Moduli spaces for quilted surfaces and Poisson structures

TL;DR

This paper explores the Poisson and quasi-Poisson structures on moduli spaces of flat connections over surfaces, extending the theory to quilted surfaces with variable structure groups and describing these structures via intersection forms and spin networks.

Contribution

It introduces a framework for describing Poisson structures on moduli spaces of flat connections for quilted surfaces, generalizing previous results and connecting to intersection forms and spin networks.

Findings

Describes quasi-Poisson structures via intersection forms on fundamental groupoids.

Extends Poisson structure descriptions to quilted surfaces with variable groups.

Uses spin networks to characterize the Poisson structures on quilted surfaces.

Abstract

Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G-bundle, P\to \Sigma, over a compact oriented surface, \Sigma, carries a Poisson structure. If we trivialize P over a finite number of points on the boundary of \Sigma, then the moduli space carries a quasi-Poisson structure instead. Our first result is to describe this quasi-Poisson structure in terms of an intersection form on the fundamental groupoid of the surface, generalizing results of Massuyeau and Turaev. Our second result is to extend this framework to quilted surfaces, i.e. surfaces where the structure group varies from region to region and a reduction (or relation) of structure occurs along the borders of the regions, extending results of the second author. We describe the Poisson structure on the moduli space for a quilted surface in terms of an operation on spin networks, i.e. graphs immersed in the surface which are endowed with some additional data on their edges and vertices. This extends the results of various authors.