Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

TL;DR

This paper introduces new KZB equations for WZW theory on complex curves with non-trivial bundles, explicitly constructs the connection for elliptic curves, and proves its flatness, advancing understanding of conformal blocks and their moduli.

Contribution

It provides explicit construction and proof of flatness of the KZB connection for non-trivial bundles over elliptic curves in WZW theory.

Findings

Constructed the KZB connection explicitly for elliptic curves.

Proved the flatness of the constructed KZB connection.

Extended the theory to bundles with different topological types.

Abstract

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint $G$-bundles of different topological types over complex curves $\Sigma_{g,n}$ of genus $g$ with $n$ marked points. The bundles are defined by their characteristic classes - elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ is a center of the simple complex Lie group $G$. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.