# What is the precise statement of the correspondence between stable Higgs bundles on a Riemann surface, solutions to Hitchin's self-duality equations on the Riemann surface, and representations of the fundamental group of the Riemann surface?

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...

• I have to run, so I'll just confine this to a short comment. Stable Higgs (resp. vector) bundles with trivial $c_1$ on a compact Riemann surface are euquivalent to irred. reps of $GL_n(C)$ (resp $U(n)$). The result in paranthesis is due to Narasimhan & Seshadri. – Donu Arapura Oct 5 '10 at 15:53
• This paper ams.org/notices/200708/tx070800980p.pdf seems to say that stable Higgs bundles correspond to $SL(n,C)$ irreps. On the other hand, this paper arxiv.org/abs/math.AG/0206012 says that they correspond to $PGL(n,C)$ irreps... whereas degree zero stable Higgs bundles (as you say) correspond to $GL(n,C)$ irreps. Confusing... – Kevin H. Lin Oct 5 '10 at 16:35
• I see what you're puzzled about. You can formulate a notion of Higgs bundle in a number of different groups. I prefer $GL(n)$ because, things are simpler to state. In addition to the references you mention, I can suggest that might look at some papers of Carlos Simpson (e.g. his ICM talk, or his IHES paper "Higgs bundles and local systens") for clear statements valid even in higher dimensions. – Donu Arapura Oct 5 '10 at 16:50
• +1 for GIANT TITLE! – Harry Gindi Oct 5 '10 at 17:13

See Ó. Garcia-Prada's appendix to the third edition of R.O. Wells' book Differential Analysis on Complex Manifolds. It addresses most of what you are asking quite explicitly, in the context of Riemann surfaces, and has references to the original papers. Also the book of Lübke and Teleman on the Kobayashi-Hitchin correspondence might be helpful.

• Excellent! Garcia-Prada's appendix does indeed address most of what I am asking. However, there is one last thing that's confusing me, which is that whereas in Hitchin's paper we have the equation $F_A + [\Phi, \Phi^\ast] = 0$, in Garcia-Prada's paper this equation has an extra term... – Kevin H. Lin Oct 6 '10 at 21:26

Stable Higgs bundles $(E,\theta)$ with vanishing Chern class over a compact smooth Riemann surface (modulo the action of $\mathbb{C}^*: \theta \mapsto t\cdot \theta$) are in bijection with irreducible ($GL(n))-$representations of $\pi_1(X)$. This result in its full generality is due to C.Simpson (for any smooth projective variety) (Higgs bundles and local systems, Moduli of representations of the fundamental group of a smooth projective variety. I and Moduli of representations of the fundamental group of a smooth projective variety. II) and by S.Donaldson for surfaces.
Also any stable Higgs bundle admits a Hermitian-Yang-Mills metric (this result is also due to Simpson Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. for general complex manifolds), that is a solution of Hitchin's equation. In the other way any Higgs bundle with a Hermitian-Yang-Mills structure is polystable (direct sum of stable bundles of the same slope).

• Thank you very much for your comments. However, I am looking at the paper "Constructing variations of Hodge structure..." now, but I do not see the Hitchin equations anywhere ...... – Kevin H. Lin Oct 6 '10 at 1:22
• The Hermitian-Yang-Mills condition on the metric implies that the Higgs field satisfies Hitchin's equations (see this paper of Donaldson: plms.oxfordjournals.org/content/s3-55/1/127.full.pdf+html, for Riemann surfaces) – Andrei Halanay Oct 6 '10 at 8:24

I'm surprised nobody has mentioned Le Potier's Bourbaki exposé :

Fibrés de Higgs et systèmes locaux, Séminaire Bourbaki, 33 (1990-1991), Exposé No. 737, 48 p.

• Merci beaucoup! – Kevin H. Lin Oct 8 '10 at 15:10

With regard to $PGL(n,{\mathbb C})$ vs $SL(n,{\mathbb C})$, you're right that the $n=2$ case generalizes. On a Riemann surface, the moduli of rank $n$ degree $d$ semistable Higgs bundles with fixed determinant is a $n^{2g}$ cover of a component of the moduli space of $PGL(n,{\mathbb C})$ representations of the fundamental group. The components are labeled by $d$ mod $n$, $d=0$ corresponding to representations that lift to $SL(n,{\mathbb C})$. Hitchin's original space is a $2^{2g}$ covering of the component of $PGL(2,{\mathbb C})$ representations consisting of representations that do not lift.