Universal KZB equations I: the elliptic case

TL;DR

This paper introduces a universal elliptic KZB connection, exploring its monodromy, relations to Eisenstein integrals, and connections to Cherednik algebras, with implications for braid groups and D-modules on sl_n.

Contribution

It defines a universal genus 1 KZB connection, analyzes its monodromy, and links it to Cherednik algebras and D-modules, extending the understanding of elliptic braid group formality.

Findings

Universal KZB connection is flat over moduli space of elliptic curves.

Monodromy relates KZ associator to Eisenstein integral series.

Connection factors through Cherednik algebras for sl_n, enabling new functors and character computations.

Abstract

We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked points. It restricts to a flat connection on configuration spaces of points on elliptic curves, which can be used for proving the formality of the pure braid groups on genus 1 surfaces. We study the monodromy of this connection and show that it gives rise to a relation between the KZ associator and a generating series for iterated integrals of Eisenstein forms. We show that the universal KZB connection realizes as the usual KZB connection for simple Lie algebras, and that in the sl_n case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant D-modules on sl_n to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over sl_n which are supported on the nilpotent cone.