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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

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