Moduli space of flat connections over a Riemann surface

Question

If I understand correctly, in the Refs below:

We can see that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\rm flat}=\mathbb{E} / {S}_N = \mathbb{P}^{N-1} $$ where $\mathbb{E}$ is given by $$ \mathbb E := \left\{ (\phi_1,\cdots, \phi_N) \equiv {(\mathbb T^2)}^N; \text{ subject to a constrain }\sum_i \phi_i=0 \right\} . $$ while the ${S}_N$ is the symmetric group usually denoted as $S_N$ (the Weyl group of SU$(N)$).

My questions

• what is the moduli space of U(1) flat connections over a genus $g$-Riemann surface?

• what is the moduli space of SU(N) flat connections over a genus $g$-Riemann surface?

Stable and Unitary Vector Bundles on a Compact Riemann Surface, M. S. Narasimhan and C. S. Seshadri, Annals of Mathematics, Second Series, Vol. 82, No. 3 (Nov., 1965), pp. 540-567

Thank you for the kind comments and helps!

Answer 1

Both of these moduli spaces are discussed in the survey "Flat connections on oriented 2-manifolds" by Lisa Jeffrey. The theme of the first section of the paper is roughly as follows. An oriented 2-manifold can be viewed topologically (i.e. up to homeomorphism), as a smooth surface (up to diffeomorphism), or holomorphically (as a Riemann surface with a complex structure). As it happens, moduli spaces of flat connections can also be given topological, smooth, and holomorphic descriptions as well. For instance, the moduli space of flat U(1)-connections (defined for smooth surfaces) on a surface of genus $g$ can be viewed topologically as $U(1)^{2g}$ or in the holomorphic setting as the Jacobian of a Riemann surface (viewed as an algebraic curve). This is described in Section 1.2 of Jeffrey's paper with some more discussion in Section 2. Similar descriptions can be given for SU(N) (Section 1.3).

"What is such-and-such a space" is a rather vague question; the answer could be kind of tautological (e.g. I could just give you another name for the space without telling you anything new), unless you specify what exactly you want to know. If you asked "how can we study the topology of these moduli spaces", then sections 3 and 4 of Jeffrey's paper give a brief introduction to some techniques for extracting topological information (computing intersection numbers of cohomlogy classes). There's a lot more information out there on this question in the literature on representation varieties. See e.g. this MO question "Cohomology of representation varieties".