Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint G-bundles of different topological types over complex curves Σg,n of genus g with n marked points. The bundles are defined by their characteristic classes - elements...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2012
Hauptverfasser:	Levin, A.M., Olshanetsky, M.A., Smirnov, A.V., Zotov, A.V.
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spelling	Levin, A.M. Olshanetsky, M.A. Smirnov, A.V. Zotov, A.V. 2019-02-18T17:37:32Z 2019-02-18T17:37:32Z 2012 Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles / A.M. Levin, M.A. Olshanetsky, A.V. Smirnov, A.V. Zotov // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 74 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H70; 32G34; 14H60 DOI: http://dx.doi.org/10.3842/SIGMA.2012.095 https://nasplib.isofts.kiev.ua/handle/123456789/148657 We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint G-bundles of different topological types over complex curves Σg,n of genus g with n marked points. The bundles are defined by their characteristic classes - elements of H²(Σg,n,Z(G)), where Z(G) is a center of the simple complex Lie group G. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness. The authors are grateful to A. Beilinson, L. Feh´er, B. Feigin, A. Gorsky, S. Khoroshkin, A. Losev, A. Mironov, V. Poberezhny, A. Rosly and A. Stoyanovsky for useful discussions and remarks. The work was supported by grants RFBR-09-02-00393, RFBR-09-01-92437-KEa and by the Federal Agency for Science and Innovations of Russian Federation under contract 14.740.11.0347. The work of A.Z. and A.S. was also supported by the Russian President fund MK-1646.2011.1, RFBR-09-01-93106-NCNILa, RFBR-12-01-00482 and RFBR-12-01-33071 mol a ved. The work of A.L. was partially supported by AG Laboratory GU-HSE, RF government grant, ag. 1111.G34.31.0023. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
spellingShingle	Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles Levin, A.M. Olshanetsky, M.A. Smirnov, A.V. Zotov, A.V.
title_short	Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
title_full	Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
title_fullStr	Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
title_full_unstemmed	Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
title_sort	hecke transformations of conformal blocks in wzw theory. i. kzb equations for non-trivial bundles
author	Levin, A.M. Olshanetsky, M.A. Smirnov, A.V. Zotov, A.V.
author_facet	Levin, A.M. Olshanetsky, M.A. Smirnov, A.V. Zotov, A.V.
publishDate	2012
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint G-bundles of different topological types over complex curves Σg,n of genus g with n marked points. The bundles are defined by their characteristic classes - elements of H²(Σg,n,Z(G)), where Z(G) is a center of the simple complex Lie group G. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/148657
citation_txt	Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles / A.M. Levin, M.A. Olshanetsky, A.V. Smirnov, A.V. Zotov // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 74 назв. — англ.
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first_indexed	2025-11-26T08:28:04Z
last_indexed	2025-11-26T08:28:04Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 095, 37 pages Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles Andrey M. LEVIN †‡, Mikhail A. OLSHANETSKY ‡, Andrey V. SMIRNOV ‡§ and Andrei V. ZOTOV ‡ † Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Str., Moscow, 117312, Russia E-mail: alevin57@gmail.com ‡ Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia E-mail: olshanet@itep.ru, asmirnov@itep.ru, zotov@itep.ru § Department of Mathematics, Columbia University, New York, NY 10027, USA Received July 14, 2012, in final form November 29, 2012; Published online December 10, 2012 http://dx.doi.org/10.3842/SIGMA.2012.095 Abstract. We describe new families of the Knizhnik–Zamolodchikov–Bernard (KZB) equa- tions related to the WZW-theory corresponding to the adjoint G-bundles of different topo- logical types over complex curves Σg,n of genus g with n marked points. The bundles are defined by their characteristic classes – elements of H2(Σg,n,Z(G)), where Z(G) is a center of the simple complex Lie group G. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the under- lying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness. Key words: integrable system; KZB equation; Hitchin system; characteristic class 2010 Mathematics Subject Classification: 14H70; 32G34; 14H60 1 Introduction The Knizhnik–Zamolodchikov–Bernard (KZB) equations [8, 9, 40] are a system of differential equations for conformal blocks in a conformal field theory. Here we consider the WZW theory of the level k, related to a simple complex Lie group G and defined on a Riemann surface Σg,n of genus g with n marked points (z1, z2, . . . , zn). To describe this model, one should define a G-bundle over Σg,n. Topologically, the G-bundles are defined by their characteristic classes. Let Z(G) be a center of G and Gad = G/Z(G). The characteristic classes are obstructions to lift the Gad-bundles to the G-bundles. They are elements of the cohomology group H2(Σg,Z(G)) ∼ Z(G) [46]1. If Ḡ is the corresponding simply connected group (the universal covering with the natural group structure) and G = Ḡ/Z∨(G), then elements from H2(Σg,Z∨(G)) are obstruction to lift the G-bundles to the Ḡ-bundles. In particular, consider G = Spin(N) and SO(N) = Spin(N)/Z2. Then H2(Σg,Z2) ∼ Z2 defines the Stiefel–Whitney classes of the SO(N)-bundles over Σg. For generic bundles the WZW theories were studied in [23, 35]. The aim of this paper is to define the KZB equations in these theories. The KZB equations have a large range of applications in mathematics. In particular, on the critical level they produce Hamiltonians of the quantum Hitchin system [30, 34, 43, 57], while in the classical limit they lead to the 1See (4.1)–(4.3) in [46]. mailto:alevin57@gmail.com mailto:olshanet@itep.ru mailto:asmirnov@itep.ru mailto:zotov@itep.ru http://dx.doi.org/10.3842/SIGMA.2012.095 2 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov monodromy-preserving equations [32, 41, 44, 62, 66]. In this way, we obtain new classes of these systems. The KZB equations are described in the following way. Consider the highest weight repre- sentations Vµa (µa are the highest weights) of G attached to the marked points. For a positive integer k define the integrable module V̂µa of level k of the centrally extended loop group D× → G, where D× = D \ za is a punctured disk around the marked point za. The conformal blocks are linear functionals V̂ [n] ≡ V̂µ1 ⊗ · · · ⊗ V̂µn → C satisfying some additional conditions (the Ward identities). Let CG(V̂ [n]) be a space of conformal blocks. This space depends on parameters – the complex structure of Σg,n, and in this way forms a bundle over the moduli space Mg,n of complex structures. There exists a projectively flat connection in this bundle (the KZB connection). Then the meaning of the KZB equations is that the conformal blocks are the horizontal sections of the KZB connection. The KZB equations were derived originally for the genus zero case by Knizhnik and Zamolodchikov [40] and were generalized later to arbitrary genus by Bernard [8, 9]. In subsequent years the KZB equations was studied in a number of works [4, 16, 22, 29, 33, 36]. If the cocenter Z∨(G) = Ker Ḡ → G is non-trivial then the integrable module is a sum of sectors, corresponding to the characteristic classes of the underlying bundles V̂µ = N−1⊕ j=0 V (j) µ , N = ordZ(G). In terms of the spectra the WZW theory this was studied essentially in [23]. Similarly, the conformal blocks are also a sum of different sectors. In each sector one can define the KZB connection. The aim of this paper is to construct explicitly the KZB connections in all sectors of con- formal blocks for the WZW theory defined on elliptic curves. The compatibility conditions (horizontality of the KZB connection) are verified explicitly. The KZB connection in the trivial sector was studied in [24]. This construction is based on the classical dynamical r-matrix with the spectral parameter living on the elliptic curve. The r-matrices of this type related to the trivial sector were classified by Etingof and Varchenko [19]. Recently, we have classified the dynamical elliptic r-matrices as sections of some bundles of an arbitrary topological type over elliptic curves [46]. It turned out that the dynamical parameters of the r-matrices are elements of the moduli spaces of the bundles. It allows us to define the KZB connection in these cases. Different approach to classification of elliptic r-matrices was proposed in [17, 18, 21] and the corresponding KZB connection was also constructed in [17, 18]. The staring point of last ap- proach is an automorphism of the extended Dynkin diagram. In our construction we considered only those automorphisms that isomorphic to elements of the center Z. In this case we come to the same r-matrices and the KZB equations as in [17, 18]. For An, Dn and E6 algebras there exists another type automorphisms. So far the underlying vector bundle structure is unclear. It should be noted that in [17, 18] the derivation of the KZB equation is based on the representa- tions of conformal blocks as twisted traces of intertwiners. We will come to this representation in the forthcoming paper where the Hecke transformation of conformal blocks will be considered (see below). For the SL(N,C) WZW model on elliptic curves the KZB equation in