I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and (irreducible) representations of the fundamental group of $\Sigma$. I am finding it a bit difficult to find a reference containing the precise statement. Mainly I'd like to know the statement for the case of stable $GL(n,\mathbb{C})$ Higgs bundles. But if anyone knows the statement for more general Higgs bundles that would be nice too.

Just at the level of say sets and not moduli spaces, I think the statement is that the following 3 things are the same, if I am reading Hitchin's original paper correctly:

stable $GL(n,\mathbb{C})$ Higgs bundles modulo equivalence,

irreducible $U(n)$ (or is it $SU(n)$?) solutions of the Hitchin equations modulo equivalence,

irreducible $SL(n,\mathbb{C})$ (or is it $GL(n,\mathbb{C})$? $PSL$? $PGL$?) representations of $\pi_1$ modulo equivalence.

Is this correct? Is there a reference?

Hitchin's original paper (titled "Self duality equations on a Riemann surface") does some confusing maneuvers; for example he considers solutions of the self-duality equations for $SO(3)$ rather than for $U(2)$ or $SU(2)$, which would seem more natural to me. Moreover, for instance, he doesn't look at all stable Higgs bundles, but only a certain subset of them - but I think this is just for the purpose of getting a *smooth* moduli space. And finally, Hitchin looks at $PSL(2,\mathbb{C})$ representations of $\pi_1$ rather than $SL(2,\mathbb{C})$ representations or $GL(2,\mathbb{C})$ representations, which confuses me as well...

Thanks in advance for any help!!

**EDIT:** Please note that I am only interested in the case of a Riemann surface. Here it appears that *degree zero* stable Higgs bundles correspond to $GL(n,\mathbb{C})$ representations. But the question remains: are stable Higgs bundles of *arbitrary degree* related to representations? If so, which representations, and how are they related? Moreover, I think that general stable Higgs bundles should correspond to solutions of the self-duality equations -- but what's the correct group to take? ("Gauge group"? Is that the correct terminology?) I think it's $U(n)$ but I am not sure.

For example, in Hitchin's paper, he considers the case of rank 2 stable Higgs bundles of *odd degree* and fixed determinant line bundle, with trace-zero Higgs field (see Theorem 5.7 and Theorem 5.8). As for the self-duality equations, he uses the group $SU(2)/\pm 1$. We get a smooth moduli space. In the discussion following Theorem 9.19, it is shown that this moduli space is a *covering* of the space of $PSL(2,\mathbb{C})$ representations. It seems that this should generalize...