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 of distinct n-tuples of points in a space , and the related natural space (the action of the symmetric group makes the points unlabeled). In the case , a point in 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, , the n-strand pure braid group; and , the full braid group (points needn’t end up in their original positions). In fact, the configuration spaces are spaces – that is, they are classifying spaces of these groups, and have no higher homotopy groups above .

Replacing here with , a surface, the same sort of thing defines the n-strand “surface braid group” for , which is . 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 ).

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 . (If , these are the more famous linear representations.) This is related to subspaces of the product , which arise from looking at the moduli space , where is a discrete group and a topological group.

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

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 , 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 , the free group on generators, and , the free abelian group on generators. These are fundamental groups of, respectively, the -punctured plane and the genus- torus. Now , and the map induces an inclusion – 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 . This is a sequence of subgroups generated by the -fold commutators . In particular, one looks at the groups , and in fact their spaces of homomorphisms:

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

Now it’s pretty standard that there are maps (by dropping the generator), and (sending the extra generator to the identity). These, thought of as face and degeneracy maps, turn the collection of spaces (for all ) into a simplicial space. This has a geometric realization, which is the classifying space (or, shifting which set is considered to be the -simplices, , where , and there’s a bundle ). BUT, each of the 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 -dimensional symplectic manifold . (That is, has an exact symplectic form – i.e. and is nondegenerate – and there’s an almost-complex structure - that is ; these give a metric ). This determines a complex derivative in a natural way.

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

Given a cohomology class (that is, 2-cocycles), one gets a form which can be integrated over . The Gromov-Witten invariant, for that choice of form, is just the total “volume” of the moduli space with respect to that form, (the form is pulled back under the map evaluating it at the 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 . 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 cocycles to be the fixed ), 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 on the space (which, since is a symplectic manifold, should preserve the form ). The symplectic geometry way to talk about this is one I’m not very familiar with, but Gonzalez referred to 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 , where is a space of connections on some bundle , and 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.

I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice – gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa.

Enxin is one of those guys who seems to really understand – it’s tempting to say grok- algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me – you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder.

The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are -algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras – the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra by new things, valued in the bigger algebra of formal power series in with coefficients in , so that the original structure you started with is just the constant part that appears when you set . (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket – in other words, algebras of observables of a physical system).

Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation – and there certainly is a lot – but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me.

So let’s get more specific.

The point is to replace a multiplication operator with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra (where is the multiplication for ) is , where the new multiplication is defined (by linearity) by its action on elements of , which works like this:

for some operators . Then there are a bunch of conditions on the that are needed to make associative. There’s one condition for each power of , since the coefficients in the associator should be zero:

The condition just says that is associative – so it’s the from the original algebra, which you get back when .

Then given an algebra , you can create the deformation category of whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero term are invertible (a consequence of the Lagrange theorem) this is actually a groupoid. Then the question is to classify the isomorphism classes of deformations – that is, . One can easily imagine that there might be no nontrivial deformations of some algebra – that is, every one is isomorphic to the deformation where all the are trivial except . So when does this happen? More generally, how can one classify the deformations up to isomorphism?

The answer has to do with Hochschild cohomology, which is related to a complex you can make from . Taking , the space of -ary multilinear operations on , you build this complex:

where the differential maps are defined by an alternating sum:

(Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that , you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.)

Then you take the Hochschild cohomology groups in the standard cohomology way: . A cohomology class in one of these groups is a class of multilinear maps from copies of to (up to a factor which is of something). As usual with cohomology, they describe obstructions to something – to exactness. Exactness, in this setting, would mean that has no interesting deformations at the level.

What does “level” mean here? Well, for example, at level 2 we’re talking about maps , such as the multiplication map. In fact, we have for an associative algebra – you can check that is twice the associator , which is zero. So is a cochain. Is it a coboundary? Sure – it’s . So is in the trivial class in . The point then is that it turns out that if this is the only class – if – then there are no interesting deformations of the multiplication of in the sense described above. The groupoid $\mathcal{D}$ has just one object. (One thing that occurs to me is that this makes it a group – which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if (as an algebra over ), there are no nontrivial deformations: .

What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to -algebras. Whereas for an associative algebra , the associator $a(x,y,z) = x(yz) – (xy)z$ is zero, in general an -algebra will have an associator map (that is, in the complex above), which might not be zero, but which is .

This is the beginning of a story relating -algebras to weak -categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity – the Pentagon identity – but in general you’d just ask for a “pentagonator” (something in ), and so on (this is where those seminar notes above help me think in pictures, by the way). An -algebra is a vector space equipped with maps at all these levels – described by Stasheff’s associahedra – satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of -algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of is a deformation not in , but in the quotient . This only needs two maps, . The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in – if , then any infinitesimal deformation can be extended to a full deformation.

All in all, I thought the talk was interesting – it tied in much more closely to things I already knew about TQFTs and higher categories than I’d expected. I’ll be really impressed if he can condense it into a 25-minute version…