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 $\Sigma_{g,n}$ of genus $g$ with $n$ marked points. The bundles are defined by their characteristic classes - elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ 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.


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 Σ g,n of genus g with n marked points (z 1 , z 2 , . . . , z n ). To describe this model, one should define a G-bundle over Σ g,n . Topologically, the G-bundles are defined by their characteristic classes. Let Z(G) be a center of G and G ad = G/Z(G). The characteristic classes are obstructions to lift the G ad -bundles to the G-bundles. They are elements of the cohomology group H 2 (Σ g , Z(G)) ∼ Z(G) [46] 1 . IfḠ is the corresponding simply connected group (the universal covering with the natural group structure) and G =Ḡ/Z ∨ (G), then elements from H 2 (Σ g , Z ∨ (G)) are obstruction to lift the G-bundles to theḠ-bundles. In particular, consider G = Spin(N ) and SO(N ) = Spin(N )/Z 2 . Then H 2 (Σ g , Z 2 ) ∼ Z 2 defines the Stiefel-Whitney classes of the SO(N )-bundles over Σ g .
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 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 representations V µa (µ a are the highest weights) of G attached to the marked points. For a positive integer k define the integrable moduleV µa of level k of the centrally extended loop group D × → G, where D × = D \ z a is a punctured disk around the marked point z a . The conformal blocks are linear functionalsV [n] ≡V µ 1 ⊗ · · · ⊗V µn → C satisfying some additional conditions (the Ward identities). Let C G (V [n] ) be a space of conformal blocks. This space depends on parameters -the complex structure of Σ g,n , and in this way forms a bundle over the moduli space M g,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) = KerḠ → G is non-trivial then the integrable module is a sum of sectors, corresponding to the characteristic classes of the underlying bundleŝ 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 conformal 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 approach 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 A n , D n and E 6 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 representations 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.
The paper has the following structure. In Section 2 we consider a general setting of the KZB equations related to arbitrary curves Σ g,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.

