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

Question

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: Moduli space of flat connections over a Riemann surface

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

Answer 1

Let $K$ be a connected compact Lie group. The moduli space of flat $K$-bundles over an $n$-torus is homeomorphic to the character variety $Hom(\mathbb{Z}^n,K)/K$.

The identity component of this space is homeomorphic to $T^n/W$ where $T$ is a maximal torus in $K$ and $W$ is the Weyl group (generalizing the example you gave when $K=SU(m)$).

Here are some examples (where $K=SU(2)$):

1. $Hom(\mathbb{Z},K)/K=[-2,2]$

2. $Hom(\mathbb{Z}^2,K)/K=$

3. $Hom(\mathbb{Z}^3,K)/K$ is a 3-dimensional orbifold. Here is video of a continuous family of slices of it.

4. $Hom^0(\mathbb{Z}^2,PSU(2))/PSU(2)=S^2$ as mentioned in Example 3.14 here. There is one other component and it is a point.

In general, the moduli space is connected iff $n=1$, or $n=2$ and $K$ is simply connected, or $n\geq 3$ and $K$ is isomorphic to a product of $SU(m)$'s and/or $Sp(k)$'s. And the identity component of the moduli space is simply connected iff $K$ is semisimple (there may be other components that are not simply connected however). Example 2. above shows that $\pi_2$ may not be trivial even if $K=SU(2)$.

Replacing $K$ with a complex (or real) reductive group $G$, we have a similar, although more complicated story.

Please see my paper: Topology of character varieties of Abelian groups (co-authored with C. Florentino) for details about the reductive case (and references for the above mentioned facts).

Also, for the fundamental group of these moduli spaces please read my paper: Fundamental Groups of Character Varieties: Surfaces and Tori (co-authored with I. Biswas and D. Ramras).