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This is the talk of the second author at the meeting ”Topological Methods in Physical Sciences”, London, November 2000. We review our work on KZB equations

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arXiv:math/0101136v1  [math.QA]  16 Jan 2001SPECIAL FUNCTIONS, CONFORMAL BLOCKS, BETHE ANSATZ,AND SL(3,Z)G. FELDER ⋆ AND A. VARCHENKO ⋄,1⋆ Departement Mathematik, ETH-Zentrum, 8092 Zürich, Switzerland,felder@math.ethz.ch⋄Department of Mathematics, University of North Carolina at Chapel Hill,Chapel Hill, NC 27599-3250, USA,av@math.unc.eduNovember, 2000Abstract. This is the talk of the second author at the meeting ”Topological Methodsin Physical Sciences”, London, November 2000. We review our work on KZB equations.KZB equations.The Knizhnik-Zamolodchikov-Bernard (KZB) equations are a system of differentialequations arising in conformal field theory on Riemann surfaces. For each g, n ∈ Z≥0, asimple complex Lie algebra g , n highest weight g -modules Vi and a complex parameterκ, we have such a system of equations. In the case of genus g = 1, they have the formκ∂v∂zj= −∑νh(j)ν∂v∂λν+∑l, l 6=jr(zj − zl, λ, τ)(j,l)v, j = 1, ..., n,4πiκ∂v∂τ= ∆λv +12∑i,js(z, λ, τ)(i,j)v.The unknown function v takes values in the zero weight space V [0] = ∩x∈hKer(x) ofthe tensor product V = V1 ⊗ · · · ⊗ Vn with respect to the Cartan subalgebra h of g . Itdepends on variables z1, ..., zn ∈ C, modulus τ of the elliptic curve and λ =∑λνhν ∈ h ,where (hν) is an orthonormal basis of h with respect to a fixed invariant bilinear form.In the equation, r, s ∈ g ⊗ g are suitable given tensor valued functions, [FW].The second equation is called the KZB-heat equation.1 Supported in part by NSF grant DMS-9801582.2 G. FELDER AND A. VARCHENKOExample. For g = slN , n = 1, V = SmNCN , h = CN/C(1, ..., 1), the weight-zerospace V [0] is one dimensional, the KZB equations are scalar equations ∂v/∂z1 = 0 and4πiκ∂v∂τ=N∑i=1∂2v∂λ2i+ 2m(m+ 1)∑1≤i<j≤Nρ′(λi − λj, τ)v .(1)Here ′ denotes the derivative with respect to the first argument and ρ is defined in termsof the first Jacobi theta function,θ(t, τ) = −∑j∈Zeπi(j+12)2τ+2πi(j+ 12)(t+ 12), ρ(t, τ) =θ′(t, τ)θ(t, τ).Notice that ρ′ = −℘+ c where ℘(t, τ) is the Weierstrass function, c = c(τ) a function ofτ , and we recover in the right hand side of the KZB-heat equation the Hamilton operatorHN,m of the elliptic Calogero-Moser quantum N -body system,−HN,m =N∑i=1∂2∂λ2i+ 2m(m+ 1)∑i<jρ′(λi − λj, τ)with coupling constant m(m+ 1).In particular, if g = sl2, then the Cartan subalgebra h can be identified with C, andλ ∈ C. For the irreducible 2m+ 1 dimensional module V , the KZB-heat equation takesthe form2πiκ∂v∂τ(λ, τ) =∂2v∂λ2(λ, τ) + m(m+ 1)ρ′(λ, τ) v(λ, τ) .