Loop algebras, loop groups and integrable modules 2.1 Loop algebras and loop groups
LetḠ be a simply-connected simple complex Lie group and Z = Z Ḡ is the center ofḠ. For all simply-connected groups (SL(N, C), Sp N , E 6 , E 7 and Spin N except N = 4n), the center is a cyclic group. For Spin 4N Z = Z 2 ⊕ Z 2 . The adjoint group is the quotient group G ad =Ḡ/Z. Assume for simplicity that Z is a cyclic group Z l of order l.
. . , γ l )} is a coweight lattice in the Cartan subalgebra h K = 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 ∼ Z l . It is generated by the element e( ∨ ) = exp(2πı ∨ ) ∈ T . For Spin 4N 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, R is an union of positive and negative roots R = R + ∪ R − with respect to some ordering in h * . Let Π = {α 1 , . . . , α l } be a basis of simple roots in R.
has a finite order poles when t → 0. In other words, L(G) is the group of Laurent polynomials 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: R aff = {α = α + n, n ∈ Z, n = 0}. Let {h α } be the basis of simple coroots in h. Then the analog of the root decomposition for the loop algebra has the following form Let −α 0 be the highest root −α 0 ∈ R + . The system of simple affine roots isΠ = Π ∪ (−α 0 + 1). It is a basis in R aff . Consider the positive loop subalgebra Each summand is a Lie subalgebra ofL(g). There are two types of the affine Weyl groups: where W is the Weyl group of g, They act on the root vectors as eα = e α t n → eŵ (α) = e w(α) t n+ γ,α . The loop groups have the Bruhat decomposition [61]. Define subgroups L + (G) = g 0 + g 1 t + · · · , g j ∈ G, g 0 = b ∈ B, is the positive Borel subgroup, (2.6) N − (G) = n − + g 1 t −1 + · · · , n − ∈ N − , is the negative nilpotent subgroup, (2.7) N + (G) = n + + g 1 t + · · · , n + ∈ N + , is the positive nilpotent subgroup. (2.8) The Bruhat decomposition takes the form For a loop g(t) in G ad denote byḡ its lift to a map from S 1 toḠ. 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(e 2πı t) = e(γ)g(t), (e(x) = e 2πıx ). 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 = G ad , but not forḠ. In this way we have the representations (2.10) If γ 1 = γ 2 + δ for any δ ∈ Q ∨ then γ 1 and γ 2 lead to the same monodromies. We say in this case that L γ 1 Ḡ and L γ 2 Ḡ are equivalent. Then from (2.10) we have In particular, if the center Z ∼ Z l is generated by a fundamental coweight ∨ , then , (2.11) and L j G ad = e(j ∨ ) L Ḡ /Z . Consider the quotient Fl aff = L G ad /L + G ad [61]. It is called the affine flag variety. Let Σŵ be an N − G ad -orbit ofŵ in Fl aff . This orbit is dipheomorphic to the intersection N − G ad ŵ = N − G ad ∩ŵN − G ad ŵ −1 . Therefore, its codimension in F l aff is the length l(ŵ) ofŵ. It is the number of negative affine roots whichŵ transforms to positive ones (Theorem 8.7.2 in [61]). The Bruhat decomposition (2.9) defines the stratification of Fl aff : (2.12)
Let E α 0 be the root subspace in g corresponding to α 0 . Consider the maximal submodule S µ of V µ generated by the singular vector (2.13) The irreducible integrable moduleV µ is the quotient (2.14) We identify the module V µ with a submodule V µ ⊗ 1 → V µ . The integrable moduleV µ can be characterized in the following way: the subspace ofV µ annihilated by the positive subalgebra The group L(G) has a central extension 1 → C * → LG → LG → 1 corresponding to (2.1). The integrable module can be described in terms of LG. The action of L + (G) on the HWV has the form where χ µ (b) is the character of the Borel subgroup B. ThenV µ is generated by the action In this way we describe only "the trivial sector" of the L(G)-module. Consider the Bruhat representation for L G ad (2.9), and letŵ = t γ , γ ∈ P ∨ . Define the Verma modules with the HWV t γ v µ , (2.16) They have the singular vectors E α 0 ⊗ t −1 k− µα 0 +1 t γ v µ (compare with (2.13)). Let S µ,γ be the maximal submodules generated by these singular vectors. Consider the quotient spaceŝ and define their direct sum We say that two subspacesV µ (γ 1 ) andV µ (γ 2 ) are equivalent if γ 1 = γ 2 + δ, where δ ∈ Q ∨ . This equivalence leads to the decomposition ofV µ (as a L(G ad )-module) into a sum of l = ord(Z(Ḡ)) sectors, Notice that (t γ v µ ) is not the HWV with respect to L + (g). However, it was proved in [23] that there exists a unique elementŵ =ŵ(γ) = t δ w ∈ W Q such that t γŵ v µ is the HWV. We demonstrate it below for L(SL(2, C)). The elementsŵ and γ represent the same element ζ ∈ Z.
Then we define the Verma module The vector E α 0 ⊗ t −1 k− µ,wα 0 +1 (γŵv µ ) is singular and corresponds to the submodule S µ,γ . As in (2.14) we identify the integrable modules V µ (ζ)/S µ,γ withV µ (ζ) (2.16). LetV * µ be the dual module. The Borel-Weil-Bott theorem for the loop group [61] states thatV * µ can be realized as the space of sections of a line bundle L µ over the affine flag variety (2.12). The line bundle is determined by the action L + (G) × C * on its sections as in (2.15), (2.21)

