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



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

  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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.