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

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!

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

## 1 Answer

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).