(2)A remarkable fact about all forms of the KZ equations is that they can be realizedgeometrically. They have solutions defined as hypergeometric integrals depending onparameters.Example. Consider the functionv(λ, τ) =∫ 10(θ(t, τ)θ′(0, τ))− 2κ θ(λ− t, τ)θ′(0, τ)θ(λ, τ)θ(t, τ)g(λ− 2κt, τ) dtwhere g(λ, τ) is any solution of the heat equation2πiκ∂g∂τ(λ, τ) =∂2g∂λ2(λ, τ).For instance, g(λ, τ) = eλµ+µ22πiκτ for µ ∈ C. Then v is a solution of the KZB-heatequation (2) with m = 1 [FV1].Remark. Simplest KZ equations have the classical Gauss hypergeometric function astheir solution. It is natural to consider solutions of all KZ type equations as generalizedhypergeometric functions. Thus there are hypergeometric functions associated withcurves of any genus. The function v(λ, τ) is the simplest elliptic hypergeometric function.SPECIAL FUNCTIONS, CONFORMAL BLOCKS, BETHE ANSATZ, AND SL(3,Z) 3The Gauss hypergeometric function isΓ(b)Γ(c− b)Γ(c)F (a, b, c; z) =∫ ∞1ta−c(t− 1)c−b−1(t− z)−a dt .The functions(θ(t, τ)θ′(0, τ))aandθ(λ− t, τ)θ′(0, τ)θ(λ, τ)θ(t, τ)are elliptic analogs of the functions ta and 1/t, respectively. The hypergeometric solu-tions to the KZB equations were discovered through this analogy.Remark. If κ tends to 0, we are dealing with the eigenfunction problem. Findeigenfunctions of the elliptic Calogero-Moser Hamiltonian,N∑i=1∂2v∂λ2i+ 2m(m+ 1)∑1≤i<j≤Nρ′(λi − λj , τ) v = E v .The stationary phase method applied to hypergeometric solutions of the KZB equationsgives eigenfunctions. For instance, application of the method to the function v(λ, τ) im-plies that for every µ ∈ C, the function v(λ) = eλµθ(λ− t0, τ)/θ(λ, τ) is an eigenfunctionof the operator∂2∂λ2+ 2ρ′(λ, τ)if t0 is a critical point of the function e−µtθ(t, τ). This is a Bethe ansatz type formula, see[FV1, FV2]. For g = sl2, the Bethe ansatz formulas reduce to Hermite’s 1872 solutionof the Lamé equation, see [WW].The fact that solutions have the form of explicitly written integrals is a tool to studysolutions as well as equations, for instance , the monodromy properties of solutions ortheir modular properties with respect to changes of τ .Modular symmetries of KZB.For anya bc d ∈ SL(2,Z), the elliptic curves with moduli τ and (aτ+b)/(cτ+d) areisomorphic, the corresponding KZB equations are related. Having a family of solutionsone can ask about monodromy properties of solutions with respect to transformationsof the lattice Z+ τZ.Example. Consider equation (2) for m = 0,2πiκ∂v∂τ=∂2v∂λ2.(3)If v(λ, τ) is a solution, then ṽ(λ, τ) = v(λ, τ + 1) andṽ(λ, τ) =1√τe−πiλ22τ v(λτ, −1τ)4 G. FELDER AND A. VARCHENKOare solutions too. If {vµ(λ, τ)} is a family of solutions depending on a parameter µ, thenone can ask about ”monodromy” relations of the three families of solutions: {vµ(λ, τ)},{vµ(λ, τ + 1)}, and { 1√τ e−πiλ22τ vµ(λτ, − 1τ)}.The KZB-heat equation as a flat connection. One can consider equation (2) asthe equation for horizontal sections of a connection on a vector bundle over the upperhalf plane H+ whose fiber F (τ) over a point τ is the space of functions of λ.If κ is a positive integer not less than 2, then the bundle has a finite dimensional sub-bundle (of conformal blocks) invariant with respect to the connection and consisting ofcertain theta functions of level κ, [EFK, FW, FV1].In this paper we address the followingProblem. Quantize the KZB heat equation, study modular properties of the quanti-zation.It turns out that the KZB-heat equation is quantized to a difference qKZB-heat equa-tion (with step p) in such a way that the SL(2,Z) symmetry of the KZB-heat equationrelated to the lattice Z+ τZ is quantized to an SL(3,Z) symmetry related to the latticeZ + τZ+ pZ.A quantization of the heat equation is a discrete connection over H+ with the samefiber, i.e. a linear operator T (τ, τ + p) : F (τ + p) → F (τ). This linear operator tends to1 + const p (∂2∂λ2+ m(m+ 1)ρ′(λ, τ) ) + ...as p → 0.For methodological reasons we describe first a quantization of equation (3). For anonzero η ∈ C, introduce linear operatorsU : f(λ) 7→ i√4iη∫ηRe−πiλµ2η f(−µ) dµ ,α : f(λ) 7→ e−πiλ24η f(λ) .DefineT (τ, τ + p) = αUα : f(λ) 7→ i√4iη∫ηRe−πi(λ+µ)24η f(−µ) dµ .(4)The translation operator T (τ, τ + p) in this case is basically just the Fourier transform.The qKZB-heat equation is the equation for flat sections,v(λ, τ) =i√4iη∫ηRe−πi(λ+µ)24η v(−µ, τ + p) dµ .(5)SPECIAL FUNCTIONS, CONFORMAL BLOCKS, BETHE ANSATZ, AND SL(3,Z) 5Set p = −2κη. Let η → 0 and λ, τ, κ fixed.Theorem. Let vη(λ, τ) be a family of solutions of (5) with asymptotics vη(λ, τ) =v0(λ, τ) + ηv1(λ, τ) + ... Then v0(λ, τ) satisfies (3).Proof. The stationary phase asymptotic expansion of the right hand side of (5) isv0(λ, τ) + ηv1(λ, τ) +iηπ(2πiκ∂v0∂τ(λ, τ)− ∂2v0∂λ2(λ, τ)) +O(η2) + ...Introduce a functionu(λ, µ, τ, p, η) = e−πiλµ2η .(6)For every µ ∈ C, the function u is a projective solution of the qKZB-heat equation (5),u(λ, µ, τ, p, η) = e−πiµ24η (T (τ, τ + p)u)(λ, µ, τ + p, p, η),in particular, v(λ, µ, τ, p, η) = eπiµ24ητp−πiλµ2η is a true solution.Remark [FV3]. Let κ be a positive integer. The functionsθj,κ(λ, τ) =∑r∈Z+j/2κe2πiκ(r2τ+rλ), j ∈ Z/2κZ,form the space Θκ(τ) of theta functions of level κ. Let E(τ) = {f ∈ Θκ(τ) | f(−λ) =−f(λ)} be the space of odd theta functions. The translation operator T (τ, τ + p) mapsE(τ + p) to E(τ) if −p/2η = κ.This statement is based on the identityθj,κ(λ, τ) =i√4iη∫2ηRe−iπ4η(λ+µ)2θj,κ(−µ, τ − 2ηκ) dµ .The space E(τ) is the quantization of the finite dimensional space of conformal blocks.Modular properties of the q-heat operator (in this case of the Fouriertransform).The group SL(3,Z) is generated by the elementary matrices ei,j , i 6= j. The ele-mentary matrix ei,j is the element of SL(3,Z) which differ from the identity matrix byhaving the i, j matrix element equal to 1. The relations can be chosen [M] to beei,jek,l = ek,lei,j, i 6= l, j 6= k,ei,jej,k = ei,kejkei,j,(e1,3e−13,1e1,3)4 = 1.6 G. FELDER AND A. VARCHENKOConsider C3 with coordinates x1, x2, x3. The group SL(3,Z) acts on C3 in the standardway. Consider the trivial bundle over C3 with the same fiber F over a point x ∈ C3. Wedefine a projectively flat connection over the orbit of a point. Namely, for any generatorei,j, we define a linear operatorϕi,j(x) : F (e−1i,j x) → F (x)so that all relations in SL(3,Z) are projectively satisfied. That means that the linearoperator corresponding to the left hand side of a relation is equal to the linear operatorcorresponding to the right hand side of the relation multiplied by a number.Remark. The operator ϕ2,1(p, τ, η) : F (p, τ − p, η) → F (p, τ, η) will correspond tothe qKZB-heat operator.Introduce linear operatorsU(x1, x2, x3) : f(λ) 7→∫x3Re−πi λµ2x3 f(−µ) dµ ,α(x3) : f(λ) 7→ e−πiλ24x3 f(λ) ,β(x1, x2, x3) : f(λ) 7→ e−πiλ24x1x2x3 f(λ) .Setϕ1,3(x1, x2, x3) = 