We know that the moduli space of SU($N$) principal bundle's 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:  https://mathoverflow.net/q/313889/27004

My questions

>-  what is the moduli space of SO(N) principal bundle's flat connections over a 2-torus?
>
>- what is the moduli space of PSU(N) principal bundle's 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!

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Some Refs I found:

[The moduli space of flat SU (2) and SO (3) connections over surfaces](https://www.sciencedirect.com/science/article/pii/S0393044098003271)

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