Scary math: Higgs bundles and Hitchin Parametrization

6 Jun

Just kidding!  I don’t know what any of the words in the title of this post mean except for “scary”, “math”, and “and.”  Today, the French postdoc Brice Loustau, visiting for a month, gave a talk on this subject to three of us (including his collaborator).  My math big brother missed the talk and asked if I took notes, so I thought I’d put up a preliminary copy here for him, and then come back and edit it for readability for whoever reads this blog who is interested in math (if you exist…)

This talk is really about the correspondence between the moduli space of the character variety and the moduli space of flat connections.  I also didn’t know what most of those words meant until the talk itself.

Generally, we might have a homomorphism \rho: \Gamma\to G, where \Gamma is a discrete group and G is a Lie group.  For instance, Teichmuller theory is the study of these homomorphisms when \Gamma is the fundamental group of a surface, and G is PSL_2(\mathbb{R}).  In Brice’s current work, he and Jonah are considering G as SL_n(\mathbb{C}).

How is Teichmuller theory connected to character varieties?  Well, if you have a surface S with negative Euler characteristic, and hence endowed with a hyperbolic structure, we have S \cong \mathbb{H}^2/\Gamma, where \Gamma is a discrete subgroup of PSL_2(\mathbb{R}) and the fundamental group of S.  A surface can have a whole bunch of hyperbolic structures, and those are represented by the character variety.

See, if I have \rho: \pi_1(S)\to PSL_2(\mathbb{R}), then the image of \rho is a subgroup \Gamma, which is discrete (given some conditions on \rho….) and is exactly the \Gamma from above, giving me a hyperbolic structure.

The character variety of a surface S is \chi(S) = \{\rho:\pi_1(S)\to PSL_2(\mathbb{R})\}/\sim, where \sim indicates equivalence up to conjugation by an element of PSL_2(\mathbb{R}).  Conjugate homomorphisms give isometric hyperbolic structures in Teichmuller space.

Let’s next talk about flat connections.  A connection on a vector bundle E is a linear operator D: \Omega^0(E)\to \Omega^1(E), that is, it takes in a smooth section in \Gamma(E) and outputs a smooth 1-form that gives values in E, subject to the Leibniz rule:

\forall s\in \Omega^0(E), f\in C^{\infty}(S,\mathbb{C}), D(fs) = df\otimes s + fDs.

Basically, connections are like derivatives, and we like them because they let us differentiate sections.  And we sort of use derivative notation with them, too.  For instance, if DS\in \Omega^1(E) and X is a vector field on S, so X\in\Gamma(TS), then Ds(X) \in \Omega^0(E), and we write it as D_Xs.

The flat adjective has to do with curvature, which has lots of details that we won’t get into.

So to build our correspondence, given a homomorphism \rho:\Gamma\to G, we need to construct a rank n complex vector bundle on S with a flat connection.

Building the vector bundle is actual pretty easy.  Let E_{\rho} = \tilde{S}\times \mathbb{C}^n / \pi_1(S).  How does \pi_1(S) act on this product?  In the only way it can: \gamma(x,v) = (\gamma x, \rho(\gamma)\cdot v)– so it acts by deck transformations in the first coordinate, and by the homomorphism and multiplication on the second.

Now we need to say it comes equipped with a flat connection.  We’ll define it.  Let s\in\Omega^0(E) be a section, and \{e_j:U\to E, j =1,\ldots, n\} be a local frame to E from U an open subset of S.  Then we can write, for any x in U, s(x) = \sum_1^n s_jk(x) e_j(x), where the s_j:U\to\mathbb{C}.  There’s a bunch of work to show that this is flat, but we’re not doing that.

So badaboom!  We got half of the correspondence between character varieties and flat connections.  Going the other way, from flat connections to character varieties, is apparently pretty easy by using the holonomy of a connectionbut we ran out of time.


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