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Quantum Knizhnik-Zamolodchikov equations and holomorphic vector bundles1

arXiv:hep-th/9303066v1 10 Mar 1993Quantum Knizhnik-Zamolodchikov equations and holomorphic vector bundles1Pavel I. EtingofYale UniversityDepartment of Mathematics2155 Yale StationNew Haven, CT 06520 USAe-mail etingof@ pascal.math.yale.eduJuly 1992IntroductionIn 1984 Knizhnik and Zamolodchikov [KZ] studied the matrix elements of inter-twining operators between certain representations of affine Lie algebras and foundthat they satisfy a holonomic system of differential equations which are now calledthe Knizhnik-Zamolodchikov (KZ) equations. It turned out that the KZ equations(and hence, representation theory of affine Lie algebras) are a source of a richvariety of special functions.

The Gauss hypergeometric function and its variousgeneralizations were obtained as solutions of the KZ system.In the recent paper [FR], Frenkel and Reshetikhin considered intertwining oper-ators between representations of quantum affine algebras. It was shown that theirmatrix elements satisfy a system of holonomic difference equations – a naturalquantum analogue of the KZ system.

It was also shown that the solutions of thisdifference system are very nontrivial special functions that generalize basic hyper-geometric series. In particular, one of the simplest solutions is expressed in terms ofthe q-hypergeometric function which was introduced by Heine in the 19th century.This is consistent with the general idea that all reasonable special functions shouldcome from representation theory, as matrix elements of certain representations.The classical KZ system can be regarded as a local system – a flat structure inthe trivial vector bundle on the space of sets of N distinct points on the complexplane.

This fact enabled Schechtman and Varchenko to apply geometric methodsto the study of the KZ equations and obtain a complete solution for a generalsimple Lie algebra[SV]. This interpretation can be extended to the quantum case[M;R] with the help of a specially designed discrete analogue of the notion of a localsystem.

Other than that, the geometric meaning of the quantum KZ equations hasso far remained unclear.The goal of this paper is to introduce a new geometric interpretation of the quan-tum KZ equations. In Chapters 1 and 2, they are linked to certain holomorphicvector bundles on a product of N equivalent elliptic curves, naturally constructedby a gluing procedure from a system of trigonometric quantum R-matrices.

Mero-morphic solutions of the quantum KZ equations are interpreted as sections of sucha bundle. This interpretation is an analogue of the interpretation of solutions of theclassical KZ equations as sections of a flat vector bundle.

It yields a non-technicalproof of consistency of the quantum KZ system.In Chapter 3 it is shown that the matrix elements of intertwiners between rep-resentations of the quantum affine algebra Uq(csl2) correspond to regular (holomor-phic) sections.In Chapter 4, Birkhoff’s connection matrices for the quantum KZ equations areinterpreted as transition matrices from one fundamental system of holomorphicsections to another. They can be used to give an alternative construction of theholomorphic vector bundle corresponding to the quantum KZ system.In Chapter 5, the topological structure of the vector bundle associated to the1TiD k M th Jl J1993

2quantum KZ equations is studied in the special case of the quantum affine algebraUq(csl2). The Chern classes of this bundle are evaluated, and it is shown that theyuniquely determine its topology.

The main tool of this evaluation is the crystallimit q →0, and its result shows that the topology of the quantum KZ equationsencodes the structure of the crystal bases in representations of the quantum groupUq(sl2).Chapter 6 treats the special case N = 2. In this case, one essentially deals witha vector bundle over an elliptic curve.

This bundle is shown to be semistable (forthe case of Uq(csl2)) if the parameters take generic values. The proof makes use ofthe crystal limit q →0.In Chapter 7, we give a vector bundle interpretation of the generalized quantumKZ equations for arbitrary affine root systems defined recently by Cherednik [Ch].For the sake of brevity, the results regarding quantum groups are formulated andproved for Uq(sl2).

Mostof them can be suitably extended to Uq(g) where g is anarbitrary simple Lie algebra.AcknowledgementsThis paper was inspired by the course on the classical and quantum Knizhnik-Zamolodchikov equations given by my adviser Igor Frenkel at Yale in the spring of1992. It is a pleasure for me to thank Professor Frenkel for helping me build up anecessary background and guiding my work on this paper.I would also like to thank Professors David Kazhdan, Nikolai Reshetikhin, andAlexandre Varchenko for useful discussions.

31. Trigonometric R-matrices and holomorphic vector bundles.Let V1, ..., VN be a collection of finite-dimensional vector spaces.

Let W = V1 ⊗V2 ⊗· · · ⊗VN.Let R ∈End(Vi⊗Vj), R = Pn an⊗bn, an ∈End(Vi), bn ∈End(Vj). Throughoutthe paper we will use the same notation R for the operator Pn 1U1 ⊗an⊗1U2 ⊗bn⊗1U3 in End(U1 ⊗Vi ⊗U2 ⊗Vj ⊗U3) and the operator Pn 1U1 ⊗bn ⊗1U2 ⊗an ⊗1U3in End(U1 ⊗Vj ⊗U2 ⊗Vi ⊗U3), where U1, U2, U3 are arbitrary vector spaces and1U ∈End(U) is the identity map.Definition 1.1.

A system of trigonometric quantum R-matrices on V1, ..., VN is acollection of rational functions {Rij(z)} with values in End(Vi⊗Vj) for 1 ≤i, j ≤N,i ̸= j such that(i) Rij(z) satisfy the quantum Yang-Baxter equation:(1.1)Rijz1z2Rikz1z3Rjkz2z3= Rjkz2z3Rikz1z3Rijz1z2in End(W), 1 ≤i, j, k ≤N;(ii) Rij(z) satisfy the unitarity condition(1.2)Rij(z) = Rji(z−1)−1in End(W), 1 ≤i, j ≤N;(iii) Rij(z) have no poles on the unit circle.Definition 1.2. A collection of linear transformations Ai ∈GL(W), 1 ≤i ≤N,is compatible to the system of matrices {Rij(z)} ifAiRjk(z) = Rjk(z)Ai, i ̸= j, k,AiAjRij(z) = Rij(z)AjAi,1 ≤i, j ≤N(1.3)in End(W).Let p ∈C∗, |p| < 1, and let Π be the subgroup of C∗generated by p. Consider thecomplex torus T = C∗/Π.

We will realize T as an annulus {z ∈C∗| |p| ≤|z| < 1}with identified boundaries. Denote by T N the direct product of N copies of T.It turns out that to any system of trigonometric quantum R-matrices on V1, ..., VNand any collection of linear transformations compatible to this system one can nat-urally associate a holomorphic vector bundle on T N with fiber W.Partition T N into N!

chambers Ds = {(z1, ..., zN) | |p| < |zs(N)| < · · · < |zs(1)| <1}, where s is a permutation from the symmetric group SN. These sets are openand disjoint, and their closures cover T N. Also, they are permuted by the naturalaction of SN on T N: s1(Ds2) = Ds2s1, where s(z1, ..., zN) = (zs(1), ..., zs(N)).We say that two chambers Ds1 and Ds2 are adjacent if ∂Ds1 ∩∂Ds2 is a hyper-surface of real codimension 1 in T N. The following trivial lemma classifies pairs ofadjacent chambers.Lemma 1.1.

Chambers Ds1 and Ds2 are adjacent if and only if s−11 s2 is eithera transposition of adjacent elements tj = (j, j + 1), 1 ≤j ≤N −1, or a cyclicpermutation c±1, c = (N, N −1, ..., 2, 1).The entire torus can be obtained by gluing the chambers together along theboundaries between themTherefore in order to define a vector bundle on T N

4it suffices to proclaim it trivial over each of the chambers and prescribe transitionmatrices on the boundaries between adjacent chambers. These boundaries areΓs,j = {(z1, ..., zN) | |p| < |zs(N)| < · · · < |zs(j+1)| = |zs(j)| < · · · < |zs(1)| < 1}andΣs = {(z1, ..., zN) | |p| = |zs(N)| < · · · < |zs(1)| < 1}.Note that Γs,j = Γstj,j.Now let {Rij(z)} be any system of rational functions with values in End(Vi ⊗Vj) satisfying condition (iii), and let {Ai} be a system of linear transformationscompatible to {Rij(z)}.On Γs,j, define the transition matrices from Ds to Dstj to be(1.4)Ts,j(z1, ..., zN) = Rs(j+1)s(j)zs(j+1)zs(j)On Σs, set the transition matrices from Ds to Dsc to be(1.5)Qs = As(N)These transition matrices take values in End(W).Theorem 1.2.

Transition matrices (4) and (5) define a holomorphic vector bundleon T N with fiber W if and only if {Rij(z)} is a system of trigonometric R-matricesand {Ai} are compatible to {Rij(z)}.Remark 1.1. This fact can be considered as a geometric interpretation of trigono-metric solutions to the quantum Yang-Baxter equation with a complex parameter.Proof.

First of all, the regularity of {Rij(z)} on the unit circle is equivalent to thesmoothness of the transition functions on the boundaries between chambers. Thus,it remains to show that the consistency conditions on transition matrices reduce torelations (1.1), (1.2) and (1.3).The consistency conditions apply at every point P ∈T N whose arbitrarily smallneighborhood intersects more than two chambers.

Let U be a small enough neigh-borhood of P. Consider the graph G(P) whose vertices are connected componentsof intersections of chambers with U.2Two vertices are connected with an edgeif the corresponding two components are adjacent inside U. Thus every edge islabeled with a transition matrix – a holomorphic End(W)-valued function in U.Hence, a holomorphic in U transition matrix can be associated to every orientedpath in the graph G(P).

This matrix is defined to be the product of the transitionmatrices labeling the edges of this path in the opposite order to that prescribedby the orientation. The consistency conditions demand that any two paths withthe same beginning and end be labeled with the same transition matrix (this isequivalent to the uniqueness of analytic continuation of sections of the bundle inU).Let P = (z1, ..., zN).

First assume that |zj| ̸= |p| for all j. Let s be a permutationlabeling one of the chambers intersecting with U.

Then there exists an increasingsequence 1 ≤j1 < j2 < ... < jr = N such that(1.6)|zs(1)| = · · · = |zs(j1)| > |zs(j1+1)| = · · · = |zs(j2| > · · · > |zs(jr−1+1)| = · · · = |zs(jr)|.2Note that although the chambers are connected, their intersections with U may consist oflt

5Obviously, vertices of G(P) correspond to permutations of the form sσ, σ ∈S(P),where S(P) = Sj1 × Sj2−j1 × · · · × Sjr−jr−1 ⊂SN. Denote such a vertex by v(σ).It is clear that v(σ1) and v(σ2) are connected with an edge if and only if σ1 = σ2tj,j ̸= j1, ..., jr−1.

It follows that a path in G(P) is a representation of an elementσ ∈S(P) as a product of the generators tj. It is known from the theory of thesymmetric group that any two such representations can be identified with the helpof relationstjtj+1tj = tj+1tjtj+1,(1.7)t2j = 1,(1.8)titj = tjti,|i −j| > 1.

(1.9)Applying this result to the transition matrices, we find that the consistency con-ditions at P are equivalent to identities (1.1) and (1.2) for the matrices {Rij(z)}.Indeed, it is easy to see that relation (1.7) reduces to the quantum Yang-Baxterequation, relation (1.8) yields the unitarity, and relation (1.9) trivially follows fromthe definition of transition matrices.Now assume that (1.6) holds and |zs(j)| = |p| for jr−1 < j ≤jr. Set k = jr−jr−1.The graph G(P) can then be represented as a union G(P) = G0(P) ∪G1(P) ∪· · ·∪Gk(P), where Gl(P) corresponds to the points in the vicinity of U with |zs(j)| closeto |p| for jr−1 < j ≤jr −l and close to 1 for jr −l < j ≤jr.

Vertices of Gl(P)are labeled by permutations from S(P) as follows: vl(σ) ∈Gl(P) corresponds tothe connected component of Dsσcl. Two vertices vl(σ1) and vl(σ2) in Gl(P) areconnected with an edge if and only if σ1cl = σ2cltj, j ̸= l, j1 + l, ..., jr−1 + l. Also,two vertices vl(σ1) ∈Gl(P) and vm(σ2) ∈Gm(P), l ̸= m, are connected if and onlyif |l −m| = 1 and σ1 = σ2.

This completely describes the structure of the graphG(P).Let k ≥1.For l ≤k −1 consider the paths π1(σ, l) : vl(σ) →vl(σtj) →vl+1(σtj) and π2(σ, l) : vl(σ) →vl+1(σ) →vl+1(σtj), j ̸= j1, ..., jr−1, jr −l, jr −l −1. These paths have a common beginning and ending, and the fact that theygive the same transition matrix is expressed by the relation Rσ(j+1)σ(j)Aσ(N−l) =Aσ(N−l)Rσ(j+1)σ(j) which is the first part of (1.3).Now let k ≥2, and for l ≤k −2 consider the pathsπ1(σ, l) : vl(σ) →vl(σtN−l−1) →vl+1(σtN−l−1) →vl+2(σtN−l−1) andπ2(σ, l) : vl(σ) →vl+1(σ) →vl+2(σ) →vl+2(σtN−l−1),with the same beginning and ending.

The fact that they should give the sametransition matrix is expressed by the relationRσ(N−l)σ(N−l−1)Aσ(N−l−1)Aσ(N−l) = Aσ(N−l)Aσ(N−l−1)Rσ(N−l)σ(N−l−1),which is the second part of (1.3).An elementary combinatorial argument shows that any two paths in the graphG(P) with the same beginning and ending can be identified with each other byreplacing subpaths of the form π1 by π2, π1 by π2, and vice versa, and using relations(1.7)–(1.9) when necessary. This means that once (1.1)–(1.3) are satisfied, any twosuch paths are forced to have the same transition matrix.

Thus, we have shownthat the consistency conditions on transition matrices are equivalent to relations(1.1),(1.2), and (1.3).■From now on we assume that the matrices {Rij(z)} and {Ai} are fixed anddenote the vector bundle constructed above by EN.Let ∆= (∆1, ..., ∆N) be a set of complex numbers. Introduce a new holomorphicvector bundle E (∆) obtained from Eby twistingFor δ ∈C define a line

6bundle Lδ on T whose meromorphic sections are functions ψ(z) meromorphic inC∗such that ψ(pz) = pδψ(z). Clearly, the isomorphism class of Lδ is determinedby δ mod(1, 2π√−1/ log p), or, eqivalently, by the image of pδ ∈C∗in T.Letθj : T N →T, 1 ≤j ≤N be the projection: θj(z1, ..., zN) = zj, and let Bj(∆j) =θ∗j (L∆j).

Bj(∆j) are topologically trivial holomorphic line bundles on T N. LetB(∆) = B1(∆1) ⊗B2(∆2) ⊗· · · ⊗BN(∆N). Now for any bundle E over T N definea new bundle E(∆) by(1.10)E(∆) = E ⊗B(∆).In the next section we will identify solutions of the quantum KZ equations withsections of EN(∆).2.

Quantum Knizhnik-Zamolodchikovequations and meromorphic sections.Let C = {(z1, ..., zN) ∈T N : 1 > |zj| > |p|, 1 ≤j ≤N}. Let φ(z1, ..., zN) be ameromorphic function in C∗N with values in W. Set ψ(z1, ..., zN) = z∆11.

. .

z∆NN φ(z1, ..., zN).To the function ψ we can associate a meromorphic section of EN(∆) over theopen set C as follows.A meromorphic section of EN(∆) over C is a collection of functions ψs(z1, ..., zN) =z∆11. .

