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For any Riemann surface with punctures $C$, and Lie group $G$, the character variety is the space of maps $\mathrm{Hom}(\pi_1(C), G)$.

I know that if $G= S_n$ (not a lie group), then $\mathrm{Hom}(\pi_1(C), S_n)//S_n$ parametrizes branched covers of $M$. Here $S_n$ acts by conjugation (permuting the various copies of $C$.)

If $G = \mathrm{GL}(n,\mathbb{C})$, is $\mathrm{Hom}(\pi_1(C), G)$ parameterizing vector bundles over $C$? What is the equivalence relation here?

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    $\begingroup$ When $G = U(n)$, the character variety parameterizes vector bundles over $C$. This is the Narasimhan-Seshadri Theorem. When $G = GL_n$, the character variety parameterizes Higgs bundles. This is part of what's known as non-Abelian Hodge theory. $\endgroup$
    – Mike Skirvin
    Commented Sep 27, 2011 at 23:18

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In general, the set $\mathrm{Hom}(\pi_1(C),G)/G$ (where $G$ acts by conjugation) naturally parameterizes $G$-local systems.

For example, if $G=GL_n(\mathbb C)$, these are just ordinary local systems of vector spaces: a map from $\pi_1(C)$ to $GL_n(\mathbb C)$ describes the monodromy around loops in $C$, and conjugate maps correspond to isomorphic local systems. If $G=U(n)$, the character variety naturally parametrizes unitary local systems (i.e. the fibres have an inner product such that the monodromy is unitary).

As Mike comments, there are various theorems generally known as non-abelian Hodge theory which relate these topological objects to holomorphic objects. For example, on a closed curve, the Narashimhan-Seshadhri theorem gives a bijection between isomorphism classes of unitary local systems and degree 0 holomorphic vector bundles (with some stability condition).

Similarly, there is a bijection between (iso classes of) all local systems and degree 0 Higgs bundles (with stability conditions).

These can be thought of as giving diffeomorphisms between the character variety and the moduli spaces which naturally parameterize these objects (note that for $G=GL_n(\mathbb C)$, the character variety has it's own complex structure, which does not pull back to the natural complex structure on the moduli of Higgs bundles under these diffeomorphisms).

There are similar results when the curve $C$ is not compact (i.e. has punctures) you have to be a bit more careful, and you may want to include extra data like a filtration of the bundle at the punctures, and constrain the monodromy around the punctures to lie in a particular conjugacy class.

I think this is discussed in the appendix to Wells' Differential Analysis on Complex Manifolds (written by Oscar Garcia-Prada), as well as in the papers of Narashimhan-Seshadhri, Hitchin, Donaldson, Corlette, Simpson etc...

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    $\begingroup$ You don't need to be careful about punctures unless you care about complex-analytical type data at the punctures. But (perhaps this is nitpicking), you do need to be careful about "instability". When $C$ is anything but a sphere with no punctures, the character variety parametrizes based maps $C\to BG$, i.e. principal $G$-bundles with chosen trivialization at the base point. (For $GL_n$, this is equivalent to vector bundles: in general, there is a standard dictionary, see en.wikipedia.org/wiki/Frame_bundle). When $C$ has $\pi_2 \neq 0$ with $G$ not discrete, the two may not be the same $\endgroup$
    – Dmitry Vaintrob
    Commented Apr 27, 2016 at 13:17

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