Conformal blocks on elliptic curves and the Knizhnik--Zamolodchikov--Bernard equations

TL;DR

This paper explicitly describes the vector bundle of WZW conformal blocks on elliptic curves, relates it to theta functions, and connects the flat connection to solutions of the generalized Knizhnik--Zamolodchikov equations.

Contribution

It provides an explicit description of conformal blocks on elliptic curves and links the Friedan--Shenker connection to Bernard's generalized KZ equations.

Findings

Explicit description of conformal blocks as theta function subbundles.

Conjectural characterization of the subbundle via hyperplane conditions.

Verification of the conjecture through explicit calculations.

Abstract

We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculation are possible and confirm the conjecture. The Friedan--Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik--Zamolodchikov equation.