X rˆαβ − ∂X
(5.13)
there is no integration constant because fˆαβ must depend on tγX or tγXX . So we get the expression for the coefficients fˆαβ and complete the proof of formula (3.9). The remaining part of the Theorem follows from Lemma 3. Theorem is proved We now proceed to prove Theorem 2. So we still assume here that G = 0 in the formulae (3.2), (3.3). Doing the same infinitesimal B¨acklund tranform (3.2) (with G(t) = 0) we obtain
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B. Dubrovin, Y. Zhang 0
γ {tα (X), tβ (Y )}2 = g αβ (t(X)) δ 0 (X − Y ) + 0αβ γ (t(X)) tX δ(X − Y ) " ∂0αβ ∂g αβ (t(X)) γ γ (t(X)) µ 2 0 w (t(X)) δ (X − Y ) + w (t(X)) tγX δ(X − Y ) −ε ∂tγ ∂tµ γ + 0αβ γ (t(X)) (∂X w (t(X)) δ(X − Y ))
∂wβ (t(Y )) ∂wβ (t(Y )) ∂wβ (t(Y )) 2 ∂wβ (t(Y )) 3 + ∂ + ∂Y + ∂Y ) Y γ ∂tγ ∂tY ∂tγY Y ∂tγY Y Y µ × (g αγ (t(X)) δ 0 (X − Y ) + 0αγ µ (t(X)) tX δ(X − Y )) ∂wα (t(X)) 2 ∂wα (t(X)) 3 ∂wα (t(X)) ∂wα (t(X)) + ∂ + ∂X + ∂X ) +( X γ ∂tγ ∂tX ∂tγXX ∂tγXXX µ 4 ×(g γβ (t(X)) δ 0 (X − Y ) + 0γβ µ (t(X)) tX δ(X − Y )) + O(ε ) . +(
(5.14)
In the Poisson bracket {tα (X), tβ (Y )}02 , the coefficient of ε2 δ (4) (X − Y ) is equal to −
∂wβ (t(X)) γα ∂wα (t(X)) γβ µβ γα µα γβ g + g = −Ft(1) g + Ft(1) g = 0, µ cγ µ cγ X X ∂tγXXX ∂tγXXX (5.15)
the last equality above is due to the associativity equation γα µα γβ cµβ γ c ν = cγ c ν ,
(5.16)
and the definition (2.22) of the intersection form. The coefficient of ε2 δ (3) (X − Y ) is equal to µα βγ ξ βµ αγ ξ αµ γβ αµ γβ ξ hαβ = 2 Ft(1) g + 3 Ft(1) tX − Ft(1) tX − 2 Ft(1) g tX µ c µ c µ ξ cγ µ cγ γξ g γξ g t X
X
X
αµ γβ µα γβ γα ξ γα ν ξ + Ft(1) + cβµ ) tX + Ft(1) + cµβ ) tX tX γ µ (c µ γ (c ξ g ξ g νξ g νξ g t t X
X X
αµ 3 Ft(1) µ cγ X
+
∂g γβ ξ βµ αγ ξ αµ γβ ξ t − Ft(1) 0ξ tX + Ft(1) 0ξ tX + S αβ , µ cγ µ cγ X X ∂tξ X
(5.17)
where βµ αγ αµ βγ βµ αγ αµ βγ g + F (1) g + Ft(1) + Ft(1) S αβ = (F (1) µ γ c µ γ c µ ξ cγ µ ξ cγ ξ g ξ g t t tX t X
−
4 F (1) tξ tµ X X
tX t X
cβµ γ
g
αγ
X X
X X
) tξXX
βµ αγ αµ βγ βµ αγ αµ βγ ξ = (Ft(1) + Ft(1) − F (1) g − F (1) g ) tXX . µ γ c µ γ c µ ξ cγ µ ξ cγ ξ g ξ g t t X X
X X
tX t X
tX t X
(5.18) Lemma 4.
S αβ = 0.
Proof. By using (2.22), (4.7) and (4.18) we have the identity βµ αγ αµ βγ g = Ft(1) g , Ft(1) µ γ cν µ ν cγ t t X X
(5.19)
X X
since both sides of the above identity are equal to −
1 X E ξ ψiα ψiβ ψiν ψiξ . 2 24 i σi2 ψi1
The lemma follows from the above identity immediately. Lemma is proved.
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337
Lemma 5. The coefficients hαβ defined in (5.17) have the expression h
αβ
1 = 12
1 µν αβ ∂ µν αβ (g cµ ) + cν cµ . ∂tν 2
(5.20)
Proof. Let’s rewrite 24 hαβ as the sum of Aαβ and B αβ , where µα βγ ξ βµ αγ ξ αµ γβ αµ γβ ξ + 3Ft(1) tX − Ft(1) tX − 2 Ft(1) g tX Aαβ = 24 2Ft(1) µ c µ c µ ξ cγ µ cγ g γξ g γξ g t X X X αµ γβ µα γβ γα ξ γα ν ξ + Ft(1) + cβµ ) tX + Ft(1) + cµβ ) tX tX , γ µ (c µ γ (c ξ g ξ g νξ g νξ g t t X
X X
and B αβ = 24
αµ 3 Ft(1) µ cγ X
∂g γβ ξ βµ αγ ξ αµ γβ ξ t − Ft(1) 0ξ tX + Ft(1) 0ξ tX µ cγ µ cγ X X ∂tξ X
.
By using the formulas given in Sect. 4 we have Aαβ =
γij σj 3 ψjγ ψiα g γβ − 3 ψiγ ψjβ g γα + ψiγ ψjα g γβ − ψjγ ψiβ g γα σi ψi1 ψj1
β ψjα ψiγ g γβ ψiα ψjγ g γβ ψj ψiγ g γα ψiα ψiγ g γβ − − 2 + 2 2 2 ψi1 ψj1 ψi1 ψi1 ψi1 ψ α ψiγ ψj1 g γβ . + i 3 ψi1
+ 2 γij
(5.21)
For B αβ , by using the formulas given in Sect. 4 and the following formulas 0αβ γ
=
1+d − qβ cαβ γ , 2
∂g αβ βα = 0αβ γ + 0γ , ∂tγ
(5.22)
we obtain γβ αµ γβ βµ γα (3 + 3 d − 4 qβ ) cαµ tξX B αβ = 24 Ft(1) µ γ cξ − 3 qγ cγ cξ + qγ cγ cξ X
= (3 + 3 d − 4 qβ )
γ ψiα ψiβ σj qγ ψiγ ψj α β β α ψ + ψ − 3 ψ ψ j i i j 2 σi ψi1 ψj1 ψi1
ψα ψβ = (3 + 3 d − 4 qβ ) i 2 i ψi1 σj d 1 α β (uj − ui ) γij + δij ψj ψi − 3 ψjβ ψiα + σi 2 ψi1 ψj1 α β ψ ψ σj (uj − ui ) γij α β ψj ψi − 3 ψjβ ψiα , (5.23) = (3 + 2 d − 4 qβ ) i 2 i + σi ψi1 ψj1 ψi1 above we have used formula (4.28). From formula (4.27) it follows that
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B. Dubrovin, Y. Zhang
24 hαβ = Aαβ + B αβ = 2 γij
ψjα ψiγ g γβ ψiα ψiγ g γβ − 2 ψi1 ψj1 ψi1
β ψ α ψjγ g γβ ψj ψiγ g γα ψiα ψiγ ψj1 g γβ −2 i 2 + + 2 3 ψi1 ψi1 ψi1
+ (3 + 2 d − 4 qβ )
!
ψiα ψiβ . 2 ψi1
(5.24)
On the other hand, for the right-hand side of (5.20) we have ∂ µν αβ (g µν cαβ µ ) + c ν cµ ∂tν ∂ ∂ µν αβ = − 2 ν (g µν cαβ (g µβ cαν µ )+4 µ ) + c ν cµ ∂t ∂tν αβ µβ αν = − 2 (1 + d − qµ − qν ) cµν ν cµ + 4 (1 + d − qµ − qβ ) cν cµ 2
αβ µν αβ + cµν cµν + 4 g µβ cαν ν cµ − 2 g µν
= (3 + 2 d − 4 qβ )
ψiα ψiβ ψiα ψiβ − 2 γ (u − u ) ij i j 2 ψi1 ψj1 ψi1
ui ψiα ψjβ ui ψiα ψiβ ψj1 ui ψiβ ψjα uj ψiα ψiβ − 2 γij − − + 2 2 3 ψi1 ψj1 ψi1 ψi1 ψi1 α ψjα ψiγ ψiα ψjγ ψiα ψiγ ψj1 ψi ψiγ + 4 γij − − + g γβ 2 2 3 ψi1 ψj1 ψi1 ψi1 ψi1 = 2 γij
ψjα ψiγ g γβ ψiα ψiγ g γβ − 2 ψi1 ψj1 ψi1
β ψ α ψjγ g γβ ψj ψiγ g γα ψiα ψiγ ψj1 g γβ −2 i 2 + + 2 3 ψi1 ψi1 ψi1
+ (3 + 2 d − 4 qβ )
!
ψiα ψiβ 2 ψi1
= 24 hαβ . Lemma is proved.
!
(5.25)
Knowing h (t) we can compute the symmetrized coefficient in front of δ 00 (X − Y ) using the skew-symmetry condition αβ
rγαβ (t) + rγβα (t) = 3 ∂γ hαβ .
(5.26)
The antisymmetrization of the same coefficients is given by the following: Lemma 6. Let’s denote r˜ αβ the coefficients before ε2 δ 00 (X − Y ) in the second Poisson 0 bracket {tα (X), tβ (Y )}2 of (5.14), then the following identity holds true: r˜ αβ − r˜ βα =
1 1 βµ γ αµ γ (d + 3 − 2 qβ ) cαν (d + 3 − 2 qα ) cβν γ cνµ tX − γ cνµ tX . 24 24 (5.27)
Bihamiltonian Hierarchies in 2D Topological Field Theory
339
Proof. From the expression (5.14) we have r˜ αβ = g γβ
β ∂wα γα ∂w γα ∂X γ −g γ + 3g ∂tX ∂tX
∂wβ ∂tγXX
2 − 6 g γα ∂X
∂wβ ∂tγXXX
∂wα µ ∂wα µ ∂wβ µ tX + 2 0βγ tX + 0αγ t γ γ µ µ ∂tXX ∂tXX ∂tγXX X µ 2 γβ ∂ 0γβ ∂X g ∂wα ∂wα ∂wβ µ tX µ αγ − 3 0µ ∂X + 3 + 3 . (5.28) t X ∂tγXXX ∂X 2 ∂tγXXX ∂X ∂tγXXX
+ 3 0γβ µ
By using the formulas given in Sect. 2 we get, through a long calculation, the following: ∂wα ∂wα ∂wα γβ γβ 2 − 3 g ∂ ∂ + 6 g X X ∂tγX ∂tγXX ∂tγXXX ∂wα µ ∂wα µ ∂wα + 3 0γβ tX + 0βγ tX + 3 0βγ tµX γ γ µ µ µ ∂X ∂tXX ∂tXX ∂tγXXX µ ∂ 0γβ ∂ 2 g γβ ∂wα ∂wα µ tX +3 + 3 γ ∂X 2 ∂tXXX ∂X ∂tγXXX − (the precedent sum with α and β changed)
r˜ αβ − r˜ βα = 2 g γβ
∂ ∂F (1) ∂ ∂F (1) ∂wα γβ γβ +2g − 3 g ∂X = 2 g ∂X ∂tγX ∂Tα,0 ∂tγ ∂Tα,0 ∂tγXX α ∂wα µ ∂wα 2 βγ ∂w + 6 g γβ ∂X t + 0 tµ + 3 0γβ γ γ γ µ µ ∂tXXX ∂tXX X ∂tXX X µ ∂ 0γβ ∂ 2 g γβ ∂wα ∂wα ∂wα µ tX µ βγ tX + 3 + 3 0µ ∂X + 3 γ γ ∂tXXX ∂X 2 ∂tXXX ∂X ∂tγXXX − (the precedent sum with α and β exchanged) γβ
(1) αµ ν αµ ν ρ αµ ν tX + Ft(1) Ft(1) + F t µ γ cνρ tX t µ γ cν µ tγ c ν XX X tX t t X (1) (1) γβ αµ ν αµ ν σ − g ∂X Ftµ tγ cν tX + Ftµ tγ cνσ tX tX X X X (1) αµ ν (1) (1) (1) γβ ν αµ ν αµ ν + g ∂X Ftµ cγν tX + 3 Ftµ tν cαµ γ tX + 3 Ftµ tν cγ tXX − Ftµ tγ cν tXX X X X X X X (1) αµ ν ρ (1) αµ ν αµ αµ ν −Ft(1) + 2 g γβ Ft(1) µ cγ µ cγν tX + F µ cνργ tX tX + F µ cνγ tXX tX tX α ∂wα µ ∂wα µ βγ ∂w βγ + 3 0γβ tµX t + 0 t + 3 0 ∂ X µ µ µ ∂tγXX X ∂tγXX X ∂tγXXX µ ∂ 0γβ ∂ 2 g γβ ∂wα ∂wα µ tX +3 + 3 ∂X 2 ∂tγXXX ∂X ∂tγXXX − (the precedent sum with α and β exchanged)
= 2 g γβ
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β 1 σj ψiα ψj ψiν ν 1 (1) αµ ν αµ ν σ u − = 2 g γβ Ft(1) c t + F c t t γ + t µ µ tγ ν j ij X tX tγ νσ X X 12 σj σi2 ψi1 ψj1 XX ! β ψiα ψjβ 1 σi ψiα ψj ∂X 2 uj γij + + (ui + uj ) γij 2 24 σj ψi1 ψj1 ψi1 αµ ν αµ ν σ + ∂X g γβ Ft(1) tX + Ft(1) µ γ cνσ tX tX µ tγ c ν t X X X (1) αµ ν αµ ν αµ ν αµ ν + g γβ ∂X Ft(1) tX + 3 Ft(1) tXX − Ft(1) tXX µ cγν tX + 3 F µ ν cγ µ ν cγ µ γ cν tX t t t X X X X X (1) αµ ν ρ (1) αµ ν αµ αµ ν −Ft(1) + 2 g γβ Ft(1) µ cγ µ cγν tX + F µ cνργ tX tX + F µ cνγ tXX tX tX α ∂wα µ ∂wα µ βγ ∂w βγ + 3 0γβ t + 0 tµX t + 3 0 ∂ X µ µ µ ∂tγXX X ∂tγXX X ∂tγXXX µ ∂ 0γβ ∂ 2 g γβ ∂wα ∂wα µ tX +3 + 3 ∂X 2 ∂tγXXX ∂X ∂tγXXX − (the precedent sum with α and β exchanged) (1) αµ ν αµ ν σ = 2 g γβ Ft(1) µ tγ cν tX + F µ γ cνσ tX tX tX t ! β ψiα ψjβ 1 σ ∂ σi ψiα ψj t 2 uj γij + (ui + uj ) γij + 2 24 X ∂tσ σj ψi1 ψj1 ψi1 (1) αµ βγ ρ ν σ αµ βγ ρ σ − 2 qγ Ft(1) t t t + F c t t µ γ cνσ cρ γ cσ µ X X ρ X X X t tX t X X n (1) αµ βγ ρ ν σ αµ βγ ρ σ + 3 (1 + d) − 4 qβ Ftµ tγ cνσ cρ tX tX tX + Ft(1) c c t t γ µt σ ρ X X X
X
X
(1) αµ βγ σ ν αµ βγ σ ν αµ βγ σ ν +qβ Ft(1) cσ tX tX − 3 qβ Ft(1) cσν tX tX µ ν cγ µ cγν cσ tX tX − 3 qβ F µ cγ tX t X X o αµ γβ σ −2 qβ Ft(1) cσ tX − (the precedent sum with α and β exchanged) µ cγ ( ψiα ψjβ ψα ψβ 1 (uj − uk ) γik γkj σk 2 + 3 (ul − uj ) γij γjl σi i 2l = 24 ψi1 ψk1 ψi1 ψj1
+ 2 (ui − uj ) γij γil σi + (uj − uk ) γik γkj σi +d γik σk γij + 24
(
ψiα ψjβ ψl1 4 ψi1
ψiα ψjβ 3 ψi1
− (ui + uj ) γij γil σi
ψjβ ψlα 3 ψi1
σj2 ψjα ψlβ 3 σi ψj1 ) ψjα ψiβ
+ 2 (uj − ul ) γij γjl
ψiα ψjβ ψiα ψkβ + 2 d γ σ − d γij σi ij i 2 ψ 2 3 ψi1 ψi1 ψj1 ψi1 k1
3 qβ σi
ψiα ψjβ 3 ψi1
+ qβ − 3 (1 + d)
σj
ψiα ψjβ 2 ψj1 ψi1
+ σi
ψiα ψjβ
!
3 ψi1
ψjα ψjβ ψjα ψiβ ψiα ψiβ ψj1 − 4 q σ − 2 q σ + 5 qβ − 6 (1 + d) σj β i β j 2 4 2 ψj1 ψi1 ψi1 ψi1 ψj1 ) σj2 ψjα ψjβ −2 qβ − (the precedent sum with α and β exchanged) 3 σi ψj1
Bihamiltonian Hierarchies in 2D Topological Field Theory
=
1 24
d − qβ 2
+γij σi
γij σj
ψiα ψjβ 3 ψi1
(
+ γij σi
ψiα ψjβ 2 ψi1
ψj1
ψiα ψjβ 3 ψi1
341
+ 3 γij σi
ψiα ψjβ 2 ψi1 ψj1
σj2 ψjα ψjβ − 2 γij 3 σi ψj1
− 2 γij σi !
ψiα ψiβ ψj1 4 ψi1
) ψiα ψjβ ψjα ψiβ ψiα ψkβ γik σk 2 + 2 γij σi − γij σi 2 3 ψi1 ψk1 ψi1 ψj1 ψi1 ( ! ψiα ψjβ ψiα ψjβ ψiα ψjβ γij 3 qβ σi + qβ − 3 (1 + d) σj + σi + 3 2 3 24 ψj1 ψi1 ψi1 ψi1
d + 24
ψjα ψjβ ψjα ψiβ ψiα ψiβ ψj1 − 4 q σ − 2 q σ + 5 qβ − 6 (1 + d) σj β i β j 2 4 2 ψj1 ψi1 ψi1 ψi1 ψj1 ) σj2 ψjα ψjβ −2 qβ − (the precedent sum with α and β exchanged) 3 σi ψj1 ! ψiα ψjβ ψiα ψjβ ψiα ψiβ ψj1 ψiα ψiβ 1 2 qβ − d − 3 γij σi + − − 2 = 2 3 4 24 ψi1 ψj1 ψi1 ψi1 ψj1 ψi1 − (the precedent sum with α and β exchanged) 1 1 βµ γ αµ γ (d + 3 − 2 qβ ) cαν (d + 3 − 2 qα ) cβν = γ cνµ tX − γ cνµ tX . 24 24
Lemma is proved.
(5.29)
Proof of Theorem 2. Let’s denote
e β,0 = 1 H 24
Z β µ ν wµν tX tX dX,
Fγβ = η ββ
0
∂2F , ∂tβ 0 ∂tγ
β where wµν are defined in (3.12), then from Lemma 3 we see that the equations in (3.8) with p = 0 can be written as
! e β,0 1 ˆ αγ ∂tα γ αβ 2 αγ δ H αγ 2 β β ˆ = cγ (t) tX + ε ∂X η + h ∂ X Fγ + ∂ X h ∂ X Fγ ∂Tβ,0 δtγ 2 γ γ µ ν 2 ˆ αβ = cαβ bγµ (t) tγX tµXX + aˆ αβ γ (t) tX + ε γµν (t) tX tX tX + γ 4 + pˆαβ (5.30) γ (t) tXXX + O(ε ),
αβ ˆ αβ where the coefficients bˆ αβ γµ (t), a γµν (t), pˆγ (t) have the expression
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1 αβ µν c c , 12 µν γ 1 β β η ασ (∂σ ∂ν wγµ = + ∂σ ∂γ wµν 72 β β β β + ∂σ ∂µ wγν − 2 ∂µ ∂ν wγσ − 2 ∂µ ∂γ wνσ − 2 ∂ν ∂γ wµσ )
pˆαβ γ = aˆ αβ γµν
β ασ β ασ β + η ξζ (6 cασ ξζ cσγµν + 3 cξζγ cσµν + 3 cξζµ cσγν
bˆ αβ γµ
β ασ β ασ β ασ β +3 cασ ξζν cσγµ + cξζγµ cσν + cξζγν cσµ + cξζµν cσγ ) , 1 β β β η ασ (−2 ∂γ wµσ = + ∂σ wγµ − ∂µ wγσ ) 12 3 ασ β 1 ασ β β +η ξζ (3 cασ c c c + c + c ) , ξζ σγµ 2 ξζγ σµ 2 ξζµ σγ
(5.31)
(5.32)
α defined by (3.12), and hˆ αβ are defined in (5.4). On the other hand, from the with wβγ bihamiltinian relation (2.82) we have Z 0 ∂tα 1+d − qβ ) = {tα , tβ (X) dX}2 + O(ε4 ), (5.33) ( 2 ∂Tβ,0 αβ which together with (5.30) leads to the expression for the coefficients pαβ γ (t), aγµν (t), bαβ γµ (t) in the formula (3.14)
1 − µβ pˆαβ = γ (t), 2 1 bαβ − µβ bˆ αβ γµ (t) = γµ (t). 2 pαβ γ (t)
aαβ γµν (t)
=
1 − µβ 2
aˆ αβ γµν , (5.34)
The expression (3.15) of the coefficients hαβ follows from Lemma 5, formulas (3.19)– (3.21) are obtained by using Lemma 6 and the anti-symmetry condition of the second Poisson bracket. In fact, if we denote r˜ αβ , f˜αβ and p˜αβ the coefficients before ε2 δ 00 (X − Y ), ε2 δ 0 (X − Y ) and ε2 δ(X − Y ) in the second Poisson bracket {tα (X), tβ (Y )}02 respectively, then the antisymmetry condition of the second Poisson bracket gives us r˜ αβ + r˜ βα = 3 ∂X hαβ , 3 αβ 2 αβ h − ∂X r˜ = p˜αβ + p˜βα . ∂X f˜αβ + ∂X
(5.35) (5.36)
Formula (3.19) follows immediately from (5.35) and Lemma 6. From (5.30) it follows that ! e β,0 1 1+d αβ αγ δ H αγ 2 β αγ β − qβ ∂X η + h ∂ X Fγ + ∂ X h ∂ X Fγ . p˜ = 2 δtγ 2 (5.37) So from (5.36) and the above expression of p˜αβ we obtain 2 αβ f˜αβ = −∂X h + ∂X r˜ αβ ! e β,0 1 1+d αγ δ H αγ 2 β αγ β − qβ + + h ∂ X Fγ + ∂ X h ∂ X Fγ . η 2 δtγ 2
(5.38)
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343
which leads to formula (3.20) and (3.21). We have thus verified the formula (3.14). The remaining part of the theorem follows from (2.82). Theorem is proved. Proof of Proposition 3. For the correction of the expression of the first Possion bracket, 2 G in the identity (5.1), then a direct calculation let’s replace the functions wα by ∂X∂∂T α,0 aided by the anti-symmetry condition of the Poisson bracket gives the expression for a˜ αβ , b˜ αβ , e˜αβ . For the correction of the expression of the second Poisson bracket, from 2 G we can easily get the expression for the identity (5.14) with wα replaced by ∂X∂∂T α,0 αβ αβ the coefficients a and b , however, it’s not easy to get the simplified expression for the coefficients q αβ and eαβ in this way. We use instead the relation 1−d + qβ {v α (X), Hβ,0 }1 = {v α (X), Hβ,−1 }2 (5.39) 2 R R (v(X)) dX and the infinitesimal B¨acklund transwith Hβ,−1 = vβ (X)dX, Hβ,0 = ∂F ∂v β (X) form t α = v α + ε2
∂2G , ∂X ∂Tα,0
(5.40)
to get the expression for the coefficient q αβ , then by using the anti-symmetry condition of the Poisson bracket we get the expression for the coefficient eαβ . Proposition is proved.
6. Genus One Gromov–Witten Invariants and G-Function of a Frobenius Manifold In the paper [21] Getzler studied recursion relations for the genus one Gromov–Witten invariants of smooth projective varieties. He derived a remarkable system of linear differential equations for a generating function G = G(t2 , . . . , tn ) of these invariants. The system can be written in the following form: X ∂2G ∂2G zα1 zα2 zα3 zα4 3 cµα1 α2 cνα3 α4 µ ν − 4 cµα1 α2 cνα3 µ α4 ν ∂t ∂t ∂t ∂t 1≤α1 ,α2 ,α3 ,α4 ≤n
∂G ∂G 1 + 2 cµα1 α2 α3 cνα4 µ ν + cµα1 α2 α3 cνα4 µν ∂tν ∂t 6 1 µ 1 + cα1 α2 α3 α4 cνµν − cµα1 α2 ν cνα3 α4 µ = 0. 24 4
− cµα1 α2 cνα3 α4 µ
(6.1)
The l.h.s. must be equal to zero identically in z1 , . . . , zn . The notations for the coefficients αβγδ are defined in (3.13). Now we solve this system for an arbitrary semisimple cα βδ , c Frobenius manifold. Proof of Theorem 3.. Let us rewrite the system (6.1) in the canonical coordinates. At this end we first do a linear change of the indeterminates zα 7→ wi :=
n X ψiα α=1
ψi1
zα .
(6.2)
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Instead of the partial derivatives of G(t) and of F (t) we substitute in (6.1) the corresponding covariant derivatives. For example, ∂2G → ∇i ∇j G, ∂tλ ∂tµ cδαβγ → ∇i ∇j ∇k ∇l F, etc. Here ∇ is the Levi–Civita flat connection for the metric < , > written in the curvilinear coordinates ui . Recall that the metric becomes diagonal in the canonical coordinates n X
< , >=
2 ψi1 d ui 2 .
(6.3)
i=1
The only nontrivial Christoffel coefficients of the connection are 0iij = γij
ψj1 ψi1 , 0jii = −γij , i 6= j, ψi1 ψj1
0iii = −
X k6=i
γik
ψk1 . ψi1
(6.4)
(6.5)
From the definition of the canonical coordinates we have ∇i ∇j ∇k F = δik δjk .
(6.6)
This simplifies the computation. Finally we obtain for the polynomial (6.1) in w1 , . . . , wn the following structure: 1) The coefficient in front of wi4 is equal to −
∂2G + Pii . ∂ui 2
2) The coefficient in front of wi3 wj for i < j is equal to −4
∂2G + 4Pij . ∂ui ∂uj
3) The coefficient in front of wi2 wj2 for i < j is equal to 6
∂2G − 6Pij . ∂ui ∂uj
4) All other coefficients of the polynomial (6.1) vanish. Here Pij = Pji is a complicated expression in u1 , . . . , un , ψ11 , . . . , ψn1 , γ12 , . . . , γn−1 n .
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From the above structure of the coefficients we immediately derive the uniqueness part of Theorem 4. Indeed, the general solution of the corresponding linear homogeneous system ∂2G =0 ∂ui ∂uj is G=
X
c i ui + c 0
i
for arbitrary constant coefficients. The quasihomogeneity equation (3.29) in the canonical coordinates reads n X j=1
uj
∂G = γ. ∂uj
(6.7)
Hence c1 = ... = cn = 0 and G = const. To find the first derivatives of G we differentiate (6.7) w.r.t. ui . This gives X ∂2G ∂G =− uj , i = 1, ..., n. ∂ui ∂ui ∂uj j So
X ∂G =− uj Pij . ∂ui j
After tedious calculations we obtain the following formula 4 2 2 2 4 X γij (uj − ui ) ψi1 − 10ψi1 ψj1 + ψj1 ∂G = 24 2 ψ2 ∂ui ψi1 j1 j 2 2 X X (ψi1 + ψj1 ) 1 X 1 + γij Vik ψk1 − Vjk ψk1 ψ ψ ψ ψ i1 j1 i1 j1 j k6=j k6=i X ψi1 ψj1 + γij − , ψ ψ i1 j1 j
(6.8)
where, we recall, Vij = (uj − ui ) γij . Using that ψi1 is an eigenvector of V we rewrite the formula in the following way: 1 X ψk1 ∂G 1 X Vij2 ψi1 = − γik − . (6.9) ∂ui 2 ui − uj 24 ψk1 ψi1 j6=i
k6=i
Using (2.19) and (4.9) we recognize in the r.h.s. the derivative 1 ∂ log(ψ11 ...ψn1 ) . log τI − ∂ui 24 It remains to observe that det
∂tα = ψ11 ...ψn1 ∂ui
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up to an inessential constant. One can easily check that X ∂G = 0, ∂ui i so
∂G = 0. ∂t1
The formula (3.30) is proved. Let us derive the formula (3.32) for the constant γ. We have n X
ui ∂i log τI =
i=1
n 1 X ui Vij2 1X 2 1 1X 2 = Vij = − Trace V 2 = − µα . 2 ui − uj 2 i<j 4 4 α=1
j6=i
The second term in the formula for G gives n X
ui ∂i log(ψ11 ...ψn1 ) =
i=1
−1 ψj1
n X
j=1
s
But ψj1 = and
n X
n X
ui ∂i ψj1 .
i=1
∂tn ∂uj
ui ∂i tn = (1 − d)tn for d 6= 1,
i=1 n X
ui ∂i tn = rn for d = 1
i=1
(the Euler identity). So
n X i=1
d ui ∂i ψj1 = − ψj1 . 2
This proves the formula (3.32). Theorem is proved.
Definition. The function (3.30) is called G-function of the Frobenius manifold. We begin our examples with the case n = 2. In the 2-dimensional case, we write the free energy F in the form 1 F = (t1 )2 t2 + f (t2 ). 2 Getzler’s equations (6.1) are reduced to 48f (3)
∂2G ∂G − 24f (4) 2 − f (5) = 0 ∂t2 ∂t2 ∂t
(cf. [25]). For the free energy F =
1 12 2 (t ) t + c (t2 )h+1 , 2
(6.10)
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347
where c is an arbitrary non-zero constant, the G-function is G=−
1 (2 − h)(3 − h) log(t2 ). 24 h
Particularly, the G-function vanishes for the A2 topological minimal model (the case h = 3). The constant γ equals d (1 − 3 d) γ= 24 since d = 1 − h2 . For the free energy of the CP 1 model F =
2 1 12 2 (t ) t + et 2
the G-function reads G=−
1 2 t . 24
The constant is γ=−
1 . 12
Observe that the G-function is analytic everywhere on the Frobenius manifold only for d = 13 (the A2 topological minimal model) and for d = 1, i.e., for the CP 1 topological sigma model. Let us consider now examples with n = 3. We will take the list of examples of Frobenius manifolds with good analytic properties from [11]. 1) For the polynomial free energy related to A3 , F =
1 12 3 1 1 22 1 22 32 1 35 (t ) t + t (t ) + (t ) (t ) + (t ) , 2 2 4 60
(6.11)
we have G = 0, γ = 0; 2) For the polynomial free energy related to B3 , F =
1 37 1 12 3 1 1 22 1 23 3 1 22 33 (t ) t + t (t ) + (t ) t + (t ) (t ) + (t ) , 2 2 6 6 210
we have G=−
1 log(2 t2 − 3 (t3 )2 ), 48
γ=−
1 . 72
3) For the polynomial free energy related to the symmetry group of icosahedron, F =
1 1 12 3 1 1 22 1 23 32 1 22 35 (t ) t + t (t ) + (t ) (t ) + (t ) (t ) + (t3 )11 , 2 2 6 20 3960
we have G=−
1 log(t2 − (t3 )3 ), 20
γ=−
3 . 100
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4) For the free energy of the CP 2 model, F =
1 1 2 3 1 1 2 2 X (0) (t3 )3 k−1 kt2 (t ) t + t (t ) + e , Nk 2 2 (3 k − 1)!
(6.12)
k≥1
where Nk(0) are the number of rational curves of degree k on CP 2 which meet 3 k − 1 generic points, for example, N1(0) = N2(0) = 1, N3(0) = 12, N4(0) = 620. The G-function has the form G=−
t2 X (1) (t3 )3 k kt2 + e , Nk 8 (3 k)! k≥1
φ000 − 27 ∂G , = ∂t2 t2 =z,t3 =1 8(27 + 2 φ0 − 3 φ00 )
3 γ=− , 8
(6.13)
where φ is defined by (1.21), and Nk(1) are the number of elliptic plane curves of degree k which meet 3 k generic points, for example, N1(1) = N2(1) = 0, N3(1) = 1, N4(1) = 225. 5) For the free energy F =
1 12 3 1 1 22 (t ) t + t (t ) + (t2 )4 φ(t3 − 2 r log(t2 )) 2 2
(6.14)
(here d = 1, r > 0) we obtain a solution of WDVV with good analytic properties only for r = 23 , 1 or 21 [11]. These solutions correspond to extended affine Weyl groups of type A˜ 2 , C˜ 2 , G˜ 2 respectively [13]. For all of them γ = −1/16. Particularly, for A˜ 2 , 1 + ez , 24
(6.15)
1 a2 2 z + a ez + e , 48 2
(6.16)
φ(z) = − 1 3 then G = − 24 t . For C˜ 2
φ(z) = −
where a is an arbitrary non-zero constant, the G-function is G=−
3 1 1 3 t − log(16 a et − (t2 )2 ). 24 48
Finally, for G˜ 2 φ(z) = −
2 1 3 9 4z + e z + e2 z + e , 72 3 2 16
the G-function is G=−
3 1 3 1 t − log(12 et − t2 ). 24 12
(6.17)
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349
6) We now take the free energy F =
1 24 1 12 3 1 1 22 (t ) t + t (t ) − (t ) φ(t3 ), 2 2 16
(6.18)
where φ(z) satisfies the Chazy equation φ000 = 6 φ φ00 − 9 (φ0 )2 ,
(6.19)
(here d = 1, r = r3 = 0). Then the G function can be obtained from the equations 1 ∂G = − 2, 2 ∂t 8t
∂G 1 = − φ(t3 ), 3 ∂t 4
γ=−
1 . 16
(6.20)
Particularly, for the case φ(t3 ) = 8 π i E2 (t3 ) = 4
d log η(t3 ), dt3
(6.21)
where η(τ ) is the Dedekind function, E2 (τ ) is the second Eisenstein series (see [11]) we obtain i h 1 G = − log η(t3 ) (t2 ) 8 .
(6.22)
We see that, for n = 3, only on the Frobenius manifold (6.11) (the free energy of the A3 topological minimal model), and on the Frobenius manifold (6.15) related to the extended affine Weyl group A˜ 2 the G-function are manifestly analytic everywhere. For the CP 2 sigma model the G-function is regular on the open subset where 27 + 2 φ0 − 3 φ00 6= 0.
(6.23)
From equations of associativity for the function (6.12) it can be seen (see [4]) that in the points x0 where 3 φ00 (x0 ) − 2 φ0 (x0 ) − 27 = 0
(6.24)
the series (6.13) diverges. Analytic properties of the G-function (6.13) deserve a separate investigation. Remark. In the cases B3 , H3 , B˜ 2 , G˜ 2 the G-function has logarithmic branching on the part of the nilpotent locus of the Frobenius manifold, where ui = uj for some i 6= j. The coefficients of our hierarchy will have singularities in these points. Probably, appearance of these singularities suggests not to select these Frobenius manifolds for a construction of a physically consistent model of 2D TFT.
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7. Some Examples We now compare the dispersion expansions of some well known examples of biHamiltonian integrable systems with those given in Theorem 1 and Theorem 2. Example 1. Let’s start with the detailed consideration of the simplest example of KdV hierarchy. We take the Lax operator in the form L=
1 (ε ∂X )2 + u(X). 2
(7.1)
Then the two compatible Poisson brackets related to this operator is given by {u(X), u(Y )}1 = δ 0 (X − Y ), {u(X), u(Y )}2 = u(X) δ 0 (X − Y ) +
(7.2) 2
1 ε uX (X) δ(X − Y ) + δ 000 (X − Y ).(7.3) 2 8
(They are derived from the formulae (7.15) and (7.16) in Example 2.) Starting from the Casimir Z (7.4) H−1 = u(X)dX of the first Poisson bracket we can construct a hierarchy of commuting Hamiltonians Hp by using the following recursion relation 1 {u(X), Hp−1 }2 = + p {u(X), Hp }1 , (7.5) 2 i.e.,
1 ε2 3 δHp−1 δHp 1 u(X)∂X + uX (X) + ∂X = + p ∂X . (7.6) 2 8 δu 2 δu Note that the factor 21 + p does not appear in the usual recursion relation for the KdV hierarchy, we use this factor here to meet the topological recursion relation of the A1 topological minimal model. Let’s list the first four Hamiltonians Z Z 1 H−1 = u(X)dX, H0 = u(X)2 dX, 2 Z 1 2 1 3 2 u(X) − ε uX (X) dX, H1 = 6 24 Z 1 2 1 4 1 4 2 2 H2 = u(X) − ε u(X) uX (X) + ε uXX (X) dX. (7.7) 24 24 480 The KdV hierarchy is then given recursively as ∂u = uX , ∂T 0 1 2 ∂u ε uXXX , = u uX + ∂T 1 12 1 1 2 2 ∂u 1 ∂u −1 −1 uX ∂ X + u + ε ∂ X = ( + p) . ∂T p 2 2 8 ∂T p−1
(7.8)
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Let’s note that each flow of the KdV hierarchy can be written as a polynomial in ε2 . The parameter ε can be introduced to the usual KdV hierarchy through the rescaling X 7→ ε X, T p 7→ ε T p . We write down explicitly the ε0 and ε2 terms in the hierarchy ∂u = uX , ∂T 0 1 2 ∂u ε uXXX , = u uX + ∂T 1 1 12 ∂u = {u(X), Hp(0) }1 + ε2 {u(X), Hp(1) }1 + O(ε4 ), ∂T p where
Z Hp(0) =
(7.9)
u(X)p+2 dX (p + 2)!
(7.10)
and (1) H−1 = H0(1) = 0, Z 1 u(X)p−1 (1) uX (X)2 dX. − Hp = 24 (p − 1)!
(7.11)
We now take the free energy to be F =
1 13 (t ) , 6
(7.12)
in this case the G-function G = 0. Plugging this free energy into Theorem 1 and Theo0 with u, T p , Hp(0) , Hp(1) respectively, we obtain, rem 2, and identify t1 , T p,1 , H1,p , δH1,p 4 modulo O(ε ), the above described KdV hierarchy and its bihamiltonian structure. Example 2. More generally, let’s consider the differential operators L = (ε ∂)N +1 + uN (X)(ε ∂)N −1 + · · · + u1 (X), where ∂ =
∂ ∂X .
(7.13)
For any pseudo-differential operator Z of the form Z = (ε ∂)−1 Z1 + (ε ∂)−2 Z2 + · · · + (ε ∂)−N ZN ,
(7.14)
define the following two Hamiltonian mappings [2]: H1 : Z 7→ [Z, L]+ ,
H2 : Z 7→ L(ZL)+ − (LZ)+ L +
1 [L, N +1
and the corresponding Poisson brackets Z δf δg ˜ {f , g} ˜ i = Res(Hi ( ) )dx, δL δL Z
for the functionals f˜ = where
Z f dx,
g˜ =
gdx,
i = 1, 2,
Z
X
Res[Z, L] dX], (7.15)
(7.16)
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δf X δf = (ε ∂)−i , δL δui N
i=1
δg X δg = (ε ∂)−i . δL δui N
(7.17)
i=1
In the case of N = 2, if we define t 1 = u1 −
1 0 εu , 2 2
t2 =
1 u2 , 3
(7.18)
then the first Poisson bracket is given as follows: {t1 (X), t1 (Y )}1 = 0, {t1 (X), t2 (Y )}1 = δ 0 (X − Y ), {t2 (X), t2 (Y )}1 = 0,
(7.19) (7.20)
and the second Poisson bracket is given as {t1 (X), t1 (Y )}2 = −6 (t2 )(X)2 δ 0 (X − Y ) − 6 t2 (X) (t2 )0 (X) δ(X − Y ) 9 15 2 0 − ε2 (t ) (X) δ 00 (X − Y ) + (t2 )00 (X) δ 0 (X − Y ) 4 4 4 5 2 1 ε + t (X)δ (3) (X − Y )+ (t1 )(3) (X)δ(X − Y ) − δ (5) (X − Y ), 2 2 6 1 {t1 (X), t2 (Y )}2 = t1 (X) δ 0 (X − Y ) + (t1 )0 (X) δ(X − Y ), 3 2 2 1 2 2 0 {t (X), t (Y )}2 = t (X) δ (X − Y ) + (t2 )0 (X) δ(X − Y ) 3 3 2ε2 (3) + δ (X − Y ). (7.21) 9 The integrable hierarchy has the form ∂tα = {tα (X), Hβ,p }1 , ∂T β,p
(7.22)
where the Hamiltonians Hβ,p are recursively defined by 1−d + p + qβ {tα (X), Hβ,p }1 (7.23) {tα (X), Hβ,p−1 }2 = 2 R with Hβ,−1 = tβ (X)dX. Up to the ε2 terms, the above Poisson brackets and the integrable hierarchy coincide with the Poisson brackets and the integrable hierarchy given in Theorem 1 and Theorem 2 with the free energy defined by F =
1 12 2 3 24 (t ) t − (t ) , 2 8
(7.24)
and with the G-function G = 0. This is the primary free energy of the A2 topological minimal model of [7]. In the case of N = 3, if we define 1 ε ε2 00 u , t1 = u1 − u23 − u02 + 8 2 12 3
t2 = u2 − ε u03 ,
t 3 = u3 ,
(7.25)
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then the Poisson brackets defined by (7.15), (7.16) and the integrable system given in the form of (7.22) and (7.23) coincide, modulo ε4 , with the Poisson brackets and the integrable system given in Theorem 1 and Theorem 2 with the free energy defined by F =
1 22 32 1 1 1 22 1 12 3 t (t ) + (t ) t − (t ) (t ) + (t3 )5 , 8 8 64 3840
(7.26)
and with the G-function G = 0. This is the primary free energy of the A3 topological minimal model [ibid]. Formulae in (7.25) coincide with formulae in (4.38) of [6]. Example 3. The explicit bihamiltonian structure related to the Lie algebra of type B2 is given, for example, in [1] in the following form: {u1 (X), u1 (Y )}1 {u1 (X), u2 (Y )}1 {u2 (X), u1 (Y )}1 {u2 (X), u2 (Y )}1
= = = =
2 u2 (X) δ 0 (X − Y ) + u02 (X) δ(X − Y ) − ε2 δ (3) (X − Y ), 2 δ 0 (X − Y ), 2 δ 0 (X − Y ), 0; (7.27)
{u1 (X), u1 (Y )}2 = 2u2 (X) u1 (X) δ 0 (X − Y ) + u02 (X) u1 (X) δ(X − Y ) 3 + u2 (X) u01 (X) δ(X − Y ) − ε2 [ u02 (X)2 δ 0 (X − Y ) 2 3 + 6 u2 (X) u02 (X) δ 00 (X − Y ) + u01 (X) δ 00 (X − Y ) 2 1 + 4 u2 (X) u002 (X) δ 0 (X − Y ) + u02 (X) u002 (X) δ(X − Y ) 2 3 00 0 + u1 (X) δ (X − Y ) + 2 u2 (X)2 δ (3) (X − Y ) 2 + u1 (X) δ (3) (X − Y ) + u2 (X) u(3) 2 (X) δ(X − Y ) 1 (3) + u1 (X) δ(X − Y )] + ε4 [8u002 (X) δ (3) (X − Y ) 2 00 0 (4) + 7 u(3) 2 (X) δ (X − Y ) + 5u2 (X) δ (X − Y ) 0 (5) + 3u(4) 2 (X)δ (X − Y ) + 2u2 (X)δ (X − Y ) 1 1 (X) δ(X − Y )] − ε6 δ (7) (X − Y ), + u(5) 2 2 2 1 {u1 (X), u2 (Y )}2 = 2u1 (X) δ 0 (X − Y ) + u01 (X) δ(X − Y ) 2 − ε2 u02 (X) δ 00 (X − Y ) + 2 u2 (X) δ (3) (X − Y ) + ε4 δ (5) (X − Y ), 3 {u2 (X), u1 (Y )}2 = 2u1 (X)δ 0 (X − Y ) + u01 (X)δ(X − Y ) − ε2 [u(3) 2 (X)δ(X − Y ) 2 0 00 (3) 00 0 + 5u2 (X) δ (X − Y ) + 2 u2 (X) δ (X − Y ) + 4 u2 (X) δ (X − Y )]
+ ε4 δ (5) (X − Y ), 1 5 {u2 (X), u2 (Y )}2 = u2 (X)δ 0 (X −Y ) + u02 (X)δ(X −Y )− ε2 δ (3) (X −Y ), 2 2
(7.28)
here we note that the above coordinates u1 , u2 should be the coordinates u1 , u0 respectively in [1], and there is a sign difference between the above first Poisson bracket and
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that of [1]. We now compare the above Poisson brackets with the Poisson brackets given by Theorem 1 and Theorem 2 with the free energy related to B2 . For this, let F =
1 12 2 1 25 (t ) t + (t ) , 2 15
(7.29)
1 log(t2 ), and the first and second Poisson then the G-function is given by G = − 48 brackets of Theorem 1 and Theorem 2 are given by
{t1 (X), t1 (Y )}1 = {t2 (X), t2 (Y )}1 = O(ε4 ), {t1 (X), t2 (Y )}1 = δ 0 (X − Y ) 20 000 ε2 (t ) (x) 00 + δ (X − Y ) − δ (X − Y ) + O(ε4 ); 24t2 (x) t2 (x) 3
(7.30)
2
{t1 (X), t1 (Y )}2 = 2 t2 (X) δ 0 (X − Y ) + 3 t2 (X) (t2 )0 (X) δ(X − Y ) 2
+ε
2
(t1 )0 (X) δ 0 (X − Y ) 2
32 t2 (X)
2
−
(t1 )0 (X) (t2 )0 (X) δ(X − Y ) 3
32 t2 (X)
2
29 (t2 )0 (X) δ 0 (X − Y ) 13 t2 (X) (t2 )0 (X) δ 00 (X − Y ) + 24 4 (t1 )0 (X) (t1 )00 (X) δ(X − Y ) 25 t2 (X) (t2 )00 (X) δ 0 (X − Y ) + + 2 12 32 t2 (X) +
2
5 (t2 )0 (X) (t2 )00 (X) δ(X − Y ) 13 t2 (X) δ (3) (X − Y ) + + 8 12 2 2 (3) t (X) (t ) (X) δ(X − Y ) + , 2 (t1 )0 (X) δ(X − Y ) {t1 (X), t2 (Y )}2 = t1 (X) δ 0 (X − Y ) + 4 1 0 00 1 2 0 ) (X) δ (X − Y ) − (t t (X) (t ) (X) δ 00 (X − Y ) + ε2 + 2 2 24 t (X) 24 t2 (X) t1 (X) δ (3) (X − Y ) , − 24 t2 (X) t2 (X) δ 0 (X − Y ) (t2 )0 (X) δ(X − Y ) + 2 4 3 ε2 δ (3) (X − Y ) + . 16
{t2 (X), t2 (Y )}2 =
(7.31)
Now if we relate the variables u1 , u2 to the variables t1 , t2 by the following relation: 1 u001 1 7 00 u01 u02 t1 = u1 − u22 + ε2 u2 + − (7.32) , t 2 = u2 , 4 24 12u2 2 12u22 then the above first Poisson brackets coincide, modulo ε4 , with the Poisson brackets given in (7.27). While for the second Poisson brackets, they coincide with the Poisson
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brackets given in (7.28) only up to ε0 , and starting from the ε2 terms, the two second Poisson brackets no longer coincide. This result is in accordance with the result of [16], where it was shown, by imposing the commutativity of the flows, that the integrable system (the tree-level one) related to the free energy (7.29) can not be extended beyond ε2 terms. Remark. In this section, the free energies corresponding to the Lie algebras of the types A2 , A3 , B2 are different from those given in Sect. 6, they are related by a rescaling. Example 4. Consider the Toda lattice equation with open boundary ∂un = vn − vn−1 , ∂t ∂vn = eun+1 − eun , ∂t
n ∈ Z.
(7.33)
If we introduce the slow variables T = t ε, X = n ε, and the new dependent variables ˜ = vn , then the Toda lattice equations lead to u(X) ˜ = un , v(X) 1 ∂ u˜ = (v(X) ˜ − v(X ˜ − ε)) , ∂T ε 1 u(X+ε) ∂ v˜ ˜ = e˜ − eu(X) . ∂T ε
(7.34)
This system has the bi-Hamiltonian structure ∂ u˜ = {u(X), ˜ H0 }1 = {u(X), ˜ H−1 }2 , ∂T ∂ v˜ = {v(X), ˜ H0 }1 = {v(X), ˜ H−1 }2 , ∂T
(7.35)
where the Poisson brackets are defined by ˜ v(Y ˜ )}1 = 0, {u(X), ˜ u(Y ˜ )}1 = {v(X), 1 {u(X), ˜ v(Y ˜ )}1 = (δ(X − Y ) − δ(X − Y − ε)) , ε 1 {u(X), ˜ u(Y ˜ )}2 = (δ(X − Y + ε) − δ(X − Y − ε)) , ε 1 ˜ {v(X), ˜ u(Y ˜ )}2 = (δ(X − Y + ε) − δ(X − Y )) v(X), ε 1 u(X+ε) ˜ e˜ δ(X − Y + ε) − eu(X) δ(X − Y − ε) , {v(X), ˜ v(Y ˜ )}2 = ε and the Hamiltonians are given by Z ˜ H−1 = v(X)dX,
Z H0 =
1 2 ˜ v(X) ˜ + eu(X) dX. 2
(7.36)
(7.37)
(7.38)
We construct the hierarchy of integrable systems ∂ u˜ = {u(X), ˜ H p }1 , ∂T p
∂ v˜ = {v(X), ˜ Hp }1 ∂T p
(7.39)
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with the Hamiltonians Hp recursively defined by {u(X), ˜ Hp−1 }2 = (p + 1){u(X), ˜ Hp }1 ,
{v(X), ˜ Hp−1 }2 = (p + 1){v(X), ˜ H p }1 . (7.40)
We identify T 0 with T . Let’s define again the following new variables: ε2 00 v˜ (X) + O(ε4 ), 24 ε ε2 ε3 000 t2 (X) = u(X) u˜ (X) + O(ε4 ), ˜ + u˜ 0 (X) + u˜ 00 (X) − 2 24 48 ˜ − t1 (X) = v(X)
(7.41) (7.42)
and expand the above Poisson brackets in Taylor series in ε, we obtain {t1 (X), t1 (Y )}1 = {t2 (X), t2 (Y )}1 = 0, ε2 (3) {t1 (X), t2 (Y )}1 = δ 0 (X − Y ) − δ (X − Y ) + O(ε4 ), (7.43) 12 2 2 {t1 (X), t1 (Y )}2 = 2 et (X) δ 0 (X − Y ) + et (X) (t2 (X))0 δ(X − Y ) 1 1 1 (3) + ε2 δ (X − Y ) + (t2 (X))0 δ 00 (X − Y ) + ((t2 (X))0 )2 δ 0 (X − Y ) 6 4 12 1 2 1 (t (X))0 (t2 (X))00 δ(X − Y ) + (t2 (X))00 δ 0 (X − Y ) + 4 12 2 1 2 (3) + (t (X)) δ(X − Y ) et (X) + O(ε4 ), 12 1 1 1 2 1 0 2 {t (X), t (Y )}2 = t (X) δ (X − Y ) − ε t (X) δ (3) (X − Y ) 12 1 + (t1 (X))0 δ 00 (X − Y ) + O(ε4 ), 12 {t2 (X), t2 (Y )}2 = 2 δ 0 (X − Y ) + O(ε4 ).
(7.44)
We also expand the Hamiltonians (7.40) and the integrable system (7.39) in Taylor series in ε. The Hamiltonians H−1 and H0 have the form Z H−1 = t1 (X)dX + O(ε4 ), Z 1 1 2 t2 (X) (t (X)) + e dX H0 = 2 Z 2 2 ε2 1 1 t2 (X) 2 − t (X) + e tX (X) dX + O(ε4 ). (7.45) 12 2 X Now if we put the CP 1 free energy F =
2 1 12 2 (t ) t + et 2
(7.46)
1 2 t , we get the Poisson brackets into Theorem 1–Theorem 3, and with G-function G = − 24 4 which coincide with those given in (7.44) modulo ε , and the Hamiltonians H2,p we get
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357
also coincide with Hp modulo ε4 . This suggests that the Toda lattice hierarchy is the appropriate hierarchy of integrable systems behind the CP 1 model, as it was suggested in [16] from the point of view of commuting flows. 8. Discussion We formulate here the conjectural shape of the integrable hierarchy to be considered starting from a Frobenius manifold and of its bihamiltonian structure in the form of genus expansion. The hierarchy must have the form X ∂t (0) (k) = Kα,p (t, tX ) + ε2 k Kα,p (t, tX , tXX , . . . ) = {t(X), Hα,p }1 , α,p ∂T
(8.1)
k≥1
where the Hamiltonians and the first Poisson bracket must have the expansions X (0) (k) + ε2 k Hα,p , Hα,p = Hα,p k≥1
{tα (X), tβ (Y )}1 = {tα (X), tβ (Y )}(0) 1 +
X
ε2 k {tα (X), tβ (Y )}(k) 1 ,
(8.2) (8.3)
k≥1
where
Z (k) = Hα,p
(k) Pα,p (t; tX , tXX , . . . )dX,
{tα (X), tβ (Y )}(k) 1 =
2X k+1
(8.4)
(s) Aα,β k,s (t; tX , tXX , . . . )|t=t(X) δ (X − Y ).
(8.5)
s=0 (k) (t; tX , tXX , . . . ) and the coefficients Aα,β The densities Pα,p k,s (t; tX , tXX , . . . ) of the Poisson bracket are quasihomogeneous polynomials in tX , tXX , . . . of the degrees (k) (t; tX , tXX , . . . ) = 2 k, deg Pα,p
(8.6)
deg Aα,β k,s (t; tX , tXX , . . . )
(8.7)
= 2 k + 1 − s,
where we assign the degrees m t=m deg ∂X
(8.8)
(k) for any m = 1, 2, . . . . The coefficients Kα,p (t; tX , tXX , . . . ) of the hierarchy are also polynomials in the same variables of the degree (k) deg Kα,p (t; tX , tXX , . . . ) = 2 k + 1,
k = 0, 1, . . . .
(8.9)
All the Hamiltonians must commute. Remark. The dispersion expansions of the known integrable hierarchies obtained by simultaneous rescaling x 7→ ε x, t 7→ ε t for any time variable t contain also odd powers of ε. However, doing an appropriate ε-dependent change of dependent variables we can reduce the hierarchy and their Poisson brackets to the form postulated in this section. (See examples above in Sect. 7).
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We expect that the quasihomogeneity (2.3) will not be involved in the construction of the hierarchy. If, however, it takes place then the coefficients of the first Poisson bracket must satisfy another quasihomogeneity condition. Let us introduce the extended Euler vector field XX ∂ m α (8.10) (1 − m − qα ) ∂X t E := E + m tα ) , ∂(∂X α m≥1
where the Euler vector field E has the form (2.10). Then the coefficients of the first Poisson bracket (8.3) must satisfy the quasihomogeneity conditions αβ LE Aαβ k,s (t; tX , tXX , . . . ) = (k(d − 3) + d + s − 1 − qα − qβ ) Ak,s (t; tX , tXX , . . . ). (8.11)
Moreover, there exists another Poisson bracket with the structure similar to (8.3), (8.5) X {tα (X), tβ (Y )}2 = {tα (X), tβ (Y )}(0) ε2 k {tα (X), tβ (Y )}(k) (8.12) 2 + 2 , k≥1
{tα (X), tβ (Y )}(k) 2 =
2X k+1
α,β Bk,s (t; tX , tXX , . . . )|t=t(X) δ (s) (X − Y ),
(8.13)
s=0 α,β (t; tX , tXX , . . . ) are polynomials in tX , tXX , . . . of the same degree 2 k + where Bk,s 1 − s in the sense of (8.8). The quasihomogeneity conditions for the coefficients of the second Poisson bracket have the form αβ αβ (t; tX , tXX , . . . ) = (k(d − 3) + d + s − qα − qβ ) Bk,s (t; tX , tXX , . . . ). LE Bk,s (8.14)
The Poisson brackets { , }1 and { , }2 must be compatible, i.e., any linear combination of them with arbitrary constant coefficients must be again a Poisson bracket. Besides ∂ αβ B = Aαβ k,s , ∂t1 k,s
∂ αβ A = 0. ∂t1 k,s
(8.15)
All the equations of the hierarchy (8.1) with the numbers (α, p) such that 1 + µα + p 6= 0 2
(8.16)
are Hamiltonian flows also w.r.t. the second Poisson bracket. Additional conjecture about the bihamiltonian structure (8.3), (8.12) is that, for d 6= 1, {tn (X), tn (Y )}(k) 1 =0
for k > 0,
{tn (X), tn (Y
for k > 1.
)}(k) 2
=0
(8.17)
Here the invariant definition of the coordinate tn is tn := η1ε tε . This conjecture means that the Virasoro algebra with the central charge (3.35) found for d 6= 1 in Corollary 1 above does not get deformations coming from the genera ≥ 2. In other words, our bihamiltonian structure is a classical W -algebra with the conformal dimensions (2.31) and the central charge (3.35).
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We recall that a Frobenius manifold M n is said to have good analytic properties if the primary free energy F (t) has the form F (t) = cubic terms + analytic perturbation
(8.18)
near some point t0 ∈ M n . (See [11].) For example, the point t0 is the origin in the topological minimal models and it is the point of classical limit in the topological sigmamodels. For Frobenius manifolds with good analytic properties we expect that all the αβ Bk,s (t; tX , tXX , . . . ) are anacoefficients of the polynomials Aαβ k,s (t; tX , tXX , . . . ), lytic in t near the point t0 . For the case t0 = 0, d < 1, i.e., the charges satisfy 0 ≤ qα ≤ d < 1,
(8.19)
the analyticity implies finiteness of all of the expansions of the Poisson bracket. Indeed, from (8.5) and (8.11) we obtain that k(d − 3) + d + s − 1 − qα − qβ ≤ k(d − 1) + d.
(8.20)
This number is nonnegative only if k≤
d . 1−d
(8.21)
But all the degrees of the variables tα are 1 − qα > 0. So all the terms { , }(k) 1 must d . Similarly, the terms in the expansion of the second Poisson bracket vanish for k > 1−d 1+d . All the examples of 1 + 1 integrable hierarchies labeled by must vanish for k > 1−d A-D-E Dynkin graphs are of this type. All the coefficients of the genus expansions are polynomials. Recall, that a polynomial Frobenius manifold can be constructed for an arbitrary finite Coxeter group [11]. For this case d=1−
2 , h
qα = 1 −
mα + 1 , h
where h is the Coxeter number and mα are the exponents of the Coxeter group. However, the bihamiltonian hierarchy (8.1) can be constructed for only simply-laced Dynkin graphs. Indeed, our formula (3.35) for the central charge coincides with the formula [20] cε2 = 12ε2 ρ2
(8.22)
of the central charge of the classical W -algebras with the same Dynkin diagram exactly for the simply-laced case! Here ρ is one half of the sum of positive roots. Our ε is equal to iα of [20]. Recall, that for the simply-laced Coxeter groups our polynomial Frobenius manifolds correspond to the topological minimal models [7]. The constant γ in (3.29) equals 0. So the G-function is identically equal to 0 for the A–D–E polynomial Frobenius manifolds. For d ≥ 1 the expansions probably are infinite. The Jacobi identity for the Poisson brackets, commutativity of the Hamiltonians etc. are understood as identities for the formal power series in ε2 . In the paper we have constructed the first terms of the expansions and showed that they are in agreement with the assumptions we formulate in this section.
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To proceed to the next order O(ε4 ) we are to compute the Poisson brackets (1) (1) (0) (1) (1) (0) (0) (1) (1) {Hα,p , Hβ,q } + {Hα,p , Hβ,q } + {Hα,p , Hβ,q } := Qα,p;β,q .
(8.23)
Then the corrections to the Hamiltonians and to the Poisson brackets are to be determined from the linear equations (2) (0) (0) (0) (0) (2) (0) (2) (0) , Hβ,q } + {Hα,p , Hβ,q } + {Hα,p , Hβ,q } = −Qα,p;β,q . {Hα,p
(8.24)
We do not expect that the deformed hierarchy and the Poisson brackets can be constructed for an arbitrary Frobenius manifold (cf. [16]). However, solvability of the linear system (8.24) together with the bihamiltonian property could give a clue to the problem of selection of “physical” solutions of WDVV equations of associativity. We plan to investigate this solvability in subsequent publications. We do not discuss in this paper the relations between the one-loop deformations of the hierarchy and the Virasoro algebra of [18, 19]. This is to be done in a subsequent publication. Another interesting problem is a relation between the hierarchy we construct and the recursion relations of [28]. Acknowledgement. The authors thank E. Getzler for fruitful discussions. We thank G. Falqui for the help with W -algebras. The work of one of the authors (B.D.) was done under partial support of the EC TMR Programme Integrability, non-perturbative effects and symmetry in Quantum Field Theories, grant FMRX-CT96-0012. The work of Y.Z. was supported by the Japan Society for the Promotion of Science, it was initiated in SISSA when he was a post-doc there; he thanks M. Jimbo for valuable discussions.
References 1. Casati, P. and Pedroni, M.: Drinfeld–Sokolov reduction on a simple Lie algebra from the bihamiltonian point of view. Lett. Math. Phys. 25, 89–101 (1992) 2. Dickey, L.: Soliton equations and Hamiltonian systems. Singapore: World Scientic, 1991 3. Di Francesco, P., Itzykson, C., Zuber, J.-B.: Classical W -algebras. Commun. Math. Phys. 140, 543–567 (1991) 4. Di Francesco, P., Itzykson, C.: Quantum intersection rings, hep-th/9412175 5. Dijkgraaf, R., Verlinde, E., Verlinde, H.: Notes on topological string theory and 2D quantum gravity. IASSNS-HEP-90/80 6. Dijkgraaf, R., Witten, E.: Mean field theory, topological field theory, and multi-matrix models. Nucl. Phys. B342, 486–522 (1990) 7. Dijkgraaf, R., Verlinde, E., Verlinde, H.: Topological strings in d < 1. Nucl. Phys. B352, 59 (1991) 8. Dijkgraaf, R.: Intersection Theory, Integrable Hierarchies and Topological Field Theory. Lectures given at the Cargese Summer School on “New Symmetry Principles in Quantum Field Theory”. July 16–27, 1991, hep-th/9201003 9. Dubrovin, B.: Integrable systems in topological field theory. Nucl. Phys. B379, 627–689 (1992) 10. Dubrovin, B.: Topological conformal field theory from the point of view of integrable systems. In: Integrable Quantum Field Theories, Edited by L. Bonora, G. Mussardo, A. Schwimmer, L. Girardello, and M. Martellini, New York: Plenum Press, NATO ASI series B310, 283–302 (1993) 11. Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups, Montecalini, Terme, 1993. Editor: M. Francaviglia, S. Greco. Springer Lecture Notes in Math. 1620, 120–348 (1996) 12. Dubrovin, B.: Painlev´e transcendents and two-dimensional topological field theory. In: Painlev´e Property, One Century Later, Carg`ese, 1996, math.AG/9803107 13. Dubrovin, B., Zhang, Y.: Extended affine Weyl groups and Frobenius manifolds. Compositio Math. 111, 167–219 (1998) 14. Eguchi, T., Kanno, H.: Toda Lattice Hierarchy and the Topological Description of the c=1 String Theory. Phys. Lett. B331, 330–334 (1994)
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15. Eguchi, T. and Yang, S.-K.: The Topological CP 1 Model and the Large-N Matrix Integral. Mod. Phys. Lett. A9, 2893–2902 (1994) 16. Eguchi, T., Yamada, Y. and Yang, S.-K.: On the Genus Expansion in the Topological String Theory. Rev. Math. Phys. 7, 279 (1995) 17. Eguchi, T., Hori, K., Xiong, C.S.: Gravitational Quantum Cohomology. Int. J. Mod. Phys. A12, 1743– 1782 (1997) 18. Eguchi, T., Hori, K., Xiong, C.S.: Quantum Cohomology and Virasoro Algebra. Phys.Lett. B402, 71–80 (1997) 19. Eguchi, T., Jinzenji, M., Xiong, C.S.: Quantum Cohomology and Free Field Representation. hepth/9709152 20. Fateev, V.A., Lukyanov,S.L.: Additional symmetries and exactly-solvable models in two-dimensional conformal field theory, parts I, II, and III. Sov. Sci. Rev. A15, 1 (1990) ¯ 1,4 and elliptic Gromov–Witten invariants, alg-geom/9612004. To 21. Getzler, E.: Intersection theory on M appear in J. Am. Math. Soc. 22. Givental, A.: Elliptic Gromov–Witten invariants and the generalized mirror conjecture. math.AG/9803053 23. Hori, K.: Constraints for topological strings in d ≥ 1. Nucl. Phys. B439, 395 (1995) 24. Jimbo, M., Miwa, T., Mori, Y., Sato, M.: Physica 1D, 80 (1980); Jimbo, M., Miwa, T.: Physica 2D, 407–448 (1981) 25. Kabanov, A., Kimura, T.: Intersection numbers and rank one cohomological field theories in genus one. alg-geom/9706003 26. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992) 27. Kontsevich, M., Manin, Yu.: Gromov–Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994) 28. Kontsevich, M., Manin, Yu.I.: Relations between the correlators of the topological sigma-model coupled to gravity. alg-geom/9708024 29. Magri, F.: A simple model of the integrable Hamiltonian systems. J. Math. Phys. 19, 1156–1162 (1978) 30. Miwa, T.: Painlev´e property of monodromy preserving deformation equations and the analyticity of τ -function. Publ. RIMS 17, 703–721 (1981) 31. Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nucl. Phys. B340, 281–332 (1990) 32. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv. in Diff. Geom. 1, 243–310 (1991) 33. Witten, E.: On the Kontsevich model and other models of two-dimensional gravity. Preprint IASSNSHEP-91/24 Communicated by T. Miwa
Commun. Math. Phys. 198, 363 – 396 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and Standard Ordering with Differential Operator Representation Martin Bordemann, Nikolai Neumaier, Stefan Waldmann Fakult¨at f¨ur Physik, Universit¨at Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg i. Br., Germany. E-mail:
[email protected];
[email protected];
[email protected] Received: 15 August 1997 / Accepted: 3 April 1998
Abstract: In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T ∗ Q by means of the Fedosov procedure using a symplectic torsionfree connection on T ∗ Q homogeneous of degree zero with respect to the Liouville vector field. By a fibrewise equivalence transformation we construct a homogeneous Fedosov star product of standard ordered type equivalent to the homogeneous Fedosov star product of Weyl type. Representations for both star product algebras by differential operators on functions on Q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex-valued functions on T ∗ Q polynomial in the momenta (where an arbitrary fixed torsion-free connection ∇0 on Q is used). Motivated by the flat case T ∗ Rn another homogeneous star product of Weyl type corresponding to the Weyl ordering prescription is constructed. The example of the cotangent bundle of an arbitrary Lie group is explicitly computed and the star product given by Gutt is rederived in our approach.
1. Introduction The concept of deformation quantization defined in [2] has now been well-established on every symplectic manifold: existence of the formal associative deformation of the pointwise multiplication of smooth functions, the so-called star product, had been shown by [11] and [15], and their classification up to equivalence transformations by formal power series in the second de Rham cohomology group is due to [18, 19, 3]. The symplectic manifolds which are mostly used by physicists are cotangent bundles T ∗ Q of a smooth manifold Q, the configuration space of the classical dynamical system, which is rather often taken to be an open subset of R2n . There is a large amount of literature concerning star products on cotangent bundles (cf. e. g. [9, 10, 20]), differential operators and their symbolic calculus ([22, 23]), and also geometric quantization on cotangent bundles (see e. g. [25] and references therein).
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The main motivation for us to write this paper was to apply the formal GNS construction in deformation quantization developed by two of us (cf. [6]) to the particular case of T ∗ Q: this method (which basically copies the standard GNS representation in the theory of C ∗ -algebras) allows to construct formal pre-Hilbert space representations of the associative algebra (over the field of formal Laurent series with complex coefficients, C((λ))) of all formal Laurent series with coefficients in the space of all smooth complex-valued functions on a symplectic manifold equipped with a star product. The basic ingredient is a formally positive C((λ))-linear functional on this algebra. In case Q = Rn the usual Weyl ordered Schr¨odinger representation could thus be reconstructed by means of an integral over configuration space (see [6]) as well as the ordinary WKB expansion by means of a certain functional with support on a projectable Lagrangian submanifold of R2n contained in a classical energy surface (see [7]). When starting to work on general cotangent bundles we realized that we had to develop first a good deal of compact practical formulas for certain star products on T ∗ Q and their possible representations as formal series with coefficients in differential operators on Q before we could start checking that even the simplest functional which consists in integration over Q (with respect to some volume in case Q is orientable) is formally positive. Therefore on one hand this paper will simply prepare the grounds for a second paper (see [4]) in which we shall define formal GNS representations on T ∗ Q and also compare our results with those obtained by analytic techniques (see e. g. [9, 20, 22, 23]). On the other hand we feel that some of our results, viz. a fairly explicit Fedosov construction on arbitrary T ∗ Q, and a rather simple closed formula relating a star product based on standard ordering and a particular star product of Weyl type (which is different from the “most natural” Fedosov star product!) based on a generalization of Weyl ordering may well be of independent interest and useful in computations because the usual techniques of asymptotic expansions of certain integrals in normal coordinates are not needed. Before summarizing our results let us first motivate our programme by the simple example Q = Rn in which everything can explicitly be computed (see e. g. [1, 6]): The quantization method which is frequently used by physicists for Q = Rn in the Schr¨odinger picturep proceeds as follows: Except for relativistic phase space functions such as the energy m2 + p2 of a free particle in Rn the great majority of classical observables occurring in physics are polynomial in the momenta p, i. e. smooth functions F : R2n → C : (q, p) 7→ F (q, p) which take the form1 (q, p) 7→ F (q, p) =
N X 1 i1 ···ik F (q)pi1 · · · pik k! k
(1)
k=0
(where the Fki1 ···ik are smooth complex-valued functions on Rn ). On the space of all ∞ these functions which we shall denote by Cpp (R2n ) (and which clearly forms a Poisson ∞ 2n subalgebra of C (R )) one establishes a linear bijection to the space of all differential operators on functions ψ on Rn according to the following rule: a smooth complexvalued function q 7→ f (q) is mapped to the multiplication with f , i. e. ψ 7→ f ψ, the coordinates pl are mapped to ~i ∂q∂ l and for a general function polynomial in the momenta a so-called ordering prescription is applied to extend the map to a bijection: An important example is the standard ordering prescription where a function of the above form (1) is mapped to its standard representation, %S (F ), in the following way: 1 From now on we shall use the Einstein summation convention where the sum over repeated coordinate indices ir where mostly 1 ≤ ir ≤ n is automatic.
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k N X ∂kψ 1 ~ %S (F )(ψ) : q 7→ Fki1 ···ik (q) i1 (q). k! i ∂q · · · ∂q ik
(2)
k=0
The standard representation, however, is unphysical in the sense that the differential operators %S (F ) are not symmetric (when F takes real values) on, say, the space D(Rn ) of all smooth complex-valued functions with compactRsupport with the standard inner product given by the Lebesgue integral (i. e. hφ, ψi := φ(q)ψ(q)dn q, where denotes pointwise complex conjugation): it is easily seen by induction and repeated partial integration that the formal adjoint %S (F )† of %S (F ) (i. e. h%S (F )† φ, ψi = hφ, %S (F )ψi) is given by the differential operator %S (F )† = %S (N 2 F ),
(3)
where ~
∂2
N := e 2i ∂qk ∂pk
(4)
is a well-defined bijective linear map on all the functions polynomial in the momenta (1). These unphysical features can be remedied by defining the Weyl representation of F by %W (F ) := %S (N F )
(5)
which is also a bijection and clearly gives %W (F )† = %W (F )
(6)
such that real-valued functions are now mapped to symmetric operators. In case F is a polynomial function in p and in q it is not hard to see using the Baker–Campbell– Hausdorff series of the Heisenberg Lie algebra spanned by 1, %S (q 1 ), . . . , %S (q n ), %S (p1 ), . . . , %S (pn ) that %W (F ) can be obtained by the so-called Weyl ordering prescription by means of which monomials are mapped to totally symmetrized operators (see e. g. [1, 6]), i. e. q i1 · · · q ia pj1 · · · pjb 7→
X 1 Aσ(1) · · · Aσ(a+b) , (a + b)!
(7)
σ∈Sa+b
where Ar := %S (q ir ) for all 1 ≤ r ≤ a and Ar := %S (pjr−a ) for all a + 1 ≤ r ≤ a + b. The usual Moyal–Weyl star product ∗W in R2n of two smooth complex-valued functions F, G polynomial in the momenta, i~ ∂2 ∂2 − 2 ∂q i ∂p0 ∂q 0 i ∂pi i (8) F (q, p)G(q 0 , p0 ) (F ∗W G)(q, p) := e 0 0 q=q ,p=p
can for instance be obtained from the multiplication of the two Weyl representations, i. e.: %W (F ∗W G) = %W (F )%W (G).
(9)
Likewise, the multiplication of the two standard representations of F and G gives rise to another star product of standard type, ∗S , in the following way (which makes sense since %S is a bijection into the space of all differential operators with smooth coefficients)
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%S (F ∗S G) := %S (F )%S (G)
(10)
and takes the following form: F ∗S G =
r ∞ X 1 ~ ∂rF ∂rG . i 1 r! i ∂pi1 · · · ∂pir ∂q · · · ∂q ir
(11)
r=0
Due to (5) we clearly have equivalence of ∗S and ∗W , F ∗S G = N ((N −1 F ) ∗W (N −1 G)).
(12)
Writing π for the canonical projection R2n → Rn : (q, p) 7→ q and i for the canonical zero section Rn → R2n : q 7→ (q, 0) we easily get the following useful formula for any smooth complex-valued function ψ on Rn %S (F )ψ = i∗ (F ∗S (π ∗ ψ)).
(13)
Considering ~ now as an additional variable on which the functions depend polynomially we define the following differential operator H := pi
∂ ∂ +~ . ∂pi ∂~
(14)
It is easy to see that both representations enjoy the following homogeneity property: ∂ (15) ~ , %S/W (F ) = %S/W (HF ). ∂~ Physically this means that the operator corresponding to the momentum component pl has also the physical dimension of a momentum which is equal to the dimension of ~ divided by length (the dimension of q l ). As a consequence, the two star products are also homogeneous in the sense that the map H is a derivation: H(F ∗S/W G) = (HF ) ∗S/W G + F ∗S/W (HG).
(16)
Both star products are associative and deform the pointwise multiplication such that the component of the commutator which is first order in ~ equals i times the canonical Poisson bracket. Moreover, as can easily be seen by the formulas (8) and (11) both star products are bidifferential in each order of ~. They are even of Vey type which means that in order ~r the corresponding bidifferential operator is of order r in each argument. One might be tempted to think that any ordering prescription may give rise to a reasonable star product, but the following example indicates that one may lose the property that the star product be bidifferential in each order of ~: For R2 define the following modified ordering prescription for a positive real number s: %perv ((q, p) 7→ f (q)pk ) if k = 6 2 %S ((q, p) 7→ f (q)pk ) . := %S ((q, p) 7→ f (q)p2 ) + s ~i %S ((q, p) 7→ f (q)p) if k = 2
(17)
A lengthy, but straightforward computation using formal symbols F (q, p) := eαq+βp , where β is considered as a formal parameter shows that in the corresponding star product (which is well-defined on all smooth functions polynomial in the momenta),
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i. e. %perv (F )%perv (G) = %perv (F ∗perv G), for each order in ~ there are infinitely many derivatives with respect to p. Hence this kind of star product is in general not extendable to the space of all formal power series in ~ with coefficients in the smooth complex-valued functions on R2 . The aim is now to construct and compute in more detail concrete star products on arbitrary cotangent bundles T ∗ Q and possible representations as formal series of differential operators on the formal series of smooth complex-valued functions on Q. The important feature of these structures will be their homogeneity in the momenta which is now defined using the Liouville vector field ξ on T ∗ Q (whose flow consists in ∂ in a bundle chart. multiplying the fibres by et ) and which takes the familiar form pi ∂p i It is interesting to note that the existence proof for star products on arbitrary cotangent bundles T ∗ Q by DeWilde and Lecomte [10] in 1983 is much easier than the general proof thanks to this notion of homogeneity (which had earlier been used by Cahen and Gutt for parallelizable manifolds, see [8]): By demanding the bidifferential operators Mr in the formal series of the star product of two smooth complex-valued functions f and g, f ∗g =
∞ X
~r Mr (f, g),
(18)
r=0
to be homogeneous of order −r with respect to the Liouville field DeWilde and Lecomte were able to show that the usual obstructions in the third de Rham cohomology which a priori occur when constructing the Mr by induction (see e. g. [12]) simply vanish due to the homogeneity requirement. The immediate generalization of the standard representation %S to an arbitrary Wk cotangent bundle T ∗ Q proceeds as follows: take the space of sections 0( T Q) of the k-fold symmetrized tangent bundle and consider the canonical linear injection Wk 1 T (α, . . . , α). We call the comb : 0( T Q) → C ∞ (T ∗ Q) : T 7→ Tb : α 7→ k! L∞ ∞ ∞ ∗ (T ∗ Q) is denoted plexification of its image Cpp,k (T Q), and the direct sum k=0 Cpp,k ∞ ∗ by Cpp (T Q) which is the obvious analogue of the smooth complex-valued functions ∞ (T ∗ Q). Moreover choose polynomial in the momenta. Clearly Lξ F = kF iff F in Cpp,k a torsion-free covariant derivative ∇0 in the tangent bundle of Q and replace partial by covariant derivatives in (2) in the following manner: for T = T1 + · · · + TN , where Wk Tk ∈ 0( T Q) (0 ≤ k ≤ N ) we define for a smooth complex-valued function ψ on Q %S (Tb)ψ : q 7→
k N X 1 ~ Tki1 ···ik (q)is (∂qi1 ) · · · is (∂qik )D0(k) ψ(q), k! i
(19)
k=0
where D0(k) is the k-fold symmetrized covariant derivative, is means symmetric substitution and the functions Tki1 ···ik are regarded as the components of the contravariant symmetric tensor field Tk on Q. This will clearly induce a star product on the commuta∞ (T ∗ Q) of C ∞ (T ∗ Q). However, since the Christoffel symbols of the tive subalgebra Cpp connection ∇0 modify the differential operators of flat space representation by terms of lower order it is a priori questionable in view of the above counterexample whether this star product is bidifferential at each order of ~. In fact it turns out that it is bidifferential (which seems to have been shown by hard analytic techniques in the past, cf. e. g. [20]) and in this paper we want to deal with this question in a more algebraic manner. Our main results are the following:
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– After giving a short generalizing review of the Fedosov construction on an arbitrary manifold in Sect. 2 (where we need the covariant derivative term in more generality and a fibrewise formulation of equivalence transformation) we then build up and compute the general Fedosov machinery for a star product of standard ordered ∗S and Weyl type ∗F on any cotangent bundle T ∗ Q (Sect. 3 and 4) based on a rather natural, seemingly well-known lift of any torsion-free covariant derivative on Q to a symplectic torsion-free covariant derivative on T ∗ Q which is homogeneous with respect to the Liouville vector field on T ∗ Q (see Appendix A for a description). It turns out that the corrections to the covariant derivative needed to construct the Fedosov derivative are “classical” in the sense that they do not depend on the formal parameter. The standard ordered type fibrewise star product is constructed by using the duality between the horizontal and the vertical subbundle of the tangent bundle of T ∗ Q. Then we can prove by constructing an equivalence transformation in Sect. 5 that the Fedosov star product of standard ordered type ∗S is equivalent to the Fedosov star product of Weyl type ∗F . – According to the general Fedosov philosophy “Whatever you plan to do on a symplectic manifold M , do it first fibrewise on the tangent spaces, and pull it then down to M by means of a nice compatible Fedosov derivative” we are then constructing a fibrewise standard representation analogous to the flat space formula (13) in Sect. 6. As a surprise it turned out that our naively constructed Fedosov derivative D0 (66) (as a conjugate of the original Weyl type Fedosov derivative DF by means of the fibrewise analogue to the operator N , see (4)) was not compatible with the fibrewise standard representation. Luckily, it could be modified by a fibrewise internal automorphism (see Thm. 7) to render it compatible. As a result we obtain exactly the above standard representation (19). Since it is easy to see that all “reasonable” star products constructed by a Fedosov type procedure automatically are bidifferential the construction shows a posteriori that the standard representation does not fall under the above-mentioned “beasty” class of ordering prescriptions. – Finally we derive a surprisingly simple analogue of the operator N (cf. (4)) for any T ∗ Q in Sect. 7: it takes the form N = exp( 2i~ 1) where the second-order differential operator 1 takes the following form in a bundle chart (q, p): (1F )(q, p) =
∂2F ∂q i ∂pi (q, p)
∂F F ∂F + 0iik (q) ∂p (q, p) + pr 0rij (q) ∂p∂i ∂p (q, p) + αr (q) ∂p (q, p), j r k 2
(20) where the 0ijk are the Christoffel symbols of the connection ∇0 and α is a particular choice of a one-form on Q such that −dα equals the trace of the curvature tensor (see the appendix for a theorem). In case ∇0 leaves invariant a volume on Q (assumed to be orientable) then α can be chosen to be zero. N can now be used as an equivalence transformation from the star product of standard ordered type ∗S to another star product ∗W (107) which is of Weyl type but which turns out to be different from the Fedosov star product of Weyl type, ∗F ! As an example, we rederive the star product on the cotangent bundle of an arbitrary Lie group constructed by means of the standard torsion-free left-invariant “half commutator” connection first given by Gutt (see [17]) and give an explicit closed formula of a star product of standard ordered type in Sect. 8.
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Convention. In what follows ~ will always denote a real number whereas λ will denote a formal parameter which in converging situations may be substituted by ~ and is considered to be real, i. e. λ := λ. 2. Fedosov Derivations and Fedosov—Taylor Series In this rather technical section we shall revisit Fedosov’s construction of star products in a slightly more general context. The notation is mainly the same as in Fedosov’s book [16] and in [5]. Let M be a smooth manifold and define Ws ∗ V ∞ C 0 T M ⊗ T ∗M [[λ]]. (21) W ⊗3(M ) := Xs=0 If there is no possibility for confusion we simply write W ⊗3 and denote by W ⊗3k the elements of antisymmetric degree k and set W := W ⊗30 . For two elements a, b ∈ W ⊗3 we define their pointwise product denoted by µ(a ⊗ b) = ab by the symmetric ∨-product in the first factor and the antisymmetric ∧-product in the second factor. Then the degree-maps degs and dega with respect to the symmetric and antisymmetric degree are derivations of this product. Therefore we shall call W ⊗3 a formally Z × Z-graded algebra with respect to the symmetric and antisymmetric degree. Moreover (W ⊗3, µ) is supercommutative with respect to the antisymmetric degree. For a vector field X we define the symmetric substitution (insertion) is (X) and the antisymmetric substitution ia (X) which are superderivations of symmetric degree −1 resp. 0 and antisymmetric degree 0 resp. −1. Following Fedosov we define δ := (1 ⊗ dxi )is (∂xi )
and
δ ∗ := (dxi ⊗ 1)ia (∂xi ),
(22)
where x1 , . . . , xn are local coordinates for M and for a ∈ W ⊗3 with degs a = ka and dega a = la we define 1 ∗ δ a if k + l 6= 0 (23) δ −1 a := k+l 0 if k + l = 0 and extend δ −1 by linearity. Clearly δ 2 = δ ∗ 2 = 0. Moreover we denote by σ : W ⊗3 → C ∞ (M )[[λ]] the projection onto the part of symmetric and antisymmetric degree 0. Then one has the following “Hodge-decomposition” for any a ∈ W ⊗3 (see e. g. [15, eq. 2.8.]): a = δδ −1 a + δ −1 δa + σ(a).
(24)
Now we consider a fibrewise associative deformation ◦ of the pointwise product which should have the form a ◦ b = ab +
∞ X
λr Mr (a, b),
(25)
r=1
where Mr (a, b) = Mri1 ...ir j1 ...jr is (∂xi1 ) · · · is (∂xir )ais (∂xj1 ) · · · is (∂xjr )b and the Mri1 ...ir j1 ...jr are the coefficients of a tensor field totally symmetric in i1 , . . . , ir and j1 , . . . , jr separately. Moreover we define dega -graded supercommutators with respect to ◦ and set ad(a)b := [a, b]. If not all Mr = 0 for r ≥ 1 then degs is no longer a derivation of the deformed product ◦ but Deg := degs + 2degλ is still a derivation and hence the
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algebra (W ⊗3, ◦) is formally Deg-graded, where degλ := λ∂λ . We shall refer to this degree as total degree. We shall treat the non-deformed case separately at the end of this section and first remember the following two theorems which can be proved completely analogously to Fedosov’s original theorems in [15, Theorem 3.2, 3.3]: Theorem 1. Let T (0) : W ⊗3 → W ⊗3 be a superderivation of ◦ of antisymmetric 2 degree 1 and total degree 0 such that [δ, T (0) ] = 0, and T (0) = 21 [T (0) , T (0) ] = λi ad(T ) with some T ∈ W ⊗32 of total degree 2, and let T satisfy δT = 0 = T (0) T . Then there exists a unique element r ∈ W ⊗31 such that δr = T + T (0) r + Moreover r =
P∞
k=3
i r◦r λ
and
δ −1 r = 0.
(26)
r(k) with Degr(k) = kr(k) satisfies the recursion formulas
r(k+3) = δ −1
r(3) = δ −1 T, ! k−1 i X (l+2) (k−l+2) (0) (k+2) + r ◦r T r . λ
(27)
l=1
In this case the Fedosov derivation D := −δ + T (0) +
i ad(r) λ
(28)
is a superderivation of antisymmetric degree 1 and has square zero: D2 = 0. Theorem 2. Let L = −δ + T : W ⊗3 → W ⊗3 be a C[[λ]]-linear map of antisymmetric degree 1 with square zero L2 = 0 such that T does not decrease the total degree. (i)
Then for any f ∈ C ∞ (M )[[λ]] there exists a unique element τL (f ) ∈ ker L ∩ W such that σ(τL (f )) = f
(29)
and τL : C ∞ (M )[[λ]] → W is C[[λ]]-linear and referred to as the Fedosov— Taylor series corresponding to L. P∞ (ii) If in addition T = k=0 T (k) such that T (k) is homogeneous of total degree k then P∞ for f ∈ C ∞ (M ) we have τL (f ) = k=0 τL (f )(k) , where DegτL (f )(k) = kτL (f )(k) which can be obtained by the following recursion formula
τL (f )(k+1)
τL (f )(0) = f, k X = δ −1 T (l) τL (f )(k−l) .
(30)
l=0
(iii) If L = D is a ◦-superderivation of antisymmetric degree 1 as constructed in Theorem 1 then ker D ∩ W is a ◦-subalgebra and a new (eventually deformed) associative product ∗D for C ∞ (M )[[λ]] is defined by pull-back of ◦ via τD .
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P∞ Let ◦0 be another fibrewise product for W ⊗3 and S = id + r=1 λr Sr , where Sr : P2r 1 i1 ···il W ⊗3 → W ⊗3 is a map of the form Sr = l=1 l! S is (∂xi1 ) · · · is (∂xil ), where S i1 ···il are the components of a symmetric tensor field. If in addition S(a ◦ b) = (Sa) ◦0 (Sb) for all a, b ∈ W ⊗3 then S is called a fibrewise equivalence transformation between ◦ and ◦0 . Note that S is clearly invertible and S −1 is a fibrewise equivalence transformation between ◦0 and ◦. If D is a ◦-superderivation as constructed in Theorem 1 and τD its corresponding Fedosov—Taylor series and ∗D the induced associative product for C ∞ (M )[[λ]] then D0 := SDS −1 is a ◦0 -superderivation of antisymmetric degree 1 of the form D0 = −δ + T 0 satisfying the conditions of part one of the preceding theorem. Hence there exists a corresponding Fedosov—Taylor series τD0 which induces an associative product ∗D0 on C ∞ (M )[[λ]]. Then ∗D and ∗D0 turn out to be equivalent too: Proposition 1. With the notation form above we define the map T : C ∞ (M )[[λ]] → C ∞ (M )[[λ]], T f := σ(SτD (f ))
(31)
for f ∈ C ∞ (M )[[λ]] which is an C[[λ]]-linear equivalence transformation between ∗D and ∗D0 , i. e. T (f ∗D g) = (T f ) ∗D0 (T g) for all f, g ∈ C ∞ (M )[[λ]] with inverse T −1 f = σ(S −1 τD0 (f )). Proof. This is a straightforward computation observing D0 SτD (f ) = 0 and applying the last theorem. At last we shall discuss the classical case with the undeformed product µ. First we restrict our considerations to the classical part W ⊗3cl of W ⊗3 which are just those elements without any positive λ-powers. Next we consider a torsion-free connection ∇ for M and define the map (using the same symbol as for the connection) ∇ := (1 ⊗ dxi )∇∂xi ,
(32)
where ∇∂xi denotes the covariant derivative with respect to ∂xi . Then clearly ∇ is globally defined and a superderivation of µ which leaves W ⊗3cl invariant. Moreover we consider Ws ∗ V ∞ C 0 T M ⊗ T ∗M ⊗ T M [[λ]] (33) W ⊗3 ⊗X (M ) := Xs=0 and define W ⊗3 ⊗X cl analogously. For % ∈ W ⊗3 ⊗X we define the symmetric substitution is (%) by inserting the vector part of % symmetrically and multiplying the form part of % by µ from the left. Then we have ∇2 = −is (R), where R is the curvature tensor viewed as an element of antisymmetric degree 2 in W ⊗3 ⊗X . The classical analogue to the Fedosov derivation is described by the following theorem which is due to Emmrich and Weinstein [14, Theorem 1]: Theorem 3. Let ∇ be defined as in (32) then ∇ is a superderivation of the undeformed product µ and there exists a uniquely determined element %0 ∈ W ⊗3 ⊗X cl of antisymmetric degree 1 such that δ −1 %0 = 0 and such that the classical Fedosov derivation D0 = −δ + ∇ + is (%0 )
(34)
has square zero: D02 = 0. Note that D0 %0 = R0 and %0 = %0 as well as D0 a = D0 a for all a ∈ W ⊗3.
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Furthermore in [14, Theorem 3 and 6] it was shown using analytical techniques that in this case the corresponding Fedosov–Taylor series τ0 is just the formal Taylor series with respect to the connection. We shall give here another more algebraic proof of this result: Theorem 4. Let τ0 be the Fedosov–Taylor series of D0 according to Theorem 2. Then for f ∈ C ∞ (M ) we have τ0 (f ) = eD f,
(35)
i
where D = dx ∨ ∇∂xi and hence τ0 (f ) is the formal Taylor series with respect to the connection ∇. Proof. Since D0 satisfies all conditions for Theorem 2 we only have to show that eD f satisfies the recursion formula (30). Note that in this particular case the total degree and the symmetric degree coincide. First we observe D = [δ ∗ , ∇] andhence applying δ and Pk )τ0 (f )(k−l) . δ ∗ to (30) implies 0 = −(δ ∗ δ+δδ ∗ )τ0 (f )(k+1) +Dτ0 (f )(k) + l=0 δ ∗ is (%(l+1) 0 Now each term in the last sum vanishes identically due to δ ∗ %0 = 0 and dega τ0 (f ) = 0, hence (k + 1)τ0 (f )(k+1) = Dτ0 (f )(k) since δ ∗ δ + δδ ∗ = degs + dega . Then (35) follows directly by induction on the symmetric degree k. 3. Homogeneous Fedosov Star Product of Weyl Type Let π : T ∗ Q → Q be the cotangent bundle of a differentiable, n-dimensional manifold Q and let θ0 be the canonical one-form, ω0 := −dθ0 the canonical symplectic form and ξ defined by iξ ω0 = −θ0 the canonical (Liouville) vector field on T ∗ Q. Moreover let i : Q → T ∗ Q be the embedding of Q in T ∗ Q as zero section. We consider now for W ⊗3 of T ∗ Q the fibrewise Weyl product defined for a, b ∈ W ⊗3 by iλ
kl
a ◦F b := µ ◦ e 2 3
is (∂xk )⊗is (∂xl )
a ⊗ b,
(36)
kl
where µ(a⊗b) = ab is the fibrewise product in W ⊗3 and 3 are the components of the canonical Poisson tensor with respect to some coordinates x1 , . . . , x2n of T ∗ Q. It will be advantageous for calculations to use local bundle (Darboux) coordinates q 1 , . . . , q n , p1 , . . . , pn induced by coordinates q 1 , . . . , q n on Q such that p1 , . . . , pn are the conjugate momenta to the q 1 , . . . , q n . In the following q 1 , . . . , q n , p1 , . . . pn shall always denote such a local bundle (Darboux) chart. Obviously ◦F is an associative deformation of µ of the form as in (25) and hence we can apply all results of Sect. 2 to this particular situation. We denote by Lξ the Lie derivative with respect to the canonical vector field ξ and define the “homogeneity derivation” ∂ . (37) ∂λ Note that H is C-linear but not C[[λ]]-linear. Then the fibrewise Weyl product is homogeneous which can be proved as in the case of R2n : H := Lξ + degλ = Lξ + λ
Lemma 1. Let a, b ∈ W ⊗3 then H is a (super-)derivation of ◦F of antisymmetric and total degree 0: H(a ◦F b) = Ha ◦F b + a ◦F Hb. (38) ∗ Moreover = Lξ , δ −1 = 0 and [H, δ] = [H, δ ∗ ] = we have Lξ , δ = Lξ , δ −1 H, δ = 0.
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According to Fedosov’s construction of a star product one needs a torsion-free and symplectic connection ∇ for T ∗ Q and the map ∇ : W ⊗3 → W ⊗3 defined as in (32). If the connection is symplectic ∇ turns out to be a superderivation of antisymmetric degree 1 and symmetric and total degree 0 of the fibrewise Weyl product ◦F . Moreover [δ, ∇] = 0 and 2∇2 = [∇, ∇] turns out to be an inner superderivation ∇2 =
i adF (R), λ
(39)
t dxi ∨ dxj ⊗ dxk ∧ dxl ∈ W ⊗32 involves the curvature of the where R := 41 ωit Rjkl connection. Moreover one has δR = 0 = ∇R as a consequence of the Bianchi identities. Hence the map ∇ as term of total degree 0 satisfies all conditions of Theorem 1 and hence there is a unique element rF ∈ W ⊗31 such that δrF = R + ∇rF + λi rF ◦F rF and δ −1 rF = 0. Moreover the Fedosov derivation
DF := −δ + ∇ +
i adF (rF ) λ
(40)
has square 0. Let τF be the corresponding Fedosov–Taylor series. Then Fedosov has shown that f ∗F g := σ(τF (f ) ◦F τF (g))
(41)
defines a star product [15, Eq. 3.14] which is of Weyl type (see e. g. [5, Lemma 3.3]). In the particular case of a cotangent bundle we consider homogeneous, symplectic and torsion-free connections (see Definition 3): Lemma 2. If the symplectic and torsion-free connection ∇ on T ∗ Q is in addition homogeneous then [Lξ , ∇] = [H, ∇] = 0,
(42)
HR = Lξ R = R.
(43)
Theorem 5. Let ∇ be defined as above with the additional property that the connection is homogeneous. Then Lξ rF = rF = HrF
and
∂ rF = 0. ∂λ
(44)
Moreover the Fedosov derivation DF (super-)commutes with H [H, DF ] = 0
(45)
and rF satisfies the following simpler recursion formulas
rF(k+3) = δ −1
rF(3) = δ −1 R, ! k−1 1 X (l+2) (k−l+2) (k+2) − rF , rF ∇rF fib , 2
(46)
l=1
where {·, ·}fib denotes the fibrewise Poisson bracket in W ⊗3. Moreover the corresponding Fedosov–Taylor series τF commutes with H
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HτF (f ) = τF (Hf )
(47)
and satisfies the usual recursion formulas for f ∈ C ∞ (T ∗ Q) with respect to the total degree
τF (f )(k+1) = δ −1
τF (f )(0) = f, ! k−1 X i ∇τF (f )(k) + adF rF(l+2) τF (f )(k−l) λ
(48)
l=1
analogously to (30). The Fedosov star product (41) is homogeneous, i. e. for f, g ∈ C ∞ (T ∗ Q)[[λ]] H(f ∗F g) = (Hf ) ∗F g + f ∗F (Hg).
(49)
Proof. The fact HrF = rF is proved by induction using the recursion formulas for rF . Then (45), (47) and (49) easily follow and (48) follows directly from (30). Since HrF = rF the section rF can depend at most linearly on λ but since it has to depend on even powers of λ only (see [5, Lemma 3.3]) it has to be independent of λ at all. Then the recursion formulas (46) follow by induction. We shall refer to ∗F as the homogeneous Fedosov star product of Weyl type induced by the homogeneous connection ∇. Using this theorem we find several corollaries. The first one is originally due to DeWilde and Lecomte [10, Proposition 4.1]: Corollary 1. On every cotangent bundle T ∗ Q there exists a homogeneous star product of Weyl type. ∞ (T ∗ Q), then τF (f ) contains only even powers of λ up to Corollary 2. Let f ∈ Cpp,k order k.
Proof. This easily follows from (47) and [5, Lemma 3.3].
Corollary 3. The Fedosov–Taylor series τF satisfies τF ◦ π ∗ = π ∗ ◦ i ∗ ◦ τ F ◦ π ∗ .
(50)
General properties of homogeneous star products are described in the following proposition: Proposition 2. Let ∗ be a homogeneous star product for T ∗ Q. ∞ (T ∗ Q)[λ] are a C[λ]The functions polynomial in λ and in the momenta Cpp submodule of C ∞ (T ∗ Q)[[λ]] with respect to ∗ and hence f ∗ g trivially converges ∞ (T ∗ Q)[λ]. for all λ = ~ ∈ R and f, g ∈ Cpp ∞ (T ∗ U )[λ] can be written as (ii) Let U ⊆ Q be a domain of a chart. Then any f ∈ Cpp ∞ ∗ ∞ (T ∗ U )[λ]. finite sum of star products of functions in Cpp,0 (T U )[λ] and Cpp,1 ω ∗ (iii) The vector space Cp (T Q)[[λ]] of formal power series with coefficients in the functions which are analytic in the fibre variables are a C[[λ]]-submodule of C ∞ (T ∗ Q)[[λ]].
(i)
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Proof. The first part is obvious using Lemma 14 and for the third part one observes that the homogeneity implies that the coefficient functions of the bidifferential operators in the star product are polynomial in the momenta. For the second part note that 1 ∞ (T ∗ U ) takes the form f (q, p) = (k+1)! F i1 ···ik+1 (q)pi1 · · · pik+1 . Subtracting f ∈ Cpp,k+1 1 ∗ i1 ···ik+1 ) ∗ pi1 ∗ · · · ∗ pik+1 results in a finite sum of functions of degree smaller (k+1)! (π F than k + 1 with respect to Lξ and polynomial in λ proving the obvious induction on k. Now we consider the Fedosov derivation DF and the section rF more closely. First we notice that δ, δ ∗ and δ −1 satisfy the following relations δπ ∗ = π ∗ δ0 ,
δ ∗ π ∗ = π ∗ δ0∗ ,
δ −1 π ∗ = π ∗ δ0−1 ,
(51)
where δ0 , δ0∗ and δ0−1 are the corresponding maps on Q defined analogously to δ, δ ∗ and δ −1 . Moreover for a homogeneous connection we get ∇π ∗ = π ∗ ∇0 ,
(52)
where ∇0 is the corresponding map on Q defined by the induced connection ∇0 on Q (see Definition 5). By direct calculation we get the following proposition: Proposition 3. Let DF and rF be given as in (40). Then there exists a unique element % ∈ W ⊗3 ⊗X cl (Q) of antisymmetric degree 1 such that DF π ∗ = π ∗ D,
(53)
where D = −δ0 + ∇0 + is (%) and clearly D2 = 0. In local coordinates the element % takes the following form % = i∗ (is (∂pi )rF ) ⊗ ∂qi
(54)
and we have δ0∗ % = 0 iff ia (X)rF = 0 for all vertical vector fields X ∈ 0(T (T ∗ Q)). In other words, if ia (X)rF = 0 for all vertical vector fields then D would coincide with the map D0 and % would coincide with %0 as in Theorem 3 applied for M = Q. In the following we shall prove that for any torsion-free connection on Q there is indeed a canonical choice for a homogeneous connection on T ∗ Q such that this is the case: Proposition 4. Consider a torsion-free connection on Q and the map ∇0 defined as in (32) and let ∇ be a homogeneous, symplectic and torsion-free connection for T ∗ Q such that ∇π ∗ = π ∗ ∇0 and let rF be the corresponding element in W ⊗31 . Then ia (X)rF(3) = 0 for every vertical vector field X ∈ 0(T (T ∗ Q)), where rF(3) is the term of total degree 3 in rF iff the connection ∇ coincides with the lifted connection ∇0 defined as in (4). Moreover in this case ia (X)rF = 0.
(55)
Proof. The fact ia (X)rF(3) = 0 for X vertical iff ∇ = ∇0 follows from Proposition 13. Then (55) follows by a lengthy but straightforward induction on the total degree using the recursion formulas (46).
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Corollary 4. Let ∇0 be a torsion-free connection for Q and ∇0 the corresponding homogeneous, symplectic, and torsion-free connection for T ∗ Q. Then the corresponding Fedosov derivation DF satisfies DF π ∗ = π ∗ D0
(56) ∞
and hence the Fedosov–Taylor series of the pull-back of functions χ ∈ C (Q)[[λ]] is just the pull-back of their Taylor series with respect to ∇0 , τF (π ∗ χ) = π ∗ τ0 (χ).
(57)
At last in this section we shall discuss the classical limit of the Fedosov derivation DF and the Fedosov–Taylor series τF . We define DFcl by setting λ = 0 in DF which is well-defined since in any case adF (rF ) starts with iλ{rF , ·}fib and analogously we define τFcl . Then clearly for f, g ∈ C ∞ (T ∗ Q), (DFcl )2 = 0
and
DFcl τFcl (f ) = 0
and
τFcl ({f, g}) = {τFcl (f ), τFcl (g)}fib (58)
which is also true for arbitrary symplectic manifolds (compare [14, Sect. 8]). Now we concentrate again on the particular case where we have chosen a torsion-free connection ∇0 for Q and the corresponding homogeneous, symplectic, and torsion-free connection ∇0 for T ∗ Q. Then for any vertical vector field X ∈ 0(T (T ∗ Q)) we have ia (X)DF + DF ia (X) = [ia (X), DF ] = LX − is (X) − (dxi ⊗ 1)is (∇0∂xi X)
(59)
which is proved by direct calculation using ia (X)rF = 0. Note that this equation is also true if DF is replaced by DFcl since the right-hand side is obviously independent of λ. Theorem 6. Let ∇0 be a torsion-free connection for Q and ∇0 the corresponding homogeneous, symplectic and torsion-free connection for T ∗ Q. Moreover let τF be the corresponding Fedosov–Taylor series and τFcl the classical part of τF and let q0 ∈ Q. If q 1 , . . . , q n are normal (geodesic) coordinates around q0 with respect to ∇0 then for any αq0 ∈ Tq∗0 Q the induced bundle coordinates are normal Darboux coordinates (see e. g. [16, Sect. 2.5, p. 67] for definition) around αq0 with respect to ∇0 and ∞ X 1 ∂rf cl dxi1 ∨ · · · ∨ dxir (60) τF (f ) αq = 0 r! ∂xi1 · · · ∂xir αq r=0
0
for any f ∈ C ∞ (T ∗ Q), where (x1 , . . . , x2n ) = (q 1 , . . . , q n , p1 , . . . , pn ). Proof. The fact that the bundle coordinates are normal Darboux coordinates is proved in Lemma 15. First we notice that it is sufficient to prove (60) for functions polynomial in the momenta only. For those functions we shall prove the theorem by induction on the order k in the momenta. For k = 0 the statement is true due to Corollary 4 and Theorem 4 since the local expression of π ∗ τ0 (·) in normal (Darboux) coordinates are clearly given ∞ (T ∗ Q). We prove the theorem by a local by (60) due to Lemma 17. Hence let f ∈ Cpp,k argument: For the coordinate function π ∗ q i we have τFcl (π ∗ q i )|αq0 = (q i + dq i )|αq0 = dq i which implies by (58) and the fact that q i is a Hamiltonian for the Hamiltonian vector field ∂f −∂pi that τFcl ( ∂p )|αq0 = is (∂pi )τFcl (f )|αq0 . On the other hand we compute L∂pi τFcl (f ) i at αq0 using (59) and (58) which implies (L∂pi τFcl (f ))|αq0 = is (∂pi )τFcl (f )|αq0 since ∇0X ∂pi |αq0 = 0 for all X ∈ 0(T (T ∗ Q)). The homogeneity of τF implies Lξ τFcl = τFcl Lξ and hence τFcl (f ) is a polynomial in pi and dpi of maximal degree k. Then the last consideration implies that at αq0 it only depends on the combination pi + dpi . Now using ∂f ∞ (60) for ∂p ∈ Cpp,k−1 (T ∗ Q) due to Lemma 14 the induction is easily finished. i
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4. Homogeneous Fedosov Star Products of Standard Ordered Type In this section we shall construct a Fedosov star product of standard ordered type. From now on we shall only use the connection ∇0 on T ∗ Q corresponding to a connection ∇0 on Q as given in (4). We define the fibrewise standard ordered product for a, b ∈ W ⊗3 by λ
a ◦S b := µ ◦ e i
is (∂pk )⊗is (∂ hk ) q
a ⊗ b,
(61)
where ∂qhk = ∂qk + (π ∗ 0lkr )pl ∂pr is the horizontal lift of ∂qk viewed as vector field on Q with respect to ∇0 to a vector field on T ∗ Q (see (130)). Clearly ◦S is globally defined and an associative deformation of µ which is of the form (25). Moreover we define the maps 1fib := is (∂pk )is (∂qhk )
and
λ
S := e 2i 1fib
(62)
which are again globally defined. Then the well-known equivalence (see e. g. [1]) of the standard ordered product and the Weyl product in R2n can easily be transferred to a fibrewise equivalence: For a, b ∈ W ⊗3 we have S(a ◦F b) = (Sa) ◦S (Sb)
(63)
and moreover, S commutes with H and Deg and thus H is a derivation of ◦S and ◦S is homogeneous, too. For the supercommutators using ◦S we have adS (a) = S ◦adF (S −1 a)◦ S −1 and adS (a) = 0 iff adF (a) = 0 iff degs a = 0. Moreover we have [δ, 1fib ] = [δ, S] = 0 [δ, S −1 ] = 0. By a direct calculation of the commutator B := iλ 2 [∇ , 1fib ] we obtain the following local expression for B B = (1 ⊗ dq i )
iλ l pl π ∗ Rjik is (∂pj )is (∂pk ), 3
(64)
l where Rjik are the components of the curvature tensor of ∇0 . Now we conjugate the ◦F -superderivation ∇0 with S to obtain a ◦S -superderivation:
S∇0 S −1 = ∇0 + B.
(65)
Note that no higher terms occur since B already commutes with 1fib . Now ∇0 + B is a superderivation of ◦S of antisymmetric degree 1 and total degree 0 which commutes again with H. Now we shall follow two ways to obtain a Fedosov derivation for ◦S : Firstly we just conjugate DF by S which will lead indeed to a Fedosov derivation for ◦S and secondly we start the recursion new with ∇0 + B as term of total degree 0 as in Theorem 1. The following proposition is straightforward: Proposition 5. The map D0 : W ⊗3 → W ⊗3 defined by D0 := SDF S −1
(66)
is a superderivation of antisymmetric degree 1 of ◦S which has square zero: D0 = 0. It commutes with H and we have 2
D0 = −δ + ∇0 + B +
i adS (SrF ), λ
(67)
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λ where SrF = rF + 2i 1fib rF . Hence D0 satisfies the conditions of Theorem 2 and the corresponding Fedosov–Taylor series τ 0 commutes with H, too. Moreover for f, g ∈ C ∞ (T ∗ M )[[λ]] (68) f ∗0 g := σ τ 0 (f ) ◦S τ 0 (g)
defines a homogeneous star product for T ∗ Q. Moreover τ 0 satisfies a recursion formula analogously to the recursion formula (48) for τF with the modification that ∇0 is replaced by ∇0 + B, adF by adS and rF by SrF . Now the star product ∗0 is no longer of Weyl type but we have the following analogue to the usual standard ordered product in R2n : Definition 1. A star product ∗ for a cotangent bundle π : T ∗ Q → Q is called of standard ordered type iff for all χ ∈ C ∞ (Q)[[λ]] and all f ∈ C ∞ (T ∗ Q)[[λ]], (π ∗ χ) ∗ f = (π ∗ χ)f.
(69)
Then the standard ordered star products can be characterized by the following easy proposition: ∗ Proposition P∞ 6.r Let ∗ be a star product for a cotangent bundle π : T Q → Q written as f ∗g = r=0 λ Mr (f, g) with bidifferential operators Mr . Then the following statements are equivalent:
(i) ∗ is of standard ordered type. (ii) For all f, g ∈ C ∞ (T ∗ Q)[[λ]] and χ ∈ C ∞ (Q)[[λ]] we have ((π ∗ χ)f ) ∗ g = (π ∗ χ)(f ∗ g). (iii) In any local bundle chart the bidifferential operators Mr , r ≥ 1 are of the form Mr (f, g) =
X l,s,t
...jl Mij11...i s k1 ,...kt
∂sf ∂ l+t g . (70) ∂pi1 · · · ∂pis ∂q j1 · · · ∂q jl ∂pk1 · · · ∂pkt
The following corollary is immediately checked using the obvious fact that τ 0 π ∗ = π ∗ i∗ τ 0 π ∗ and the particular form of ◦S and Lemma 14: Corollary 5. On every cotangent bundle there exists a homogeneous star product of standard ordered type, namely the homogeneous Fedosov star product ∗0 . Now we shall discuss the other important alternative: starting the recursion new with ∇0 + B as the term of total degree 0. First we have to show that ∇0 + B satisfies indeed the conditions of Theorem 1. Clearly [δ, ∇0 + B] = 0 and moreover 1 0 ∇ + B, ∇0 + B 2 = S∇0 S −1 S∇0 S −1 = S λi adF (R)S −1 = λi adS (SR) = λi adS (R) λ 1fib R and degs 1fib R = 0 which implies adS (1fib R) = 0. Hence (∇0 +B)2 since SR = R+ 2i is an inner superderivation with the element R. It remains to show that (∇0 +B)R = 0 but this is clear since Lξ R = R and thus BR = 0 due to Lemma 14 and (64) and ∇0 R = 0 anyway. Thus we can apply indeed Theorem 1 and obtain the following proposition completely analogously to Proposition 5 and Corollary 5:
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Proposition 7. There exists a unique element rS ∈ W ⊗31 such that δrS = R + (∇0 + B)rS + λi rS ◦S rS and δ −1 rS = 0. Moreover HrS = rS . Then the corresponding Fedosov derivation DS = −δ + ∇0 + B +
i adS (rS ) λ
(71)
has square zero and DS as well as the corresponding Fedosov–Taylor series τS commute with H. Then f ∗S g := σ (τS (f ) ◦S τS (g)) ,
(72)
where f, g ∈ C ∞ (T ∗ Q)[[λ]] defines a homogeneous star product of standard ordered type. Again we have a recursion formula for τS (f ) analogously to (48) resp. (30) with the obvious modifications. We shall refer to ∗S as the homogeneous Fedosov star product of standard ordered type. At last we consider the element rS more closely and notice first that BrS = 0 which follows immediately from the local expression (64) for B and HrS = rS and Lemma 14. This implies that the recursion formula for rS as proposed by Theorem 1 can be simplified to k−1
(k+3)
rS
=δ
−1
∇ rS
0 (k+2)
1 X (l+2) (k−l+2) − rS , r S fib 2
!
l=1
since HrS = rS and using the explicit expression for ◦S and Lemma 14. Thus rS satisfies the same recursion formula as rF with the same first term rS(3) = δ −1 R = rF(3) which implies that they coincide: Lemma 3. Let rS be given as in Proposition 7 then rS = rF .
(73)
Finally we compute the Fedosov–Taylor series τ 0 and τS of the pull-back of a function on Q: Proposition 8. Let χ ∈ C ∞ (Q)[[λ]] and D0 , τ 0 and DS , τS be given as above. Then D 0 π ∗ = DS π ∗ = π ∗ D 0 ,
(74)
τ 0 (π ∗ χ) = τS (π ∗ χ) = π ∗ τ0 (χ),
(75)
where D0 as in (34) and τ0 is the formal Taylor series with respect to ∇0 as in (35). Proof. Clearly Bπ ∗ = 0 and adS (SrF ) as well as adS (rF ) applied to pull-backs by π ∗ reduce to fibrewise Poisson brackets due to Lemma 14 since Lξ rF = rF . This implies D0 π ∗ = DS π ∗ = DF π ∗ and by (56) the first equation is proved which implies the second equation.
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5. Equivalence of ∗F , ∗0 , and ∗S In this section we shall construct equivalence transformations of the star products ∗F , ∗0 and ∗S . Though DeWilde and Lecomte had shown in [10, Proposition 4.2.] that any homogeneous star products of Weyl type on a cotangent bundle are equivalent this is in so far a non-trivial problem since the star products ∗0 and ∗S are evidently not of Weyl type. First we define for f ∈ C ∞ (T ∗ Q)[[λ]] T f := σ S −1 τ 0 (f ) , (76) then the fibrewise equivalence of ◦F and ◦S induces via T an equivalence of ∗F and ∗0 : Proposition 9. The map T is homogeneous [H, T ] = 0 and an equivalence transformation between ∗0 and ∗F T (f ∗0 g) = T f ∗F T g
(77)
∞ (T ∗ Q)[[λ]]. and we have τF (T f ) = S −1 τ 0 (f ) and T −1 f = σ (SτP F (f )), where f, g ∈ C ∞ r Moreover T can be written as formal series T = id+ r=1 λ Tr , where Tr is a differential operator of order 2r given by s r X 1 i σ 1fib s τ 0 (f )(2r) , (78) Tr f = 2s s! 2 s=0
where τ 0 (f )(k) l denotes the term of total degree k and symmetric degree l. Proof. The first part follows from Proposition 1 and the order of differentiation follows from the fact that τ 0 (f )(2r) is a differential operator of order 2r which follows directly from the recursion formula (30) for τ 0 . The construction of an equivalence transformation between ∗0 and ∗S is more involved. First we prove that the Fedosov derivations D0 and DS are conjugated to each other by a fibrewise automorphism of ◦S : Theorem 7. Let DS and D0 be the Fedosov derivations constructed as in Sect. 4 then there exists an element h ∈ W(Q)cl such that DS = eadS (π
∗
h)
D0 e−adS (π
∗
h)
= eadS (π
∗
h)
SDF S −1 e−adS (π
∗
h)
.
(79)
Proof. Let h ∈ W(Q)cl be an arbitrary element. Then by direct calculation using the properties of ◦S and Lemma 14 we obtain ∗ ∗ eadS (π h) (−δ)e−adS (π h) = −δ + adS π ∗ (δ0 h ), ∗ ∗ eadS (π h) (∇0 + B)e−adS (π h) = ∇0 + B − adS π ∗ ∇0 h , ∗ ∗ i λ λ i eadS (π h) adS (SrF )e−adS (π h) = adS rF − π ∗ (is (%0 )h) + π ∗ (tr%0 ) , λ λ i 2i where tr := is (∂qk )i(dq k ) and %0 is given as in Theorem 3. Collecting these results we obtain due to rS = rF 1 adS (π ∗ h) 0 −adS (π ∗ h) ∗ e De = DS − adS π D0 h − tr%0 , 2
Homogeneous Fedosov Star Products on Cotangent Bundles I
381
where D0 as in Theorem 3 applied for ∇0 on Q. Hence we have to find an element h such V1 that the second term vanishes which is the case iff we find a one-form α ∈ 0( T ∗ M ) such that D0 h =
1 (tr%0 + 1 ⊗ α) 2
(80)
since in this case D0 h − 21 tr%0 is a central element. Note that we only consider elements h ∈ Wcl . A necessary condition for (80) to have a solution is that the right-hand side is D0 -closed since D0 2 = 0. We have D0 %0 = R0 due to Theorem 3 and apply tr on both sides leading to trD0 %0 = trR0 , where R0 is the curvature tensor of ∇0 . Now a straightforward V1 computation leads to trD0 %0 = D0 tr%0 . On the other hand if α ∈ 0( T ∗ Q) then D0 (1 ⊗ α) = 1 ⊗ dα since ∇0 is torsion-free. Since the trace of the curvature tensor R0 is an exact two-from (see Lemma 16) we find always a one-form α such that trR0 = −dα and hence D0 (tr%0 + 1 ⊗ α) = 0 iff α satisfies trR0 = −dα and thus the necessary condition is fulfilled. But this is also sufficient since the D0 -cohomology is trivial on forms: Indeed we define W ⊗3+ := {a ∈ W ⊗3|σ(a) = 0} then we have the following lemma (see [16, Theorem 5.2.5]): Lemma 4. Let D0−1 : W ⊗3(Q) → W ⊗3(Q) be defined by D0−1 := −δ0−1
1 . 1 − [δ0−1 , ∇0 + is (%0 )]
(81)
Then for any a ∈ W ⊗3+ (Q) the following “deformed Hodge decomposition” holds D0 D0−1 a + D0−1 D0 a = a
(82)
and [δ0−1 , ∇0 + is (%0 )] commutes with δ0−1 and D0 and D0−1 a = D0−1 a. Note that D0−1 is a well-defined formal series in the symmetric degree. Now choose a one-form α with trR0 = −dα. Then we define h :=
1 −1 D (tr%0 + 1 ⊗ α) 2 0
(83)
and notice that σ(h) = 0 since D0−1 raises the symmetric degree. Moreover σ (tr%0 + 1 ⊗ α) = 0 hence we can apply the lemma and find D0 h = 1 2 (tr%0 + 1 ⊗ α) and thus the theorem is proved. Corollary 6. If h is a solution of (80) for a fixed one-form α then every other solution h0 is obtained by h0 = h + τ0 (ϕ) with ϕ ∈ C ∞ (Q). For a fixed one-form α satisfying trR0 = −dα there exists a unique solution of (80) with σ(h) = ϕ for any ϕ ∈ C ∞ (Q) namely h = 21 D0−1 (tr%0 + 1 ⊗ α) + τ0 (ϕ). The one-form α is determined up to a closed one-form and can be chosen to be real α = α which leads to a real h = h iff ϕ is real. Corollary 7. If the connection ∇0 is unimodular then there exists a canonical solution h of (79) uniquely determined by D0 h = 21 tr%0 and σ(h) = 0, namely h = 21 D0−1 tr%0 . In this case h = h is real. Corollary 8. Let h be an arbitrary solution of (80) then eadS (π morphism of ◦S satisfying (79) and commuting with H.
∗
h)
is a fibrewise auto-
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M. Bordemann, N. Neumaier, S. Waldmann ∗
Now we can use the fibrewise automorphism eadS (π h) to construct an equivalence transformation between ∗S and ∗0 analogously to the construction of T in (76). We define for f ∈ C ∞ (T ∗ Q)[[λ]] ∗ (84) V f := σ e−adS (π h) τS (f ) and get the following proposition completely analogously to Proposition 9: Proposition 10. For any solution h of (79) the map V is homogeneous [H, V ] = 0 and an equivalence transformation between ∗S and ∗0 : V (f ∗S g) = V f ∗0 V g and we have τ 0 (V f ) = e−adS (π C ∞ (T ∗ Q)[[λ]].
∗
h)
τS (f ) and V −1 f = σ eadS (π
∗
h) 0
(85)
τ (f ) , where f, g ∈
To compute the orders of differentiation in V we first need the following lemma obtaining by the way that ∗S is a Vey product: Lemma 5. Let τS be the Fedosov–Taylor series as in Proposition 7 and let τS (·)(k) l be (2r) (2r+1) the term of total degree k and symmetric degree l. Then τS (·)2s resp. τS (·)2s+1 is a differential operator of order r + s resp. r + s + 1 for all r, s ∈ N. Moreover this implies that the homogeneous Fedosov star product of standard ordered type ∗S is a Vey product. Proof. This lemma is proved by a straightforward induction on the total degree using the recursion formula (30) for τS . Then the Vey type property of ∗S follows directly. Corollary 9. For any solution P∞h of (80) the equivalence transformation V can be written as formal series V = id + r=1 λr Vr , where Vr is a differential operator of order r. 6. The Standard Representation Now we shall construct a canonical representation of the fibrewise algebra (W, ◦S ) and of the star product algebra (C ∞ (T ∗ Q)[[λ]], ∗S ) reproducing the well-known standard order quantization rule for cotangent bundles. First we define the representation space H := W(Q)
(86)
and define the fibrewise standard ordered representation for a ∈ W and 9 ∈ H by (87) %˜S (a)9 := i∗ a ◦S π ∗ 9 and notice that %˜S : W → End(H) is indeed a representation of W with respect to ◦S : Lemma 6. Let ◦S be the fibrewise standard ordered product and %˜S be defined as in (87). Then %˜S is a ◦S -representation of W on H, i. e. for a, b ∈ W, %˜S (a ◦S b) = %˜S (a)%˜S (b).
(88)
%˜F (a) := %˜S (Sa)
(89)
Furthermore defines a representation with respect to the fibrewise Weyl product ◦F of W on H given by (90) %˜F (a)9 = i∗ S a ◦F π ∗ 9 .
Homogeneous Fedosov Star Products on Cotangent Bundles I
383
Proof. The C[[λ]]-linearity of %˜S is obvious and the representation property is proved by straightforward computation. Then the fibrewise equivalence (63) of ◦S and ◦F implies that %˜F is a representation with respect to ◦F . Now we shall construct a representation %S for the star product ∗S induced by %˜S . First we notice that the restriction of %˜S to ker DS ∩ W is still a representation of ker DS ∩ W and hence it induces via τS a representation of C ∞ (T ∗ Q)[[λ]] on H. But we have in mind to construct a representation of ker DS ∩ W on the smaller representation space i∗ (ker DS ∩ π ∗ H) ⊂ H which is via τ0 in bijection with C ∞ (Q)[[λ]]. In the case of the standard ordered product ∗S this can be done directly: Theorem 8. Let DS be the Fedosov derivation constructed as in Proposition 7 and let τS be the corresponding Fedosov–Taylor series and ∗S the homogeneous Fedosov star product of standard ordered type. Then D S π ∗ i ∗ = π ∗ i ∗ DS
(91)
%S (f )ψ := i∗ (f ∗S π ∗ ψ) = σ(%˜S (τS (f ))τ0 (ψ)),
(92)
and
where f ∈ C ∞ (T ∗ Q)[[λ]] and ψ ∈ C ∞ (Q)[[λ]] defines a representation of C ∞ (T ∗ Q) [[λ]] with respect to ∗S on C ∞ (Q)[[λ]]. Proof. One immediately checks that δ, ∇0 and B commute with π ∗ i∗ . Moreover adS (rS ) commutes with π ∗ i∗ too due to the particular form of ◦S and Lξ rS = rS and Lemma 14. Hence (91) is shown. This ensures that %S defines indeed a representation which is now a straightforward computation using (88) and Proposition 8. Note that (91) is crucial for the representation property of %S and that neither DF nor D0 commute with π ∗ i∗ . We shall refer to %S as standard representation with respect to ∗S . A representation with respect to the homogeneous Fedosov star product of Weyl type ∗F can be constructed by use of the equivalence transformations V and T but we shall see in the next section that there is another star product ∗W of Weyl type with a corresponding representation. Now we compute an explicit formula for the representation %S and rediscover the well-known standard order quantization rule for cotangent bundles (see (19)): Theorem 9. Let ∗S be the homogeneous Fedosov star product of standard ordered type and let %S be the corresponding standard representation. Then for f ∈ C ∞ (T ∗ Q)[[λ]] and ψ ∈ C ∞ (Q)[[λ]] we have %S (f )ψ =
r ∞ X 1 λ ∂rf is (∂qi1 ) · · · is (∂qir )D0(r) ψ, i∗ r! i ∂pi1 · · · ∂pir
(93)
r=0
where D0(r) ψ := r!1 (dq k ∨∇0 ∂qk )r ψ is the rth symmetrized covariant derivative of ψ with ∞ (T ∗ Q) polynomial in the momenta respect to ∇0 . In particular for a function f ∈ Cpp,k of order k we have 1 %S (f )ψ = k!
k D E λ F, D0(r) ψ , i
(94)
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M. Bordemann, N. Neumaier, S. Waldmann
Wk where F ∈ 0( T Q) is the symmetric tensor field such that f = Fb and h·, ·i denotes b is linear in the momentum where X ∈ 0(T Q) then the natural pairing. If X %S (f )ψ =
λ LX ψ. i
(95)
Proof. We compute %S (f )ψ using (92) and Proposition 8 and obtain %S (f )ψ r ∞ X 1 λ = i∗ σ is (∂pi1 ) · · · is (∂pir )τS (f ) σ is (∂qi1 ) · · · iS (∂qir )τ0 (ψ) . r! i r=0
But since τ0 is the Taylor series with respect to ∇0 we find that the last part σ(is (∂qi1 ) · · · is (∂qir )τ0 (ψ)) equals is (∂qi1 ) · · · is (∂qir )D0(r) ψ and thus we only have to r f prove that i∗ σ(is (∂pi1 ) · · · is (∂pir )τS (f )) coincides with i∗ ( ∂pi ∂···∂ ). But this follows pi 1 r directly from the following lemma: Lemma 7. Let f ∈ C ∞ (T ∗ Q), then in any local bundle chart there exist locally defined elements τ˜ir (f ) such that for any total degree r ≥ 0, τS (f )(r) = D(r) f + dq i ∨ τ˜ir (f ).
(96)
Proof. This lemma is proved by a straightforward induction on the total degree using the recursion formulas (30) for τS . ∞ (T ∗ Q)[λ] as well as the Corollary 10. The restriction of %S to the C[λ]-submodule Cpp ω ∗ restriction to the C[[λ]]-submodule Cp (T Q)[[λ]] is injective.
7. Two Different Homogeneous Star Products of Weyl Type In this section we shall construct an analogue to the operator N mentioned in the introduction. This operator allows us to define a star product of Weyl type ∗W equivalent to ∗S which corresponds to the Weyl ordering prescription of flat R2n and turns out to be different from ∗F in general. In this section we sometimes denote the complex conjugation by C, i. e. Ca := a for a ∈ W ⊗3 and define CS := eadS (π
∗
h)
SCS −1 e−adS (π
∗
h)
,
(97)
where h ∈ W(Q)cl is constructed as in Theorem 7 with σ(h) = 0 incorporating a V1 particular but fixed choice of a real one-form α ∈ 0( T ∗ Q) such that −dα = trR0 . Then we have the following lemmata: Lemma 8. Let a, b ∈ W ⊗3 with dega a = ka and dega b = lb. Then C(a ◦F b) = (−1)kl (Cb) ◦F (Ca)
(98)
CS (a ◦S b) = (−1)kl (CS b) ◦S (CS a),
(99)
[DS , CS ] = 0 = [H, CS ] .
(100)
and [DF , C] = 0 and
Homogeneous Fedosov Star Products on Cotangent Bundles I
385
Proof. Equation (98) and the fact [DF , C] = 0 are well-known (see e. g. [5, Lemma 3.3]) and imply together with (79) the other statements of the lemma. Lemma 9. Let f, g ∈ C ∞ (T ∗ Q)[[λ]]. Then the map CS f := σ (CS τS (f ))
(101)
defines an involutive anti-C[[λ]]-linear anti-automorphism of the star product ∗S , i. e. CS 2 = id and CS (f ∗S g) = (CS g) ∗S (CS f )
(102)
and we have τS (CS f ) = CS τS (f ) and [CS , H] = 0. Moreover CS C is a formal series of differential operators CS C = id +
∞ X
λr CS (r) ,
(103)
r=1
where CS (r) is a differential operator of order 2r and CS (1) = 1i 1 and 2
1 := ∇0 (∂p
i
,∂ hi ) q
+ ∇0(αv )
(104)
which is clearly globally defined and takes the following form in a bundle chart 1 = ∂qi ∂pi + pr π ∗ (0rij )∂pi ∂pj + π ∗ (0iij )∂pj + π ∗ (αj )∂pj .
(105)
Proof. Equation (102) is proved in a completely analogous manner as Proposition 1. Since the term τS (·)(k) of total degree k is easily seen to be of order k as a differential operator and since CS does obviously not decrease the total degree (S being of total degree 0 and π ∗ h contains only terms of positive total degree) Eq. (103) follows. Moreover CS (1) = −i1 follows by straightforward computation using τS (f )(2) = 21 (dxk ∨ ∇0 ∂xk )2 f due to (30) and h1 = − 21 α ⊗ 1, where h1 is the term of symmetric degree 1 in h due to (83). Now we come to a simple analogue of the operator N mentioned in (4): λ 1 N := exp 2i
(106)
which clearly is a formal power series in λ of differential operators and [N, H] = 0. Motivated by Eq. (12) we define an equivalent star product to ∗S using N as an equivalence transformation by f ∗W g := N −1 ((N f ) ∗S (N f ))
(107)
for f, g ∈ C ∞ (T ∗ Q)[[λ]] which is clearly bidifferential and homogeneous. Furthermore we have the following theorem: Theorem 10. (i) The operator N 2 C coincides with CS . (ii) The star product ∗W is a star product of Weyl type where in particular f ∗ W g = g ∗W f for f, g ∈ C ∞ (T ∗ Q)[[λ]].
(108)
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M. Bordemann, N. Neumaier, S. Waldmann
(iii) A representation %W of (C ∞ (T ∗ Q)[[λ]], ∗W ) on C ∞ (Q)[[λ]] is given by the following analogue of Eq. (5) %W (f ) := %S (N f ).
(109)
Proof. (i) We proceed in two steps: Let us first prove that N 2 C is an involutive antilinear anti-automorphism of (C ∞ (T ∗ Q)[[λ]], ∗S ). This is equivalent to the identity N 2 C(f ∗S g) − (N 2 Cg) ∗S (N 2 Cf ) = 0 for all f, g ∈ C ∞ (T ∗ Q)[[λ]]. Since ∗S consists in bidifferential operators this identity is an identity of bidifferential operators for each power of λ. Hence it suffices to check it on an arbitrarily small neighbourhood of each point m ∈ T ∗ Q on ∞ ∞ (T ∗ Q). Since the standard representation %S is injective on Cpp (T ∗ Q)[λ] f, g ∈ Cpp it is sufficient to prove this identity after having applied %S to the left-hand side. Now let q ∈ Q be arbitrary and choose a contractible open neighbourhood U around q which lies in the domain of a chart. Suppose that the supports of f, g lie both in π −1 (U ). Then there is a local volume form µ on U such that ∇0 X µ = α(X)µ for each vector field X on U : indeed, choose an arbitrary local volume form µ0 on U . Then ∇0 X µ0 = α0 (X)µ0 for a certain locally defined one-form α0 on U . Since dα0 (X, Y )µ0 = ([∇0 X , ∇0 Y ] − ∇0 [X,Y ] )µ0 = −(trR)(X, Y )µ0 = dα(X, Y )µ0 it follows from the Poincar´e lemma that there is a local real-valued smooth function φ on U such that α0 = α + dφ. Then µ := e−φ µ0 will clearly do the job. Consider next the space D(U ) of all smooth complex-valued functions on Q whose support lies in U and is compact. This R space is an inner product space with respect to the Lebesgue integral hφ, ψiU := U φψµ. We define the following covariant divergence operator Wk Wk−1 Wl divα : 0( T Q) → 0( T Q) with 0( T Q) := {0} for negative integers l: divα S := divS + is (α)S,
where
divS := is (dq i )∇∂qi S
(110)
Wk+1 which is clearly globally defined. Let T be in 0( T Q). Using the global coordinates (q 1 , . . . , q n ) in U and D0 := dq i ∨ ∇0 ∂i we get for any ψ ∈ D(U ): 1 T i1 ···ik+1 is (∂i1 ) · · · is (∂ik+1 )D0k+1 ψ (k + 1)! 1 i1 ···ik+1 j k = L∂j T is (∂i1 ) · · · is (∂ik+1 ) dq ∨ D0 ψ (k + 1)! 1 (∇0 ∂j T )i1 ···ik+1 is (∂i1 ) · · · is (∂ik+1 ) dq j ∨ D0k ψ − (k + 1)! 1 0rrj T i1 ···ik+1 is (∂i1 ) · · · is (∂ik+1 ) dq j ∨ D0k ψ + (k + 1)! 1 ji1 ···ik k = L∂j T is (∂i1 ) · · · is (∂ik )D0 ψ k! 1 − (∇0 ∂j T )ji1 ···ik is (∂i1 ) · · · is (∂ik )D0k ψ k! 1 + 0rrj T ji1 ···ik is (∂i1 ) · · · is (∂ik )D0k ψ. k! Since for any vector field X on Q we obviously have LX µ = (divα X)µ it can easily be seen by partial integration and induction that the following identity is true:
Homogeneous Fedosov Star Products on Cotangent Bundles I
Z U
φ%S (Tb)ψ µ =
Z = U
Z = U
Z U
1 k!
