On the variation of the Poisson structures of certain moduli spaces

TL;DR

This paper constructs a Poisson algebra on a space of surface group representations into a Lie group, describing how symplectic structures vary with boundary conditions, advancing understanding of moduli space Poisson structures.

Contribution

It introduces a Poisson algebra framework for representation spaces with boundary constraints, elucidating the variation of stratified symplectic structures in moduli spaces.

Findings

Constructed a Poisson algebra on a subspace of representations.

Described how Poisson structures vary with boundary conjugacy classes.

Connected Poisson algebra to stratified symplectic structures.

Abstract

Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the fundamental group of a compact connected orientable topological surface with finitely many boundary circles; when G is compact and connected, R may be taken dense in the space of all representations. The space R contains spaces of representations where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and, on these representation spaces, the Poisson algebra restricts to stratified symplectic Poisson algebras constructed elsewhere earlier. Hence the Poisson algebra on R gives a description of the variation of the stratified symplectic Poisson structures on the smaller representation spaces as the chosen conjugacy classes move.