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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.