387
! k λ k (divα ) (φT ) ψ µ i
! k X k i1 ···ik−s k 1 λ k−s s (divα ) T is (∂i1 ) · · · is (∂ik−s )D0 φ ψµ s k! i s=0 ! ! Z k X 1 λ b φ ψ µ, \ (divα )s T %S N 2 T φ ψµ= %S s! i U s=0
sb s \ where in the last equality we have used the identity (div α ) T = 1 T for s ∈ N which follows from
\ b div α T = 1T
(111)
which can easily be proved by calculation. It follows that on U the differential b ). By the well-known operator %S (Tb) has a formal adjoint %S (Tb)† given by %S (N 2 T rules of multiplying adjoints, viz. † b b † %S (Tb)† %S (Tb)%S (S) = %S (S) we finally get that N 2 C is an involutive anti-linear anti-automorphism since clearly N 2 CN 2 C = id. Secondly, since now both CS and N 2 C are involutive anti-linear anti-automorphisms the composition CS −1 N 2 C will be a linear automorphism of the star product ∗S of standard ordered type which again is a formal series in λ whose coefficients consist of differential operators. Hence it suffices to prove the desired identity CS −1 N 2 C = ∞ (T ∗ Q). Take again the above neighbourhood U . By id locally on functions in Cpp Proposition 2 every such function can be written as a finite linear combination ∞ (T ∗ Q)[λ] + over terms consisting of star products of functions belonging to Cpp,0 ∞ ∗ Cpp,1 (T Q)[λ]. By the automorphism property it thus suffices to check our identity on these functions at most linear in the momenta. An easy computation using Lemma 9 shows that this is the case. (ii) This is straightforward: using the fact that N 2 C is an involutive anti-linear antiautomorphism of ∗S it follows that complex conjugation is an involutive anti-linear anti-automorphism of ∗W . Moreover since all constructions depend only on the combination λi it follows that the parity transformation generated by λ 7→ −λ is an anti-automorphism which shows that ∗W is indeed of Weyl type. (iii) This is a straightforward computation. Remark. The two star products ∗W and ∗F are not the same in general as we shall see evaluating both star products on functions linear in the momenta: Let X, Y ∈ 0(T Q) and b =X b + λ divα X b Yb ∈ C ∞ (T ∗ Q). Then one gets by direct calculation N X consider X, pp,1 2i as well as b Yb ) = Xdiv b α Y + Yb divα X + ∇ \ \ 1(X 0 X Y + ∇0 Y X,
(112)
388
M. Bordemann, N. Neumaier, S. Waldmann
b Yb ) = 2(divα X)(divα Y ) + LX (divα Y ) + LY (divα X) 12 (X +divα (∇0 X Y ) + divα (∇0 Y X) . b = λ (LX + 1 divα X) and Using Theorem 9 one obtains %S (N X) i 2 λ 2 λ \ b Yb = %S X L X L Y − %S ∇ 0X Y . i i
(113)
(114)
b S (N Yb ) and using the injectivity of %S on functions This enables us to calculate %S (N X)% polynomial in the momenta (Corollary 10) and the representation property of %S we obtain λ \ 1b 1b b b b b ∇0 X Y + Y divα X + Xdivα Y (N X) ∗S (N Y ) = X Y + i 2 2 2 1 1 λ LX (divα Y ) + (divα X)(divα Y ) . + 2 i 2 b ∗S (N Yb )) and obtain b ∗W Yb = N −1 ((N X) Finally we calculate X n o 2 b Yb + iλ M W (X, b Yb ), b Yb + iλ X, b ∗W Yb = X X 2 2 2 where b Yb ) = M2W (X,
(115)
1 LX (divα Y ) + LY (divα X) − divα (∇0 X Y ) − divα (∇0 Y X) 2 (116)
observing that LX (divα Y ) − LY (divα X) − divα (∇0 X Y ) + divα (∇0 Y X) = 0 due to dα(X, Y ) = −trR0 (X, Y ). Writing the coordinate expression X|lk Y|kl for trace(Z 7→ ∇0 ∇0 Z X Y ) to avoid clumsy notation the operator M2W can be simplified to b Yb ) = −X k Y l − 1 Ric0 (X, Y ) + Ric0 (Y, X)−(∇0 X α) (Y )−(∇0 Y α) (X) , M2W (X, |l |k 2 (117) where |k denotes the covariant derivative with respect to ∂qk and Ric0 denotes the Ricci b and Yb tensor of ∇0 . On the other hand we compute the Fedosov star product ∗F of X using [5, Theorem 3.4] and obtain here the following expression for the second order b Yb ): term M2F (X, b Yb ) = −X k Y l . M2F (X, |l |k
(118)
For general manifolds Q with torsion-free connection ∇0 these two star products do not coincide for any choice of α: for example, let Q be equal to S 2 with the standard metric. Its Levi–Civita connection ∇0 is clearly unimodular whence every possible α is a closed one-form, hence exact (α = dφ) since the first de Rham cohomology group of the two-sphere vanishes. If the two expressions (117) and (118) were the same for any X, Y then the Ricci tensor would be equal to the second covariant derivative of φ. In particular upon contracting with the inverse metric we would obtain that the Laplacian of φ were equal to a positive multiple of the scalar curvature of S 2 which is positive and constant. But this differential equation has no smooth solution since the integral (with respect to the Riemannian volume) over S 2 of the Laplacian of φ would vanish as opposed to the integral over S 2 of a positive constant.
Homogeneous Fedosov Star Products on Cotangent Bundles I
389
8. Example: The Cotangent Bundle of a Lie Group This section shall be dedicated to finding explicit formulae for the homogeneous Fedosov star product ∗S of standard ordered type on the cotangent bundle of a connected Lie group G, that is equipped with the natural torsion-free connection defined by ∇0 U V := 21 [U, V ] for left-invariant vector fields U, V on G, obtained using the standard representation %S for functions polynomial in the momentum variables. To express the bidifferential operators defining the star product we make use of the natural set of basis sections in the tangent bundle of T ∗ G given by Yi resp. Z j that are the horizontal resp. vertical lifts of a basis of left-invariant vector fields Xi resp. the dual left-invariant one-forms θj with respect to the e U V = 0 for left-invariant flat connection on G (see Definition 2) which is defined by ∇ vector fields U , V . Instead of using Darboux coordinates it will be convenient to use natural fibre-variables Pi (1 ≤ i ≤ dim(G)) given by Pi : T ∗ G → R : αg → αg (Xi ) for αg ∈ Tg∗ G. The obtained expression for ∗S shall moreover be related to a star product ∗G of Weyl type by means of an equivalence transformation N and we shall prove that ∗G coincides (up to a rescaling of the formal parameter) with the star product obtained by Gutt in [17] using cohomological methods instead of our purely algebraic approach. As a first step we shall find an expression for the star product ∗S of polynomial functions in the momenta on T ∗ G that are invariant under the canonical lift T ∗ (lg ) of the lefttranslations lg : G → G to T ∗ G. Since those polynomials are generated by functions ∨k for left-invariant vector fields U on G, we may restrict our calculations of the form Ud to functions ∞ X r 1 U (αg ) ∈ Cpω (T ∗ G)[[λ]]. eU : αg 7→ r! r=0
Lemma 10. Let U, V be left-invariant vector fields on G, then we have the formula [13, 17] e U ∗S e V = e i
λH
λ
i
U, λi V
,
(119)
at which H denotes the Baker–Campbell–Hausdorff series. Moreover this equation uniquely determines bidifferential operators Mrinv defined by eU ∗S eV =:
∞ r X λ r=0
i
Mrinv (eU , eV ),
(120)
that are of order r in every argument, homogeneous of degree −r and only containing derivatives with respect to vertical directions and multiplications with polynomials in the momenta on T ∗ G, that are invariant under T ∗ (lg ). Proof. We just have to notice that %S (eU )χ is given by exp(◦ λi U )χ for χ ∈ C ∞ (G), at which ◦ denotes the composition of the left-invariant vector fields viewed as differential operators on C ∞ (G). Using this fact yields λ λ χ %S (eU ∗S eV )χ = (%S (eU ) ◦ %S (eV )) χ = exp ◦ U ◦ exp ◦ V i i λi λ λ χ. = exp ◦ H U, V χ = %S e i λ U, λi V iλ i i λH i
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Now both arguments eU ∗S eV and e i
λH
λ
i
U, λi V
are elements in the submodule
Cpω (T ∗ Q)[[λ]] and hence Corollary 10 ensures that they coincide. The assertions about the bidifferential operators Mrinv are obvious consequences of properties of the Baker– Campbell–Hausdorff series resp. of Proposition 7. Proposition 11. The star product ∗S of two functions f, g ∈ C ∞ (T ∗ G) is given by ∞ r X r X 1 inv j1 λ M (Z . . . Z jt f, Yj1 . . . Yjt g). (121) f ∗S g = i t! r−t r=0
t=0
By Lemma 10 it is obvious that this star product is of Vey type. Proof. First we notice that it is sufficient to prove (121) for functions polynomial in b = 1 λ l S j1 ...jl Xj . . . Xj χ for Sb ∈ C ∞ (T ∗ G) and the momenta. Using %S (S)χ 1 l pp,l l! i ∞ an analogous formula for Tb ∈ Cpp,k (T ∗ G),which are obtained from D0k χ = (θl ∨ ∇0 Xl )k χ = Xi1 . . . Xik χ θi1 ∨ . . . ∨ θik and the symmetry of S resp. T , we get for χ ∈ C ∞ (G) b ∗S Tb)χ %S ( S (a)
l X
=
t=0 (b)
= %S
1 k!t!(l − t)!
= %S
λ i
l ∞ X λ rX r=0
(c)
t
i
t=0
r ∞ X λ rX r=0
i
t=0
S j1 ...jl (Xj1 . . . Xjt T i1 ...ik )%S (Pjt+1 · · · Pjl ∗S Pi1 · · · Pik )χ
1 t!(l − t)!
t λ i
!
Mrinv
π ∗ S j1 ...jl Pjt+1 · · · Pjl , Yj1 . . . Yjt Tb
χ
!
1 inv b, Yj1 . . . Yjt Tb Z j1 . . . Z jt S M t! r−t
χ.
Equation (a) is a consequence of the Leibniz rule and k 1 X λ %S (Pi1 · · · Pik )χ = Xiσ(1) · · · Xiσ(k) χ. i k! σ∈Sk
In (b) we used Lemma 10 and the fact that %S (π ∗ ψF ) = ψ%S (F ) for ψ ∈ C ∞ (G) and the π-relatedness of the vector fields Xi and Yi . In (c) the sum over t was extended to ∞ since the terms Z j1 . . . Z jt Sb vanish for t > l due to Lemma 14. Then (121) follows ∞ by the injectivity of %S restricted to Cpp (T ∗ G) proved in Corollary 10. Now according to the last section we consider a Weyl ordered star product defined by f ∗G g := N −1 (N f ∗S N g) andN := exp
λ 1 2i
(122)
with 1 = Yi Z i + 21 Clil Z i +π ∗ (αi )Z i (which is easily computed using Eq. (104)) denoting by αi the components of a one-form fulfilling dα = −trR0 = 0 due to the particular k := θk ([Xi , Xj ]) are the structure constants of the choice of the connection ∇0 , where Cij Lie algebra g of G. Choosing α = (t − 21 )Clil θi for t ∈ R it is easily verified that dα = 0 λ λ Yi Z i and At := exp 2i tdS with dS := Clil Z i , since and N = N0 At with N0 := exp 2i Yi Z i and dS commute. For any choice of t ∈ R we shall prove the coincidence of ∗G with the one constructed in [17].
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Lemma 11. For α = (t − 21 )Clil θi (t ∈ R) and N defined as above we have eU ∗G eV = eU ∗S eV for all left-invariant vector fields U, V on G.
(123)
Moreover At is a one-parameter-group of automorphisms of ∗S , i. e. At As = At+s and A0 = id and ∀f, g ∈ C ∞ (T ∗ G)[[λ]], ∀t ∈ R,
At (f ∗S g) = (At f ) ∗S (At g)
(124)
implying that dS is a derivation of ∗S . Moreover this implies that all choices of t ∈ R lead to the same star product ∗G . Proof. Since N = N0 At we prove the first assertion in two steps using N0 and At separately. Obviously we have N0 eU = eU and therefore using Eq. (119) N0 −1 ((N0 eU ) ∗S (N0 eV )) = N0 −1 (eU ∗S eV ) = N0 −1 e i
λH
λ
i
U, λi V
=e i
For the second part we compute %S (At eU ) = exp %S ((At eU ) ∗S (At eV )) = exp
λH
λ
i
λ l i 2i tCli U
U, λi V
= e U ∗S e V .
%S (eU ) yielding
λ l i i tC (U + V ) %S (eU ∗S eV ). 2i li
On the other hand we have again by Eq. (119), λ l i %S (At (eU ∗S eV )) = exp tCli (U + V i ) %S e i λ U, λi V 2i λH i λ l i = exp tC (U + V i ) %S (eU ∗S eV ) 2i li using the shape of the Baker–Campbell–Hausdorff series and the fact that Clil [W, X]i = 0 for any left-invariant vector fields W, X on G. By Corollary 10 this implies A−1 t ((At eU ) ∗S (At eV )) = eU ∗S eV . Since dS only contains derivatives with respect to vertical directions this implies that At is even an automorphism of ∗S for all t ∈ R, proving the second assertion. The derivation property of dS is an immediate consequence of (124). Observe that an equation analogous to (124) does not hold for N0 , instead we have the following lemma: Lemma 12. For any f ∈ C ∞ (T ∗ G) and χ ∈ C ∞ (G) we have π ∗ χ ∗G f =
r ∞ X 1 iλ π ∗ (Xi1 . . . Xir χ)Z i1 . . . Z ir f. r! 2
(125)
r=0
Proof. The proof is a lengthy but straightforward computation using Proposition 11 and the fact that all terms containing Z-derivations applied to π ∗ χ vanish due to Lemma 14.
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Lemma 13. For any left-invariant vector fields U, V on G we have b ∗G eV = eV U Expressing the term Wr of order
λ r i
V, ·] [ λi\
exp([ λi V, ·]) − 1
U.
(126)
in components this reads
r
l k1 kr b , eV ) = (−1) Br Pl (Z j U b )C j1 · · · C jr−1 Wr ( U jk1 jr−2 kr−1 Cjr−1 kr (Z . . . Z eV ), r! (127)
where Br denotes the rth Bernoulli number. Proof. By Lemma 11 eU ∗S eV = eU ∗G eV and therefore we have V, ·] [ λi\ d d b U etU ∗G eV = e i H(t λ U, λ V ) = eV U ∗G eV = λ i i dt t=0 dt t=0 λ exp([ i V, ·]) − 1 using that for a, b ∈ g the following is valid adb d a. H(ta, b) = dt t=0 exp(adb) − 1 Then Eq. (127) is a direct consequence of the Taylor series expansion of
x ex −1 .
Collecting our results we get the following proposition. Proposition 12. The star product ∗G being related to the Fedosov star product of standard ordered type ∗S by means of the equivalence transformation N coincides with the product constructed in [17]. Proof. The assertion follows from Lemma 12 and Lemma 13 and the theorem proven in Chapter 4 in [17]. A. Homogeneous Connections on Cotangent Bundles We now shall briefly recall some well-known basic definitions concerning horizontal and vertical lifts of vector fields, homogeneous connections and normal Darboux coordinates on cotangent bundles. Definition 2. Let ∇0 be a connection on Q, then we consider the connection mapping K : T (T ∗ Q) → T ∗ Q of ∇0 defined by K(α(0)) ˙ := ∇0 ∂t α |t=0 ∗
for a curve α in T Q. It turns out that (T π × K) : T (T ∗ Q) → T Q ⊕ T ∗ Q is a fibrewise isomorphism. Because of this fact the notion of horizontal resp. vertical lifts with respect to ∇0 is well-defined by: X h ∈ 0(T (T ∗ Q)) is called the horizontal lift of X ∈ 0(T Q) iff T πX h = X ◦ π and K(X h ) = 0
(128)
resp. β v ∈ 0(T (T ∗ Q)) is called the vertical lift of β ∈ 0(T ∗ Q) iff K(β v ) = β ◦ π and T πβ v = 0.
(129)
Homogeneous Fedosov Star Products on Cotangent Bundles I
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Working in a local bundle Darboux chart (q 1 , . . . , q n , p1 , . . . , pn ) one finds that and β v = π ∗ βi ∂pi , (130) X h = π ∗ X i ∂qi + π ∗ (X k 0jki )pj ∂pi at which 0jki denotes the Christoffel symbols of ∇0 and X i resp. βi the components of X resp. β. The following lemma should be well-known and is crucial throughout this paper. It can be proved easily in a local bundle chart: Nl ∗ ∗ T (T Q)), where l ≥ 0, be a homogeneous tensor field of Lemma 14. Let T ∈ 0( degree k, i. e. Lξ T = kT . Then k ∈ Z and T = 0 if k < 0 and T = π ∗ T˜ with T˜ ∈ Nl ∗ 0( T Q) iff k = 0. Moreover [Lξ , Lβ v ] = −Lβ v and [Lξ , im (β v )]T = −im (β v )T , where im denotes the substitution into the mth argument of T . Hence Lξ T = kT implies Lξ (Lβ v T ) = (k − 1)T as well as Lξ (im (β v )T ) = (k − 1)im (β v )T . Definition 3. A connection ∇ on T ∗ Q is said to be homogeneous iff Lξ ∇ = 0, i. e. Lξ ∇X Y − ∇Lξ X Y − ∇X Lξ Y = 0 for all X, Y ∈ 0(T (T ∗ Q)). Definition 4. The remark about (T π ×K) being a fibrewise isomorphism justifies defining a connection on T ∗ Q by disposing of ∇0 for X, Y ∈ 0(T Q), β, γ ∈ 0(T ∗ Q) in the following manner: ∇0X h Y h αq v := (∇0 X Y )h + αq 21 R(X, Y ) · − 16 (R( · , X)Y + R( · , Y )X) , (131) αq
αq
∇0X h β v αq := (∇0 X β)v |αq , ∇0β v X h α := ∇0β v γ v α := 0. q
(132) (133)
q
It is straightforward to check that ∇0 is homogeneous, torsion-free and symplectic. A simple calculation using a bundle Darboux chart yields the following local expressions xk
xk
for the Christoffel symbols 00 xi xj and the components of the curvature tensor R0 xi xj xl = dxk (∇0∂ j ∇0∂ l ∂xi − ∇0∂ l ∇0∂ j ∂xi ) of the connection ∇0 : x
x
x
x
q
k
pj
pi
00 qi qj = −00 qi pk = −00 pk qj = π ∗ 0kij ,
pa ∗ π 20ajs 0ski − ∂qj 0aki + cycl.(ijk) , 3 l 1 j q p pl k j l R0 qk qi pj = π ∗ Rlki + Rkli R0 qk qi qj = −R0 pl qi qj = π ∗ Rkij , 3 pa ∗ pi a s s s π Rjlk|i − 30ais Rjlk − 0als Rijk + 0aks Rijl + (i ↔ j) . R0 q j q k q l = 3 All other combinations vanish and . . .|i denotes the covariant derivative with respect to ∂qi . At this instance we want to refer to a question that was brought to our interest by M. Cahen, namely whether the Ricci tensor Ric0 corresponding to R0 defined by Ric0 (X, Y ) := tr(Z → R0 (Z, X)Y ) for X, Y ∈ 0(T (T ∗ Q)) enjoys the property that (∇0X Ric0 )(Y, Z) + cycl.(X, Y, Z) = 0 for all X, Y, Z ∈ 0(T (T ∗ Q)). It turns out by a direct but lengthy calculation that this is the fact iff the Ricci tensor Ric0 corresponding to R0 does so with respect to ∇0 . In the next definition we shall briefly explain the concept of a connection on Q that is induced by a connection on T ∗ Q. pk
00 q i q j =
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Definition 5. Let ∇ be a torsion-free homogeneous connection on T ∗ Q. Choose any connection ∇Q on Q and define for X, Y ∈ 0(T Q): T i (∇0 X Y ) := ∇X h Y h ◦ i, where the horizontal lift is taken with respect to ∇Q . Then ∇0 X Y is a well-defined vector field on Q, and does not depend on the choice of ∇Q . We refer to ∇0 as the connection induced by ∇. Using this notion we can give a characterization of ∇0 as follows: Proposition 13. Let ∇0 be a torsion-free connection on Q. Among all homogeneous, torsion-free, symplectic connections ∇ on T ∗ Q which induce ∇0 on Q the connection ∇0 is uniquely characterized by the condition ia (X)δ −1 R = 0
(134) t
for all vertical vector fields X on T ∗ Q, where R := 41 ωit Rjkl dxi ∨ dxj ⊗ dxk ∧ dxl . Proof. The proof mainly relies on the fact that there is a uniquely determined tensor W3 ∗ T Q) such that field B ∈ 0(T Q ⊗ ∇X h Y h |αq = ∇0 X h Y h |αq +αq (B(X, Y, · ))v |αq for all X, Y ∈ 0(T Q), the horizontal lifts being taken with respect to ∇0 . Then the assertion follows by comparing R to R0 and the fact that (134) is fulfilled iff B = 0. Finally we shall prove the following three lemmata: Lemma 15. Let (q 1 , . . . , q n ) be a normal chart around q0 with domain Dq of Q with respect to ∇0 . Then for any αq ∈ π −1 (Dq ) we have l 00 ijk (x)xi xj xk α := ωil 00 jk (x)xi xj xk q
αq
= 0,
where (x1 , . . . , x2n ) = (q 1 , . . . , q n , p1 , . . . , pn ). Hence the bundle Darboux coordinates are normal Darboux coordinates with respect to ∇0 around αq0 . Proof. By direct calculation the assertion follows easily using the fact that (q 1 , . . . , q n ) xk
are normal coordinates on Q and the local expressions for 00 xi xj as stated above.
Lemma 16. Let M be a differentiable manifold and ∇ a connection on M . Then the trace of the curvature tensor R is exact, i. e. (trR)(X, Y ) := dxi (R(X, Y )∂i ) = dα(X, Y )
∀X, Y ∈ 0(T M )
for an α ∈ 0(T ∗ M ). If trR = 0 then the connection ∇ is called unimodular.
Homogeneous Fedosov Star Products on Cotangent Bundles I
395
Proof. First observe that the Levi–Civita connection ∇LC of a Riemannian metric is unimodular. Now since every manifold admits such a connection we shall compare this one to ∇. Let S ∈ 0(T M ⊗ T ∗ M ⊗ T ∗ M ) be the uniquely determined tensor field such that ∇LC X Y = ∇X Y + S(X, Y ). Straightforward computation using this equation and trRLC = 0 yields (trR)(X, Y ) = −tr (Z 7→ (S(T (X, Y ), Z) + S(X, S(Y, Z)) − S(Y, S(X, Z)) +(∇X S)(Y, Z) − (∇Y S)(X, Z))) , where T is the torsion of ∇. Since tr (Z → S(X, S(Y, Z)) − S(Y, S(X, Z))) = 0 and since ∇ commutes with contractions we get (trR)(X, Y ) = ((∇X α)(Y ) − (∇Y α)(X)) + α(T (X, Y )) = dα(X, Y ) for α(Y ) := −tr (Z → S(Y, Z)). The last lemma should be well-known and is used in Theorem 6: Lemma 17. Let ∇ be a torsion-free connection on a connected manifold M . Considering normal coordinates (x1 , . . . , xn )(dim(M ) = n) around an arbitrary point q ∈ M we have the following identity for the Taylor series of a function f ∈ C ∞ (M ) with respect to ∇, ∞ ∞ X X 1 r 1 ∂rf dxi1 ∨ . . . ∨ dxir , D f = τ0 (f )|q := i i 1 · · · ∂x r r! r! ∂x q q r=0
r=0
where D := dxk ∨ ∇∂xk . Proof. For the proof one has to observe that in any local coordinates Dr S = (dxi ∨ L∂xi − dxi ∨ dxk ∨ 0jik is (∂xj ))r S, where S is an arbitrary symmetric covariant tenPNr sor field on M . By induction this is equal to (dxi ∨ L∂xi )r S + a=1 Ca (Ka (S)), where Nr is some integer, Ka are certain differential operators and Ca = dxi1 ∨ · · · ∨ dxis+2 (∂xi1 · · · ∂xis 0jis+1 is+2 )Dj , where either Dj = L∂xj or Dj = is (∂xj ) and s ≥ 0. Finally note that in normal coordinates around q we have 0 = (∂xi1 · · · ∂xir 0akl )(q)dxk ∨ dxl ∨ dxi1 ∨ . . . ∨ dxir at q which is obtained by r-fold differentiation (with respect to the affine parameter) of the geodesic equation for a geodesic emanating at q. Hence Ca equals 0 at q which proves the lemma after having set S = f . Acknowledgement. The authors would like to thank D. Arnal, M. Cahen, C. Emmrich, S. Gutt, B. Kostant, and A. Weinstein for useful discussions.
References 1. Agarwal, G. S., Wolf, E.: Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators. Phys. Rev. D 2, 10, 2161–2188 (1970) 2. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation Theory and Quantization. Annals of Physics 111, part I: 61–110, part II: 111–151 (1978) 3. Bertelson, M., Cahen, M., Gutt, S.: Equivalence of Star Products. Class. Quant. Grav. 14, A93–A107 (1997) 4. Bordemann, M., Neumaier, N., Waldmann, S.: Homogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and Applications. Preprint Univ. Freiburg FRTHEP-97/23, November 1997, and q-alg/9711016
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5. Bordemann, M., Waldmann, S.: A Fedosov Star Product of Wick Type for K¨ahler manifolds. Lett. Math. Phys. 41, 243–253 (1997) 6. Bordemann, M., Waldmann, S.: Formal GNS Construction and States in Deformation Quantization. Commun. Math. Phys. 195, 549–583 (1998) 7. Bordemann, M., Waldmann, S.: Formal GNS Construction and WKB Expansion in Deformation Quantization. In: Sternheimer, D., Rawnsley, J., Gutt, S.: Deformation Theory and Symplectic Geometry. Mathematical Physics Studies 20, Dordrecht: Kluwer, 1997, pp. 315–319 8. Cahen, M., Gutt, S.: Regular ∗ Representations of Lie Algebras. Lett. Math. Phys. 6, 395–404 (1982) 9. Connes, A., Flato, M., Sternheimer, D.: Closed Star Products and Cyclic Cohomology Lett. Math. Phys. 24, 1–12 (1992) 10. DeWilde, M., Lecomte, P. B. A.: Star Products on Cotangent Bundles. Lett. Math. Phys. 7, 235–241 (1983) 11. DeWilde, M., Lecomte, P. B. A.: Existence of star products and of formal deformations of the Poisson Lie Algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7, 487–496 (1983) 12. DeWilde, M., Lecomte, P. B. A.: Formal Deformations of the Poisson Lie Algebra of a Symplectic Manifold and Star Products. Existence, Equivalence, Derivations. In: Hazewinkel, M., Gerstenhaber, M. (eds): Deformation Theory of Algebras and Structures and Applications. Dordrecht: Kluwer, 1988 13. Drinfel’d, V. G.: On constant quasiclassical solutions of the Yang–Baxter quantum equation. Soviet Math. Dokl. 28, 667–671 (1983) 14. Emmrich, C., Weinstein, A.: The differential geometry of Fedosov’s quantization. In: Brylinski, J.-L., Brylinski, R., Guillemin, V., Kac, V. (eds): Lie Theory and Geometry, in Honor of B. Konstant. Boston, Basel: Birkh¨auser, 1994 15. Fedosov, B.: A Simple Geometrical Construction of Deformation Quantization. J. Diff. Geom. 40, 213– 238 (1994) 16. Fedosov, B.: Deformation Quantization and Index Theory. Berlin: Akademie Verlag, 1996 17. Gutt, S.: An explicit ∗-Product on the cotangent bundle of a Lie Group. Lett. Math. Phys. 7, 249–258 (1983) 18. Nest, R., Tsygan, B.: Algebraic Index Theorem. Commun. Math. Phys. 172, 223–262 (1995) 19. Nest, R., Tsygan, B.: Algebraic Index Theorem for Families. Adv. Math. 113, 151–205 (1995) 20. Pflaum, M. J.: Local Analysis of Deformation Quantization. Ph.D. thesis, Fakult¨at f¨ur Mathematik der Ludwig-Maximilians-Universit¨at, M¨unchen, 1995 21. Pflaum, M. J.: The Normal Symbol on Riemannian Manifolds. New York J. Math. 4, 97–125 (1998) 22. Underhill, J.: Quantization on a manifold with connection. J. Math. Phys. 19, 1932–1935 (1978) 23. Widom, H.: Families of Pseudodifferential Operators. In: Gohberg, I., Kac, M. (eds): Topics in Functional Analysis. New York: Academic Press, 1978 24. Widom, H.: A Complete Symbol Calculus for Pseudodifferential Operators. Bull. Sci. Math. 104, 19–63 (1980) 25. Woodhouse, N.: Geometric Quantization. Oxford: Clarendon Press, 1980 Communicated by A. Connes
Commun. Math. Phys. 198, 397 – 406 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Monotonicity of the Lozi Family Near the Tent-Maps Yutaka Ishii1 , Duncan Sands2 1 Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan. E-mail:
[email protected] 2 Institute for Mathematical Sciences, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA. E-mail:
[email protected]
Received: 22 September 1997 / Accepted: 6 April 1998
Abstract: We show the monotonicity of the Lozi family when the Jacobian determinant is close to zero. The main ingredients of our proof are the “pruning pair method” and a detailed analysis of the parameter dependence of the kneading invariant of the tent-map family. 1. Introduction and Statement of Results In this article we study some dynamical properties of the two-parameter family of piecewise affine homeomorphisms of the plane known as the Lozi family: x 1 − a|x| + by 7−→ , a, b ∈ R, b 6 = 0. (1.1) L = La,b : y x This is like the H´enon family, but with |x| replacing x2 . R. Lozi [7] observed numerically that this simple mapping may give rise to very complicated dynamics: he found a strange attractor for the parameter values a = 1.7 and b = 0.5. A mathematical proof of the existence of attractors was given by Misiurewicz [9]. The purpose of the present paper is to show the monotonicity of the topological entropy and bifurcations of the Lozi family with respect to the parameter a when the Jacobian determinant −b is fixed sufficiently close to zero. Theorem 1 (Monotonicity of the topological entropy). For every a∗ > 1 there exists b∗ > 0 such that, for any fixed b with |b| < b∗ , the topological entropy of La,b is a non-decreasing function of a > a∗ . Here we define the topological entropy of La,b as follows: if b 6 = 0, then we extend La,b to a continuous map of the one-point compactification of R2 , and take the topological entropy of this map. If b = 0, then La,b maps R2 into the curve C ≡ {(x, y) | x = 1−a|y|}.
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Y. Ishii, D. Sands
We extend La,b to a continuous map of the one-point compactification of C, and take the topological entropy of this map. (Topological entropy is defined for continuous maps of compact spaces). We also consider the bifurcation problem. Theorem 2 (Monotonicity of bifurcations). Under the same assumptions as in Theorem 1, the bifurcations of the periodic orbits of La,b are monotone increasing in a (there is only orbit creation, and no annihilation, as a increases). Remark. Actually one can say more. For example, all new periodic points are created by tangencies of stable and unstable manifolds on the x-axis in this monotone region. Before that bifurcation the corresponding periodic orbit does not exist, and after that bifurcation it never bifurcates. See [6] for details. This extends some results proved by the first author in [6], where these monotonicity properties were established near the horseshoe parameters (these correspond to a ≥ 2 when b = 0). There are two main ingredients in our proof. The first is the “pruning pair method” for the Lozi mapping as developed in [5, 6]. In these papers, the first author constructed a symbolic dynamics for Lozi mappings similar to Milnor and Thurston’s kneading theory for unimodal maps [8]. He defined a two-dimensional analogue of the kneading sequence consisting of the pruning front Pa,b and the primary pruned region Da,b , together called the pruning pair (Pa,b , Da,b ), and constructed a topological model of the dynamics of La,b in terms of the pruning pair (Pa,b , Da,b ) alone (see Sect. 2 and [5]). This pruning method allows us to reduce our two-dimensional problem to an analysis of the one-dimensional tent-map family – the second main ingredient in our proof. When b = 0, the dynamics of La,b on R2 essentially reduces to that of the tent-map Ta : R → R, x 7→ 1 − a|x|. The topological entropy of Ta is min{log a, log 2} when a > 1, so it is a strictly increasing function of a when 1 < a < 2. The kneading invariant of the tent-family also increases strictly monotonically with respect to the parameter. In our proof we show that the kneading invariant increases “with positive speed”. This and the pruning pair method imply the claimed results. We assume throughout this paper that a > 1 + |b|. This ensures the absolute convergence of all expressions.
2. The Pruning Pair We use 6 to denote the symbol space {−1, +1}Z equipped with the standard product topology. By the shift map σ : 6 → 6 we mean the continuous map σ(. . . , ε−1 · ε0 , ε1 , . . . ) = (. . . , ε0 ·ε1 , ε2 , . . . ). For every element ε = (. . . , ε−2 , ε−1 ·ε0 , ε1 , . . . ) ∈ 6 we call εu ≡ (. . . , ε−2 , ε−1 ) the tail of ε, and εs ≡ (ε0 , ε1 , . . . ) the head of ε. Let C u (resp. C s ) be the set of all tails (resp. heads) of the elements of 6, so 6 = C u × C s . We define a two-dimensional analogue of the Milnor-Thurston kneading sequence [8] as follows [5]. The idea, following Cvitanovi´c et al. [3, 4], is to regard a Lozi mapping as an “incomplete horseshoe” and to measure its incompleteness with respect to the full shift (6, σ). Define p(. . . , ε−2 , ε−1 )(a, b) ≡ 1 − bs−2 + b2 s−2 s−3 − b3 s−2 s−3 s−4 + . . . , where
Monotonicity of the Lozi Family
399
1
sn ≡
,
b
−aεn + −aεn−1 +
b −aεn−2 + .
(2.1)
..
and q(ε0 , ε1 , . . . )(a, b) ≡ rˆ0 − rˆ0 rˆ1 + rˆ0 rˆ1 rˆ2 − . . . , where 1
rˆn ≡
.
b
aεn + aεn+1 +
(2.2)
b aεn+2 + .
..
See [5] for the dynamical interpretation of these quantities. We can consider p and q as functions p, q : 6 × R2>1+ → R. Here R2>1+ = {(a, b) ∈ R2 | a > 1 + |b|}. Lemma 3. For fixed ε ∈ 6, the functions p(ε) : R2>1+ → R, (a, b) → p(ε)(a, b) and q(ε) : R2>1+ → R, (a, b) → q(ε)(a, b) are real analytic; p, q and their partial derivatives with respect to a and b are continuous as functions 6 × R2>1+ → R. Proof. See [5].
Definition. We call Pa,b ≡ {ε ∈ 6 | (p − q)(ε)(a, b) = 0} the pruning front of La,b and Da,b ≡ {ε ∈ 6 | (p − q)(ε)(a, b) < 0}
(2.3)
the primary pruned region. The pair (Pa,b , Da,b ) is known as the pruning pair of La,b . We call Aa,b ≡ 6 \ ∪n∈Z σ n Da,b the admissible set. These definitions were introduced in [5]. In this paper it will be important to distinguish between Pa,b and the so-called admissible pruning front: Definition. The set Pˆ a,b ≡ Pa,b ∩ Aa,b is the admissible pruning front. When b = 0, the admissible pruning front Pˆ a,b is closely connected to Milnor and Thurston’s notion of a kneading invariant [8], see Proposition 9 below. The admissible set Aa,b corresponds to their admissible itineraries. The connection with Lozi dynamics is given by the following result. Suppose b 6 = 0. Let K = Ka,b be the set of all points whose forward and backward orbits remain bounded, so Ka,b is completely invariant under the application of La,b , non-empty and compact. For ε ∈ Aa,b (resp. ε0 ∈ Aa,b ), let M (resp. M0 ) be the set of all integers m such that σ m ε ∈ Pˆ a,b (resp. σ m ε0 ∈ Pˆ a,b ). We write ε ∼a,b ε0 if M = M0 and εn−1 = ε0n−1 for n ∈ Z \ M. Let σ/∼a,b : Aa,b /∼a,b → Aa,b /∼a,b be the factor of σ by the quotient map.
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Theorem 4 (Ishii). Suppose a > 1 + |b| and b 6 = 0. Then La,b : Ka,b → Ka,b is topologically conjugate to σ/∼a,b : Aa,b /∼a,b → Aa,b /∼a,b . Proof. See [5], Theorem 5.5. 3. Symbolic Dynamics for Tent-Maps The main result of this section is: Theorem 5. Suppose a > 1. Then ∂(p − q)(ε) (a, 0) > 0 ∂a
(3.1)
for all ε ∈ Pˆ a,0 . Since p(ε) and q(ε) depend only on the head εs when b = 0, we will use the following shorthand: q(ε)(a, 0) ≡ q(εs )(a), p(ε)(a, 0) ≡ p(εs )(a), so q : C s × R>1 → R, ∞ X ε0 · · · ε i (−1)i i+1 q(εs )(a) = a
(3.2) (3.3)
i=0
and p : C s × R>1 → R, p(εs )(a) = 1. Here R>1 = {a ∈ R | a > 1}. The proof of Theorem 5 is based on an analysis of tent-map combinatorics. Recall that the tent-map Ta : R → R is given by Ta (x) = 1 − a|x|. Definition. The itinerary of x ∈ R under Ta is ιa (x) ≡ {εs ∈ C s | εi Tai (x) ≥ 0 for all i ≥ 0}. We call κ(a) ≡ ιa (1) the kneading invariant of Ta . Remark. Alternatively, ιa : R → C s is the multivalued function given by: +1 if Tai (x) > 0 ∗ if Tai (x) = 0 ιa (x)i = −1 if Tai (x) < 0.
(3.4)
Here ∗ plays the role of “joker”, i.e. ∗ is both +1 and −1. Lemma 6 (Inverse of ιa ). Suppose 1 < a ≤ 2, x ∈ [−1/(a − 1), 1/(a − 1)] and εs ∈ ιa (x). Then x = q(εs )(a). The significance of the interval [−1/(a − 1), 1/(a − 1)] is that the forward orbit of x ∈ R under Ta is bounded if and only if x ∈ [−1/(a − 1), 1/(a − 1)]. Proof. Since εi Tai (x) ≥ 0 for all i ≥ 0, Tai+1 (x) = 1 − a|Tai (x)| = 1 − aεi Tai (x) for all i ≥ 0. This rearranges to Tai (x) = εi (1 − Tai+1 (x))/a, yielding ε0 ε 1 ε0 ε0 − 2 (1 − Ta2 (x)) = . . . (3.5) x = (1 − Ta (x)) = a a a n X ε0 · · · ε i ε0 · · · ε n (−1)i i+1 + (−1)n+1 n+1 Tan+1 (x) (3.6) = a a i=0
0. Since Tan+1 (x) s
for all n ≥ ∈ [−1/(a − 1), 1/(a − 1)] is bounded for all n ≥ 0, Eq. 3.6 converges to q(ε )(a) in the limit n → ∞.
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401
Corollary 7. Suppose 1 < a ≤ 2 and εs ∈ κ(a). Then ∞ X
(−1)i
i=0
ε0 · · · εi−1 = 0. ai
(3.7)
We define the empty product ε0 · · · ε−1 to equal 1. Proof. Take x = 1.
Definition. We call A+a,0 ≡ 6 \ ∪n<0 σ n Da,0 the forward admissible set. Lemma 8 (Admissibility). Suppose 1 < a ≤ 2. Then ε ∈ A+a,0 if and only if εs ∈ ιa (x) for some x ∈ [−1/(a − 1), 1/(a − 1)], and then x = q(εs )(a). Proof. Let us first prove: ε ∈ A+ ⇔ εi q(σ i εs )(a) ≥ 0, ∀i ≥ 0 ⇔ q(σ i εs )(a) = Tai (q(εs )(a)), ∀i ≥ 0. (3.8) Observe that q(σεs )(a) = 1 − aε0 q(εs )(a), so (p − q)(σεs )(a) ≥ 0 if and only if ε0 q(εs )(a) ≥ 0 if and only if q(σεs )(a) = Ta (q(εs )(a)). Now ε ∈ 6 \ σ −1 Da,0 if and only if (p − q)(σεs )(a) ≥ 0, so (3.8) follows from the above. If εs ∈ ιa (x) for some x ∈ [−1/(a − 1), 1/(a − 1)], then Lemma 6 implies that x = q(εs )(a) and moreover that Tai (q(εs )(a)) = Tai (x) = q(σ i εs )(a) for all i ≥ 0. Therefore ε ∈ A+ by (3.8). If ε ∈ A+ , then εi Tai (q(εs )(a)) = εi q(σ i εs )(a) ≥ 0 for all i ≥ 0 by (3.8), which implies that εs ∈ ιa (q(σ i εs )(a)). Moreover, q(εs )(a) ∈ [−1/(a − 1), 1/(a − 1)] because |q(εs )(a)| ≤ 1/a + 1/a2 + . . . = 1/(a − 1). Take x = q(εs )(a). The two-dimensional analogues of Lemmas 6 and 8 can be found in [5]. Denote by πs : 6 → C s the map ε 7→ εs . The following result gives the precise relationship between the one-dimensional pruning front and the kneading invariant: Proposition 9. Suppose 1 < a ≤ 2. Then πs (Pˆ a,0 ) = κ(a). Proof. If ε ∈ Pˆ a,0 then in particular ε ∈ A+ , so εs ∈ ιa (q(εs )(a)) by Lemma 8. Also (p − q)(εs )(a) = 0, so q(εs )(a) = 1 and εs ∈ κ(a). If εs = (ε0 , ε1 , . . . ) ∈ κ(a), then put ε = (. . . , +1, +1 · ε0 , ε1 , . . . ). We have q(ε)(a, 0) = q(εs )(a) = 1 by Corollary 6, so ε ∈ Pa,0 . Take . . . , x−1 , x0 in [0, 1] such that Ta (x−(i+1) ) = x−i for all i ≥ 0 and x0 = 1. Since (σ −i ε)s ∈ ιa (x−i ) for all i ≥ 0, from Lemma 8 we obtain ε ∈ ∩i≥0 σ i A+ = Aa,0 , so ε ∈ Pˆ a,0 . a,0
Remark. It is not difficult to show that Pˆ a,0 = ∅ for a > 2. The remainder of this section is dedicated to proving that (p − q)(εs )0 (a) > 0 when ε ∈ κ(a) (Corollary 13). s
Lemma 10. Suppose 1 < a ≤ 2 and εs ∈ κ(a). Then (p − q)(εs )0 (a) =
∞
1X ε0 · · · εi−1 i (−1)i Ta (1). a ai i=0
We define the empty product ε0 · · · ε−1 to equal 1.
(3.9)
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Proof. Note that p(εs )0 (a) = 0. We have s 0
q(ε ) (a) = −
∞ X
∞
∞
1 XX · · · εi ε0 · · · εi+j = − (−1)i+j . (3.10) ai+2 a ai+j+1
i ε0
(i + 1)(−1)
i=0
i=0 j=0
P∞
By Lemma 6, Tai (1) = q(σ i εs )(a) = j=0 (−1)j εi · · · εi+j /aj+1 . But this in turn is equal P∞ to ai (−1)i ε0 · · · εi−1 j=0 (−1)i+j ε0 · · · εi+j /ai+j+1 , which rearranges to give ∞ X
(−1)i+j
j=0
Thus q(εs )0 (a) = (−1/a)
ε0 · · · εi+j ε0 · · · εi−1 i = (−1)i Ta (1). i+j+1 a ai
P∞
i i=0 (−1) ε0
· · · εi−1 Tai (1)/ai .
(3.11)
Remark. If the critical point 0 is not periodic under Ta and εs ∈ κ(a), then (p − q)(εs )0 (a) =
∞ X ∂Ta /∂a(T i (0)) a
DTai (1)
i=0
dTan (1)/da . n→∞ DTan (1)
= lim
(3.12)
That is, (p − q)(εs )0 (a) gives the asymptotic ratio of the parameter and space derivatives at the critical value. The first equality follows from (3.9) and the second by a calculation. Several authors [2, 1, 10] have shown that the right-hand side of (3.12) is always positive if the critical point is not periodic. Our approach has the advantage of working regardless of whether the critical point is periodic or not. Lemma 11. Suppose 1 < a ≤ 2 and εs ∈ κ(a). Then a3 + 2a2 − 6a + 4 a3 + 2a2 − 6a + 2 ≤ (p − q)(εs )0 (a) ≤ . 2 2a (a − 1) 2a2 (a − 1) √ √ In particular, (p − q)(εs )0 (a) ≥ ( 2 − 1)/2 > 0 if a ≥ 2.
(3.13)
Proof. Lemma 10 and Corollary 7 show that (p − q)(εs )0 (a) =
∞
1X ε0 · · · εi−1 i (−1)i (Ta (1) − α) a ai
(3.14)
i=0
for any α. Put α = (1 + Ta (1))/2 and δ = (1 − Ta (1))/2 = a/2. Now Tai (1) ∈ [Ta (1), 1] for all i ≥ 0, so −δ ≤ Tai (1) − α ≤ δ for all i ≥ 0. Note that ε0 = +1, ε1 = −1, 1 − α = δ, Ta (1) − α = −δ and Ta2 (1) − α = −δ(2a − 3). It follows that (p − q)(εs )0 (a) =
Ta (1) − α Ta2 (1) − α 1 (1 − α + + a −a −a2 ∞ X ε0 · · · εi−1 i + (−1)i (Ta (1) − α)) ai
(3.15) (3.16)
i=3
≥
1 δ δ(2a − 3) 1 1 (δ + + − δ( 3 + 4 + . . . )), a a a2 a a
which gives the lower bound. The upper bound is obtained similarly.
(3.17)
Monotonicity of the Lozi Family
403
√ Lemma 12 (Renormalization). Suppose 1 < a ≤ 2 and εs = (ε0 , ε1 , . . . ) ∈ κ(a). Then εˆs ≡ (−ε1 , −ε3 , . . . ) ∈ κ(a2 ) and (p − q)(εs )0 (a) = 2(a − 1) · (p − q)(εˆs )0 (a2 ). Proof. We use a renormalization argument. Define : [−1/(a + 1), 1/(a + 1)] → [−1/(a2 − 1), 1/(a2 − 1)] by (x) = −x/(a − 1). It is well known and easy to check that Ta2 ◦ = ◦ Ta2
on [−1/(a + 1), 1/(a + 1)].
(3.18)
In particular, (Ta (1)) = 1, from which it follows that εˆs ∈ κ(a2 ) (note that is orientation reversing). Moreover, Ta2i (1) > 0 for all i ≥ 0, so ε0 = ε2 = · · · = +1. We therefore have (p − q)(εs )0 (a) = = =
∞
1X ε0 · · · εi−1 i (−1)i Ta (1) a ai 1 a
i=0 ∞ X
1 a2
(−1)i
i=0 ∞ X
Ta2i+1 (1) εˆ0 · · · εˆi−1 2i ) (T (1) − a a2i a
(−1)i
i=0
(3.19)
εˆ0 · · · εˆi−1 (1 − 2Ta2i+1 (1)), (a2 )i
(3.20) (3.21)
2i+1 (1) = 1 − aTa2i (1) to calculate Ta2i (1) (recall where we have used the identity TP a ∞ that Ta2i (1) > 0). By Corollary 7, i=0 (−1)i εˆ0 · · · εˆi−1 /(a2 )i = 0 since εˆs ∈ κ(a2 ). Equation 3.18 implies Ta2i+1 (1) = −1 (Tai2 (1)) = −(a − 1)Tai2 (1). Thus
(p − q)(εs )0 (a) = 2(a − 1)
∞ 1 X εˆ0 · · · εˆi−1 i (−1)i Ta2 (1) 2 a (a2 )i
(3.22)
i=0
= 2(a − 1) · (p − q)(εˆs )0 (a2 ).
(3.23)
Corollary 13. Suppose 1 < a ≤ 2 and εs ∈ κ(a). Then (p − q)(εs )0 (a) > 0. Proof. This follows from Lemmas 11 and 12. Remark. If the critical point 0 is periodic under Ta , then κ(a) is a Cantor set rather than a single point. This corresponds to a jump in Fig. 1. It is well known (cf. Theorem 16) that the tent-map family is monotone: if 1 < a1 < a2 ≤ 2, εs1 ∈ κ(a1 ), εs2 ∈ κ(a2 ), then εs1 <s εs2 . The dynamical order <s is given by: εs <s δ s if (−ε0 ) · · · (−εi−1 )εi < (−δ0 ) · · · (−δi−1 )δi where i ≥ 0 is minimal such that εi 6 = δi . The map (κ(a), <s ) → (R, <), εs 7→ (p − q)(εs )0 (a) can be shown to be injective and order reversing. It follows from these facts that, as a increases, the jumps in Fig. 1 are always downward. Proof of Theorem 5. If a > 2 then Pˆ a,0 = ∅, so we may assume that 1 < a ≤ 2. If ε ∈ Pˆ a,0 then εs ∈ κ(a) by Proposition 9, so ∂(p − q)(ε)/∂a(a, 0) ≡ (p − q)(εs )0 (a) > 0 by Corollary 13.
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Fig. 1. Graph of (p − q)()0 (a) and the bounds from Lemma 11 for 1.1 ≤ a ≤ 2 and ∈ κ(a)
4. Proof of Monotonicity In this section we prove a general criterion for local monotonicity in the Lozi family (Theorem 15), which combined with Theorem 5 implies global monotonicity for the Lozi family near b = 0 (Theorem 16), as well as the results announced in the introduction. We expect Theorem 15 to prove useful in showing monotonicity of the Lozi family for other parameter ranges. ˆ Suppose a > 1 + |b|. Then for every neighbourhood U of Lemma 14 (Stability of P). ˆ ∈V. ˆ a, b) Pa,b , there exists a neighbourhood V of (a, b) such that Pˆ a, ˆ bˆ ⊂ U for every (ˆ Proof. We may suppose that U is open, so B ≡ Pa,b ∩ U c is compact. Every ε ∈ B is non-admissible, so there exists i = i(ε) ∈ Z such that (p − q)(σ i ε)(a, b) < 0. By the continuity of p − q, we can find an open neighbourhood W (ε) of ε and a neighbourhood ˆ < 0 for all δ ∈ W (ε) and (ˆa, b) ˆ V (ε) of (a, b) such that (p − q)(σ i δ)(ˆa, b) S ∈ V (ε). Due to the compactness of B, there exist W (ε1 ), . . . , W (εk ) such that B ⊆ i W (εi ) ≡ W . T ˆ ∈ V1 . Now U ∪ W is a Put V1 ≡ j V (εj ). It follows that W ∩ Aa, a, b) ˆ bˆ = ∅ for all (ˆ ˆ in a neighbourhood a, b) neighbourhood of Pa,b and 6 is compact, so Pa, ˆ bˆ ⊂ U ∪W for (ˆ V2 of (a, b). Take V = V1 ∩ V2 . ˆ A) ≺ (Pˆ 0 , A0 ) if A ⊂ A0 and Pˆ 0 ∩ A = ∅. Definition. We say that (P, This definition is weaker than that in [6], which was unnecessarily strong.
Monotonicity of the Lozi Family
405
Theorem 15 (Local monotonicity). Suppose f : (−δ, δ) 7→ R2>1+ is C 1 and d(p − q)(ε)(f (t)) | t=0 > 0 dt
(4.1)
for all ε ∈ Pˆ f (0) . Then there exists a C 1 -neighbourhood F of f and a neighbourhood I of 0 such that, for any C 1 -curve g ∈ F , the map t ∈ I 7→ (Pˆ g(t) , Ag(t) ) is order preserving: if t1 , t2 ∈ I and t1 < t2 , then (Pˆ g(t1 ) , Ag(t1 ) ) ≺ (Pˆ g(t2 ) , Ag(t2 ) ). Proof. It follows from Lemma 3 and the compactness of Pˆ f (0) that there exists an open neighbourhood U of Pˆ f (0) , a C 1 -neighbourhood F of f and a convex neighbourhood I1 of 0 such that d(p − q)(ε)(g(t))/dt > 0 for all g ∈ F , t ∈ I1 and ε ∈ U . It follows that if g ∈ F , t, t0 ∈ I1 and t < t0 , then (p − q)(ε)(g(t)) < (p − q)(ε)(g(t0 ))
(4.2)
for all ε ∈ U . By Lemma 14, there exists a convex neighbourhood I2 of 0 such that Pˆ g(t) ⊂ U for all t ∈ I2 . Thus by the compactness of 6 \ U , (p − q)(ε)(g(t)) > 0
(4.3)
for all ε ∈ Ag(t0 ) \ U and any t, t0 ∈ I2 . Put I = I1 ∩ I2 . Suppose that g ∈ F, t1 , t2 ∈ I, t1 < t2 , and take any ε ∈ Ag(t1 ) , so (p − q)(σ n ε)(g(t1 )) ≥ 0 for all n ∈ Z. It follows from (4.2) and (4.3) that (p−q)(σ n ε)(g(t2 )) > 0 for all n ∈ Z. In particular, ε ∈ Ag(t2 ) and ε 6∈ Pˆ g(t2 ) . Theorem 16 (Global Monotonicity). For every a∗ > 1 there exists b∗ > 0 such that the map a ∈ (a∗ , ∞) 7→ (Pˆ a,b , Aa,b ) is order preserving for all |b| < b∗ . Proof. We have ∂(p − q)(ε)/∂a(a, 0) > 0 for all a > 1 by Theorem 5, so the result follows from Theorem 15. (By [6], Aa,b = 6 and Pa,b = ∅ whenever a > 2 + 2|b|, so there exists a fixed b∗ > 0 that works as a → ∞). Remark. If b = 0 then it follows from this that the kneading invariant of Ta is a strictly increasing function of a ∈ (1, 2]. Proof of Theorem 1. If b = 0 then htop (La,b ) = min{log a, log 2}. If b 6 = 0 then one can show that htop (La,b ) = htop (La,b Ka,b ). Theorem 4 implies that htop (La,b Ka,b ) ≤ htop (L ˆ ) if (Pˆ a,b , Aa,b ) ≺ (Pˆ ˆ , A ˆ ). The result therefore follows from Theoa, ˆ b Ka, ˆ bˆ
rem 16.
a, ˆ b
a, ˆ b
Theorem 2 can be deduced as in [6]. Acknowledgements. These results were obtained while the authors were visiting the Equipe de Topologie et Dynamique at the Universit´e de Paris-XI, in Orsay, France. We would like to thank the Equipe for their hospitality.
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References 1. Brucks, K. and Misiurewicz, M.: Trajectory of the turning point is dense for almost all tent maps. Ergodic Theory Dynamical Systems 16, 1173–1183 (1996) 2. Coven, E.M., Kan, I. and Yorke, J.A.: Pseudo-orbit shadowing in the family of tent maps. Trans. Am. Math. Soc. 308 (1), 227–241 (1988) 3. Cvitanovi´c, P.: Periodic orbits as the skeleton of classical and quantum chaos. Phys. D 5, 138–151 (1991) 4. Cvitanovi´c, P., Gunaratne, G.H., and Procaccia, I.: Topological and metric properties of H´enon-type strange attractors. Phys. Rev. A 38, 1503–1520 (1988) 5. Ishii, Y.: Towards a kneading theory for Lozi mappings I: A solution of the pruning front conjecture and the first tangency problem. Nonlinearity 10, 731–747 (1997) 6. Ishii, Y.: Towards a kneading theory for Lozi mappings II: Monotonicity of the topological entropy and Hausdorff dimension of attractors. Commun. Math. Phys. 190, 375–394 (1997) 7. Lozi, R.: Un attracteur e´ trange(?) du type attracteur de H´enon. J. Phys. (Paris), 39, 69–77 (1978) 8. Milnor, J. and Thurston, W.: On iterated maps of the interval. In: Dynamical Systems: Proceedings 1986–87, Lecture Notes in Mathematics 1342, Berlin–Heidelberg–New York: Springer-Verlag, 1988, pp. 465–563 9. Misiurewicz, M.: Strange attractors for the Lozi mappings. In: R. G. Helleman, editor, Nonlinear Dynamics, New York: The New York Academy of Sciences, 1980, Volume 357, pp. 348–358 10. Sands, D.: Topological conditions for positive Lyapunov exponent in unimodal maps. Preprint 95-59, Universit´e de Paris-XI (Orsay), 1995 Communicated by Ya. G. Sinai
Commun. Math. Phys. 198, 407 – 425 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Semiclassical Representation of the Scattering Matrix by a Feynman Integral Setsuro Fujii´e Mathematical Institute of Tohoku University, 980 Sendai, Japan. E-mail:
[email protected] Received: 14 November 1997 / Accepted: 6 April 1998
Abstract: We study the scattering problem for one-dimensional Schr¨odinger equations in the semiclassical limit when the energy level is close to the quadratic maxima of the potential. Starting from the formula of the scattering matrix obtained by the exact WKB method, we represent its principal term in a reduced form of the Feynman integral, an absolutely convergent sum of suitably defined probability amplitudes over countably many trajectories on R generated by classical trajectories and tunneling effects.
1. Introduction In this paper, we study the scattering problem for the Schr¨odinger equation in one dimension in the semiclassical limit: P u = −h2
d2 u + V (x)u = Eu. dx2
The energy level E is assumed to be in the disk D(rh) of radius rh centered at the maximum V0 of the short range potential V (x) in the complex plane for a positive constant r and sufficiently small h’s. Let n be the number of barriers whose height is exactly V0 . We assume that V (x) is quadratic at the tops of these barriers. [Ra] calculated the asymptotic expansion of the scattering matrix in the case n = 1 by applying on one hand the exact WKB method developed by [Ge-Gr] and on the other hand, the semiclassical microlocal method of [He-Sj] in order to reduce the Schr¨odinger operator to a normal form at the top of the barrier. [Fu-Ra]1 extended these methods to complex energies and made clear the location of resonances created by a heteroclinic orbit in the case n = 2. The existence of a heteroclinic orbit means, from the physical point of view, the long life-time of particles. Our aim is to associate the semiclassical scattering matrix with the motion of particles in the case where n ∈ Z.
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S.Fujii´e
First we calculate the scattering matrix S(E, h) recalling the method of [Fu-Ra]1 . We connect the Jost solutions with the exact WKB solutions by the so-called transition matrices. Then we obtain a formula of the transmission matrix T(E, h), from which one gets S(E, h) (see Eq. (7)), as a product of the transition matrices (see Eq. (10)). Let Sj (E, h), Sj,j+1 (E, h), Sl (E, h), Sr (E, h) be the classical action integrals associated to the j th and the j + 1th barriers and ±∞ the j th barrier top, the potential well between √ respectively (see Definition 1) and let N (z) = 2π exp{z log(z/e)}/0(1/2 + z). Then the principal term of each transition matrix is written in terms of ej (E, h) = exp(−Sj /h), ej,j+1 (E, h) = exp(iSj,j+1 /h), el (E, h) = exp(−iSl /h), er (E, h) = exp(−iSr /h) and of what is called in [L-T-M] barrier penetration factors Nj (E, h) = N (iSj /(πh)) (see ˆ and Sˆ be the matrices defined like T and S but with the principal Proposition 3). Let T ˆ term of the transition matrices (see Eq. (28), Eq. (29)). We will show that T(E, h) (resp. ˆS(E, h)) is a single-valued holomorphic (resp. meromorphic) function of E in D(rh) ˆ and that S(E, h) = S(E, h) + O(h log h) at least on the real axis of the energy plane. Remark that this is no longer true if |(V0 − E)/h| tends to infinity as h tends to 0. In this case, the asymptotic form of T, S are given sectorwise (see [Fu-Ra]1 for the case n = 2). Each component sjk (j, k = 1, 2) of S(E, h) is thus obtained asymptotically as a rational function of ej , ej,j+1 , el , er , Nj and their conjugates (e∗ (E, h) = e(E, h)). On the other hand, sjk is defined to be the scattering amplitude and describes the motion of particles on R coming from (−1)k ∞ and going away to (−1)j−1 ∞. So it is natural to expect to find another expression in accordance with such a physical meaning. Consider the trajectory of the motion of a particle. In classical mechanics, it always reflects at barriers if E < V0 and always transmits if E > V0 . But semiclassically, tunneling effects are not small when |V0 − E| = O(h). In other words, we must take into account both the transmission and the reflection effects at each barrier whether E is greater or smaller than V0 . Then we arrive at a natural definition of the semiclassical trajectory (see Definition 2). It is important to notice that there exist at most countably many semiclassical trajectories. Each semiclassical trajectory is generated by three mo± (0 ≤ j ≤ n) in tions: the motion along classical trajectories (which we call arc σj,j+1 Sect. 5), transmissions (denoted by τj ) and reflections (ρj ) at barriers oj (1 ≤ j ≤ n). We associate to each of these motions a complex number φ depending on E and h; ± ) = {e2j,j+1 /(Nj Nj+1 )}1/2 , φ(τj ) = ej , φ(ρj ) = i. Then we define the probability φ(σj,j+1 amplitude 8 for each semiclassical trajectory γ as the product of all φ’s corresponding to the motions which constitute γ (see Eq. (46)). Our main result (Theorem 1) shows that the principal term of sjk is equal to the sum of probability amplitudes for all (countably many) semiclassical trajectories coming from (−1)k ∞ and going away to (−1)j−1 ∞. This interpretation as a Feynman integral was suggested to the author by several physical works, for example [L-T, Ko and Br], on the three turning point problem for the radial Schr¨odinger equation reduced in one dimension where the scattering matrix is given by a sum of the so-called internal and external waves (see also [Fu-Ra]2 ).
2. Definition of the Scattering Matrix We study the one-dimensional Schr¨odinger equation − h2
d2 u + V (x)u = Eu. dx2
(1)
Representation of the Scattering Matrix by a Feynman Integral
409
In this paper, we work with dilation analytic potentials. More precisely, we suppose that there exist two real numbers 0 < θ0 < π/2 and δ > 0 such that: (H1) V is real valued on R and analytic in the complex domain S = {x ∈ C; | Im x| < | Re x| tan θ0 } ∪ {| Im x| < δ}. We will suppose that the potential V goes to 0 at infinity in S, and in order to simplify the statements, suppose that V is of short range, that is (H2) there exist C > 0, > 0 such that |V (x)| ≤ C(1 + |x|)−1− in S. Set 5θ = {E ∈ C\{0}; −2θ < arg E < 2θ}. Then for E ∈ 5θ0 , there exist Jost solutions fr± , fl± of Eq. (1) uniquely defined by the following asymptotic conditions: √
fr± ∼ e±i
√
fl± ∼ e±i
Ex/h
as
Re x → +∞ in
Ex/h
as
Re x → −∞
S,
in S.
Let W(f, g) = f 0 g − f g 0 be the wronskian. Then, the two couples of solutions (fl+ , fl− ), (fr+ , fr− ) form two bases of solutions of Eq. (1) since √ 2i E − + + − 6= 0. (2) W(fl , fl ) = W(fr , fr ) = h Let T(E, h) be the matrix of change of bases: ! ! fl+ fr+ =T . fr− fl−
(3)
Given a holomorphic function f (x, E), we define its conjugate by ¯ ¯ E). f (x, E)∗ = f (x, − + ∗ Then fl,r = (fl,r ) and T is of the form
T=
a b
!
b∗ a∗
.
(4)
The components a(E, h), b(E, h) (and so a∗ (E, h), b∗ (E, h)) are holomorphic with respect to E ∈ 5θ0 . The formula (2) means that det T = aa∗ − bb∗ = 1. One rewrites (3) in the following form by means of (5): ( ∗ + a fl = fr+ + bfl− , a∗ fr− = −b∗ fr+ + fl− . The scattering matrix S(E, h) is defined as 1 S= ∗ a
1 −b∗ b1
(5)
(6)
! .
(7)
410
S.Fujii´e
Its components sjk (E, h) are therefore meromorphic in 5θ0 . Then poles, called resonances, are the zeros of the a∗ (E, h) and are located in the lower half plane. If E is positive, S is unitary. In particular, if we set |s11 (E, h)|2 = |s22 (E, h)|2 = T (E, h),
|s12 (E, h)|2 = |s21 (E, h)|2 = R(E, h), (8)
then we have T (E, h) + R(E, h) = 1,
(9)
where T (E, h) and R(E, h) are called transmission coefficient and reflection coefficient respectively.
3. Action Integrals and Exact WKB Solutions Let V0 be the maximum of the potential V (x). The equation V (x) = V0 admits on the real axis a finite number of solutions o1 , o2 , · · · , on . We suppose that p (H3) V (x) is quadratic at oj , that is kj = −V 00 (oj )/2 > 0 (j = 1, · · · , n). Example 1. The following potential is an example which satisfies (H1), (H2) and (H3): V (x) =
V0 . 1 + (x − o1 )2 · · · (x − on )2
We see immediately from the definition of the scattering matrix that S(E, h) is invariant under parallel translations of the potential. We will then assume in the sequel that o1 ≤ 0 ≤ on . We will take energies E < V0 as reference points. Therefore we set µ = V0 − E and consider it as a new energy parameter. When µ is positive and sufficiently small, the equation V (x) − E = 0 has 2n real roots {αj (µ), βj (µ)}nj=1 satisfying αj (µ) < oj < βj (µ) < αj+1 (µ). We then define the action integrals between these turning points and ±∞ as follows: Definition 1. For µ = V0 − E positive and sufficiently small, we set Z βj p Sj (µ) = V (x) − Edx (1 ≤ j ≤ n), αj
Z Sj,j+1 (µ) =
αj+1
βj
Z Sl (µ) =
α1
−∞
Z Sr (µ) =
∞
βn
p E − V (x)dx
(1 ≤ j ≤ n − 1),
p √ √ { E − E − V (x)}dx − Eα1 , p √ √ { E − E − V (x)}dx + Eβn .
These actions can be extended analytically in the complex plane around µ = 0. More ˆ precisely, let us denote by D(R) and D(R) the disk {µ ∈ C; |µ| < R} and the universal covering space of the punctured disk {µ ∈ C; 0 < |µ| < R} respectively.
Representation of the Scattering Matrix by a Feynman Integral
411
Proposition 1. There exists R > 0 such that Sj (µ), Sj,j+1 (µ), Sl (µ), Sr (µ) are all positive for 0 < µ < R and 1. Sj (µ) is holomorphic in D(R) and Sj (µ) =
π µ + O(µ2 ) 2kj
(j = 1, · · · , n).
2. There exist functions Aj,j+1 (µ), Al (µ) and Ar (µ) holomorphic in D(R) and positive for 0 < µ < R such that Sj,j+1 (µ) =
1 (Sj (µ) + Sj+1 (µ)) log µ + Aj,j+1 (µ), 2π
Sl (µ) = −
1 S1 (µ) log µ + Al (µ), 2π
Sr (µ) = −
1 Sn (µ) log µ + Ar (µ), 2π
where log µ > 0 when arg µ = 0. Proof. The positivity of Sl and Sr for µ positive follows from the assumption o1 ≤ 0 ≤ o2 . See [Fu-Ra]1 for the rest of the proof. In the classically allowed domain, that is in each interval of the set {x ∈ R; V (x) < E}, the exact WKB method enables us to prove the following proposition due to ([GeGr]): Proposition 2. Let δ > 0. Then there exist R > 0 and exact solutions wlj,± , wrj,± (j = 1, · · · , n) of Eq. (1) whose asymptotic expansions as h tends to 0 are given by wlj,± (x, µ, h)
1 1 =√ exp{±i( 4 h E − V (x)
Z
p π E − V (y)dy − )}(1 + O(h)), 4 αj (E) x
ˆ and uniformly in every compact subset of ]oj−1 + δ, oj − δ[×D(R) wrj,± (x, µ, h) = √ 4
1 1 exp{±i( h E − V (x)
Z
p π E − V (y)dy − )}(1 + O(h)), 4 βj (E) x
ˆ uniformly in every compact subset of ]oj + δ, oj+1 − δ[×D(R). Remark 1. The two couples (wlj,+ , wlj,− ) and (wrj,+ , wrj,− ) (1 ≤ j ≤ n) form bases in the space of solutions. Indeed we have the wronskian formulae (see [Ge-Gr]): 2 W(wlj,+ , wlj,− ) = W(wrj,+ , wrj,− ) = − (1 + O(h)). h
412
S.Fujii´e
4. Calculation of the Scattering Matrix In this section, we present a formula for the scattering matrix S by using the connection formulae given in [Fu-Ra]1 . Then the transmission matrix T is expressed as a product of some matrices called transition matrices. Here we do not calculate the explicit form of each component of neither T nor S but show by induction with respect to the number n of barriers the following fact (see Proposition 4): If µ is in the region D(rh), the ˆ (resp. S) ˆ of the asymptotic expansion of T (resp. components of the principal part T S) with respect to h are single-valued holomorphic (resp. meromorphic) functions of µ and the poles of Sˆ are away from the real axis. This is no longer true if |µ/h| tends to infinity as h tends to 0. In such a case, the principal terms of T and S are given sectorwise and there exist poles of the principal part of S on the real axis (shape resonances, see [Fu-Ra]1 for the case n = 2). Recall that T is the matrix which connects two couples of Jost solutions (fl+ , fl− ) and (fr+ , fr− ). We connect these bases through the medium of the exact WKB solutions given in the preceding section. Let us define matrices which connect two bases whose asymptotic expansions are given side by side: + 1,+ n,+ + fl fr wr wl −1 = Tl (µ, h) , = T , (µ, h) − − r fr wrn,− fl wl1,− j+1,+ j,+ j,+ j,+ wl wr wl wr = Tj,j+1 (µ, h) , = Tj (µ, h) . wrj,− wrj,− wlj,− wlj+1,− We call Tl , Tr , Tj,j+1 , Tj transition matrices. Then T is equal to their product: T(µ, h) = Tl · T1 · T1,2 · T2 · T2,3 · · · Tn−1,n · Tn · Tr .
(10)
We now introduce the function called barrier penetration factor (see[L-T-M]). Before giving the asymptotic expansions of these matrices, let R(C\{0}) be the universal covering space of C\{0} and N be the function defined on R(C\{0}) by √ 2π ez log(z/e) , (11) N (z) = 0(1/2 + z) where log z is real when arg z=0. A simple calculation shows Lemma 1. For z ∈ R(C\{0}), we have N (e2nπi z) = e2nπiz N (z),
(n ∈ Z),
N (eπi/2 z)N (e−πi/2 z) = 1 + e−2πz .
(12) (13)
If z = re±iπ/2 , (r > 0), we have |N (re±iπ/2 )|2 = 1 + e−2πr . If arg z ∈] − π, π[, we have N (z) →
√ 2
N (z) → 1
as as
(14)
z → 0,
(15)
|z| → ∞.
(16)
Representation of the Scattering Matrix by a Feynman Integral
413
The principal terms of the transition matrices are represented in terms of exponential functions of classical action integrals defined in the preceding section: ej (µ, h) = exp(−Sj (µ)/h),
ej,j+1 (µ, h) = exp(iSj,j+1 (µ)/h),
el (µ, h) = exp(−iSl (µ)/h),
er (µ, h) = exp(−iSr (µ)/h),
Sj (µ) ). πh In the definition of the function Nj (µ, h), we assume arg Sj (µ) = 0 if arg µ = 0. These functions are multi-valued around µ = 0 except at ej . By Proposition 1, we see that a tour around µ = 0 in the positive sense is equivalent to a multiplication by some ej ’s: Nj (µ, h) = N (i
Lemma 2. For µ ∈ C sufficiently small, we have ej,j+1 (e2πi µ, h) = ej (µ, h)ej+1 (µ, h)ej,j+1 (µ, h), el (e2πi µ, h) = e1 (µ, h)el (µ, h),
er (e2πi µ, h) = en (µ, h)er (µ, h),
(17)
Nj (e2πi µ, h) = ej (µ, h)2 Nj (µ, h). Remark also that each transition matrix is of the form Eq. (4). Recall that we defined f ∗ (µ) = f (µ) for uniform holomorphic functions. We are now concerned with functions ramified around µ = 0. For such functions, we define µ = |µ|e−i arg µ . Then we have the Lemma 3. For µ ∈ R(C\{0}) sufficiently small, we have e∗j = ej ,
e∗j,j+1 = e−1 j,j+1 ,
e∗l = e−1 l ,
e∗r = e−1 r ,
Nj Nj∗ = 1 + e2j .
(18) (19)
ˆ Proof. Notice that if f (µ) is a holomorphic function in D(R) and real for arg µ = 0, then f ∗ (µ) = f (µ). Indeed, this is true for arg µ = 0 and since both f and f ∗ are holomorphic ˆ ˆ in D(R), this is true in D(R) by the theorem of identity. Now Sj (µ), Sj,j+1 (µ), Sl (µ) and Sr (µ) are such functions. So we have Eq. (18). The last identity Eq. (19) follows from Eq. (13) together with the fact Nj∗ (µ, h) = N (e−πi/2
Sj ). πh
(20)
To check Eq. (20), we first remark that Sj (µ) and N (z) are real and analytic for µ ∈ R and arg z = 0 respectively and hence Sj (µ) = Sj (µ), N (z) = N (z). Therefore we have Nj∗ (µ, h) = N (eπi/2 Sj (µ)/(πh)) = N (e−πi/2 Sj (µ)/(πh)) = N (e−πi/2 Sj (µ)/(πh)). Finally we evaluate the absolute value of the above functions for arg µ = 0 and arg µ = π.
414
S.Fujii´e
Lemma 4. The function ej is always positive for real µ. When arg µ = 0, we have |el |2 = |er |2 = |ej,j+1 |2 = 1,
|Nj |2 = |Nj∗ |2 = 1 + e2j ,
(21)
and when arg µ = π, we have |el |2 = e1 ,
|er |2 = en ,
|Nj |2 = e2j (1 + e2j ),
|ej,j+1 |2 = ej ej+1 ,
(22)
|Nj∗ |2 = 1 + e−2 j .
We are now ready to present the asymptotic formulae for transition matrices. Proposition 3. ([Fu-Ra]1 ) 1. There exist R > 0 and > 0 such that Tl =
√ 4
E
1 Tr = √ 4 E
Tj,j+1 =
!
eiπ/4 el (1 + O(h)) O(e−/h ) O(e−/h )
e−iπ/4 e∗l (1 + O(h))
,
eiπ/4 e∗r (1 + O(h))
(24)
!
ej,j+1 (1 + O(h)) O(e−/h ) O(e−/h )
(23)
!
e−iπ/4 er (1 + O(h)) O(e−/h ) O(e−/h )
,
e∗j,j+1 (1 + O(h))
,
(25)
ˆ uniformly with respect to µ in every compact subset of D(R). 2. For any r > 0, one has Tj = e−1 j
!
Nj∗ (1 + O(h log h)) 1 + O(h) 1 + O(h)
,
Nj (1 + O(h log h))
ˆ uniformly with respect to µ in every compact subset of D(rh). Remark 2. We see immediately by this proposition that det Tl =
√
E(1 + O(h)),
1 det Tr = √ (1 + O(h)), E
and that det Tj = 1 + O(h log h), thanks to Eq. (19).
det Tj,j+1 = 1 + O(h),
(26)
Representation of the Scattering Matrix by a Feynman Integral
415
We now calculate the asymptotic expansion of the product Eq. (10). Set ! ! e−iπ/4 er 0 eiπ/4 el 0 √ 1 4 ˆ ˆ Tl = E , Tr = √ , 4 E 0 e−iπ/4 e∗l 0 eiπ/4 e∗r Tˆj,j+1 =
ej,j+1 0 0
!
e∗j,j+1
,
Tˆj =
e−1 j
Nj∗ 1
! ,
(27)
1 Nj
ˆ ˆ which are the principal parts of transition matrices, and let T(µ, h), S(µ, h) be the matrices in Eq. (10) and Eq. (7) obtained by replacing Tl , Tr , Tj,j+1 , Tj by Tˆl , Tˆr , Tˆj,j+1 , Tˆj respectively: ˆ T(µ, h) = Tˆl · Tˆ1 · Tˆ1,2 · Tˆ2 · Tˆ2,3 · · · Tˆn−1,n · Tˆn · Tˆr , 1 Sˆ = ∗ aˆ
1 −bˆ ∗ bˆ 1
(28)
! .
(29)
Remark that Tˆl , Tˆr , Tˆj,j+1 , Tˆj are multi-valued (see Lemma 2 for their multi-valuedness). Remark also that Sˆ has aˆ ∗ as denominator and that Sˆ does not give the principal part of S at zeros of aˆ ∗ . ˆ ˆ Proposition 4. The matrix T(µ, h) (resp. S(µ, h)) is single-valued and holomorphic (resp. meromorphic) on D(rh) for each h and if µ is real, we have s11 (µ, h) = s22 (µ, h) = sˆ11 (µ, h)(1 + O(h log h)), s12 (µ, h) = sˆ12 (µ, h) + O(h log h),
s21 (µ, h) = sˆ21 (µ, h) + O(h log h).
(30)
ˆ ˆ Proof. In order to prove that T(µ, h) and S(µ, h) are single-valued, it is enough to prove ˆ ˆ respectively, are singlethat aˆ (µ, h) and b(µ, h), the (1, 1) and (1, 2) components of T valued. We do it by induction with respect to n the number of barriers. Notice that only the expression of er depends on n. We put the induction number n as superscript when it is necessary. ∗ ˆ (1) = iel e(1)∗ When n = 1, we simply compute that aˆ (1) = el e(1) r N1 , b r . We see by Lemma 2 that these functions are single-valued. Next we suppose that aˆ (n−1) and bˆ (n−1) (n ≥ 1) are single-valued. From Eq. (27) and Eq. (28), one has ˆ (n) = T ˆ (n−1) (Tr(n−1) )−1 Tn−1,n Tn Tr(n) , T and hence ∗ (n−1) ˆ (n−1) , )∗ e(n) ˆ − ie∗n−1,n e(n−1) e(n) en aˆ (n) = en−1,n (e(n−1) r r Nn a r r b ∗ (n−1) ∗ ˆ (n−1) . en bˆ (n) = ien−1,n (e(n−1) )∗ (e(n) ˆ + e∗n−1,n e(n−1) (e(n) r r ) a r r ) Nn b
(31)
Again by Lemma 2, we see that the coefficients of aˆ (n−1) and bˆ (n−1) in Eq. (31) are all single-valued. Therefore, the same fact holds for aˆ (n) , bˆ (n) . Similarly, we can prove
416
S.Fujii´e
inductively that aˆ (n) and bˆ (n) are bounded functions of µ in D(rh) for every fixed h. This implies that aˆ (n) (µ, h) and bˆ (n) (µ, h) are holomorphic in D(rh). We next prove Eq. (30). The energy µ is assumed here to be in the interval [−rh, rh] and h ∈]0, h0 ] for a positive constant h0 . It is enough to check that 1 ≤ |ˆa(µ, h)| ≤ C,
ˆ h)| ≤ C, |b(µ,
(32)
for some constant C > 1 independent of (µ, h) ∈ [−rh, rh]×]0, h0 ] and that a(µ, h) = aˆ (µ, h)(1 + O(h log h)),
ˆ h) + O(h log h). b(µ, h) = b(µ,
(33)
Indeed, one has then s11 =
1 1 1 = = (1 + O(h log h)) = sˆ11 (1 + O(h log h)), a∗ aˆ ∗ (1 + O(h log h)) aˆ ∗
s21 = bs11 = (bˆ + O(h log h))sˆ11 (1 + O(h log h)) = bˆ sˆ11 + O(h log h) = sˆ21 + O(h log h), and so on. The estimate of |ˆa(µ, h)| from below in Eq. (32) follows from the fact that for µ ∈ R, ˆ 2 = aˆ aˆ ∗ − bˆ bˆ ∗ = det T ˆ = 1. |ˆa|2 − |b| ˆ h)| from above, it suffices to evaluate the For the estimate of |ˆa(µ, h)| and |b(µ, absolute value of el , er , ej , ej,j+1 , Nj and their ∗-conjugates which constitute aˆ (µ, h) ˆ h). In fact, aˆ (µ, h) and b(µ, ˆ h) are polynomials of and b(µ, n−1 ∗ n ∗ n (el , e∗l , er , e∗r , {ej }nj=1 , {ej,j+1 }n−1 j=1 , {ej,j+1 }j=1 , {Nj }j=1 , {Nj }j=1 ).
For ej , one has
(34)
Cj−1 ≤ ej = exp(Sj /h) ≤ Cj
for some constant Cj > 1 when µ ∈ [−rh, rh] because Sj (µ) ∼ πµ/(2kj ) (see Proposition 1). The other functions enumerated in Eq. (34) are also seen to be bounded by Lemma 4 and Eq. (18). The formula Eq. (33) follows immediately from what we have shown above. This completes the proof. Let us calculate the scattering matrix for n = 1, 2. Corollary 1. In the case n = 1, one has: sˆ11 (µ, h) = er el e1 /N1 ,
sˆ12 (µ, h) = ie2r /N1 ,
(35)
and in the case n = 2, one has sˆ11 (µ, h) = er el e1 e2 /(e12 + N1 N2 e−1 12 ), −1 sˆ12 (µ, h) = ie2r (N2∗ e12 + N1 e−1 12 )/(e12 + N1 N2 e12 ).
For real E, which is of physical interest, one can express the semiclassical transmission and reflection coefficients in a simple form:
Representation of the Scattering Matrix by a Feynman Integral
417
Corollary 2. If −rh < µ < rh for some r > 0, one has in the case n = 1, |sˆ11 (µ, h)|2 =
1 , 1 + exp(2S1 /h)
|sˆ12 (µ, h)|2 =
exp(2S1 /h) . 1 + exp(2S1 /h)
(36)
When E = V0 (that is µ = 0), one has |sˆ11 (0, h)|2 = |sˆ12 (0, h)|2 =
1 . 2
(37)
In the case n = 2 one has for E = V0 , 1 4(1 + cos(2A12 (0)/h)) , |sˆ12 (0, h)|2 = , (38) 5 + 4 cos(2A12 (0)/h) 5 + 4 cos(2A12 (0)/h) Ro √ where 2A12 (0) = 2 o12 E − V (x)dx is the area of the domain 612 in the phase space defined by {(x, ξ) ∈ R2 ; o1 < x < o2 , ξ 2 < V0 − V (x)} (see Eq. (41)). |sˆ11 (0, h)|2 =
Remark 3. From Corollary 1, we get the quantization rule of resonances: Indeed, the resonance µ, which is a pole of s11 , must satisfy N1 (µ, h) = O(h),
(in the case
1 + N1 N2 e−2 12 = O(h log h),
n = 1),
(in the case
n = 2).
These conditions yield the following quantization rule of resonances: 1 S1 (µ) = (n + )πih + O(h2 ), 2 S12 (µ) +
(in the case
n = 1),
1 ih X Sj (µ) ) = (n + )πih + O(h2 log h), log N (i 2 πh 2 j=1,2
(in the case
n = 2).
Remark 4. The formulae Eq. (37) and Eq. (38) yield an interesting fact: consider particles coming from −∞ with energy E = V0 . The formula Eq. (37) means the transmission coefficient T (V0 , h) and the reflection coefficient R(V0 , h) (see Eq. (8)) are both 1/2 in the semiclassical limit if there is only one barrier. Suppose then that the motion of particles obeys a law of a Markov chain where particles move on R straightforward except at oj ’s and transmit or reflect at oj with the same probability 1/2. Then in the case n = 2, the transmission coefficient, namely the probability for particles to go away j to +∞, is equal to 1/3. Indeed there are countably many trajectories {γ j }∞ j=0 , where γ first transmits at o1 , then reflects j times at each barrier o1 and o2 one after the other and finally transmits o2 (see Example 2). The probability for particles to go along γ j is then (1/2)2j+2 and the sum for all j is equal to 1/3. Evidently, this does not coincide with our result Eq. (38), where T (V0 , h) varies between 1/9 and 1 depending on 2A12 (0)/h. However, this idea of pursuing the trajectories of particles is not completely false. As we will see in the next section (Remark 7), there exist complex numbers 8[γ j ] with |8[γ j ]|2 = (1/2)2j+2 called probability amplitude such that the sum of all 8[γ j ] is equal to sˆ11 (0, h).
418
S.Fujii´e
5. Physical Interpretation In this section, we are interested in the principal term Sˆ of the scattering matrix. Recall the definition of the scattering matrix. From Eq. (6) and Eq. (7), we have fl+ = s11 fr+ + s21 fl− ,
(39)
fr− = s12 fr+ + s22 fl− .
(40)
The Jost solutions fl+ and fr− on the left-hand side represent incoming particles from −∞ and +∞ respectively and fr+ and fl− on the right-hand side represent outgoing particles to +∞ and −∞ respectively. The coefficients sjk (j, k = 1, 2) are so-called scattering amplitudes of particles coming from (−1)k ∞ and going to (−1)j−1 ∞. According to the idea of the Feynman path integral, it is natural to expect that there exist a class Cjk of trajectories γ starting from (−1)k ∞ and ending at (−1)j−1 ∞ and a complex valued function 8 on Cjk called probability amplitude such that sjk is given as the sum of 8[γ] for all γ ∈ Cjk . In the quantum case, Cjk must contain all continuous trajectories and the sum should be considered as the so-called path integral. Our result of this section is that in the semiclassical case, i.e., if we replace sjk by sˆjk , the path integral reduces to a countable sum. In other words, Cjk consists only of countably many trajectories ˆ (Theorem 1). We shall derive the previous statement by starting from the formula of T obtained as a product of transition matrices. Let us first define the notion of semiclassical trajectory and a class of semiclassical trajectories Cjk . We define them on the microsupport of the operator P − E. Recall that µ = V0 − E = O(h). The microsupport M of the operator P − E consists of 2n + 2 arcs ± }nj=0 in the (x, ξ) plane (see for example [Fu-Ra]1 ): {σj,j+1 − − + + M = σl+ ∪ σl− ∪ σ1,2 ∪ σ1,2 ∪ · · · ∪ σn−1,n ∪ σn−1,n ∪ σr+ ∪ σr− , ± σj,j+1 = {(x, ξ); ξ = ±
p V0 − V (x),
oj < x < oj+1 } (1 ≤ j ≤ n − 1), p ± σl± = σ0,1 = {(x, ξ); ξ = ± V0 − V (x), x < o1 }, p ± σr± = σn,n+1 = {(x, ξ); ξ = ± V0 − V (x), on < x}.
Fig. 1. Microsupport M and the motion of particles
Notice that lim
x→−∞,(x,ξ)∈σl±
ξ=±
p V0 ,
lim
x→+∞,(x,ξ)∈σr±
ξ=±
p V0 .
Representation of the Scattering Matrix by a Feynman Integral
419
− + For each j (1 ≤ j ≤ n − 1), σj,j+1 and σj,j+1 enclose a bounded domain 6j,j+1 whose area is Z oj+1 p V0 − V (x)dx = 2Aj,j+1 (0). (41) 2 oj
− − + + and σ01 (resp. σn,n+1 and σn,n+1 ) enclose an unbounded domain 601 (resp. The arcs σ01 √ √ 6n,n+1 ) and the area of {x < 0, |ξ| < V0 }\601 (resp. {x > 0, |ξ| < V0 }\6n,n+1 is given by Z o1 p p p { V0 − V0 − V (x)}dx − 2 V0 o1 = 2Al (0), (42) 2 −∞
Z (resp.
∞
2 on
p p p { V0 − V0 − V (x)}dx + 2 V0 on = 2Ar (0)).
(43)
The consecutive domains 6j−1,j and 6j,j+1 intersect at the point (oj , 0) (1 ≤ j ≤ n). Definition 2. Given a positive integer m, γ(t) = (x(t), ξ(t)) is said to be a semiclassical trajectory of length m if γ(t) : ]0, m[→ M is a continuous function with lim γ(t), lim γ(t) ∈ {(±∞, ±
t→0+
t→m−
γ(k) ∈ {{(oj , 0)}nj=1 },
p
V0 ), {(oj , 0)}nj=1 },
(k = 1, · · · , m − 1)
− + and decreasing on σj,j+1 (j = 0, · · · , n). The length m and x(t) is increasing on σj,j+1 of γ is denoted by m(γ).
Definition 3. Let γ(t) = (x(t), ξ(t)) be a semiclassical trajectory and k an integer with 1 ≤ k ≤ m(γ) − 1. We say that γ transmits at t = k if x(t) is monotone in ]k − 1, k + 1[. Otherwise we say that γ reflects at t = k. We denote these motions by τj and ρj respectively if γ(k) = j. Remark 5. The parameter t of γ here is different from the time variable of the time dependent Schr¨odinger equation. We normalize the time to pass each arc to be 1 and identify two semiclassical trajectories if limt→0+ γ(t), γ(k) (k = 1, · · · , m − 1) and limt→m− γ(t) coincide. Any semiclassical trajectory γ is identified with a finite sequence of arcs, transmissions and reflections γ = (γ1 , γ1,2 , γ2 , γ2,3 , · · · , γm(γ)−1,m(γ) , γm(γ) ), ± }nj=0 , γk ∈ {σj,j+1
γk,k+1 ∈ {τj , ρj }nj=0 ,
satisfying the following conditions: i)
+ (j = 1, · · · , n), If γk = σj−1,j
then
420
S.Fujii´e
(
+ γk+1 = σj,j+1 − γk+1 = σj−1,j
ii)
γk,k+1 = τj ,
in the case
γk,k+1 = ρj .
− If γk = σj,j+1 (j = 1, · · · , n), then ( − in the case γk,k+1 = τj , γk+1 = σj−1,j + γk+1 = σj,j+1
iii)
in the case
If
− γk = σ01
or
in the case + σn,n+1 ,
γk,k+1 = ρj .
then k = m(γ).
With this identification, the image of γ(t), t ∈]k − 1, k[ is the arc γk (k = 1, · · · , m(γ)). Let C be the set of all semiclassical trajectories γ. The following four subsets of C are the sets of γ = (γ1 , γ1,2 , · · · , γm(γ)−1,m(γ) , γm(γ) ) starting from x = +∞ or −∞ and ending at x = +∞ or −∞: C11 = {γ ∈ C; γ1 = σl+ , γm(γ) = σr+ },
C12 = {γ ∈ C; γ1 = σr− , γm(γ) = σr+ },
C21 = {γ ∈ C; γ1 = σl+ , γm(γ) = σl− },
C22 = {γ ∈ C; γ1 = σr− , γm(γ) = σl− }.
Example 2. If n = 1, Cjk consists of only one semiclassical trajectory: C11 = {(σl+ , τ1 , σr+ )},
C12 = {(σr− , ρ1 , σr+ )},
C21 = {(σl+ , ρ1 , σl− )},
C22 = {(σr− , τ1 , σl− )}.
If n ≥ 2, Cjk consists of countably many semiclassical trajectories. For example, when n = 2, − + k , ρ1 , σ12 ) , τ2 , σr+ )}∞ C12 = {(σr− , ρ2 , σr+ )} ∪ {(σr− , τ2 , (σ12 k=1 , where
− − + 1 + (σ12 , ρ1 , σ12 ) = (σ12 , ρ1 , σ12 ), − − − + k + k−1 + (σ12 , ρ1 , σ12 ) = ((σ12 , ρ1 , σ12 ) , ρ2 , (σ12 , ρ1 , σ12 ))
(k ≥ 2).
Secondly we introduce a complex-valued function 8 on the space of semiclassical trajectories C and we call 8[γ] probability amplitude for each γ = (γ1 , γ1,2 , · · · , γm(γ) ) ∈ C. ± For this, we define a complex-valued function φ on the set ({σj,j+1 }nj=0 , {τj , ρj }nj=1 ) depending on µ and h. ± Definition 4. For each arc σj,j+1 , we define ± φ[σj,j+1 ]={
± ]={ φ[σl± ] = φ[σ0,1
e2j,j+1 1/2 } Nj Nj+1
e2l 1/2 } , N1
(1 ≤ j ≤ n − 1),
± φ[σr± ] = φ[σn,n+1 ]={
e2r 1/2 } . Nn
(44)
For transmissions and reflections, we define φ[τj ] = ej (µ, h),
φ[ρj ] = i.
(45)
Representation of the Scattering Matrix by a Feynman Integral
421
We see by Lemma 2 that e2j,j+1 /(Nj Nj+1 ), e2l /N1 , e2r /Nn are holomorphic functions of µ in D(rh) if r is small enough in order to get that√|Sj (µ)/h| < 1/2√for 1 ≤ j ≤ n. Their values at the origin are Aj,j+1 (0)2 /2, Al (0)2 / 2 and Ar (0)2 / 2 respectively, which are all positive, and the square roots in the definition are taken so that they are positive at µ = 0. Definition 5. Let γ = (γ1 , γ1,2 , · · · , γm(γ) ) be a semiclassical trajectory. We define the probability amplitude 8[γ] by 8[γ] = φ[γ1 ]φ[γ1,2 ] · · · φ[γm(γ)−1,m(γ) ]φ[γm(γ) ].
(46)
Remark 6. If we introduce the following integer-valued functions on C, we can rewrite Eq. (46) as a product with respect to the number j of barriers. Set ± }, sj,j+1 (γ) = #{k; 1 ≤ k ≤ m(γ), γk = σj,j+1
(0 ≤ j ≤ n),
tj (γ) = #{k; 1 ≤ k ≤ m(γ) − 1, γk,k+1 = τj },
(1 ≤ j ≤ n),
rj (γ) = #{k; 1 ≤ k ≤ m(γ) − 1, γk,k+1 = ρj },
(1 ≤ j ≤ n).
We easily see that n X
sj,j+1 (γ) = m(γ),
j=0
n X
{tj (γ) + rj (γ)} = m(γ) − 1,
(47)
j=1
sj−1,j (γ) + sj,j+1 (γ) = 2(tj (γ) + rj (γ)).
(48)
The probability amplitude 8[γ] is then given by 8[γ] =
n Y j=1
n
{(
i rj (γ) ej tj (γ) Y sj,j+1 (γ) ) ( ) } ej,j+1 . Nj Nj
(49)
j=0
Our main result is the following: Theorem 1. If µ is real and |µ| < rh for a positive constant r and sufficiently small h, we have X 8[γ](µ, h), (50) sˆjk (µ, h) = γ∈Cjk
where the series converges absolutely and uniformly. Proof. We prove this for sˆ(n) 12 . If n = 1, C12 consists only of the semiclassical trajectory γ = (σr− , ρ1 , σr+ ) of length 2 (see Example 2). So we have − + sˆ(1) 12 = 8[γ] = φ[σr ]φ[ρ1 ]φ[σr ] =
2 i(e(1) r ) . N1
This coincides with Eq. (35) and hence Eq. (50) holds for n = 1. Suppose that Eq. (50) holds for n − 1. From Eq. (31) and Eq. (7), we see that the reflection coefficient sˆn12 is given by the following recurrence formula:
422
S.Fujii´e
(n) 2 sˆ(n) 12 = i(er )
e2n−1,n Nn∗ sˆ(n−1) + i(e(n−1) )2 r 12 e2n−1,n sˆ(n−1) + iNn (e(n−1) )2 r 12
.
(51)
Next we eliminate Nn∗ from (51) by making use of (19). Multiplying both the numerator and the denominator of Eq. (51) by Nn , we obtain sˆ(n) 12 = i
=i
2 (n−1) 2 e2 ˆ12 + i(e(n−1) )2 N n (e(n r n−1,n (1 + en )s r )) Nn e2n−1,n sˆ(n−1) + iNn (e(n−1) )2 r 12
2 is˜(n−1) (e(n) r ) 12 {1 − e2n }, Nn 1 − is˜(n−1) 12
where s˜(n−1) = 12
e2n−1,n Nn (e(n−1) )2 r
sˆ(n−1) . 12
(52)
In terms of probability amplitudes, we can write 2 (e(n) r ) = φ[σr(n)− ]φ[σr(n)+ ], Nn
s˜(n−1) = sˆ(n−1) 12 12
− + φ[σn−1,n ]φ[σn−1,n ]
φ[σr(n−1)− ]φ[σr(n−1)+ ]
.
(53)
/(1 − is˜(n−1) ) in geometric series, we have finally Expanding is˜(n−1) 12 12 (n)− ]φ[ρn ]φ[σr(n)+ ] sˆ(n) 12 = φ[σr P∞ )k+1 }φ[τn ]φ[σr(n)+ ]. +φ[σr(n)− ]φ[τn ]{ k=0 φ[ρn ]k (s˜(n−1) 12
(54)
The right-hand side of Eq. (54) has a physical interpretation: the first term is equal to the probability amplitude corresponding to the semiclassical trajectory (σr(n)− , ρn , σr(n)+ ) which starts from +∞, reflects at the nth barrier on and goes back to +∞. The second term represents all contributions by other semiclassical trajectories which transmit the can be considered as nth barrier on . From Eq. (53) and the hypothesis of induction, s˜(n−1) 12 the sum of probability amplitudes corresponding to all semiclassical trajectories starting (n−1) = from P on to the left and stopping as soon as they come back to the same point: s˜12 8[γ], where the sum is extended over all semiclassical trajectories of the form γ = − ± + (γ1 , γ1,2 , · · · , γm(γ)−1,m(γ) , γm(γ) ) with γ1 = σn−1,n , γm(γ) = σn−1,n and γj 6= σn−1,n (n−1) for 1 < j < m(γ). The exponent k + 1 of s˜12 in the sum counts the number of times where the particle comes back to on and the exponent k of φ[ρn ] is the number of reflections at on . This finishes a formal justification of Eq. (50) by induction. Finally we prove the absolute convergence of the series Eq. (50) by recurrence. For n = 1, it is trivial since C12 consists of only one semiclassical trajectory. Suppose the also has an absolutely sum of Eq. (50) converges absolutely for n − 1. Then s˜(n−1) 12 convergent expansion of the same type by Eq. (52) and hence it suffices to prove the | < 1. absolute convergence of the sum in the right-hand side of Eq. (54), that is, |s˜(n−1) 12 In fact we have |en−1,n |2 < 1. | < 1 and |sˆ(n−1) 12 (n−1) 2 |er | |Nn |
Representation of the Scattering Matrix by a Feynman Integral
423
The right-hand side of Eq. (51) is a linear fractional transformation on the unit disk when µ is real. Indeed if we set sˆ(n) ˆ(n−1) = z, Eq. (51) is with respect to sˆ(n−1) 12 12 = ζ, s 12 rewritten in the form z−a ζ=ω 1 − az with 2 2 ∗ (e(n) (e(n−1) )2 r ) en−1,n Nn a = −i 2 r , ω= . (n−1) ∗ en−1,n Nn (er )2 Nn We first see by Lemma 2 that a and ω are single-valued holomorphic functions in D(rh). So we calculate their absolute values on the p real axis for arg µ = 0 and arg µ = π by using Lemma 4. Then we see that |a| = 1/ 1 +pe2n < 1 and |ω| = 1 when µ is real. sˆ(1) 12 (n) (1) 2 2 < 1), so is s being in the unit disk (|sˆ(1) | = |e | /|N | = 1/ 1 + e ˆ for all n. 1 r 1 12 12 Similarly we can verify that |en−1,n |2 (n−1) 2 |er | |Nn |
1 =p <1 1 + e2n
for real µ. The proof for other sˆjk ’s can be done almost in the same manner. The only difference ˆ(n−1) but also on sˆ(n−1) : is that sˆ(n) 11 , for example, depends not only on s 11 12 sˆ(n) ˜(n−1) { 11 = s 11
∞ X
φ[ρn ]k (s˜(n−1) )k }φ[τn ]φ[σr(n)+ ], 12
k=0
where = sˆ(n−1) s˜(n−1) 11 11
+ φ[σn−1,n ]
φ[σr(n−1)+ ]
.
Finally we calculate the asymptotic form of the probability amplitudes. ± Proposition 5. The asymptotic form of φ[σj,j+1 ] (0 ≤ j ≤ n) is given by
µ ± ](µ, h) = Rj,j+1 ( )eiθj,j+1 (µ,h)/h (1 + O(h log h)), φ[σj,j+1 h with Rj,j+1 (z) =
√1 2π
Q q=j,j+1
0( 21 +
exp[i{A0j,j+1 (0) +
iz 1/2 2kq )
P q=j,j+1
log(2ekq )−πi/2 }z], 4kq
1 X 1 )µ log h, θj,j+1 (µ, h) = Aj,j+1 (0) + ( 4 kq q=j,j+1
for 1 ≤ j ≤ n − 1 and 1 iz 1/2 1 log(2ek1 ) − πi/2 0( + ) exp[i{−A01 (0) + }z], R0,1 (z) = √ 4 2 2k 4k1 2π 1 θ0,1 (µ, h) = −Al (0) +
1 µ log h, 4k1
424
S.Fujii´e
1 iz 1/2 1 log(2ekn ) − πi/2 0( + Rn,n+1 (z) = √ ) exp[i{−A0n (0) + }z], 4 2 2k 4kn 2π n 1 µ log h. θn,n+1 (µ, h) = −Ar (0) + 4kn If µ = 0, the probability amplitude associated with a semiclassical Pntrajectory γ = (γ1 , γ1,2 , · · · , γm(γ)−1,m(γ) , γm(γ) ) becomes much simpler: let r(γ) = j=1 rj (γ) be the number of reflections and m(γ) X 2(γk ), 2(γ) = k=1
where
± ) = Aj,j+1 (0) 2(σj,j+1 2(σl± ) = −Al (0), 2(σr± ) = −Ar (0).
(1 ≤ j ≤ n − 1),
Recall that 2Aj,j+1 (0)√(1 ≤ j ≤ n − 1), 2Al (0) and 2Ar (0) √ are the areas of 6j,j+1 , {(x, ξ); x < 0, |ξ| < V0 }\601 and {(x, ξ); x > 0, |ξ| < V0 }\6n,n+1 respectively. We then have Proposition 6. For any semiclassical trajectory γ ∈ C , we have 8[γ](0, h) =
ir(γ) 2(m(γ)−1)/2
ei2(γ)/h .
Remark 7. In particular, we have 1 |8[γ](0, h)|2 = ( )m(γ)−1 . 2 P∞ The number m(γ) − 1 is equal to j=1 {tj (γ) + rj (γ)}, the total number of transmissions and reflections. Therefore the square of the absolute value of the probability amplitude associated with a semiclassical trajectory γ ∈ Cjk coincides with the ordinary probability calculated as if it obeyed a law of a Markov chain on M whose trajectory is semiclassical and the probability of transmission and reflection at barriers oj are both 1/2 (see Remark 4). References [Br] [Fu-Ra1] [Fu-Ra2] [Ge-Gr] [He-Sj] [Ko]
Brink, D.M.: Semi-classical methods in nucleus-nucleus scattering. Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press Fujii´e, S. and Ramond, T.: Matrice de scattering et r´esonances associ´ees a` une orbite h´et´erocline. To appear in Annales de l’I.H.P. Physique Th´eorique Fujii´e, S. and Ramond, T.: Semiclassical resonances for the radial Schr¨odinger equation at fixed angular momentum. In preparation G´erard, C. and Grigis, A.: Precise estimates of tunneling and eigenvalues near a potential barrier. J. Diff. Eqs. 72, 149–177 (1988) Helffer, B. and Sj¨ostrand, J.: Semiclassical analysis of Harper’s equation III, Cantor structure of the spectrum. M´em. Soc. Math. France (N.S.) 39, 1–124 (1989) Korsch, H.J.: Semiclassical theory of resonances. Lecture Notes in Physics No. 211, Berlin– Heidelberg–New York: Springer, 1987
Representation of the Scattering Matrix by a Feynman Integral
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[L-T-M] Lee, S.Y., Takigawa, N. and Marty, C.: A semiclassical study of optical potentials: Potential resonances. Nucl. Phys. A308, 161–188 (1978) [L-T] Lee, S.Y. and Takigawa, N.: A wave propagation matrix method in semiclassical theory. Nucl. Phys. A308, 189–209 (1978) [Ra] Ramond, T.: Semiclassical study of quantum scattering on the line. Commun. Math. Phys. 177, 221–254 (1996) Communicated by B. Simon
Commun. Math. Phys. 198, 427 – 468 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
The Many-Electron System in the Forward, Exchange and BCS Approximation Detlef Lehmann? University of British Columbia, Department of Mathematics, Vancouver, B.C., V6T 1Z2, Canada Received: 9 October 1997 / Accepted: 13 April 1998
Abstract: The nonrelativistic many-electron system in the forward, exchange and BCS approximation is considered. In this approximation, the model is explicitly solvable for arbitrary space dimension d. The partition function and the correlation functions are given by finite-dimensional integral representations. Renormalization effects as well as symmetry breaking can be seen explicitly. It is shown that the usual mean field approach, based on approximating the Hamiltonian by a quadratic expression, may be misleading if the electron-electron interaction contains higher angular momentum terms and the space dimension is d = 3. The perturbation theory of the solvable model is discussed. There are cases where the logarithm of the partition function has positive radius of convergence but the sum of all connected diagrams has radius of convergence zero implying that the linked cluster theorem is not applicable in these cases.
1. Introduction In this paper we consider the nonrelativistic many-electron system in the forward, exchange and BCS approximation. In this approximation, which is still quartic in the annihilation and creation operators, the model can be solved explicitly. The partition function and the correlation functions are given by finite-dimensional integral representations. We work in the quantum grand canonical ensemble and start with positiv temperature T = β1 > 0 and finite volume Ld < ∞. The standard model of the many-electron system in d space dimensions is given by the Hamiltonian (1.1) H = H0 + Hint , where ? Present address: TU Berlin, FB Mathematik Ma 7-2, Strasse des 17. Juni 136, D-10623 Berlin, Germany. E-mail:
[email protected]
428
D. Lehmann
H0 =
1 Ld
X
ek a+kσ akσ
(1.2)
k,σ
X
and Hint =
1 L3d
X
U (k − p) a+kσ apσ a+q−kτ aq−pτ .
(1.3)
σ,τ ∈{↑,↓} k,p,q
We assume U to be short range, that is U is L1 in coordinate space. The energy momentum relation ek is given by ek =
k2 2m
−µ
(1.4)
or may be substituted by a more general expression which satisfies ek = e−k . The parameter µ > 0, the chemical potential, is present since we are working in the grand canonical ensemble and is determined by the density of the system. Since we are in d finite volume, the spatial momenta k range over some subset of 2π L Z . The physics of the nonrelativistic many-electron system is determined by momenta close to the Fermi surface d | ek = 0 } (1.5) F = { k ∈ 2π LZ so we impose a fixed ultraviolet cutoff and choose d k ∈ Mω = { k ∈ 2π | |ek | ≤ ω }. LZ
(1.6)
In the context of conventional superconductivity the cutoff ω is referred to as the Debye frequency. The normalizations chosen in the definition of the creation and annihilation operators are such that the anticommutation relations read {akσ , a+k0 σ0 } = Ld δk,k0 δσ,σ0 .
(1.7)
In particular, if F = 5a+kσ 1 is the zero temperature ground state of the noninteracting kσ ek <0
P system, then (F , a+kσ akσ F ) = Ld θ(−ek ) such that (F , H0 F ) = k,σ ek θ(−ek ) ≈ R dd k Ld (2π) d ek θ(−ek ) is indeed an extensive quantity as it should be. Here θ(ε) is the step function being one for ε > 0 and zero otherwise. We are interested in the grand canonical partition function Z = Z(β, L, U ) = T r e−βH ,
(1.8)
which may be normalized by T r e−βH0 such that Z(U = 0) = 1, and in particular in the two point function T r e−βH a+kσ ak0 σ0 , (1.9) ha+kσ ak0 σ0 iβ,L = T r e−βH which gives the momentum distribution of the system. Recall that in the free (U = 0) system, the ideal Fermi gas, lim ha+kσ ak0 σ0 iβ,L = Ld δk,k0 δσ,σ0 θ(−ek ).
β→∞
(1.10)
Since we are in the quantum grand canonical ensemble, the traces in (1.8,9) are to be taken over the Fock space F = ⊕∞ n=0 Fn where
Many-Electron System in Forward, Exchange and BCS Approximation
Fn =
429
n Fn (xπ1 σπ1 , · · · , xπn σπn ) = sgnπ Fn (x1 σ1 , · · · , xn σn ) .
Fn ∈ L2
[0, L]d × {↑, ↓}
The fact that the physical system has a fixed number of particles N is expressed by P requiring that the expectation value hNiβ,L of the number operator N = L1d kσ a+kσ akσ (which is extensive, see above) is equal to the number of particles, hNiβ,L = N , which determines µ as a function of the density LNd . As usual, the quartic part Hint (1.3) of the Hamiltonian H (1.1) may be represented by the following four legged diagram: q − k, τ
k, σ
U (k−p) ∼∼∼∼∼∼∼∼∼∼
Hint :
q − p, τ
p, σ
Because of conservation of momentum, there are three independent momenta here labelled with k, p and q. Then one can consider the following three limiting cases with only two independent momenta: forward
exchange
p, τ
k, σ
U (0) ∼∼∼∼∼∼∼∼∼∼
U (k−p) ∼∼∼∼∼∼∼∼∼∼
p, τ
k, σ
p, τ
k, σ
p, σ
k, τ
Hforw
Hex
BCS −k, τ
k, σ
U (k−p) ∼∼∼∼∼∼∼∼∼∼
−p, τ
p, σ HBCS
That is, one may consider the approximation Hint ≈ Hforw + Hex + HBCS ,
(1.11)
where Hforw =
1 L3d
X
X
U (k − p) δk,p a+kσ apσ a+q−kτ aq−pτ
σ,τ ∈{↑,↓} k,p,q
=
1 L3d
X
X
σ,τ ∈{↑,↓} k,p
U (0) a+kσ akσ a+pτ apτ ,
(1.12)
430
D. Lehmann
Hex =
1 L3d
=
1 L3d
X
X
U (k − p) δk,q−p a+kσ apσ a+q−kτ aq−pτ
σ,τ ∈{↑,↓} k,p,q
X
X
U (k − p) a+kσ apσ a+pτ akτ ,
(1.13)
σ,τ ∈{↑,↓} k,p
HBCS =
1 L3d
X
X
U (k − p) δq,0 a+kσ apσ a+q−kτ aq−pτ
σ,τ ∈{↑,↓} k,p,q
=
1 L3d
X
X
U (k − p) a+kσ apσ a+−kτ a−pτ .
(1.14)
σ,τ ∈{↑,↓} k,p
Let us shortly make a comment on the volume factors. In (1.12–14), we only introduced some Kroenecker delta’s but we did not cancel a volume factor Ld . That this is the right thing to do, that is, that the left-hand side as well as the right-hand side of (1.11) is indeed proportional to the volume may be seen in the easiest way for the forward term. On a fixed n particle space Fn the interacting part Hint is a multiplication operator given by n X U (xi − xj ). (1.15) Hint |Fn = 21 i,j=1 i6=j
Let δy (x) = δ(x−y). Then ϕ(x1 , · · · , xn ) = δy1 ∧· · ·∧δyn (x1 , · · · , xn ) is an eigenfunction of Hint with eigenvalue E=
1 2
n X
U (yi − yj ) =
1 2Ld
i,j=1 i6=j
n X X i,j=1 i6=j
ei(yi −yj )q U (q)
(1.16)
q
which is, for U ∈ L1 , proportional to n or to the volume Ld for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforw ϕ = Eforw ϕ, where Eforw is obtained from (1.16) by putting q = 0 without cancelling a volume factor, Eforw =
1 2Ld
n X
U (q = 0),
(1.17)
i,j=1 i6=j
which is also proportional to the volume. We now come to the exact definition of the model which is solved in this paper. To do so, we need the functional integral representation of the perturbation series for the partition function. It is summarized in the following theorem which is fairly standard. One may look in [FKT1] for a nice, clean proof. Theorem 1.1. Let H = H0 + Hint be the Hamiltonian (1.1–3), let Z = Z(β, L) = T r e−β(H0 +Hint ) /T r e−βH0
(1.18)
be the normalized grand canonical partition function and let ha+kσ akσ iβ,L =
T r e−βH a+kσ akσ . T r e−βH
(1.19)
Many-Electron System in Forward, Exchange and BCS Approximation
431
a) The partition function Z has the following perturbation series Z=
∞ X (−
1 )n βLd
n!
σ1 ···σ2n
n=0
Here k = (k0 , k) ∈ given by
X
π β (2Z
n X Y
δk2i−1 +k2i ,p2i−1 +p2i U (k2i − p2i ) ×
k1 ···k2n p1 ···p2n
i=1
+ 1) × { k ∈
det δσi ,σj δki ,pj C(ki ) 1≤i,j≤2n (.1.20) d 2π LZ
C(k0 , k) =
| |ek | ≤ ω } and the covariance C is
1 . ik0 − ek
(1.21)
If kU (x)kL1 ≤ c (βLd )−1 or kU (x)kL∞ ≤ c (βLd )−2 , c a constant, then (1.20) converges. b) Let P 1 ¯ d ¯ = 5 βL e− βLd k,σ (ik0 −ek )ψk,σ ψk,σ 5 dψk,σ dψ¯ k,σ dµC (ψ, ψ) (1.22) ik0 −e k
k,σ
k,σ
be the Grassmann Gaussian measure with covariance C. Then the perturbation series (1.20) can be rewritten as Z ¯ ¯ (1.23) Z = e−Vint (ψ,ψ) dµC (ψ, ψ), where ¯ = Vint (ψ, ψ)
1 (βLd )3
X X σ1 σ2
δk1 +k2 ,p1 +p2 U (k2 − p2 )ψ¯ k1 σ1 ψ¯ k2 σ2 ψp1 σ1 ψp2 σ2 . (1.24)
k1 ,k2 p1 ,p2
c) The momentum distribution (1.19) at temperature T = ha+kσ akσ iβ,L = lim →0 <0
1 β
hψ¯ kσ ψkσ iβ,L, = Here C (k) =
e−ik0 ik0 −ek
and Z =
1 Z
R
is given by
hψ¯ k0 kσ ψk0 kσ iβ,L, ,
(1.25)
k0 ∈ π β (2Z+1)
Z
where
X
1 β
¯ ¯ ψ¯ kσ ψkσ e−Vint (ψ,ψ) dµC (ψ, ψ).
(1.26)
e−Vint dµC .
Remarks. (i) A bound from which convergence of the perturbation series follows for sufficiently small U is easiest obtained in coordinate space. There the perturbation series reads Z R ¯ ¯ ¯ Z = e− dξ1 dξ2 (ψψ)(ξ1 )U (ξ1 −ξ2 )(ψψ)(ξ2 ) dµC (ψ, ψ) =
∞ X n=0
Z 1 n!
dξ1 · · · dξ2n
n Y i=1
U (ξ2i−1 − ξ2i ) det C(ξi , ξj ) 1≤i,j≤2n , (1.27)
432
D. Lehmann
R R R P d where ξ = (x0 , x, σ), dξ = σ∈{↑,↓} [0,β] dx0 [0,L]d d x, U (ξ) = U (x) = 0 0 δ(x0 )U (x), C(ξ, ξ ) = δσ,σ0 C(x − x ) where X C(x0 , x) = L1d eikx e−ek x0 [θ(−x0 )θβ (−ek ) − θ(x0 )θβ (ek )] (1.28) k∈Mω
and θβ is an approximate step function given by θβ (ε) = 1+e1−βε . First observe that, since supx0 ∈[0,β] |e−ek x0 [θ(−x0 )θβ (−ek ) − θ(x0 )θβ (ek )]| ≤ 1 for all k, one has kCk∞ ≤ P 1 k∈Mω 1 < ∞ if we choose a fixed ultraviolet cutoff in (1.6). The determinant can Ld be bounded using | det[~a1 · · · ~ar ] | ≤ 5ri=1 k~ai k2 , where the ~ai are r component vectors. That is 2n n 2n o 21 2n 1 ≤ 5 (2nkCk2∞ ) 2 = (2nkCk2∞ )n , det C(ξi , ξj ) 1≤i,j≤2n ≤ 5 6 |C(ξi , ξj )|2 i=1 j=1
i=1
(1.29) which gives |Z| ≤
∞ X
e n (4βLd )n kU knL1 (2nkCk2∞ )n n
=
n=0
∞ X
8ekCk2∞ βLd
n
kU knL1 .
(1.30)
n=0
(ii) The -limit shows up in (1.25) because of X x0 6=0 eikx e−ek x0 θβ (−ek )θ(−x0 ) − θβ (ek )θ(x0 ) C(x0 , x) = L1d k
=
1 βLd
X
1 eikx−ik0 x0 ik0 −e
(1.31)
k
k,k0
but C(0, x) =
1 Ld
X
eikx θβ (−ek ) = lim →0 <0
k
1 βLd
X
1 eikx−ik0 ik0 −e , k
(1.32)
k,k0
see the proof of Lemma A1 in the appendix for a more detailed discussion. (iii) Whereas the momenta in the Hamiltonian are d dimensional, the variables in the perturbation series are d + 1 dimensional. The additional k0 variables in (1.20) are the Fourier transform of the x0 variables in (1.27) which in turn enter the perturbation series because of Rβ d −β(H0 +λHint ) = 0 e−x0 H0 Hint e−(β−x0 )H0 dx0 , (1.33) dλ |0 e that is, because Hint does not commute with H0 . As remarked under (iii), the momenta in Hint =
1 L3d
X
X
U (k − p) a+kσ apσ a+q−kτ aq−pτ
(1.3)
σ,τ ∈{↑,↓} k,p,q
are d dimensional, but the variables in X X U (k − p) ψ¯ kσ ψpσ ψ¯ q−kτ ψq−pτ Vint = (βL1d )3
(1.34)
σ,τ ∈{↑,↓} k,p,q
are d + 1 dimensional. On the Hamiltonian level, the forward, exchange and BCS approximation (1.11) does not lead to an explicitly solvable model, but on the functional integral level it does. This is the main result of this paper.
Many-Electron System in Forward, Exchange and BCS Approximation
433
Theorem 1.2. Let ¯ {Vforw + Vex + VBCS }(ψ, ψ) XX = (βL1d )3 U (k − p) [δk,p + δk,q−p + δq,0 ] ψ¯ kσ ψpσ ψ¯ q−kτ ψq−pτ
(1.35)
στ kpq
Then the approximation
Vint ≈ Vforw + Vex + VBCS
(1.36)
in (1.23,26) makes the model explicitly solvable. The partition function, the two point functions and the generating functional for the connected amputated Greens functions are given by finite-dimensional integral representations. The dimension of these representations is proportional to J if the electron electron interaction is given by j X λ0 1 if d=2 λ|`| ei`ϕk e−i`ϕp + J 2 2 X `=−j =: λl yl (k) y¯l (p) (1.37) U (k − p) = j ` X X l=0 0 0 ¯ λ` Y`m k Y`m p if d=3 `=0 m=−`
Here k = |k|(cos ϕk , sin ϕk ) for d = 2 and k0 = k/|k| for d = 3. The general solution is written down in Theorem 2.1. In the following, we summarize the results for Vint ≈ VBCS and for an electron-electron interaction given by (1.37). This case is treated in Sect. 3.1. In that case the two point function (1.26) becomes (we suppress the ): R hψ¯ p↑ ψp0 ↑ iβ,L = βLd δp,p0
Fp↑↑ (φ) e−βL R
e
−βLd V
d
J
l 5 5 dulστ dvστ
V (φ)
στ l=0
(φ)
.
J
(1.38)
l 5 5 dulστ dvστ
στ l=0
l Here φlστ = ulστ + ivστ , στ ∈ {↑↑, ↓↓, ↑↓},
Fp↑↑ (φ) = 8k↑↓ =
J X
¯ −p↑↓ +8p↓↓ 8 ¯ p↓↓ ] (ip0 +ep )[p20 +e2p +8−p↑↓ 8 (p20 +e2p ++p )(p20 +e2p +− p )
1
λl2 φl↑↓ yl (k0 ) ,
¯ k↑↓ = 8
l=0
8kσσ =
J X l=0
1
λl2 φlσσ [yl (k0 ) − yl (−k0 )] ,
J X
,
(1.39)
1
λl2 φ¯ l↑↓ y¯l (k0 ) ,
(1.40)
l=0
¯ kσσ = 8
J X
1
λl2 φ¯ lσσ [y¯l (k0 ) − y¯l (−k0 )]
l=0
(1.41) are the solutions of the quadratic equation and ± k ¯ k↑↓ 8k↑↓ + 8 ¯ −k↑↓ 8−k↑↓ + 8k↑↑ 8 ¯ k↑↑ + 8k↓↓ 8 ¯ k↓↓ + 8k↑↑ 8k↓↓ 8 ¯ k↑↓ 8 ¯ −k↑↓ 2 − 8 ¯ k↑↑ 8 ¯ k↑↓ 8k↑↓ 8 ¯ k↓↓ 8k↑↓ 8−k↑↓ + 8k↑↑ 8k↓↓ 8 ¯ k↑↑ 8 ¯ k↓↓ + 8 ¯ −k↑↓ 8−k↑↓ = 0. (1.42) +8 ¯ kστ is not necessarily the complex conjugate of 8kστ , depending on the Observe that 8 signs of the coupling constants λl . The effective potential Vβ is given by
434
D. Lehmann
Vβ (φ) =
J X
|φl↑↓ |2
+
|φl↑↑ |2
+
|φl↓↓ |2
l=0
X Z
−
∈{+,−}
Mω
dd k 1 (2π)d β
log
cosh( β2
√
cosh
e2k +k )
β 2 ek
.
(1.43) For a pure even interaction, that is, if λ` = 0 for all odd angular momentum `, one has, if yl (−k) = (−1)l yl (k), 8k↑↑ = 8k↓↓ = 0 and the expectation value simplifies to R hψ¯ p,σ ψp0 ,σ iβ,L = βLd δp,p0
−ip0 −ep ¯ p↑↓ p20 +e2p +8p↑↓ 8
R
e−βL
e−βL
dV
d
Vβ (φ↑↓ )
J
l 5 dul↑↓ dv↑↓
l=0 J
β (φ↑↓ )
(1.44)
l 5 dul↑↓ dv↑↓
l=0
with an effective potential Vβ (φ↑↓ ) =
J X
Z |φl↑↓ |2 −
l=0
M
dd k 1 (2π)d β
log
cosh( β2
√
¯ k↑↓ ) e2k +8k↑↓ 8
cosh
2
β 2 ek
.
(1.45)
Recall that the free two point function is given by hψ¯ p,σ ψp0 ,σ iβ,L = βLd δp,p0 1/(ip0 − ep ) = −βLd δp,p0 (ip0 + ep )/(p20 + e2p ). In particular, for a delta function interaction one obtains the two dimensional integral representation R ip0 +ep −βLd Vβ (u,v) dudv 2 +e2 +λ(u2 +v 2 ) e p p 0 R −βLd V (u,v) hψ¯ p,σ ψp0 ,σ iβ,L = −βLd δp,p0 , (1.46) β dudv e where
Z Vβ (u, v) = u + v − 2
2
Mω
dd k 1 (2π)d β
log
cosh( β2
√
e2k +λ(u2 +v 2 ))
cosh
β 2 ek
2 .
(1.47)
A positive λ corresponds to an attractive interaction. Since the only place where the volume Ld shows up in the integral representations d (1.44,46) is the prefactor in the exponential e−βL V (φ) , the computation of the infinite volume limit comes down to the determination of the global minimum of the real part of the effective potential V as a function of the φlστ ’s.
Figure 1
For repulsive coupling λ < 0 Vβ (1.47) has a global minimum at u = v = 0 which results in d e−βL Vβ (u,v) dudv = δ(u)δ(v) (1.48) lim R −βLd V (u,v) β L→∞ dudv e
Many-Electron System in Forward, Exchange and BCS Approximation
and
435
lim hψ¯ p,σ ψp0 ,σ iβ,L = −βδp0 ,p00 δ(p − p0 )
L→∞
For attractive coupling λ > 0 and sufficiently small T = has the form of a mexican hat. This results in
1 β
ip0 +ep . p20 +e2p
(1.49)
the effective potential (1.47)
lim hψ¯ p,σ ψp0 ,σ iβ,L = −βδp0 ,p00 δ(p − p0 )
L→∞
ip0 +ep , p20 +e2p +12
(1.50)
where 12 = λρ20 ∼ e− λ and ρ20 = u20 +v02 is the value where Vβ takes its global minimum which lies on a circle in the u, v plane. ¯ can also be computed. To make them nonzero, The expectation values hψψi and hψ¯ ψi we introduce a small external field r = |r|eiα . That is, we substitute VBCS by (we do not consider here ha↑ a↑ i expectations) X VBCS,r = VBCS + βL1 d [r ψk↑ ψ−k↓ − r¯ ψ¯ k↑ ψ¯ −k↓ ]. (1.51) 1
k
One obtains again a finite-dimensional integral representation. For a delta function interaction, that is for J = 0 in (1.37), it reduces to a two dimensional integral: R
R
hψ¯ p,↑ ψ¯ −p0 ,↓ iβ,L = βL δp,p0 d
where 2
Vβ,r (u, v) = u + v +
|r| √ λ
2
√ −i λ eiα (u+iv) 2 p0 +e2p +λ(u2 +v 2 )
−
M
dd k (2π)d
d
Vβ,r (u,v)
dudv ,
d e−βL Vβ,r (u,v) dudv
Z 2 β
e−βL
log
cosh( β2
√
e2k +λ(u2 +v 2 ))
cosh
β 2 ek
(1.52)
.
(1.53)
Consider an attractive λ > 0 and sufficiently small T = β1 such that the global minimum of the effective potential moves away from zero. For r = 0, the global minimum of Vβ,r is degenerated and lies on a circle in the u, v plane. In particular, Vβ,0 is an even function of u and v and hψ¯ p↑ ψ−p↓ i vanishes by symmetry.
Figure 2
For r 6= 0, Vβ,r has a unique global minimum at (u, v) = (0, v0 ) where v0 is given by the negative solution of Z √2 2 β ek +λv0 ) 2 dd k tanh(√ − 1 = 2|r|, (1.54) v0 λ 2 2 (2π)d M
2
ek +λv0
436
D. Lehmann
which, in the limit r → 0, becomes the BCS equation for |1|2 = λv02 . Thus lim lim R
|r|→0 L→∞
e−βL e
dV
−βLd V
= lim lim R |r|→0 L→∞
β,r (u,v)
β,r (u,v) dudv
e−βL e
d [V
= δ(u)δ v +
β,r (u,v)−Vβ,r (0,v0 )]
−βLd [Vβ,r (u,v)−Vβ,r (0,v0 )]
dudv
|1| √ λ
(1.55)
and hψ¯ p↑ ψ¯ −p↓ i becomes nonzero. For repulsive λ < 0, Vβ,r is complex and the real part of Vβ,r has a unique global minimum at (u, v) = (0, 0) which results in lim|r|→0 limL→∞ R
e−βL
dV
β,r (u,v)
−βLd Vβ,r (u,v)
e
dudv
=
δ(u)δ(v) and lim|r|→0 limL→∞ hψ¯ p↑ ψ¯ −p↓ i = 0. These results of course are also obtained if one applies the usual mean field formalism which is based on approximating the quartic Hamiltonian by a quadratic expression. However, the situation is different if the electron-electron interaction contains higher angular momentum terms and the space dimension is 3. In that case the standard mean field formalism [AB,BW] predicts an angle dependent gap but, if ek has SO(3) symmetry, this is not the case. Suppose that λ` > 0 is attractive and U (k − p) = λ`
` X
Y¯`m (k0 )Y`m (p0 ).
(1.56)
m=−`
Then the Anderson Balian Werthammer mean field formalism gives h tanh( β √e2 +1∗ 1 ) i + 1 1 √2 2 k ∗ k k , lim Ld hakσ akσ i = 2 1 − ek
(1.57)
where the 2 × 2 matrix 1k , 1Tk = −1−k , is a solution of the gap equation Z √ 1 ) tanh( β2 e2k +1∗ dd k 0 0 k k √ U (p − k ) 1 . 1p = k 2 ∗ (2π)d
(1.58)
ek +1k 1k
L→∞
2
Mω
σσ
ek +1k 1k
In 3 dimensions, it has been proven [FKT2] that for all ` ≥ 2 (1.58) does not have unitary isotropic (1∗k 1k = const Id) solutions. That is, the gap in (1.57) is angle dependent. However in Theorem 3.3 it is shown that, for even `, Z ik0 +ek d(x) , (1.59) lim βL1 d hψ¯ kσ ψkσ iβ,L = k2 +e2 +λ` ρ2 | 6m α0 Y`m (x)|2 4π L→∞
S2
0
k
m
0
which gives, using (1.25),
Z lim 1d ha+kσ akσ i L→∞ L 1
= S2
1 2
1 − ek
tanh( β2
√
√
e2k +|1(x)|2 )
e2k +|1(x)|2
d(x) 4π .
(1.60)
0 Y`m (x) and ρ0 and α0 are values at the global minimum. The if 1(x) = λ`2 ρ0 6m αm point is that for SO(3) symmetric ek also the effective potential has SO(3) symmetry which means that also the global minimum has SO(3) symmetry. Since in the infinite volume limit the integration variables are forced to be at the global minimum, the integral over the sphere in (1.59,60) is the averaging over all global minima.
Many-Electron System in Forward, Exchange and BCS Approximation
437
There is some physics literature [B,BZT,BR,G,Ha,W] which investigates the relation between the reduced but still quartic and not solvable BCS Hamiltonian X X U (k − p) a+kσ a+−kτ apσ a−pτ (1.61) HBCS = H0 + L13d σ,τ ∈{↑,↓} k,p
and the quadratic, explicitly diagonalizable mean field Hamiltonian X X U (k − p) a+kσ a+−kτ hapσ a−pτ i + ha+kσ a+−kτ iapσ a−pτ HMF = H0 + L13d σ,τ ∈{↑,↓} k,p
− ha+kσ a+−kτ ihapσ a−pτ i ,
(1.62a)
where the numbers hapσ a−pτ i have to be determined by the condition hapσ a−pτ i =
T r e−βHMF apσ a−pτ , T r e−βHMF
(1.62b)
P which is equivalent to the gap Eq. (1.58) if one defines 1pστ = L1d k U (p − k) L1d hakσ a−kτ i. One has, if H 0 := HBCS − HMF , X X U (k − p) a+kσ a+−kτ − ha+kσ a+−kτ i apσ a−pτ − hapσ a−pτ i . H 0 = L13d σ,τ ∈{↑,↓} k,p
(1.63) It is claimed that, in the infinite volume limit, the correlation functions of both models should coincide. More precisely, it is claimed that 1 d L→∞ L
lim
−βHBCS
log TTrr ee−βHMF
(1.64)
vanishes. To this end it is argued that each order of perturbation theory (with respect 0 to H 0 ) of T r e−β(HMF +H ) /T r e−βHMF is finite as the volumeRgoes to infinity. The Haag paper argues that spatial averages of field operators like L1d [0,L]d dd xa↑ (x)a↓ (x) may be substituted by numbers in the infinite volume limit, since commutators with them have an extra one over volume factor, but there is no rigorous control of the error. At least for a more complicated electron-electron interaction (1.56), this reasoning cannot be correct in view of (1.57–60). Namely, consider the ha+kσ akσ iβ,L expectation. In terms of Grassmann variables it is given by (1.25,26). Assume first a delta function interaction U (k − p) = λ. For the full model (1.3), by making a Hubbard Stratonovich transformation, (1.26) can be rewritten as R F (φ) e−V (φ) 5 dφq0 q dφ¯ q0 q q0 q ¯ R , (1.65) hψpσ ψpσ i = −V (φ) 5 dφq0 q dφ¯ q0 q e q0 q
where the integrand is given by
"
1
λ 2 A + S↑ i( βL d) φ ∂ 1 λ ∂spσ |s=0 det i( βLd ) 2 φ¯ A¯ + S↓ # " F (φ) = 1 λ 2 A i( βL d) φ det 1 λ 2 ¯ A¯ i( βL d) φ
# (1.66)
438
D. Lehmann
and the effective potential reads "
A det 1 λ X i( βLd ) 2 φ¯ 2 |φq0 q | − log V (φ) = A q0 q det 0
# 1 λ 2 i( βL d) φ A¯ . 0 A¯
(1.67)
Here A and Sσ are diagonal matrices with entries Ak = ik0 − ek and skσ respectively and φ is a short notation for the matrix (φk−p )k,p , where k = (k0 , k) ∈ πβ (2Z + 1) × Mω . h i 0 2 2 Of course, for example det A 0 A¯ = 5k (k0 + ek ) does not make sense, but the quotients in (1.66,67) are well defined. The correlation functions of the reduced, but still quartic BCS Hamiltonian (1.61) of V (φ) are obtained from (1.65–67) by first assuming that the global minimum φmin q is proportional to δq,0 φq0 and second by suppressing the quantum fluctuations around φmin qq0 = 0 for all q 6= 0. That is, for the model (1.61) one finds R hψ¯ pσ ψpσ i =
Fp (φ) e−L V (φ) 5 dφq0 dφ¯ q0 q0 R , e−Ld V (φ) 5 dφq0 dφ¯ q0 d
(1.68)
q0
where the integrand is given by "
1
Ap + Sp↑ i( βλ ) 2 φ det 1 i( βλ ) 2 φ¯ A¯ p + Sp↓ # " 1 Ap i( βλ ) 2 φ det 1 i( λ ) 2 φ¯ A¯ p
#
∂ ∂sp0 σ |s=0
Fp (φ) =
(1.69)
β
and the effective potential reads "
V (φ) =
X q0
|φq0 |2 −
1 Ld
Ak det λ 21 ¯ X i( β ) φ log A k det k 0
# 1 i( βλ ) 2 φ A¯ k . 0 A¯ k
(1.70)
The volume factor L− 2 in the determinant in (1.66,67) has been transformed away by a substitution of variables such that it shows up in the exponent in front of the effective potential in (1.68). The matrices in (1.69,70) are labelled only by the k0 , p0 variables, that is, Ak is the diagonal matrix with entries Ak,k0 = ik0 − ek and φ is a short notation for the matrix (φk0 −p0 )k0 ,p0 . Contrary to the full model, the volume dependence of the model (1.61) or (1.68–70) is such that in the infinite volume limit the integration variables φq0 in (1.68) are forced to be at the global minimum of (1.70). The model discussed in this paper (that is, only the BCS part) is obtained from (1.68-70) by assuming that the global minimum of (1.70) is proportional to δq0 ,0 . In that case, the only integration variable which is left in (1.68) is the q0 = 0 mode φ = φ0 and the expressions (1.68-70) reduce to the integral representation (1.46,47). Now assume d
Many-Electron System in Forward, Exchange and BCS Approximation
439
that the elctron electron interaction is given by (1.56) and suppose for simplicity that ` is even which suppresses ↑↑↑↑ and ↓↓↓↓ contributions in (1.3) and (1.61). In that case the model (1.61) gives R hψ¯ pσ ψpσ i =
Fp (φ) e−L R
d
`
¯m 5 5 dφm q 0 d φq 0
V (φ)
m=−` q0
,
`
(1.71)
¯m e−Ld V (φ) 5 5 dφm q 0 d φq0 m=−` q0
where the integrand is given by
"
1
Ap + Sp↑ i( βλ ) 2 8p det 1 ¯ p A¯ p + Sp↓ i( βλ ) 2 8 " # 1 Ap i( βλ ) 2 8p det 1 ¯p A¯ p i( βλ ) 2 8
#
∂ ∂sp0 σ |s=0
Fp (φ) =
and the effective potential reads
V (φ) =
` X X m=−` q0
2 |φm q0 | −
(1.72)
"
1 Ld
# 1 Ak i( βλ ) 2 8k det 1 ¯ k −A¯ k X i( βλ ) 2 8 . log Ak 0 k det 0 A¯ k
(1.73)
P` Here 8k denotes the matrix with entries 8k,p0 p00 = m=−` φm p0 −p00 Y`m (k), labelled by 0 the p0 , p0 variables. The model discussed in this paper is obtained from (1.71–73) by assuming that the global minimum of (1.73) has only nonzero φm q0 =0 modes. In that case (1.71–73) reduce to the integral representation (1.44,45) which further can be reduced, by Theorem 3.3, to (1.59). However, the argument used in Theorem 3.3 that the hψ¯ pσ ψpσ i expectation has to be SO(3) invariant if ek is SO(3) invariant still applies to (1.71), since (1.73) is invariant under simultaneous transformation of the φm q0 ’s P 0 m 0 to m0 U (R) Pmm φq0 , where U (R) is the unitary representation of SO(3) given by Y`m (Rk) = m0 U (R)mm0 Y`m0 (k). That is, if the BCS Eq. (1.58) or (1.62b) of the quadratic mean field model (1.62) has a solution such that |1k |2 is not SO(3) invariant (and, by [FKT2], this is necessarily the case for any nonzero solution for d = 3 and ` ≥ 2), then the ha+kσ akσ i expectation of the quadratic mean field model (1.62) does not coincide with the corresponding expectation of the quartic reduced BCS Hamiltonian (1.61). In [AB], Anderson and Brinkmann used the quadratic mean field Hamiltonian with an ` = 1 interaction to describe the properties of superfluid Helium 3. The basic quantities in their analysis are the hakσ a−kτ i expectations or the matrix 1kστ which is obtained as a solution of the gap equation of the quadratic model. In view of the discussion above, one may regard as the more natural approach to take the quartic BCS Hamiltonian (1.61), to add in a symmetry breaking term which breaks the U(1) particle symmetry as well as m the spatial SO(3) symmetry and then to compute (in the approximation φm q0 = δq0 ,0 φ ) the infinite volume limit followed by the limit symmetry breaking term → 0. In particular, besides the usual U(1) symmetry breaking one may expect SO(3) symmetry breaking in the sense that probably also for the ha+ ai expectations the above two limits do not commute. That is, if B denotes the SO(3) symmetry breaking term, whereas
440
D. Lehmann
limL→∞ limB→0 ha+kσ akσ i has SO(3) symmetry, limB→0 limL→∞ ha+kσ akσ i may have not. However, to what extent the quantities of the [AB] paper are related to these expectations is not clear. For example, for ` ≥ 2 ([AB] has ` = 1, but still uses the quadratic mean field formalism) any nontrivial solution of the quadratic model (1.62) is necessarily anisotropic, without any external SO(3) symmetry breaking field at all. The quantities limB→0 limL→∞ ha+kσ akσ i for the quartic model (1.61) or the model discussed in this paper of course would depend on the direction of B, if they become anisotropic. A more careful analysis of this question we defer to another paper.