the similar to our form was described in [42]. The authors considered a particular type of bundles that lead to the Belavin–Drinfeld classical r-matrix. In this case the corresponding KZB equation has not dynamical parameter and similar to the KZ equation. However, if N is not a prime number there exist r-matrices and the corresponding KZB equations intermediate between Felder and Belavin–Drinfeld cases. Hecke Transformations of Conformal Blocks in WZW Theory. I 3 In the subsequent paper we will describe the transformation operators that intertwine the different sectors (the Hecke transformations). It is worthwhile to notice that in the classical case these transformations provide a passage from the elliptic Calogero–Moser system to integrable Euler–Arnold top [48, 69, 70] (see also [47, 59, 70, 73]). For arbitrary characteristic classes these type of models were described in [45]. Different aspects and applications of the Hecke transformations to integrable systems and related topics (such as Painlevé–Schlesinger equations [2, 15, 50, 51, 60, 67, 68, 71], monopoles [14, 25, 31, 37, 39, 49, 58], quadratic Poisson structures [13, 74], applications to AGT conjecture [53, 54, 55] etc.) can be found in wide range of literature. The paper has the following structure. In Section 2 we consider a general setting of the KZB equations related to arbitrary curves Σg,n and arbitrary characteristic class of the bundles. In Section 3 the space of conformal blocks is described. In Section 4 we consider the genus one case in detail. The proofs of main relations (Propositions 1 and 2) and information about the special basis in simple Lie algebras as well as the elliptic functions identities are given in the appendices. 2 Loop algebras, loop groups and integrable modules 2.1 Loop algebras and loop groups Let Ḡ be a simply-connected simple complex Lie group and Z = Z ( Ḡ ) is the center of Ḡ. For all simply-connected groups (SL(N,C), SpN , E6, E7 and SpinN except N = 4n), the center is a cyclic group. For Spin4N Z = Z2 ⊕ Z2. The adjoint group is the quotient group Gad = Ḡ/Z. Assume for simplicity that Z is a cyclic group Zl of order l. Let K be a maximal compact subgroup of Ḡ and T is the Cartan torus of K. Consider the homomorphisms of S1 → T e(ϕ)→ ( e(γ1ϕ), e(γ2ϕ), . . . , e(γlϕ) ) ∈ T, e(ϕ) = exp(2πıϕ). P∨ = {γ = (γ1, . . . , γl)} is a coweight lattice in the Cartan subalgebra hK = Lie(T ) and in h ⊂ g = Lie ( Ḡ ) . Let Q∨ be the coroot lattice (Q∨ ⊆ P∨). The center Z ( Ḡ ) is isomorphic to the quotient group Z ∼ P∨/Q∨. In particular, if $∨ ∈ P∨ is a coweight such that l$∨ ∈ Q∨, then the Z ∼ Zl. It is generated by the element e($∨) = exp(2πı$∨) ∈ T . For Spin4N the center is generated by two coweights, corresponding to the left and right spinor representations. Let h be a Cartan subalgebra of g and {α} = R ∈ h∗ is the root system [12]. There is the root decomposition of g, g = h⊕ ∑ α∈R gα, adX gα = 〈X,α〉gα, X ∈ h. R is an union of positive and negative roots R = R+ ∪R− with respect to some ordering in h∗. Let Π = {α1, . . . , αl} be a basis of simple roots in R. The dual system Π∨ = {α∨1 , . . . , α∨l } (〈αj , α∨k 〉 = δjk) forms a basis in h. Let t be a coordinate in C. Define the loop group L(G) = {C∗ → G} = {g(t)} such that g(t) has a finite order poles when t→ 0. In other words, L(G) is the group of Laurent polynomials L(G) = G⊗ C[[t, t−1]. There is a central extension L̂(g) of L(g) = g⊗ C[[t, t−1] L̂(g) = L(g)⊕ CK, (2.1) defined by a two-cocycle c(X ⊗ f, Y ⊗ g) = (X,Y ) Res(gf ′dt). The set of the affine roots if of the form: Raff = {α̂ = α + n, n ∈ Z, n 6= 0}. Let {hα} be the basis of simple coroots in h. Then the analog of the root decomposition for the loop algebra 4 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov has the following form L(g) = g + ∑ n6=0 ∑ α∈Π hαt n + ∑ α̃∈Raff gα̂, gα̂ = xαeα̂ = xαeαt n. Let −α0 be the highest root −α0 ∈ R+. The system of simple affine roots is Π̂ = Π∪ (−α0 +1). It is a basis in Raff . Consider the positive loop subalgebra L+(g) = ( b + g⊗ tC[[t]] ) , (2.2) where b = h⊕ ∑ α∈R+ gα is the positive Borel subalgebra. Let also L−(g) = ( n− + g⊗ t−1C[t−1] ) , n− = ∑ α∈R− gα. (2.3) Then L̂(g) (2.1) is the direct sum L̂(g) = L−(g)⊕ L+(g)⊕ CK. (2.4) Each summand is a Lie subalgebra of L̂(g). There are two types of the affine Weyl groups: WP = W n P∨ and WQ = W nQ∨, where W is the Weyl group of g, WP = { ŵ = wtγ , w ∈W, γ ∈ P∨ } , WQ = { ŵ = wtγ , w ∈W, γ ∈ Q∨ } . (2.5) They act on the root vectors as eα̂ = eαt n → eŵ(α̂) = ew(α)t n+〈γ,α〉. The loop groups L(G) = G⊗ C[[t, t−1]] have the Bruhat decomposition [61]. Define subgroups L+(G) = { g0 + g1t+ · · · } , gj ∈ G, g0 = b ∈ B, is the positive Borel subgroup, (2.6) N−(G) = { n− + g1t −1 + · · · } , n− ∈ N−, is the negative nilpotent subgroup, (2.7) N+(G) = { n+ + g1t+ · · · } , n+ ∈ N+, is the positive nilpotent subgroup. (2.8) The Bruhat decomposition takes the form L ( Gad ) = ⋃ ŵ∈WP N− ( Gad ) ŵL+ ( Gad ) , L ( Ḡ ) = ⋃ ŵ∈WQ N− ( Ḡ ) ŵL+ ( Ḡ ) . (2.9) For a loop g(t) in Gad denote by ḡ its lift to a map from S1 to Ḡ. This map can be multivalued, after turning along the circle the value can be multiplied by some element of the center which we call the monodromy: g(e2πıt) = e(γ)g(t), (e(x) = e2πıx). If γ /∈ Q∨ then ζ = e(γ) is a non-trivial element of the center Z and the map g(t) is well defined for G = Gad, but not for Ḡ. In this way we have the representations L ( Gad ) = ⋃ γ∈P∨ Lγ ( Gad ) , Lγ ( Gad ) = { g ( e2πıt ) = e(γ)g(t) } . (2.10) If γ1 = γ2 + δ for any δ ∈ Q∨ then γ1 and γ2 lead to the same monodromies. We say in this case that Lγ1 ( Ḡ ) and Lγ2 ( Ḡ ) are equivalent. Then from (2.10) we have L ( Gad ) = ⋃ ζ∈Z Lζ ( Gad ) . Hecke Transformations of Conformal Blocks in WZW Theory. I 5 In particular, if the center Z ∼ Zl is generated by a fundamental coweight $∨, then L ( Gad ) = l−1⋃ j=0 Lj ( Gad ) , Lj ( Gad ) = { g ( e2πıt ) = e(j$∨)g(t) } , (2.11) and Lj ( Gad ) = e(j$∨) ( L ( Ḡ ) /Z ) . Consider the quotient Flaff = L ( Gad ) /L+ ( Gad ) [61]. It is called the affine flag variety. Let Σŵ be an N− ( Gad ) -orbit of ŵ in Flaff . This orbit is dipheomorphic to the intersection N− ( Gad ) ŵ = N− ( Gad ) ∩ ŵN− ( Gad ) ŵ−1. Therefore, its codimension in Flaff is the length l(ŵ) of ŵ. It is the number of negative affine roots which ŵ transforms to positive ones (Theorem 8.7.2 in [61]). The Bruhat decomposition (2.9) defines the stratification of Flaff : Flaff = L ( Gad )/ L+ ( Gad ) = ⋃ ŵ∈WP Σŵ. (2.12) 2.2 Integrable modules Consider a subset of dominant weights P+ = {µ ∈ P \|〈µ, α∨〉 ≥ 0 for α∨ ∈ Π∨}. Each dominate weights define a g-module Vµ. It contains the highest weight vector (HWV) vµ such that Xvµ = 〈X,µ〉vµ for X ∈ h, gαvµ = 0 for α ∈ R+. Define the Verma module Vµ of L̂(g) associated with Vµ [38]. Let Ik = {µ ∈ P+\|〈µ, α∨0 〉 ≤ k} be a subset of dominant weights. Define the action of L+(g) (2.2) on Vµ: (g ⊗ tC[[t]])Vµ = 0, KVµ = k Id, and b acts on Vµ as described above. Then Vµ = U(L̂(g))⊗U(L(g)+) Vµ is induced, where µ ∈ Ik. There is the isomorphism Vµ ∼ U(L−(g))⊗C vµ. Let Eα0 be the root subspace in g corresponding to α0. Consider the maximal submodule Sµ of Vµ generated by the singular vector( Eα0 ⊗ t−1 )k−〈µ,α0〉+1 vµ. (2.13) The irreducible integrable module V̂µ is the quotient V̂µ = Vµ/Sµ. (2.14) We identify the module Vµ with a submodule Vµ ⊗ 1 ↪→ Vµ. The integrable module V̂µ can be characterized in the following way: the subspace of V̂µ annihilated by the positive subalgebra g⊗ C[[t]] is isomorphic to the finite-dimensional g-module Vµ Vµ ∼ { v ∈ V̂µ\|(g⊗ tC[[t]]) · v = 0 } . The group L(G) has a central extension 1→ C∗ → L̂G→ LG→ 1 corresponding to (2.1). The integrable module can be described in terms of L̂G. The action of L+(G) on the HWV has the form L+(G)vµ = χµ(b)vµ, λvµ = e(k)vµ, λ ∈ the center C∗, (2.15) where χµ(b) is the character of the Borel subgroup B. Then V̂µ is generated by the action of N−(G) (2.7) on Vµ. 6 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov In this way we describe only “the trivial sector” of the L(G)-module. Consider the Bruhat representation for L ( Gad ) (2.9), and let ŵ = tγ , γ ∈ P∨. Define the Verma modules with the HWV tγvµ, Vµ(γ) = U(L−(g))⊗C t γvµ. (2.16) They have the singular vectors ( Eα0 ⊗ t−1 )k−〈µα0〉+1 tγvµ (compare with (2.13)). Let Sµ,γ be the maximal submodules generated by these singular vectors. Consider the quotient spaces V̂µ(γ) = Vµ(γ)/Sµ,γ , (2.17) and define their direct sum V̂µ = ⊕ γ∈P∨ V̂µ(γ). (2.18) We say that two subspaces V̂µ(γ1) and V̂µ(γ2) are equivalent if γ1 = γ2 + δ, where δ ∈ Q∨. This equivalence leads to the decomposition of V̂µ (as a L(Gad)-module) into a sum of l = ord(Z(Ḡ)) sectors, V̂µ = ⊕ ζ∈Z V̂µ(ζ). (2.19) Notice that (tγvµ) is not the HWV with respect to L+(g). However, it was proved in [23] that there exists a unique element ŵ = ŵ(γ) = tδw ∈ WQ such that tγŵvµ is the HWV. We demonstrate it below for L(SL(2,C)). The elements ŵ and γ represent the same element ζ ∈ Z. Then we define the Verma module Vµ(ζ) ∼ U(L(g)−)⊗C (tγŵvµ). (2.20) The vector ( Eα0 ⊗ t−1 )k−〈µ,wα0〉+1 (γŵvµ) is singular and corresponds to the submodule Sµ,γ . As in (2.14) we identify the integrable modules Vµ(ζ)/Sµ,γ with V̂µ(ζ) (2.16). Let V̂ ∗µ be the dual module. The Borel–Weil–Bott theorem for the loop group [61] states that V̂ ∗µ can be realized as the space of sections of a line bundle Lµ over the affine flag varie- ty (2.12). The line bundle is determined by the action L+(G)× C∗ on its sections as in (2.15), Lµ = { (g, ξ) ∼ ( gb, χµ ( b−1 ) ξ ) , g ∈ L(G), b ∈ L+(G) } . (2.21) 3 Conformal