1,ϕ2,3(x1, x2, x3) = 1,ϕ1,2(x1, x2, x3) = α(x3),ϕ3,2(x1, x2, x3) = β(x1, x2 − x3, x3),ϕ2,1(x1, x2, x3) = α(x3)U(x2, x2 − x1, x3)α(x3),ϕ3,1(x1, x2, x3) = β(x1 − x3,−x3, x2)−1 U(x1 − x3,−x3, x2)−1 β(x3, x2, x3 − x1)−1.Theorem. The operators ϕi,j define a projectively flat connection over orbits of theSL(3,Z) action.Remark. Consider C3 and the projectivization of the dual space, P (C3)∗. ConsiderX ⊂ (C− 0)3 × P (C3)∗ whereX = {((x1, x2, x3), (y1 : y2 : y3)) | x1y1 + x2y2 + x3y3 = 0 } .The natural projection X → C3−0 is a projective line bundle. The group SL(3,Z) actson X , g : (x, y) 7→ (gx, (gt)−1y) for any g ∈ SL(3,Z). Fix an affine coordinate on fibers,t : (y1 : y2 : y3) 7→ y2/y1. Then for any i, j, i 6= j, we haveei,j : (e−1i,j x, t) 7→ (x, fi,j(x, t))SPECIAL FUNCTIONS, CONFORMAL BLOCKS, BETHE ANSATZ, AND SL(3,Z) 7where f1,3(x, t) = f2,3(x, t) = t, f1,2(x, t) = t − 1, f3,2(x, t) = (tx3 + x1)/(x3 − x2),f2,1(x, t) = t/(1− t), f3,1(x, t) = t(x3 − x1)/(tx2 + x3).The SL(3,Z)-action on X is closely related to the projectively flat connection de-scribed in the Theorem. Namely, setG(λ; x1, x2, x3; t) = eπi4λ2 tx3 .Call a Gaussian in the fiber F (x1, x2, x3) a function of the form const·G(λ; x1, x2, x3; t) forsome number t. The linear operators ϕi,j(x) : F (e−1i,j x) → F (x) preserve the Gaussians.Moreover, for all i, j, we haveϕi,j(x) : G(λ; x1, x2, x3; t) 7→ const ·G(λ; x1, x2, x3; fi,j(t, x)) .Remark. The Theorem easily follows from the following two main equations satisfiedby the Fourier transform, which we callthe q-heat equation,α(x3)U(x1, x1 + x2, x3)α(x3)U(x1 + x2, x2, x3)α(x3) = U(x1, x2, x3) ,(7)and the modular equation,U(x3, x2,−x1) β(−x3, x2, x1)U(x1,−x3, x2) =(8)β(x2, x3, x1)U(x1, x2, x3) β(x1, x2, x3) .Proof of the q-heat equation.e−πi λ24x3∫e−πi λν2x3(e−πi ν24x3∫eπi νµ2x3 e−πi µ24x3 f(−µ) dµ)dν =∫(∫e−πi (λ+µ−ν)24x3 dν)e−πi λµ2x3 f(−µ) dµ =const(x3)∫e−πi λµ2x3 f(−µ) dµ .Proof of the modular equation.∫eπi λν2x1 eπi ν24x3x1x2(∫eπi νµ2x2 f(−µ) dµ)dν =∫(∫eπi 14x1x2x3(x1µ+x2λ+x3ν)2 dν)e−πi4(λ2x2x1x3+2λµx3+µ2x1x2x3)f(−µ) dµ =const(x1, x2, x3) e−πiλ24x2x1x3∫e−πi λµ2x3 e−πiµ24x1x2x3 f(−µ) dµ .8 G. FELDER AND A. VARCHENKOSimilarly, one can quantize equation (2) for m = 1,2πiκ∂v∂τ(λ, τ) =∂2v∂λ2(λ, τ) + 2ρ′(λ, τ) v(λ, τ) .(9)Introduce a functionu(λ, µ, τ, p, η) = e−πiλµ2η∫ 10Ω2η(t, τ, p)θ(λ+ t, τ)θ(µ + t, p)θ(t− 2η, τ)θ(t− 2η, p) dt(10)whereΩa(t, τ, p) =∞∏j,k=0(1− e2πi(t−a+jτ+kp))(1− e2πi(−t−a+(j+1)τ+(k+1)p))(1− e2πi(t+a+jτ+kp))(1− e2πi(−t+a+(j+1)τ+(k+1)p)) .This is the analog for m = 1 of function (6).Define the translation operator for m = 1 asT (τ, τ + p) : f(λ) 7→ − 14π√iηe−πiλ24η∫ηRu(λ, µ, τ, τ + p, η) ×θ(4η, τ + p)θ′(0, τ + p)θ(µ− 2η, τ + p)θ(µ+ 2η, τ + p) e−πiµ24η f(−µ) dµ .The qKZB-heat equation is the equationv(λ, τ) = (T (τ, τ + p)v)(λ, τ + p) .