Moduli space of holomorphic G-bundles
Let P be a principle G-bundle over a curve Σ g,n of genus g with n marked points z = (z 1 , . . . , z n ), (n > 0), V is a G-module and E G = P × G V is the associated bundle. We consider the set of isomorphism classes of holomorphic G-bundles M G,g,n over Σ g,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 D a around the marked points and over Σ g,n \ z. Therefore, P is defined by the transition holomorphic functions on D × = n a=1 (D × a ) and D × a = D a \ z a . If G(X) are the holomorphic maps from X ⊂ Σ g to G, then the isomorphism classes are defined as the double coset space (3.1) Let t a be a local coordinate in the disks D a . Then G(D) = 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(D a ) = 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 g out = Lie(G(Σ g,n \ z)) below.
Consider the one-point case z = z 0 in (3.2). Let g(t) ∈ G[[t, t −1 ] = G(D × z 0 ) be the transition function on the punctured disc D × z 0 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)) Introduce a new transition matrixg(t) = t γ g(t), where γ ∈ P ∨ is an element of the coweight lattice. It defines a new bundleẼ G . The passage from E G toẼ G is called the modification of the bundle E G at the point z 0 . The modification amounts to the passage between different sectors of the integrable module attached at z 0 (see (2.16), (2.17), (2.18)). Since t γ ∈ 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 z 0 . In general, it can have an arbitrary direction. It means that t γ may be replaced by Ad f (t γ ), where f ∈ G. As it was mentioned in Section 2.2 there is a unique modification that preserves the HWV of the integrable moduleV µ attached at z 0 .
To be aḠ-bundle over Σ g the transition matrix g should have a trivial monodromy g te 2πi = g(t) around w. If g(t) has a trivial monodromy and γ belongs to the coroot sublattice Q ∨ , theng(t) also has a trivial monodromy. Otherwise, the monodromy is an element of the center Z Ḡ . For example, let γ = j ∨ , where ∨ generate the group Z l , i.e. l ∨ ∈ Q ∨ , while j ∨ / ∈ Q ∨ for j = 0, mod(l). In this case is not a transition matrix for theḠ-bundle. But it can be considered as a transition matrix for the G ad -bundle, since G ad = G/Z. In this case the G-bundle is topologically non-trivial and ζ represents the characteristic class of E G . The characteristic class is an obstruction to lift G ad -bundle to G-bundle. It is represented by an element H 2 (Σ g , Z) [46]. Letg(t) = g j (t) = t j ∨ . Then the multiplication by g j (t) provides a passage in (2.11) from the trivial sector to the non-trivial sectors In general, we have a decomposition of the moduli space (3.1) into sectors In particular, for Σ 0,1 (CP 1 ∼ C ∪ ∞) and the marked point z 1 = 0 this representation is related to the Grothendieck description of the vector bundles over It means that any vector bundle E G over CP 1 is isomorphic to the direct sum of the line bundles . . , γ l ). If γ / ∈ Q ∨ E G then has a non-trivial characteristic class. In fact, the bundle with γ = 0 are unstable.
Two subsets M (γ 1 ) G,g,1 and M (γ 2 ) G,g,1 of the moduli space correspond to the vector bundles with the same characteristic class if γ 1 = γ 2 + β, β ∈ Q ∨ . Then the topological classification of the moduli spaces of the vector bundles by their characteristic classes follows from (3.5) Similar representation exists for the space M G,g,n .

