Symplectic and Poisson geometry of the moduli spaces of flat connections over quilted surfaces

TL;DR

This paper explores the symplectic and Poisson structures of moduli spaces of flat connections on quilted surfaces, introducing a new Dirac geometry-based reduction method that unifies various important geometric examples.

Contribution

It develops a novel moment-map reduction framework in Dirac geometry, enabling analysis of moduli spaces with varying structure groups and boundaries in a unified way.

Findings

Constructed moduli spaces include Poisson Lie groups and homogeneous spaces.

Unified approach to symplectic and Poisson geometry of these moduli spaces.

Provided new geometric insights into important examples like meromorphic connections.

Abstract

In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective. The moduli spaces we construct in this way include a number of important examples, including Poisson Lie groups and their Homogeneous spaces, moduli spaces for meromorphic connections over Riemann surfaces (following the work of Philip Boalch), and various symplectic groupoids. Realizing these examples as moduli spaces for quilted surfaces provides new insights into their geometry.