A Tannakian Interpretation of the Elliptic Infinitesimal Braid Lie Algebras

Abstract

Let n ≥ 1. The pro-unipotent completion of the pure braid group of n points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve E, and an appropriate flat connection on the configuration space of n points in E (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space E[superscript #] of an affine line bundle over E, which identifies with the moduli space of line bundles over E equipped with a flat connection.