Moduli of complex structures of curves
Let M g be the moduli space of complex structures of compact curves Σ g of genus g. The moduli space M g,n of the complex structures of curves with marked points is foliated over M g with fibers U ⊂ C n corresponding to the moving marked points. An infinitesimal deformation of the complex structures is represented by the Beltrami (−1, 1) differential µ(z,z) = µ ∂ dz ⊗ dz on Σ g,n . In this way µ is (0, 1) form on Σ taking values in T (1,0) (M g,n ) and vanishing at the marked points. The basis in the tangent space T (M g,n ) is represented by the Dolbeault cohomology group H 1 (Σ g , Γ(Σ g \ z) ⊗K), whereK is the anticanonical class.
Let us compare it with theČech like construction of T M g,n as a double coset space. As above, consider small disks D a around marked points with local coordinates ]∂ ta be vector fields on D × a while D a and Γ (Σg\ z) is a space of vector fields on Σ g \ z. The vector fields from the latter space can have poles of finite orders at the marked points. Then This construction has the following relation to the Dolbeault description. We establish cor- D × a and the Beltrami differential µ. Let On D × a∂ (ς out − ς int ) = 0 and, therefore, ς out − ς int represents a Dolbeault cocycle. The first equation has solutions that can be continued on Σ g \ z and the second -on n a=1 D a . If ς ∈ ⊕ n a=1 C[[t a , t −1 a ]∂ ta has continuations ς out and ς int then it corresponds to a trivial element of T M g,n . On the other hand,∂ς = µ globally and, therefore, µ represents an exact Dolbeault cocycle. In this way the non-trivial vector fields

Def inition of conformal blocks and coinvariants
Let us associate with Σ g,n the following set: integer k and the weights µ = (µ 1 , . . . , µ n , µ a ∈ I k ) attached to the marked points z = (z 1 , . . . , z n ). TheL(g)-module (2.4) According to (2.19) Coming back to (3.1) we define a Lie algebra g out = Lie(G(Σ g \ D) as a Lie algebra of meromorphic functions on Σ g,n with poles at z = (z 1 , . . . , z n ) taking values in g.
In this way g out acts onV [n] z, µ as This is a Lie algebra action. Due to the residue theorem this homomorphism is lifted to the diagonal central extension In what follows we need a relation ofV µ 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 g out , z, µ gout , The space of conformal blocks C V [n] z, µ is the dual space to the coinvariants. In other words, z, µ is the space of linear functionals onV [n] z, µ , invariant under g out : Put it differently, the conformal blocks are g out -invariant elements of the contragradient modulê z, µ . For a single marked point case the conformal blocks are g out invariant sections of the line bundle L µ over the affine flag variety (2.21).
According to (3.9) and (3.10) the spaceV µ has the representation In a similar way the conformal blocks are decomposed in subspaces corresponding to the characteristic classes of the bundles

Variation of the moduli space of complex structures
The space of conformal blocks C V [n] z, µ is a bundle over M g,n . This bundle is equipped with the KZB connection that can be described as follows.
A stress-tensor T (z,z) in general theories, defined on a surface Σ g,n , generates vector fields on Σ g,n . A dual object to T (z,z) is the Beltrami differential µ(z,z). It means that there is a connection on the bundle of fields over M g,n (the Friedan-Shenker connection) In conformal field theories the stress-tensor is a meromorphic projective structure on Σ g,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 ⊂ Σ g and γ a ⊂ D × a is a small contour and ς a is a vector field in D × a . Then (3.12) can be written as and the KZB equation assumes the form At the marked points T has the second order poles, while ς a ∈ C[[t a ]]∂ ta (3.8). Thereby, this integral produces ∂ za F . On the other hand, the product T F is non-singular outside the disks D a . Then for ς a ∈ Γ (Σg\ z) the integrals vanish. It means that the conformal blocks F are defined on M g,n .
Consider a one point case and let t be a local coordinate on a punctured disk D × . The stresstensor in the local coordinate has the Fourier expansion T (t) = n∈Z L n t −n−2 . The coefficients obey the Virasoro commutation relations [L n , L m ] = (n − m)L n+m + c 12 n(n 2 − 1). In the WZW model the stress-tensor is obtained from the currents by means of the Sugawara construction (see [7]). Let {t α } be a basis in g, {t β } is the dual basis, and where h ∨ is the dual Coxeter number. The Fourier coefficients of T (t) take the form The normal ordering means placing to the right t α n (t α,n ) with n > 0. The Virasoro central charge is This action is well defined because the action of the Sugawara tensor is well defined on the integrable modules. In particular, it follows from (3.16) that for the moving points equation (3.14) assumes the form The restriction of ∇ ς on C a (3.11) yields a family of the KZB equations In next section we construct these equations explicitly for the bundles over elliptic curves.

