Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix

TL;DR

This paper explores the Poisson structure on the moduli space of flat connections on Riemann surfaces with boundary, using graph connections and Poisson-Lie groups, linking to r-matrix theory.

Contribution

It demonstrates how the moduli space can be realized as a quotient of graph connection space via a Poisson action of a Poisson-Lie gauge group.

Findings

Moduli space obtained as a Poisson quotient of graph connections.

Poisson structure described using ciliated fat graphs.

Connection to r-matrix formalism.

Abstract

We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex.) Our aim is however to study the Poisson structure on the moduli space of locally flat vector bundles on a Riemann surface with holes (i.e. with boundary). It is shown that this moduli space can be obtained as a quotient of the space of graph connections by the Poisson action of a lattice gauge group endowed with a Poisson-Lie structure. The present paper contains as a part an updated version of a 1992 preprint ITEP-72-92 which we decided still deserves publishing. We have removed some obsolete inessential remarks and added some newer ones.