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][1] 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?


  [1]: http://www.math.ucdavis.edu/~osserman/rfg/290W/branched-covers.pdf