General construction
The moduli space of holomorphic bundles M G,g,n = Bun G (3.1) is foliated over the moduli space of complex structures M g,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 Bun G . For simplicity consider the one-point case. Let t a be a local coordinate in D × a , and G out = G(Σ g \ z a ). Define the quotient This space is the moduli space of G-bundles with a trivialization around z a (see (3.1)). LetV µa be an integrable module (2.14) attached to z a . Recall thatV * µa is the space of holomorphic sections Γ(L µa ) 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]. SinceV µa is the integrable representation, the group G(D × ) acts onV µa . Thereby, the subgroup G out acts onV µa also. Due to (3.19) G(D × a ) acts on M G from the right. Therefore, G(D × a ) acts on the Consider the space of the coinvariantŝ In particular, Stab x = g out for x corresponding to G out . The spaces of coinvariants are isomorphic for different choices of x. The dual space Γ(L µ )/G out is the space of conformal blocks. The quotient Γ(L µ )/G out is a space of sections of the line bundle over Bun G (3.1). It means that the space of conformal blocks is a non-Abelian generalization of the theta line bundles over the Jacobians.

The form of connection
For conformal blocks we have (see (3.20)) . A local version of (3.20) is defined by the operator where t α is a generator of g and u α is a coordinate of the tangent vector to Bun G . The action of ∇ uα on the conformal blocks is well defined because the conformal blocks are g out -invariant. Therefore, they are horizontal with respect to this connection

Moduli space of holomorphic G-bundles over elliptic curves
For G = GL N 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 Σ τ using an approach of [46]. Let G be a complex simple Lie group. An universal coverḠ of G in all cases apart G 2 , F 4 and E 8 has a non-trivial center Z(Ḡ). The adjoint group is the quotient G ad =Ḡ/Z(Ḡ). For the cases A n−1 (when n = pl is non-prime) and D n the center Z(Ḡ) has non-trivial subgroups Z l ∼ µ l = Z/lZ. Assume that (p, l) are co-prime. There exists the quotient-groups where Z(G l ) is the center of G l and Z(G l ) ∼ µ p = Z(Ḡ)/Z l . Following [56] we define a G-bundle E G = P × G V by the transition operators Q and Λ j acting on the sections of s ∈ Γ(E G ) as where Q(z) and Λ(z) take values in End(V ). Going around the basic cycles of Σ τ we come to the equation It follows from [56] that it is possible to choose the constant transition operators. Then we come to the equation Replace (4.4) by the equation where ζ is a generator of the center Z Ḡ . In this case (Q, Λ) are the clutching operators for G adbundles, but not forḠ-bundles, and ζ plays the role of obstruction to lift the G ad -bundle to thē G-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 If j and N are co-prime numbers then ζ j generates Z Ḡ . In this case Q and Λ j can serve as transition operators only for a G ad =Ḡ/Z-bundle, but not forḠ-bundle and ζ j is an obstruction to lift G ad -bundle toḠ-bundle.
The element ζ has a cohomological interpretation. It is called the characteristic class of E G . It can be identified with elements of the group H 2 (Σ g,n , Z Ḡ ). This group classifies the of the characteristic classes of the bundles [46]. Consider, as above, a non-prime N = pl and put j = p. Then ζ j is a generator of the group Z l . In this case Q and Λ j are transition operators for G l =Ḡ/Z l -bundles (see (4.1)) and ζ j is an obstruction to lift a G l -bundle to aḠ-bundle.
The moduli space of stable holomorphic over Σ τ with the sections (4.5) is defined as 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 Assume that Q is a semi-simple element and Q ∈ HḠ is a fixed Cartan subgroup ofḠ. It means that we consider an open subset In this case the solutions of (4.4) have the form [46] where ρ ∨ is a half-sum of positive coroots, h is the Coxeter number, Λ 0 is an element of the Weyl group defined by ζ j : The element Λ 0 preserves the extended system of simple roots Π ext = Π ∪ (α 0 ), where −α 0 is a maximal root [46, Proposition 3.1]. In this way Λ 0 is a symmetry of the extended Dynkin diagram of g = Lie Ḡ , generated by ∨ [12]. LetH 0 ⊂ HḠ be the Cartan subgroup commuting with Λ 0 . To describe V j consider the adjoint action λ = Ad(Λ 0 ) on the Cartan subalgebra h = Lie(HḠ). Leth 0 = Lie(H 0 ) be the invariant subalgebra (λ(h 0 ) =h 0 ). Then V j = exp(2πıu)(u ∈h 0 ) is an arbitrary element fromH 0 defining the moduli space M (j) 0 (G). There exists a basisΠ ∨ j inh 0 such thatΠ is a system of simple roots for a simple Lie subalgebrã g 0 ⊂ g. For the list of these subalgebras see [46]. If j = N , we come to the trivial bundles (4.4). In this case Λ 0 = Id,h 0 = h andg 0 = g.
LetQ ∨ andP ∨ be the coroot and the coweight lattices inh 0 , andW is the Weyl group corresponding toΠ. Define the Bernstein-Schwarzman type groups [10,11]. They are constructed by means of the latticesQ ∨ orP ∨ . In the first case it is the semidirect products Then the moduli space of non-trivialḠ-bundles with the characteristic class ζ j is the fundamental domain inh is the moduli space of non-trivialḠ-bundles. Consider G ad -bundles. Define the semidirect product A fundamental domain of this group inh is the moduli space of the non-trivial G ad -bundles. It is the moduli space of E G ad -bundles with characteristic class defined by ζ j . In other words u ∈ C sc j for EḠ-bundles, C ad j for E G ad -bundles.

