On the universal ellipsitomic KZB connection
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On the universal ellipsitomic KZB connection Damien Calaque1,2 · Martin Gonzalez3 Accepted: 20 September 2020 © Springer Nature Switzerland AG 2020
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 -structured elliptic curves with marked points, where = Z/MZ × Z/N Z, and M, N ≥ 1 are two integers. It restricts to a flat connection on -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. Mathematics Subject Classification 32G34 · 14H52 · 14H15 · 14H30 · 14F35 · 20C08
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bundles with flat connections on -twisted configuration spaces 2 Lie algebras of derivations and associated groups . . . . . . . . 3 Bundles with flat connections on moduli spaces . . . . . . . . . 4 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Formality of subgroups of the pure braid group on the torus . . 6 Representations of Cherednik algebras . . . . . . . . . . . . . Appendix A: Conventions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Damien Calaque [email protected] Martin Gonzalez [email protected]
1
IMAG, Univ Montpellier, CNRS, Montpellier, France
2
Institut Universitaire de France, Paris, France
3
I2M, Aix-Marseille Université, 39 rue F. Joliot Curie, 13453 Marseille, France 0123456789().: V,-vol
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D. Calaque, M. Gonzalez
List of symbols Groups PBn PBnM Gn ¯ n G SL2 (Z) PB1,n B1,n PB1,n
Pure braid group on the complex plane M-decorated pure braid group on the cylinder Structure group of the principal bundle over M1,n ¯ Structure group of the principal bundle over M 1,n -level principal congruence subgroup of SL2 (Z) -decorated pure braid group on the torus Braid group on the torus Pure braid group on the torus
Spaces Conf(C, n) Configuration space of n points in C Conf(C× , n) Configuration space of n points in C× Conf(C× , n, M)M-decorated configuration space of n points in C× Conf(T, n) Configuration spa
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