.z∆NN φs(z1, ..., zN), s ∈SN, where φs is a W-valued meromorphic functionon Dcs ∩C (Dcs denotes the closure of the chamber Ds), satisfying the consistencyconditions(2.1)φstj = Ts,jφs on Γs,j.Let w be the element of maximal length in SN (w(i) = N −i + 1). To constructa section of EN(∆) from ψ, set φw = φ, and then extend it to the whole set C byapplying rule (2.1).

That means, if s = tj1 . .

.tjn and sl = tj1 . .

. tjl, 1 ≤l ≤n,then set(2.2)φws = Twsn−1,jnTwsn−2,jn−1 .

. .Tws1,j2Tw,j1φin Dws.

It follows from relations (1.1) and (1.2) that the result of this extensiondoes not depend on the decomposition of s in the product of tj. Clearly, functions(2.2) satisfy conditions (2.1).

Thus, we have constructed a section of the bundleEN(∆) over C. Denote this section by ˜ψ.Now let us define the quantum KZ equations – the main subject of this paper.Definition 2.1. The difference equations on a W-valued function ψψ(z1, ..., pzj, ..., zN) = Rj,j−1 zjzj−1pRj,j−2 zjzj−2p.

. .

Rj,1zjz1p×(2.3)AjRj,N zjzNRj,N−1 zjzN−1. .

. Rj,j+1 zjzj+1ψ(z1, ..., zj, ..., zN),1 ≤j ≤N,are called the quantum Knizhnik-Zamolodchikov equations.The following theorem gives a new geometric interpretation of the quantum KZequations

7Theorem 2.1. The section ˜ψ of the bundle EN(∆) over C extends to a globalmeromorphic section of this bundle if and only if the function ψ(z1, ..., zN) satisfiesthe quantum KZ equations.Proof.

In order for ˜ψ to extend, it is necessary and sufficient that ψs defined abovesatisfy the additional consistency conditions on the surfaces Σs. Obviously, it isenough to require consistency only for s = wtjtj+1 .

. .

tN−1, 1 ≤j ≤N. In thiscase, the conditions are(2.4)φsc(z1, ..., pzj, ..., zN) = Ajφs(z1, ..., zj, ..., zN).Using the decomposition c = tN−1tN−2 .

. .

t1 and equations (2.2) and (1.4), weobtain from (2.4)R1,j z1zjp. .

. Rj−2,jzj−2zjpRj−1,jzj−1zjpψ(z1, ..., pzj, ..., zN) =(2.5)AjRj,N zjzNRj,N−1 zjzN−1.

. .

Rj,j+1 zjzj+1ψ(z1, ..., zj, ..., zN),which is equivalent to (2.3).■Corollary 2.2. The quantum KZ equations are consistent.Proof.

Any holomorphic vector bundle on a compact complex manifold has nonzeromeromorphic sections. Therefore, equations (2.3) have nonzero solutions, whichimplies that they are consistent.■Definition 2.2.

Let us say that a system of trigonometric R-matrices {Rij(z)} isregular if for every pair i, j such that i ̸= j(i) Rij(z) is regular and nondegenerate at the origin and infinity;(ii) Rij(z) is regular outside the unit circle and R−1ij (z) is regular inside the unitcircle.From now on we assume that {Rij(z)} is regular and use the notationMj(z1, ..., zN) =Rj,j−1 zjzj−1p. .

. Rj,1zjz1pAjRj,N zjzN.

. .Rj,j+1 zjzj+1(2.6)Fix s ∈SN.

Let(2.7)M sj =limzs(i)/zs(i+1)→∞,1≤i≤N−1 Mj(z1, ..., zN).It follows from the consistensy of the quantum KZ equations that(2.8)[M si , M sj ] = 0for any pair i j

8Theorem 2.3. [FR](i) There exists a matrix solution of the quantum KZ equations of the form(2.9)Ls(z1, ..., zN) = zlog Ms1log p1.

. .zlog MsNlog pNFs(z1, ..., zN),such that Fs is an End(W)-valued meromorphic function in C∗N regular in theregion |zs(1)| > |zs(2)| > ... > |zs(N)| with limzs(i)/zs(i+1)→∞,1≤i≤N−1 Fs = 1W (thisfunction will be homogeneous of degree 0).

(ii) Any vector solution ψ of the quantum KZ equations regular in the region|zs(1)| > |zs(2)| > ... > |zs(N)| has the form Lsu where u ∈W.Idea of proof. The solution Ls is given by the following limit:Ls(z1, ..., zN) =limks(j1)−ks(j2)→∞,j1>j2NYj=1kj−1Yi=0M −1j(z1pk1, ..., zj−1pkj−1, zjpi, zj+1, ..., zN)NYj=1(M sj )kj.

(2.10)The existence of this limit follows from the results of [Ao].Theorem 2.4. Let φ and ψ be as above.

If {Rij(z)} is regular and φ(z1, ..., zN) isholomorphic in the region 1 ≥|zs(1)| ≥|zs(2)| ≥· · · ≥|zs(N)| ≥|p|, s ∈SN, then ˜ψis a holomorphic section of the bundle EN(∆).Proof. We will assume that s = id.

For an arbirary permutation s, the proof issimilar.First of all, let us show that φ(z1, ..., zN) is holomorphic in the annulus A ={(z1, ..., zN) ∈CN : |p| ≤|z1|, ..., |zN| ≤1}. Let J be an integer such that all Rij(z)are holomorphic whenever |z| ≤|p|J.

We will use the notation ˆzj = pJ(j−1)zj. Sinceψ satisfies the quantum KZ equations, we have(2.11)φ(z1, ..., zN) = constNYj=2(j−1)J−1Yi=0M −1j(z1, ..., zj−1, pizj, ˆzj+1, ..., ˆzN)φ(ˆz1, ..., ˆzN)Since {Rij(z)} is regular, M −1j(z1, ..., zN) is holomorphic if zjp ≤zi for i < j andzj ≤zi for i > j.

This implies that all the factors M −1jin (2.11) are holomorphicin z1, ..., zN in A. The function φ(ˆz1, ..., ˆzN) is also holomorphic in the annulus,since |ˆz1| < · · · < |ˆzN| whenever (z1, ..., zN) ∈A.

This shows that φ(z1, ..., zN) isholomorphic in the annulus.Now we are in a position to prove the holomorphicity of the section ˜ψ. Since φis holomorphic in A, clearly ˜ψ is holomorphic in Dw.

Let us analytically continue˜ψ into the region Dws, s ∈SN, along some path. Let s = tm1 .

. .

tml be a minimallength decomposition of s. Then over Dws ˜ψ is represented by a function of theform(2.12)ψws(z1, ..., zN) = Riljlzilzjl. .

. Ri1j1zi1zj1ψ(z1, ..., zN).The key property of this decomposition is that |zim| is always greater than |zjm|in DIt follows from the fact that the decomposition of s we used had minimal

9length. This property and the regularity of the R-matrices imply that ψws is aholomorphic (multivalued) function in Dws which proves that ˜ψ is a holomorphicsection of EN(∆) .■Let s ∈SN be a permutation.

Let d = dimW, and let us1, ..., usd be the basisof W such that M sj ui = ∆(s,i)jui, 1 ≤j ≤N, 1 ≤i ≤d (we assume the genericsituation when such a basis exists). Let(2.13)ψ(s,i) = Lsui, and let ˜ψ(s,i) be the corresponding holomorphic sections of EN(∆(s,i)).The following proposition is a corollary of formula (2.10).Proposition 2.5.

The sections ˜ψ(s,i)(z1, ..., zN) form a basis of the fiber W every-where except points wherezs(j1)zs(j2) pn is a pole of Rs(j1)s(j2)(z) for a suitable n ∈Zand pair of indices j1, j2 such that j1 > j2. At such points, ˜ψ(s,i)(z1, ..., zN) arelinearly dependent.Remark 2.1.