blocks and KZB equation in general case 3.1 Moduli space of holomorphic G-bundles Let P be a principle G-bundle over a curve Σg,n of genus g with n marked points ~z = (z1, . . . , zn), (n > 0), V is a G-module and EG = P ×G V is the associated bundle. We consider the set of isomorphism classes of holomorphic G-bundles MG,g,n over Σg,n with the quasi-parabolic structures at the marked points [64]. They are defined in the following way. A G-bundle can be trivialized over small disjoint disks D = n⋃ a=1 Da around the marked points and over Σg,n \~z. Therefore, P is defined by the transition holomorphic functions on D× = n⋃ a=1 (D×a ) and D×a = Da \ za. If G(X) are the holomorphic maps from X ⊂ Σg to G, then the isomorphism classes are defined as the double coset space BunG = G(Σg,n \ ~z) \G(D×)/G(D) ∼MG,g,n. (3.1) Hecke Transformations of Conformal Blocks in WZW Theory. I 7 Let ta be a local coordinate in the disks Da. Then G(D) = n∏ a=1 G(Da) = n∏ a=1 G⊗ C[[ta]] and G(D×) = n∏ a=1 La(G), La(G) = G⊗ C[[ta, t −1 a ]. (3.2) Let us fix G-flags at fibers over the marked points. The quasi-parabolic structure of the G-bundle means that G(D) preserves these G-flags. In other words, G(Da) = L+ a (G) (2.6). At the level of the Lie algebra Lie(G(D)) = ⊕n a=1 L + a (g) (2.2). We discuss the Lie algebra gout = Lie(G(Σg,n \ ~z)) below. Consider the one-point case ~z = z0 in (3.2). Let g(t) ∈ G[[t, t−1] = G(D×z0) be the transition function on the punctured disc D×z0 with the local coordinate t. This transition function defines a G-bundle. Its Lie algebra Lie(G(D×)) = g⊗ C[[t, t−1] assumes the form (see (3.1)) Lie(G(D×)) = gout ⊕ T BunG⊕L+(g). (3.3) Introduce a new transition matrix g̃(t) = tγg(t), where γ ∈ P∨ is an element of the coweight lattice. It defines a new bundle ẼG. The passage from EG to ẼG is called the modification of the bundle EG at the point z0. The modification amounts to the passage between different sectors of the integrable module attached at z0 (see (2.16), (2.17), (2.18)). Since tγ ∈ B, where B is the Borel subgroup (b = Lie(B) ⊂ L+(g)) (2.2), we say that modification is performed in the “direction”, consistent with the quasi-parabolic structure at z0. In general, it can have an arbitrary direction. It means that tγ may be replaced by Adf (tγ), where f ∈ G. As it was mentioned in Section 2.2 there is a unique modification that preserves the HWV of the integrable module V̂µ attached at z0. To be a Ḡ-bundle over Σg the transition matrix g should have a trivial monodromy g ( te2πi ) = g(t) around w. If g(t) has a trivial monodromy and γ belongs to the coroot sublattice Q∨, then g̃(t) also has a trivial monodromy. Otherwise, the monodromy is an element of the cen- ter Z ( Ḡ ) . For example, let γ = j$∨, where $∨ generate the group Zl, i.e. l$∨ ∈ Q∨, while j$∨ /∈ Q∨ for j 6= 0, mod(l). In this case g ( te2πi ) = ζjg(t), ζ = e($∨). (3.4) If j 6= 0 then g(t) is not a transition matrix for the Ḡ-bundle. But it can be considered as a transition matrix for the Gad-bundle, since Gad = G/Z. In this case the G-bundle is topologically non-trivial and ζ represents the characteristic class of EG. The characteristic class is an obstruction to lift Gad-bundle to G-bundle. It is represented by an element H2(Σg,Z) [46]. Let g̃(t) = gj(t) = tj$ ∨ . Then the multiplication by gj(t) provides a passage in (2.11) from the trivial sector to the non-trivial sectors gj(t) · L0 ( Gad ) = Lj ( Gad ) . In general, we have a decomposition of the moduli space (3.1) into sectors MGad,g,1 = ⋃ γ∈P∨ M(γ) Gad,g,1 , M(γ) Gad,g,1 = G ( Σg,n \ w ) \Gad γ ⊗ C[[t, t−1]/G⊗ C[[t]], Gad γ ⊗ C[[t, t−1] = Gad ⊗ tγC[[t, t−1]. (3.5) In particular, for Σ0,1 (CP 1 ∼ C ∪ ∞) and the marked point z1 = 0 this representation is related to the Grothendieck description of the vector bundles over CP 1. Let g− ∈ G ⊗ C[z−1], g+ ∈ G⊗ C[z]. Then g(z) ∈ L(G) = (C∗ → G) = {g(z)} has the Birkhoff decomposition [61] g(z) = g−z γg+, γ ∈ P∨. (3.6) 8 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov It means that any vector bundle EG over CP 1 is isomorphic to the direct sum of the line bundles ⊕li=1Lγi , where Lγi is defined by the transition function zγi , γ = (γ1, . . . , γl). If γ /∈ Q∨EG then has a non-trivial characteristic class. In fact, the bundle with γ 6= 0 are unstable. Two subsets M(γ1) G,g,1 and M(γ2) G,g,1 of the moduli space correspond to the vector bundles with the same characteristic class if γ1 = γ2 + β, β ∈ Q∨. Then the topological classification of the moduli spaces of the vector bundles by their characteristic classes follows from (3.5) MG,g,1 = ⋃ ζ∈Z M(ζ) G,g,1, M(ζ) G,g,1 = G ( Σg,n \ w ) \Gζ ⊗ C[[t, t−1]/G⊗ C[[t]], G(ζ) ⊗ C[[t, t−1] = G⊗ tj$∨C[[t, t−1], ζ = e ( j$∨ ) . (3.7) Similar representation exists for the space MG,g,n. 3.2 Moduli of complex structures of curves Let Mg be the moduli space of complex structures of compact curves Σg of genus g. The moduli space Mg,n of the complex structures of curves with marked points is foliated over Mg with fibers U ⊂ Cn corresponding to the moving marked points. An infinitesimal deformation of the complex structures is represented by the Beltrami (−1, 1) differential µ(z, z̄) = µ ∂ dz ⊗ dz̄ on Σg,n. In this way µ is (0, 1) form on Σ taking values in T (1,0)(Mg,n) and vanishing at the marked points. The basis in the tangent space T (Mg,n) is represented by the Dolbeault cohomology group H1(Σg,Γ(Σg \ ~z) ⊗ K̄), where K̄ is the anti- canonical class. Let us compare it with the Čech like construction of TMg,n as a double coset space. As above, consider small disks Da around marked points with local coordinates ta. Let C[[ta, t −1 a ]∂ta , C[[ta]]∂ta be vector fields on D×a while Da and Γ(Σg\~z) is a space of vector fields on Σg \ ~z. The vector fields from the latter space can have poles of finite orders at the marked points. Then TMg,n = Γ(Σg\~z) ∖ n⊕ a=1 C[[ta, t −1 a ]∂ta / n⊕ a=1 C[[ta]]∂ta . (3.8) This construction has the following relation to the Dolbeault description. We establish cor- respondence between ς ∈ ⊕na=1C[[ta, t −1 a ]∂ta on n⋃ a=1 D×a and the Beltrami differential µ. Let ςout ∈ Γ(Σg\~z), ςint ∈ ⊕na=1C[[ta]]∂ta . Consider two equations on n⋃ a=1 D×a ∂̄ςout = µ, ∂̄ςint = µ, where ∂̄\|D×a = ∂t̄a . On D×a ∂̄(ςout − ςint) = 0 and, therefore, ςout − ςint represents a Dolbeault cocycle. The first equation has solutions that can be continued on Σg \ ~z and the second – on n⋃ a=1 Da. If ς ∈ ⊕na=1C[[ta, t −1 a ]∂ta has continuations ςout and ςint then it corresponds to a trivial element of TMg,n. On the other hand, ∂̄ς = µ globally and, therefore, µ represents an exact Dolbeault cocycle. In this way the non-trivial vector fields ς ∈ ⊕na=1C[[ta, t −1 a ]∂ta correspond to elements of H1(Σg,Γ(Σg \ ~z)⊗ K̄). Hecke Transformations of Conformal Blocks in WZW Theory. I 9 3.3 Definition of conformal blocks and coinvariants Let us associate with Σg,n the following set: integer k and the weights ~µ = (µ1, . . . , µn, µa ∈ Ik) attached to the marked points ~z = (z1, . . . , zn). The L̂(g)-module (2.4) V̂ [n] ~z,~µ = n⊗ a=1 V̂µa , ~z = (z1, . . . , zn). (3.9) According to (2.19) V̂µa = ⊕ ζa∈Z V̂µa(ζa). (3.10) Coming back to (3.1) we define a Lie algebra gout = Lie(G(Σg \ D) as a Lie algebra of mero- morphic functions on Σg,n with poles at ~z = (z1, . . . , zn) taking values in g. Let (t1, . . . , tn) are local coordinates in D. There is a homomorphism O(Σg \ ~z)→ C[[ta, t −1 a ] for each za providing the homomorphism of the Lie algebras gout → g⊗ C[[ta, t −1 a ]. In this way gout acts on V̂ [n] ~z,~µ as (X ⊗ f) · (v1 ⊗ · · · ⊗ vn) = ∑ a v1 ⊗ · · · ⊗ (X ⊗ f(ta)) · va ⊗ · · · ⊗ vn. This is a Lie algebra action. Due to the residue theorem this homomorphism is lifted to the diagonal central extension gout ↪→ n⊕ a=1 L̂a(g), L̂a(g) = ( g⊗ C[[ta, t −1 a ] ) ⊕K, K → k. In what follows we need a relation of V̂µ with the space of coinvariants. In general setting the coinvariants are defined in the following way. Let W be a module of a Lie algebra k. The space of coinvariants [W]k is the quotient-space [W ]k = W/k ·W . In the case at hand we define the space of coinvariants with respect to the action of gout, H(~z, ~µ) = [ V̂ [n] ~z,~µ ] gout , ( [V ]g = V/g · V ) . The space of conformal blocks C ( V̂ [n] ~z,~µ ) is the dual space to the coinvariants. In other words, C ( V̂ [n] ~z,~µ ) is the space of linear functionals on V̂ [n] ~z,~µ , invariant under gout: F : V̂ [n] ~z,~µ → C, F (X · v) = 0 for any X ∈ gout. Put it differently, the conformal blocks are gout-invariant elements of the contragradient module V̂ ∗[n] ~z,~µ . For a single marked point case the conformal blocks are gout invariant sections of the line bundle Lµ over the affine flag variety (2.21). According to (3.9) and (3.10) the space V̂ [n] ~µ has the representation V̂ [n] ~z,~µ = n⊗ a=1 ⊕ ζa∈Z V̂µa(ζa). In a similar way the conformal blocks are decomposed in subspaces corresponding to the cha- racteristic classes of the bundles C ( V̂ [n] ~z,~µ ) = n⊗ a=1 ⊕ ζa∈Z Ca ( V̂µa(ζa) ) , where Ca = { F (ζa) : V̂µa(ζa)→ C } . (3.11) 10 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov 3.4 Variation of the moduli space of complex structures The space of conformal blocks C ( V̂ [n] ~z,~µ ) is a bundle over Mg,n. This bundle is equipped with the KZB connection that can be described as follows. A stress-tensor T (z, z̄) in general theories, defined on a surface Σg,n, generates vector fields on Σg,n. A dual object to T (z, z̄) is the Beltrami differential µ(z, z̄). It means that there is a connection on the bundle of fields over Mg,n (the Friedan–Shenker connection) ∇µF = δµF + ∫ Σ µTF. (3.12) In conformal field theories the stress-tensor is a meromorphic projective structure on Σg,n. The connection acting on the space of conformal blocks is projectively flat. The conformal blocks are horizontal sections of this bundle. The horizontality conditions are nothing else but the KZB equations for the conformal blocks. In general setting these equations are discussed in [33] (for the smooth curves) and in [22]. They have the form of non-stationary Schrödinger equations [36]. The connection (3.12) can be rewritten in a local form based on the representation (3.8). Let n⋃ a=1 D×a ⊂ Σg and γa ⊂ D×a is a small contour and ςa is a vector field in D×a . Then (3.12) can be written as ∇ςaF = ∂ςaF + ∮ γa ςaTF (3.13) and the KZB equation assumes the form ∇ςF = 0. (3.14) At the marked points T has the second order poles, while ςa ∈ C[[ta]]∂ta (3.8). Thereby, this integral produces ∂zaF . On the other hand, the product TF is non-singular outside the disks Da. Then for ςa ∈ Γ(Σg\~z) the integrals vanish. It means that the conformal blocks F are defined on Mg,n. Consider a one point case and let t be a local coordinate on a punctured disk D×. The stress- tensor in the local coordinate has the Fourier expansion T (t) = ∑ n∈Z Lnt −n−2. The coefficients obey the Virasoro commutation relations [Ln, Lm] = (n−m)Ln+m + c 12n(n2 − 1). In the WZW model the stress-tensor is obtained from the currents by means of the Sug- awara construction (see [7]). Let {tα} be a basis in g, {tβ} is the dual basis, and Iα(t) =∑ m tα,mt −m−1 ∈ g⊗ C[[t, t−1]. Then T (t) = 1 2(k + h∨) ∑ α :Iα(t)Iα(t):, where h∨ is the dual Coxeter number. The Fourier coefficients of T (t) take the form Lm = 1 2(k + h∨) ∑ p∈Z :tα,−pt α p+m:. (3.15) The normal ordering means placing to the right tαn (tα,n) with n > 0. The Virasoro central charge is c = dim g k + h∨ . Hecke Transformations of Conformal Blocks in WZW Theory. I 11 The Virasoro algebra acts on g⊗ C[[t, t−1] as Ln 7→ tn+1 d dt . (3.16) This action is well defined because the action of the Sugawara tensor is well defined on the inte- grable modules. In particular, it follows from (3.16) that for the moving points equation (3.14) assumes the form( ∂za − La−1 ) F = 0. (3.17) The restriction of ∇ς on Ca (3.11) yields a family of the KZB equations∑ a ∇aF (ζa) = 0. (3.18) In next section we construct these equations explicitly for the bundles over elliptic curves. 3.5 Variation of the moduli space of holomorphic bundles 3.5.1 General construction The moduli space of holomorphic bundlesMG,g,n = BunG (3.1) is foliated over the moduli space of complex structures Mg,n. Let us consider the dependence of the space of coinvariants H(~z, ~µ) (conformal blocks C(V [n])) on the variations of the moduli of the bundles BunG. For simplicity consider the one-point case. Let ta be a local coordinate in D×a , and Gout = G(Σg \ za). Define the quotient MG = Gout \G(D×a ), G(D×a ) = G[[ta, t −1 a ]. (3.19) This space is the moduli space of G-bundles with a trivialization around za (see (3.1)). Let V̂µa be an integrable module (2.14) attached to za. Recall that V̂ ∗µa is the space of holomorphic sections Γ(Lµa) of the line bundle (2.21) over the affine flag variety (2.12). In these terms the space of conformal blocks has the following interpretation [6, 20]. Since V̂µa is the integrable representation, the group G(D×) acts on V̂µa . Thereby, the subgroup Gout acts on V̂µa also. Due to (3.19) G(D×a ) acts on MG from the right. Therefore, G(D×a ) acts on the sections V̂µa ⊗O(MG) of the trivial vector bundle V̂µa ×MG g · v(x) = (gv)(xg), g ∈ G[[ta, t −1 a ], v ∈ V̂µa . (3.20) Consider the space of the coinvariants V̂µa ⊗O(MG) /( V̂µa ⊗O(MG) ) · Stabx, where Stabx is Lie(Gx(D×), Gx(D×) = {g\|x · g = x, x ∈MG}. In particular, Stabx = gout for x corresponding to Gout. The spaces of coinvariants are isomorphic for different choices of x. The dual space Γ(Lµ)/Gout is the space of conformal blocks. The quotient Γ(Lµ)/Gout is a space of sections of the line bundle over BunG (3.1). It means that the space of conformal blocks is a non-Abelian generalization of the theta line bundles over the Jacobians. 3.5.2 SL(2,C)-bundles over CP 1 It is instructive to consider this construction for Σ = CP 1 = C ∪∞. This case was analyzed in details in [5] for the trivial G-bundles and γ = 0 in (3.6). Here we consider G = SL(2,C)-bundles with γ ∈ P∨. Let tα = h, e, f be the Cartan–Chevalley basis in the Lie algebra sl(2,C) [h, e] = 2e, [h, f ] = 2f, [e, f ] = h, 12 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov and tα(n) = tαt n. The Verma module Vµ is generated by L−(sl(2,C)) = c · f + g ⊗ t−1C[t−1] (2.20), (2.3). The HV vµ with the weight µ ∈ P (sl(2,C)) is defined by the conditions hvµ = 2svµ, s = 1 2〈µ, h〉 ∈ 1 2Z, evµ = 0, tα(n)vµ = 0 for n > 0. The singular vector (et−1)k+1−2s generates the submodule Sµ ⊂ Vµ, and the integrable module is the quotient V̂µ = Vµ/Sµ. This form of V̂µ defines a trivial sector in (2.19). Note that Z(SL(2,C)) = Z2 = {ζ = (0, 1)}. Therefore, there are two sectors in the integrable module (2.19). Consider the non-trivial sector corresponding to ζ = 1. Let WP = {ŵ} be the Weyl group (2.5), ŵ = Z2 n tγ , where γ belongs to the weight lattice γ ∈ P∨(sl(2,C)) = P (sl(2,C)) = 1 2Z. Since Q∨(sl(2,C)) = Q(sl(2,C)), Z = P/Q ∼ Z2, and ζ = 1 corresponds to γ /∈ Q. It means that 〈γ, h〉 is odd. Then according to (2.19) Vµ(ζ = 1) = U ( L−(sl(2,C)) ) (tγvµ), γ /∈ Q, V̂µ(ζ = 1) = Vµ(ζ = 1)/Sµ. As it was mentioned above, tγvµ is not the HWV. In other words, it is not annihilated by the positive nilpotent loop subalgebra (2.8) Lie(N+(SL(2,C))) = {n(t) = b · e+ g⊗ tC[[t]], b ∈ C}. In fact, we have n(t)tγvµ = tγ Ad−1 tγ (n(t))vµ. Let 〈γ, h〉 = 2s > 0, and s ∈ 1 2 + Z. Then Ad−1 tγ (n(t)) = ∑ m≥0 ( am+1 · htm+1 + bm · et2s+m + cm+1 · ft−2s+m+1 ) . The terms cm+1 ·ft−2s+m+1) for m ≤ 2s−1 do not belong to Lie(N+(SL(2,C))). Multiply tγ by wtγ1 ∈ WQ, where 〈γ1, h〉 = −2s + 1 and w : e ↔ f . This transformation preserves the sector. Now n(t) annihilates the vector wtγ1tγvµ. Note (wtγ1) is uniquely defined by γ. Thus, for any vector tγvµ we define a unique HWV from the same sector. It is a particular case of general theorem proved in [23]. Consider the trivial G-bundles over CP 1. It was proved in [5] that the conformal blocks are G-invariant functionals on the module V̂ [n] µ satisfying some additional conditions. In particular, for n = 1 dim(C(G)) = { 0, µ 6= 0, 1, µ = 0, (3.21) and for V̂ [2] (0,∞)(µ0,µ∞) dim(C(G)) = { 0, µ0 6= µ∗∞, 1, µ0 = µ∗∞. Let us analyze the case of SL(2,C)-bundles. It follows from the Bruhat decomposition (2.9) that there are two types of the SL(2,C)-bundles over CP 1 – the trivial, when γ ∈ Q in (3.5) is an element of the root lattice γ ∈ Q, and non-trivial, when γ /∈ Q. Note that the stable bundles correspond γ = 0. In the first case we deal with the adjoint bundles that can be lifted to the SL(2,C)-bundles. In the second case there is an obstruction to lift these bundles to the SL(2,C)-bundles. Let z−1 be a local coordinate in a neighborhood of ∞. The Lie algebra gout assumes the form gout = sl(2,C) + z−1sl(2,C) ⊗ C[z−1]. Let n = 1 and z = 0 is the marked point with the attached integrable L(sl(2,C))-module V̂µ. We have gout(vµ) = V̂µ for µ 6= 0. But v0 /∈ gout(v0) and v0 is the coinvariant confirming (3.21). Hecke Transformations of Conformal Blocks in WZW Theory. I 13 Consider the integrable module generated by zγvµ, where, as above, 〈γ, h〉 = 2s > 0, and s ∈ 1 2 + Z. Then goutz γvµ = Ad−1 zγ (n(t)) = zγ ∑ m≥0 ( a−mz −m · h+ b−mz −2s−m · e+ c−mz 2s−m) · fvµ. Then the elements b−mz −2s−m · e(vµ) for 0 ≤ m < 2s are not generated by gout. Thus, if γ 6= 0, the space of coinvariants (and the space of conformal blocks) is non-empty for an arbitrary weights µ. Its dimension depends on γ: dim(C(SL(2,C))) = 2s (compare with (3.21)). 3.5.3 The form of connection For conformal blocks we have (see (3.20)) F (x) = (gF )(xg), g ∈ Stabx . Define the current J(ta) = (g−1dg)(ta) ∈ g⊗C[[ta, t −1 a ])⊗Ω1(D×a ) for g(ta) ∈ G⊗C[[ta, t −1 a ]. A local version of (3.20) is defined by the operator ∇uα = ∂uα + ∮ γa 〈 ( g−1dg ) , tα〉, where tα is a generator of g and uα is a coordinate of the tangent vector to BunG. The action of ∇uα on the conformal blocks is well defined because the conformal blocks are gout-invariant. Therefore, they are horizontal with respect to this connection ∂uαF + ∮ γa 〈 ( g−1dg ) (ta), tα〉F = 0. (3.22) If one takes u from M(ζ) G,g,n (3.7) then (3.22) takes the form ∂uα(ζa)F (ζa) + ∮ γa 〈 ( g−1dg ) (ta), tα〉F (ζa) = 0, where F (ζa) ∈ Ca (3.11). 