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 \(M,N\ge 1\) 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.

我们构造了Calaque-Enriquez-Etingof引入的一种通用的Knizhnik-Zamolodchikov-Bernard（KZB）连接的扭曲版本，我们将其称为椭圆KZB连接。这是主束上位于具有标记点的\（\ Gamma \）结构的椭圆曲线的模空间上的平坦连接，其中\（\ Gamma = {{\ mathbb {Z}}} / M {{\ mathbb { Z}}} \ times {{\ mathbb {Z}}} / N {{\ mathbb {Z}}} \）和\（M，N \ ge 1 \）是两个整数。它限制为\（\ Gamma \）上的平面连接椭圆曲线上的点的加捻配置空间，可用于为圆环上纯辫子组的一些有趣子组构造过滤形式同构。我们证明了通用的椭圆KZB连接实现为与具有谱参数的椭圆动力学r矩阵相关联的常规KZB连接，最后还产生了环原子Cherednik代数的表示。