We can legitimately talk about linear dependence or independenceof ˜ψ(s,i) despite they are sections of different bundles, because the projectivizationsof all these bundles are isomorphic to each other.Remark 2.2. Proposition 2.5 shows that ˜ψ(s,i) play the role of a fundamentalsystem of sections: they give a coordinate frame of the fiber at all points of the baseexcept those lying on a finite set of hypersurfaces in T N of complex codimension 1.3.

Matrix elements of intertwining operatorsfor Uq(csl2) and holomorphic sections.The quantum affine algebra Uq(csl2) is a Hopf algebra obtained by a standardq-deformation of the universal enveloping algebra of the Kac-Moody algebra csl2[D;J1]. As an associative algebra, it is generated by elements ei, fi, K±1i, i = 0, 1which satisfy the following relations:eifi −fiei = Ki −K−1iq −q−1 ,eifj −fjei = 0, i ̸= j,KieiK−1i= q2ei, KiejK−1i= q−2ej, i ̸= jKifiK−1i= q−2fi, KifjK−1i= q2fj, i ̸= jKiKj −KjKi = 0,e3i ej −q3 −q−3q −q−1 e2i ejei + q3 −q−3q −q−1 eieje2i −eje3i = 0, i ̸= j,f 3i fj −q3 −q−3q −q−1 f 2i fjfi + q3 −q−3q −q−1 fifjf 2i −fjf 3i = 0, i ̸= j.

(3.1)The comultiplication in Uq(csl2) is defined by∆(Ki) = Ki ⊗Ki,∆(ei) = ei ⊗Ki + 1 ⊗ei,∆(f )f ⊗1 + K−1 ⊗f(3 2)

10and the antipode acts according to(3.3)S(ei) = −eiK−1i,S(fi) = −Kifi,S(Ki) = K−1i.Here q is a complex number. We will assume that q is not 0 and not a root of unity.The algebra Uq(csl2) can be extended by adding elements D±1 which satisfy therelations(3.4)DKi = KiD, De0 = qe0D, Df0 = q−1f0D, De1 = e1D, Df1 = f1DThe algebra obtained by this extension is denoted by Uq(fsl2).Two kinds of representations are defined for Uq(csl2): Verma modules Vλ,k andfinite dimensional evaluation representations Vµ(z).The Verma module Vλ,k is generated by a highest weight vector v satisfying therelations(3.5)eiv = 0,K1v = qλv,K0v = qk−λv,k, λ ∈C.This module is free over the subalgebra generated by fi and should be regarded asa quantum deformation of the Verma module over csl2.The evaluation representation Vµ(z) is defined with the help of the quantumgroup Uq(sl2) (see e.g.[CP]).

Uq(sl2) is a Hopf algebra generated by elementse, f, K±1 satisfying the relationsef −fe = K −K−1q −q−1 ,KeK−1 = q2e,KfK−1 = q−2f,(3.6)in which the comultiplication and the antipode are given by∆(K) = K ⊗K,∆(e) = e ⊗K + 1 ⊗e,∆(f) = f ⊗1 + K−1 ⊗f,(3.7)S(e) = −eK−1,S(f) = −Kf,S(K) = K−1. (3.8)For z ∈C∗Jimbo J2 defined the canonical algebra homomorphismspz : Uq(csl2) →Uq(sl2)as follows:pz(e1) = e,pz(f1) = f,pz(K±11 ) = K±1,p (e )zfp (f )z−1ep (K±1)K∓1(3 9)

11Let Vµ be the finite dimensional irreducible representation of Uq(sl2) with thehighest weight µ – a nonnegative integer. Such a representation is unique for any µ.The homomorphisms pz allow us to define the action of Uq(csl2) in Vµ.

The obtainedµ + 1-dimensional representation of Uq(csl2) is called the evaluation representationand is denoted by Vµ(z).The Verma module Vλ,k can be made a Uq(fsl2)-module by setting Dv = v, wherev ∈Vλ,k is the highest weight vector. This condition uniquely determines the actionof D±1 in Vλ,k.

Let us say that a vector w ∈Vλ,k is at level n if Dw = qnw. Denotethe space of all such vectors by Vλ,k[n].

The subspace of top level vectors Vλ,k[0]is a Uq(sl2)-subrepresentation isomorphic to the Verma module Mλ with highestweight λ over Uq(sl2).We are interested in formal expressions of the form(3.10)Φ(z) =Xn∈ZΦ[n]z−n,where Φ[m] : Vλ,k[n] →Vν,k[n + m] ⊗Vµ(z) are linear maps such that(3.11)Φ(z)au = ∆(a)Φ(z)u,a ∈Uq(csl2), u ∈Vλ,k.Affording a slight abuse of terminology, we can say that Φ(z) is a Uq(csl2) inter-twining operator Vλ,k →Vν,k ⊗Vµ(z).3 It is easy to show that any such operatoris uniquely determined by its action on top level vectors. Moreover, it is clear thatif w ∈Vλ,k is top level then it is enough to know the top level component of Φ(z)win order to retrieve Φ(z)w. This shows that Φ(z) is uniquely determined by themap Φ[0] : Vλ,k[0] →Vν,k[0] ⊗Vµ(z).

This map must be an intertwining operatorMλ →Mν ⊗Vµ over Uq(sl2). Moreover, for generic values of q, k this conditionis necessary and sufficient for Φ[0] to extend to the entire module Vλ,k [FR].

Fromnow on we consider this generic situation.Clebsch-Gordan formula tells us that there is a unique nonzero intertwiner Φλ,ν,µof form (3.10) up to a constant if ν + µ −λ is even, nonnegative, and not greaterthan 2µ, and no nonzero intertwiners of this form otherwise.Let u ∈Vµ(z)∗. Define the operator Φλ,ν,µ(u, z) : Vλ,k →ˆVν,k, Φλ,ν,µ(u, z)w =u(Φλ,ν,µ(z)w).Despite the range of the operator Φλ,ν,µ(u, z) lies in the completion of the highestweight module Vν,k, one can form products of such operators.

If |z1| > |z2| > · · · >|zN| then the product(3.12)Φλ1,λ0,µ1(u1, z1) . .

. ΦλN,λN−1,µN (uN, zN)is a well defined linear map: VλN,k →ˆVλ0,k.Let vλN ,k and v∗λ0,k be the highest weight vector of VλN,k and the lowest weightvector of V ∗λ0,k, respectively, and let Λ = (λ0, λ1, ..., λN).

Form the scalar product(3.13)ϕΛ(u1, ..., uN, z1, ..., zN) =< v∗λ0,k, Φλ1,λ0,µ1(u1, z1) . .

.ΦλN ,λN−1,µN (uN, zN)vλN,k > .3This statement is not quite precise because if u ∈Vλ,k and z ∈C∗then Φ(z)u lies inˆVλ,k ⊗Vµ(z), where ˆVλ,k is the completion of Vλ,k which allows infinite sums of homogeneoustf i fiit ldid

12This scalar product is a matrix element of the intertwiner (3.12). We will regardit as a holomorphic function in z1, ..., zN in the region |z1| > |z2| > · · · > |zN| withvalues in the finite dimensional space Vµ1 ⊗· · ·⊗VµN and write it as ϕΛ(z1, ..., zN).Let h(λ) =λ2+2λ2(k+2).

Let ∆i(Λ) = h(λi−1) −h(λi), 1 ≤i ≤N. Define a new(multivalued) function(3.14)ΨΛ(z1, ..., zN) = z∆1(Λ)1. .

. z∆N (Λ)NϕΛ(z1, ..., zN).It turns out that the function ΨΛ is a product of a scalar function and a solutionof the quantum KZ equations associated with a certain system of trigonometricR-matrices which is described explicitly as follows.Let m, n ≥0 be integers.

