Moduli spaces for quilted surfaces and Poisson structures

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 with boundary, $\Sigma$, carries a Poisson structure. If we trivialize $P$ over a finite number of points on $\partial \Sigma$ 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 citeMassuyeau:2012uw,Turaev:2007jh. 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 citeSevera:2011ug,Severa98,Severa:2005vla. 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 citeGoldman:1986eh,Goldman:1984hr,Roche:2000ws,Andersen:1996ur.