4 KZB equations related to elliptic curves and non-trivial bundles 4.1 Moduli space of elliptic curves We consider in details the genus one case Σ1,n. Let Στ = C/〈τ, 1〉 be the elliptic curve with the modular parameter in the upper half-plane H = {Immτ > 0}. For n ∈ Z, n ≥ 1 define the set of marked points ~z = (z1, . . . , zn). Due to the C action on Στ (z → z + c), we assume that∑ a za = 0. A big cell M0 1,n in the Teichmüller space M1,n is defined as M0 1,n = { (z1, . . . , zn), ∑ a za = 0, zk 6= zj , mod(〈τ, 1〉) } ×H. 14 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov 4.2 Moduli space of holomorphic G-bundles over elliptic curves For G = GLN the moduli space of holomorphic bundles was described by M. Atiyah [3]. For the trivial G-bundles, where G is a complex simple group, it was done in [10, 11, 52]. Non-trivial G-bundles and their moduli spaces were considered in [26, 27, 28, 63]. We describe the moduli space of stable non-trivial holomorphic bundles over Στ using an approach of [46]. Let G be a complex simple Lie group. An universal cover Ḡ of G in all cases apart G2, F4 and E8 has a non-trivial center Z(Ḡ). The adjoint group is the quotient Gad = Ḡ/Z(Ḡ). For the cases An−1 (when n = pl is non-prime) and Dn the center Z(Ḡ) has non-trivial subgroups Zl ∼ µl = Z/lZ. Assume that (p, l) are co-prime. There exists the quotient-groups Gl = Ḡ/Zl, Gp = Gl/Zp, Gad = Gl/Z(Gl), (4.1) where Z(Gl) is the center of Gl and Z(Gl) ∼ µp = Z(Ḡ)/Zl. Following [56] we define a G-bundle EG = P ×G V by the transition operators Q and Λj acting on the sections of s ∈ Γ(EG) as s(z + 1) = Q(z)s(z), s(z + τ) = Λ(z)s(z), (4.2) where Q(z) and Λ(z) take values in End(V ). Going around the basic cycles of Στ we come to the equation Q(z + τ)Λ(z)Q(z)−1Λ−1(z + 1) = Id . (4.3) It follows from [56] that it is possible to choose the constant transition operators. Then we come to the equation QΛQ−1Λ−1 = Id . (4.4) Replace (4.4) by the equation QΛQ−1Λ−1 = ζ Id, where ζ is a generator of the center Z ( Ḡ ) . In this case (Q,Λ) are the clutching operators for Gad- bundles, but not for Ḡ-bundles, and ζ plays the role of obstruction to lift the Gad-bundle to the Ḡ-bundle. Here ζ = e($∨) is a generator of the center Z ( Ḡ ) , where $∨ ∈ P∨ is a fundamental coweight such that N$∨ ∈ Q∨ and N = ord(Z ( Ḡ ) ).2 Let 0 < j ≤ N . Consider a bundle with the space of sections with the quasi-periodicities s(z + 1) = Qs(z), s(z + τ) = Λjs(z) (4.5) such that QΛjQ−1Λ−1 j = ζj Id . (4.6) If j and N are co-prime numbers then ζj generates Z ( Ḡ ) . In this case Q and Λj can serve as transition operators only for a Gad = Ḡ/Z-bundle, but not for Ḡ-bundle and ζj is an obstruction to lift Gad-bundle to Ḡ-bundle. The element ζ has a cohomological interpretation. It is called the characteristic class of EG. It can be identified with elements of the group H2(Σg,n,Z ( Ḡ ) ). This group classifies the of the characteristic classes of the bundles [46]. 2For the simplicity we assume here and in what follows that Z ∼ Zl. The case Z(Spin(4n)) = Z2 ⊕ Z2 can be considered in a similar way. Hecke Transformations of Conformal Blocks in WZW Theory. I 15 Consider, as above, a non-prime N = pl and put j = p. Then ζj is a generator of the group Zl. In this case Q and Λj are transition operators for Gl = Ḡ/Zl-bundles (see (4.1)) and ζj is an obstruction to lift a Gl-bundle to a Ḡ-bundle. The moduli space of stable holomorphic over Στ with the sections (4.5) is defined as M(j) G,1 = (solutions of (4.6))/(conjugation). (4.7) For the stable bundles this description of the moduli space is equivalent to (3.1). In fact, the monodromy of s(z) around z = 0 is the same as in (3.4). Similar to (3.7) we have MG,1 = N⋃ j=1 M(j) G,1. (4.8) Assume that Q is a semi-simple element and Q ∈ HḠ is a fixed Cartan subgroup of Ḡ. It means that we consider an open subset M(j) G,1 ⊃ ( M(j) G,1 )0 ≡M(j) 0 (G) = { (Q ∈ HḠ,Λj)/(conjugation) } . In this case the solutions of (4.4) have the form [46] Q = exp ( 2πi ρ∨ h ) , Λj = Λ0Vj , (4.9) where ρ∨ is a half-sum of positive coroots, h is the Coxeter number, Λ0 is an element of the Weyl group defined by ζj : ζj → Λ0, (Λ0)l = Id . The element Λ0 preserves the extended system of simple roots Πext = Π ∪ (α0), where −α0 is a maximal root [46, Proposition 3.1]. In this way Λ0 is a symmetry of the extended Dynkin diagram of g = Lie ( Ḡ ) , generated by $∨ [12]. Let H̃0 ⊂ HḠ be the Cartan subgroup commuting with Λ0. To describe Vj consider the adjoint action λ = Ad(Λ0) on the Cartan subalgebra h = Lie(HḠ). Let h̃0 = Lie(H̃0) be the invariant subalgebra (λ(h̃0) = h̃0). Then Vj = exp(2πıu)(u ∈ h̃0) is an arbitrary element from H̃0 defining the moduli space M(j) 0 (G). There exists a basis Π̃∨j in h̃0 such that Π̃ is a system of simple roots for a simple Lie subalgebra g̃0 ⊂ g. For the list of these subalgebras see [46]. If j = N , we come to the trivial bundles (4.4). In this case Λ0 = Id, h̃0 = h and g̃0 = g. Let Q̃∨ and P̃∨ be the coroot and the coweight lattices in h̃0, and W̃ is the Weyl group cor- responding to Π̃. Define the Bernstein–Schwarzman type groups [10, 11]. They are constructed by means of the lattices Q̃∨ or P̃∨. In the first case it is the semidirect products W̃BS = W̃ n ( τQ̃∨ ⊕ Q̃∨ ) . (4.10) Then the moduli space of non-trivial Ḡ-bundles with the characteristic class ζj is the fundamental domain in h̃ (j) 0 under the action of W̃BS M(j) 0 (Ḡ) = Csc j = h̃ (j) 0 /W̃BS (4.11) is the moduli space of non-trivial Ḡ-bundles. Consider Gad-bundles. Define the semidirect product W ad BS = W̃ n ( τP̃∨ ⊕ P̃∨ ) . (4.12) 16 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov A fundamental domain of this group in h̃ (j) 0 is Cad = h̃ (j) 0 /W̃ ad BS and M(j) 0 ( Gad ) = Cad j = h̃ (j) 0 /W̃ ad BS (4.13) is the moduli space of the non-trivial Gad-bundles. It is the moduli space of EGad-bundles with characteristic class defined by ζj . In other words u ∈ { Csc j for EḠ-bundles, Cad j for EGad-bundles. 4.3 The gauge Lie algebra for elliptic curves Here we define the moduli space of holomorphic G-bundles coming back to the double coset construction (3.1). Recall, that the Lie algebra gout = Lie(G(Στ,n\~z)) is a Lie algebra of mero- morphic functions on Στ,n with poles at ~z = (z1, . . . , zn) and the quasi-periodicities (4.2), (4.3). Let us take for simplicity the case (4.4) and apply the decomposition (A.1) corresponding the characteristic class defined by ζ to the Lie algebra gout: gout = ⊕l−1 k=0gk, g0 = g′0 ⊕ g̃0, AdΛ0(gk(z)) = e ( k l ) gk(z). Consider the quasi-periodicity conditions (4.2). The GS-basis is diagonal under AdΛ and AdQ actions (A.6)–(A.8). We should find functions on Στ \D that have the same phase-factors and pole singularities at ~z. To define gk and g′0 we use the functions φ (B.1), and ϕk,mα (B.4). They have the needed quasi-periodicities (B.16), (B.17) and poles at z = 0 (B.7), (B.9). Then we find gk = { n∑ a=1 K(a,k)∑ m=0 (∑ α∈Π xkα,m,a∂ m z φ ( k l , z − za ) hkα + ∑ α∈R ykα,m,aϕ k,m α (u, z − za)tkα )} , (4.14) g′0 = { n∑ a=1 ∑ α∈R K(a,α)∑ m=0 y′α,m,aϕ 0,m α (u, z − za)t0α } . (4.15) Similarly, from (B.10), (B.11), (B.12), (B.14), (B.15) we have g̃0 = { n∑ a=1 (∑ α∈Π̃ ( x0 α,0 + K(a,α)∑ m=1 x0 α,m,aEm(z − za) ) hα + ∑ α∈R̃ K(a,α)∑ m=0 y0 α,m,aϕ 0,m α (u, z − za)Eα )} . (4.16) Then gout has the correct quasi-periodicities and has poles of orders K(a,m), K(a, α) at za, a = 1, . . . , n. In this last expression (due to the residue theorem) from (B.14) we assume that n∑ a=1 x0 α,1,a = 0. (4.17) Let us unify the last two expression (4.15) and (4.16) in a single formula, g0 = { n∑ a=1 (∑ α∈Π̃ ( x0 α,0 + K(a,α)∑ m=1 x0 α,m,aEm(z − za) ) hα Hecke Transformations of Conformal Blocks in WZW Theory. I 17 + ∑ α∈R K(a,α)∑ m=0 y0 α,m,aϕ 0,m α (u, z − za)t0α )} . (4.18) We will act on the coinvariants by gout. In what follows we need the limit z → za of these expressions. Notice that gout is the filtered Lie algebra. The filtration is defined by the orders of poles. The behavior of gout is defined by the asymptotics (B.7)–(B.9), (B.12). As it will become clear below we need the least singular terms in gout. In this way we take m = 0 in (4.14), (4.18) and m = 1 (E1(z − za)) in (4.18): gk ∼ n∑ a=1 ∑ α∈Π xkα,0,a ( · · ·+ 1 z − za + E1 ( k l ) + ∑ b6=a φ ( k l , za − zb ) + ) Hkα + ykα,0,a ( · · ·+ 1 z − za + E1 ( 〈u + κτ, α〉+ k l ) + 2πı〈κ, α〉 + ∑ b 6=a ϕk,0α (u, zb − za) + · · · ) tkα, (4.19) g0 ∼ ∑ α∈Π̃ ( · · ·+ x0 α,1,a ( 1 z − za + ∑ b6=a E1(zb − za) ) + x0 α,0 + · · · ) hα + ∑ α∈R̃ y0 α,0,a ( · · ·+ 1 z − za + E1 ( 〈u + κτ, α〉 ) + 2πı〈κ, α〉 + ∑ b6=a ϕ0 α(u, zb − za) + · · · ) t0α. (4.20) Here “· · · ” means the terms of order o ( z− za )−1 and o(1). For gint = Lie(G(UD)) we have local expansions in neighborhoods of the marked points gint = { X = n∑ a=1 ( ba + ∑ j>0 yaj (z − za)j ) , ba ∈ b0, y a j ∈ g } . Define the Lie algebra with the loose condition (4.17)) g′out = gout with n∑ a=1 x0 α,1,a ∈ C and let n− = ∑ α∈R+ g−α. Then the Lie algebra Lie(G(D×)) has the form (compare with the general case (3.3)) Lie ( G(D×) ) = g′out ⊕ ( ⊕na=1 n − a ) ⊕ gint = gout ⊕ ( n∑ a=1 ∑ α∈Π̃ x0 α,1,ahα ) ⊕ ( ⊕na=1 n − a ) ⊕ gint. (4.21) Notice that the constant terms n−a come from the constant terms c(m, k) in (B.9). We can conclude from (4.21) that locally the action on G(D×) by Gout = G(Σ1,n \ ~z) from the left and by Gint = n∏ a=1 G(Da) from the right absorbs almost all negative and positive modes of G(D×) except the two types of modes describing the moduli space: 18 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov • The vector u = n∑ a=1 ∑ α∈Π̃ x0 α,1,ahα ∈ h̃0. It defines an element of the moduli spaceMG,1 (4.8). • The Lie algebras n−a , a = 1, . . . n. They are the tangent spaces to the flag varieties attached the marked points coming from the quasi-parabolic structure of the bundle. 