# On the universal ellipsitomic KZB connection

## TL;DR

This paper introduces a new twisted elliptic KZB connection on moduli spaces of elliptic curves with additional structure, linking it to dynamical r-matrices and cyclotomic Cherednik algebras, expanding the mathematical framework of flat connections.

## Contribution

It constructs the ellipsitomic KZB connection, a novel flat connection generalizing the universal KZB connection to include $ ext{Z}/M ext{Z} imes ext{Z}/N ext{Z}$-structures, and relates it to dynamical r-matrices and Cherednik algebras.

## Key findings

- Realizes the ellipsitomic KZB connection as the usual KZB connection with elliptic dynamical r-matrices.
- Provides a filtered-formality isomorphism for certain subgroups of the pure braid group on the torus.
- Produces representations of cyclotomic Cherednik algebras.

## Abstract

We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of $\Gamma$-structured elliptic curves with marked points, where $\Gamma=\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$, and $M,N\geq1$ are two integers. It restricts to a flat connection on $\Gamma$-twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subgroups of the pure braid group on the torus. We show that the universal ellipsitomic KZB connection realizes as the usual KZB connection associated with elliptic dynamical $r$-matrices with spectral parameter, and finally, also produces representations of cyclotomic Cherednik algebras.