Commentarii Mathematici Helvetici
Full-Text PDF (277 KB) |
Table of Contents | CMH summary
Multicurves and regular functions on the representation variety of a surface in SU(2)
Laurent Charles (1) and Julien Marché (2)
(1) nstitut de Mathématiques de Jussieu, UMR 7586I, Université Pierre et Marie Curie VI, 4, place Jussieu , 75252, PARIS CEDEX 05, FRANCE(2) Centre de Mathématiques Laurent Schwartz, École Polytechnique, Route de Saclay, 91128, PALAISEAU CEDEX, FRANCE
Given a compact surface $\Sigma$, we consider the representation space $$ \mathcal{M}(\Sigma)= \operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2). $$ We show that the trace functions associated to multicurves on $\Sigma$ are linearly independent as functions on $\mathcal{M}(\Sigma)$. The proof relies on the Fourier decomposition of the trace functions with respect to a torus action on $\mathcal{M}(\Sigma)$ associated to a pants decomposition of $\Sigma$. Consequently the space of trace functions is isomorphic to the Kauffman skein algebra at $A=-1$ of the thickened surface.
Keywords: Representation variety, multicurve, skein algebra, Dehn coordinates, topological quantum field theory