# Moduli spaces of connections as representation spaces

It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi_1(M), G) / G$.

Furthermore, Atiyah and Bott (1983) showed that over a closed surface the moduli space of (central) Yang-Mills connections can be seen as the space of representations of a group $\Gamma_R$ (which is a central extension of the fundamental group by $\mathbb{R}$) to $G$.

Are there other moduli spaces which can be identified with representation spaces?

Narasimhan and Seshadri proved that the moduli space of semi-stable vector bundles of rank $r$ on a compact Riemann surface is canonically isomorphic to the moduli space of unitary representations of degree $r$. This has been extended to projective manifolds by Donaldson, then to compact Kähler manifolds by Uhlenbeck-Yau; and to the case of principal $G$-bundles (for $G$ complex reductive) and representations of a maximal compact subgroup by Ramanathan and Subramanian.

• what does isomorphic mean? Dec 11 '14 at 18:56
• Good question: as real analytic (possibly singular) varieties.
– abx
Dec 11 '14 at 21:46
• although the mathoverflow website itself discourages this, I would like to use this comment box for the sole purpose of expressing gratitude for your comment above. So, thanks. Dec 11 '14 at 23:31

There are all sorts of variants of the answer of abx and of the example in the question:

By Hitchin's self-duality paper, you can identify the $SL(2,\mathbb C)$-representation space with the moduli space of Higgs bundles (holomorphic bundles together with Higgs fields). Note that in this case, the spaces are only isomorphic as real analytic varieties, but the corresponding complex structures differ. In fact, one gets a hyper-kaehler structure from the two different complex structures.

As another example, you can consider a punctured Riemann surface $\tilde M=M\setminus \{p_1,..,p_n\}$ and consider the subspace of the representation space $Hom(\pi_1(\tilde M),SU(2))/SU(2)$ given by fixed local conjugacy classes $C_i$ around $p_i.$ Then, this space is isomorphic (as a real analytic variety) to the moduli space of parabolic structures on $\tilde M,$ where the parabolic weights are determined by $C_i.$ Generalizing to $SL(2,\mathbb C)$ gives the moduli space of parabolic Higgs bundles.

There is another example in Hitchin's paper on the self-duality equations, namely the Teichmüller space, which can be identified with the space of certain Fuchsian representations of the fundamental group of the surface modulo conjugation. On the other hand, from Hitchin, this space can be identified with a subspace of the moduli space of Higgs bundles.

Another interesting example, not mentioned yet, where character varieties come up is the moduli space of polygons.

Moduli spaces of $(G,X)$-structures on compact manifolds are also related to character varieties via the Ehresmann--Thurston Theorem. See here for example.

They are also related to spaces of phylogenetic trees and spaces of spin networks.