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 g out = Lie(G(Σ τ,n \ z)) is a Lie algebra of meromorphic functions on Σ τ,n with poles at z = (z 1 , . . . , z n ) 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 g out :

Consider the quasi-periodicity conditions (4.2). The GS-basis is diagonal under Ad
Then g out has the correct quasi-periodicities and has poles of orders K(a, m), K(a, α) at z a , a = 1, . . . , n. In this last expression (due to the residue theorem) from (B.14) we assume that n a=1 x 0 α,1,a = 0. (4.17) Let us unify the last two expression (4.15) and (4.16) in a single formula, We will act on the coinvariants by g out . In what follows we need the limit z → z a of these expressions. Notice that g out is the filtered Lie algebra. The filtration is defined by the orders of poles. The behavior of g out is defined by the asymptotics (B.7)-(B.9), (B.12). As it will become clear below we need the least singular terms in g out . In this way we take m = 0 in (4.14), (4.18) and m = 1 (E 1 (z − z a )) in (4.18): Here "· · · " means the terms of order o z − z a −1 and o(1). For g int = Lie(G(U D )) we have local expansions in neighborhoods of the marked points Define the Lie algebra with the loose condition (4.17)) g out = g out with n a=1 x 0 α,1,a ∈ C and let n − = α∈R + g −α . Then the Lie algebra Lie(G(D × )) has the form (compare with the 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 G out = G(Σ 1,n \ z) from the left and by G int = n a=1 G(D a ) from the right absorbs almost all negative and positive modes of G(D × ) except the two types of modes describing the moduli space: • The vector u = n a=1 α∈Π x 0 α,1,a h α ∈h 0 . It defines an element of the moduli space M G,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.

