Category: Poisson geometry

Characters, Brackets, and Skeins

The character varieties are a remarkable family of spaces that lie at the center of many different active strands of research of the past 30 or 40 years. The basic definition is actually quite simple. We start with a group \pi , and consider the space of representations

\mathcal{R}_{n, \pi} = \mathrm{Hom}(\pi, \mathrm{GL}_{n}) .

This space is an algebraic variety, possibly with singularities, and there is an action of \mathrm{GL}_{n} given by conjugation. We can think of this as a change of basis in the representation. The character variety is the quotient

\mathcal{M}_{n, \pi} = \mathrm{Hom}(\pi, \mathrm{GL}_{n}) / \mathrm{GL}_{n} .

There are a few things to note here. First, we can easily replace \mathrm{GL}_{n} by any Lie group G in the definition, giving what we can call the G -character variety. Second, the group \pi is usually the fundamental group of a manifold \pi_{1}(M) . In this case, the Riemann-Hilbert correspondence says that the character variety is (analytically) isomorphic to the moduli space of flat connections on M . Roughly, the flat connections are certain kinds of differential equations on the manifold, and by solving them we obtain monodromy representations of the fundamental group. These give the corresponding points of the character variety.

Arguably, the most interesting examples occur when the manifold M has dimensions two or three. For example, in dimension two the character variety is a Poisson manifold, whereas, in dimension three, it is closely related to things like Chern-Simons theory and knot invariants. Recently, I’ve been trying to learn a bit more about the relations to low-dimensional topology: things like the Goldman-Turaev Lie bialgebra, string topology, and skein algebras. Below I’ll try to summarize a bit of what I’ve understood so far, along with some questions and confusion.

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