Mapping class group actions in Chern-Simons theory with gauge group $G⋉{\mathfrak{g}}^{\ast }$

Abstract

We study the action of the mapping class group of an oriented genus g surface with n punctures and a disc removed on a Poisson algebra which arises in the combinatorial description of Chern-Simons gauge theory when the gauge group is a semidirect product $G⋉{\mathfrak{g}}^{\ast }$. We prove that the mapping class group acts on this algebra via Poisson isomorphisms and express the action of Dehn twists in terms of an infinitesimally generated G-action. We construct a mapping class group representation on the representation spaces of the associated quantum algebra and show that Dehn twists can be implemented via the ribbon element of the quantum double D(G) and the exchange of punctures via its universal R-matrix.

arXiv comment30 pages, 5 eps figures; corrections concerning the mapping class group which acts on the Poisson algebra discussed in the paper

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• Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
• Combinatorial quantisation of the Euclidean torus universe
• Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant
• Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry
• The Hilbert space of 3d gravity: quantum group symmetries and observables
• Gauge fixing and classical dynamical r-matrices in ISO(2,1)-Chern-Simons theory
• Dual generators of the fundamental group and the moduli space of flat connections
• Mapping class group and U(1) Chern-Simons theory on closed orientable surfaces