Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, like for examples $U(1)$ or $SU(2)$?
Thanks.
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2$\begingroup$ What do you mean by "known"? The moduli space of flat $G$-connections on a torus is the space $\{ (g, h) \in G^2 : gh = hg \}$ of commuting pairs of elements of $G$ ($g, h$ are given by monodromy around a pair of generators of $\pi_1$) modulo the action of $G$ given by simultaneous conjugation. What does it mean to know this space? $\endgroup$– Qiaochu YuanCommented Mar 25, 2015 at 23:50
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$\begingroup$ @QiaochuYuan: Thanks. What I meant is, is it a known manifold? From your answer I guess that that is indeed the case. Can you give a reference about this result? $\endgroup$– BilateralCommented Mar 25, 2015 at 23:53
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1$\begingroup$ It is not a manifold in general. Lots of stuff is known about these sorts of spaces; one keyword you can use is "character variety." $\endgroup$– Qiaochu YuanCommented Mar 25, 2015 at 23:57
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2$\begingroup$ For the examples $U(1)$ and $SU(2)$, the answer is pretty easy to work out by hand. In general, this moduli space is homeomorphic to Hom($\mathbb{Z}^2, G)/G$ (via the holonomy map) and for $G$ abelian (e.g. $U(1)$) we just get $G\times G$. For $SU(2)$, any commuting pair of matrices is simultaneously diagonalizable, and from this you can explicitly describe Hom$(\mathbb{Z}^2, SU(2))/SU(2)$ as a 2-dimensional sphere (up to homeomorphism). Working this out explicitly in terms of eigenvalues of the matrices is a nice exercise. $\endgroup$– Dan RamrasCommented Mar 26, 2015 at 1:55
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1$\begingroup$ I put a brief discussion of the unitary case Section 5.1 of my paper "The stable moduli space of flat connections over a surface," Trans. Amer. Math. Soc. 363 (2011), no. 1, 1061-1100. arXiv:0810.4882. (It's probably written out in other places also.) For the case of $SU(n)$ there is a discussion in Adem-Cohen-Gomez "Stable splittings, spaces of representations and almost commuting elements in Lie groups," Mathematical Proceedings of the Cambridge Philosophical Society / Volume 149 / Issue 03 / November 2010, pp 455-490 (arxiv.org/abs/1010.0735). $\endgroup$– Dan RamrasCommented Mar 26, 2015 at 20:32
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1 Answer
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Check out Almost commuting elements in compact Lie groups by Borel, Freedman, and Morgan.
"We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group."
answered Mar 26, 2015 at 2:49