A few days ago, John Mangual requested that one of us secret blogging seminarians write a post explaining what Gromov-Witten invariants are all about. I volunteered to do this, which puts me in an awkward position. Gromov-Witten theory is a big subject, and there are a lot of good introductions to the subject (e.g., Behrend’s Algebraic Gromov-Witten Invariants). It’s not clear that I can do much in a single blog post.
So I’m going to limit the scope of this post somewhat. I’d like to explain in what sense Gromov-Witten theory is a topological field theory. Caution: This may involve some rambling. If you’re curious, come below the fold.
Let’s start with the following observation. If is a smooth projective complex variety, then the cohomology algebra is a supercommutative Frobenius algebra. The inner product comes from integration.
Poincare duality implies that the inner product is non-degenerate. So we have a finite-dimensional Frobenius algebra on our hands. One of Folk’s theorems tells us that commutative Frobenius algebras are the same thing as two-dimensional TQFTs, so it’s natural to wonder if we can build up some sort of 2d topological field theory by assigning to the circle, and multiplication to the pair of pants. (If the supercommutativity bothers you, just take the even subalgebra.)
I want to reformulate this statement in a way that suggests further structure.
Consider the moduli stack , which classifies stable genus curves having marked points indexed by a set . (A marked curve is stable if its automorphism group is finite.) This is a pretty nice stack. It’s smooth, its dimension is what you’d expect from deformation theory, it’s Deligne-Mumford, and it’s compact. And it’s got a really interesting natural stratification.
The different strata in classify curves having fixed topological types. The open top-dimensional stratum classifies smooth curves, and the strata of higher codimension classify curves with varying numbers of nodal singularities. If you want to be precise about this, you use modular graphs to classify the topological types. The modular graph of a curve consists of
- a set of vertices, one for every irreducible component of the curve
- a set of edges, one for every node in the curve. Edges connect vertices if the corresponding components meet at a node
- a set of tails, one for each marked point on the curve
- an integer label on each vertex , the genus of the associated component.
What makes this stratification so remarkable is that each boundary stratum in is a (finite quotient of a) product of simpler moduli stacks, which classify smooth curves. For example, the substack of curves which have the topological type depicted in the example above is . The first factor parametrizes complex structures and the positions of marked points and nodes on the left component; the second does the same thing on the right. And essentially the same thing is true if we take closures. Continuing with the example, there is a natural surjection
More formally, the collection of all moduli stacks of stable marked curves is a modular operad. For any collection of graphs and any way of pairing tails in these graphs to make a new graph , we get a morphism
This operad structure induces an operad structure on the collection of homology groups , with a product
for every way of making a graph out of the .
If you have an operad, the first thing you should do is ask what it would mean for a vector space to be an algebra over your operad. I’ll assume that , like , comes equipped with an isomorphism to its dual. In this case, saying that is an algebra over the homology operad of the moduli stack of curves means that we get an operation
for every homology class , and that we can compose these morphisms in ways that correspond to gluing curves together. You should imagine that we have a copy of living at each marked point on our curves, and that when we glue curves at marked points, we compose through these copies of , after using the duality on to relabel some points as outgoing.
What does this mean when every homology class has degree zero? Degree zero homology classes are represented by points in , i.e. by curves with marked points. Since is connected, we can always assume that the curve representing is smooth. Explicitly, we can resolve the singularity to ; this is obviously a homotopy, and you should think of it as the opposite of contracting a cycle to a point.
The operation on homology classes sends a collection to the homology class in represented by the curve you get by gluing the together at marked points. Of course, we can always replace this curve with a homologous smooth curve. So, up to homotopy, the multiplication on degree zero homology classes reproduces the obvious composition on curves with boundary circles. Which means that an algebra over the degree zero homology operad is essentially the same thing as a Frobenius algebra.
In particular, is an algebra over the degree zero homology operad. Hopefully, now everyone still reading is primed to wonder, can we make into an algebra over the full homology operad?
The answer, of course, is yes. (Wouldn’t you be annoyed if I’d taken you all this way to say “no”?) This is what Gromov-Witten theory is all about.
To get this operadic structure, we use Kontsevich’s stack of stable maps from marked curves to . This stack classifies stable triplets consisting of a nodal (not necessarily stable) marked curve and a map . A triplet is stable if it has no automorphisms, i.e. if every unstable component in carries a non-trivial maps. The label is a homology class in ; a map has degree if .