2. Solution of the Model in the Forward, Exchange and BCS Approximation Let H = H0 + Hint , where H0 =
1 Ld
X
X
ek a+kσ akσ
(1.2)
σ∈{↑,↓} k∈M
X
and Hint =
1 L3d
X
U (k − p) a+k,σ ap,σ a+q−k,τ aq−p,τ .
(1.3)
σ,τ ∈{↑,↓} k,p,q
Then Z=
T r e−β(H0 +Hint ) = T r e−βH0
Z
¯ ¯ e−Vint (ψ,ψ) dµC (ψ, ψ)
(1.23)
where the exponent in the fermionic integral is given by (κ = βLd ) X X U (k − p) ψ¯ k,σ ψp,σ ψ¯ q−k,τ ψq−p,τ Vint (ψ) = κ13
(1.24)
σ,τ ∈{↑,↓} k,p,q
and k = (k0 , k) ∈ πβ (2Z + 1) × M . The forward, exchange and BCS approximation is obtained by restricting the above sum to the following terms: forward : δk,p ,
exchange : δk,q−p ,
BCS : δq,0 .
(1.35)
¯ That is, We consider the approximation Vint ≈ Vforw + Vex + VBCS ≡ U(ψ, ψ). X X ¯ = 13 U (k−p) δk,p +δk,q−p +δq,0 ψ¯ k,σ ψp,σ ψ¯ q−k,τ ψq−p,τ (2.1) U (ψ, ψ) κ σ,τ ∈{↑,↓} k,p,q
and
U (k − p) =
j λ0 1 X λ|`| ei`ϕk e−i`ϕp + 2 2
`=−j − j ` X X
0
λ` Y¯`m k Y`m p
if d = 2 0
=: − if d = 3
J X
λl yl (k0 ) y¯l (p0 ),
l=0
`=0 m=−`
(1.37) and we abbreviated κ = βLd .
(2.2)
Many-Electron System in Forward, Exchange and BCS Approximation
441
This model can be solved explicitly. Before we write down the general solution we shortly indicate the computation for Vint ≈ VBCS and U (k − p) = −λ. In that case it comes down to the usual effective potential computation with “constant φ”. That is, using the identity (φ = u + iv, φ¯ = u − iv ∈ C) R 2 1 ¯ 1 eaφ+bφ e− 2 |φ| dudv, (2.3) e2ab = 2π R2 one obtains Z P P 1 1 ψ¯ ψ¯ −( λ ) 2 1 ×( λ ) 2 1 ψ ψ ¯ Z = e κ κ k k↑ −k↓ κ κ p p↑ −p↓ dµC (ψ, ψ) Z Z P P 1 1 λ 2 1 λ 2 ¯ 1 ¯ ¯ ¯ 1 e− 21 |φ|2 dudv ei( κ ) φ κ k ψk↑ ψ−k↓ + i( κ ) φ κ k ψk↑ ψ−k↓ dµC (ψ, ψ) = 2π 2 R Z 1 ik0 −ek iλ 2 φ¯ 1 κ −κ|φ|2 1 5 k2 +e dudv = 2 det πe iλ 2 φ −ik0 −ek 0 k 2 k R Z e−κVβ (u,v) dudv. (2.4) = πκ R2
In the following theorem the ξ variables take care of the exchange contributions, the w variable makes the forward contribution quadratic in the fermion fields and the φ variables, as in (2.3,4) above, sum up BCS contributions. ¯ be given by (2.1) and let Theorem 2.1. Let U (ψ, ψ) Z P ¯ 1 −U (ψ,ψ)+ sk,σ ψ¯ k,σ ψk,σ κ ¯ k,σ dµC (ψ, ψ), Z(β, L, {sk,σ }) = e hψ¯ pσ ψpσ i = κ
∂ ∂spσ |s=0
log Z(β, L, {skσ }).
(2.5) (2.6)
l l Let w ∈ R, ξστ = alστ + iblστ , φlστ = ulστ + ivστ ∈ C for 0 ≤ l ≤ J, (στ ) ∈ {↑↑, ↓↓, ↑↓} and define the fields
4k↑↓ =
J X
l (2λl ) 2 ξ↑↓ yl (k0 ) , 1
¯ k↑↓ = 4
l=0
4kσσ =
J X
1
l λl2 ξσσ yl (k0 ) ,
1
¯ kσσ = 4
J X
(2λl ) 2 φl↑↓ yl (k0 ) , 1
1
l λl2 ξ¯σσ y¯l (k0 ) ,
(2.7)
1
¯ k↑↓ = 8
J X
(2λl ) 2 φ¯ l↑↓ y¯l (k0 ), 1
(2.8) (2.9)
l=0
1
λl2 φlσσ [yl (k0 ) − yl (−k0 )] ,
l=0
where U0 = U (k = 0) = symmetric matrix
J X
¯ k↓↓ , 0k↓ = U02 w + i4k↓↓ + i4
l=0 J X
1
l=0
¯ k↑↑ , 0k↑ = U02 w + i4k↑↑ + i4
8kσσ =
l (2λl ) 2 ξ¯↑↓ y¯l (k0 ),
l=0
l=0
8k↑↓ =
J X
¯ kσσ = 8
J X
1
λl2 φ¯ lσσ [y¯l (k0 ) − y¯l (−k0 )] ,
l=0
R
(2.10) dd x U (x). For each k = (k0 , k), let Sk be the 8 × 8 skew
442
D. Lehmann
¯ k↑↓ 0 − 1i Ak↑ 8 0 0 8k↑↓ 0 0 − 1i A−k↓ 1 −8k↑↓ i A−k↓ 0 0 0 8k↓↓ ¯ k↓↓ 4k↑↓ 0 8 −8k↑↑ −4−k↑↓ 0 ¯ −k↑↓ 0 0 4
0 1 Ak↑ i −8 ¯ k↑↓ 0 Sk = −i −4 ¯ k↑↓ 0 0 ¯ k↑↑ −8
¯ k↑↓ ¯ k↑↑ 4 0 0 8 0 −4k↑↓ 8k↑↑ 0 ¯ k↓↓ 4−k↑↓ 0 −8 0 ¯ −k↑↓ −8k↓↓ 0 0 −4 , 1 0 0 i Ak↓ −8−k↑↓ ¯ −k↑↓ − 1i Ak↓ 0 0 −8 1 8−k↑↓ 0 0 i A−k↑ ¯ −k↑↓ − 1 A−k↑ 0 8 0 i
where Akσ = ak − skσ − 0kσ = ik0 − ek − skσ − 0kσ and let PfSk be the Pfaffian of Sk given by Lemma 2.2 below. Then: a) Z(β, L, {sk,σ }) =
Z Y
1 ak a−k
2
PfSk dνκ (w, ξ, φ),
(2.11)
k0 >0 k∈M
where dνκ (w, ξ, φ) =
κ 4π
21
J
e− 4 w dw 5 5 κ
2
l=0 στ
n
κ 2 π
e−κ(|φστ | l
2
l +|ξστ |2 )
o l dulστ dvστ dalστ dblστ . (2.12)
b) R
∂ ∂spσ
PfSp
PfSp
hψ¯ pσ ψp0 σ i = βL δp,p0 d
R
e−κV (w,ξ,φ) dw 5 dξdφ
e−κV (w,ξ,φ) dw 5 dξdφ
,
(2.13)
where the effective potential V is given by V (w, ξ, φ) = 41 w2 +
J XX στ
l 2 (|φlστ |2 + |ξστ | )−
1 βLd
X k∈M
l=0
log 5
PfSk 2 2 k0 >0 ak a−k
(2.14)
J
l and 5 dξdφ = 5 5 dξστ dφlστ . Here στ ∈ {↑↑, ↓↓, ↑↓}. στ l=0
Remark. The product
5
PfSk 2
2
k0 >0 ak a−k
in the effective potential (2.14), where PfSk
= PfSk (ak , a−k ) has to be computed according to the rule Y PfSk (eik0 ak , e−ik0 a−k ) Y PfSk = lim . →0 a2 a2 a2k a2−k <0 k >0 k >0 k −k 0
(2.15)
0
That there are cases where it is necessary to make the phase factors explicit can be seen from the discussion given in the proof of Lemma A1 in the Appendix. Proof. Since we assume U (p − k) = U (k − p), one has
Many-Electron System in Forward, Exchange and BCS Approximation
¯ U(ψ, ψ) XX U (0) ψ¯ k,σ ψ¯ p,τ ψk,σ ψp,τ + = κ13
XX
1 κ3
σ,τ k,p
+
1 κ3
XX
=
U (k − p) ψ¯ p,σ ψ¯ k,τ ψk,σ ψp,τ
σ,τ k,p
U (k − p) ψ¯ p,σ ψ¯ −p,τ ψk,σ ψ−k,τ
σ,τ k,p
− κ12 Uκ0
443
X
ψ¯ k↑ ψk↑ + ψ¯ k↓ ψk↓
ψ¯ p↑ ψp↑ + ψ¯ p↓ ψp↓
k,p
+
1 κ3
+
1 κ3
X
n o U (k − p) ψ¯ p↑ ψp↑ ψ¯ k↑ ψk↑ + ψ¯ p↓ ψp↓ ψ¯ k↓ ψk↓ + 2ψ¯ p↑ ψp↓ ψ¯ k↓ ψk↑
k,p
n o U (k−p) ψ¯ p↑ ψ¯ −p↑ ψk↑ ψ−k↑ + ψ¯ p↓ ψ¯ −p↓ ψk↓ ψ−k↓ +2ψ¯ p↑ ψ¯ −p↓ ψk↑ ψ−k↓ .
X k,p
We substitute U (k − p) =
PJ l=0
λl yl (k0 )y¯l (p0 ) and use the identities 2
1
e2X = e−2XY =
1 2π
√1 2π
R
R
eXw e− 2 w dw,
eiXφ+iY φ e− 2 |φ| dudv ¯
R2
2
1
R
2
1
to obtain eZ−U (ψ,ψ) = ¯
n
exp
2U0 κ
n Xh exp
i
21
λl 2κ
l
n Xh exp
i
λl 2κ
l
n Xh exp
i
λl κ
l
n Xh exp
i
λl 2κ
l
n Xh exp
i
λl 2κ
l
n Xh exp
i
λl κ
w
21 21 21 21 21 21
1 κ
X k
l ξ¯↑↑
1 κ
ψ¯ k↑ ψk↑ + ψ¯ k↓ ψk↓
X
o
× λl 2κ
y¯l (k0 ) ψ¯ k↑ ψk↑ + i
k
l ξ¯↓↓
1 κ
X
λl 2κ
y¯l (k0 ) ψ¯ k↓ ψk↓ + i
k
l ξ¯↑↓
1 κ
X
λl κ
y¯l (k0 ) ψ¯ k↑ ψk↓ + i
21 21 21
k
φ¯ l↑↑
1 κ
X
y¯l (k0 ) ψ¯ k↑ ψ¯ −k↑ + i
λl 2κ
k
φ¯ l↓↓
1 κ
X
y¯l (k0 ) ψ¯ k↓ ψ¯ −k↓ + i
λl 2κ
k
φ¯ l↑↓
1 κ
X
l
y¯l (k0 ) ψ¯ k↑ ψ¯ −k↓ + i
λl κ
l ξ↑↑
1 κ
X
yl (k0 ) ψ¯ k↑ ψk↑
k
l ξ↓↓
1 κ
X
yl (k0 ) ψ¯ k↓ ψk↓
k
l ξ↑↓
21 21 21
1 κ
X
yl (k0 ) ψ¯ k↓ ψk↑
io io io
× ×
×
k
φl↑↑
1 κ
X
yl (k0 ) ψk↑ ψ−k↑
k
φl↓↓
1 κ
X
yl (k0 ) ψk↓ ψ−k↓
k
φl↑↓
1 κ
X
k
yl (k0 ) ψk↑ ψ−k↓
io io
× ×
io dν(w, ξ, φ),
k
where dν(w, ξ, φ) =
√1 2π
e− 2 w dw 5 1
2
l
5
(στ )∈ {↑↑,↓↓,↑↓}
1 (2π)2
e− 2 (|φστ | 1
l
2
l +|ξστ |2 )
By a substitution of variables and collecting terms, one obtains
l dulστ dvστ dalστ dblστ .
444
D. Lehmann
e−U (ψ,ψ) = ¯
Z
n Xh 1 κ
exp
n Xhk exp
1 κ
1
U02 w + i
1 2
U0 w + i
nXkh exp
n Xh
i(2λl )
l 1 ξ¯↑↓ κ
n Xh
1
iλl2 φ¯ l↑↑
1 κ
n Xh
i
1
l l λl2 ξ¯↓↓ y¯l (k0 ) + ξ↓↓ yl (k0 )
X
o
ψ¯ k↓ ψk↓ ×
1
l y¯l (k0 ) ψ¯ k↑ ψk↓ + i(2λl ) 2 ξ↑↓
X y¯ (k0 )−y¯ (−k0 ) l
1 κ
X
l
2
k 1
ψ¯ k↑ ψ¯ −k↑ + iλl2 φl↑↑
1 κ
k
1
iλl2 φ¯ l↓↓
1 κ
X y¯ (k0 )−y¯ (−k0 ) l
l
2
l
exp
l
o
ψ¯ k↑ ψk↑ ×
k
l
exp
X
i
1
l l λl2 ξ¯↑↑ y¯l (k0 ) + ξ↑↑ yl (k0 )
l 1 2
l
exp
X
k 1 i(2λl ) 2 φ¯ l↑↓
1 κ
X
l
yl (k0 ) ψ¯ k↓ ψk↑
io
X y (k0 )−y (−k0 ) l
l
2
×
io ψk↑ ψ−k↑
k
1
ψ¯ k↓ ψ¯ −k↓ + iλl2 φl↓↓
1 κ
X y (k0 )−y (−k0 ) l
l
2
io ψk↓ ψ−k↓
k
1 y¯l (k0 ) ψ¯ k↑ ψ¯ −k↓ + i(2λl ) 2 φl↑↓
1 κ
k
X
yl (k0 ) ψk↑ ψ−k↓
× ×
io dνκ (w, ξ, φ),
k
where dνκ (w, ξ, φ) is defined in the statement of the theorem. Using the definition of the fields 4, 0 and 8, the above expression reads e−U (ψ,ψ) ¯
Xh
Z =
¯ k↑↓ ψ¯ k↑ ψk↓ + i4k↑↓ ψ¯ k↓ ψk↑ 0k↑ ψ¯ k↑ ψk↑ + 0k↓ ψ¯ k↓ ψk↓ + i4
1 κ
exp
k
¯ k↑↓ ψ¯ k↑ ψ¯ −k↓ + i8k↑↓ ψk↑ ψ−k↓ + i8 +
i 2
X
¯ kσσ ψ¯ kσ ψ¯ −kσ + 8kσσ ψkσ ψ−kσ 8
i
dνκ .
σ
We now rewrite the exponent in order to perform the fermionic functional integral. Since, if the set of spatial momenta satisfy k ∈ Mω = −Mω , X k
0k↑ ψ¯ k↑ ψk↑ X
X
=
k0 ∈ π β (2Z+1) k∈Mω
=
1 2
Xh
0k↑ ψ¯ k↑ ψk↑ =
X Xh
0k↑ ψ¯ k↑ ψk↑ + 0−k↑ ψ¯ −k↑ ψ−k↑
k0 >0 k∈Mω
0k↑ ψ¯ k↑ ψk↑ − ψk↑ ψ¯ k↑ + 0−k↑ ψ¯ −k↑ ψ−k↑ − ψ−k↑ ψ¯ −k↑
k k0 >0
¯ kσσ , one obtains, using the antisymmetry of the 8kσσ , 8
i
i
Many-Electron System in Forward, Exchange and BCS Approximation
Xn k
=
1 2
445
¯ k↑↓ ψ¯ k↑ ψ¯ −k↓ + i8k↑↓ ψk↑ ψ−k↓ 6 0kσ ψ¯ kσ ψkσ + i8 σ
¯ k↑↓ ψ¯ k↑ ψk↓ + i4k↑↓ ψ¯ k↓ ψk↑ + i 6 8 ¯ kσσ ψ¯ kσ ψ¯ −kσ + 8kσσ ψkσ ψ−kσ + i4 2 σ X h i 6 0kσ ψ¯ kσ ψkσ − ψkσ ψ¯ kσ + 0−kσ ψ¯ −kσ ψ−kσ − ψ−kσ ψ¯ −kσ k k0 >0
o
σ
¯ −k↑↓ ψ¯ −k↑ ψ¯ k↓ − ψ¯ k↓ ψ¯ −k↑ ¯ k↑↓ ψ¯ k↑ ψ¯ −k↓ − ψ¯ −k↓ ψ¯ k↑ + i8 + i8 + i8k↑↓ ψk↑ ψ−k↓ − ψ−k↓ ψk↑ + i8−k↑↓ ψ−k↑ ψk↓ − ψk↓ ψ−k↑ ¯ k↑↓ ψ¯ k↑ ψk↓ − ψk↓ ψ¯ k↑ + i4 ¯ −k↑↓ ψ¯ −k↑ ψ−k↓ − ψ−k↓ ψ¯ −k↑ + i4 + i4k↑↓ ψ¯ k↓ ψk↑ − ψk↑ ψ¯ k↓ + i4−k↑↓ ψ¯ −k↓ ψ−k↑ − ψ−k↑ ψ¯ −k↓ h i ¯ ¯ ¯ ¯ ¯ + i 6 8kσσ ψkσ ψ−kσ − ψ−kσ ψkσ + 8kσσ ψkσ ψ−kσ − ψ−kσ ψkσ . σ
Since, if ak = ik0 − ek , ¯ = 5 dµC (ψ, ψ)
κ k,σ ak
e
− κ1
P k,σ
ak ψ¯ k,σ ψk,σ
5 dψk,σ dψ¯ k,σ ,
k,σ
one obtains Z Z(β, L, {sk,σ }) =
¯ 1 −U (ψ,ψ)+ κ
P
e
k,σ
sk,σ ψ¯ k,σ ψk,σ
¯ dµC (ψ, ψ)
Z Z P P −1 1 h9k ,Sk 9k i e 2 κ k0 >0 k 5 dψkσ dψ¯ kσ dνκ (w, ξ, φ) = 5 aκk k,σ k,σ Z Y 2 Z 1 1 κ2 = e− 2 κ h9k ,Sk 9k i dψ¯ k↑ dψk↑ dψ¯ −k↓ dψ−k↓ × ak a−k k0 >0
k∈Mω
Z =
¯ ¯ dψk↓ dψk↓ dψ−k↑ dψ−k↑ dνκ (w, ξ, φ) 2 Y 1 PfSk dνκ (w, ξ, φ), ak a−k
k0 >0
k∈Mω
where
9k = ψ¯ k↑ , ψk↑ , ψ¯ −k↓ , ψ−k↓ , ψk↓ , ψ¯ k↓ , ψ−k↑ , ψ¯ −k↑ ,
and PfSk is the Pfaffian of the 8 × 8 skew symmetric matrix Sk defined in the statement of the theorem. We used that n o 5 dψkσ dψ¯ kσ = 5 5 + dψ¯ k↑ dψk↑ dψ¯ −k↓ dψ−k↓ dψk↓ dψ¯ k↓ dψ−k↑ dψ¯ −k↑ . k,σ
k0 >0 k
Part b) of the theorem follows from 1 ¯ κ hψpσ ψpσ i
=
∂ ∂spσ |s=0
log Z(β, L, {skσ }).
446
D. Lehmann
Lemma 2.2. Let Sk be the skew symmetric 8 × 8 matrix of Theorem 2.1. Then the Pfaffian of Sk is given by ¯ k↑↓ 8k↑↓ A−k↑ Ak↓ + 8 ¯ −k↑↓ 8−k↑↓ PfSk = Ak↑ A−k↓ + 8 ¯ k↓↓ + Ak↓ A−k↓ 8k↑↑ 8 ¯ k↑↑ + Ak↑ A−k↑ 8k↓↓ 8 ¯ ¯ ¯ ¯ + 8k↑↑ 8k↓↓ 8k↑↓ 8−k↑↓ + 8k↑↑ 8k↓↓ 8k↑↓ 8−k↑↓ ¯ k↑↑ 8 ¯ k↓↓ + 8k↑↑ 8k↓↓ 8 ¯ k↑↓ A−k↑ 8k↑↓ 8 ¯ ¯ k↓↓ + i4k↑↓ A−k↑ 8k↑↓ 8k↓↓ + i4 ¯ k↑↓ A−k↓ 8 ¯ k↑↑ − i4 ¯ −k↑↓ 8k↑↑ − i4k↑↓ A−k↓ 8−k↑↓ 8 ¯ ¯ ¯ + i4−k↑↓ Ak↓ 8k↑↓ 8k↑↑ + i4−k↑↓ Ak↓ 8k↑↓ 8k↑↑ ¯ −k↑↓ Ak↑ 8−k↑↓ 8 ¯ −k↑↓ 8k↓↓ − i4 ¯ k↓↓ − i4−k↑↓ Ak↑ 8 ¯ ¯ + 4k↑↓ 4k↑↓ A−k↑ A−k↓ + 4−k↑↓ 4−k↑↓ Ak↑ Ak↓ ¯ −k↑↓ 8 ¯ k↑↓ 8−k↑↓ + 4−k↑↓ 4 ¯ k↑↓ 8 ¯ −k↑↓ 8k↑↓ + 4k↑↓ 4 ¯ ¯ ¯ ¯ k↓↓ − 4k↑↓ 4−k↑↓ 8k↑↑ 8k↓↓ − 4k↑↓ 4−k↑↓ 8k↑↑ 8 ¯ k↑↓ 4−k↑↓ 4 ¯ −k↑↓ . + 4k↑↓ 4 P Proof. The Pfaffian of Sk is given by the sum of all contractions 5hψψi of the fields ψ¯ k↑ , ψk↑ , ψ¯ −k↓ , ψ−k↓ , ψk↓ , ψ¯ k↓ , ψ−k↑ , ψ¯ −k↑ , where the value hψψi is given by the corresponding matrix element. That is, the Pfaffian can be evaluated by using Wick’s Theorem or integration by parts. Since Sk is an 8 × 8 matrix, one has Pf[−Sk ] = PfSk and PfSk = hψ¯ k↑ ψk↑ ψ¯ −k↓ ψ−k↓ ψk↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk ¯ k↑↓ hψk↑ ψ−k↓ ψk↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk = −Ak↑ hψ¯ −k↓ ψ−k↓ ψk↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk −i8 ¯ k↑↓ hψk↑ ψ¯ −k↓ ψ−k↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iS + i8 ¯ k↑↑ hψk↑ ψ¯ −k↓ ψ−k↓ ψk↓ ψ¯ k↓ ψ−k↑ iS − i4 k
¯ k↓↓ hψ−k↓ ψk↓ ψ−k↑ ψ¯ −k↑ iSk = +Ak↑ A−k↓ hψk↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk + iAk↑ 8 ¯ ¯ + Ak↑ 4−k↑↓ hψ−k↓ ψk↓ ψk↓ ψ−k↑ iS k
¯ k↑↓ 8k↑↓ hψk↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk − 8 ¯ k↑↓ 4k↑↓ hψ−k↓ ψk↓ ψ−k↑ ψ¯ −k↑ iSk +8 ¯ k↑↓ 8k↑↑ hψ−k↓ ψk↓ ψ¯ k↓ ψ¯ −k↑ iS −8 k
¯ k↑↓ 8k↑↓ hψ¯ −k↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk − 4 ¯ k↑↓ 4k↑↓ hψ¯ −k↓ ψ−k↓ ψ−k↑ ψ¯ −k↑ iSk −4 ¯ k↑↓ 8k↑↑ hψ¯ −k↓ ψ−k↓ ψ¯ k↓ ψ¯ −k↑ iSk −4 ¯ k↑↑ 8k↑↓ hψ¯ −k↓ ψk↓ ψ¯ k↓ ψ−k↑ iS − 8 ¯ k↑↑ 4k↑↓ hψ¯ −k↓ ψ−k↓ ψk↓ ψ−k↑ iS +8 k
k
¯ k↑↑ 8k↑↑ hψ¯ −k↓ ψ−k↓ ψk↓ ψ¯ k↓ iSk −8 ¯ k↑↓ hψk↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk = + Ak↑ A−k↓ + 8k↑↓ 8 ¯ k↑↓ 8k↑↑ hψ−k↓ ψk↓ ψ¯ k↓ ψ¯ −k↑ iSk + Ak↑ i4−k↑↓ − 8 ¯ k↓↓ − 8 ¯ k↑↓ 4k↑↓ hψ−k↓ ψk↓ ψ−k↑ ψ¯ −k↑ iS + Ak↑ i8 k
¯ k↑↓ 8k↑↓ hψ¯ −k↓ ψ¯ k↓ ψ−k↑ ψ¯ −k↑ iSk − 4 ¯ k↑↓ 4k↑↓ hψ¯ −k↓ ψ−k↓ ψ−k↑ ψ¯ −k↑ iSk −4 ¯ k↑↓ 8k↑↑ hψ¯ −k↓ ψ−k↓ ψ¯ k↓ ψ¯ −k↑ iSk + 8 ¯ k↑↑ 8k↑↓ hψ¯ −k↓ ψk↓ ψ¯ k↓ ψ−k↑ iSk −4 ¯ k↑↑ 4k↑↓ hψ¯ −k↓ ψ−k↓ ψk↓ ψ−k↑ iS − 8 ¯ k↑↑ 8k↑↑ hψ¯ −k↓ ψ−k↓ ψk↓ ψ¯ k↓ iS −8 k
k
k
Many-Electron System in Forward, Exchange and BCS Approximation
447
¯ k↑↓ Ak↓ A−k↑ + 8−k↑↓ 8 ¯ −k↑↓ = + Ak↑ A−k↓ + 8k↑↓ 8 ¯ k↑↓ 8k↑↑ −8k↓↓ 8 ¯ −k↑↓ Ak↓ ¯ −k↑↓ − i4 + Ak↑ i4−k↑↓ − 8 ¯ k↓↓ − 8 ¯ k↑↓ 4k↑↓ −i8k↓↓ A−k↑ − 4 ¯ −k↑↓ 8−k↑↓ + Ak↑ i8 ¯ k↑↓ 8k↑↓ −i8 ¯ k↓↓ A−k↑ − 4−k↑↓ 8 ¯ −k↑↓ −4 ¯ k↑↓ 4k↑↓ −A−k↓ A−k↑ − 4−k↑↓ 4 ¯ −k↑↓ −4 ¯ k↑↓ 8k↑↑ A−k↓ i8 ¯ −k↑↓ + 8 ¯ k↓↓ 4 ¯ −k↑↓ −4 ¯ k↑↑ 8k↑↓ 8 ¯ k↓↓ 8−k↑↓ + i4−k↑↓ Ak↓ +8 ¯ k↑↑ 4k↑↓ A−k↓ i8−k↑↓ + 4−k↑↓ 8k↓↓ −8 ¯ k↑↑ 8k↑↑ − A−k↓ Ak↓ − 8 ¯ k↓↓ 8k↓↓ . −8
By multiplying out the brackets one obtains the stated formula.
Before we specialize Theorem 2.1 in Sect. 3 where the effective potential and the two point functions are computed more explicitly, in the following theorem we write down the integral representation for the generating functional of the connected amputated Greens functions. ¯ be given by (2.1) and let V be the effective potential (2.14). Theorem 2.3. Let U(ψ, ψ) Let Z ¯ ¯ ¯ dµC (ψ, ψ) (2.16) G(η) = log e−U (ψ+η,ψ+η) be the generating functional for the connected amputated Greens functions. Then, if ζ k denotes the eight component vector ζ k = (ak ηk↑ , −ak η¯k↑ , a−k η−k↓ , −a−k η¯−k↓ , −ak η¯k↓ , ak ηk↓ , −a−k η¯−k↑ , a−k η−k↑ ) and Sk = Sk (w, ξ, φ) is the 8 × 8 matrix of Theorem 2.1, one has the following integral representation: G(η) − G(0) = −
1 κ
X
R
ak η¯kσ ηkσ + log
e− 2 κ 1 1
P k
R
hζ ,Sk−1 ζ i −κV (w,ξ,φ) k k
kσ
e
dw 5 dξdφ
e−κV (w,ξ,φ) dw 5 dξdφ
.
(2.17)
Proof. By a substitution of Grassmann variables, Z ¯ ¯ e−U (ψ+η,ψ+η) dµC = e− κ 1
Z
P kσ
ak η¯ kσ ηkσ
e−U (ψ,ψ) 5 ¯
κ kσ ak
1
eκ
P kσ
ak [ψ¯ kσ ηkσ +η¯ kσ ψkσ −ψ¯ kσ ψkσ ]
5 dψkσ dψ¯ kσ .
kσ
As in the proof of Theorem 2.1, one has Z P P P 1 − 21 κ1 h9k ,Sk 9k i ¯ −U (ψ,ψ)− ak ψ¯ kσ ψkσ k0 >0 k κ kσ = e dνκ (w, ξ, φ) e such that
448
Z
D. Lehmann
e−U (ψ,ψ) 5 ¯
P
1
κ kσ ak
eκ Z Z
kσ
5
=
ak [ψ¯ kσ ηkσ +η¯ kσ ψkσ −ψ¯ kσ ψkσ ]
k0 >0,k
Z Z
κ4 k0 >0,k PfSk
5
5
=
k0 >0,k
e
κ4 k0 >0,k PfSk
5
k0 >0,k
e
PfSk a2k a2−k
− 21
PfSk a2k a2−k
5
Z =
PfSk a2k a2−k
κ
− 21
e
P
1
eκ P 1
kσ
P k
κ
×
dνκ (w, ξ, φ) 5 dψkσ dψ¯ kσ kσ
k0 >0,k
P k0 >0
1 κ
h9k ,Sk 9k i
P
e P 1 − 21
ak [ψ¯ kσ ηkσ +η¯ kσ ψkσ ]
k0 >0
1 κ
5 dψkσ dψ¯ kσ
kσ
k
h9k ,ζ i k
×
h9k ,Sk 9k i
5 d9k dνκ (w, ξ, φ)
k0 >0,k
P k0 >0,k
hζ ,Sk−1 ζ i k
k
dνκ (w, ξ, φ),
where d9k = dψ¯ k↑ dψk↑ dψ¯ −k↓ dψ−k↓ dψk↓ dψ¯ k↓ dψ−k↑ dψ¯ −k↑ . By definition of V 5
k0 >0,k
PfSk a2k a2−k
dνκ (w, ξ, φ) = e−κV (w,ξ,φ) dw 5 dξdφ
which proves the theorem.
3. Solution for Pure BCS We consider the model
Z
Z(β, L, {sk,σ }) =
¯ 1 −U (ψ,ψ)+ κ
e
P k,σ
sk,σ ψ¯ k,σ ψk,σ
¯ dµC (ψ, ψ)
with ek = k2 /2m − µ, C(k) = 1/(ik0 − ek ) and X X ¯ = 13 U (k − p) ψ¯ k,σ ψ¯ −k,τ ψp,σ ψ−p,τ . U(ψ, ψ) κ
(3.1)
(3.2)
σ,τ ∈{↑,↓} k,p
The electron-electron interaction U is given by (1.37). To write down the effective potential and the two point functions in this case, one first has to compute the Pfaffian of the matrix Sk of Theorem 2.1. Since we consider only a BCS interaction, the 4 and 0 fields are zero. With Lemma 2.2 one obtains ¯ k↑↓ 8k↑↓ a−k↑ ak↓ + 8 ¯ −k↑↓ 8−k↑↓ PfSk = ak↑ a−k↓ + 8 ¯ k↓↓ + ak↓ a−k↓ 8k↑↑ 8 ¯ k↑↑ + ak↑ a−k↑ 8k↓↓ 8 ¯ k↑↑ 8 ¯ k↑↓ 8 ¯ −k↑↓ + 8 ¯ k↓↓ 8k↑↓ 8−k↑↓ +8k↑↑ 8k↓↓ 8 ¯ k↑↑ 8 ¯ k↓↓ , (3.3) + 8k↑↑ 8k↓↓ 8 where akσ = ak − skσ . In particular, for skσ = 0 , PfSk = (ak a−k + +k )(ak a−k + − k ), where ± k are the solutions of the quadratic equation
(3.4)
Many-Electron System in Forward, Exchange and BCS Approximation
449
¯ k↑↓ 8k↑↓ + 8 ¯ −k↑↓ 8−k↑↓ +8k↑↑ 8 ¯ k↑↑ +8k↓↓ 8 ¯ k↓↓ +8k↑↓ 8 ¯ k↑↓ 8−k↑↓ 8 ¯ −k↑↓ 2 − 8 ¯ k↑↑ 8 ¯ k↑↓ 8 ¯ −k↑↓ + 8 ¯ k↓↓ 8k↑↓ 8−k↑↓ +8k↑↑ 8k↓↓ 8 ¯ k↑↑ 8 ¯ k↓↓ = 0. + 8k↑↑ 8k↓↓ 8
(3.5)
The effective potential becomes V (φ) =
X
N X
|φlστ |2
−
1 βLd
στ ∈{↑↑,↓↓,↑↓} l=0
X
log 5
k0 >0
k∈M
− ak a−k ++k ak a−k +k ak a−k ak a−k
. (3.6)
We start in Sect. 3.1 with the case of a delta function interaction, that is, J = 0 in (1.37). This reproduces the usual mean field results. In Sect. 3.2, we consider a more general electron-electron interaction of the form (1.37) with arbitrary J. One finds that the standard approach based on approximating the Hamiltonian by quadratic terms and imposing a self consistency equation may be misleading. 3.1. BCS with Delta Function Interaction. Corollary 3.1. Let sk,↑ , sk,↓ , rk , r¯k be some real or complex numbers, let Z Z(β, L, {sk }, {rk }) = exp − κλ3 6 ψk↑ ψ−k↓ ψ¯ p↑ ψ¯ −p↓ + κ1 6[sk↑ ψ¯ k↑ ψk↑ k,p k ¯ ¯ ¯ ¯ (3.7) + sk↓ ψk↓ ψk↓ + rk ψk↑ ψ−k↓ − r¯k ψk↑ ψ−k↓ ] dµC (ψ, ψ) and let 1 ¯ κ hψpσ ψpσ iβ,L,r
=
∂ ∂spσ
1 κ hψp↑ ψ−p↓ iβ,L,r
=
∂ ∂rp
log Z(β, L, {sk }, {rk })|sk =0,rk =r ,
(3.8)
log Z(β, L, {sk }, {rk })|sk =0,rk =r .
(3.9)
Wβ,k λ(u2 + v 2 ) ,
(3.10)
Define the effective potential Vβ,r (u, v) = u2 + (v +
|r| √ )2 λ
−
where
Wβ,k (y) =
1 β
log
R
dd k M (2π)d
cosh( β2
√
cosh
e2k +y)
β 2 ek
2 .
a) Let ap = ip0 − ep . There are the two dimensional integral representations R Z(β, L, r) = πκ e−κVβ,r (u,v) dudv, R −a−p −κVβ,r (u,v) dudv ap a−p +λ(u2 +v 2 ) e 1 ¯ R , h ψ ψ i = p,σ p,σ β,L,r κ e−κVβ,r (u,v) dudv R −i√λ e−iα (u−iv) −κV (u,v) β,r dudv ap a−p +λ(u2 +v 2 ) e 1 R hψ ψ i = p,↑ −p,↓ β,L,r κ −κV (u,v) β,r dudv e R −i√λ eiα (u+iv) −κV (u,v) β,r dudv ap a−p +λ(u2 +v 2 ) e 1 ¯ ¯ R . ψ h ψ i = p,↑ −p,↓ β,L,r κ −κV (u,v) β,r dudv e
(3.11)
(3.12) (3.13)
(3.14)
450
D. Lehmann
R dd k 2 2 b) Let Vβ (ρ) := ρ2 − (2π) = λρ20 be given by Vβ (ρ0 ) = d Wβ,k (λρ ) and |1| −iα minρ≥0 Vβ (ρ) Then, if 1 = 1(λ, β) = |1| e , one has lim lim 1 d |r|→0 L→∞ βL
log Z(β, L, r) = −Vβ (|1|),
(3.15)
−a−p ap a−p +|1|2 ,
lim lim 1 d hψ¯ p,σ ψp,σ iβ,L,r |r|→0 L→∞ βL
=
lim lim 1 d hψp,↑ ψ−p,↓ iβ,L,r |r|→0 L→∞ βL
=
1 ap a−p +|1|2 ,
(3.17)
lim lim 1 d hψ¯ p,↑ ψ¯ −p,↓ iβ,L,r |r|→0 L→∞ βL
=
¯ −1 ap a−p +|1|2 ,
(3.18)
(3.16)
but lim lim 1 hψp,↑ ψ−p,↓ iβ,L,r L→∞ |r|→0 κ
1 ¯ hψp,↑ ψ−p,↓ iβ,L,r L→∞ |r|→0 κ
= lim lim
= 0.
(3.19)
Proof. a) For a δ function interaction one has J = 0 in Theorem 2.1 and the fields 1 1 (2.10) are zero, 8kσσ = 0. The fields (2.9) become 8k↑↓ = (2λ) 2 φ0↑↓ ≡ (2λ) 2 φ. Since we added the rψψ and r¯ψ¯ ψ¯ terms to the exponent in (3.7) √ which were not present in Theorem 2.1, these fields have to be substituted by (g = 2λ) g φ¯ → g φ¯ + ir¯k .
gφ → gφ − irk ,
The Pfaffian (3.3) becomes PfSk = ak↑ a−k↓ + [g φ¯ + ir¯k ][gφ − irk ] a−k↑ ak↓ + [g φ¯ + ir¯−k ][gφ − ir−k ] (3.20) such that
∂ ∂spσ
PfSp
PfSp
and
∂ ∂rp
|sk =0,rk =r
PfSp
PfSp
=
|sk =0,rk =r
−a−p ¯ r] ap a−p +[gφ−ir][g φ+i ¯
(3.21)
¯ r] −i[g φ+i ¯ . ¯ r] ap a−p +[gφ−ir][g φ+i ¯
(3.22)
=
The effective potential (3.6) is given by (the |φ|2 term is not shifted by r) n o2 X ¯ r] φ+i ¯ log 5 ak a−k +[gφ−ir][g . V (φ) = |φ|2 − βL1 d ak a−k k0 >0
k∈M
The product over k0 ∈ [Hn]
(3.23)
π β (2N + 1) in (3.23) can be computed explicitly using the formula ∞
5 1+
n=0
ξ (2n+1)2 +a2
=
π 2
cosh
√
cosh
a2 +ξ
π 2 a
.
(3.24)
One obtains 5
k0 >0
n
ak a−k +ξ ak a−k
o2 =
5
k0 ∈ π β (2N+1)
1+
ξ k02 +e2k
2 =
which gives, if one approximates the Riemannian sum
cosh
β 2
cosh 1 Ld
√
e2k +ξ
β 2 ek
P k∈Mω
2 (3.25)
by an integral
Many-Electron System in Forward, Exchange and BCS Approximation
Z V (φ) = |φ|2 − Mω
451
Wβ,k [gφ − ir][g φ¯ + ir] ¯ .
dd k (2π)d
(3.26)
Thus one arrives at the integral representations R Z(β, L, r) = πκ e−κV (φ) dudv, R 1 ¯ κ hψp,σ ψp,σ iβ,L,r
= R
1 κ hψp,↑ ψ−p,↓ iβ,L,r
=
(3.27)
−a−p e−κV (φ) dudv ¯ r] ap a−p +[gφ−ir][g φ+i ¯ R , e−κV (φ) dudv ¯ r] −i[g φ+i ¯ e−κV (φ) dudv ¯ r] ap a−p +[gφ−ir][g φ+i ¯ R , e−κV (φ) dudv
(3.28)
(3.29)
where V is given by (3.26). Part (a) then follows from the substitution of variables Z 2 F gφ − ir, g φ¯ + ir¯ e−κ|φ| dudv = Z (3.30) −κ u2 +(v+ |r| )2 g F eiα gφ, e−iα g φ¯ e dudv which holds for both signs of λ. To obtain part (b), one has to compute the limit of δκ,r (u, v) = R
e−κVβ,r (u,v) e−κVβ,r (u,v) dudv
where Vβ,r (u, v) = u2 + (v +
|r| 2 g )
,
(3.31)
− Wβ λ(u2 + v 2 )
(3.32)
if we define Z Wβ (ξ) = M
In particular,
dd k (2π)d
Z Wβ,k (ξ) =
1 β
M
dd k (2π)d
log
cosh
β 2
cosh
√
e2k +ξ
β 2 ek
2 .
(3.33)
signWβ λ(u2 + v 2 ) = signλ.
For positive λ and nonzero r, Vβ,r (u, v) is real and has a unique global minimum determined by 2u − 2uλWβ0 λ(u2 + v 2 ) = 0, 2(v + |r|) − 2vλWβ0 λ(u2 + v 2 ) = 0. Since λWβ0 = 1 does not solve the second equation, the only solution of the first equation is u = 0 and one is left with v λWβ0 (λv 2 ) − 1 = |r| which has a solution v = O(|r|) which is a local maximum and two nontrivial solutions λv 2 = |1|2 + O(|r|) where the positive one is only a local minimum and the negative one, v0 , is the global minimum. Therefore
452
D. Lehmann
e−κ[Vβ,r (u,v)−Vβ,r (0,v0 )] = δ(u) δ(v − v0 ) e−κ[Vβ,r (u,v)−Vβ,r (0,v0 )] dudv
lim δκ,r (u, v) = lim R
κ→∞
κ→∞
lim lim δκ,r (u, v) = δ(u) δ v +
and
|1| g
|r|→0 κ→∞
.
(3.34)
This proves the formulae under (b) for attractive λ. Since
lim Vr,β (u, v) = u2 + v 2 − Wβ λ(u2 + v 2 )
|r|→0
is an even function in u and v, limr→0 hψp,↑ ψ−p,↓ iβ,L,r =limr→0 hψ¯ p,↑ ψ¯ −p,↓ iβ,L,r = 0. The limit of the logarithm of the partition function becomes Z 1 1 κ e−κVβ,r (u,v) dudv log Z = log κ κ π Z = κ1 log πκ e−κ[Vβ,r (u,v)−Vβ,r (0,v0 )] dudv + κ1 log e−κVβ,r (0,v0 ) . The first term on the right-hand side may be approximated by (Vuv = Z 2 2 κ 1 κ e− 2 (Vuu u +Vvv (v−v0 ) ) dudv log κ π Z 2 2 1 κ→∞ e− 2 (Vuu u +Vvv v ) dudv → 0 = κ1 log π1
∂ 2 Vβ,r ∂u∂v (0, v0 )
= 0)
which results in log Z = −Vβ,r (0, v0 ).
lim 1 κ→∞ κ
Now let λ be negative. In that case the effective potential (3.32) is complex:
|r|2 |λ|
Vβ,r (u, v) = u2 + v 2 −
− 2iv √|r| − Wβ λ(u2 + v 2 )
= ρ2 + |Wβ λρ2 | −
|λ|
|r|2 |λ|
− 2iρ cos ϕ √|r| . |λ|
Since the real part Uβ,r = ReVβ,r has a global minimum at u = v = 0 one has, since U 00 =
∂ 2 Uβ,r ∂u2 (0, 0)
δκ,r (u, v) =
=
=R
=R
e
u2
00 −κ U2
u2 du
e
e−κ
U 00 2
|λ|
|r|
00 −κ U2
00 −κ U2
|r|
−κ Uβ,r (u,v)−2iv √
−κ Uβ,r (u,v)−2iv √
e e
> 0,
e R
∂ 2 Uβ,r ∂v 2 (0, 0)
e R
u2 u2 du
e
e
−κ
−κ
e R
|λ|
e
≈
R
dudv
U 00 2
v 2 −2iv √
U 00 2
|r| v 2 −2iv √ |λ|
00 −κ U2
00
−κ U2
e
|r|
v−i √
v−i √
|λ|
2|r| |λ|U 00
2|r| |λ|U 00
−κ
−κ
U 00 2
U 00 2
|r|
(u2 +v 2 )−2iv √
|λ|
|r| (u2 +v 2 )−2iv √ |λ|
dudv
dv 2
→ δ(u) δ v − i √2|r| 00
κ→∞
2
|λ|U
dv
Many-Electron System in Forward, Exchange and BCS Approximation
453
which results in lim lim δκ,r (u, v) = δ(u) δ(v)
(3.35)
|r|→0 κ→∞
and 1 ¯ hψp,↑ ψ−p,↓ iβ,L,r |r|→0 κ→∞ κ
lim lim
=
R
lim lim 1 hψp,↑ ψ−p,↓ iβ,L,r |r|→0 κ→∞ κ
−a−p ap a−p +λ(u2 +v 2 )
δ(u)δ(v) dudv =
1 ¯ hψp,↑ ψ¯ −p,↓ iβ,L,r |r|→0 κ→∞ κ
= lim lim
−a−p ap a−p
= − a1p ,
= 0.
Using (2.17) and (3.34,35) the infinite volume limit for the generating functional for the connected Greens functions can be computed in a similar way. One finds Corollary 3.2. Let r = |r|eiα , γ = geiα (u + iv), γ¯ = ge−iα (u − iv) and let Z ¯ ¯ ¯ dµC (ψ, ψ) G(η) = Gr (β, L, η) = log eUr (ψ+η,ψ+η)
(3.36)
be the generating functional for the connected amputated Greens functions where X X ¯ = − λ3 ψk↑ ψ−k↓ ψ¯ p↑ ψ¯ −p↓ + κ1 [rψk↑ ψ−k↓ − r¯ψ¯ k↑ ψ¯ −k↓ ]. (3.37) Ur (ψ, ψ) κ k,p
k
Let Vβ,r be the effective potential (3.10) and let 1 be given as in Corollary 3.1. Then there is the integral representation X 1 G(η) − G(0) = − κ
R
ak η¯ k↑ ηk↑ + a−k η¯ −k↓ η−k↓ +
k
n P 1
dudv e−κVβ,r (u,v) exp
+log
κ
(a η¯ ,−a−k η−k↓ ) a k k k↑
R
(3.38)
1
¯ k a−k +γ γ
a−k −iγ¯ iγ −ak
ak ηk↑ −a−k η¯ −k↓
o
dudv e−κVβ,r (u,v)
.
For attractive λ > 0 the infinite volume limit of the generating functional is given by lim lim {G(β,L, r, η) − G(β, L, r, 0)} = Xh |1|2 − κ1 ak ak a−k ¯k↑ ηk↑ + a−k +|1|2 η
(3.39)
|r|→0 L→∞
k
|1|2 ak a−k +|1|2
η¯−k↓ η−k↓
i ¯ ak a−k 2 η¯k↑ η¯−k↓ . + 1 ak aak−ka−k η η + 1 2 −k↓ k↑ +|1| ak a−k +|1|
For repulsive λ < 0 one obtains lim lim {G(β, L, r, η) − G(β, L, r, 0)} = 0 .
|r|→0 L→∞
(3.40)
3.2. BCS with higher angular momentum terms. We now consider the case where the electron-electron interaction contains higher angular momentum terms. In this case one finds that the usual mean field approach, based on approximating the Hamiltonian by quadratic terms, may be misleading. To simplify the algebra we assume that only even ` terms contribute in (1.37). In that case the fields 8kσσ with equal spin (2.10) are still zero. So let
454
D. Lehmann
0
0
U (k − p ) =
j λ0 1 X λ|`| ei`ϕk e−i`ϕp + 2 2 `=−j
if d=2
` even
=:
j ` X X λ` Y¯`m k0 Y`m p0 if d=3 `=0 ` even
J X
λl yl (k0 ) y¯l (p0 ),
l=0 l even
m=−`
(3.41) ¯ expectations, so we let r = 0. The where yl (−k) = (−1)l yl (k). We only consider the ψψ Pfaffian (3.3) becomes ¯ k↑↓ 8k↑↓ a−k↑ ak↓ + 8 ¯ −k↑↓ 8−k↑↓ , (3.42) PfSk = ak↑ a−k↓ + 8 where 8k↑↓ =
J X
(2λl ) 2 φl↑↓ yl (k0 ) , 1
¯ k↑↓ = 8
l=0 l even
J X
1 (2λl ) 2 φ¯ l↑↓ y¯l (k0 )
(3.43)
l=0 l even
and the effective potential is given by, using (3.25) again to compute the k0 product, V (φ↑↓ ) =
J X
Z |φl↑↓ |2
− Mω
l=0
dd k (2π)d
¯ k↑↓ 8k↑↓ ), Wβ,k 8
(3.44)
where Wβ,k is given by (3.11). By Theorem 2.1, the two point function is given by R 1 ¯ κ hψpσ ψpσ i
=−
J
l e−κV (φ↑↓ ) 5 dul↑↓ dv↑↓
ip0 +ep ¯ k↑↓ 8k↑↓ p20 +e2p +8
R
l=0
J
.
(3.45)
l e−κV (φ↑↓ ) 5 dul↑↓ dv↑↓ l=0
To compute the infinite volume limit, one has to find the global minimum of the real part of V (φ↑↓ ). This is easier for d = 2. However, using symmetry arguments, it is possible to give a rather explicit expression also for d = 3 without knowing the exact location of the global minimum. Consider first the two dimensional case. An analysis done by Albrecht Schuette in his Diploma thesis [Sch] shows the following result: Suppose that λm > 0 is attractive and λm > λ` for all ` 6= m. Then lim 1 hψ¯ pσ ψpσ iβ,L κ→∞ κ
=
ip0 +ep , p20 +e2p +λm ρ2m
where ρm is determined by the BCS equation Z √ tanh( β2 e2k +λm ρ2m ) λm √2 2 d|k||k| = 0. 1 − 4π ek +λρm
(3.46)
(3.47)
The form of the two point function (3.46) can still be obtained by applying the standard mean field formalism [AB,BW]. However, the situation is different in 3 dimensions. Before we state the corresponding theorem, we briefly recall the mean field equations (1.57, 58) The ha+ ai expectations are given by h tanh( β √e2 +1∗ 1 ) i + 1 √2 2 k ∗ k k , (1.57) hakσ akσ i = 2 1 − ek ek +1k 1k
σσ
Many-Electron System in Forward, Exchange and BCS Approximation
455
where the 2 × 2 matrix 1k , 1Tk = −1−k , is a solution of the gap equation Z √ 1 ) tanh( β2 e2k +1∗ dd k 0 0 k k √ U (p − k ) 1 . 1p = k d 2 ∗ (2π) 2
M
ek +1k 1k
(1.58)
i`(ϕp −ϕk ) In 2 dimensions, if one substitutes U (p − k) by a single term , attractive λ` e 0 ei`ϕk then (1.58) has the unitary isotropic solution 1k = 1 −ei`ϕk 0 for even ` and `ϕk sin `ϕk ∗ 2 1k = 1 cos sin `ϕk − cos `ϕk if ` is odd. In that case 1k 1k = |1| Id and (1.57) coincides with (3.46) (after integration of the latter over p0 ). In 3 dimensions, it is proven [FKT2] that for all ` ≥ 2 (1.58) does not have unitary isotropic (1∗k 1k = const Id) solutions. In view of that result, the symmetry considerations below indicate that in 3 dimensions for ` ≥ 2 the standard mean field approach is misleading since one would no longer expect SO(3) invariance for the ha+kσ akσ i expectations according to (1.57). But this is indeed the case if there is no external SO(3) symmetry breaking term (which is also not present in the quadratic mean field model, see the discussion following (1.61) at the end of the introduction).
Theorem 3.3. Let ` be even, λ` > 0 be attractive and let ¯ = U(ψ, ψ)
λ` (βLd )3
` X X
Y¯`m (k0 )Y`m (p0 ) ψ¯ k↑ ψ¯ −k↓ ψp↑ ψ−p↓ .
(3.48)
k,p m=−`
Then, if eRk = ek for all R ∈ SO(3), one has Z ¯ −U (ψ,ψ) 1 ¯ ¯ lim κ1 hψ¯ pσ ψpσ iβ,L = lim dµC (ψ, ψ) κ ψpσ ψpσ e κ→∞ κ→∞ Z ip0 +ep d(x) , = p2 +e2 +λ` ρ2 | 6m α0 Y`m (x)|2 4π S2
0
p
0
(3.49)
m
0 2 where ρ0 ≥ 0 and α0 ∈ C2`+1 , 6m |αm | = 1, are values at the global minimum (which is degenerated) of 2 Z √ cosh( β2 e2k +λ` ρ2 | 6m αm Y`m (k0 )|2 ) 2 dd k 1 V (ρ, α) = ρ − log . (2π)d β cosh β e 2
M
k
In particular, the momentum distribution np is given by Z √ tanh( β2 e2p +|1(x)|2 ) d(x) + 1 1 √2 np = lim Ld hapσ apσ iβ,L = 2 2 1 − ep 4π L→∞
ep +|1(x)|
S2
(3.50)
1
0 and has SO(3) symmetry. Here 1(x) = λ`2 ρ0 6m αm Y`m (x).
Proof. Substituting ul↑↓ by um and ul↑↓ by −vm in (3.45), one has to compute the infinite volume limit of R a−p e−κV (φ) 5`m=−` dum dvm R4`+2 ap a−p +|8p |2 1 ¯ R , (3.51) κ hψpσ ψpσ i = − e−κV (φ) 5`m=−` dum dvm R4`+2 where
456
D. Lehmann ` X
V (φ) =
Z |φm | − 2
M
m=−`
and 1
8k = λ`2
dd k 2 (2π)d β
` X
log
cosh( β2
√
cosh
e2k +|8k |2 )
β 2 ek
φ¯ m Y`m (k0 ).
m=−`
Let U(R) be the unitary representation of SO(3) given by X Y`m (Rk0 ) = U(R)mm0 Y`m0 (k0 ) m0
P and let m (U φ)m Y`m (k0 ) =: (U 8)k . Then for all R ∈ SO(3) one has U(R)8 k = 8R−1 k and Z √ X cosh( β2 e2k +|[U 8]k |2 ) dd k 2 |[U(R)φ]m |2 − log V U(R)φ = (2π)d β cosh β e m
=
X
|φm |2 −
m
=
X m
M
Z Z
M
|φm | − 2
M
dd k 2 (2π)d β
log log
cosh( 2
cosh( β2
√
cosh
k
e2k +|8R−1 k |2 )
cosh
dd k 2 (2π)d β
2
√ β
β 2 ek
e2k +|8k |2 ) β 2 ek
= V (φ). (3.52)
P Let S 4`+1 = {φ ∈ C2`+1 | m |φm |2 = 1}. Since U(R) leaves S 4`+1 invariant, S 4`+1 can be written as the union of disjoint orbits, S 4`+1 = ∪[α]∈O [α], where [α] = {U(R)α | R ∈ SO(3)} is the orbit of α ∈ S 4`+1 under the action of U(R) and O is the set of all orbits. If one chooses a fixed representant α in each orbit [α], that is, if one chooses a fixed section σ : O → S 4`+1 , [α] → σ[α] with [σ[α] ] = [α], every φ ∈ C2`+1 can be uniquely written as ρ = kφk ≥ 0, α =
φ = ρ U(R)σ[α] ,
φ kφk ,
[α] ∈ O and R ∈ SO(3)/I[α] ,
σ = {S ∈ SO(3) | U (S)σ[α] = σ[α] } is the isotropy subgroup of σ[α] . where I[α] = I[α] Let R R R R 5m dum dvm f (φ) = R+ Dρ O D[α] [α] DR f ρ U(R)σ[α] R4`+2
be the integral in (3.51) over R4`+2 in the new coordinates. That is, for example, Dρ = ρ4`+1 dρ. In the new coordinates 2 X X 2 U(R)σ[α] m Y`m (p0 ) = λ` ρ2 σ¯ [α],m Y`m R−1 p0 |8p |2 = λ` ρ2 m
m
such that −a−p ap a−p +|8p |2
=
−a−p ap a−p +λ` ρ2 | 6m σ¯ [α],m Y`m (R−1 p0 )|2
Since V (φ) = V (ρ, [α]) is independent of R, one obtains
≡ f (ρ, [α], R−1 p).
Many-Electron System in Forward, Exchange and BCS Approximation
R 1 ¯ κ hψpσ ψpσ i
=
−a−p ap a−p +|8p |2
457
`
e−κV (φ) 5 dum dvm m=−`
R
`
e−κV (φ) 5 dum dvm m=−` R R R Dρ D[α] DR f (ρ, [α], R−1 p) e−κV (ρ,[α]) R+ O R [α] R R = Dρ O D[α] [α] DR e−κV (ρ,[α]) R+ R DRf (ρ,[α],R−1 p) R R D[α] vol([α]) [α] R e−κV (ρ,[α]) + Dρ R
=
O
R R+
Dρ
R
[α]
O
DR
D[α] vol([α]) e−κV (ρ,[α])
. (3.53)
It is plausible to assume that at the global minimum of V (ρ, [α]) ρ is uniquely determined, say ρ0 . Let Omin ⊂ O be the set of all orbits at which V (ρ0 , [α]) takes its global minimum. Then in the infinite volume limit (3.53) becomes R DR f (ρ,[α],R−1 p) R [α] R D[α] vol([α]) Omin DR [α] 1 ¯ R . (3.54) lim κ hψpσ ψpσ i = κ→∞ D[α] vol([α]) Omin Consider the quotient of integrals in the numerator of (3.54). Since 2 f (ρ, [α], R−1 p) = f ρ2 |6 σ¯ [α],m Y`m R−1 p0 | m 2 2 = f ρ |6 U (S)σ[α] m Y`m R−1 p0 | m 2 2 = f ρ |6 σ¯ [α],m Y`m (RS)−1 p0 | = f (ρ, [α], (RS)−1 p) m
for all S ∈ I[α] , one has, since [α] ' SO(3)/I[α] , R R R DR f (ρ, [α], R−1 p) I[α] DS DR f (ρ, [α], R−1 p) SO(3)/I[α] [α] R R R = DR DR I[α] DS [α] SO(3)/I[α] R R DR I[α] DS f (ρ, [α], (RS)−1 p) SO(3)/I[α] R R = DR DS SO(3)/I[α] I[α] R DR f (ρ, [α], R−1 p) SO(3) R = DR SO(3) R R d(x) SO(3)x→p DR f (ρ, [α], R−1 p) S2 R R = d(x) SO(3)x→p DR S2 R R d(x) f (ρ, [α], x) SO(3)x→p DR S2 R R , (3.55) = d(x) SO(3)x→p DR S2 where SO(3)x→p = {R ∈ SO(3) | Rx = p}. If Rone assumes that DR has the usual invariance properties of the Haar measure, then SO(3)x→p DR does not depend on x such that it cancels out in (3.55). Then (3.54) gives
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D. Lehmann
R
R lim 1 hψ¯ pσ ψpσ i κ→∞ κ
d(x) f (ρ,[α],x)
D[α] vol([α]) S2 R d(x) S2 R D[α] vol([α]) Omin
Omin
=
.
(3.56)
Now, since the effective potential, which is constant on Omin , may be written as Z 2 d(x) 2 (x)| G ρ |6 σ ¯ Y V (ρ, [α]) = m [α],m `m 4π with G(X) = ρ2 − R S2
R
dk k2 2π 2
log
S2
cosh( β2
√
cosh
d(x) f (ρ, [α], x) R = d(x) S2
e2k +λ` X) β 2 ek
Z S2
, it is plausible to assume that also
ip0 +ep d(x) 4π p20 +e2p +λ` ρ2 | 6m σ¯ [α],m Y`m (x)|2
is constant on Omin . In that case also the integrals over Omin in (3.56) cancel out and the theorem is proven. 4. Solution with Delta Function Interaction and Its Perturbation Theory 4.1. Solution with Delta Function. We consider the model Z P ¯ 1 −U (ψ,ψ)+ sk,σ ψ¯ k,σ ψk,σ κ ¯ k,σ dµC (ψ, ψ), Z(β, L, {sk,σ }) = e where ¯ = U (ψ, ψ)
λ κ3
=
λ κ3
X
X
(4.1)
δk,p + δk,q−p + δq,0 ψ¯ kσ ψ¯ q−kτ ψpσ ψq−pτ
σ,τ ∈{↑,↓} k,p,q
X
δk,p + δk,q−p + δq,0 ψ¯ k↑ ψ¯ q−k↓ ψp↑ ψq−p,↓ .
(4.2)
k,p,q
It is useful to explicitly cancel the ψ¯ ↑ ψ¯ ↑ ψ↑ ψ↑ and ψ¯ ↓ ψ¯ ↓ ψ↓ ψ↓ terms before directly applying Theorem 2.1 because then the integral representation becomes 4 dimensional instead of 5 dimensional. Both representations are of course equivalent. One obtains R hψ¯ pσ ψp0 σ i = βLd δp,p0
∂ ∂spσ
PfSp e−κVβ (z,x,ρ) dv dw xdx ρdρ PfSp R −κV (z,x,ρ) , β dv dw xdx ρdρ e
(4.3)
where z = v + iw, PfSk = (Ak↑ A−k↓ + λρ2 )(A−k↑ Ak↓ + λρ2 ) + (Ak↑ Ak↓ + A−k↑ A−k↓ )λx2 + 2λ2 x2 ρ2 + λ2 x4 , Ak↑ = ak − sk↑ −
(4.4) √
λ z,
Ak↓ = ak − sk↓ −
as before ak = ik0 − ek , and the effective potential is given by X Vβ (z, x, ρ) = |z|2 + x2 + ρ2 − βL1 d log 5 k∈Mω
√
λ z, ¯
PfSk . 2 2 k0 >0 ak a−k
(4.5)
Many-Electron System in Forward, Exchange and BCS Approximation
459
The product over k0 is computed in Lemma A1 in the appendix. One finds Vβ (v, w, x, ρ) = v 2 + w2 + x2 + ρ2 2 X cosh 1 − L1d log β
β 2
√
√ (ek + λv)2 +λρ2 − sin2 cosh2
k∈Mω
β 2
√
λ(w2 +x2 )
e−β
β 2 ek
√
(4.6) λv .
The z variable sums up forward contributions, x = |ξ| sums up exchange contributions and ρ = |φ| collects the BCS contributions. Pure BCS is given by z = x = 0 and (4.6) coincides with (1.47) or (3.10) (for r = 0). To compute the infinite volume limit of (4.3) one has to find the global minimum of the effective potential (4.6). Consider first the case of an attractive coupling λ = g 2 > 0. To make (4.6) small, the numerator in the logarithm has to be large. Hence for λ positive w2 + x2 has to be zero. This gives ∂ ∂sp↑
PfSp
a−p −gv [ap −gv][a−p −gv]+g 2 ρ2 .
=
PfSp
(4.7)
As in Sect. 3.1 ρ is positive for sufficiently small T = β1 and v may or may not be nonzero to renormalize ep . Thus for attractive coupling the two point function becomes lim 1 hψ¯ p↑ ψp↑ i L→∞ κ
ip +epg 2 pg +|1|
= − p2 +e02 0
(4.8)
if epg = ep + gv0 , |1|2 = λρ20 and v0 , ρ0 are the (λ dependent) values where V takes its global minimum. Now consider a repulsive coupling λ = −g 2 < 0. In that case the p √ numerator in the 2 β 2 β 2 2 2 logarithm in (4.6) reads cosh ( 2 (ek + igv) − g ρ ) + sinh ( 2 g w2 + x2 ) and the value ρ = 0 is favourable. Furthermore v = 0 seems favourable. In that case ∂ ∂sp↑
PfSp
PfSp
=
ap −gw a2p −g 2 (w2 +x2 )
(4.9)
and the two point function becomes, if δg2 = g 2 (w2 + x2 ) ≥ 0, 1 ¯ hψp↑ ψp↑ i L→∞ κ
lim
=
ap a2p −δg2
=
1 2
h
1 ip0 −ep −δg
+
1 ip0 −ep +δg
i ,
(4.10)
since the effective potential is even in w and therefore the w term in (4.9) cancels. At zero temperature, this results in a momentum distribution Z h i dk0 1 + 1 1 1 + nk = lim Ld hak ak iβ,L = lim ik ik 2π 2 e 0 (ik −e )−δ e 0 (ik −e )+δ L→∞ β→∞
=
1 2
→0 <0
R
h
i χ(ek − δg < 0) + χ(ek + δg < 0) =
0
1 1 2
0
k
g
0
k
if ek < −δg if − δg < ek < δg . if δg < ek
g
(4.11)
Thus, if δg is nonzero, the exchange and forward contributions lead to a splitting of the Fermi surface. A nonzero δg can be achieved by making the coupling g 2 sufficiently big: Let K =
|S d−1 | mkFd−2 (2π)d
∼ kFd /µ and let
460
D. Lehmann
Z Vβ (y) = y 2 − Mω
Then
dd k 1 (2π)d β
h log
cosh2 ( β2 ek )+ sinh2 ( β2 gy) cosh2 β2 ek
lim Vβ (y) = (1 − Kg 2 )y 2 + K[gy − ω]2+ ,
β→∞
i .
(4.12)
(4.13)
where [x]+ = x for positive x and 0 otherwise. The global minimum of V∞ is at ymin where ( 0 if g 2 < 1/K . (4.14) ymin = Kωg if g 2 > 1/K 4.2. Perturbation theory. In this section we consider the perturbation theory of the model Z ¯ ¯ Zex/BCS (β, L) = e−Vex/BCS (ψ,ψ) dµC (ψ, ψ). (4.15) For BCS, one obtains a power series in C(k)C(−k) and for Vex one gets a power series in C(k)2 . We will find that the linked cluster theorem, log Z is given by the sum of all connected diagrams, cannot be applied to (4.15). More specifically, whereas the series for Z(λ), the sum of all diagrams, converges for sufficiently small λ (for finite β and L) with Z(0) = 1 which implies that also log Z is analytic for λ sufficiently small, the sum of all connected diagrams has radius of convergence zero (for finite β and L). We start with BCS. First we show how the integral representation (3.12) (for r = 0) β√ 2 2 Z −κ ρ2 − 1d P log cosh( 2 βek +λρ ) k L cosh e 2 k ρ dρ (4.16) ZBCS = 2κ e is obtained by direct summation of the diagrams without making a Hubbard Stratonovich transformation. That is, without using the identity Z P λ P 1 P 2 ¯ ¯ ¯ 1 − κ3 ψk↑ ψ−k↓ ψ¯ p,↑ ψ−p↓ k,p = eiφ g κ k ψk↑ ψ−k↓ +iφ g κ k ψk↑ ψ−k↓ πκ e−κ|φ| dudv. e One has Z P ψ¯ ψ¯ −λ ψ ψ ¯ ZBCS = e κ3 k,p k↑ −k↓ p↑ −p↓ dµC (ψ, ψ) (4.17) ∞ λ n X X κ3 det κδki ,pj C(ki ) 1≤i,j≤n det κδki ,pj C(−ki ) 1≤i,j≤n = n! k ,··· ,kn n=0
=
∞ X n=0
1 p1 ,··· ,pn
λ n κ
X π∈Sn
π
X
C(k1 )C(−k1 ) · · · C(kn )C(−kn ) δk1 ,kπ1 · · · δkn ,kπn .
k1 ,··· ,kn
This is the expansion into Feynman diagrams. It can be summed up by collecting the fermion loops: Say that the permutation π is of type t(π) = 1b1 · · · nbn if the decomposition into disjoint cycles contains br r-cycles for 1 ≤ r ≤ n. Necessarily one has 1b1 +· · ·+nbn = n. The number of permutations which have br r-cycles for 1 ≤ r ≤ n is
Many-Electron System in Forward, Exchange and BCS Approximation
461
n! . b1 ! · · · bn ! 1 b 1 · · · n b n The sign of such a permutation is given by π = (−1)(1−1)b1 +(2−1)b2 +···+(n−1)bn = (−1)n−
Pn r=1
br
.
Therefore one obtains Z=
∞ X
n X
λ n κ
n=0
b1 ,··· ,bn =0 1b1 +···+nbn =n
Pn n! n− br r=1 (−1) × b 1 ! · · · bn ! 1 b 1 · · · n b n X
b1 [C(k)C(−k)]
1
k
=
∞ X
n X
n=0
b1 ,··· ,bn =0 1b1 +···+nbn =n
···
X
bn [C(k)C(−k)]
n
k
n Y X r br 1 1 λ −r n! . − κ C(k)C(−k) br ! r=1
(4.18)
k
The only factor which prevents us from an explicit summation of the above series is the n!. Therefore we substitute Z ∞ e−x xn dx n! = 0
and obtain Z
∞
e
Z=
−x
0
∞ X
n X
n=0
b1 ,··· ,bn =0 1b1 +···+nbn =n
n Y X r br 1 1 λ −r dx − κ x C(k)C(−k) br ! r=1
k
∞ ∞ X Y X r br 1 − r1 dx − λκ x C(k)C(−k) br ! 0 r=1 br =0 k ir P P∞ 1 h λ Z ∞ − − κ x C(k)C(−k) k r=1 r −x e e dx = 0 i P h λx Z ∞ log 1+ κ C(k)C(−k) k −x e e = dx 0 h i Z ∞ −κ ρ2 − 1 P log 1+ λρ2 2 2 κ k k +e 0 k ρ dρ e = 2κ 0 √ 2 2 2 β Z ∞ −κ ρ2 − 1d P β1 log cosh( 2 βek +λρ ) k L cosh e 2 k = 2κ e ρ dρ Z
=
∞
e−x
(4.19)
0
which coincides with (4.16). In the last line we used (3.25) again. The case of an exchange interaction is treated in the same way:
462
D. Lehmann
Z
−
Zex = =
e ∞ X
λ κ3
P k,p
=
X
λ n κ
n=0
ψk↑ ψp↓ ψ¯ p↑ ψ¯ k↓
π∈Sn
∞ X
n X
n=0
b1 ,··· ,bn =0 1b1 +···+nbn =n
Z
∞
e−x e
= 0
n Y X r br 1 − r1 − λκ C(k)2 br ! r=1
P P∞ k
r=1
Z
∞ −κ y 2 −
e
= 2κ
C(k1 )2 · · · C(kn )2 δk1 ,kπ1 · · · δkn ,kπn
k1 ,··· ,kn
n!
−
X
π
¯ dµC (ψ, ψ)
1 Ld
(4.20)
k
h
− λκ x C(k)2
1 r
P
h 1 log k β
cosh2 (
ir dx
β β√ e )−sin2 ( λ y) 2 k 2 β 2 e cosh 2 k
i2 y dy,
(4.21)
0
where in the last line Lemma A1Pin the appendix has been used toR compute the k0 1 0 product. Observe that limβ→∞ κ1 k C(k)2r = 0 for all r ≥ 1 since dk 2π (ik0 −e)j = 0 for all j ≥ 2, but for large repulsive coupling lim β→∞ βL1 d log Zex = −V∞ (ymin ) > 0 by (4.13,14). We now consider the linked cluster theorem. It states that, if the partition function ∞ X
Z(λ) =
λn
n=0
X
val(G)
(4.22)
G∈0n
is given by a sum of diagrams, 0n being the set of all nth order diagrams, then the logarithm ∞ X X λn val(G) (4.23) log Z(λ) = n=0
G∈0cn
is given by the sum of all connected diagrams. This theorem is easily illustrated for a quadratic perturbation. Namely, if Z P λ ¯ ¯ Z = e− κ kσ ψkσ ψkσ dµC (ψ, ψ) =
∞ X (−λ)n X
n!
n=0
=
∞ X
n X
n=0
b1 ,··· ,bn =0 1b1 +···+nbn =n
P =e
k
det[δσi ,σj δki ,kj C(ki )]1≤i,j≤n
k1 ···kn σ1 ···σn
br n Y X 1 r 1 −r [−λC(k)] br ! r=1
(4.24)
k
log[1+λC(k)]
(4.25)
then the sum of all connected diagrams is obtained by summing all the terms with b1 = · · · = bn−1 = 0 and bn = 1 in (4.24). That is, one gets 1 X ∞ X X 1 n 1 −n = log [1 + λC(k)] (4.26) [−λC(k)] 1! n=1
k
k
Many-Electron System in Forward, Exchange and BCS Approximation
463
which coincides with log Z. Now consider ZBCS . The sum of all connected diagrams is given by the sum of all the terms with b1 = · · · = bn−1 = 0 and bn = 1 in (4.18). That is, one obtains ∞ X X n (−1)n+1 λκ n! n1 (4.27) [C(k)C(−k)]n n=0
k
which has radius of convergence zero, at finite temperature and finite volume. However, ZBCS (λ) has positive radius of convergence and ZBCS (0) = 1 which means that also log ZBCS (λ) is analytic for sufficiently small (volume and temperature dependent) λ. That is, in this case the linked cluster theorem does not apply since the right-hand side of (4.23) is infinite. We remark that we think that this is an artefact of the specific model at hand for the following reason. Suppose for the moment that the k sums in (4.17,18) are finite with N different values of k. Then ZBCS (λ) is a polynomial in λ of degree (at most) N . In particular, the coefficients n X b1 ,··· ,bn =0 1b1 +···+nbn =n
br n Y X 1 r 1 −r = 0 [C(k)C(−k)] br ! r=1
if n > N .
(4.28)
k
That is, there are strong cancellations between fermion loops of different orders. However, for Vint ≈ VBCS (or forward or exchange), an nth order connected diagram is necessarily a single nth order fermion loop such that there are no cancellations at all. In a more realistic model a connected diagram contains fermion loops of different orders and cancellations are present. In fact, a careful diagrammatic analysis [FT,FKLT,FST] shows that the renormalized sum of all connected diagrams of the two dimensional electron system with anisotropic dispersion relation ek (such that Cooper pairs are suppressed) has positive radius of convergence (which is, in particular, independent of volume and temperature).
Appendix Lemma A.1. a) Let ak = ik0 − ek , e−k = ek and let b, c, d be some complex numbers. Then Y k0 ∈ π β (2Z+1)
ak a−k +bak +ca−k +d ak a−k
Y
:= lim →0 <0
=
cosh
k0 ∈ π β (2Z+1) β 2
√
cosh
ak a−k +beik0 ak +ce−ik0 a−k +d ak a−k
e2k +ω+ cosh
β 2 ek
β 2
√
cosh
e2k +ω−
β 2 ek
β
e 2 (b+c) ,
(A.1)
√ p± p2 −4q where ω± = with p = (b + c − ek )2 − e2k − 4bc + 2d and q = 4bce2k − 2 2 2(b + c)ek d + d . In particular, for b = c and d = c2 , Y k0 ∈ π β (2Z+1)
1+
c ak
:= lim →0 <0
Y k0 ∈ π β (2Z+1)
1 + e−ik0 ack
=
1+e−β(ek −c) . 1+e−βek
(A.2)
464
D. Lehmann
b) Let Ak↑ = ak −
√
λ z, Ak↓ = ak −
√
λ z, ¯ and let
PfSk = PfSk (ak , a−k ) = (Ak↑ A−k↓ + λρ2 )(Ak↓ A−k↑ + λρ2 ) 2
(A.3) 2
2 4
+ (Ak↑ Ak↓ + A−k↑ A−k↓ + 2λρ )λx + λ x . Then
Y k0 ∈ π (2Z+1) β k0 >0
PfSk (ak ,a−k ) a2k a2−k
=
cosh2
β 2
√
Y
:= lim
→0 <0 k0 ∈ π (2Z+1) β k0 >0
PfSk (eik0 ak ,e−ik0 a−k ) a2k a2−k
√ (ek + λ v)2 +λρ2 −sin2 cosh2
β 2
√
λ(w2 +x2 )
β 2 ek
e−β
√
λv
.
(A.4)
Proof. We first explain why the definition of the product given in the lemma is the right one. Consider the quadratic perturbation X X ek a+kσ akσ + Lλd a+kσ akσ . H = H0 + λN = L1d kσ
kσ
One finds by explicit computation of the trace Z(λ, β) =
T r e−β(H0 +λN ) T r e−βH0
=5
kσ
1+e−β(ek +λ) . 1+e−βek
On the other hand, by explicit summation of the perturbation series or equivalently, by performing the fermionic functional integral, one obtains Z P λ ψ¯ ψ Z(λ, β) = e βLd kσ kσ kσ dµC Z P P λ d ψ¯ ψ − 1 (ik −e )ψ¯ ψ e βLd kσ kσ kσ e βLd kσ 0 k kσ kσ 5 dψkσ dψ¯ kσ = 5 βL kσ ik0 −ek
=
5 ik0 −ek −λ kσ ik0 −ek
= 55 1− kσ k0
λ ak
kσ
.
(A.5)
In the perturbation expansion the product in the last equation results from P P∞ λr P 1 log 1− aλ − r kσ ak . r=1 r k 5 1 − aλk = e kσ =e kσ
P 1 is not absolutely converThe term for r = 1 requires special attention since k0 ik0 −e k gent. This sum enters the perturbation series as the propagator T r e−βH0 T ψ(x0 x)ψ + (x00 x0 ) T r e−βH0
C(x0 − x00 , x − x0 ) =
evaluated at x0 − x00 = 0. Here T ψ(x0 x)ψ + (x00 x0 ) is defined to be ψ(x0 x)ψ + (x00 x0 ) if x0 ≥ x00 and −ψ + (x00 x0 )ψ(x0 x) if x0 < x00 . By explicit computation one finds that for x0 6= x00 , C(x0 − x00 , x − x0 ) X 0 0 = L1d eik(x−x ) e−ek (x0 −x0 ) θβ (−ek )θ(x00 − x0 ) − θβ (ek )θ(x0 − x00 ) k
=
1 βLd
X
0
0
1 eik(x−x )−ik0 (x0 −x0 ) ik0 −e
k
k,k0
Many-Electron System in Forward, Exchange and BCS Approximation
and for x0 = x00 , C(0, x − x0 ) =
1 Ld
X
0
eik(x−x ) θβ (−ek ) = lim →0 <0
k
1 βLd
465
X
0
1 eik(x−x )−ik0 ik0 −e . k
k,k0 −βv
e Here θ is a step function being 1 for positive arguments and θβ (−v) = 1+e −βv is an approximate step function. Therefore the k0 -sums have to be evaluated according to X X −ik0 1 1 1 e−βek ik0 −e = ik0 −e := lim e ik0 −e = βθβ (−ek ) = β 1+e−βek . k
k0
k
k
→0 <0
k0
We emphasize this point because the following computation leads to a wrong result i X Xsym Xh X −βek −ek 1 1 1 1 1 = = − β2 1−e . ik0 −e = ik0 −e := 2 ik0 −e + −ik0 −e k2 +e2 1+e−βek k
k
k0
k0
k
k
k0
k
0
k0
Therefore the product in (A.5) is found to be P∞ λr P 1 − r r=1 r k0 (ik0 −ek ) 5 1 − ik0λ−e = 5 1 − ik0λ−e = e k k k0 k0 P∞ λr Psym P Psym λ 1 − ik0 −ek − (ik0 −ek )r r=1 r k0 k0 k0 =e e = 5sym 1 − k0
(3.25) cosh β2 (ek +λ) = cosh β2 ek
k02 +(ek +λ)2 k02 +e2k k0 >0
β
e− 2 λ = 5
λ ik0 −ek
β
e− 2 λ =
λ ik0 −ek β
e− 2 λ
1+e−β(ek +λ) , 1+e−βek
(A.6)
which proves (A.2). In the general case one may proceed similarly to obtain (using ak + a−k = −2ek ) ak a−k +bak +ca−k +d beik0 ak +ce−ik0 a−k +d 1 + =: 5 · · · = lim 5 5 ak a−k ak a−k π k0 ∈ β (2Z+1)
= 5sym 1 + k0
= 5
k0 >0
→0 <0
k0
bak +ca−k +d ak a−k
e
b
P k0
−
Psym k0
1 a−k
(ak a−k +bak +ca−k +d)(ak a−k +ba−k +cak +d) (ak a−k )2
+c
P k0
k0
−
Psym k0
1 ak
β
e 2 (b+c)
(ak a−k )2 +ak a−k (−2ek (b+c)+2d+b2 +c2 )+bc(a2k +a2−k )−2ek d(b+c)+d2 (ak a−k )2 k0 >0
= 5
β
e 2 (b+c) .
Since a2k + a2−k = −2k02 + 2e2k = −2ak a−k + 4e2k one obtains ak a−k +bak +ca−k +d ak a−k k0 ∈ β (2Z+1) 5 π
(ak a−k )2 +pak a−k +q (ak a−k )2 k0 >0
= 5 = 5
k0 >0
1+
ω+ ak a−k
1+
β
e 2 (b+c) ω− ak a−k
β
e 2 (b+c) ,
where p = −2ek (b + c) + 2d + (b + c)2 − 4bc and q = 4bce2k − 2ek d(b + c) + d2 . Application of (3.25) proves part (a) of the lemma. √ Part b) First, if bk = ak − λ v, one computes that PfSk = (bk b−k + λρ2 )2 + (b2k + b2−k + 2λρ2 )λ(w2 + x2 ) + λ2 (w2 + x2 )2 .
(A.7)
466
D. Lehmann
Namely, since Ak↑ A−k↓ = ak a−k −
√
√ λ v(ak + a−k ) + i λ w(ak − a−k ) + λ(v 2 + w2 )
⇒
h i2 √ (Ak↑ A−k↓ + λρ2 )(Ak↓ A−k↑ + λρ2 ) = ak a−k − λ v(ak + a−k ) + λ(v 2 + w2 + ρ2 ) + λw2 (ak − a−k )2 2 = bk b−k + λ(w2 + ρ2 ) + λw2 (bk − b−k )2
and Ak↑ Ak↓ = (ak −
√
λ v)2 + λw2 = b2k + λw2
one has PfSk 2 = bk b−k + λ(w2 + ρ2 ) + λw2 (bk − b−k )2 + b2k + b2−k + 2λ(w2 + ρ2 ) λx2 + λ2 x4 2 = bk b−k + λρ2 + (b2k + b2−k + 2λρ2 )λ(w2 + x2 ) + λ2 (w2 + x2 )2 . Now let y 2 = w2 + x2 and 2 Fk (ak , a−k ) = bk b−k + λρ2 + (b2k + b2−k + 2λρ2 )λy 2 + λ2 y 4 . Then part (b) follows from the following formula: √ Y F (eik0 a ,e−ik0 a ) cosh2 β2 (ek +√λ v)2 +λρ2 −sin2 k k −k lim = a2 a2 cosh2 β e →0 >0 k0 ∈ π (2Z+1) β k0 >0
k −k
2
Proof of (A.8). One has Ak A−k = ak a−k −
√
β 2
√
λy 2
e−β
√
λv
.
k
(A.8) λ v(ak + a−k ) + λv 2 and
√ A2k + A2−k = a2k + a2−k − 2 λ v(ak + a−k ) + 2λv 2 = 2Ak A−k + (ak − a−k )2 which gives Fk = (Ak A−k + λρ2 + λy 2 )2 + (ak − a−k )2 λy 2 1
1
= [Ak A−k + λρ2 + λy 2 + iλ 2 y(ak − a−k )] [Ak A−k + λρ2 + λy 2 − iλ 2 y(ak − a−k )] √ √ 1 1 = ak a−k − ( λ v − iλ 2 y)ak − ( λ v + iλ 2 y)a−k + λv 2 + λρ2 + λy 2 × √ √ 1 1 ak a−k − ( λ v + iλ 2 y)ak − ( λ v − iλ 2 y)a−k + λv 2 + λρ2 + λy 2 . Therefore one obtains Fk (ak ,a−k ) a2k a2−k k0 >0
5
= lim 5 ak a−k −( →0 <0
k0
(A.) cosh
=
where
β 2
√
cosh
√
√ 1 1 λ v−iλ 2 y)eik0 ak −( λ v+iλ 2 y)e−ik0 a−k +λv 2 +λρ2 +λy 2 ak a−k
e2k +ω+ cosh
β 2 ek
β 2
√
cosh
e2k +ω−
β 2 ek
e−β
√
λv
,
Many-Electron System in Forward, Exchange and BCS Approximation
467
√ p = (ek + 2 λ v)2 − e2k − 4(λv 2 + λy 2 ) + 2λv 2 + 2λρ2 + 2λy 2 , √ = 4 λ vek + 2λv 2 + 2λρ2 − 2λy 2 , √ q = 4(λv 2 + λy 2 )e2k + 4 λ vek (λv 2 + λρ2 + λy 2 ) + (λv 2 + λρ2 + λy 2 )2 √ = (2 λ vek + λv 2 + λρ2 + λy 2 )2 + 4λy 2 e2k such that p2 4
√ √ − q = −4(2 λ vek + λv 2 2 + λρ2 )λy 2 − 4λy 2 e2k = −4 (ek + λ v)2 + λρ2 λy 2 ,
and therefore √ √ 1 1 e2k + ω± = (ek + λ v)2 + λρ2 − λy 2 ± 2iλ 2 y (ek + λ v)2 + λρ2 2 n o2 √ 1 1 = (ek + λ v)2 + λρ2 2 ± iλ 2 y , which results in 5
k0 >0
Fk (ak ,a−k ) a2k a2−k
cosh
= =
n
β 2
√ 1 1 [(ek + λ v)2 +λρ2 ] 2 +iλ 2 y cosh
o
n cosh
β 2
√ 1 1 [(ek + λ v)2 +λρ2 ] 2 −iλ 2 y
β 2 ek
√ √ √ sinh2 ( β2 i λ y) + cosh2 ( β2 (ek + λ v)2 +λρ2 ) β cosh2 2 ek
cosh
e−β
√
λv
.
β 2 ek
o e−β
√
λv
Acknowledgement. I am grateful to Horst Kn¨orrer and Eugene Trubowitz and to the Forschungsinstitut f¨ur Mathematik at ETH Z¨urich for the hospitality and the support during the summer of 1996. Furthermore I would like to thank Joel Feldman who made it possible for me to visit the University of British Columbia in Vancouver in the academic year 1996/97.
References [AB]
Anderson, P.W., Brinkman, W.F.: Theory of Anisotropic Superfluidity. In: Basic Notions of Condensed Matter Physics. P.W. Anderson (ed.), New York: Benjamin/Cummings, 1984 [B] Bogoliubov, N.N.: On Some Problems of the Theory of Superconductivity. Physica 26, Supplement, 1–16 (1960) [BZT] Bogoliubov, N.N., Zubarev, D.B., Tserkovnikov, Iu.A.: On the Theory of Phase Transitions. Sov. Phys. Doklady 2, 535 (1957) [BR] Bardeen, J., Rickayzen, G.: Ground State Energy and Green’s Function for Reduced Hamiltonian for Superconductivity. Phys. Rev. 118, 936 (1960) [BW] Balian, R., Werthamer, N.R.: Superconductivity with Pairs in a Relative p Wave. Phys. Rev. 131, 1553–1564, 1963 [FKLT] Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Fermi Liquids in Two Space Dimensions. In: Constructive Physics. Springer Lecture Notes in Physics, Bd. 446, Berlin–Heidelberg–New York: Springer, 1994 [FKT1] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Mathematical Methods of Many Body Quantum Field Theory. Lecture Notes, ETH Z¨urich [FKT2] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Remark on Anisotropic Superconducting States. Helv. Phys. Acta 64, 695–699 (1991) [FST] Feldman, J., Salmhofer, M., Trubowitz, E.: Perturbation Theory around Non-nested Fermi Surfaces I. Keeping the Fermi Surface Fixed. J. Stat. Phys. 84, 1209–1336 (1996)
468
[FT]
[G] [Ha] [Hn] [Sch] [W]
D. Lehmann
Feldman, J., Trubowitz, E.: Perturbation Theory for Many Fermion Systems. Helv. Phys. Acta 63, 156–260 (1990); The Flow of an Electron-Phonon System to the Superconducting State. Helv. Phys. Acta 64, 214–357 (1991) Girardeau, M.D.: Variational Method for the Quantum Statistics of Many Particle Systems. Phys. Rev. A 42, 3303 (1990) Haag, R.: The Mathematical Structure of the Bardeen Cooper Schrieffer model. Nuovo Cimento 25, 287 (1962) Hanson, E.R.: A Table of Series and Products. New York: Prentice-Hall, 1975, Sect. 89.5. Sch¨utte, A.: The Symmetry of the Gap in the BCS Model for Higher l-Wave Interactions. Diploma thesis, ETH Z¨urich, 1997 Wentzel, G.: Phys. Rev. 120, 1572 (1960)
Communicated by M. E. Fisher
Commun. Math. Phys. 198, 469 – 491 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
On Hyper-K¨ahler Manifolds of Type A∞ and D∞ Ryushi Goto? Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, UK Received: 18 July 1997 / Accepted: 14 April 1998
Abstract: We shall use an infinite dimensional hyper-K¨ahler quotient method to obtain hyper-K¨ahler 4 manifolds of type A∞ and D∞ . Hyper-K¨ahler manifolds of type A∞ and D∞ are constructed in terms of Dynkin diagrams of type A∞ and D∞ respectively. A hyper-K¨ahler manifold of type D∞ is the minimal resolution of the quotient space of a hyper-K¨ahler manifold of type A∞ by an involution. Finally we shall show that a hyper-K¨ahler manifold of type A∞ can be considered as the universal cover of elliptic fibre space of type Ib .
1. Introduction A hyper-K¨ahler manifold is, by definition, a Riemannian manifold equipped with three complex structures I, J, K, satisfying the quaternionic relations, with respect to all of which the metric is K¨ahlerian. Hyper-K¨ahler manifolds have been studied extensively in geometry and mathematical physics. Moduli spaces of certain important geometric objects have hyper-K¨ahler structures such as the moduli spaces of instantons, monopoles and Higgs bundles, [ADHM, AH, H1]. Hyper-K¨ahler manifolds have been used to check the S-duality hypothesis in N = 4 supersymmetric Yang–Mills theory, [LWY, GG] and have also been applied to the N = 4 supersymmetric σ model to obtain an invariant of 3 manifolds, [RW]. An important family of hyper-K¨ahler manifolds arises as hyperK¨ahler quotients from the quaternionic vector space Hm , [HKLR]. This family includes ALE gravitational instantons, which consists of hyper-K¨ahler manifolds of type Ak , Dk and E6 , E7 , E8 (k = 1, 2, 3, . . . ), [EH, GH, H3, K1, K2]. In a paper [AKL], Anderson, Kronheimer and Lebrun constructed a hyper-K¨ahler manifold of infinite topological type by Gibbons–Hawking Ansatz. The intersection matrix of the middle homology ? Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560, Japan. E-mail:
[email protected]
470
R. Goto
of this manifold is (−1)× Cartan matrix of type A∞ . (We call this manifold a hyperK¨ahler manifold of type A∞ .) Anderson, Kronheimer and Lebrun also conjectured that there exists a hyper-K¨ahler manifold of type D∞ . One of the purposes of this paper is to provide an explicit construction of a hyper-K¨ahler manifold of type D∞ as the hyper-K¨ahler quotient in terms of Dynkin diagram of type D∞ : E0+ -&
−→ −→ −→ −→ E1 ←− E2 ←− E3 ←− E4 ←− · · · %.
E0− Fig. 1. Dynkin diagram of type D∞
Each vertex is an irreducible representation of the normalizer of a maximal torus of Sp(1) and each edge means a homomorphism between representation spaces. From ˆ D with the the diagram of type D∞ , we shall construct a certain Hilbert affine space M ˆ D preserving hyper-K¨ahler structure and a Hilbert Lie group GD . Then GD acts on M the hyper-K¨ahler structure. Then a hyper-K¨ahler manifold of type D∞ is obtained as the hyper-K¨ahler quotient XD . Each vertex of the Dynkin diagram also corresponds to a simple root. When k goes to ∞, two vertices of the Dynkin diagram of type Dk s t goes to infinity. We introduce two certain roots of infinite type θ∞ , θ∞ in type D∞ such that we can control periods of exceptional curves of XD . The reason we can use roots of infinity type is that a metric of the Hilbert Lie algebra gD of GD is not invariant under the adjoint action of GD . In Sect. 2, we shall develop a hyper-K¨ahler quotient construction in the case of infinite dimensional affine spaces. A Hilbert metric of our Lie group is compatible with the one of an orbit in the affine space. Then we can construct a hyper-K¨ahler quotient by using the invariance of the index of Fredholm operator under any compact perturbation. In Sect. 3, we shall construct a hyper-K¨ahler Affine space ˆ A and a Hilbert Lie group GA in terms of the Dynkin diagram of type A∞ . Then a M hyper-K¨ahler 4 manifold of type A∞ can be obtained as the hyper-K¨ahler quotient XA . When we transform the Dynkin diagram of type A∞ as in Fig. 2, we have an involution σ on XA preserving the hyper-K¨ahler structure, α−3
α−2
α−1
β−3
β−2
β−1
α
α
α
β0
β1
β2
0 1 2 −→ −→ −→ −→ −→ −→ · · · ←− V−2 ←− V−1 ←− V0 ←− V1 ←− V2 ←− · · ·
⇓ β
β
β
−β−1
−β−2
−β−3
−α2
−α1
−α0
α−1
α−2
α−3
0 2 1 −→ −→ −→ −→ −→ −→ · · · ←− V−2 ←− V−1 ←− V0 ←− V1 ←− V2 ←− · · ·
Figure 2
Then the quotient XA /σ is an orbifold with two quotient singularities of type A1 . In Sect. 4, we shall construct hyper-K¨ahler manifolds of type D∞ . We shall show that XD is a minimal resolution of the orbifold XA /σ in Sect. 5. The Dynkin diagram of type A∞ has another transform, i.e., a shifting. This transform is explained by Fig. 3:
Hyper-K¨ahler Manifolds of Type A∞ and D∞
471
α−3
α−2
α−1
α
α
α
β−3
β−2
β−1
β0
β1
β2
α−4
α−3
α−2
α−1
α
α
β−4
β−3
β−2
β−1
β0
β1
0 1 2 −→ −→ −→ −→ −→ −→ · · · ←− V−2 ←− V−1 ←− V0 ←− V1 ←− V2 ←− · · ·
⇓ 0 1 −→ −→ −→ −→ −→ −→ · · · ←− V−2 ←− V−1 ←− V0 ←− V1 ←− V2 ←− · · ·
Figure 3.
By using the shifting, we shall show that a hyper-K¨ahler manifold of type A∞ is the universal cover of the fibre space of elliptic curves of type Ib in Sect. 6. (We use the same notation as in [KK].) 2. Hyper-K¨ahler Quotient Definition 2.1. A hyper-K¨ahler structure on a Riemannian manifold (X, g) consists of three almost complex structures I, J, K which satisfy the following conditions. (1) g(u, v) = g(Iu, Iv) = g(Ju, Jv) = g(Ku, Kv). (2) I 2 = J 2 = K 2 = −1, IJ = −JI = K. (3) Denote by ωI , ωJ , ωK associated symplectic 2 forms w. r. t. I, J, K . Then dωI , dωJ , dωK = 0. Let (X, g, I, J, K) be a hyper-K¨ahler manifold. We assume that a Lie group G acts on X preserving the hyper-K¨ahler structure of X. Each element ξ ∈ g of the Lie algebra of G defines a vector field ξˆ on X by the action of G. Definition 2.2. A hyper-K¨ahler moment map for the action of G on M is a map µ = iµI + jµJ + kµK : M −→ ImH⊗ g∗ which satisfies x ∈ M, g ∈ G, α = 1, 2, 3, µIα (gx) = Ad∗g (µIα )(x), ˆ Iα , hξ, dµIα i = i(ξ)ω ξ ∈ g, α = 1, 2, 3, where (I1 , I2 , I3 ) = (I, J, K), g∗ the dual space of g, Ad∗g : g∗ → g∗ the coadjoint map, ˆ the interior product. h , i the dual pairing between g and g∗ , and i(ξ) We shall explain an infinite dimensional hyper-K¨ahler quotient. Let M be a Hilbert ˆ whose model space with a hyper-K¨ahler structure. Consider the Hilbert affine space M ˆ space is M : there is an element 3 ∈ M such that ˆ = {3 + x; x ∈ M }. M
(2.3)
Denote by Sp(M ) the Banach Lie group of invertible bounded linear operators of M preserving hyper-K¨ahler structure of M : Sp(M ) = { g ∈ U (M ) ;
[g, I] = [g, J] = [g, K] = 0 }.
(2.4)
ˆ be the Banach Lie group of affine transformations of M ˆ preserving the hyperLet Sp K¨ahler structure with an exact sequence of smooth maps: p ˆ −→ Sp(M ) −→ 0, 0 −→ M −→ Sp
(2.5)
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ˆ . For an element q ∈ M ˆ we define the where M is a group of parallel translations of M ˆ smooth map 8q : Sp → M by g 7→ ( g(q) − q ) . Let G be a Hilbert Lie group with an injective smooth map ˆ i : G → Sp.
(2.6)
ˆ so as to preserve the hyper-K¨ahler structure: From (2.6), G acts on M g(3 + x) = g(3) + p(g)(x),
(2.7)
where p(g) ∈ Sp(M ). Denote by g the Hilbert Lie algebra of G. We set Z = { η ∈ g∗ ;
Ad∗g η = η,
for all g ∈ G},
(2.8)
where g∗ is the dual space of g and Ad∗ is the coadjoint action. From now on we assume the following conditions (1), (2), (3) and (4): ˆ for the action of (1) We assume that there exists a hyper-K¨ahler moment map µ on M G: ˆ → ImH ⊗ g∗ . µ:M (2) Define a map f by f : G × µ−1 (ζ) → µ−1 (ζ) × µ−1 (ζ), f (g, q) 7→ ( g(q), q ). Then we assume that f is injective and proper for some ζ ∈ ImH ⊗ Z. (3) Let p ◦ i be the composite map from G to Sp(M ): p ◦ i : G → Sp(M ). Denote by sp(M ) the Lie algebra of Sp(M ). Then we assume that the differential d (p ◦ i) is a compact operator from the Hilbert space g to the Banach space sp(M ). (4) Let 9q be the composite map 8q ◦ i: 9q : G → M. Denote by d9q the differential of 9q . If q = 3, then we assume d93 : g → T3 G(3), is an isomorphism between Hilbert spaces, where T3 G(3) is the tangent space of ˆ. G-orbit at 3 with the metric given by M
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473
Our argument is explained by a diagram: 8
3 p ←− ˆ −→ 0 −→ M −→Sp Sp(M ) −→ 0 93 - i ↑ % G
(2.9)
We consider another diagram: d9q q ˆ −−dµ 0 −−−−→ g −−−−→ Tq M −−→ ImH ⊗ g∗ −−−−→ 0.
(2.10)
ˆ → H ⊗ g∗ by We define an operator Dq : T M Dq (x) := d9∗q (x) + dµq (x),
ˆ, x ∈ Tq M
(2.11)
ˆ → g is the adjoint of d9q and we settle the image of d9∗q on the real where d9∗q : Tq M ∗ part of H ⊗ g . Note that g is identified with g∗ by the HIlbert metric. Lemma 2.12. Let ζ be as in Condition (2). Then the operator Dq satisfies (i), (ii) for q ∈ µ−1 (ζ): (i) Dq is a linear operator over the quaternion H. (ii) There exists a bounded operator Lq such that Dq ◦ Lq = 1H⊗g∗ . Proof of Lemma 2.12. (i) From our definition of Dq , we have < Dq (x), ξ >g = < x, Vξ (q) >M +i < x, IVξ (q) >M +j < x, JVξ (q) >M +k < x, KVξ (q) >M ,
(2.13)
where ξ ∈ g and Vξ (q) = d9q (ξ). Lemma 2.4 (i) follows from the description of Dq (2.13). Next consider the diagram: d9q ˆ g −→ Tq M q & ↓d9∗ q g∗
(2.14)
Denote by q the composite map d9∗q ◦ d9q : g → g∗ ∼ = g. Then 3 is an isomorphism from Assumption (4): 3 is a Fredholm operator of index 0. We can show that the ˆ from (3). It implies that difference q − 3 is a compact operator of g for any q ∈ M q is also a Fredholm operator of index 0 for any q. Since G acts on µ−1 (ζ) freely from ˜ q the H-linear map idH ⊗q : (2), q is an isomorphism for any q ∈ µ−1 (ζ). Denote by ˜ q : H ⊗ g → H ⊗ g.
(2.15)
˜ q : H ⊗ g → Tq M ˆ by Similarly we define d9 ˜ ξ0 + iξ1 + jξ2 + kξ3 ) := d9q (ξ0 ) + Id9q (ξ1 ) + Jd9q (ξ2 ) + Kd9q (ξ3 ). d9( Define Lq by
˜q ◦ ˜ −1 Lq = d9 q .
(2.16) (2.17)
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Then from definition of q , we have Dq ◦ Lq = 1H⊗g , where Lq is a bounded operator. The next theorem is obtained by [HKLR] in the case where M is finite dimensional. In gauge theory, the quotient construction has been developed extensively, [K, DK]. Our point of view is different from these results since we stress a general affine Hilbert space for an infinite dimensional quotient construction. Our quotient method is suitable for the construction of hyper-K¨ahler manifolds of type A∞ and D∞ . ˆ be an affine Hilbert space with a hyper-K¨ahler structure. Denote Theorem 2.18. Let M ˆ the group of affine transformations of M ˆ preserving the hyper-K¨ahler structure. by Sp ˆ If G acts on M ˆ satisfying Let G be a Hilbert Lie group with an injective map i : G → Sp. Conditions (1), (2), (3) and (4), then the quotient µ−1 (ζ)/G is a hyper-K¨ahler manifold. Proof of Theorem 2.18 . Define a function d by d( q1 , q2 ) := inf k q1 − g(q2 ) ||, g∈G
ˆ . Then from Assumption (2), d defines a distance of the orbit space where q1 , q2 ∈ M µ−1 (ζ)/G. Hence µ−1 (ζ)/G is Hausdorff. From Lemma 2.12 dµq has a right inverse ˆ from the implicit operator for any q ∈ µ−1 (ζ). Hence µ−1 (ζ) is a submanifold of M function theorem. The slice Sq of µ−1 (ζ) for the action of G is defined as Sq = { q + x ∈ µ−1 (ζ) ; d9∗q (x) = 0 },
(2.19)
ˆ ∼ where x ∈ Tq M = M . If we take x sufficiently small, then Sq defines a coordinate −1 system of µ (ζ)/G from Lemma (2.12). Hence µ−1 (ζ)/G is a Hilbert manifold. Denote by X the quotient µ−1 (ζ)/G. Let π : µ−1 (ζ) → X be the natural projection. Then each tangent space Tπ(q) X is identified with KerDq from (2.19). Since Dq is a H-linear operator, Ker Dq has a hyper-K¨ahler Hilbert structure. Furthermore symplectic structures ωI , ωJ and ωK are defined on each slice Sq by a pull back of the symplectic structures ˆ . Hence these symplectic forms are closed. of M Remark 2.20. If a hyper-K¨ahler quotient X is finite dimensional, then X is complete. 3. Hyper-K¨ahler Manifolds of Type A∞ We shall remember the diagram of type A∞ : α−3
α−2
α−1
α
α
α
β−3
β−2
β−1
β0
β1
β2
0 1 2 −→ −→ −→ −→ −→ −→ · · · ←− V−2 ←− V−1 ←− V0 ←− V1 ←− V2 ←− · · · ,
(3.1)
where each Vn is the representation of S 1 given by einθ . We fix an unitary basis en = einθ of Vn . We define an infinite dimensional complex vector space HA by M Hom(Vn , Vn+1 ) ⊕ Hom(Vn , Vn+1 ). (3.2) HA = n∈Z
We denote by (α, β) an element of HA and each component is written as (αn , βn ) ∈ Hom(Vn , Vn+1 ) ⊕ Hom(Vn , Vn+1 ). Define a Hilbert space MA by
Hyper-K¨ahler Manifolds of Type A∞ and D∞
475
( MA =
X
(α, β) ∈ HA ;
) |αn | + |βn | < ∞ . 2
2
(3.3)
n∈Z
We have a complex structure I on M by the multiplication i. Another complex structure J is given by J(α, β)n∈Z = (−β ∗ , α∗ )n∈Z , where α∗ , β ∗ are adjoint maps of α, β respectively. If we set K = I ◦J, then (M, I, J, K) L is a Hilbert space with the hyper-K¨ahler structure. An element 3A = (3R n , 3 n )n ∈ H A is given by ( for n ≥ 0 λn en+1 R (3.4) 3n (en ) = λn+1 en+1 for n < 0, ( for n ≥ 0 λn en , L (3.5) 3n (en+1 ) = −λn+1 en , for n < 0, √ ˆ A by where λn = 1 + n2 . Then we define a hyper-K¨ahler affine space M L ˆA = M (α, β) ∈ HA ; (α − 3R (3.6) A , β − 3A ) ∈ MA . Let U (Vn ) be the unitary group of Vn . Then define an infinite group G∞ A by G∞ A = × U (Vn ).
(3.7)
n∈Z
There is a natural action of G∞ A on HA , −1 ). (αn , βn ) → (gn+1 αn gn−1 , gn βn gn+1
(3.8)
We denote by g = (gn )n∈Z an element of G∞ A , where gn ∈ U (Vn ). Let us define a group GA by ˆ GA = g ∈ G∞ (3.9) ; lim < g e , e >= 0, g(3 ) ∈ M n n n A A , A |n|→∞
where < , > is the inner product of Vn and −1 L −1 g(3A ) = ( gn+1 3R n gn , gn 3n gn+1 )n .
Then we see that GA is a Hilbert Lie group. From the definition of GA (3.9), GA acts ˆ A preserving the hyper-K¨ahler structure. on M ˆ A for the action GA . Theorem 3.10. There exists a hyper-K¨ahler moment map µA on M −1 Furthermore the quotient XA = µA (ζ)/GA is a 4 dimensional complete noncompact hyper-K¨ahler manifold for generic ζ ∈ ImH ⊕ g∗ . Theorem 3.10 is essentially similar to Theorem 2.9 in [G2], pp. 432. For the sake of argument, we shall give a proof of Theorem 3.10. Proof of Theorem 3.10. The Hilbert Lie algebra gA of GA is given by ( ) M X R R 2 u(Vn ) ; k ξn+1 3n − 3n ξn k < ∞, lim ξn = 0, . gA = (ξn )n ∈ n∈Z
n∈Z
|n|→∞
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Hence gA satisfies Condition (4) as in Theorem 2.18. From (3.4) and (3.5), we see that any element ξ of gA is bounded. Let Sp be invertible bounded operators of MA preserving the hyper-K¨ahler structure. Then the differential of the inclusion GA ,→ Sp(M ) is a ˆ A , GA ) satisfies Condition (3). We can see that (M ˆ A , GA ) compact operator. Hence (M also satisfies Conditions (1) and (2) for generic ζ. Hence the result follows from Theorem 2.18. We shall describe XA by invariant functions. Let us define x, y, z by Y
x=
α0 n ,
y=
Y
β0n,
z=
n∈Z
n∈Z
(
where 0
αn= ( β
0
n
=
λ−1 n αn (λn+1 )−1 αn
1 (α0 β0 + α−1 β−1 ), 2
for n ≥ 0, for n < 0,
λ−1 n βn −(λn+1 )−1 βn
for n ≥ 0, for n < 0.
(3.11)
(3.12)
(3.13)
Note that these infinite products are well defined. We denote by µC the holomorphic moment map of µA w.r.t. I and ζC the value of µC (XA ). Theorem 3.14. If XA is smooth then XA is biholomorphic to the hypersurface (x, y, z) ∈ C3 ; xy = p(z) with respect to a certain complex structure I on XA , where Y Y (z + c + ηn ) (z − c − ηn ) p(z) = n≥0
and 2c = (ζ0 )C and
(3.15)
n≤0
P n −2 m=1 λm (ζm )C ηn = 0 Pn −2 m=−1 λm (ζm )C
for n > 0, for n = 0, for n < 0.
Proof of Theorem 3.14. Since (αn , βn )n ∈ µ−1 C (ζC ), we have βn αn − αn+1 βn+1 = (ζn )C ,
for all n ∈ Z.
From (3.11), xy =
Y n∈Z
α0 n β 0 n = (
Y
α0 n β 0 n )(
Y
α0 n β 0 n ) = p(z).
n<0
n≥0
Hence we have a biholomorphic map from XA to the hypersurface {xy = p(z)}. We can show that this map is bijective. From now on we assume that ζ = (ζI , ζC ) with ζn = −ζ−n ,
n 6= 0.
We define an involution σ of XA preserving its hyper-K¨ahler structure by
(3.16)
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477
σ(α, β) = (σ(β), σ(α)), (
where σ(α)n =
( σ(β)n =
−β−n−1 β−n−1
for n ≥ 0, for n < 0,
−α−n−1 αn−1
for n < 0, for n ≥ 0.
(3.17)
(See Fig. 2 in the Introduction.) Then we see that µ−1 A (ζ) is invariant under the action of σ. Furthermore the action σ is compatible with the action of GA . Hence we have an action on XA by σ preserving the hyper-K¨ahler structure. Theorem 3.18. Let XA /σ be the quotient of XA for the action σ. Then XA /σ is a hyperK¨ahler orbifold with two singularities of type A1 . Furthermore XA /σ is biholomorphic to the hypersurface ˆ =0 (3.19) (x, ˆ y, ˆ z) ˆ ∈ C3 ; xˆ 2 + yˆ 2 zˆ + q(z) with respect to a certain complex structure I, where q(z) ˆ = p(z). Proof of Theorem 3.18. From definition of x, y, z, σ(x, y, z) = (y, x, −z).
(3.20)
Hence xˆ = (x − y)z, yˆ = i(x + y), zˆ = z 2 are invariant functions under the action of σ. Then we see that xˆ 2 + yˆ 2 z + q(z) ˆ = 0, where Y (zˆ − (c + ηn )2 ). (3.21) q(z) ˆ = n≥0
Let Cn be (−2)-curves in XA , n ∈ Z. Then σ(Cn ) = (C−n ). Fixed point sets of XA under the action of σ are only two points in C0 . Hence XA /σ is a orbifold with two singularities of type A1 Theorem 3.22. Let X˜ be the minimal resolution of XA /σ. Then the intersection matrix ˜ Z) is (−1)× Cartan matrix of type D∞ . of H 2 (X, Proof of Theorem 3.22. It follows from Theorem 3.18.
4. Hyper-K¨ahler Manifolds of Type D∞ We shall recall unitary representations of group D∞ . Let D∞ be the normalizer of the maximal torus of Sp(1). This group D∞ is written as D∞ := z1 + z2 j ∈ Sp(1); |z1 |2 + |z2 |2 = 1, z1 z2 = 0 , (4.1) where the maximal torus is given by z2 = 0. Irreducible unitary representations of D∞ consist of E0+ , E0− and En , n = 1, 2, · · · , where E0+ is the trivial representation, dim
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E0− = 1, and dim En = 2. Then each irreducible representation is described by the following table: eiθ
j
E0+
1
1
E0−
1
−1
h
En
einθ 0 0 e−inθ
i
h
0 (−1)n 1 0
i
Table 1. Irreducible representation of D∞
Since the maximal torus S 1 is the subgroup of D∞ , each irreducible representation of D∞ can be regarded with a representation of the maximal torus. Indeed E0+ and E0− are trivial representations of S 1 . Each En is decomposed into Vn ⊕ V−n , where n ≥ 1. Let en = einθ be the unitary basis of Vn . Then {en , e−n } can be considered as a unitary basis of En , where En has Hermitian metric. Define another unitary basis of En by 1 fn+ := √ (en + e−n ), 2
1 fn− := √ (en − e−n ), 2
(4.2)
where n ≥ 1. Then we have the decomposition En = En+ ⊕ En− , where En+ (resp. En− ) is one dimensional space generated by fn+ (resp. fn− ). We denote by f0+ (resp. f0− ) the unitary basis of E0+ (resp. E0− ). Finally we denote by E0 the representation E0+ ⊕ E0− . We shall start our construction of hyper-K¨ahler manifolds of type D∞ . Let En be the representation of D∞ as before. Consider the following diagram (This is called the Dynkin diagram of type D∞ ). E0+ -&
−→ E1 ←− E2 %.
−→ ←− E3
−→ ←− E4
−→ ←− · · ·
(4.3)
E0− Fig. 4. Diagram of type D∞
where each arrow implies a homomorphism between representation spaces. Let us define the vector space HD by M Hom(En , En+1 ) ⊕ Hom(En+1 , En ), (4.4) HD := n≥0
where E0 = E0+ ⊕ E0− . Note that Hom(E0 , E1 ) ∼ = Hom(E0+ , E1 ) ⊕ Hom(E0− , E1 ), Hom(E1 , E0 ) ∼ = Hom(E1 , E0+ ) ⊕ Hom(E1 , E0− ).
Hyper-K¨ahler Manifolds of Type A∞ and D∞
479
We denote by (A, B) an element of H, where each component of (A, B) is (An , Bn ) ∈ Hom(En , En+1 ) ⊕ Hom(En+1 , En ). A Hilbert space MD is defined by ( ) ∞ X 2 2 (A, B) ∈ H ; k (A, B) k2 = |An | + |Bn | < ∞ . (4.5) MD := n=0
Since MD is a complex vector space, we have a complex structure I on MD by the multiplication of i. Another complex structure J is defined as J(An , Bn )n≥0 = (−Bn∗ , A∗n )n≥0 , where A∗n , Bn∗ are adjoint maps of An , Bn respectively. Then we see that I ◦ J = −J ◦ I. Hence (MD , I, J, K) is a hyper-K¨ahler Hilbert space where K = I ◦ J. We shall define a divergent element 3D ∈ H by L 3D = (3R n , 3n )n≥0 ,
where
+ + 3R n (fn ) = λn fn+1 ,
(4.6)
− − 3R n (fn ) = λn fn+1 ,
L ∗ 3R n = (3n ) ,
√ ˆ D is given by 1 + n2 . A Hilbert affine space M ( ) ∞ X R 2 L 2 := (A, B) ∈ HD ; |An − 3n | + |Bn − 3n | < ∞ .
and λn = ˆD M
(4.7)
n=0
ˆD ∼ ˆ D × M, M ˆ D is a hyper-K¨ahler affine space. Let U (En ) be the unitary Since T M =M group of En . Then consider an infinite group G∞ D as − + G∞ D := × U (En ) × (U (E0 ) × U (E0 )). n>0
We denote by g = (gn )n≥0 an element of G∞ D , where gn ∈ U (En ). Then a group GD is defined as ( ) lim < gn (fn+ ), fn− >= 0 ∞ n→∞ GD := g ∈ GD , (4.8) k [ g, 3D ] k2 < ∞ where < , > is the inner product of En and [ g, 3D ] = g3D − 3D g is an element of HD whose norm k[ g, 3D ]k2 is given by X R 2 kgn+1 3R (4.9) k [ g, 3D ] k2 = 2 n − 3n gn k . n≥0
Define g∞ by g∞ :=
M
u(En ) ⊕ (u(E0+ ) ⊕ u(E0− )).
n>0
We denote by ξ = (ξn )n≥0 an element of g∞ , where ξn ∈ u(En ).
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Proposition 4.10. Let GD be the group as in (4.8). Then GD is a Hilbert Lie group and the Hilbert Lie algebra g of GD is given by lim < ξn (fn+ ), fn− >= 0 n→∞ , (4.11) g := ξ ∈ g∞ k [ ξ, 3 ] k2 + lim kξ k2 < ∞ D
2
n→∞
n
where k[ ξ, 3D ]k22 is given by (4.9) and the metric of g is given by < ξ, η >=< [ ξ, 3D ], [ η, 3D ] >2 + < ξ∞ , η∞ > .
(4.12)
Note that there exists ξ∞ = limn→∞ ξn since k [ ξ, 3D ] k2 < ∞. We shall denote the norm of ξ ∈ g by kξkg . Lemma 4.13. Let g be as in (4.11). Then we have sup kξn k ≤ Ck ξ kg , n≥0
where constant C does not depend on ξ. Proof of Lemma 4.13. We regard ξn with an element of u(2) using the basis {fn , f−n } of En . Then from definition of 3D (4.8), k[ ξ, 3D ]k2 is given by X λ2n kξn+1 − ξn k2 . k [ ξ, 3D ] k2 = n≥0
From the Schwartz inequality, n1 X
kξn+1 − ξn k <
n=n0
where
λ2n
n1 X 1 2 λ n n=n
! 21
n1 X
! 21 λ2n kξn+1
− ξn k
2
,
n=n0
0
2
= 1 + n . Hence we have
kξn1 − ξn2 k < C
n2 X
! 21 λ2n kξn+1 − ξn k2
< Ck [ ξ, 3D ] k2 ,
∀n1 , n2 ≥ 0.
n=n1
This implies that {ξn }n≥0 is a Cauchy sequence. When n goes to infinity, ξn converges to ξ∞ ∈ u(2). Then we have k ξn k ≤ (
∞ X
k ξm+1 − ξm k ) + kξ∞ k
m=n 0
≤ C k [ ξ, 3D ] k2 + k ξ∞ k ≤ C 00 k ξ kg .
Proof of Proposition 4.10. This follows from Lemma 4.13. We define a group G0 by G0 := GD /U (1).
(4.14)
G0 is a Hilbert Lie group whose Lie algebra is g0 . Let HD be the space of homomor∞ phisms as in (4.4). From definition of G∞ D , there is the action of GD on HD . Since GD ∞ is a subgroup of GD , we can consider the action of GD on HD .
Hyper-K¨ahler Manifolds of Type A∞ and D∞
481
ˆ D as in (4.7). Then M ˆ D is invariant under Lemma 4.15. Let GD be as in (4.8) and M the action of GD . ˆ D : (4.7), (4.8). Proof of Lemma 4.15. This result follows from definition of GD and M ˆ D from Lemma 4.15. Then this Proposition 4.16. We can define the action of GD on M ˆ D. action is smooth and preserving hyper-K¨ahler structure of M Proof of Proposition 4.16. This result follows from definition of the action of GD on ˆ D. M ˆ D →ImH ⊗ g∗ . Proposition 4.17. There exists a hyper-K¨ahler moment map µ : M ˆ D is an affine space, a hyper-K¨ahler moment map µ Proof of Proposition 4.17. Since M is constructed explicitly: i [ A, A∗ ] + [ B, B ∗ ] , µI (A, B) = 2 µC (A, B) = [ A, B ] − [ 3R , 3L ].
(4.18)
Define Z by Z = { ζ ∈ g∗ ;
< ζ, Adg η >=< ζ, η >, ∀g ∈ G,
∀η ∈ g
}.
(4.19)
Theorem 4.20. Let µ be the hyper-K¨ahler moment map as in (4.17). Then the quotient µ−1 (ζ)/G0 is a 4-dimensional complete, noncompact hyper-K¨ahler manifold if G0 acts on µ−1 (ζ) freely for ζ ∈ImH ⊗ Z. In particular, µ−1 (ζ)/G0 is smooth for generic ζ. R
L
ˆ = (3 ˆ n,3 ˆ n ) by Proof of Theorem 4.20. We shall define an element 3 ˆR 3 n (en ) = λn en+1 , + ˆR 3 0 (f0 ) = λ0 e1 ,
where λn =
√
ˆR 3 n (e−n ) = λn e−n−1 , for n > 0, − ˆR 3 0 (f0 ) = λ0 e−1 ,
ˆ L )∗ , ˆ R = (3 3 ± 1 + n2 . Note the e± 0 = f0 and
fn± = R
(4.21)
1 (en ± e−n ). 2
R
R ˆ ˆ 0 6= 3R Note that 3 ˆ G0 be the tangent space of G0 -orbit 0 and 3n = 3n for n > 0. Let T3 ˆ T3ˆ G0 is a Hilbert space by the metric of M . By using the basis {en , e−n }, we at 3. ˆ the identify u(En ) with u(2). We also use a basis {f0+ , f0− } of E0 . Denote by Vξ (3) ˆ tangent vector at 3 corresponding to ξ. Then
ˆ k2 = k Vξ (3)
∞ X
ˆ n ξ n k2 , ˆn−3 k ξn+1 3
n=0
=
∞ X n=0
k λn (ξn+1 − ξn ) k2u(2) .
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Under the identification u(En ) ∼ = u(2), we have ˆ ] k2 = C kVξ (3) ˆ k. k ξ0 − ξ∞ k < Ck [ ξ, 3 Since tr ξ∞ = 0, we see that each diagonal component of ξ vanishes with respect to a basis {en , e−n }. (Note that ξ∞ is diagonal if we identify u(En ) with u(2) with respect to the basis {fn+ , fn− }.) Because ξ0 is diagonal with respect to the basis {f0+ , f0− }, k ξ0 − ξ∞ k = k ξ0 k + k ξ∞ k. Hence we have
ˆ k. k ξ kg < C 0 k Vξ (3)
We also have an inequality: ˆ k < C 00 k ξ kg . k Vξ (3) Hence we have an isomorphism ˆ ∼ T3ˆ G0 (3) = g0 .
(4.22)
ˆ is equivalent to the metric of g0 . Hence we can So that is, the metric of T3ˆ G0 (3) ˆ apply Theorem 2.12 to MD and G0 . Therefore the quotient µ−1 (ζ)/G is a hyper-K¨ahler manifold if G0 acts on µ−1 (ζ) smoothly. Furthermore we can show that µ−1 (ζ) is not empty for generic ζ. Finally we shall show that µ−1 (ζ)/G is 4 dimensional. We use the same notation as in Sect. 2. It is sufficient to prove that Ker D3ˆ is 4-dimensional. A ˆ D belongs to ker D ˆ if and only if vector (A, B) ∈ T3ˆ M 3 B0 = −(B0 )t , A0 = −(A0 )t , λn Bn = λn+1 Bn+1 , λn An = λn+1 An+1 , ± with respect to the basis {en , e−n }. (A key point is to consider an element of π∞ of infinite type as in (4.23).) Hence Ker D3ˆ is 4 dimensional.
We shall describe a criterion of a smoothness for µ−1 (ζ)/G. Let π0+ (resp. π0− ) be the projection on E0+ (resp. E0− ). Denote by πn the projection on En . Then iπ0± , iπn can be considered as elements of g. From definition of g there exists a limit ξ∞ = limn→∞ ξn . Then we have a diagram: 0 −→ f −→g−→t2 −→ 0 ξ 7−→ ξ∞ , where ξ∞ ∈ t2 , f is Ker of {g → t2 }. When we choose a splitting map, g is identified + − with f ⊕ t2 . We consider elements arising from t2 . We define iπ∞ , iπ∞ ∈ g by ( 0 if n = 0 + (π∞ )n = πn+ otherwise , ( − (π∞ )n
=
0 πn−
if n = 0 otherwise ,
where πn+ (resp. πn− ) is the projection from En to En+ (resp. En− ).
Hyper-K¨ahler Manifolds of Type A∞ and D∞
483
Lemma 4.23. Let Z ∗ be the dual space of Z ⊂ g∗ as in (4.19). We have the map τ : τ : g → Z ∗. Let H be the closure of linear combination of {
n X i πi ; 2
+ − iπ0+ , iπ0− , iπ∞ , iπ∞ ,
n≥1 }
i=1
with respect to the metric of g. Then τ gives a isomorphism H ∼ = Z ∗. Proof of Lemma 4.23. We identify Z ∗ with Z by the metric. By the basis {fn+ , fn− } of En , u(En ) can be regarded with u(2). If ξ belongs to Z ∗ then ξ satisfies the following: + x 0 , ξ10 = 0 x− a+ 0 , 0 a−
λ20 ξ01 + ξ∞ =
−λ2n ξn+1,n + λ2n−1 ξn,n−1 = cn 1, where ξn+1,n = ξn − ξn−1 , x+ , x− , a+ , a− ∈ u(1). From these equations, we have < ξ, iπ0± >= ix± , < ξ,
± < ξ, π∞ >= ia± ,
n n X X i πm >= cm . 2
m=1 +
−
+
m=1
−
We see that if x , x , a , a = 0 then ξ = 0. Hence the coupling between H and Z is perfect. Therefore τ gives an isomorphism. ± ) for We denote by θn the image 2i τ (πn ) for n > 0. We also write θ0± (resp. θ∞ ± ± ∗ iτ (π0 ) (resp. iτ (π∞ )). Hence from Lemma 3.23 we have a basis of Z : ) ( n X + − , θ∞ , θi , n > 0 . (4.24) θ0+ , θ0− , θ∞ i=1
We define a root of finite type by a finite linear combination: − m+0 θ0+ + m− 0 θ 0 + mn θ n ,
where m± 0 = 0, 1 and mn = 0, 1, 2. Furthermore we need roots of “infinite type”. Denote s t , θ∞ by by roots θ∞ s + t − = θ0+ + θ∞ , θ∞ = θ0+ + θ∞ . (4.25) θ∞ A root θn± is given by ± − θn± = θ∞
n−1 X
θm .
m=1 s t , ±θ∞ We call { ±θ∞ , ±θn± , n > 0 } roots of infinite type.
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Theorem 4.26. Let ζ be an element of ImH ⊗ Z. If < ζ, η >6= 0, for every root of finite and infinite type η ∈ Z ∗ , then G0 acts on µ−1 (ζ) freely. Proof of Theorem 4.26. If G0 does not act freely, then there exists g ∈ G0 such that g(A, B) = (A, B), for (A, B) ∈ µ−1 (ζ). Let Fn (s) be an eigenspace for each gn ∈ U (En ) whose eigen value is s. Then ⊕n Fn (s) is invariant under maps A, B. If there exists n such that dim Fm (s) = 0, 2 for all m > n then we call ⊕n Fn (s) an invariant subspace of finite type. In a case of finite type we can show that there exists a root θ of finite type such that < ζ, θ >= 0. (If dim Fm (s) = 2 for all m > n then consider the complement of ⊕n Fn (s).) If ⊕n Fn (s) is not of finite type, then ζ is in the kernel of some root of infinite type. Hence the result follows. Remark 4.27. The metric of g is not invariant under the adjoint action of G. Hence Z is not the metrical dual space of the center of g. This is a reason why we need roots of infinite type. 5. Hyper-K¨ahler Manifolds of Type A∞ and D∞ Let XA be a hyper-K¨ahler 4-manifold of type A∞ as in Theorem 3.10. Then we see that there exists an involution σ on XA from (3.17). The quotient XA /σ is an orbifold which has two singularities of type A1 from Theorem 3.18. Theorem 5.1. Let XD be a hyper-K¨ahler manifold of type D∞ as in Theorem 4.20. Then XD is a minimal resolution of singularity of XA /σ if XD is smooth. We shall give the proof of Theorem 5.1 at the end of this section. Let 0 be a subgroup SU(2). Then consider a module: V ⊗ End (L2 (0) ), where V ∼ = C2 is the natural representation of 0 and L2 (0) is the complex right regular representation. Then V ⊗ End (L2 (0)) is a representation of 0. We denote by ( V ⊗ End (L2 (0) ) )0 a set of invariant elements under the action of 0. From (3.2) and (4.4), we have 1 HA ∼ = ( V ⊗ End (L2 ( S 1 )) )S , HD ∼ = ( V ⊗ End (L2 ( D∞ )) )D∞ .
When we take a basis of C2 , an element of V ⊗ End (L2 ( 0 )) can be regarded with a pair of End(L2 (0) ). From 3.1 (definition of the group D∞ ), D∞ = S11 ∪ S21 , where S11 is the maximal torus of D∞ and S21 = { z2 j ∈ Sp(1) ; k z2 k = 1 }. Since L2 (D∞ ) is decomposed into L2 (S11 ) ⊕ L2 (S21 ), an element of End written as 2 × 2 matrix. Then define the map 8 by 8 : V ⊗ End (L2 ( S 1 )) −→ V ⊗ End (L2 ( D∞ )),
L2 (D∞ ) is
Hyper-K¨ahler Manifolds of Type A∞ and D∞
485
(α, β) 7−→
α 0 β 0 , . 0 −β 0α
(5.2)
Then we see that 8 gives a map from HA to HD from (5.2). In Sect. 3 (resp. 4), type ˆA A∞ (resp. D∞ ) is constructed as the quotient XA (resp. XD ) from the affine space M ˆ D ) by the group GA (resp. GD ). From the definition of 3A , 3D (3.4, 3.5 and (resp. M 4.6 ), we see that (5.3) 8(3A ) = 3D . ˆ A to M ˆ D preserving hyper-K¨ahler structures. From (4.2), 8 defines the map from M A A (ζ )/G From now on we assume that XA = µ−1 A , where ζ satisfies A A ζnA = −ζ−n for n > 0,
ζIA = 0.
(5.4)
ˆA → M ˆ D be as before. Denote by ζ D the image µD ◦ Lemma 5.5. Let 8 : M A D (ζ )). Then ζ is an element of Im H ⊗ Z satisfying 8 (µ−1 A s t < ζ D , θ∞ >=< ζ D , θ∞ >= 0, s t , θ∞ are roots of infinity type as in (4.25). Furthermore 8 gives a map from where θ∞ −1 XA to XD = µD (ζ0 )/GD .
Proof of Lemma 5.5. We shall describe the map 8 explicitly: When we choose a basis (e+n , e− n ) of En for n ≥ 0, then (A, B) = 8(α, β), where 1 1 β0 −α−1 α0 α 0 A0 = √ , B0 = √ , (5.6) 2 −β−1 β−1 2 β0 α−1 α β 0 0 An = n , Bn = n , 0 −β−n−1 0 α−n−1 for n > 0. Hence from (4.17), we see that ζ D is an element of Im H ⊗ Z. Since the map 8 is equivalent under the action of GA and GD , 8 gives a map from XA to XD . Next we shall consider the following two cases: Case (1) α0 = −β−1 , β0 = α−1 , αn = −β−n−1 , βn = α−n−1 for n > 0, Case (2)
α0 = β−1 , β0 = −α−1 , αn = −β−n−1 , βn = α−n−1 for n > 0.
From Case (1), we see that s >= 0. < ζ D , θ∞
From Case (2), we see that t >= 0. < ζ D , θ∞
(Note that these two cases correspond to singularities of XA /σ.) Lemma 5.7. Let σ be the involution of type XA as in (3.17). (We assume that ζ A = (0, ζC ), i.e., ζI = 0.) Then 8 defines an isomorphism between XA /σ and D D XD = µ−1 D (ζ )/GD , where ζ is an element of ImH ⊗ Z as in Lemma 5.5.
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Proof of Lemma 5.7. From (3.17) and (4.8), we see that 8 defines the map from XA /σ to XD . Since ζD = µD ◦ 8(µ−1 A (ζA )), we see from (5.5) that < ζD , e+0 >=< ζD , e− 0 >.
(5.8)
From our assumption, (ζD )I = 0. Then it turns out that every one parameter subgroup − ∗ + −1 of GC D -orbit in µ (ζD ) is closed because 3D = (3D ) and (4.18). This means that A0 B0 ∈End(E1 ) is diagonalizable. Hence we see that 8 is surjective from (5.6) and (5.8). We shall use invariant functions to show an injectivity. Let us recall invariant functions of type A∞ (3.11): x=
Y
α0 n ,
Y
y=
n∈Z
β0n,
z = α−1 β−1 + β0 α0 .
n∈Z
The involution σ satisfies σ : (x, y, z) 7−→ (y, x, −z). Then invariant functions of the quotient XA /σ are xˆ = z(x − y),
zˆ = z 2 .
yˆ = i(x + y),
Then from (3.18), the quotient is isomorphic to a hypersurface {
(x, ˆ y, ˆ z) ˆ ∈ C3
;
xˆ 2 + yˆ 2 zˆ + q(z) ˆ =0
},
where q(z) ˆ is an analytic function as in Theorem 3.18. We also consider invariant functions of type D∞ . Let πn+ be the orthogonal projection from En to En+ . Denote by A+n + + the composite map πn+1 ◦ An . We write Bn+ for Bn ◦ πn+1 . Define P by 0 Bn0 A0n · · · A02 A01 , P = lim B10 B20 · · · Bn−1 +
+
n→∞
where A0m = λ−1 m Am , functions X, Y, Z by
0 Bm = λ−1 m Bm , and λm =
X = 2Tr [ C0− , C0+ ] P,
√
1 + m2 . Then we define invariant
Y = iTr (C0+ − C0− ) P,
Z = Tr C0+ C0− ,
(5.9)
− where C0+ (resp.C0− ) = A+0 B0+ (resp. A− 0 B0 ). If (A, B) = 8(α, β) then we see that X = x, ˆ Y = y, ˆ Z = z. ˆ Hence 8 is injective from Lemma 5.5 and Theorem 4.18.
Lemma 5.10. Let ζ D = (0, ζC ) be as in Lemma 5.5. If ζˆD = (ζI , ζC ) satisfies s >, < ζˆD , θ∞
t < ζˆD , θ∞ >6= 0,
ˆD then the corresponding hyper-K¨ahler manifold µ−1 D (ζ )/G is a minimal resolution of −1 D the orbifold µD (ζ )/G.
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487
Proof of Lemma 5.10. We denote by Xζˆ (resp. Xζ ) the hyper-K¨ahler manifold ˆD µ−1 D (ζ )/GD
(resp. µ−1 (ζ D )/GD ).
Then we see that Xζˆ is smooth from Theorem 4.26. We can show that invariant functions X, Y and Z in (5.9) satisfies X 2 + Y 2 Z + q(Z) = 0. From Lemma 5.7, Xζ ∼ = XA /σ. Then we have a map 9 from Xζˆ to Xζ from Theorem 3.18: 9 : Xζˆ → Xζ . ˆ If A0 B0 is diagonalizable, we see that there exists Let (A, B) be any element of µ−1 (ζ). such that g ∈ GC D g(A, B) ∈ µ−1 I (0). Hence 9(A, B) is a nonsingular point of Xζ . If A0 B0 is not diagonalizable, then the GC D -orbit through (A, B) is not closed and 9(A, B) is a singular point in Xζ . Since 9 is surjective, 9 gives a bijective map from 9−1 (Xζreg ) to Xζreg , where Xζreg is the non-singular part of Xζ . Furthermore the inverse image of a singular point in Xζ is a rational curve. Hence 9 is a resolution of singularity. Proof of Theorem 5.1. If XD is smooth, then a diffeormorphism type XD does not depend on ζ. Hence the result follows from Lemma 5.10. Corollary 5.11. Let XD as in 5.1. Then intersection matrix of H2 (XD , Z) is (−1)× Cartan matrix of type D∞ . Proof of Corollary 5.11. The result follows from Theorem 3.22 and Theorem 5.1.