4.4 Conformal blocks In this section we define connections on the space of conformal blocks and derive the KZB equations in a similar way as it was done for the trivial characteristic classes in [24]. The derivation is based on the representation of the moduli space of bundles as the double coset space (3.1) in a given sector of the decomposition (3.5). In other words, the characteristic class (defined by j = 0, . . . , l−1 in (4.7)) is fixed and we deal with Gout \Gad γ ⊗C[[ta, t −1 a ]/Gint, where Gout = G(Στ,n \ ~z) and Gint = G⊗ C[[ta]] were described above. Let us write down the Virasoro generators (3.15) using the GS-basis (A.3), (A.4), (A.5) tkα ⊗ tma ≡ tkα(m) (tkα(0) ≡ tkα) for the generators of the loop algebra Lam = 1 2(k + h∨) ∑ p∈Z l−1∑ q=0 (∑ α∈R :\|α\|2tqα(−p)t−q−α(p+m): + ∑ α∈Π̃ :Hqα(−p)h−qα (p+m): ) . (4.22) Consider the integrable modules attached to the marked points V̂ [n] ~z~µ (3.9) and the corresponding conformal blocks. They satisfy the equations (3.14), (3.18), (3.22). For elliptic curve they assume the form: • The moving points (3.17): ( ∂a − La−1 ) F = 0, ∂a = 1 k + h∨ ∂za , ta = z − za. (4.23) • The vector field corresponding to the deformation of the moduli τ of the elliptic curve Στ,n:( ∂τ − 1 2πi E1(z)∂z ) F = 0. (4.24) This action follows from (3.13) and the operator algebra T (z′)F (z) = E1(z′ − z)∂zF (z) + analitic part. • The invariance with respect to the action of gout (3.22):( l∂u−α + E1(z)H0 α ) F = 0, α ∈ Π̃, u = {uα}, α ∈ Π̃, (4.25) where H0 α are the Cartan generators (A.5). Notice that this operator is well defined on MG (3.19). The vector field (4.24) is defined on the universal curveH×C/〈τ, 1〉\H×0, since it is invariant under the lattice shifts 〈τ, 1〉. The τ deformation can be defined in the non-holomorphic form as ∂τ + z−z̄ τ−τ̄ ∂z. The invariance with respect to Lie(Gout) (4.14), (4.18) means that ϕkα(uα, z − za)tkαF = 0, α ∈ R, ∀ k, ϕkα(0, z − za)HkαF = 0, α ∈ Π̃, k 6= 0. (4.26) Hecke Transformations of Conformal Blocks in WZW Theory. I 19 Now using (4.19), (4.20) and (4.26) we write down the annihilation condition goutF = 0 in in the basis tk,cα (m) = 1⊗ · · · ⊗ 1⊗ tk,cα (m)⊗ 1⊗ · · · ⊗ 1 (on the c-th place):( tk,aα (−1) + ( E1 ( uα + 〈κ, α〉τ + k l ) + 2πi〈κ, α〉 ) tk,aα (0) + ∑ c 6=a ϕkα(uα, zc − za)tk,cα (0) ) F = 0, ( Hk,aα (−1) + ( E1 ( 〈κ, α〉τ + k l ) + 2πi〈κ, α〉 ) Hk,aα (0) + ∑ c 6=a ϕkα(0, zc − za)Hk,cα (0) ) F = 0 (4.27) for α ∈ R, ∀ k and α ∈ Π̃, k 6= 0 correspondingly. In the same way (4.24) and (4.25) assume the form ( ∂τ + 1 2πi La−2 + 1 2πi ∑ c 6=a E1(zc − za)La−1 ) F = 0, ( l∂u−α + H0,a α (−1) + ∑ c 6=a E1(zc − za)H0,c α (0) ) F = 0, α ∈ Π̃. (4.28) Now we are ready to evaluate the Virasoro generators, i.e. to express them in terms of zero modes of the loop algebra tk,cα (0) ≡ tk,cα only. As we have found above the positive modes of the loop algebra act on F by zero tk,aα (m)F = 0, m ∈ Z+. Therefore, from (4.22) we have (k + h∨)La−1 = l−1∑ q=0 (∑ α∈R tq,aα (−1)t−q,a−α (0) + ∑ α∈Π̃ Hq,aα (−1)h−q,aα (0) ) on F (4.29) and (k + h∨)La−2 = l−1∑ q=0 (∑ α∈R tq,aα (−2)t−q,a−α (0) + ∑ α∈Π̃ Hq,aα (−2)h−q,aα (0) ) (4.30) + 1 2 l−1∑ q=0 (∑ α∈R tq,aα (−1)t−q,a−α (−1) + ∑ α∈Π̃ Hq,aα (−1)h−q,aα (−1) ) on F. In order to find La−1 one need to substitute tk,aα (−1), Hk,aα (−1) from (4.27) and H0,a α (−1) from (4.28) into (4.29) −(k + h∨)La−1 = l ∑ α∈Π̃ h0,a α (0)∂uα + l−1∑ q=0 ∑ α∈R \|α\|2 (( E1 ( uα + 〈κ, α〉τ + q l ) + 2πi〈κ, α〉 ) tq,aα (0)t−q,a−α (0) + ∑ c 6=a ϕqα(uα, zc − za)tq,cα (0)t−q,a−α (0) ) 20 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov + l−1∑ q=0 ∑ α∈Π̃ (( E1 ( 〈κ, α〉τ + q l ) + 2πi〈κ, α〉 ) Hq,aα (0)h−q,a−α (0) + ∑ c 6=a ϕqα(0, zc − za)Hq,cα (0)h−q,a−α (0) ) , where ϕ0 α(0, zc−za) = E1(zc−za). The first term in the last line vanishes due to skew-symmetry with respect to α, q → −α,−q. The similar term in the second line does not vanish because [tq,aα (0), t−q,a−α (0)] = pα√ l exp ( −2πi ql ) h0,a α [46]. Therefore, l−1∑ q=0 ∑ α∈R \|α\|2 ( E1 ( uα + 〈κ, α〉τ + q l ) + 2πi〈κ, α〉 ) tq,aα (0)t−q,a−α (0) = 1 2 l−1∑ q=0 ∑ α∈R \|α\|2 ( E1 ( uα + 〈κ, α〉τ + q l ) + 2πi〈κ, α〉 ) pα√ l exp ( −2πi q l ) h0,a α = l ∑ α∈Π̃ h0,a α (0)∂uα log l−1∏ q=0 ∏ α∈R ϑ ( uα + 〈κ, α〉τ + q l ) pα\|α2\| 2l √ l exp(−2πi q l )  . The term 1 2 l−1∑ q=0 ∑ α∈R \|α\|22πi〈κ, α〉 pα√ l exp ( −2πi q l ) h0,a α vanishes because of summation over q. Notice also that the obtained scalar expression does not depend on {zc}. Then, the equation (4.23) gives( ∂a + l ∑ α∈Π̃ h0,a α (0)∂uα + l−1∑ q=0 ∑ c 6=a (∑ α∈R \|α\|2ϕqα(uα, zc − za)tq,cα (0)t−q,a−α (0) + ∑ α∈Π̃ ϕqα(0, zc − za)Hq,cα (0)h−q,a−α (0) )) F̃ = 0, where F̃ = F l−1∏ q=0 ∏ α∈R ϑ ( uα + 〈κ, α〉τ + q l )− pα\|α2\| 2l √ l exp(−2πi q l ) . This is the first set of equations in (4.41). In order to obtain the second one (the KZB connec- tion ∇τ along τ) one should use (4.30). It is needed to compute La−2. The later arises from the local expansion of (B.4) for k = 1. Then the following identities should be used ∂zφ(u, z) = φ(u, z) ( E1(z + u)− E1(z) ) = f(u, z) + ( E1(u)− E1(z) ) φ(u, z), where f(u, z) = ∂uφ(u, z) for t(−2)t(0)-terms and φ(u, z − za)φ(−u, z − zb) = −φ(u, z − za)φ(u, zb − z) = −φ(u, zb − za) ( E1(u) + E1(z − za) + E1(zb − z)− E1(u+ zb − za) ) = f(u, zb − za) + φ(u, zb − za) ( E1(z − zb)− E1(z − za) ) for t(−1)t(−1)-terms. On the other hand ∇τ is a unique flat connection for given ∇a (4.37). The final answer is given below in Section 4.6. This answer is verified in Appendix C. Hecke Transformations of Conformal Blocks in WZW Theory. I 21 4.5 Classical r-matrix The construction of the KZB connection is based on the classical dynamical elliptic r-matrix defined as sections of bundles over elliptic curves [13, 47, 74]. For trivial G-bundles our list coincides with the elliptic r-matrices were defined in [19]. A more general class of elliptic r-mat- rices was constructed in [17, 18]. The latter classification includes our list though it was derived from different postulates. 4.5.1 Axiomatic description of r-matrices The classical dynamical r-matrix is a meromorphic one form r = r(u, z)dz, (u ∈ h̃0) on C taking values in g⊗ g that satisfies the following conditions: 1. r(z) has a pole at z = 0 and Res ∣∣ z=0 r(z) = C2 = 1 2 l−1∑ k=0 ∑ α∈R \|α\|2tkα ⊗ t−k−α + l−1∑ k=0 ∑ α∈Π Hkα ⊗ h−kα , where tkα, Hkα, h−kα are generators of the GS basis in g (see Appendix A). If V is a g-module, then C2 acts by the permutation on V ⊗ V . 2. Behavior under the shifts by the generators of the lattice Z⊕ τZ: r(z + 1) = AdQ r(z), r(z + τ) = 2πı ∑ α∈Π H0 α ⊗ h0 α + AdΛj r(z), (4.31) where the Ad-action is taken with respect to the first factor in g ⊗ g. Here Q = e(κ), Λ(j) = Λ0e(u) (see (4.9)), {h0 α} ({H0 α}) is the simple coroot basis (the dual basis) in the invariant subalgebra g̃0 (see Appendix A). It means that r is a connection in the g ⊗ g- bundle over Στ . 3. The classical dynamical Yang–Baxter equation (CDYBE). It follows from 1 that r(z) can be represented as r(z) = 1 2 l−1∑ k=0 ∑ α∈R Φk α(z)\|α\|2tkα ⊗ t−k−α + l−1∑ k=0 ∑ α∈Π Ψk α(z)Hkα ⊗ h−kα . Then r(z) is a solution of CDYBE: [r12(z12), r13(z13)] + [r12(z12), r23(z23)] + [r13(z13), r23(z23)] − √ l l−1∑ k=0 ∑ α∈R \|α\|2 2 tkα ⊗ t−k−α ⊗ h̄0 α∂1Φk α(u, z − w) − \|α\| 2 2 tkα ⊗ h̄0 α ⊗ t−k−α∂1Φk α(u, z − x) + \|α\|2 2 h̄0 α ⊗ tkα ⊗ t−k−α∂1Φk α(u, w − x) = 0, zij = zi − zj , (4.32) where ∂1 is the differentiation with respect to the first argument. 4. The unitarity r12(u, z) + r21(u,−z) = 0. 22 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov 5. The zero weight condition [X ⊗ 1 + 1⊗X, r(u, z)] = 0, X ∈ h. Lemma 1. • Any r′-matrix satisfying 1–5 has the form r′(u, z) = r(u, z) + δr(u), where r(u, z) = rH(u, z) + rR(z), (4.33) rR(u, z) = 1 2 l−1∑ k=0 ∑ α∈R rkα(u, z), rkα(u, z) = \|α\|2ϕkα(u, z)tkα ⊗ t−k−α, rH(z) = l−1∑ k=0 ∑ α∈Π r0 α(z), r0 α(z) = ϕk0(z)Hkα ⊗ h−kα , satisf ies 1–5 and ϕkβ(x, z) is def ined in (B.3), ϕk0(z) = φ(k/l, z), ϕ0 0(z) = E1(z). • δr(u) ∈ h̃0 ⊗ h̃0 δr(u) = ∑ α,β∈Π̃ AαβH 0 α ⊗ h0 β, Aαβ = −Aβα. • δr(u) is generated by the gauge transformation δr(u) = −l ∑ α∈Π̃ ( ∂uα̂f ) f−1 ⊗ h0 α, ( ∂uα̂f ) f−1 ∈ h̃0, f = f(u) ∈ H̃0, where H̃0 is a Cartan subgroup of the invariant subgroup G̃0 ⊂ G (see Table 1 in [46]). Proof. It follows from the properties of the functions ϕkα(u, z), ϕk0(u, z) described in the Ap- pendix B that r(u, z) satisfy 1 and 2. It was proved in [46] that it is a solution of the CDYBE. This sum is a classical dynamical r-matrix corresponding to a non-trivial characteristic class defined by (4.31). The conditions 4 and 5 can be checked as well. The conditions 1–3, 5 define the r-matrix up to a constant (z-independent) Cartan term δr. Then it follows from 4 that Aαβ is antisymmetric. Next we wish to prove that locally Aαβ = −l(∂uα(f)f−1)β for some f ∈ H̃0. The twisted r-matrix must satisfy the CDYB equation. Plugging r+ δr into (4.32) we see that the “commu- tator” part vanishes identically since [rab, δrac] + [rab, δrbc] ≡ 0 due to[ tk,aα , h0,a β ] ⊗ t−k,a−α + tk,aα ⊗ [ t−k,a−α , h0,a β ] = 0. The “derivative” part of (4.32) yields ∂uα̂Aβγ + ∂uγ̂Aαβ + ∂uβ̂Aγα = 0 or dA = 0, A = ∑ α,β∈Π Aαβduα ∧ duβ ∈MG. The term δr is called the dynamical twist of the r-matrix. The statement follows from the Poincaré lemma. � Hecke Transformations of Conformal Blocks in WZW Theory. I 23 4.5.2 r-matrices as sections of bundles over moduli spaces Consider the behavior of the r-matrix (4.33) under the action of latices τQ̃∨ ⊕ Q̃∨ (4.10) and τP̃∨⊕ P̃∨ (4.12) on the dynamical parameter u. It follows from (B.2), (B.3) and (B.16) that the r-matrices has distinct type of quasi-periodicities with respect u ∈ h̃0. Let β∨ ∈ Π̃∨ be a simple coroot, corresponding to the invariant algebra g̃0. For α ∈ R define the integers nα,β = 〈α, β∨〉. Then we find rkα ( u + β∨, z ) = rkα(u, z), rkα ( u + τβ∨, z ) = e ( −nα,βz ) rkα(u, z). Let Ξ∨ be a basis of fundamental co-weights dual to the basis Π, and $̃∨ is a fundamental coweight in P̃∨. Since P̃∨ is a sublattice of P∨, the weight $̃∨ can be decomposed in the basis of the fundamental co-weights $̃∨ = ∑ ν∨∈Ξ∨ n $ ν ν ∨, where n$ν ∈ Z. As above we find rkα(u + $̃∨, z) = rkα(u, z), rkα(u + τ$̃∨, z) = e ( −n$ν δ〈ν∨,α〉z ) rkα(u, z). On the other hand, due to the Λ-invariance of Q̃∨, we have 〈β∨, λm(α)〉 = 〈β∨, α〉. Therefore, Adexp 2πıβ∨ Eλm(α) = e(〈α, β∨〉)Eλm(α). Then from (A.3) we find that Adexp(−2πıβ∨z) t a α = e(−nα,βz)taα. Similarly, due to Λ-invariance of P̃∨, we have also Adexp(−2πı$̃∨z) t a α = e ( −n$ν δ〈ν∨,α〉z ) taα. Since the Cartan part rH of the r-matrix does not depend on u we come to the relations r ( u + β∨, z ) = r(u, z), r ( u + τβ∨, z ) = Adexp(−2πıβ∨z) r(u, z), (4.34) r ( u + $̃∨, z ) = r(u, z), r ( u + τ$̃∨, z ) = Adexp(−2πı$̃∨z) r(u, z). (4.35) In all cases the adjoint actions Adh act on the first component of the tensor product and play the role of the clutching operators. Let x(k, α) = 〈u, α+κτ〉+k/l. Then r(u, z) is singular when x(k, α)→ 0 (see (B.7) and (B.2)) r(u, z) = \|α\|2e(〈κ, α〉z) ( 1 x(k, α) +O(1) ) tkα ⊗ t−k−α. (4.36) It means that r(u, z) are sections of the bundles over the moduli spaces Csc j (4.11), or Cad j (4.13) with sections taking values in g ⊗ g with the quasi-periodicities (4.34), (4.35) and with the singularities (4.36). 