Character variety

In the mathematics of moduli theory, given an algebraic, reductive, Lie group $G$ and a finitely generated group $\pi$ , the $G$ -character variety of $\pi$ is a space of equivalence classes of group homomorphisms

\mathfrak{R}=\operatorname{Hom}(\pi,G).

More precisely, $G$ acts on $\mathfrak{R}$ by conjugation and two homomorphisms are defined to be equivalent if and only if their orbit closures intersect. This is the weakest equivalence relation on the set of conjugation orbits that yields a Hausdorff space.

Formulation

Formally, and when the algebraic group is defined over the complex numbers $\mathbb{C}$ , the $G$ -character variety is the spectrum of prime ideals of the ring of invariants (i.e., the GIT quotient)

\mathbb{C}[\operatorname{Hom}(\pi,G)]^G.

Here more generally one can consider algebraically closed fields of prime characteristic. In this generality, character varieties are only algebraic sets and are not actual varieties. To avoid technical issues, one often considers the associated reduced space by dividing by the radical of 0 (eliminating nilpotents). However, this does not necessarily yield an irreducible space either. Moreover, if we replace the complex group by a real group we may not even get an algebraic set. In particular, a maximal compact subgroup generally gives a semi-algebraic set. On the other hand, whenever $\pi$ is free we always get an honest variety; it is singular however.

Examples

For example, if $G=\mathrm{SL}(2,\mathbb{C})$ and $\pi$ is free of rank two, then the character variety is $\mathbb{C}^3$ since by the Fricke-Klein-Vogt theorem its coordinate ring is isomorphic to the complex polynomial ring in 3 variables $\mathbb{C}[x,y,z]$ . Restricting to $G=\mathrm{SU}(2)$ gives a closed real three-dimensional ball (semi-algebraic, but not algebraic).

Another example, also studied by Vogt and Fricke-Klein is the case with $G=\mathrm{SL}(2,\mathbb{C})$ and $\pi$ is free of rank three. Then the character variety is isomorphic to the hypersurface in $\mathbb{C}^7$ given by the equation $a^2+b^2+c^2+d^2 + x^2+y^2+z^2 -(ab+cd)x-(ad+bc)y-(ac+bd)z +abcd+xyz-4 = 0.$

Variants

This is not necessarily the same construction as the Culler-Shalen character variety (generated by evaluations of traces), although when the $G=\mathrm{SL}(n,\mathbb{C})$ they do agree since Procesi has shown that in this case the ring of invariants is in fact generated by only traces. Since trace functions are invariant by all inner automorphisms, the Culler-Shalen construction essentially assumes that we are acting by

G=\mathrm{SL}(n,\mathbb{C})

\mathfrak{R}=\operatorname{Hom}(\pi,H)

even if

G\not=H.

For instance, for a free group of rank 2 and $G=\mathrm{SO}(2)$ the conjugation action is trivial and the $G$ -character variety is the torus

S^1\times S^1.

But the trace algebra is a strictly small subalgebra (there are less invariants). This provides an involutive action on the torus that needs to be accounted for to yield the Culler-Shalen character variety. The involution on this torus yields a 2-sphere. The point is that up to $\mathrm{SO}(2)$ -conjugation all points are distinct, but the trace identifies elements with differing anti-diagonal elements (the involution).

Connection to geometry

There is an interplay between these moduli and the moduli of principal bundles, vector bundles, Higgs bundles, and geometric structures on topological spaces, given generally by the observation that, at least locally, equivalent objects in these categories are parameterized by conjugacy classes of holonomy homomorphisms. In other words, with respect to a base space $M$ for the bundles or a fixed topological space for the geometric structures the holonomy homomorphism is a group homomorphism between $\pi_1(M)$ and the structure group $G$ of the base space.

Connection to skein modules

The coordinate ring of the character variety has been related to skein modules in knot theory.^[1]^[2] The skein module is roughly a deformation (or quantization) of the character variety. It is closely related to topological quantum field theory in dimension 2+1.