Hodge theory of the Goldman bracket

TL;DR

This paper demonstrates that the Goldman bracket on loops on a complex algebraic curve, after I-adic completion, respects mixed Hodge structures, linking algebraic topology with Hodge theory and addressing the Kashiwara--Vergne problem.

Contribution

It establishes the compatibility of the Goldman bracket and related structures with mixed Hodge theory, providing new torsors and partial solutions to the Kashiwara--Vergne problem.

Findings

Goldman bracket respects mixed Hodge structures after completion

Constructs torsors of isomorphisms linking Goldman Lie algebra and its graded version

Provides partial solutions to the Kashiwara--Vergne problem in all genera

Abstract

In this paper we show that, after completion in the I-adic topology, the Goldman bracket on the space spanned by homotopy classes of loops on a smooth, complex algebraic curve is a morphism of mixed Hodge structure. We prove similar statements for the natural action (defined by Kawazumi and Kuno) of the loops in X on paths from one "boundary component" to another. These results are used to construct torsors of isomorphisms of the the completed Goldman Lie algebra with the completion of its associated graded Lie algebra. Such splittings give torsors of partial solutions to the Kashiwara--Vergne problem (arXiv:1611.05581) in all genera. Compatibility of the cobracket with Hodge theory is established in arXiv:1807.09209.