Dual generators of the fundamental group and the moduli space of flat connections - Meusburger, Cathérine (Journal of Physics A: Mathematical and General, 2006)
We define the dual of a set of generators of the fundamental group of an oriented 2-surface S of genus g with n punctures and the associated surface Swith a disc D removed. This dual is another set of generators related to the original generators via an involution and has the properties of a dual graph. In particular, it provides an algebraic prescription for determining the intersection points of a curve representing a general element of the fundamental group π(S) with the representatives of the generators and the order in which these intersection points occur on the generators. We apply this dual to the moduli space of flat connections on S and show that when expressed in terms of both, the holonomies along a set of generators and their duals, the Poisson structure on the moduli space takes a particularly simple form. Using this description of the Poisson structure, we derive explicit expressions for the Poisson brackets of general Wilson loop observables associated with closed, embedded curves on the surface and determine the associated flows on phase space. We demonstrate that the observables constructed from the pairing in the Chern-Simons action generate infinitesimal Dehn twists and show that the mapping class group acts by Poisson isomorphisms. © 2006 IOP Publishing Ltd.