In the last couple of weeks of the winter term, there were two series of talks here at UWO, by different speakers, from very different points of view, which bear on the subject of moduli spaces of connections.

There seem to be several schools of thought approaching the subject of moduli spaces, and in particular how to handle the reduction by symmetries without losing too much – three approaches I know of are the symplectic point of view (thinking of the moduli space as a symplectic space, or perhaps orbifold, and reduction by taking whole “leaves” to points), the algebraic-geometric (describing them using Deligne-Mumford stacks), and the groupoid point of view (which is the one I’m most familiar with). I suppose, in light of my previous note, that there must be a noncommutative-geometry view of the subject, though if anyone is using NCG to look at these moduli spaces in particular I don’t know who. Before talking about reducing the moduli spaces, there’s already a lot to say about them which people have studied in some detail.

The first speaker here who touched on this was Fred Cohen, who gave a series of three talks about special subspaces of products (and talked a lot about about stable homotopy theory). The second was Eduardo Gonzalez, who gave a seminar and a colloquium talk on equivariant Gromov-Witten theory. I’ll try to briefly give an overview of what they each had to say, mainly focusing on this common element.

Part 1 – Talks by Fred Cohen

Fred Cohen was speaking about various subspaces of products. He was summarizing a number of different projects, including for example this (on loop spaces of configuration spaces) and this (about spaces of homomorphisms). The first talk dealt with the seemingly simple space $Conf(X,n) = \{(x_1, \dots, x_n) | x_i \neq x_j \text{ when } i \neq j\}$ of distinct n-tuples of points in a space $X$, and the related natural space $Conf(X,n)/S_n$ (the action of the symmetric group makes the points unlabeled). In the case $X= \mathbb{R}^2$, a point in $Conf(\mathbb{R}^2,n)$ is a list of n distinct points. So a loop in this space is a motion of the n points which returns them to their original locations – considered up to homotopy, this is just a braid. In fact, $\pi_1(Conf(\mathbb{R}^2,n)) = PB_n$, the n-strand pure braid group; and $\pi_1(Conf(\mathbb{R}^2,n)/S_n) = B_n$, the full braid group (points needn’t end up in their original positions). In fact, the configuration spaces are $K(\pi,1)$ spaces – that is, they are classifying spaces of these groups, and have no higher homotopy groups above $\pi_1$.

Replacing $X = \mathbb{R}^2$ here with $X=S$, a surface, the same sort of thing defines the n-strand “surface braid group” for $S$, which is $P_n(S) = \pi_1(Conf(S,n))$. We heard how this decomposes in terms of the “Borromean” braid group – the subgroup of braids which become disconnected when you remove one strand (this is the kernel of a map induced by the projections into $Conf(S,n-1)$).

There was more about the homotopy type of these spaces, and a second talk covered “moment-angle complexes”, but here I’m interested in Cohen’s third talk about subspaces of products. This was on “representations” of a discrete group, which in this context means – almost – homomorphisms into a chosen group $G$. (If $G = GL(n)$, these are the more famous linear representations.) This is related to subspaces of the product $G^N$, which arise from looking at the moduli space $Hom(\pi,G)$, where $\pi$ is a discrete group and $G$ a topological group.

In particular, if $\pi = \pi_1(X)$, for a space $X$, such a representation can be thought of as a $G$-connection. In this picture, a connection is just a way of assigning an element of the gauge group $G$ to each path in $X$.) Actually, I mentioned this is “almost” the space of representations, which is actually $Rep(\pi,G) = Hom(\pi,G)/G$ – the moduli space of flat connections modulo gauge transformations. A gauge transformation (assuming $X$ is connected) acts by conjugation: $g(\gamma) \rightarrow h g(\gamma) h$, for a class $\gamma$ of loops in $X$.

This is the usual way of looking at this moduli space of geometric structures – I’ve mentioned here the alternative view that a flat connection is a functor $g : \Pi_1(M) \rightarrow$, and a gauge transformation is a natural transformation. Then the moduli space becomes a moduli stack, which as mentioned above I tend to think of as a groupoid. But the moduli spaces of homomorphisms (the objects) and representations (isomorphism classes of objects) carry a lot of information. Particular cases which Cohen discussed were $\pi = F_n$, the free group on $n$ generators, and $\mathbb{Z}^n$, the free abelian group on $n$ generators. These are fundamental groups of, respectively, the $n$-punctured plane and the genus-$n$ torus. Now $Hom(F_n,G) \cong G^n$, and the map $F_n \rightarrow \mathbb{Z}^n$ induces an inclusion $Hom(\mathbb{Z}^n,G) \stackrel{i}{\rightarrow} Hom(F_n,G)$ – in fact it’s a subvariety – so this is a subset of a product, and techniques for dealing with these were Cohen’s real subject.

One that he discussed (described in the paper linked above by Adem, Cohen and Torres-Giese) uses the “descending central series” of $F_n$. This is a sequence of subgroups $\Gamma^q$ generated by the $q$-fold commutators $[\dots[g_1,g_2],g_3],\dots, g_q]$. In particular, one looks at the groups $F_n/\Gamma^q$, and in fact their spaces of homomorphisms:

$Hom(F_n/\Gamma^2,G) \subset Hom(F_n/\Gamma^3,G) \subset \dots \subset G^n$

So there’s a filtration of spaces associated to $F_n$ and $G$.

