A Tannakian interpretation of the elliptic infinitesimal braid Lie algebras

TL;DR

This paper provides a Tannakian interpretation of the elliptic infinitesimal braid Lie algebras, connecting the algebraic structures with geometric and categorical frameworks via the Riemann-Hilbert correspondence.

Contribution

It offers a novel Tannakian perspective on the elliptic infinitesimal braid Lie algebras, linking algebraic and geometric approaches through the Riemann-Hilbert framework.

Findings

Isomorphism between pro-unipotent completion of pure braid group and explicit pro-unipotent group

Interpretation of the isomorphism via Riemann-Hilbert correspondence

Identification of the moduli space of line bundles with flat connections

Abstract

Let $n\geq 1$. 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 $E^\#$ of an affine line bundle over $E$, which identifies with the moduli space of line bundles over $E$ equipped with a flat connection.