Universal KZB Equations: The Elliptic Case

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 is realized as the usual KZB connection for simple Lie algebras, and that in the $$\mathfrak {sl}_n$$ case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant D-modules on $$\mathfrak {sl}_n$$ to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over $$\mathfrak {sl}_n$$ that are supported on the nilpotent cone.