Goldman flows on a nonorientable surface

TL;DR

This paper defines and analyzes Goldman flows on nonorientable surfaces, relating them to flows on their orientable double covers, and shows that certain flows preserve Lagrangian submanifolds.

Contribution

It introduces a gauge theoretic definition of Goldman flows on nonorientable surfaces and relates these flows to flows on their orientable double covers.

Findings

The Goldman flow on a nonorientable surface is related to the composite flow on its double cover.

The composite flow preserves a specific Lagrangian submanifold.

Explicit descriptions of the flow dynamics are provided.

Abstract

Given an embedded cylinder in an arbitrary surface, we give a gauge theoretic definition of the associated Goldman flow, which is a circle action on a dense open subset of the moduli space of equivalence classes of flat SU(2)-connections over the surface. A cylinder in a compact nonorientable surface lifts to two cylinders in the orientable double cover, and the "composite flow" is the composition of one of the associated flows with the inverse flow of the other. Providing explicit descriptions, we relate the flow on the moduli space of the nonorientable surface with the composite flow on the moduli space of the double cover. We prove that the composite flow preserves a certain Lagrangian submanifold.