Twisted Moduli Spaces and Duistermaat-Heckman Measures

TL;DR

This paper develops a framework for analyzing twisted moduli spaces of flat connections on bordered surfaces using quasi-Hamiltonian geometry, introducing a new Duistermaat-Heckman measure invariant under twisted conjugation and providing explicit localization formulas.

Contribution

It constructs and characterizes a new Duistermaat-Heckman measure for twisted moduli spaces, extending the quasi-Hamiltonian approach to twisted local systems and automorphism-invariant measures.

Findings

Defined a twisted DH measure invariant under automorphism-based conjugation

Derived a localization formula for Fourier coefficients of the measure

Computed DH measures for specific twisted moduli spaces

Abstract

Following Boalch-Yamakawa and Meinrenken, we consider a certain class of moduli spaces on bordered surfaces from a quasi-Hamiltonian perspective. For a given Lie group $G$, these character varieties parametrize flat $G$-connections on "twisted" local systems, in the sense that the transition functions take values in $G\rtimes\mathrm{Aut}(G)$. After reviewing the necessary tools to discuss twisted quasi-Hamiltonian manifolds, we construct a Duistermaat-Heckman (DH) measure on $G$ that is invariant under the twisted conjugation action $g\mapsto hg\kappa(h^{-1})$ for $\kappa\in\mathrm{Aut}(G)$, and characterize it by giving a localization formula for its Fourier coefficients. We then illustrate our results by determining the DH measures of our twisted moduli spaces.