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 Knizhnik–Zamolodchikov–Bernard equations [1][9]. Our system completely determines the $N$-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 $\widehat{\mathfrak{sl}}(2,\mathbb{C})$-characters.