6. Hyper-K¨ahler Manifolds of Type A∞ and Elliptic Fibrations In this section we use the same notations as in Sect. 3. Let HA be the infinite dimensional vector space as in (3.2). We define a map 4 : HA → HA by ( √1 for n ≥ 0, βn + αn∗ , −αn + βn∗ , 2 (6.1) 4(α, β)n = 1 ∗ ∗ √ αn − β n , βn + α n for n < 0. 2 ˜ A the image 4(M ˆ A ): Denote by M ˜ A = 4(M ˆ A ). M
(6.2)
From (3.4), (3.5) and (6.1) we see
(√ 2( λn , 0 ) 4(3A )n = √ 2( 0, λn , )
for n ≥ 0, for n < 0.
˜ A is a Hilbert affine space with the hyper-K¨ahler structure. Let GA be the Hilbert M ˜ A preserving the hyper-K¨ahler Lie group as in (3.9). Then we see that GA acts on M structure. Denote by µ˜ the hyper-K¨ahler moment map: ˜ A → ImH ⊗ g∗ . µ˜ : M
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Theorem 6.3. Let ζ˜ be the image µ(4(3 ˜ A )). Then we have a biholomorphic map ˜ ( µ−1 (ζ)/GA , K ) ∼ = ( µ˜ −1 (ζ)/G A , I ), with respect to the complex structure K and I respectively. Proof of Theorem 6.3. The map 4 defines a holomorphic map ˜ ( µ−1 (ζ)/GA , K ) −→ ( µ˜ −1 (ζ)/G A , I ). We see that this is biholomorphic.
˜ We shall give a holomorphic description of µ˜ −1 (ζ)/G A . From now on we assume that ζ˜C = 0,
(6.4)
and ζ˜I is generic satisfying (ζ˜I )n < 0,
for all n ∈ Z.
(6.5)
˜ µ˜ −1 C (ζC ).
We define an open subset N0 of N by Denote by N the inverse image (6.6) N0 = (α, β) ∈ N ; |αn |2 + |βn−1 |2 6= 0, ∀n ∈ Z . Theorem 6.7. Let GC A be the complexification of GA . Then we have an isomorphism C ∼ ˜ ( µ˜ −1 (ζ)/G A , I ) = N0 /GA ,
with respect to holomorphic symplectic structures. Proof of Theorem 6.7. From (6.6), we have a map C ˜ µ˜ −1 (ζ)/G A −→ N0 /GA .
This map is injective from a general theory of moment map. The real part of the hyperK¨ahler moment map µ˜ is given by i µ˜ n = − { ( |αn |2 − |βn |2 ) − ( |αn−1 |2 − |βn−1 |2 ) }, 2 where we identify g∗ with g by the metric. Hence we can see that this map is surjective. Since N0 /GC A is a holomorphic symplectic quotient, there exists a holomorphic symplectic structure ωC on N0 /GC A . Then we see that our map is preserving symplectic structure. We consider the weighted map W : HA → HA : ( ( λ−1 n α n , λn β n ) W (α, β) = ( λn αn , λ−1 n βn )
for n ≥ 0, for n < 0.
(6.8)
The map W is equivalent to the action of GA from (2.8). Hence W defines the isomorphism: C (6.9) W : N0 /GC A → W (N0 )/GA . We denote by XA the holomorphic symplectic manifold W (N0 )/GC A . (We hope that readers do not confuse our slightly abused notation.) We define an action of C∗ on XA by
Hyper-K¨ahler Manifolds of Type A∞ and D∞
( λ (α, β) =
489
( λα0 , λ−1 β0 ) ( αn , βn )
for n = 0, for n 6= 0,
where λ ∈ C∗ . This action preserves the holomorphic symplectic form ωC . Hence we have the moment map φ with respect to ωC : φ : XA → C, φ( α, β ) = α0 β0 .
(6.10)
We also define the action of the additive group Z on XA by a shifting ( αn , βn ) 7−→ ( αn−1 , βn−1 ).
(6.11)
(See diagram 2 in the Introduction.) This action of Z is holomorphic and preserving the holomorphic symplectic structure. Note that this action can not be defined as a hyperK¨ahler isometry. The holomorphic moment map φ is invariant under the action of Z since βn αn = αn−1 βn−1 from µC (α, β) = 0 (6.4). For any positive integer b, φ define a map XA /bZ → C. Then we have a diagram: XA −→XA /bZ ←- φ−1 (4) ↓ & ↓φ C ←- 4,
(6.12)
where 4 = { z ∈ C ; |z| ≤ 1 }. Theorem 6.13 . φ−1 (4) → 4 is the elliptic fibre space of type Ib . (We use the same notation as in [KK].) Proof of Theorem 6.13. First we consider the singular fibre φ−1 (0). Define a subspace Lm of HA by Lm = { (α, β) ∈ HA ; βn = 0 for n ≥ m,
αn = 0 for n < m } .
(6.14)
Lm is given by a subdiagram of type A∞ : α
α
m m+1 · · · ←− Vm−1 ←− Vm −→ Vm+1 −→ ··· .
βm−2
βm−1
1 Then ( Lm ∩ N0 )/GC A is a rational curve CPm . Hence from (6.6), we see that [ 1 CPm . φ−1 (0) =
(6.15)
m∈Z
This infinite chain is sifted by the action of Z from (6.14): 1 1 → CPm+1 . CPm
(5.16)
Hence on the quotient space XA /bZ , the singular fibre φ−1 (0) becomes a circle of b rational curves from (6.15) and (6.16). Next we consider a general fibre φ−1 (z), z 6= 0. When we remove the singular fibre from XA , we can define a coordinate (z, t) by Y Y z = α0 β0 , t = ( W (αn ) )( W (βn ) )−1 , (6.17) n≥0
n<0
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where W is the map as in (6.8) and t ∈ C∗ . Note that αn , βn 6= 0 from z = α0 β0 6= 0. Our coordinate (z, t) is shifted by the action Z: (z, t) → (z, zt).
(6.18)
Hence a general fibre is an elliptic curve C∗ /Z. Furthermore the holonomy around the singular fibre is given by 1b . (6.19) 01 Hence we see that φ−1 (4) → 4 is an elliptic fibration of type Ib from Kodaira’s classification theorem of elliptic fibrations [KK]. Acknowledgement. The author is grateful to Prof. N. Hitchin for helpful observations and hospitality during his stay in Cambridge. The author wishes to thanks Prof. G. Gibbons for energetic and stimulative discussions.
References [ADHM] Atiyah, M.F., Drinfeld, D.G., Hitchin, N.J. and Manin, Y.I.: Construction of instantons. Phys. Lett. 65A, 185–187 (1978) [AH] Atiyah, M.F. and Hitchin, N.J.: The Geometry and Dynamics of Magnetic Monopoles. Princeton, NJ: Princeton University Press, 1988 [AKL] Anderson, T., Kronheimer, P., Lebrun, Claude: Complete Ricci-flat K¨ahler manifolds of infinite topological type. Commun .Math. Phys 125, 637-642 (1989) [BD] Bielawski, R. and Dancer, A.: The Geometry and Topology of Toric hyperK¨ahler manifolds. Preprint [C] Calabi, E.: M´etriques K¨ahleriennes et fibr´es holomorphes. Ann. Ecol. Norm. Sup. 12, 269–294 (1979) [DK] Donaldson, S.K and Kronheimer, P.B: The Geometry of Four Manifolds. Oxford Mathematical Monographs, Oxford: Oxford Science Publications, 1990 [EH] Eguchi, T. and Hanson, A.: Asymptotically flat solutions to Euclidean gravity. Phys Lett. 74B, 249–251 (1978) [F] Fujiki, A.: Hyper-K¨ahler structure on the moduli space of flat bundles. Preprint [G1] Goto, R.: On toric hyper-K¨ahler manifolds given by the Hyper-K¨ahler quotient method. In: Infinite Analysis. Advanced Series in Mathematical Physics, Vol. 16 [G2] Goto, R.: On hyper-K¨ahler manifolds of type A∞ . Geom. Funct. Anal. 4, No. 4, 424–454 (1994) [GG] Gibbons, G.W.: The Sen conjecture for fundamental monopoles of distinct types. Phys. Lett. 382 B, 93 (1996) [GM] Gibbons, G.W. and Manton, N.S.: The modili space metric for well-separated BPS monopoles. Phys. Lett. 356 B, 32–38 (1995) [GRG] Gibbons, G.W., Rychenkova, P. and Goto, R.: Hyper-K¨ahler Quotient Construction of BPS Monopole Moduli Spaces. Commun. Math. Phys. 186, 581–599 (1997) [GH] Gibbons, G.W. and Hawking, S.W.: Gravitational multi-instantons. Phys Lett. 78B, number 4, 430–432 (1978) [H1] Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London. Math. Soc. 55, 59–126 (1987) [H2] Hitchin, N.J.: Metrics on moduli spaces. Proceedings of Lefshetz Centennial Conference, Mexico city 1984, Contemporary Mathematics 58, Part 1. Providence, R.I.: American Mathematical Society, 1986, pp. 157–178 [H3] Hitchin, N.J. Polygons and Gravitons. Math. Proc. Camb. phil. Soc. 85, 465–476 (1979) [H4] Hitchin, N.J.: On the construction of monopoles. Commun. Math. Phys. 89, 145–190 (1983) [HKLR] Hitchin, N.J., Karlhede, A., Lindstr¨om, U. and Roˇcek, M.: Hyper-K¨ahler metrics and supersymmetry. Commun. Math. Phys. 108, 535–589 (1987) [IS] Intriligator, K. and Seiberg, N.: Mirror Symmetry in Three Dimensional Gauge Theories. hepth/9607207
Hyper-K¨ahler Manifolds of Type A∞ and D∞ [K] [Kir] [KK] [K1] [K2] [K3] [KN] [LWY] [PP] [RW]
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Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Publications of the Mathematical Society of Japan 15, Iwanami Shoten Kirwan, F.: Cohomology of quotients in symplectic and algebraic geometry. Princeton: Princeton University Press, 1985 Kodaira, K.: On compact analytic surfaces: II. Ann. of Math. 77, 563–626 (1963) Kronheimer, P.B.: The construction of ALE spaces as hyper-K¨ahler quotients. J. Differ. Geom. 29, 665–683 (1989) Kronheimer, P.B.: A Torelli-type theorem for gravitational instantons. J. Differ. Geom. 29, 685–697 (1989) Kronheimer, P.B.: Instantons and the geometry of the nilpotent variety. J. Differ. Geom. 32, 473–490 (1990) Kronheimer, P.B. and Nakajima, H.: Yang–Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990) Lee, K., Weinberg, E.J. and Yi, P.: Electric magnetic duality and SU (3) monopoles. hep-th 9601097 Pedersen, H. and Poon, Y.S: Hyper-K¨ahler metrics and a Generalization of the Bogomolny Equation. Commun. Math. Phys. 117, 569–580 (1988) Rozansky, L. and Witten, E.: Hyper-K¨ahler Geometry and Invariant of Three manifolds. hepTh/9612216
Communicated by A. Connes
Commun. Math. Phys. 198, 493 – 534 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
The Large Field Renormalization Operation for Classical N -Vector Models Tadeusz Balaban Rutgers University, Department of Mathematics, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA Received: 2 February 1998 / Accepted: 12 February 1998
Abstract: A renormalization operation of the large field densities for the classical N vector models is constructed in this paper. It allows to improve bounds of these densities, and thus to prove all inductive hypotheses. This completes the construction and analysis of the full renormalization group flow for these models. The results will be used in the next paper to analyze the correlation functions. 1. Introduction In the previous papers [4–6] we have completed the analysis of the k th renormalization transformation S (k) T (k) applied to a density ρk satisfying all the inductive hypotheses (H.1)–(H.7) [4,5], and we have obtained a new density ρk+1 satisfying all the hypotheses for the index k + 1, with the possible exception of (H.6) [5]. This result is described in Theorem 1 [5,6]. Let us begin by making a few comments on it. The density ρk+1 has the representation X c c ρ0k+1 (Zk+1 , Ak+1 )χk+1 (Zk+1 ) exp Ak+1 (Zk+1 ) ρk+1 = Zk+1 ,Ak+1
c +Fk+1 (Zk+1 ) + R(k+1) (Zk+1 ) =
X
ρk+1 , (Zk+1 , Ak+1 ),
(1.1)
Zk+1 ,Ak+1
where the densities ρk+1 (Zk+1 , Ak+1 ) are defined by the last equality; the multi-indices Ak+1 and the “large field” densities ρ0k+1 (Zk+1 , Ak+1 ) have been defined inductively by (3.1)–(3.3) [6]. The first comment is that the “small field” parts of the above density, that is the characteristic and exponential functions in (1.1), satisfy all the inductive hypotheses concerning them, i.e. the hypotheses (H.2)–(H.4) [5] together with (H.2)– (H.7) [4]. These hypotheses control all relevant properties of the small field densities; in particular they allow one to perform and control the final integration in the last step
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of the procedure, and to obtain a final representation of the generating functional for correlation functions. This is the subject of Sect. 4 of this paper. The second comment is on the large field densities ρ0k+1 , and it is much more elaborate. The crucial property of these densities is their smallness. The basic idea behind this property is that the large field restrictions provide sufficiently small bounds to control all other bounds coming from integrals and all other expressions with localizations intersecting the large field regions. The estimates connected with the large field restrictions have been discussed in Sect 3. [6], see (3.7)–(3.9) [6], and they indeed provide such very small factors. This leads to the inductive assumptions on the densities formulated in (H.5)–(H.7) [5], and in particular to the bounds (1.26), (1.29)–(1.31) [5]. Let us explain the meaning and implications of these bounds, since they have not been discussed before. For example, we would like to integrate a density ρk with respect to the new variable ψk and we would like to estimate contributions to this integral coming from the large field densities. These contributions can be obtained by integrating the densities ρ0k (Zk , Ak ) with respect to ψk restricted to the large field regions Zk . Let us fix Zk and assume that it is a union of n connected components Xi , i = 1. . . . , n. From the above hypotheses and bounds we obtain n Y XZ dψk Zk ρ0k (Zk , Ak ) ≤ exp(−2p2 (βk ) − Hk |Xi |) · Ak
i=1
· sup χ exp |B (k) | ≤ Ak ,ψ,ψ 0
· exp
X
n Y
exp(−2p2 (βk ) − Hk |Xi |) ·
i=1
B0 exp(−κdk (Xmodck ))
(1.2)
X∈Dk (modck ):X∩Zk6 =∅,X∩Zkc6 =∅
≤
n Y
exp(−2p2 (βk ) − Hk |Xi | + B0 K0 M −d |Xi |)
i=1
≤
n Y i=1
[ 1 exp(−2p2 (βk ) − Hk |Xi |), Zk = Xi , 2 n
i=1
if we take into account that Hk = H0 log βk , and H0 can be taken arbitrarily large, for example H0 > 2B0 K0 M −d . Thus these contributions are indeed very small. Moreover the bounds have a form that is needed in constructing a convergent “polymer” expansion with small “activities”. Of course much more properties are needed for the expression on the left-hand side above to construct such a polymer expansion. For example we need the localization properties and the exponential decay properties of the terms in B (k) (Zk , Ak ), so that we can interpret this expression as a part of a partition function of a “gas” of weakly interacting polymers {Xi } with exponentially decaying “interaction potentials”. It is fairly simple. We will construct and analyze such expansions in Sect. 3, 4 of this paper, and the bound (1.2), or related bounds always provide a basis for convergence of the expansions. Let us repeat again that (1.2) is based on the inductive hypotheses and the bounds (1.26), (1.29)–(1.31) [5]. The hypotheses (H.5), (H.7), and the bounds (1.26), (1.29), (1.31) [5] are simple consequences of the constructions in [5,6], in particular the inductive definitions (3.1)–(3.3) [6], and the inequality (1.30) [5] and its strengthened form formulated in the hypothesis (H.6) [5], that is really crucial in the bound (1.2). This inequality is based on properties of the function κκ (Z, Ak ∩ Z). Let us consider the inductive definition (3.10), (3.9) [6] of this function. We have analyzed this definition
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in the rest of Sect. 3 [6], especially from the point of view of the inequality (1.30) [5]. Let us summarize the results of this analysis. We have noticed that the worst case in this inductive definition is the case when there are no new large field regions, because then the function is decreasing most rapidly. The new contribution κ(κ) (Ak+1 ∩ (Zkc ∩ Z)) defined in (3.9) [6] has only negative terms. More precisely it is equal to −(O(1) log βk + const.)|Zkc ∩Zk+1 |. Obviously, if this situation continues in a sufficiently large number of subsequent steps, the function will eventually become negative, so the inequality (1.30) [5] certainly will not hold. Then the inequality (1.2) does not hold also, and we lose any control over the convergence of the polymer expansions, therefore over a bound of the whole effective density. We cannot allow this to happen, so at some moment we have to change appropriately the procedure. The further analysis of Sect. 3 [6], and in particular the results formulated in Proposition 3.2 [6], show that we may lose the inequality (1.30) [5] only after a very large number of steps, so large that the initial large field region is shrunk by the scaling operations to a region of a very small size, and on all layers added to it by the procedure there are only small field restrictions. This allows us to stop the previously described procedure when we still have both properties, the inequality (1.30) [5], and the structure described above as the “dangerous” large field regions. We can now treat, or better attempt to treat, the corresponding part of the large field density in a way similar to the procedure of [5] leading to the fluctuation integrals. Thus at first to extract the small field expression from it, and then to construct the polymer expansion for the remainder, and to exponentiate it. This way we replace the large field density on those regions by a small field density with an additional term in the exponential. On the remaining large field regions we still have the inequality (1.30) [5] holding for some number of subsequent steps. After these preliminary comments and explanations let us describe now in a more precise way the idea sketched above. For simplicity of notation we replace the index “k + 1” by “k”, so we consider densities ρk in the image of the operation S (k−1) T (k−1) applied to densities in the space Rk−1 (β, α, λ, ν; B, ν), ¯ and we construct for them the large field renormalization operation R(k) . We start with a description of the structure and the main formal aspects of this operation. For the purpose of this description it is convenient to use the second representation in (1.1), i.e. to write X ρk (Zk , Ak ). (1.3) ρk = Zk ,Ak
The large field regions Zk are unions of some number of components Z, and we classify them according to values of the function K(Z, Ak ∩ Z). The components for which this function is positive are called “components of the first kind”, and those for which the function is equal to 0 are called “components of the second kind”. On components of the first kind we leave the corresponding large field densities unchanged. On components of the second kind we perform several operations, like introducing some new characteristic functions, new fluctuation variables, expanding the main actions in (3.2) [6], etc. The main idea behind these operations is to try to perform again a kind of fluctuation integral. Most of the next section is concerned with their definitions and properties. They transform the set of multi-indices in (1.3) into a new set of multi-indices denoted by A00k , and the densities on the right-hand side of (1.3) into new densities ρ00k . In particular they introduce new large field restrictions on some components, so they transform them into components of the first kind. Let us change the notation in (1.3) and denote by Yk the union of components of the first kind after the operations, and by Zk the union of components of the second kind. The whole density ρk is unchanged by the
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operations, and we represent it in the following transformed form: X ρ00k (Yk , Zk , A00 ). ρk =
(1.4)
Yk ,Zk ,A00 k
The small field parts of the densities in (1.3) are also unchanged; they are the same for the densities in the above representation. On the domain Zk we would like to perform the same operations as in Sect. 2 [5], namely to extract the corresponding small field action, and to exponentiate the sum of remaining large field expressions. Unfortunately there are serious obstacles against doing it directly for (1.4), in particular it seems impossible to control any restrictions on variables ψk inside the domain Zk . This difficulty may be overcome by an additional integral operation which introduces the necessary restrictions. To define such an operation we introduce the following definition. For a component Z of Zk , which is a rectangular parallelepiped by the remark at the end of Sect. 3 [6], we define in a unique way its subdomain Z0 , and a characteristic function χZ0 giving some restrictions on the variable ψk on Z0 ; the definitions will be given later on. We define the densities (1.5) ρk,0 (Z) = χZ0 exp −βk A∗k (Z; ψk , φk (Bk (U ); ψ˜ k )) , where U = Z ≈ . Finally, we define the operation R(k) by the formula R X dψk Zk,0 ρ00k (Yk , Zk , A00k ) (k) , ρk,0 (Zk ) R R ρk = dψk Zk,0 ρk,0 (Zk ) 00
(1.6)
Yk ,Zk ,Ak
where Zk,0 is the union of the subdomains Z0 over components Z of Zk , and ρk,0 (Zk ) is the product of ρk,0 (Z) over the components. Let us make a few comments on this operation. It is based on the representation (1.4), which is uniquely defined starting from (1.3), so R(k) is uniquely defined also. It changes the density ρk on the domain Zk only, in particular it does not change the small field part of the density on (Yk ∪ Zk )c . It introduces some small field restrictions on the variable ψk on Zk,0 through the characteristic functions χZ0 , which is its main purpose, and it satisfies the fundamental normalization property Z dψk R(k) ρk = R Z X Z dψk Zk,0 ρ00k (Yk , Zk , A00k ) R c = = dψk Zk,0 dψk Zk,0 ρk,0 (Zk ) dψk Zk,0 ρk,0 (Zk ) Yk ,Zk ,A00 k Z Z X Z c = dψk Zk,0 dψk Zk,0 ρ00k (Yk , Zk , A00k ) = dψk ρk . Yk ,Zk ,A00 k
(1.7) The definition (1.6) is quite simple, but to write the result of this operation as a density satisfying all the inductive hypotheses is quite a difficult problem. It requires introducing several auxiliary operations and connected with them long and rather tedious considerations. Most important operations are of the same type as the ones connected with introducing fluctuation integrals in Sect. 2 [5]. In particular we have to extract expressions determined by the main actions in the numerator and denominator in (1.6)
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in order to see their cancellation. This cancellation is a crucial problem in the analysis of (1.6), and most operations and considerations are centered around it. Another important problem is analyticity properties of the expressions in (1.6), in particular the expressions localized in components of Zk . These expressions are exponentiated and become a part of the function R(k) m , so the analyticity properties are essential for future renormalizations and localizations, yet they are difficult to obtain, mainly because of the non-linear and non-local restrictions introduced by the characteristic functions. This leads to quite elaborate constructions and estimates. Final results of this analysis can be formulated in the following theorem. Theorem 2. Under the assumptions of Theorem 1 [5,6], and for k < k0 , there exists a transformation R(k) defined on the densities ρk = S (k−1) T (k−1) ρk−1 in the image of S (k−1) T (k−1) considered on the space (1.42) [5] for the index k − 1 instead of k, defined in a general way by (1.3) - (1.6), and such that R(k) ρk satisfies all the inductive hypotheses (H.1)–(H.7) [5], i.e. it belongs to the space (1.42) [5]. This transformation satisfies the basic normalization property (1.7), i.e. Z Z (k) (1.8) dψk R ρk = dψk ρk , and its general effect is to remove the terms from the sum in (1.3), for which the large field regions Zk have components on which the hypothesis (H.6) [5] is not satisfied, i.e. the components Z for which K(Z, Ak ∩ Z) = 0. It satisfies also a much stronger normalization property, it determines uniquely a transformation R(k) of the extended space (1.6) [4] of the effective actions and generating functionals, such that R(k) is equal to the identity transformation on {E (j) }j≤k , {M(j) }j≤k , {R(k) n }n<m , and it generates a (k) new function R00(k) (ψ , g) which is added to R (ψ , g). This new function satisfies the k k m m (ψ , g), and the sum of the two satisfies all the properties and same properties as R(k) k m bounds required by the hypothesis (H.4) [5]. For k ≥ k0 we take R(k) as the identity operator, so R00(k) m = 0 in these cases. The above theorem is a reformulation of Theorem 2 [5]; the only essential change is the change of the index “k + 1” there by “k”. Combining Theorem 1 [5,6], Theorem 2 [5, and above], and Theorem 4 [6], we obtain the following description of the renormalization group flow. Theorem 5. Under the assumptions of Theorem 1 [5,6], and for k < k0 , the transformation R(k+1) S (k) T (k) maps the space (1.42) [5] into the space determined by ¯ 2 , with the remaining constants unchanged. The transformation k + 1, BLd−2 , νL R(k+1) S (k) T (k) satisfies all the conclusions of Theorem 1 [4,5,6]. For k ≥ k0 the transformations applied to an initial density ρk0 in the corresponding space (1.42) [5] yield the densities given by (1.1) [6] and satisfying all the inductive hypotheses (H.1)–(H.5), (H.7) [5], with the integral operators Tk (Z, Ak ∩ Z) satisfying the inequality (1.31) [5] with ` = 1, as long as νk ≤ 1. Thus the densities of the whole renormalization group flow {ρk }, obtained by applying the renormalization transformations to the initial density ρ0 defined by (1.1) [1], satisfy all the above conditions with the appropriate constants, e.g. ¯ 2k , with B sufficiently with the constants β = β0 , a = 1, λ = λ0 , ν = ν0 , BLk(d−2) , νL large and ν¯ sufficiently small. We will prove Theorem 2 in the following two sections. In the next section we introduce in detail the representation (1.4), we study the integrals in (1.6) and the cancellation problem. In the third section we construct the polymer expansion, its exponentiation, we
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discuss properties of the obtained expressions, the inductive hypotheses, and we finish the proof of Theorem 2. In the last section we discuss the last step of the whole procedure, which is the final integration of the last density ρkf in the flow with respect to the variable ψkf . The definition of the index kf is recalled at the beginning of that section, and a discussion of the results of this operation is given at the end of the section. A more detailed description is given in the following paper [7] on properties of correlation functions, written together with M. O’Carroll. 2. The Detailed Definition of R(k) and the Cancellation Problem In this section we consider only expressions and operations connected with one component Z of Zk , a component of the second kind. In order to avoid writing the intersections Zj ∩ Z, Ak ∩ Z, etc. every time, we assume that Zk = Z, so that these intersections are equal to Zj , Ak , etc. We start with some geometric constructions, in particular with a definition of Z0 . Let us recall that for the component Z all conditions of the statement (ii) in Proposition 3.2 [6] are satisfied. Hence the domains Zj for k1 ≤ j ≤ k are rectangular parallelepipeds with diameters in the L−j -scales smaller than the critical sizes 12M [Rj−1 ]. Also, there are no large field regions Pj+1 , Qj+1 , Rj+1 , Sj+1 , so 0j+1 = j+1 = 00j+1 , and the equalities and inclusions (2.4), (2.9), (2.46) [5], (2.5) [6] take on a much simpler form. From (2.4) [5] 0∼4+[Rj ]+ L−1 2
we have Uj cj+1 , and
∼([Rj−1 ]+1+4L+L[Rj ]+ L−1 2 L)
⊂ cj+1 , where Uj = Zj≈ , hence Zj
distj (Zj , j+1 ) > 2(L + 1)M + M [Rj−1 ] + LM [Rj ], distk (Zj , j+1 ) > L−(k−j) LM [Rj ].
⊂
(2.1)
The sequence L−(k−j) LM [Rj ] is decreasing with j decreasing, so there is a smallest index k2 such that the corresponding term is still ≥ 1. Then L−(k−k2 ) LM [Rk2 ] < 98 L, and putting n2 = k − k2 we have diamk (Zk2 ) < L−n2 12M [Rk2 −1 ] < 12 ·
9 27 = , distk (Zk2 , k2 +1 ) > 1. 8 2
(2.2)
We define Z0 as the smallest rectangular parallelepiped which is a union of the unit cubes with centers in T1(k) , and which contains Zk2 , i.e. [ Z0 = {1k (y) : y ∈ T1(k) , 1k (y) ∩ Zk2 6 = ∅}. (2.3) From this definition, (2.1), (2.2) we obtain diamk (Z0 ) < 15, Zk2 ⊂ Z0 ⊂ ck2 +1 , distk2 (Z0 , ck2 +1 ) > 2(L + 1)M + M [Rk2 −1 ], distk2 (Z0c , ck2 ) ≥ distk2 (Zkc2 , ck2 ) ≥ 6M + 2M [Rk2 −1 ].
(2.4)
We introduce also an auxiliary domain Z000 by taking a smallest index k3 such that L−(k−k3 ) L[Rk3 ] ≥ 1. Then L−n3 L[Rk3 ] < 98 L, diamk (Zk3 ) < 12 · 98 M = 27 2 M , and we define [ (2.5) Z000 = { ∈ πk : ∩ Zk3 6 = ∅}, hence diamk (Z000 ) < 15M. We could introduce the domain Z000 only and use it in the definition (1.6), but then we would obtain worse bounds. This does not really matter of course, but the introduction
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and use of the two domains is no more complicated. It will even have some notational advantages. Our first goal is to determine the representation (1.4). It is done in several steps. In each step we start with some operation changing the form of the main action in the large field density, and then we introduce some new characteristic functions. All this is done for the integral operator Tk (Z, Ak ), so from now on we consider this operator only. This whole section is concerned basically with various transformations and decompositions of this operator. Using the conclusions discussed above on Ak , and the inductive definitions (2.47) [5], (3.2), (3.3) [6] simplified in this case, we obtain that the operator has the following structure: Tk (Z, Ak ) =
k1 Y
(j) c 0 ∼3 c ∼3 χj+1 (∼−3 j+1 ∩ Zj+1 )T (j+1 )χj+1 (j+1 ∩ (j+1 ) )
j=k−1
·
χj (Zjc
−
∩
Ej |Zjc
c (∼3 j+1 ) ) exp
∩
cj+1 |ξ
−
˜ −βj+1 A∗j+1 (Zjc ∩ Zj+1 , Bk ; ψ, φj+1 (Bj+1 (∂j+1 ); ψ))
Ej000 |j+1
∩ Zj+1 |ξ
X k1
Tk1 (Zk1 , Ak1 ) exp
B
(j)
(Zjc
∩ Zj+1 ) .
j=k−1
(2.6) Consider the operator given by the product of the first k − k1 operators in the curly brackets above. The constants in the exponentials are unimportant, so we separate them from these operators and combine them with corresponding boundary terms in the last exponential. The remaining product is denoted by Tk,k1 (Z, Zk1 ), or simply by Tk,k1 , and it is uniquely determined by Zk1 and k. We move all characteristic functions and exponentials to the right in the product, and the operator begins with the product of the renormalization transformations T (j) (cj+1 ). Let us recall now the discussion of the renormalization conditions (1.36) [4], and the choice of the constants a defining the transformations, leading to the equalities (1.37) [4]. These are exactly the equalities we need to use the composition formulas in Sect. 4 [2]. The composition of the above transformations yields the transformation described in Lemma 4.1 [2] and corresponding to the one generating set equal to (Bk ∩ Zkc1 ) ∪ (T (k1 ) ∩ Zk1 ), and another, “coarser” one, equal to T1(k) , i.e. the composition is equal to the transformation T (T1(k) , (Bk ∩ Zkc1 ) ∪ (T (k1 ) ∩ Zk1 )). We factorize it into a product of two transformations localized correspondingly in Z0 , Z0c ∩ ck , and we obtain the following, a bit awkward formula: Tk,k1 (Z, Zk1 ) = T (T1(k) ∩ Z0 , (Bk ∩ Zkc1 ∩ Z0 ) ∪ (T (k1 ) ∩ Zk1 )) · k1 Y (k) c c c χj+1 (∼−3 · T (T1 ∩ Z0 ∩ k , Bk ∩ Z0 ∩ k ) j+1 ∩ Zj+1 ) · j=k−1
·
χ0j+1 (∼3 j+1
∩
(cj+1 )∼3 )χj (Zjc
∩
c (∼3 j+1 ) )
(2.7)
·
X k−1 ∗ c ˜ βj+1 Aj+1 (Zj ∩ Zj+1 , Bk ; ψ, φj+1 (Bj+1 (∂j+1 ); ψ)) . · exp − j=k1
For later applications let us notice that the integration (2.6) acts on the first operator above, and it yields
R
dψk Z0 applied to the operator
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Z dψk Z0 T (T1(k) ∩ Z0 , (Bk ∩ Zk1 ∩ Z0 ) ∪ (T (k1 ) ∩ Zk1 ))· = Z Z dψ B∩Zkc ∩Z0 ·. = dψk1 Zk1
(2.8)
1
Let us recall also that the second operator on the right-hand side of (2.7) has the kernel X 1 2 (2.9) βk a(1k (z), Bk )|ψk (z) − Q(1k (z), Bk )ψ| , const. exp − 2 (k) c c z∈T1 ∩Z0 ∩k
where the averaging operation and the constant are defined by the formulas (4.6), (4.7) [2]. Now we would like to transform the sum of the main actions in (2.7) into one main action on the domain Zkc1 ∩ Zk . In order to do this we have to perform at first several preliminary operations. A j th term in the sum is determined by the coefficients βj+1 , aj+1 , λj+1 , νj+1 obtained after j + 1 steps, and we need to replace them by the corresponding coefficients determined by the renormalized coefficients obtained after k steps, i.e. by 1 − L−2k , λj+1 = λk (Lj+1 η)2 , νj+1 = νk (Lj η)2 . 1 − L2(j+1) (2.10) This means that we have to renormalize the actions in (2.7). In order to use the previous results we do it in k − j − 1 steps for the j th term. In the first step we transform this term into the term determined by the coefficients (2.10) with k, η replaced by j + 2, L−(j+2) , applying the formula (2.31) [6] with k + 1 replaced by j + 1. We continue these steps until we reach the term with the coefficients (2.10), each time applying the formula (2.31) [6]. Notice that we do not renormalize the spin variables. We have discussed in Sects. 4, 5 [4], and in Sect. 2 [6], bounds for the corresponding functions C (j+1) and the surface terms. We conclude from this discussion that in the step in which we obtain the coefficients (2.10) with k, η replaced by n, L−n , they can be bounded by βj+1 = βk (Lj+1 η)d−2 , aj+1 = ak
const.βj+1 ε2j+1
1 |Z c ∩ Zj+1 |ξ ≤ const.L−2γ(n−j−1) |Zjc ∩ Zj+1 |ξ . βn ε2n j
(2.11)
We have assumed here that the configuration ψ is in the space (1.32) [5] localized to the domain Wj ∩ Uj+1 . Combining the above formulas we obtain ˜ = βj+1 A∗j+1 (Zjc ∩ Zj+1 , Bk ; ψ, φj+1 (Bj+1 (∂j+1 ); ψ)) ˜ − C (j+1) (Zjc ∩ Zj+1 , Bk ; ψ), = βk A∗k (Zjc ∩ Zj+1 , Bk ; ψ, φk (B(∂j+1 ); ψ))
(2.12)
where the last function is the sum of all the previous functions and surface terms. It can be bounded by the sum of the bounds (2.11), thus by const.|Zjc ∩ Zj+1 |ξ . Notice that the main action on the right-hand side above is in the η-scale. Now we would like to replace the background function in this action by a function common for all terms in the exponential in (2.7). At first we have to define a proper generating set. We take the following simple generalization of definition (2.15) [6] of the sets Bj+1 (∂j+1 ), Bk1 ,k = (Bk1 (Wk1 ) ∩ Zk1 ) ∪ (Bk ∩ Zkc1 ∩ Zk ) ∪ (Bk (Uk ) ∩ Zkc ∩ Uk∼ ).
(2.13)
This definition means simply that we take the set Bk on Wk1 ∩ Uk , and complete it by the minimal generating sets in a neighbourhood of the boundaries. We take the corresponding ˜ and the expansion function φk1 ,k = φk (Bk1 ,k ; ψ),
Large Field Renormalization for Classical N -Vector Models
˜ = φk (Bj+1 (∂j+1 ); ψ(Bj+1 (∂j+1 ), Bk1 ,k ; ψ)) ˜ φk (Bj+1 (∂j+1 ); ψ) ˜ = φk1 ,k + δφk (∂j+1 ). + δφk (Bj+1 (∂j+1 ); ψ˜ − ψ(Bj+1 (∂j+1 ), Bk1 ,k ; ψ))
501
(2.14)
The operations we are discussing now are inverse to the ones done in Sect. 2 [6], in the paragraph including (2.11)–(2.17) [6]. In particular the function δφk (∂j+1 ) satisfies the bound (2.13) [6]. For bounds it is convenient to rescale the main action in (2.12) back to the L−1 ξ-scale. We apply again formula (2.14) [6], with the obvious modifications, and the sum of the last three terms on the right-hand side is very small, it can be bounded by const.exp(−Rj−1 )|Zjc ∩ Zj+1 |ξ . We denote by Ck1 ,k (Zkc1 ∩ Zk , Bk ; ψ), or simply Ck1 ,k (ψ), the sum of these terms and the functions C (j+1) in (2.12) over all j 0 s in (2.7). It is an analytic function of ψ defined on the space (1.32) [5] with X = Wk1 ∩ Uk , and depending on ψ restricted to Wk∼1 ∩ Uk∼ . On this space it satisfies the bound |Ck1 ,k (Zkc1 ∩ Zk , Bk ; ψ)| < const.
k−1 X
|Zjc ∩ Zj+1 |ξ < const.M d R0d (log βk−1 a)2d+1 ,
j=k1
because |Zj+1 |ξ < (12LM [Rj ])d and k − 1 < log βk−1 a. We have transformed the exponential in (2.7) into the form ∗ c c exp −βk Ak (Zk1 ∩ Zk , Bk ; ψ, φk1 ,k ) + Ck1 ,k (Zk1 ∩ Zk , Bk ; ψ) ,
(2.15)
(2.16)
which is a basis for further constructions and bounds. Now we are interested in bounds we can obtain for the sum of the quadratic form in (2.9) and the main action above restricted to the domain Z0c ∩ Zk . Using again the composition formulas from Sect. 4 [2] we have X 1 a(1k (z), Bk )|ψk (z) − Q(1k (z), Bk )ψ|2 2 (2.17) z∈T1(k) ∩Z0 ∩ck + A∗k (Z0c ∩ Zk , Bk ; ψ, φk1 ,k ) ≥ A∗k (Z0c ∩ Zk ; ψk , φk1 ,k ), where the action on the right-hand side is defined “homogeneously” on the whole domain Z0c ∩Zk , i.e. assuming that the generating set is equal to T1(k) . To this action we can apply Proposition 3.1 [6], and we obtain the inequality (3.5) [6], or (3.6) [6], in this case. They allow us to introduce new restrictions on the variables ψk on Z0c ∩ Zk . We have already ∩ Zk ). Now we introduce the function χ˜ k (Z0c ∩ (ck )∼3 ), whose the functions χk (∼−3 k ˜ domain is the space 9(Z0c ∩ (ck )∼3 ; 3δk ), or rather we introduce the decomposition of unity (2.18) 1 = χ˜ k (Z0c ∩ (ck )∼3 ) + χ˜ ck (Z0c ∩ (ck )∼3 ). ˜ k (X; δ), where X is a union of the unit cubes and δ > 0, Let us recall that the space 9 in the case k ≤ k0 considered here, is defined by the conditions |(∂ 1 ψk )(b)| < δ for unit bonds b ⊂ X, and ||ψk (z)| − 1| < δ for points z of the unit lattice, z ∈ X. The decomposition of unity introduced into the operator (2.6), or rather (2.7), yields two operators. The one with the function χ˜ ck involves some new large field restrictions, so in the bound we obtain some new small factors. Indeed, we have either one of the derivatives (∂ 1 ψk )(b) large, i.e. |(∂ 1 ψk )(b)| ≥ 3δk , and then the inequality (3.5) [6] yields the small factor exp(− 29 γ0 p20 (βk )), or all the derivatives are small, and then there exists a point z in the domain such that ||ψk (z)| − 1| ≥ 3δk . Using the fact that the domains
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T. Balaban
Z0 , (ck )∼3 are rectangular parallelepipeds we can choose a point z0 ∈ ∂(∼−3 ∩ T1(k) ) k such that the segment [z, z0 ] is contained in the domain Z0c ∩ (ck )∼3 , except the last bond. A length of this segment is certainly smaller than 6M [Rk−1 ], and the derivatives of ψk are smaller than 3δk on bonds of this segment, also on the last bond, because ∩ Zk ). They imply also of the restrictions introduced by the functions in χk (∼−3 k that ||ψk (z0 )| − 1| < 3δk , and we have |ψk (z) − ψk (z0 )| < 6M [Rk−1 ] · 3δk , hence −1
||ψk (z)| − 1| < 18M Rk δk = βk 2 18M A0 R0 (log βk a)p0 +2 < c5 for βk large enough. This reasoning can be applied to any point z in the domain, so the assumptions of the second part of Proposition 3.1 [6] are satisfied and we obtain the inequality (3.6) [6] in the considered case. This inequality and the assumption ||ψk (z)| − 1| ≥ 3δk yield the small factor exp(− 29 γ0 p20 (βk )) again. Thus the characteristic function χ˜ ck (Z0c ∩ (ck )∼3 ) introduced into the integral operator yields this small factor, times a combinatoric factor, which can be taken as equal to exp O(1)|Z0c ∩ (ck )∼3 |. We have also the exponential factor with the same bound as on the right-hand side of (2.15), and the three factors combined can be bounded by exp(−4γ0 p20 (βk )) for 2p0 > 2d + 1 and A0 large enough. Let us make now the following important remark. Inspecting the way we have obtained the bound (3.9) [6], and the definition of the function κk , it is easy to see that the new integral operator, with the additional large field characteristic function from the decomposition (2.18), can be estimated by the old bound in (1.26) [5] modified by the above additional small factor. The new bound exp(−κk (Z, Ak ∩ Z) − 4γ0 p20 (βk )) controls obviously a very large number of additional steps, hence the component Z becomes a component of the first kind. More precisely we modify the multi-index Ak introducing into it the new large field region, and the component Z for the modified multi-index becomes a component of the first kind. The modified multi-index becomes one of the new multi-indices A00k . Similar remarks should be made for other operations discussed below, but usually we will omit them, discussing only how to get some new small factors. For the integral operator (2.6), or (2.7), with the new small field characteristic function from (2.18) we introduce a second decomposition of unity of the form (2.1) [6], replaced by but with the index “k” instead of “k + 1”, and with the domain 00∼−3 k+1 Z000c ∩ (ck )∼3 . We write the last sum in (2.1) [6] as a sum over Sk00 ⊂ Z000c ∩ (ck )∼3 , so we obtain the corresponding sum of the integral operators, and in their bounds we obtain the additional factors exp(− 2·6γd0K 2 p20 (βk ) M1 d |Sk00 |), the same as in (3.8) [6]. For 1 the operators with non-empty regions Sk00 the component Z becomes a component of the first kind, with properly modified multi-indices including the new large field regions. We obtain one operator with the small field characteristic functions, for which Z is still a component of the second kind. Notice that the newly introduced functions yield stronger restrictions than the one in (2.18), by Lemma 3.1 [1], so we can write their product in the form χ˜ k (Z0c ∩Z000 )χk (Z000c ∩Zk ), where we have included also the previous characteristic ∩ Zk . functions on ∼−3 k To introduce next characteristic functions we need a “reference vector” in the spin space RN . We take it as a normalized average of the spin configuration ψk on the boundary ∂(Z0c ∩ T1(k) ), thus ˜ ˜ 0 = ψ , ψ˜ = ψ0 = (ψ) ˜ |ψ|
X |∂(Z0c z∈∂(Z0c ∩T1(k) )
1 ∩ T1(k) )|
ψk (z).
(2.19)
Large Field Renormalization for Classical N -Vector Models
503
For two arbitrary points z 0 , z 00 at the boundary we have |ψk (z 0 ) − ψk (z 00 )| < ˜ < 90(d − 1)δk , 1 < |ψ| ˜ < 3 for δk 30(d − 1)3δk = 90(d − 1)δk , hence |ψk (z) − ψ| 2 2 2 ˜ 2 2 5 1 2 ˜ small enough, ||ψ| − 1| < 3 ||ψ| − 1| < 3 ( 2 · 3δk + 2 (90(d − 1)δk ) ) < 6δk , and |ψk (z) − ψ0 | < 90(d − 1)δk + 6δk < 90dδk for z ∈ ∂(Z0c ∩ T1(k) ).
(2.20)
The characteristic functions in (2.7) depend in a complicated way on the “new” variables ψk and the integration variables ψ. As in the case of the fluctuation integrals we would like to separate these variables and to remove the dependence on ψk from the characteristic functions. In this case it is not possible to do it completely, but we can replace the dependence on ψk by a dependence on one vector “representing” it. We choose ψ0 as such a vector. This replacement is achieved in two steps, by introducing , two kinds of characteristic functions. A first one is introduced on the domain Ukc1 ∩ Uk∼6 1 where we consider Uk1 as a domain from Dk1 , and the operation “∼” is defined in terms of cubes of the partition πk1 . We define a configuration ψk0 1 by the equalities ψk0 1 = ψk1 on 3k1 ∩Uk∼3 , ψk0 1 = ψ0 on 3k1 ∩(Uk∼3 )c , and we introduce new characteristic 1 1 0 functions χk , 1 () defined in the same way as χk1 (), except that the constant δk1 1 2
in the bounds is replaced by 21 δk1 , and the configuration ψk1 by ψk0 1 . The functions are introduced by the usual decomposition of unity, which can be written as a sum 0,c 00 of the functions χ0k , 1 (Ukc1 ∩ Uk∼6 ∩ Q00c over regions Q00k1 ⊂ Ukc1 ∩ Uk∼6 k1 )χk , 1 (Qk1 ). 1 1 1 2
1 2
Notice that these characteristic functions depend on the configuration ψk0 1 restricted to , which is contained in k1 ∩ ck1 +1 . Consider a function the domain (Ukc1 )∼3 ∩ Uk∼9 1 () for ⊂ Q00k1 . We have to show that a bound of the exponential (2.16) yields χ0,c k1 , 21 a proper small factor for it. This is now more difficult than in the previous cases, and we have to consider it carefully. At first we notice that at least one of the inequalities |(∂ 1 ψk0 1 )(b)| ≥ K1−1 21 δk1 , ||ψk0 1 (y)| − 1| ≥ K1−1 21 δk−1 must hold at a bond or point of ∼3 ∩ 3k1 , because otherwise we would have χ0k , 1 () = 1. This means that we must 1 2 have at least one of the inequalities 1 1 ∩ 3k1 , |(∂ 1 ψk1 )(b)| ≥ K1−1 δk1 , ||ψk1 (y)| − 1| ≥ K1−1 δk1 for b or y in ∼3 ∩ Uk∼3 1 2 2 1 ∩ T (k1 ) ). |ψk1 (y) − ψ0 | ≥ K1−1 δk1 for y in the boundary ∼3 ∩ ∂(Uk∼3 1 2 (2.21) If one of the first two inequalities above holds, then the inequality (3.6) [6] applied to the main action in (2.16) yields immediately the small factor exp(− 18 γ0 K1−2 p0 (βk1 )). The difficult part is to obtain a small factor if the third inequality holds. We obtain it by a reasoning which will be applied a couple of more times later on, so we describe it in detail here for future reference. We start with the remark that the configurations ψ in the domain of integration in (6.7) satisfy bounds |∂ 1 ψj | < cδj on 3j , |ψj+1 (b+ ) − (Qψj )(b− )| < cδj+1 for b ⊂ T (j+1) , b+ ∈ 3j+1 , b− ∈ 3j , hence |ψj+1 (b+ ) − ψj (y)| < cδj+1 + d(L − 1)cδj < dLcδj for y ∈ B(b− ), j = k1,..., k − 1, (2.22) if c is sufficiently large, for example if c = 3dL by (1.24) [5]. If we take a small c > 0, then the above bounds are not assured by the characteristic functions in (2.7), but if
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T. Balaban
one of them is not satisfied, then the inequality (3.5) [6] applied to the main action in (2.16) yields a small factor exp(− 21 γ0 c2 p20 (βj )) ≤ exp(− 21 γ0 c2 p20 (βk1 )). Again, from the equalities and inclusions (2.4), (2.9), (2.46) [5], (2.5) [6] we obtain cj ⊂ cj+1 ⊂ (cj )∼6+3[Rj−1 ]+L[Rj ]+5L+ 2 (L 1
2
−1)
⊂ (cj )∼3L[Rj ] ,
hence distj (cj , j+1 ) < 3LM Rj . Take now a point y for which the third inequality in (2.21) holds, and denote by yk0 1 a point on the boundary ∂(ck1 +1 ∩ T (k1 ) ) which is closest to y. The points y, yk0 1 are connected by a segment of a length < 3LM Rk1 . The point yk0 1 determines a block to which it belongs, and we take a point yk1 +1 on the boundary ∂3k1 +1 , which is a nearest neighbour to the center of the block. By (2.22) we have |ψk1 (y) − ψk1 (yk0 1 )| < 3LM Rk1 cδk1 , |ψk1 (yk0 1 ) − ψk1 +1 (yk1 +1 )| < dLcδk1 .
(2.23)
We continue this construction until we reach the boundary ∂3k , obtaining a sequence 0 , yk on the corresponding of points yk1 +1 , yk0 1 +1 , yk1 +2 , . . . , yj , yj0 , yj+1 , . . . , yk−1 , yk−1 boundaries, for which the following inequalities hold: |ψj (yj ) − ψj (yj0 )| < 3LM Rj cδj , |ψj (yj0 ) − ψj+1 (yj+1 )| < dLcδj
(2.24)
for j = k1 + 1, . . . , k − 1. We would like to replace them by ones written in terms of the constants on the L−k1 -scale. There is the following general inequality: 1 (d − 2) log L δk (k − j))p0 = L− 2 (d−2)(k−j) (1 + δj log βj a 2p
− 21 (d−2)(k−j)(1− log β0 a ) j
− 13 (k−j)
(2.25)
,
which holds for βj large enough, e.g. βj a > e6p0 . We may take also arbitrary p instead of p0 in the definition of the constants δj . From this and the inequalities (2.23), (2.24) we obtain |ψk1 (y) − ψk (yk )| <
k−1 X
(3LM Rj cδj + dLcδj ) < 4LM c
j=k1
< 4LM Rk1 δk1 c
k−1 X
L
k−1 X
Rj δ j
j=k1
− 13 (j−k1 )
(2.26)
< 16LM Rk1 δk1 c,
j=k1
where we have used the bound L− 3 < 43 . Now take a point z0 on the boundary ∂(Z0c ∩ T1(k) ) which is closest to yk . It is connected with yk by a segment of a length < 6M Rk+1 contained in Z0c ∩ Zk , and by the introduced characteristic functions we have 1
|ψk (yk ) − ψk (z0 )| < 6M Rk−1 · 3δk < 18M Rk δk < 18M Rk1 δk1 L− 3 n1 . 1
(2.27)
Finally by (2.20) we have |ψk (z0 ) − ψ0 | < 90dδk < 90dδk1 L− 3 n1 . 1
Combining the bounds (2.26), (2.27), (2.28) we obtain
(2.28)
Large Field Renormalization for Classical N -Vector Models 1 1 |ψk1 (y) − ψ0 | < K1−1 δk1 (32K1 LM Rk1 c + 36K1 M Rk1 L− 3 n1 2 1 1 + 180dK1 L− 3 n1 ) ≤ K1−1 δk1 , if 2
505
(2.29)
1 , or c = (64K1 LM Rk1 )−1 , and 2 (2.30) 1 1 1 1 1 36K1 M Rk1 L− 3 n1 ≤ , 180dK1 L− 3 n1 ≤ or L 3 n1 ≥ 144K1 M Rk1 . 4 4 We shall assume later a stronger condition on n1 than the above ones, so we assume now that they are satisfied. Then the inequalities (2.29) hold if we take c given by (2.30), but this gives a contradiction with the third inequality in (2.21). This means that at least one of the inequalities (2.22) is not satisfied, and by the remark after (2.22) this yields the small factor exp(− 21 γ0 (64K1 LM Rk1 )−2 p20 (βk1 )). It is the largest small factor obtained in the above bounds. We need to express all the factors in terms of βk , so we have to make a more precise assumption on k1 , or rather n1 . Until now we have not specified the index k1 . We know that there exists the index satisfying the statements (i),(ii) of Proposition 3.2 [6]. Now we take in a sense the largest index satisfying these statements, more precisely we take the smallest n1 such that 32K1 LM Rk1 c =
n1 log L >
1 1 log βk1 a, hence n1 log L ≤ log βk1 a + log L. α α
(2.31)
With this choice of n1 , k1 we can express the above factors in terms of βk using the inequality p p log βk1 a log βk1 a = log βk a log βk1 a + n1 (d − 2) log L (2.32) p α log βk1 a α p ≥ > . (d − 1) log βk1 a + (d − 2) log L d We can estimate finally the term corresponding to a region Q00k1 in the discussed decomposition by the usual bound multiplied by the additional factor γ0 α2p0 −4 A20 1 (log βk a)2p0 −4 d |Q00k1 |k1 exp − d 2p 2 · 6 d 0 −4 (64K1 LM R0 )2 M (2.33) 1 + d |(Zkc1 )∼3 ∩ Zk∼3 | , 1 k1 M where we have included also the combinatoric factor controlling the sum over Q00k1 . For the terms of the decomposition with non-empty regions Q00k1 we simplify the above factor by estimating the first rescaled volume from below by 1, and the second from above 12d d−1 d−1 < 12d (log βk a)2d−2 . Assuming 2p0 − 4 ≥ 2d − 2 by M d (15M Rk1 ) M (15R0 ) 2 and A0 large enough we estimate the factor (2.33) by exp(−A03 (log βk a)2p0 −4 ), where
γ0 α2p0 −4 A20 . This gives the strongest condition on A0 up to now. We have 4·6d d2p0 −4 (64K1 LM R0 )2 3d6d d2p0 −4 2 to assume that A0 ≥ γo α2p0 −4 M (16K1 LM )2 (15R0 )d+1 . Assuming that 2p0 −4 ≥ 2d+2, and that the constant A03 is still large enough, we can control with this additional factor a large number of additional steps, certainly larger than log βk a. Thus we include the component Z with non-empty regions Q00k1 into the class of components of the first kind.
A03 =
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T. Balaban
The only term for which Z is still a component of the second kind is the term with the ). We consider this term only from now on. characteristic function χ0k , 1 (Ukc1 ∩ Uk∼6 1 1 2 Our basic goal is to represent the integral operator (2.7) in a form analogous to the representation (2.48) [5], that is to introduce proper background configuration and fluctuation variables, to separate the exponential with the main action calculated at the background configuration, and to write the remaining integral operator in terms of the fluctuation variables as an analytic function of the background configuration. This means that we have to introduce also additional restrictions on the fluctuation variables. A natural candidate for the background configuration would be φk (Bk (Uk ); ψ˜ k ). Unfortunately we cannot use it because we do not have any restrictions on ψk considered on the domain Z0 . In fact we cannot obtain any new restrictions on this domain, or rather on Zk1 , because the main actions localized on it could have been already “used” to bound previous large field characteristic functions. A way out of this difficulty is to extend the configuration ψk restricted to Z0c ∩Uk∼ , in some “minimal” way, onto the domain Z0 . The corresponding variational problem is suggested by the expression in the denominator on the right-hand side of (1.6). This expression yields also the exponential with the main action calculated at some background configuration, and the two exponentials, from the numerator and the denominator in (1.6), must cancel, so we have to choose the same background configuration for both of them. A natural background configuration for the integral in the denominator in (1.6) is given by a solution of the following variational problem: ˜ νk = 0). (2.34) inf Ak (Bk (Uk ); ψ˜ k , φk (Bk (Uk ); ψ), ψk Z0
This problem is different from the ones studied in the paper [2]. It is closest to the variational problem for the effective main action in (2.16) [5], but it does not involve the additional quadratic form there, so there is no “effective mass” term. Fortunately, the domain Z0 is quite small, diamk (Z0 ) < 15 by (2.4), hence the Dirichlet boundary conditions in (2.34) should give such an “effective mass”. We analyze now the problem in detail, and prove existence and uniqueness statements based on this idea. As in [2] we start with a sufficiently good approximation of a solution and expand around it. One possible approximation is obtained by taking the configuration ψk,0 defined by the equalities ψk,0 = ψk on Uk∼ ∩ Z0c , ψk,0 = ψ0 on Z0 . Expanding around it we obtain ψk = ψk,0 + δψ, δψ = 0 on Z0c , δψ = ψk − ψ0 on Z0 , φk (Bk (Uk ); ψ˜ k ) = φk (Bk (Uk ); ψ˜ k,0 ) + δφk (Bk (Uk ); δψ), δφk = 0 on 1 (Uk )c , and the same equalities for αk (Bk (Uk ); ψ˜ k ).
(2.35)
We expand the action in (2.34) applying the above expansions and the formula (2.23) [5], in which we take for example = 1 (Uk )∼ , so the surface term vanishes. Then we apply the formulas (2.28)–(2.30) [5], and the problem (2.34) is reduced to the variational problem
1
δψ, 1(k) δψ + V (k) (δψ) + ak δψ, ψk,0 − Qk φk (Bk (Uk ); ψ˜ k,0 ) . (2.36) inf δψ 2 ˜ k (Uk ); 90dδk ). We extend this configuNotice that from (2.20) we obtain ψ˜ k,0 ∈ 9(B ration to complex ψk satisfying the corresponding restrictions on Uk∼ ∩ Z0c with 3δk replaced by a small positive number δ. Then by the same resoning we obtain that ˜ c (Bk (Uk ); 30dδ). The derivatives with respect to δψ of the last two terms ψ˜ k,0 ∈ 9
Large Field Renormalization for Classical N -Vector Models
507
above are of the order O(δ). The quadratic form is given by the second equality in (2.23) [4], with the operator Gk (α) given by (2.27) [5] for the generating set Bk (Uk ). From the inequality (2.24) [4] together with the following discussion we obtain hδψ, 1(k) δψi ≥ γ0 hδψ, (−11 )δψi − O(δ)kδψk2 , X X hδψ, (−11 )δψi ≥ |δψ(b+ ) − δψ(b− )|2 + b⊂Z0
|δψ(x)|2 .
(2.37)
x∈∂(Z0 ∩T1(k) )
Take a point x ∈ Z0 ∩ T1(k) and let 0x,x0 be the shortest segment connecting x with a point x0 ∈ ∂(Z0 ∩ T1(k) ). Then |0x,x0 | ≤ 7, and X |δψ(x)|2 ≤ (|0x,x0 | + 1) |δψ(b+ ) − δψ(b− )|2 + |δψ(x0 )|2 , b⊂0x,x0
X
|δψ(x0 )|2 ≤
x0 ∈0x,xo
+ |δψ(x0 )|2
X 1 (|0x,xo | + 1)(|0x,x0 | + 2) |δψ(b+ ) − δψ(b− )|2 2 b⊂0x,xo
1 kδψk2 , and finally , hence hδψ, (−11 )δψi ≥ 36
γo kδψk2 for δ sufficiently small. (2.38) 72 The variational problem (2.36) is a simple low-dimensional problem. The corresponding equation can be written in the form ∂ V (k) (δψ) + ak (ψk,0 − Qk φk (Bk (Uk ); ψ˜ k,0 )) = 0, δψ + (1(k) Z0 )−1 ∂(δψ) (2.39) and by the inequality (2.38) and the remarks after (2.36) it is obvious that it has a unique small solution if δ is small enough. The solution is O(δ), and we construct a solution of the original problem (2.34) taking ψ 0 = ψ0 + δψ, where δψ is the solution of (2.39). The solution ψ 0 = ψ 0 (ψ˜ k Z0c ) is defined on Z0 ∩ T1(k) , but we extend it on the whole subregion Uk∼ ∩ T1(k) putting it equal to ψk on Uk∼ ∩ Z0c ∩ T1(k) , and we denote it by ψk0 . We obtain the following lemma. hδψ, 1(k) δψi ≥
˜ k∼ ∩ Lemma 2.1. There exist positive constants c11 , K6 such, that if ψk Uk∼ ∩Z0c ∈ 9(U Z0c ; δ), δ ≤ c11 , then the variational problem (2.34) has a unique solution ψk0 in the ˜ k∼ ; K6 δ). This solution has an analytic extension which determines a mapping space 9(U between the corresponding complex spaces. This lemma holds for an arbitrary rectangular parallelepiped, but then the inequalities (2.20), (2.38), and the constants c11 , K6 , depend on its size. With the size smaller than 15 we obtain absolute constants. We will apply the lemma in the cases δ = 3δk and δ = 3εk , and this gives additional restrictions on βk and α0 . We take ψk0 as the basic background configuration. It has the important property that it depends on the variables ψk restricted to Uk∼ ∩ Z0c , where they satisfy the regularity conditions discussed above. It does not depend on these variables, or any other variables, on the domain Z0 , so we do not need any regularity conditions on this domain. This way
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T. Balaban
we have avoided the difficulty mentioned at the beginning of the previous paragraph. The configuration ψk0 is substituted into other functions of ψk to determine other background configurations, in particular we define φ0k = φk (Bk (Uk ); ψ˜ k0 ), ψk0 1 ,k = ψ(Bk1 ,k , Bk (Uk ); ψ˜ k0 ).
(2.40)
Properties of these configurations are important in the following considerations, so we formulate them in the lemma. Lemma 2.2. The above defined configurations satisfy the bounds |Q∗k ψk0 − φ0k |, |∂ η φ0k |, |1η φ0k |, |αk0 | < 3K1 K6 δk on Uk , hence on Z000 , |ψk − Qk φ0k |, |∂ η φ0k |, |1η φ0k |, |αk0 | <
9 δk on Z000c ∩ Uk , 8
10 · 3K1 K6 δk < 4K1 K6 δk on 1k (z) ⊂ ck , 9 ψk0 (z) − ψk0 1 ,k = 0 on 1k (z) ⊂ k ∩ Uk∼ ,
|ψk0 (z) − ψk0 1 ,k | <
(2.41) (2.42) (2.43) (2.44)
|ψk0 1 ,k (y 0 ) − ψk0 1 ,k (y 00 )| < 4K1 K6 δk Lj η for hy 0 , y 00 i ⊂ 3j (Bk1 ,k ), |ψk0 1 ,k (y 0 ) − ψk0 1 ,k (y)| < 3(d + 2)K1 K6 δk Lj+1 η for y ∈ B(y 00 ), y 0 , y 00 are nearest neighbours in T (j+1) , y ∈ ∂3j (Bk1 ,k ), y 0 ∈ ∂3j+1 (Bk1 ,k )
(2.45)
||ψk0 1 ,k | − 1| < 4K1 K6 δk (Lj η)2 for y ∈ 3j (Bk1 ,k ), j = 0, 1, . . . , k, assuming the restrictions connected with the characteristic functions introduced until now. The analytic extensions of the functions (2.40) satisfy the above bounds with δk replaced by 2εk . To prove the lemma let us notice at first that (2.43)-(2.45) are the usual consequences of (2.41) and the basic formula (4.21) [2] for ψk0 1 ,k , and also of the definitions of the generating sets, in particular the equality Bk1 ,k ∩ k = Bk (Wk ) ∩ k . The bounds (2.41) follow from Proposition 1.1 [2] and Lemma 2.1. The bounds (2.42) are obtained by comparing the configuration φ0k on ∈ πk , ⊂ Z000c ∩ Uk , with the configuration φk (; ψ˜ k ). It is the same argument as in the proof of Lemma 2.1, and it will be discussed in the proof of the most general Lemma 2.5. The main action in the exponential (2.16) depends on the integration variables ψ through ψ˜ = Q∗ (Bk1 ,k , Bk )ψ. The second background configuration ψk0 1 ,k in (2.40) determines the difference (2.46) δψ = ψ˜ − ψk0 1 ,k , which plays now a role of a fluctuation variable. We will introduce some additional restrictions on it, but we have to discuss at first bounds on these configurations following from the existing characteristic functions and Lemma 2.2. They are formulated in the following lemma. Lemma 2.3. The configurations δψ defined by (2.46) satisfy the equality δψ = 0 on Bk1 ,k ∩ k ,
(2.47)
and the bounds |δψ| < 54dK1 K6 L2 M Rj δj on Bk1 ,k ∩ j ∩ cj+1 , j = k1,..., k − 1
(2.48)
on the domain determined by all the characteristic functions introduced until now.
Large Field Renormalization for Classical N -Vector Models
509
To prove the lemma we notice at first that the equality (2.47) follows from the ˜ at some point x ∈ Bk1 ,k ∩ ck . definition of the generating set Bk1 ,k . Take the value ψ(x) This value is equal to a value ψ(y) = ψj (y) at the point y ∈ 3j such that x ∈ 1j (y), k1 ≤ j ≤ k−1. Now we apply again the reasoning between (2.22) and (2.26), and we obtain the ˜ inequality |ψ(x) − ψk (yk )| < 48dL2 M Rj δj . We have taken c = 3dL by (1.24) [5], and j instead of k1 . Consider the value ψk0 1 ,k (x). By (2.43) we have |ψk0 1 ,k (x) − ψk0 (z(x))| < 4K1 K6 δk , where z(x) is the point of T1(k) such that x ∈ 1k (z(x)). The points z(x), yk certainly belong to Z, hence from Lemma 2.1 we have |ψk0 (z(x)) − ψk (yk )| < 45K6 δk + 36dM Rk δk . Combining the three bounds we obtain |δψ| < 48dL2 M Rj δj + 4K1 K6 δk + 45K6 δk + 36dM Rk δk < 54dK1 K6 L2 M Rj δj , which ends the proof of the lemma. ˜ around the configuration ψ 0 , and we will We will expand the function φk1 ,k (ψ) k1 ,k need a bound for the remainder of this expansion. It follows from the above lemma and Proposition 2.1 [3], and it is given in the following lemma. Lemma 2.4. The function δφk1 ,k = δφ(Bk1 ,k ; δψ), where δψ is given by (2.46), satisfies the bounds |χ1j (y) δφk1 ,k |, |χ1j (y) ∂ ξ δφk1 ,k |, |χ1j (y) 1ξ δφk1 ,k |, |χ1j (y) δαk1 ,k | d+2 1 c ∼ < 108dK1 K3 K6 L 2 M Rj δj exp − γ0 d(y, (k ) ) on j ∩ cj+1 ∩ (Zkc1 )∼ . 4 (2.49) Let us notice that here we actually apply the proof of Proposition 2.1 [3] using the fact d−2 that the numbers Rj δj in the bounds (2.48) differ at most by 2L 2 for two consecutive indices. We may introduce this factor in the conditions in (2.11) [3], and it yields also d+2 the factor L 2 in the above bounds. Notice also that the exponential factor is different . from 1 only on the domain ∼−1 k Now we finally come to the main point in the considerations of this section. We ˜ around ψ 0 , and we have expand the function φk1 ,k (ψ) k1 ,k ˜ = φk1 ,k (ψk0 ,k ) + δφk1 ,k (δψ) = φ0k + δφk1 ,k . φk1 ,k (ψ) 1
(2.50)
The configuration φ0k is determined by the following variational problem: inf Ak (Bk (Wk ); ψ˜ k , φk (Bk (Wk ); ψ˜ k )) = inf inf Ak (Bk (Wk ); ψ˜ k , φ),
ψk Z
ψk Z φ
(2.51)
hence φ0k , ψk0 satisfy the usual variational equations (1.10) [2], and the equation ψk0 − Qk φ0k = 0 on Z0 , hence ψk0 1 ,k − Q(Bk1 ,k )φ0k = 0 on Z0 .
(2.52)
We expand the action in (2.16) using (2.50), (2.23) [5] and the variational equations, and we obtain A∗k (Zkc1 ∩ Zk , Bk ; ψ, φk1 ,k ) c = A∗k (Zkc1 ∩ Zk , Bk ; ψk0 1 ,k , φ0k ) + hδψ, a(Bk1 ,k )(ψk0 1 ,k − Q(Bk1 ,k )φ0k )iZK
+ +
νk hδφk1 ,k , φ0k
− hiZkc
1
∩Zk
+
δA∗k (Zkc1
1 (δφk1 ,k )+ + (δφk1 ,k )− , ∂φ0k 2
∩ Zk , Bk ; δψ, δφk1 ,k ) ,
st(Zkc ∩Zk ) 1
1
∩Zk
(2.53)
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T. Balaban
where the “fluctuation” action δA∗k is defined by the formula (2.24) [5]. Consider the second term on the right-hand side of (2.53). The part of the term restricted to Zkc1 ∩ Z0 is equal to 0 by the second equation (2.52). The part restricted to k ∩ Zk is equal to 0 because δψ = 0, so we have only the part of the term restricted to Z0c ∩ ck . Using the representation (4.21) [2] for ψk0 1 ,k we obtain that the term is equal to X ak Q(1k (z), Bk )δψ · (ψk (z) − (Qk φ0k )(z)), (2.54) z∈Z c ∩ck ∩T1(k)
where we have used also the fact that the set Bk1 ,k restricted to Z0c ∩ ck is equal to Bk . We expand also the quadratic form in (2.9), and we obtain X 1 a(1k (z), Bk )|ψk (z) − Q(1k (z), Bk )ψk0 1 ,k |2 2 (k) c z∈Z c ∩k ∩T1
− a(1k (z), Bk )Q(1k (z), Bk )δψ · (ψk (z) − Q(1k (z), Bk )ψk0 1 ,k ) 1 2 + a(1k (z), Bk )|Q(1k (z), Bk )δψ| . 2
(2.55)
Using again the representation (4.21) [2] in the second term above, and using the equality k (z),Bk ) k) (4.18) [2], which in this case takes the form Q(1k (z), Bk ) a(1a(B = −1 + a(1ka(z),B , k) k we obtain that the second term above is equal to (2.54) with the minus sign, so their sum is equal to 0. Take the quadratic form determined by the renormalization transformations in the first term on the right-hand side of (2.53). It can be written in the following way: k X
X
j=k1 y∈3j ∩Zkc ∩Zk
1 aj (Lj η)d−2 |ψk0 1 ,k (y) − (Qj φ0k )(y)|2 . 2
(2.56)
1
The terms in the sum with y ∈ Z0 are equal to 0, again by the second equation (2.52). The remaining terms combined with the first quadratic form in (2.55) yield the form X 1 ak |ψk (z) − (Qk φ0k )(z)|2 . (2.57) 2 (k) c z∈Z0 ∩Zk ∩T1
This follows from the composition formula for the corresponding renormalization transformations, as in (4.11)[2], and from the fact that ψk0 1 ,k is a minimal configuration of the sum of the corresponding quadratic forms. This follows also from the explicit representation (4.21) [2] for ψk0 1 ,k , and from the equality (4.18) [2], as above. Notice that we can extend the form (2.57) to the whole set Zk ∩ T1(k) writing ψk0 instead of ψk , again by the first equation in (2.52). Thus the main action in the exponential (2.16) plus the quadratic form in (2.9) are equal to A∗k (Zk ; ψk0 , φ0k ) − A∗0,k (Zk1 ; φ0k ) X 1 a(1k (z), Bk )|Q(1k (z), Bk )δψ|2 +δA∗k (Zkc1 ∩Zk , Bk ; δψ, δφk1 .k ) + 2 (k) c z∈Z0 ∩Zk ∩T1
1 (δφk1 ,k )+ + (δφk1 ,k )− , ∂φ0k + 2 st(Z c
k1
, ∩Zk )
(2.58)
Large Field Renormalization for Classical N -Vector Models
511
where A∗0,k means the main action without the quadratic form connected with the renormalization transformations. This is a fundamental formula, the main formula of this section. Let us discuss properties of the terms in (2.58). The first two terms are the main actions which do not depend on any integration variables, therefore the corresponding exponentials can be written in front of the integral operator (2.6). The first term is cancelled by the same term in the denominator of the quotient in (1.6), as will be discussed later. The second term yields the exponential exp βk A∗0,k (Zk1 ; φ0k ), which remains in the large field expression. The exponent is quite small because the volume of the domain Zk1 in the η-scale is very small. This is true even for its analytic extension in ψk . The functions φ0k , αk0 satisfy the bounds (2.41) with δk replaced by 2εk , and we have |βk A∗0,k (Zk1 ; φ0k )|
< βk
1 9 1 2 2 (6K1 K6 εk ) + (6K1 K6 εk ) + νk |Zk1 |η 2 2λk 2
< (6K1 K6 )2 βk ε2k L−n1 d (12M Rk1 )d ≤ const.βk1 ε2k1 Rkd1 L−2(1+α)n1 1 −2( α +1)
< const.βk1 ε2k1 Rkd1 βk1
(2.59)
< 1,
where we have used the scaling properties of βk , εk , and the condition (2.31). Notice that we could obtain an arbitrarily small bound for β large enough. By the same reasoning we prove that one of the surface terms in (2.58), corresponding to the “surface” st(Zk1 ), is bounded by 1, or is arbitrarily small for β large. The second surface term corresponding to st(Zk ) is small because of the exponential decay property in (2.49), which yields the factor exp(− 21 γ0 M Rk ). Obviously this factor can be estimated by an arbitrarily large power of βk−1 , as in the other bounds of such terms in Sect. 2 [6]. We will come back to these estimates later on, in connection with a discussion of analyticity properties of large field expressions. The most important term in (2.58) is the expression in the curly brackets, which is a generalization of the effective action for the fluctuation variables in the curly brackets on the right-hand side of the last equality in (2.26) [5]. We need to establish some positivity properties of this expression which would allow us to introduce new restrictions on the variables δψ. For simplicity we use only the first two terms in βk δA∗k (Zkc1 ∩ Zk , Bk ; δψ, δφk1 ,k ), for which we have the inequality (3.5) [6]. The last two terms in it, and the first quadratic form in the curly brackets, are non-negative. We have to bound the third term, which is 1 1 | βk hδφk1 ,k , αk0 δφk1 ,k iZkc ∩Zk | ≤ βk h|δφk1 ,k |2 , |αk0 |iZkc ∩ck +1 1 1 1 2 2 k−1 X + h|δφk1 ,k |2 , |αk0 |ij ∩cj+1 + h|δφk1 ,k |2 , |αk0 |ik ∩Zk j=k1 +1 (2.60) k X1 d+2 βj (108dK1 K3 K6 L 2 M Rj δj )2 · 3K1 K6 δk · (10LM Rj )d L−2(k−j) < 2 j=k1
− 21
< const.Rkd+2 p30 (βk )βk
< 1.
We have used the bounds (2.41), (2.49), and the estimate diamj (cj+1 ) < 10LM Rj following from the estimate distj (cj , j+1 ) < 3LM Rj obtained between (2.22) and (2.23). It follows from the above statements that the expression in the curly brackets
512
T. Balaban
in (2.58), considered as a function of δψ, and multiplied by βk , can be bounded from below by the right-hand side of (3.5) [6] for X = Zkc1 ∩ Zk , minus the number 1. With the above bound we can introduce and control new restrictions on the configurations δψ restricted to Zkc1 ∩ ck . We define ψ 00 =δψ on Zkc1 ∩ ck ∩ Bk ,
χ00 =χ({|ψ 00 | = |ψj00 | < δj,1 on 3j ∩ Zkc1 , j = k1 , . . . , k − 1}),
χ
00c
00
=1 − χ , δj,1 =
(2.61)
−1 βj 2 A1 (log βj a)p0 .
The decomposition of unity 1 = χ00 + χ00c is introduced into the integral operator (2.7), and for the operator with the function χ00c we have to obtain a new small factor. This function implies that there is a point y ∈ 3j ∩ Zkc1 for some j, such that |ψj00 (y)| = |δψ(y)| ≥ δj,1 . Let us assume that δψ satisfy the bounds (2.22) with δj replaced by δj,1 and with a properly chosen small c. We apply again the reasoning between (2.22) and (2.26), and we obtain the bound |δψ(y)| = |δψ(y) − δψ(yk )| < 16LM Rj δj,1 c. Taking c = (16LM Rj )−1 we obtain the contradictory bound |δψ(y)| < δj,1 , so at least one of the bounds (2.22) cannot be satisfied. By the inequality discussed at the end of the previous paragraph we obtain the small factor exp(− 21 γ0 (16LM Rj )−2 p21 (βn ) + 1) for some n, k1 ≤ n ≤ k. Using the inequalities (2.32) and Rj < Rk we can bound it by 1 α exp(− γ0 ( )2p0 (16LM Rk )−2 p21 (βk ) + 1) 2 d γ0 α2p0 A21 < exp(− (log βk a)2p0 −4 ) = exp(−A003 (log βk a)2p0 −4 ), (32dp0 LM R0 )2
(2.62)
assuming again that A003 is still large enough. This is an assumption on A1 , which can be translated into one on A0 , because these constants are connected by a simple relation, which will be discussed in a moment. Actually we will see that the constants A03 , A003 are almost equal, so the assumptions on them are practically the same. For the operator with the function χ00c we include the component Z into the class of components of the first kind, and we consider further the operator with χ00 . The function χ00 together with the characteristic functions previously introduced in this section imply that some characteristic functions in the operator (2.7) can be removed. We have used already several times such statements in this and the previous paper in this series. Now we formulate and prove the most general statement of this type. Lemma 2.5. The “small field” characteristic functions introduced in this section imply that the characteristic functions in (2.7) with localization cubes contained in Ukc1 ∩ c ∩ (∼3 (ck )∼3 are equal to 1, i.e. χj () = 1, χ0j+1 (0 ) = 1 for ∈ πj , ⊂ ∼−3 j+1 ) ∩ j ∼3 c c 0 ∼3 Uk1 , ⊂ j+1 ∩ (j+1 ) , j = k1 , . . . , k − 1. We prove now this lemma. We analyze functions χj (), mainly for simplicity of notations, and later we discuss changes necessary in the analysis of functions χ0j+1 (0 ). Consider at first satisfying the conditions of the lemma and such that ∼3 ∩ Uk∼3 = ∅. 1 The analysis in this case covers in particular a proof of Lemma 2.1 [5]. Notice that ∼3 ⊂ j ∩ cj+1 , and that in this case δψ = ψj00 on ∼3 . We start writing the equality φj (; ψ˜ j ) = φj (; ψ˜ k0 1 ,k ) + δφj (; ψ˜ j00 ),
(2.63)
Large Field Renormalization for Classical N -Vector Models
513
with φj () = φj (Bj (∼2 )) and the usual interpretation of the symbol “∼” over the configurations above, e.g. ψ˜ j = Q∗ (Bj (∼2 ))ψj on ∼3 . Let us recall that in this section k ≤ k0 , so the characteristic functions do not contain the conditions involving the vectors h. For the second function on the right-hand side above we have the bounds |δφj |, |∂ ξ δφj |, |1ξ δφj |, |δαj | < K2 δj,1 =
K2 A1 1 δj ≤ δj on ∼ , A0 8
(2.64)
by Proposition 3.1 [2] and the restrictions (2.61) on ψj00 . The last inequality does not give the strongest condition on the constants involved. In the case of Lemma 2.1 [5] there is the additional constant B5 in the numerator, and we can take 4K2 B5 A1 = A0 . This is the above mentioned relation between the constants A0 and A1 . Consider the first function on the right-hand side of (2.63). We would like to replace it eventually by φ0k , and to show that the difference is sufficiently small. There are two problems with this. The first is that the function ψk0 1 ,k is determined by the coefficients after k steps, and the function φj () by the coefficients after j steps, so they do not match and no composition formula can be applied directly. Such a formula cannot be applied also because the generating sets of the two functions do not match, which is the second problem. The first problem is dealt with in the same way as the corresponding problem connected with the actions in (2.7). We apply k − j times the corresponding decompositions (4.12)[4], each time “renormalizing” the coefficients by one step, until we obtain the function determined by the renormalized coefficients after k steps. Summing up the decompositions we obtain φj,u (; ψ˜ k0 1 ,k ) = φj (; ψ˜ k0 1 ,k ) + δφj,u (; ψ˜ k0 1 ,k ),
(2.65)
where we have denoted now the function φj () in (2.63) by φj,u (). Notice that from the properties formulated in (2.45) and Proposition 1.1 [2] we obtain ˜ j (∼2 ); 4K1 K6 Lj ηδk ) ⊂ 4j (Bj (∼2 ); 4K12 K6 Lj ηδk ). ψ˜ k0 1 ,k ∈ 9(B
(2.66)
The inequalities (4.11), (5.31) [4] yield the following bounds for the second function on the right-hand side of (2.65): |δφj,u |, |∂ ξ δφj,u |, |1ξ δφj,u |, |δαj,u | < 28K12 K2 K6 Lj ηδk · 2B6 K5
k X n=j+1
βn−1
k 1 1 −1 X −2γn j 2 < 112B K K K K β L L ηδk 6 2 5 6 1 ε2n α02
< 56B6 K12 K2 K5 K6
n=j+1
(2.67)
1 1 −1 −2γj j L ηδk < δj on ∼ , β L 8 α02 γ
for β large enough. Obviously we could get much better bounds with the same conditions on β, but it is unimportant here. Let us remark that a similar, only simpler argument, with k + 1 instead of j and one term on the right-hand side of (2.67), and the constant 4K1 K6 Lj ηδk in the spaces (2.66) replaced by 3δk+1 , yields the equalities (2.2) [6]. Consider again the first function on the right-hand side of (2.65). It is a composition of two functions with matching coefficients, and we write φj (; ψ˜ k0 1 ,k ) = φj (Bj (∼2 ); ψ(Bj (∼2 ), Bk (Uk ); ψ˜ k0 )) + δφj (Bj (∼2 ); ψ˜ k0 ,k − ψ(Bj (∼2 ), Bk (Uk ); ψ˜ k0 )). 1
(2.68)
514
T. Balaban
The first configuration on the right-hand side above is equal to φ0k on ∼3 . To estimate the second configuration we have to estimate the difference inside this function. Obviously this difference is equal to 0 on ∼2 . Take a point x ∈ 3n (∼2 ), where the set is a layer of the generating set Bj (∼2 ) with n < j. We have x ∈ 1j (y) ⊂ 1k (z) for some y ∈ 3j ∩ ∼3 , z ∈ T1(k) ∩ Z c , and we estimate |ψ˜ k0 1 ,k (x) − ψ(x; Bj (∼2 ), Bk (Uk ); ψ˜ k0 )| = |(Qj φ0k )(y) +
ak j 2 0 (L η) (ψk (z) aj
ak n 2 0 (L η) (ψk (z) − (Qk φ0k )(z))| an ≤ (Qj−n |(Qn φ0k )(·) − (Qn φ0k )(x)|)(y) + (Lj η)2 |ψk0 (z) − (Qk φ0k )(z)|
− (Qk φ0k )(z)) − (Qn φ0k )(x) −
(2.69)
< 3dK1 K6 Lj ηδk + 3K1 K6 (Lj η)2 δk < 3(d + 1)K1 K6 Lj ηδk . Applying Proposition 2.1 [3] we estimate the last function in (2.68), and the other expressions defined by it as on the left-hand sides of (2.64), (2.67), by 1 1 K3 exp(− γ0 (M − 2M1 ))3(d + 1)K1 K6 Lj ηδk < 3(d + 1)K1 K3 K6 exp(− γ0 M ) · 4 5 1 j 1 j ∼ · L ηδk ≤ L ηδk < δk on , 8 8 (2.70) for M large enough. Let us remark that the second inequality above yields one of the strongest conditions on M . Combining now the expansions (2.63), (2.65), (2.68) with the bounds (2.64), (2.67), (2.70) we obtain 3 |∂ ξ φj |, |1ξ φj |, |δαj | < max |∂ ξ φ0k |, |1ξ φ0k |, (Lj η)2 |αk0 | + δj on ∼ . 8
(2.71)
We have to consider two cases. At first assume that ∼ ∩ Z000 6 = ∅, then ∼ ⊂ ck3 +1 , and by the inequalities (2.41) in Lemma 2.2 we bound the maximum above by 3K1 K6 Lj ηδk < L− 3 (k−j) 3K1 K6 δj ≤ L− 3 (n3 −1) 3K1 K6 δj < L−n3 3K1 K6 δj 9 1 30K1 K6 30K1 K6 5 3K1 K6 δj < < δj < δj ≤ δj , 2 8 [Rk3 ] 8Rk3 8R0 (log βa) 8 4
4
where the last inequality yields either a restriction on R0 , e.g. R0 ≥ 6K1 K6 , or on β. We have used here also the inequality on L−n3 formulated before the definition (2.5) of the domain Z000 . Assume that ∼ ⊂ Z000c , then by the inequalities (2.42) in Lemma 2.2 we bound the maximum in (2.71) by 3 3 9 j L ηδk ≤ δk < δj , where we have used the fact that j < k. 8 8 8 In both cases the left-hand side of (2.71) is bounded by δj , so the three conditions in the definition of the function χj () are satisfied. We have yet to estimate the difference ψj − Qj φj . We use again the above expansions and bounds, and we obtain A1 3 δj + |ψk0 1 ,k − Qj φ0k | + δj |ψj − Qj φj (; ψ˜ j )| < A0 8 1 < (Lj η)2 |ψk0 − Qk φ0k | + δj on ∼ . 2
(2.72)
Large Field Renormalization for Classical N -Vector Models
515
For the first term on the right hand side we consider again the two cases. In the first case we bound it by 3K1 K6 (Lj η)2 δk < L− 3 (n3 −1) 3K1 K6 δj < L−2n3 3K1 K6 δj < ( 7
<
10 2 ) 3K1 K6 δj 8Rk3
10 5 1 · δj < δj , 8Rk3 8 2
where we have used the previous restrictions. In the second case we bound it by (Lj η)2 89 δk ≤ 18 δk < 18 δj , hence in both cases the left-hand side of (2.72) is bounded by δj , so the remaining condition in the definition of χj () is satisfied. Thus we have proved the statement of the lemma for the functions χj (), under the additional assumption = ∅. that ∼3 ∩ Uk∼3 1 Consider now the case when ∼3 ∩ Uk∼3 6 = ∅, so j = k1 , and ⊂ Ukc1 ∩ Uk∼6 . 1 1 Instead of comparing the configuration φj () with φ0k we have to use here the function χ0k , 1 (). Expanding in the difference ψk1 − ψk0 1 we obtain 1 2
φk1 (; ψ˜ k1 ) = φk1 (; ψ˜ k0 1 ) + δφk1 (; ψ˜ k1 − ψ˜ k0 1 ),
(2.73)
and the first configuration satisfies all the conditions in the definition of χk1 () with 21 δk1 instead of δk1 on the right-hand sides. By the definition of ψk0 1 we have ψk1 − ψk0 1 = 0 on ∼3 ∩ Uk∼3 , and ψk1 − ψk0 1 = ψk1 − ψ0 on ∼3 ∩ (Uk∼3 )c , hence ψ˜ k1 − ψ˜ k0 1 = 0 1 1 on Bk1 (∼2 ) ∩ Uk∼3 and ψ˜ k1 − ψ˜ k0 1 = ψ˜ k1 − ψ0 on Bk1 (∼2 ) ∩ (Uk∼3 )c . Take a point 1 1 (k) x in the last set, a point z ∈ T1 such that x ∈ 1k (z), and a point z0 on the boundary ∂(Z0c ∩ T1(k) ), closest to z. We have ψ˜ k1 (x) − ψ0 = ψ˜ k001 (x) + (ψ˜ k0 1 ,k (x) − ψk0 (z)) + (ψk0 (z) − ψk (z0 )) + (ψk (z0 ) − ψ0 ), hence 1 A1 δk1 + 5K1 K6 e− 3 n1 δk1 |ψ˜ k1 − ψ˜ k0 1 | < δk1 ,1 + 4K1 K6 δk + 7K6 δk + 90dδk < A0 1 1 1 A 1 − 3 n1 − 3α − 3α + 90dL δ k1 < + 5K1 K6 (βk1 a) + 90d(βk1 a) δk1 A0 1 A1 − 1 < max , β 3 5K1 K6 , β − 3 90d δk1 . A0
From this it follows that the second configuration on the right-hand side of (2.73) satisfies all the necessary bounds to have χk1 () = 1, if 1 A1 − 1 − 13 3 , β 5K1 K6 , β 90d ≤ . (K2 + 1) max A0 2 These conditions are weaker than other conditions introduced until now, so we have proved the lemma for all the required cubes. The equalities χ0j+1 (0 ) = 1 are proved in obviously the same way. There are only small changes in some constants, for example in the considerations from (2.65) to (2.67) the constant 4K1 K6 Lj ηδk is replaced by 3(d + 2)K1 K6 Lj+1 ηδk . These changes lead only to unessential modifications of some conditions, for example on β in (2.67), but we still obtain the corresponding inequality (2.71). The remaining inequalities hold as they are written, or with some obvious minor modifications, except the case j + 1 = k
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T. Balaban
in the inequality (2.72), where we have to be more careful with bounds. On ∼ ∩ ck the first inequality holds without any changes, and we bound the right-hand side by A1 1 3 3 ∼ A0 δk−1 + 8 δk + 8 δk−1 < 4 δk−1 . On ∩ k we bound the left-hand side by |ψk −
Qk φ0k |+ 38 δk−1 < 98 δk + 38 δk−1 < ( 98 L− 3 + 38 )δk−1 < 45 δk−1 . Thus we obtain the required inequalities in this case also. This ends the proof of the lemma. Let us remark that the characteristic functions χ00 obviously improve the bounds )c , but we cannot improve them essentially on Bk1 ,k ∩ Uk∼3 . The (2.48) on Bk1 ,k ∩ (Uk∼3 1 1 same is true for the bound (2.49) on δφk1 ,k . We do not formulate the improved bounds because we will not need them. Finally we have completed all the preparatory operations and we can write the new represention (1.4) of the effective densities. We have obtained a new set of mult-indices and a new classification of components of the large field region. Let us summarize briefly the main results of the previous analysis. In the “old” components of the first kind nothing has been changed, we have the same restricted multi-indices and the same integral operators. The only differences are connected with the “old” components of the second kind. We have considered such a component Z and a multi-index Ak ∩ Z. This multiindex gave rise to a set of new multi-indices, which we denote by A00k ∩Z. Their structure should be quite obvious, for each operation introducing new large field characteristic functions we join the corresponding supporting domain to Ak . For example for the decomposition (2.18) we join the domain Z0c ∩(ck )∼3 to Ak . For the next decomposition we join Sk00 to the previous multi-index, then we join Z000c ∩ (ck )∼3 , Z0c ∩ Z000 , Q00k1 to Ak . , (Uk∼3 )c ∩ck ∩Bk Finally we join the previous first two domains together with Ukc1 ∩Uk∼6 1 1 to Ak for the term of the decomposition (2.1) with the large field function. All these multi-indices make the component Z a component of the first kind. It is also clear what form has the corresponding operators T00k (Z, Ak ∩ Z). For Z, Ak ∩ Z, there is exactly one term, after all the above operations, which has only the small field characteristic functions. The corresponding multi-index A00k ∩Z can be obtained by joining the supports of the new characteristic functions to Ak . The component Z remains still a component of the second kind. On this component the operator Tk,k1 (Z, Zk1 ) has been transformed into the new operator T00k,k1 (Z, Zk1 ) including the new characteristic functions obtained by application of Lemma 2.5, and the exponential with the new action obtained from (2.7) by applying (2.16), (2.58). This operation will be transformed further, so we write it explicitly for future reference. We have 1
h
T00k,k1 (Z, Zk1 ) = χk (Z000c ∩ Z)χ˜ k (Z000 ∩ Z0c ) exp −βk A∗k (Z; ψk0 , φ0k )
i
+ βk A∗k,0 (Zk1 ; φ0k ) T (T1(k) ∩ Z0 , (Bk ∩ Zkc1 ∩ Z0 ) ∪ (T (k1 ) ∩ Zk1 ))
Z
·
dψ 00 Bk ∩Z0c ∩ck χk1 (Zkc1 ∩ Uk1 )χ0k1 , 1 (Ukc1 ∩ Uk∼6 )χ00 1
h
· exp −βk
n
X
z∈T1(k) ∩Z0c ∩Z
2
1 a(1k (z), Bk )|Q(1k (z), Bk )ψ 00 |2 2
o
+ δA∗k (Zkc1 ∩ Z, Bk ; δψ, δφk1 ,k ) − βk νk δφk1 ,k , φ0k − h − βk +
D h
i
1 (δφk1 ,k )+ + (δφk1 ,k )− , ∂φ0k 2
Ck1 ,k (Zkc1
∩
Zk , Ak ; ψk0 1 ,k
i
+ δψ) ,
E st(Z c ∩Zk ) k1
(2.74)
Z c ∩Zk k1
Large Field Renormalization for Classical N -Vector Models
T00k (Z, A00k
∩ Z) =
T00k,k1 (Z, Zk1 )Tk1 (Zk1 , Ak1
517
∩ Zk1 ) exp
k−1 X
B (j) (Zjc ∩ Zj+1 ), (2.75)
j=k1
where Ak1 ∩Zk1 = A00k ∩Zk1 = Ak ∩Zk1 , and B (j) (Zjc ∩Zj+1 ) depends on the configuration (ψ U ∼3 , (ψk0 1 ,k + ψ 00 ) (U ∼3 )c ) instead of ψ. k1 k1 The above discussed operators determine a density ρ00k (Yk , Zk , A00k ) in (1.4), where Yk is a union of components of the first kind, and Zk is a union of the components Z of the second kind. The density is defined by the same formulas (1.1), (1.2) [5] in the hypotheses (H.1), (H.5) [5], only the operators Tk (Zk , Ak ) are replaced by T00k (Yk ∪ Zk , A00k ). This new representation of ρk , and in particular the form of the operators (2.74), (2.75), form a basis of the further analysis of the renormalization operation R(k) in the next section. 3. A Completion and a Discussion of the Operation R(k) In the last section we have done all the preparatory operations for R(k) , which form really its technical core. Now we perform the operation R(k) according to definition (1.6). The first important problem is the cancellation mentioned at the beginning of the last section. We have to cancel the main actions on Zk in the numerator and the denominator in (1.6). We have already separated the corresponding action in the formula (2.58), and it gives the contribution from the numerator. For the denominator we have to repeat the operations leading to the fluctuation integral in Sect. 2 [5], only in a much simplified form because we do not have to perform any localization operation. We have to define yet the characteristic function χZ0 in (1.5). We take the background configuration ψk0 and we expand ψk around ψk0 , where ψk0 is the solution of the variational problem (2.34), −1
ψk = ψk0 + βk 2 ψ, hence ψ = 0 outside Z0 .
(3.1)
The characteristic function is defined as χZ0 = χ({|ψ| < C6 p0 (βk ) on Z0 }),
(3.2)
where C6 is a sufficiently large constant. We will see that it can be taken as C6 = 21(K6 +1). The integral in the denominator in (1.6) obviously factorizes into the integrals over the components of Zk , and we consider the one corresponding to Z. We expand it around the background configuration ψk0 , and we use again formulas (2.23)–(2.25) [5]. In this particular case we have −1
−1
A∗k (Z; ψk , φk (ψ˜ k )) = A∗k (Z; ψk0 , φ0k ) + δA∗k (Z; βk 2 ψ, δφk (βk 2 ψ)) 1 − 21 − 21 − 21 0 0 (δφk )+ (βk ψ) + (δφk )− (βk ψ) , ∂φk , + νk hδφk (βk ψ), φk − hiZ + 2 st(Z) (3.3) where φk , δφk are defined by Bk (U ), U = Z ≈ , and we have used the equalities ψk0 − Qk φ0k = 0 on Z0 , ψ = 0 on Z0c ∩ Z, to cancel the second term on the right-hand side of (2.23) [5]. We expand the function δA∗k (Z) above into the sum of the quadratic form and the higher order function as in (2.28)-(2.30) [5]. The integral is now equal to
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Z 1 1
exp − N log βk |Z0 | dψ Z0 χZ0 exp − ψ, 1(k) (Z)ψ 2 2 D E −1 −1 − βk V (k) (Z; βk 2 ψ) − βk νk δφk (βk 2 ψ), φ0k − h Z 1 − 21 − 21 0 (δφk )+ (βk ψ) + (δφk )− (βk ψ) , ∂φk − βk . 2 st(Z) (3.4) − 21 3 The second term in the underintegral exponential is bounded by O(1)βk p0 (βk )|Z0 |, −βk A∗k (Z; ψk0 , φ0k )
−1
which can be seen by expanding in βk 2 ψ up to the third order. This is analyzed in detail in the section on the localization expansions of the fluctuation integral. The third term −3
−1
is bounded by O(1)βk 2 p20 (βk )|Z0 |, as can be seen by expanding in βk 2 ψ up to the first order, using (2.41) and the condition βk νk ≤ 1. The fourth term can be bounded by a high power of βk−1 multiplied by |Z0 |, and this can be seen again by expanding −1
in βk 2 ψ up to the first order and using the exponential decay properties of δφk . All the expressions are analytic functions of ψk0 , hence of ψk restricted to U ∼ , and the analyticity domain is quite “large”, determined by the absolute constants in Proposition 1.1 [2], Propositions 2.1, 2.2 [3] and Lemma 2.1. The bounds discussed above hold on the whole analyticity domain. The quadratic form has been analyzed in the proof of Lemma 2.1; in particular it has been shown there that it is bounded from below, and we have the inequality (2.38). Finally let us recall that |Z0 | ≤ 14d , so the integral in (3.4) is very simple and elementary. We do not have to do any more operations on it, like cluster expansions, etc. We define E˜0(k) (Z0 ; ψk0 ) as the logarithm of this integral. It is a bounded analytic function of ψk0 , hence of ψk restricted to U ∼ , defined on the “large” domain and satisfying the usual symmetry properties. This is all the information we need, no other more detailed properties are necessary, in particular a localization expansion is meaningless. Thus the integral in the denominator in (1.6), restricted to the component Z, is equal to 1 exp −βk A∗k (Z; ψk0 , φ0k ) − N log βk |Z0 | + E˜0(k) (Z0 ; ψk0 ) . 2
(3.5)
Taking into account the formulas (2.73) and (3.5) we see that the main actions are cancelled. We write the definition (1.6) of the operation R(k) in the following explicit form: (R(k) ρk )(ψk ) =
X
χk ((Yk ∪ Zk )c )T00k (Yk ∪ Zk , A00k ) exp B (k) (Yk ∪ Zk , A00k )
Yk ,Zk ,A00 k
· exp −βk A∗k (Zk ; ψk , φk (Bk (Zk≈ ); ψ˜ k )) + Ak ((Yk ∪ Zk )c ; ψk ) c (k) c + Fk ((Yk ∪ Zk ) ; ψk , g) + R ((Yk ∪ Zk ) ; ψk , g) . (3.6) The integral operator T00k restricted to a component of Yk is defined as discussed at the end of the last section, but restricted to a component Z of Zk is defined now as follows:
Large Field Renormalization for Classical N -Vector Models
T00k (Z, A00k ∩ Z) = χk (Z ∩ Z000c )χ˜ k (Z000 ∩ Z0c )χZ0 1 · exp N log βk |Z0 | − E˜0(k) (Z0 ; ψk0 ) + βk A∗k,0 (Zk1 ; φ0k ) 2 Z Z )χk1 (Zkc1 ∩ Uk1 ) · dψ 00 dψk1 U ∼3 χ00 χ0k1 , 1 (Ukc1 ∩ Uk∼6 1 k1 2 X 1 a(1k (z), Bk )|Q(1k (z), Bk )ψ 00 |2 · exp −βk 2 z∈T (k) ∩Z c ∩c 1 0 k
+ δA∗k (Zkc1 ∩ Z, Bk ; δψ, δφk1 ,k ) − βk νk δφk1 ,k , φ0k − h Z c ∩Z k1 1 (δφk1 ,k )+ + (δφk1 ,k )− , ∂φ0k − βk 2 st(Zkc ∩Z) 1 + Ck1 ,k (Zkc1 ∩ Z, A00k ∩ (Zkc1 ∩ Z)) Tk1 (Zk1 ∩ Z, Ak1 ∩ Z) k−1 X · exp B (j) (Zjc ∩ Zj+1 ∩ Z, A00k ∩ Z) .
519
(3.7)
j=k1
This formula has been obtained by taking the operator defined by (2.73), (2.74), integrating it over ψk restricted to Z0 , replacing the integral of the composition of the , and renormalization transformations in (2.73) by the integral over ψk1 restricted to Uk∼3 1 c ψ 00 restricted to (Uk∼3 ) ∩ Z , by using the identity (2.8), and then dividing the obtained 0 1 expression by the exponential (3.5). Notice that the function Ck1 ,k and the boundary terms in the last exponential depend now on ((ψ, ψ 0 ) U ∼3 , (ψk0 1 ,k + ψ 00 ) (U ∼3 )c ∩U ∼ ). k1 k1 These configurations belong to the original integration domain by Lemma 2.5, hence to (1.24) [5]. We have to estimate the above operator, because it is changed considerably in comparison with the original inductive form. It is enough to use the inductive assumption on the operator Tk1 (Zk1 ∩ Z, Ak1 ∩ Z), in particular the fact that it controls the next n1 renormalization steps. This means that it can be bounded by exp(−2p2 (βk ) − Hk |Z|η −
k−1 X
(O(1) log βj + const.)|Zjc ∩ Zj+1 |ξ )
(3.8)
j=k1
times the inverse of the combinatoric factor. Notice that A00k ∩ Zk1 = Ak1 , and the remaining part A00k ∩ (Zkc1 ∩ Z) of the multi-index A00k is determined uniquely by the large field region Zk1 , in particular S n (Zk1 ) = Zk1 +n . We estimate the operator (3.7) in a simplest possible way by a volume of the integration domain times a supremum of the integrated function, except the integration over ψk1 restricted to Zk1 , which is applied to the operator Tk1 . We have discussed already bounds for the expressions in the exponentials, e.g. in (2.33) [6], (2.15), (2.59), (2.60). The result is that the product of the exponentials in (3.7) can be estimated by X 1 (O(1) log βj + const. )|Zjc ∩ Zj+1 |ξ ). exp( N log βk |Z0 | + O(1) + 2 k−1
j=k1
(3.9)
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T. Balaban
The value of the integration can be estimated by ∼3
N |(Uk∼3 )c ∩3k1 ∩Z|k1
(3δk )N |Z∩Z0 | (C6 δk )N |Z0 | (3δk1 )N |Zk1 ∩Uk1 ∩Z|k1 δk1 ,1 c
c
·
k−1 Y
N |3j ∩Z|j
δj,1
1
,
j=k1 +1
hence obviously by 1. This volume could be used to improve the bound (3.9), in fact a more careful analysis would show a cancellation of the terms proportional to log βj , but it is unimportant. We can take the constants in the sum (3.8) large enough, so that the sum in (3.9) is cancelled by the sum in (3.8) in a complete bound. Even this is unimportant, because the whole expression in the exponential (3.9) can be estimated by const.(log βk )2d+2 , and we can use the term −p2 (βk ) in (3.8) to cancel it. In fact we will include later the factor exp Ek |Z| into the integral operator, and |Ek | < O(1) log βk + const., hence |Ek |Z|| < (O(1) log βk + const.)(12M Rk )d ≤ const.(log βk )2d+1 . We use −p2 (βk ) to cancel this bound, and also some other factors in later constructions, so we may use it as well to cancel (3.9). We assume that p2 ≥ 2d + 2, and that A2 is large enough, which means A0 large enough. Thus we obtain the estimate X Z dψk Z |T00k (Z, A00k ∩ Z) exp Ek |Z|F | A00 ∩Z k
≤ exp(−p2 (βk ) − Hk |Z|η )
(3.10) sup
A00 ∩Z,ψ,ψ 0 k
|F |.
We estimate in the same way the operators constructed in the last section and involving some large field restrictions, but then with an additional small factor, which restores the factor 2 in the last exponential and controls a large number of steps. The integral operator (3.7) has the usual symmetry properties, including the orthogonal group invariance and the worst possible Euclidean invariance compatible with all the above constructions, namely the invariance with respect to the transformations preserving the partition πk into the large cubes. We consider now the integral operator (3.7) without the characteristic functions in front of the first exponential, and we are going to prove that it can be analytically extended with respect to (ψk , h), g onto the space 4ck (U ; 1, εk ) × {g : kgk`1 < 1}, and that the supremum of the values of this operator on the space satisfies the inequality (3.10). This property is a new feature of these operators, there was no need for it in the previous constructions, like in the analysis leading to the fluctuation integral, in fact generally it does not hold for the large field integral operators, mainly because of the characteristic functions involved in their definitions. The operations of the last section were done just for the purpose of assuring this property by removing a dependence on ψk from the characteristic functions, as in the fluctuation integral. We have succeeded in ) which this almost completely, except for the characteristic functions χ0k , 1 (Ukc1 ∩ Uk∼6 1 1 2 depend on the vector ψ0 , hence still indirectly on ψk restricted to the boundary of Z0c . To show the analyticity we remove at first the dependence on ψ0 from the characteristic functions. We do it by making the change of variables ψk1 → Rψk1 in the integral in (3.7), where R ∈ O(N ) is such that Re1 = ψ0 . Together with this we make the change of all the integration variables in the integral operator Tk1 (Zk1 , Ak1 ), in particular
Large Field Renormalization for Classical N -Vector Models
521
(ψ, ψ 0 ) → (Rψ, Rψ 0 ) on Uk∼3 . The integral operators with the characteristic functions 1 forming the above one are invariant with respect to the orthogonal group, so they have the same form in the new variables. Some expressions in the exponentials in the inductive definition of the integral operator depend on the vector variables h, g, so they are not invariant, but they depend analytically on (ψ, ψ 0 ), hence on (Rψ, Rψ 0 ). Notice that now we use again the explicit inductive definition of the integral operator, and in particular properties of the expressions in the exponentials. The effect of the change of variables is that the integral operator (3.7) is written in the same form, but with the following changes: ) the vector ψ0 is replaced by e1 ; the in the characteristic functions χ0k , 1 (Ukc1 ∩ Uk∼6 1 1 2 configuration δψ is given by the formulas δψ = Rψ˜ k1 − ψ 0 on Bk1 (Wk1 ) ∩ U ∼3 , δψ = k1 ,k
k1
)c ∩ ck ; the function Ck1 ,k and the boundary terms depend on the ψ 00 on Bk ∩ (Uk∼3 1 configuration ((Rψ, Rψ 0 ) U ∼3 , (ψk0 1 ,k + ψ 00 ) (U ∼3 )c ) properly restricted, and the same k1 k1 is true for the terms in the exponentials inside the operator Tk1 (Zk1 , Ak1 ). Now we do not have any dependence on ψk in the characteristic functions. The only dependence is through the analytic functions in the exponentials. The problem with this representation is that R depends obviously on ψ0 , but it is impossible to construct an analytic function R of ψ0 on the whole sphere S N −1 , we can do it only locally, and non-uniquely. A way out of this difficulty is to consider all R satisfying the equality Re1 = ψ0 , and to integrate over them. They form the left coset Oψ0 (N ) = {R ∈ O(N ) : Re1 = ψ0 } of the subgroup O1 (N ) = Oe1 (N ) = {R ∈ O(N ) : Re1 = e1 }. We integrate the obtained expression (3.7) after the change of variables over the coset Oψ0 (N ), and we divide the integral by the volume of the coset, which is equal to the volume of the subgroup. Of course we do it in each component of Z independently, each component having its own vector ψ0 . The reason for doing this is that the obtained expression is analytic in ψ0 . To see this we notice that locally we can always choose a function R(ψ0 ) depending analytically on ψ0 . Then Oψ0 (N ) = R(ψ0 )O1 (N ), the integration over the cosets may be replaced by the integration over the subgroup, which by reversing the above argument with the change of variables can be actually omitted from the expression, so only the dependence on R(ψ0 ) is left. We obtain the above discussed expression for the operator (3.7), with R replaced by R(ψ0 ). Of course we could do it from the beginning. The purpose of introducing the above integral expression is to obtain a globally defined analytic function of ψ0 . It is very easy to construct a locally analytic function R(ψ0 ), and then it is also an analytic function of ψk restricted to ∂Z0c . Extending it to complex configurations ψk + ψk0 , |ψk0 | < 2εk , we obtain an analytic matrix-valued function R(ψ0 + ψ00 ) which is slightly non-orthogonal, but with the norm |R(ψ0 + ψ00 )| < eO(εk ) < 23 .
Thus we have “moved” the whole dependence on ψk into the expressions in the exponentials in (3.7). Consider the analyticity properties of these expressions. We have discussed already some of them. For example the bounds for the expressions in the first exponential in (3.7) have been obtained for the expressions extended onto the whole space 4ck (U ; 1, εk ). The only more delicate problem is with the main expression in the second exponential, the one in the curly brackets. Of course there is no problem with the analyticity, in fact it is analytic again on the “large” space determined by the absolute constants; the problem is with a bound. We cannot estimate it directly using bounds for δψ, δφk1 ,k , because this way we obtain inevitably a very large bound, of the order of the ones coming from the large field restrictions, or even larger. We have used before, in the proof of (3.10), the positivity, or rather non-negativity properties of the expression, and we should use them again properly for the analytically extended one. The first sum in the curly brackets does not depend on ψk and is non-negative, so we
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bound it by 0. We consider now the “fluctuation action” δA∗k (Zkc1 ∩ Z, Bk ; δψ, δφk1 ,k ) given by the formula (1.27) [5], with φ0 , α0 , δφ replaced by φ0k , αk0 , δφk1 ,k . Let us recall that δψ = R(ψ0 )ψ˜ k1 − ψk0 1 ,k on Bk1 (Wk1 ) ∩ Uk∼3 , δψ = ψ 00 on Bk ∩ (Uk∼3 )c ∩ ck , so δψ 1 1 has an analytic extension in ψk . Let us recall also that the functions in the above action are determined by the variational problems without the external field, i.e. for νk = 0, hence they do not depend on h. The only analyticity for these functions is with respect to ψk . We extend δψ, δφk1 ,k to complex configurations ψk + ψk0 ∈ 4ck (U ; 1, εk ), and we expand them in ψk0 up to the first order. This yields an expansion of δA∗k . The idea is to use the non-negativity properties of the real part of δA∗k , i.e. the part depending on ψk only. The remainder in the expansion has a bound with the additional factor εk , because |ψk0 | < 2εk , hence it should be already small. To prove this statement we have to reconsider the bounds on δψ, δφk1 ,k obtained in Lemmas 2.3, 2.4, because we have different restrictions on the variables ψk now. Consider δψ at first. On Bk1 (Wk1 ) ∩ Uk∼3 1 it is given by the formula δψ = R(ψ0 )ψ˜ k1 − ψk0 1 ,k = R(ψ0 )ψ˜ k1 − Q(Bk1 (Wk1 ))φ0k , so ˜ c (U ∼ ; c01 , c00 ) with some it is an analytic function of ψk + ψk0 defined on a large space 9 absolute constants, e.g. c01 = K6−1 c1 , c00 = K6−1 c0 , where c1 , c0 are the constants in Proposition 1.1 [2]. Expanding in ψk0 we obtain δψ(ψk + ψk0 ) = δψ(ψk ) + δψ 0 (ψk0 ), and |δψ 0 (ψk0 )| ≤ O(1)|ψk0 | + K2 |ψk0 | = const.|ψk0 |, hence |δψ 0 (ψk )| < const. εk for ψk + ψk0 in the space. To bound δψ(ψk ) we apply again the argument used before in the proof of by the characteristic functions Lemma 2.3. We have still |∂ 1 ψk0 1 | < 3δk1 on Wk∼1 ∩ Uk∼4 1 c ˜ ˜ in (3.7), but now ψk0 1 = e1 on (Uk∼3 ) , hence |R(ψ ) ψ 0 k1 (x) − ψ0 | = |ψk1 (x) − e1 | < 1 (k) ∼3 3δk1 (3M Rk1 + 8M ) for x ∈ Bk1 (Wk1 ) ∩ Uk1 . Take z ∈ T1 ∩ Z0 such that x ∈ 1k (z), and take z0 as a point closest to z on the boundary ∂(T1(k) ∩ Z0c ). Then |z − z0 | ≤ 7, |ψ0 −ψk (z0 )| < 90dεk , |ψk (z0 )−ψk0 (z)| < K6 7·3εk , |ψk0 (z)−ψk0 1 ,k (x)| < K1 K6 3εk , but εk = εk1 L−αn1 < εk1 βk−1 by the condition (2.31), hence 1 |R(ψ0 )ψ˜ k1 (x) − ψk0 1 ,k (x)| < (9M Rk1 + 24M )δk1 + (90d + 21K6 + 3K1 K6 )εk1 βk−1 1 < 10M Rk1 δk1 , and |δψ| < 10M Rk1 δk1 on Bk1 (Wk1 ) ∩ Uk∼3 , |δψ| < δk1 ,1 1 on 3k1 ∩ (Uk∼3 )c , |δψ| < δj,1 on 3j , j = k1 + 1, . . . , k − 1, where δψ = δψ(ψk ). 1 (3.11) Notice that the above bounds hold for the whole configuration δψ(ψk + ψk0 ), because −1
const.εk < const.εk1 βk−1 < const.εk1 βk1 2 , but they are not sufficient for our purposes, 1 except for a generalization of the bound (2.60) discussed again below. Consider now the function δφk1 ,k (δψ). It is an analytic function of φ0k , αk0 , δψ, hence of ψk +ψk0 , and we write this dependence explicitly in δφk1 ,k (ψk +ψk0 , δψ). By the results of Sect. 3 [2], in particular Proposition 3.1 [2], this function is analytic on the large ˜ c (U ∼ ; c01 , c00 ) × {|δψ| < c06 }, c06 = K −1 c6 , and it satisfies the bounds (3.47) [2], space 9 2 or the corresponding bounds with K2 δ1 replaced by K2 |δψ|. We substitute the function δψ(ψk + ψk0 ) = δψ(ψk ) + δψ 0 (ψk0 ) = δψ + δψ 0 in place of the variable δψ, and we expand in δψ 0 , i.e. we write 0 0 δφk1 ,k (ψk + ψk0 , δψ + δψ 0 ) = δφk1 ,k (ψk + ψk0 , δψ) + δφ(1) k1 ,k (ψk + ψk , δψ; δψ ). (3.12)
We assume that δψ, δψ 0 are small enough, e.g. |δψ|, |δψ 0 | < 13 c06 , which obviously holds for β large enough, and from the analyticity and the above bounds we obtain −1 0 c c |δφ(1) k1 ,k | ≤ 3K2 |δψ | < const.εk < const.εk1 βk1 on the domain Zk1 ∩ k1 +1 ,
Large Field Renormalization for Classical N -Vector Models
523
and the same bound for the remaining expressions on the left-hand side of (2.49). On the remaining layers j ∩ cj+1 we obtain better bounds using the exponential decay . properties of δφk1 ,k , and the fact that δψ 0 has a support contained in Bk1 (Wk1 ) ∩ Uk∼3 1 We write 0 = δφ (δψ + δψ ) − δφ (δψ) = δφk1 ,k (δψ + δψ 0 ; s = 1) δφ(1) k1 ,k k1 ,k k1 ,k ∼3 − δφk1 ,k (δψ + ψ 0 ; s = 0 on Uk∼3 ) + δφ (δψ; s = 0 on U ) − δφ (δψ; s = 1) , k1 ,k k1 ,k k1 1 and on the domain k1 +1 ∩ ck1 +2 we obtain the bound 1 −R0 log βk1 a < βk−2 < εk1 +1 βk−1 , |δφ(1) k1 ,k | < c6 exp(− γ0 M Rk1 ) < c6 βk1 1 +1 1 +1 4 the same bound for the remaining expressions. On the successive layers we obtain the corresponding additional exponential factors, hence bounds become very small, and we certainly have −1 c |δφ(1) k1 ,k | < εj βj on j ∩ j+1 , j = k1 + 1, . . . , k, k+1 is replaced by Z.
Take now the function δφk1 ,k (ψk + ψk0 , δψ) on the right-hand side of (3.12). From the bounds (3.11) we obtain the inequalities (2.49) in Lemma 2.4, but with the constants d−2 on the right-hand sides replaced by 20K3 L 2 M Rj δj . In fact we can replace them by d−2 2K3 L 2 δj,1 for j > k1 , but it is not important. These inequalities hold on the whole large analyticity domain in ψk + ψk0 , and we expand in ψk0 , 0 δφk1 ,k (ψk + ψk0 , δψ) = δφk1 ,k (ψk , δψ) + δφ(2) k1 ,k (ψk , δψ; ψk ).
(3.13)
By the analyticity the function δφ(2) k1 ,k satisfies the above discussed bounds for δφk1 ,k , with the right-hand side multiplied by 6K6 |ψk0 |, or by 12K6 εk . Thus we have obtained the expansion δφk1 ,k (ψk + ψk0 , δψ + δψ 0 ) = δφk1 ,k (ψk , δψ) + δφ0k1 ,k (ψk0 , δψ 0 ),
(3.14)
(2) where the function δφk1 ,k (ψk , δψ) is real, and δφ0k1 ,k = δφ(1) k1 ,k + δφk1 ,k is generally a complex function satisfying the bounds
|δφ0k1 ,k |, Lj η|∂ η δφ0k1 ,k |, (Lj η)2 |1η δφ0k1 ,k |, (Lj η)2 |δαk0 1 ,k | < const.(Rj δj εk + εj βj−1 ) on the domain j ∩ cj+1 ∩ Zkc1 ∩ Z, j = k1 , . . . , k.
(3.15) These bounds hold also with the exponential factor in (2.49) on the right-hand sides, but it is not important here. The real function δφk1 ,k = δφk1 ,k (ψk , δψ) satisfies the above d−2 bounds with the constants 20K3 L 2 M Rj δj , or const.Rj δj , on the right-hand sides. Finally we are ready to estimate the analytically extended action βk δA∗k (Zkc1 ∩ Z, Bk ; δψ, δφk1 ,k ). We separate the term βk hδφk1 ,k , (νk + αk0 )δφk1 ,k i from the action, and we consider it below. The rest of the action is denoted by the same symbol in this paragraph. We expand δψ, δφk1 ,k into the sums of the real and the complex functions, as above. This yields the corresponding expansion of the action
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T. Balaban
βk δA∗k (Zkc1 ∩ Z, Bk ; δψ + δψ 0 , δφk1 ,k + δφ0k1 ,k ) = βk δA∗k (Zkc1 ∩ Z, Bk ; δψ, δφk1 ) c ∗ c + βk δA0∗ k (Zk1 ∩ Z, Bk ), where δAk (Zk1 ∩ Z, Bk ; δψ, δφk1 ,k ) ≥ 0.
(3.16) The action is the sum of the three quadratic terms in (2.24) [5], hence δA0∗ k is the sum of the corresponding six quadratic terms. They all can be estimated in the same way, for example consider the term βk h∂ η δφk1 ,k , ∂ η δφ0k1 ,k i(Zkc ∩Z)∗ . We estimate it as in (2.60). 1 At first we write it as the sum of the terms corresponding to the layers j ∩j+1 ∩Zkc1 ∩Z, and on such a layer we rescale the corresponding term to the ξ-scale. This yields the factor (Lj η)d−2 , which combined with βk yields βj . We estimate the term using the bounds (3.15), and the bounds for δφk1 ,k . This yields
|βk ∂ η δφk1 ,k , ∂ η δφ0k1 ,k (Z c
k1
· (10LM Rj )d ≤ const.0 k X
∩Z)∗
|<
k X
βj ( const.)2 Rj δj (Rj δj εk + εj βj−1 )
j=k1
k X
− 21
εk (log βj a)2p0 +2d+4 + εj (log βj a)p0 +2d+2 βj
j=k1
1 d −1 ( )2p0 +2d+4 εk1 (log βk1 a)2p0 +2d+4 βk1 2 L− 2 αn1 α j=k1 1 1 1 −1 +εj (log βj a)p0 +2d+2 βj 2 < const.00 (L− 2 αn1 (n1 + 1) + 1)β − 4 ≤ const.000 β − 4 , (3.17) where we have used the inequality (2.32) in the third bound above. It is easy to see that 0 all the terms in δA0∗ k can be estimated in the same way, the terms quadratic in δφk1 ,k have even stronger bounds. The result is that for this function we obtain the bound
< const.0
c −4 ≤1 |βk δA0∗ k (Zk1 ∩ Z, Bk )| < const.β 1
(3.18)
for β large enough. The last inequality is unimportant, it is enough to have a bound by a constant. Let us finally reconsider the bounds (2.60) in the present situation. They hold as they are written, in fact with better constants, but with the bound 3K1 K6 δk of (Lj η)2 αk0 replaced by the bound 6K1 K6 εk + (Lj η)2 βk−1 of (Lj η)2 (αk0 + νk ). It is easy to see that we obtain the bound O(1), for example 1 as in (2.60), if we estimate the last sum over j in the same way as in (3.17). All the above results and bounds lead to the following statement: the function δA∗k (Zkc1 ∩ Z, Bk ; δψ, δφk1 ,k ) has an analytic extension onto the space 4ck (U ; 1, εk ), and it satisfies the inequality | exp −βk δA∗k (Zkc1 ∩ Z, Bk ; δψ, δφk1 ,,k ) < eO(1) , or simply < e, (3.19) on this space. This ends the analysis of the most difficult expression in the exponentials in (3.7). The next two terms in the second exponential are bounded in the now standard way, in fact they are particularly simple because of the inequality βk νk ≤ 1 and the exponential decay properties of δφk1 ,k yielding a large power of βk−1 . The function Ck1 ,k and the boundary terms have the same bounds as before. We have to notice only that values of the function ((R(ψ0 )ψ, R(ψ0 )ψ 0 ) U ∼3 , (ψk0 1 ,k + ψ 00 ) (U ∼3 )c ) belong to the k1 k1 analyticity domains of the functions, for example to the domain (1.32) [5]. It is easy to
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see that this is a simple consequence of the proof of Lemma 3.5, and the bounds leading to the bounds (3.11). Let us formulate the obtained results in the following lemma. Lemma 3.1. The kernel of the integral operator (3.7), without the characteristic functions in front, has an analytic extension in (ψk , h), g onto the space 4ck (U ; 1, εk )×{g Z : kgk`1 (Z) < 1}, and the operator satisfies the bound (3.10) with the integral over ψk restricted to Z replaced by the supremum of values of the operator over this space. The above result is very important and it will be used at the end of this section. Some expressions involving the integral operators will be “exponentiated” and will become a part of the function R(k) m , hence eventually a part of the effective action after the additional m steps. The lemma assures that they have the analyticity properties required by the inductive hypotheses. Consider now the expression on the right-hand side of (3.6). We would like to transform it in such a way that we obtain the whole small field expression on the domain Ykc in the exponential. To achieve this we apply several operations almost identical to ones performed and discussed previously. At first in each component Z of Zk we introduce the decomposition of unity. 1=
Y
(χk () + χck ()) =
∈πk :⊂Z000
X
χk (Z000 ∩ Sk00c )χck (Sk00 ).
(3.20)
Sk00 ⊂Z000
A term in this decomposition with Sk00 6 = ∅ gives new large field restrictions, and the corresponding main action in the exponential in (3.6) generates the small field factor (3.8) [6] with βk , Sk00 instead of βk+1 , Sk+1 . This makes the component Z a component of the first kind again, because the new small factor “cancels” the bound (3.8) and controls a large number of additional steps. Thus we include all the components of ZK with non-empty regions Sk00 into the large field region Yk , modifying also the multi-indices A00k by including Sk00 . This yields the final definition of the large field region Yk and the multi-indices. The integral operator corresponding to such a new component of Yk is defined by (3.7) multiplied by a corresponding term on the right-hand side of (3.20), and the exponential with the corresponding main action from the first term in the exponential in (3.6). In the remaining components, whose union is now equal to the final region Zk , there are the characteristic functions χk (Z000 ). Combining them with the functions in (3.7) we obtain χk (Z000 )χk (Z ∩ Z000c )χ˜ k (Z000 ∩ Z0c )χZ0 = χk (Z).
(3.21)
The product of the first two functions above is equal to χk (Z), and then obviously χ˜ k (Z000 ∩Z0c ) = 1. By the restrictions introduced by the characteristic functions χk and by Lemma 2.1 we obtain that |ψk − ψk0 | < 21(K6 + 1)δk , hence χZ0 = 1 if C6 ≥ 21(K6 + 1). We combine the characteristic functions χk (Z) for all components of the final region Zk with χk ((Yk ∪ Zk )c ), and we obtain χk (Ykc ). We consider the integral operator (3.7) for a component Z of Zk without the characteristic functions in front of it, so it is the operator to which Lemma 3.1 applies. Our next goal is to transform the expression in the exponential in (3.6) into the complete small field expression on the region Ykc , plus some inevitable boundary terms. We do it using formulas and operations applied already several times previously, so we discuss them very briefly here. At first we expand various background configurations around the configurations determined by the small field region Ykc , for example
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T. Balaban
φk Bk (((Yk ∪ Zk )c )≈2 ); ψ˜ k ) = φk (Bk ((Ykc )≈2 ); ψ˜ k (3.22) + δφk Bk (((Yk ∪ Zk )c )≈2 ); ψ˜ k − ψ(Bk (((Yk ∪ Zk )c )≈2 ), Bk ((Ykc )≈2 ); ψ˜ k ) . The configuration inside the second function on the right-hand side is equal to 0 on the set Bk ((Yk ∪ Zk )c )≈2 ∩ (Zkc )≈2 , it may not be equal to 0 only on the boundary layers Bk ((Yk ∪ Zk )c )≈2 ∩ ((Zkc )≈2 )c ∩ ((Zkc )≈2 )∼ = Bk ((Zkc )≈2 ) ∩ ((Zkc )≈2 )c ∩ [ ∩ ((Zkc )≈2 )∼ = Bk ((Z c )≈2 ) ∩ ((Z c )≈2 )c ∩ ((Z c )≈2 )∼ . Z⊂Zk
We expand the main action on (Yk ∪ Zk )c using again the formulas (2.23)-(2.35) [5], (2.14) [6], and the analysis after (2.14) [6], i.e. we “pre-localize” the remainders in large cubes of (Yk ∪ Zk )c , and we construct their localization expansions. Localization domains of the expansions intersect at least one of the components Z of Zk , so they determine domains of the class Dk (modck ), and we resum terms of the expansions corresponding to the same localization domain. The obtained terms are new boundary terms, they satisfy the conditions of (H.7) [5], but with B0 replaced by a small constant, because of the factors exp(− 41 γ0 (2Rk − 2M1 )) coming from the exponential decay properties (2.55), (2.56) [3]. We apply the same operations to the first main action in (3.6), or rather to the terms of the corresponding sum over the components Z of Zk , and we obtain this action at the new background configuration plus a sum of new boundary terms with properties described above. Thus we obtain that the sum of the main actions in (3.6) is equal to X B 0(k) (X; ψk ), (3.23) −βk A∗k (Ykc ; ψk , φk (Bk ((Ykc )≈2 ); ψ˜ k )) + X∈Dk (modck )
where the boundary terms depend on the k th spin variable ψk only, besides h, and satisfy all the conditions in the inductive hypothesis (H.7) [5], with the constant B0 replaced by an arbitrarily small constant, in fact by an arbitrary power of βk−1 for βk large enough. Consider now the effective action in (3.6). It is the sum of the actions E (j) ((Yk ∪ Zk )c ; ψk(j) (ψ˜ k )), and we expand the background configuration ψk(j) (ψ˜ k ) = ψ(Bj (((Yk ∪ Zk )c )≈2 ), Bk (((Yk ∪ Zk )c )≈2 ); ψ˜ k ) around the new configuration ψ(Bj ((Ykc )≈2 ), Bk ((Ykc )≈2 ); ψ˜ k ) using (3.22) and the standard formula (4.21) [2]. We write the expansion E (j) ((Yk ∪ Zk )c ; ψk0(j) (ψ˜ k )) = E (j) ((Yk ∪ Zk )c ; ψk(j) (ψ˜ k )) Z 1 d dt E (j) ((Yk ∪ Zk )c ; ψk(j) (ψ˜ k ) + δψk0(j) (tδψ)), + dt 0
(3.24)
where ψk0(j) is the “old” background configuration, ψk(j) is the “new” one, and δψ is the difference inside the second function on the right-hand side of (3.22). Basically the same expansion has been studied in Sect. 2 [6], see (2.19) [6] and Proposition 2.1 [6], and we have the same conclusion here, namely the sum over j of the second terms on the right-hand side of (3.24) is a sum of boundary terms having the same properties as the terms of the sum in (3.23). The same is true for the generating functional. We do not need even to write a formula identical to (3.24). Thus we have replaced the “old” background configurations by the “new” ones in all the expressions in the exponential in (3.6).
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Next we extend all these expressions to the whole small field region Ykc . For the effective action it means that we write it in the form c c c (3.25) Ek (Yk ; ψk ) + Ek ((Yk ∪ Zk ) ; ψk ) − Ek (Yk ; ψk ) . It is easy to see that the difference above is a sum of terms with localization domains intersecting one of the components Z of the large field region Zk . Actually we have to take the representation (3.127)[1] discussed in Corollary 3.10 [1] for both terms in the difference, and then the difference is represented as a sum of corresponding “irrelevant” terms with localization domains intersecting a component Z. We construct localization expansions of these terms using Proposition 4.3 [3], and we resum terms with the same final localization domain from Dk (modck ). This resummation is controlled in the same way as in the proof of Corollary 3.10 [1], as it will be discussed in the last section, and we obtain again a sum of the form (3.23), with terms having all the properties discussed above, except that now the constant replacing B0 in the bounds (1.35) [5] is not small; in fact we can take the constant (3.129)[1] in the inequality (3.128)[1]. We write the remaining two expressions Fk ((Yk ∪ Zk )c ), R(k) ((Yk ∪ Zk )c ) as in (3.25), and the differences contribute again to the boundary terms as in (3.23). We have to do similar operations for Fk as for Ek , except that bounds are simpler; we do not need any special representation of the generating functional. For R(k) we need only to resum over domains determining the same domain in Dk (modck ). The whole analysis above is the same as in Sect. 2 [6]. Thus we have obtained the whole small field expression on the domain Ykc plus the sum of the boundary terms (3.23). Notice that these new boundary terms satisfy the bounds (1.35) [5] with the constant independent of B0 . To write R(k) ρk in a form convenient for further analysis we change slightly the notations. We write Zk as the union Z1 ∪ · · · ∪ Zn of its connected components. We have X χk (Ykc ) exp Ak (Ykc ) + Fk (Ykc ) + R(k) (Ykc ) R(k) ρk = Yk ,A00 ∩Yk k
· T00k (Yk , A00k ∩ Yk ) exp B (k) (Yk , A00k ∩ Yk ) n X X X Y · T00k (Zi , A00k ∩ Zi ) exp Ek |Zi | S n≥0 {Z1 ,...,Zn } (A00 ∩Z ) i=1 i k [ [ (k) 00 (k) 00 0(k) · exp B (Yk ∪ Zi , Ak ) − B (Yk , Ak ∩ Yk ) + B (Yk ∪ Zi ) .
(3.26)
S Notice that in the difference B (k) (Yk ∪ S Zi , A00k ) − B (k) (Yk , A00k ∩ Yk ) all the terms with localization domains disjoint with (ck ∩ Zi ) are cancelled, so the remaining terms have the domains intersecting at least one of the components ck ∩ Zi , similarly S 0(k) as the terms of B (Yk ∪ Zi ). We combine these terms together by taking S sums of terms with the same localization domains, and we denote them by B 00(k) (Yk ∪ Zi , A00k ). Notice also that for n = 0 this is equal to 0, so the last exponential in (3.26) is equal to 1. Our final goal is to “exponentiate” the sum in the curly brackets. This is done in two steps. At first we represent S this sum as a polymer expansion by taking the Mayer expansion of exp B 00(k) (Yk ∪ Zi , A00k ) and performing proper resummations. The Mayer expansion yields a sum over sets of “clusters”; each cluster {X1 , . . . , Xp } is determined
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T. Balaban
S by the condition that Xi is a connected domain. This domain intersects at least one component Zi . Let us denote by {Zi1 , . . . , Ziq } all the components it intersects. The S S )∼2 . For a given polymer X ∈ cluster determines the “polymer” X = Xi ∪ (Zi≈2 r c Dk (modk ∩ Yk ) we sum over all clusters and all sets {Zi1 , . . . , Ziq } determining the polymer. A bound of this sum is almost completely standard, and we will discuss a similar bound in the section on fluctuation integrals. The only difference here is that the bounds of the boundary terms are not small, and we have to use the exponential decay factors and the bounds (3.10) to obtain small factors controlling the sum. From (3.10) we obtain also an overall small factor for the “activity” of the polymer. Notice that we use here the exponential decay properties obtained after the last fluctuation integral and the scaling operation, and they are better than the properties formulated in the inductive hypothesis (H.7) [5]. The Mayer expansion and the resummation yield a representation of the expression in the curly brackets in (3.26) as a polymer expansion of the form X X ρ00k (X1 ) · . . . · ρ00k (Xn ), (3.27) 1+ n≥1 {X1 ,...,Xn }
where the sum is over sets {X1 , . . . , Xn } of disjoint domains Xi ∈ Dk (modck ). The activities ρ00k (X) are functions of ψ, ψ 0 , h, g restricted to X, where ψ, ψ 0 are now determined by the multi-index A00k ∩ Yk , in particular ψ = ψk on T1(k) ∩ ((k ∩ Yk ) ∪ Ykc ). By the inductive assumptions on the boundary terms, and by Lemma 3.1, they can be analytically extended onto the corresponding spaces, and they satisfy the bounds (1.35) [5] with κ and B0 replaced, for example, by 23 κ, βk−m−1 . Applying a theorem on “cluster expansions”, e.g. applying results of [9,11,13-15], we exponentiate the polymer expansion (3.27), and we obtain that the expression in the curly brackets in (3.26) is equal to exp P (k) (k ∪ Ykc ; ψ, ψ 0 , h, g), where X (3.28) P (k) (k ∪ Y c ; ψ, ψ 0 , h, g) = P (k) (X; ψ, ψ 0 , h, g). k
X∈Dk (modck ∩Yk )
The terms of the last sum have the same properties as the activities of the polymer expansion (3.27), but the bounds (1.35) [5] hold with B0 replaced by const.βk−m−1 ≤ 1 −m c 2 βk , for βk or β large enough. For domains X ⊂ Yk they depend on ψk , h, g restricted to X, and they satisfy all the conditions formulated in (H.4) [5] for terms of the function (k) is defined on the whole lattice, and we take R(k) m . If Yk = ∅, then the function P (k) R00(k) m (ψk , h, g) = P (T ; ψk , h, g).
(3.29)
The expansion in (3.28) of the last function is the localization expansion of R00(k) m described in (H.4) [5], or (H.4) [4], and its term for X ∈ Dk is equal to the corresponding term in the expansion (3.28) constructed for any Y such that X ⊂ Ykc , as it follows from the standard formula for a term in the cluster expansion. The function (3.29) satisfies also the symmetry properties formulated in (H.4) [5], so it satisfies all the conditions of the hypotheses (H.4) [5]. Proof of Theorems 2, 3 [5], 5 . Obviously we have been proving Theorem 2 since the beginning of Sect. 2. To finish the proof we have to introduce a few final definitions, or rather identifications. We take Ak = A00k ∩ Yk , and we denote again Yk by Zk . For components Z of Zk we have of course K(Z, Ak ∩ Z) > 0, by the construction of the operation R(k) . The sum of terms of the expansion (3.28) with domains X ⊂ Ykc = Zkc
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529
c c is equal to R00(k) m (Zk ), the remaining terms with domains X ∈ Dk (modk ∩ Yk ) = Dk (modk ), X ∩ Zk 6 = ∅, are included into the boundary terms. This yields the final definition of the terms, they satisfy all the conditions of the hypothesis (H.7) [5], with a constant in the bound (1.35) [5] independent of B0 . We can fix B0 as equal to this constant. It is clear that the corresponding operation R(k) on the level of the small field 0(k) , to which it adds the expression is equal to the identity on all of them except Rm satisfying all the conditions of (H.4) [5]. This ends the proof of Theorem function R00(k) m 2. Theorems 3[5], 5 are simple conclusion of Theorems 1,2 [5], 4 [6].