4.6 KZB connection related to elliptic curves As it was established the part of connection related to the moving points coincides with the introduced above r-matrix. Here we prove that this connection is flat. Consider the following differential operators ∇a = ∂za + ∂̂a + ∑ c 6=a rac, (4.37) ∇τ = 2πi∂τ + ∆ + 1 2 ∑ b,d f bd, 24 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov with rac = l−1∑ k=0 ∑ α∈R \|α\|2ϕkα(u, za − zc)tk,aα t−k,c−α + l−1∑ k=0 ∑ α∈Π ϕk0(u, za − zc)Hk,aα hk,cα , fac = l−1∑ k=0 ∑ α∈R \|α\|2fkα(u, za − zc)tk,aα t−k,c−α + l−1∑ k=0 ∑ α∈Π fk0 (u, za − zc)Hk,aα hk,cα , where tk,aα = 1 ⊗ · · · ⊗ 1 ⊗ tkα ⊗ 1 ⊗ · · · ⊗ 1 (with tkα on the a-th place) and similarly for the generators Hk,aα and hk,aα .3 The following short notations are used here ∂̂a = l ∑ α∈Π h0,a α ∂α̂, ∆ = l 2 ∑ α∈Π l−1∑ s=0 ∂uα∂uλsα̂ and ϕkα(u, z) = e2πi〈κ,α〉zφ ( 〈u + κτ, α〉+ k l , z ) , fkα(u, z) = e2πi〈κ,α〉zf ( 〈u + κτ, α〉+ k l , z ) . (4.38) From the definition it follows that rac = −rca and fac = f ca. Following (B.6) and (B.7) we put ϕ0 0(z) = E1(z), (4.39) f0 0 (z) = ρ(z) = 1 2 ( E2 1(z)− ℘(z) ) . (4.40) Notice that fkα(u, 0) = −E2 ( 〈u+ κτ, α〉+ k l ) = −℘ ( 〈u+ κτ, α〉+ k l ) − 2η1 and, therefore f cc = − l−1∑ k=0 ∑ α∈R \|α\|2℘kαtk,cα t−k,c−α − l−1∑ k=0 ∑ α∈Π Hk,cα h−k,cα − 2lη1C c 2, where Cc2 is the Casimir operator acting on the c-th component. Recall that we study the following system of differential equations ∇aF = 0, a = 1, . . . , n, ∇τF = 0. (4.41) There are two types of the compatibility conditions of KZB equations (4.41) [∇a,∇b]F = 0, a, b = 1, . . . , n, [∇a,∇τ ]F = 0, a = 1, . . . , n. (4.42) It is important to mention that the solutions of (4.41) F are assumed to satisfy the following condition( n∑ c=1 h0,c α ) F = 0, for any α ∈ Π̃. (4.43) 3For brevity we write tk,aα , hk,aα instead of representations of these generators in the spaces Vµa . Hecke Transformations of Conformal Blocks in WZW Theory. I 25 Proposition 1. The upper equations in (4.42) [∇a,∇b] = 0 are valid for the r-matrix (4.33) on the space of solutions of (4.41) satisfying (4.43). They follow from the classical dynamical Yang–Baxter equations[ rab, rac ] + [ rab, rbc ] + [ rac, rbc ] + [ ∂̂a, rbc ] + [ ∂̂c, rab ] + [ ∂̂b, rca ] = 0. (4.44) Proposition 2. The lower equations in (4.42) [ ∇a,∇τ ] = 0 are valid for the r-matrix (4.38) on the space of solutions of (4.41) satisfying (4.43). The proofs of these statements are given in the Appendix C. Let us also remark that the non-trivial trigonometric and rational limits of the above formulae can be obtained via procedures described in [1, 65, 72]. A Generalized Sine (GS) basis in simple Lie algebras Let Z be a subgroup of the center Z(Ḡ) of Ḡ, and consider a quotient group G = Ḡ/Z. Assume for simplicity that Z ( Ḡ ) is cyclic. The case Spin(4n) where Z(G) = µ2 × µ2 can be treated similarly. Let us take an element ζ ∈ Z(Ḡ) of order l, generating Z. It defines uniquely an element Λ0 from the Weyl group W (see [12, 46]). It is a symmetry of the corresponding extended Dynkin diagram and (Λ0)l = Id. Λ0 generates a cyclic group µl = ( Λ0, (Λ0)2, . . . , (Λ0)l = 1 ) isomorphic to a subgroup of Z(Ḡ). Note that l is a divisor of ord(Z(Ḡ)). Consider the action of Λ0 on g. Since (Λ0)l = Id we have a l-periodic gradation g = ⊕l−1 a=0ga, λ(ga) = ωaga, ω = exp 2πi l , λ = AdΛ0 , (A.1) [ga, gb] = ga+b mod l, where g0 is a subalgebra g0 ⊂ g and the subspaces ga are its representations. Since Q and Λ commute in the adjoint representations the root subspaces ga are their common eigenspaces. GS-basis. Here we shortly reproduce the construction of the GS-basis following [46]. Since Λ0 ∈W it preserves the root system R. Define the quotient set Tl = R/µl. Then R is represented as a union of µl-orbits R = ∪TlO. We denote by O(β̄) an orbit starting from the root β O(β̄) = { β, λ(β), . . . , λl−1(β) } , β̄ ∈ Tl. The number of elements in an orbit O (the length of O) is l/pα = lα, where pα is a divisor of l. Let να be a number of orbits Oᾱ of the length lα. Then ]R = ∑ ναlα. Notice that if O(β̄) has length lβ(lβ 6= 1), then the elements λkβ and λk+lββ coincide. First, transform the root basis E = {Eβ, β ∈ R} in L. Define an orbit in E Eβ = { Eβ, Eλ(β), . . . , Eλl−1(β) } corresponding to O(β̄). Again E = ∪β̄∈TlEβ̄. For O(β̄) define the set of integers Jpα = { a = mpα ∣∣ m ∈ Z, a is defined mod l } , pα = l lα . (A.2) Let Eα (α ∈ R) be the root basis of g. “The Fourier transform” of the root basis on the orbit O(β̄) is defined as taβ = 1√ l l−1∑ m=0 ωmaEλm(β), ω = exp 2πi l , a ∈ Jβ. (A.3) 26 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov Almost the same construction exists in H. Again let Λ0 generates the group µl. Since Λ0 preserves the extended Dynkin diagram, its action preserves the extended coroot system Π∨ext = Π∨ ∪ α∨0 in H. Consider the quotient Kl = Π∨ext/µl. Define an orbit H(ᾱ) of length lα = l/pα in Π∨ext passing through Hα ∈ Π∨ext H(ᾱ) = { Hα, Hλ(α), . . . ,Hλl−1(α) } , ᾱ ∈ Kl = Π∨ext/µl. The set Π∨ext is a union of H(ᾱ): (Π∨)ext = ∪ᾱ∈KlH(ᾱ). Define “the Fourier transform” hcᾱ = 1√ l l−1∑ m=0 ωmcHλm(α), ω = exp 2πi l , c ∈ Jα (see (A.2)). The basis hcα (c ∈ Jα, ᾱ ∈ Kl) is over-complete in H. Namely, let H(ᾱ0) be an orbit passing through the minimal coroot { Hα0 , Hλ(α0), . . . ,Hλl−1(α0) } . Then the element h0 ᾱ0 is a linear combination of elements h0 −ᾱ, (α ∈ Π) and we should exclude it from the basis. We replace the basis Π∨ in H by hcᾱ, c ∈ Jα, { α ∈ K̃l = Kl \ H(ᾱ0), c = 0, ᾱ ∈ Kl, c 6= 0. (A.4) As before there is a one-to-one map Π∨ ↔ {hcᾱ}. The elements (haᾱ, t a ᾱ) form GS basis in g(l−a) (A.1). The dual basis is generated by elements Haᾱ( Haᾱ, h b β̄ ) = δ(a+b,0(mod l))δα,β, Haᾱ = ∑ β∈Π (Aaα,β)−1h−a β̄ , haβ̄ = ∑ α∈Π (A−aα,β)H−aᾱ , (A.5) where Aaα,β = 2 (β, β) l−1∑ s=0 ω−saaβ,λs(α) and aα,β is the Cartan matrix of g. The λ-invariant subalgebra g0 contains the subspace V = { ∑ β̄∈T ′l aβ̄t 0 β̄, aβ̄ ∈ C } . Then g0 is a sum of g̃0 and V g0 = g̃0 ⊕ V. In the invariant simple algebra g̃0 instead of the basis (h0 ᾱ, t 0 β̄ ) we can use the Chevalley basis and incorporate it in the GS-basis{ h0 ᾱ, t 0 β̄ } → { g̃0 = ( Hα̃, α̃ ∈ Π̃, Eβ̃, β̃ ∈ R̃ ) , V = ( t0β̄, β̄ ∈ T ′)}, where Π̃ is a system of simple roots constructed by the averaging of the λ action on Πext, and R̃ is a system of roots of g̃0 generated by Π̃. We have the following action of the adjoint operators on the GS basis: AdΛ ( tcβ̄ ) = e ( 〈ũ, β〉 − c l ) tcβ̄, AdΛ ( hcβ̄ ) = e ( −c l ) hcβ̄, e(x) = exp(2πix). (A.6) Hecke Transformations of Conformal Blocks in WZW Theory. I 27 In addition, AdQ ( hcβ̄ ) = hcβ̄, AdQ(Hα̃) = Hα̃, (A.7) AdQ ( tcβ̄ ) = e(〈κ, β〉)tcβ̄, AdQ(Eα̃) = e〈κ, α̃〉Eα̃. (A.8) There are also the evident relations AdΛ(Eα̃) = e(〈ũ, α̃〉)Eα̃, AdΛ(Hα̃) = Hα̃, ũ ∈ h̃. In particular, AdΛ ( tcβ̄ ) = e ( 〈ũ, β〉 − c l ) tcβ̄, AdΛ ( hcβ̄ ) = e ( −c l ) hcβ̄, e(x) = exp(2πix). Commutation relations in the GS basis: [ taα, t b β ] =  1√ l l−1∑ s=0 ωbsCα,λsβt a+b α+λsβ, α 6= −λsβ, pα√ l ωsbha+b α , α = −λsβ, [ hkα, t m β ] = 1√ l l−1∑ s=0 ω−ks 2(α, λsβ) (α, α) tk+m β , [ Hkα, t m β ] = 1√ l l−1∑ s=0 ω−ks (α, α) 2 (α̂, λsβ)tk+m β . B Elliptic functions The basic function is the theta-function ϑ(z\|τ) = q 1 8 ∑ n∈Z (−1)neπi(n(n+1)τ+2nz). It is a holomorphic function on C with simple zeroes at the lattice τZ + Z and the quasi- periodicities ϑ(z + 1) = −ϑ(z), ϑ(z + τ) = −q− 1 2 e−2πizϑ(z). Define the ration of the theta-functions φ(u, z) = ϑ(u+ z)ϑ′(0) ϑ(u)ϑ(z) . (B.1) Then φ(u, z) = φ(z, u), φ(−u,−z) = −φ(u, z). (B.2) Related functions: ϕmβ (u, z) = e(〈κ, β〉z)φ ( 〈u + κτ, β〉+ m l , z ) , (B.3) ϕm,kβ (u, z) = ∂kz ( e(〈κ, β〉z)φ ( 〈u + κτ, β〉+ m l , z )) , ϕm,0β = ϕmβ , (B.4) f(u, z) = ∂uφ(u, z), (B.5) f(u, z) = φ(u, z)(E1(u+ z)− E1(u)). (B.6) 28 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov φ(u, z) has a pole at z = 0 and φ(u, z) = 1 z + E1(u) + z 2 ( E2 1(u)− ℘(u) ) + · · · . (B.7) Similarly, ϕmβ (u, z) = 1 z + E1 ( 〈u + κτ, β〉+ m l ) + 2πı〈κ, β〉+ z 2 ( E2 1(u)− ℘(u) ) + · · · , (B.8) where E1 is (B.10). It follows from this expansion that ϕm,1β (u, z) = − 1 z2 + 1 2 ( E2 1(u)− ℘(u) ) + · · · , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B.9) ϕm,kβ (u, z) = (−1)k zk+1 + c(m, k) + · · · . In other words ϕm,kβ (u, z) has not poles of order less than k + 1. The Eisenstein functions: E1(z\|τ) = ∂z log ϑ(z\|τ), E1(z\|τ) ∼ 1 z − 2η1z + · · · , (B.10) where η1(τ) = 3 π2 ∞∑ m=−∞ ∞′∑ n=−∞ 1 (mτ + n)2 = 24 2πi η′(τ) η(τ) , η(τ) = q 1 24 ∏ n>0 ( 1− qn ) , E2(z\|τ) = −∂zE1(z\|τ) = ∂2 z log ϑ(z\|τ), E2(z\|τ) ∼ 1 z2 + 2η1, (B.11) and more general for k > 2 Ek(z\|τ) = (−∂z)k+1 log ϑ(z\|τ), Ek(z\|τ) ∼ 1 zk + · · · . (B.12) Relation to the Weierstrass functions: ζ(z, τ) = E1(z, τ) + 2η1(τ)z, ℘(z, τ) = E2(z, τ)− 2η1(τ). Quasi-periodicity: ϑ(z + 1) = −ϑ(z), ϑ(z + τ) = −q− 1 2 e−2πizϑ(z), (B.13) E1(z + 1) = E1(z), E1(z + τ) = E1(z)− 2πi, (B.14) Ek(z + 1) = Ek(z), Ek(z + τ) = Ek(z), k > 1, (B.15) φ(u, z + 1) = φ(u, z), φ(u, z + τ) = e−2πıuφ(u, z), (B.16) ϕm,kβ (u, z + 1) = e(〈κ, β〉)ϕm,kβ (u, z), ϕm,kβ (u, z + τ) = e ( −〈u, β〉 − m l ) ϕm,kβ (u, z), (B.17) f(u, z + 1) = f(u, z), f(u, z + τ) = e−2πıuf(u, z)− 2πıφ(u, z). The following identities are also used here 2πi∂τφ(u, z) = ∂z∂uφ(u, z) = ∂zf(u, z) Hecke Transformations of Conformal Blocks in WZW Theory. I 29 and for the functions (4.38) this identity takes the form 2πi∂τϕ m α (z) = ∂zf k α(z). (B.18) Fay identity: φ(u1, z1)φ(u2, z2)− φ(u1 + u2, z1)φ(u2, z2 − z1)− φ(u1 + u2, z2)φ(u1, z1 − z2) = 0. Differentiating over u2 we find φ(u1, z1)f(u2, z2)− φ(u1 + u2, z1)f(u2, z2 − z1) = φ(u2, z2 − z1)f(u1 + u2, z1) + φ(u1, z1 − z2)f(u1 + u2, z2). Substituting here u1 = 〈u+ κτ, α+ β〉+ k +m l , u2 = −〈u+ κτ, β〉 − m l , z1 = za − zc = zac, z2 = zb − zc = zbc, and multiplying by appropriate exponential factor we can rewrite it in the form ϕkα(zac)f m β (zab)− ϕmβ (zab)f k α(zac) + ϕk+m α+β (zab)f k α(zcb)− ϕk+m α+β (zac)f −m −β (zbc) = 0. (B.19) Taking the limit m = 0, β = 0 and using the expansion φ(z, u) ∼ 1 u + E1(z) + uρ(z) + · · · , we find ϕkα(zac)ρ(zab)− E1(zab)f k α(zac) + ϕkα(zab)f k α(zbc)− ϕkα(zac)ρ(zcb) = 1 2 ∂uf k α(zac). (B.20) More Fay identities: ϕkα(zac)f m β (zac)− ϕmβ (zac)f k α(zac) = ϕk+m α+β (zac) ( ℘kα − ℘mβ ) , (B.21) ϕmβ (zac)f −m −β (zac)− ϕ−m−β (zac)f m β (zac) = E′2 m β , (B.22) ϕkβ(zac)℘ k β − ϕkβ(zac)ρ(zac) + E1(zac)f k β (zac) = 1 2 ∂uf k β (zac). (B.23) The last one follows from ∂uφ(u, z) = φ(u, z) ( E1(z + u)− E1(u) ) and ( E1(z + u)− E1(u)− E1(z) )2 = ℘(z) + ℘(u) + ℘(z + u). C Proofs of Propositions 1 and 2 Proof of Proposition 1. [∇a,∇b] = [ ∂za , r ba ] + [ ∂̂a, rba ] (C.1) + ∑ c 6=a,b [ ∂̂a, rbc ] − [ ∂zb , r ab ] − [ ∂̂b, rab ] − ∑ c 6=a,b [ ∂̂b, rac ] + ∑ d6=b ∑ a6=c [ rac, rbd ] . 30 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov First, notice that[ ∂za + ∂zb , r ab ] = 0. Secondly, [ ∂̂a, rba ] − [ ∂̂b, rab ] = − [ ∂̂a + ∂̂b, rab ] = ∑ c 6=a,b [ ∂̂c, rab ] − [∑ c ∂̂c, rab ] . Thirdly,∑ d6=b ∑ a6=c [ rac, rbd ] = ∑ c 6=a,b ([ rac, rba ] + [ rab, rbc ] + [ rac, rbc ]) . Therefore, [ ∇a,∇b ] = ∑ c6=a,b CDYBabc − [∑ c ∂̂c, rab ] , (C.2) where CDYBabc= [ rab, rac ] + [ rab, rbc ] + [ rac, rbc ] + [ ∂̂a, rbc ] + [ ∂̂c, rab ] + [ ∂̂b, rca ] (4.44) = 0. (C.3) and [∑ c ∂̂c, rab ] (4.43) = 0. � Proof of Proposition 2. 1. [ ∂zaf ac ] − 2πi [ ∂τr ac ] (B.18) = 0. 2. [ ∆, ∑ c 6=a rac ] = l 2 ∑ c 6=a ∑ β∈R l−1∑ m=0 \|β\|2∂ufmβ (za − zc) [ h0,a β , tm,aβ ] + t−m,c−β + l ∑ c 6=a ∑ β∈R ∑ α∈Π l−1∑ s=0 \|β\|2〈λsα̂, β〉fmβ (za − zc)∂uαt m,a β t−m,c−β . 3. Terms [ ∂̂a, 1 2f bc ] for b, c 6= a and b 6= c:[ ∂̂a, 1 2 ∑ b,c 6=a b 6=c f bc ] = l 2 ∑ b,c 6=a b 6=c l−1∑ k=0 ∑ α∈R \|α\|2∂ufkα(zb − zc)h0,a α tk,bα t−k,c−α . (C.4) 3.1. Terms [ ∂̂a, 1 2f ac ] for c 6= a:[ ∂̂a, 1 2 ∑ c 6=a (fac + f ca) ] = [ ∂̂a, ∑ c 6=a fac ] = l ∑ c 6=a l−1∑ m=0 ∑ β∈R \|β\|2∂ufmβ (za − zc)h0,a β tm,aβ t−m,c−β + l ∑ c 6=a ∑ β∈R ∑ α∈Π l−1∑ s=0 \|β\|2〈λsα̂, β〉fmβ (za − zc)∂uαt m,a β t−m,c−β . Hecke Transformations of Conformal Blocks in WZW Theory. I 31 Therefore we get[ ∂̂a, ∑ c6=a fac ] − [ ∆, ∑ c 6=a rac ] = l 2 ∑ c6=a ∑ α∈R l−1∑ m=0 \|α\|2∂ufmα (za − zc) [ h0,a α , tm,aα ] + t−m,c−α . (C.5) 3.2. Terms [ ∂̂a, 1 2f aa ] : [ ∂̂a, 1 2 faa ] = − l 2 l−1∑ m=0 ∑ β∈R \|β\|2∂u℘mβ h 0,a β tm,aβ t−m,a−β = − l 4 l−1∑ m=0 ∑ β∈R \|β\|2E′2 m β h 0,a β [ tm,aβ t−m,a−β ] + . (C.6) 3.3. Terms [ ∂̂a, 1 2f cc ] for c 6= a:[ ∂̂a, ∑ c6=a 1 2 f cc ] = − l 4 ∑ c 6=a l−1∑ m=0 ∑ β∈R \|β\|2E′2 m β h 0,a β [ tm,cβ t−m,c−β ] + . (C.7) 4. Terms [r, f ]:[∑ c 6=a rac, 1 2 ∑ b,d f b,d ] = 1 2 ∑ b,c6=a b 6=c ([ rac, fab ] + [ rac, f bc ] + [ rab, fac ] + [ rab, f bc ]) + ∑ c 6=a ([ rac, fac ] + 1 2 [ rac, faa ] + 1 2 [ rac, f cc ]) . 4.1. Terms [ r, f ] for b, c 6= a and b 6= c: 1 2 ∑ b,c 6=a b 6=c ([ rac, fab ] + [ rac, f bc ] + [ rab, fac ] + [ rab, f bc ]) = 1 2 ∑ b,c6=a b6=c l−1∑ k,m,s=0 ∑ α,β∈R \|α\|2\|β\|2ωksCλsα,β ( ϕkα(zac)f m β (zab)− ϕmβ (zab)f k α(zac) + ϕk+m α+β (zab)f k α(zcb)− ϕk+m α+β (zac)f −m −β (zbc) ) tk+m,a α+λsβt −m,b −β t−k,c−α − l 2 ∑ b,c6=a b6=c l−1∑ k,m=0 ∑ α∈R \|α\|2 [( ϕkα(zac)f m 0 (zab)− ϕm0 (zab)f k α(zac) + ϕk+m α (zab)f k α(zcb)− ϕk+m α (zac)f −m 0 (zbc) ) tk+m,a α h−m,bα t−k,cα + ( ϕkα(zab)f m 0 (zac)− ϕm0 (zac)f k α(zab) + ϕk+m α (zac)f k α(zbc)− ϕk+m α (zac)f −m 0 (zcb) ) tk+m,a α t−k,bα h−m,cα + 2 ( ϕkα(zbc)f m 0 (zba)− ϕm0 (zba)f k α(zbc) + ϕk+m α (zba)f k α(zba)− ϕk+m α (zba)f −m 0 (zac) ) tk+m,b α h−m,aα t−k,cα ] . (C.8) 32 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov Almost all terms in this big sum vanish due to the Fay identity (B.19), except the terms with m = 0. Using (B.20) for these terms in the expression (C.8) we get − l 4 ∑ b,c 6=a b 6=c l−1∑ k=0 ∑ α∈R \|α\|2∂ufkα(zac)t k,a α h0,b α t−k,c−α − l 4 ∑ b,c6=a b 6=c l−1∑ k=0 ∑ α∈R \|α\|2∂ufkα(zab)t k,a α h0,c α t−k,b−α + l 2 ∑ b,c6=a b 6=c l−1∑ k=0 ∑ α∈R \|α\|2∂ufkα(zbc)t k,b α h0,a α t−k,c−α . Finally, using the symmetry in summation over c and b we obtain 1 2 ∑ b,c 6=a b 6=c ([ rac, fab ] + [ rac, f bc ] + [ rab, fac ] + [ rab, f bc ]) = l 2 ∑ b,c6=a b6=c l−1∑ k=0 ∑ α∈R \|α\|2∂ufkα(zac)t k,a α h0,b α t−k,c−α − l 2 ∑ b,c6=a b6=c l−1∑ k=0 ∑ α∈R \|α\|2∂ufkα(zbc)t k,b α h0,a α t−k,c−α . (C.9) Notice that the second term here cancels the expression (C.4). 4.2. Terms [ rac, fac ] : ∑ c6=a [ rac, fac ] = 1 4 ∑ c6=a l−1∑ k,m,s=0 ∑ α,β∈R \|α\|2\|β\|2 ( ϕkα(zac)f m β (zac) − ϕmβ (zac)f k α(zac) ) ωmsCα,λsβt k+m,a α+λsβ [ t−k,c−α , t−m,c−β ] + + 1 4 ∑ c6=a l−1∑ k,m,s=0 ∑ α,β∈R \|α\|2\|β\|2 ( ϕkα(zac)f m β (zac) − ϕmβ (zac)f k α(zac) ) ω−msCα,λsβt −k−m,c −α−λsβ [ tk,aα , tm,aβ ] + − l 4 ∑ c6=a l−1∑ k,m=0 ∑ β∈R \|β\|2 ( ϕ−m−β (zac)f m+k β (zac) − ϕm+k β (zac)f −m −β (zac) ) hk,aβ [ t−k−m,c−β , tm,cβ ] + + l 4 ∑ c6=a l−1∑ k,m=0 ∑ β∈R \|β\|2 ( ϕ−m−β (zac)f m+k β (zac) − ϕm+k β (zac)f −m −β (zac) ) h−k,cβ [ tk+m,a β , t−m,a−β ] + + l 2 ∑ c6=a l−1∑ k,m=0 ∑ β∈R \|β\|2 ( − ϕ−m0 (zac)f m+k β (zac) + f−m0 (zac)ϕ m+k β (zac) )[ h−m,aβ , tm+k,a β ] + t−k,c−β + l 2 ∑ c6=a l−1∑ k,m=0 ∑ β∈R \|β\|2 ( ϕ−m0 (zac)f m+k β (zac) Hecke Transformations of Conformal Blocks in WZW Theory. I 33 − f−m0 (zac)ϕ m+k β (zac) )[ hm,cβ , t−m−k,a−β ] + tk,cβ . (C.10) 4.3. Terms [ rac, faa ] and [ rac, f cc ] for c 6= a:[∑ c 6=a rac, faa ] = −1 4 ∑ c 6=a l−1∑ k,m=0 ∑ α,β∈R \|α\|2\|β\|2ϕk+m α+β (zac)(℘ k α − ℘mβ ) × ω−msCα,λsβ [ tk,aα , tm,aβ ] + t−k−m,c−α−λsβ − l 4 ∑ c 6=a l−1∑ k,m=0 ∑ β∈R \|β\|2ϕk0(℘mβ − ℘k+m β )h−k,cβ [ tk+m,a β , t−m,a−β ] + + l 2 ∑ c 6=a l−1∑ k,m=0 ∑ β∈R \|β\|2ϕkβ(zac)(℘ m 0 − ℘m+k β ) [ h−m,aβ , tm+k,a β ] + t−k,c−β , (C.11) [∑ c 6=a rac, f cc ] = −1 4 ∑ c 6=a l−1∑ k,m=0 ∑ α,β∈R \|α\|2\|β\|2ϕk+m α+β (zac)(℘ k α − ℘mβ ) × ωmsCα,λsβ [ t−k,a−α , t−m,a−β ] + tk+m,c α+λsβ + l 4 ∑ c 6=a l−1∑ k,m=0 ∑ β∈R \|β\|2ϕk0(℘mβ − ℘k+m β )hk,aβ [ t−k−m,c−β , tm,cβ ] + − l 2 ∑ c 6=a l−1∑ k,m=0 ∑ β∈R \|β\|2ϕkβ(zac)(℘ m 0 − ℘m+k β ) [ hm,cβ , t−m−k,c−β ] + tk,aβ . (C.12) The first two lines in (C.10) are canceled by first lines in (C.11) and (C.12) due to identi- ty (B.21). Next, the sum of the third line in (C.10), the second line in (C.12), the sum of fourth line in (C.10) and the second line in (C.11) are vanished due to (B.21) for all values of summation parameters except k = 0. For k = 0 these sums give − l 4 ∑ c 6=a l−1∑ m=0 ∑ β∈R \|β\|2 ( ϕ−m−β (zac)f m β (zac)− ϕmβ (zac)f −m −β (zac) ) × ( h0,a β [ t−m,c−β , tm,cβ ] + − h0,c β [ tm,aβ , t−m,a−β ] + ) (B.22) = l 4 ∑ c6=a l−1∑ m=0 ∑ β∈R \|β\|2E′2 m β ( h0,a β [ t−m,c−β , tm,cβ ] + − h0,c β [ tm,aβ , t−m,a−β ] + ) and this is exactly what we need to compensate (C.6) and (C.7). (Note that (C.7) cancels by the first term here and (C.6) cancels by the second one due to (4.43).) Finally, the last two lines in (C.10) are canceled by the list lines in (C.11) and (C.12) for all values of summation parameters except m = 0. For m = 0 the sum of these terms equals − l 2 ∑ c 6=a l−1∑ k=0 ∑ β∈R \|β\|2 ( ϕ0 0(zac)f k β (zac)− f0 0 (zac)ϕ k β(zac) + ϕkβ(zac)℘ k β ) × ([ h0,a β , tk,aβ ] + t−k,c−β − [ h0,c β , t−k,c−β ] + tk,aβ ) . (C.13) Notice that due to (4.39) and (4.40) f0 0 (zac) = ρ(zac) and ϕ0 0(zac) = E1(zac). Then using (B.23) we can simplify the expression (C.13) − l 4 ∑ c 6=a l−1∑ k=0 ∑ β∈R \|β\|2∂ufkβ (zac) ([ h0,a β , tk,aβ ] + t−k,c−β − [ h0,c β , t−k,c−β ] + tk,aβ ) . (C.14) 34 A.M. Levin, M.A. Olshanetsky, A.V. Smirnov and A.V. Zotov Finally, we have nonzero terms from (C.5), first term in (C.9) and (C.14). All these terms are proportional to ∂uf k β . Summing them up we find l ∑ c 6=a l−1∑ k=0 ∑ β∈R \|β\|2∂ufkβ (zac) ( −1 4 [ h0,a β , tk,aβ ] + t−k,c−β + 1 4 [ h0,c β , t−k,c−β ] + tk,aβ + 1 2 ∑ b 6=a,c tk,aβ t−k,c−β h0,b β + 1 2 [ h0,a β , tk,aβ ] + t−k,c−β ) = l ∑ c 6=a l−1∑ k=0 ∑ β∈R \|β\|2∂ufkβ (zac) ( 1 4 [ h0,a β , tk,aβ ] + t−k,c−β + 1 4 [ h0,c β , t−k,c−β ] + tk,aβ + 1 2 ∑ b 6=a,c tk,aβ t−k,c−β h0,b β ) (4.43) = l ∑ c 6=a l−1∑ k=0 ∑ β∈R \|β\|2∂ufkβ (zac) ( 1 4 [ h0,a β , tk,aβ ] t−k,c−β + 1 4 [ h0,c β , t−k,c−β ] tk,aβ ) = 0. � Acknowledgements The authors are grateful to A. Beilinson, L. Fehér, B. Feigin, A. Gorsky, S. Khoroshkin, A. Losev, A. Mironov, V. Poberezhny, A. Rosly and A. Stoyanovsky for useful discussions and remarks. The work was supported by grants RFBR-09-02-00393, RFBR-09-01-92437-KEa and by the Federal Agency for Science and Innovations of Russian Federation under contract 14.740.11.0347. 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Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

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