Now it’s pretty standard that there are maps $d_i : Hom(F_n,G) \rightarrow Hom(F_{n-1},G)$ (by dropping the $i^{th}$ generator), and $s_j : Hom(F_n,G) \rightarrow Hom(F_{n+1},G)$ (sending the extra generator to the identity). These, thought of as face and degeneracy maps, turn the collection of spaces $G^n$ (for all $n$) into a simplicial space. This has a geometric realization, which is the classifying space $BG$ (or, shifting which set is considered to be the $n$-simplices, $EG$, where $BG = EG/G$, and there’s a bundle $EG \rightarrow BG$). BUT, each of the $Hom(F_n,G)$ has the filtration above – so it turns out there’s a filtration of simplicial spaces, and in fact of bundles. The paper above uses this to find the cohomology, fundamental group, and so on of the spaces I just mentioned – including the moduli space of connections.

(Then Cohen talked about a generalization of this to arbitrary “transitively commutative” groups, but that takes us away from the geometry I started off talking about).

Part 2 – Talks by Eduardo Gonzalez

The second set of talks which touched on moduli spaces of connections was by Eduardo Gonzalez, related to stuff in this paper by Gonzalez and Chris Woodward speaking about gauged (or equivariant) Gromov-Witten invariants. These are discussed in this paper by Givental, and Gonzalez referenced several other people who’ve worked on related things, including Chen and Ruan (see this on GW theory for orbifolds), and Abramovich, Graber and Vistoli (see this, on GW theory for stacks). Strictly speaking, this doesn’t address just the moduli space of flat connections, but actually a more complex moduli space for a theory involving a choice of connection (on a bundle), and also a section of the bundle. It is called the moduli space of symplectic vortices, and is very much involved with symplectic geometry as you might expect.

The usual Gromov-Witten invariants, roughly, count the number of holomorphic curves on a $2k$-dimensional symplectic manifold $X$. (That is, $X$ has an exact symplectic form $\omega$ – i.e. $d \omega = 0$ and $\omega$ is nondegenerate – and there’s an almost-complex structure $J : TX \rightarrow TX$- that is $J^2 = -1$; these give a metric $g(u,v) = \omega(u, Jv)$). This $J$ determines a complex derivative $\partial_J$ in a natural way.

A curve is a map $u : \Sigma \rightarrow X$, where $\Sigma$ is a Riemann surface (i.e. complex curve), which is holomorphic if $\partial_J(u) = 0$. The moduli space $\mathcal{M}(\Sigma, X, J)$ of these holomorphic curves – which is also the space of sections of suitable bundles over $\Sigma$ – each one amounts to a choice of a particular bundle over $\Sigma$, and a connection and holomorphic section of the bundle. This is where the Gromov-Witten invariants come from. Actually, it comes from a compactification $M$ of the space of maps from $\Sigma$ with $n$ “marked” (distinguished) points (so here actually we start to circle back around to the configuration space $Conf(\Sigma, n)$ Fred Cohen talked about).

Given a cohomology class $\alpha \in H^2(X,\mathbb{Q})^n$ (that is, $n$ 2-cocycles), one gets a form which can be integrated over $M$. The Gromov-Witten invariant, for that choice of form, is just the total “volume” of the moduli space with respect to that form, $\int_M ev^(\alpha)$ (the form $\alpha$ is pulled back under the map evaluating it at the $n$ marked points). This is sometimes described (rather roughly) as “counting” the pseudoholomorphic maps.

One thing people seem to be quite interested in is how this is related to so-called “quantum cohomology” for the space $X$. Since the GW invariants take some forms and give numbers, the idea is that they can be used to define a “three point function” on cohomology classes (by taking all but three of the $n$ cocycles to be the fixed $\omega$), which in turn can be taken to be the structure coefficients for a deformation of the cup product for cohomology. (Take the cohomology ring, take its tensor product with a ring of power series, and write the new product as a power series whose first terms give the usual cup product).

However, what Gonzalez was talking about was “gauged” Gromov-Witten invariants, where spaces are replaced by stack – in particular, stacks that come from an action of a group $G$ on the space $X$ (which, since $X$ is a symplectic manifold, should preserve the form $\omega$). The symplectic geometry way to talk about this is one I’m not very familiar with, but Gonzalez referred to $X\/\!\!\/G$ as the “categorical quotient” (i.e. the transformation groupoid, in the language I’m more used to) or the “symplectic reduction” (here‘s a brief note on the subject, and here a long paper on the relevance to physics which I’m linking so I can find it later). Roughly, this is a two-step process, the second stage being a reduction to a quotient by a group action. The result, in general, will be a symplectic orbifold (if the action is free on orbits, it’ll be a manifold – otherwise, some orbits have extra symmetry, which give the special points of the orbifold).

In particular – and here we really get to the point of contact with the groupoid picture I’m more familiar with, the gauged GW invariants are associated to a space $M(P,X) = \mathcal{A}(P,X) \/\!\!\/ \mathcal{G}(P)$, where $\mathcal{A}(P,X)$ is a space of connections on some bundle $P \rightarrow X$, and $\mathcal{G}$ is the group of gauge transformations. Now, these aren’t the space of flat connections, which I’ve thought more about, but rather connections satisfying another equation, namely that the curvature plus a certain volume form should be zero (defining the volume form takes a while and I don’t get it in enough detail to try to sort it out here). Connections satisfying this equation are called vortices, for reasons which escape me.

But in any case, the invariants amount to some geometry-aware generalization of the groupoid cardinality of this orbifold, thought of as an (equivalence class of) groupoid(s), defined by the integral above. There is much more to say here, but it’s taken me long enough to write this up as is, so maybe I’ll return to those things in a separate post some time.