Flat $GL(1|1)$-connections and fatgraphs

TL;DR

This paper explores the moduli space of flat $GL(1|1)$-connections on punctured surfaces using fatgraphs, providing explicit formulas for transformations and discussing Poisson brackets.

Contribution

It introduces a coordinate system for the moduli space involving bosonic and fermionic variables and derives explicit formulas for Whitehead moves in trivalent graphs.

Findings

Explicit formulas for Whitehead moves on trivalent graphs.

A coordinate system involving bosonic and fermionic variables.

Discussion of the invariant Poisson bracket.

Abstract

We study the moduli space of flat $GL(1|1)$-connections on a punctured surface from the point of view of graph connections. To each fatgraph, a system of coordinates is assigned, which involves two bosonic and two fermionic variables per edge, subject to certain relations. In the case of trivalent graphs, we provide a closed explicit formula for the Whitehead moves. In addition, we discuss the invariant Poisson bracket.