Kontsevich’s stack has two really interesting morphisms. There are evaluation maps , gotten by evaluating at the marked point . And there is a morphism
which forgets the map to and contracts any unstable components in. So, if we have a collection of cohomology classes in , we can pull them back via the evaluation maps, wedge/cup them together, and then (virtually) push them forwards along the integration map. This gets us a cohomology class . Using Poincare duality on , we can reassemble these data into a morphism
which gives us an operation for every homology class on the stack of curves. And the geometry of will guarantee that we can compose these morphisms, making into an algebra over the homology operad. The key point is that you can specify a map on a curve with multiple components by saying what the map is on each component; then you can glue these maps together if they match at the nodes.
Rather than going into detail about exactly how becomes an algebra over the homology operad, I want to comment on two interesting features of the construction.
First, for some reason, I said “virtual” pushforward. That”virtual” guy looks a bit shady, lurking there in the parentheses. What’s his deal? And second, it’s probably worth pointing out explicitly: Gromov-Witten theory isn’t really a topological field theory. The operations coming from the degree zero homology are topological, but the higher operations do actually depend on a conformal structure, albeit very weakly. One way of understanding these phenomena is to look at the physical theories which inspired the development of Gromov-Witten theory.
This means that we should study the A-twisted nonlinear sigma model. I don’t want to get into gory details here, so let me just review the key points.
- The spacetime of the sigma model is a Riemann surface with boundary circles, which we’ll denote by to make you think it’s an algebraic curve. The Riemann surface carries a metric .
- The space of fields in the twisted sigma model is the parity-reversed tangent bundle of the vector bundle on the space of smooth maps whose fiber at a given map is the space of odd 1-forms on valued in the pullback of the parity reversed tangent sheaf of .
- Functional integration over this space looks a lot like integrating differential forms, because we use Berezin integration for the odd fields.
- This space of fields carries a natural odd vector field , which is basically the de Rham differential reinterpreted; this vector field generates a supersymmetry.
- The local observables at a point (i.e., those which depend only on the jets of the fields at ) which are annihilated by the vector field are in correspondence, via evaluation maps at the point , with closed forms on the target space $X$.
- Most importantly, functional integration in this theory involves a really amazing localization phenomenon. Roughly speaking, the path integral measure only gets contributions from an infinitesimal neighborhood of the subspace of holomorphic maps to . (Holomorphic with respect to the complex structure determined by the metric .) Everything else cancels out.
What this means is that the functional integral reduces to an integral on some nilpotent neighborhood of the space of holomorphic maps. In particular, the integral of -closed local operators can be computed by integrating cohomology classes on the space of holomorphic maps. (One can show, at a physical level of rigor, that -exact classes give vanishing functional integrals, so we can descend to -cohomology, which reduces to ordinary deRham cohomology once we restrict to the space of holomorphic maps.)
There’s a correction factor here, accounting for the contributions from the normal bundle to the locus of holomorphic maps. This is the source of the “virtual” integration; when we integrate over the space of holomorphic sections, we naturally pair with the “corrected” fundamental class .
Mathematically, what this means is that, if we want to construct the virtual fundamental class for the fiber of the map-forgetting morphism , then we need to introduce a correction factor. In symplectic geometry, this is usually done by deforming the -operator, so that the locus of holomorphic maps has the expected size. In algebraic geometry, people do something similar, using Behrend & Fantechi’s intrinsic normal cone, which is a stack that knows what the infinitesimal neighborhood of a scheme looks like when we embed inside any ambient scheme.
Now what about the metric dependence? How strong should we expect it to be? Not very strong, obviously, since the equation above indicates that it only depends on the conformal class of the metric. But what if we vary the conformal structure? Well, it turns out that the exponentiated action is -exact, i.e., , for some function . In particular, infinitesimal shifts in change the integrand by a -exact cohomology class; hence don’t produce any changes when we pass to the cohomology of .
One way of formally realizing this idea is to say that Gromov-Witten theory is a topological conformal field theory, which is a concept that lies midway between topological field theory and conformal field theory.
Mathematically, a topological field theory is a functor to vector spaces from a category whose objects are integers (counting the number of circles) and whose morhisms are cobordisms between these circles. A CFT is supposed to be a similar gadget, except that the space of morphisms consists of cobordisms equipped with complex structures. In topological conformal field theories, one instead considers functors from a category whose objects are integers $# of circles) and whose morphisms are chains on the moduli stack of complex structures on bordisms between these circles. This functor maps to the category of differential graded vector spaces, and has to send the boundary operator on chains to the differential on the vector spaces.
I’m about done at this point, so I’ll finish with one last thought: It’s these quantum field theory structures that make Gromov-Witten theory so useful for studying problems in enumerative geometry. Most of the useful equations in Gromov-Witten theory, such as the WDVV equations or the string and dilaton equations and the rest of the Virasoro hierarchy, arise directly from the fact that Gromov-Witten theory is a QFT.