Dual generators of the fundamental group and the moduli space of flat connections

Abstract

We define the dual of a set of generators of the fundamental group of an oriented two-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 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 to 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 of infinitesimal Dehn twists and show that the mapping class group acts by Poisson isomorphisms.

arXiv comment54 pages, 13 .eps figures

References in corpus (8)

• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
• Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
• Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity
• Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
• Mapping class group actions in Chern-Simons theory with gauge group $G⋉{\mathfrak{g}}^{\ast }$
• Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant

Cited by in corpus (4)

• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction
• Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
• Superintegrable Systems on Moduli Spaces of Flat Connections