Character variety

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In the mathematics of moduli theory, given a Lie group G, a G-character variety is a construction on a manifold that generalizes both moduli space and Teichmüller space when G is the manifold's structure group. Formally, if X is the manifold and $π$ represents its fundamental group, then the character variety is $\operatorname{Hom}(\pi,G)//G$ with respect to the action of simultaneous conjugation, where $/ / G$ represents the categorical quotient. Simply quotienting by G will usually not produce a variety since orbits of the G-action may not be closed. The categorical quotient 'eliminates' enough points so the quotient remains an algebraic variety.

[edit] Connection to invariant theory

The character variety is essentially an algebraic construction, since it depends only on the fundamental group of the manifold. The coordinate ring of the character variety is the ring of G-invariants: $R[\operatorname{Hom}(\pi,G)//G]\cong R[\operatorname{Hom}(\pi,G)]^G$ , and so represents a way in which classical invariant theory can be applied to geometric questions.

For example, if $G=SL(2,\mathbb{C})$ and $π$ is free of rank two, then the character variety is affine 3-space since by the Fricke-Vogt Theorem its coordinate ring is isomorphic to $\mathbb{C}[x,y,z]$ .

[edit] Connection to Skein modules

The coordinate ring of the character variety has recently been related to skein modules in knot theory. The skein module is a deformation (or quantization) of the character variety.