Symplectic structure of the moduli space of flat connections on a Riemann surface

TL;DR

This paper derives an explicit formula for the symplectic structure on the moduli space of flat connections on a Riemann surface, linking it to Kirillov forms and Poisson-Lie group structures.

Contribution

It provides a new explicit formula for the symplectic form on the moduli space, connecting it to Poisson-Lie groups and their symplectic structures.

Findings

Explicit formula for the symplectic form on the moduli space.

Representation of the form as sums of Kirillov forms and structures on the Heisenberg double.

Connection between the moduli space structure and Poisson-Lie group theory.

Abstract

We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum of $n$ copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie group $G^{*}$ and $g$ copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group $G$ (the pair ($G,G^{*}$) corresponds to the Lie algebra ${\g}$).