Moduli spaces of flat connections and Morita equivalence of quantum tori

TL;DR

This paper explores the geometric structure of moduli spaces of flat connections with boundary conditions, linking them to Poisson-Lie groups and extending Morita equivalence to quantum tori.

Contribution

It provides a geometric interpretation of symplectic structures on moduli spaces and extends Morita equivalence of quantum tori within the framework of Poisson-Lie groups.

Findings

Explicit formula for symplectic form via holonomies

Connection between moduli spaces and Poisson-Lie group structures

Extension of Morita equivalence to quantum tori in Poisson-Lie context

Abstract

We study moduli spaces of flat connections on surfaces with boundary, with boundary conditions given by Lagrangian Lie subalgebras. The resulting symplectic manifolds are closely related with Poisson-Lie groups and their algebraic structure (such as symplectic groupoid structure) gets a geometrical explanation via 3-dimensional cobordisms. We give a formula for the symplectic form in terms of holonomies, based on a central extension of the gauge group by closed 2-forms. This construction is finally used for a certain extension of the Morita equivalence of quantum tori to the world of Poisson-Lie groups.