Commentarii Mathematici Helvetici


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Volume 87, Issue 2, 2012, pp. 409–431
DOI: 10.4171/CMH/258

Published online: 2012-04-17

Multicurves and regular functions on the representation variety of a surface in SU(2)

Laurent Charles[1] and Julien Marché[2]

(1) Université Pierre et Marie Curie VI, Paris, France
(2) École Polytechnique, Palaiseau, 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

Charles Laurent, Marché Julien: Multicurves and regular functions on the representation variety of a surface in SU(2). Comment. Math. Helv. 87 (2012), 409-431. doi: 10.4171/CMH/258