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On the universal ellipsitomic KZB connection

MPS-Authors
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Gonzalez,  Martin
Max Planck Institute for Mathematics, Max Planck Society;

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Fulltext (public)

arXiv:1908.03887.pdf
(Preprint), 561KB

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Citation

Calaque, D., & Gonzalez, M. (2020). On the universal ellipsitomic KZB connection. Selecta Mathematica, 29(5): 73. doi:10.1007/s00029-020-00601-6.


Cite as: http://hdl.handle.net/21.11116/0000-0007-83A5-A
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.