4. The Last Step of the Renormalization Group Procedure The analysis of the last step in the general case is very similar to the analysis in the small field case in Sect. 6 of [4]. We finish the previously described renormalization steps when the coefficient νk satisfies for the first time the inequality νk > 78 L−2 . The sequence {νk } is strictly increasing, more precisely νk+1 ≈ νk L2 , or νk+1 > νk 78 L2 , so the corresponding index k = kf is uniquely defined. To simplify formulas in this section we denote this index by k again. Consider the density ρk . It is given by the formula (1.1) [5] and it satisfies the hypotheses (H.1)–(H.5), (H.7) [5]. The term corresponding to a large field region Zk contains the characteristic function χk (Wk ), a part of it in the large field ˜ k (W ∼3 ; K −1 δk ), density ρ0k (Zk ), and the domain of this function contains the space 9 1 k ˜ k (Wk∼ ; 3δk ). The conditions defining the last space imply again and is contained in 9 the restriction (6.2) [4], i.e. the restriction |ψk − h| < (5L + 3)δk on Wk∼ . Expanding the main action around ψk = h we obtain a quadratic form with a positive lower bound on Zkc , as in (6.5) [4] and the discussion following that inequality, the lower bound of the same order as the lower bounds for the previous quadratic forms in the expansions (2.28), (2.29) [5] connected with the fluctuation integrals. Because of this we can finally perform the integration with respect to ψk without any further renormalization transformations, as in Sect. 6 of [4]. Thus we consider the integral Z XZ 0 c dψk ρk (Zk )χk (Zk ) exp Ak (Zkc ) dψk ρk = Zk
+
Fk (Zkc )
+R
(k)
(Zkc )
(4.1) .
In order to use the formulas and the results of the previous sections we follow as close as possible the procedure used there, which means a slight departure from the one used in Sect. 6 [4]. Let us go over the steps of this procedure, but let us discuss only new aspects involved. We do not introduce now the decomposition of unity (2.6) [5], so k+1 = 0k+1 by (2.9) [5]. Consider the integral over k+1 for the term of (4.1). The integration does not involve the integral operators in the large field density, and omitting them we obtain Z ∗ c c dψk k+1 χk (∼3 k+1 ) exp −βk Ak (Zk ; ψk , φ) + Ek (Zk ) (4.2) + Fk (Zkc ) + R(k) (Zkc ) + B (k) (Zk , Ak ) − Ek |Zkc | . The above integrals correspond to the integrals (2.16) [5], and the variational problem in Sect. 2 [5] is replaced by the problem
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T. Balaban
inf
ψk k+1
Ak Bk (Wk ); ψ˜ k , φk (ψ˜ k ) .
(4.3)
Formally it is a different problem than the one in Sect. 2 [5], which has been analyzed in the papers [2, 3]. It appears to be the same as (3.39) [5], but in fact it is the same as the problem (4.1) in [2] because the basic quadratic form is “massive”, i.e. it has an absolute positive lower bound determined now by the bound νk > 78 L−2 . In particular Proposition 4.1 [2] is valid for the problem (4.3), and it has a unique solution given by the formula ψ0(k) (ψ˜ k ck+1 ) = Qk φk,0 (ψ˜ k ck+1 ), ψ˜ k = Q∗ (Bk (Wk ))ψk , where φk,0 is the solution of the variational problem 1 hψ − Q(Bk )φ, a(Bk )(ψ − Q(Bk )φ)ic + Ak,0 (φ; h) inf k+1 φ1 2 1 hψ − Q(Bk )φ, a(Bk )(ψ − Q(Bk )φ)ic + Jk,0 (φ, α; h) , = inf sup k+1 φ1 α 2
(4.4)
(4.5)
1
and where Bk = Bk (Wk ), the definition of Jk,0 is given in (1.7)[2]. Notice that we do not need the term with the averaging operations on k+1 to generate a “mass” because there is the mass term in the actions Ak,0 , Jk,0 . Proposition 1.1 [2] is valid for the above problem also, and we obtain the unique solution φk,0 , or φk,0 , αk,0 defined on the corresponding space (1.11), (1.12) [2]. By the remark made at the beginning of this section the space is a small neighbourhood of the constant configuration equal to h. We take finally h as the unit vector, i.e. |h| = 1. To better understand properties of the solution φk,0 , αk,0 let us notice that we can take the approximate solutions φ0 , α0 in Sect. 3 [2] given by φ0 = h, α0 = 0. Hence φk,0 = h + δφk,0 , αk,0 = δαk,0 , and δφk,0 , δαk,0 are solutions of the variational problem 1 hδψ − Q(Bk )δφ, a(Bk )(δψ − Q(Bk )δφ)ic + δAk,0 (δφ) inf k+1 δφ1 2 1 hδψ − Q(Bk )δφ, a(Bk )(δψ − Q(Bk )δφ)ic + δJk,0 (δφ, δα) , = inf sup k+1 δφ1 δα 2 1 (4.6) where Bk = Bk (Wk ), δψ = (ψ˜ k − h) ck+1 . Propositions 3.1 [2], 2.1 [3] are valid for the above problem, and we obtain unique solutions δφk,0 (δψ), δαk,0 (δψ), which are exponentially decaying according to (2.55), (2.56) [3], as we move away from ck+1 . Similarly, the configurations (2.12) [5] are replaced by φk,0 (0 ; ψ˜ k ck+1 ) = h + δφk,0 (0 ; (ψ˜ k − h) ck+1 ), where the last one is the solution of (4.6) with Bk = Bk (0∼2 ), δψ = (Q∗ (Bk (0∼2 ))ψk − h) ck+1 . Instead of the functions χ0k+1 (0 ) defined by (2.13) [5] we take functions χk,0 (0 ) defined by obvious modifications of (2.13) [5]. Notice that χk,0 (0 ) = 1 for 0 ∩ (ck+1 )∼3 = ∅, so the decomposition of c ∼3 unity (2.14) [5] is introduced on the layer ∼3 modifications k+1 ∩ (k+1 ) . With these R (k) we obtain the equalities (2.15), (2.16) [5] with T replaced by dψk and without the first term in the exponential in (2.16) [5]. All the subsequent steps are done with the configuration ψ0(k) replacing ψ (k) . Notice in particular that the local configuration ψ0(k) (0 ) = ψ0(k) (0 , s Rk+1 (0 )c = 0) defined between (2.17) [5] and (2.18) [5] satisfies the equality ψ0(k) (0 ) = h if Rk+1 (0 ) ⊂ k+1 . Finally we obtain the equalities (2.47),
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(2.48) [5] with the corresponding changes, in particular the main action in (2.48) [5] is equal to
2 1
βk A∗k (Zkc ; ψ0(k) , φk,0 ) = βk ak ψk − Qk φk,0 c c + βk A∗k,0 (Zkc ; φk,0 ). (4.7) 2 Zk ∩k+1 The fluctuation integral is analyzed then in exactly the same way as in Sect. 3 [5], and we obtain the formula (3.18) [5]. The second and third terms in the exponential in (3.18) [5] is depend on ψ0(k) , and the remaining terms depend on (ψ ck+1 , ψ 0 ), where ψk0 k+1 ∩00c k+1 included also in ψ 0 . We do not renormalize anymore the obtained effective action, so we apply only a part of the analysis of Sect. 2 [6]. We do not introduce the decomposition of unity (2.1) [6], so the equalities (2.2)–(2.7) [6] have a simpler form with Sk+1 = ∅. c The definitions (2.8), (2.9) [6] determine the constant E0(k+1) (Zk+1 ; h), similarly the (k+1) c (k+1) c following analysis yields the functions F0 (Zk+1 ; h, g), Rn (Zk+1 ; h, g) for n < 0(k+1) c (Zk+1 ; h, g), plus the boundary terms. m, Rm We transform the main action (4.7) as in (2.11), (2.14)–(2.16) [6], using the expansion φk,0 = h + δφk,0 instead of (2.12) [6], and we obtain that it is equal to (2.17) [6] plus the boundary terms. The expression (2.17) [6] is now equal to
2 1
βk ak ψk − Qk φk,0 (∂k+1 ; ψ˜ k ck+1 ) c c 2 Zk ∩k+1 (4.8) ∗ c ˜ c + βk Ak,0 (Zk ∩ Zk+1 ; φk,0 (∂k+1 ; ψk k+1 )), c because the second term in (2.17) [6] is βk A∗k,0 (Zk+1 ; h) = 0. Next we apply the expanth sions (2.19), (2.20) [6] to the k effective action and the generating functional, and we c c ; h) = 0, Fk (Zk+1 ; h, g), plus the boundary terms. Thus the expression obtain Ek (Zk+1 in the exponential in (3.18) [5] is transformed into the sum of the above written terms c , (4.8), the boundary terms and the constants in (2.21) [6]. With the localized in Zk+1 corresponding definitions (3.1)–(3.3) [6] we obtain the representation Z X T0k+1 (Zk+1 , A0k+1 ) exp B 0(k+1) (Zk+1 , A0k+1 ) dψk ρk = Zk+1 ,A0k+1
c c c c ; h) − Ek000 |Zk+1 | + Fk (Zk+1 ; h, g) + F0(k+1) (Zk+1 ; h, g) · exp E0(k+1) (Zk+1
+
m−1 X
(4.9)
c 0(k+1) c R(k+1) (Z ; h, g) + R (Z ; h, g) , n k+1 m k+1
n=1
where the “primes” indicate the changes in the definitions, for example the renormalR ization transformation T (k) (ck+1 ) is replaced by dψk ck+1 , and so on. The integral operator T0k+1 is actually an integration, its value is independent of any spin variable, and it satisfies directly the inequality (1.26) [5] without the integration over ψk+1 Zk+1 on the left-hand side. By the inequality (1.31) [5] we have X |T0k+1 (Z, A0k+1 ∩ Z)F | ≤ exp(−p2 (βk ) − Hk |Z|) sup χ|F |. (4.10) A0k+1 ∩Z
ψ,ψ 0
Now we analyze the expression (4.9) in the same way as the operator R(k) in Sect. 3 starting with (3.25). We complete all the expressions in the exponential in (4.9) to the whole lattice expressions, and we obtain
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T. Balaban
Z
dψk ρk = exp −Ek+1,0 |TL−1 η | + Fk+1,0 (h, g) + R X · 1+
X
n≥0 {Z1 ,...,Zn }
X S
Y n
(A0k+1 ∩Zi )
0(k+1)
(h, g) ·
T0k+1 (Zi , A0k+1 ∩ Zi )
(4.11)
i=1
[ Zi , A0k+1 , · exp Ek+1,0 |Z| exp B 00(k+1) c replaced by the whole torus. where R0(k+1) is the sum of the “R-terms” in (4.9) with Zk+1 00(k+1) c The boundary terms B are obtained by adding the differences Fk+1,0 (Zk+1 ; h, g) − 0(k+1) c 0(k+1) 0(k+1) Fk+1,0 (h, g), R (Zk+1 ; h, g) − R (h, g) to the boundary terms B in (4.9), and then separating the terms with localization domains contained in one of the comabove there ponents Zi , so in the last exponential S S are boundary terms with localization domains intersecting the both regions Zi , ( Zi )c . The above representation is almost the same as (3.26), actually it is simpler because it corresponds to a special case in (3.26) when Yk = ∅ and ψk = h. We proceed in the same way as in Sect. 3 after (3.26). We construct the Mayer expansion and write it as a polymer expansion of the form (3.27). Let us discuss very briefly a bound ofSthe activity of this expansion. We write now a S local“polymer” Xas the union X = Xi ∪ Zir , where {X1 , . . . , Xp } is a cluster of S ization domains, and {Zi1 , . . . , Ziq } are the components intersecting the union Xi . The activity corresponding to the domain X is the sum of the expressions corresponding to all possible {Xi }, {Zir } determining X. A term in this sum can be bounded by q Y 1 1 exp −p2 (βk ) − Hk |Zir | + Ek+1,0 |Zir | + 2 B0 K0 d |Zir | ε M r=1 (4.12) p Y 3 2 c · ε exp −( κ − κ0 )dk+1 (Xi mod k+1 ) , 2 i=1
where ε can be chosen arbitrarily. We have introduced this parameter because the boundary terms are not small, the constant B0 is greater than any other constant in this procedure. The only difference now in comparison with the bound discussed in Sect. 3 is that we have to use for the first time the term −Hk |Z| in (1.31) [5]. Until now we have been using only the term −p2 (βk ), because the large field regions Z were small. Here they can be arbitrarily large and we use the S second term to obtain the missing exponential decay factors in the large field region Zir . We take ε12 = 3d K0 exp 23 dLκ, and we assume that the constant H0 in the equality Hk = H0 log βk is sufficiently large, so that we can estimate the sum of the bounds (4.12) by 3 1 exp(−p2 (βk )) exp −( κ − 2κ0 − 1)dk+1 (X) . (4.13) ε2 2 We have used the fact that there is at least one large field region in (4.12), or q ≥ 1. The constant in front of the exponential above can be bounded by an arbitrary power of βk−1 for βk large enough, hence the activities of the polymer expansion are very small again. We exponentiate the expansion, and we obtain that the expression in the curly brackets in (4.11) is equal to X P (k+1) (X; h, g), (4.14) exp P (k+1) (h, g), P (k+1) (h, g) = X∈Dk
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0(k+1) where the terms of the last expansion can be bounded in the same way as terms of Rm , 1 −m−1 0(k+1) (k+1) (k+1) exp(−κdk+1 (X)). We denote Rm + P = Rm , and we obtain i.e. by 2 βk finally Z dψk ρk = exp −Ek+1,0 |TL−1 η | + Fk+1,0 (h, g) + R(k+1) (h, g) Z 1 (4.15) ∂ (k+1) R dt = exp Fk+1,0 (h, g) + (h, tg), g , ∂g 0 1
where we have used the normalization property Z dψk ρk (g = 0) = 1,
(4.16)
following from the basic normalization properties (1.12) [1], (1.43) [5]. The expression in the last exponential is the generating functional of the correlation functions F (h, g). By the above formula and the inductive hypothesis (H.3) [5] it has the following form F(h, g) = hg, hi1 +
k X j=1
hg, M(j) (h, g)i1 +
m X
hg, M(k+1) (h, g)i1 . n
(4.17)
n=0
The functions M(j) satisfy the assumptions of the hypothesis (H.7) [4]. The functions satisfy the corresponding assumptions following from the hypothesis (H.4) [5], M(k+1) n which are the same as for the functions M(j) , except that the Euclidean symmetry is invariant with respect to the transformations properties are worse. The function M(k+1) n of the lattice TL(k+1+n) , but the bounds of the localization expansions are better. The n constant in front of the exponential factor can be taken as equal to βk−n for n > 0, and as in (H.7)[4] for n = 0. Notice that the term with n = 0 corresponds to F0(k+1) . As it has been mentioned in the Introduction we will give a more precise description of the generating functional in the following paper [7] on the correlation functions, so we postpone a discussion of the representation (4.17) to that paper.
References 1. Balaban, T.: A Low Temperature Expansion for Classical N -vector Models. I. A. Renormalization Group Flow. Commun. Math. Phys. 167, (1995) 2. Balaban, T.: The Variational Problems for Classical N -Vector Models. Commun. Math. Phys. 175, 607–642 (1996) 3. Balaban, T.: Localization Expansions. I. Functions of the “Background” Configurations. Commun. Math. Physics 182, 33–82 (1996) 4. Balaban, T.: A Low Temperature Expansion for Classical N -Vector Models. II. Renormalization Group Equations. Commun. Math. Phys. 182, 675–721 (1996) 5. Balaban, T.: A Low Temperature Expansion for Classical N -Vector Models III. A Complete Inductive Description, Fluctuation Integrals. Commun. Math. Phys. To appear 6. Balaban, T.: Renormalization and Localization Expansions II. Expectation Values of the “Fluctuation” Measures. Commun. Math. Phys. To appear 7. Balaban, T., O’Carroll, M.: Properties of correlation functions in N -vector models. Preprint 8. Balaban, T.: Commun. Math. Phys. a) 89, 571–597 (1983), b) 119, 243–285 (1988), c) 122, 175–202 (1989), d) 122, 355–392 (1989)
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9. Brydges, D.: A Short course in Cluster Expansions. In: Critical Phenomena, Random Systems, Gauge Theories. Les Houches, (1984) K. Osterwalder, ed. London–New york: Elsevier Science Publishers, 1986 10. Brydges, D., Dimock, J., Hurd, T.: Weak Perturbations of Gaussian Measures. In: Mathematical Quantum Theory I. Field Theory and Many Body Theory. Feldman, J., Froese. R., Rosen, L., eds. CRM Proceedings Lecture Notes 11. Cammarota, C.: Commun. Math. Phys. 85, 517–528 (1982) 12. Feldman, J., Magnen, J., Rivasseau, V., S´em´eor. R.: Commun. Math. Phys. 109, 473 (1987) 13. Abdessalam, A., Rivasseau, V.: An Explicit Large Versus Small Field Multiscale Cluster Expansion. Rev. Math. Phys. To appear 14. Rivasseau, V.: Cluster Expansions with Small/Large Field Conditions, In: Mathematical Quantum Theory I: Field Theory and Many Body Theory. Feldman, J., Froese, R., Rosen, L., eds., CRM Proceedings Lecture Notes 15. Glimm, J., Jaffe, A.: Quantum Physics: Functional Integral Point of View. New York: Springer, 1989 Communicated by D. C. Brydges
Commun. Math. Phys. 198, 535 – 590 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
On the Connectivity of Cobordisms and Half-Projective TQFT’s Thomas Kerler? Institute for Advanced Study, Princeton, NJ, USA Received: 20 October 1997 / Accepted: 18 March 1998
Abstract: We consider a generalization of the axioms of a TQFT, the so-called halfprojective TQFT’s, where we inserted an anomaly, xµ0 , in the composition law. Here µ0 is a coboundary (in a group cohomological sense) on the cobordism categories with non-negative, integer values. The element x of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be zero. This modification makes it possible to extend quantum-invariants, which vanish on S 1 × S 2 , to non-trivial TQFT’s. Note, that a TQFT in the ordinary sense of Atiyah with this property has to be trivial all together. We organize our discussions such that the notion of a half-projective TQFT is extracted as the only possible generalization under a few very natural assumptions. Based on separate work with Lyubashenko on connected TQFT’s, we construct a large class of half-projective TQFT’s with x = 0. Their invariants all vanish on S 1 × S 2 , and they coincide with the Hennings invariant for non-semisimple Hopf algebras and, more generally, with the Lyubashenko invariant for non-semisimple categories. We also develop a few topological tools that allow us to determine the cocycle µ0 and find numbers, %(M ), such that the linear map associated to a cobordism, M , is of the form x%(M ) fM . They are concerned with connectivity properties of cobordisms, as for example maximal non-separating surfaces. We introduce in particular the notions of “interior” homotopy and homology groups, and of coordinate graphs, which are functions on cobordisms with values in the morphisms of a graph category. For applications we will prove that half-projective TQFT’s with x = 0 vanish on cobordisms with infinite interior homology, and we argue that the order of divergence of the TQFT on a cobordism, M , in the “classical limit” can be estimated by the rank of its maximal free interior group, which coincides with %(M ). ? Present address: The Ohio State University, Columbus, OH, 43210 USA. E-mail:
[email protected]
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Contents 1 1.1 2 2.1 2.2 2.3 2.4 3 3.1 3.2 3.3 3.4 3.5 4 4.1 4.2 4.3 4.4 4.5 A.1 A.2 A.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 Survey of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Cobordism Categories, and Half-Projective TQFT’s . . . . . . . . . . . . . . . . 540 Categories of cobordisms, the structure of Cob3 (∗) . . . . . . . . . . . . . . . . 541 Elementary compositions, and the cocycle µ0 . . . . . . . . . . . . . . . . . . . . 542 Half-pojective TQFT’s, and generalizations . . . . . . . . . . . . . . . . . . . . . . 545 TQFT’s for cobordisms with corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Non-Separating Surfaces, Interior Fundamental Groups, and CoordinateGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 r-diagrams of non-separating surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 Interior fundamental groups, and an a-priori estimate on %(M ) . . . . . . . 552 The graph-category 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Coordinate graphs of cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 Existence of coordinate graphs from interior groups . . . . . . . . . . . . . . . . 559 Construction of Half-Projective TQFT’s . . . . . . . . . . . . . . . . . . . . . . . . . 561 Surface-connecting cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 Basic constraints on generalized TQFT’s . . . . . . . . . . . . . . . . . . . . . . . . 565 The example of extended TQFT’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Integrals, semisimplicity, and x = V (S 1 × S 2 ) . . . . . . . . . . . . . . . . . . . . 574 Main result, and hints to further generalizations and applications . . . . . 579 Proofs of subsection 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 The spaces H1int (M, G), the numbers βjint (M ), and further anomalies . 584 Summary of tangle presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
1. Introduction Although physical motivations were at its origin, the notion of topological quantum field theories (TQFT’s) has become a part of algebraic topology, since it was axiomatically defined by Atiyah [A]. In the same way as for example homology or homotopy, it is given as a functor from a topological category into an algebraic category. More precisely, it is a functor of the following symmetric tensor categories: V : Cobd+1 −→ R − mod ,
(1.1)
where Cobd+1 has as objects d-manifolds and as morphisms d + 1-dimensional cobordisms between them, and R−mod is the usual category of (free) R-modules and R-linear maps. Their tensor products are the disjoint union t and ⊗R , respectively. (In the original definition R is a field, but quite often we shall only require R to be a ring). We may think of V as a representation of the algebra of cell-attachments to the boundaries of d + 1-manifolds. But unlike, e.g., homology it detects algebraically much more involved relations between the cells than just their intersection numbers. In this paper we shall be exclusively concerned with the case d = 2, where this algebra corresponds to quantum groups. We will make extensive reference to known TQFT’s in 2+1 dimensions, and use results of three-dimensional, geometric topology. The original motivation of this paper is to resolve a paradox that occurs in several different examples of “quantum-invariants” of 3-manifolds and concrete quantum field theories. It is about a degeneracy that at first sight appears to prevent us from constructing a TQFT in the rigorous, axiomatic sense. The problem is resolved by a seemingly
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minor modification of the axioms of a TQFT, yielding what we shall call half-projective TQFT’s. The generalization naturally leads us to several questions about the connectivity of cobordisms, for which we will develop several tools that should have applications also to other topological problems. Specifically, the phenomenon that we are interested in is that sometimes the invariant of a “quantum-theory” vanishes on the product of sphere and circle, i.e., x := V(S 1 × S 2 ) = 0.
(1.2)
It is an elementary implication of the axioms of a TQFT, observed in [Wi] but also [A], that for a surface, 6, the invariant of the circle product is the dimension of the associated vector space, i.e., (1.3) V(S 1 × 6) = dim V(6) . Hence (1.2) entails also triviality of the vector space, V(S 2 ) = 0. A dramatic consequence of this for a TQFT is that V ≡ 0 on all surfaces and cobordisms. The reason is easily seen, if we remove from a general cobordism M : 6s → 6t a ball so that M ∗ := M − D3 : 6s → 6t t S 2 . Expressing the regluing of D3 as a composition of cobordisms, we obtain M∗ D3 M : 6s −−−−→ 6t t S 2 −−−−→ 6t .
(1.4)
An application of V to this yields the assertion, since the middle surface is mapped to the zero dimensional vector space. Clearly, this means that a (non-trivial) invariant can be extended to a TQFT only if it does not vanish on S 1 × S 2 . However, the examples, in which the degeneracy of (1.2) is encountered, do appear to have a lot of the structural properties of a TQFT, and are very closely related to situation where non-trivial TQFT’s actually exist. An example of a more algebraic nature is the Hennings invariant of 3-manifolds, see [H], for a finite dimensional, quasi-triangular Hopf algebra, A. Its construction is analogous to that of the invariant of Reshetikhin-Turaev in [RT], also [T], except that special elements of A are used directly instead of the representation theory of A. The two invariants can be put on the same footing [Ke3] via the Lyubashenko invariant [L2], which is defined for abelian braided tensor categories. The invariant of [RT] can be extended to a TQFT, which is usually identified with the Chern Simons quantum field theory. Nevertheless, it is easy to see that the (non-trivial) Hennings invariant vanishes on S 1 × S 2 if (and only if) A is not semisimple. Subsequent studies in [L1,L2,KL], and [Ke3] showed that the latter invariants can still be associated to representations of mapping class groups and, more generally, TQFT’s for cobordisms of connected surfaces. Indeed in [Ke3] we show very explicitly how the Hennings algorithm is extended to tangles representing cobordisms between connected surfaces. This reveals that the vanishing paradox in (1.2) has to have its origin in basic connectivity properties of cobordisms. Similarly, there are arguments that suggest that for the Kuperberg invariant [Ku] for a Hopf algebra B is the same as the Hennings invariant for the double D(B). If B is not semisimple, also the Kuperberg invariant vanishes on S 1 × S 2 , for similar reasons as for the Hennings invariant. Nevertheless it is constructed in a similar way as (and specializes for semisimple B to) the Turaev-Viro invariant [TV], for which we always
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have TQFT’s. Although implicitly contained in the Hennings description, an explicit extension of Kuperberg’s prescription to cobordisms is still missing. A second example in the concrete physical framework of quantum field theory is given by conformal field theories and their corresponding Chern Simons theories, whose gauge groups are supergroups. The case of U (1, 1)-WZW-models has been worked out in [RS]. In this model we have the same vanishing for the ratios of partition functions V(S 1 × S 2 ) := Z(S 1 × S 2 )/Z(S 3 ) = 0. In the attempt to construct the operators of a TQFT (in some regularization with parameter ε → 0) one is faced in [RS] with serious singularities of matrix elements in ε that can only be removed at the price of having degenerate inner products for the physical state spaces V(6). We show in this article that there is essentially only one way to modify the axioms of a TQFT, such that we preserve the tensor product rule for V, and ensure that V respects gluings of cobordisms over connected surfaces. The answer is provided by the notion of a half-projective TQFT. By this we mean a map between the category of cobordisms to the category of R-modules, which is a functor as in (1.1), except that the composition law is of the form (1.5) V M ◦ N = xµ0 (M,N ) V(M )V(N ). Here µ0 is a “coboundary” on Cob3 in the sense of group cohomology, when we view categories as generalizations of groupoids. µ0 (M, N ) can be computed from basic connectivity data of M and N , and it has values only in the non-negative integers, µ0 ∈ Z0,+ . If the number x ∈ R is invertible (e.g., x 6= 0 and k = R is a field) then the anomaly can of course be removed by rescaling V. Also, if M and N are composed over only one connected component, we find that µ0 (M, N ) = 0 so that V behaves like an honest functor. This is consistent with the connected TQFT-functors in [KL]. The identity (1.3), however, is now modified. Repeating the original derivation with (1.5) we find (1.6) V(S 1 × 6) = x dim V(6) . Hence the above examples are not in contradiction with extensions to half-projective TQFT’s, if we set x = 0. One of the main results of this paper is the construction of a large class of non-trivial, half-projective TQFT’s with x = 0. Our starting point here are the connected TQFT’s from [KL], but as in the U (1, 1)-WZW-models we have to deal with degenerate pairings of the spaces V(6). The TQFT’s we find extend, in particular, the Hennings invariant for an arbitrary non-semisimple, finite-dimensional, modular Hopf algebra. In the quantum-algebraic framework, the element x (and, especially, the fact whether it is trivial or not) is intimately related to semisimplicity and cointegrals of the respective categories or Hopf algebras. In a concrete quantum field theory x may be seen as a parameter for the renormalization of the product of field operators. It is interesting to observe that the two non-semisimple examples from above share a few more common features beyond (1.2). In both cases we find that the representations of the mapping class group SL(2, Z) of the torus contains algebraic summands and tensor factors (the semisimple ones only produce finite representations), and that the invariants of lens spaces and Seifert-manifolds are proportional to the order of the first homology group, see [RS] and [Ke3], and references therein. We will investigate these properties in the general, axiomatic setting in separate work. We will also see that for any half-projective TQFT with x = 0, the associated invariant must vanish on any closed manifold with non-zero first Betti number. This vanishing
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property is shared by certain normalizations in the construction of the Casson and other finite type invariants, which makes extensions of these invariants beyond homology three spheres difficult. This naturally leads to the question whether the finite type invariants can be reproduced in a non-perturbative fashion, via a special class of non-semisimple braided tensor categories. The vanishing of invariants, if manifolds are connected in certain ways, is also observed in one dimension higher, e.g., for the Seiberg-Witten invariant. It is nearby to speculate that behind these phenomena analogous mechanisms are at work such as the ones discussed in this paper. 1.1. Survey of contents. In Sect. 2 we provide the definitions of the cobordism category Cob3 (∗) (Subsect. 2.1), the special cocycle µ0 (Subsect. 2.2), the notion of half-projective TQFT’s (Subsect. 2.3), and various versions of extended (half-projective) TQFT’s (Subsect. 2.4). In Subsect. 2.2 we also give an explicit formula (see Lemma 2) and an algorithm as in (2.16) for the computation of µ0 . Subsect. 2.3 includes a discussion of the basic implications of the axioms of a half-projective TQFT. In particular, it is shown in Corollary 1 that, generically, the only properly half-projective, indecomposable TQFT’s are those with x = 0, even for general rings, R. The purpose of Sect. 3 is to develop the topological means that allow us to treat the connectivity properties of cobordisms relevant to the formalism of half-projective TQFT’s. In this we are mainly motivated by the result in Lemma 6, which asserts that V(M ) = x%(M ) fM for some “regularized” R-linear map fM , where %(M ) is given by the maximal number of non-separating surfaces in M . Note, that in general x%(M ) may generate non-trivial ideals in the respective space of linear maps, seen as an Rmodule. In order to be able to compute the number %(M ), we show in Theorem 4 of Subsect. 3.5 that it is identical with the maximal rank of a free interior group. The notion of interior fundamental groups, where we divide by the subgroup coming from the surfaces, is introduced in Subsect. 3.2 , and a basic gluing-property under compositions of cobordisms is described in Lemma 8. In Subsect. 3.4 we define coordinate graphs of manifolds with boundary, which is a rather useful tool to the end of encoding the connectivity properties of a cobordism in a combinatorial way. Coordinate graphs are given by (Morse) functions on cobordisms with values in graphs that belong to the graph category from Subsect. 3.3. A result of particular interest is Lemma 13, which ensures that to a decomposition of coordinate graphs we can always find a corresponding connected decomposition of the cobordisms. An interesting application of the results in Sect. 3 is Corollary 5, which asserts that if V is a general, half-projective TQFT with x = 0, then V(M ) = 0 for any cobordism, for which β1int (M ) 6= 0, i.e., with infinite, interior homology. For the special case of the Uq (s`2 )-Hennings invariant (q a root of unity), evaluated on closed manifolds, M : ∅ → ∅, this vanishing phenomenon was also observed by a direct calculation in [O]. In Sect. 4 we show how non-trivial, half-projective TQFT’s can be constructed from connected ones as, e.g., those in [KL]. We start in Subsect. 4.1 with the discussion of an algebra of special cobordisms between a surface and the connected sum of its components. Using the existence of decompositions as in Lemma 19, these cobordisms allow us to express any cobordism by one that cobords only connected surfaces. In Subsect. 4.2 we start with the list of Axioms V1–V5 for a generalized TQFT, V, which essentially state that V respects tensor products as well as compositions over connected surfaces. We show that V necessarily has to be a half-projective TQFT. Moreover, we exhibit a list of eight properties that have to hold for a connected TQFT,
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if it should extend to a disconnected one satisfying V1-V5. The latter is true if it is, e.g., a specialization from a half-projective one. In Theorem 6 we show that these properties are in fact also sufficient, in order to guarantee the existence of such a half-projective TQFT. The purpose of Subsect. 4.3 is to show that all but one of these properties are automatically fulfilled, if the connected TQFT descends from an extended TQFT. In particular, we will use that V is a factorization of a functor V 1 : Cob3 (1) → C, where the objects of Cob3 (1) are surfaces with one hole, and C is an abelian, braided tensor category. For the connected TQFT’s the vector spaces V(6) are thus identified with invariances of special objects in C. When we pass to the disconnected case we have to divide by the null spaces of the pairing with the respective coinvariance, as it is stated in the summary in Lemma 22. In the derivation we actually first construct in (4.68) the linear morphism spaces, in which the τ -move of closed surfaces is realized, and then identify these in Lemma 20 as linear maps between the quotiented vector spaces. In Subsect. 4.4 we tie the last remaining property, regarding the projective factor, to the existence of a natural transformation of the identity functor of C, whose image for each object is a multiple of the unit, see (4.81). In Lemma 23 we show that non-triviality of the value of such a transformation on the unit object itself (which will be the same as x) is a necessary and sufficient condition for the semisimplicity of C. In the remainder of this section we establish the existence of such a transformation by identification with R × X, assuming that this is contained in C. the cointegral of the coend F = X ∨ Finally, in Subsect. 4.5 we combine the results of the previous sections and of [KL] in Theorem 9, in which we establish the existence of a large class of truly half-projective TQFT’s. We also use the last section to speculate on the possibility of constructing generalized TQFT’s, where we consider besides the tensor products also derived functors, like Tor, whose contributions may allow us to salvage some of the TQFT data that is lost in the division by the null spaces. As a further possible application of the formalism of half-projective TQFT’s we give a brief discussion of classical limits, for which x −→ ∞. We check for circle products the quality of the estimate kV(M )k ≥ const. x%(M ) , which is suggested by Lemma 6 and the normalizations used in the canonical construction of the invariants. The estimates turn out to hold in all of the considered cases, and they are roughly half of the true order of divergence. The proofs for the basic, technical lemmas on coordinate graphs are delivered in Appendix A.1. In Appendix A.2 we compute formulas for the coboundaries µ1 and µ∂ := µ1 − µ0 , which are generalizations in homology of µ0 . We find µ∂ ∈ Z0,+ . The corresponding anomaly in homotopy µπ (see Subsect. 3.2) counts the number of additional, non-separating surfaces in a product of cobordisms that do not stem from the composites. Thus we pick up an additional factor, xµπ , besides the one from the usual anomaly from (1.5). The tangle presentations of cobordisms from [Ke2], which we refer to in Sections 4.1 and 4.4, are summarized in Appendix A.3.
2. Cobordism Categories, and Half-Projective TQFT’s In this Sect. we shall define and discuss generalizations of the TQFT-axioms of Atiyah. To this end we first introduce in Subsect. 2.1 the cobordism category Cob3 (∗) , whose objects are compact Riemann surfaces with boundaries, and whose morphisms are homeomorphism classes of cobording 3-manifolds. Decompositions into connected components are expressed in Subsect. 2.2 in terms of the symmetric tensor structure of Cob3 (∗). Here we also introduce a coboundary, µ0 := −δβ0int , on Cob3 (∗), where the coboundary
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operator δ is again to be understood in the generalized, group cohomological sense, see (2.14). The value of µ0 (M, N ) is always a non-negative integer, and it is K −1, if M and N are two connected cobordisms that are glued together along K connected surfaces. This allows us in Subsect. 2.3 to define the notion of half-projective TQFT’s, V : Cob3 (0) −→ R − mod, on the category of closed surfaces, Cob3 (0), by inserting an anomaly of the form xµ0 into the composition law for V, where x does not have to be invertible. We show that - except for specified, exceptional situations - a half-projective TQFT-functor, V, is the sum of functors, V j , where each V j maps into the free modules of a summand, Rj ⊂ R, and the component of x in Rj is either zero or invertible. Finally, in Subsect. 2.4 we also discuss the various formalisms of extended TQFT’s, and how the notion of half-projective TQFT’s can be extended to Cob3 (∗). 2.1. Categories of cobordisms, the structure of Cob3 (∗). The cobordism categories, which we wish to consider here, are slightly more general than the ones defined, e.g., in [Ke1] or [KL]. The objects are as usual given by a set of inequivalent, compact, oriented Riemann surfaces, 6. Here we are only interested in 6 as a topological manifold. Moreover, we assume that 6 is equipped with an ordering of its components. We also fix parametrizations of the boundary components, ∂6 ∼ = tn S 1 . A morphism, M : 6s → 6t , is now defined between any two such surfaces with ns and nt holes, respectively, if the total number, ns + nt , of boundary components of ∂M is even. We may organize the set of holes into pairs, such that only holes of different surfaces are matched. For any such choice we glue in cylinders connecting the boundary components of a pair so that we obtain a closed surface as follows: nG ns +nt o 2 S 1 × [0, 1] ttnt S 1 6t . (2.7) 6cl = −6s ttns S 1 A cobordism consists now of an oriented, compact 3-manifold, M , and an orieng →∂M . Let us denote also the resulting tation preserving homeomorphism, ψ 0 : 6cl − inclusion of the source and target surfaces into the cobordism: ψ : −6s t 6t ,−−→ M.
(2.8)
These maps will be sometimes called charts or parametrizations. We shall denote the set of homeomorphie classes of such cobordisms with fixed numbers, ns and nt , of source and target surface components by Cob3 (ns , nt ). Below we depict a typical situation of how the boundary 6cl ∼ = ∂M is built up, where M : 6s → 6t is a cobordism in Cob3 (3, 5). z
61,s }| { α1
6s :
β2
'
|
β1
α2
{z 61,t
α4
{
α3
%
' 6t :
62,s }|
z
}
$
|
β4
β3
{z 62,t
}
(2.9)
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In this example the source and target surface both have two components, i.e., (1) (2) 6(3) s = 61,s t 62,s
and
(3) (2) 6(5) t = 61,t t 62,t ,
where the superscript at a surface displays the number of its holes. A surface component is indicated in diagram (2.9) by a fat, horizontal line with an interruption for each hole. We also introduced a labelling of holes so that the hole with label αj is connected to the hole with label βj by the j th cylindrical piece from (2.7), with j = 1, . . . , 4. Each of these pieces is depicted in (2.9) by a pair of thinner lines. For simplicity we have omitted in our example the possibility of crossings, although the cylinders may be arbitrarily braided and knotted inside the cobordism. Note that in this definition we are allowed to have a cylinder connect two holes from two different target (or source) surfaces, as for example the fourth cylinder between 61,t and 62,t in (2.9). For these we shall also specify a direction so that there is a distinguished start- (or end-) hole. In the above example we thus have to decide whether α4 or β4 is the start hole. The union of all Cob3 (n, m) shall constitute Cob3 (∗) as a set. The composition in Cob3 (∗) is given by gluing the two 3-manifolds along the intermediate surface, requiring that end-holes are glued to start-holes. In the process we may encounter a situation, where several cylindrical parts combine to form a closed (connected) surface in ∂M , which can only be a torus, T 2 . By construction this torus has a distinguished long meridian, with a given direction, and a distinguished short meridian, which carries also a direction due to the induced orientation. If we have similar data fixed on the respective surface, T 2 , that we chose as an object of Cob3 (∗), then there is (up to isotopy) a unique homeomorphism between the two tori. Hence we can add T 2 to either the start- of target-surface of the composite cobordism in a well defined manner. See also Subsect. 2.4, where these tori are interpreted as “horizontal objects” of a 2-morphism. On the morphism set we can also define a filling map φ0 : Cob3 (ns , nt ) → Cob3 (0),
(2.10)
which behaves nicely under compositions. It is given by gluing a full tube D2 ×[0, 1] into the cylindrical parts, such that the holes of −6s t 6t are closed with discs D2 × {0, 1}. The cobordism φ0 (M ) is thus between the same surfaces without punctures. In the definitions of [Ke2] and [KL] we only considered the case ns = nt , and the cylinders had to connect a source hole to a target hole. There φ is a true functor. Also, we considered a central extension of Cob3 (∗) by the cobordism-group 4 . This was naturally constructed by considering first also four-folds bounding the cobordism, and then retaining their signature as an additional structure to the 3-cobordism, see [Ke2] for details. We shall tacitly assume this extension here, too. 2.2. Elementary compositions, and the cocycle µ0 . In this section we shall introduce the coboundary µ0 on the cobordism category Cob3 (∗), which enters the definition of a half-projective TQFT. We start with the basics of the symmetric tensor structure of Cob3 (∗). We make the following straightforward observations: A category of cobordisms between closed surfaces admits a natural tensor product given by the disjoint union. This tensor product extends to the morphisms by disjoint unions in a functorial way, and it is obviously strictly associative.
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Recall that for Cob3 (0) we also assumed the connected components to be ordered. Thus if 6j and 60j are connected surfaces the tensor product of their ordered union 61 t. . .t6a ⊗ 601 t. . .t60b shall be the ordered union 61 t. . .t6a t601 t. . .t60b . In particular, this means that 6 ⊗ 60 and 60 ⊗ 6 are different objects. However, we g →60 ⊗ 6, in between them. In Cob3 (0) this can find a natural isomorphism, γ : 6 ⊗ 60 − means that γ is a cobordism, whichhasa two-sidedinverse. As a three-manifold with boundary it is given by 6 × [0, 1] t 60 × [0, 1] and the boundary identifications −→6 t 60 and 6 × {1} t 60 × {1}g −→60 t 6 are the obvious 6 × {0} t 60 × {0}g canonical maps. Also, it is clear that γ ◦ γ = 11. The system of morphisms γ = γ(6, 60 ) has in fact all the properties of the commutativity contraint of a symmetric tensor category. Specifically, it is natural with respect to both arguments, and the triangle equality is readily verified. Let us summarize in the following lemma the basic terminology that is natural from a categorial point of view and which provides a convenient language to organize reorderings of surface components and decompositions of cobordisms. In the statements for Cob3 (∗) we actually have to apply the adequate generalizations that take care of the non-trivial vertical 1-arrows, like the extra tori discussed in the previous section or renumberings of holes, see Subsect. 2.4. Lemma 1. 1. Cob3 (∗) and Cob3 (0) are strict, symmetric tensor categories. 2. For any set, {6j : j = 1, . . . , K}, of surfaces and any permutation, π ∈ SK , we have morphisms π ∗ : 61 ⊗ . . . ⊗ 6K → 6π−1 (1) ⊗ . . . ⊗ 6π−1 (K) , with π ∈ SK , which are natural in every argument, and which represent SK . M , such that there 3. For every morphism there is a unique, maximal number, b = β 0 is a decomposition in the form M = π1∗ ◦ M1 ⊗ . . . ⊗ Mb ◦ π2∗ . The cobordisms Mj are then all connected. The second remark in the lemma follows from the observation that cobordisms of the form σ ∗ = 11 ⊗ γ ⊗ 11, where γ is the commutativity constraint on the union of two consecutive, connected components, automatically fulfill the relations of the usual generators of the symmetric group. Hence for a permutation π ∈ SK we can define the cobordism π ∗ as the respective composition of the σ ∗ ’s. The last remark is simply expressing in categorial terms the fact that, up to reordering of the boundary components, every cobordism can be given as the union of its connected components. In later chapters we will return to the notation t instead of ⊗. Adding a sufficient number of cylinders to the right and left of each Mj in the formula ˘ j = 11X ⊗ Mj ⊗ 11Y , such in Part 3 of Lemma 1, we obtain commuting morphisms M that the tensor product M can be rewritten as the composite: ˘1 ◦ ... ◦ M ˘ b ◦ π2∗ . M = π1∗ ◦ M (2.11) With this it follows that the compositions of two morphism can be obtained as an iteration of two types of elementary composites. The first are of the form M ◦ π ∗ , where M is connected. The second is given by products as follows: (2.12) M = 11 ⊗ M2 ◦ M1 ⊗ 11 ,
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where M1 and M2 are both connected, and are glued over K ≥ 1 boundary components. After having made a gluing along one connected component, the identifications of the remaining (K − 1) boundary components are among boundary components of the same connected manifold. We will relate them to (K − 1) uncancellable one-handle attachments in either a direct decomposition in three-dimensions, or to a four-manifold bounding M . We wish to assign the excess number of connected boundary components over which we glue two cobordisms as a penalty in the form of a cocycle of Cob3 (∗). To this end. let us introduce the interior Betti-numbers 1 (2.13) βjint M = βj φ0 (M ) − βj ∂φ0 (M ) , 2 where βj = dim Hj (X) are the usual Betti-numbers. In this section we are interested only in the case j = 0, where we can omit the filling functor φ0 from (2.10) in the formula. Some computations for j = 1 are given in Appendix A.2. It is easily seen that any number β0int (M ) ∈ 21 Z is realized. They define a coboundary on Cob3 (∗) with coefficients a-priori in the half integers by µ0 M2 , M1 := −δβ0int M1 , M2 = β0int M2 ◦ M1 − β0int M1 − β0int M2 . (2.14) It is readily seen that (2.14) actually defines an integer cocycle, which is, in fact, also an integer coboundary of, e.g., β˜0int (M ) = β0 (M ) − β0 (6t ). The property that makes µ0 still an interesting quantity is that it is non-negative on all pairs of cobordisms. Indeed, for an elementary composition over K ≥ 1 connected boundary components as in (2.12), we find that this integer is the desired excess number of boundary-component over which we glue: (2.15) µ0 11⊗M2 , M1 ⊗ 11 = K − 1. Also, it is easy to see that µ0 (π ∗ , M ) = 0 for a permutation, and µ0 (11X ⊗ M2 , M1 ⊗ 11Y ) = 0, if Y is the source of M2 and X the target of M1 , so that µ0 M1 ⊗ M2 , N = µ0 11X ⊗ M2 , (M1 ⊗ 11Y ) ◦ N + µ0 (M1 ⊗ 11Y , N ). (2.16) This formula allows us to compute µ0 recursively from a presentation as in Lemma 1, and also implies µ0 ≥ 0 for general compositions. For a more systematic computation of the cocycle, let us introduce the spaces W2 := ker H0 (ψ2s ) . (2.17) W1 := ker H0 (ψ1t ) , s/t
Here the maps ψj surfaces,
are the restrictions of the charts in (2.8) to source and target
ψ t : 6t ,→ M ,
ψ s : −6s ,→ M,
(2.18)
for a cobordism M : 6s → 6t . We now have the following: Lemma 2. For two cobordisms, M1 : 6s,1 → 6 and M2 : 6 → 6t,2 , and spaces Wj as above, the cocycle from (2.14) is given as follows: µ0 (M2 , M1 ) = dim W1 ∩ W2 . (2.19)
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Proof. The exact sequence 0 → W1 ∩ W2 → H0 (6) → H0 (M2 ) ⊕ H0 (M1 ) → H0 (M2 ◦ M1 ) → 0, implies the dimension-formula β0 (M2 ◦ M1 ) − β0 (M2 ) − β0 (M1 ) = dim(W1 ∩ W2 ) − β0 (6). Counting also boundary components, this yields the asserted formula for µ0 . A basic property of this cocycle is that it vanishes on invertible cobordisms, i.e., (2.20) µ0 (M, G) = µ0 (G, M ) = 0 if G ∈ π0 Diff (6)+ . (Here we identify G with a cobordism by picking a representing automorphism of −→6. The associated cobordism is then given as a 3-manifold by the surface, ψG : 6g −→6. 6×[0, 1], and the boundary identification are id : 6×{0}g −→6 and ψG : 6×{1}g It is a basic fact that this actually establishes an isomorphism between the mapping class group of 6 and the group of invertible cobordisms from 6 to itself). Another property of µ0 is found in Subsect. 3.2 and Appendix A.2: If µ0 (M2 , M1 ) > 0, then the composite M2 ◦M1 will contain paths or 1-cocycles that give rise to additional, infinite generators of the fundamental group or the first homology group, respectively, besides those of M2 and M1 . 2.3. Half-pojective TQFT’s, and generalizations. As we remarked in the introduction it is not possible to construct non-trivial TQFT’s in the classical sense, which vanish on S 1 × S 2 . The purpose of this section is to define the modification that allows such a construction and discuss a few basic implications. Definition 1. Suppose R is a commutative ring with unit, and R−mod the symmetric tensor category of free R-modules. Further, let x be an element in R, and µ a 2-cocycle on Cob3 (0), which takes values only in Z+,0 . We call V : Cob3 (0) → R−mod a half-projective TQFT (with respect to x and µ), if it fulfills all the requirements for a functor of symmetric tensor categories, except for the preservation of compositions. Instead of this we shall assume the relaxed condition: (2.21) V M2 ◦ M1 = xµ(M2 ,M1 ) V(M2 )V(M1 ). As usual a projective TQFT is one for which x ∈ R is invertible and the nonnegativity of µ is dropped. An example of the latter is the well known signature-extension of the 2+1-dimensional cobordisms, which we avoided here by passing to a central extension of Cob3 (∗) by 4 . In this case µ is the Wall-cocycle. The number x ∈ C× for TQFT’s that are associated to Chern-Simons theory with Lie-algebra g and level ` may be obtained from the representation theory of Kac-Moody algebras, see [KW], and is a phase depending on `, the dimension, and the dual Coxeter number of g. For quantum-group constructions starting from a double D(B) it is the pairing of square roots of the modulus and comodulus of B, see [Ke1]. In general, if µ is a coboundary over Z, we may rescale a projective TQFT-functor, and obtain a functor in the ordinary sense. However, for a half-projective TQFT and a trivial cocycle we can only replace x by xy−1 for invertible y ∈ R. This is the situation, which we are interested in here, as the non-semisimple invariants will be associated to half-projective TQFT’s with respect to the connectivity cocycle µ0 defined in the previous section. It is completely separated from the signature-extension. E.g., it does not lead to any extensions of the mapping class group. In particular, we have the property expressed in (2.20), and that µ0 is invariant under the natural 4 action.
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Moreover, in the usual constructions of invariants, R is assumed to be a field so that the only non-trivial, half-projective TQFT occurs when x = 0, (with the usual convention 00 = 1). Let us, however, continue to assume more general R in the following discussion, in order to give more insight into the underlying structures and show directions of possible, further generalizations. Let us recall a general implication about the dimension of the R-modules associated to a surface that has already been observed by Witten [Wi] in the context of ordinary TQFT’s. Lemma 3. Suppose V is a half-projective TQFT w.r.t. x ∈ R and µ0 . If V6 = V(6) is an R-module associated to a surface 6, then V(S 1 × 6) = x dim(V6 ). Proof. For every connected surface 6 let us fix an orientation reversing involution − . χˆ 6 ∈ π0 Diff (6, ∂6) 2
If we consider (disjoint unions of) the χˆ 6 ’s as boundary charts of the cylinder 6×[0, 1], we obtain morphisms χ†6 : ∅ → 6 ⊗ 6
and
χ6 : 6 ⊗ 6 → ∅,
(2.22)
that are inverses of each other, and hence define a rigidity structure on Cob3 (∗). With χˆ 26 = 1 they are also symmetric, in the sense that χ6 ◦ γ = χ6 . Since V preserves also the symmetric tensor structure ∅ has to be associated to R and γ is mapped to the transposition of tensor factors. Thus, if we apply V to the χ(†) 6 we obtain maps † θ6 : R → V 6 ⊗ R V6
and
θ6 : V6 ⊗R V6 → R
(2.23)
that are symmetric with respect to the ordinary transposition, and are inverses of each other. Since we assumed the V6 to be free R-modules, we find by straightforward algebra † is the dimension of V6 . We also know that S 1 × 6 = χ6 χ†6 . The anomaly of that θ6 θ6 this product is µ0 = 1 so that we find the asserted formula. From the above proof we see that, in fact, we do not have to assume that the V6 are free-modules. morphisms in (2.23) implies an isomorphism V ∼ = The existence of the P † HomR V, R , which applied to θ6 gives an element ν eν ⊗ lν ∈ V ⊗R HomR V, R that inverts the canonical pairing. Suppose X and Z are R-modules and f : V6 → X and p : Z− → →X are R-morphisms, where p is onto. ForP xν := f (eν ) and p(zν ) = xν , we can define a map h : V6 → Z by the formula h(v) = zν lν (v) so that f = p ◦ h. Hence V6 is projective and is therefore a direct summand of a free R-module. Note also, that if R = R0 ⊕ R1 then we have a direct sum decomposition V6 = 0 V6 ⊕ V61 using the idempotents that are given by the units in the Rj . Moreover, we have V ⊗R W = V 1 ⊗R1 W 1 ⊕ V 0 ⊗R0 W 0 , etc. In summary, we find the following: L Lemma 4. Suppose V is a TQFT into possibly non-free R-modules, and R = j Rj , where Rj are indecomposable. L Then V = j V j , where each V j is a functor into the category of free Rj -modules.
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A modification of the prerequisites that would be consistent with different values of V(S 1 × 6) and hence x, is to allow the symmetry-structure of R−mod to be different from that induced by V. Thus the θ6 are now symmetric only up to isomorphism, i.e., we have θ6 T = θ6 (11 ⊗ P6 ), where T is the ordinary transposition and P6 ∈ AutR (V6 ). Instead of the dimension of V6 we then obtain the trace over P6 , which may even be zero. The induced symmetry structure yields in place of the transposition the morphisms, γˆ 6 ∈ End(V6 ⊗ V6 ), which are the images of the γ as in Lemma 1 with 6 = 6j . It is not hard to see that for R = C the structure is equivalent to the canonical one if and only if tr(γˆ 6 ) = dim(V6 ). It is also apparent from Lemma 3 that we should not write the anomaly-term to the other side of the equation in Definition 1. For h“= x−1 ” this would imply that h divides the dimensions of the vector spaces for every genus. Under the usual assumption (see also V3 of Subsect. 4.2) that the vector space of the sphere is R this would imply that h is invertible. The next lemma only uses the composition rule and applies also to the more general settings alluded to above: Lemma 5. Suppose V is a half-projective TQFT w.r.t. x ∈ R and µ0 . For a connected surface 6 of genus g we then have: V(S 1 × 6)
∈
xmax(g,1) R.
Proof. Consider the two-dimensional four-holed sphere as a 1+1-dimensional cobordism H4 : S 1 t S 1 → S 1 t S 1 , and, further, let φ† : ∅ → S 1 t S 1 and φ : S 1 t S 1 → ∅ be given by two-holed spheres. For g ≥ 1 we have 6 = φ ◦ H4 )g−1 ◦ φ† , and thus g−1 S 1 × 6 = S 1 × φ ◦ S 1 × H4 ◦ S 1 × φ† . Here, every one of the g compositions is over two tori in the boundaries of two connected cobordisms. Their anomalies are thus always µ0 = 1 and the assertion follows from the definition of a half-projective TQFT. If we set dg = dim(V6g ), where 6 has genus g ≥ 1, the combination of Lemmas 3 and 5 yields that dg x ∈ xg R. Suppose for some g ≥ 2, we have already x ∈ xg R. Then there is y ∈ R with x(1 − xy) = 0. In particular, e = xy is an idempotent in R, which can be used to write R as a direct sum eR ⊕ (1 − e)R. Now x lies in the first summand, and y is an inverse of this sub-ring with identity e. In summary, we have the following strong restriction on the element x and the dimensions of the vector-spaces. Corollary 1. Suppose V , x, and dg are as above. Then at least one of the following two has to be true: 1. The dimensions dg are zero-divisors in R/xg R for every g ≥ 2, or 2. The ring is a direct sum R = R1 ⊕ R0 , where the component of x in R0 is zero, and the component in R1 is invertible (in R1 ). As in Lemma 4 the second possibility implies for the TQFT-functor that V = V 1 ⊕ V 0 , and we have that V 1 can be rescaled to an ordinary TQFT. The only non-trivial half-projective TQFT we can therefore get if the dimension condition fails to hold (and if we stay strictly within the framework of Definition 1) is one with x = 0.
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2.4. TQFT’s for cobordisms with corners. There are several ways of defining extended TQFT’s, which represent categories of cobordisms with corners, like Cob3 (∗). Most of them are consistent, although not always precisely equivalent. In this section we shall give a brief survey over the structures that are of interest to us. To begin with the Kazhdan-Reshetikhin ladder is defined on a series of categories, Cob3 (n) ⊂ Cob3 (n, n), for which all of the cylinders in (2.7) start at a hole in the source surface, and end at a hole in the target surface. The extended TQFT is then defined, for a given abelian category C, as a series of functors V n : Cob3 (n) −→ C . . . C , | {z }
(2.24)
n times
where is Deligne’s tensor product of categories, see [D]. In particular, to a surface, 6, with ∂6 = tn S 1 , this associates an object X6 ∈ C n . We also require compatibility with the topological tensor product, i.e., X61 t62 = X61 X62 . Moreover, cobordisms are mapped to morphisms in the respective category, C n . Quite often it is more convenient to consider fiber functors that depend on a coloration. By this we mean an assignment of objects, {Xj : j = 1, . . . , n}, to the special, cylindrical pieces in the boundary of a cobordism, M : 6s → 6t , in Cob3 (n). Instead of f := V n (M ) : X6s → X6t , we then consider the following linear spaces and maps: 1 ,... ,Xn ) (M ) : HomC n X1 . . . Xn , X6s V (X n J Hom Xj , f (2.25) j −−−−−−−−−−→ HomC n Xπ(1) . . . Xπ(n) , X6t , where π ∈ Sn is the permutation of the holes, defined by the cylinders in ∂M , with respect to standard orderings of the holes in 6s and 6t . Notice, that the maps in (2.25) fulfill an obvious naturality condition, w.r.t. any given object, Xj , which appears both in the source and the target linear space. Conversely, suppose any functor, C → R − mod, of abelian categories (i.e., not necessarily tensor) is representable. Then any such system of maps with the naturality property stems from a functor like the one in (2.24). Examples ofR categories with representable fiber functors are all those, for which the coend IF := X ∨ X ∈ C 2 exists. See [M,L1], and [Ke3] for definitions. If we distinguish between in- and out-holes among the boundary components of cobordisms thema surface, we can view the objects of Cob3 (n) as 1+1-dimensional S selves. Thus it is quite natural to define Cob1+1+1 := nin ,nout Cob3 (nin + nout ) as a 2-category, where the objects are one-manifolds, the 1-morphisms are 1+1-cobordisms, and the 2-morphisms are 2+1-cobordisms between them. An extended TQFT is now a 2-functor of 2-categories: V : Cob1+1+1 −→ AbCat.
(2.26)
Here, AbCat is the 2-tensor-category of abelian categories. I.e., the object associated to a one-fold, S, is as usual an abelian category, C (S) = C β0 (S) , but to a surface we associate a functor between the category of the in-holes and the category of the out-holes. To a cobordism between two surfaces V then assigns a natural transformation between the respective functors. This picture may be extracted from the previous one, if we construct from an object X6 ∈ C (in) C (out) the functor
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F6 : C (in) −→ C (out) : X 7→ HomC (in) X6 , X , •
and from a morphism f : X61 → X62 a transformation F62 −−→ F61 in the obvious way. Furthermore, the 2-categorial description imposes more constraints on V than the Kazhdan-Reshetikhin picture, as V has to be compatible not only with compositions of 3-dimensional cobordisms but also with those of the 2-cobordisms. The observant reader might have noticed that we have suppressed here the permutation that appears in (2.25). Indeed, for a precise description we need to consider a slightly more complicated structure for Cob1+1+1 than that of a 2-category. Specifically, in the more general formalism the objects of two cobordant 1-morphisms are not simply the same but are connected by an arrow from a special category. Hence the 2-arrow-diagram of a 2-morphism is not simply given by a 2-gon, but by a square as below: 6s Ss(in) −−−−−−−→ α M ⇓ y
Ss(out) β y
(2.27)
6s St(in) −−−−−−−→ St(out) The new vertical arrows are associated to the special cylindrical pieces of ∂M , and are defined by the permutation they induce on the numbering of the holes, i.e., we have α ∈ Snin and β ∈ Snout . Horizontal compositions are only allowed if the adjacent vertical morphisms are identical. For vertical (ordinary) compositions of 2-morphisms the special, vertical arrows are also multiplied. The TQFT functor V shall now assign to a vertical permutation in Sn the obvious functor on C n that implements the respective permutation of tensor factors. A cobordism M is then mapped by V to a natural transformation between the two composites of functors that start at the category of the upper, left corner of the square in (2.27) and end in the lower, right corner. Recall, that in Subsect. 2.1 we actually defined a more general class of cobordisms, for which the cylindrical pieces are allowed to run from a hole in a component of, e.g., the source surface, 6s , to a hole in another component of 6s . A natural way to incorporate this possibility in our description is to enlarge the category, from which we may take the vertical arrows, from the symmetric groupoid to the category of singular tangles (i.e., strands for which an overcrossing can be changed to an undercrossing). A closed component of such a tangle, which in this category can be isolated as a circle, corresponds to an interior torus that can be added in a unique way as a closed component to either 6s or 6t as explained in Subsect. 2.1. In order to define V on a singular tangle it suffices to give the action on a maximum or minimum: S × Y ), V : C C −→ R − mod, X Y 7→ HomC (1, X (2.28) T R V : R − mod −→ C C, R 7→ IF = X ∨ X , where IF is the coend as above. Here × is a (braided) R tensor product in C. Note that × X. (2.28) also implies V( ) : R 7→ Inv(F ), where F = X ∨
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As an alternative to this direct functorial description, we may consider also here the assignments of linear maps depending on a given coloration, as in (2.25). The difference is now that a cylindrical piece, which starts and ends in the source surface results in a dependence of the source vector space on Xj∨ Xj (instead of only Xj ) and no dependence of the target linear space. The first then fulfills a di-naturality condition (instead of a naturality condition), which also plays an important role in liftings to the coend IF . The notion of a half-projective, extended TQFT is most conveniently defined for the version of functors into R − mod that depend on colorations. The generalization from Definition 1 is then literally the same. It is also not hard to generalize the construction and discussion of half-projective TQFT for closed surfaces to the case of punctured surfaces, since all that needs to be checked in this picture is the preservation of naturality. We shall thus content ourselves in this article with a construction of half-projective functors V 0 : Cob3 (0) → R − mod. Only in Subsect. 4.3 will we return to the direct description of an extended TQFT in terms of functors as in (2.24). 3. Non-Separating Surfaces, Interior Fundamental Groups, and Coordinate-Graphs In Lemma 5 we used a decomposition of a manifold to show that it is mapped by V to a multiple of x% for some % ∈ Z+,0 . In this sectionwe wish to generalize this result, and identify for any type of cobordism, M , orders in x for V(M ), that are maximal for this decomposition argument.(It will turn out that the one in Lemma 5 is indeed maximal.) From the discussion in Subsect. 2.3 it seems that x=0 is the only case we should be worried about, i.e., the only relevant question would be whether % = 0 or not, but not the precise order for % > 0. Nevertheless, we shall stay within the more general framework, not only because of the possible modifications we outlined in Subsect. 2.3, but also because of anticipated applications to “classical limits”, which we will sketch in Subsect. 4.5. In Subsect. 3.1 we relate the orders, %, to the maximal number of non-separating surfaces in a cobordism. The subsequent sections are devoted to computing these numbers from the fundamental groups of M and ∂M . Specifically, we will find in Subsect. 3.5 that % is the maximal rank of a free group, F , for which there is an exact sequence of the form, π1 (∂M ) −→ π1 (M ) −→ F −→ 1. For a special case see Lemma 6.6 in [He]. As an application we find that a half-projective TQFT, V, with x = 0, vanishes on cobordisms, M , with nontrivial “interior Betti-number”, i.e., V(M ) = 0 if β1int (M ) 6= 0. An important tool in this discussion are the coordinate graphs, which reduce the relevant connectivity information of cobordisms in Cob3 (∗) to that of morphisms in a graph-category 0. A useful result, proven in Appendix A.1, is that decomposition along such graphs can also be realized as decompositions of the corresponding cobordisms. 3.1. r-diagrams of non-separating surfaces. Let us begin with a definition of the systems of non-separating surfaces we are interested in: Definition 2. For a manifold M with boundary an r-diagram is an embedding of r Riemann surfaces 6j ,→ M with j = 1, . . . , r, such that 1. the surfaces are disjoint from each other, i.e., 6i ∩ 6j = ∅ for i 6= j, 2. they lie in the interior of M , i.e., ∂M ∩ 6j = ∅, 3. the surfaces are closed so that the embeddings are proper,
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4. every 6j is two-sided, and 5. their union is non-separating, i.e., M − trj=1 6j is connected. Let us reformulate the existence of an r-diagram for a cobordism, M : 6s → 6t , in Cob3 (0) in a more categorial language. If we remove thin, two-sided collars, χj := 6j × [−ε, ε], from M , we obtain a connected manifold M ∗ , which has 2r additional boundary components, 6± j = 6 × {±ε}. Hence, as a morphism in Cob3 (0), it can be written as follows: + − M ∗ : 62 −→ 6t ⊗ 6+1 ⊗6− 1 ⊗ . . . ⊗6r ⊗6r .
In terms of the rigidity cobordisms from (2.22) the original manifold is given by (3.29) M = 116t ⊗ χ61 ⊗ . . . ⊗χ6r ◦ M ∗ . It is easy to see that the total anomaly of this product is µ0 = r. Let us also introduce the quantity %(M ) := max{r : M admits an r−diagram}.
(3.30)
Note at this point that an r-diagram of some M with r < %(M ) can usually not be completed to a maximal diagram. An example is S 1 × 6g . Here we can find a g-diagram from the lower curves of a Heegaard diagram on 6g , which contains one torus from every composition in the proof of Lemma 5. We will, however, see in the next section that after removing {1} × 6g we have for the complement %([0, 1] × 6g ) = 0. From (3.29) we find immediately for the following: Lemma 6. Suppose V is a half-projective TQFT w.r.t. x ∈ R and µ0 . Then we have for a cobordism M : 6s → 6t from Cob3 (0), V(M ) ∈ x%(M ) HomR V6s , V6t . T Here, V6 = V(6), and we mean r xr Hom(. . . ) if %(M ) = ∞.
(3.31)
The generalization of Lemma 3.31 to extended TQFT’s as in Subsect. 2.4 also holds true, if we have M ∈ Cob3 (∗) and replace HomR by HomC ⊗N on the objects assigned by V to the punctured source and target surfaces. For a generalization of the previous arguments to surfaces with punctures let us make the following observations: The definition of an r-diagram of a cobordism in Cob3 (∗) is simply an r-diagram of φ0 (M ), where φ0 is as in (2.10). In order to obtain the generalization of the composition in (3.29) we first have to make all the surfaces transversal to the external strands, τ = D2 ×I, that connect holes of the surfaces 6s t6t to each other. A given strand, τ , is then divided by its intersections with the 6j into several components. If one such piece, δ, connects a surface 6 to itself from the same side, we can surger 6 along a slightly thickened δ, and obtain a two-sided surface 60 = (6 − D2 × S 0 ) ∪ (S 1 × I), which is also disjoint from the other surfaces and, together with them, is non-separating. We can thus assume that strands never connect a surface to itself from the same side. By transversality we may also assume that an external strand, τ , meets a collar, χj = 6j × [−ε, ε], in a vertical, cylindrical piece, Dτ2 × [−ε, ε], where Dτ2 ⊂ 6j . Hence, if we remove all external strands, we have presented M as the composite of the unit-morphisms χ0j = 60j × [−ε, ε], where 60j is obtained from 6j by removing the discs Dτ2 , and an admissible, connected cobordisms M ∗ in Cob3 (∗), as in (3.29).
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3.2. Interior fundamental groups, and an a-priori estimate on %(M ). The maximal number %(M ) of non-separating surfaces in a cobordism can be obtained from the fundamental groups of the cobordism M and its boundary ∂M . It will be given by the maximal rank of a free group onto which the interior fundamental group factorizes. In this Subsect. we give the definition of π1int (M ) and a first implication for %(M ). If M : 6s → 6t is a cobordism in Cob3 (∗) and 6ν , with ν = 1, . . . , K are the connected components of −6s t 6t we define the interior fundamental group, π1int (M ) as the fundamental group of M with the cones of the boundary components glued to it: (3.32) π1int (M ) = π1 M ∪ C61 ∪ . . . ∪ C6K . The following shows that it is enough to consider cobordisms in Cob3 (0). Lemma 7. The inclusion M ,→ φ0 (M ) induces an isomorphism g → π1int φ0 (M ) . π1int (M ) − Proof. Suppose a cobordism M 0 is obtained by filling a tube in M that connects a hole in the component 6i to one in the component 6j . Since for M ∈ Cob3 (∗) we have i 6= j, the cones C6i and C6j are disjoint. Thus M 0 with cone-attachments is obtained from M with boundary cones by gluing in a ball along a sphere, which does not affect fundamental groups. For the practical computation of π1int (M ), assume that we have marked points p0 ∈ M and pν ∈ 6ν . Let us call a spider, hγi, of M a collection of paths γν inside M , with ν = 1, . . . , K, that start at p0 and end at pν . Thus X = hγi ∪ C61 ∪ . . . ∪ C6K is a contractible space such that M ∪ X is the union of M with its boundary cones as in (3.32), and X ∩ M ' 61 ∨ . . . ∨ 6K . The interior group π1int (M ) is then given by Seifert-van Kampen (see, e.g., Theorem 7.40 in [Ro]) as the pushout of the respective fundamental groups, i.e., it is universal among the solutions, ζ and G, of the following diagram: QK f ree −−−−→ 0 ν=1 π1 (6ν , pν ) − . I∗ y (3.33) y π1 (M, p0 )
ζ −−−−→ →G
The image of I∗ in π1 (M ) generally depends on the choice of the spider, but their generators lie in the same conjugacy classes. Hence the smallest normal subgroup N [im(I∗ )] ⊂ π1 (M ) that contains the image of I∗ does not depend on the choice of a spider. This yields the formula: π1int (M ) := π1 (M )/N [im(I∗ )].
(3.34)
We also introduce the notion of a free interior group, F , of M . By this we mean a solution to the diagram (3.33), where F = G is a free group (non-abelian for rank > 1 ), and ζ an epimorphism. Clearly, by universality it may also be defined by the existence of an epimorphism → F. (3.35) ζ˜ : π1int (M ) −−→ Let us denote by F (k) the free group in k generators. The role of the anomaly of Subsect. 2.2 for internal fundamental groups can be described as follows:
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Lemma 8. Suppose M and N are cobordisms with anomaly µ0 = µ0 (M, N ). Then there is an epimorphism ξ, such that the following diagram commutes: ξ → π1int (M ) ∗ π1int (N ) ∗ F (µ0 ) π1int M ◦ N −−−→ x x . π1 (M ◦ N )
←−
(3.36)
π1 (M ) ∗ π1 (N ) ∗ F (µ0 )
Proof. It is enough to consider only connected cobordisms M and N that are connected over µ0 + 1 surfaces. Moreover, the assertion is the same, if we glue in the cones for the remaining boundary components of M and N , i.e., we may assume that N : ∅ → 60 t . . . t 6µ0 , and M is a cobordism in reverse direction. Fix a point x0 ∈ 60 and choose spiders hγ M i and hγ N i of M and N respectively that originate in x0 . Hence a leg, γjM , is a path in M that connects x0 ∈ 60 to a point x j ∈ 6j . For N˜ = N ∪ hγ M i we have a natural isomorphism π1 (N, x0 ) ∗ F (µ0 ) ∼ = π1 N˜ , x0 , −1 M in which the j th free generator aj of F (µ0 ) is mapped to γjN γj . M ˜ ˜ Since M ◦N = M ∪ N and M ∩ N = hγ i∪60 ∪. . .∪6µ0 , the group π1 (M ◦N ) = π1int (M ◦ N ) can be computed as the push-out of the following diagram: f ree
Qµ
1 π1 (6j , xj ) M I∗ y
Iˆ∗N −−−−→ π1 (N ) ∗ F (µ0 )
π1 (M )
(3.37) ,
M/N N are defined as in (3.33). where Iˆ∗N (g) = a−1 j I∗ (g)aj if g ∈ π1 6j , xj , and the I∗ int int Clearly, π1 (M ) ∗ π1 (N ) ∗ F (µ0 ) is a solution of (3.37), since im Iˆ∗N is also in the kernel for the map onto π1int (N ) ∗ F (µ0 ). Hence a surjection ξ exists. The remainder of the diagram, expressing that ξ is defined naturally, can be completed in the obvious way. In the lower horizontal morphism the free generator aj ∈ F (µ0 ) is −1 M mapped to the closed path γjN γj in the composite M ◦ N . In analogy to %(M ) from (3.30) let us define the maximal rank of a free interior group: ϕ(M ) := max{µ : M has F (µ) as free interior group} . The following are easily found from Lemma 8 and (3.29): Corollary 2. 1. ϕ(M ◦ N ) ≥ ϕ(M ) + ϕ(N ) + µ0 (M, N ). 2. ϕ(M ) ≥ %(M ).
(3.38)
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For ϕ(M ) and ϕ(N ) the first part implies the existence of another trivial cocycle with values in the non-negative integers, given by: µπ := −δϕ − µ0 .
(3.39)
Its computation can be quite intricate, and shall not be attempted here. Instead we shall give the analogous computation for homology in Appendix A.2. The second part of the corollary implies for one that %(M ) ≤ ϕ(M ) < ∞ , since the fundamental group of a compact manifold is finitely generated. This also renders the convention made at the end of Lemma 6 superfluous. Another consequence →π1 (M ) being onto for invertible cobordisms, we have is that with π1 (6)−−→ ϕ(G) = %(G) = 0 if G ∈ π0 Diff (6)+ , (3.40) which was used in the counter-example in Subsect. 3.1. In the remaining sections of this section we shall see that in fact ϕ ≡ %. 3.3. The graph-category 0. In this section we shall define a category of graphs. It will be used to encode the basic connectivity properties of Cob3 (0) . To begin with let us fix a label set, Sl , that is in one-to-one correspondence with the Riemann surfaces, used as objects for Cob3 (0). The objects of the category 0 are then given by strings of (possibly repeated) labels, [a1 , . . . , aK ], aj ∈ Sl . The morphisms, γ : [a1 , . . . , aK ] → [b1 , . . . , bL ], are given by one-dimensional cell-complexes, γ, taken up to homotopy type, for which ∂γ contains K + L special points, that are labeled by a1 , . . . , bL . The composition is, as for cobordisms, given by gluings along the respective boundary components, i.e., end-points with the same labels. A representing cell-complex can be visualized by a graph, with K + L distinguished vertices of edge-degree one. A generic example of a representing graph is depicted below. In this form the composition of two graphs is defined by placing them on top of each other. ar1 @
ar 4 ar2 ar3 ar5 r J ##
r J r @r r# Jr r bb @ r @r b b D r r r br Dr b1 b2 b3 b4 b5
ar6 r r (3.41)
In analogy to Lemma 1 we also have a natural symmetric tensor structure on 0: Lemma 9. 1. 0 is a strict, symmetric tensor category. 2. For any π ∈ SK , there is a morphism, π ∗ : [a1 , . . . , aK ] → [aπ−1 (1) , . . . , aπ−1 (K) ]. They are natural in 0 and give rise to a representation of SK . Any invertible morphism of 0 is of this form. 3. For every morphism there is a unique b, such that there is a de maximal number, composition in the form γ = π1∗ ◦ γ1 ⊗ . . . ⊗ γb ◦ π2∗ . A representing graph of a component, γj , is connected.
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The tensor product of 0 is given by the juxtapositions of both labels and graphs. The permutations are given by joining labels aj and aπ−1 (j) on top and bottom by straight lines. Using the triangle identity, this also defines the commutativity constraint γ on strings of arbitrary length, as the obvious crossing of two sets of parallel strands. A graph, γ, that is invertible cannot connect two different source (or target) labels to each other, and cannot contain internal loops. It follows for 0, that up to homotopy the permutations are the only possibilities. A similar statement holds true also for Cob3 (∗), if we include the action of the mapping class groups on the surfaces. A decomposition as in the last part of Lemma 9 is given for the k = 5 component graph in (3.41), by the permutations π1 = (3, 4) and π2 = (2, 3), and the connected morphisms γ1 : [a1 , a3 ] → [b1 ], γ2 : [a2 ] → [b2 , b4 ], γ3 : [a4 ] → [b3 ], γ4 : [a5 , a6 ] → [∅], and γ5 : [∅] → [b5 ]. As opposed to Cob3 (∗) it is easy to give a list of the homotopy-inequivalent, connected graphs, thus giving a complete description of the category. The class of a connected morphism, γ, is clearly determined by the number of source labels, K, the number of target labels, L, and the first Betti number β1 (γ) = dim H1 (γ) . Canonical representatives can be found by shrinking all of the internal edges, until we have at most one internal vertex. The results are given in the next diagram. ar K . . . PP PP PPr ` ... ` P B P` PP P β 1 (γ) . . . B Br r bL b1 ar 1 P PP
K+L≥3
or
β1 (γ)6=0
r
r
r
r
r
r
K+L=2, β1 (γ)=0
r
r (3.42)
K+L=1, β1 (γ)=0
Examples for the first type of graphs, which contain an internal vertex, are γ1 , γ2 , and γ4 with β1 = 2, β1 = 0, and β1 = 1, respectively. The graph γ3 is of the second type with only one edge, and γ5 is represented by simply one external vertex without edges. Although we will not always need the graphs to be one of the representatives, we shall always assume below that we have no internal vertices of valency one (as, e.g., γ2 ). Hence we have ∂γ = ∂γs t ∂γt , with |∂γs | = K and |∂γt | = L. As for the cobordisms in (2.14) we also have an anomaly for the Betti-numbers of graphs: β1 (γ2 ◦ γ1 ) = β1 (γ1 ) + β1 (γ2 ) + µ0 (γ2 , γ1 ).
(3.43)
Here, µ0 is defined exactly as in Lemma 27, where the Wj are now given with respect to the inclusions ∂γs,t ,→ γ. There is no µ∂ -contribution, and β1int = β1 for graphs, since H1 (∂γ) = 0. For the composite of two connected graphs over k ≥ 1 end-points, we obtain as in (2.15) that µ0 = k − 1. It will be convenient to introduce a natural partial order on the morphisms of 0. For two graphs, γ1 and γ2 , we say that γ1 ≺ γ2 ,
(3.44)
iff the γj belong to the same morphism set, and there is an embedding γ1 ,→ γ2 of some representatives, such that the corresponding maps H0 (γ1 )g −→H0 (γ2 ) and H1 (γ1 ) ,→ H1 (γ2 ) are an isomorphism and a monomorphism, respectively. It is clear that γ2 is
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obtained by adding internal edges to a given component of γ1 . I.e., for representatives as in (3.42) γ2 differs from γ1 only by adding loops to the internal vertices. An inequality as in (3.44) obviously also implies β1 (γ1 ) < β1 (γ2 ), γ1 ◦ γ ≺ γ2 ◦ γ, as well as γ1 ⊗ γ ≺ γ2 ⊗ γ. 3.4. Coordinate graphs of cobordisms. The relation between the categories Cob3 (∗) and 0 cannot be given precisely by a functor because the composition anomaly, µ0 + µπ , of Cob3 (∗) is greater than the anomaly µ0 of 0. We shall, however, attempt to relate the morphisms of the two categories in a way that we can conclude from the decomposition of a graph also the decomposition of a cobordism. We begin with the definition of the notion of a (faithful) coordinate graph, which will be our principal tool in the description of the connectivity of cobordisms. Definition 3. A coordinate-graph of a cobordism M : 6s → 6t , is a graph γ ∈ 0, together with a continuous function f : (M, 6s , 6t ) −→ (γ, ∂γs , ∂γs ) ,
(3.45)
such that we have a one-to-one correspondence between boundary components, (i.e., −→H0 (∂γs/t ) are isomorphisms), and the interior of M is the induced maps H0 (6s/t )g also mapped to the interior of γ. We say that f : M → γ is a faithful coordinate graph, if there is an embedding: J : (γ, ∂γ) ,→ (M, ∂M ), which maps again connected components to each other, and for which the composite f ◦ J is homotopic to the identity (with fixed endpoints). An immediate consequence of the correspondence of boundaries is that we can always write the morphisms of a coordinate graph as unions M = M 0 tN and γ = γ 0 tκ, where N and κ have no boundaries, and the coordinate map f induces an isomorphism g → H0 (γ 0 ) . H0 (M 0 ) − For this reason we shall often consider only the case of connected cobordisms with connected coordinate graphs. Next, let us state some obvious facts about the composition and collapse of coordinate graphs: Lemma 10. 1. If M1 and M2 have (faithful) coordinate graphs γ1 and γ2 , respectively, then γ2 ◦ γ1 is a (faithful) coordinate graph of M2 ◦ M1 . 2. Suppose γ = γ2 ◦ γ1 , where the γj ’s are coordinate graphs of the Mj ’s as above, is a maximal (faithful) coordinate graph of M = M2 ◦ M1 . Then γj has to be a maximal (faithful) coordinate graph of Mj , for both j = 1, 2. 3. If γ is a coordinate graph of M and ξ : γ → γ : a continuous map that preserves endpoints, then γ : is also a coordinate graph of M . If γ is in addition faithful and there is an inclusion γ : ,→ γ, whose composition with ξ is homotopic to the identity on γ : , then γ : is also faithful. Note that the converse of Part 2 is not true. A typical application of the observation in part 3) is given when γ : ≺ γ is a subgraph, missing one edge of γ, and the map γ → γ : given by collapsing the additional edge into another path in γ : , as for example in the following picture:
Connectivity of Cobordisms and Half-Projective TQFT’s
r − , −−−→
r
557
ξ −−−−→ →
r r .
(3.46)
If γ is as in (3.42) and γ : ≺ γ is of the same form with β1 (γ) − k inner loops, then ξ may be defined by collapsing the k outer loops of γ to the interior vertex. Next, we assure the existence of (maximal and minimal) coordinate graphs. Lemma 11. 1. Every cobordism, M , admits a (faithful) coordinate graph, γ M , which is minimal among all (faithful) coordinate graphs. 2. Every cobordism has a maximal, faithful coordinate graph. Proof. For a connected cobordism, M , we can choose γ M , to be the spider with β1 (γ M ) = 0 as in Subsect. 3.2. To define the coordinate map choose a map g : 6 = 6s t 6t → ∂γ, which maps different components to different points. Let p : M → v be the constant map to a point v. We set p t (g × id[0,1] ) f : M ∼ = γM , = M t6×{0} 6 × [0, 1] −−−−−−−−−−−→ v t∼ ∂γ M × [0, 1] ∼ where we identified ∂γ M × {0} ∼ v. In the proof of the second part it is clear that a faithful coordinate graph with β1 (γ) = →Zk . For a compact M we know, e.g., from a Heegaard k, implies a surjection H1 (M )−−→ decomposition, that H1 (M ) is finitely generated so that k must be bounded. In Definition 3 we assumed that for a faithful coordinate graph, h = f ◦ J is only homotopic to the identity. The next lemma asserts that we may assume that in this case h is also equal to the identity. Lemma 12. Suppose f : M → γ is a generic faithful coordinate graph with embedding J : γ ,−−→M . Then there exists f $ : M → γ, such that f $ ◦ J = id, and f $ coincides with f outside a neighborhood of J(γ). The proof, although fairly standard, is rather technical and is thus deferred to Appendix A.1. An application lies in the proof of the following lemma, asserting that if a coordinate graph is a composite, so is the associated cobordism. Lemma 13. Let f : M → γ be a generic coordinate graph of a connected cobordism, M , and γ = γ2 ◦ γ1 a decomposition in 0. 1. There is a graph, γˆ γ, with a collapse map, c : γˆ → γ, such that γˆ = γˆ 2 ◦ γˆ 1 , and c(γˆ j ) = γj . Moreover, γˆ has the property that there exists a coordinate map fˆ : M → γˆ – with f = c ◦ fˆ – such that the Mj := fˆ−1 (γˆ j ) are cobordisms with graphs γˆ j and M = M 2 ◦ M1 .
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2. If a composed coordinate graph, γ = γ2 ◦ γ1 , is faithful and γ2 and γ1 are connected, then there exists a coordinate graph, γˆ γ, and a map, fˆ, as above (except that f may be different from c ◦ fˆ ), such that the γˆ j are also faithful, the Mj are connected, and the embedding of γˆ j extends that of γj . Proof. We shall prove here only the first part of the lemma. Due to its technical nature the proof of the second statement is again deferred to Appendix A.1. It relies heavily on Lemma 12. By genericity we may assume that f is differentiable, and that P = γ2 ∩ γ1 consists only of regular values of f . For a point, p ∈ P , we can find a neighborhood Ip , p) ∼ = [−ε, ε], 0 , such that f −1 Ip = 6p1 × [−ε, ε] t . . . t 6pnp × [−ε, ε], where each 6pj is a connected Riemann surface, and f acts on this space as the projection on the interval [−ε, ε]. In order to obtain a coordinate-graph, for each p ∈ P we insert into γ np − 1 additional edges, epj ∼ = [−ε, ε], identifying their boundary points, {±ε}, with those in ∂Ip . For the resulting graph γˆ we also have a coordinate map fˆ : M → γ, ˆ which maps 6pj × [−ε, ε] → epj for j = 1, . . . , np − 1, through a projection onto [−ε, ε], and which coincides with f outside these regions. It is clear that fˆ is continuous, and if we define a collapse c : γˆ → γ by mapping epj to I p with fixed endpoints, then c ◦ fˆ = f . Now, Mj = f −1 (γj ), for j = 1, 2, are cobordisms with M = M2 ◦ M1 , where we glue over the union of all 6pj . For the subgraphs γˆ j = fˆ(Mj ) ⊂ γˆ we have of course γˆ = γˆ 2 ◦ γˆ 1 , and a one-to-one correspondence between the set of boundary points of, e.g., γˆ 1 and the set of 6pj ’s, since (∂ γˆ 1 )t contains in addition to the points of (∂γ1 )t the interior points 0 ∈ epj . The endpoints of γˆ 1 hence correspond to the boundary components of M1 so that fˆ : M1 → γˆ 1 is a coordinate graph. The following are useful applications of the decomposition along a graph: Corollary 3. 1. Every cobordism is given by the composite of cobordisms, whose maximal faithful coordinate graphs are spiders (with at most three end-points). 2. If M has a faithful coordinate graph with β1 (γ) = r, then M admits an r-diagram (see Subsect. 3.1). Proof. Pick a maximal (see Lemma 11) faithful coordinate-graph γ of M , and write it as a composite of two trees over β1 (γ) + 1 points as in the following diagram: ... @\ @ \ @ \r @ ... @ @ ... @\ @ \ @ \r @ . . . @ @
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Since γ is already maximal we only have to vary the coordinate map in order find the corresponding decomposition for the cobordisms according to Part 2) of Lemma 13. By Lemma 10 the sub-graphs are also maximal. The inner vertex of each tree can of course be resolved, such tat we obtain a homotopic tree with vertices that have valencies of at most three. Repeated application of Part 2) of Lemma 13 yields then the decomposition intoelementary pieces. If we use the decomposition over r + 1 points as above and reconnect the two cobordisms over only one surface we obtain a connected manifold. Hence the remaining surfaces form an r-diagram in M . 3.5. Existence of coordinate graphs from interior groups. The existence of faithful coordinate graphs and – hence r-diagrams – can be derived from projections onto the fundamental groups of graphs. The first observation regarding this connection follows immediately by picking transversal, closed paths in M , that represent preimages of the generators of π1 (γ) : Lemma 14. Suppose f : M → γ is a coordinate graph of connected M , and → π1 (γ) π1 (f ) : π1 (M ) −−−−→ is onto. Then γ is also a faithful coordinate graph. Note, that since f is constant on the boundaries, π1 (f ), also factors through π1 (M ) → π1int (M ). Hence, with π1 (γ) = F β1 (γ) we have that π1 (γ) is in fact a free interior group in the sense of (3.35) of Subsect. 3.2. Next, we show that, conversely, a free interior group also implies the existence of a coordinate graph. Lemma 15. Suppose that for connected M ζ : π1int (M ) −−−−→ → F (k) ∼ = π1 (γ) is a free interior group, with k = β1 (γ). Then there exists a faithful coordinate graph f : M → γ, such that ζ = π1 (f ). Proof. Assume that γ is as in diagram (3.42), and let γ & ⊂ γ be the bouquet of circles without the exterior edges so that π1 (γ) = π1 (γ & ). Since πj (γ & ) = 0 for j ≥ 2 it follows, e.g., from Theorem 6.39.ii) in [Sz], that there is a continuous map f & : M → γ & , which ζ induces the map π1 (M ) → π 1int (M ) −−−→ → γ & . By construction we have π(f∂& ) = 0 for the restriction f∂& := f & ∂M , which in turn implies that f∂& is homotopic to the constant map ∂M → {v}, where v ∈ γ & is the interior vertex of the bouquet. Let F∂& : ∂M × [0, 1] → γ & be a corresponding homotopy, with F∂& (x, 0) = f∂& (x) and F∂& (x, 1) = v. Given ψ : ∂M × [0, 1] t∂M ×{0} M ∼ = M , whose restriction to M is isotopic to the identity, we can define a function: f c = F∂& tf & f & ◦ ψ −1 : M, ∂M −→ (γ & , v) . ∂
As in the proof of Part 1) of Lemma 11 we can define from this (using again ψ) a coordinate map f : M → γ, with π1 (f ) = π1 (f c ). Together with Lemma 14 this implies the assertion. Let us summarize in the next theorem the results of Lemma 8, Corollary 3, and Lemma 15:
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T. Kerler
Theorem 4. Suppose M is a connected cobordism. Then the following are equivalent: 1. M admits an r-diagram. 2. M admits a free interior group of rank r. 3. M has a faithful coordinate graph, γ, with β1 (γ) = r. In particular Theorem 4 implies the converse inequality of Part 2) of Corollary 2. We find %(M ) = ϕ(M ) ,
(3.47)
i.e., the maximal number of non-separating surfaces is given by the rank of the maximal free interior group of a cobordism. It is thus possible to compute the order in x of a linear map associated to a cobordism, M , by a half-projective TQFT as it appears in Lemma 6 using only the fundamental group of M and ∂M . If x = 0 it suffices to consider only homology, since we only have to know whether there is a non-trivial interior group or not. More precisely, with H1int and β int as defined in Appendix A.2, we have: Lemma 16. For a connected cobordism M ∈ Cob3 (0), β int (M ) = 0
if and only if
%(M ) = 0.
Proof. By naturality of the Hurewicz map the first square of the following diagram commutes. → π1int (M ) π1 (∂M ) −→ π1 (M ) −−−−→ α y y y y y
(3.48)
→ H1int (M, Z) H1 (∂M, Z) −→ H1 (M, Z) −−−−→ Since the lower sequence is exact, and the Hurewicz maps are surjective, we can infer the existence of a surjection α of the interior groups, such that all of (3.48) commutes. →Z, which, From H1int (M, Q) 6= 0 we know that there is an epimorphism H1int (M, Z)−−→ composed with α, gives rise to a free interior group of rank one. → Z, the corresponding map Conversely, if there is an epimorphism π1int (M )−−→ → Z has the commutator sub-group in its kernel and thus factors into hoπ1 (M )−−→ mology. Since the kernel also contains the image of π1 (∂M ), it follows from diagram (3.48) that the epimorphism on homology factors through H1int so that β int (M ) ≥ 1. We immediately find from this and Lemma 6 the following. Corollary 5. Suppose V is a half-projective TQFT w.r.t. x = 0 and µ0 , and let M ∈ Cob3 (0) is a cobordism with β int (M ) 6= 0. Then V(M ) = 0
.
In the special case of the “Hennings-invariant” for Uq (s`2 ), with q at a root of unity, this vanishing property was observed by Ohtsuki, see [O]. The result there, however, is found more-or-less from a direct computation of the invariant. From the discussion in Sect. 2 we have seen that x = 0 and invertible x are probably the only possibilities so that – from an algebraic point of view – we only have to worry
Connectivity of Cobordisms and Half-Projective TQFT’s
561
whether there are non-trivial interior groups as, e.g., in Lemma 16, but not about the exact order, as suggested in Lemma 6. Still, as we shall discuss in Subsect. 4.5, the precise order can be of interest, if we consider “classical limits” of TQFT’s. In this case we have x → ∞, and %(M ) may yield an estimate on the order in x, by which kV(M )k diverges. 4. Construction of Half-Projective TQFT’s The aim of this section is to show how half-projective TQFT’s can be constructed from connected ones, as for example those found in [KL]. We shall organize our discussions in a deductive way, including additional assumptions only where needed. Specifically, we shall begin in Subsect. 4.2 by introducing a set of Axioms, V1–V5, on a map V : Cob3 (0) → R − mod, extract from this a list of properties, P1–P8, that ensure the existence of such a map, and conclude that V necessarily has to be a half-projective TQFT. In Subsect. 4.3 we show that the existence of extended structures implies all but one of the properties automatically. The missing Property P7 of the projectivity x is discussed in Subsect. 4.4 as a consequence of the closely related concepts of cointegrals, semisimplicity, and the invariant on S 1 × S 2 . In the discussions of these sections we attempt to give a clear picture of how the given assumptions influence the existence and uniqueness of the properties we derive for V. The last section then summarizes the possible variations of our axioms that might lead to more general definitions of V, in particular the tensor product rule. We also discuss a possible application of the formalism of half-projective TQFT’s to the study of “classical limits”, where the exact orders %(M ) = ϕ(M ) become relevant. 4.1. Surface-connecting cobordisms. For two closed, connected Riemann surfaces 6j , with j = 1, 2, we can think of their connected sum 61 #62 as being the result of a 1-surgery on 61 t 62 , i.e., we cut away a disc from each surface and reglue the cylinder S 1 × I along the boundaries. The corresponding morphism 5 : 61 t 62 → 61 #62 is constructed by attaching a one-handle to the cylinder over 61 t 62 , i.e., 5 = 61 × [0, 1] tD2 ×{1} D2 × I tD2 ×{1} 62 × [0, 1] , or, equivalently, a boundary-connected sum of the 6j ×[0, 1]. More generally, we obtain for every surface 6 with K ordered connected components 6j a morphism 56 : 6 = 61 t . . . t 6K −→ 6# := 61 # . . . #6K . The obvious associativity condition for these morphisms is readily verified. Changing orientations we obtain a cobordism in the opposite direction: 5†6 = −56 : 6# = 61 # . . . #6K −→ 6 = 61 t . . . t 6K . In the next lemma we evaluate the composites of 56 and 5†6 : Lemma 17. Suppose 6 is a closed Riemann surface with K components, 6j , and the cobordisms 56 and 5†6 are as above. Then we have 1.
5†6 ◦ 56 = 1161 # . . . #116K , i.e., the (interior) connected sum of the cylinders 6j × [0, 1].
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T. Kerler
2. For we have
36 := 56 ◦ 5†6 : 6# −→ 6# , 36 ◦ 56 = 56 # (S 1 × S 2 )# . . . #(S 1 × S 2 ) | {z } K−1 times
and the analogous equation for 5†6 ◦ 36 . (#
is the interior connected sum of 3-folds.)
Proof. For the proof of the first part it is enough to consider only the case N = 2. If we think of 5 = 61 × [0, 1] ∪D2 62 × [0, 1] as a boundary-connected sum of two parts, the composite 5† ◦5 is glued together from four pieces. The cylinders 61 ×[0, 1] ⊂ 5 and 61 × [1, 2] ⊂ 5† are glued together along 61 × {1} except at the disc D2 ⊂ 61 . Their union is thus homeomorphic to 61 × [0, 1] − D3 , where the union D2 ∪S 1 D2 ∼ = S12 3 of the discs, that are not identified, bounds the removed ball D . In the same way the cylinders over 62 are glued to give 62 × [0, 1] − D3 with an additional boundary component, S22 . In order to find the total composite, we have to glue the discs together as in the definition of 5, which amounts to gluing S12 onto S22 . This shows the first part of the lemma. For the second part we find 36 ◦56 = 56 ◦ 1161 −D3 ∪ S 2 ×I ∪ 1162 −D3 ∪. . .∪ S 2 ×I ∪ 116K −D3 , where we rewrote the connected sum as a (3-dimensional) index-1-surgery on the union of the cylinders. Clearly, we can view these in the composite also as index-1-surgeries on 56 ∼ = 56 ◦ 1161 t . . . t 116K . Since 56 is connected, the surgery-points can be moved together without changing the homeomorphism type of 36 ◦ 56 . The assertion follows now from the fact that an index-1-surgery in a contractible neighborhood is the same as connected summing with S 1 × S 2 . The proof for 5†6 ◦ 36 is analogous. Besides the formulas in Lemma 17 and associativity we shall consider another type of relations among the connecting morphisms. The basic example is given next: Lemma 18. Suppose 61 , 62 , and 63 are closed, connected surfaces. Then ◦5 5(61 t62 ) t 1163 ◦ 1161 t 5†(62 t63 ) = 5† (61 #62 )t63
61 t(62 #63 )
. (4.49)
•• 2 2 1 1 Proof. Denote by 6•1/3 = 61/3 −D2 , and 6•• 2 = 62 − D tD , with ∂62 = SL tSR , 2 1 D . the corresponding surfaces with holes, such that, e.g., 61 #62 = 6•1 ∪SL1 6•• 2 ∪ SR 2 1 D The morphism 561 t62 , for example, can then be seen as 6•1 ∪SL1 6•• ×[1, 2] 2 ∪ SR 2 1 ∼ with a 2-cell, C12 = D × I, attached alongthe connecting SL of the source surface. † • 1 6 Similarly, 562 t63 is D2 ∪SL1 6•• 2 ∪ SR 3 × [0, 1], with a 2-cell, C23 , glued along 1 the We construct the composite by first gluing the pieces SR of the target surface. • •• •• • 61 ∪SL1 62 × I and 62 ∪SR1 63 × I together along the respective 6•• 2 boundary • ∼ pieces. The result is homeomorphic to the cylinder over 6•1 ∪ 6•• ∪ 6 61 #62 #63 . = 2 3 2 Attaching the remaining pieces, the 2-cell C12 combines with D × [0, 1] ⊂ 5†62 t63 to one thickened disc, as does C23 with the D2 × I-piece of 561 t62 . The composite
Connectivity of Cobordisms and Half-Projective TQFT’s
563
1 is thus 61 #62 #63 × I with a 2-cell attached along the SR to the target surface and another 2-cell glued along SL1 to the source surface. If we split the middle cylinder of the cobordisms into two, the result is precisely the composite on the right of (4.49).
The connecting morphisms allow us to express a general, connected cobordism by a cobordism between connected surfaces. For this purpose let us introduce the notations (∗) ⊂ Cob3 (∗) Cobconn 3
and
Cobconn (0) ⊂ Cob3 (0) 3
for the subcategories, whose objects are connected surfaces. We have the following general presentation: Lemma 19. For any connected cobordism M : 6s → 6t in Cob3 (0) there exists a ˆ : 6#s → 6#t , in Cobconn (0) , such that morphism, M 3 ˆ ◦ 56s . M = 5†6t ◦ M Proof. The proof is immediate from tangle presentations of cobordisms as in Appendix A.3 or [Ke2] . A direct proof is given by choosing a Morse function f : M → [0, 1] with 6s = f −1 (0) and 6t = f −1 (1). It follows from the general theory of stratified function spaces that f can be deformed such that it does not have any index-zero-singularities, and the index-one-singularities have values below all other critical values. Also the order of the critical values of index-one can be freely permuted so that we can assume that the one with values in an interval, [0, δ], are all fusing singularities. This means the one-handle attachment given by passing through such singularities is between different components of the surface. As M is connected there will be exactly β0 (6s ) − 1 such handles. The handles of X = f −1 [0, δ] can be freely slid using isotopies of maps on the upper surfaces f −1 (δ) of X. Hence we can find a boundary chart, for which X is equivalent to 56s . The arguments for splitting off 5†6t are analogous. A useful application of Lemma 19 is the presentation of the symmetric group action on the 56 . As in Lemma 1 a permutation of the connected components of 6 can be given in terms of a cobordism π ∗ . It is not hard to see that they can be induced from a corresponding braid group action on the connected sum, 6# . More precisely, for K = β0 (6) we have a homomorphism ρ : BK −→ π0 Diff (6# )+ , such that ρ(b) ◦ 56 = 56 b)∗ , (4.50) where b 7→ b is given by the natural map BK −−→ →SK . It is quite helpful to consider choices of the connecting morphisms in the framework of tangle presentations, see Appendix A.3. For simplicity we consider only closed surfaces, denoting by 6g a connected standard surface of genus g. Both 6 = 6g1 t. . .t6gK and 6# = 6(g1 +...+gK ) are represented in a tangle category by g1 + . . . + gK pairs of end-points. However, in the first case they are organized in K groups (i.e., we have a τ -move for each group), and for 6# all end-points belong to the same group. Indicating groups by braces and admitting through-strands the connecting morphisms can be presented as follows: 2(g1 +}| 2g{1 . 2gK . . . + gK ) . . z }| { z }| { z 5†6
=
.
.
... | {z } 2g1
...
. ...
.
.
} . | {z 2gK
56 =
... .
|
.
.
{z 2(g1 + . . . + gK )
}
(4.51)
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T. Kerler
We can reproduce the first assertion of Lemma 17 using the fact that tangles are composed in the na¨ıve way, if the intermediate object consists of only one group. The resulting tangle diagram of 5†6 ◦ 56 consists thus again of 2(g1 + . . . + gK ) vertical strands. However, now both the top and bottom end-points are divided into K groups. From the three-dimensional interpretation of the tangles, the K − 1 dividing lines of the diagram can be seen as K − 1 dividing spheres of the corresponding cobordism. Since the composites with connected objects are easily identified as the punctured identitycobordisms, we find the connected sum 1161 # . . . #116K . If we consider the opposite composite, we glue over K components, and thus, by the rules given in [Ke2], we have to insert K − 1 zero-framed annuli. The first will surround the first 2g1 strands, the second the next 2g2 strands, etc. Only the last group of 2gK strands is not surrounded by an annulus, as depicted in the following diagram: 2(g1 + . . . + gK ) }|
z ... 36 =
.
.
{ ...
.
...
{z 2(g1 + . . . + gK )
|
} (4.52)
Instead of the last group we could have also chosen any other group as the one without an annulus. The second assertion of Lemma 17 follows easily from (4.52), since the composition of a connecting morphism from either side has simply the effect of creating K groups, and hence introduces the τ -move at a particular group. It is clear that the τ -move at the j th group with 2gj strands allows us to push the annulus through the strands, and thus separate it from the rest of the diagram. The resulting formula for 36 ◦ 56 follows now from the fact that an isolated, unframed unknot represents a connected sum with S1 × S2. Moreover, the relation (4.49) in Lemma 18 is readily verified in the framework of tangles. We find that in both cases the composition yields vertical strands going from two groups with g1 and g2 + g3 pairs to two groups with g1 + g2 and g3 pairs of strands. ˆ as in Lemma 19 is also easily found from a given tangle, representing A cobordism M M , by simply interpreting top and bottom strands as one group. The ambiguity in ˆ is expressed by the additional τ -moves that arise, when we compose with choosing M the connecting morphisms. Finally, let us illustrate the composites in (4.50) for K = 2 in the following tangle diagram: 2g2 z { }| { Q Q Q . . . Q Q Q Q Q Q Q . . . Q Q Q . . . Q Q Q Q} Q | {z z
ρ(σ1 ) ◦ 56 =
2g1 }|
2(g1 + g2 )
(4.53)
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565
The cobordism ρ(σ1 ) of the generator σ1 ∈ B2 is presented by the crossing of 2g1 parallel strands with the remaining set of 2g2 parallel strands. As ρ(σ1 ) ⊂ Cobconn (0), 3 all 2(g1 + g2 ) belong to the same group. However, if we multiply a connecting morphism, they belong to different groups at one end of the composite. The additional τ -move allows us to change the collective overcrossing to a collective undercrossing. Thus ρ(σ1 )2 ◦ 56 = 56 , and we recover the action of the symmetric group. 4.2. Basic constraints on generalized TQFT’s. The purpose of this section is to list a number of natural assumptions on a map from Cob3 (0) to the category of R modules, such that they necessarily imply the notion of a half projective TQFT as the only possible generalization over an ordinary TQFT. We shall also give a list of conditions in terms of some elementary morphisms and relations that allow us to construct a unique halfprojective TQFT from a given connected one. Let us begin with the axioms we shall require of an assignment V : Cob3 (0) −→ R − (f ree)mod , which maps objects to objects and morphism-sets to the corresponding morphism-sets. V1) V(M )V(N ) = V(M ◦ N ), if M and N are connected, and if they are composed over a connected surface. V2) V respects the symmetric tensor structure. 2 V3) dim V(S ) = 1. V4) V(116 ) = 11V(6) . For the remainder of the section we shall define for a given V: x ≡ xV := V(S 1 × S 2 ) .
(4.54)
Note that in a (generalized) TQFT with V1 and V2, U := V(S 2 ) is a commutative algebra over R, which acts on V . I.e., if U = U0 ⊕ U1 , then V also decomposes into a sum in a similar way in Lemma 4. Hence V3 is, e.g., the same as assuming that U is semisimple and V is indecomposable. Furthermore, together with V1 Axiom V3 implies V(M #N ) = V(M t N ) .
(4.55)
It is, by V2, clearly enough to require V4 only for connected surfaces. Also, it is easy to see [A] that V(116 )is in any case a projector on V(6), and that by a reduction red to V (6) = im V(116 ) we can define a consistent, effective TQFT V red which obeys V4. Hence the last axiom simply assumes that we have already carried out this reduction. Axiom V1 also implies that we have an honest functor (0) −→ R − mod , V conn : Cobconn 3
(4.56)
(0) is the category of cobordisms between connected surfaces as in dewhere Cobconn 3 scribed in the previous section. Using the decomposition in Lemma 19 and the fact that by V1 the composition in (19) of the lemma should be respected by V, it follows that ˆ ) and elementary maps, to be V(M ) for general M ∈ Cob3 (0) is given by V conn (M associated to the surface connecting cobordisms:
566
T. Kerler
i6 := V(56 ) : V61 ⊗ . . . ⊗ V6K −→ V6# p6 := V(5†6 ) : V6# −→ V61 ⊗ . . . ⊗ V6K ,
(4.57)
with 6 = 61 t . . . t 6K and 6# = 61 # . . . #6K . Next, let us derive from V1-V3 a set of constraints on these maps, from which we infer the existence of a generalized TQFT for disconnected surfaces. To begin with, observe that the composite in Part 1 of Lemma 17 is over a connected surface so that we have p6 i6 = V(1161 # . . . #116K ). By (4.55) the connected sums can be replaced by disjoint unions, which yields the identity. N Hence, by V4, the i6 and p6 must be injections and projections that identify V6 = j V6j as a complemented subspace in V6# . They also satisfy associativity constraints as in i6A t6B = i6#A t6#B i6A ⊗ i6B , (4.58) where 6A and 6B are possibly disconnected surfaces, and 6#A and 6#B are as usual the connected sums of their components. Given a connected TQFT, V conn , we define ∈ EndR V6# . (4.59) L6 := V conn 36 If V were an ordinary TQFT, the definition of 36 in Lemma 17 would imply that this map is identical with the following projector: (4.60) IP6 := i6 p6 ∈ EndR V6# . However, the composition for 36 is over disconnected 6 so that we cannot apply V1. Still, the findings in the second part of Lemma 17 allow us to derive from Axioms V1 K−1 p6 , where K = β0 (6). This and (4.55) the relations L6 i6 = xK−1 i6 and p6 L6 = x means that L6 = xK−1 IP6 + Lc6 11 − IP6 , where [Lc6 , IP6 ] = 0, and Lc6 2 = xK−1 Lc6 . So, in principle we could have a composition anomaly of iN 6 and p6 in the form of an operator Lc6 acting on the complement of the space V6 = j V6j , through which the injection and projection normally map. At this point we cannot find a reasonable way to incorporate such an anomaly into our formalism, and it does not appear in the known constructions. We shall thus add its absence to the list of axioms: Lc6 = 0.
V5)
In the case, where each V(6) is generated by all V(M ), with M : ∅ → 6, Axiom V5 can also be inferred from the condition that for any collection of Mj : ∅ → 6, X X cj V(Mj ) = 0 implies cj V conn (56 ◦ Mj ) = 0. j
j
Since the equations in Lemma 18 and in (4.50) involve only compositions over connected surfaces, we shall impose by V1 the corresponding formulas for the maps i6 and p6 as ˜ j , for additional conditions. Moreover, suppose that we have cobordisms, Mj : 6j → 6 # # ˜ j , and M : 6 → 6 ˜ for their connected sums, such that connected surfaces 6j and 6 56˜ ◦ M1 t . . . t MK = M ◦ 56 . (4.61) Then V1 and V2 imply that this relation is respected by V. Let us summarize the conditions on V that we have derived so far from the Axioms V1–V5, with x as in (4.54):
Connectivity of Cobordisms and Half-Projective TQFT’s
567
P1) There is a connected TQFT, V conn , as in (4.56). p6 i6 → V6 , with P2) There are injections and projections, V6 ,−−−→ V6# and V6# −−−→ p6 i6 = 11. ˆ )i6s only depends on M = 5† ◦ M ˆ ◦ 56s . P3) τ -Invariance: V(M ) = p6t V conn (M 6t . P4) Associativity: See (4.58) and analogously for p 6 P5) Symmetry: V conn ρ(b) i6 = i6 b)∗ , N(see (4.50)), where SK acts on V6 = j V6j by canonical permutation (same for p6 ). P6) Naturality: i6˜ V conn (M1 ) ⊗ . . . ⊗ V conn (MK ) = V conn (M )i6 , where M and Mj are as in (4.61). (same for p6 ). 36 = xβ0 (6)−1 IP6 . (See P7) Projectivity: V conn (4.60).) i P8) Commutation: i(6 t6 ) ⊗ 11 11 ⊗ p(6 t6 ) = p 1
2
2
3
with each 6j connected, etc.
(61 #62 )t63
61 t(62 #63 )
This list of properties is, in fact, also sufficient for the existence of a generalized TQFT. Furthermore, this generalization comes out to be precisely the one defined and discussed in Subsect. 2.3, and, conversely, implies Axioms V1–V5. Theorem 6. Suppose there is a functor, V conn , and maps i6 and p6 , such that properties P1–P8 are fulfilled. Then there exists a unique, half-projective TQFT, w.r.t. x = V conn (S 1 × S 2 ) and µ0 , V : Cob3 (0) −→ R − mod , (0). such that i6 = V(56 ), p6 = V(5†6 ), and V specializes to V conn on Cobconn 3 Proof. The assignment of a map V(M ) to a connected cobordism, M , is uniquely determined and well defined by Axiom V1 and Property P2. Using V2 and P5 its extension to a disconnected cobordisms, M , is found from the decompositions in Lemma 1. From the discussion in Subsect. 2.2 it follows that compatibility of V with the symmetric tensor structure allows us to consider only elementary compositions as in (2.12). Assume that the cobordisms in this formula are M1 : 6A −→ 6B t 6C
and
M2 : 6C t 6D −→ 6E
so that (11 ⊗ M2 )(M1 ⊗ 11) : 6A t 6D → 6B t 6E . The connectivity cocycle is given ˘ 2, M ˘ 1 ) = β0 (6C ) − 1. We find by µ0 = µ0 (M ˘ 2 )V(M ˘ 1 ) = xβ0 (6C )−1 11 ⊗ {p6E V conn (M ˆ 2 )i6C t6D } xµ0 V(M ˆ 1 )i6A } ⊗ 11 × {p6B t6C V conn (M (by P 4) ˆ 2 )i6# t6# (i6C ⊗ i6D )} = xβ0 (6C )−1 11 ⊗ {p6E V conn (M C D conn ˆ (M1 )i6A } ⊗ 11 × {(p6B ⊗ p6C )p6#B t6#C V ˆ 2 )i6# t6# } = xβ0 (6C )−1 (p6B ⊗ p6E ) 11 ⊗ {V conn (M C D conn ˆ (M1 )} ⊗ 11 (i6A ⊗ i6D ) ×(11 ⊗ IP6C ⊗ 11) × {p6#B t6#C V
568
T. Kerler by (P 7)
by (P 6)
by (P 8)
ˆ 2 )i6# t6# (V conn (36C ) ⊗ 11)} = (p6B ⊗ p6E ) 11 ⊗ {V conn (M C D conn ˆ (M1 )} ⊗ 11 (i6A ⊗ i6D ) ×(11 ⊗ IP6C ⊗ 11) {p6#B t6#C V ˆ 2 ◦ 3% = (p6B ⊗ p6E ) 11 ⊗ V conn (M 6C ) ˆ 1 ) ⊗ 11 (i6A ⊗ i6D ) × 11 ⊗ i6#C t6#D p6#B t6#C ⊗ 11 V conn (M ˆ 2 ◦ 3% = (p6B ⊗ p6E ) 11 ⊗ V conn (M ) p6#B t(6#C #6#D ) 6C ˆ 1 ) ⊗ 11 (i6A ⊗ i6D ) × i(6#B #6#C )t6#D V conn (M
by (P 6)
= (p6B ⊗ p6E )p6#B t6#E V conn (X2 )V conn (X1 )i6#A t6#D (i6A ⊗ i6D )
by (P 4)
= p6B t6E V conn (X2 ◦ X1 )i6A t6D =V M with M = 5†6B t6E ◦ X2 ◦ X1 ◦ 56A t6D .
by (P 3)
In order to apply Naturality P6 in this calculation we had to choose cobordisms, 3% 6C : # # # # # # # # # # # # 6C #6D → 6C #6D , X1 : 6A #6D → 6B #6C #6D , and X2 : 6B #6C #6D → 6#B #6#E , such that they fulfill the following equations
116#
D
56#C t6#D ◦ 36C t 116#D = 3% 6C ◦ 56#C t6#D , ˆ 1 t 116# = X1 ◦ 56# t6# , 5(6#B #6#C )t6#D ◦ M D A D † † % ˆ 2 ◦ 36 ) ◦ 5 # t (M = 56# t6# ◦ X2 . C 6 t(6# #6# ) B
C
D
B
E
(N 1) (N 2) (N 3)
In order to complete the proof that V is a half-projective functor, we still need to show ˆ = X2 ◦X1 , is in fact M ˘ 2 ◦M ˘ 1 . This is accomplished that the above cobordisms M , with M by basically the same calculation, only now for cobordisms and in reverse order: = 5†6B t6E ◦ X2 ◦ X1 ◦ 56A t6D , assoc. = 5†6B t 5†6E ◦ 5†6# t6# ◦ X2 ◦ X1 ◦ 56#A t6#D ◦ 56A t 56D , B E † † % N2 & N3 ˆ 2 ◦ 36 = 56B t 56E ◦ M ◦ 5†6# t(6# #6# ) ◦ 5(6#B #6#C )t6#D C B C D ˆ ◦ M1 ◦ 56A t 56D , † Lemma 18 ˆ 2 ◦3% # # # # = 5†6B t 5†6E ◦ M 1 1 t 5 ◦ ◦ 5 t 1 1 6B 6C t6D 6D 6C 6#B t6#C ˆ 1 ◦56A t 56D ◦ M ˆ 2 ◦ 3% # # ◦ 5 = 5†6B t 5†6E ◦ M 6C t6D 6C † ˆ 1 ◦ 56A t 56D , ◦ 56# t6# ◦ M M
B
C
Connectivity of Cobordisms and Half-Projective TQFT’s
569
ˆ 2 ◦ 56# t6# ◦ 36C t 56D } = 116B t {5†6E ◦ M C D † † ˆ 1 ◦ 56A } t 116D , ◦ { 56B t 116#C ◦ 56# t6# ◦ M B C † Lemma 17 ˆ = 116B t {56E ◦ M2 ◦ 56#C t6#D ◦ 56C t 56D } ˆ 1 ◦ 56A } t 116D , ◦ { 5†6B t 5†6C ◦ 5†6# t6# ◦ M B C † assoc. ˆ ˆ 1 ◦ 56A } t 116D = 116B t {56E ◦ M2 ◦ 56C t6D } ◦ {5†6B t6C ◦ M ˘ 1. ˘2 ◦M = 116B t M2 ◦ M1 t 116D = M N1
This completes the proof of the theorem.
In the remaining sections we use Theorem 6 to construct half-projective TQFT’s from known ones. We will verify the necessary and sufficient properties P1–P8 for a very general class of examples. 4.3. The example of extended TQFT’s. In this section we show that if the connected TQFT from (4.56) originates from an extended structure, most of the properties entering Theorem 6 are already fulfilled. Thus for the remainder of this and the next section we as in (2.24), with Cob3 (n) replaced shall require that there be a series of functors V conn n (n). by Cobconn 3 Connected, extended TQFT’s exist for a quite general class of abelian categories: Theorem 7 ([KL]). Suppose C is an abelian, rigid, balanced, modular, braided tensor R × X exists. category over a field R = k, for which the coend F = X ∨ , on the categories Cobconn (n) , as in Then there is a series of functors, V conn n 3 (2.24), which respects both types of tensor products and 2-categorial compositions, and for which (6g ) = Inv F × ... ×F , (4.62) V conn 0 | {z } g times
where × is the (braided) tensor product of C, and 6g is the closed surface of genus g. (We denote Inv(X) ≡ HomC (1, X) .) As explained in [Ke3] this specializes to the Reshetikhin-Turaev [RT] invariant (often identified with the Chern-Simons quantum field theory) if C is semisimple,L and to the × j, Hennings-invariant [H] if C = A − mod. The coend is in the first case F = j j ∨ where j runs over a representative set of simple objects, and in the second case F = A∗ , equipped with the coadjoint action ad∗ . The part of this generalization that will be relevant for this section is that the con(1) → C, nected functor needed in Theorem 6 descends from the functor V 1 : Cobconn 3 and that the latter is a functor of braided tensor categories. Including also the intermediate tangle presentations this is made precise in the following commutative diagram: ∼ = V conn : Cobconn (1) −−−−−−→ Tgl∞ (1)conn −−−−−−→ C 1 3 φ 0 / τ -move Inv (4.63) y y y ∼ = conn : Cob (0) − − − − −−→ Tgl(0)conn −−−−−−→ R − mod V conn 0 3
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T. Kerler
All of the vertical arrows are surjections. The functor φ0 is the filling functor that was described in (2.10) of Subsect. 2.2. It assigns to a once punctured surface, 6• , a corresponding closed surface, 6, by gluing in a disc, and to a cobordism, M • , with (0), by filling in a tube, D2 × I. The τ -move is corners a morphism, M ∈ Cobconn 3 described in Appendix A.3. As in [KL] we denote by Tgl∞ (1)conn the tangle category, which has the same generators as Tgl(0)conn , but which is not subject to the (1), which generalizes the presentation of τ -move. It represents isomorphically Cobconn 3 the mapping class groups of punctured surfaces, 6• , from [MP]. The τ -move accounts (0). The for isotopies over the puncture so that Tgl(0)conn actually represents Cobconn 3 is finally constructed by assigning to a tangle a system of morphisms functor V conn 1 with naturality properties, and lifting those to a morphism between tensor powers of the coend, F , see [L1,L2,KL], and also us [Ke3] for a less technical summary. Let conn • 6 ∈ obj C for the object assigned by V use the shorthand X6 := V conn 1 1 to a punctured surface 6• . In order to describe the condition imposed by the τ -move suppose that the strand crossing all strands emerging from one boundary of the tanglediagram carries a representation Y . The crossing itself is described by the morphism ×Y → Y × X6 , where ε is the braid constraint of C. τ -invariance ε(X6 , Y ) : X6 implies that µ(X6 , Y ) = ε(Y, X6 )ε(X6 , Y ) is represented trivially. This can be done by passing to the invariance, since for any f ∈ Inv(X6 ) we have by naturality × 11Y ) = (f × 11Y )µ(1, Y ) = (f × 11Y ). µ(X6 , Y )(f
(4.64)
for 6 or M by first choosing punctured representatives 6• Thus we can construct V conn 0 conn • or M , apply to these V 1 , and then map the result into R − mod by the Inv-functor (6) = Inv(X6 ). The assignment so that, e.g., V conn 0 × ... ×F X6g = F {z } | g times
then explains the formula for the vector spaces in Theorem 7. The benefit of this description lies in the fact that the horizontal arrows in the top row (1) is of (4.63) are functors of braided tensor categories. The tensor product of Cobconn 3 given by gluing the boundaries of two surfaces to a three-holed sphere, 60,3 = S 2 − D2 t D 2 t D2 , and accordingly the two cylindrical boundary components of two cobordisms with corners to two respective pieces in 60,3 × I. We shall make identifications with standard surfaces that are compatible with those for the ordered connected sums in Subsect. 4.1, using isomorphisms • × 6•2 ∼ (4.65) 6•1 = 61 #62 . The category Tgl∞ (1)conn also admits a natural tensor product, given by the (opposite) juxtaposition of two tangles. The presentation functor can be chosen, such that the tensor is such that it is also structure is strictly respected. Moreover, the construction of V conn 1 a strict tensor functor into C, i.e., × X62 X61 #62 = X61
.
(4.66)
A connected cobordism between surfaces with several components, M : 61 t . . . t ˜ 1 t ... t 6 ˜ L may be similarly first described by a morphism 6K → 6 ˆ • : 61 # . . . #6K • −→ 6 ˜ 1 # . . . #6 ˜ L •, M
Connectivity of Cobordisms and Half-Projective TQFT’s
571
which is presented by a tangle T (M ) ∈ Tgl∞ (1)conn . The original cobordism M is then presented by the image of T (M ) in the tangle category Tgl(0)conn , where we have introduced K + L additional τ -moves, one for every group of strands representing a boundary component. In order to guarantee invariance under the τ -moves at the source ends of the tangle, ˆ • to the tensor product of the M we may proceed analogously and restrict V conn 1 invariances of the objects associated to the individual groups. If we start by carrying out ˆ of closed, connected surfaces as in (4.63) we find a the reduction to the cobordism M first candidate for the inclusion from Theorem 6. Specifically, we have that ˆ )i06 (M V conn 0
only depends on
ˆ ◦ 56 , M
where we use the canonical injection: i06 : Inv(X61 ) ⊗ . . . ⊗ Inv(X6K ) ,−−−→ Inv X61 × ... × X6K .
(4.67)
The difficulty that remains is to find a projection in reverse direction in the case that the target surface is also disconnected, i.e., L > 1. In general, if the vector spaces are given by the invariances as above, a canonical map with these desired properties does not exist. Still, we can define canonical matrix elements. More precisely, for every choice of invariances, fj ∈ Inv(X6j ), and coinvariances, gj ∈ Cov(X6˜ j ), (denoting Cov(Y ) ≡ HomC (Y, 1) ) we have that also ˆ • (f1 (g1 × ... × gL )V conn × ... × fK ) M 1 ˆ • ) ◦ 56 . This circumstance naturally leads us to first only depends on M = 5†6˜ ◦ φo (M construct V 0 on the morphisms spaces, and then reconstruct the vector spaces. Generally, let us define for a set of objects Aj , Bj ∈ obj(C) the null space: n × ... × A K → B1 × ... × BL : H 0 := h : A1 o (g1 × ... × gL )h(f1 × ... × fK ) = 0 for all fj ∈ Inv(Aj ), gj ∈ Cov(Bj ) . From this we define the space of matrices: × ... × AK , B 1 × ... × BL /H 0 . H A1 , . . . , AK |B1 , . . . , BL := HomC A1 For a morphism, I, between the tensor products of the Aj ’s and Bj ’s, let us also denote its image in the above space (i.e., its class modulo H0 ) by [I]. A natural definition of the TQFT-functor for disconnected surfaces on only the morphism spaces is thus ˆ• ] M ∈ H(X61 , . . . , X6K |X6˜ 1 , . . . , X6˜ L . (4.68) V 0 (M ) := [V conn 1 Even in the connected case H(A|B) is usually going to be smaller than HomC (A, B), if C is not semisimple. This is due to the fact that the canonical pairing Cov(X) ⊗ Inv(X) −→ R
(4.69)
is degenerate for most objects X. Let us denote the null spaces of this pairing by Cov 0 (X) and Inv 0 (X), respectively. It is easily seen that, e.g, Inv 0 (X) is mapped to Inv 0 (Y ) by Inv(f ) for a morphism f : X → Y , and that [f ] = 0, if all of Inv(X) is mapped into
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T. Kerler
Inv 0 (Y ). Still, we can think of the morphisms in the H( | )-spaces as maps between vector spaces, if we pass to the quotients Inv(X) :=
Inv(X) Inv 0 (X)
and
Cov(X) :=
Cov(X) . Cov 0 (X)
(4.70)
Assuming that R acts nicely on these spaces this assertion is made more precise in the following lemma: Lemma 20. Suppose that for an abelian tensor category, C, over R the spaces Inv(X), Inv 0 (X), Inv(X), Cov(X), etc., are free R-modules, and that the exact sequence → Inv(X) −→ 0, 0 −→ Inv 0 (X) ,−−−→ Inv(X) −−−→
(4.71)
as well as the analogous one for Cov(X), are split over R. Then we have ∼ 1. Duality: Cov(X) = HomR Inv(X), R and vice versa. 2. H A1 , . . . , AK B1 , . . . , BL ∼ = HomR Inv(A1 ) ⊗ . . . ⊗ Inv(AK ), Inv(B1 ) ⊗ . . . ⊗ Inv(BL ) . The proof is standard, and makes use of the fact that the split sequence in (4.71) allows us to choose a dual basis in Inv(X) and Cov(X). If R is a field the prerequisites of Lemma 20 are of course always fulfilled (assuming finite dimensions). Yet, as usual let us consider a more general situation, in order to indicate the extent to which our constructions are the only possible ones. The identity in the second part of Lemma 20 show that we have to modify the vector spaces of the connected TQFT from Theorem 7 or, more generally, the diagram in (4.63) by assigning V 0 (6) := Inv(X6 ) ,
if 6 is connected ,
(4.72)
and extending this to disconnected surfaces by tensor products. This is then compatible ˆ•∈ with the assignment of linear maps to a morphism, M , where we first choose a M conn Cob3 (1), compute its image in the H( | )-space as in (4.68), and then apply the above (0) the functor isomorphism into the corresponding space of linear maps. On Cobconn 3 , where we divided out the null V o will thus be given by the factored version of V conn 0 spaces as in (4.70) Using the easily verified property that a canonical injection, as in (4.67), maps, e.g., × Y ), we can factorize i06 from (4.67) into a map Inv 0 (X) ⊗ Inv(Y ),−→Inv 0 (X i6 : Inv(X61 ) ⊗ . . . ⊗ Inv(X6K ) ,−−−→ Inv X61 × ... × X6K . (4.73) Since Inv and Cov are now dual spaces, we may define canonical projections, p6 , associated to the connecting cobordisms 5†6 . They shall be the adjoints of the corresponding inclusion of coinvariances, i.e., × ... × X6K . p6 ∗ : Cov(X61 ) ⊗ . . . ⊗ Cov(X6K ) ,−−−→ Cov X61 (4.74) ˆ )i6 , It is also straightforward to see that with the definitions in (4.68), V 0 (M ) = p6˜ V 0 (M ˆ ˜ if M : 6 → 6, and M is a corresponding morphism between the connected sums, such
Connectivity of Cobordisms and Half-Projective TQFT’s
573
ˆ ◦ 56 . It is also clear from the construction that p6 i6 = id so that we that M = 5†6˜ ◦ M have now the ingredients entering Theorem 6, which fulfill properties P1, P2, and P3. Moreover, associativity P4 follows immediately from the associativity of the canonical inclusions of invariances and coinvariances. Compatibility with symmetry as in P5 can be inferred from the fact that ε(X61 , X62 ) acts by naturality as the transposition on Inv(X61 ) ⊗ Inv(X62 ) and hence also on Inv(X61 ) ⊗ Inv(X62 ). Also, property P8 is evident, if we consider it for matrix elements. Specifically, choose f ∈ Inv(X1 ), × X2 ), α ∈ Cov(X1 × X2 ), and β ∈ Cov(X3 ), and denote by f , g, α, and g ∈ Inv(X1 β, the images in the quotient spaces. We then have the obvious identities E D α ⊗ β, i(1t2) ⊗ 11 11 ⊗ p(2t3) f ⊗ g = (α × β)(f × g) D E i f ⊗ g . = α ⊗ β, p (1#2)t3
1t(2#3)
, as in (4.63) allows us to Finally, the construction of V 0 from the tensor functor, V conn 1 infer naturality P6 from the following relation between cobordisms: ˜ is the (1), 6 ( 6) Lemma 21. Suppose Mj• , with j = 1, 2, are morphisms in Cobconn 3 union of the closed source (target) surfaces, and the choice of the 56 ’s is as in Subsect. 4.1. Then φ0 M1• ⊗ M2• ◦ 56 = 56˜ ◦ φ0 (M1• ) t φ0 (M2• ) . Proof. The relation is readily verified, given the tangle presentation in (4.51), the fact that the tensor product in Tgl∞ (1)conn is given by juxtaposition, and that φ0 only introduces another relation, but may be chosen as identity on representing tangles. The formula may also by understood directly, by considering both sides of the equation as M1• tM2• to which certain elementary manifolds are attached along the two cylin2 2 2 drical boundary pieces. On the left-hand side we glue∼in B2 = S 2−(D2 tD )×[0, 1] to get • • φ0 M1 ⊗M2 , and attach a 2-cell, C2 , to the piece, = S −(D tD ), of X in the source surface, in order to realize the composition with 56 . On the right-hand side we first glue in two tubes, D2 × J, to get φ0 (M1• ) t φ0 (M2• ) and then describe the composition with 56˜ by attaching a 1-cell, C1 , at the discs, D2 , in the target surface that belong to the tubes. In both cases the combined glued-in piece, X ∪ C2 ∼ = (D2 × J) ∪ C1 ∪ (D2 × J) ∼ = D3 , • • 3 is a ball, and the cylindrical boundary pieces of M1 t M2 are glued to D along two annuli that are embedded in its boundary ∂D3 = S 2 . Let us summarize the findings of this section in the next lemma: that descends from Lemma 22. Suppose we have a connected TQFT functor V conn 0 a functor V conn of braided tensor categories as in (4.63) (e.g., the one proposed in 0 Theorem 7). Then we can construct a map V 0 : Cob3 (0) → R, which satisfies the properties P1–P6, and P8 from Subsect. 4.2. The vector spaces are the quotient spaces as in (4.72). The only thing left to investigate, in order to complete the construction of a halfextended TQFT, is thus the projectivity property P7. This will be done in the next section, as it relates to more specific properties of the constructions starting from an abelian, braided tensor category C over a field k. The triviality of x will turn out to determine completely the semisimplicity of C. Let us conclude this section with an example of how vector spaces change, when we divide out the null spaces as in (4.70) for a non-semisimple category. The result in the
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T. Kerler
case of C = Uq (s`2 ) − mod, with q a primitive 2m + 1st root of unity, is described, e.g., in [Ke3]. The dimensions of the vector spaces for a torus are given by dim Inv(F ) = 3m + 1 and dim Inv(F ) = 2m . It is a quite remarkable fact that at least for prime 2m + 1, the space Inv 0 (F ) is not only an invariant subspace of the representation of the mapping class group derived from , but that it is also a direct summand. I.e., the sequence in (4.71) is also split as a V conn 0 sequence of SL(2, Z)-modules. Notice also that Inv(F ) is naturally identified with an invariance of the semisimtr ple trace sub-quotient, C , which is the starting point for the TQFT extending the Reshetikhin-Turaev invariant. The vector space of the torus of the latter is however only tr m-dimensional since the coend of C is smaller than the image of the coend of C. 4.4. Integrals, semisimplicity, and x = V(S 1 × S 2 ). In view of the tangle presentation in (4.52) it is clear that the key to understanding the representation of 36 in a TQFT, and hence the projectivity P7, is to explain the effect of a trivially framed annulus around a vertical strand as in diagram (4.75) in algebraic terms. S
A
(4.75) Geometrically, the surgery along the annulus A is equivalent to connecting an S 1 × S 2 to the manifold we surger on. The strand S passing through A then indicates a path that generates π1 (S 1 × S 2 ). It also has a natural interpretation in the language of cobordism categories, if we consider the tangle in (4.75) as a morphism in Tgl∞ (2), i.e., the tangle category with one external strand, S, and no τ -move. Here the diagram from (4.75) represents a cobordism, (2). It is explicitly given by the cylinder 60,2 × [0, 1], λ•• : 60,2 → 60,2 , in Cobconn 3 inside of which we have performed a surgery along the meridian of one of the cylindrical boundary pieces. It is clear that if we apply a filling functor, which glues a tube to one of these boundary pieces, the result in Cobconn (1) will be φ1 (λ•• ) = D2 × [0, 1] #(S 1 × 3 S 2 ). Moreover, we have λ•• ◦ λ•• = λ•• #(S 1 × S 2 ). (Here # is always the sum with the interior.) At this point it turns out to be rather instructive to include into our discussion the 2-categorial picture of cobordisms and TQFT’s, that was outlined in Subsect. 2.4. For example, we can think of a surface with a hole as a 1+1-cobordism 6• : ∅ → S 1 , and of a cobordism M • ∈ Cob3 (1) as a 2-morphism between two such 1-morphisms. Also, 60,2 : S 1 → S 1 may be seen as the identity 1-morphism 11S 1 on the circle, and λ•• : 11S 1 ⇒ 11S 1 is a 2-endomorphism. A 2-category also implies a composition operation, •1 , which is the usual composition on the 1+1-cobordisms, and which is naturally extended to the cobording 2morphisms. Since 11S 1 can be composed with any 6• we can form the •1 -composite of
Connectivity of Cobordisms and Half-Projective TQFT’s
575
any M • ∈ Cob3 (1) with λ•• . It is clear that the result can be obtained from M • also by doing a surgery along a meridian that is pushed off the special cylindrical piece of (1)g −→Tgl∞ (1)conn ∂M • . From this and the rules of the tangle presentation Cobconn 3 • • it follows easily that if T (M ) presents M the tangle for the composite is given by placing a trivially framed annulus around the entire tangle T (M • ). If we introduce also λ•6 := λ•• •1 11•6 , we easily verify the following identities from the 2-categorial distributive law: λ•• •1 M • = λ•6t ◦ M • = M • ◦ λ•6s .
(4.76)
(1). It is thus both natural and useful to think of λ•• as a natural transformation on Cobconn 3 In particular, the associated morphism λ•6g , for the connected surface 6g of genus g, is presented in Tgl∞ (1) by 2g vertical strands with a trivially framed annulus around (1) them. Comparing this to (4.52) and using the braided tensor structures of Cobconn 3 and Tgl∞ (1), we find × ... × λ•6K−1 × 116K , 3•6 = λ•61
(4.77)
where the 6j ’s are the connected components of 6. From the definition of an extended TQFT as in (2.26) in Subsect. 2.4 - as well as what we expect from the property expressed in (4.76) - it follows that λ•• is represented by a natural transformation of the identity functor of C to itself. In particular we have the following: λ•6 . (4.78) λ(X6 ) := V conn 1 (Recall that a natural transformation λ ∈ N at idC , idC consists of an endomorphisms λ(X) ∈ EndC (X) for every object, such that f λ(X) = λ(Y )f , if f : X → Y .) Let us determine a few constraints on the transformation X 7→ λ(X). To begin with, × 6•2 → 6•2 × 6•1 of Cobconn (1), can note that the braid morphisms, ε(6•1 , 6•2 ) : 6•1 3 ∞ conn be presented in Tgl (1) by the diagram in (4.53), except that we combine the top × 1162 is given by diagram (4.52), with K = 2. In ends into one group. Moreover, λ•61 the composite of these two tangles we can slide the 2g2 lower strands one by one over (1) the annulus and hence turn the overcrossing into an undercrossing. I.e., in Cobconn 3 we have the following identity: × 1162 ) = ε(6•2 , 6•1 )−1 (λ•61 × 1162 ). ε(6•1 , 6•2 )(λ•61
(4.79)
The corresponding conditions on the transformation of C is given by × 11Y µ(X, Y )(λ(X) × 11Y ) = λ(X)
and
× λ(Y )) = 11X µ(X, Y )(11X × λ(Y ), (4.80)
where µ(X, Y ) is the square of ε, as in (4.64). To be precise, we should impose this relation only for the special objects, X6 . However, they will be sufficiently big so that this implies the general statement by naturality. Other topological considerations lead us to impose conjugation invariance λ(X)t = λ(X ∨ ). These two properties also correspond to generating, elementary moves in a “Bridged Link Calculus”, which replace the 2-handle slides of the conventional Kirby calculus, see [Ke2]. Equation (4.64) also shows that (4.80) can be derived from a stronger condition, namely that each λ(X) has a decomposition into a monomorphism and an epimorphism, going through a multiple of the unit object:
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T. Kerler
pλX iλX λ(X) : X −−−−−→ → 1 ⊕ . . . ⊕ 1 ,−−−−−→ X | {z }
(4.81)
nX times
(see, e.g., [M] for existence and uniqueness of monic-epic-decompositions). Instead of the injection and projection in (4.81) we may also consider the injections iνX ∈ Inv(X) and projectionsP pνX ∈ Cov(X), with ν = 1, . . . , nX , for the individual summands ∼ = 1 nX ν ν iX pX . Its action on invariance is determined by naturality, i.e., we so that λ(X) = ν=1 have λ(X)f = λ(1)f for f : 1 → X, (using λ(1) ∈ k ∼ = EndC (1) ). We thus find the following linear dependence between the iνX and f : nX X
hpνX , f i iνX = λ(1) f
for all f ∈ Inv(X).
