Dual generators of the fundamental group and the moduli space of flat connections. (English) Zbl 1107.53057
Summary: We define the dual of a set of generators of the fundamental group of an oriented 2-surface \(S_{g,n}\) of genus g with n punctures and the associated surface \(S_{g,n}\setminus D\) with 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 \(\pi _{1}(S_{g,n}\setminus D)\) 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 \(S_{g,n}\) 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.
MSC:
53D30 | Symplectic structures of moduli spaces |
20M99 | Semigroups |
53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
81T13 | Yang-Mills and other gauge theories in quantum field theory |