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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# Which bundles does the character vareity parameterize?

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?