As a Uq(sl2)-module, Vm ⊗Vn decomposes asVm ⊗Vn = ⊕min(m,n)r=0Vm+n−2r.Let um be a highest weight vector of Vm, and let ωrmn be the highest weight vectorsof the components Vm+n−2r such that (e ⊗1)ωrmn = ωr−1mn for r ≥1, and ω0mn =um ⊗un. Let P rmn : Vn ⊗Vm →Vm ⊗Vn be the Uq(sl2)-invariant map such thatP rmnωrnm = ωrmn and P rmnωlnm = 0 if r ̸= l.Define the map Rqij(z) : Vµi ⊗Vµj →Vµi ⊗Vµj by(3.15)Rqij(z) =min(µi,µj)Xr=0r−1Yl=01 −zqµi+µj−2lz −qµi+µj−2l P rµiµjσµiµj,where σµiµj : Vµi ⊗Vµj →Vµj ⊗Vµi is the permutation of factors.Let W = Vµ1 ⊗· · · ⊗VµN .

Fix a complex number λ0. Let vi ∈Vµi, 1 ≤i ≤N,and let Kvi = qmivi.

Set m = PNj=1 mj. Define the operators Aq,λ0i: W →W by(3.16)Aq,λ0i(v1 ⊗v2 ⊗· · · ⊗vN) = q(2λ0+2+m)miv1 ⊗v2 ⊗· · · ⊗vN.Proposition 3.1.

[CP](i) {Rqij(z)} is a system of trigonometric R-matrices on Vµ1, ..., VµN, regular if|q| < 1. (ii) {Aq,λ0i} are compatible to {Rqij(z)}.Thus, the matrices {Rqij(z)} and {Aq,λ0i} define a holomorphic vector bundlewith fiber W. We will denote this bundle by Eq,λ0N.Observe that Eq,λ0N= ⊕µr=0Eq,λ0N,r , where Eq,λ0N,r is the subbundle of Eq,λ0Nwhosefiber is the subspace Wr of vectors of weight µ −2r in W, µ = PNj=1 µj.Because the transition matrices of the bundle Eq,λ0N,r depend only on the ratioszi/zj, this bundle can be obtained from a bundle over T N−1.

Indeed, let Θ ⊂T N bethe diagonal: Θ = {(z, z, ..., z)|z ∈T}, and let η be the projection: η : T N →T N/Θ(the space T N/Θ is isomorphic to T N−1). Denote by ξ the map T N →T whichacts according to the formula ξ(z1, ..., zN) = z1z2...zN.Let ∆0 =(2λ0+2+r)r2N(k+2) .Then Eq,λ0N,r = ξ∗(L∆0) ⊗η∗( ˆEq,λ0N,r ), where ˆEq,λ0N,r is a holomorphic vector bundle onT N/ΘT N−1

13Proposition 3.2. [FR] The function ΨΛ(z1, ..., zN) can be represented in the form(3.17)ΨΛ(z1, ..., zN) =Yi

In thispaper, they will not be of further interest.Now let s = tj1 . .

. tjn be a permutation.

Let sl = tj1 . .

. tjl, 1 ≤l ≤n.

Definethe functionϕs,Λ(u1, ..., uN, z1, ..., zN) = Rqsn−1(jn)sn−1(jn+1) . .

. Rqs1(j2)s1(j2+1)Rqj1,j1+1×< v∗λ0,k, Φλ1,λ0,µs(1)(us(1), zs(1)) .

. .ΦλN ,λN−1,µs(N)(us(N), zs(N))vλN,k > .

(3.18)(Abusing notation, we write Rij instead of Rij(zi/zj)). Define Ψs,Λ by(3.19)Ψs,Λ(z1, ..., zN) = z∆1(Λ)s(1).

. .

z∆N (Λ)s(N)ϕs,Λ(z1, ..., zN)and ψs,Λ by(3.20)Ψs,Λ(z1, ..., zN) =Ys−1(i)

Using Proposition 3.1 and the results of Section 2, we deduceProposition 3.4. (i) ˜ψs,Λ is a holomorphic section of the bundle Eq,λ0N(s−1∆(Λ)).

(ii) For any λ ∈C, there exist exactly d = dimW vectors Λ such that λ0 = λand ϕΛ ̸= 0. These vectors can be arranged in an order Λ1(s), ..., Λd(s) so that˜ψs,Λi(s) = ˜ψs,i, 1 ≤i ≤d, where ˜ψs,i, s ∈SN, is defined by (2.13).Thus, we have shown that matrix elements of intertwining operators between rep-resentations of the quantum affine algebra Uq(csl2) can be geometrically interpretedas holomorphic sections of a certain holomorphic vector bundle.

This interpretationremains valid if Uq(csl2) is replaced with Uq(ˆg) where g is an arbitrary simple Liealgebra.4. Connection matrices as clutching transformations.Let s1, s2 ∈SN.

Then the systems of sections ˜ψ(s1,i1) and ˜ψ(s2,i2) are related bya connection matrix Cs1s2 = {cs1s2i1i2 }, where cs1s2i1i2 is a meromorphic section of thebundle B(∆(s1,i1) −∆(s2,i2)):(4.1)˜ψs1,i1 =dXcs1s2i1i2 ⊗˜ψs2,i2.

14The connection matrices will have poles since the systems of solutions ψ(s,i),1 ≤i ≤d, are not everywhere linear independent. According to Proposition 2.5,the poles will be at points wherezs(j1)zs(j2) pn is a pole of Rs(j1)s(j2)(z) for a suitablen ∈Z and pair of indices j1, j2 such that j1 > j2.Thus, any matrix element of the connection matrix can be written as a product ofpowers of z1, ..., zN and a rational expression of elliptic functions in log z1, ..., log zN.Let s ∈SN and s′ = s · (ij), where (ij) is the transposition of i and j.

Let Rqijand Ai be defined by (3.15) and (3.16).Proposition 4.1. [FR] There exists a system of meromorphic functions Bij(ζ)with values in Matd(C), 1 ≤i, j ≤N such that Cs,s′(z1, ..., zN) = Bs(i)s(j)(logzs(i)zs(j) ).These matrices satisfy the conditions:(i) the quantum Yang-Baxter equation:(4.2)Bij (ζ1 −ζ2) Bik (ζ1 −ζ3) Bjk (ζ2 −ζ3) = Bjk (ζ2 −ζ3) Bik (ζ1 −ζ3) Bij (ζ1 −ζ2) ;(ii) unitarity:(4.3)Bij(ζ) = B−1ji (−ζ);(iii) double periodicity:(4.4)Bij(ζ + log p) = Bij(ζ);Bij(ζ + 2π√−1) = LBij(ζ)L′,where L, L′ ∈End(W) are constant diagonal matrices.This statement shows that the connection matrices provide elliptic solutions tothe quantum Yang-Baxter equation (elliptic quantum R-matrices).Now we can give an alternative construction of the bundle Eq,λ0N, using theconnection matrices as clutching transformations.Let X be a complex analytic space, and let {Ui, i ∈I} be an open cover of X.Let Ei →Ui be holomorphic vector bundles, and let βij : Ei |Ui∩Uj→Ej |Ui∩Uj beisomorphisms of holomorphic bundles such that the consistency conditions βijβji =id in Ui ∩Uj, βijβjkβki = id in Ui ∩Uj ∩Uk are satisfied.

Then one can constructa holomorphic vector bundle E on X by setting E |Ui= Ei |Ui and defining theclutching transformation from Ui to Uj to be βij.Assume that pn/qm ̸= 1 for any nonzero integers n and m. Let s ∈SN, and letHs be the set of all points (z1, ..., zN) ∈T N such that if n ∈Z and j1 > j2 thenzs(j1)zs(j2) pn is not a pole of Rqs(j1)s(j2)(z). The sections { ˜ψs,i, 1 ≤i ≤d} are linearlyindependent over Hs.Lemma 4.2.