(4.82)
ν=1
This formula allows us to establish a relation between the number on the right side and semisimplicity of the category: Lemma 23. Suppose C is an abelian, rigid, balanced tensor category over a field k = R, with finite dimensional morphism sets. Assume further that there is a transformation λ ∈ N at idC , idC , with λ 6= 0, such that (4.81) holds for all objects X. Then C
is semisimple, if and only if
λ(1) 6= 0.
Proof. In a balanced category we can construct traces, trX : EndC (X) → k, that are generally cyclic, and respect the tensor product. As in [Ke4], we may then define a category to be semisimple, iff all pairings of the form composition trX HomC (Y, X) ⊗ HomC (X, Y ) −−−−−−−−−−−−−→ EndC (X) −−−−−−→ k,
are non-degenerate. Rigidity allows us to reduce this to non-degeneracy of the pairings of invariance and coinvariance, as in (4.68), for all objects. Now, if λ(1) = 0, (4.82) implies that pνX f = 0 for all ν, X, and f ∈ Inv(X), and hence a degeneracy of the pairing, if λ(X) 6= 0. Since we assumed λ 6= 0, this proves onenimplication. o n o 1 pνX are If λ(1) 6= 0 and R = k is a field, it follows from (4.82) that iνX and λ(1) dual basis so that we have non-degeneracy. For a semisimple category it not hard to see from the proof that such a transformation always exists, and that it is (up to a total scaling) uniquely given by the projection IPX ∈ End(X) onto the maximal, trivial sub-object. More precisely, we have λ(X) = λ(1)IPX .
(4.83)
Next, we shall evaluate V 0 (36 ) for a surface 6 with K connected components, 6j . (We shall often use the abbreviation Xj ≡ X6j .) Since V conn is a functor of tensor categories, we can find from this an expression 1 for 3•6 using (4.77): (3•6 ) = λ(X61 ) × ... × λ(X6K−1 ) × 11X6K . V conn 1
(4.84)
If we use the factorization (4.81), we can write the action of this morphism on an × ... × XK ), as follows: element, f ∈ Inv(X1
Connectivity of Cobordisms and Half-Projective TQFT’s
V conn (3•6 )f = 1
X
577
ν
iνX11 ⊗ . . . ⊗ iXK−1 ⊗ ξ (ν1 ,... ,νK−1 ) , K−1
(4.85)
{νj }
where
ν ξ (ν1 ,... ,νK−1 ) = pνX11 ⊗ . . . ⊗ pXK−1 f ⊗ 1 1 X K K−1
∈ Inv(XK ).
× ... ×1 × XK . In the semisimple case we find from (4.85) that Here we used Xk ∼ = 1 conn (3 )f = ( 1 1 × . . . × I P )V V conn Xk 1 (36 )f so that with (4.83) we have 6 1 K−1 V conn × . . . × I P I P f. (3 )f = λ(1) 6 X X 1 1 k Semisimplicity also implies Inv = Inv and Cov = Cov so that we can use the bases consisting of the iνX and pνX , in order to express the injections and projections in (4.73) and (4.74). It follows immediately that the projection IP6 in (4.60) is precisely given by the above tensor product of projections, restricted to invariance. Hence, V 0 (36 ) = λ(1)K−1 IP6 .
(4.86)
In the case λ(1) = 0 it follows from (4.82) that the vectors iνX and pνX all lie in the null spaces Inv 0 (X) and Cov 0 (X), respectively. If K > 1, this and (4.85) imply that × ... × XK . (36 )f ∈ Inv 0 (X1 )⊗. . .⊗Inv 0 (XK−1 )⊗Inv(XK ) ⊂ Inv 0 X1 V conn 1 It follows that V 0 (36 ) = 0, i.e., (4.86) also holds in the non-semisimple case. (For the case K = 1 we have 36 = 116 , and thus V 0 (36 ) = 11X6 for either case.) In order to prove that the property P7 follows from the assignment of a natural transformation with a decomposition as in (4.81), we still have to make the identification λ(1) = x ≡ V 0 (S 1 × S 2 ).
(4.87)
To this end we shall make the assumption that the unit object in C is irreducible, and also preserves unit objects, i.e., XS 2 = 1. Equation (4.87) follows now from that V conn 1 λ•S 2 = (11•S 2 )#(S 1 × S 2 ). For semisimple categories existence and uniqueness of transformations as in (4.87) is obvious. Still, we wish to understand this assertion also in the general case, and make sure that the construction, e.g, in Theorem 7 actually assigns such a transformation to λ•• . For this purpose it will be both useful and instructive to attribute another algebraic interpretation to λ•• . This has its origins in the case C = A − mod, where A is a finite dimensional Hopf algebra. Here a natural transformation, λ, of the identity functor is uniquely identified with a central element, λ ∈ Z(A), of the Hopf algebra. As the trivial representation of A is given by its counit , the relation (4.81) translates to yλ = λy = (y)λ
for all
y∈A,
and λ(1) = (λ).
(4.88)
Elements λ satisfying this relation, which we will call (two-sided) cointegrals, are well known in the theory of Hopf algebras. (In a more common convention λ is actually the cointegral of A∗ , which is the algebra used in the categorial description). Existence and uniqueness of integrals and cointegrals has been proven for finite dimensional Hopf algebras in [Sw]. In [LS] it is also shown that A is semisimple, if and only if (λ) 6= 0. Thus Lemma 23 is just the categorial generalization of this result. The action of the associated natural transformation X 7→ λ(X) of the identity on A − mod to itself, is given by the canonical application of λ to the A-module, VX .
578
T. Kerler
We have nX = dim im(λ(X)) . For example, if A = Uq (s`2 )0 , with q a root of unity, it follows from its representation theory, see [Rp], that nX is given by the number of times P0 occurs as a direct summand in VX , where P0 is the indecomposable, projective representation that contains an invariant vector. In particular, nX = 0 if VX is fully reducible. The notion of cointegrals can be generalized toR the type of tensor categories, C, considered in Theorem 7, for which the coend F = X ∨ ⊗ X exists. In [L1] it is shown that F has the structure of a categorial, braided Hopf algebra in C. Moreover, there is a natural Hopf algebra pairing ω : F × F −→ 1, for which the left and right null space ker(ω),−→F coincide. Let us call ω balanced, if ker(ω) is preserved by the balancing v ∈ N at(11C ), where the action of v is given by × X). This property has been introduced in [L1] as × v(X) ∈ End(X ∨ the lifting of 11 axiom (M2). We shall also say that C is strictly modular if ker(ω) = 0. The algebra of natural transformations of the identity of C is canonically isomorphic to the elements in Cov(F ). Hence a cointegral of F is defined to be a morphism,
λ : F → 1,
such that
1 F −−−−→ λ y
F ×F 11F y ×λ
e 1 −−−−→
commutes,
(4.89)
F
where 1 is the coproduct and e is the unit of F . ( The previously discussed case of an ∗ ∗ ∼ ordinary Hopf algebra is implied by the identity Cov(F ) = Cov A , ad = Z A ). We also have the dual notion of a categorial integral for F , which is given by an invariance,
µ : 1 → F ,
such that
F 1∗ y 1
µ × 11F −−−−−−→ F ×F m y µ −−−−−−→
commutes,
(4.90)
F
with m being the multiplication, and 1∗ the counit of F . The existence and uniqueness, proven for ordinary Hopf algebras in [Sw], is generalized to the categorial versions in the following theorem: Theorem 8 ([L1]). Suppose C is a category as above, which contains the coend F . Then there exist (up to scalings) a unique integral, µ : Iµ → F , and a unique cointegral, λ : F → Iλ , where Iµ and Iλ are invertible. If the pairing, ω, is balanced, then Iµ ∼ = Iλ ∼ = 1, and λ and µ are as in (4.89) and (4.90), respectively.
Connectivity of Cobordisms and Half-Projective TQFT’s
579
In the construction in [L2] and [KL] the transformation that is a-priori associated to the diagram in (4.75) is given as a coinvariance of F by the composite ×µ 11F ω × F −−−−→ 1. λ+ : F −−−−−−−→ F
(4.91)
The following lemma compels us to add non-degeneracy of ω to our list of requirements on C. Lemma 24 ([Ke3]). Suppose C is as above, and ω is balanced. Then λ = λ+ , if and only if C is strictly modular. This now establishes the fact that the construction used in Theorem 7 does in fact associate the cointegral of F to the diagram in (4.75). Still, we need to show that the image of λ(X) is actually of the form 1 ⊕ . . . ⊕ 1, in order to complete our construction. For categories over fields, k, with char(k) = 0, this is implied by a result of P. Deligne, which asserts that an object, on which the coaction of F is trivial, necessarily has to be the direct sum of units. It is found [D2] as a corollary to the existence of tensor products of abelian categories. The same result has been proven independently by V. Lyubashenko [L3] by the use of squared coalgebras: Lemma 25 ([L3, D2]). The cointegral of a braided tensor category over a field k, with char(k) = 0, and with coend F factors for each object as in (4.81). It thus follows that the construction in Theorem 7 also implies Property P7. 4.5. Main result, and hints to further generalizations and applications. In this concluding section we shall summarize the various possible deviation from our axioms, that we pointed out throughout this chapter, in order to find more general types of non-semisimple TQFT’s for disconnected surfaces than half-projective TQFT’s. We also discuss possible applications of the half-projective formulation to “classical limits”, where the x has the role of a renormalization parameter with x → ∞. To begin with, let us state the main result of this chapter, which follows from results of the preceding three sections, and which completes the construction in [KL]: Theorem 9. Suppose that C is a abelian, rigid, balanced, strictly modular, braided tensor category, which is defined over a field k with char(k) = 0, and which contains the coend F . Then there exists a half-projective TQFT-functor, w.r.t. µ0 and x ∈ k, V 0 : Cob3 (0) −→ V ect(k)
with
V 0 (6g ) = Inv F × ... ×F , | {z } g times
such that x 6= 0
if and only if
C
is semisimple.
(0) is the null space quotient Moreover, for x = 0 the functor V 0 restricted to Cobconn 3 , and for x = 6 0 it is the TQFT extending the Reshetikhin–Turaev invariant. of V conn 0
580
T. Kerler
The arguments given in Subsect. 4.2 that lead up to P2, and the ones given in Subsect. 4.3, yielding Lemma 20, essentially necessitate the division of the original invariances by the null spaces Inv 0 (X6 ), in order to achieve compatibility of the TQFT with the tensor structures in Cob3 (0) and R − mod, as required in the axioms V1-V5. Nevertheless, by Theorem 7 these spaces do carry quite interesting representations of mapping (0). class groups, or other features of cobordisms in Cobconn 3 In Subsect. 2.3 we alluded to the possibility of circumventing the alternative of Corollary 1 by admitting non-canonical symmetry structures on R − mod so that x ∈ R may be, e.g., nilpotent. This would yield a richer filtration of Inv(X) with respect to the action of λ(X) on X. A more promising approach is to relax the Axioms V1-V5. Recall for example that V5 has already been put in by hand for mostly technical reasons. In order to explain what we might anticipate as a generalization of the tensor product rule, let us show that, in some cases, we can think of the Inv-spaces as cohomologies. In the non-semisimple 2 case (R = ka field) we have that λ acts like a differential, i.e., λ(X) = 0. Note also that im λ(X) ⊂ Inv 0 (X) for all X, and that Inv(X) ⊂ ker λ(X) . For C = Uq (s`2 )−mod, and if X is the sum of only projective and irreducible representations, it follows from the representation theory of Uq (s`2 ), see [Rp], that the first inclusion for the image is in fact an isomorphism. Moreover, if we restrict the action of λ(X) to the trivial weight space of X, then the second inclusion for the kernel is an isomorphism for X = 1 and for X = P0 (i.e., the unique indecomposable, projective representation that contains an invariant vector). It has been worked out in [Os] that the adjoint representation does in fact contain only projective and irreducible summands. Thus we conclude for the vector space of the torus V(S 1 × S 1 ) ≡ Inv(F ) = H ∗ F00 , λ(F ) , where the coend F is as usual given by the functions on Uq (s`2 ) with coadjoint representation, and F00 is the intersection of the summand with trivial Casimir value and the trivial weight space. This formula for the vector space of a torus suggests to consider besides the tensor product also a derived functor, analogous to Tor, in order to retain some information of the Inv 0 -spaces. The formalism of half-projective TQFT’s may also have applications in the case, where C is semisimple so that x ∈ k is invertible. For a fixed, finite x we recover an ordinary TQFT by rescaling the canonically constructed TQFT functor by V(M ) −→ int x−β0 (M ) V(M ). Yet, for “classical limits”, in which the renormalization parameter x will tend to infinity, the anomaly may give us some estimates on the divergence of the canonically defined V(M ). Unlike the non-semisimple case not only triviality but the exact value of %(M ) should be of interest. In the construction from [KL] the integral and cointegrals admit a canonical normalization: X d(j)coev gj and λ = Dev1 , µ = D−1 j∈J
where ev and coev g are the evaluation and flipped coevaluation, d(j) are the quantumdimensions for simple objects j ∈ J , and
Connectivity of Cobordisms and Half-Projective TQFT’s
x = D := ±
sX
581
d(j)2 .
j∈J
The notation for D is the same as in [T]. The normalization is determined by λ · µ = 1, and relation (4.91), which are imposed by invariance under the local, interior moves of cancellation (or 0
-move) and modification, see [Ke3]. For the Chern–Simons quantum field theory with a simple, connected, and simply connected gauge group, G, the set J is identified with the (highest) weights in an elementary, truncated alcove of the weight space of G, whose size depends on the level of the theory (see, e.g., [KW]). In the classical limit the level goes to infinity so that eventually every dominant weight will belong to J . Hence |J | → ∞, and we have x = V(S 1 × S 2 ) −→ ∞ . Evidently, this limit of TQFT’s is ill-defined on most cobordisms. Still, there are cases, in which the limit exists in a given sense. For example, if G = SU (2), and J = {1, . . . , N }, where the labels are given by the dimension of the irreducible SU (2)-representation, we can consider the (projective) representation of 1 1 + SL(2, Z) = π0 Diff (S × S ) on V(S 1 × S 1 ) = CN , induced by the TQFT. If we identify the labels with points in R+ , via xj = √jN , then we can define the limits of the generators of SL(2, Z) as unitary operators. In particular, S is identified with the Fourier transform on V ∞ S 1 × S 1 = L2 (R+ , dx), and the limit of T is given by the 2 multiplication operator of eix on the same space, where 0 1 10 . S := and T := −1 0 11 For a connected cobordism, M , with maximal, free, interior group of rank ϕ(M ), we find analogous to Lemma 6 from Theorem 4 and Lemma 13 that V 0 (M ) = xϕ(M ) V 0 (M2 )V 0 (M1 ).
(4.92)
Here, M2 and M1 are connected cobordisms, for which we have ϕ(Mj ) = 0. Note that the latter also holds true for the invertible cobordisms, for which the classical limit existed. For general M we may ask by which order kV 0 (M )k diverges in the limit with J , x → ∞. Assuming that the composition of V 0 (M1 ) and V 0 (M2 ) does not degenerate we expect to find from (4.92) that %(M ) is at least a lower bound on this order. It is still quite crude, as we can see in the example of S 1 × 6, where %(M ) is roughly half of the true order. More precisely, if we use the Verlinde formula, dim(V6g ) = P 3 D2g−2 j d(j)2−2g , (see, e.g., [T]), and that for SU (2) we have x = D ∼ N 2 , we compute from Lemma 3: V 0 (S 1 × 6g )
∼ x if g = 0 ,
5
∼ x 3 if g = 1 , and
∼ x2g−1 if g > 1
.
In contrast to that we have that for g > 0 the number %(S 1 × 6g ) is given precisely by the maximal number of non-separating curves on 6g , which is exactly g. Also, %(S 1 × S 2 ) = 1 so that %(S 1 × 6g ) = max(g, 1) and the decompositions in Lemma 5 are in fact maximal.
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T. Kerler
Appendix
A.1. Proofs of subsection 3.4. A.1.1. Proof of Lemma 12. Generically we may assume that f : M → γ is a differentiable function, which is transversal to the embedded graph J(γ). This implies that there are no critical points in a vicinity of the graph, and that the edges of J(γ) are tangential to the level sets of f only at a finite number of interior points. Let us add the latter as vertices of valency two to γ. By genericity we may also assume that for the vertices vj ∈ γ, the images h(vj ) ∈ γ (where h = f ◦ J ) are distinct points in the interiors of the edges. We can choose disjoint open intervals Ij∗ and Ij ⊂ Ij∗ , such that vj ∈ Ij , and Ij∗ lies inside of an edge of γ. Hence we can find a function ψ : γ → γ, which is the identity outside of all of the Ij∗ , which is strictly monotonous outside of Ij , and which collapses Ij to the vertex vj , i.e., ψ(Ij ) = {vj }. The function f ∼ := ψ ◦ f has the property that it collapses an entire neighborhood of a vertex of the embedded graph J(γ) to a point. Since we can choose f ∼ arbitrarily close to f by making the intervals Ij∗ smaller and smaller, we shall assume this property, by genericity, already for f . Let us thus consider the compact regions Bjo = f −1 h(vj ) in M , for which the J(vj ) ∈ Bjo are interior points. At points outside of the Bjo and in a vicinity of the edges f is regular, and the level surfaces are transversal to the edges. Hence we may assume that for each edge e(j, k) ⊂ γ, joining the vertex vj to the vertex vk , there is an embedding, k)) ⊂ M , ρe(j,k) : D2 × [0, 1] ,−−→ M , such that t 7→ ρe(j,k) (0,t) parametrizes J(e(j, and f is constant on the disc-fibers, i.e., f ρe(j,k) (p, t) = f ρe(j,k) (0, t) . Moreover, we o 2 2 may choose the parametrization, such that ρ−1 e(j,k) Bj = D × [0, ε], and D is a disc of radius ε. Next, we choose vicinities of the vertices, given by embeddings, δj : D3 , 0 ,−−→ Bjo , vj , such that the discs ρe(j,k) D2 × {ε} are (disjointly) contained in δj (S 2 ) for every edge joining vj , and ρe(j,k) D2 × [0, ε] ⊂ im(δj ). The union of the images of the δj and the ρe(j,k) forms a neighborhood U (γ) of J(γ) in M . It may be given as a disjoint union of regions Re(j,k) := {ρe(j,k) (p, t) : |p| < t < 1 − |p|, |p| ≤ ε } , and their complement, tj Yj . Here, the Yj are the images of the δj with the cones ∨ Re(j,k) := {ρe(j,k) (p, t) : |p| < t < ε, |p| ≤ ε} removed, and are therefore homeomorphic to a cone over a sphere, Sj ∼ = S 2 − te D2 , which has a hole for every edge, e, at vj . We may choose a parametrization, δˆj : Sj × [0, ε], Sj × {0} → CSj , ∗ ∼ = Yj , ∗ ⊂ M , such that the second parameter is equal to t = |p| at the common boundary with Re(j,k) . We define a function κa : γ → γ, which maps an edge, parametrized by [0, 1] → γ : t 7→ s(t), to itself, such that t−a if a < t < 1 − a , κa s(t) = s 1 − 2a
Connectivity of Cobordisms and Half-Projective TQFT’s
583
κa s([0, a]) = s(0), and κa s([1 − a, 1]) = s(1). Also we define the projection
5e(j,k) : Re(j,k) −→ γ : ρe(j,k) (p, t) 7→ J −1 ρe(j,k) (0, t) ,
and from this the composite
κ ρe(j,k) (p, t)
:= κ|p| ◦ 5e(j,k) ρe(j,k) (p, t) = J
−1
ρe(j,k)
t − |p| 0, 1 − 2|p|
.
Moreover, if we define κ im(δˆj ) = vj , we obtain a continuous map κ : U (γ) −→ M . Now, h : γ → γ collapses, in the same way as κε , the ε-neighborhoods of a vertex onto a point. Hence we can write h = h$ ◦ κε , for some h$ : γ → γ, which is again homotopic to one. Let H : [0, ε] × γ → γ be such a homotopy, i.e., Hε = h$ and H0 = id. We then define f $ : M → γ by for x ∈ M − U (γ), f $ (x) = f (x) $ f ρe(j,k) (p, t) = H|p| κ ρe(j,k) (p, t) on Re(j,k) , $ ˆ f δj (s, r) = Hr (vj ) on Yj . For the |p| = ε -piece of Re(j,k) this gives Hε κ ρe(j,k) (p, t) = h$ ◦ κε ◦ 5e(j,k) ρe(j,k) (p, t) = h J −1 ρe(j,k) (p, t) = f ρe(j,k) (p, t) , = f ρe(j,k) (0, t) since f is constant along the disc-fibers. Also, H|p| κ ρe(j,k) (p, t) = Hr vj , if |p| = t = r, and Hε (vj ) = h$ (vj ) = h(vj ) = f (Yj ), so that f $ is continuous. Finally, on J(γ) we have f $ ρe(j,k) (0, t) = H0 κ ρe(j,k) (0, t) = J −1 ρe(j,k) (0, t) , so that f $ ◦ J = id.
A.1.2. Proof of Part 2) of Lemma 13. By Lemma 12 we may assume that f ◦ J is the identity. With Mno = f −1 (γn ), for n = 1, 2, we thus obtain, as in the proof of the first part, cobordisms with M = M2o ◦ M1o , and γn embeds into Mno , such that γn ,−−→ Mno −→ γn is the identity. As a first step let us alter f , such that the f −1 (γn ) are connected: Denote by M ! the component of M1o , which contains γ1 , and by B1ν the other components. Each of these is a cobordism B1ν : ∅ −→ 6, where 6 6= ∅, and 6 ⊂ f −1 (P ), i.e., it is a union of the connected components 6pj from the proof of the first part. For given B1ν , denote by Qν the set of labels (p, j) that occur in this union. Choose ν (e.g., as in Part 1 of Lemma 11), and for every ν a coordinate graph g ν : B1ν −→ γB extend this to
584
T. Kerler ν
g ν : B1 = B1ν t
a
6pj × [0, ε] −→ γB ν .
(p,j)∈Qν
Here we have added the collars of the boundary components, 6pj , of B1ν in M2 , and ν by intervals [0, ε] to obtain γB ν . Each 6pj × {ε}, with elongated the ends of γB (p, j) ∈ Qν , is mapped to an interior point, p# , in γ2 . Since γ2 is connected, there are maps hν : γB ν −→ int(γ2 ), such that (p, j) is assigned to the respective p# . We can ν thus define a continuous function f˜ : M → γ, which is given by hν ◦ g ν on each B1 , ˜ 1 := f˜−1 (γ1 ) is connected. In the same and by f on the rest of M . We now have that M way we can define from this coordinate map fˆ : M → γ, for which also fˆ−1 (γ2 ) is ˜ 1 and the extra components of f˜−1 (γ2 ) connected, and fˆ−1 (γ1 ) is the composite of M – hence also connected. We can thus assume that the Mj∗ = f −1 (γj ) are connected, yet f −1 (p), with p ∈ P = γ1 ∩ γ2 , may still contain several components. As before we have to add an additional edge to γ for each components disjoint from the embedded γ. For such a component 6 choose a path t 7→ q(t) in M1∗ , which has starting point q(0) ∈ 6, and which ends in a path r(t) := f (q(t)) in γ1 , we define the compact point in J(γ1 ). For the corresponding subsets Mt := f −1 r([0, t]) , with Mt ⊂ Ms ⊂ f −1 (γ1 ) for t < s. We set T
T := inf{ t : J(γ1 ) and 6 are connected in Mt } .
Since MT = t>T Mt is the intersection of compact sets, in which J(γ1 ) and 6 are connected, they are also connected in MT . Let Mt6 be the component of 6 in Mt for t < T , and Mtγ its complement in Mt . S ? ˇ6 ˇγ ˇ T := follows. It shall be On M t
Connectivity of Cobordisms and Half-Projective TQFT’s
585
The coboundary µ0 = −δβ0int turned out to be non-negative and entered the definition of half-projective TQFT’s. In this appendix we shall be interested in the properties of the coboundary µ1 = −δβ1int , for which we find µ∂ := µ1 − µ0 ≥ 0. If we consider homotopy instead of homology the analog for β1int is ϕ(M ), which has been introduced in Subsect. 3.2. There we also defined in (3.39) the coboundary µπ , which is in analogy to µ∂ . Since the explicit computation of µπ is quite hard, we shall give here instead a derivation of a formula for µ∂ . We shall first identify the numbers β1int (M ) ∈ Z+,0 with the dimensions of the spaces H1int M, G := coker H1 ψ, G) , (A.93) where we usually consider coefficients G = Z or Q, and ψ is the inclusion of the punctured boundary pieces, −6s t6t , into M . In the case, where we have no punctures, i.e., M = φ0 (M ), we know that H1 (ψ) is half-rank, and the dimension is in fact given by β1int . Hence int H1int M, Z = Zβ1 (M ) ⊕ T orsion, (A.94) which, (by naturality of universal coefficients) implies int H1int M, Q = H1int M, Z ⊗ Q = Qβ1 (M ) .
(A.95)
For punctured surfaces these formulae are still correct, since we have in analogy to Lemma 7 the following: Lemma 26. Suppose M ∈ Cob3 (∗) is a cobordism, φ0 (M ) ∈ Cob3 (0) is as in (2.10), and iφ : M ,→ φ0 (M ) is the respective inclusion. 1. The inclusion induces an isomorphism (for both types of coefficients): H1int (iφ ) : H1int M − g → H1int φ0 (M ) . 2. The first interior Betti-numbers are computed directly from β1int (M ) = β1 (M ) −
1 β1 (−6s t 6t ) + dim ker(iM ) , 2
where iM : H1 tN S 1 ×I → H1 (M ) is induced by the inclusion of the cylindrical boundary pieces into the cobordism. Proof. It is enough to prove the identities for gluing a tube D2 × I into only one cylinder S 1 × I of a cobordism M . If 6 = −6s t 6t , the corresponding surface 60 of the filled cobordism is obtained by gluing discs into the holes of 6. Since, the tube and the discs are contractible, the inclusions of boundary pieces induce the following transformation of Mayer-Vietoris sequences: p6 i6 , −−−→ H1 (6) −−−−→ → H1 (60 ) → 0 0 → H1 (S 1 t S 1 ) − 0 ψ1 y ψ1 y p1 y pM iM H1 (S 1 × I) −−−−→ H1 (M ) −−−−→ → H1 (M 0 ) → 0
(A.96)
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T. Kerler
In the top row p6 is onto because the discs were glued to different components of 6 (see definition of Cob3 (∗)), and i6 is always into. As p1 is onto, the two sequences factor into the isomorphism in Part 2 of the lemma. The analog of (A.96) also holds for the union of all cylindrical pieces tN S 1 × I and 0 M replaced by φ0 (M ). The formula for β1int in Part 2 is now only a matter of counting dimensions. For a cobordism M : 6s → 6t there are natural projections pt : H1 (−6s t 6t ) → H1 (6t ), and ps analogously. For a composite of two cobordisms, M2 ◦ M1 , we can define with 6 = 6t,1 = 6s,2 the following sub-spaces of H1 (6) : V2 := ps2 ker H1 (ψ2 ) . (A.97) V1 := pt1 ker H1 (ψ1 ) , From these we can find µ1 as follows: Lemma 27. For two cobordisms, M1 : 6s,1 → 6 and M2 : 6 → 6t,2 , and spaces Vj as above, the cocycles from (2.14) are given by: (A.98) µ1 (M2 , M1 ) − µ0 (M2 , M1 ) = codim V1 + V2 . also introduce the spaces t H1int (M ) := Proof. In order to compute µ1 let sus int t coker H1 (ψ ) , and, correspondingly, H1 . We have the following exact sequence: 0 → V1 → H1 (6) → s H1int (M1 ) → H1int (M1 ) → 0.
(A.99)
The fact that V1 appears here as the kernel is seen, when we divide 0 → ker(ψ1 ) → H1 (6s,1 t 6) → H1 (M1 ) by the sub-sequence through H1 (6s,1 ). If we divide the Mayer-Vietoris sequence for the gluing of M2 and M1 , by the homologies of the outer surface 6s,1 t 6t,2 , we obtain an exact sequence: 0 → V1 ∩ V2 → H1 (6) →s H1int (M1 ) ⊕ t H1int (M2 ) → H1int (M2 ◦ M1 ) → Gµ0 → 0.
(A.100)
Combining this with (A.99) and its analogue for V2 , we find the formula for µ1 by counting dimensions. Lemma 27 implies that not only µ0 , and µ1 , but also µ∂ := µ1 − µ0 are nonnegative integers. This means that the interior Betti-numbers βjint , and also β∂int := β1int − β0int can only increase under compositions. An increase in β0int (M ) indicates new 1-cycles in H1int (M ) that are created by connecting two manifolds over several boundary components as in (2.15). An increase in β∂int indicates 1-cycles from the intermediate surfaces 6 that are not killed in the gluing process. The simplest example for this are two copies of a full torus that are glued together along the same meridians to give S 1 × S 2 . We easily see that in this case µ∂ = 1. A.3. Summary of tangle presentations. We shall summarize here the tangle presentations of the cobordism category Cob3 (0) of cobordisms between parametrized, connected surfaces, with central extension by 4 , as in [Ke2]. There are several versions, and we shall choose here the one that involves ordinary, framed tangles, as opposed to bridged tangles. The bridged versions are more practical for the construction of invariants
Connectivity of Cobordisms and Half-Projective TQFT’s
587
and connected TQFT’s, as in [KL], but are less common, and involve another type of surgeries. The presentation is given by an isomorphism functor: g → Tgl(0). T : Cob3 (0) − On the level of objects we assign to a surface 6 = 6g1 t . . . t 6gK , with K ordered, connected components, a configuration of intervals, called groups, and points on the real line as follows. To every component 6gj corresponds to a group Gj ⊂ R, and to 6 we associate the disjoint, ordered union G1 t. . .tGK ⊂ R. Furthermore, in each interval Gj we pick 2gj points, which we denote in increasing order {aj,1 , a0j,1 , aj,2 , . . . , a0j,gj } ⊂ Gj . A connected morphism in Cob3 (0), s −→ 6t = 6g1t t . . . t 6gLt , M : 6s = 6g1s t . . . t 6gK
is then represented by a generically projected tangle in a strip R × I, where we have marked the groups and points of 6s on the upper boundary, Rs , of the strip, and the data of 6t on the lower real line, Rt . The tangle diagram has thus a general form as below, where we indicated groups by braces. Gs1 GsK . . . z }| { z }| { Rs s . . . . . . 2g1s . . . . . . 2gK ...
Tangle t . . . . . . 2g1t . . . . . . 2gL ... | {z } | {z } . . . Gt1 GtL The tangle consists of possibly linked and knotted strands that end in the 2(g1s + . . . + s t ) points at Rt . To each strand we assign in addition gK ) points at Rs , and the 2(g1t +. . .+gL a framing of its normal bundle, which is the same as considering ribbons. In diagrams we tacitly assume the framing to be in the plane of projection (blackboard-framing). A component of a tangle in Tgl(0) has to be one of the following four types:
Rt
1. Source-ribbon: A strand, that connects a pair in Rs , i.e., it starts at a point aj,ν ∈ Gsj and ends in a0j,ν ∈ Gsj . 2. Target-ribbon: A strand, that connects a pair in Rs , i.e., it starts at a point aj,ν ∈ Gsj and ends in a0j,ν ∈ Gsj . 3. Closed ribbon: A strand (∼ = S 1 × [−ε, ε]) that is disjoint from the boundaries of the diagram. 4. Through-ribbon-pair: A pair of strands, connecting a pair in Rs to a pair in Rt . I.e., one strand connects aj,ν ∈ Gsj to ak,µ ∈ Gtk and the other a0j,ν ∈ Gsj to a0k,µ ∈ Gtk . (Flipped version also admissible). Furthermore, we require that in all cases the ribbon, as a surface with boundary, has to be oriented, and the orientation shall be compatible with a fixed one at Rs and Rt , if it ends in the boundary of the strip. An element in Tgl(0) is then an equivalence class of tangles, where we consider two tangles equivalent, if one can be transformed into the other by the application of a sequence of the following five moves:
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1. Isotopies with fixed endpoints. 2. The O2 -move or 2-handle slide of any type of ribbon over a closed ribbon, see [Ki].
-move, in which we add or remove an isolated Hopf link, for which one 3. The 0 component has 0-framing, and the other either 1- or 0-framing. s/t 4. The τ -move (at a group Gj ), which allows us to push any type of ribbon through s/t
s/t
the 2gj parallel strands emerging at the group Gj very close to Rs/t , as in the following diagram: aj,1 a0j,1 a0j,gj a0j,gj aj,1 a0j,1 . . .
τ -move -
. . .
5. The σ-move (at a pair {aj,ν , a0j,ν } ⊂ Gj ), in which we replace (say for Rs ) the two parallel strands at a pair by the chain of an upward arc, a 0-framed annulus, and a downward arc as in the following diagram: s/t
aj,ν
a0j,ν
aj,ν
σ-move -
a0j,ν
In general the τ -move has to be assumed for all groups both at Rs and Rt . Yet, if (0), where we have only one source and one target we consider cobordisms in Cobconn 3 group, it is easy to see that one τ -move (say at the source) also implies the other. We denote the respective sub-category of tangles with only one group at top and bottom by Tgl(0)conn . (n) are represented by the It has been shown in [KL] that the cobordisms Cobconn 3 tangle category Tgl(n)conn . The tangles in there contain in addition to the top-, bottom-, closed, and through-Ribbons, n so called exterior ribbons. Each of these starts at a fixed point at Rs and ends at another fixed point in Rt . They can also be isotoped, and slid over closed ribbons. However, there is no analog of a σ-move, and the groups, which are associated to one-to-one to the components of ∂φ0 (M ), and onto which we impose the τ -move, include the external strands that represent the punctures in the respective surface. In the special case of Tgl(1)conn the external ribbon can be isolated, and the class of tangles is determined by an ordinary tangle without external strand, see [KL]. We therefore introduce the category, Tgl∞ (1)conn , which has the same tangles as Tgl(0)conn as generators, but where we omitted the τ -move, since we cannot isotop through the extra strand. Although their presentations are different we have Tgl(1)conn ∼ = Tgl∞ (1)conn , which gives us the presentation of Cobconn (1) we used in Subsect. 4.3, as well as the 3 relation in diagram (4.63). For mapping class groups this coincides with the presentation in [MP].
Connectivity of Cobordisms and Half-Projective TQFT’s
589
The composition law in Tgl(0), which makes T into a functor, has mostly been described already in the discussion given in Subsect. 4.1. If we have two cobordisms, M : 6A −→ 6B t 6C
N : 6C t 6D −→ 6E ,
and
where M and N are connected, but the surfaces 6A , 6B etc., may be disconnected, then the elementary composition (11 ⊗ N )(M ⊗ 11) : 6A t 6D → 6B t 6E is represented by the following tangle: 6 6 z }|A { z }|D {
.
.
.
T (M )
. .
.
.
.
. 36C
.
.
.
.
.
. T (N )
. |
{z 6B
}
|
. {z 6E
. }
Here T (M ) and T (N ) are tangles representing M and N , and the tangle 36C is as in (4.52) of Subsect. 4.1. The braces in this diagram enclose the union of the groups in the indicated surface, i.e., we have for example β0 (6A ) + β0 (6D ) groups at Rs , with β1 (6A ) + β1 (6D ) emerging strands. The action of the symmetric group is as described in (4.53). The presentation of a disconnected cobordism is given simply by the presentations of its connected components, if those are in a compatible order. The interior groups can also be computed from the diagrams, in an analogous way as homotopy and homology groups are computed for link presentations of closed manifolds. Using the explicit form of the functor T , see [Ke2], it is not hard to see that, e.g., an a-cycle in H1 (6s ) corresponds to a small meridian around the corresponding sourceribbon, and the b-cycle of the same handle corresponds to a path that is pushed off the same source-ribbon. Acknowledgement. I am indebted to P. Deligne and, especially, V. Lyubashenko for many useful discussions, and, in particular, for contributing Lemma 25. I also thank L. Rozansky for making me aware of the example of supergroups. Further, I wish to thank A. Beilinson, L. Crane, L. Kauffman, D. Kazhdan, T. Kimura, and D. Yetter for interest and encouraging discussions, and J. Stasheff for reading the manuscript and making suggestions for improvements. The author is supported by NSF-grant DMS-9304580.
References [A] [D] [D2] [H]
Atiyah, M.: Publ. Math. Inst. Hautes Etudes Sci. 68, 175–186 (1989) Deligne, P.: In: The Grothendieck Festschrift. II, Progr. in Math. Basel–Boston: Birkh¨auser, 1990 Deligne, P.: Private communication Hennings, M.: Invariants of Links and 3-Manifolds obtained from Hopf Algebras. Preprint (1990); Kauffman, L., Radford, D.: J. Knot Theory and its Rami. 4, no. 1, 131–162 (1995); Ohtsuki, T.: see [O]
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[He] [KW] [Ke1] [Ke2] [Ke3]
[Ke4]
[Ki] [KL] [Ku] [L1] [L2] [L3] [LS] [M] [MP] [O] [Os] [Ro] [Rp] [RS] [RT] [Sw] [Sz] [T] [TV] [Wi]
T. Kerler
Hempel, J.: 3-Manifolds. Annals of Mathematical Studies 86, Princeton, NJ.:Princeton University Press, 1976 Kac, V., Wakimoto, M.: Adv. Math. 40, no. 1, 156–236 (1988) Kerler, T.: Commun. Math. Phys. 168, 353–388 (1994) Kerler, T.: Bridged Links and Tangle Presentations of Cobordism Categories. Adv. Math. To appear. http://www.math.ohio-state.edu/∼kerler/papers/BL/ Kerler, T.: Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family. In: Geometry and Physics, Lecture Notes in Pure and Applied Physics, 184, New York: Marcel Dekker, 1996, http://www.math.ohio-state.edu/∼kerler/papers/Gen/ Kerler, T.: Non-Tannakian Categories in Quantum Field Theory. New Symmetry Principles in Quantum Field Theory, Cargese, 1991. Physics 295, New York: Plenum Press, http://www.math.ohiostate.edu/∼kerler/papers/nonT/ Kirby, R.: Invent. Math. 45, 35–56 (1978) Kerler, T., Lyubashenko, V.: Non-Semisimple TQFT’s for Connected Surfaces. Preprint Kuperberg, G.: Duke Math. J. 84, no.1, 83–129 (1996) Lyubashenko, V.: J. Pure Appl. Alg. 98, 279–327 (1995) Lyubashenko, V.: Commun. Math. Phys. 172, 467–516 (1995) Lyubashenko, V.: Private communication, see also [KL] Larson, R., Sweedler, M.: Am. J. Math. 91, No 1., 75–94 (1969) MacLane, S.: Categories for the Working Mathematician. GTM 5. Berlin–Heidelberg–New York:: Springer-Verlag, 1971 Matveev, S., Polyak, M.: Commun. Math. Phys. 160, 537 (1994) Ohtsuki, T.: Math. Proc. Cambridge Philos. Soc. 117, 259–273 (1995) Ostrik, V.: Decompos. of the Adjoint Repres. of the Small Quantum sl2 . q-alg/9512026 Rotman, J.: An Introduction to Algebraic Topology. GTM 119. Berlin–Heidelberg–New York: SpringerVerlag, 1993 Kerler, T.: Diploma thesis: Darstellungen der Quantengruppen, 1989 Rozansky, L., Saleur, H.: Nucl. Phys. B389, 461 (1993) Reshetikhin, N., Turaev, V.: Invent. Math. 103, 547–597 (1991) Sweedler, M.: Annals of Math. 89, 323–335 (1969) Switzer, R.: Algebraic Topology – Homotopy and Homology. Grundlehren. Berlin–Heidelberg–New York: Springer-Verlag, 1975 Turaev, V.: Quantum Invariants of 3-Manifolds. New York: Walter de Gruyter, 1994 Turaev, V., Viro, O.: Topology. 31, 865–902 (1992) Witten, E.: Commun. Math. Phys. 121, 351–399 (1989)
Communicated by G. Felder
Commun. Math. Phys. 198, 591 – 606 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Odd Invariant Semidensity and Divergence-like Operators on an Odd Symplectic Superspace O. M. Khudaverdian? Laboratory of Computing Techniques and Automation, Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia. E-mails:
[email protected];
[email protected];
[email protected] Received: 19 August 1997 / Accepted: 27 March 1998
Abstract: The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hypersurfaces in Euclidean space. 1. Introduction In this paper we construct a differential first order divergence-like operator on a superspace endowed with an odd symplectic structure. This operator is applied for constructing invariant differential objects in this superspace. In particular an odd semidensity invariant under the transformations preserving the odd symplectic structure of the superspace and the volume form is constructed in a geometrically clear way. The odd symplectic structure plays an essential role in Lagrangian formulation of the BRST formalism (Batalin–Vilkovisky formalism) [2]. In supermathematics it is a natural counterpart of even symplectic structure [3, 10]. On the other hand it has some “odd” features which have no analogues in usual mathematics. Canonical transformations preserving odd symplectic structure (the non-degenerate odd Poisson bracket) do not preserve any volume form. This fact, indeed, is the reason why in the case of odd symplectic structure some invariant geometrical objects have no natural homologues as for the even symplectic structure, for which the superconstructions are rather straightforward generalizations of those for the symplectic structures in the usual spaces. We consider some examples. In order to construct geometrical integration objects one needs to consider a pair, the volume form and the odd symplectic structure. This pair is in fact the geometrical ? On leave of absence from Department of Theoretical Physics, of Yerevan State University, 375049 Yerevan, Armenia
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background for the formulation of Batalin–Vilkovisky formalism (see [2, 5, 13, 7]). The so called 1 operator which plays an essential role in this formalism can be defined in the following way: its action on the function f is equal to the divergence with respect to the given volume form (defined by the action of the theory) of the Hamiltonian vector field, corresponding to the function f . It is a second order differential operator. In the case of even symplectic structure the volume form exists which naturally corresponds to this structure, and the 1 operator is evidently equal to zero (Liouville Theorem). Even if the volume form is arbitrary, one arrives at a first order differential operator [5]. The second example is the invariant volume form (density) which can be defined on the Lagrangian supersurfaces embedded in the superspace endowed with an odd symplectic structure and a volume form. It is nothing but the integrand for the partition function in the space of fields and antifields in the BV formalism [13, 7]. We focus on the third example, on the problem of finding an analogue of Poincar´e– Cartan integral invariants for the odd symplectic structure. In usual mathematics to the Poincar´e–Cartan integral invariant (invariant volume form on the embedded surfaces) corresponds the wedged power of a differential two-form which defines the symplectic structure. In case of even symplectic structure in spite of the fact that differential form has nothing in common with invariant integration objects, the superdeterminant of the two-form induced on the embedded supersurface from the two-form which defines the symplectic structure leads us to the Poincar´e–Cartan invariant [8]. For the odd symplectic structure the situation is essentially different. In [6] the problem of constructing of the invariant densities was considered for the superspace endowed with an odd symplectic structure. Density is the object which defines a volume form on embedded (super)surfaces. If z = z(ζ) is the local parametrization of a (super)surface then a density A is a function ∂2z ∂z ∂ 2 z ∂k z , , . . . A = A z(ζ), , , . . . , subject to the condition that of z(ζ), ∂z ∂ζ ∂ζ∂ζ ∂ζ ∂ζ∂ζ ∂ζ...∂ζ ˜ A → A · sdet ∂ζ . A is a density of the rank k if under reparametrization ζ → ζ(ζ), ∂ ζ˜ it depends on the tangent vectors of the order ≤ k (i.e. on the derivatives of order ≤ k). In usual mathematics densities are the natural generalization of differential forms (if k = 1). In supermathematics even in the case of k = 1 this object is of more importance, since differential forms in supermathematics are no longer integration objects [3, 4, 1, 14]. In the case of symplectic structure in usual mathematics as well as in the case of even symplectic structure invariant densities are exhausted by densities of the rank k = 1 which correspond to Poincar´e–Cartan integral invariants [8]. This is not the case for the odd symplectic structure. One can show that on (p.p)-dimensional non-degenerate supersurfaces embedded in a superspace E n.n which is endowed with a volume form and an odd symplectic structure there are no invariant densities of rank k = 1 (except the volume form itself), and in the class of densities of rank k = 2 and of weight σ (i.e. which are multiplied by the σ th power of the superdeterminant of the reparametrization) a unique (up to multiplication by a constant) semidensity (σ = 21 ) exists in the case of p = n − 1 [6]. In fact, in [6] this semidensity was constructed in a non-explicit way in terms of dual densities: If the (n − 1.n − 1)-dimensional supersurface M is given by the equations f = 0, ϕ = 0, where f is an even function and ϕ is an odd function, then this semidensity corresponds to the function {f, {f, ϕ}} {f, f } {f, f } 1 1ϕ − − 1f − =p {ϕ, {f, ϕ}} A˜ 2{f, ϕ} {f, ϕ} 2{f, ϕ}2 f =ϕ=0 {f, ϕ} (1.1)
Divergence-like Operators on Odd Symplectic Superspace
593
which depends on the second derivatives (k = 2), is invariant under the transformations preserving the odd symplectic structure and the volume form, and is multiplied by the square root of the corresponding Berezinian (superdeterminant) under the transformation f → af +αϕ, ϕ → βf +bϕ, which does not change the supersurface M . The semidensity (1.1) takes odd values. It is an exotic analogue of the Poincar´e–Cartan invariant: the corresponding density A˜ 2 = 0, so it cannot be integrated nontrivially over supersurfaces. The formula (1.1) was obtained in [6] by straightforward calculations. To clarify its geometrical meaning in this paper a special geometrical object for odd symplectic superspace is considered. As it was mentioned above, to a symplectic structure in usual space corresponds a volume form (Liouville form): p (1.2) dv = ρ(x)dx1 ...dx2n = det (ik )dx1 ...dx2n , where = ik dxi ∧ dxk is the closed non-degenerated two-form which defines the symplectic structure. The volume form (1.2) is preserved under canonical transformations (i.e., the trans∂ formations preserving the two-form ). If X = X i ∂x i is an arbitrary vector field one can consider its divergence: divX =
LX dv ∂X i ∂ log ρ = + Xi . i dv ∂x ∂xi
(1.3)
In a symplectic space the canonical transformations preserve not only the volume form ˆ (1.2) but also its projection on an arbitrary symplectic subspace. Moreover, if L(z) is ˆ a projector-valued function such that ImL(z) is a symplectic subspace in Tz E, then it ˆ is easy to see that to L(z) one can associate a divergence-like invariant operator whose action on an arbitrary vector field X is given by the expression k ˆ X) = ∂X + 1 X m ∂ip pk Lik . (1.4) ∂(L, ∂xi 2 ∂xm (compare with (1.3) in case of Lˆ = id). The formulae (1.2)–(1.4) have the straightforward generalization to the case of even symplectic structure in superspace (by changing the determinant to a superdeterminant and adding the powers of (−1) wherever necessary). It is not the case for the odd symplectic structure, where there is not any invariant volume form related with this structure. Nevertheless, it turns out that the analogue of formula (1.4) can be considered for odd symplectic structure in a case where Lˆ is a projector on (1.1)-dimensional symplectic subspace and X is the odd vector field which belongs to this subspace and is symplectoorthogonal to itself. In the next two sections we perform the corresponding constructions which are essentially founded on the following remark. Let E 1.1 be (1.1)-dimensional odd symplectic superspace and (x, θ) be Darboux coordinates on it: {x, θ} = 1, {x, x} = 0, where { , } is the odd Poisson bracket (Buttin bracket) corresponding to this symplectic structure. Let 9 be an odd vector field in this superspace which is equal to 9(x, θ)∂/∂θ in these Darboux coordinates. Then this vector field has the same form and its divergence (∂9(x, θ)/∂θ) remains the same in arbitrary Darboux coordinates, in spite of the fact that there is no invariant volume form. (See Example 2.) In Sect. 4 we consider a (n − 1.n − 1)-dimensional supersurface embedded in an odd symplectic superspace E n.n endowed with a volume form. The differential
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operator described above can be naturally applied to the odd vector field which is defined only at the points of this supersurface and is symplecto-orthogonal to itself and to this supersurface. In spite of the fact that this field is not defined on the whole superspace one can define the invariant “truncated divergence” of this vector field. The analogue of this construction for usual symplectic structure is trivial. On the other hand this construction can be considered as an analogue of the corresponding operator acting on the vector field which is defined in a Euclidean (Riemannian) space at the points of embedded surface. But the essential difference is that the group of transformations which preserve metrics in Euclidean (Riemannian) space is exhausted by linear transformations (the linear part of transformation defines uniquely all higher terms) and this is not true for symplectic case where the group of canonical transformations is infinite-dimensional. In Sect. 5 we apply our geometrical construction to obtain the formula for odd semidensity (1.1) in a geometrically clear way: it turns out that on (n − 1.n − 1)dimensional supersurfaces embedded in a (n.n)-dimensional odd symplectic superspace endowed with a volume form one can define in a natural way the odd semidensity whose values are odd vectors, symplecto-orthogonal to itself and to these supersurfaces. The “truncated divergence” of this vector-valued semidensity is an invariant semidensity (1.1). It has to be noted that our formula for this semidensity is very similar to the formula for the density corresponding to the mean curvature of hypersurfaces in the Euclidean space. In the last section to discuss this point we consider invariant operators in Riemannian geometry which can be treated as analogues of the geometrical constructions of this paper.
2. Odd Symplectic Superspace Let E n.n be a superspace with coordinates z A = x1 , . . . , xn , θ1 , . . . , θ n ; p(xi ) = 0, p(θj ) = 1, where p is a parity. We say that this superspace is an odd symplectic superspace if it is endowed with odd symplectic structure, i.e., if an odd closed nondegenerate 2-form = AB (z)dz A dz B ,
(p() = 1,
d = 0)
(2.1)
is defined on it [3, 10]. To the differential form (2.1) on the superspace E n.n one can relate a function which at every point defines the following skewsymmetric (in a supersense) odd bilinear form on tangent vectors: ˜˜
(X(z), Y(z)) = −(Y(z), X(z))(−1)XY , (X(z), λY(z)) = (X(z)λ, Y(z)), (λY(z) + µZ(z), X(z)) = λ(Y(z), X(z)) + µ(Z(z), X(z)), p ((X, Y)) = 1 + p(X) + p(Y).
(2.2)
More precisely, a point of superspace E n.n is 3-point– 2n-plet (a1 , . . . , an , α1 , . . . , αn ), where (a1 , . . . , an ) are even and (α1 , . . . , αn ) are odd elements of an arbitrary Grassmann algebra 3. (We use the most general definition of superspace suggested by A.S. Schwarz as the functor on the category of Grassmann algebras [12].)
Divergence-like Operators on Odd Symplectic Superspace
In the coordinates:
˜ ˜
AB = −BA (−1)AB =
∂ ∂ , B A ∂z ∂z
595
, (p(AB ) = 1 + p(A) + p(B)) .
(2.3)
We use the notation ˜ for the parity of corresponding object. X(z), Y(z), Z(z) are the vector fields X A (z) ∂z∂A , Y A (z) ∂z∂A , Z A (z) ∂z∂A , the left derivations of functions on E n.n . (The differential form is usually considered as an element of an algebra generated by z A and dz A , where the parity of dz A is opposite to the parity of z A . In this case ˜ ˜ AB = BA (−1)(A+1)(B+1) instead of (2.3). The slight difference is eliminated by the ˜ transformation AB → AB (−1)B ). From (2.2, 2.3) it follows that ∂ ∂ ˜ ˜ ˜ A B (2.4) X (z) A , Y (z) B = X A (z)AB (z)Y B (z)(−1)YB+Y . ∂z ∂z In the same way as in the standard symplectic calculus one can relate to the odd symplectic structure (2.1) the odd Poisson bracket (Buttin bracket) [3, 10]: ∂g ∂f ˜ ˜ ˜ (−1)f A+A AB B , A ∂z ∂z
{f, g} = ˜
(2.5)
˜
where AB = −BA (−1)(A+1)(B+1) = {z A , z B } is the inverse matrix to AB : A AC CB = δB . To a function f via (2.5) there corresponds the Hamiltonian vector field Df = {f, z A }
∂ ∂z A
and
Df (g) = {f, g},
(Df , Dg ) = −{f, g}.
(2.6)
The condition of the closedness of the form (2.1) leads to the Jacoby identities: ˜
˜
{f, {g, h}}(−1)(f +1)(h+1) + cycl.
permutations = 0.
(2.7)
Using the analog of the Darboux Theorem [11] one can consider the coordinates in which the symplectic structure (2.1) and the corresponding Buttin bracket have the canonical expressions. We call the coordinates wA =(x1 , . . . , xn , θ1 , ..., θn ) Darboux coordinates if in these coordinates ∂ ∂ = 0, , = IAB dwA dwB : ∂xi ∂xj (2.8) ∂ ∂ ∂ ∂ , , = 0, = −δ ij ∂θi ∂θj ∂xi ∂θj holds, respectively {xi , xj } = 0, {θi , θj } = 0, {xi , θj } = δ ij , n X ∂f ∂g f˜ ∂f ∂g . + (−1) {f, g} = ∂xi ∂θi ∂θi ∂xi
(2.9)
i=1
Now on the odd symplectic superspace E n.n endowed with an odd symplectic structure (2.1) we consider the following geometrical constructions: Let 9(z) be an odd non-degenerate vector field symplecto-orthogonal to itself:
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p(9(z)) = 1,
(9(z), 9(z)) = 0.
(2.10)
(We call the vector non-degenerate if at least one of its components is not nilpotent.) For example, to the even function f such that {f, f } = 0 there corresponds the Hamiltonian vector field Df defined by (2.6), subject to condition (2.10). Let 5(z) be a field of (1.1)-dimensional subspaces (5(z) ∈ T Ezn.n ) which contain the vector field 9(z) and the symplectic structure induced on these subspaces is not degenerate. It means that there exists an even vector field H(z) such that 9(z), H(z) ∈ 5(z)
and
(H(z), 9(z)) = 1.
(2.11)
To this field of subspaces 5(z) there corresponds the symplecto-orthogonal projector ˆ 5(z) of the vectors in the tangent space on these subspaces: ˆ (z) = id, 5X ˆ = 0 if (X, 5) = 0. ˆ : Tz E → 5(z), 5 (2.12) 5 5
ˆ there corresponds the matrix-valued function In the coordinates {z A } to the projector 5 ∂ ∂ ∂ ∂ B A ˆ ˆ (z) : 5 (z) , so 5 X (z) . = 5 = X A (z)5B 5B A A A (z) ∂z A ∂z B ∂z A ∂z B (2.13) ˆ Later on we call (5(z), 9(z)) or equivalently (5(z), 9(z)) an odd normal pair if 9(z) and 5(z) are defined by (2.10–2.12). 3. Divergence-like Operator on Odd Normal Pairs In this section for an odd normal pair (5(z), 9(z)) in an odd symplectic superspace we construct a first-order divergence-like differential operator which transforms it to a function on this superspace. ˆ In a superspace E, let X(z) and L(z) be a vector field and a linear operator field A ˆ X) respectively, defined on Tz E. If {z } are arbitrary coordinates then for the pair (L, A one can consider the function which depends on the coordinate system {z }: ˆ X){z} = ∂(L,
∂X A (z) B ˜ ˜ ˜ LA (z)(−1)XB+B . ∂z B
(3.1)
Expression (3.1) is invariant under linear transformations of the coordinates {z A }. In the general case if {wA } and {z A } are two different coordinate systems on E then for ˆ X) we consider the pair (L, {w}
{w} ˆ X) ˆ ˆ X){z} . 0(L, − ∂(L, {z} = ∂(L, X)
(3.2)
From (3.1) and (3.2) it follows that {w}
Q A B ˆ X) 0(L, {z} = X (z)0QB (z |{w}, {z})LA (z),
0A BC (z |{w}, {z}) =
∂ 2 wK (z) ∂z A (w) ˜ (−1)C . ∂z B ∂z C ∂wK
(In (3.3) the components of Lˆ and X are in the coordinates {z A }.)
(3.3)
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ˆ X){w} it follows that for three different coordinate From definition (3.2) of the 0(L, {z} systems {wA }, {z A } and {uA }, {w}
{z}
{u}
ˆ ˆ ˆ X) 0(L, {z} + 0(L, X){u} + 0(L, X){w} = 0.
(3.4)
ˆ X), Let F be a class of some coordinate systems such that for a given pair (L, ∀{w}, {w} ˜ ∈F
{w} ˜
ˆ X) 0(L, {w} = 0.
(3.5)
Then to the class F one can relate the first-order divergence-like differential operator D: ˆ X){w} , ˆ X) = ∂(L, ˆ X){z} + 0(L, (3.6) D(L, {z} where {z A } are arbitrary coordinates on E and {wA } are arbitrary coordinates from the class F. From (3.2, 3.5) it follows that the r.h.s. of (3.6) does not depend on the choice of these coordinates. Before going to the considerations for an odd symplectic superspace we consider an example where we come to the standard definition of the divergence in superspace using (3.1–3.6). Example 1. Let E be a superspace with a volume form dv which in coordinates {w0A } = {x1 , . . . , xn , θ1 , . . . , θ m } on E is equal to dv = dx1 . . . dxn dθ1 . . . dθm .
(3.7)
We define F as a class of coordinate systems in which the volume form dv is given by (3.7): ∂w = 1 }. (3.8) F = { {w} : Ber ∂w0 (BerA is the superdeterminant of A.) It is easy to see that if Lˆ = id is the identity operator and X is an arbitrary vector field then in arbitrary coordinates {z A } {w}
Q ˆ X) 0(L, {z} = X (z)
∂ 2 wK (z) ∂z A (w) ∂ log ρ(z) ˜ (−1)A = X Q (z) , ∂z Q ∂z A ∂wK ∂z Q
(3.9)
where {wA } are arbitrary coordinates from the class (3.8) and ρ(z)dz 1 . . . dz m+n is the volume form (3.7) in the coordinates {z A } (ρ = Ber(∂w/∂z)). The condition (3.5) is fulfilled and we come to the standard definition of the divergence in a superspace endowed with volume form. For the pair (id, X) and for class (3.8) the operator D(id, X) is the divergence of the vector field X corresponding to the volume form dv: D(id, X) =
∂X A (z) ∂ log ρ(z) ˜ ˜ ˜ (−1)XA+A + X A = divdv X. ∂z A ∂z A
(3.10)
Now we return to the considerations of Sect. 2. For the superspace E n.n endowed with the odd symplectic structure we consider a ˆ ˆ field (5(z), 9(z)), where (5(z), 9(z)) is an odd normal pair in a vicinity of some point. (See the end of the previous section.) We denote by FD the class of Darboux coordinates (2.8, 2.9) on E n.n and apply constructions (3.1–3.6) in this case.
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ˆ Lemma. If (5(z), 9(z)) is an odd normal pair in E n.n then {w} ˜
ˆ 9) 0(5, {w} = 0.
∀{w}, {w} ˜ ∈ FD ,
(3.11)
Using the statement of the lemma we consider the action of the operator Dcan corresponding to the class FD of Darboux coordinates by (3.6), on the odd normal pair ˆ (5(z), 9(z)): A ˆ 9) = ∂9 (z) + 9Q (z)0A (z |{w}, {z}) 5B (3.12) Dcan (5, QB A (z), ∂z B where {wA } are arbitrary Darboux coordinates. From (3.6) and the lemma follows ˆ 9) is an invariant geometrical ˆ Theorem. For an odd normal pair (5(z), 9(z)), Dcan (5, object. In particular if {wA } are arbitrary Darboux coordinates then A ˆ 9) = ∂9 (w) 5B Dcan (5, A (w) ∂wB
(3.13)
does not depend on the choice of Darboux coordinates {wA }. Before proving the lemma we will consider Example 2. Let E 1.1 be a (1.1)-dimensional superspace endowed with odd symplectic structure (2.1). Let w = (x, θ) be some Darboux coordinates on it: {x, θ} = 1, {x, x} = 0. It is easy to see that in this case the odd vector field obeying (2.10) is of the form ∂ , (3.14) ∂θ where 9(x, θ) is a non-nilpotent even function. The projector operator (2.12) is evidently an identity operator. Hence a normal pair is of the form (id, 9(x, θ)∂θ ) and 9 = 9(x, θ)
Dcan (id, 9) =
∂9(x, θ) . ∂θ
(3.15)
One can see that if x0 , θ0 are some other Darboux coordinates then they are related with coordinates x, θ by canonical transformation x0 = f (x),
θ0 =
θ + β(x), df (x)/dx
(3.16)
where f (x) and β(x) are even and odd functions on E 1.1 respectively. (To obtain (3.16) one has to note that in E 1.1 {x0 , x0 } = 0 → x0θ = 0.) It is easy to see from (3.16) that 9 = 9(x, θ)∂/∂θ = (9(x, θ)/fx )∂/∂θ0 and (3.15) does not change under transformation (3.16), so in this case the statements of the lemma and of the theorem hold. Indeed, for this case one can say more about (3.15). Let (y, η) be an arbitrary coordinates on the E 1.1 and a volume form dv on E 1.1 is defined by the equation dv =
dydη . {y, η}
(3.17)
Then one can check using (3.10) that Dcan (id, 9) in (3.15) is the divergence of the vector field 9 by the volume form (3.17). For a vector field (3.14) and a volume form (3.17) divdv 9 does not depend on the choice of coordinates (y, η).
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Proof of the Lemma. Let B C A 2 z A = w B LA B + w w TBC + o(w )
(3.18)
be an arbitrary canonical transformation from Darboux coordinates {z} to Darboux ˆ coordinates {w} in a vicinity of the point z = 0. Let (5(z), 9(z)) be an odd normal pair which is defined also in the vicinity of the point z = 0. For proving the lemma we have to show that for transformation (3.18), ˆ 9){w} = 0. (3.19) 0(5, {z} z=0
(We can consider (3.18) without loss of generality, since in Darboux coordinates the translation is obviously the canonical transformation.) We include the Darboux transformation (3.18) in the chain of Darboux transformations: linear
linear
˜ −→ {w}−→{w} ˜ {z}−→{z}
(3.20)
which obey the following conditions: a) The transformation {z} → {z} ˜ is the linear canonical transformation such that in n.n ˆ the Darboux coordinates z˜ A = (x˜ 1 , . . . , x˜ n , θ˜1 , . . . , θ˜n ) 5| z=0 projects T Ez |z=0 on 1 1 ˜ the subspace which is generated by the vectors ∂/∂ x˜ , ∂/∂ θ : ∂ ∂ ∂ ∂ ˆ ˆ 5 = , 5 = 1. (3.21) 1 1 ˜ ∂ x˜ z=0 ∂ x z=0 ∂ θ˜1 ∂ θ˜ b) The transformation {w} −→ {w} ˜ is the linear canonical transformation such that ˜ w˜ A = z˜ A + o(z).
(3.22)
From (3.2, 3.4) it follows that {w}
{w}
{w} ˜
{z} ˜
ˆ ˆ ˆ ˆ 9) 0(5, {z} = 0(5, 9){w} ˜ + 0(5, 9){z} ˜ + 0(5, 9){z} . {w}
(3.23)
{z} ˜
ˆ 9) ˆ But 0(5, {w} ˜ and 0(5, 9){z} are zero because the corresponding transformations ˜ ˆ 9){w} are linear. To prove that 0(5, {z} ˜ |z=0 is also zero we note that from (3.21) and (2.10)
˜ it follows that 9{z} |z=0 = 9∂/∂θ˜1 (compare with (3.14)). Hence from (3.3), from the property (3.22) of the transformation w˜ → z˜ and from (2.9) it follows that ∂ 2 x01 9 ∂ ˜ 01 01 ˆ 9){w} 0(5, = 9 = {x , x } (3.24) = 0, {z} ˜ z=0 z=0 ∂ θ˜1 ∂ x˜ 1 z=0 2 ∂ x˜ 1 where w˜ A = (x01 , . . . , x0n , θ01 , . . . , θ 0n ). Hence (3.19) is obeyed. The Lemma is proved.
ˆ Considering a pair (L(z), X(z)) we constructed in this section the divergence-like operator (3.12, 3.13) in an odd symplectic superspace in the case when Lˆ is a projector on an (1.1)-dimensional symplectic subspace and X is an odd vector in it which is symplecto-orthogonal to itself. In the case of even symplectic structure this construction can be carried out in a more general case and it is trivial because in this case there exists a volume form corresponding to the symplectic structure. Indeed if in a superspace ˆ X(z)), where E 2m.n endowed with even symplectic structure there is given a pair (L(z), ˆ L(z) is a symplecto-orthogonal projector on (2p.q)-dimensional symplectic subspaces in Tz E 2m.n and X(z) is an arbitrary vector field then to the class F of the Darboux ˆ X) defined by (3.6) which is a coordinates on this superspace there corresponds D(L, straightforward generalization of (1.4).
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4. Truncated Divergence We consider in this section an odd vector field which is defined at the points of (n − 1.n − 1)-dimensional nondegenerate supersurface embedded in the (n.n)-dimensional odd symplectic superspace with a volume form. In the case of this vector field being symplecto-orthogonal to this supersurface and to itself using the geometrical object ˆ 9) for an odd normal pair we define the linear operator on it (truncated diverDcan (5, gence) whose action on this field gives the function on this supersurface. Let M be an arbitrary nondegenerate (n − 1.n − 1)-dimensional supersurface embedded in a superspace E n.n which is endowed with odd symplectic structure (2.1 ) and the volume form dv: (4.1) dv = ρ(z)dz 1 . . . dz 2n . (The supersurface M is nondegenerate if the symplectic structure of E n.n induces nondegenerate symplectic structure on M .) Let z A = z A (ζ α ) be a local parametrization of the supersurface M . The induced two-form αβ dζ α dζ β on M according to (2.2-2.4) is given by the following equation: ˜ ˜ ˜ A ∂ B ∂ αβ (z(ζ)) = ∂α z , ∂β z = ∂α z A (ξ)AB (z(ξ))∂β z B (ξ)(−1)B β+β . A B ∂z ∂z (4.2) A ∂f Hereafter we use notations ∂α z A = ∂z , ∂ f = , . . . for derivatives along the α ∂ζ α ∂ζ α supersurface. The induced Poisson bracket structure on M { , }M according to (2.5, 2.6) is defined by the matrix αβ which is inverse to the matrix αβ . Using this induced symplectic structure one can construct the symplecto-orthogonal projector on T M , Pˆ (ζ) : Tz(ζ) E n.n −→ Tz(ζ) M which can be expressed in terms of AB and αβ : ˜
˜ α ˜ αβ PAB (ζ) = AK (z(ζ)) · {z K (ζ), z B (ζ)}M = AK (z(ζ))∂α z K (−1)K α+ ∂β z B . (4.3) The operator ˆ 5(ζ) = id − Pˆ (ζ) (4.4)
at every point z(ζ) is a symplecto-orthogonal projector on the (1.1)-dimensional subspace 5(z(ζ)) in Tz(ζ) E which is symplecto-orthogonal and transversal to Tz(ζ) M because M is non-degenerate. (See 2.10–2.12.) We consider an odd vector field 9(ζ) defined at the points of M which is symplectoorthogonal to M , to itself and which is non-degenerate: 9(ζ) ∈ 5(z(ζ)), Pˆ 9 = 0, (9, 9) = 0, ˜
˜ α ˜ = 0, ∂α z A PAB = ∂α z B . 9A PAB = 0, 9A AK ∂α z K (−1)K α+
(4.5) (4.6)
For example if the supersurface M is defined by equations f (z) = 0, ϕ(z) = 0, then for an arbitrary point z0 on this supersurface, the vectors Dϕ (z0 ) and Df (z0 ) (see 2.6) are the basis vectors of the subspace 5(z0 ) and the odd vector 9 = (2Df +({f, f }/{f, ϕ})Dϕ ) z0
is subject to the conditions (4.5). One can see that the vector field obeying the conditions (4.5) is fixed uniquely up to the multiplication by an even non-nilpotent function of ζ, 9(ζ) → f (ζ)9(ζ). In other words (4.5) define the linear subbundle SN (M ) in T E n.n |M .
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ˆ form an odd normal pair A field 9 obeying these conditions and the projector 5 ˆ 9) at the points z(ζ) of the supersurface M . (5, Now on odd vector fields, sections of the bundle SN (M ), we will define the linear operator “truncated divergence". For this purpose we consider in the vicinity of a given ˜ˆ ˜ point z(ζ) an odd normal pair (5(z), 9(z)) in E n.n which is a prolongation of the odd ˜ˆ ˜ ˆ ˆ normal pair (5(ξ), 9(ξ)): 9| M = 9, 5|M = 5 and define the truncated divergence in the following way: ˜ˆ 9) ˜ − Dcan (5, ˜ . Divtrunc (9(ζ)) = divdv 9 (4.7) M
˜ˆ 9) ˜ˆ 9), ˜ is ˜ is given by (3.12, 3.13) for the odd normal pair (5, ˜ divdv 9 In (4.7) Dcan (5, the divergence (3.10) of the vector field corresponding to the volume form dv defined by (4.1). One can see that the r.h.s. of Eq. (4.7) indeed does not depend on the prolongation ˜ˆ 9) ˜ of the odd normal pair (5, ˆ 9), moreover (4.7) is a zeroth order linear differential (5, operator: (4.8) Divtrunc (f (ζ)9(ζ)) = f (ζ) · Divtrunc (9(ζ)) , because 9 is symplecto-orthogonal to M . ˆ and Pˆ , Eq. (4.3) for Pˆ , the conditions Using the relation (4.4) between operators 5 (4.6) we arrive by straightforward calculations at the following expression for (4.7): Divtrunc (9(ζ)) = ∂ log ρ(w) ˜ α ˜ ˜ β)+ ˜ − 9A (ζ)IAB ∂β ∂α wB (ζ)αβ (w(ζ))(−1)B(α+ . (4.9) 9A (ζ) · ∂wA w(ζ) Here 9(ζ) = 9A (ζ)∂/∂wA is an odd vector field obeying (4.5), {wA } are arbitrary Darboux coordinates on E n.n , IAB is the matrix of the symplectic two-form in Darboux coordinates (see 2.8), ρ(w) is the density of the volume form (4.1) in these coordinates and wA (ζ) is the parametrization of the non-degenerate supersurface M n−1.n−1 . In the case if there exist Darboux coordinates {wA } in which the volume form (4.1) obeys condition (4.10) dv = dw1 . . . dw2n , i.e. its density ρ(w) ≡ 1 then ˜
˜
˜ β)+α ˜ Divtrunc (9(ζ)) = −9A (ζ)IAB ∂β ∂α wB (ζ)αβ (w(ζ))(−1)B(α+ .
(4.11)
The analogous construction in Riemannian geometry in the case if a vector field is tangent to a surface is reduced to the divergence along a surface and in the case if a vector field is orthogonal to a surface is reduced to a linear operator related with the second quadratic form of this surface (see Sect. 6). 5. The Odd Invariant Semidensity Now we are well prepared for writing the formula for odd invariant semidensity using constructions (4.7–4.11) for truncated divergence. The constructions of this section are founded on the following remark. Let M n−1.n−1 be an arbitrary nondegenerate (n − 1.n − 1)-dimensional supersurface embedded in the odd symplectic superspace E n.n
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and a field 9 on this supersurface obeys conditions (4.5). Then the r.h.s. of (4.7, 4.9) by its definition is invariant under coordinate transformations of the superspace E n.n and does not depend on the parametrization z A = z A (ζ α ) of the supersurface M n−1.n−1 . So Eqs. (4.7, 4.9) define a density of the weight σ = 0 on this supersurface. Moreover, if 9 is a density of an arbitrary weight σ which is defined on non-degenerate (n − 1.n − 1)dimensional non-degenerate supersurfaces and takes values in the odd vector fields obeying (4.5) then the truncated divergence (4.9) of this density is the density of the same weight σ which is defined on (n − 1.n − 1)-dimensional nondegenerate supersurfaces and takes numerical values. Let, as in Sect. 4, E n.n be a superspace which is endowed with odd symplectic structure (2.1) and volume form (4.1), and M n−1.n−1 be an arbitrary nondegenerate supersurface in this odd symplectic superspace. Now we will construct the semidensity on M which takes values in the odd vectors 9 obeying conditions (4.5). Let the vectors (e1 , ..., en−1 ; f1 , ..., fn−1 ) constitute a basis of the tangent space Tz(ζ) M at an arbitrary point z(ζ) of the supersurface M n−1.n−1 (ei are even vectors and fi are odd ones). Let 9(z(ζ)) and H(z(ζ)) be respectively an odd and an even vector field, which belong to 5(z(ζ)) (see (4.5)) such that (5(z(ζ)), 9(z(ζ)) form an odd normal pair: H, 9 ∈ 5,
(9, 9) = 0
and
(H, 9) = 1.
(5.1)
These conditions fix the vector fields H and 9 up to the transformation H→
1 H + β9, λ
9 → λ9,
(5.2)
where λ is an arbitrary even function (taking values in non-nilpotent numbers) and β is an arbitrary odd function (compare with (3.16)). Using transformation (5.2) one can choose the vector field 9 (but not the vector field H) in the unique way by imposing the normalization condition via volume form (4.1): dv(e1 , ..., en−1 , H; f1 , ..., fn−1 , 9) = 1.
(5.3)
We arrive at the function 9 = 9(z(ζ), e1 , ..., en−1 ; f1 , ..., fn−1 ),
(5.4)
which depends on points z(ζ) of the supersurface M n−1.n−1 and the basis (e1 , ..., en−1 ; f1 , ..., fn−1 ) in the Tz(ζ) M n−1.n−1 and which takes values in odd vectors obeying condition (4.5). This function is defined uniquely by conditions (5.1, 5.3). It is easy to see that under the change of the basis the function (5.4) is multiplied by the square q root of
the corresponding Berezinian. For example if e1 → λe1 and f1 → µf1 then 9 → µλ 9. Hence (5.4) defines semidensity. (It is interesting to note that these considerations for obtaining the formula for invariant vector-valued semidensity are similar to the considerations for obtaining the formula for the invariant density on the Lagrangian surfaces in E n.n suggested by A.S. Schwarz [13].) If z A = z A (ζ α ) is any parametrization of a supersurface M n−1.n−1 , where {ζ α } = (ξ 1 , . . . , ξ n−1 ; ν 1 , . . . , ν n−1 ) are even and odd parameters of this supersurface then considering as the basis vectors e1 =
∂z A ∂ ∂z A ∂ ∂z A ∂ ∂z A ∂ , . . . , e = ; f = , . . . , f = , n−1 1 n−1 ∂ξ 1 ∂z A ∂ξ n−1 ∂z A ∂ν 1 ∂z A ∂ν n−1 ∂z A
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603
we come from (5.4) to odd vectors valued semidensity 9(z(ζ), ∂z ∂ζ ) of rank k = 1 on nondegenerate (n − 1.n − 1)-dimensional supersurfaces. The truncated divergence of this semidensity is the odd semidensity of rank k = 2. Using formula (4.9) we arrive at this odd invariant semidensity: ∂w ∂w ∂ 2 wA , = Divtrunc 9 w(ζ), = (5.5) A w(ζ), ∂ζ ∂ζ∂ζ ∂ζ ∂w ∂ log ρ(w) ˜ α ˜ ˜ β)+ ˜ 9C w(ζ), · −ICB ∂β ∂α wB (ζ)αβ (w(ζ))(−1)B(α+ + , ∂ζ ∂wC w(ζ) where {wA } are arbitrary Darboux coordinates in E n.n , ρ(w) is the density of the volume form in these coordinates and w(ζ) is parametrization of the supersurface M n−1.n−1 . In a case if there exist Darboux coordinates {wA } in which the volume form dv obeys condition (4.10) (ρ(w) ≡ 1) formula (5.5) according to(4.11) is reduced to ∂w ∂ 2 w , A w(ζ), ∂ζ ∂ζ∂ζ (5.6) ∂w ˜ α ˜ ˜ β)+ ˜ = −9C w(ζ), · ICB ∂β ∂α wB (ζ)αβ (w(ζ))(−1)B(α+ . ∂ζ The semidensity (5.5) is nothing but the semidensity (1.1) obtained in [6] because of its uniqueness. (See Ref. [6].) To compare (5.5) with (1.1) we do the following. Let M be a (n − 1.n − 1)-dimensional arbitrary non-degenerate supersurface in E n.n and z0 be any point in M . One can consider Darboux coordinates {wA } = (x1 , . . . , xn , θ1 , . . . , θ n ) on E n.n and parameters {ζ α } = (ξ 1 , . . . , ξ n−1 , ν 1 , . . . , ν n−1 ) such that this supersurface in the parametrization wA (ζ α ) is flat up to second order derivatives in the vicinity of z0 : xn = o(ζ 2 ), θn = o(ζ 2 ), xi = ξ i + o(ζ 2 ), θi = ν i + o(ζ 2 ) if 1 ≤ i ≤ n − 1 and ρ(z0 ) = 1. The vector valued semidensity (5.4) at the point z0 in these coordinates and in this parametrization is equal to ∂/∂θn , ∂α ∂β wA |z0 = 0, hence semidensity (5.5) at the point z0 is equal to ∂ log ρ . (5.7) ∂θn z=z0 On the other hand to this supersurface M there correspond the functions f = xn + o(z 2 ) and ϕ = θn + o(z 2 ) in (1.1), hence the dual density A˜ in (1.1) at the point z0 is equal to ∂ log ρ = divdv Df = 1f ∂θn z=z0 z=z0 z=z0 and coincides with (5.7).
6. Discussion In this section we consider analogues of the geometrical constructions (3.6), (4.7) and (5.5) in Riemannian geometry. 1) Let E n be n-dimensional Riemannian space endowed with Riemannian metrics G. ˆ ˆ Then for a pair (L(x), h(x)), where L(x) and h(x) are a linear operator and vector field respectively, defined on Tx E n , one can consider the divergence-like operator
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ˆ h) = T r(Lˆ · ∇h) = Lki ∇k hi = D(L,
∂hi i p + 0kp h Lki , ∂xk
(6.1)
where ∇ is a covariant derivative corresponding to the metrics G and 0ikm are the components of the corresponding connection in the coordinates {xi }. In a case if E is locally Euclidean, i.e., there exists a class Feuc of local coordinates in which ˆ h) in (6.1) is nothing but Deuc (L, ˆ h) in Gik ≡ δik , then it is easy to see that D(L, (3.6) corresponding to the class Feuc and (3.3) defines the components of the trivial connection in the coordinates {xi }. Here like in a case of even symplectic structure (1.4) we do not need to put special conditions like (2.10, 2.11). 2) The construction analogous to (4.7) can be also carried out without special conditions on the dimension of the embedded surface and on the vector field. Let M be an arbitrary surface embedded in the Riemannian space E n and h be an arbitrary vector field which is defined at the points of M . It is useful to recall here the following standard formulae from Riemannian differential geometry [9]: if u1 and u2 are arbitrary vector fields which are defined on M and are tangent to M then Pˆ (∇u1 u2 ) = (∇M )u1 u2 , ˆ u1 u2 ) = A(u1 , u2 ), 5(∇
Pˆ is orthogonal projector T E n |M → T M, ˆ 5 is orthogonal projector T E n |M → T M ⊥ ,
(6.2)
where ∇M is the covariant derivative on M induced from E n and A(u1 , u2 ) is the bilinear symmetric form on the T M which takes values in the vectors orthogonal to M . To an arbitrary vector field h which is defined at the points of M there correspond number-valued symmetric bilinear form ah and related to it the linear operator Ah on T M such that ah (u1 , u2 ) = G (u1 , Ah (u2 )) = G (h, A(u1 , u2 )) .
(6.3)
In the case if M is a hypersurface and n is the field of unit orthogonal vectors then an is nothing but its second quadratic form. The trace of An defines the mean curvature at the points of M and the determinant of An defines the Gaussian curvature. Using (6.1) we consider the following expression as an analogue of the formula (4.7) for truncated divergence: ˜ˆ · ∇h) ˜ = T r(P˜ˆ · ∇h)| ˜ (6.4) Divtrunc h = div h˜ − T r(5 M, M
˜ˆ P˜ˆ are prolongations of h, 5, ˜ 5, ˆ Pˆ respectively, in the vicinity where like in (4.7) h, of M . From (6.4) and (6.2) it follows that ( if h is tangent to M a) divM h Divtrunc h = (6.5) b) − T rAh if h is orthogonal to M In the components the orthogonal projector Pˆ is given by the expression (compare it with (4.3)) (6.6) Pik = Gij ∂α xj g αβ ∂β xk , where x(ξ) is the parametrization of the surface M , gαβ = ∂α xm Gmn ∂β xn is the metrics induced on the surface M and g αβ = (g)−1 αβ is the inverse metrics tensor.
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So if h is orthogonal to M then (6.5b) in components has the form Divtrunc (h) = −hi (ξ)Gij ∂α ∂β xj (ξ)g αβ (x(ξ))
(6.7)
in the case if E n is Euclidean and {xi } are cartesian coordinates on it (Gij = δij ). In the case if E n is not Euclidean Eq. (6.7) also defines (6.5b) at any given point ξ0 if coordinates {xi } are Euclidean at this point, i.e. 0ikm = 0. x(ξ0 )
We see that in case (6.5a) the truncated divergence is reduced to the induced divergence operator on M and in case (6.5b) it is reduced to a zeroth order linear differential operator which corresponds to (4.9, 4.11). 3) Now we compare the odd semidensity constructed in Sect. 5 with the mean curvature of the hypersurfaces. In analogy with considerations (5.1–5.4) of Sect. 5 one can consider on hypersurfaces in E n invariant density N(x(ξ), ∂x/∂ξ) of the rank k = 1 which takes values in the vectors orthogonal to the hypersurfaces. By these conditions it is fixed uniquely (up to multiplication by a constant): p ∂x ∂x = n x(ξ), · det(gαβ ), (6.8) N x(ξ), ∂ξ ∂ξ p p where det(gαβ ) = det(∂α xi Gik ∂β xk ) is the density of the volume form dvM induced on the hypersurface x(ξ) and n(x(ξ), ∂x ∂ξ ) is the unit vector orthogonal to the vectors ∂x/∂ξ which are tangent to the hypersurface x(ξ) at the point ξ. Applying (6.4) to the vector-valued density (6.8) and using (6.5b) we come in analogy with (5.5) to the density of the second rank corresponding to the mean curvature: p ∂x ∂ 2 x , = −T rAn · det(gαβ ). (6.9) H x(ξ), ∂ξ ∂ξ∂ξ In the case if E n is Euclidean and the coordinates {xi } are cartesian then according to (6.7), p ∂x ∂ 2 x , = −ni Gij ∂α ∂β xj (ξ)g αβ det(g αβ ), (Gij = δij ). (6.10) H x(ξ), ∂ξ ∂ξ∂ξ (Compare it with (5.6).) Here we want to note that in spite of the fact that the odd semidensity (5.5) cannot be integrated over surfaces (A2 = 0, so it leads to trivial volume form) one can consider the equation A|M ≡ 0 which extracts the class of (n − 1.n − 1) supersurfaces in an odd symplectic superspace endowed with a volume form. For example, from (5.5–5.7) it follows that to this class there belong supersurfaces which can be defined by equations xn = θn = 0, where (x1 , . . . , xn , θ1 . . . , θn ) are Darboux coordinates in which the density of the volume form obeys condition (4.10) (if such Darboux coordinates exist). The analogous condition for mean curvature (6.10) H|M ≡ 0 is the solution of the variational problem for the minimal hypersurface. Acknowledgement. I want to express my deep gratitude to my teacher A. S. Schwarz. Many years ago he taught me geometry and I tried to answer some of his questions in Ref. [6] and also in this paper. I am grateful to D. A. Leites for interest in my work on odd symplectic geometry. When the paper [6] had appeared he asked me a question about the meaning of the invariant semidensity constructed therein. This paper can be
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considered as an answer to this question. I am deeply grateful to J.-C. Hausmann for valuable discussions and to A. V. Karabegov, O. Piguet and Th. Voronov who encouraged me during the preparation of this work. The work was supported in part by the Suisse National Science Foundation and by INTAS Grant No 95-0829.
References 1. Baranov, M.A., Schwarz, A.S.: Characteristic Classes of Supergauge Fields. Funkts. Analiz i ego pril. 18, No.2, 53–54 (1984); Cohomologies of Supermanifolds. Funkts. analiz i ego pril. 18, No.3, 69–70 (1984) 2. Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and Quantization. Phys.Lett. 102B, 27–31 (1981); Closure of the gauge algebra, generalized Lie equations and Feynman rules. Nucl.Phys. B234, 106–124 (1984) 3. Berezin, F.A.: Introduction to Algebra and Analysis with Anticommuting Variables. Moscow, MGU, 1983 (in English – Introduction to Superanalysis. Dordrecht-Boston: D.Reidel Pub. Co., 1987) 4. Gayduk, A.V., Khudaverdian, O.M., Schwarz, A.S.: Integration on surfaces in Superspace. Teor. Mat. Fiz. 52, 375–383 (1982) 5. Khudaverdian, O.M.: Geometry of Superspace provided by Poisson brackets of different gradings. J. Math. Phys. 32, 1934–1937 (1991) 6. Khudaverdian, O.M., Mkrtchian, R.L.: Integral invariants of Buttin bracket. Lett. Math. Phys. 18, 229– 234 (1989) 7. Khudaverdian, O.M., Nersessian, A.P.: On the geometry of Batalin–Vilkovisky formalism. Mod. Phys. Lett. A8, 2377–2385 (1993); Batalin–Vilkovisky Formalism and Integration Theory on Manifolds. J. Math. Phys. 37, 3713–3724 (1996) 8. Khudaverdian, O.M., Schwarz, A.S., Tyupkin, Yu.S.: Integral invariants for Supercanonical Transformations. Lett. Math. Phys. 5, 517–522 (1981) 9. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. 1963 10. Leites, D.A.: The new Lie superalgebras and Mechanics. Docl. Acad. Nauk SSSR 236, 804–807 (1977); The theory of Supermanifolds. Karelskij Filial AN SSSR (1983) 11. Shander, V.N.: Analogues of the Frobenius and Darboux Theorems for Supermanifolds. Comptes rendus de l’ Academie bulgare des Sciences 36, n.3, 309–311, (1983) 12. Schwarz, A.S.: Supergravity, Complex Geometry and G-structures. Commun. Math. Phys. 87, 37–63 (1982) 13. Schwarz, A.S.: Geometry of Batalin–Vilkovisky Formalism. Commun. Math. Phys. 155, 249–260 (1993) 14. Voronov, T.: Geometric Integration Theory on Supermanifolds. Sov. Sci. Rev. C Math. 9, 1–138 (1992) Communicated by R. H. Dijkgraaf
Commun. Math. Phys. 198, 607 – 688 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Quantum Chaos and Dynamical Entropy Fabio Benatti1 , Thomas Hudetz2,? , Andreas Knauf3 1 Dipartimento di Fisica Teorica, Universit` a di Trieste, Strada Costiera 11, Trieste, Italy. E-mail:
[email protected] 2 Institut f¨ ur Mathematik, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail:
[email protected] 3 Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig, Germany. E-mail:
[email protected]
Received: 26 June 1997 / Accepted: 13 April 1998
Abstract: We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More specifically, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may reduce the dynamical entropy of the quantum gas in comparison to free motion. Contents 1 2 3 3.1 3.2 3.3 3.4 3.5 4 5 5.1 5.2 5.3 5.4 A
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 C*-Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 CNT Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 Entropy of subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 Non-commutative dynamical entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 Classical dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 Systems with purely discrete spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 622 Quantum mechanical randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Non-commutative Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 628 Motion in a Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 The classical motion (one particle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 The quantum motion (one particle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 The classical electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 The quantum electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 The Classical Discretized Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 681
? Supported by “Osterreichische ¨ Forschungsgemeinschaft” (1995/96) and “Austrian Science Foundation” (FWF; Dec. 1996/Febr. 1997)
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1. Introduction In his 1990 book “Chaos in Classical and Quantum Mechanics” Gutzwiller wrote “While the idea of an entropy is of great help in understanding classical mechanics, nobody has been able to define its analog in quantum mechanics; therein lies the great unresolved mystery of quantum chaos.” [20] Here entropy of a system of classical mechanics means its Kolmogorov-Sinai (KS) metric entropy. Since this quantity is a measure of the loss of information per time unit about the dynamical system, it is also called dynamical entropy. In 1987 Connes, Narnhofer and Thirring described in [13] a generalization of that dynamical entropy to quantum systems (henceforth abbreviated as CNT entropy). Since then, this quantum dynamical entropy has been applied to infinite particle systems of statistical mechanics, and its general mathematical properties were studied. However, the above citation suggests that it could be useful to study in some depth the dynamical entropy of finite quantum systems which exhibit “quantum chaos”. In classical mechanics the KS entropy can be regarded as an order parameter which is (strictly) positive if the system is chaotic, and thus may even be used to define what chaotic means. Now for many years physicists looked for order parameters indicating (and defining!) the occurrence of quantum chaos. We prove in Prop. 3.4 that CNT entropy is not such an order parameter, and argue that there is no other order parameter for finite localized quantum systems. Thus we support the operational definition of a quantum chaotic system as one whose semiclassical limit ~ & 0 leads to a classical system with positive dynamical entropy. So quantum chaos is an emergent phenomenon which can only be seen in the semiclassical limit. Entropy measures information (or ignorance). Thus it is a probabilistic notion. But unlike the classical theory, quantum statistical mechanics is not a probabilistic theory. Since information about a system is gained by measuring it and since a measurement may change the state of a quantum system, one has to be careful if one wants to define a physically relevant notion of quantum dynamical entropy. In our opinion, CNT entropy describes a property of the quantum system which is as little as possible influenced by the repeated measurements necessary to define it. We have to emphasize that the more recent work by Alicki, Fannes, and coauthors, beginning with [2], explores a different approach to a physically relevant quantum dynamical entropy, with which we will not be concerned in this paper. One purpose of our paper is to give a pedagogical introduction into quantum dynamical entropy in the sense of Connes, Narnhofer and Thirring, starting from the more familiar notion of classical dynamical entropy. To our opinion the notion of quantum dynamical entropy has not yet really become popular within the mathematical physics community, not to speak of theoretical physics. We see two reasons for this deplorable fact: • As mentioned above, it does not help to discern quantum chaos for finitely many particles. • The very notion of CNT entropy is involved, and it is quite complicated to estimate its basic functional. Thus it could up to now only be applied to systems of statistical mechanics which are relatively simple from a physical point of view (e.g. space translations).
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Conceptually speaking, we think that what is needed is the development of techniques which allow to localize in configuration space and, more far-fetched, to microlocalize in phase space. In the theory of partial differential equations the pseudodifferential calculus has been developed in the last decades and applied to the Schr¨odinger equation (see Robert [42] for a general introduction and Helffer [21], and [30] for reviews on the application to quantum chaos). In recent years, there have been successful attempts to apply this calculus to problems of statistical mechanics. Our paper is also aimed to promote the development of such localization techniques for CNT entropy, in order to widen its field of potential applications. In order to describe in which sense CNT entropy generalizes KS entropy one needs a common framework for classical and quantum mechanics. This is given by the C ∗ dynamical systems, which are discussed in Sect. 2. In Sect. 3 the notion of CNT entropy is introduced and explained. Proposition 3.4 says that in a situation typical for finitely many particles (algebra of bounded operators on a Hilbert space, time evolution implemented by unitaries, state represented by a density matrix) the CNT entropy equals zero. Classically topological entropy is the supremum over all phase space probability measures of the (metric) KS entropy. Section 4 is devoted to a brief introduction to the approach, developed by one of the present authors [23, 24], to a physically relevant non-commutative generalization of topological entropy, which approach had been formally inspired by the CNT entropy theory. Again, we have to emphasize that there are alternative approaches, by Thomsen and by Voiculescu, to topological entropy for non-commutative operator algebras [49, 47]. In Sect. 5 we work out in detail the example of a particle in a periodic potential (crystal). We shortly describe the main results. Periodic Potentials. One Particle (Classical). The notion of measure theoretic entropy is not directly applicable to the description of a classical particle in a periodic potential, since Liouville measure is invariant under lattice translation and thus infinite. However one may divide by the lattice L and thus consider motion on the configuration torus T = Rn /L, using Liouville measure. In that case the KS entropy is zero if the potential is separable, √and should be positive in the generic case. If the potential is smooth, it is of order 1/ E as the energy E of the particle goes to infinity (Prop. 5.6). Classical topological entropy has a generalization, due to Bowen, to non-compact phase spaces. So it is directly applicable to the motion in the crystal but coincides with the topological entropy of the motion on the configuration torus. The topological entropy of smooth potentials has an E-independent upper bound, and for ≥ 3 freedoms there exist smooth potentials with a similar E-independent lower bound (Prop. 5.5). As a special class of examples we analyze planar periodic potentials with attractive singularities of Coulombic (−1/r) form. For these potentials the motion is diffusive above a threshold energy. In the high √ energy limit the topological entropy is a smooth increasing function asymptotic to c E (Prop. 5.8). One Particle (Quantal). In the quantum case the motion in periodic potentials is ballistic, but for the Coulombic case the speed is known to vanish in the semiclassical limit ~ & 0 as a consequence of quantum chaos, see Prop. 5.11.
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The CNT entropy is undefined for the motion in configuration space and zero for motion on the configuration torus. Classical Gas. Although second quantization (or Poisson construction, as it is called in the case of classical dynamical systems, see Def. 5.12) is a purely functorial construction, the consideration of a gas of non-interacting particles sheds new light on the single– particle dynamics. Classically the unbounded system is now described by a probability measure. As noted by Goldstein, the total entropy of an ideal gas is infinite. This was a motivation for him to introduce space time entropy which, as he showed, equals entropy density. This is a finite quantity for the motion in a periodic potential, since it equals the entropy per particle multiplied by the density of particles (Thm. 5.14). However, if one considers only a bounded configuration space region B, then for large energies and smooth potentials the local entropy associated with that region is infinite, since every incoming particle carries an infinite amount of information. For energies below the delocalization threshold that entropy is finite (Thm. 5.14). Above an energy threshold, the gas of classical particles in a Coulombic periodic potential forms a space-time K-system with strictly positive entropy densities (Prop. 5.15). Thus it has extremely good ergodic properties. Fermi Gas. In Sect. 5 we consider a noninteracting Fermi gas in Rn with a latticeinvariant quasi-free state. We prove that for a shift by a lattice vector v the local CNT entropy in region B is finite and for large B asymptotically proportional to the surface area of ∂B in the v direction times the speed |v| (Thm. 5.19). Here we adapt the method of the paper [39] by Park and Shin to our situation, which is only invariant under lattice translations, not under general space translations. We simplify their estimate for the entropy defect. At the same time we obtain a more localized estimate which shows that entropy defect is only due to the boundary of the region of observation. The finiteness of local CNT entropy is a consequence of coarse–graining of phase space with cells of volume (2π~)n . So each particle carries only a finite amount of information. In a forthcoming paper we will apply these techniques to the motion of the noninteracting quantum electron gas in a periodic potential. This needs phase space microlocalization, and we indicate in an appendix the usefulness of local algebras of Wannier functions in this context. The growth of the local CNT entropy with the surface area instead with the volume is a clear sign of the fact that quantum chaos is less chaotic than classical chaos. For it is not the dynamics of the particles in the bulk, but the information carried by the electron entering the region, that leads to that growth of entropy. In the case of the Coulombic potential the abovementioned vanishing of speed in the semiclassical limit has the curious consequence that, apart from a “kinematical factor” (2π~)−n corresponding to coarse-graining of phase space, the local CNT entropy associated with B decreases as Planck’s constant goes to zero. The physical reason is simply that less particles enter B per unit time. Perspectives. • At a first glance the vanishing of the CNT entropy (density) is somewhat disappointing, seen from the point of view of quantum chaology. On the other hand, it fits well with many other observations made in that field.
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The obvious answer is not to perform the large time limit but to ask in which sense and up to which time the quantum time evolution imitates the classical one. Mathematically, for this study semiclassical techniques are used (e.g. Egorov’s theorem on the convergence of quantum to classical time evolution). We think that the analysis leading to the notion of CNT entropy can also be useful in that context. • The connection between the CNT definition of the multi-algebra entropy (Def. 3.2) and quantum information theory should be studied further. The latter is a rapidly growing field with new experimental and theoretical content. As has been shown recently in [7] the basic quantum generalization of Shannon information is directly related to the CNT one-algebra entropy 3.1. • We mention that our use of Wannier functions resembles techniques in wavelet analysis. Namely, the Wannier functions of a given band are related by lattice translations and orthonormal. Moreover, going to sublattices and subbands, as we do, is similar to the dilations used in the case of wavelets. • Concerning infinite particle systems, an obvious question is whether the introduction of an interaction between the electrons leads to a positive configuration space density of quantum dynamical entropy. This question is probably out of reach of presently available techniques. However it is a challenging problem to develop some intuition. The case of Fermi liquids is of particular interest, since there the one-particle concept of the Fermi surface is still important, and since their mathematical understanding has progressed much in recent years. Notation. Whenever it is clear from the context which measure µ we use (Lebesgue, Liouville, counting etc.), we denote µ(S) by |S|. 2. C*-Dynamical Systems Instead of using a Hilbert space picture, we develop our considerations about the dynamical properties of quantum systems in the language of the algebra A of observables. This makes it easier to compare with classical dynamical systems, where A is an Abelian algebra. Anyhow, one cannot observe everything in statistical mechanics, so that the quantum algebra of observables need not be A = B(H), the bounded operators on some Hilbert space H. On A the (discrete or continuous) time evolution σ t : A → A (with t ∈ Z resp. R) acts as a group of automorphisms. We have then two possibilities: either we focus on the C*-dynamical system (A, σ), independently of any reference state, or we fix a time-invariant (ω ◦ σ t = ω) state ω and concentrate on the triple (A, σ, ω). In the first instance we will speak of (noncommutative) “topological”, in the second of state-dependent properties. The algebra A may be a C* or a von Neumann algebra. • In quantum mechanics of finitely many particles A is typically the algebra B(H) (both a C* and a von Neumann algebra) of all bounded linear operators on some (separable) Hilbert space H. The dynamics is usually defined by a one-parameter group {Uˆ t } of ˆ such that σt : A → A is given by unitaries generated by a Hamiltonian H, ˆ = Uˆ t∗ Xˆ Uˆ t , Uˆ t := exp −i Hˆ t , (Xˆ ∈ B(H)). (1) σt (X)
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Typical invariant states ω on B(H) are given by density matrices ρˆ ∈ B(H), with ρˆ ≥ 0, Tr ρˆ = 1 and ρ, ˆ Uˆ t = 0, ˆ := Tr ρˆXˆ , (Xˆ ∈ B(H)). ω(X) (2) If a state is pure (i.e., cannot be written as a non-trivial convex combination), then ρˆ is a rank-one projection. We will sometimes use the same notation for a density matrix and its state. • In the more general context of quantum statistical mechanics, A will not necessarily be the whole of B(H), maybe not yet represented on some Hilbert space H. There may be no Hamiltonian Hˆ ∈ A of the system as a whole. Thus the automorphisms σ might not be implemented by unitaries in A and the states ω over A will be general, normalised (that is, ω(1ˆl) = 1) positive linear functionals ω : A → C, not necessarily representable in the form (2) (see Bratteli and Robinson [11, 12]). Example. As an instance of this state of affairs let us consider a system of infinitely many A is the CAR algebra of the Canonical Anticommutation Relations † Fermions. aˆ (f ), aˆ (g) + = hg, f i. So A is the norm closure (with respect to the norm kˆa] (f )k = kf k) of polynomials in the creation resp. annihilation operators aˆ † (f ) resp. aˆ (f ) of oneparticle states f ∈ H (here and in the following, ] is a dummy index taking the two possible values “†” or “void”). ˆ = h, Xi ˆ The Fock vacuum |i gives rise to a state ω(X) which belongs to a class of states, the so-called quasi-free states, that are specified by assigning two-point functions: ˆ i, (3) ωA aˆ † (f )ˆa(g) := hg, Af where Aˆ is a One-Particle operator on H and 0 ≤ Aˆ ≤ 1ˆl. The higher moments are then given by (4) ωA (ˆa† (fn ) . . . aˆ † (f1 )ˆa(g1 ) . . . aˆ (gm )) := δmn det ωA (ˆa† (fi )ˆa(gj )) . In this class we find the KMS-states for inverse temperature β among which the Fock state (zero temperature) corresponds to Aˆ = 0, while the tracial state (infinite 1ˆl temperature) to Aˆ = . For general β we have 2 Aˆ :=
ˆ exp(−β H) , ˆ1l + exp(−β H) ˆ
Hˆ denoting the one-particle Hamiltonian. Establishing an interaction among fermions capable of leading to an automorphism σt of the algebra is rather a difficult task, cf. [33, 34]. Much easier to handle are the so-called quasi-free evolutions that map aˆ ] into themselves. For instance, if Uˆ : H 7→ H is a One-Particle unitary evolution, then σ : aˆ ] (f ) 7→ aˆ ] (Uˆ f ) defines an automorphism of A. Given A, an automorphism σ and a σ-invariant state ω, we can recover a Hilbert space representation via the so-called GNS construction [11, 12]. We are ensured of the existence of a state-dependent Hilbert space Hω , a state-dependent representation πω (A) of A as a subalgebra of B (Hω ) and a state-dependent unitary implementation
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Uˆ ω of the automorphism σ. Moreover the state ω corresponds to an invariant vector state |i ∈ Hω : ˆ (5) πω σ(ˆa] (f )) = Uˆ ω∗ πω aˆ ] (f ) Uˆ ω , ω Xˆ = h, πω (X)i. Finally, the strong-operator closure of πω (A) as a C*-algebra of operators on Hω is a von Neumann algebra, denoted by πω (A)00 that coincides with the bicommutant (commutant of the commutant) of πω (A) ⊂ B(Hω ). As to the CAR algebra A we last notice that it has the structure of a UHF C*algebra, namely there exists an increasing sequence of finite-dimensional subalgebras to 2n × 2n matrix algebras) and A is the norm closure of (the inductive An (isomorphic [ limit) An [12, Par. 5.2.2]. Once represented via the GNS construction based on some n [ πω (An ) (i.e., a state ω, πω (A)00 is the strong-operator closure of the corresponding n
hyperfinite von Neumann algebra). 3. CNT Entropy In this section we introduce and apply the notion of CNT entropy [13] (quantum dynamical entropy) of a triple (A, σ, ω), where (A, σ) is a C*-dynamical system with σ-invariant state ω. 3.1. Entropy of subalgebras. Since the CNT entropy generalizes the KS entropy, we follow the classical case as far as possible. The key problem consists in finding a suitable definition for the entropy of subalgebras. The starting point in the definition of the KS entropy of a map S : X → X on phase space X with probability measure µ is to partition phase space into disjoint measurable atoms Pi ⊂ X (X = ∪di=1 Pi , up to µ-measure zero), and to define the entropy Hµ (P) :=
d X
η(µ(Pi ))
i=1
of the partition P, with the positive function on [0, 1], η(x) := −x log x for 0 < x ≤ 1,
η(0) := 0.
By concavity of η the entropy increases if the partition is refined. The C*-algebraic version of the entropy of a partition works with the Abelian C*algebra (actually, von Neumann algebra) A := L∞ (X , µ), the state ω : A → C, ω(f ) := R f dµ and the characteristic functions χi ∈ A of the atoms Pi of the partition P. X These span a finite-dimensional (here, d-dimensional) subalgebra A ⊂ A, and can be recovered from A as its unique minimal projections. Then the entropy Hµ (P) of the partition equals the entropy Hω (A) :=
d X i=1
of the associated subalgebra.
η(ω(χi ))
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In the non-Abelian case of a general C*-algebra A with subalgebra A and state ω it is tempting to use the von Neumann entropy S (ω) := −Tr (ρˆ log ρ) ˆ
(7)
of a state ω with density matrix ρˆ to define the entropy of the subalgebra A ⊂ A by S ω |`A . Indeed, although the state ω : A → C does not necessarily have a density ˆ = Tr (ρˆX)), ˆ the state ω |` : A → C has one, since dim(A) < ∞ matrix ρˆ (with ω(X) A (and hence A can be included in a full matrix algebra). Moreover, S ω |`A = Hω (A) for A Abelian, so that we have a generalization of the classical definition. of a subalgebra of A need not However, unlike Hω (A), the von Neumann entropy be smaller than the von Neumann entropy S ω |`A of A. Example. We consider a spin 21 particle with algebra A = A = B(C2 ). Thus the density 1 ˆ ~ 3 ~ matrices have the form ρˆ~b := 2 (1l + b · ~σ), with b in the unit ball B ⊂ R and the vector 0 i 1 0 of Pauli matrices. The pure states constitute ~σ = (σˆ 1 , σˆ 2 , σˆ 3 ) := 01 01 , −i 0 , 0 −1 the sphere ∂B and have zero von Neumann entropy, whereas for mixed states S ~b := S ρˆ~b |`A = η( 21 (1 + |~b|)) + η( 21 (1 − |~b|)) > 0. The Abelian subalgebra B ⊂ A generated by σˆ 3 is used to measure the polarization in the 3-direction. Now ρˆ~b |`B = ρˆ~bk with ~bk := (0, 0, b3 ) so that S ρˆ~b |`B = η( 21 (1 + b3 )) + η( 21 (1 − b3 )) ≥ S ~b = S ρˆ~b |`A . (8) The two von Neumann entropies are only equal if ~bk = ~b, since only then the measurement of the 3-component does not produce additional uncertainty. In particular, S ρˆ~b |`B = ln(2) if ~bk = 0, since then both polarizations have probability 21 , even if ρˆ~b is pure. As the example clearly shows, the von Neumann entropy of a subalgebra does not measure the information gained by a suitable experiment using the observables of that subalgebra. Rather, it measures our uncertainty about the outcome of that experiment. Whereas these quantities coincide classically, in quantum mechanics we generally disturb the state by the experiment. So we have to subtract from the von Neumann entropy S ω |`A a term which describes the information loss through the experiment and thus leads P to a zero entropy of ˆ i in a convex the algebra if ω is pure. This can be done by decomposing ω = i∈I µi ω P > 0, µ = 1) of normalized states ω ˆ and subtracting the weighted combination (µ i i i i P sum i∈I µi S ωˆ i |`A from S ωˆ |`A . Using the relative entropy S (ρˆ1 , ρˆ2 ) := Tr ρˆ2 (log ρˆ2 − log ρˆ1 ) , of ρˆ1 with respect to ρˆ2 [37], we thus consider the positive bounded functional X µi S ω |`A , ωˆ i |`A 0 ≤ Hω, {µi ωˆ i } (A) :=
(9)
(10)
i∈I
X = S ω|`A − µi S ωˆ i|`A i∈I
≤ S ω|`A ≤ log(rank(ω |`A ))
(11)
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on the set of finite dimensional subalgebras A (where rank(ω |`A ) denotes the rank of the corresponding density matrix). Hω, {µi ωˆ i } (A) is also monotonically increasing with the subalgebra A, since the relative entropy has this property: if A ⊆ B. (12) S ρˆ1 |`A , ρˆ2 |`A ≤ S ρˆ1 |`B , ρˆ2 |`B Again the restricted states have density matrices so that the definition makes sense. Positivity, boundedness and monotonicity (with respect to the subalgebra) appear to be necessary properties of any measure of the amount of information contained in a given finite-dimensional subalgebra with respect to a given state. The following non-commutative generalization of (6) was proposed in [13] (and already before that, see the references in [13]): Definition 3.1 (Entropy of a Subalgebra). Given a C* or a von Neumann algebra A and a state ω on it, we will call entropy of a finite-dimensional subalgebra A ⊆ A with respect to ω, the following functional: Hω (A) :=
Psup
ω=
i∈I
Hω, {µi ωˆ i } (A) ,
(13)
µi ω ˆi
P where the supremum is computed over all decompositions ω = i∈I µi ωˆ i of ω as linear convex combinations of states ωˆ i on A. P Even if in the above definition only decompositions ω = i∈I µi ωˆ i into pure states ωˆ i were taken into consideration, unlike in the Abelian case these decompositions would be in general non-unique (e.g., on B(C2 ) we may decompose the tracial state ρ~0 = 21 1ˆl into ρ~0 = 21 (ρ~b + ρ−~b ) for arbitrary ~b ∈ ∂B). So it is in any case necessary to perform the supremum over decompositions. Notation. In the above definition and in the sequel we refer to states as to positive normalised linear functionals over algebras that might or might not be represented as density matrices acting on some Hilbert space. Furthermore, we will also use not normalized states when decomposing a given state ω: In this case P we will omit P the hat, so that ωˆ i means ωˆ i (1ˆl) = 1, whereas ωi (1ˆl) = µi < 1 and ω = i∈I µi ωˆ i = i∈I ωi . Definition 3.1 is indeed a generalization of (6), since in the Abelian case we may take R Pd the decomposition ω = i=1 ωi with ωi (f ) := X f (x)χi (x)dµ(x). Then the restricted Pd states ωˆ i |`A are pure so that Hω (A) = S ω| ˆ `A = i=1 η(ω(χi )). Example. In the above case of the spin 21 particle with state ρˆ~b it was shown by Benatti and Grava [8] that the optimal decomposition leading to Hω (B) consists in writing ρˆ~b as q ± ~ ~ a convex combination of ρˆ~b+ and ρˆ~b− , with the vectors b := b⊥ ± 1 − |~b⊥ |2 (0, 0, 1) ∈ ∂B (and ~b⊥ := ~b − ~bk ). Using formula (8), S ρˆ~b±|`B = η
1 2
q 1+
1 − |~b⊥ |2
+η
1 2
q 1−
1 − |~b⊥ |2
and Hω (B) = S ρˆ~b|`B − S ρˆ~b+|`B . Thus the entropy of B only coincides with the von Neumann entropy S ρˆ~b|`B if ~b⊥ = 0, that is, if σˆ 3 and ρˆ~b commute. Figure 1 shows a contour plot of Hω (B).
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1
3
0.5
0
-0.5
-1 -1
-0.5
0
0.5
1
1
Fig. 1. Entropy of the subalgebra A for a spin σ ), with b2 = 0 matrix ρˆ~b = 21 (1ˆl + ~b · ~
1 2
particle as a function of the parameters b1 and b3 of the density
In the Abelian case of classical C*-dynamical systems the entropy (6) of one subalgebra can be directly used to define the KS dynamical entropy. However, Def. 3.1 is not sufficient for the introduction of the quantum dynamical entropy. Instead, one must first give a definition of the entropy of n subalgebras. • In the classical case the measure-preserving map S : X → X on some phase space X induces via σ(f ) := f ◦ S −1 an automorphism σ on the algebra A = L∞ (X , µ) of phase space functions. Then the algebra A ⊂ A of a finite partition P is mapped into k the algebras σ k (A) ⊂ A of the partitions σ (P), which due to the S-invariance of µ k all have the same entropy Hµ σ (A) = Hµ (A). The σ l (A), . . . , σ r (A) (with l ≤ r ∈ Z) generate a subalgebra Arl := W algebras k l≤k≤r σ (A) ⊂ A which is still finite-dimensional. Its entropy is subadditive and bounded, that is + Hµ AT0 −t−1 ≤ T Hµ (A) . Hµ AT0 −1 ≤ Hµ At−1 0
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1 Hµ AT0 −1 exists. The KolmoT →∞ T gorov–Sinai entropy is then defined as the maximal information per unit time coming from finite-dimensional partitions: Thus, the average information hµ (S, A) := lim
hµ (S) := sup hµ (S, A) . A
• In the non-commutative case, the idea of replacing the refinement of two finite partitions by the subalgebra A ∨ σ(A) generated by a finite dimensional subalgebra A ⊆ A and its image σ(A) under the evolution (still finite-dimensional) does not in general lead to meaningful results, since A ∨ σ(A) might be infinite-dimensional and Hω (A ∨ σ(A)) undefined. See [46] for a nice example. The idea [13] (see also references there) is to replace the classical refinement of the algebras by (multiindexed) decompositions of the state ω. Then one adds the entropies of the algebras, and subtracts from that sum a quantity describing the correlation of the decomposition. That is, when dealing with n finite-dimensional subalgebras A1 , . . . , An , we consider the decompositions of the state which are of the form X µ(i) ωˆ (i) , ω= (i)∈I
with the multi-index (i) = (i1 , . . . , in ) ∈ I := I1 × · · · × In . Such a decomposition gives rise to the n subordinate decompositions (or partial sums) X ω= µkik ωˆ ikk (k = 1, . . . , n) (14) ik ∈Ik
with µkik ωˆ ikk :=
X
µ(j) ωˆ (j)
and
ωˆ ikk (1ˆl) = 1 .
(15)
(j)∈I, jk =ik
If we denote by P the natural partition of the index set I into the atoms (i) ∈ I and by P k the partition into subsets of the form Pik := {(j) ∈ I | jk = ik }, then P n k k=1 Hµ (P ) − Hµ (P) measures the deviation of the probability measure µ on I from the product measure µ1 × . . . × µn . Namely by telescoping we obtain n X k=1
Hµ (P ) − Hµ (P) = k
n X
l Hµ (P k ) − Hµ (P k | ∨k−1 l=1 P )
k=2
with the conditional entropy Hµ (A|B) := Hµ (A ∨ B) − Hµ (B) ≤ Hµ (A). The Wk−1 terms Hµ (P k ) − Hµ (P k | l=1 P l ) ≥ 0 measure how much information about P k is gained by measuring the “previous” partitions P l , l < Pk. These quantities have to be subtracted from the entropies Hω, {µki ωik } (Ak ) = ik ∈Ik µkik S ω |`Ak , ωˆ ikk |`Ak k k of the single subalgebras Ak . We are thus lead to consider the functional which extends (10) to the case of n subalgebras [13] (see also references there): Hω, {µ(i) ωˆ (i) } (A1 , . . . , An ) :=
n X
Hω, {µki
k=1
+ Hµ (P) −
k
ω ˆ ik } (Ak )
n X k=1
k
Hµ (P k ).
(16)
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Then, the quantum version of the classical entropy Hµ (P) is: Definition 3.2 (n-Subalgebra Entropy). Given the finite-dimensional subalgebras A1 , . . . , An ⊂ A, their entropy Hω (A1 , . . . , An ) with respect to the state Pω, is given by taking the supremum of (14) over all multiindexed decompositions ω = (i) µ(i) ωˆ (i) : Hω (A1 , . . . , An ) :=
Psup
ω=
(i)
Hω, {µ(i) ωˆ (i) } (A1 , . . . , An )
.
(17)
µ(i) ω ˆ (i)
Below we list some properties of these n-subalgebra functionals. 1. The monotonicity property (12) of the relative entropy carries over to the entropy of n subalgebras, i.e. for Ai ⊆ Bi ⊆ A Hω (A1 , . . . , An ) ≤ Hω (B1 , . . . , Bn )
.
(18)
2. If the state ω is σ-invariant, then
Hω σ j (A), . . . , σ j+n−1 (A) = Hω A, . . . , σ n−1 (A)
.
(19)
3. The inequality Hµ (A|B) ≤ Hµ (A) for the classical conditional entropy ensures that the second line in (16) is negative so that we deduce subadditivity and, using (11), boundedness, while positivity is almost obvious: 0 ≤ Hω (A1 , . . . , An ) ≤ ≤
n X k=1 n X k=1
Hω (Ak ) n X S ω|`Ak ≤ log(rank(ω |`Ak )).
(20)
k=1
4. Permuting or repeating subalgebras does not alter the n-subalgebra functionals [13]: (21) Hω Aπ(1) , . . . , Aπ(n) = Hω (A1 , . . . , An , An ) = Hω (A1 , . . . , An ) . 5. The n-subalgebra entropy (17) is indeed a generalization of the classical (Abelian) entropy Hω (A), A := A1 ∨ . . . ∨ An , of subalgebras A1 , . . . , An ⊆ A := L∞ (X , µ). The Ak are generated by minimal projections χkik with indices ik ∈ Ik := {1, . . . , dim(Ak )} which are characteristic functions on theRphase space X . These P give rise to decompositions ωˆ = ik ∈Ik ωikk with ωikk (f ) := X f (x)χkik (x)dµ(x). As in the above case of a single algebra, the restricted states ωˆ ikk |`Ak are pure so that Hω, {µki ωˆ ik } (Ak ) = S ω|`Ak = Hµ (P k ). Thus we deduce from (16) that for k k this decomposition Hω, {µ(i) ωˆ (i) } (A1 , . . . , An ) = Hµ (P) = Hω (A). That the above decomposition leads to the supremum in (17) is a consequence of the inequality Hω (A1 , . . . , An ) ≤ Hω (A) . This inequality, following from Hω (A1 , . . . , An ) ≤ Hω (A, . . . , A) = Hω (A), is not very useful in quantum mechanics, where A may be of infinite dimension. But in our Abelian case A is finite-dimensional so that we have seen that the n-subalgebra entropy (17) is also ≤ Hω (A).
Quantum Chaos and Dynamical Entropy
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6. A non-commutative situation similar to the classical one is when the reference state ˆ for all X, ˆ Yˆ in A. Suppose that the n ω is tracial in the sense that ω(Xˆ Yˆ ) = ω(Yˆ X) finite-dimensional subalgebras Aj , 1 ≤ j ≤ n, commute with each other. Let Pj be maximally Abelian subalgebras of the corresponding Aj , with minimal projections d {Pˆijj }ijj=1 . P Because ω is tracial, we may decompose it into ω = (i) µ(i) ωˆ (i) with the multiindex (i) = (i1 , . . . , in ), where ! Qn n ˆk ˆ Y ω k=1 Pik X k ˆ ˆ , µ(i) = ω Pik , (22) ωˆ (i) (X) := Qn ˆk ω k=1 Pik k=1 and we obtain ω =
X
µjij ωˆ ijj , where µjij = ω Pˆijj
ij
ω Pˆijj Xˆ ˆ = . and ωˆ ijj (X) ω Pˆijj
Therefore, as in Property 5 above, Hω, {µj
ij
ω ˆ ij } (Pj ) j
X = S ω|`Pj = η(µjij ) ij
and, consequently, Hω (P1 , . . . , Pn ) = S ω|`P , where P = P1 ∨ . . . ∨ Pn is the finiten Y dimensional Abelian algebra with minimal projections Pˆik , which is maximally k=1 Abelian in A = A1 ∨ · · · ∨ An . Then, Hω (A1 , . . . , An ) ≥ S ω|`P follows from (18). On the other hand, in full generality, S ω|`A = min{S ω|`P | P ⊆ A maximally Abelian}. Thus itfollows, again e.g. from (18), (20), (21), that Hω (A1 , . . . , An ) ≤ Hω (A) ≤ S ω|`A ≤ S ω|`P , whence Hω (A1 , . . . , An ) = S ω|`P . 7. In the above examples we have seen how to optimally decompose a state with respect to the n-subalgebra entropy functionals in the Abelian case and when the reference state is tracial. In general quantal situations when neither the algebra A is commutative, nor the state ω tracial, the way one gets all possible decompositions of ω is by considering the von Neumann algebra M := πω (A)00 arising from the GNS construction based on ω, touched upon at the end of the introductory Sect. 2. The commutant M0 of M over Hω contains all sets of positive operators Xˆ i0 ∈ M0 X Xˆ i0 = 1ˆl are in one-to-one correspondence with decompositions of such that ω=
X
i∈I
µi ωˆ i (using the ω-representing vector |i ∈ Hω from Sect. 2):
i∈I
h, Xˆ i0 πω Yˆ i ˆ , ωˆ i Y = h, πω Xˆ i
(Yˆ ∈ A).
(23)
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F. Benatti, T. Hudetz, A. Knauf
8. The above point is best exemplified by referring to the C* algebra A = B(C2 ) and to a state on it described by an invertible density matrix ρ. ˆ The GNS construction based on ρˆ has A represented as A ⊗ 1ˆl on the Hilbert space C4 . Its commutant is 1ˆl ⊗ A P2 √ √ and the GNS cyclic vector is | ρi ˆ = i=1 ρi |ei i ⊗ |ei i, with |ei i the eigenvectors of ρ. ˆ Thus, formula (23) concretely reads √ρˆXˆ √ρˆ i Yˆ , ωˆ i Yˆ = Tr Tr ρˆXˆ i where the Xˆ i are positive operators of A such that 3.2. Non-commutative dynamical entropy. Kolmogorov–Sinai entropy, the limit
P i∈I
Xˆ i = 1ˆl.
Because of (19) and (20), as for the
1 Hω A, . . . , σ T −1 (A) T →∞ T
hω (σ, A) := lim
(24)
exists. It is interpreted as the information gained per time unit by repeatedly performing suitable experiments with observables in the subalgebra A ⊆ A. The Quantum Dynamical Entropy of A. Connes, H. Narnhofer and W. Thirring is then defined as the maximal average information contributed by finite-dimensional subalgebras [13] (see also references there): hω (σ) := sup hω (σ, A) .
(25)
A⊆A
Some properties and comments. i. By the above Property 5 of the n-subalgebra entropy, the CNT entropy hω (σ) is a generalization of the Kolmogorov–Sinai entropy. ii. Generally, the entropy for a fixed subalgebra A is sublinear with respect to taking positive powers of the automorphism σ: hω (σ n , A) ≤ n · hω (σ, A)
n∈N .
(26)
This follows from the properties of the n-subalgebra entropy [13], partly listed after Definition 3.2. iii. Like classically, we can benefit from a Kolmogorov–Sinai type theorem [13] which states that, if {Aj } is an increasing sequence, Aj ⊆ Aj+1 , of finite-dimensional subalgebras that generate A as a C*-algebra or as a von Neumann algebra and the state ω is compatible with the corresponding topologies, then . (27) hω (σ) = lim hω σ, Aj j→∞
In this case, also the converse of the inequality (26) above can be proved [13] for the full CNT entropy: hω (σ n ) = n · hω (σ) then scales linearly with the power n of σ, as for the classical KS entropy. Even more generally, a Kolmogorov–Sinai type theorem can be proved [13] for the class of so-called nuclear C∗ -algebras A, see also the brief remark after Def. 4.5 in Sect. 4 below. This is achieved in [13] by extending the entropy functional (24) above from a finite-dimensional subalgebra A ⊂ A (then equivalently viewed as the corresponding canonical inclusion ∗-homomorphism of
Quantum Chaos and Dynamical Entropy
621
A into A) to general completely positive, unital (i.e., unity-preserving) maps from abstractly given finite-dimensional C∗ -algebras into A. For our applications, the definition (24) of entropy for proper subalgebras of A will be sufficient, whereas we will have to use the concept of positive unital maps of finite rank into A for the non-commutative generalization of the classical topological entropy [23] to be introduced in Sect. 4 below. Remark 3.3. If A is a C*-algebra, the generating sequence {Aj } must yield A as a uni[ un form limit Aj . The state ω, as a positive linear functional over A, is automatically j
compatible with the uniform topology. The case of a von Neumann algebra A is different. We can think of it as an algebra of operators acting on some Hilbert space [ H. An increasing sequence of subalgebras Aj is dense in A in the strong-operator sense. {Aj } is generating for it, if the union j
As the strong-operator topology is coarser than the uniform one, not all positive linear functionals (states) are continuous with respect to it. Those states that enjoy such a propery are called normal. When A is generated by {Aj } as a von Neumann algebra, it is for normal ω that the Kolmogorov–Sinai like theorem is proved [13]. iv. As the Kolmogorov–Sinai entropy is already conjugacy-invariant [50], here also two isomorphic quantum dynamical systems have the same CNT-entropy. More concretely, let (A1 , σ1 , ω1 ), (A2 , σ2 , ω2 ) be two quantum dynamical systems and i12 : A1 → A2 an isomorphism between the two algebras such that 1) ω2 = ω1 ◦ i−1 12 and 2) σ2 = i12 ◦ σ1 ◦ i−1 12 . Then hω1 (σ1 ) = hω2 (σ2 ) .
(28)
3.3. Classical dynamical systems. As already in Sect. 3 (Property 5 of the final list of properties there), we now consider a discrete classical dynamical system (X , S, µ) with phase space X , probability measure µ and a measure-preserving invertible map S : X → X . We associate to it the C*-dynamical system (Acl , σcl , ωcl ) with Acl = L∞ (X , µ), R −1 σcl (f ) = f ◦ S for f ∈ Acl and ωcl (f ) = X f dµ, and we noticed in 3 above that the CNT entropy hωcl (σcl ) coincides with the Kolmogorov-Sinai entropy hµ (S). Now we will follow a von Neumann (Hilbert space) approach instead of the above C*-algebraic approach. The Hilbert space Hµ := L2 (X , µ) of square integrable phase space functions is then unitarily mapped onto itself by (29) Uˆ ψ (x) = ψ(S −1 x), (ψ ∈ Hµ ), Uˆ : Hµ 7→ Hµ since S leaves µ invariant. The algebra Acl is a maximally Abelian subalgebra of the von Neumann algebra B(Hµ ) of bounded operators, and the functions f ∈ Acl act by multiplication on Hµ . The identity function on X , identified with the vector |1ˆlµ i ∈ Hµ , defines the pure state ω : B(Hµ ) → C, on B(Hµ ) whose restriction
ˆ := h1ˆlµ , Xˆ 1ˆlµ i ω(X)
ω |`Acl = ωcl .
(30)
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F. Benatti, T. Hudetz, A. Knauf
If we now consider the automorphism σ : B(Hµ ) → B(Hµ ),
ˆ := Uˆ ∗ Xˆ Uˆ σ(X)
induced by Uˆ , then σ restricts to the automorphism σ |`Acl = σcl . So the CNT entropy of the restricted state ωcl is larger than the CNT entropy of the pure state ω: hµ (S) = hωcl (σcl ) ≥ hω (σ) = 0, In particular the former CNT entropy is strictly positive if the motion is chaotic, since it coincides with the KS entropy (and since we call a system chaotic if its KS entropy is > 0). There is no contradiction in the fact that hωcl (σcl ) > 0, while hω (σ) = 0. In fact, if N ⊆ M, then the commutant of N contains that of M. Therefore, according to (23), if ω is a state on M, the restriction ω |`N has more possibilities of being decomposed. The case of a pure state is extremal in the sense that, in the GNS construction, the commutant of M is trivial, whereas the commutant of a proper subalgebra of M is not trivial. In our case the restriction ωcl of the pure state ω to Acl is in general not pure. As in Sect. 3 we only need to find a partition P of the phase space X into Pi of measure Patoms n µi := µ(Pi ) ∈R (0, 1). The corresponding decomposition ωcl = i=1 µi ωˆ i of the state with ωˆ i (f ) := Pi f dµ/µ(Pi ) is non-trivial which then shows that ωcl is not pure. Suppose now that the classical dynamical system (X , S, µ) corresponds to a socalled K-system. Then hωcl (σcl ) > 0, since by definition the Kolmogorov–Sinai entropy hµ (S, P) of a K-system is strictly positive for all non-trivial finite-dimensional measurable partitions P of X [15]. On the other hand in the above situation, there are no other normal, σ-invariant states. In fact, if the classical dynamical system is a K-system, the spectrum of Uˆ has only 1 as an eigenvalue and its multiplicity is 1, with |1ˆlµ i as a unique eigenvector. Thus, in spite of the high randomness of the classical motion, the non-commutative dynamical system naturally arising from the Koopman formulation loses sight of that randomness when measured (on the whole of B (H)) from the point of view of normal, invariant states. While we cannot renounce σ-invariance, we might want to work with invariant states that are not strong-operator continuous, that is not matching the von Neumann structure of B(H), but only its C*-structure: see at the end of Remark 3.3 of Sect. 3, why this does not work for computing the CNT entropy in this situation. 3.4. Systems with purely discrete spectrum. 1) AN typical application of the CNT-entropy construction is to the quantum spin chain A := n∈Z Md (C), with the same full d × d matrix algebra Md (C) at each integer site. A is understood as a quasi-local C* algebra (inductive uniform limit) with O the generating system of increasing, finite-dimensional, Md (C). subalgebras A[−n,n] := −n≤k≤n
ˆ := ˆ i := Aˆ i+1 and the tracial state ω(A) Equipping A with the shift σ(A)
1 d2n+1
ˆ Tr A,
Aˆ ∈ A[−n,n] , the CNT entropy of the dynamical system (A, σ, ω) equals the entropy density [13]
Quantum Chaos and Dynamical Entropy
623
1 S ω|`A[−n,n] = log d. n→∞ 2n + 1
s(ω) := lim
(31)
Since S ω|`A[−n,n] = (2n + 1) log d, hω (σ) ≤ log d is easily proved. Indeed, hω (σ) = lim hω σ, A[−n,n] from (27) n→∞
1 Hω A[−n,n] , . . . , A[−n+k−1,k+n−1] k 1 from (18) ≤ lim lim Hω A[−n,k+n−1] n→∞ k→∞ k 1 from (9). ≤ lim lim S ω|`A[−n,n+k−1] n→∞ k→∞ k = lim lim
n→∞ k→∞
(32)
The inequality hω (σ) ≥ s(ω) comes from choosing the subalgebra A0 at the 0th site and invoking property 6. of the n-subalgebra entropy (see Sect. 3): 1 Hω A0 , A1 , . . . , Ak−1 k→∞ k 1 = lim S ω|`A[0,k−1] = log d. k→∞ k
hω (σ) ≥ hω (σ, A0 ) = lim
(33)
Inequality (32) is independent from the tracial properties of the translation invariant state ω, whereas the characteristic absence of correlations is crucial in getting the equality [13]. 2) More physically, let A be the CAR (CCR) algebra of Fermions (Bosons) over the single particle Hilbert space L2 (Rν , dν x) and θn (ˆa] (f )) := aˆ ] (Uˆ (n)f )
(Uˆ (n)f )(x) := f (x − n), n ∈ Zν
with
(34)
the ν-dimensional space translation group. Finally, let Aˆ be a positive One-Particle operator, ≤ 1ˆl for Fermions, such that Z Z ν ν ˆ i= ωA aˆ † (f )ˆa(g) := hg, Af g(x)K ¯ (35) A (x − y)f (y)d x d y Rν
Rν
ˆ A (k) := defines a translation invariant quasi-free state, and the Fourier transform K Z e−i k·x KA (x)dν x enjoys sufficiently rapid decreasing properties [38]. Rν
Then, the CNT entropy of the full-translation group is again proved to coincide with the entropy density (upper signs for Fermions, lower ones for Bosons) [39]: Z 1 ˆ A (k)) ± η(1 ∓ K ˆ A (k)) dν k. η(K (36) hωA (θ) = ν (2π) 3) In [35] the case of the CAR algebra A over L2 (R, dx) is studied, with the quasi-free time evolution (37) σ aˆ ] (f ) = aˆ ] (Uˆ f ), (Uˆ f )(p) = e−i (p) f (p), and the σ -invariant quasi-free state
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F. Benatti, T. Hudetz, A. Knauf
ω aˆ † (f )ˆa(g) =
Z dp g(p)f ¯ (p)
1 . 1 + e(p)} | {z ρ(p)
(38)
Under suitable conditions on the One-Particle energy (p), the CNT entropy is computed to be Z d(p) 1 [η(ρ(p)) + η(1 − ρ(p))] dp. (39) hω (σ ) = 2π dp 4) Finally, Størmer and Voiculescu prove in [45] that, given the quasi-free state ωA in (3), if Aˆ has pure point spectrum and commutes with the One-Particle evolution Uˆ , then only the absolutely continuous component of the spectrum of Uˆ contributes to the CNT entropy which equals Z 2π 1 m(θ) [η(A(θ)) + η(1 − A(θ))] dθ, (40) hωA (σU ) = 2π 0 where A(θ) is the diagonalization of Aˆ according to the spectral representation of Uˆ and m(θ) is its spectral multiplicity. The fact that the singular components (both discrete and continuous) of the spectrum of Uˆ have zero CNT entropy is obtained by different arguments in [44]. 5) It is none the less useful to examine the result from a less abstract point of view than in [45, 44]. Indeed, most of the quantum systems under investigation as instances of chaotic behaviour have finite degrees of freedom, while the CNT entropy seems to be nontrivial only for systems containing infinitely many particles. It is thus interesting to understand what role is played by the invariant state, respectively the evolution in fixing the value of the CNT entropy. The case when the algebra A is finite-dimensional, say a matrix algebra of bounded operators over a finite-dimensional Hilbert space H, is clear. The CNT entropy of any automorphism with respect to any invariant state ω is zero, as all n-subalgebra entropy functionals are bounded by the entropy Hω (A) ≤ S ω|`A ≤ log dim(H). Let us now examine a typical quantum mechanical situation where H is not finiteˆ −iHt ˆ and dimensional, A = B(H) and the evolution is given as in Eq. (1) with Ut = e ˆ assume that there is an invariant normal state ρ as in (2), with ρ, ˆ U = 0. Then, ˆ −iHˆ , and any ˆ = eiHˆ Xe Proposition 3.4. Given the time evolution B(H) 3 Xˆ 7→ σ(X) invariant normal state ρ on B(H), then hρ (σ) = 0. Proof. By assumption ρ is by the trace with respect to a density matrix ρˆ represented ˆ ˆ such that ρ, ˆ H = 0. Let Pj j∈N be any resolution of the identity by one-dimensional, pairwise orthogonal eigenprojectors of ρ, ˆ ρˆPˆj = ρj Pˆj , the eigenvalues ρj being repeated according to their multiplicities and enumerated in decreasing order (if there are zero eigenvalues, also possibly with infinite multiplicity, we order them alternatingly with the decreasing non-zero eigenvalues in the sequence ρj ). Ln Let Pˆn] := j=0 Pˆj , Qˆ n] := 1ˆl − Pˆn] . Then, consider Bn] := Pˆn] B(H)Pˆn] ⊕ C · Qˆ n] . These are finite-dimensional subalgebras of B(H) with the same identity, whose elements are finite-rank operators (of rank at most (n + 1)). Moreover, the sequence is increasing
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625
S and the strong-operator closure of n∈N Bn] is the whole of B(H). Furthermore, due to the assumption that ρ, ˆ Uˆ = 0, the subalgebras Bn] are σ-invariant: σ Bn] ⊆ Bn] . Therefore, we can argue as in (32): hρ (σ) = lim hρ σ, Bn] n→+∞
1 Hρ Bn] , σ Bn] , . . . k 1 ≤ lim lim Hρ Bn] , Bn] , . . . n→∞ k→∞ k 1 1 ≤ lim lim Hρ Bn] ≤ lim lim S ρ|`Bn] = 0. n→∞ k→∞ k n→∞ k→∞ k = lim lim
n→∞ k→∞
Remark 3.5. Notice that the normal states over the von Neumann algebra B(H) coincide with the density matrices and, according to Remark 3.3, the above result holds for them and not necessarily for other states that are only S continuous in the norm-topology of B(H): with respect to the latter, the closure of n∈N Bn] would give the compact operators over H (with the identity operator adjoined) and not all the bounded ones. 3.5. Quantum mechanical randomness. In this section we would like to deepen the question addressed in Sect. 3, there concerning the Koopman formulation of classical mechanics. Namely, we will consider cases where a classically chaotic behaviour can be directly transferred to the quantum level, and we will investigate whether there is a way to put this quantum chaoticity into evidence by entropic means. However, if we decide to signal the presence of quantum chaos by means of the positivity of the CNT-entropy, then according to Proposition 3.4 of the preceding section, at least all systems of finitely many degrees of freedom, even with significantly “erratic” quantum evolution in finite time-intervals, will not match anyway the time average in the entropy definition, and will thus not be chaotic according to the chosen criterion. 3.5.1. Finite degrees of freedom. Sometimes one hears the wrong argument that quantum mechanics, by the mere linearity of the Schr¨odinger equation, should be incompatible with chaos. As Berry noticed in [9], that reasoning would equally apply to classical mechanics in the Koopman formulation (29). As a counterexample, we discuss in this subsection a Schr¨odinger equation with random properties. ~ on a manifold X which is divergenceIn a general setting we start with a vector field V free for some phase space volume µ. This vector field then induces the ordinary differential equation ~ (x) x˙ = V
(41)
on X whose flow transforms a phase space density ρ on X according to ∂ ~ ). ρt = dρt (V ∂t
(42)
On the Hilbert space L2 (X , µ) this Liouville equation has the form i~
∂ ˆ t ρt = Hρ ∂t
(43)
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Pdim(X ) ~ of a Schr¨odinger equation, with the “Hamiltonian” operator Hˆ := a=1 V a πˆ a anda a ∂ “momenta” πˆ a := −i~ ∂xa . These momenta have the usual commutator xˆ , πˆ b = i~δb ~ has no with the coordinates xˆ a , and Hˆ is formally hermitian since the vector field V divergence. Clearly, the solutions of (43) will follow classical trajectories and, in case the latter behave chaotically, the support of the former will develop characteristic features like whorls and tendrils, no ~ smoothing effect being at work. ~ := J 5 H on One class of such systems is given by Hamiltonian vector fields V ˆ n n a phase space X := Rq × Rp with Hamilton function H : X → R and J := −01ˆl1l0 . Then, however, the Hamiltonian operator Hˆ acts on L2 (X , µ), that is, on the phase space functions rather than on the configuration space functions, so that we definitely do not consider a quantization of the classical dynamical system. If we want to consider the quantization of a classical motion, we should thus start with a vector field on configuration space. Indeed, in spite of its abstractness, the above construction indicates the possibility that quantum Hamiltonians, linear in the momentum operators, be used to transfer erratic classical behaviours to the quantum case. This is exactly what has been done in a paper [40] by Pasmanter, where, given an incompressible velocity field u(x), div u(x) = 0, he constructs the Schr¨odinger-type equation: ∂ψt = (u(x) · pˆ + σˆ 3 E0 + σˆ 2 κ) ψt , where ∂t 0 i 1 0 ~ pˆ = −i~5, σˆ 2 = , σˆ 3 = . −i 0 0 −1 i~
(44) (45)
The incompressibility of u(x) ensures the hermiticity of the Hamiltonian in (44). The model describes a two-particle system (ψ 1 , ψ 2 ) whose components are advected by the velocity field u(x) and spontaneously converted one into the other at a rate κ/~. From the linearity of the Hamiltonian it follows that the total density ρt (x) := |ψt1 (x)|2 + |ψt2 (x)|2 satisfies the advection equation: ∂ ρt (x) = −div(u(x) ρt (x)). ∂t
(46)
The interest of the above model lies in the fact that there are choices of the incompressible velocity field giving rise to chaotic patterns in the dynamics of planar and tridimensional fluids. Again, these features would be transmitted to the solutions of (44). Moreover, from possible eigenfunctions of (46) with real eigenvalues ωn , − 5 · [u(x)ψn (x)] = ωn ψn (x), one obtains eigenvalues n and eigenfunctions 9n of (44) [40]. The latter possibility renders the non-commutative situation somewhat richer than the one arising from the Koopman approach of Subsect. 3.3. In fact, convex linear combinations of eigenprojectors would provide non-trivial invariant states. However, their normality would permit to follow exactly the argument of Proposition 3.4 and to show that the CNT-entropy for the above dynamical system is zero. As before, the persistence (less abstract in such a case) of classically erratic behaviour affecting the supports of the solutions of the quantum dynamics (44) is not reflected by the strict positivity of the corresponding CNT-entropy with respect to normal invariant states.
