Introduction to moduli spaces and Dirac geometry

Abstract

Let $G$ be a Lie group, with an invariant metric on its Lie algebra $\mathfrak{g}$. Given a surface $\hat{I}\pounds$ with boundary, and a collection of base points $\mathcal{V}\subset \hat{I}\pounds$ meeting every boundary component, the moduli space (representation variety) ${\mathcal{M}}_G(\hat{I}\pounds ,\mathcal{V})$ 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.

50 pages