Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, like for examples $U(1)$ or $SU(2)$?


Thanks.



Moduli space of flat connections over a torus
Check out [Almost commuting elements in compact Lie groups](http://arxiv.org/abs/math/9907007) by Borel, Freedman, and Morgan.


"We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group."