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3.5.2. Infinite degrees of freedom. Typically, one goes from a one-particle system to infinitely many degrees of freedom through second quantization. As a consequence, there is an abundance of non-normal states that remain invariant under a given evolution and thus allow us to avoid the consequences of Proposition 3.4. Moreover, the possibility that information can get lost is greatly enhanced by a mechanism that we describe by considering a one-particle system evolving according to a discrete group of unitaries Uˆ (n) on a Hilbert space H. Further, we assume that Uˆ has homogeneous Lebesgue spectrum. This means that there exists an increasing \sequence ˆ Hk = ∅, Hk ⊆ Hk+1 ⊆ H, k ∈ Z, of Hilbert spaces such that 1) U Hk = Hk+1 , 2) while 3) the closure of the linear span
[
k
Hk is either H or the orthogonal subspace to
k
the unique Uˆ -invariant state |i, if the latter exists. It turns out that, by considering the orthogonal complements Kk ⊂ Hk of Hk−1 , we can choose a basis |e0,j i, 1 ≤ j ≤ J, in K0 such that the vectors |ek,j i := Uˆ (k)|e0,j i form a basis in Kk , hek1 ,j1 , ek2 ,j2 i = δk1 ,k2 δj1 ,j2 ,
(47)
and {|ek,j i} is a basis in H (or in |i⊥ ), J being the multiplicity of the spectrum. The spectrum of Uˆ on the orthogonal subspace |i⊥ is absolutely continuous; thus, according to Proposition 3.4, the absence at all of normal invariant states or of nonpure normal invariant states means that either we cannot use a Kolmogorov–Sinai type approximation as in (27) or that the result is trivially zero as in Sect. 3.3. Nevertheless, for future purposes (see around Eq. (74) in Sect. 4), we elaborate on that argument. That is, we keep for a moment the One-Particle description and thus consider the C*-algebra B(H) over the Hilbert space H. For sake of simplicity we suppose |i not to exist (otherwise our considerations will be valid with reference to the Hilbert space |i⊥ ) and that the multiplicity J < ∞. Thus, let Eˆ k,j be the one-dimensional, pairwise orthogonal projections corresponding to the basis vectors |ek,j i and construct M M Eˆ [−n,n] := Eˆ k,jk , Qˆ [−n,n] := 1ˆl − Eˆ [−n,n] . (48) −n≤k≤n 1≤jk ≤J
In this way, the strong-operator limit of the double sequence of finite rank orthogonal projections Eˆ [−n,n] is the identity operator on H and the sequence of finite-dimensional subalgebras B[−n,n] := Eˆ [−n,n] B(H)Eˆ [−n,n] ⊕ C · Qˆ [−n,n] [ is such that the strong-operator closure of B[−n,n] = B(H).
(49)
n∈N
The subalgebras B[−n,n] , as linear spaces, have dimension (2n + 1)J + 1. Moreover, Uˆ |ei,j i = |ei+1,j i, whence σ l B[−n,n] = B[−n+l,n+l] . It follows that the finitedimensional subalgebras B[−n+l,n+l] , for 0 ≤ l ≤ k − 1, can be embedded into B[−n,n+k−1] which has dimension (2n + k)J + 1 that increase linearly with k. Thus, even if we could find a normal invariant state, the proof of Proposition 3.4 would give zero CNT entropy, for such an increase would not be enough to match the asymptotic 1 average lim in (24). n n
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Let us the CAR algebra A over H with the quasi-free evolution now] consider ] ˆ σ aˆ (f ) = aˆ U f as in (34), (37) and the unitary Uˆ considered above. The situation radically changes. Because of (47), the creation and annihilation operators aˆ ]k,j := aˆ ] (ek,j ) associated with the basis vectors |ek,j i fulfill aˆ †k1 ,j1 aˆ k2 ,j2 + aˆ k2 ,j2 aˆ †k1 ,j1 = δk1 ,k2 δj1 ,j2 . Therefore, by introducing (see [12, Par. 5.2.2]) ˆ 0,j aˆ †0,j , eˆ0,j 11 := a
0 eˆ0,j ˆ 0,j Vˆj−1 , 12 := a
ˆ 0 ˆ† , eˆ0,j 21 := Vj−1 a 0,j where Vˆj0 :=
j Y
1ˆl − aˆ †0,i aˆ 0,i
eˆ0,j ˆ †0,j aˆ 0,j , 22 := a and
(50)
Vˆ00 := 1ˆl,
i=1
we have a complete set of 2 × 2 matrix units, one for each 1 ≤ j ≤ J with which to establish an isomorphism between the subalgebra A0 ⊆ A generated by the aˆ ]0,j , X † 1 ≤ j ≤ J and 1ˆl − aˆ aˆ 0,j , and the full 2J × 2J matrix algebra M2J (C). 0,j
1≤j≤J
same argument holds for the subalgebras σ l (A0 ), 1 ≤ l ≤ k − 1, generated by The ] σ aˆ 0,j = aˆ ]l,j . Let us denote by Ml , l = 0, . . . , k − 1 the corresponding images under the_ isomorphism. Because of (47), the matrix algebras Ml pairwise commute. Therefore, σ l (A0 ) is isomorphic to the full matrix algebra M2kJ (C) whose dimension l
0≤l≤k−1
increases now exponentially with k, unlike the linear increase on the One-Particle level. Therefore, the possibility remains that the asymptotic average might be matched and that there can be states such that the CNT entropy of the quasi-free automorphism σ is not zero. For instance, let us take the tracial state τ on A. Thus we can apply the result (40) which gives hτ (σ) = J log 2, the multiplicity of the spectrum being constant and equal to J by assumption.
4. Non-commutative Topological Entropy What emerges from the previous sections is that states of ordinary quantum mechanical systems either are too regular to keep track of irregular dynamics or they are too irregular and good approximation theorems cannot be used. Therefore, one might hope to get hold of quantum randomness by avoiding reference to invariant states and by considering the C*-algebraic properties of the evolution only. In classical ergodic theory this corresponds to passing from the measure-theoretic point of view to the topological one, the latter being based on the analysis of classical dynamical systems on topological spaces X (that we shall assume to be compact and Hausdorff resp. even metrizable) equipped with continuous (usually even invertible continuous) maps S : X → X representing the evolution. Such couples may be viewed in algebraic terms by considering the algebra C(X ) of continuous functions x 7→ f (x) over X evolving under the automorphism σ : C(X ) → C(X ), (σf )(x) = f (S −1 x). C(X ) is a C*-algebra with respect to the sup-norm. Central in the classical theory of the topological entropy (cf. [50]) is the notion of open cover of X by a set U of open subsets Ui , i ∈ IU . Since by our compactness
Quantum Chaos and Dynamical Entropy
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assumption we can always reduce to finite open subcovers, we will consider the class O(X ) of U with card IU < ∞. Obviously, U ∈ O(X ) may contain many more open subsets Ui than needed to cover X . Let then denote N (U ) := min {card U 0 | U 0 ⊂ U , U 0 ∈ O(X )} H (U) := log N (U) .
and
(51) (52)
The latter will define the entropy of U ∈ O(X ). Supposing that S is a homeomorphism, the image S (U) := {S(Ui )}i∈I of a finite open cover is still in O(X ) and we can define the joining of U and S(U) as the finite open cover U ∨ S(U ) whose elements are all the intersections Ui ∩ S(Uj ). Clearly, N (S(U )) = N (U) since S is assumed invertible, N (U ∨ S(U)) ≤ N (U)2 ⇒ H (U ∨ S(U )) ≤ 2H (U ) , n−1 _ 1 S j (U ) exists. h (S, U) := lim H n→∞ n
and
(53) (54) (55)
j=0
In analogy with the Kolmogorov–Sinai entropy, the maximal state-independent average information coming from the open covers of X , namely h (S) :=
sup h (S, U) ,
(56)
U ∈O(X )
is defined as the topological entropy of (X , S), cf. [50]. Let V, U ∈ O(X ). If for all V ∈ V there is U ∈ U with V ⊂ U , we will say that V is a refinement of U. Then, H (U) ≤ H (V). Moreover, if a sequence of Un ∈ O(X ) is refining and “cofinal”, in the sense that: each open cover is a refinement of the previous one, and that for all V ∈ O(X ) there is n ∈ N such that Un is a refinement of V, then we have a Kolmogorov–Sinai type theorem which states that h (S) = lim h (S, Un ) n→∞
.
(57)
The topological entropy of (X , S) is related by a so-called variational principle to the Kolmogorov–Sinai entropy hµ (S) of (X , S, µ), where µ is any S-invariant Borel measure on X . Let M (X , S) denote the convex, compact set of such measures. Then: h (S) =
sup
hµ (S)
.
(58)
µ∈M (X ,S)
We will now concentrate on quantum dynamical systems (A, σ) without referring to any invariant state, thus to the intrinsic properties of the dynamics, and as before we will deal with unital C*-algebras A and with their automorphisms σ : A → A. Following [23, 24], in the non-commutative case (A, σ) we start by replacing open covers U ∈ O(X ), introducing the so-called (positive) operator covers. Definitions 4.1. 1. An operator cover for A is any finite subset α = Aˆ i i∈I of A X Aˆ ∗i Aˆ i is an invertible operator, namely strictly > 0. If this is such that the sum i∈I
the case, we will write α ∈ O (A).
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2. X A positive operator cover for A is any finite set of operators 0 ≤ Aˆ i ≤ 1ˆl such that Aˆ i > 0. We will denote their class by O+ (A). i∈I
3. Let α, β ∈ O (A). The ordered join of α, β is the operator cover ~ β := Bˆ i · Aˆ j | Bˆ i ∈ β, Aˆ j ∈ α . α∨
(59)
4. We will denote by N (α) the minimum number of elements of α ∈ O(A) such that they still constitute an operator cover: N (α) := min {card α0 | α0 ⊆ α, α0 ∈ O(A)} .
(60)
Notice: 1. The non-commutative join generalizes the classical join of two open covers by specifying which operator cover is superfluous in the X coming first (this X X has its elements ∗ ˆ ∗ ˆ ˆ ˆ Ai Ai > 0, Bj Bj > 0, both Bˆ j∗ Aˆ ∗i Aˆ i Bˆ j and commutative case). If P
i
j
i,j
ˆ∗ ˆ ∗ ˆ ˆ i,j Ai Bj Bj Ai are invertible. ~ β ∈ O (A) if α, β ∈ O (A), the ordered join of α, β ∈ O+ (A) ⊂ O (A) While α∨ need not be in O+ (A). ~ β , but N (β) might be bigger. 2. N (α) ≤ N α∨ ~ α∨ ~ ···∨ ~ α = N (α) for all α ∈ O+ (A), but this is not necessarily true for 3. N α∨ operator covers α ∈ O (A) \ O+ (A). 4. However, like in the commutative situation the following inequality holds: ~ β ≤ N (α) · N (β) . N α∨ (61) 5. Furthermore, denoting by σ(α), for an automorphism σ of A, the (positive) operator cover with elements σ(Aˆ i ), Aˆ i ∈ α, then N (σ(α)) = N (α) . 6. If B is a finite-dimensional C*-algebra isomorphic to
(62) n M k=1
Mdk (C) and d =
n X
dk ,
k=1
then N (α) ≤ d for all α ∈ O (B). Given any partition of the identity into orthogonal one-dimensional projections of the matrix algebra, the isomorphic image in B is a positive cover α∗ ∈ O (B) for which (63) max N (α) | α ∈ O (B) = N α∗ = d. Since C*-algebras need not contain proper finite-dimensional subalgebras, like C(X ) for a connected space X , and we want finite-dim. C*-algebras, together with their (positive) operator covers, to replace the classical open covers, we will, in general, use positive (i.e., positivity-preserving) unital maps γ : B 7→ A, γ(1ˆlB ) = 1ˆl, where B with unit 1ˆlB is a finite-dimensional C*-algebra. We will also need to identify as equivalently covering (γ1 ' γ2 ) two positive unital maps γk : Bk → A, k = 1, 2, such that γ1 = γ2 ◦ θ12 , where θ12 : B1 → B2 is any *-isomorphism. Notice that positive unital maps γ : B → A map β ∈ O+ (B) into γ(β) ∈ O+ (A).
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Definition 4.2. Given positive unital maps γk : Bk → A from finite-dimensional C* algebras Bk into A, k = 1, . . . , n, their entropy is defined by ~ ···∨ ~ γn (αn ) , H (γ1 , . . . , γn ) := (64) log N γ1 (α1 )∨ max αj ∈O + (Bj ), j=1,... ,n with the restriction that, if γk ' γl for k < l, then the trivial cover αl = 1ˆl ∈ Bl is to be used for the later map with index l > k. Some comments and properties: 1. The restriction on the evaluation of the maximum in (64) is necessary in view of the third observation after (60). In fact, we want the entropy not to be sensitive of repeated unital maps in different places. 2. For a single γ : B → A, from (63) it follows that H (γ) ≤ log D(B), where D(B) is what is d in (63). If B is a finite-dimensional subalgebra of A, we will denote by H (B) := log D(B) the entropy of the inclusion map (which is hence independent of the inclusion into A). 3. For positive unital maps θk : Bk → Ak and γk : Ak → A, k = 1, . . . , n on finite dimensional algebras Ak , Bk , such that if γi ' γi also γi ◦ θi ' γj ◦ θj , we have: H (γ1 ◦ θ1 , . . . , γn ◦ θn ) ≤ H (γ1 , . . . , γn ) .
(65)
4. If σ : A 7→ A is an automorphism, then from (62) we deduce H (σ ◦ γ1 , . . . , σ ◦ γn ) = H (γ1 , . . . , γn ) .
(66)
5. It follows from the definition and (61) : H γi1 , . . . , γil ≤ H (γ1 , . . . γn ) ≤ H (γ1 , . . . , γk ) +H (γk+1 , . . . , γn ) ,
(67)
where 1 ≤ i1 ≤ . . . ≤ il ≤ n, l < n and none of the maps equivalently covering. 6. Additivity rather than mere subadditivity for all the single arguments γ1 , . . . , γn in Property 5 above should follow from some algebraic independence of these n arguments. For such a result, we need some preliminary definitions for the general case of positive unital maps into A: Definitions 4.3. 1. A positive unital map γ : B → A from a finite-dimensional algebra 0 + into A is called faithfully 0covering, if there exists α ∈ O (B) such that H (γ) = 0 log N γ(α ) = log card α . 2. Given n positive unital maps γk : Bk → A, k = 1, . . . , n, they are called independently covering, if for all sets of 0 ≤ Bk ∈ Bk , k = 1, . . . , n, such that γk (Bk ) are not invertible, there exists a single state ω on A such that ω ◦ γk (Bk ) = 0, k = 1, . . . , n. 3. Two positive unital maps γk : Bk → A, k = 1, 2, are called commuting if γ1 (B1 )γ2 (B2 ) = γ2 (B2 )γ1 (B1 ) for all Bk ∈ Bk , k = 1, 2. The above definitions generalize the case when one is dealing with pairwise commuting, algebraically independent, finite-dimensional subalgebras Bk ⊆ A whose natural inclusion ∗-homomorphisms are thus commuting, faithfully and independently covering [23].
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Further, if the positive unital maps γk : Bk → A, k = 1, . . . , n, are pairwise commuting, each one faithfully and all together independently covering, then [23] H (γ1 , . . . , γn ) =
n X
H (γk ) .
(68)
k=1
Using properties 5 and 4, we define as for the Kolmogorov–Sinai entropy in the classical case and for the CNT-entropy in the non-commutative one (Sect. 3.2 above): Definition 4.4. For a positive unital map γ : B → A from a finite-dimensional C*algebra B and for an automorphism σ : A → A, the entropy of σ with respect to γ is defined by: 1 H γ, σ ◦ γ, . . . , σ n−1 ◦ γ . n→∞ n
~ (σ, γ) := lim
(69)
Among the other properties of ~ (σ, γ) we quote [23, 24]: ~ (σ, γ) ≤ H (γ) ~ θ ◦ σ ◦ θ −1 , θ ◦ γ = ~ (σ, γ) ,
θ : A → θ(A) a ∗-isomorphism.
(70) (71)
In analogy with the measure-theoretic Kolmogorov–Sinai entropy, with the classical topological entropy and with the CNT-entropy, a straightforward definition of noncommutative topological entropy would be: ~ (σ) := sup ~ (σ, γ) ,
(72)
γ
where γ : A → A is any linear, positive, unital map from a finite-dimensional C*-algebra A into A. However, a major difference between the above definition and that of the CNTentropy lies in that (72) is not supported by a Kolmogorov–Sinai like theorem as (27), or (57) for the classical topological entropy. Indeed, there is no control over the continuity of the entropy functionals (64) with respect to the appropriate (norm-) topology on the space of γ: there are instances of γ1 , γ2 from B into A, arbitrarily close to each other, with different entropy functionals [23, 24]. A more workable definition which explicitly depends on the existence of a finite approximation of the C*-algebra A by means of an increasing sequence of finitedimensional C*-algebras included or positively mapped into it, is contained in the following: Definitions 4.5. 1. Suppose that a sequence Bn of finite-dimensional C*-algebras is given together with positive unital maps τn : Bn → A. The sequenceτ := {τn }n∈N will be called “cover increasing” if τn O+ (Bn ) ⊆ τn+1 O+ (Bn+1 ) . 2. In this case, the τ -topological entropy of an automorphism σ : A → A is defined by [23]: ~τ (σ) = lim ~ (σ, τn ) , n→∞
(73)
where the increasing limit exists due to Assumption 1, see Definitions 4.2 and following ones.
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This is typically the case when A is the C*-inductive limit of an increasing sequence [ An , in which case we choose as τn the of finite-dimensional subalgebras An , A = n
corresponding inclusions. More in general, the structure of A might be that of a nuclear C*-algebra, that is there exists an increasing sequence {An }n∈N of finite-dimensional C*-algebras and completely positive maps σn : A 7→ An , τn : An 7→ A such that ˆ − Xk ˆ → 0 for all Xˆ ∈ A when n → ∞. This is not sufficient, however, kτn ◦ σn (X) to ensure that ~ (σ) = lim ~ (σ, τn ); see [23] for an alternative definition instead of 4.4 n for this case. The above theory does not distinguish between Abelian and nonabelian algebras. Therefore we might ask whether, when A is the Abelian C*-algebra C(X ) of continuous functions over a compact metric space X , (72) resp. (73) coincides with (56). At least one inequality between (72) and (56) is indeed easy to show: Namely let U = {Ui }i∈I ∈ O (X ) be given. Correspondingly, we have a partition of the identity by positive functions fi ∈ A = C(X ), i ∈ I supported by the open sets Ui ∈ U. Clearly, αU = {fi }i∈I ∈ O (A) and N (U) = N (αU ). Given the automorphism σ : A → A, defined by σf (x) = f (S −1 x), according to (51), Definitions 4.1 andowing to commutativity of the operator algebra A, it is also true that _n n _ ~ σ j (αU ) . Therefore, h (S, U) = ~ (σ, αU ). Comparing N S j (U ) = N j=0
j=0
definitions (56) and (72), we thus deduce h (S) ≤ ~ (σ). See [23], in particular Remark (2.12),(5) resp. Lemma (2.10) there, for the more complicated converse inequality. Remark 4.6. In the above example one could start off with an abstract Abelian algebra A and represent it via the Gelfand transform as an algebra of continuous functions over the space of its characters (see [11, Par. 2.3.5]) which plays the role of X . However, the fact that A is assumed to contain an identity forces X to be compact as a topological space. Indeed, the quantum topological entropy defined by (72) mimicks the construction of the classical topological entropy (56) that is based on compact topological spaces. However, if X is endowed with a metric, then a metric dependent definition of topological entropy was given by Bowen (see Def. 5.3 and [50]). On compact metrizable spaces, equivalent metrics give the same Bowen’s entropy which in turn equals the expression (56). For a noncompact metrizable space X it is Bowen’s construction that must be used and to different, not uniformly equivalent, metrics there can correspond different topological entropies. One then wonders whether an algebraic version of Bowen’s entropy can be envisaged, to be possibly extended to non-unital C*-algebras. We come now to some concrete examples that aim at comparing the use of the quantum topological entropy introduced here versus the CNT quantum dynamical entropy. From (63) and inequality (70) it trivially follows that the topological entropy of whatever evolution over finite dimensional C*-algebras vanishes exactly as does the CNT entropy. Let now H be an infinite dimensional Hilbert space and let U : H → H be the unitary evolution considered in Sect. 3.5.2. We have seen that, despite the presence of homogeneous Lebesgue spectrum (of multiplicity J), the CNT entropy of the corresponding automorphism vanishes with respect to normal states and is out of control for more general states because of lack of continuity.
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With the topological entropy the situation is more or less equivalent in the following sense. By using the finite rank subalgebras B[−n,n] in (49) as a sequence τ generating ˜ the C*-algebra B(H) :== C ∗ (B∞ (H) ⊕ C · 1ˆl) of compact operators plus the identity, we get ~τ (σ) = 0 (see (73)) for the automorphism σ on B(H) defined by U . In fact, σ l (B[−n,n] ), l = 0, . . . , k−1, are all included in B[−n,n+k−1] . Therefore, considering the ˜ inclusions γnl : σ l (B[−n,n] ) → B[−n,n+l−1] and γn : B[−n,n+l−1] → B(H), from (65) and (63) we get (74) H γn ◦ γn1 , . . . , γn ◦ γnk ≤ H (γn ) = log(J(2n + k)) . ˜ Thus, ~ (σ, τn ) = 0 for all inclusions τn : B[−n,n] → B(H). However, we do not know whether the result extends to the whole of B(H) which is the strong-operator closure of the sequence τ . Again, as in the case of the CNT entropy we can pass to the infinitely many degrees of freedom by second quantization. We shall work not with the CAR algebra A itself, but with an isomorphic dynamical system and use (71). In fact, the isomorphism constructed in Sect. 3.5.2 by means of the system of matrix extends to an isomorphism between Nunit +∞ the CAR algebra A and the C*-closure C = −∞ Ml of the increasing sequence τ of Nk finite tensor products l=−k Ml , that is the spin-chain algebra with a J×J matrix algebra at each site. Clearly, the automorphism σ on A becomes isomorphic to the bilateral shift σ˜ : Ml 7→ Ml+1 as automorphism on C. Now it is easy to show, applying (65), that [23] ˜ = ~ (σ, ˜ M0 ) = H (M0 ) = J log 2, ~τ (σ)
(75)
where by ~ (σ, ˜ M0 ) we denote the entropy defined in Def. 4.4 for the inclusion map of M0 into C. Notice: It is very doubtful if the equality between this result and that obtained at the end of Sect. 3.5.2 reflects the presence of a general quantum variational principle linking the quantum topological entropy constructed in this section and the CNT entropy (which would then generalize the classical relation 58); but here this is revealing the peculiar properties of the tracial state from an entropic point of view, too. Despite the previous result, it is not necessary to go to infinitely many degrees of freedom in order to get nontrivial results. Indeed, unlike the CNT-entropy, the cases dealt with in Subsects. 3.3 and 3.5.1 show that the non-commutative topological entropy is capable of keeping track of “erratic” behaviour in these cases. Given the Koopman suspension of a classical dynamical system with strictly positive topological entropy, the next, simple lemma follows: Lemma 4.7. Let (X , S, µ) be a classical dynamical system (with X compact, metric and µ an invariant Borel probability measure) with a positive topological entropy, h (S) > 0. system emerging Let (Mcl = L∞ (X , µ), σcl , ωcl ) be the algebraic classical dynamical from the Koopman approach as in Sect. 3.3, and B(Hµ ), σ, ωµ be the non-commutative “Hilbert space” dynamical system naturally arising from the previous one. Then, the full, non-commutative topological entropy of the latter, defined as in (72), is also positive, ~ (σ) > 0. Proof. By assumption, there exists an open cover U = {Ui } ∈ O (X ) such that h (S, U ) > 0. The characteristic functions Xˆ i := χUi , as multiplication operators, belong to the von Neumann subalgebra Mcl = L∞ µ (X ) ⊂ B(Hµ ) and constitute a
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positive operator cover αU ∈ O (Mcl ),
X
Xˆ i = 1ˆl. As σ(Xˆ i ) = χS(Ui ) , we can re-
i
peat the argument used above for the Abelian case A = C(X ) after 4.5, to show that ~ (σ) ≥ h (S, U ). As to the more physical model (44), let a proper advecting velocity field u on R3 be chosen that develops at least one (bounded) chaotic region X ⊂ R3 , R3 (3 x) mapped into itself by the dynamics S˙t (x) = u(St (x)). Given the C*-algebra B (H) and the evolution ˆ = ei H Xe ˆ −i H , σ(X) ˆ
ˆ
Hˆ = u(x) · pˆ + βE0 + ακ,
(76)
the multiplication operators by functions supported within this region provide the necessary tool to get ~ (σ) > 0. In fact, due to our assumptions we can concentrate on the invariant chaotic region only, where an open cover U(X ) = {Ui } can be found such that the classical h S1 |`X > 0. It is clear that, for the set of multiplication operators Xˆ i := χUi by the corresponding characteristic functions, we can repeat the argument of Lemma 4.7. 5. Motion in a Crystal In this section we illustrate the various relations between classical and quantum chaos by the example of an electron in a n-dimensional crystal. The dynamical properties of this electron will be compared with the ones of a free particle. In the one-electron approximation of solid state physics the effect of the nuclei and the other electrons is taken into account in the form of a potential V . We assume the potential to be L-periodic with a lattice n X zj `j | z1 , . . . , zn ∈ Z L := j=1
spanned by a basis `1 , . . . , `n ∈ Rnq of configuration space Rnq . The fundamental domain n X xj `j | 0 ≤ x1 , . . . , xn < 1 D := j=1
is then naturally isomorphic as a set to the n-dim. configuration space torus T := Rnq /L, and V can be considered to be a function V : T → R. In the quantum mechanical context the dynamics exists for more singular potentials than in the classical case. However, we will discuss one example in both cases, namely the one of motion in a planar crystal with attracting Coulombic forces. There in the fundamental domain m ≥ 1 nuclei are situated at the positions s1 , . . . , sm ∈ D which attract the electron with the charges Z1 , . . . , Zm > 0. That is, we assume the potential of the form V (q) ∼ −Zi /|q − si | for q near si . Now by the periodicity of the crystal the potential is singular at the points of S := {q ∈ R2 | q = si + `, ` ∈ L} ˜ := R2 \ S. Sometimes we and thus only defined in the punctured configuration plane M 2 identify the plane R with C. We assume that V has the following form
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˜ , R) which is L-periodic Definition 5.1. A potential V ∈ C ∞ (M V (q + `) = V (q),
˜ , ` ∈ L) (q ∈ M
is called Coulombic if for ε > 0 small the functions f1 , . . . , fm , V (si + Q2 ) · QQ¯ , 0 < |Q| < ε fi (Q) := ,Q = 0 −Zi are C ∞ (where Q ∈ C is a so-called regularizing complex coordinate, related to the complex coordinate q in the plane by the Levi–Civita transformation Q2 = q − si ). Obviously we have a finite Vmax := supq∈M˜ V (q). We assume without loss of generality that Vmax ≤ 0. Example (Periodic Yukawa potentials). We choose “masses” µ1 , . . . , µm > 0 and set V (q) :=
m XX `∈L i=1
−Zi
exp(−µi · |q − si − `|) , |q − si − `|
˜ ). (q ∈ M
Due to the positivity of the µi this sum is absolutely converging together with its derivatives and thus leads to a smooth, L-periodic V . The fi have the form ¯ + gi (Q) · QQ¯ fi (Q) = −Zi · exp(−µi · QQ) with gi smooth and are thus smooth, too. The reason for the somewhat odd-looking definition is that we want to regularize the Coulomb singularities by using the so-called Levi–Civita transformation. 5.1. The classical motion (one particle). Now we consider the topological and metric entropy of a classical particle of energy E in the periodic potential V . If V : Rn → R is smooth (C 2 being sufficient), then the Hamiltonian flow 8t : P → P on the phase space P := T ∗ Rn ∼ = Rnp × Rnq generated by the Hamiltonian function 1 2 H : P → R, H(p, q) := 2 p + V (q) exists for all times t. Moreover for E > Vmax the energy shell 6E := {x ∈ P | H(x) = E} is a smooth 2n − 1-dimensional submanifold, and we denote the restricted flow 8t |`6E by 8tE . In the case of a Coulombic potential the classical motion is generated by the Hamiltonian function ˜ ). ˜ H(p, q) := 21 p2 + V (q), ((p, q) ∈ T ∗ M Due to collisions with the singularities in S the Hamiltonian flow on the cotangent bundle ˜ of the punctured plane M ˜ does not exist for all times. T ∗M However, as described in Lemma 5.2 below, the flow can be smoothly regularized. Lemma 5.2. There exists a unique smooth extension (P, ω, H) of the Hamiltonian sys˜ , dq1 ∧ dp1 + dq2 ∧ dp2 , H), ˜ where the phase space P is a smooth fourtem (T ∗ M dimensional manifold with [ ˜ ∪ P := T ∗ M R × S1 (77) S
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as a set, ω is a smooth symplectic two-form on P with ω|T ∗ M˜ = dq1 ∧ dp1 + dq2 ∧ dp2 , ˜ and H : P → R is a smooth Hamiltonian function with H|T ∗ M˜ = H. The smooth Hamiltonian flow 8t : P → P,
(t ∈ R)
(78)
generated by H is complete. For all energies E which are regular values of V , the energy shell 6E := {x ∈ P | H(x) = E}
(79)
is a smooth, three-dimensional manifold, and we write 8tE := 8t |6E . Proof. The construction works locally near the singularities s ∈ S. We shortly explain the method by considering the simplest case. For more details see Prop. 2.3 of [26], where a scattering potential is considered. One may linearize the Kepler flow with Hamiltonian function ˜ K (p, q) := 1 p2 − Z/|q| H 2 in a suitable neighbourhood of R2p × {0} in the phase space T ∗ (R2q \ {0}), using the ˜ q) := p∧q is the angular momentum, ˜ K , T˜ , L˜ and ϕ, ˜ where L(p, canonical coordinates H T˜ (p, q) is the time needed to come from the phase space point (p, q) to the pericenter of the Keplerian conic section and ϕ(p, ˜ q) is the angle between the direction of that pericenter and, say, the 1-direction. Except T˜ these phase space functions are constant along the Kepler flow, and the collision points correspond to T˜ = 0, L˜ = 0. The remaining ˜ take values in a cylinder R × S 1 . coordinates (H˜ K , ϕ) Similarly, because the singularities of V are of the Kepler form, one may thus com˜ by gluing one cylinder for each singularity in S. plete the phase space T ∗ M So after this regularization the Hamiltonian flow exists for all time, and we are in a similar situation as in the case of a smooth periodic potential. However, in both cases the energy shells extend to spatial infinity and are thus noncompact. Therefore the Liouville measure µE on 6E is not normalizable and we cannot define a metric entropy of the flow 8tE w.r.t. that measure. But it is natural to use the invariance of H under the symplectic diffeomorphisms 9` : P → P,
9` (p, q) := (p, q + `),
(` ∈ L)
(80)
of the phase space P generated by the lattice translations. So H descends to a function Hˆ on the phase space Pˆ := P/L ∼ = T ∗ T over the configuration torus T, and generates t ˆ ˆE : 6 ˆE → 6 ˆ E , t ∈ R on its energy shells 6 ˆ E := {x ∈ Pˆ | H(x) a flow 8 = E}. For regular values E of Hˆ (and thus in particular for E > Vmax ) the Liouville measure µˆ E ˆ E can be normalized. Then we will consider the metric entropy of 8 ˆ tE w.r.t. the on 6 probability measure µˆ E . In the Coulombic case the flow is defined as 8tE in Lemma 5.2. The topological entropy of our flow 8tE on the non-compact energy shell 6E can be defined directly. Namely we use the definition by Bowen of topological entropy which
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also applies to non-compact phase spaces X. This definition is not (and cannot be) purely topological, and one needs a metric d on X. The topological entropy will then be invariant under uniformly continuous changes of the metric d. Therefore, for compact metrizable spaces X the topological entropy will only depend on the topology generated by the metric. Definition 5.3 (Bowen). Let (X, d) be a metric space and T : X → X a uniformly continuous map. Then for m ∈ N, ε > 0, a subset F ⊂ X is said to (m, ε)-span a compact K ⊂ X if ∀x ∈ K ∃y ∈ F : dm (x, y) ≤ ε, with the metric dm (x, y) := max0≤i≤m−1 d(T i x, T i y). Let rm (ε, T, K) be the smallest cardinality of an (m, ε)-spanning set F of K, hr (ε, T, K) := lim sup m→∞
1 ln(rm (ε, T, K)), m
and htop (T, K) := limε→0 hr (ε, T, K). Then the topological entropy of T is htop (T ) := sup {htop (T, K) | K ⊂ X compact} .
Note that hr (ε, T, K) is monotonically increasing in ε, so that its limit htop (T, K) exists (it can be ∞). In estimating the topological entropy, the relation htop (T m , K) = m · htop (T, K),
(m ∈ N)
(81)
for the iterated map is useful. The topological entropy of a flow 8t : X → X is defined to be htop (81 ). In our case it is natural to use a metric dE on 6E which is compatible with the topology on that energy shell and invariant under the lattice translations 9` , ` ∈ L. Lemma 5.4. The topological entropy for energy E of the flow on the configuration space Rnq coincides with the one of the flow on the configuration torus T: ˆ 1E ). htop (81E ) = htop (8 1
ˆ E ) ≤ htop (81E ). In fact, considering the projection Proof. Clearly htop (8 ˆ E, 56E ,6ˆ E : 6E → 6 it follows that d(56E ,6ˆ E (x), 56E ,6ˆ E (y)) ≤ d(x, y). This implies 1
ˆ E , 56 ,6ˆ (K)) ≤ rm (, 81E , K) rm (, 8 E E for all > 0 and compact K ⊂ Rnq . Therefore, ˆ 1E , 56 ,6ˆ (K)) ≤ htop (81E ) sup htop (8 E E
K⊂Rn q
(82)
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ˆ E is the image of a compact in 6E and the inequality follows, for every compact K ⊂ 6 under the projection 56E ,6ˆ E . On the other hand, any (m, ε)-spanning set F of a compact K ⊂ 6E is contained in the compact ε-neighbourhood Kε := {x ∈ 6E | ∃y ∈ K : d(x, y) ≤ ε} , of K, and the maximal multiplicity n o (5 (x)) ∩ K Mmax := sup 5−1 ˆ ε 6 , 6 ˆ E E 6E ,6E x∈Kε
of the projection 56E ,6ˆ E is finite on Kε . Now if F is a (m, ε)-spanning set of 56E ,6ˆ E (K), then it is contained in 56E ,6ˆ E (Kε ), ˆ 1E ) and thus and 5−1 (F ) ∩ K ≤ Mmax · |F |, which implies htop (81E ) ≤ htop (8 ε ˆ 6E ,6E (82). Proposition 5.5. If the periodic potential V is twice continuously differentiable, the topological entropy for the flow is bounded above: 1
htop (81 ) ≤ 2n · maxq kD2 V (q)k 2 .
(83)
On the other hand for n ≥ 3 dimensions there exist smooth periodic potentials for which the topological entropy is bounded below above a threshold energy Eth , htop (81E ) ≥ C > 0
(E > Eth ).
(84)
Proof. Pn Phase space P is a Riemannian manifold w.r.t. the canonical Riemannian metric i=1 dpi ⊗ dpi + dqi ⊗ dqi . If the potential is constant, the flow is integrable and (83) is trivially true. Otherwise 1
b := maxq kD2 V (q)k 2 > 0. By Thm. 7.15 of [50] we have on the 2n-dim. manifold P the bound htop (8t ) ≤ 2n · ln maxx∈P kU t (x)k , where kU t (x)k is the operator norm of the linearization U t (x) := D8t (x) of the flow at x. But U t (x) is the solution of the initial value problem d t 0 −D2 V (q(t, x)) U t (x), U 0 (x) = 1ˆl. U (x) = ˆ 1l 0 dt 1ˆl 0 t t We set W (x) := BU (x) with B := . Thus W t (x) solves 0 b · 1ˆl d t 0 −D2 V (q(t, x))/b W t (x), W 0 (x) = B W (x) = b · 1ˆl 0 dt or
Z t W t (x) = B + 0
0 −D2 V (q(s, x))/b b · 1ˆl 0
W s (x)ds.
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Thus Gronwall’s inequality says kW t (x)k ≤ kBk exp(bt) so that kU t (x)k ≤ kB −1 k kBk exp(bt). Using the formula htop (81 ) = htop (8t )/|t| in the t → ∞ limit, we prove (83). Concerning the lower bound, we use the fact that there exist two-dimensional smooth L0 -periodic potentials V 0 with positive entropy h := htop (81Eth ) > 0 for some energy Eth . Examples can be found in the paper [16] by Donnay and Liverani (we even expect it to be a generic property). Let V : Rn → R be the potential V (q) := V 0 (q1 , q2 ) and let L := L0 × Zn−2 ⊂ Rn = R2 × Rn−2 be a n-dim. lattice based on the lattice L0 of V 0 . Then F 1 : P 1 → R, F 1 (p, q) = 21 (p21 + p22 ) + V (q) is a constant of the motion generated ˆ For E > Eth the submanifold by H. ˆ E | F 1 (x) = Eth } X := {x ∈ 6 is thus flow-invariant, and the flow on this product manifold is the product flow of the flow in V 0 and a free flow. But the topological entropy of a product flow is the sum of the entropies of the factors (Thm. 7.10 of [50]) so that the flow on X has entropy equal to h. But htop (81E ) ≥ htop (81E |X ) which proves (84). However, the metric entropy of smooth periodic potentials decays in the high energy limit: Proposition 5.6. For 3n times continuously differentiable periodic potentials on Rn the p ˆ 1E ) w.r.t. Liouville measure µE is of order O( 1/E). metric entropy hµE (8 Proof. In [4] it is shown, using p KAM theory, that the µE -measure of the complement of ˆ E is O( 1/E). Since on the KAM tori the motion is conditionally the invariant tori in 6 periodic, only that complement contributes in the Shannon–McMillan–Breiman formula Z 1 ˆ ˆ 1E , x)dµE (x), hµ E ( 8 hµE (8E ) = ˆE 6
ˆ 1E , x) (see, e.g. Ma˜ne´ [32], Chapter IV). But by (83) and the variational principle hµE (8 is bounded above by a constant independent of E. Remark 5.7. If the potential is separable, i.e. if the lattice P L is rectangular (say, oriented n in the coordinate directions) and V of the form V (q) = j=1 Vj (qj ), then the motion is integrable and has zero topological entropy. In particular the topological entropy vanishes for n = 1 dim. However, there does not seem to exist another example of a smooth periodic potential leading to integrable motion. For these potentials the motion is ballistic in the sense that for almost all initial points x0 ∈ 6E on an energy shell with E > Vmax the asymptotic velocity v(x) ¯ := lim
t→∞
q(t, x0 ) t
is non-zero. In the simplest case of classical free motion 8t (p, q) = (p, q + tp)
(85) √ with Hamiltonian function H(p, q) := 21 p2 , one has |v(x)| ¯ = 2E. Of course a ballistic form of the motion does not contradict positive entropy of the flow, see [4].
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We now calculate the topological entropy for the Coulombic potentials in the high energy limit. Although Lemma 5.2 solves the problem of collision orbits and regularizes the motion without changing it otherwise, we will now introduce a second regularization which is more useful to calculate quantities like the topological entropy, whereas it leads to a completely different phase space and a new time parametrization. It is well-known that in general the solution curves for a Hamiltonian function of the form “kinetic + potential” and energy E coincide with the geodesics of the Jacobi metric which is the Euclidean metric in configuration space, multiplied at point q with the conformal factor (1 − V (q)/E), the only difference consisting in a modified time parametrization. In our case, the Jacobi metric becomes singular near the positions S of the nuclei. Nevertheless one may consider for the Kepler flow the regularizing complex coordinate Q2 = q, that is, the Levi–Civita transformation. It turns out that in this coordinate the Jacobi metric, originally defined only for Q 6= 0, can be smoothly extended to a nonsingular Riemannian metric on the whole complex Q plane. Note that the Q plane is a two-fold branched covering of the original q plane, with a branch point at the position q = 0 of the singularity. In [27] this local construction was globalized using the toral Riemann surfaces T = M/L with projection 5M,T : M → T, its four-fold covering torus T2 := M/(2L) (with projections 5M,T2 : M → T2 , 5T2 ,T : T2 → T) and the compact Riemann surface Qm σ(q − si + `1 )σ(q − si + `1 + `2 ) , (86) M4 := (q, Q) ∈ T2 × P Q2 = i=1Qm i=1 σ(q − si )σ(q − si + `2 ) Y 1 − z/w exp z/w + 21 (z/w)2 σ(z) := z w∈2L\{0}
being the Weierstrass σ-function and P := C∪∞ the Riemann sphere. The map 5M4 ,T2 : M4 → T2 , (q, Q) 7→ q is a two-fold branched covering with branch points at the singularities, all branch numbers equaling one. Thus by the Riemann–Hurwitz relation the genus G(M4 ) of M4 equals G(M4 ) = 2m + 1 (m being the number of singularities in the fundamental domain). R Since thus the genus is ≥ 3, by Gauss-Bonnet the integrated Gaussian curvature M4 KE dM4 = −4π · (G(M4 ) − 1) of the lifted Jacobi metric g4,E on M4 becomes negative. Due to the branched covering construction the metric g4,E , originally not defined at the branch points, is be smoothly extended to these points by taking limits. So the geodesic flow φt4,E on the unit tangent bundle T1 M4 of the surface is smooth and defined for all times. In terms of the potential V the Gaussian curvature KE equals KE (q) =
(1 − V (q)/E)1V (q)/E + (∇V (q)/E)2 1 ln(E − V (q)) =− 2(1 − V (q)/E)3 2(E − V (q))
(87)
with 1 and ∇ denoting the Euclidean Laplacian and gradient, respectively. For many potentials V the Gaussian curvature becomes strictly negative (that is, KE (x) < 0 for all x ∈ M4 ) if E is large enough. Clearly this can only happen if 1V < 0 in (87), and the negativity is then preserved if one enlarges the energy E. In our example of Yukawa potentials Nobbe [36] estimated analytically an upper bound for the threshold energy Eth above which KE < 0 of Eth < 5.17, if one has a square unit lattice L and m = 1 singularity with µ = 1 in the fundamental domain. The actual Eth is probably even much smaller.
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Furthermore, as shown in [27], the negative curvature property is preserved by convex combinations of our Coulombic potentials. So it is a quite common property. As is wellknown, it leads to a geodesic flow on the unit tangent bundle T1 M4 which is a transitive Anosov flow and is mixing w.r.t. the Liouville measure. Moreover the autocorrelation of this flow decays rapidly enough that for many phase space functions a central limit theorem holds true. This made it possible to show in [27] that for probability measures νE of initial conditions x0 ∈ 6E whichRare absolutely continuous w.r.t. Liouville measure µE on the energy shell 6E (and with q 2 dνE < ∞) the motion in the crystal is diffusive, i.e. R D(E) := lim
6E
q 2 (t, x0 )dνE (x0 ) t
t→∞
>0
(88)
if E > Eth . In fact the diffusion constant D only depends on the energy E and not on the probability measure νE of initial conditions. Nevertheless, there are many νE not absolutely continuous w.r.t. µE which do not lead to diffusive motion. For example, this is the case if νE is supported on a periodic orbit. Such a diffusive behaviour does not follow from good ergodic properties alone. Like our system, the Sinai billard is Bernoulli, but the corresponding motion in the plane (Lorenz gas) is diffusive only if the so-called finite horizon condition is met, that is, if all trajectories hit the scattering centers. In [36] Nobbe gave analytical and numerical evidence for a divergence of the diffusion constant as D(E) ∼ c · E 3/2 . Theorem 5.8. If there exists a threshold energy Eth for which 1 ln(Eth − V ) > 0, then ˆ 1E ) is a smooth positive monotone increasing on (Eth , ∞) the topological entropy htop (8 function of E asymptotic to htop (81E ) ∼ h∞ top ·
√
2E
(E → ∞),
and the constant h∞ top > 0 depends only on the set S of singularities. Remark 5.9. Below Eth the topological entropy is not expected to be a smooth function of the energy but to exhibit a singular continuous (devil’s staircase) dependence on E. The reason for this kind of behaviour is a transition from regular (small E) to chaotic (E > Eth ) dynamics, see e.g. Troll [48]. Proof. By the formula (87) for Gaussian curvature the condition 1 ln(Eth − V ) > 0 implies that for E ≥ Eth curvature is strictly negative. As shown in [27], then the ˆ tE on the compact manifold 6 ˆ E is Bernoulli for each energy E > Eth . Thus flow 8 1 htop (8E ) > 0 (see [50]). Moreover it was shown in [27] that this flow is Anosov. Since Eth > Vmax , all the 6E for E > Eth are diffeomorphic (the diffeomorphisms leaving the configuration space point q invariant and rescaling momentum p). Thus 8tE can be considered as a family of flows on a fixed manifold smoothly depending on E. But by Thm. 2 of Katok et al. [25] for smooth families of Anosov flows the topological entropy varies smoothly, too (this is a deep result since their invariant (un)stable foliations are very non-smooth in general).
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ˆ 1E ) by comparing it with the topological entropy We will now estimate htop (8 htop (814,E ) of the geodesic flow 8t4,E on unit tangent bundle T1 M4 of the branched covering surface M4 . The limit lim htop (814,E )
E→∞
(89)
exists, since htop (814,E ) is increasing in E. This can be seen by noticing that the topological entropy of the geodesic flow equals the asymptotic exponential growth rate in L of the number of closed geodesics whose length is ≤ L. Now the geodesic flow 8t4,E , being Anosov, is structurally stable so that for two given energies E, E 0 > Eth there exists a homeomorphism HE 0 ,E : T1 M4 → T1 M4 which maps oriented orbits on oriented orbits. In particular the closed orbits of the two flows are conjugated. If E ≥ E 0 , then the length lE 0 (γE 0 ) of each closed orbit γE 0 of 0 ) of 8t4,E . 8t4,E 0 is larger than the length lE (γE ) of the closed orbit γE := HE 0 ,E (γE This follows since 0 • the length lE (γE 0 ) w.r.t. the Jacobi metric g4,E is smaller than the length lE 0 (γE ) w.r.t. g4,E 0 , since the conformal factor (1 − V /E) decreases in E (we assumed Vmax ≤ 0), • the closed geodesics are unique in their conjugacy class, due to the negativity of the Gaussian curvature KE , • and we can thus find the closed geodesic γE by shortening the closed loop γE 0 in the metric g4,E .
The decrease in length of the closed orbits then leads to an increase of the topological entropy, proving our claim. We will now show that (89) is finite. This is not immediately clear since as E → ∞, the Gaussian curvature KE concentrates near the singularities, so that the geodesic flow becomes singular there. On the other hand, using the relation htop (814,E ) = lim
L→∞
1 ln(NE (L)), L
NE (L) being the number of closed geodesics of length not exceeding L, we are done if we can show the existence of a c ≥ 1 with lE (γE ) ≥ lE 0 (γE 0 )/c
(90)
0 ) for all closed geodesics γE 0 and all energies E ≥ E 0 . Then with γE := HE 0 ,E (γE NE (L) ≤ NE 0 (c · L) so that htop (814,E ) ≤ c · htop (814,E 0 ). To estimate the length of the geodesics, we consider ε-neighbourhoods of the singularities with Euclidean radius ε = 1/E (see also [26], Chap. 5). Since the singularities of V are Coulombic, outside these neighbourhoods the conformal factor of the Jacobi metric is uniformly bounded by
1 ≤ 1 − V (q)/E ≤ C(E 0 ) for all energies E ≥ E 0 .
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For E 0 large the 1/E-neighbourhoods, E ≥ E 0 , are not intersecting each other, and by the above remark there exists a E-independent minimal length Lmin of a geodesic segment which connects two different disks. The boundaries of these disks are geodesically convex. Because of the √ negative Gaussian curvature each geodesic entering a disk has a maximal length of π · C(E 0 )/E inside that disk (it must be shorter than half the circumference). Moreover, after leaving a disk D, it cannot reenter the same disk after a geodesic segment which is homotopic to one of the circle segments of ∂D between the points x1 , x2 ∈ ∂D of leave and re-entrance. This is a consequence of the Gauss-Bonnet theorem, the negativity of KE and the geodesic convexity of ∂D. Thus the length lE (γE ) of a closed geodesic γE which has k encounters with a disk is lE (γE ) ≥ k · Lmin . On the other hand the length lE 0 (γE 0 ) of the homotopic closed geodesic γE 0 is p lE 0 (γE 0 ) ≤ k C(E 0 ) · (Lmin + π/E 0 ), since this is an upper bound for the length lE 0 (γ) of the loop γ which coincides outside the disks with γE 0 but whose points of entrance and departure of the disks D are connected along the shorter circle segment of ∂D. Thus √
lE (γE ) ≥ lE 0 (γE 0 )/c 0
with c := C(E 0 ) · (1 + π/(E Lmin )), showing (90). So the topological entropy of the geodesic flow 8t4,E has a high energy limit h∞ top . In that limit the length of the geodesic segments between consecutive encounters of disks is given by the Euclidean distance between the two corresponding points in the set S of singularities. So h∞ top depends only on the geometry of S. ˆ 1E ) to that limit. We now relate htop (8 √The proportion of the total time spent by the orbits of that flow inside disks of radius 1/ E goes to zero as E → ∞, since the speed inside that disk is larger than the speed outside, multiplied by a suitable E-independent constant C, 0 < C ≤ 1. ˆ tE covered n times by a closed geodesic The period T (γ˜ E ) of a closed orbit γ˜ E of 8 ˆ tE is related to the length lE (γE ) by the reparametrization formula flow line γE of 8 Z T (γE ) 1 1 · T (γ˜ E ) = n · lE (γE ) · √ (1 − V (q(t))/E)−1 dt, (91) 2E T (γE ) 0 √ which takes the speed 2(E − V (q)) and the geodesic length element p 1 − V (q)/E into account. The disks shrink to zero, and for the region outside√the disks the integrand goes to one as E → ∞. Thus (91) is asymptotic to n · lE (γE )/ 2E. ˆ E by 64,E has finite multiplicity (namely 8, a factor of two Since the covering of 6 coming from the double branched covering of 5M4 ,T2 : M4 → T2 in (86) and a factor of four coming from 5T2 ,T : T2 → T), n is bounded and we obtain the statement of the theorem. 1
ˆ ) is continuously differentiable in Remark 5.10. The measure theoretic entropy hµˆ E (8 the energy E > Eth . This can be derived from a result by Knieper and Weiss [31] for geodesic flows of negative curvature.
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5.2. The quantum motion (one particle). Now we describe the quantum motion and its semiclassical limit as Planck’s constant ~ & 0. So we analyze the Hamiltonian operator Hˆ ~ := − 21 ~2 1 + V
(92)
on L2 (Rn ), with an L periodic potential V . In n ≥ 2 dimensions H~ is self-adjoint, and the spectrum of H~ is known to be . purely absolutely continuous and bounded below, if V ∈ Ln−1+ε loc This condition is also met by the Coulomb potentials, although they are so singular in two dimensions that H~ is only defined as a quadratic form, see [28]. Using the dual basis `∗1 , . . . , `∗n ∈ Rn,∗ , (`∗i , `j ) = 2πδi,j
(93)
spanning the dual lattice L∗ in momentum space, we define the dual torus T∗ := Rn,∗ /L∗ (a fundamental domain of T∗ adapted to the symmetries of V is called the Brillouin zone). Denoting by dk := dn k = dk1 · · · dkn the elementary volume in cartesian coordinates and by |T∗ | = det(`∗1 , . . . , `∗n ) the volume of the Brillouin zone, the unitary transformation Z ⊕ dk 2 n ˆ (94) L2 (T) ∗ , U : L (R ) → H := |T | T∗ which on wave functions f ∈ S(Rn ) of rapid decrease is given by X exp (−ik · (q + `)) f (q + `) , (Uˆ f )k (q) = `∈L
defines the Bloch decomposition. The Hamiltonian is transformed into a direct integral Z ⊕ dk −1 ˆ ˆ ˆ Hˆ ~ (k) ∗ U H~ U = |T | T∗ of Hamiltonians Hˆ ~ (k) = 21 ~2 (−i∇ + k)2 + V on L2 (T). The operators Hˆ ~ (k) have compact resolvent and spectrum {E~,l (k)}∞ l=1 . Now by this decomposition we have ballistic motion in the plane (see Asch and Knauf [4]): (ψ, q 2 (t)ψ) =: C > 0 t→∞ t2 lim
(95)
for ψ in the form domain of Hˆ ~ with (∇ψ, ∇ψ) + (qψ, qψ) < ∞. This is qualitatively different from the classical diffusive behaviour (88). On the other hand the constant C for the squared speed of the ballistic motion is given by ∞ Z X −1 ~ ∇E~,l (k) 2 dµψ (l, k), C= l=1
T∗
µψ being the spectral measure of the wave function ψ. The gradient ∇E~,l of the band function appearing in the above expression is physically interpreted as the group velocity. In [28] it was shown for Coulombic periodic potentials that
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Proposition 5.11. Above Eth in each compact energy interval [E1 , E2 ] and for each ε > 0 the number of bands with −1 ~ ∇E~,n < ε is of density one as ~ & 0. This can be interpreted in the sense that due to quantum chaotic effects the typical wave function is ballistic but very slow in the semiclassical limit. 2 For the quantum free particle with Hamiltonian Hˆ ~ := − ~2 1 the ballistic constant C equals 2(ψ, Hˆ ~ ψ), independent of the value of Planck’s constant ~. In both the free and the potential case, the CNT entropy is undefined, as there is no normal state on B(L2 (R2 )) invariant under the time evolution Uˆ t := exp(−iHˆ ~ t/~). On the other hand, by Proposition 3.4, the CNT entropy for the operators Hˆ ~ (k), which generate the motion on the configuration torus Rn /L, equals zero, given any density matrix describing the time-invariant state. 5.3. The classical electron gas. We consider now the ergodic properties of the system of infinitely many classical non-interacting electrons in the crystal. Of course by our noninteraction assumption the dynamics of the infinite system is determined by the dynamics of a single electron. However, ergodic properties involve dynamics and measure. A third aspect comes in when we discuss entropy, namely the question of the kinds of observation which we are able to make. Due to the possibility of an arbitrarily large particle number within a given compact region of the one-particle phase space P , the space P describing a many-particle system is non-compact for usual choices of the topology, even if P is compact. Whereas Bowen’s Definition 5.3 of topological entropy needs no compactness of the phase space, some metric on it must be chosen. Since we know no physically natural choice of such a metric on P , we restrict ourselves to a discussion of measure-theoretical entropy. Although the Liouville measure of a phase space of finitely many particles is infinite, the state of the infinite ensemble will be described by a probability measure. That measure describes a Poisson distribution of particles on the one-particle phase space P w.r.t. the invariant measure µ on P which one considers (as an example, µ may be Liouville measure on one energy shell or the measure of a Boltzmann distribution for a given temperature). Now it turns out that even an ideal gas, i.e. an ensemble of free particles, is a Bernoulli system w.r.t. time evolution and has infinite total entropy. The reason for these optimal stochastic properties is the fact that the particles move away to infinity and are thus not observed any longer. These questions are analyzed in the articles [18] and [1] by Aizenman, Goldstein and Lebowitz. As a consequence of Ornstein’s theorem (see Walters [50], Thm. 4.28) all infinite entropy Bernoulli systems are isomorphic. So from the measure-theoretic point of view the ideal gas cannot be discerned from, say, the classical electron gas in a Coulombic periodic potential whose particle energies are above the energy threshold Eth . On the other hand, unlike for the ideal gas, in our example of Coulombic potentials there exists a local mechanism leading to positive entropy for the motion of one particle. One would like to have a measure-theoretical notion which reflects that distinction between the two dynamical systems. A clue to that problem is the observation by Goldstein in [17] that systems of statistical mechanics typically have a larger symmetry group than finite particle systems,
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since in addition to time evolution they are invariant under a group of space translations. This enabled Goldstein to define what he called space-time entropy (entropy density) and other related notions. So the general setting starts with an Abelian group G acting on the measure space (P, 6, µ) of a single particle by automorphisms 8g : P → P , g ∈ G. The measure space is assumed to be totally σ-finite and non-atomic. If µ is a probability measure (P, 6, µ, 8) is called a generalized dynamical system. Such systems are discussed by Conze in [14]. We first describe the Poisson construction, the classical analog of second quantization. Definition 5.12. The Poisson system (P , 6, µ, 8) built over (P, 6, µ, 8) has phase space P which is the set of countable subsets of P . The σ-algebra 6 of P is the smallest σ-algebra on P w.r.t. which for all A ∈ 6 the “particle number” functions NA : P → N ∪ {∞},
NA (x) := |A ∩ x|,
are measurable. Its probability measure µ is fixed by setting µ
x ∈ P | NA (x) = m
:= exp(−µ(A)) ·
µ(A)m . m!
g
For g ∈ G the automorphism 8 : P → P is given by g
8 (x) := {8g (x) | x ∈ x}. In the case of L-periodic potentials V we take • the one-particle phase space P := T∗ Rnq if V is smooth. In the Coulombic case P will be the four-dimensional manifold defined in (77). • In both cases the one-particle σ-algebra 6 consists of the Borel subsets of the phase space P . 1 ω ∧ . . . ∧ ω on P , • Measures: Starting from the phase space volume form µ := n! we will consider the Liouville measures µE induced by µ which have support on the energy shells 6E ⊂ P . Moreover, we will use the Boltzmann measures νβ which are given by dνβ := exp(−β · H)dµ for inverse temperatures β > 0. In the Coulombic case dνβ := θ(H − Eth ) exp(−β · H)dµ, θ being the Heaviside function. The main reason why we give zero measure to the phase space region below the threshold energy Eth is that otherwise the Boltzmann measures for β > 0 would be concentrated at the negative energies where the particles are bounded by the Coulombic singularities. A partial physical justification for this cutoff is given by the fact that quantum mechanically the Pauli exclusion principle forbids the electrons to be in those low-energy states which are already occupied by electrons bounded to the nuclei of the crystal. In all cases we may adjust the particle density by multiplying the one-particle measure by a constant.
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• The group actions of interest are the discretized flow 8t , t ∈ Z, on P generated by the Hamiltonian function H, and the Abelian space-time translations 4t,` : P → P,
4t,` := 8t ◦ 9`
(t ∈ R, ` ∈ L)
with the lattice translations 9` on P generated by the maps q 7→ q + ` on the configuration plane M , see (80). The notion of the measure-theoretic entropy of a generalized dynamical system is a direct generalization of the case of one iterated map (see [17]): Definition 5.13. The entropy hµ (8) of a generalized dynamical system (P, 6, µ, 8) with the action 8g : P → P of an Abelian group G with finitely many generators g1 , . . . , gn ∈ G is given by hµ (8) := sup hµ (8, X), X
where the supremum is w.r.t. all finite partitions X of P . Here W Hµ ( kgk≤k 8g (X)) hµ (8, X) := lim k→∞ |{g ∈ G | kgk ≤ k}| isP the µ-entropy P for the partition X, and the norm on G is n n k i=1 ki gi k := i=1 |ki |. The definition does not depend on the choice of the generators. We will also discuss the information gain by local measurements in a region B ⊂ Rnq of configuration space. Denoting the phase space over this region by B := {(p, q) ∈ P | q ∈ B}, we thus consider the dynamical entropy of B given by hµ,B (8) := sup hµ (8, A),
(96)
A⊂A(B)
where the supremum is taken over all finite-dimensional subalgebras A of the algebra A(B) of classical observables over B. Note that by the inclusion A(B1 ) ⊂ A(B2 ) for B1 ⊂ B2 this entropy is monotonic: hµ,B1 (8) ≤ hµ,B2 (8)
(B1 ⊂ B2 ).
(97)
For simplicity we state our result for a compact configuration space region B ⊂ Rnq which we assume to be an n-manifold with boundary (or a finite transversal intersection B = B1 ∩ . . . ∩ Bl of such manifolds). However (97) immediately implies statements about unbounded B, in particular B = Rnq . Theorem 5.14. If the L-periodic potential V is smooth (V ∈ C 3n (Rn , R) suffices), then above some threshold energy Eth the classical microcanonical subregion entropy is infinite hµE ,B (8) = ∞
(E > Eth ).
(98)
The same holds for the canonical subregion entropy for inverse temperature β > 0: hνβ ,B (8) = ∞. On the other hand
(99)
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hµE ,B (8) < ∞
(100)
for low energies E > Vmin for which the components of 6E are compact. The microcanonical space-time entropy equals the density of particles multiplied by the time-entropy per particle: hµE (4) = µˆ E (Pˆ ) · hµE (8).
(101)
Similarly the canonical space-time entropy meets hνβ (4) = νˆ β (Pˆ ) · hν β (8)
(102)
(thus in particular both quantities are finite). Proof. Let v¯ : P → Rn be the asymptotic velocity of the classical particle Z 1 T v(x ¯ 0 ) := lim p(t, x0 )dt. T →∞ T 0 By Birkhoff’s ergodic theorem (applied to the phase space Pˆ over the configuration torus T) this quantity exists µ-almost ˆ everywhere for an invariant measure µ, ˆ (and we set v(x ¯ 0 ) := 0 otherwise for definiteness). See [4] for an extensive discussion. We consider the restriction of this classical observable to the energy shell 6E . By KAM theory we know that for energies E > Eth above some threshold there exists a ˆ E of positive Liouville measure set Bˆ E ⊂ 6 kE := µˆ E (Bˆ E ) > 0 composed of invariant tori. On each of these KAM tori the asymptotic velocity v¯ is ˆ E the deviation of the configuration constant. By compactness of the energy shell 6 space trajectory from a straight line in the direction v¯ is uniformly bounded by some constant CE > 0: ¯ 0 ) · tk ≤ CE kq(t, x0 ) − q0 − v(x
(x0 = (p0 , q0 ) ∈ BE , t ∈ R).
Moreover for initial conditions on these tori the motion is ballistic: ¯ > 0. vmin := inf |v(x)| x∈BE
Using the asymptotic velocity as an observable we partition the region BE ∩ B of BE which projects to the observation region B into r parts of equal size. Namely for arbitrary r ∈ N we can find a partition (U1 , . . . , Ur ) of the space Rnv¯ of asymptotic velocities such that the preimages (P1 , . . . , Pr ) ¯ ∈ Uj } Pj := {x ∈ BE ∩ B | v(x) have equal volume µE (P1 ) = . . . = µE (Pr ) =
kE >0 r
(103)
(this is actually possible since the image of νE |`BE is non-atomic, every invariant torus having its specific asymptotic velocity). This partition of BE ∩ B induces an infinite
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partition Q of the many-particle phase space P whose atoms Qm1 ,... ,mr , mj ∈ N0 , are of the form Qm1 ,... ,mr := x ∈ P | NP1 (x) = m1 , . . . , NPr (x) = mr . Observe the crucial fact that the partition ignores the number of particles which do not belong to the KAM component. For times t larger than diam(B) + CE +1 T := vmin ([x] denoting the integral part of x) 8t (BE ∩ B) ∩ B = ∅. Thus the partitions l
Q, 9(Q) , . . . , 9 (Q) , . . .
(104)
T
are independent for the time T evolution 9 := 8 : P → P . We will show that hµE ,B (9) = ∞ which then implies hµE ,B (8) = ∞, since in general hµ (81 , ξ) ≥ hµ (8T , ξ)/T for any partition ξ. By independence of the partitions (104) and by independence and equidistribution of the r events {NP1 (x) = m1 }, . . . , {NPr (x) = mr }, ˜ hµE (9, Q) = HµE (Q) = r · HµE (Q),
(105)
the partition Q˜ consisting of the atoms {NP1 (x) = m}, m ∈ N0 . By (103) these atoms have measure cm µE {x ∈ P | NP1 (x) = m} = exp(−c) m! with c := kE /r. Thus (105) diverges as r → ∞. Note that technically Definition 5.13 uses finite partitions. However, our infinite partitions can be approximated by finite ones (merging the atoms with large particle numbers), so that we obtain a proof of (98). The corresponding statement (99) for the canonical distribution is obtained along the same lines. Next we show that the microcanonical subregion entropy (100) is finite for low energies E, such that connected components of the energy shell 6E are compact. By the monotonicity property (97) we can estimate (100) from above by hµE ,B2 (8), where the compact configuration space region B2 ⊂ Rnq contains B but additionally contains the projections of full energy shell components. That is, we assume the energy shell region {(p, q) ∈ 6E | q ∈ B2 } t
to be flow-invariant. But this implies that the particle numbers NB2 (8 (x)) are tinvariant, too. Since the microcanonical measure µE ({(p, q) ∈ 6E | q ∈ B2 }) < ∞, that particle number has finite expectation
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Z NB2 (x)dµE (x) < ∞.
(106)
P
The total entropy of distinguishable independent particles equals the entropy per particle, multiplied by the expectation (106). But the dynamical entropy per particle is finite (it was first shown by Kushnirenko that a C 1 transformation on a compact manifold has finite entropy), so that hµE ,B2 (8) < ∞. Equations (101) and (102) follow from Thm. 6.8 of [17], since the systems (P , 6, µE , 8) and (P , 6, νβ , 8) meet the condition of that theorem, namely to be of periodic-σ-bounded type. Roughly speaking this means: 1. The one-particle system has a Zn lattice symmetry commuting with time evolution. 2. Although the particle speed |p| is unbounded in phase space P , it is bounded in each region H −1 ([Vmin , E]) ⊂ P , which exhausts P as the maximal energy E goes to infinity. Now we turn to the case of Coulombic potentials. The motion in these potentials is diffusive for high energies and thus qualitatively different from the ballistic case of smooth potentials discussed above. Proposition 5.15. For E > Eth and β > 0 the infinite particle systems (P , 6, µE , 4)
resp. (P , 6, νβ , 4)
are space-time K-systems with infinite (time) dynamical entropy hµE (8) = ∞ = hνβ (8)
(107)
and the (micro)canonical space-time entropies are positive: hµE (4) = µˆ E (Pˆ ) · hµE (8) > 0,
(108)
hνβ (4) = νˆ β (Pˆ ) · hν β (8) > 0.
(109)
Proof. Although the phase space Pˆ = P/L over an elementary configuration torus T = R2 /L is non-compact, the projected measures µˆ E and νˆ β on it are finite for all energies E ∈ R resp. inverse temperature β > 0. Moreover, although in the Coulombic case the maximal speed |p| of a particle is unbounded on the energy shells 6E , the maximal distance |q(1) − q(0)| traversed by that particle within one time unit is bounded. Namely, outside a ε-neighbourhood Sε ⊂ R2 of the singularity set S the maximal speed sup{|p| | (p, q) ∈ T∗ (R2 \ Sε ), H(p, q) = E} is finite, and for small ε the disks of Sε are disjoint so that trajectories with |q(1)−q(0)| > 2ε intersect R2 − Sε . Thus the system (P , 6, µE , 8) is of periodic-bounded type in the language of [17]. Furthermore, although the motion of the particle in the configuration plane R2 is not a K-flow (since the Liouville measures µE are not finite so that ergodic notions do not apply), it is closely related to a family of K-flows. Namely for r ∈ N we may consider the sublattice rL ⊂ L and periodize phase space w.r.t. that lattice in the way we did using L. rL Then for energies E > Eth the dynamical system (P rL , 6rL , µrL E , 8 ) is a Bernoulli system which implies the K-property. So (P , 6, µE , 8) is of periodic Ktype in the sense of [17], and Thm. 6.1 of [17] implies that (P , 6, µE , 4) is a space-time K-system.
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On the other hand, the Boltzmann measure νβ is not supported on a single invariant energy shell, so that the systems (P rL , 6rL , νβrL , 8rL ) cannot even be ergodic. But, given the partition of phase space P into energy shells and the fact that for E > Eth rL the dynamical systems (P rL , 6rL , µrL E , 8 ) are Bernoulli, (P , 6, νβ , 8) is of periodic σ − K-type (see [17]). The K property for the space-time system (P , 6, νβ , 4) follows immediately from Theorem 6.3 of [17], since (P , 6, νβ , 8) is of σ − K type. The formulas (108), (109) for the space-time entropies are consequences of Theorem 6.8 of [17]. Since by the K property these are strictly positive, the dynamical entropies (107) must be infinite. Remark 5.16. Whereas the ideal gas (i.e. the Poisson system built over the free flow (85) with the canonical distribution νβ ) is a K system w.r.t. to time, it is not a space-time K system [17]. Although its time entropies are infinite, the entropy densities are zero in that case. So from the measure-theoretical point of view the ideal gas and the classical gas in a periodic array of attracting Coulombic potentials are really different. It is also possible that the Coulomb case is qualitatively different from the case of a smooth potential as described in Prop. 5.14 in that the finite region entropy could be finite for large energies. An argument in favour of that idea is that the motion on long time scales resembles the random walk so that each particle should recur to B with probability one (as we are in dimension two). Conversely, because of the finite density of particles, the number of particles visiting B for the first time should go to zero as time goes to infinity. So one could argue that in the T → ∞ limit the dynamical one-particle mechanism of entropy production could be dominant over the fresh information about the phase space position of incoming particles. 5.4. The quantum electron gas. As shown in Subsect. 5.3, the (time) dynamical entropy of noninteracting classical electrons has a density which may well be positive. Our conjecture is that the quantum (time) dynamical entropy of electrons within some bounded configuration space region B is not proportional to the volume but to the surface of B. Thus in n = 1 dimension the total quantum dynamical entropy of the gas, being a finite positive quantity, has density zero, whereas for n > 1 the entropy of the scaled region λ · B scales like λn−1 as λ → ∞, see (134). The general setting will be as follows: we will second-quantize the Schr¨odinger operator Hˆ ~ = − 21 ~2 1 + V with the L-periodic potential V ∈ Ln−1+ε (Rn ) on the loc 2 n Hilbert space L (R ). In addition to the spectral properties discussed in Subsect. 5.2, Wilcox [52] showed that the Bloch generalized eigenfunctions ψm,k ∈ L∞ (Rn ), m ∈ N, k ∈ T∗ , Hˆ ~ ψm,k = Em (k)ψm,k , ψm,k (x + `) = exp(ik · `)ψm,k (x),
(` ∈ L),
(110) (111)
can be chosen in such a way that up to zero measure sets in T∗ they are analytic functions of the quasi-momentum vector k, and have L2 -norm one on each elementary cell. The bands are assumed to be enumerated so that Em+1 (k) ≥ Em (k). The functions um,k ∈ L∞ (Rn ),
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um,k (x) := ψm,k (x) · exp(−ik · x)
(112)
are L-periodic and thus may be considered as functions in L2 (T). We shall consider the CAR algebra A(Rn ) generated by creation and annihilation operators aˆ † (f ), aˆ (g) of single-particle Fermionic states f , g ∈ L2 (Rn ) satisfying aˆ † (f ) aˆ (g) + aˆ (g) aˆ † (f ) = hg, f i, †
†
†
†
aˆ (f ) aˆ (g) + aˆ (g) aˆ (f ) = 0, aˆ (f ) aˆ (g) + aˆ (g) aˆ (f ) = 0.
(113) (114) (115)
The observables measuring the electrons in a region 3 ⊂ Rn are elements of the CAR subalgebra of the whole CAR algebra A(Rn ) based on the Hilbert space L2 (3) which we denote by A(3). In a natural way A(31 ) ⊂ A(32 ) if 31 ⊂ 32 . Though the general aim is the study of the entropic properties of the quasi-free automorphism ˆ t αH (116) aˆ † (f ) = aˆ † e−i/~H~ t f , in this paper we shall limit ourselves, as a preliminary step, to the study of the CNTentropy of the group of translations by multiples of a lattice vector. For an arbitrary vector v ∈ Rn we denote by Vˆ : L2 (Rn ) → L2 (Rn ), the unitary map Vˆ f (q) := f (q − v), and by αv the Bogoliubov transformation αVˆ on A(Rn ): αv (ˆa(f )) := aˆ (Vˆ f ),
(f ∈ L2 (Rn )).
If v is a lattice vector, then αv leaves the state ωρ (introduced below) invariant. So αv incorporates the shift of the lattice gas by v. We are inconsistent to use in one case the generator H and in the other case the unitary Vˆ (or rather, the velocity vector v) as an index. However, the use will be clear from the context. In sight of our later application to the transformation αH we are interested in αv invariant states on A(Rn ) which are also αH -invariant. Now by Theorem XIII.98 of Reed and Simon we can write a wave function f ∈ L2 (Rn ) as the superposition ∞ Z X dk f˜(m, k)ψm,k (x) (117) f (x) = 1 ∗ m=1 T |T∗ | 2 of Bloch functions ψm,k with coefficients f˜(m, k) := um,k , (Uˆ f )k L2 (T) , Uˆ being defined in (94). Thus suitable densities ρ : N × T∗ → R+ of Bloch functions define quasifree states ω of non-interacting electrons which are fixed by their two-point functions ∞ Z X f˜(m, k)g(m, ˜ k)ρ(m, k)dk. (118) ω(ˆa† (f )ˆa(g)) := m=1
T∗
Assumption 5.17. We assume nondegeneracy of the bands and decay and smoothness of the density ρ: ρ ∈ C n+1 (N × T∗ , (0, 1)),
(119)
and for some α > 1 + 1/n, max ∂ki11 , . . . , ∂kinn ρ1/2 (m, k) = O(m−α )
k∈T∗
(i1 + . . . + in ≤ n + 1).
(120)
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In particular we may consider the inverse temperature β state ωρβ , ρβ (m, k) := (1 + exp(β(Em (k) − µ)))−1 ,
(121)
whose chemical potential µ ∈ R is determined by the prescribed density D > 0 of particles in configuration space. Under the nondegeneracy assumption k 7→ Em (k) is a real-analytic function, and it even decays exponentially. The nondegeneracy assumption is imposed in order to avoid technicalities alien to our question. One way to relax it is to consider ρ as a function on the Fermi surface and to formulate the assumptions accordingly. The Bogoliubov automorphism αv leaves the state ω invariant, since the latter derives from the One-Particle Hamiltonian H which commutes with Vˆ . It proves convenient to introduce the Wannier functions φm,` ∈ L2 (Rn ) defined by Z dk φm,` := (122) e−ik·` ψm,k ∗ , (m ∈ N, ` ∈ L), |T | ∗ T (see Ashcroft and Mermin [5]). The index m refers to the band. Using that X ei(k1 =k2 )` = |T∗ | δ(k1 − k2 ),
(123)
`∈L
we obtain the inversion formula ψm,k =
X
eik` φm,` .
(124)
`∈L
Furthermore, from the completeness of the Bloch functions ψm,k one concludes that the Wannier functions form an orthonormal basis of L2 (Rn ). By (112) the Wannier functions of the mth band are translates of each other: φm,` (q + `) = φm,0 (q)
(q ∈ Rn , ` ∈ L).
Thus they are somewhat localized in configuration space Rn near l. Comparing the defining relation of the Wannier functions with the decomposition (117), one sees that their expectation (125) ω aˆ † (φm1 ,`1 )ˆa(φm2 ,`2 ) = δm1 ,m2 ρ˜m1 (`1 − `2 ), where
Z ρ˜m (`) :=
ρ(m, k) eik·` T∗
dk |T∗ |
(m ∈ N, ` ∈ L)
(126)
is proportional to the `th Fourier coefficient of the density on the mth band. As all quasi-free states, ω in (118) has its two-point functions determined by a single-particle operator 0 ≤ Aˆ < 1ˆl: ˆ i. ω(ˆa† (f )ˆa(g)) = hg, Af In our case, by means of the Wannier functions, Aˆ reads
(127)
Quantum Chaos and Dynamical Entropy
Aˆ =
655
∞ X X
ρ˜m (`1 − `2 ) · |φm,`1 ihφm,`2 |.
(128)
m=1 `1 ,`2 ∈L
We now consider a bounded configuration space region B ∈ Rn and assume that B is a n-dimensional submanifold with boundary (e.g. a ball) or a finite intersection B = B 1 ∩ . . . ∩ Bl
(129)
of such regions Bi whose boundaries ∂Bi are intersecting transversally (like a cube). Then we introduce (using Def. (24)) a measure of the information loss per time unit within B. Definition 5.18. Let ω : A → C be a state invariant under an automorphism α : A → A. Then we call sup hω (α, A)
hω,B (α) :=
(130)
A⊂A(B)
the volume-B (or local) quantum dynamical entropy of α w.r.t. ω Enlarging B by a factor λ and considering the scaling of hω,λB (αv ) with λ we have the following Theorem 5.19. If B ⊂ Rn is a finite, convex volume, then Z 1 |v · dn| · s(ω), lim n−1 hω,λB (αv ) = 21 λ→∞ λ ∂B
(131)
where s(ω) :=
∞ Z X m=1
T∗
dk ηF (ρ(m, k)) |T∗ |
(132)
with the fermionic function ηF : [0, 1] → R
given by ηF (x) := η(x) + η(1 − x)
is the mean entropy of ω. More generally, for arbitrary (not necessarily convex) regions B of the form (129) lim
λ→∞
1 hω,λB (αv ) = |Pr v (B)| · |v| · s(ω), λn−1
(133)
where Pr v (B) denotes the shadow of the volume B with respect to projection in the v direction. Proof. Equation (131) reduces to (133) if B is convex, since then each line of projection through B (parallel to v) is either tangential to the surface ∂B and thus does not contribute to both expressions (see Remark 5.28 below), or intersects ∂B in exactly two points. The proof of (133) will be split into an upper bound (Prop. 5.30) and a lower bound (Prop. 5.35).
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F. Benatti, T. Hudetz, A. Knauf
Remarks 5.20. 1. We plan to prove in a second paper that the local CNT-entropy for the non-interacting electron gas in a periodic potential, too, scales with the boundary: 1
hω,λB (αH ) λ→∞ λn−1 Z ∞ Z X ∇k Em (k) 1 =2 ~ ∗ T ∂B m=1 lim
dk · dn ηF (ρ(m, k)) ∗ . |T |
(134)
To see how our theorem on the shift is related to this question, consider the Appendix, where we introduce a family of Hamiltonians Hˆ ~r (defined in (196)) which approximate Hˆ ~ in the r → ∞ limit. These Hamiltonians act as shifts on the subbands. 2. It should be inessential for the result (but simplify the proof) that we increase the size of B without changing its form. What is certainly essential is a van Hove type bound on the expanding region which ensures that ultimately most points are far from the boundary (see Def. 5.24 below). On the other hand we do not believe that a van Hove condition is sufficient. As an ˙ 2 ⊂ Rn , where B1 is n-dimensional example one may consider a region B := B1 ∪B (say, a ball) and B2 is of dimension n − 1, (say, a disk). Now λ 7→ λB forms a van Hove sequence, since in that limit the volume of λB1 dominates the surface of λB2 . However, the r.h.s. of (134) is enlarged by the existence of B2 . Although there is no conceptual contradiction (particles can in principle also be detected at B2 ), we do believe that due to a large entropy defect that formula becomes wrong for this B. Our interpretation of (134) is that the contribution of the individual electrons to the local CNT entropy is negligible if λ is large. Then the electrons typically spend a long time within λB. Thus an initial quasiclassical behaviour (ascertained by Egorov’s Theorem) with an eventually exponential separation rate is followed by a spreading of the wave packet and a quasifree ballistic behaviour. Hence for large λ the only remaining entropy contribution comes from incoming particles measured for the first time near the boundary of λB. We work out this interpretation in the simplest case of a vanishing potential: Example (The free electron gas). In the low density limit (and in the semiclassical limit ~ & 0) µ → −∞ in (121) so that the Fermi–Dirac distribution converges to the Boltzmann distribution. For example, for the 2-dim. free electron gas with mass m (and two spin states) one has µ = ln(exp(π D~2 β/m) − 1)/β, 2
since the nonrelativistic dispersion relation E(k) = (~k) 2m leads to the density Z ρβ (k)d2 k/(4π 2 ) = D. from the equation R ∞ More generally, for the n-dimensional free electron gas one obtains−n exp(−β(E−µ))d(E)dE = D with density of states d(E) = (2π~) ·|B n | n E n/2−1 0 that βµ ∼ ln cD(2π~)n (β/m)n/2
Quantum Chaos and Dynamical Entropy
657
in the low density (resp. ~) limit (c being a numerical constant depending on the dimension d). Then in ρ & 0 and only the ρ ln(ρ)-term gives a contribution to (134) , which is asymptotic to r T |∂B| ln cD−1 (2π~)−n (T m)n/2 . D 2πm q T is the number of particles per unit surface The interpretation is as follows: D 2πm element and time unit entering (and leaving) the region λB of surface area λn−1 |∂B| so that the total number per time unit equals r N := λ
n−1
|∂B| · D
T . 2πm q
These incoming particles are detected in a mean distance λB and thus may be in one of q ≈
T n−1 |∂B| · 2πm λ (2π~)n
T 2πm
from the boundary of
(T m)n/2
phase space cells. The logarithm of that expression gives the contribution of each electron to the quantum dynamical entropy. But the particles are indistinguishable, so that one has to subtract from the entropy the term N ln N . Of course the interpretation of formula (134) for general states and non-constant periodic potentials is exactly analogous to the above one for the free electron gas and the Fermi–Dirac distribution. 5.4.1. One-particle estimates. We first prove that the state ω is locally normal. That is, its restriction ω |`3 to any finite volume 3 is a state with a density matrix, and this belongs to the local algebra A3 with eigenvectors in the Fock space H3 generated by A3 acting on the Fock vacuum [38]–[39]. In the following we shall basically use restrictions of the global state ω to finite volumes 3 ⊂ Rn . It is thus convenient to introduce at this point some useful notations and comments. Remarks 5.21. 1. By the restriction of the state ω to a finite volume 3 ⊂ Rn we shall mean taking expectation (with respect to ω) of observables from the CAR algebra A(3) localized within 3. The assumed quasi-freeness of ω relates it to the restriction Aˆ 3 := χ3 Aˆ χ3 , ˆ where χ3 is the characteristic function of the volume of the single-particle operator A, 3. This means that Aˆ 3 will be considered to act on the square-summable functions supported within 3.
658
F. Benatti, T. Hudetz, A. Knauf
2. Since 0 ≤ Aˆ < 1ˆl, the single-particle Bˆ =
Aˆ ˆ1l − Aˆ
is positive. We shall consider the finite-volume restrictions Bˆ 3 := χ3 Bˆ χ3 and also Bˆ 3 :=
Aˆ 3 . ˆ1l − Aˆ 3
The latter operator Bˆ 3 is quite different from the restriction Bˆ 3 . Indeed, we have [39]: Bˆ 3 = χ3
p Bˆ
p 1 p p Bˆ χ3 . 1ˆl + Bˆ χRn \3 Bˆ
Lemma 5.22. Under the above assumptions on ρ the state ω is locally normal, and for bounded regions 3 the operator Bˆ 3 of Remark 5.21.2 meets the estimate Tr (Bˆ 3 ) = O(|3|). The (unnormalized!) energy expectation XZ dn k ρ(m, k)Em (k) ∗ |T | T∗
(135)
(136)
m∈N
is finite. Proof. Local normality follows from the estimate Tr (Aˆ 3 ) < ∞ for Aˆ 3 in Remark 5.21.1 ˆ 1ˆl − A) ˆ ≥ Aˆ so that Bˆ 3 ≥ Aˆ 3 and we need only show (135). (see [12]). But Bˆ = A/( By our assumptions on ρ we conclude that ρmax :=
sup
m∈N,k∈T∗
ρ(m, k) < 1.
Indeed by (120) a density ρ(m, k) = 1 can only occur on finitely many bands m, by (119) ρ(m, k) < 1, ρ is continuous and T∗ is compact. From (118) and (127) we see that the unitary Bloch transformation defined in (94) conjugates Aˆ with multiplication by ρ. So in fact Aˆ ≤ ρmax 1ˆl which implies ˆ Bˆ ≤ A/(1 − ρmax ).
(137)
Therefore we can equally well show that Tr (Aˆ 3 ) = O(|3|) to prove the a priori stronger (135). After expressing the one-particle operator Aˆ in terms of the Bloch functions, by means of (122), (123) and (124), we get the kernel
Quantum Chaos and Dynamical Entropy
A(x, y) = =
∞ Z X m=1 ∞ Z X m=1
659
dk ρ(m, k)ψ¯ m,k (x)ψm,k (y) ∗ |T | T∗ T∗
ρ(m, k)u¯ m,k (x)um,k (y) exp(ik(y − x))
dk , |T∗ |
(the last form being manifestly L-invariant). By an L∞ -bound on the Wannier functions (see [52]), the Weyl estimate Em (k) ∼ C · mn/2
(m → ∞)
and the decay assumption (120) for ρ that expression converges. So Z A(x, x)dn x = O(|3|) Tr (Aˆ 3 ) =
(138)
3
(actually one sees from the L2 -normalization of the Wannier functions ψm,k on Rn /L that ∞ Z X dk ρ(m, k) ∗ + O(|(∂3)ε |), Tr (Aˆ 3 ) = |3| |T | ∗ T m=1
where (∂3)ε is a thickening of the boundary of 3). Finiteness of (136) follows from the same argument, since by Assumption (120) α − 1 is strictly larger than the exponent n/2 in the Weyl estimate. Lemma 5.23. There exists a constant c > 0 such that for all bounded regions 3 ⊂ Rn with finite surface area |∂3| (defined as in (168)) Tr χ3 (Bˆ 1/2 − (Bˆ 3 )1/2 )2 χ3 ≤ c|∂3|, with Bˆ 3 as in 5.21, 2. Proof. We use the abstract inequality Tr χ3 (Bˆ 1/2 − (Bˆ 3 )1/2 )2 χ3 ≤ ˆ Rn \3 Bχ ˆ 3 + 2Tr χ3 Bˆ 1/2 χRn \3 Bˆ 1/2 χ3 Tr χ3 Bχ
(139)
from the proof of Lemma 5.2 in [39]. Here we need decay estimates for the kernel of ˆ Similar to A, ˆ the kernel of Bˆ can be written in the form B. ∞ Z X dk ρB (m, k)ψ¯ m,k (x)ψm,k (y) ∗ , B(x, y) = |T | ∗ T m=1
with the density ρB = ρ/(1 − ρ). By functional calculus, the representation of the kernel of B 1/2 is similar, but with 1/2 density ρB . Note that by 0 < ρ < 1, ρB > 0, too, so that no smoothness is lost when taking the 1/2 1/2 root ρB . Moreover the decay rate for ρB is the same as for ρ1/2 :
660
F. Benatti, T. Hudetz, A. Knauf
max ∂ki11 , . . . , ∂kinn ρB (m, k) = O(m−α ) 1/2
k∈T∗
(i1 + . . . + in ≤ n + 1)
(140)
and double as much for ρB . Our nondegeneracy assumption implies that the phases of the Bloch functions can be chosen in a way that for each x ∈ Rn the maps T∗ → C,
k 7→ ψm,k (x)
(m ∈ N)
are analytic. Thus the Fourier coefficients Z dk exp(+ik · `)ψ¯ m,k (x)ψm,k (y) p Fx,y,m (`) := |T∗ | T∗
(` ∈ L)
decay faster than any power of k`k. In fact, Pn by writing the vector ` ∈ T in terms of the direct lattice basis vectors `i as ` = i=1 li `i and k ∈ T∗ in terms of the dual lattice basis vectors `∗j (see (93)) as Pn k = j=1 sj `∗j we have, using partial integration and dual lattice periodicity Z 1 i ∂ 2π i Pn si li ¯ i=1 ds1 · · · dsn e ψm,k (x)ψm,k (y) l Fx,y,m (`) = − 2π 0 ∂k j Z 1 Pn i i ∂ ¯ m,k (x)ψm,k (y) . ds1 · · · dsn e2π i i=1 s l ψ =i ∂k j 0 j
This equality can be generalized to powers of the components li of ` ∈ L and the decay properties result from application of the Riemann–Lebesgue Lemma. The Fourier coefficients of ρB can be handled in an analogous manner, Z dn k , (141) ρB (m, k) exp(+ik · `) p ρ˜B,m (`) = |T∗ | T∗ that, due to the assumed decay properties of ρ, can be proved to obey the decay rate |ρ˜B,m (`)| ≤ Cm−α klk−n−1
(m ∈ N, ` ∈ L).
Now we assume w.l.o.g. (using L-translation invariance of B), that the first argument x of the kernel belongs to the fundamental domain spanned by the basis vectors `1 , . . . , `n , and write the second argument in the form y = y 0 + ` with ` ∈ L and y 0 , too, in the fundamental domain. Then ∞ Z X dk B(x, y) = ρB (m, k)ψ¯ m,k (x)ψm,k (y 0 ) exp(ik`) ∗ |T | ∗ T m=1
∞
1 XX = ∗ ρ˜B,m (` − `0 ) Fx,y0 ,m (`0 ), |T | 0 m=1 ` ∈L
since the Fourier transform converts the product into convolution. By the above decay properties and uniformity of estimates for x and y 0 in the compact fundamental domain, we obtain B(x, y) = O(kx − yk−n−1 )
(kx − yk → ∞)
(142)
Quantum Chaos and Dynamical Entropy
661
and a similar estimate for B 1/2 (x, y), using (140). The estimate of the r.h.s. of (139) is now performed for the first term, the second being of the same kind. First we need the following geometrical statement on regions 3 ⊂ Rn , using our definition 3− (a) = {q ∈ 3 | dist(q, Rn − 3) ≥ a}: The function a 7→ 3− (a) (a ≥ 0) is convex, and d2 3− (a) /da2 ≥ 0 in the smooth case. This need only be proved at a=0, since (3− (a1 ))− (a2 ) = 3− (a1 + a2 ). Here for 3 being ∂-manifolds, one has by Steiner’s formula [19] Z d2 3− (a) = (k1 (x) + . . . + kn (x))dσ(x), da2 ∂3 a=0
where k1 , . . . kn are the principal curvatures of the boundary manifold. This integral over the mean curvature of the boundary manifold is known to be bounded below by the surface area of S n−1 . Thus in this case our assertion holds true. Using this uniform bound, we may prove it in the general case by an approximation argument. Now we use the convexity property to prove that ˆ Rn \3 Bχ ˆ 3 ≤ C |∂3| Tr χ3 Bχ for
Z
!
Z
∞
n
2
B (x, y)d y dr.
C := sup
x∈Rn
ky−xk≥r
0
Because of periodicity we need to take the sup only for x within the compact fundamental domain, so that the constant C is finite. ˆ Rn \3 Bχ ˆ 3 Tr χ3 Bχ ! Z Z B 2 (x, y)dn y dn x = Z
Rn \3
3
Z
∞
= 3
0
Z
∞
≤− 0
Z
∞
≤− 0
δ dist(x, R \ 3) − a n
d 3− (a) · da d 3− (r) dr
Z
r=0
n
2
B (x, y)d y Rn \3
d x
da
!
ky−xk≥a
!
Z
·
! n
B 2 (x, y)dn y da
sup x∈3
!
Z
2
B (x, y)d y da
sup
x∈Rn
n
ky−xk≥a
= C |∂3| . In the last inequality we used the above convexity property (this was also implicitly used d|3− (a)| before since it allowed us to integrate over da which is then defined for almost all a). The last equality is another application of Steiner’s formula, which yields
662
F. Benatti, T. Hudetz, A. Knauf
d 3− (r) − dr
= |∂3| .
r=0
5.4.2. Mean entropy. In order to calculate the local (finite-volume) dynamical entropy of a lattice shift, we have to consider the mean entropy s(ω) (introduced in (132)) for the L-invariant quasifree state ω ≡ ωρ of the form (118). By (120) s(ω) < ∞,
(143)
since ηF (ρ) ≤ cε ρ1−ε . s(ω) can be interpreted as the entropy per unit volume of our electron gas, since it turns out to be the van Hove limit of the von Neumann entropy for a configuration space region 3, divided by the volume |3|. For a > 0 such a bounded measurable region 3 ⊂ Rn is contained in 3+ (a) := {q ∈ Rn | dist(q, 3) ≤ a} and contains
3− (a) := {q ∈ 3 | dist(q, Rn − 3) ≥ a},
so that
δ3(a) := 3+ (a) \ 3− (a)
is a thickened boundary region. Definition 5.24. A sequence {3j }j∈N of bounded measurable subsets 3j ⊂ Rn tends to infinity in the sense of van Hove if [12] lim |3j | = ∞
j→∞
and
lim
j→∞
|δ3j (a)| = 0. |3j |
For simplicity we use the shorthand ω |`3 := ω |`A(3) for the restriction of our quasi-free state to the local CAR algebra over 3. Proposition 5.25. Let the density function ρ meet the above estimates. Then mean entropy s(ω) equals the van Hove limit lim
3→∞
S(ω |`3 ) = s(ω) |3|
(144)
of the von Neumann entropy density. The deviation S(ω |`3 ) |3| − s(ω) has an upper bound depending on 3 only through the quotient
(145) |δ3(a)| . |3|
Quantum Chaos and Dynamical Entropy
663
Proof. Following Park and Shin [39], Prop. A.1, from the definition of the quasi-free state ω, of the single-particle operator Aˆ in (128) and of its restriction Aˆ 3 in Remark 5.21,1, we have S(ω |`3 ) = Tr (ηF (Aˆ 3 )).
(146)
We first consider parallelotopes 3r ⊂ Rn . So let (`1 , . . . , `n ) be a basis of L and for r ∈ N, n X xj `j | xj ∈ [0, r) . (147) 3r := j=1
We claim that (144) holds for the van Hove sequence {3r }r∈N . By subadditivity of ˆ entropy and L-invariance of A, inf r
The first observation is that Aˆ 3r = |3r | ·
S(ω |`3r ) S(ω |`3r ) = lim . r→∞ |3r | |3r | XZ m∈N
T∗
(148)
r r ρ(m, k)|ψm,k ihψm,k |
dk |T∗ |
(149)
with the truncated rescaled Bloch functions r ψm,k :=
χ3r · ψm,k p |3r |
of L2 -norm one. Moreover by Def. (112), Z 0 r r |3r |hψm,k , ψm,k u¯ m,k (q)um,k0 (q)ei(k −k)q dn q 0 iL2 (3r ) = 3 Z r 0 u¯ m,k (q)um,k0 (q)ei(k −k)q fr (k 0 − k)dn q = 31
with fr (s) :=
r−1 X
exp is · (
m1 ,... ,mn =0
n X
mj `j )
(s ∈ T∗ ).
j=1
As r → ∞, fr tends weakly to the delta function. Thus r r 0 ∗ , ψm,k |3r |hψm,k 0 iL2 (3r ) → δ(k − k ) |T | ,
(150)
too. We evaluate the trace w.r.t. the basis r ψm,k
the group
(m ∈ N, k ∈ G∗r ),
(151)
664
F. Benatti, T. Hudetz, A. Knauf
, n X G∗r := L∗ /(rL)∗ ≡ cj `∗j r cj ∈ {0, r − 1} j=1
being isomorphic to Znr . These vectors coincide with the (orthonormalized) eigenfunc2 tions of the Schr¨odinger operator − ~2 1 + V on L2 (3r ) with periodic boundary conditions. We substitute (149) and expand ηF in X S(ω |`3r ) X r r = |3r |−1 hψm,k , ηF (Aˆ 3r )ψm,k i. |3r | ∗ m∈N
k∈Gr
By (150) we obtain lim
r→∞
S(ω |`3r ) = s(ω), |3r |
and by (148) the lower bound S(ω |`3r ) ≥ s(ω). |3r |
(152)
We denote the deviation from mean entropy by δ1 (r) :=
S(ω |`3r ) − s(ω). |3r |
From now on we consider a general bounded measurable 3 ⊂ Rn and more or less follow the strategy of proof which was used in the lattice case in [12], Prop. 6.2.38. However, we need an apriori bound of the following form which makes up for the lack of monotonicity of von Neumann entropy. We claim that there exists a C ≥ 0 independent of r such that for all measurable 3 ⊂ 3r , S(ω |`3 ) ≤ C|3r |.
(153)
To prove (153), we remark that, since 3 ⊂ 3r , the positive operator Aˆ 3 meets the operator inequality 0 ≤ Aˆ 3 ≤ Aˆ 3,r ≤ 1ˆl. Thus, by the minmax theorem, the number of eigenvalues of Aˆ 3 in the interval [ 21 , 1] is smaller than the number c0 (r) of corresponding eigenvalues of Aˆ 3r . Therefore, Tr (ηF (Aˆ 3 )) ≤ Tr (ηF (Aˆ 3r )) + c0 (r) ln(2), which, using (146), (149) and the fact that, for suitable 1 > > 0 and c > 0, ηF (ρ) ≤ c ρ1− , we derive (153) for r = 1. For the sublattice rL the set of shifted parallelotopes 3r,` := {q + ` | q ∈ 3r }
(` ∈ rL)
forms a tessellation of Rn . We show (153) for arbitrary r ∈ N by dividing the parallelotope 3r into rn shifted parallelotopes 31,l and using subadditivity. Thus we control 3 ⊂ 3r by estimating
Quantum Chaos and Dynamical Entropy
!
r−1 X
S(ω |`3 ) ≤
665
ω |`3∩3 P
S
1,
c1 ,... ,cn =0
j
cj lj
≤ rn C · |31 | = C · |3r |,
proving (153). For a general bounded measurable 3 ⊂ Rn we denote by A+3,r := {` ∈ rL | 3r,` ∩ 3 6= ∅} and
A− 3,r := {` ∈ rL | 3r,` ⊂ 3}
the index sets of those shifted parallelotopes which have intersection with resp. are contained in 3. Then for a ≥ diam(3r ) = r · diam(31 ), [ [ 3r,` ⊂ 3+ (a), 3r,` ⊃ 3− (a) (154) `∈A− 3,r
`∈A+3,r
and
[
3r,` ⊂ δ3(a) := 3+ (a) \ 3− (a).
(155)
`∈A+3,r −A− 3,r
By subadditivity, translation invariance of ω and (153) we get from (154) and (155), X − + S(ω |`3 ) ≤ S(ω |`3r,` ∩3 ) ≤ |A− 3,r | · S(ω |`3r ) + C · |3r | · |A3,r − A3,r | `∈A+3,r
≤ |3| ·
S(ω |`3r ) + C · |δ3(a)|. |3r |
(156)
On the other hand by subadditivity (155) implies ! X S(ω |` ) ≥ S ω |`S − 3
`∈A+
3,r
3r,`
! ≥S
ω |`S
`∈A+
3,r
3r,`
`∈A+3,r −A− 3,r
S ω |`(3r,` \3)
− C · |δ3(a)|.
We use strong subadditivity [51] S ω |`3I ∪3II ∪3III − S ω |`3I ∪3II ≤ S ω |`3I ∪3III − S ω |`3I
(157)
(158)
of von Neumann entropy for obtaining a lower bound for the first term in (157). We denote by 0j (j = 1, . . . , |A+3,r |) the union of the first j translates 3r,l w.r.t. lexicographic ordering of ` ∈ A+3,r and want to prove for all j, (159) S ω |`0j+1 − S ω |`0j ≥ s(ω) · |3r |. Assume to the contrary that for some j and ε > 0, S ω |`0j+1 − S ω |`0j < (s(ω) − ε) · |3r |.
666
F. Benatti, T. Hudetz, A. Knauf
For t ∈ N let 1k denote the union of the first k translates (in lexicographic order) of parallelotopes 3r contained in the large parallelotopes 3r·t . Then, by (158) and the invariance of ω under lattice-translations S ω |`1k+1 − S ω |`1k < (s(ω) − ε) · |3r | for all k such that there exists a translation of 0j+1 into 1k+1 which maps 0j+1 \ 0j to 1k+1 \ 1k . This is possible if the distance between 1k+1 \ 1k and the boundary of 3r·t is larger than the diameter of 0j . For the remaining set of indices k subadditivity gives the a priori estimate S ω |`1k+1 − S ω |`1k ≤ S ω |`3r . This subset is of order O(tn−1 ). Thus for t large,
S ω |`3rt =
n tX −1
k=0
ε , S ω |`1k+1 − S ω |`1k ≤ |3rt | · s(ω) − 2
contradicting our above proof of the van Hove limit for the parallelotopes. Thus we may use (159) to estimate the first term in (157) by ! 3r,` ≥ s(ω) · |A+ | · |3r | ≥ s(ω) · |3|. S ω |`S 3,r
`∈A+
(160)
3,r
To collect our estimates, we set δ2 (a) :=
|δ3(a)| . |3|
Then (156) and (152) gives us an upper bound S ω |`3 − s(ω) ≤ δ1 (r) + δ2 (a) C |3| for the deviation from mean entropy. Equations (157) and (160) give the lower bound S ω |`3 ≤ δ2 (a) C. s(ω) − |3| We couple a and r by setting a := diam(31 ) · r and choose r ∈ N so that δ1 (r) + δ2 (a) C is minimized. This shows our proposition. 5.4.3. Volume estimates. We now consider a bounded, not necessarily convex, configuration space region B ⊂ Rn of the form (129). For v ∈ Rn , v 6= 0 we denote by Pr v : Rn → Ev⊥ the orthogonal projection to the subspace Ev⊥ := {x ∈ Rn | x·v = 0}. For a Borel subset B ∈ Rn we consider the volume |Pr v (B)| of its shadow Pr v (B) (and set |Pr v (B)| := 0 if v = 0).
Quantum Chaos and Dynamical Entropy
667
After t ∈ Z time steps the region λB ⊂ Rn is shifted to Bλt := λB + t · v, and we are interested in estimates of the regions Bλ
T[ −1
Bλt .
t=0
Lemma 5.26. Let the new volume at time t ∈ N be 1tλ := Bλ
(161)
1tλ = 1tλ0 + (t − t0 ) · v
(162)
(so that |1tλ | becomes independent of t), with |1tλ | ≤ |v| · λn−1 · |Pr v (B)|
(163)
|1tλ | − |v| · λn−1 · |Pr v (B)| = O(λn−2 ).
(164)
and
Thus uniformly in t ∈ N,
|Bλ
(165)
Proof. 1tλ = Bλt \
t [
Bλt ∩ Bλt−i
i=1
from which one sees that t 7→
|1tλ |
is monotone decreasing, and for t ≥ t0 ,
1tλ = Bλt \
t0 [
Bλt ∩ Bλt−i
i=1
so that (162) is valid. For a bounded ∂-manifold Bj the set δBj (a) = Bj+ (a) \ Bj− (a) of points of distance ≤ a from ∂Bj is of volume |δBj (a)| = O(a)
(a & 0),
and the same holds for B = B1 ∩ . . . ∩ Bl . Thus |δ(λB)(|v|)| = λn |δB(|v|/λ)| = |v| · O(λn−1 )
(λ → ∞),
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F. Benatti, T. Hudetz, A. Knauf
which implies (161), since |1tλ | ≤ |11λ | and 11λ = λB + v \ λB ⊂ |δ(λB)(|v|)|. Equation (163) follows from (162). Namely the sets 1tλ are disjoint, so that for each t s ∈ S := Pr v (B) (the shadow of B) the intersection between the line Pr −1 v (s) and 1λ must be of total length ≤ |v|. To prove (164), we consider for ε > 0 the subset Sε of the shadow given by Sε := s ∈ S | Pr −1 v (s) ∩ B contains an interval of length ≥ ε . For s ∈ Sε , ε ≥ |v|/λ and t ≥ t0 the set
contains an interval of length ≥ (t − t0 ) · |v|, since the shifted intervals overlap. Together with the translation property (162) this implies that for t ≥ t0 , t Pr −1 v (λs) ∩ 1λ
is of length |v|. Otherwise, due to the fixed and finite length of the intersections with the volume 1tλ0 , intersections of length ≤ |v| could not make for intervals of length ≥ (t − t0 )|v| when t >> 1. This implies that −1 Pr v (λSε ) ∩ 1tλ = |v| · |λSε | = |v| · λn−1 · |Sε |. Next we show that most points of 1tλ project to the subset λSε of the shadow. The main step is to see that for ε > 0 small, 2 |B − Pr −1 v (Sε ) ∩ B| = O(ε ).
(166)
To prove (166), we first claim that (P) There exists a L > 1 such that for ε > 0 small and all s ∈ S − Sε the end points b0 , b1 ∈ ∂B of each interval in Pr −1 v (s) ∩ B can be connected by a curve of length ≤ ε · L lying on the boundary ∂B. By definition of Sε these end points have a distance |b1 − b0 | < ε. By compactness of ∂B we could otherwise find a limit point b ∈ ∂B of such pairs of end points which is isolated in Pr −1 v (Pr v (b)). 1. If b ∈ ∂Bi ∩ ∂Bj with i 6= j, then the nonexistence of a common L would contradict transversality of intersections of the ∂-manifolds Bj , j = 1, . . . , l. 2. Otherwise there is only one j ∈ {1, . . . , l} with b ∈ ∂Bj , that is, b is contained in a smooth component of the boundary ∂B. But in that case the curvature of ∂B in a neighbourhood of b is bounded, so that for some L > 1 any two points b0 , b1 in that neighbourhood can be connected by a curve of length ≤ L · |b1 − b0 | in that neighbourhood. Now having proved property (P), we consider the end points b0 , b1 ∈ ∂B of such an interval in Pr −1 v (s) ∩ B. 1. If b0 ∈ ∂Bi and b1 ∈ ∂Bj with i 6= j, the points of the interval between b0 and b1 belong to a (L + 1) · ε-neighbourhood Y1ε of the codimension two (dimension n − 2) subset X1 := {x ∈ ∂B | x ∈ ∂Bi ∩ ∂Bj with i 6= j}. The size of that neighbourhood is thus |Y1ε | = O(ε2 ), compatible with (166).
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669
2. Otherwise b0 , b1 ∈ ∂Bj . Then uniformly for all points b on these interval dist(b, ∂B) ≤ C · ε2
(167)
for a suitable C > 0. To prove (167), we take a curve c : [0, 1] → ∂Bj of length ≤ L · ε which connects c0 = b0 with c1 = b1 on the boundary component ∂Bj . If n(b) denotes the unit normal at a point b ∈ ∂Bj , then the sign of n(b0 ) · v is different from sign(n(b1 ) · v), since b0 and b1 are the end points of an interval on a line parallel to v (more precisely, sign(n(b0 ) · v) · sign(n(b1 ) · v) 6= +1). Thus by continuity there is a t ∈ [0, 1] with n(c(t)) ⊥ v. By compactness the curvature of ∂Bj and thus the change of the normal vector n along c is bounded. Thus remembering the length estimate Lε for c, we obtain n(bi ) · v = O(ε), i = 0, 1. In words, at bi the interval between b0 and b1 , (which by assumption is of length ≤ ε) makes an small angle of order ε with a suitable tangential direction in Tbi (∂Bj ). But this implies that each point b ∈ B of the interval has only a distance of order ε2 from the boundary, proving (167). The measure of these points Y2ε := b ∈ B \ Y1ε | dist(b, ∂B) ≤ Cε2 is thus |Y2ε | = O(ε2 ), too. This shows (166). But (166) implies that for λ large and ε := |v|/λ t −1 t n t t n−2 1λ − Pr −1 ) v (λSε ) ∩ 1λ = λ 11 − Pr v (Sε ) ∩ 11 = O(λ and thus (164).
Remark 5.27. If B is convex, then (162) holds for t0 = 1, since then Bλt ∩ Bλt−i ⊂ Bλt ∩ Bλt−1 . The volume of 1tλ = Bλt \ (Bλt ∩ Bλt−1 ) then equals t 1 0 1λ = Bλ 1Bλ1 , 2 where 1 denotes the symmetric difference. Our next task is to control under which conditions the above sets form van Hove sequences as the scaling λ goes to infinity. Remark 5.28. Analytically one problem consists in controlling the shadow S := Pr v (M ) ⊂ Ev⊥ ∼ = Rn−1 of a subset M ⊂ Rn . In our case M := int(B). If B is a strictly convex ∂-manifold, then it follows from a transversality argument that the topological boundary ∂S ⊂ Ev⊥ of its shadow is an (n − 2)-dimensional smooth submanifold. If one does not assume convexity, ∂S may lose smoothness (think of the similar geometrical situation of the V -shaped skyline between the two contours of two smooth mountains). If B is a manifold with boundary ∂B, then ∂S is contained in the set of critical values of (Pr v )|`∂B , Thus in that case it follows from the Morse-Sard Theorem that ∂S has (n − 1-dimensional) measure zero (see, e.g., Hirsch [22]).
670
F. Benatti, T. Hudetz, A. Knauf
In [29] one of us shows the following theorem: “Let M ⊂ Rn be a closed embedded (n − 1)-dimensional manifold with maximal exterior curvature R > 0. Then the (n − 2)-dimensional measure |∂S(M )| := lim
1 s&0 2
|Uε | ε
(168)
of the shadow boundary is bounded above by Vol(M ) n−1 ) · , |∂S(M )| ≤ ∂S(SR n−1 Vol(SR ) n−1 ⊂ Rn is a sphere with radius R.” where SR
In particular this geometrical estimate implies that |∂S(M )| < ∞. In order to obtain the lower bound in our theorem on shifts, we rescale speed, that is, we go from v to rv, with r ∈ N. So we consider for time t ∈ N the sets r,
where
Bλr,
(169)
Bλrt := λB + rtv,
are the configuration space subsets for rv (compare Lemma 5.26). By (162) for t ≥ t00 := [t0 /r] + 1, where t0 has been defined in Lemma 5.26, these sets are shift-related: 1r,t+l = 1r,t λ λ + lr · v. For simplicity we enforce shift invariance for all t ∈ N0 by setting r,t ˜ r,t 1 λ := 1λ
if t ≥ t00 ,
r,t00
0 ˜ r,t ˜ 1 λ := 1λ − (t0 − t)r · v
Then for all t ∈ N0 ,
otherwise.
(170)
r(t−1) rt ˜ r,t 1 . λ ⊂ Bλ \ Bλ
Lemma 5.29. As λ → ∞ the family of sets Bλ
0, δ B
uniformly in T ∈ N. ˜ r,0 For t = 0 (and thus for all t ∈ N0 ) and a r λ the family of sets 1 λ forms a van Hove sequence, with ˜ r,0 δ 1λ (a) (172) = O(r−1 ) ˜ r,0 1λ for all a > 0.
Quantum Chaos and Dynamical Entropy
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Proof. In both cases the volumes go to infinity in the λ → ∞ limit. So we need only check the second condition in (5.24). This clearly follows from (171) resp. (172). We now prove (171) and start by rescaling. So equivalently we must show that for ε > 0 small, ε, 0 with each ni (b), i ∈ I(b). By compactness for some minimal cosine C > 0 we may cover ∂B by finitely many neighbourhoods U1 , . . . , Us and choose unit vectors n1 , . . . , ns such that ni (b) · nj > C
(j = 1, . . . , s; b ∈ Uj ; i ∈ I(b)).
We thus may obtain an upper estimate of the surface areas |Uj | by dividing the area |Pr nj (Uj )| by C. Now we consider the subsets Vj , j = 1, . . . , s of points on the surface ∂B1ε, 0, |Pr nj (Vj )| ≤ |Pr nj (B1ε,
(174)
so that we obtain (173). To prove (172), we use the estimate ˜ r,0 1λ = rλn−1 |Pr v (B)| · (1 + O(r/λ)) r,t ˜ r,0 which is obtained from (164) by rescaling v and using |1 λ | = |1λ | for t large. Thus (172) follows from an estimate of the form ˜ r,0 δ 1λ (a) = O(λn−1 )
for each a > 0. After rescaling, this is a consequence of ˜ ε,0 δ 11 (ρ) lim = O(ε0 ), ρ&0 ρ
(175) ε,0
˜ 1 . By Def. (170) that is, uniform boundedness w.r.t. ε of the surface area of ∂ 1 0
ε,t0 ˜ ε,0 − (t00 − 1)εv 1 1 = 11
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F. Benatti, T. Hudetz, A. Knauf
for t00 = [t0 /ε] + 1 so that we may instead estimate the surface of 1ε,t 1 . But according to Def. (169), ε,t00
11
ε,
= B1
ε,
− B1
,
ε,t00 ε,
so that
By (174) the r.h.s. is bounded by 2c2 (1 + ε(t00 + 1)) < 2c2 (1 + (t0 + 1)) so that we have shown (175). 5.4.4. Upper bound for the shift. Proposition 5.30. For v ∈ L, lim
λ→∞
1 hω ,λB (αv ) ≤ s(ω) · |Pr v (B)| · |v|. λn−1 ρ
Proof. As assumed at the beginning of this section, the state ω is locally normal. That is the restriction ω |`λB corresponds to a density matrix ρˆλ belonging to the local CAR algebra A(λB). We shall write its eigenvalues ρλi in decreasing order and denote by Pˆλ≤N the orthogonal projection onto the Hilbert space spanned by the first N eigenvectors |9λi i. That is N X |9λi ih9λi |, ρˆλ |9λi i = ρλi |9λi i, Pˆλ≤N = i=1
where the vectors |9λi i belong to the Fock space HλF built over the single-particle Hilbert space L2 (λB). These eigenvectors are obtained by acting on the Fock vacuum with creation operators aˆ (ψiλ ), where the |ψiλ i are the eigenvectors of the restriction Aλ of the operator A in (128) to the given volume λB. The eigenvectors |ψiλ i are thus supported within λB and ∞ X |ψiλ ihψiλ | = χλB , i=1
namely, the characteristic function of the considered volume λB. Using the orthogonal projections PNλ ] we construct the finite dimensional subalgebra := Pˆλ≤N A(λB)Pˆλ≤N A≤N λ
M
C 1ˆl − Pˆλ≤N
that form an increasing sequence uniformly converging to A(λB). As shown by Park and Shin in [39], Prop. 4.4 (compare also the proof of Thm. 3.2), the finite volume CNT entropy of the translation automorphism is given by . hω (αv , A(λB)) = lim hω αv , A≤N λ N
So we can continue to use this finite-dimensional subalgebra. The join T_ −1 t=0
αvt A≤N ⊆ A Bλ
Quantum Chaos and Dynamical Entropy
673
generated by the first T translates of A≤N is finite dimensional. In fact, it is conλ tained in the C*-algebra B(HT ) of bounded operators on the Hilbert space HT := PT −1 t λ λ t=0 αv (AN ] )HF , which is finite dimensional: dim(HT ) ≤ T N . Thus ! T_ −1 ≤N ≤N ≤N Hω Aλ , . . . , αvT −1 (Aλ ) ≤ Hω αvt Aλ t=0
≤ Hω A Bλ
≤ S ω |`B
which is a consequence of monotonicity under inclusions, invariance under repetitions and boundedness by the von Neumann entropy of the entropy functionals. We divide on both sides of the above inequality by T and perform the T → ∞ and N → ∞ limits. Then we obtain 1 S ω |`B
The proposition follows from (165), together with the van Hove estimate (171). 5.4.5. Lower bound for the shift. In this section we shall prove the lower bound lim
1
λ→∞ λn−1
hω,λB (αv ) ≥ s(ω) · |Pr v (B)| · |v|
for a shift by a lattice vector v ∈ L. ˆ ˜ r,0 in decreasing order. We Let ψjλ , j ∈ N, be the orthonormal eigenstates of A|` 1λ denote by Pj1 := aˆ † (ψjλ )ˆa(ψjλ )
(176)
the projection operators on multi-particle states which include the j th eigenfunction, and set Pj0 := 1ˆl − Pj1 = aˆ (ψjλ )ˆa† (ψjλ ). For I ⊂ {1, ..., N } let PI≤N :=
Y
Pi1
i∈I
Y
Pi0 .
i6∈I
Similarly for a finite subset J ⊂ {N + 1, N + 2, . . . } let Y Y Pi1 Pi0 . PJ>N := i∈J
i6∈J
More generally we consider for I = (I0 , . . . , IT −1 ), It ⊂ {1, ..., N } the operators PI≤N :=
TY −1
αvrt (PI≤N ) t
t=0
and for finite J = (J0 , . . . , JT −1 ), Jt ⊂ {N + 1, . . . } the operators
(177)
674
F. Benatti, T. Hudetz, A. Knauf
>N
PJ
:=
TY −1
αvrt (PJ>N ). t
(178)
t=0 1 2 ˜ r,t ˜ r,t Since the regions 1 and 1 are disjoint for t1 6= t2 , the operators in the prodλ λ ucts (177) and (178) commute so that the PI≤N and PJ>N form a family of commuting orthoprojectors. Observe that we do not only have the resolution of identity X PI≤N = 1ˆl,
I,It ⊂{1,... ,N }
X
but also
PJ>N = 1ˆl,
J ,Jt ⊂{N +1,... }
with the convention of summing only over finite J , since our construction is based on Fock space: That is, only finitely many single-particle states are occupied. The operators PI≤N , I ⊂ {1, ..., N }, generate a 2N -dimensional Abelian subalgebra ab Nλ (N ) ⊂ A(λB). We will use this algebra and its translates in two ways: • as the subalgebra whose entropy we estimate and • as a means to decompose the state. Thus we consider the decomposition X ωI ω= I,It ⊂{1,...,N }
into the linear functionals
ωI (a) := ω σ i/2 (PI≤N )a ,
where σ i/2 is the modular automorphism of ω which ensures positivity of the ωI . So by normalization we obtain the states ωˆ I :=
ωI . ωI (1ˆl)
Lemma 5.31. For all r, N ∈ N the local dynamical entropy of the shift is bounded below by hω,λB (αv ) ≥ lim Eλr,N (T ) − Dλr,N
(179)
T →∞
with Eλr,N (T ) :=
1 rT
X
η ω(PI≤N )
(180)
I,It ⊂{1,... ,N }
and Dλr,N :=
1 r
X I⊂{1,... ,N }
ω PI≤N ·
X J⊂{1,... ,N }
η ωˆ I (PJ≤N ) .
(181)
Quantum Chaos and Dynamical Entropy
675
Remarks 5.32. 1. The modular group σ t of the state ω is defined by σ t aˆ ] (f ) = aˆ ] e−iht f , where the modular Hamiltonian is related to the operator Aˆ in (128) by 1 . 1ˆl + eh It proves useful to write the positive operator B introduced in 5.21, 2 as Aˆ =
B=
Aˆ ˆ = e−h , ˆ1l − Aˆ
so that the celebrated KMS-relations ω aˆ † (f )ˆa(g) = ω aˆ (g)σ −i (ˆa† (f )) read
ˆ ) . ω aˆ † (f )ˆa(g) = ω aˆ (g)ˆa† (Bf
(182)
2. By considering the positive single-particle operator B 3 of 5.21, 2, we can similarly write ˆ Bˆ 3 = e−h3 , and thus define a local modular group as in Point 1 above (compare [39], Sect. 4.). Clearly, the eigenvectors of the restricted state ω |`3 that are all the way used in this section are also eigenstates of Aˆ 3 , Bˆ 3 and hˆ 3 . 3. The term (180) is the entropy of the restriction of the state ω to the Abelian algebra which is the direct sum of Nλab (N ) and its translates. It will be shown to be related to mean entropy. 4. The entropy defect (181) is a boundary term. This is related to the fact that ωˆ I (PJ≤N ) = δI,J if the modular automorphism leaves PI invariant, and the definition of both σ and PI in terms of the state. Proof. We apply the definitions and properties of finite-volume CNT-entropy and of the entropy functionals, in particular the property (26) together with the subalgebramonotonicity of the entropy (24), to estimate from below as follows: hω (αv , A(λB)) 1 ≥ hω αvr , Nλab (N ) r X 1 1 lim ≥ r T →∞ T
η ω(PI≤N )
(183)
I,It ⊂{1,... ,N }
−
T −1 X
X
η ω(αvrt (PI≤N ))
(184)
t=0 I⊂{1,... ,N }
+
T −1 X
S ω|`αvrt Nλab (N )
(185)
t=0
−
T −1 X
X
t=0 I⊂{1,... ,N }
ω(αvrt (PI≤N ))
S
! ω σ i/2 (αvrt (PI≤N )) · rt ab |`αv Nλ (N ) . (186) ω(PI≤N )
676
F. Benatti, T. Hudetz, A. Knauf
So the first term (183) coincides with limT →∞ Eλr,N (T ). Moreover, ω is αv -invariant, and Nλab (N ) is Abelian. Thus, ω αvrt (PI≤N ) = ω PI≤N and S ω|`αvtr(λ) Nλab (N )
X
=−
ω(PI≤N ) log ω(PI≤N ),
I⊂{1,... ,N }
so that (184) and (185) cancel. By the same argument, and since the automorphism αv commutes with the modular automorphism σ i/2 , (186) reduces to (181), because ! ω σ i/2 (αvrt (PI≤N )) · rt ab S |`αv Nλ (N ) ω(PI≤N ) ! X ω σ i/2 (αvrt (PI≤N ))αvrt (PJ≤N ) = η ω(PI≤N ) J⊂{1,... ,N } ! X ω σ i/2 (PI≤N )PJ≤N . = η ω(PI≤N ) J⊂{1,... ,N }
Next we relate the main term Eλr,N (T ) in Lemma 5.31 to von Neumann entropy. Lemma 5.33. For ε > 0 there is an N0 (ε) so that for all N ≥ N0 (ε), 1 r,N S Eλ (T ) ≥ S ω |` T −1 1˜ r,t − ε. λ T t=0 Proof. From local normality and the other assumptions on ω we required at the beginning of the section, the restricted state := ω |`ST −1 1˜ r,t ρr,
λ
is a density matrix with finite von Neumann entropy. The Abelian subalgebra Nλr,
T_ −1
T[ −1 ˜ r,t αvrt Nλ (N ) ⊂ A 1 λ
t=0
t=0 ≤N
generated by the spectral projections PI is not maximally Abelian. , but the difference can be handled Therefore ω |`Nr, 0 the entropy defect is bounded above by Dλr,N ≤ C(ε)λn−1+ε /r, uniformly in N ∈ N.
Quantum Chaos and Dynamical Entropy
677
Proof. Let us rewrite (181) as follows: X rDλr,N =
X
ω(PI≤N ) · η(xI,J )
I⊂{1,... ,N } J⊂{1,... ,N }
with xI,J := ωˆ I (PJ≤N ) =
ω σ i/2 (PI≤N )PJ≤N
.
ω(PI≤N )
For a given I we interpret J 7→ xI,J as a probability measure on the power set of {1, . . . , N }. For l ∈ {1, . . . , N } and e ∈ {0, 1} we set xeI,l := ωˆ I (Ple ) X
so that x1I,l =
xI,J
X
and x0I,l =
J⊂{1,... ,N }:l∈J
xI,J .
J⊂{1,... ,N }:l6∈J
In particular we have x0I,l + x1I,l = 1. Then by subadditivity of the entropy w.r.t. joins of partitions for all I ⊂ {1, . . . , N }, N X
ηF (x1I,l ) ≡
1 N X X
so that rDλr,N ≤
X
ω(PI≤N ) ·
N X
I⊂{1,... ,N }
η(xI,J )
J⊂{1,... ,N }
l=1 e=0
l=1
X
η(xeI,l ) ≥
ηF (x1I,l ) =
1 N X X
l=1 e=0
l=1
X
ω(PI≤N )η(xeI,l ) .
I⊂{1,... ,N }
Setting ωˆ le (·) :=
ω(σ i/2 (Ple )·) , ω(Ple )
these states are the convex combinations ωˆ l1
=
X
ω(PI≤N 0 ∪{l} )
I 0 ⊂{1,... ,N }−{l}
ω(Pl1 )
ωˆ I 0 ∪{l}
and
ωˆ l0 =
X I 0 ⊂{1,... ,N }−{l}
So by concavity of x 7→ η(x), X
ω(PI≤N )η(xeI,l ) ≤
1 X
0 0 ω(Ple ) · η ωˆ le (Ple )) .
e0 =0
I⊂{1,... ,N }
Thus we have the N -independent upper bound rDλr,N ≤
∞ X 1 X l=1 e=0
for the entropy defect.
ω(Ple ) η ωˆ le (Ple )) + η ωˆ le (Pl1−e ))
ω(PI≤N 0 ) ωˆ I 0 . ω(Pl0 )
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F. Benatti, T. Hudetz, A. Knauf
By concavity of η, η(1) = 0 and η 0 (1) = −1 we have η(x) ≤ 1 − x
(x ∈ [0, 1]).
On the other hand for 0 < β < 1 and a suitable positive constant c0β , η(x) ≤ c0β xβ
and x ≤ xβ
(x ∈ [0, 1]).
Thus, with cβ := 2 + c0β , rDλr,N ≤ cβ
∞ X 1 X
β
ω(Ple ) ωˆ le (Pl1−e ))
≤ cβ
l=1 e=0
∞ X 1 X
β
ω(σ i/2 (Ple )Pl1−e )
.
l=1 e=0
Finally the KMS identity (see Remarks 5.32, 2 and 5.21, 2) ω σ i/2 (Pl1 )Pl0 = ω σ i/2 (Pl0 )Pl1 implies rDλr,N ≤ 2cβ
∞ X
β
ω(σ i/2 (Pl1 )Pl0 )
.
l=1
Let Bˆ λ be the operator associated with the restricted state ω |`1˜ r,0 according to Reλ ˆ ˜ r,0 and thus of Bˆ λ itself. Then, mark 5.32, 2 and let ψ λ be the eigenvectors of Aˆ λ := A|` 1λ
l
using Remark 5.32, 2, ω σ i/2 (Pl1 ) · Pl0 = ω aˆ (Bˆ 1/2 ψlλ )Pl0 aˆ † (Bˆ 1/2 ψlλ ) = ω aˆ (Bˆ 1/2 − (Bˆ λ )1/2 )ψlλ Pl0 aˆ † (Bˆ 1/2 − (Bˆ λ )1/2 )ψlλ
(187)
ˆ l ) · (ψlλ , Aψ ˆ lλ ) − (gl , Aψ ˆ lλ ) · (ψlλ , Ag ˆ l) = (gl , Ag ≤ (gl , gl )
(188)
with gl := (Bˆ 1/2 − (Bˆ λ )1/2 )ψlλ . Equation (187) holds since 1) ψlλ is an eigenvector of (Bˆ λ )1/2 , 2) by Def. (176) of 0 Pl = aˆ (ψjλ )ˆa† (ψjλ ), and 3) since aˆ (ψjλ )ˆa(ψjλ ) = 0. We now write β = 1 − uε, with u ∈ (0, 21 ) and ε ∈ (0, 1). Then trivially ∞ (1−ε) (1−u)ε X r Dλr,N ≤ ω σ i/2 (Pl1 ) · Pl0 · ω σ i/2 (Pl1 ) · Pl0 . 2cβ l=1
We use the H¨older inequality kf gk1 ≤ kf kp · kgkq with p, q ≥ 1 and 1/p + 1/q = 1, setting p := 1/(1 − ε), q = 1/ε. Thus r Dr,N ≤ 2cβ λ
∞ X l=1
!1−ε k(B
ˆ 1/2
− (B )
ˆ λ 1/2
)ψlλ k2
·
∞ 1−u X ω σ i/2 (Pl1 ) · Pl0 l=1
!ε . (189)
Quantum Chaos and Dynamical Entropy
679
The first sum in the right-hand side of (189) is ∞ X
2 kgl k2 = Tr χλ Bˆ 1/2 − (Bˆ λ )1/2 χλ ,
l=1
˜ r,0 where χλ is the characteristic function of the volume 1 λ . So by Lemma 5.23 this gives a contribution of order !1−ε ∞ X 1/2 λ 1/2 λ 2 ˆ ˆ k(B − (B ) )ψl k ≤ C1 · λ(1−ε)·(n−1) l=1
to (189). The second sum in (189) is estimated using ˆ lλ ) = (ψlλ , Aˆ λ ψlλ ) ω σ i/2 (Pl1 ) · Pl0 ≤ (ψlλ , Aψ and the fact that ψlλ is an eigenfunction of Aˆ λ . Thus ∞ ∞ 1−u X X i/2 1 0 ω σ (Pl ) · Pl ≤ (ψlλ , (Aˆ λ )1−u ψlλ ) = Tr ((Aˆ λ )1−u ). l=1
l=1
˜ r,0 Now Aˆ λ is defined as the restriction of Aˆ to 1 λ . As long as the scaling of velocity is smaller than the scaling of the region (that is, r ≤ λ), the diameter scales like ˜ r,0 ≤k·λ diam 1 λ r,0
˜λ for a suitable k > 0. We can thus find a k 0 ∈ N, k 0 ≥ k such that, after shifting 1 with a lattice vector ` = `(λ) ∈ L, if necessary, ˜ r,0 1 λ + ` ⊂ 3s ,
the parallelotopes 3r ⊂ Rn being defined in (147) and s := k 0 [λ] ∈ N. The L-invariance of the state guarantees that our estimate is unaffected by the shift. Moreover, we have the operator inequality ˆ ˜ r,0 ≤ χ3s Aχ ˆ 3s χ(1˜ r,0 +`) Aχ (1 +`) λ
λ
which by monotonic increase of x 7→ x and positivity of both operators implies 1−u ˆ ˜ r,0 ˆ 3s 1−u . ≤ χ3s Aχ χ(1˜ r,0 +`) Aχ (1 +`) 1−u
λ
Thus
λ
ˆ 3s )1−u ). Tr ((Aˆ λ )1−u ) ≤ Tr ((χ3s Aχ
But for (1 − u) · (1 + 2/n) > 1, ∞ Z ˆ 3r )1−u X Tr (χ3r Aχ dn k lim = ρ(m, k)1−u < ∞. r→∞ |3r | (2π)n T∗ m=1
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F. Benatti, T. Hudetz, A. Knauf
The equality is shown by the same argument leading to the limit mean entropy formula (148) for parallelograms. The finiteness of the right hand expression is shown like the finiteness (143) of s(ω), observing that by our assumption (120) on ρ, ∞ Z ∞ X X dn k ρ(m, k)1−u ≤ C m−α(1−u) < ∞. (2π)n T∗ m=1
m=1
So the second term in (189) is bounded by ∞ 1−u X ω σ i/2 (Pl1 ) · Pl0
!ε ≤ C2 λnε
l=1
which together gives an upper estimate r Dr,N ≤ C1 (ε)C2 (ε) · λn−1+ε 2cβ λ for arbitrarily small ε > 0.