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 for the generators of the loop algebra Consider the integrable modules attached to the marked pointsV 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): (4.23) • The vector field corresponding to the deformation of the moduli τ of the elliptic curve Σ τ,n : This action follows from (3.13) and the operator algebra • The invariance with respect to the action of g out (3.22): where H 0 α are the Cartan generators (A.5). Notice that this operator is well defined on M G (3.19).
The vector field (4.24) is defined on the universal curve H×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−z τ −τ ∂ z . The invariance with respect to Lie(G out ) (4.14), (4.18) means that Now using (4.19), (4.20) and (4.26) we write down the annihilation condition g out F = 0 in in the basis t k,c α (m) = 1 ⊗ · · · ⊗ 1 ⊗ t k,c α (m) ⊗ 1 ⊗ · · · ⊗ 1 (on the c-th place): for α ∈ R, ∀ k and α ∈Π, k = 0 correspondingly. In the same way (4.24) and (4.25) assume the form Now we are ready to evaluate the Virasoro generators, i.e. to express them in terms of zero modes of the loop algebra t k,c α (0) ≡ t k,c α only. As we have found above the positive modes of the loop algebra act on F by zero t k,a α (m)F = 0, m ∈ Z + . Therefore, from (4.22) we have In order to find L a −1 one need to substitute t k,a α (−1), H k,a α (−1) from (4.27) and H 0,a α (−1) from (4.28) into (4.29) 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 [t q,a α (0), t −q,a −α (0)] = pα √ l exp −2πi q l h 0,a α [46]. Therefore, The term This is the first set of equations in (4.41). In order to obtain the second one (the KZB connection ∇ τ along τ ) one should use (4.30). It is needed to compute L a −2 . The later arises from the local expansion of (B.4) for k = 1. Then the following identities should be used where f (u, z) = ∂ u φ(u, z) for t(−2)t(0)-terms and 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.

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-matrices was constructed in [17,18]. The latter classification includes our list though it was derived from different postulates.

Axiomatic description of r-matrices
The classical dynamical r-matrix is a meromorphic one form r = r(u, z)dz, (u ∈h 0 ) on C taking values in g ⊗ g that satisfies the following conditions: 1. r(z) has a pole at z = 0 and where t k α , H k α , h −k α are generators of the GS basis in g (see Appendix A). If V is a g-module, then C 2 acts by the permutation on V ⊗ V .
2. Behavior under the shifts by the generators of the lattice Z ⊕ τ Z: where the Ad-action is taken with respect to the first factor in g ⊗ g.
3. The classical dynamical Yang-Baxter equation (CDYBE). It follows from 1 that r(z) can be represented as Then r(z) is a solution of CDYBE: where ∂ 1 is the differentiation with respect to the first argument.
Proof . It follows from the properties of the functions ϕ k α (u, z), ϕ k 0 (u, z) described in the Appendix 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 A αβ is antisymmetric.
Next we wish to prove that locally A αβ = −l(∂ uα (f )f −1 ) β for some f ∈H 0 . The twisted r-matrix must satisfy the CDYB equation. Plugging r + δr into (4.32) we see that the "commutator" part vanishes identically since [r ab , δr ac ] + [r ab , δr bc ] ≡ 0 due to The "derivative" part of (4.32) yields ∂ uα A βγ + ∂ uγ A αβ + ∂ uβ A γα = 0 or The term δr is called the dynamical twist of the r-matrix. The statement follows from the Poincaré lemma.

