Differential Equations Associated to the Su(2) Wznw Model on Elliptic Curves

We study the SU(2) WZNW model over a family of elliptic curves. Starting from the formulation developed in [TUY], we derive a system of differential equations which contains the Knizhnik-Zamolodchikov-Bernard equations[Be1][FW]. Our system completely determines the N-point functions and is regarded as a natural elliptic analogue of the system obtained in [TK] for the projective line. We also calculate the system for the 1-point functions explicitly. This gives a generalization of the results in [EO2] for ŝl(2, C)-characters. §0. Introduction. We consider the Wess-Zumino-Novikov-Witten (WZNW) model. A mathematical formulation of this model on general algebraic curves is given in [TUY], where the correlation functions are defined as flat sections of a certain vector bundle over the moduli space of curves. On the projective line P, the correlation functions are realized more explicitly in [TK] as functions which take their values in a certain finite-dimensional vector space, and characterized by the system of equations containing the Knizhnik-Zamolodchikov (KZ) equations[KZ]. One aim in the present paper is to have a parallel description on elliptic curves. Namely, we characterize the N -point functions as vector-valued functions by a system of differential equations containing an elliptic analogue of the KZ equations by Bernard[Be1]. Furthermore we write down this system explicitly in the 1-pointed case. To explain more precisely, first let us review the formulation in [TUY] roughly. Let g be a simple Lie algebra over C and ĝ the corresponding affine Lie algebra. We fix a positive integer l (called the level) and consider the integrable highest weight modules of ĝ of level l. Such modules are parameterized by the set of highest weight Pl and we denote by Hλ the left module corresponding to λ ∈ Pl. By Mg,N we denote the moduli space of N -pointed curves of genus g. For X ∈ Mg,N and ~λ = (λ1, . . . , λN ) ∈ (Pl) , we associate the space of conformal blocks V† g(X;λ). The space V† g(X;λ) is the finite dimensional subspace of H ~λ := HomC(Hλ1 ⊗ · · ·⊗HλN ,C) defined by “the gauge conditions”. Consider the vector bundle Ṽ† g(λ) = ∪X∈Mg,NV g(X;λ) over Mg,N . On this vector bundle, projectively flat connections are defined through the Kodaira-Spencer theory, and flat sections of Ṽ† g(λ) with respect to these connections are called the N -point correlation functions Typeset by AMS-TEX