Character variety

In the mathematics of moduli theory, given an algebraic, reductive, Lie group

is a space of equivalence classes of group homomorphisms from

by conjugation, and two homomorphisms are defined to be equivalent (denoted

This is the weakest equivalence relation on the set of conjugation orbits,

Formally, and when the reductive group is defined over the complex numbers

-character variety is the spectrum of prime ideals of the ring of invariants (i.e., the affine GIT quotient).

Here more generally one can consider algebraically closed fields of prime characteristic.

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.

In particular, a maximal compact subgroup generally gives a semi-algebraic set.

An interesting class of examples arise from Riemann surfaces: if

, or Betti moduli space, is the character variety of the surface group

is the Riemann sphere punctured three times, so

is free of rank two, then Henri G. Vogt, Robert Fricke, and Felix Klein proved[1][2] that the character variety is

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

is the Riemann sphere punctured four times, so

given by the equation This character variety appears in the theory of the sixth Painleve equation,[3] and has a natural Poisson structure such that

are Casimir functions, so the symplectic leaves are affine cubic surfaces of the form

This construction of the character variety is not necessarily the same as that of Marc Culler and Peter Shalen (generated by evaluations of traces), although when

they do agree, since Claudio 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

-character variety is the torus But the trace algebra is a strictly small subalgebra (there are fewer invariants).

This provides an involutive action on the torus that needs to be accounted for to yield the Culler–Shalen character variety.

There is an interplay between these moduli spaces and the moduli spaces 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 of flat connections.

In other words, with respect to a base space

for the bundles or a fixed topological space for the geometric structures, the holonomy homomorphism is a group homomorphism from

[citation needed] The coordinate ring of the character variety has been related to skein modules in knot theory.

[4][5] 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.