# Moduli space of flat connections of Lie group over a 2-torus

We know 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$$(N)$$).

I learned the answer from the post and Lisa Jeffrey's note: Moduli space of flat connections over a Riemann surface

My questions

• what is the moduli space of SO(N) flat connections over a 2-torus?

• what is the moduli space of PSU(N) flat connections over a 2-torus?

If this is too general, we can focus on the case: SO(3)=PSU(2).

Thank you for the kind comments and helps!

Some Refs I found:

The moduli space of flat SU (2) and SO (3) connections over surfaces

https://faculty.math.illinois.edu/~michiel2/docs/thesis.pdf