The real locus of an involution map on the moduli space of flat connections on a Riemann surface

TL;DR

This paper explores the relationship between moduli spaces of flat connections on nonorientable surfaces and their orientable double covers, revealing conditions for isomorphism and constructing minimal Lagrangian submanifolds.

Contribution

It identifies the fixed point set of an involution on the moduli space and establishes when it is isomorphic to the original moduli space, providing new geometric insights.

Findings

M is isomorphic to the fixed point set of alM if and only if the center of G has odd order.

A method to construct minimal Lagrangian submanifolds in the moduli space alM.

Relation between nonorientable surfaces and their orientable double covers in the context of flat G-connections.

Abstract

It is known that every nonorientable surface $\Sigma$ has an orientable double cover $\tilde{\Sigma}$. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat $G$-connections on $\tilde{\Sigma}$. We identify the relation between the moduli space $\M$ and the fixed point set of the moduli space $\tilde{\M}$. In particular, $\M$ is isomorphic to the fixed point set of $\tilde{\M}$ if and only if the order of the center of $G$ is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space $\tilde{\M}$.