{Hs, s ∈SN} is an open cover of T N.Proof. Let P = (z1, ..., zN) ∈T N. We need to show that there exists s ∈SN suchthat P ∈Hs.

Let Q be the infinite cyclic subgroup in T generated multiplicativelyby q. Let x1, ..., xr ∈T/Q be the distinct images of the points z1, ..., zN ∈T underthe homomorphism h : T →T/Q, and let Xj = h−1(xj), 1 ≤j ≤r.

Inside Xj,the elements are naturally ordered: for a, b ∈Xj, we say that a ≼b if and only ifa/b = qmpn, where m, n ∈Z and m > 0. Let s ∈SN be a permutation such thatzs(i) ≼zs(j) implies i ≤j.

Then P ∈Hs.■Now define a holomorphic vector bundle Eq,λ0Non T N as follows. Set Eq,λ0N|Hsto be isomorphic to ⊕di=1B(−∆(s,i)) |Hs, and define the clutching transformationsβs1s2 : ⊕di=1B(−∆(s1,i)) |Hs1∩Hs2 →⊕di=1B(−∆(s2,i)) |Hs1∩Hs2by βCs2s1It is trivial to check that the consistency conditions are satisfied

15Proposition 4.3. The bundles Eq,λ0Nand Eq,λ0Nare isomorphic to each other.Proof.

The system of sections { ˜ψs,i, 1 ≤i ≤d} defines an isomorphism between thebundles ⊕di=1B(−∆(s,i)) |Hs and Eq,λ0N|Hs. Transition from one system of sectionsto another is performed by the connection matrix Cs1s2.■5.

Topology of the quantumKnizhnik-Zamolodchikov equations and crystal bases.In this section we study the vector bundle Eq,λ0Ntopologically, disregarding itsholomorphic structure. It turns out that the topological structure of Eq,λ0Nis quitenontrivial and can be described in terms of the combinatorics of crystal bases inrepresentations of quantum groups.Clearly, we may assume, without loss of generality, that dimVµj ≥2 for all j.Then rank(Eq,λ0N) ≥2N.Recall some properties of characteristic classes.

Let E be a vector bundle on T N.Let ck(E) ∈H2k(T N, Z), 0 ≤k ≤N, be the Chern classes of E (by conventionc0 = 1). The sum c(E) = PNk=0 ck(E) ∈⊕Nk=0H2k(T N, Z) is called the total Chernclass of E. It has the property c(E ⊕F) = c(E)c(F) for any two vector bundles Eand F. Also, if E and F are one-dimensional then c1(E ⊗F) = c1(E) + c1(F) [Mi].Let fj(x1, ..., xN) be the elementary symmetric polynomials:NXj=0fjtN−j =NYj=1(t + xj).Let Qk(y1, ..., yk) be the Newton polynomials defined by Qk(f1, ..., fk) = PNj=1 xkj .The Chern character of a vector bundle E is defined by(5.1)Ch(E) = r +NXk=11k!Qk(c1, ..., ck),where r is the rank of E.It has the properties Ch(E ⊕F) = Ch(E) + Ch(F),Ch(E ⊗F) = Ch(E)Ch(F).Proposition 5.1.

A complex vector bundle over T N of rank r ≥N is uniquelydetermined by its Chern classes.Proof. Two bundles of rank r ≥N over T N are equivalent if and only if they arestable equivalent.

Therefore, a bundle of rank ≥N is uniquely determined by itsclass in the ring K(T N). A well known theorem of K-theory ([K, Theorem 5.3.25]),asserts that the Chern character induces an isomorphism of rings: K(T N) ⊗Q →H2∗(T N) ⊗Q.

The complex K-ring of the torus is isomorphic to the even part ofits cohomology ring. This fact follows from the rule of evaluation of the K-ringof a product of two spaces (see Prop 4.3.24 in [K]).

Therefore, K(T N) is a torsionfree abelian group, which implies that the natural map K(T N) →K(T N) ⊗Q is amonomorphism. Thus, the composed map Ch : K(T N) →H2∗(T N, Q) is injective,Q.E.D.■Since rank(Eq,λ0N) > N, in order to describe the bundle Eq,λ0Ntopologically it isenough to calculate c(Eq,λ0N).

This turns out to be quite simple.When q goes to 0, the operators P rmn defined in Chapter 3 tend to finite limits– a remarkable phenomenon known as crystallization [Ka]. This implies that theR matrices Rq (z) have finite limits as q →0

16It is clear that the topological structure of the bundle Eq,λ0N,r is stable underdeformations and hence independent of the values of q and λ0 (as long as |q| < 1).Therefore, Eq,λ0Nis topologically equivalent to the bundle ˜EqN = ⊕µr=0Eq,−1−r/2N,r.As q →0, the transition matrices of the bundle ˜EqN have finite limits: Rqij →R0ij,Aq,−1−r/2i→1 on Wr. Therefore, there exists a limiting bundle ˜E0N, with the fiberW, defined by the matrices Rij(z) = R0ij(z) and Ai = 1 according to (1.4) and(1.5), and the bundle Eq,λ0Nis topologically equivalent to ˜E0N for any q and λ0.Let us now describe the limiting R-matrix R0ij.

Let ulm = f lum ∈Vm, 0 ≤l ≤m.Proposition 5.2. [Ka] Let q = 0.

Then P rmnσ(uim⊗ujn) = 0 unless min(i, n−j) =r. If min(i, m −j) = r then(5.2)P rmnσ(uim ⊗ujn) =ujm ⊗uin,i + j ≤mum−im⊗u2i+j−mn,m < i + j ≤nun−m+jm⊗um−n+in,n < i + j ≤n + mif m ≤n, and(5.3)P rmnσ(uim ⊗ujn) =ujm ⊗uin,i + j ≤nui+2j−nm⊗un−jn,m < i + j ≤nun−m+jm⊗um−n+in,m < i + j ≤n + mif m > n.Corollary 5.3.

The R-matrix at q = 0 has the form(5.4)R0ij(z)(ukµi ⊗ulµj) =z−min(k,µj−l)ulµi ⊗ukµj,k + l ≤µiz−min(k,µj−l)uµi−kµi⊗u2k+l−µiµj,µi < k + l ≤µjz−min(k,µj−l)uµj−µi+lµi⊗uµi−µj+kµj,µj < k + l ≤µi + µjif µi ≤µj, and(5.5)R0ij(z)(ukµi ⊗ulµj) =z−min(k,µj−l)ulµi ⊗ukµj,k + l ≤µjz−min(k,µj−l)uk+2l−µjµi⊗uµj−lµj,µj < k + l ≤µiz−min(k,µj−l)uµj−µi+lµi⊗uµi−µj+kµj,µi < k + l ≤µi + µjif µi > µj.Let L = (l1, ..., lN), 0 ≤lj ≤µj. Let UL = ul1µ1 ⊗· · · ⊗ulNµN ∈W.

The vectors{UL} form a basis of W (the crystal basis).Corollary 5.4. In the basis {UL}, the operator Mj(z1, ..., zN) defined by (2.7) canbe represented in the form of a product(5.6)Mj(z1, ..., zN) = Yj(z1, ..., zN)πjwhere πj is a permutation matrix (πj ∈Sd, d = dimW), and Yj is a diagonalmatrix whose eigenvalues are products of integer powers of z1, ..., zN.It is clear that the permutations πj commute with each other.

Let αj be theorder of π in SLet ˆT N be the torus QN(C∗/Γ ) where Γ is the infinite cyclic

17group multiplicatively generated by pαj. Let β : ˆT N →T N be the natural covering.Consider the holomorphic vector bundle β∗( ˜E0N) on ˆT N. Corollary 5.4 implies thatthis bundle breaks up into a direct sum of line bundles, and each of these linebundles is a tensor product of line bundles over factors C∗/Γj.

