Dual generators of the fundamental group and the moduli space of flat connections

TL;DR

This paper introduces a dual set of generators for the fundamental group of a punctured surface, providing a new algebraic framework to analyze the moduli space of flat connections and its Poisson structure, with applications to Wilson loops and mapping class group actions.

Contribution

It defines a dual generator set related by an involution, simplifying the Poisson structure on the moduli space and enabling explicit computation of Poisson brackets and mapping class group actions.

Findings

Dual generators relate to intersection points of curves on surfaces.

Poisson structure simplifies when expressed in terms of generators and their duals.

Mapping class group acts by Poisson isomorphisms on the moduli space.

Abstract

We define the dual of a set of generators of the fundamental group of an oriented two-surface $S_{g,n}$ of genus $g$ with $n$ punctures and the associated surface $S_{g,n}\setminus D$ with a disc $D$ removed. This dual is another set of generators related to the original generators via an involution and has the properties of a dual graph. In particular, it provides an algebraic prescription for determining the intersection points of a curve representing a general element of the fundamental group $\pi_1(S_{g,n}\setminus D)$ with the representatives of the generators and the order in which these intersection points occur on the generators.We apply this dual to the moduli space of flat connections on $S_{g,n}$ and show that when expressed in terms both, the holonomies along a set of generators and their duals, the Poisson structure on the moduli space takes a particularly simple form. Using this description of the Poisson structure, we derive explicit expressions for the Poisson brackets of general Wilson loop observables associated to closed, embedded curves on the surface and determine the associated flows on phase space. We demonstrate that the observables constructed from the pairing in the Chern-Simons action generate of infinitesimal Dehn twists and show that the mapping class group acts by Poisson isomorphisms.