Differential Equations Associated to the $SU(2)$ WZNW Model on Elliptic Curves

Abstract

We study the SU(2] WZNW model over a family of elliptic curves. Starting from the formulation developed in [13], we derive a system of differential equations which contains the Knizhmk-Zamolodchikov-Bernard equations [1] [9]. Our system completely determines the AT-point functions and is regarded as a natural elliptic analogue of the system obtained in [12] for the projective line. We also calculate the system for the 1-point functions explicitly. This gives a generalization of the results in [7] for si (2, C)-characters. § 0. IntroductionWe consider the Wess-Zumino-Novikov-Witten (WZNW) model. A mathematical formulation of this model on general algebraic curves is given in [13], 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 1 , the correlation functions are realized more explicitly in [12] 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 [11]. 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 [1]. Furthermore we write down this system explicitly in the 1-pointed case.To explain more precisely, first let us review the formulation in [13] roughly. Let g be a simple Lie algebra over C and g the corresponding affine Lie algebra. We fix a positive integer / (called the level) and consider the integrable highest weight modules of g of level /. Such modules are parameterized by the set of highest weight P £ and we denote by J-f A the left module corresponding to leP^. By M 9tN we denote the moduli space of Communicated by T. Miwa, February 9, 1995. 1991 Mathematical Subject Classification (s):

Spin-–spin- two-point functions on the torus

“…the characters, were determined in [11]. The one-point functions [12] vanish as they are not singlets under SU (2). The two-point functions were determined at level k = 1, 2 by the identification of the SU (2) WZW model with free field theories [13], by solving appropriate differential equations [14,15,16], and by pinching genus-2 characters [17,18].…”

On non-stationary Lamé equation from WZW model and spin-1/2 XYZ chain

“…Later on, the authors of [7,8,9] further extended this direction to the elliptic case. Certainly, their works are based on important investigations on both XYZ Gaudin model [10] and conformal field theory (CFT) on elliptic curves [11,12,13].…”