This allows us tocalculate c(β∗( ˜E0N)) using the properties of Chern classes that we discussed above.Applying the map (β∗)−1 : H∗( ˆT N, Q) →H∗(T N, Q) to c(β∗( ˜E0N)), we obtain thesought-for total Chern class c(E0N).For the sake of brevity, we implement this calculation under the assumption thatall Vµj are the same: µj = m for all j. Then formulas (5.4) and (5.5) undergo amajor simplification: R0ij(z) = R0(z), i ̸= j, and(5.7)R0(z)(ukm ⊗ulm) = z−min(k,m−l)ulm ⊗ukm.In the formulas below, indices i, j are allowed to take values between 1 and N,and operations on them are fulfilled modulo N. For instance, N + 1 is identifiedwith 1, and 0 is identified with N.Introduce the notation I(k, l) = min(k, m −l), J(L, j) = Pj−1i=1 I(li, li+1), andJ(L) = PNi=1 I(li, li+1).Substituting (5.7) into (2.6), we obtainMj(z1, ..., zN)ul1m ⊗· · · ⊗ulN−1m⊗ulNm =z−J(L)+I(lj−1,lj)jYi̸=jzI(li−1,li)ip−J(L,j)ulNm ⊗ul1m ⊗· · · ⊗ulN−1m(5.8)This shows that for all j πj is the cyclic permutation (123...N).Therefore,αj = N for every j.

Let(5.9)ˆMj(z1, ..., zN) = Mj(z1, ..., pN−1zj, ..., zN) . .

.Mj(z1, ..., pzj, ..., zN)Mj(z1, ..., zj, ..., zN)The matrices ˆMj are the transition matrices for the bundle β∗( ˜E0N). On the otherhand, as we have mentioned, these matrices are diagonal:(5.10)ˆMj(z1, ..., zN)UL = z−(N−1)J(L)jYi̸=jziJ(L)pK(L)UL,where K(L) is an integer.Let xi =1log |p|Re log zi, yi =12π√−1Im log zi.Let [Ω] ∈H2(T N, Z) be thecohomology class of the differential form(5.11)Ω= NNXj=1dxj ∧dyj −(NXj=1dxj) ∧(NXj=1dyj)Formula (5.10) immediately yields an expression for c(β∗( ˜E0N)):(5.12)c(β∗( ˜E0N)) =YL(1 + J(L)[Ω]),where the product is taken over all L = (l1, ..., lN) with 1 ≤lj ≤m.

This, in turn,gives a formula for c( ˜E0N):(5.13)c( ˜E0N) =YL(1 + J(L)N[Ω]).

18Remark 5.1. Formula (5.12) contains fractions, but it always gives an integralcocycle – the total Chern class of a vector bundle.Thus, we have provedProposition 5.5.

If Vµi = Vm for 1 ≤i ≤N thenc(Eq,λ0N,r ) =YL:P lj=r(1 + J(L)N[Ω]),0 ≤r ≤Nm,(5.14)c(Eq,λ0N) =YL(1 + J(L)N[Ω]),(5.15)for all values of q and λ0.6. Special case N = 2.According to Chapter 3, in the special case N = 2 the bundle Eq,λ0N,r (∆) can berepresented as a product:(6.1)Eq,λ02,r (∆) = ξ∗(L∆0+ 12 (∆1+∆2)) ⊗η∗( ˆEq,λ02,r (∆1 −∆22)),where ξ, η : T 2 →T are defined by ξ(z1, z2) = z1z2, η(z1, z2) = z1/z2, ∆0 =(2λ0+2+r)r4(k+2), and ˆEq,λ02,ris a bundle over T. Clearly, it is enough to understand ˆEq,λ02,rin order to understand Eq,λ02,r .Proposition 6.1.rank( ˆEq,λ02,r ) = ρ0 =r + 1,r ≤min(µ1, µ2)min(µ1, µ2) + 1,min(µ1, µ2) < r ≤max(µ1, µ2)µ1 + µ2 −r + 1,max(µ1, µ2) < r ≤µ1 + µ2deg( ˆEq,λ02,r ) = δ0 = 12ρ0(ρ0 −1)(6.2)This proposition follows directly from formulas (4.4) and (4.5).

It completelydescribes the topological structure of the bundle ˆEq,λ02,r .The bundle ˆEq,λ02,ris a holomorphic vector bundle over an elliptic curve. Suchbundles were completely classified by M.Atiyah in 1957 [A].

M.Atiyah showed thatthe set of indecomposable bundles of a fixed rank and degree over an elliptic curveT can be identified with T: if E is an indecomposable bundle then any otherindecomposable bundle F of the same rank and degree can be represented in theform F = E ⊗L, where L is a line bundle of degree 0.In 1962 D.Mumford introduced the notions of stability and semistability for vec-tor bundles over curves [Mu;NS]. A vector bundle E over a curve X is called stableif for every proper nonzero subbundle F ⊂Edeg(F)rank(F)

However, if X is an elliptic curve,stable bundles of rank ρ and degree δ over X exist if and only if ρ and δ are rela-tively prime. On the contrary, semistable bundles over X exist for any values of ρand δ.

Thus, a generic vector bundle over an elliptic curve is semistable.Here are some properties of stable and semistable bundles:

19Lemma 6.2. (i) If ρ and δ are relatively prime, every semistable bundle of rank ρ and degreeδ is stable.

(ii) If E is a stable (semistable) bundle and L is a line bundle then E ⊗L is stable(semistable). (iii) If L1, ..., Ln are line bundles of the same degree then the bundle E = L1 ⊕· · · ⊕Ln is semistable.

(iv) Let E be a vector bundle over a curve X, and let γ : ˆX →X be a finitecovering. If γ∗E is a stable (semistable) vector bundle over ˆX then E is also stable(semistable).Proof.

Properties (i) and (ii) are obvious. (iii) Let deg(Lj) = δ.

Let F be a subbundle of rank ρ in E. Then ΛρF is aline bundle embedded in ΛρE. Obviously, ΛρE = ⊕iBi, where Bi are line bundlesof degree ρδ.

This implies that for at least one j the projection ΛρF →Bj is notzero. This projection is a nonzero regular section of the bundle ΛρF∗⊗Bj.

Thus,deg(ΛρF∗⊗Bj) ≥0, which implies that deg(F) ≤ρδ. (iv) Pulling back by γ does not change the rank of the bundle and multiplies itsdegree by the number of sheets of the covering.

Therefore, if F ⊂E violates thestability (semistability) condition for E, so does γ∗F for γ∗E.■Proposition 6.3. The bundle ˆEq,λ02,ris semistable for generic values of q and λ0.Remark 6.1.

If ˆEq,λ02,ris semistable then, according to Lemma 6.2, (ii), so isˆEq,λ02,r (∆) for every complex number ∆.Proof. First let us show that for a generic value of q the bundle ˆEq,−r−1/22,rissemistable.

Since the set of such values is automatically Zariski open in the unitdisk, we only need to show that there exists at least one such value. We will proveit for q = 0.Let ˆT = C∗/Π2, where Π2 is the infinite cyclic subgroup multiplicatively gen-erated by p2.

Let β : ˆT →T be the natural 2-sheeted covering. Formulas (5.4)and (5.5) show that the bundle β∗ˆE0,−r−1/22,ris a direct sum of ρ0 line bundles ofdegree ρ0 −1.

By Lemma 6.2, (iii), this bundle is semistable. Then Lemma 6.2,(iv) ensures the semistability of ˆE0,−r−1/22,r.We know that the set of pairs (q, λ0) for which the bundle ˆEq,λ02,ris semistableis Zariski open in D0 × C, D0 being the open unit disk punctured at 0.

This setis also nonempty because it contains almost every point of the form (q, −r −1/2).Hence, it contains almost every point of D0 × C.■Corollary 6.4. If r = 1 then ˆEq,λ02,ris stable, and hence indecomposable for genericvalues of q and λ0.Proof.