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 =a r ac , (4.37) where t k,a α = 1 ⊗ · · · ⊗ 1 ⊗ t k α ⊗ 1 ⊗ · · · ⊗ 1 (with t k α on the a-th place) and similarly for the generators H k,a α and h k,a α . 3 The following short notations are used herê From the definition it follows that r ac = −r ca and f ac = f ca . Following (B.6) and (B.7) we put and, therefore where C c 2 is the Casimir operator acting on the c-th component. Recall that we study the following system of differential equations ∇ a F = 0, a = 1, . . . , n, ∇ τ F = 0. It is important to mention that the solutions of (4.41) F are assumed to satisfy the following condition 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) ofḠ, and consider a quotient group G =Ḡ/Z. Assume for simplicity that Z Ḡ is cyclic. The case Spin(4n) where Z(G) = µ 2 × µ 2 can be treated similarly.
Let us take an element ζ ∈ Z(Ḡ) of order l, generating Z. It defines uniquely an element Λ 0 from the Weyl group W (see [12,46]). It is a symmetry of the corresponding extended Dynkin diagram and (Λ 0 ) l = Id. Λ 0 generates a cyclic group µ l = Λ 0 , (Λ 0 ) 2 , . . . , (Λ 0 ) l = 1 isomorphic to a subgroup of Z(Ḡ). Note that l is a divisor of ord(Z(Ḡ)). Consider the action of Λ 0 on g. Since (Λ 0 ) l = Id we have a l-periodic gradation where g 0 is a subalgebra g 0 ⊂ g and the subspaces g a are its representations. Since Q and Λ commute in the adjoint representations the root subspaces g a are their common eigenspaces. GS-basis. Here we shortly reproduce the construction of the GS-basis following [46]. Since Λ 0 ∈ W it preserves the root system R. Define the quotient set T l = R/µ l . Then R is represented as a union of µ l -orbits R = ∪ T l O. We denote by O(β) an orbit starting from the root β O(β) = β, λ(β), . . . , λ l−1 (β) ,β ∈ T l .
The number of elements in an orbit O (the length of O) is l/p α = l α , where p α is a divisor of l. Let ν α be a number of orbits Oᾱ of the length l α . Then R = ν α l α . Notice that if O(β) has length l β (l β = 1), then the elements λ k β and λ k+l β β coincide. First, transform the root basis Let E α (α ∈ R) be the root basis of g. "The Fourier transform" of the root basis on the orbit O(β) is defined as Almost the same construction exists in H. Again let Λ 0 generates the group µ l . Since Λ 0 preserves the extended Dynkin diagram, its action preserves the extended coroot system Π ∨ext = Π ∨ ∪ α ∨ 0 in H. Consider the quotient K l = Π ∨ext /µ l . Define an orbit H(ᾱ) of length l α = l/p α in Π ∨ext passing through H α ∈ Π ∨ext H(ᾱ) = H α , H λ(α) , . . . , H λ l−1 (α) ,ᾱ ∈ K l = Π ∨ext /µ l .
The set Π ∨ext is a union of H(ᾱ): Define "the Fourier transform" The basis h c α (c ∈ J α ,ᾱ ∈ K l ) is over-complete in H. Namely, let H(ᾱ 0 ) be an orbit passing through the minimal coroot H α 0 , H λ(α 0 ) , . . . , H λ l−1 (α 0 ) . Then the element h 0 α 0 is a linear combination of elements h 0 −ᾱ , (α ∈ Π) and we should exclude it from the basis. We replace the basis Π ∨ in H by As before there is a one-to-one map Π ∨ ↔ {h c α }. The elements (h ā α , t ā α ) form GS basis in g (l−a) (A.1). The dual basis is generated by elements H ā and a α,β is the Cartan matrix of g.
The λ-invariant subalgebra g 0 contains the subspace Then g 0 is a sum ofg 0 and V In the invariant simple algebrag 0 instead of the basis (h 0 α , t 0 β ) we can use the Chevalley basis and incorporate it in the GS-basis whereΠ is a system of simple roots constructed by the averaging of the λ action on Π ext , andR is a system of roots ofg 0 generated byΠ. We have the following action of the adjoint operators on the GS basis: In addition, Ad Q t c β = e( κ, β )t c β , Ad Q (Eα) = e κ,α Eα.
In particular, Commutation relations in the GS basis:
The following identities are also used here and for the functions (4.38) this identity takes the form 2πi∂ τ ϕ m α (z) = ∂ z f k α (z).

C Proofs of Propositions 1 and 2
Proof of Proposition 1.