(11)It turns out that the translation operator for m = 1 has properties analogous to theproperties of the Fourier transform, see [FV3, FV4, FV5]. Namely,1. The semiclassical limit of equation (11) is equation (9).2. For every µ ∈ C, the function u in (10) is a projective solution of the qKZB-heatequation (11),u(λ, µ, τ, p, η) = e−πiµ24η (T (τ, τ + p)u)(λ, µ, τ + p, p, η),in particular, v(λ, µ, τ, p, η) = eπiµ24ητpu(λ, µ, τ, p, η) is a true solution.3. The integral operator T has an SL(3,Z) symmetry similar to the SL(3,Z) symmetryof the Fourier transform.Remark. The fact that u is a ( projective ) solution of the qKZB-heat equation is arelation of the formu(λ, ν, τ, p) = u(λ, µ, τ, τ + p) ∗ u(µ, ν, τ + p, p)(12)where ∗ is a suitable convolution. If τ, p tend to infinity, then the function u has atrigonometric limit. In this limit, equation (12) becomes a simplest example of theMacdonald-Mekhta identity which has the form u(λ, ν) = u(λ, µ) ∗ u(µ, ν). The theoryof the qKZB-heat equation is an elliptic analog of Macdonald’s theory, see [EV1].SPECIAL FUNCTIONS, CONFORMAL BLOCKS, BETHE ANSATZ, AND SL(3,Z) 9Remark. We haveΩa(t, τ, p) =Γ(t+ a, τ, p)Γ(t− a, τ, p)whereΓ(t, τ, p) =∞∏j,k=01− e2πi(−t+(j+1)τ+(k+1)p)1− e2πi(t+jτ+kp)is the elliptic gamma function. The elliptic gamma function has an SL(3,Z) symmetry[FV5] based on the following two main equations, which we callthe q-heat equation,Γ(t+ τ, τ, τ + p) Γ(t, τ + p, p) = Γ(t, τ, p),and the modular equation,Γ(tp,τp,−1p) = eiπQ(t;τ,p) Γ(t− pτ,−1τ,−pτ) Γ(t, τ, p),whereQ(t; τ, p) =13τpt3 − τ + p− 12τpt2 +τ 2 + p2 + 3τp− 3τ − 3p+ 16τpt+112(τ + p− 1)(τ−1 + p−1 − 1),cf. (7), (8).The q-heat and modular equations for the gamma function imply the SL(3,Z) sym-metry of the translation operator for m = 1.References[EFK] P. Etingof, I. Frenkel, A. Kirillov, Spherical Functions on Affine Lie Groups,Duke Math. J. 80(1), (1995) 59-90.[EV1] P. Etingof, A. Varchenko, Traces of Intertwiners for Quantum Groups and Dif-ference Equations, I, QA/9907181.[FV1] G.Felder and A.Varchenko, Integral representation of solutions of the ellipticKnizhnik-Zamolodchikov-Bernard equations, Int. Math. Res. notices, 1995, N. 5,221-233.[FTV] G.Felder, V.Tarasov, A.Varchenko, Monodromy of solutions of the elliptic quan-tum Knizhnik-Zamolodchikov-Bernard difference equations, preprint, 1997, 1-26,accepted for publication in Int. Journal of Math.10 G. FELDER AND A. VARCHENKO[FV2] G.Felder, A.Varchenko, Three formulas for eigenfunctions of integrableSchrodinger operators, Compositio Math. 107 (1997), 143- 175.[FV3] G.Felder, A.Varchenko, The q-Deformed KZB-heat Equation, math.QA/9809139[FV4] G. Felder and A. Varchenko, The q-KZB heat equation, modular transformations,and GL(3,Z), in preparation.[FV5] G.Felder, A.Varchenko, Resonance Relations for Solutions of the Elliptic QKZBEquations, Fusion Rules, and Eigenvectors of Transfer Matrices of RestrictedInteraction-round-a-face Models, Communications in Contemporary Mathematics,Vol. 1, No. 3 (1999) 335-403.[FV6] G.Felder, A.Varchenko, The Elliptic Gamma Function and SL(3,Z) × Z3,math.QA/9907061[FW] G. Felder and C. Wieczerkowski, Conformal blocks on elliptic curves and theKnizhnik-Zamolodchikov-Bernard equation, Commun. Math. Phys. 176 (1996),133-162.[M] J. Milnor, Introduction to algebraic K-theory, Annals of Mathematics Studies, No.72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo,1971[WW] E.T. Whittaker and G.N. Watson, Modern Analysis, Cambridge Univ. Press,Cambridge, 1927.

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