The bundle ˆEq,λ02,1has degree 1 and rank 2, and it is semistable for genericvalues of parameters.Hence, by Lemma 6.2, (i), it is (generically) stable andtherefore indecomposable.■Proposition 6.5. For generic q and λ0, the vector bundle ˆEq,λ02,ris a direct sumof line bundles if ρ0 is odd, and a sum of indecomposable 2-dimensional bundles ifρ0 is even.

Thus, ˆEq,λ02,ris almost always a direct sum of stable bundles.Proof. Fix a complex number a and consider the family of vector bundles ˆEq,−1−r2 +alog q2,r,|q| < 1 This family is analytic in the unit disc and at q0 the limiting bundle is

20prescribed by (4) and (5) with R12(z) = R012(z), Ai(um1µ1 ⊗um2µ2 ) = e2amium1µ1 ⊗um2µ2 .As before, β∗limq→0 ˆEq,−1−r2 +alog q2,ris a direct sum of line bundles. Moreover, it iseasy to check that for a generic a, all these line bundles are pairwise non-isomorphic.This shows that such a decomposition takes place for β∗ˆEq,λ02,rfor generic values ofq and λ0.Now assume that ˆEq,λ02,ris semistable but not a direct sum of stable bundles.

LetˆEq,λ02,r= I1 ⊕· · · ⊕Is, where Ik are indecomposable. Suppose that Ik is unstablefor some k. Then, according to [A], Ik = E ⊗F where E is a stable bundle (of rank1 if ρ0 is odd and of rank 2 if ρ0 is even), and F is an indecomposable degree zerobundle on T with rank(F) > 1.

Therefore, we have β∗Ik = β∗E ⊗β∗F. We knowthat β∗E is either a line bundle or a direct sum of two line bundles, whereas β∗Fis obviously indecomposable.

Thus, β∗Ik is not a direct sum of line bundles. Thisdoes not happen for generic values of the parameters.■Remark 6.2.

These statements indicate that the trigonometric R-matrices aris-ing from Uq(csl2), as a rule, give rise to ’generic’ holomorphic bundles over T, i.e.ones corresponding to regular points of the moduli space of bundles. It would beinteresting to check if this property holds for other simple Lie algebras and forN > 2.Remark 6.3.

It seems plausible that propositions 6.3 – 6.5 should hold for ar-bitrary values of q and λ0, not only for generic ones.However, a proof of thisstatement would probably have to involve a more delicate argument than what isused above.Remark 6.4. It seems to be an interesting problem to separate the matrix ele-ments of intertwining operators from the whole variety of holomorphic sections ofthe bundles ˆEq,λ0N,r (∆) by some geometric condition.

It is not clear how to approachthis question since, for instance, the bundle ˆEq,λ02,r (∆), being generically semistable,has as many linearly independent regular sections as its degree, for all values of ∆.7. Arbitrary root systemsIn the recent paper [Ch] a quantum R-matrix R was interpreted as a 1-cocycleon the semidirect product ˜W = W ⋉P of a Weyl group W of the type An andits weight lattice P with coefficients in a certain group G on which ˜W operates byautomorphisms.

The quantum KZ system can then be viewed as a condition on anelement g ∈G (regarded as a 0-cochain on ˜W with coefficients in G) requiring thatdg(b) = R(b), b ∈P, where d denotes the coboundary operator.This interpretation is very useful since it can be made a definition of a quantumR-matrix and the KZ system if An is replaced with an arbitrary Dynkin diagram.Our purpose now is to link the 1-cocycle and the vector bundle interpretations ofthe quantum KZ equations together and to extend the vector bundle interpretationto arbitrary root systems.Let W be a (finite) Weyl group, and let P be its dual root lattice. Denote by Kthe field of P-periodic trigonometric functions on P ⊗C (i.e.rational expressions ofmonomials e2π√−1λ(z), z ∈P ⊗C, where λ ∈Q∨, Q∨being the dual lattice to P).Let G = G(K) where G = GL(H) and H is a finite-dimensional representation of˜W = W ⋉P.Let τ ∈C+.

We define the action of ˜W on G by (wb◦g)(z) = wg(w−1z−bτ)w−1,b ∈P, z ∈P ⊗C, w ∈W.Let S0 ⊂P ⊗R be the set of points with a nontrivial stabilizer in ˜W.LetSP ⊗R ⊕τS ⊂P ⊗C Let Gbe the subgroup of the elements of G that are

21regular at S. Clearly, the action of ˜W maps elements of G0 to (other) elements ofG0.Definition 7.1. A trigonometric quantum R-matrix on H is a W-invariant 1-cocycle R on ˜W with coefficients in G0 with the described action of ˜W: for x, y ∈˜WR(xy) = (y−1 ◦R(x))R(y) in G0.Trigonometric R-matrices defined in Chapter 1 (for Vi = V , 1 ≤i ≤N) areobtained in the special case H = V ⊗N, W = SN, and the action of ˜W in Hfactorizes through the action of W in H by permutations of factors.Let ˜K be the field of meromorphic P-periodic functions in P ⊗C, and let ˜G =G( ˜K).

Let ˜G0 be the subgroup of the elements of ˜G that are regular at S. Theaction of ˜W in ˜G0 is the same as in G0.Definition 7.2. Let g ∈˜G0.

The equations (dg)(b) = R(b) with respect to g, forall b ∈P, where d is the coboundary operator, are called the quantum KZ systemfor R.Again, the quantum KZ equations considered in Chapter 2 come about in thespecial case when H = V ⊗N and W = SN.Now, given a 1-cocycle R, we will construct a W-equivariant holomorphic vectorbundle on the torus T(W) = P ⊗C/(P ⊕τP).The set S partitions the space P ⊗C into infinitely many bounded chambers Dulabeled by elements u ∈˜W, so that u2 ∈˜W maps Du1 to Du1u2. If two chambersDu1 and Du2 are adjacent (i.e the intersection of their boundaries has codimension1) then u2 = u1s where s ∈˜W is a reflection.

Consider the open cover of P ⊗C bythe union of small neighborhoods of the chambers Du. Let us define the transitionmatrix in the vicinity of the boundary between Du1 and Du2 (from Du1 to Du2) tobe(7.1)Tu1,s(z) = (u−11◦R(u1su−11 ))(z),z ∈P ⊗C,where the right hand side is regarded as a GL(H)-valued function.

By construction,this matrix is regular near the boundary. Because R is a 1-cocycle on ˜W, the systemof matrices Tu,s is a holomorphic GL(H)-valued Cech 1-cocycle on P ⊗C.

It is easyto check that this Cech cocycle is invariant under the action of ˜W ⋉(P ⊕τP). Sinceevery Cech cocycle automatically defines a vector bundle, we obtain a holomorphicvector bundle with fiber H on P ⊗C, which descends to a holomorphic bundle overthe torus T(W) = P ⊗C/(P ⊕τP).In this setting, meromorphic solutions of the quantum KZ system are interpretedas meromorphic sections of the constructed vector bundle, since geometrically rep-resentation of a 1-cocycle as a coboundary of something is equivalent to constructinga global fundamental system of sections of the corresponding bundle.In particular, in the special case H = V ⊗N and W = SN, this scheme gives thevector bundle EN constructed in Chapter 1 (assuming A1A2 .

. .

AN = 1) pushedforward to the N −1-dimensional torus T N/Θ, where Θ is the main diagonal.This push-forward is possible since if A1A2 . .

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[NS] Narasimhan, M.S., and Seshadri, C.S., Stable and unitary vector bundles on a compactRiemann surface, Ann.Math. 82 (1965), 540-567.

[R] Reshetikhin, N.Yu., Jackson-type integrals, Bethe vectors, and solutions to a difference analogof the Knizhnik-Zamolodchikov system, preprint (1992). [SV] Schechtman, V.V., and Varchenko, A.N.,, Arrangements of hyperplanes and Lie algebrahomology, Inv.

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