Thus we conclude with Proposition 5.35. For v ∈ L the local dynamical entropy of the shift has the lower bound λ1−n · hω,λB (αv ) ≥ s(ω) · |Pr v (B)| · |v|. Proof. We use Lemma 5.31 in order to get a lower bound for the local entropy hω,λB (αv ) of the shift. Note that by that lemma hω,λB (αv ) ≥ lim Eλr,N (T ) − Dλr,N T →∞
(190)
for all N and r. However, the estimate reduces to the trivial statement hω,λB (αv ) ≥ 0 if r λ, whereas the D-term can only be made small if r 1. Thus we couple r := r(λ) to λ so that 1 r λ. In Lemma 5.33 the E-term is related to von Neumann entropy: For ε > 0 and N ≥ N0 (ε), 1 r,N S (191) Eλ (T ) ≥ S ω |` T −1 1˜ r,t − ε. λ T t=0 0 ˜ r,t By Def. (170) of 1 λ for T > t0 the region
X(λ, r) := Bλr,
T[ −1
n ˜ r,t 1 λ ⊂R
t=0
becomes independent of T . The size of this region is of order |X(λ, r)| = O(λn ), r,
0 using the inclusions X(λ, r) ⊂ Bλ 0 ⊂ Bλ
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681
By subadditivity of S we have S ω |`ST −1 1˜ r,t ≥ S(ω |`B r,
λ
λ
(192)
Another application of (165) gives us by the van Hove property (171) and Prop. 5.25, (193) S(ω |`B r,
The a priori upper bound (153) for von Neumann entropy gives us S(ω |`X(λ,r) ) = O(λn ).
(194)
Putting (192), (193) and (194) together, and performing the time T → ∞ limit, we get for the first term in (190), lim Eλr,N (T ) ≥ |v| · λn−1 · |Pr v (B)| + o(λ0 ) · s(ω) − ε. T →∞
We make ε vanish by performing the N → ∞ limit. Finally this gives the result lim λ1−n lim
λ→∞
lim Eλr,N (T ) ≥ |v| · |Pr v (B)| · s(ω).
N →∞ T →∞
In the λ → ∞ limit with 1 r λ the contribution lim λ1−n lim Dλr,N
λ→∞
vanishes, using Lemma 5.34.
N →∞
Acknowledgement. A. K. thanks the members of the IHES (Bures) for their hospitality during his visit in May 1997.
A. The Classical Discretized Electron Gas Similar to [35] we consider now a caricature of the quantum electron gas which is classical in the sense that the C ∗ -algebra is one of the functions on a phase space, but quantum in the sense that this phase space is discretized, the discretization depending implicitly on ~. In fact we have a family of these caricatures, the elements being indexed by r and connected with the sublattices rL. The C ∗ -algebras of these classical discretized electron gases will then turn out to coincide with maximal Abelian subalgebras Ar ⊂ A generated by aˆ † (φrs,l )ˆa(φrs,l ) and aˆ (φrs,l )ˆa† (φrs,l ). Below we introduce a family r 7→ H r of operators converging in the norm resolvent sense to H. Their Bogoliubov automorphism αr 2 H r : A → A restricts to Ar . Since the potential is also rL-periodic, we obtain a decomposition of each original band into rn subbands. However, in order not to introduce additional points of nonanalyticity of the band functions, we do not order these subbands with increasing energy.
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Instead we remember that the Brillouin zone T∗ can be identified with a domain of the form n X xj `∗j , xj ∈ [0, 1) , k ∈ Rn,∗ k = j=1 `∗1 , . . . , `∗n ∈ Rn,∗ being the basis dual (`∗j (`k ) = 2πδjk ) to a basis `1 , . . . , `n ∈ Rn of L. Then we define the subbands Esr of Em using the rn subsets n X k ∈ Rn,∗ k = xj `∗j , xj ∈ [aj /r, (aj + 1)/r) , j=1 indexed by a := (a1 , . . . , an ), aj ∈ {0, . . . , r − 1}. We order the bands by energy, and the subbands lexicographically w.r.t. a, so that Esr is a subband of Em if r2 (m − 1) < s ≤ r2 m. We denote the Wannier functions of the sublattice rL by φrs,` . They are centered in space near ` ∈ rL, their delocalisation in configuration space growing asymptotically linearly with r. But as the sizes of the subbands are bounded above by λ/r, λ being the Lipschitz constant of the original band, and the band function is real-analytic, up to a measure zero set, the variation of the group velocities ~−1 ∇k Esr (k) in those subbands which do not contain a point of nonanalyticity is of the order r−1 . The set of nonanalyticity of Em is an analytic variety of degree ≤ n − 1 (see [52]) in the Brillouin zone T∗ . Thus the subbands which meet that measure zero set are of density zero (compared to their total number rn ) as r → ∞. Conversely the group velocity of a density one subset of nice subbands has a variation smaller than C/r as r → ∞. We now attribute to the subbands certain mean group velocities, starting with the true mean Z ~−1 ∇k Esr (k)dk/|T∗,r |. v¯ sr := T∗,r
We want instead a velocity vector which is rational w.r.t. the lattice L. Let L = (`1 , . . . , `n ) be the matrix composed of a basis of L. Then the discretization (195) vˆ sr := r−1 L rL−1 v¯ sr , [·] denoting the componentwise integral part, is L-rational, the denominators dividing √ r, and kvˆ sr − v¯ sr k ≤ nkLk/r. So there exists an r-independent constant c ≡ c(m) > 0 such that for the nice subbands of the mth band, −1 ~ ∇k Esr (k) − vˆ sr ≤ c/r. max n,∗ k∈T
Instead of considering directly the Schr¨odinger operator Z ⊕ H~ = U −1 H~ (k)dk U T ∗,r
with
Quantum Chaos and Dynamical Entropy
683 ∞ X
H~ (k) =
r Esr (k)Ps,k ,
s=1
we consider its approximation H~r := U −1 with H~r (k) :=
∞ X
Z
⊕ T ∗,r
H~r (k)dk U
(196)
r . Esr (0) + ~vˆ sr · k Ps,k
s=1
The useful property of the Hamiltonian H~r is that its time evolution permutes the Wannier functions: (197) exp −iH~r t/~ φrs,l = exp(−iEsr (0)t/~)φrs,l+tvˆ sr if tvˆ sr ∈ rL, as can be seen from Def. (196). The latter condition holds certainly true if t is a multiple of r2 . We consider the one-particle classical dynamical system (P r , 6r , µr , 8r ) whose phase space P r := N × rL enumerates the Wannier functions of the sublattice, the σ-algebra 6r being the power set of P r . The `-independent measure µr (s, `) of a phase space point (s, `) ∈ P r is given by the expectation µr (s, `) := ω(ˆa† (φrs,` )ˆa(φrs,` )) = ρ˜rs (0)
(198)
of φrs,l , see (125). The motion 8r,t : P r → P r , at times t ∈ Z is defined by 8r,t (s, l) := (s, l + tr2 vˆ sr ). Here we rescaled time by a factor r2 in order to stay in accordance with (197). The many-particle system is described as a Poisson system in the sense of Def. (5.12) built over (P r , 6r , µr , 8r ). However, the Poisson construction automatically takes the Pauli principle into account, since the One-Particle phase space P r is discrete (and thus the non-atomicity assumption does not hold). That is, we consider (P r , 6r , µr , 8r ) with multi-particle phase space P r := {A ⊂ P r }. A ∈ P r is interpreted as the set of phase space cells occupied by electrons. We set Y Y (1 − µr (b)) . µr P(A,B) := µr (a) · a∈A
b∈B
The non-interacting dynamics is given by t
8r (A) := {8r,t (x) | x ∈ A}
(A ∈ P r ).
The C ∗ -algebra C(P r ) of phase space functions is generated by the orthogonal projections
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F. Benatti, T. Hudetz, A. Knauf
P(A,B) ∈ C(P r ), indexed by finite disjoint subsets A, B ⊂ P r , whose value at a phase space point C ∈ P r is given 1, A ⊂ C, B ∩ C = ∅ P(A,B) (C) := . 0, otherwise So P(A,B) (C) = 1 iff the phase space region A is fully occupied by electrons, whereas there is no electron in B. We define the isomorphism I r : C(P r ) → Ar on the projections by Y aˆ † (φrs,` )ˆa(φrs,` ) I r P(A,B) :=
Y
(s,`)∈A
(s0 ,`0 )∈B
aˆ (φrs0 ,`0 )ˆa† (φrs0 ,`0 )
and then extend linearly. Here the orthonormality property of the Wannier functions φrs,` is used. As r → ∞, the measure µr becomes more and more related to the restriction of the state ω to Ar . This may be seen as follows: The expression X X ρ˜rs (`1 − `2 )|φrs,`1 ihφrs,`2 | A= s∈N `1 ,`2 ∈rL
with the Fourier coefficients Z ρ˜rs (`) :=
T∗,r
exp(+ik · `)ρr (s, k)
dk |T∗,r |
(` ∈ rL)
shows that, similar to (125) the two-point function ω aˆ † (φrs1 ,`1 )ˆa(φrs2 ,`2 ) = δs1 ,s2 · ρ˜rs (`1 − `2 ).
(199)
Now if we assume ρ to be smooth (C 1 suffices here), then by partial integration the non-zero Fourier coefficients are seen to decay: ρ˜rs (r`) = O(r−1 ),
(` ∈ L \ {0}),
whereas ρ˜rs (0) is bounded below by mink∈T∗ ρ(m, k) if s enumerates a subband of band m. The higher moment formula (4) for the quasifree state ω together with (199) then shows as r → ∞ that µr P(A,B) ∼ ω I r P(A,B) . A finite phase space region R ⊂ P r carries an 2|R| -dimensional subalgebra AR ⊂ A generated by the family {P(A,R−A) | A ⊂ R} of projections. Clearly r
AR1 ∨ AR2 = AR1 ∪R2 .
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The entropy of such a partition equals Sµr (AR ) =
X
ηF (µr (x)).
(200)
x∈R
Since the algebra Ar is Abelian, the CNT and KS definitions of dynamical entropy coincide and can be easily calculated by lattice counting. In particular for a phase space region R := {(s, `) ∈ P r | ` ∈ B} over a convex observation region B one has for the automorphism σ r := αr2 H r |`Ar the expression hµr (σ r , AR ) = Sµr (AR\8r 1 (R) ), since by Remark 5.27 one has the geometrical contraction R \ ∪nt=1 8r,t (R) = R \ 8r,1 (R).
(201)
(202)
The r.h.s. of (201) can then be calculated using (200). Non-convex regions B can also be treated, although then more effort is needed. The above classical caricature of the Fermi gas can (at least) be used as a testing ground for ideas which one would like to apply to the non-Abelian case. One idea to prove our conjectured formula (134) for the local entropy of the Fermi gas is the following. The automorphisms αH and αH r , r ∈ N all commute, since the one-particle Hamiltonians H and H r have this property. So if a triangle inequality hω,B (α1 ◦ α2 ) ≤ hω,B (α1 ) + hω,B (α2 )
(203)
for commuting automorphisms would hold, we could estimate (using time inversion invariance) |hω,B (αH ) − hω,B (αH r )| ≤ hω,B (αH ◦ α−Hr ). It is known (see Sinai [43]) that (203) does not hold in general. For our classical caricature one can prove (203) using (202) if B is convex. However as the following example shows, no weakened form hω,B (α1 ◦ α2 ) ≤ C(hω,B (α1 ) + hω,B (α2 )) of (203) with C ≥ 1 holds for general non-convex B. Example. By a theorem of Hurwitz there exist irrational numbers ξ ∈ (0, 1) which are badly approximable by rationals in the sense that for c > 0 small there exists no rational approximation p/q with ξ − p ≤ c . q q2 √ The best example is the golden mean ξ = 21 ( 5 − 1).
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We now consider planar shifts with velocities v1 := BN :=
N X
1 0
and v2 :=
0 ξ
and regions
D(l1 ,l2 ) (r)
l1 ,l2 =1
√ with r := c/( 8N ), where Da (r) is the disk of radius r centered at a ∈ R2 . Now the shadow of BN in the v1 - and in the v2 -direction is of size 2rN . In the v1 + v2 -direction it is of size 2rN 2 , since by the above property of ξ no disk shadows any other disk. Thus the entropy for the shift in the v1 + v2 -direction is (at least, since the speed is augmented) N times larger than each of the entropies in the v1 - and v2 -direction. This counterexample shows that one needs good geometrical control if one wants to compare two time evolutions. Another idea for proving formula (134) is to compare the algebras t (A), αH
(A ⊂ A(λB))
at time t with the algebra C t generated by the set of Wannier functions r φs,` | s ∈ N, ` ∈ rL : ` − tvˆ sr ∈ λB
(204)
for some r with 1 r λ (technically one introduces an energy cutoff, uses an enlarged region λBε in (204), etc.). t Then as λ and r go to infinity, one expects a δ-inclusion of the algebra αH (A) by C t , since the deviation of the group velocity ~−1 ∇k Es (k) from the mean velocity vˆ sr on the subbands with Brillouin zone T∗,r becomes smaller so that the direction of the moving particle is fixed with higher precision. The basic idea here is to have some algebraic version of phase space microlocalization. References 1. Aizenman, M., Goldstein, S., Lebowitz, J.: Ergodic Properties of Infinite Systems. In: Dynamical Systems, Theory and Applications. J. Moser, ed. Springer Lecture Notes in Physics 38. Berlin, Heidelberg, New York: Springer, 1975 2. Alicki, R., Fannes, M.: Defining Quantum Dynamical Entropy. Lett. Math. Phys. 32, 75–82 (1994) 3. Arnold, A., Avez, A.: Ergodic Problems of Classical Mechanics. New York: Benjamin, 1968 4. Asch, J., Knauf, A.: Motion in Periodic Potentials. Nonlinearity 11, 175–200 (1998) 5. Ashcroft, N., Mermin, D.: Solid State Physics. New York: Holt, Rinehart and Winston, 1976 6. Benatti, F.: Deterministic Chaos in Infinite Quantum Systems. Trieste Notes in Physics. Berlin: Springer, 1993 7. Benatti, F.: Entropy of a Subalgebra and Quantum Estimation. J. Math. Phys. 37, 5244 (1996) 8. Benatti, F., Grava, T.: The Entropy of a Subalgebra. In: C. Cecchini ed.: Contributions in Probability, in Memory of A. Frigerio, 1996 9. Berry, M.: True Quantum Chaos? An Instructive Example. Proceedings of Yukawa Symposium, 1990 10. Benatti, F., Narnhofer, H., Uhlmann, A.: Optimal Decompositions of Quantum States with Respect to Entropy. Rep. Math. Phys. 38, 123 (1996) 11. Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics, Vol. 1. New York: Springer, 1987 12. Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics, Vol. 2. New York: Springer, 1987
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13. Connes, A., Narnhofer, H., Thirring, W.: Dynamical Entropy for C* algebras and von Neumann Algebras. Commun. Math. Phys. 112, 691 (1987) 14. Conze, J.P.: Entropie d’un groupe ab´elien de transformations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 25, 11–30 (1972) 15. Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. New York: Springer, 1982 16. Donnay, V., Liverani, C.: Potentials on the Two-Torus for Which the Hamiltonian Flow is Ergodic. Commun. Math. Phys. 135, 267–302 (1991) 17. Goldstein, S.: Space-Time Ergodic Properties of Systems of Infinitely Many Independent Particles. Commun. Math. Phys. 39, 303–327 (1975) 18. Goldstein, S., Lebowitz, J.: Ergodic Properties of an Infinite System of Particles Moving Independently in a Periodic Field. Commun. Math. Phys. 37, 1–18 (1974) 19. Gomkrelidze, R.V.: Geometry I. Encyclopaedia of Mathematical Sciences, Vol. 28. Berlin: Springer, 1991 20. Gutzwiller, M.: Chaos in Classical and Quantum Mechanics. Berlin, Heidelberg, New York: Springer, 1990 21. Helffer, B.: ~-pseudodifferential operators and applications: an introduction. In: Rauch ed. Quasiclassical Methods. New York: Springer IMA Vol. Math Appl. 95, 1–49 (1997) 22. Hirsch, M.: Differential Topology. Graduate Texts in Mathematics, Vol. 33. Berlin, Heidelberg, New York: Springer, 1988 23. Hudetz,T.: Topological Entropy for Appropriately Approximated C* algebras. J. Math. Phys. 35, 4303 (1994) 24. Hudetz, T.: Quantum Topological Entropy: First Steps of a Pedestrian Approach. In: Quantum Probability and Related Topics III, L. Accardi et al. eds., Singapore: World Scientific, 1993 25. Katok, A., Knieper, G., Pollicott, M., Weiss, H.: Differentiability and Analyticity of Topological Entropy for Anosov and Geodesic Flows. Inventiones Math. 98, 581–597 (1989) 26. Klein, M., Knauf, A.: Classical Planar Scattering by Coulombic Potentials. Lecture Notes in Physics m 13. Berlin, Heidelberg, New York: Springer, 1993 27. Knauf, A.: Ergodic and Topological Properties of Coulombic Periodic Potentials. Commun. Math. Phys. 110, 89–112 (1987) 28. Knauf, A.: Coulombic Periodic Potentials: The Quantum Case. Ann. Phys. 191, 205–240 (1989) 29. Knauf, A.: The Size of the Shadow Boundary Projection. Manuscripta Math. 95, 519–527 (1998) 30. Knauf, A., Sinai, Ya. G.: Classical Nonintegrability, Quantum Chaos. With a contribution by V. Baladi. DMV Seminar, Band 27, Basel: Birkh¨auser, 1997 31. Knieper, G., Weiss, H.: Regularity of measure theoretic entropy for geodesic flows of negative curvature, I. Invent. Math. 95, 579–589 (1989) 32. Ma˜ne´ , R.: Ergodic Theory and Differentiable Dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Vol. 8. Berlin: Springer, 1987 33. Narnhofer, H., Thirring, W.: Quantum Field Theories with Galilei-Invariant Interactions. Phys. Rev. Lett. 64, 1863–1866 (1990) 34. Narnhofer, H., Thirring, W.: Galilei-Invariant Quantum Field Theories with Pair Interactions: A Review. Int. J. Mod. Phys. 17, 2937–2970 (1991) 35. Narnhofer, H., Thirring, W.: Dynamical Entropy of Quantum Systems and their Abelian Counterpart. In: Emch, Hegerfeldt, Streit eds.: On Klauder’s Path. Singapore: World Scientific, 1994 36. Nobbe, B.: Classical Motion in Two-Dimensional Crystals. J. Stat. Phys. 78, 1591–1605 (1995) 37. Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin: Springer, 1993 38. Park, Y. M., Shin, H.H.: Dynamical Entropy of Quasi-Local Algebras in Quantum Statistical Mechanics. Commun. Math. Phys. 144, 149–161 (1992) 39. Park, Y. M., Shin, H.H.: Dynamical Entropy of Space Translations of CAR and CCR Algebras with Respect to Quasi-Free States. Commun. Math. Phys. 152, 497–537 (1992) 40. Pasmanter, R.: From Chaotic Advection to Chaotic Schr¨odinger-Type Equations. Phys. Rev. A 42, 3622 (1990) 41. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I: Functional Analysis. Vol. II: Fourier Analysis, Self Adjointness. Vol. IV: Analysis of Operators. New York, San Francisco, London: Academic Press, 1972, 1975, 1978 42. Robert, D.: Autour de l’approximation semi-classique. Boston, Basel, Stuttgart: Birkh¨auser, 1987 43. Sinai, Ya. G.: Topics in Ergodic Theory. Princeton, NJ: Princeton University Press, 1994 44. Sauvageot, J. P., Thouvenot, J. P.: Une nouvelle definition de l‘entropie dynamique des syst`emes non commutatifs. Commun. Math. Phys. 145, 411–423 (1992)
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45. Størmer, E., Voiculescu, D.: Entropy of Bogoliubov Automorphisms of the Canonical Anticommutation Relations. Commun. Math. Phys. 133, 521–542 (1990) 46. Thirring, W.: The Dynamical Entropy of Quantum Systems. In: Recent Developments in Mathematical Physics, Proceedings of the XXVI Int. Universit¨atswochen f¨ur Kernphysik, H. Mitter, L. Pittner eds., Berlin–Heidelberg–New York: Springer, 1987 47. Thomsen, K.: Topological Entropy for Endomorphisms of Local C∗ -Algebras. Commun. Math. Phys. 164, 181–193 (1994) 48. Troll, G.: How to Escape a Sawtooth – Transition to Chaos in a Simple Scattering Model. Nonlinearity 5, 1151–1192 (1992) 49. Voiculescu, D. V.: Dynamical Approximation Entropies and Topological Entropy in Operator Algebras. Commun. Math. Phys. 170, 249–281 (1995) 50. Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, Vol. 79, Berlin, Heidelberg, New York: Springer, 1982 51. Wehrl, A.: General Properties of Entropy. Rev. Mod. Phys. 50, 221–260 (1978) 52. Wilcox, C.: Theory of Bloch Waves. J. d’Analyse Math. 33, 146–167 (1978) Communicated by A. Connes
Commun. Math. Phys. 198, 689 – 703 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
Instantons on Noncommutative R4 , and (2, 0) Superconformal Six Dimensional Theory Nikita Nekrasov1,2 , Albert Schwarz3 1
Institute of Theoretical and Experimental Physics, 117259, Moscow, Russia Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected] 3 Department of Mathematics, UCDavis, Davis, CA 95616, USA. E-mail: [email protected]
2
Received: 19 March 1998 / Accepted: 14 April 1998
Abstract: We show that the resolution of moduli space of ideal instantons parameterizes the instantons on noncommutative R4 . This moduli space appears to be the Higgs branch of the theory of k D0-branes bound to N D4-branes by the expectation value of the B field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of (2, 0) superconformal theories in six dimensions. 1. Introduction The appearence of noncommutative geometry [1] in the physics of D-branes has been anticipated with the very understanding of the fact that the gauge theory on the worldvolume of N coincident D-branes is a non-abelian gauge theory [2], [3]. In this theory the scalar fields Xi in the adjoint representation are the non-abelian generalizations of the transverse coordinates of the branes. It is known that the compactifications of Matrix theory [4], [5] on tori Td exhibit richer structures as the dimensionality d of the tori increases [6], [7]. The compactification on a torus implies that certain constraints are imposed on the matrices Xi : Xi + 2πRi δij = Uj Xi Uj−1 .
(1.1)
It seems natural to study all possible solutions Ui to the consistency equations for the compactification of the matrix fields1 [9]. Recently, the noncommutative torus emerged as one of the solutions to (1.1), [9]. It has been argued that the parameter of noncommutativity is related to the flux of the B-field through the torus. It has been further shown in [10], that the compactification on a noncommutative torus can be thought of as a T -dual to a limit of the conventional 1 The conjecture of [8] is that the non-abelian tensor fields in six dimensions would also appear as such solutions.
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N. Nekrasov, A. Schwarz
compactification on a commutative torus. See [11] for further developments in the studies of compactifications on low-dimensional tori. On the other hand, the modified self-duality equations on the matrices in the Matrix description of fivebrane theory has been used in [12] in the study of quantum mechanics on the instanton moduli space. The modification is most easily described in the framework of ADHM equations. It makes the moduli space smooth and allows to define a six dimensional theory decoupled from the eleven-dimensional supergravity and all other M -theoretic degrees of freedom. The heuristric reason for the possibility of such decoupling is the fact that the Higgs branch of the theory is smooth and there is no place for the Coulomb branch to touch it. In this paper we propose an explanation of the latter construction in terms of noncommutative geometry. We show that the solutions to modified ADHM equations parameterize (anti-)self-dual gauge fields on noncommutative R4 . 2. Instantons on a Commutative Space Let X be a four dimensional compact Riemannian manifold and P a principal G-bundle on it, with G = U (N ). The connection A is called anti-self-dual (ASD), or an instanton, if its curvature obeys the equation: F + := 21 (F + ∗F ) = 0, where ∗ : k → 4−k is the Hodge star. The importance of ASD gauge fields in physics stems from the fact that they R minimize the Yang–Mills action in a given topological sector, i.e. for fixed k = − 8π1 2 TrF ∧ F . In supersymmetric gauge theories the instantons are the configurations of gauge fields which preserve some supersymmetry, since the self-dual part of F appears in the right-hand side of susy transformations. For the same reason they play a major role in Matrix theory. The space of ASD gauge gauge fields modulo gauge transformations is called the moduli space of instantons Mk and in a generic situation it is a smooth manifold of dimension N2 − 1 (χ + σ), 4N k − 2 where χ and σ are the Euler characteristics and signature of X respectively. The moduli space M is non-compact. The lack of compactness is due to the so-called point-like instantons. What can happen is that for a sequence of ASD connections Ai R the region Di where some topological charge − 8π1 2 Di TrF ∧ F is concentrated can ¯ k due to Uhlenbeck [13] which shrink to zero size. There exists a compactification M simply adds the centers of the point-like instantons: ¯ k = Mk ∪ Mk−1 × X ∪ . . . ∪ Mk−l × Syml X . . . M ¯ k has which is suitable for certain purposes but not for all. In particular, the space M orbifold singularities. One can also study non-compact spaces, X = R4 being the first example. In posing a moduli problem one has to specify the conditions on the behavior of the gauge fields at infinity. The natural condition is : Aµ ∼ g −1 ∂µ g + O( r12 ) as r → ∞. One may also restrict the allowed gauge transformations to those which tend to 1 at infinity.
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2.1. Review of ADHM construction. Atiyah–Drinfeld–Hitchin–Manin describe [14] a way of getting the solutions obeying the asymptotics stated above to the instanton equations on R4 in terms of solutions to some quadratic matrix equations. More specifically, in order to describe charge N instantons with gauge group U (k) one starts with the following data: 1. A pair of complex hermitian vector spaces V = CN and W = Ck . 2. The operators B0 , B1 ∈ Hom(V, V ), I ∈ Hom(W, V ), J ∈ Hom(V, W ), which must obey the equations µr = 0, µc = 0, where: µr =[B0 , B0† ] + [B1 , B1† ] + II † − J † J, µc =[B0 , B1 ] + IJ.
(2.1)
There is also a non-degeneracy condition which must be imposed by hand, namely, the set (B, I, J) should have trivial stabilizer in the U (V ) group. For z = (z0 , z1 ) ∈ C2 ≈ R4 define an operator Dz† : V ⊕ V ⊕ W → V ⊕ V by the formula: τ Dz† = z† , σz (2.2) B 1 − z1 τz = B0 − z0 −B1 + z1 I , σ z = B 0 − z0 . J Given the matrices obeying all the conditions above, the actual instanton solution is determined by the following rather explicit formulae: Aµ = ψ † ∂µ ψ,
(2.3)
where ψ : W → V ⊕ V ⊕ W solves the equations: D† ψ = 0 and is normalized : ψ † ψ = 1. The moduli space of instantons with fixed framing at infinity is identified with −1 M = µ−1 (2.4) r (0) ∩ µc (0) /U (V ), where U (V ) is the group of unitary transformations of V acting on the matrices B, I, J in a natural way. ¯ k is a 2.2. Regularization of ADHM data. As we noted above the compactification M singular manifold. One may resolve it to a smooth variety by deforming the equations µr = µc = 0 to µr = ζr Id, µc = 0. One may add a constant to µc as well but this modification is equivalent to the one already considered by a linear transformation of † the data B0,1 , B0,1 , I, I † , J, J † . The modification has been studied mathematically by various people and we recommend the beautiful lectures by H. Nakajima [15] for a review. The deformed data form a moduli space: −1 Mζ = µ−1 (2.5) r (ζId) ∩ µc (0) /U (V ) and they parameterize the torsion free sheaves on CP2 with fixed framing at the line at infinity. As we shall show in the next section, the deformed moduli space parameterizes the instantons on the noncommutative R4 .
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3. Instantons on Noncommutative Spaces The paradigm of noncommutative geometry is to describe the geometry of ordinary spaces in terms of the algebra of (smooth, continuous, ...) functions and then generalizing to the noncommutative case. In this sense, the noncommutative R4 is the algebra Aω generated by xα , α = 1, 2, 3, 4 which obeys the relations: [xα , xβ ] = ωαβ ,
(3.1)
where ωαβ is a constant antisymmetric matrix. There are three distinct cases one may consider: 1. ω has rank 0. In this case Aω is isomorphic to the algebra of functions on the ordinary R4 . 2. ω has rank 2. In this case Aω is the algebra of functions on the ordinary R2 times the noncommutative R2 = {(p, q)|[p, q] = −i}. 3. ω has rank 4. In this case Aω is the noncommutative R4 . Let π αβ be the inverse matrix to ωαβ and xα = π αβ xβ . Let ~x = (xα ) and ~x∨ = (xα ). The algebra depends essentially on one number ζ (which can be scaled away, but we shall keep it in order to be able to take a limit to the commutative case). We shall denote it as Aζ . Introduce the generators: z0 = x1 + ix2 , z1 = x3 + ix4 , then2 ζ [z0 , z¯0 ] = [z1 , z¯1 ] = − . 2
(3.2)
The commutation relations (3.2), have an obvious group of automorphisms of the form xα 7→ xα + βα · 1, βα ∈ R. We denote the Lie algebra of this group by g. For the algebra (3.2) to represent the algebra of real-valued functions (and be represented by Hermitian operators) we need ζ ∈ R. ( In the language of mathematics we can say that this condition means that the algebra at hand has an involution.) We choose ζ > 0. Of course, the algebra of polynomials in z, z¯ should be completed in some way. P We propose to start with the algebra EndH of operators acting in the Fock space H = (n0 ,n1 )∈Z2+ C|n0 , n1 i, where z, z¯ are represented as creation-annihilation operators: √ √ z0 |n0 , n1 i = ζ2 n0 + 1|n0 + 1, n1 i z¯0 |n0 , n1 i = ζ2 n0 |n0 − 1, n1 i, (3.3) ζ√ ζ√ z1 |n0 , n1 i = 2 n1 + 1|n0 , n1 + 1i z¯1 |n0 , n1 i = 2 n1 |n0 , n1 − 1i. The algebra EndH has a subalgebra End0 H of operators A which have finite norm; we 1
will take the Hilbert–Schmidt norm: Tr H (AA† ) 2 . We consider an algebra Aζ defined as a subalgebra of End0 H which consists of smooth operators, i.e. those A for which the function fA : g → End0 H, fA (~t) = Ad~t·~x A is smooth. Notice that 1 does not belong to this algebra. This is a consequence of non-compactness of R4 . One can represent the elements of the Fock space H as L2 functions in two variables q0 = x1 and q1 = x3 (oscillator wave functions in real polarization). It is actually possible to prove using the results of [16] that smooth operators are those whose matrix elements A(q00 , q10 ; q0 , q1 ) belong to the Schwartz space S(R2 × R2 ). We consider representations of the involutive algebra Aζ by means of operators in Hilbert space (Hilbert modules with involution in mathematical terminology). A vector bundle over a noncommutative space A is a 2
So, π µν is proportional to ζ, while ωαβ is proportional to
limits.
1 ζ
in accordance with the standard quasiclassical
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projective module E, i.e. such a module for which another module E ⊥ exists with the property E ⊕ E ⊥ = A ⊕ . . . ⊕ A. (The last module is called free. In the commutative case free modules correspond to trivial bundles.) In our description of instantons over R4 we shall be dealing with free modules only. The notion of connection in the bundle E has several definitions. The most convenient for us at the moment will be the following one. Let g ⊂ Aut(A) be a Lie sub-algebra of the algebra of automorphisms of A. Then the connection ∇ is an operator ∇ : g×E → E which obeys the Leibnitz rule: ∇ξ (f · s) = f · ∇ξ s + ξ(f ) · s,
f ∈ A,
ξ ∈ g,
s ∈ E.
(3.4)
The curvature of ∇ is an operator: F (∇) : 32 g × E → E,
F (∇)(ξ, η) = [∇ξ , ∇η ] − ∇[ξ,η] .
(3.5)
3.1. Instantons on noncommutative R4 . Just like in the commutative case one may define the Yang–Mills action, the instanton equations, etc. See [1] for the definitions and [9] for recent discussion. We are looking at the solutions of the ASD conditions F+ = 0 on the connection in the module E over Aζ , where the + sign means the self-dual part with respect to the natural Hodge star operator acting on the 32 g where g ≈ R4 is the abelian Lie algebra of automorphisms of Aζ . Now we are going to show that the resolution (2.5) gives rise to the ADHM description of instantons on the noncommutative R4 . Indeed, the core of the ADHM construction are the equations τz σz = 0,
τz τz† = σz† σz ,
(3.6)
where the operators σz , τz are constructed as above. Now suppose that the matrices B, B † , I † , I, J, J † obey the modified equations (2.5). Then (3.6) are no longer valid but they will be valid if the coordinate functions zi , z¯i will not commute! In fact, by imposing the commutation law [z0 , z¯0 ] = [z1 , z¯1 ] = − ζ2 we fulfill (3.6) as the term with ζ from commutators of B’s is now compensated by the commutators of z’s! We now follow the steps of the ordinary ADHM construction. We define an operator: Dz† : (V ⊕ V ⊕ W ) ⊗ Aζ → (V ⊕ V ) ⊗ Aζ
(3.7)
by the same formula (2.2). We look for the solution to the equation Dz† ψ = 0,
ψ : W ⊗ Aζ → (V ⊕ V ⊕ W ) ⊗ Aζ
(3.8)
which is again normalized: ψ † ψ = IdW ⊗Aζ . These are defined up to unitary gauge transformations g acting on ψ on the right. Again, Aµ = ψ † ∂µ ψ, where the derivative is understood as the action of g = R4 on Aζ by translations. We may now derive the formula for the curvature of the gauge field A. The derivation is very similar to the commutative case and it yields: 1 (3.9) F = ψ † dDz † dDz† ψ. Dz D z
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The operator Dz† Dz : (V ⊕ V ) ⊗ Aζ → (V ⊕ V ) ⊗ Aζ is again block-diagonal: 1z 0 Dz† Dz = , 1z = τz τz† = σz† σz . (3.10) 0 1z Hence, just as in the commutative case: 1 † † 1 † F =ψ dτz dτz + dσz dσ ψ = 1z 1z z dz¯0 11z dz0 + dz1 11z dz¯1 −dz¯0 11z dz1 + dz1 11z dz¯0 = ψ † −dz¯1 11 dz0 + dz0 11 dz¯1 dz¯1 11 dz1 + dz0 11 dz¯0 z z z z 0
0
0
0 ψ
(3.11)
0
which is anti-self-dual. We should warn the reader that here dzi , dz¯i are just the generating anti-commuting parameters for representing the matrix F∇ (ξ, η) in short print. They commute with zi , z¯i . 3.2. Commutative interpretation of the noncommutative equations. The construction above is not yet completely rigorous. We must prove that ψ exists and that 1z is invertible. A useful technique is to represent the equations over Aζ in terms of the (perhaps differential) equations on ordinary functions. In this way the multiplication of operators a · b is mapped to 1
a ? b(x) = e 2 π
µν
∂2 ∂ξµ ∂η ν
a(x + ξ)b(x + η)|ξ=η=0 .
(3.12)
Now we can study the questions we posed in the beginning of the section. It follows from (3.10) that the condition that υ is a zero mode of 1z is equivalent to the conditions: τz† υ = 0, σz υ = 0, where υ is an element of our algebra that is considered as an algebra of Hilbert–Schmidt operators in Fock space. Let us remark that an operator equation KL = 0 is equivalent to the condition that the image of the operator L is contained in the kernel of K. In other words if we can solve the equation Kv = 0, where v is a vector then we can also solve the operator equation KL = 0, where L is an unknown operator. This remark allows us to say that in the conditions above we can consider υ as an element of Fock space; if a non-zero solution does not exist in this new setting it does not exist in the old setting either. The elements of the Fock space H can be represented either as polynomials in z0 , z1 (this representation we use below) or as L2 functions on R2 with coordinates q0 , q1 . The equations for the vector υ can be written in holomorphic representation as follows: ∂ υ, B0 υ = z0 υ, B0† υ = ζ4 ∂z 0 † ζ ∂ (3.13) B1 υ = z1 υ, B1 υ = 4 ∂z1 υ, † Jυ = 0, I υ = 0. The right column suggests that: υ∼e
4z0 B0 ζ
z 0 B0
υ0 (z1 ),
which is inconsistent with the left column equations. We also say that the left column equations imply that the creation operators have a finite-dimensional invariant subspace which is impossible. In the case N = 1 the operator 1z is explicitly positive definite.
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The question of existence of ψ is addressed similarly: write ψ = ψ+ ⊕ ψ− ⊕ ξ, then (3.8) assumes the form: ζ ∂ ζ ∂ − z0 )ψ+ − (B1 + − z1 )ψ− + Iξ = 0, 4 ∂ z¯0 4 ∂ z¯1 ζ ∂ ζ ∂ − z¯1 )ψ+ + (B0† − − z¯0 )ψ− + J † ξ = 0. (B1† − 4 ∂z1 4 ∂z0 (B0 +
(3.14)
Here ψ is a function which corresponds to an operator from our algebra. Equation (3.14) can be rewritten in the form: (3.15) DA 9 = −4 ψ+ , ψ−
for 9=
4=
−J † ξ Iξ
and DA being the Dirac operator in the gauge field Aµ dxµ , where 4 −B1† dz1 − B0† dz0 + B0 dz¯0 + B1 dz¯1 . Aµ − 21 ωµν xν dxµ = ζ The gauge field A has constant curvature. Now let us write down Eq. (3.14) for vectors in the Fock space, this time using the L2 representation: v = v+ (q) ⊕ v− (q) ⊕ w(q), ζ ∂ ζ ∂ − q0 )v+ − (B1 − + q1 )v− + Iw = 0, 4 ∂q0 4 ∂q1 ζ ∂ ζ ∂ − q1 )v+ + (B0† − − q0 )v− + J † w = 0. (B1† − 4 ∂q1 4 ∂q0 (B0 +
(3.16)
It has again the form (3.15) with DA being a Dirac-like operator in two dimensions. Let us assume that the operator DA has no zero modes (we hope that this assumption can be justified in the framework of perturbation theory with respect to ζ). Then we can write a solution to (3.15) in terms of the Green’s function of the operator DA . The space of solutions to (3.16) is identified with the space of w’s, i.e. the space of W -valued L2 functions on R2 . Now, given the solution v+ (w) ⊕ v− (w) ⊕ w to (3.16) we construct a solution to (3.15) as follows: ψ is supposed to map W ⊗H⊗H → (V ⊕V ⊕W )⊗H⊗H (where we used the fact that the algebra of Hilbert–Schmidt operators can be identified with the Hilbert tensor product H ⊗ H and that Aζ ⊂ H ⊗ H). Now, ψ˜ = w ⊗ w0 7→ v+ (w) ⊕ v− (w) ⊕ w ⊗ w0 g does the job for any non-degenerate g ∈ GLk (A). What remains is to normalize: ψ = − 1 ˜ This normalization ψ † ψ = 1 reduces the freedom of the choice of solutions ψ˜ † ψ˜ 2 ψ. to (3.16) to the unitary gauge transformations in U (k). One can also rewrite the ASD equations for the gauge field on the noncommutative R4 in terms of commutative functions which are Wick symbols of the gauge fields. Let Aµ (x) be four functions on R4 and consider the equations: + Fµν =0
with
(3.17)
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N. Nekrasov, A. Schwarz i Fµν,j = ∂µ Aiν,j − ∂ν Aiµ,j + Aiµ,k ? Akν,j − Aiν,k ? Akµ,j .
(3.18)
Thus, the noncommutative ASD equations can be thought of as the deformation of the ordinary ASD equations. The solutions to (3.17) are automatically the solutions to the deformed Yang–Mills equations: ∂µ Fµν + Aµ ? Fµν = 0.
(3.19)
3.3. The completeness of the ADHM construction in the noncommutative case. Just like in the case of ordinary R4 one faces the question – whether the full set of solutions of ASD equations on noncommutative R4 is enumerated by the solutions to the matrix equation (3.6). It is natural to try to imitate the arguments of [17] and express the matrices Bα , I, J in terms of solutions to the massless Dirac equations in the instanton background field. We simply sketch the relevant steps of the construction without giving any proofs. The setup is similar to the commutative case. Given a projective module E over Aζ one studies the associated modules S± = E ⊗Aζ S± , S± = C2 ⊗ Aζ . Given a connection ∇ in the module E we form the Dirac-Weyl operators D† : S+ → S− ,
D : S− → S+
(3.20)
by the standard formulae: D† = ∇α ⊗ σ α , D = ∇α ⊗ σ¯ α , where σ α : S+ → S− , σ¯ µ : S− → S+ are essentially the ordinary Pauli matrices and σ4 = σ¯ 4 = 1. One proves the identities: − DD† = 1∇ ⊗ 1 + Fµν ⊗ σ µν ,
+ D† D = 1∇ ⊗ 1 + Fµν ⊗ σ¯ µν .
(3.21)
Here 1∇ is the covariant Laplacian E → E. Since there are no normalizable (see below) solutions to the equations 1∇ φ = 0 then for the ASD ∇ one concludes that there are no solutions to the equation Dψ = 0. On the other hand, the index arguments predict, just like in the commutative case, the existence of N normalizable (in the sense, described in the next paragraph) zero modes of D† . Let ψ i ∈ E ⊗ S+ be a solution to D† ψ i = 0, i = 1, . . . , N . Then one may define a projection (just like in the commutative case) of xα ψ i onto the space of zero modes of D† : (3.22) xα δji − Biα,j ψ j = D(. . . ), where Bα is some matrix (with C-valued entries). We call B0 = B1 + iB2 , B1 = B3 + iB4 , etc. The matrices I, J are recovered from the large ~x2 asymptotics of ψ i . In order to explain what it means in the noncommutative setting we represent the coordinates on the noncommutative R4 as creation-annihilation operators acting in the auxiliary Fock space H. Then the operator ψ i has the corresponding Wick symbol ψ˜ i which is simply a function on R4 whose large ~x2 limit is well-defined and is independent on the ordering prescription since for large ~x2 >> ζ one may neglect the noncommutativity of the coordinates. 4. Examples 4.1. Abelian instantons. It is very interesting to study the case k = 1. In the commutative case there are no solutions to the abelian instanton equations except for the trivial ones. We shall see that for every N there are non-trivial abelian instantons in the noncommutative case.
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We need to solve the equations µr = ζId, µc = 0. It can be shown that for ζ > 0 the solution must have J = 0 [0]. Moreover, on a dense open set in Mζ the matrices B0 , B1 can be diagonalized by a complex gauge transformation3 : Bα → diag βα1 , . . . , βαN . (4.1) The eigenvalues βαi parameterize a set of N points on R4 . The actual moduli space is the resolution of singularities of the symmetric product. Now suppose (B0 , B1 , I) is a solution to the ADHM equations. If we write the element of V ⊕V ⊕W as ψ = ψ0 ⊕ψ1 ⊕ξ then the equation D† ψ = 0 is solved as: ψα = −(Bα − zα )† δ −1 Iξ, δ=
1 X
(Bα − zα )(Bα† − z¯α ).
(4.2)
α=0
The operator ξ is determined from the equation ψ † ψ = 1: ξ = 1 + I † δ −1 I
− 1 2
.
(4.3)
¯ − ∂ξξ −1 , A = ξ −1 ∂ξ
(4.4)
The connection A = ψ † dψ has Yang’s form:
¯ where ∂ = dzα ∂z∂α , ∂¯ = dz¯α¯ ∂∂z¯α¯ . Explicitly:
A = ξ −1 dξ + ξ −1 αξ, where the “gauge transformed” connection α is equal to: α = ξ 2 ∂ξ −2 = 1 I † δ −1 (Bα† − z¯α )dzα δ −1 I. 1 + I † δ −1 I
(4.5)
1 2 , d = z0 z¯0 +z¯1 z1 , and the gauge field α is explictly non-singular For N = 1: ξ = d−ζ/2 d+ζ/2 if the correct ordering is used: α=
1 (z¯0 dz0 + z¯1 dz1 ) . d(d + ζ/2)
(4.6)
One can also compute the curvature: ζ f3 (dz0 dz¯0 − dz1 dz¯1 ) + f+ dz¯0 dz1 + f− dz¯1 dz0 , (d − ζ/2)d(d + ζ/2) f3 = z0 z¯0 − z1 z¯1 , f+ = 2z0 z¯1 , f− = 2z1 z¯0 . FA =
(4.7)
ζ has a singularity at the state |0, 0i but it is projected out since The factor (d−ζ/2)d(d+ζ/2) f3,± always have z¯0 or z¯1 on the right. The action density is given by 3
We thank D. Bernard for pointing out an error in the earlier version of this paper.
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1 ζ2 1 Sˆ = − 2 FA FA = 5, 8π 4π 2 d2 (d − ζ/2)(d + ζ/2)
(4.8)
where 5 = 1 − |0, 0ih0, 0|. We may define the total action as (ζπ) Tr H Sˆ = 4 2
∞ X N =1
1 = 1. N (N + 1)(N + 2)
4.2. ’t Hooft solutions. It is also relatively easy to describe the noncommutative analogues of ’t Hooft solutions for k = 2: Aµ = i6µν 8−1 ∂ν 8,
(4.9)
where 6µν is self-dual in µν and takes values in traceless two by two Hermitian matrices. In the commutative case the ASD conditions boil down to the Laplace equation on 8 [18]. In the noncommutative case the potential trouble comes from the term in the curvature Fµν : {6µα , 6νβ }[Jα , Jβ ] with Jα = 8−1 ∂α 8. It is easy to show that the problematic piece is equal to [Jµ , Jν ] − 21 µναβ [J α , J β ] which is explicitly anti-self-dual. Hence we have shown that the ansatz (4.9) works in the noncommutative case if 8 obeys the Laplace equation which is now to be solved in the noncommutative setting. A solution looks exactly like the commutative ansatz: 8=1+
N X i=1
ρ2i , |~x − β~i |2
(4.10)
where now the components of ~x = (xµ ) represent the noncommuting coordinates. One might wonder what are the properties of 8 viewed as an operator in a Fock space, where xα ’s are realized as the creation and annihilation operators. By acting on a vacuum state |0, 0i in the occupation number representation 8 creates a sum of the coherent states. We may represent 8 as follows: 8=1+
N X
ρ2i υi ,
i=1 ∨
~
∨
~
where υi = e−~x ·βi ϒe~x ·βi , ϒ = ~x12 . Let us now prove that each υi obeys the Laplace equation. Since the automorphisms in g = R4 are internal and they commute it is sufficient to prove that ϒ obeys the Laplace equation. For the latter it is enough to check that ~x2 ϒ + ϒ~x2 = 2~xϒ~x which is true for ϒ = ~x12 . The operator ϒ is diagonal in the occupation number repre1 sentation and its eigenvalue on a state |n0 , n1 i is equal to ζ (n +n . +1) 2
0
1
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5. Relation to Matrix Description of Six Dimensional (2, 0) Theory In the recent papers [12], [19] the proposal for Matrix description of six dimensional (2, 0) superconformal theory of k coincident fivebranes has been made. The theory which arises on the worldvolume of k coincident fivebranes has the property that when it is compactified on a circle it becomes U (k) gauge theory in 4 + 1 dimension which the coupling which goes to zero as the radius goes to zero. So, by compactifying the theory on a light-like circle one gets the gauge theory with zero coupling. Of course, in a given instanton sector the only surviving gauge configurations are instantons. The theory becomes a quantum mechanics on the moduli space of instantons. But, the latter space has singularities coming from the point-like instantons and at these singularities a second branch of the theory, corresponding to the emission or absorption of D0-branes develops. The theory becomes interacting with the bulk degrees of freedom which makes it harder to study. The proposal of [12] was to deform the instanton moduli space by turning on a B field along four dimensions which serves as a FI term ζ in (2.5). The interpretation of our paper is that turning on the B field effectively makes R4 noncommutative and the instantons live on it. Now, the problem of instantons shrinking to zero size is cured by quantum fluctuations! Indeed, the position of the point-like instanton is smeared over a region of size ∼ ζ which makes it no longer point-like. So the Higgs branch becomes a smooth hyperkahler manifold and the theory becomes six dimensional (at least decoupled from the bulk degrees of freedom). 6. Future Directions In this section we briefly sketch a few directions of future research. 6.1. Nahm’s transform. Let us define an n-dimensional noncommutative torus Aθ as an associative involutive algebra with unit generated by the unitary generators U1 , ..., Un obeying Uµ Uν = eiθµν Uν Uµ . Here θµν is an antisymmetric tensor; we can also consider it as a 2-form on Rn . The infinitesimal automorphisms δα Uβ = δαβ Uβ generate the abelian Lie algebra Lθ acting on Aθ . As before we use Lθ to define the notion of connection in a Aθ -module. We always consider modules equipped with Hermitian inner product and Hermitian connections. We will construct a generalization of Nahm’s transform [20] relating connections on Aθ modules to connections on Aθˆ -modules [21]. Here Aθ and Aθˆ are two four-dimensional noncommutative tori. To define a noncommutative generalization of Nahm’s transform ˆ µ . The we need a (Aθ , Aθˆ )-module P with Aθ -connection ∇α and Aθˆ -connection ∇ fact that P is a (Aθ , Aθˆ )-module means that P is a left Aθ -module and a right Aθˆ module; we assume that (ax)b = a(xb) for a ∈ Aθ , b ∈ Aθˆ , x ∈ P. In other words, P can be considered as a Aθ⊕θˆ -module, where Aθ⊕θˆ is an eight-dimensional noncommutative torus corresponding to the 2-form θ ⊕ θˆ on R8 . Assume that the commutators ˆ µ, ∇ ˆ ν ], [∇α , ∇ ˆ µ ] are c-numbers: [∇α , ∇β ], [∇ ˆ µ, ∇ ˆ ν ] = ωˆ µν , [∇α , ∇ ˆ µ ] = σαµ . [∇α , ∇β ] = ωαβ , [∇
(6.1)
ˆ µ on A ˆ is The condition (6.1) implies that the curvature of connection ∇α ⊕ ∇ θ⊕θ constant. One more assumption is that ∇α commutes with multiplication by elements ˆ α commutes with multiplication by elements of Aθ . Of course, the module of Aθˆ and ∇
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ˆ µ obeying the above requirements exist only under certain P and the connections ∇α , ∇ ˆ conditions on θ, θ. For any right Aθ -module R with connection ∇R α we consider Dirac operator D = 0α (∇R α + ∇α ) acting in the tensor product (R ⊗Aθ P) ⊗ S. (To define 0-matrices we introduce an inner product in Lθ and in Lθˆ .) This operator commutes with multiplication by the elements of Aθˆ , hence the space of zero modes ˆ The connection ∇ ˆ µ induces a of D can be regarded as Aθˆ -module; we denote it as R. 0 ˆ connection ∇µ on (R ⊗Aθ P) ⊗ S. ˆ ˆ ˆ0 We define a connection ∇R µ on R as P ∇µ , where P is the orthogonal projection:
ˆ (R ⊗Aθ P) ⊗ S → R. The above construction can be regarded as a generalized Nahm’s transform. To prove that its properties are similar to the properties of the standard Nahm’s transform ˆ µ . It is we should impose additional conditions on module P and connections ∇α , ∇ sufficient to assume that σαµ determines a non-degenerate pairing between Lθ and Lθˆ . Then we can use this pairing to define an inner product in Lθˆ . A connection ∇R α is + + ωαβ · 1 = 0. (This an analogue of ASD connection if its curvature Fαβ obeys Fαβ condition is equivalent to antiselfduality of the connection ∇R α + ∇α .) Then one can + − ωˆ µν · 1 = 0. prove that the curvature Fˆµν satisfies Fˆµν ˆ We plan The forms ω, ωˆ obey certain quantization conditions which depend on θ, θ. to return to this issue elsewhere. Thus, Nahm’s transform maps the modified instantons on one noncommutative torus to the modified instantons on the other noncommutative torus. It can be thought of as an analogue of Morita equivalence. The last notion is usually used in the context of equivalence of algebras and the categories of left- and/or right- modules over them.4 In the context of gauge theories we study the modules with connections. Nahm’s construction gives correspondence between such modules; modules with ASD connections are mapped to each other. The instantons on the noncommutative torus appear in the problem of D0-branes bound to D4-branes wrapping the four-torus A = T 4 with the B-field turned on. T duality maps the torus A to its dual and the natural conjecture [22] is that on the level of low-energy fields it maps the instantons on A to those on A∨ . There exist stronger conjectures about the moduli space of instantons on the four-torus which fulfill the base of the constructions of six dimensional interacting string theories [23] and also provide a heuristic test of U -dualities [24]. One assumes that MN,U (k) = SymkN T 4 and studies the two dimensional sigma model with target MN,U (k) . Since the symmetric product is an orbifold of the hyperkahler space one may perturb the theory by the marginal operators responsible for blowing up the singularities of the orbifold. As was argued in [25] such a perturbation breaks the natural symmetries of the problem (and cannot be 4 Two algebras A, B are called Morita equivalent if there exist the (A, B) module E and (B, A) module F s.t. E ⊗B F ≈ A, F ⊗A E ≈ B. For example, every algebra A is Morita equivalent to the algebra MatN (A) of matrices with coefficients in A. Given two Morita equivalent algebras one may construct for any A-module E a B-module F and vice versa: F = F ⊗A E. The fact that the holomorphic counterpart of Nahm’s transform – the Fourier–Mukai transform – can be understood in the framework of generalized Morita equivalence has been pointed out to us by M. Kontsevich.
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interpreted as responsible for interactions of little strings). We have argued that given a noncommutative torus (or noncommutative R4 ), where the relevant SU (2) symmetry is already broken by the noncommutative deformation, the moduli space of instantons is already smooth, and the decoupling arguments can be applied. 6.2. Instantons on noncommutative ALE spaces. The asymptotically locally euclidean (ALE) manifolds X0 are the hyperkahler resolutions of singularities of the orbifold C2 /0 for 0 ⊂ SU (2) being a discrete subgroup. The subgroups correspond to ADE Lie groups G via McKay’s correspondence [26]. The space X0 depends on r = rkG parameters ζ~i ∈ R3 , i = 0, . . . , r, X ζ~i = 0. i
The spaces X0 as well as the moduli spaces of U (N ) instantons on X0 can be constructed via hyperk¨ahler reductions of vector spaces [27][28]. These constructions were interpreted in [29] as originating from the gauge theory on v Dp-branes, put at the orbifold point of C2 /0 inside the w D(p + 4)-branes. It has been noticed in [29] that in principle P the condition i ζ~i = 0 can be relaxed by turning on a self-dual part of Bµν along C2 /0. As before, we interpret this as the process of going to the noncommutative ALE space with instantons on it. The ADHM construction of [28] generalizes straightforwardly to this case provided the original vector space V = Hom(C0 , C0 ⊗ C2 )0 whose reduction yields the ALE space is replaced by its noncommutative deformation: V q = Hom(C0 , C0 ⊗ Aζ )0 . 6.3. Instantons and holomorphic bundles on noncommutative surfaces. It is well-known that solutions to the instanton equations on the complex surface determine the holomorphic structure in the bundle where the gauge field is defined. In fact, DonaldsonUhlenbeck-Yau theorem establishes an equivalence between the moduli space of stable in a certain sense holomorphic bundles over a surface S and solutions to the instanton equations on S [30][31]. The holomorphic bundle E defines a sheaf of its sections, and can be replaced by this sheaf for many purposes. In fact, not every sheaf F comes from a bundle - for this it must be what is called a locally-free sheaf (the term free means that the sections of F form a free module over the sheaf O of holomorphic functions on S). By relaxing the condition of being locally free but insisting on being torsion free, one gets a generalization of the notion of a holomorphic bundle. However, it was not clear how to obtain the corresponding generalization of the notion of instanton. Of course, the consideration of torsion free sheaves allows us to compactify the moduli space of instantons [32], [15]. But it is by no means clear whether the compactification can be achieved within a gauge theory. It has been conjectured in [33] that such a compactification occurs in string theory. We claim that there exists a generalization of the instanton field for the torsion free sheaves. This is simply the gauge field on the noncommutative surface S. The importance of the spaces Mζ has been appreciated in the context of string duality and field theory a while ago (see [34], [35], [24], [12]). We hope that the proposed interpretation will help to find further applications as well as justify various assertions needed for establishing string/field dualities. In our presentation the noncommutativity of R4 entered only through the construction of the connection in a given bundle. The holomorphic bundle underlying the instanton could be described in purely commutative terms. This relies on the fact that
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with our choice of the Poisson structure the subalgebra O of holomorphic functions on C2 is commutative. It is interesting to study the picture in other complex structures. In fact, a slightly redundant but interesting deformation of ADHM equations is the three-parametric one: (6.2) µr = ζr Id, µc = ζc Id, where ζr ∈ R, ζc ∈ C. It is customary to combine (ζr , ζc , ζ¯c ) into a three vector ζ~ ∈ R3 . The specifics of R4 is that one can always get rid of ζc by appropriate rotation in the SU (2)R global group. Let us not do it but rather look at the complex equation µc = ζc Id: τz σz = 0,
iff
[z0 , z1 ] = − 21 ζc
(6.3)
Thus the ζc -deformation allows one to construct a coherent sheaf over the noncommutative C2 as the cohomology of the complex of sheaves: σz τz 0 → V ⊗ Oζc → V ⊗ C2 ⊕ W ⊗ Oζc → V ⊗ Oζc → 0,
(6.4)
where Oζc is the associative algebra generated by z0 , z1 obeying the relation [z0 , z1 ] = − 21 ζc and τz , σz are given by the same formulae (2.2). Acknowledgement. We are grateful to the organizers of the Workshop on “Dualities in String Theory” at ITP at Santa Barbara and especially to M. Douglas, D. Gross, W. Lerche and H. Ooguri for providing a very stimulating atmosphere. We are grateful to A. Astashkevich, T. Banks, D. Bernard, V. Fock, A. Gerasimov, M. Kontsevich, A. Losev, G. Moore , M. Rieffel, A. Rosly, K. Selivanov, S. Sethi and S. Shatashvili for useful discussions. The research of N. Nekrasov was supported by Harvard Society of Fellows, partially by NSF under grant PHY-92-18167, partially by RFFI under grant 96-02-18046 and partially by grant 96-15-96455 for scientific schools. The research of A. Schwarz was partially supported by NSF under grant DMS-9500704. As our paper was ready for publication we have learned about the paper [36] which will adress certain issues concerning gauge theories on noncommutative T 4 .
References 1. Connes, A.: Noncommutative Geometry. London–New York: Academic Press, 1994 2. Witten, E.: Bound states of strings and p-branes. Nucl. Phys. B460, 335 (1990), hep-th/9510135 3. Douglas, M.: Superstring dualities, Dirichlet branes and the small scale structure of space. hepth/9610041 4. Banks, T., Fischler, W., Shenker, S. and Susskind, L.: M theory as a Matrix Model: A Conjecture. hep-th/9610043, Phys. Rev. 55, 112 (1997) 5. Banks, T.: Matrix Theory. hep-th/9710231 6. Rozali, M.: Matrix Theory and U Duality in Seven Dimensions. hep-th/9702136, Phys. Lett. 400B, 260 (1997); Berkooz, M., Rozali, M. and Seiberg, N.: Matrix Description of M theory on T 4 and T 5 . hep-th/9704089, Phys. Lett. 408B, 105 (1997) 7. Seiberg, N.: New Theories in Six Dimensions and Matrix Description of M-theory on T 5 and T 5 /Z2 . hep-th/9705221 8. Shatashvili, S.: Unpublished 9. Connes, A., Douglas, M.R. and Schwarz, A.: Noncommutative Geometry and Matrix Theory: Compactification on Tori. hep-th/9711162 10. Douglas, M., Hull, C.: D-branes and noncommutative geometry. hep-th/9711165 11. Leigh, R. and Rozali, M.: hep-th/9712168; Ho, P.-M., Wu, Y.-Y. and Wu, Y.-S.: hep-th/9712201; Ho, P.-M., Wu, Y.-S.: hep-th/9801147; Li, M.: hep-th/9802052 12. Aharony, O., Berkooz, M., Seiberg, N.: Light-cone description of (2, 0) supersonformal theories in six dimensions. hep-th/9712117 13. Uhlenbeck, K.: Removable Singularities in Yang–Mills Fields. Commun. Math. Phys. 83, 11 (1982)
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14. Atiyah, M., Drinfeld, V., Hitchin, N. and Manin, Y.: Construction of Onstantons. Phys. Lett. 65B, 185 (1978) 15. Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. alggeom/9507012; Lectures on Hilbert schemes of points on surfaces. H. Nakajima’s homepage 16. Rieffel, M.: Deformation quantization for actions of Rd . Memoirs of AMS, vol. 106, N 506 (1993) 17. E. Corrigan, Goddard, P.: Constructions of instanton monopole solutions and reciprocity. Ann. Phys. (NY) 154, 253–279 (1984) 18. Rajaraman, R.: Solitons and Instantons. Amssterdam: North-Holland, 1987 19. Aharony, O., Berkooz, M.,Kachru, S., Seiberg, S., Silverstein, E.: Matrix Description of Interacting Theories in Six Dimensions. hep-th/9707079 20. van Baal, P., Braam, P.: Nahm’s transformation for instantons. Commun. Math. Phys. B 122, 267–280 (1989) 21. Astashkevich, A., Nekrasov, N., Schwarz, A.: In preparation 22. Douglas, M., Moore, G., Nekrasov, N., Shatashvili, S.: Unpublished 23. Dijkgraaf, R., Verlinde, E., Verlinde, H.: BPS spectrum of the five-brane and black hole entropy. hep-th/9603126; Dijkgraaf, R., Verlinde, R., Verlinde, H.: BPS quantization of the five-brane. hepth/96040055; Schwarz, J.: Self-dual string in six dimensions. hep-th/9604171; Dijkgraaf, R., Verlinde, E., Verlinde, H.: 5D black holes and matrix strings. hep-th/9704018; Losev, A., Moore, G., Shatashvili, S.: M & m’s. hep-th/9707250 24. Vafa, C.: Instantons on D-branes. hep-th/9512078, Nucl.Phys. B463, 435–442 (1996) 25. Witten, E.: On the conformal field theory of the Higgs branch. hep-th/9707093 26. McKay, J.: Graphs, Singularities and Finite groups. Proc. Sympos. Pure Math. 37, Providence, RI: AMS 1980, pp. 183–186 27. Kronheimer, P.: The construction of ALE spaces as hyper-kahler quotients. J. Diff. Geom. 28, 665 (1989) 28. Kronheimer, P. and Nakajima, H.: Yang–Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263 (1990) 29. Douglas, M., Moore, G.: D-branes, Quivers and ALE Instantons. hep-th/9603167 30. Donaldson, S.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc, 50, 1 (1985) 31. Donaldson, S.K. and Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford: Clarendon Press, 1990 32. For Gieseker’s compactification, see Okonek, C, Schneider, M., Spindfler, H.: Vector bundles on complex projective spaces. Progress in Math., Basel–Boston: Birkhauser, 1980 33. Vafa, C., Witten, C.: A strong coupling test of S-duality. Nucl. Phys. B431, 3–77 (1994) 34. Losev, A., Moore, G., Nekrasov, N., Shatashvili, S.: Four-Dimensional Avatars of 2D RCFT. hepth/9509151, Nucl. Phys. Proc. Suppl. 46:130–145 (1996) 35. Harvey, J.A., Moore, G.: On the algebras of BPS states. hep-th/9609017 36. Berkooz, M.: To appear Communicated by A. Jaffe
Commun. Math. Phys. 198, 705 – 709 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
The Multi-form Generalised Gleason Theorem Oliver Rudolph1 , J. D. Maitland Wright2 1 2
Theoretical Physics Group, Blackett Laboratory, Imperial College, London SW7 2BZ, UK Mathematics Department, University of Reading, Reading RG6 6AX, UK
Received: 10 April 1998 / Accepted: 17 April 1998
Abstract: Arising from investigations of decoherence functionals in the “histories” approach to quantum mechanics, the following result is given by the methods of [14]. Let A1 and A2 be von Neumann algebras without Type I2 direct summands and let P (Aj ), (j = 1, 2) be their lattices of projections. Let m : P (A1 ) × (P (A2 ) → C be a bounded quantum bi-measure. Then there is a unique bounded bilinear functional M on A1 × A2 which extends m. In this note we use a different approach to establish a generalisation of this result to k-fold, vector valued, quantum multi-measures. This tool is needed for further investigations of decoherence functionals in quantum theory.
1. Introduction Throughout this note A will be a von Neumann algebra and P (A) the set of projections in A. For an arbitrary Banach space X, an X-valued quantum measure on P (A) is a map m : P (A) → X such that m(p + q) = m(p) + m(q) whenever p and q are orthogonal projections. The Generalised Gleason Theorem states: Theorem 0. Let A be a von Neumann algebra with no direct summand of Type I2 and let X be a Banach space. If m is an X-valued quantum measure on P (A), taking its values in a bounded subset of X, then there exists a unique bounded linear operator M : A → X such that M is an extension of m. This is proved in [1,2], making essential use of the work of [3]. See, also, the interesting work of Ylinen [19]. Gleason’s original result [7] may be recovered from Theorem 0 by first specialising A to L(H), the algebra of all bounded operators on a Hilbert space H (where H is of dimension greater than 2), secondly by putting X = C, and, thirdly, by demanding that the quantum measure m be completely additive and take only positive real values.
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In Mackey’s monograph on the mathematical foundations of quantum mechanics [10], he sets out a system of axioms. He indicates why most of these axioms are natural and physically plausible. However he points out that his Axiom VII (where it is demanded that the underlying vector space be L(H)) has no natural physical justification but is made for mathematical convenience. Mackey emphasises the importance of Gleason’s Theorem. One of the mathematical difficulties which Axiom VII circumvents, is that, in 1963 no version of Gleason’s Theorem was available for von Neumann algebras other than L(H). Many years later, the combination of the powerful insights of Christensen [4] and Yeadon [17,18] showed that Theorem 0 was true when X was one dimensional provided that m took only positive values. See [5] and its extensive bibliography for a lucid account of Gleason’s Theorem and its ramifications. Arising from their investigations of the “histories” approach to quantum mechanics, Isham et al. [8] classified continuous decoherence functionals when the underlying Hilbert space was of finite dimensions greater than 2. They made a conjecture on the significance of boundedness for decoherence functionals when the underlying Hilbert space was infinite dimensional. When Wright [14] established their conjecture, the crucial step was to obtain a “bilinear” analogue of Gleason’s Theorem. This is also a key tool for the investigation of decoherence functionals representable by trace class operators undertaken in [12]. See also [15,16]. Recently, Edwards and Rüttimann applied this “bilinear” result in groundbreaking work on JB ∗ -triples [6]. For our current programme of investigation into decoherence functionals and quantum histories we need a k-fold analogue of the Generalised Gleason Theorem. For the intended applications (which will appear elsewhere in due course) we only require this result for complex valued multi-measures. (See below for definitions.) The proof, for k = 2, given in [14], does not seem to readily extend to higher dimensions so that a somewhat different approach has been adopted here. Furthermore, it was only when we considered Banach space valued multi-measures that we were able to construct an efficient proof of our main theorem, Theorem 3. By putting k = 1 and k = 2 in Theorem 3, we recover, respectively, Theorem 0 and Theorem 1 [14]. Our standard references for operator algebras are [9,11,13].
2. The k-fold Analogue of the Generalised Gleason Theorem In this section we generalise Theorem 1 [14] from bilinear forms to vector valued klinear forms. Our proof differs from the approach of [14] but, as before, a fundamental tool is the Generalised Gleason Theorem [1,2]. Let A1 , A2 . . . Ak be arbitrary Banach spaces and let X be a Banach space. We recall that a k-linear form f : A1 × A2 × · · · × Ak → X is bounded if there is a constant M such that, for each zj in Aj , kf (z1 , z2 , . . . )k ≤ M kz1 k kz2 k k · · · kzkk. The smallest constant M for which this inequality holds is defined to be the norm of f . We denote the Banach space of all such X valued, bounded k-linear forms by Lin(A1 , A2 . . . Ak ; X). From now onward A1 , A2 . . . Ak are von Neumann algebras and X is an arbitrary Banach space. For any von Neumann algebra A, the set of all projections in A is denoted by P (A).
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Lemma 1. Let f : A1 × A2 × · · · × Ak → X be a bounded k-linear form and let K be a constant such that, for each projection pj in Aj , kf (p1 , p2 , . . . )k ≤ K. Then kf k ≤ 4k K. Proof. It suffices to show that kf (z1 , z2 , . . . )k ≤ 4k K if kzj k ≤ 1 for j = 1, 2 . . . k. Each zn is of the form xn + iyn , where xn and yn are self-adjoint and lie in the closed unit ball of An . By spectral theory each xn and yn is the difference of two positive self-adjoint elements in the closed unit ball. Hence it suffices to show that kf (a1 , a2 , . . . )k ≤ K whenever a1 , a2 , . . . are positive and in the unit balls of, respectively, A1 , AP 2 .... The result now follows from the observation that each aj is of the form 1/2n pn , where p1 , p2 , . . . are pairwise commuting projections in Aj . (For a proof of the latter assertion see Pedersen [11], p. 27.) Corollary 2. Let f and g be in Lin(A1 , A2 . . . Ak ; X) and such that f (p1 , p2 , . . . pk ) = g(p1 , p2 , . . . pk ) for all projections p1 , p2 , . . . pk in, respectively, A1 , A2 . . . Ak . Then f = g. Proof. Apply Lemma 1 to f − g.
Let us define an X-valued k-measure on P (A1 ) × (P (A2 ) × · · · × P (Ak ) to be a function m : P (A1 ) × P (A2 ) × · · · × P (Ak ) → X such that, for each j, whenever q and q # are orthogonal projections in Aj , m(. . . pj−1 , q +q # , p−j + 1, . . . ) = m(. . . pj−1 , q, pj+1 , . . . )+m(. . . pj−1 , q # , pj+1 , . . . ). Theorem 3. Let X be a Banach space. Let k be a positive integer. For j = 1, 2 . . . k let Aj be a von Neumann algebra with no direct summand of Type I2 . Let m be an X-valued k-measure on P (A1 ) × P (A2 ) × · · · × P (Ak ) whose values are bounded in norm by a constant K. Then there exists a unique f in Lin(A1 , A2 . . . Ak , ; X) such that f (p1 , p2 , . . . pk ) = m(p1 , p2 , . . . pk ) for all projections p1 , p2 , . . . pk in, respectively, A1 , A2 . . . Ak . Also kf k ≤ 4k K. Proof. Corollary 2 shows that if f exists it is unique. Lemma 1 shows that if f exists then its norm satisfies the required inequality. When k = 1 the existence of f follows from Theorem 0. Let us now suppose that the theorem is valid for k = n. For j = 0, 1, 2 . . . n, let Aj be a von Neumann algebra with no direct summand of Type I2 . Let m be an X-valued (n + 1)-measure on P (A0 ) × P (A1 ) × · · · × P (An ) which is bounded by a constant K. For each e in P (A0 ), let me (p1 , p2 , . . . pn ) = m(e, p1 , p2 , . . . pn ) for each pj in P (Aj ) (j = 1, 2 . . . ). Then me is an n-measure bounded by K. So, by hypothesis, there exists a unique fe in Lin(A1 , A2 . . . An; X) corresponding to me . Moreover kfe k ≤ 4n K. Let Y = Lin(A1 , A2 . . . An ; X). Let M : P (A0 ) → Y be defined by M (e) = fe . Let e1 be orthogonal to e2 . Then, for each (p1 , p2 , . . . pn ) in P (A1 ) × . . . × P (An ), M (e1 + e2 )(p1 , p2 , . . . pn ) = m(e1 + e2 , p1 , p2 , . . . pn) = m(e1 , p1 , p2 , . . . pn ) + m(e2 , p1 , p2 , . . . pn ) = M (e1 )(p1 , p2 , . . . pn ) + M (e2 )(p1 , p2 , . . . pn ).
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So M (e1 + e2 ) − M (e1 ) − M (e2 ) is in Lin(A1 , A2 . . . An , X) and vanishes on P (A1 ) × · · · × P (An). So, by Corollary 2, M (e1 + e2 ) = M (e1 ) + M (e2 ). Moreover, for each e, kM (e)k ≤ 4n K. Thus M is a bounded Y -valued measure on the projections of A0 . Since A0 does not have a direct summand of Type I2 , it follows from the Generalised Gleason Theorem that M extends to a bounded linear map M # from A0 to Y . Let F (x0 , x1 , . . . xn ) = M # (x0 )(x1 , x2 , . . . xn ). Then F is in Lin(A0 , A1 . . . An ; X) and F is an extension of m. By Lemma 1, F satisfies the required inequality. Thus the theorem is valid for k = n + 1. It follows by induction that it is valid for all k. Remark. If, in the above theorem, X = C and the map e → m(e, p2 , p3 , . . . pk ) is completely additive for each p2 , p3 , . . . pk in, respectively, P (A2 ), P (A3 ), . . . P (Ak ), then, see Corollary III. 3.11 [13], x → f (x, p2 , p3 , . . . pk ) is a normal functional on A1 . Since the normal functionals form a norm closed subspace of the dual of A1 , it follows by standard arguments that x → f (x, x2 , x3 , . . . xk ) is a normal functional for each choice of x2 , x3 , . . . xk . Acknowledgement. Oliver Rudolph is a Marie Curie Research Fellow and carries out his research at Imperial College as part of a European Union training project financed by the European Commission under the Training and Mobility of Researchers (TMR) programme.
References 1. Bunce, L.J. and Wright, J.D.M.: The Mackey-Gleason Problem. Bull. Amer. Math. Soc. 26, 288–293 (1992) 2. Bunce, L.J. and Wright, J.D.M.: The Mackey-Gleason Problem for vector measures on projections in von Neumann algebras. J. London Math. Soc. 49, 133–149 (1994) 3. Bunce, L.J. and Wright, J.D.M.: Complex measures on projections in von Neumann algebras. J. London Math. Soc. 46, 269–279 (1992) 4. Christensen, E.: Measures on projections and physical states. Commun. Math. Phys. 86, 529–538 (1982) 5. Dvurechenskij, A.: Gleason’s Theorem and its applications. Dordrecht/Boston/London: Kluwer, 1993 6. Edwards, C.M. and Rüttimann, G.T.: The lattice of weak*-closed inner ideals in a W ∗ -algebra. Commun. Math. Phys. (In press) 7. Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957) 8. Isham, C.J., Linden, N. and Schreckenberg, S.: The classification of decoherence functionals: an analogue of Gleason’s Theorem. J. Math. Phys. 35, 6360–6370 (1994) 9. Kadison, R.V. and Ringrose, J.R. Fundamentals of the Theory of Operator Algebras. Vol. I & II Orlando: Academic Press, 1983 & 1986 10. Mackey, G.W. Mathematical Foundations of Quantum Mechanics. New York: Benjamin, 1963 11. Pedersen, G.K.: C*-algebras and their automorphism groups. London: Academic Press, 1979 12. Rudolph, O. and Wright, J.D.M.: On tracial operator representations of quantum decoherence functionals. J. Math. Phys. 38, 5643–5652 (1997) 13. Takesaki, M.: Theory of Operator Algebras. New York: Springer, 1979 14. Wright, J.D.M.: The structure of decoherence functionals for von Neumann quantum histories. J. Math. Phys. 36, 5409–5413 (1995) 15. Wright, J.D.M.: Linear representations of bilinear forms on operator algebras. Expo. Mathematicae 16, 75–84 (1998) 16. Wright, J.D.M.: Decoherence functionals for von Neumann quantum histories: Boundedness and countable additivity. Commun. Math. Phys. 191, 493–500 (1998) 17. Yeadon, F.J.: Measures on projections in W ∗ -algebras of type II1 . Bull. London Math. Soc. 15, 139–145 (1983)
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18. Yeadon, F.J.: Finitely additive measures on projections in finite W ∗ -algebras. Bull. London Math. Soc. 16, 145–150 (1984) 19. Ylinen, K.: Vector measures on the projections of a W ∗ -algebra. Ann. Univ. Turkuensis, ser. A, 186, 129–135 (1984) Communicated by H. Araki
Commun. Math. Phys. 198, 711 – 736 (1998)
Communications in
Mathematical Physics © Springer-Verlag 1998
The Relativistic Dynamics of the Combined Particle–Field System in Renormalized Classical Electrodynamics Hans-Peter Gittel1 , Jerzy Kijowski2 , Eberhard Zeidler1,3 1 Department of Mathematics, University of Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany. E-mail: [email protected] 2 Centrum Fizyki Teoretycznej PAN, Aleja Lotnik´ ow 32/46, PL-02-668 Warsaw, Poland and Department of Mathematical Methods in Physics, University of Warsaw, ul. Ho˙za 74, PL-00-682 Warsaw, Poland. E-mail: [email protected] 3 Department of Mathematics, University of Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany and Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig, Germany. E-mail: [email protected]
Received: 10 December 1997 / Accepted: 21 April 1998
Abstract: This paper develops a general theory for the nonlinear, renormalized interaction between charged particles and electromagnetic fields. For the combined “particle + field” system, a fundamental relativistically invariant dynamical equation is derived from first principles. This theory was first obtained in an alternative way by one of us (J.K.) in an earlier paper. Here, we prove that the initial-value problem for the “particle + field” system is well-posed. The existence and uniqueness result is based on a careful analysis of the singularites of the electromagnetic field along the trajectory of the moving charged particle. Furthermore, the Banach fixed-point principle is used. The theory improves the classical Dirac theory for the motion of electrons. In particular, it is shown how to deal with the Dirac paradox of runaway solutions.
1. Introduction From first principles, we rigorously derive the relativistic equation of motion for a charged particle in an electromagnetic field, taking fully into account the self-interactions of the particle with its radiation field. Our nonlinear equation of motion is formulated in a relativistic covariant way. In addition, we formulate and solve a well-posed initial-value problem for the combined “particle + field” system. We show that, for a given external electromagnetic field together with initial data of the particle and of the electromagnetic field at time t = 0, there exist locally both a unique trajectory of the particle and a unique total electromagnetic field produced by the moving particle. We want to stress the important fact that the initial data of the particle and of the electromagnetic field cannot be given independently. In particular, we show that in our approach the value of the acceleration of the particle at time t = 0 is uniquely encoded in the initial data of the electromagnetic field. This is a crucial point which was first discussed in the paper [9].
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The derivation of the equation of motion for the “particle + field” system is based on the following four facts. In the particle’s comoving frame 6+ , the particle rests. Naturally enough, we assume that the leading singularity of the electromagnetic field on 6+ is equal to the Coulomb singularity. (ii) By a careful mathematical analysis, we show that the leading singularity uniquely determines the total singularity of the electromagnetic field near the trajectory of the moving particle. We explicitly compute this singularity. (iii) Using (ii), we calculate the renormalized, total energy and momentum of the combined “particle + field” system (cf. [9]) in a simple way. (iv) We postulate conservation of renormalized energy and momentum. (i)
Throughout the paper we use the decomposition F = Fsing + Freg of the electromagnetic field F . This decomposition is defined locally, in a neighbourhood of the particle’s trajectory. The singular part Fsing is uniquely determined by our method within a reasonable large class of functions. The regular part Freg is continuous across the trajectory of the particle. Our decomposition is justified by the observation that the field F − Fsing is continuous for each solution F of the initial-value problem for the Maxwell system in the sense of generalized functions. Our approach improves the classical nonlinear Dirac theory for the motion of electrons. However, in contrast to the ill-posed Dirac theory, we get an initial-value problem which is well-posed from the physical point of view. In particular, runaway solutions do not appear for appropriately chosen inital data of the “particle + field” system. In contrast to the classical paper [5] by Dirac, our derivation of the equation of motion is completely independent of the use of the retarded or advanced solutions of Maxwell equations. This way, the symmetry of both the electrodynamics and mechanics with respect to time reversal is not violated by our theory. However, we shall see that the retarded (resp. advanced) Green functions for Maxwell equations provide a technical tool for solving the initial-value problem. Using this tool, the equation of motion can be reduced to a nonlinear operator equation for the particle’s trajectory. The existence and uniqueness proof follows then by means of the Banach fixed-point theorem. Here, a detailed analysis of the field singularities in the vicinity of the particle’s trajectory is necessary. The paper is based on ideas published in [9, 10]. It is organized as follows: 1 Introduction 2 Notation 3 The basic equations of motion for the particle-field system 4 The main result 5 The retarded electromagnetic field 6 An equivalent formulation of the equation of motion 7 Fermi–Walker transport 8 Computation of the singular part of the electromagnetic field 9 Derivation of the equation of motion for the particle 10 Solution of the initial-value problem for the basic equations.
2. Notation Tensors. We will work on a 4-dimensional flat (oriented) Minkowski space-time manifold M4 with the metric tensor G := gαβ dxα ⊗ dxβ . Greek (resp. latin) indices run from
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0 to 3 (resp. from 1 to 3). Moreover, greek (resp. latin) indices are lowered by gαβ (resp. gjk ). We assume that the metric (gαβ ) has signature (−, +, +, +). We shall use also curvilinear coordinates, but the coordinate x0 is always time-like, whereas the coordinates xj are space-like. The natural basic vectors ∂0 , ∂1 , ∂2 , ∂3 are also denoted by b0 , b1 , b2 , b3 . There exists a natural inner product vw on M4 . If v = v α bα and w = wα bα , then vw = gαβ v α wβ ,
bα bβ = gαβ .
The vector v is called space-like (resp. time-like) iff v 2 > 0 (resp. v 2 < 0). Let F = F αβ bα ⊗bβ be an antisymmetric tensor, i.e., F αβ = −F βα . Since bα ∧bβ := b α ⊗ bβ − bβ ⊗ bα , 1 F = F αβ bα ∧ bβ . 2 The dual tensor is given by ∗F = (∗F )αβ bα ⊗ bβ =
1 (∗F )αβ bα ∧ bβ . 2
Let T = T αβ bα ⊗ bβ . Then the tensor T becomes a linear operator if we define T n := nγ gγα T αβ bβ for all vectors n = nα bα . This definition is independent of the chosen basis. Hence the trace of T equals tr T = gαβ T αβ . Let a, b, c be 4-vectors. Then (a ⊗ b)c = (ac)b,
(a ∧ b)c = (ac)b − (bc)a.
(2.1)
Inertial systems. By an inertial (or laboratory) system 6 of coordinates we understand a system for which the metric tensor is given by (gαβ ) = diag(−1, 1, 1, 1). Here, the natural basis vectors will be denoted by e0 , e1 , e2 , e3 . Then eα eβ = gαβ . In particular, the three position vectors e1 , e2 , e3 form an orthonormal system. In the system 6, the 3-dimensional position vector and time are denoted by x = xj ej and t = x0 , respectively. Furthermore, we use the 4-position vector x = xα eα = x + te0 . Maxwell equations. The classical vacuum Maxwell equations read as follows in the inertial system 6 : div E = ρ, div B = 0, curl E = −Bt . curl B = Et + j, Here, E ρ
= =
electric field vector, electric charge density,
B j
= =
magnetic field vector, electric current density vector.
We use a system of units where ε0 = c = 1, and hence µ0 = 1. The tensor of the electromagnetic field is given by F = 21 F αβ bα ∧ bβ . In an inertial system 6, we get bα = eα , E = E j ej , B = B j ej along with 0 −E 1 −E 2 −E 3 E 1 0 −B 3 B 2 F αβ = E 2 B 3 0 −B 1 . E 3 −B 2 B 1 0
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Moreover, Fjk = F jk as well as F0j = −F 0j . This means that F = E ∧ e0 − Alt(B),
(2.2)
where we introduce the convenient notation Alt(B) := B 1 (e2 ∧ e3 ) + B 2 (e3 ∧ e1 ) + B 3 (e1 ∧ e2 ).
(2.3)
The dual tensor ∗F is obtained from F by replacing E with −B and B with E. Hence ∗F = −B ∧ e0 − Alt(E). If the magnetic field can be expressed by a vector product, i.e., if B = c × d, then Alt(B) = c ∧ d and F = E ∧ e0 − c ∧ d.
(2.4)
Note that Alt(E)b = E × B for b = B 0 e0 + B. In order to translate expressions for the electromagnetic field tensor from the special comoving system to general systems of reference, we will use (2.4) later on. Furthermore, let jext denote the 4-vector of the external sources. In an inertial system 6, we put jext = jext + ρext e0 , where ρext and jext denote the external electric charge density and the external electric current density vector, respectively. Trajectories of particles. On M4 , the 4-trajectory denotes a time-like curve x = q(τ ) which is parametrized by the eigentime τ . Furthermore, the 4-velocity u(τ ) and the 4-acceleration a(τ ) are defined through u(τ ) := q(τ ˙ ) and a(τ ) := q(τ ¨ ). In the sequel, the dot will always indicate the derivative with respect to τ . In particular, let x = q(t) describe the motion of a particle in an inertial system 6. Suppose that the particle moves with underlight speed, i.e., |q0 (t)| < 1. Then the eigentime τ of the particle is given by Zt q
1 − q0 (σ)2 dσ.
τ (t) := 0
The inverse function is denoted by t = t(τ ). Set q(τ ) := q(t(τ )). Then ˙ ) = q0 (t(τ ))t˙(τ ), q(τ
2 ¨ ) = q00 (t(τ ))t˙(τ ) + q0 (t(σ))t¨(τ ). q(τ
The 4-trajectory of the particle is given by x = q(τ ) = q(τ ) + t(τ )e0 . Furthermore, ˙ ) + t˙(τ )e0 , u(τ ) = q(τ
¨ ) + t¨(τ )e0 . a(τ ) = q(τ
(2.5)
0
¨ The relations above In particular, if q(0) = q (0) = 0, then u(0) = e0 , and a(0) = q(0). immediately imply u2 = −1 (i.e., u is time-like), au = 0 (i.e., a is space-like), a˙ u = −a2 .
(2.6)
The comoving system 6+ . To simplify substantially our analysis, we will use an (accelerated) comoving system of reference, uniquely defined by the particle’s trajectory. For this purpose, we consider the rest-frame hyperplane 6+ (τ ), i.e. the 3-dimensional plane through the point q(τ ) of the trajectory which is orthogonal to the tangent vector u(τ ) (Fig. 1). Analytically, the hyperplane 6+ (τ ) consists of precisely all the points x ∈ M4 which satisfy the equation (x − q(τ ))u(τ ) = 0.
(2.7)
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Definition 2.1. The comoving system 6+ consists of all the points (τ, y) with y ∈ 6+ (τ ). That is, the time coordinate in 6+ equals the eigentime of the particle. By the Fermi transformation, we understand the passage from (τ, y) to the corresponding event x, i.e. x = q(τ ) + y.
(2.8)
Fig. 1. The Fermi system
The map (τ, y) 7→ x is not bijective. However, the following argument shows that this map is locally injective. Let us fix a point x0 := q(τ0 ) of the trajectory at a fixed eigentime τ0 . Proposition 2.2. Let k ≥ 2. Suppose that the 4-trajectory x = q(τ ) is C k on a neighbourhood of τ0 . Then, the following are true. (i)
There exist both a neighbourhood U of the point x0 in M4 and a neighbourhood V of τ0 in R such that, for each given point x ∈ U , Eq. (2.7) possesses a unique solution τ (x) ∈ V.
(2.9)
The function τ = τ (x) is C k−1 on V. (ii) For all x points and all 4-vectors h, τ 0 (x)h = −u(τx )h + (q(τx ) − x)a(τx )τ 0 (x)h.
(2.10)
Here, we set τx := τ (x). In particular, τ 0 (x0 ) = −u(τ0 ).
(2.11)
Proof. Ad (i). Define g(x, τ ) := (x − q(τ ))u(τ ). The real function g is C k−1 on U × V. ∂ g(x0 , τ0 ) = −u(τ0 )2 = 1. Thus, statement (i) Obviously, g(x0 , τ0 ) = 0. Moreover, ∂τ follows from the implicit function theorem (cf. [20], Theorem 4.B). Ad (ii). Replace x with x + σh in Eq. (2.7), and differentiate with respect to the real parameter σ at the point σ = 0. Definition 2.3. Let τx := τ (x). The inverse Fermi transformation is given by the local transformation τ = τx ,
y = x − q(τx ),
(2.12)
from Proposition 2.2(i). We call (τx , y) the local Fermi coordinates of the event x. By Proposition 2.2(i), the local map x 7→ (τx , y) is C k−1 . From the physical point of view, the event x ∈ M4 is observed in the comoving system at the point y at time τx .
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3. The Basic Equations of Motion for the Particle–Field System In this section we formulate results which will be derived later. In particular, points (i)–(iii) are consequences of the Maxwell equations (see Sect. 8), whereas point (iv) is implied by conservation of the renormalized energy-momentum (see Sect. 9). 3.1. The basic equations in an arbitrary system of reference. Let F denote the electromagnetic tensor field, and let jext = (ρext , jext ) be a regular 4-vector field of the external sources. We assume that the continuity equation Div jext = 0
(3.1)
is satisfied. The total electromagnetic current is a sum of jext and a singular δ-like current carried by the particle. It will be proved that the singular current uniquely implies the singular part of the field in the vicinity of the particle. Then, both the field and the particle’s trajectory have to fulfill the following basic equations: (i)
The Maxwell equations for the electromagnetic field Div F = jext ,
Div(∗F ) = 0
(3.2)
must be valid outside the trajectory. (ii) The electromagnetic field in a neighbourhood of the trajectory may be decomposed as F = Fsing + Freg ,
(3.3)
where the regular part Freg is continuous across the trajectory. (iii) The singular part Fsing of the electromagnetic field has the following structure Fsing = F−2 + F−1 + F0 ,
(3.4)
where QN ∧ u(τx ), 4πR2 Q F−1 (x) := − ((aN)N + a) ∧ u(τx ), (3.5) 8πR Q Q F0 (x) := (3(aN)2 N + 6(aN)a + a2 N) ∧ u(τx ) − N ∧ a˙ (τx ). 32π 8π
F−2 (x) :=
For the sake of brevity, we write p R := y2 , N := y/R,
and
a := a(τx )
(3.6)
(y and τx are as in the previous section). (iv) The following equation of motion for the particle m0 a(τ ) = QFreg (q(τ ))u(τ )
(3.7)
must be satisfied. Here, m0 and Q denote the (renormalized) observed rest mass and the charge of the particle, respectively (see the discussion on renormalization in Sect. 9). In addition, x = q(τ ) represents the particle’s trajectory, and τ denotes its eigentime.
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We stress that all the expressions above have an invariant (geometric) meaning. Therefore, the basic equations (i) through (iv) do not depend on the choice of a reference system. In the present paper, we consider the following Problem. Given an external 4-current jext and an appropriate set of initial conditions, find both (i) the electromagnetic field tensor F and (ii) the 4-trajectory x = q(τ ) of the particle fulfilling the above system of equations and satisfying the initial conditions. Remark 3.1. The equation of motion (3.7) is equivalent to m0 a(τ ) = Q F − Fsing (q(τ ))u(τ ).
(3.8)
This can be regarded as a renormalized Lorentz equation. Remark 3.2 (regularity). Both the singular part Fsing and the regular part Freg are smooth outside the trajectory. The continuity of Freg across the trajectory will be proved in Sect. 10. Remark 3.3. In coordinate language, we have: (i)
The Maxwell equations for the electromagnetic field 1 1 1 1 β , |g|− 2 ∂α |g| 2 (∗F )αβ = 0 |g|− 2 ∂α |g| 2 F αβ = jext
for xα 6= q α (τ ).
(ii) The local structure of the electromagnetic field in a neighbourhood of the trajectory αβ αβ + Freg . F αβ = Fsing
(iii) The singular part of the electromagnetic field αβ αβ αβ Fsing = F−2 + F−1 + F0αβ .
The expressions of Fkαβ for k = −2, −1, 0 follow from (3.5). (iv) The equation of motion for the particle αβ m0 aβ = QFreg uα
along the trajectory xα = q α (τ ).
3.2. The basic equations in an inertial system. Let 6 be an inertial system with position vector x and time t = x0 . Then, the basic equations read as follows: (i)
The Maxwell equations for the electromagnetic field div E = ρext , −∂t E + curl B = jext ,
div B = 0 , ∂t B + curl E = 0
(3.9)
for x 6= q(t). (ii) The local structure of the electromagnetic field in a neighbourhood of the trajectory E = Esing + Ereg ,
B = Bsing + Breg .
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(iii) The singular part of the electromagnetic field Esing = E−2 + E−1 + E0 ,
Bsing = B−2 + B−1 + B0 ,
(3.10)
where E−2 (t, x) :=
Q ˙ x) , t˙(τx )(x − q(τx )) − (t − t(τx ))q(τ 3 4πR
B−2 (t, x) := −
Q ˙ x ), (x − q(τx )) × q(τ 4πR3
E−1 (t, x) := −
Q ¨ x )) t˙(τx ) ((aN)(x − q(τx )) + Rq(τ 8πR2 Q ˙ x ), (aN)(t − t(τx )) + Rt¨(τx ) q(τ + 2 8πR
Q ¨ x )) × q(τ ˙ x ), ((aN)(x − q(τx )) + Rq(τ 8πR2 ... Q E0 (t, x) := t˙(τx )(3(aN)2 + a2 ) − 4t(τx ) (x − q(τx )) 32πR Q ˙ x) − (3(aN)2 + a2 )(t − t(τx )) + 6R(aN)t¨(τx ) q(τ 32πR ... Q ¨ x ) + 4(t − t(τx ))q(τx ) , + 6R(aN)t˙(τx )q(τ 32πR
B−1 (t, x) :=
Q ¨ x ) × q(τ ˙ x) (3(aN)2 + a2 )(x − q(τx )) + 6R(aN)q(τ 32πR ... Q + (x − q(τx )) × q(τx ). 8πR Here, we use the notation introduced in (3.6). Moreover, x := x + te0 . (iv) The equation of motion for the particle B0 (t, x) := −
d m0 v(t) p = QEreg (t, q(t)) + Qv(t) × Breg (t, q(t)), dt 1 − v(t)2
(3.11)
where x = q(t) (resp. v := q0 (t)) denotes the trajectory (resp. the 3-dimensional laboratory-frame velocity) of the particle. The above explicit expressions of the fields Ek , Bk for k = −2, −1, 0 follow from (3.5) along with (2.2) through (2.5). Remark 3.4. Let ρext , jext be smooth everywhere. Moreover, suppose that Eext (0, x) = Bext (0, x) = 0 for all x ∈ R3 . In this special case, the unique (smooth) solution to (3.9) is given by the following well-known formulas: Zt Eext (t, x) = −
Mxt−σ {jext }(σ) + (t − σ)Mxt−σ {∂r jext + grad ρext }(σ) dσ ,
0
(3.12) Zt Bext (t, x) = −
(t − σ)Mxt−σ {curl jext }(σ) dσ 0
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(cf., e.g. [18], §§15,46). Here, for a function f = f (t, x), we use the mean value over a sphere of radius R centered at x: Z 1 R f (σ, x + Rz) dSz , (3.13) Mx {f }(σ) := 4π S
where S := {z : |z| = 1}. The radial derivative is given by ∂r j = (z grad)j. The expressions (3.12) follow from the fact that each component of the fields Eext and Bext satisfies the inhomogeneous wave equation. Solutions of this equation are given by a mean value formula. For our purposes, the solution (3.12) is not satisfactory because 1. it does not take into account the singular current carried by the particle, and 2. it does not satisfy the (usually inhomogeneous) initial conditions for the field. Nevertheless, the integral representation (3.12) of the field will be an important tool for the construction of the solution of our problem. 4. The Main Result Let us consider a fixed inertial system 6. Suppose that, at the initial time t = 0, we 0 0 (x) + Ereg (x) and B(0, x) = B0sing (x) + are given the field initial data E(0, x) = Esing 0 Breg (x) compatible with our basic equations formulated above. Under specific regularity 0 and B0reg we will prove that the initial-value problem has a conditions imposed on Ereg unique local solution. Observe that the singular part of the field data contains the information about the initial position, velocity, and acceleration of the particle. Hence, no additional initial conditions for the particle’s trajectory are necessary. To simplify the formulation of the regularity conditions, we will pass to the particle’s comoving frame in which the initial velocity of the particle vanishes. By a suitable choice of the origin, the initial position of the particle also may be trivialized: q(0) = q0 (0) = 0. Putting r := |x| and n := x/r, we thus assume the following: (i)
There exist two vectors a0 and a˙ 0 such that the singular part of the initial data has the structure 3Q Qn Q (a0 n)n + a0 + (a0 n)2 n + 2(a0 n)a0 − (a0 )2 n , − 4πr2 8πr 32π (4.1) Q B0sing (x) = n × a˙ 0 . 8π 0 (x) = Esing
Note that a0 = q00 (0) = a(0) and a˙ 0 = q000 (0) = a0 (0). 0 (ii) The regular part of the initial data Ereg and B0reg is C 2 in a neighbourhood U of the origin and satisfies the compatibility conditions 0 (x) = ρext (0, x), div Ereg
div B0reg (x) = 0
for all x ∈ U ,
(4.2)
for given external sources ρext , jext . They are assumed to be C ∞ on [0, t0 ] × R3 for a certain t0 > 0.
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(iii) The following constraint between the singular and the regular part of the initial data 0 m0 a0 = QEreg (0),
(4.3)
is fulfilled. This follows from (3.7). Theorem 4.1. The above initial-value problem for the basic equations of motion has a unique local solution. This solution has the following properties: (a) The trajectory x = q(t) of the particle is C 4 on a sufficiently small time interval [0, t1 ]. (b) The regular part E − Esing , B − Bsing of the electromagnetic field is continuous across the trajectory for all times t ∈ (0, t1 ]. 0 , B0reg are C m , m ≥ 3. Corollary 4.2. In addition, suppose that the initial data Ereg Then the trajectory is C m+2 .
The complete proof of this theorem will be given in the sequel.
5. The Retarded Electromagnetic Field Formula (3.12) enables us to deal with external sources. To handle singular (δ-like) sources of the field, carried by the particle’s trajectory, we may use e.g. the classical retarded field Fret or the advanced field Fadv . For physical background see, for instance, the monograph [15] by Rohrlich (Chapters 4 and 6) or the monograph [16] by Ruei (Chapter XVII). Those fields are solutions of the inhomogeneous Maxwell system (5.1) Div F ret = Qδζ , Div ∗F ret = 0, adv
adv
where Q is the electric charge of the particle, ζ := {x = q(τ ), τ ∈ R} its trajectory, and δζ denotes the Dirac distribution concentrated on ζ. For our purposes, the specific form of both the solutions (cf. [15], (4–98) and [16], (XVII-26)) will not be necessary. What we need is the following approximation (cf. [15], (6–62)): F ret (x) = adv
Q Q ((aN)N + a) ∧ u N∧u− 2 4πR 8πR Q 3(aN)2 N + 6(aN)a + a2 N ∧ u + 32π Q Q − N ∧ a˙ ∓ u ∧ a˙ + O(R) 8π 6π Q = Fsing (x) ∓ u ∧ a˙ + O(R), as R → 0 . 6π
(5.2)
This formula contains the standard singular part, common to any solution of (5.1) (cf. (3.4), (3.5), and the proof in Sect. 8). But formula (5.2) contains also the value of the regular part of the field on the particle’s trajectory Freg (q(τ )) = ∓ (Q/6π) u ∧ a˙ , being characteristic for the special solution chosen here. These properties may be summarized as follows.
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Lemma 5.1. (a) Let the trajectory x = q(τ ) with q(0) = 0 , u(0) = e0 be C m , m ≥ 2, on a neighbourhood of τ = 0. Then, the fields F ret are C m−2 outside the trajectory. adv (b) The field Fret − Fsing can be extended to a field which is continuous across the trajectory. Proof. Ad (a). The assertion follows from the expressions for retarded and advanced fields containing second derivatives of the trajectory (cf. [15], (4–98)). Ad (b). This follows from (5.2) above. In the existence and uniqueness proof for the initial-value problem below, we will use the following construction. C 3 -Prolongation of the trajectory. Let us suppose that the trajectory is given only for nonnegative times t. We prolong the trajectory to the past in such a way that the prolonged trajectory is C 3 at the initial time t = 0. For this purpose, let us make the ansatz ˜ = f + gt + ht2 + kt3 exp(λt) if t < 0. (5.3) q(t) Here, the four vectors f, g, h, k, and the scalar λ are chosen in such a way that they match the corresponding derivatives of the trajectory at t = 0 and that the trajectory remains time-like. Explicitly, in our fixed inertial system, the prolonged trajectory is given by the formula ( q(t) for t ≥ 0 2 (5.4) x= 1 0 1 0 0 ˙ a + − 3λa exp(λt) for t < 0. a t t 2 6 Depending on the given vectors a0 and a˙ 0 , the value of λ > 0 can be calculated from the relation q˜ 0 (t)2 < 1 for all t < 0 . (In fact, we need the prolongation of the trajectory only for small negative times. Thus, the trajectory can be appropriately altered for negative t where |t| is sufficiently large.) To solve the initial value problem, we shall use the retarded field associated with the 0 represents the prolonged trajectory (5.4). We denote this field by F˜ret . In addition, F˜ret ˜ initial value of F˜ret at time t = 0. This initial value depends only on the part x = q(t) 0 is of the trajectory corresponding to times t < 0 . Therefore, (5.4) implies that F˜ret uniquely determined by the initial acceleration a0 and its derivative a˙ 0 . Furthermore, by (5.2), we get Q 0 0 e0 ∧ a˙ 0 + O(r) , (x) = Fsing (x) − F˜ret 6π
as r → 0 ,
(5.5)
0 (x) = Fsing (0, x) for x 6= 0. where r := |x| and Fsing
6. An Equivalent Formulation of the Equation of Motion 6.1. Derivation of a Dirac–Lorentz type equation. For solving the initial-value problem, let us express the regular part of the field in Eq. (3.7) in terms of the field F˜ which arises in the following decomposition F = F˜ret + F˜ . Using (3.3) and (5.2) we get
(6.1)
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Q F˜ = Freg + u ∧ a˙ + O(R) , 6π
as R → 0 ,
R=
p y2
for time τ > 0. Hence, Eq. (3.7) can be rewritten in the form Q2 (u ∧ a˙ ) u(τ ) + QF˜ ((q(τ ))u(τ ) 6π Q2 = a˙ − a2 u (τ ) + QF˜ ((τ ))u(τ ) , 6π
m0 a(τ ) = −
(6.2)
by (2.1) and (2.6). Eq. (6.2) represents an equation of the so-called Dirac–Lorentz type1 with the nonlinear term QF˜ ((τ ))u(τ ) which is sometimes called “external force”, whereas the first term is often called a “self-interaction force”. We stress that only the sum of the two has a physical interpretation as a necessary constraint which must be fulfilled by the “particle + field” system. The decomposition (6.2), based on (6.1), has been chosen only because it is useful in solving the initial value problem. To solve the final-value problem, (6.1) should be replaced by an analogous decomposition using the advanced field. Moreover, given a trajectory, any combination αFadv + βFret , α + β = 1, may be used. This proves that the notion of a “self-interaction force” and an “external force” is purely conventional and has no physical meaning (these problems were thoroughly discussed in [9] and [10]). From the point of view of the initial-value problem, the “external force”, given by the last term of (6.2), depends in a nonlocal way on the particle initial data q, u, a, and the field initial data at τ = 0. The mathematical structure of Eq. (6.2) will be carefully studied in Sect. 10. 6.2. The solution of the Dirac paradox. Let us consider the very special case where the electromagnetic field is given by F = F˜ret . That is, F˜ ≡ 0. From (6.2) we obtain the classical (nonlinear) Dirac equation of motion m0 a(τ ) =
Q2 a˙ − a2 u (τ ), 6π
q(0) = 0,
q(0) ˙ = e0 .
(6.3)
The initial condition tells us that the particle rests at time τ = 0. Eq. (6.3) is identical to the following equation ¨ )= m0 q(τ
Q2 ... q (τ ) − q(τ ¨ )2 q(τ ˙ ) , 6π
q(0) = 0,
q(0) ˙ = e0 .
(6.4)
In the purely mechanical framework (i.e. regarded as the dynamical equation for the particle alone), this equation represents an ill-posed initial-value problem because the initial data for q¨ are missing. Moreover, this equation admits self-accelerated (so-called runaway) solutions which approach the speed of light as time goes to infinity. In order to avoid such unphysical solutions, Dirac added ad hoc an additional, artificial condition. 2
In the context of our theory, the missing initial data q(0) ¨ for the third order differential Eq. (6.4) are provided by the field initial data (in fact, all the initial data are carried by the field). Consider, for example, the following quite natural initial condition: 1
Such equations are considered in [15], (6–57). In this connection, we refer to [5, 7, 15] and [16], Vol. 2. A detailed, very critical discussion of the Dirac theory can be found in the monograph [14] by Parrott. 2
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F = F−2
723
at time t = 0 outside the point x = 0.
(6.5)
This condition means that the total electromagnetic field equals the Coulomb field at the initial time t = 0. Let us discuss the consequences of condition (6.5). From (3.3),(3.4) it follows that F−1 = F0 = Freg = 0 at t = 0 outside the point x = 0. Hence E(0, x) =
Qn , 4πr2
B(0, x) = 0.
By (4.1) and (4.3), this implies a0 = 0, i.e., q00 (0) = 0. Thus, we have to add the condition q(0) ¨ = 0 to the Dirac Eq. (6.3). The modified Eq. (6.4) has the unique solution q(τ ) ≡ τ e0 (particle at rest for all times). This way, runaway solutions have been excluded. 7. Fermi–Walker Transport In order to simplify the Maxwell equations in the comoving system, let us introduce a special orthonormal frame on the Fermi hyperplane. Let 6 be a fixed inertial system along with the orthonormal vector basis e0 , e1 , e2 , e3 (cf. Sect. 2). For m ≥ 2, let x = q(t),
0 ≤ t ≤ t0 ,
(7.1)
q0 (0) = 0.
(7.2)
be a C m -trajectory of a particle such that q(0) = 0,
Suppose that |q0 (t)| < 1 on [0, t0 ], i. e., the velocity of the particle is less than the speed of light. Definition 7.1. By Fermi–Walker transport along the trajectory x = q(τ ), we understand four vectors b+0 , b+1 , b+2 , b+3 depending on τ which satisfy the ordinary differential equation (7.3) b˙ +α = ab+α u − ub+α a along with the initial condition b+α (0) = eα . We call b+0 (τ ), . . . , b+3 (τ ) the Fermi frame at eigentime τ . Proposition 7.2. The Fermi frame is orthonormal, i.e., b+α (τ )b+β (τ ) = eα eβ for all τ . Proof. By (7.3), d + + ˙+ + b b = bα bβ + b+α b˙ +β dτ α β = ab+α ub+β − ab+β ub+α + ab+β ub+α − ab+α ub+β = 0. In addition, b+α (0)b+β (0) = eα eβ .
Proposition 7.3. The 4-vector b+0 (τ ) of the Fermi frame is identical to the 4-velocity u(τ ). In addition, we obtain the differential equation (7.4) b˙ +j = ab+j u along with the initial condition b+j (0) = ej .
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Proof. By (2.6), au = 0 and u2 = −1. Hence, u˙ = (au)u − u2 a. Furthermore, u(0) = e0 . Since the solution of (7.3) is unique for α = 0, b+0 (τ ) = u(τ ). Finally, it follows from Proposition 7.2 that ub+j = b+0 b+j = 0. This yields (7.4). Since b+1 (τ ), b+2 (τ ), b+3 (τ ) are orthogonal to b+0 (τ ), these vectors form an orthonormal basis of the Fermi hyperplane 6+ (τ ). Proposition 7.4. The natural (holonomic) basis of the comoving system at the point (τ, y) is given by bj = b+j (τ ).
b0 = (1 + ay)u,
∂x + Proof. Note that x = q(τ ) + y j b+j . Thus, from bα = ∂y α we get bj = bj , and b0 = j ˙+ j + q(τ ˙ ) + y bj = u + y (abj )u = u + (ay)u, where (7.4) is used. + is given by Corollary 7.5. In the comoving system 6+ , the metric tensor gαβ
+ = diag(−(1 + ay)2 , 1, 1, 1) . gαβ
(7.5)
Proof. Relation gαβ = bα bβ and Proposition 7.4 imply (7.5). For a given vector d = dj (τ, y)bj , define ∂τ+ d = ∂τ dj bj . Physically, this is the time derivative of d observed in the comoving system. We have ∂τ+ d = d˙ − (ad)u.
(7.6)
This follows from d˙ = ∂τ+ d + dj b˙ j along with b˙ j = (abj )u. In particular, we define aF0 := ∂τ+ a,
aF(n) := ∂τ+
n
a.
(7.7)
8. Computation of the Singular Part of the Electromagnetic Field In this section, we work in the comoving system 6+ . Set R := |y|, and N := y/R. Proposition 8.1. In the comoving system 6+ , the Maxwell equations read as follows: div E = Lρext , −∂τ+ E + curl(LB) = Ljext , for R > 0, where L :=
q
div B = 0, ∂τ+ B + curl(LE) = 0
(8.1)
+ ) = 1 + ya(τ ). − det(gαβ
Proof. This follows from F = 21 F αβ bα ∧ bβ , (3.2) and Corollary 7.5. The system (8.1) corresponds to the tensor equation (3.2), where jext = jext + ρext L−1 b0 , F
ij
= −ε
ijk
Bk ,
F
0j
= −E L j
−1
,
ε
ijk
1 2 3 = sgn . i j k
(8.2)
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725
Remark 8.2. In the comoving system 6+ , the tensor of the electromagnetic field F αβ is explicitly given by 0 −E 1 L−1 −E 2 L−1 −E 3 L−1 E 1 L−1 0 −B 3 B2 . F αβ = 3 E 2 L−1 B 0 −B 1 E 3 L−1 −B 2 B1 0 This follows immediately from (8.2). Furthermore, F = L−1 E ∧ b0 − Alt(B) = E ∧ u − Alt(B).
(8.3)
Suppose that the electromagnetic field in 6+ has the following expansion as a sum of homogeneous components: E = E−2 + E−1 + · · · + En + . . . ,
B = B−2 + B−1 + · · · + Bn + . . . ,
(8.4)
where Ek = fk (N, a, aF0 , aF00 , . . . )Rk and Bk = gk (N, a, aF0 , aF00 , . . . )Rk for all k. We assume that the functions fk , gk , and a = a(τ ) are sufficiently smooth with respect to all their arguments. By (3.5) and (8.3), E−2 =
QN , 4πR2
B−2 = 0.
Observe that div E−2 = 0 and curl E−2 = 0 for R 6= 0. Definition 8.3. By F we denote the smallest space of functions of comoving variables (τ, y), |y| = 6 0, which satisfies the following conditions: (i) (ii) (iii) (iv)
The functions N, a, aF(n) , n = 1, 2, 3, . . . , belong to F . If A ∈ F, then Rk A ∈ F for k = ±1, ±2, . . . . If A, B ∈ F , then A × B ∈ F and αA + βB ∈ F for all α, β ∈ R. If A, B, C ∈ F , then (AB)C ∈ F .
Furthermore, let Fk denote the linear subspace of F consisting of precisely all the functions from F which have the form fk (N, a, aF0 , . . . )Rk ,
k = 0, ±1, ±2, . . . .
Lemma 8.4. (a) If A ∈ Fk , then ∂τ+ A ∈ Fk . (b) If A ∈ Fk , B ∈ F0 , then the functions curl A, (B grad)A, and B div A belong to Fk−1 . (c) If A ∈ Fk , then R(aN)A ∈ Fk+1 . (d) If A ∈ Fk , then A is C k−1 with respect to the space variables provided k ≥ 1. Proof. The proof proceeds by induction. Note that N2 = 1 and use vector calculus. For example, ∂j Rk = kRk−1 Nj and ∂j Nk = R−1 (δjk − Nj Nk ). Hence grad Rk = kRk−1 N,
div N = 2R−1 ,
curl N = 0.
Furthermore, use the product rule. For example, div(Rk N) = N grad Rk + Rk div N.
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Let jext = 0 and ρext = 0. Observing the structure of the Maxwell system (8.1) along with Lemma 8.4, it is quite natural to assume that Ek , Bk ∈ Fk for all k. Substituting the ansatz (8.4) into the Maxwell equations (8.1), a comparison of coefficients with respect to Rk yields the following recursive system for k = −1, 0, . . . , n: div Ek = 0, div Bk = 0, curl Bk = −curl (RaN)Bk−1 − ∂τ+ Ek−1 , curl Ek = −curl (RaN)Ek−1 + ∂τ+ Bk−1 .
(8.5)
This system can be solved for all k (which are not too large) via computer algebra 3 . Motivated by the decomposition (3.3) of the electromagnetic field, we make the ansatz E = E−2 + E−1 + E0 + Ereg , B = B−2 + B−1 + B0 + Breg ,
(8.6)
where the regular field is assumed to be at least continuous for all y in a neighbourhood of y = 0. Again, Ek and Bk are determined by the recursive system (8.5) for k = −1 and k = 0. Explicitly, E−2 =
QN , 4πR2
E−1 = − E0 =
Q ((aN)N + a), 8πR
B−2 = 0, B−1 = 0,
(8.7)
3Q Q ((aN)2 N + 2(aN)a − a2 N), B0 = N × aF0 . 32π 8π
One checks easily by hand that these fields satisfy (8.5) for R 6= 0. Note the rules given in Lemma 8.4. Remark 8.5. To get the expressions (3.5) from (8.7), set Fk := Ek ∧ u − Alt(Bk ) , k = −2, −1, 0 , by (2.3) and (8.3). Observe that aF0 = a˙ − a2 u (cf. (7.6), (7.7)). 3 We thank Bernd Fiedler (Department of Mathematics, University of Leipzig) who did the calculations using the computer algebra system ’MATHEMATICA’. He developed a package ’VECTAN’ and two associated extension packages to determine Ek ,Bk from (8.5). In particular, solutions for k = −1, 0, . . . , 6 were calculated. Some details are contained in the description of ’VECTAN’ which is presented at the World Wide Web pages http://www.mathematik.uni-leipzig.de/MI/fiedler/vectan.html and http://www.wolfram.com/cgibin/msitem?0208-594.
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9. Derivation of the Equations of Motion The energy-momentum vector of the pure field system (no particles!) is given by Z T u dV, P(τ ) := − 6+ (τ )
where u is the normal unit vector of the Fermi hyperplane 6+ (τ ). In addition, dV denotes the volume element on 6+ (τ ) induced by the metric on M4 . By (7.5), this is the Euclidean metric. The energy-momentum tensor of the electromagnetic field equals T := −F 2 +
1 tr F 2 I, 4
(9.1)
where F is regarded as a linear operator. Explicitly, tr F 2 = Fγµ F µγ , and 1 Tαβ = −Fαγ F γβ + Fγµ F µγ δαβ . 4
(9.2)
Proposition 9.1. If E, B denotes the electromagnetic field observed in the comoving system, then the energy-momentum vector is given by P(τ ) = H(τ )u(τ ) + P(τ ), where
Z
H(τ ) := R3
1 E(τ, y)2 + B(τ, y)2 dy, 2
(9.3)
Z (E(τ, y) × B(τ, y)) dy. (9.4)
P(τ ) := R3
From the physical point of view, H (resp. P) represents the total energy (resp. the total momentum) carried by the field and observed in the comoving reference system 6+ . Proof. In the comoving system, we have the representations (7.5)and (8.3) which imply Fjk = F jk , and Fj0 = −L2 F j0 . Hence, tr F 2 = 2 E2 − B2 . By (2.1) and (2.3), F u = (Eu)u − u2 E = E. Consequently, F 2 u = F (F u) = E2 u − (Eu)E − Alt(B)E = E2 u + E × B. Hence −T u = E2 u + E × B −
1 2 1 2 E − B2 u = E + B2 u + E × B. 2 2
Proposition 9.2. For a sufficiently smooth situation, conservation of the energy-momentum vector P is equivalent to the system ∂τ+ H + Pa = 0,
∂τ+ P + Ha = 0.
Proof. Differentiation of (9.3) with respect to τ yields ˙ )u + Ha + P˙ = ∂τ+ H u + Ha + ∂τ+ P + (Pa)u, P˙ = H(τ by (7.6). Thus, P˙ = 0 is equivalent to (9.5).
(9.5)
(9.6)
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H.-P. Gittel, J. Kijowski, E. Zeidler
Renormalization. For the “field + particle” system, the integrals (9.4) are divergent because of the singular E−2 component of the field. Representing the total field as the sum of the Coulomb “tail” F−2 of the particle and the remaining part F − F−2 , the integrals (9.4) decompose as the sum of the three terms: the one quadratic in F−2 , the one quadratic in F − F−2 and, finally, the mixed term. It is precisely the first term which is divergent. Consequently, the total energy-momentum is a priori not well defined. In the papers [9] and [10], a detailed analysis of this physical situation was given. Assume that the point-like particle is merely a mathematical idealization of an extended “soliton-like” object built from other physical matter fields strongly concentrated within a tiny region around zero and interacting (possibly in a highly nonlinear way) with the electromagnetic field. For the extended particle, the term quadratic in F−2 represents correctly the Coulomb field surrounding the particle only when we are outside the particle. Inside the particle, we have to regard the total energy-momentum tensor of all the field-constituents of the particle. Such an energy-momentum tensor is no longer singular. The integrals (9.4) applied to this field give us the total energy and the total momentum of the particle at rest: H(τ ) = m0 and P(τ ) = 0. We stress the fact that the quantity m0 represents the total rest mass of the particle (including already the mass carried by the electromagnetic field surrounding the particle). On the other hand, the mixed terms vanish when integrated over each sphere (at least outside the particle) because F−2 contains only the monopole part, i.e. the spherically symmetric Coulomb singularity, whereas F − F−2 is monopole-free. We should like to define the theory of point-like particles as an approximation of the above situation valid in the regime when the internal structure of the extended particle may be neglected. Hence, the above heuristics leads to the mathematically correct definition of the renormalized energy-momentum for the “point-particle + field” system. Here, we calculate the integrals (9.4) only for the component F − F−2 but we add the energymomentum m0 u carried by the particle at rest. Therefore, the renormalization consists in replacing the integral containing F−2 with the “renormalized” energy-momentum m0 u. More precisely, we put R := {y : |y| ≥ R}, SR := {y : |y| = R} and define Z Z 2 1 HR := E − E−2 + B2 dy, PR := (E × B) dy. (9.7) 2 R
R
Since E − E2 behaves like O(|y|−1 ), as |y| → 0, the limits Hren := m0 + lim HR , R→0
Pren := lim PR R→0
exist and define the renormalized four-momentum vector Pren := Hren u + Pren . We may also put PR := (HR + m0 ) u + PR . Then, Pren = lim PR . It would be quite natural R→0
to require its conservation P˙ ren = 0.
(9.8)
For mathematical reasons, let us replace condition (9.8) with the following stronger assumption: (A) W e postulate that lim P˙ R = 0. R→0
This condition would imply (9.8) if we could interchange the differentiation with the limit. The following result is crucial. Proposition 9.3. From (A) we obtain the equation of motion (3.7).
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729
Remark 9.4. Since F u = E, the relativistically invariant equation (3.7) is equivalent to m0 a(τ ) = QEreg (τ, 0)
(9.9)
in the comoving system. Proof. Step 1. By (9.6),
P˙ R = ∂τ+ HR + PR a u + ∂τ+ PR + HR a + m0 a .
We will show below that the following are met: (i) ∂τ+ HR + PR a = 8(R), (ii) ∂τ+ PR + HR a + m0 a = 9(R), where Z L E − E−2 × B N dS, 8(R) :=
(9.10)
SR
9(R) := m0 a −
Q 4πR2
Z Ereg dS + O(R) ,
as R → 0 .
(9.11)
SR
Taking this for granted, let us show that (i) and (ii) imply the statement. Since E − E−2 = O(R−1 ), we get lim 8(R) = 0. The continuity of Ereg at y = 0 gives us R→0
lim 9(R) = m0 a(τ ) − QEreg (τ, 0). From these relations and assumption (A), Eq. (9.9)
R→0
follows. Therefore, it only remains to prove (i) and (ii). To this end, we will use the Maxwell Eq. (8.1) in the comoving system along with integration by parts. Step 2. To simplify the considerations, let us assume that ρext = 0 and jext = 0. The general case follows by replacing F with F + Fext , where Fext is explicitly given by the representation formulas (3.12) in the inertial system. Observe that the field Fext is smooth everywhere. Ad (i). Using L = 1 + ay, grad L = a , and curl E−2 = 0, we obtain the identity 41 := E − E−2 curl(LB) − B curl(LE) = −(E × B)a − div L E − E−2 × B) . The Maxwell equations (8.1), with ρext = 0, jext = 0, imply 41 = E − E−2 ∂τ+ E + + B∂τ B. Therefore, by (9.7), Z Z + 41 dy = −PR a + L E − E−2 × B N dS. ∂ τ HR = R
SR
Ad (ii). Note the identity 42 := curl(LB) × B − E × curl(LE) 1 1 2 2 E + B id − a E2 + B2 . = div L E ⊗ E + B ⊗ B − 2 2 From the homogeneous Maxwell system associated with (8.1), we deduce 42 = ∂τ+ E × B + E × ∂τ+ B. Hence, (9.7) gives
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H.-P. Gittel, J. Kijowski, E. Zeidler
Z 42 dy
∂τ+ PR = R
1 =− a 2
Z
E2 + B2 dy +
R
and HR =
1 2
1 = 2
Z
1 2 E + B2 N dS, L (EN)E + (BN)B − 2
SR
Z R
1 E2 + B2 dy + 2
Z
2
E +B R
2
1 dy + 2
Z
E−2 E−2 − 2E dy
R
Z
R(E−2 N)
E−2 − 2E N dS.
SR
To get the last relation, observe that E−2 = −grad(E−2 y) along with div E−2 = div E = 0. Consequently, ∂τ+ PR + HR a = χ(R), where Z R χ(R) = (E−2 N) E−2 − 2E N a dS 2 SR Z (9.12) 1 2 2 (1 + R(aN)) (EN)E + (BN)B − E + B N dS. − 2 SR
Use the representations (8.6) and (8.7) of E, B, Ek , and Bk in order to collect all the terms which are odd with respect to N. In (9.12), the integrals over these terms vanish. Finally, we obtain Z R (E−2 N)2 a + (E−2 N) E−1 + Ereg + E−1 + Ereg N E−2 χ(R) = − 2 SR 1 2 − E−1 + Ereg E−2 N + R(aN) (E−2 N)E−2 − E−2 N + O(R−1 ) dS 2 Z Q Ereg dS + O(R) , as R → 0. =− 4πR2 SR
This yields the desired expression for 9(R) in (9.11).
10. Solution of the Initial–Value Problem for the Basic Equations In this section we prove Theorem 4.1. 10.1. Outline of the proof. The basic idea of the existence proof. It is sufficient to solve the equation of motion for the particle ¨ ) = QFreg (q(τ ))q(τ ˙ ), m0 q(τ
q(0) = 0,
q(0) ˙ = e0 ,
q(0) ¨ = a0 ,
(10.1)
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731
where Freg = F − Fsing , by (3.3). The point is that we use the decomposition F = F˜ret + F˜ of the electromagnetic field from (6.1). Thus, Freg = F˜ + F˜ret − Fsing .
(10.2)
Hence the equation of motion (10.1) is equivalent to ¨ )= m0 q(τ
Q2 ... q (τ ) − q(τ ¨ )2 q(τ ˙ ) + QF˜ (q(τ ))q(τ ˙ ), 6π
q(0) = 0, q(0) ˙ = e0 , q(0) ¨ = a0 , (10.3)
(cf. (6.2)). Here F˜ satisfies the Maxwell system Div F˜ = jext ,
Div(∗F˜ ) = 0.
(10.4)
Let us consider eigentimes τ ≥ 0. The following facts are crucial for our approach. (i) The fields Fsing and F˜ret depend upon the unknown trajectory in an explicitly known manner (cf. (3.4), (3.5) for the first one and formula (4-98) in [15] for the latter). (ii) By construction, F˜ret satisfies the homogeneous Maxwell system (5.1) outside the trajectory. (iii) The field Freg is smooth outside the trajectory and continuous across the trajectory. The same holds true for the difference F˜ret − Fsing (cf. (5.2)). Hence the Maxwell equations (10.4) are satisfied on a neighbourhood of the trajectory in the sense of generalized functions. (iv) The field F˜ is uniquely determined by its initial value 0 0 0 + Fsing − F˜ret F˜ 0 := Freg on a space-like hyperplane at τ = 0. Here, F˜ 0 depends only upon the initial data 0 , and also F 0 , i. e. upon the acceleration a0 and its derivative a˙ 0 contained in Fsing 0 0 ˜ ˜ upon Freg . Explicit representation formulas for F depending on F will be given in (10.7) below. ... Remark 10.1. Note that the initial value a˙ 0 (i.e., the value q(0)) is compatible with Eq. (10.3). In fact, this equation is equivalent to (10.1), and we have assumed the latter to be fulfilled by our initial data. We remind the reader that the first term on the right hand side of (10.3) comes from the value of F˜ret − Fsing at zero. This value has been added here and, at the same time, subtracted from Freg , converting it into F˜ , by (10.2). To simplify notation, we assume that the external sources vanish, i.e., let jext = 0. By superposition, the general case can be easily reduced to this special one. For this purpose, we replace F with F + Fext , where Fext is a smooth field satisfying the inhomogeneous Maxwell system (3.2) (cf. Remark 3.4), and observe that both the Maxwell system and the equation of motion for the particle are linear expressions in Freg . The uniqueness proof.. Suppose that we are given two solutions of the initial-value problem for the particle–field system having the same initial values. Furthermore, suppose that the two solutions possess the smoothness properties as indicated in Theorem 4.1. Then, the two trajectories satisfy the same equation of motion (10.3). The fixed-point argument below will show that the solution of (10.3) is unique for small times t > 0. Thus, the two trajectories coincide. Hence, by (i) and (iv) above, the two electromagnetic fields are identical.
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10.2. Mathematical preparations. For the proof, let us use a fixed inertial system. In such a system, the equation of motion (10.3) for the particle is transformed the following equation: Q2 d d m0 v(t) p (1 − v(t)2 )−2 (1 − v(t)2 ) id + v(t) ⊗ v(t) v0 (t) = 2 dt 1 − v(t) 6π dt Q2 (v(t)v0 (t))2 + (1 − v(t)2 )(v0 (t))2 v(t) (10.5) − 6π(1 − v(t)2 )3 ˜ q(t)) + Qv(t) × B(t, ˜ q(t)) + QE(t, which we have to solve along with the initial conditions q(0) = 0,
q0 (0) = 0,
q00 (0) = a0
(10.6)
for the unknown trajectory q = q(t), where v(t) := q0 (t). Furthermore, according to (2.2), ˜ The fields E, ˜ B ˜ are solutions of homogeneous Maxwell ˜ ∧ e0 − Alt(B). we set F˜ = E equations and, therefore, may be expressed in terms of initial data as follows (cf., e.g. [18], §§15, 46): Z ˜0 +E ˜ 0 (x + tz) dSz , ˜ 0 + t curl B ˜ x) = 1 t(z grad)E E(t, 4π S
˜ x) = 1 B(t, 4π
Z ˜0 +B ˜ ˜ 0 − t curl E t(z grad)B
0
(10.7) (x + tz) dSz ,
S
˜ 0 for these fields at time t = 0 are ˜ 0, B where S := {z : |z| = 1}. The initial values E known: 0 0 0 ˜ ret ˜ 0ret . ˜ 0 = B0reg + B0sing − B ˜ 0 = Ereg + Esing −E , B (10.8) E 0 0 0 ˜ ret ˜ 0ret depend , B0reg are given. Moreover, Esing , B0sing , E , and B Note that the fields Ereg 0 0 on the initial acceleration a of the particle and its derivative a˙ in an explicitly known manner (cf. (3.4), (3.5) and formula (4-98) in [15]). ˜ 0 of the field ˜ 0, B Let us first study some regularity properties of the initial values E ˜E, B ˜ at time t = 0. 0 , B0reg of the electromagnetic field Proposition 10.2. Suppose that the initial values Ereg m 0 0 ˜ ,B ˜ are continuous in a neighbourhood of the origin x = 0. are C , m ≥ 2. Then, E Furthermore, they are C m outside the origin. Moreover,
˜0 ∂E = O(1), ∂xi ˜0 ∂2E = O(r−1 ), ∂xi ∂xj where r := |x|.
˜0 ∂B = O(1), ∂xi
as r → 0,
˜0 ∂2B = O(r−1 ), ∂xi ∂xj
as r → 0,
(10.9)
(10.10)
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733
˜ 0, B ˜ 0 follow from Lemma 5.1 (b). To prove the Proof. Continuity and smoothness of E desired estimates for the derivatives, we use the explict formula for the retarded field 0 (x) in terms of the (prolonged backwards) trajectory (e.g. formula (4-98) in [15]). F˜ret Remark 10.3. The estimates (10.9) show that the integral expressions (10.7) make sense for all (t, x) in a neighborhood V of 0 ∈ R3 × [0, t1 ]. As it is known from the theory of linear hyperbolic systems, jumps in the data at x = 0 are transported along the bicharacteristics emanating from the origin, i.e., along the rays t = |x|. These rays form the forward cone associated with the Maxwell system at x = 0. At those points of V that ˜ B ˜ are C m−1 . This can be immediately seen from are away from this cone, the fields E, formulas (10.7). 10.3. The fixed-point argument. It is our aim to transform the equation of motion (10.5) into an operator equation for the acceleration w(t) := q00 (t) of the particle. To this end, we express the trajectory q = q(t) and its velocity v = q0 (t) through the acceleration by means of Zt
Zt w(s) ds,
v(t) =
v(s) ds.
q(t) =
0
(10.11)
0
Recall the initial conditions (10.6). To eliminate the third derivative q000 , we integrate the equation of motion (10.5) on the time interval [0, t]. We obtain an equivalent operator equation w = G[w].
(10.12)
Explictly, G[w](t) := G1 [w](t) G2 [w](t) +
Zt
(G3 [w] − G4 [w]) (s) ds
(10.13)
0
for t ∈ [0, t1 ], where −1 G1 [w] := (1 − v2 )2 (1 − v2 ) id + v ⊗ v , 6πm0 v 0 √ G2 [w] := +a , Q2 1 − v2 G3 [w] := (vw)2 + (1 − v2 )w2 (1 − v2 )−3 v, 6πm0 ˜ ˜ . E+v×B G4 [w] := Q
(10.14)
˜ and B ˜ stand for E(t, ˜ q(t)) and B(t, ˜ q(t)), respectively. The corresponding fields Here, E are given in (10.7). Furthermore, q and v are related to w by (10.11). We study the operator Eq. (10.12) on the Banach space X := C[0, t1 ], where t1 > 0 will be chosen sufficiently small. This space consists of all continuous vector functions w = w(t) on the time interval [0, t1 ]. The norm on X is given by
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kwk := max |w(t)|. t∈[0,t1 ]
Consider now the ball B := w ∈ X : kw − a0 k ≤ R ,
(10.15)
where the radius R > 0 is fixed. Proposition 10.4. The operator equation w = G[w], w ∈ B has a unique solution. Proof. This follows from the Banach fixed point-theorem (cf., e.g. [20], Vol. 1, Theorem 1.A) by using Lemma 10.6 ahead. Corollary 10.5. The trajectory q = q(t), related to the solution w ∈ B, is C 2 on [0, t1 ]. Proof. For any given w ∈ B, set W := G[w]. By the definitions (10.13) and (10.14), the function W = W(t)) is C 1 on [0, t1 ]. Similarly, if w is C 1 on [0, t1 ], then W is C 2 on this interval. Finally, observe that w = G[w]. The higher regularity of the trajectory stated in Corollary 4.2 follows by a repeated application of the boot-strap argument of the preceding proof. Lemma 10.6. Let the operator G be defined by (10.13),(10.14). Then, for sufficiently small t1 > 0, the following are met: (a) G maps the ball B into itself. (b) G is a contraction on B, i.e., ˆ ≤ Lkw − wk ˆ kG[w] − G[w]k ˆ ∈ B and a fixed real number L with 0 ≤ L < 1. for all w, w Proof. Let w ∈ B. From (10.11), we get the estimates |v(t)| ≤ t(R + ka0 k),
|q(t)| ≤
t2 (R + ka0 k) 2
(10.16)
for t ∈ [0, t1 ]. Thus, the expression G1 [w](t) in (10.14) is well-defined for all such times t provided t1 is sufficiently small. Ad (a). Let w ∈ B. Set W := G[w]. By (10.13) and (10.14), the function W = W(t) is continuous on [0, t1 ], i.e., W ∈ X . To prove that W ∈ B, use the estimate |G[w](t) − a0 | ≤ C tC + tka0 k ≤ R R . Here and in what follows, the same symbol C denotes C C + ka0 k different positive constants. In this connection, observe that 1 v ⊗ v a0 O(1) G1 [w]G2 [w] − a0 = (1 − v2 )a0 − id + 1 − v2 for all t, 0 ≤ t ≤
= a0 O(t2 ), by (10.16).
as t → 0,
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Ad (b). First let n = 1, 2, 3. It is easy to see that the operators Gn are Fr´echetdifferentiable on B. Moreover, it follows from (10.11) and (10.14) that ˆ ˆ ≤ C|v(t) − vˆ (t)| ≤ tCkw − wk, |Gn [w](t) − Gn [w](t)|
(10.17)
ˆ ∈ B and all t ∈ [0, t1 ]. for all w, w ˜ q(t)) − The argument for G4 is more involved. Let us consider the difference E(t, ˜ q(t)). ˆ E(t, By the representation formulas (10.7), it is sufficient to estimate the quantity 0 ˜0 ˜ ∂E ∂E ˆ + tz) , (q(t) + tz) − (q(t) D := ∂xi ∂xi where |z| = 1. Let t > 0 be fixed. For some ϑ ∈ (0, 1), the mean value theorem along with (10.10) implies −1 ˆ |q(t) − q(t)| ˆ D ≤ C tz + ϑq(t) + (1 − ϑ)q(t) −1 t2 ˆ ˆ ≤ C |tz| − ϑ|q(t)| − (1 − ϑ)|q(t)| kw − wk 2 ˆ ≤ tCkw − wk. Here, the relations (10.11) and (10.16) have been used. Observe that for estimating the ˆ denominator, the time t has to be chosen small enough to ensure, say, |q(t)|, |q(t)| < 21 t. ˜ 0 and By the same argument, we can derive similar inequalities for the derivatives of B ˜ 0 itself (use (10.9) instead of (10.10)). Hence for E E(t, ˜ q(t)) − E(t, ˜ q(t)) ˆ ≤ tCkw − wk. ˆ ˜ is replaced with B. ˜ Consequently, the definition of G4 in An analog estimate holds if E (10.14) implies the validity of the inequality (10.17) for n = 4, too. Summarizing we get ˆ ˆ ≤ Lkw − wk ˆ |G[w](t) − G[w](t)| ≤ tCkw − wk for all t ∈ [0, t1 ], where L := t1 C. Choose t1 < C −1 to obtain L < 1.
Remark 10.7 (Mathematical generalizations). A refined analysis gives more detailed information about the regularity of the fields under consideration. Moreover, similar results can be derived if the function spaces C m are replaced with appropriate Sobolev spaces (cf., e.g. [18], §§5,40 and [20], Vol. 2A, Sect. 21.2). Acknowledgement. This research was supported in part by the Deutsche Forschungsgemeinschaft (DFG) via Forschergruppe "Nichtlineare Funktionalanalysis und Mathematische Methoden der Kontinuumsmechanik". Moreover, one of the authors (J.K.) is grateful to the Polish National Committee for Scientific Research (KBN), Warsaw, for financial support (Grant 2P 302 189 07).
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