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Universal KZB equations I: the elliptic case

D. Calaque·B. Enriquez·P. Etingof
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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.

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D. Calaque, B. Enriquez, P. Etingof. 2009-09-21. Universal KZB equations I: the elliptic case. https://arxiv.org/abs/math/0702670

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