Introduction to moduli spaces and Dirac geometry

TL;DR

This paper introduces the construction and properties of a quasi-symplectic 2-form on moduli spaces of flat G-connections on surfaces with boundary, utilizing quasi-Hamiltonian and Dirac geometric techniques.

Contribution

It provides a finite-dimensional construction of the quasi-symplectic form on moduli spaces using quasi-Hamiltonian and Dirac geometry, expanding on previous approaches.

Findings

Explicit construction of the quasi-symplectic 2-form

Analysis of its basic properties using Dirac geometry

Application of quasi-Hamiltonian techniques to moduli spaces

Abstract

Let $G$ be a Lie group, with an invariant metric on its Lie algebra $\mathfrak{g}$. Given a surface $\Sigma$ with boundary, and a collection of base points $\mathcal{V}\subset \Sigma$ meeting every boundary component, the moduli space (representation variety) $\mathcal{M}_G(\Sigma,\mathcal{V})$ carries a distinguished `quasi-symplectic' 2-form. We shall explain the finite-dimensional construction of this 2-form and discuss its basic properties, using quasi-Hamiltonian techniques and Dirac geometry. This article is an extended version of lectures given at the summer school 'Poisson 2024' at the Accademia Pontaniana in Napoli, July 2024.