Generalized classical dynamical Yang-Baxter equations and moduli spaces of flat connections on surfaces

TL;DR

This paper introduces a method to derive generalized dynamical r-matrices via (quasi-)Poisson reduction, leading to new Poisson structures and a novel finite-dimensional description of the Atiyah-Bott symplectic structure on moduli spaces.

Contribution

It presents a new approach to obtain generalized dynamical r-matrices and applies it to describe the symplectic structure on moduli spaces of flat connections.

Findings

New examples of Poisson structures and groupoid actions

Finite-dimensional description of Atiyah-Bott symplectic structure

Connection between gauge fixing and dynamical r-matrices

Abstract

In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new finite dimensional description of the Atiyah-Bott symplectic structure on the moduli space of flat connections on a surface.