Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix
V. V. Fock, A. A. Rosly
TL;DR
This work constructs a finite-dimensional, combinatorial framework for the Poisson geometry of moduli spaces of flat $G$-connections on Riemann surfaces with boundary. By decorating a ciliated fat graph with $r$-matrices and equipping the graph-connection space ${\cal A}^l$ with a compatible Poisson structure, the authors realize ${\cal M}$ as the Poisson quotient ${\cal A}^l/{\cal G}^l$, with leaves labeled by boundary monodromies and a clear link to affine/Kac–Moody structures. The paper also provides explicit Poisson brackets, shows the preservation of Poisson structure under key graph operations, and connects to representation-theoretic bases for functions on the moduli space, offering a robust route toward quantization and connections to WZNW conformal blocks. Overall, it develops a rigorous, graph-based reduction framework for moduli spaces that integrates Poisson–Lie group actions, r-matrix data, and topological surface features.
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.