Twisted moduli spaces and Duistermaat-Heckman measures

Following Boalch-Yamakawa, Li-Bland-Severa 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 (sic) 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 bar right arrow hg kappa (h(-1)) for kappa is an element of 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. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

The article introduces a method of considering a certain class of moduli spaces on bordered surfaces from a quasi-Hamiltonian perspective, by parameterizing flat G-connections to describe twisted local systems, constructing a Duistermaat-Heckman measure that is invariant under twisted conjugation, and characterizing it using a localization formula for its Fourier coefficients.