Universal KZB Equations for arbitrary root systems

TL;DR

This paper constructs a universal KZB connection for any finite root system, extending previous work and revealing new algebraic structures and monodromy properties related to elliptic braid groups and Cherednik algebras.

Contribution

It introduces a universal flat connection D_R for arbitrary root systems, explicitly determines its Lie algebra, and links monodromy to elliptic braid groups and Cherednik algebras.

Findings

D_R is a flat connection with explicit Lie algebra relations.

Monodromy induces an isomorphism between elliptic pure braid groups and t_R.

Establishes a connection between double affine Hecke algebras and Cherednik algebras.

Abstract

Generalising work of Calaque-Enriquez-Etingof, we construct a universal KZB connection D_R for any finite (reduced, crystallographic) root system R. D_R is a flat connection on the regular locus of the elliptic configuration space associated to R, with values in a graded Lie algebra t_R with a presentation with relations in degrees 2, 3 and 4 which we determine explicitly. The connection D_R also extends to a flat connection over the moduli space of pointed elliptic curves. We prove that its monodromy induces an isomorphism between the Malcev Lie algebra of the elliptic pure braid group P_R corresponding to R and t_R, thus showing that P_R is not 1-formal and extending a result of Bezrukavnikov valid in type A. We then study one concrete incarnation of our KZB connection, which is obtained by mapping t_R to the rational Cherednik algebra H_{h,c} of the corresponding Weyl group W. Its monodromy gives rise to an isomorphism between appropriate completions of the double affine Hecke algebra of W and H_{h,c}.