Today AJ’s talking in the Grand Unified Seminar (representation theory, geometry, and combinatorics) on his this (joint work with his collaborator C. Teleman and his advisor E. Frenkel). The title of the talk is “Gromov-Witten Theory for a point/C*.” As AJ points out with delight (after the title was misintroduced without the mod C*, “It’s negative 2 dimensional!”

The outline of the talk is:

- Gromov-Witten Invariants
- The stack pt./C*
- Integration on quotient stacks
- (the unfortunately named) admissible classes
- Bundles on nodal curves
- Invariants are well-defined

The first question (from Eric Babson) is where these invariants live, “are they numbers.” AJ says “they’ll live in the K-theory of a point, which you can think of as Z.”

Ok, so first we need to learn about a whole zoo of moduli stacks of stable curves. To describe which moduli stack you’re referring to you should specify: a) the gens, b) how many marked points you’re allowing, c) whether you’re allowing nodes (allowing nodes compactifies the space). Hopefully the details of the construction are unimportant (or AJ will explain them in comments) because I missed them.

Now we review Gromov-Witten theory. You have some space X and a divisor d, you look at the moduli space of stable maps from some curve (again fix the genus and the marked points) such that the the pushforward of the curve is the divisor d. Given a collection of (where i ranges over the set of marked points) we can use an integral over the moduli space of a certain form build from the to get GW invariants. Perhaps someone in comments can clarify a bit for me what this has to do with counting curves. But if I understand things right it’s an invariant of X together with the choice of .

Alright, what AJ wants to do is modify this construction to the situation where X is the point mod C*. What is the point mod C*? Well, there’s a map point->point/C*. Since the point is contractible, you can think of point/C* as an algebraic analogue of the classifying space of C*. However, pt/C* behaves rather differently than the usual model of the classifying space , for example the former has real dimension -2 while the latter has infinite real dimension.

Now we turn to integration. There’s a bit of a problem trying to use the usual cohomological approach in GW theory. By a simple degree argument you can show that all the integrals occuring in the usual GW theory invariant you’ll get 0. So we’ll replace cohomology with K-theory. This means we need to understand something AJ is calling K-theory integration. I’m starting to get scared.

So suppose we have two spaces X and Y with a proper map f:X->Y. Suppose that V is a vector bundle over X, if Y is smooth and quasi-projective we can define V’s “pushforward in K-theory” by the Euler characteristic of the derived pushforward. This pushforward is going to play the role of integration.

If the Euler characteristic of derived pushforward is playing the role of integration, then what are we integrating? There’s several classes AJ defines here, “evaluation bundles,” “index classes,” and “determinant lines.” The first of these is the “one we care about” but the various modifications are going to be necessary in order to make “the integral converge.” If integration is push-forward, what on earth does “making the integral converge” mean? Well remember that above f needs to be proper for “pushforward in K-theory” to be well-defined. What if the map isn’t proper? Then only certain pushforwards are going to be well-defined, if the pushforward is well-defined we think of that as “the integral converging.”

After I asked a question clarifying this issue AJ says “Noah are you liveblogging this?” I sheepishly have to admit that I am. Unfortunately he’s lost me in the last 10 minutes of the first hour of the talk, so I’m not going to be liveblogging the second hour. Hopefully this will motivate AJ to write his own post explaining what’s going on here…

On pushing forward in K-theory:

People usually define K-cohomology, made out of vector bundles. These pull back. If you want to push forward, you have to have some kind of K-homology. The right way to do this in the algebro-geometric context is to make your classes using more general sheaves, not just vector bundles. K-equivalence is defined using short exact sequences.

Sheaves push forward, but K-equivalent sheaves may not push forward to K-equivalent ones. This is why one must use the derived pushforward.

One can tensor a sheaf with a vector bundle and get a sheaf, in a way that descends to K-theory. So this is a cap product. The obvious sheaf to tensor with is the structure sheaf, i.e. taking a vector bundle to its underlying sheaf; this is the Poincare morphism. On smooth varieties one can resolve sheaves by a finite sequence of vector bundles; this is (part of) Poincare duality.

Anyway where I’m heading with this is this “making the integral converge” business. Say I have a point in a noncompact space, giving me a K-homology class [pt], That sounds like something I could reasonably push to a point, getting the class [pt] = 1. But it’s very confusing (at least, initially) to think of this class cohomologically; you have to resolve the sheaf by a sequence,

0 -> Fun(line) -mult-by-z-> Fun(line) -> Fun(point) -> 0.

If you’re careless, that looks like [Fun(point)] = 0. So how can it push to 1? It means that on this noncompact space, you can’t just think of K-cohomology classes as being made out of vector bundles; you have to make it out of sequences of vector bundles, that are exact outside a compact set. This is the analogue of being integrable.

The dimension is (-1). Or are you still thinking in terms of real numbers? :-)

It’s a really neat idea. Is the paper already online?

I wondered that too, until I read

“for example the former has real dimension -2”

Yep, real dimension. It’s what matters when you’re pushing forward

cohomology classes; the degree shifts by the real dimension. So for

instance, if we try to integrate in cohomology, we get into trouble,

because we need a map

The only such map is the zero map, because for , while for .

This is why we work in K-theory, and it’s why the 2 matters. Bott

periodicity gives us a chance at a non-zero integration map.

The paper is not online yet, but it will be quite soon.

“Perhaps someone in comments can clarify a bit for me what this has to do with counting curves.”

No one has picked up on this yet, so I will.

Let X be the moduli space of degree d maps of a genus zero curve to P^2. Let Y be the same moduli space, except that we mark 3d-1 points on the curve. Then we have maps from Y to X (forget the points) and Y to P^2 (evaluate at the points).

Now, take the top class in H^*(P^2) and pull it back to Y 3d-1 times, once along each of the different evaluation maps. Cup the results together and pushforward to X. I claim that what you get is the number of degree d, genus zero, curves passing through 3d-1 given points.

To understand this, it is best to let your mind blur a little. Pretend all of our spaces are smooth and compact (as they are in this case, but not AJ’s). Then we can use Poincare duality to confuse H^k and H_{n-k} (where n is the dimension of the ambient space) as both are dual vector spaces to H^{n-k}. So I’ll describe a class in H^k(Z) by describing a codimenson k subvariety of Z. In particular, the top class in P^2 corresponds to a point. Under this dictionary, if everything is transverse enough, pull back honestly means take the preimage, cup means intersect and push forward hnestly means map forward. So the question I am asking above is “how many points in X are images of points in Y that map down to a given 3d-1 points of P^2 under the evaluation map?” Which, if you think about it, is the same as “How many degree d genus zero curves pass through 3d-1 given points in P^2?”

Does this help?

If these K-theoretic Gromov-Witten invariants live in the K-theory of a point, shouldn’t they live in , with u degree two, rather than just the integers?

Depending on where your confusion lies,

1) They live in one (even) degree, and it’s so easy to figure out which that nobody bothers making reference to it; instead they just set u = 1.

2) This was (so far) nonequivariant K-theory of a point; the ring you wrote down is the -equivariant K-theory of a point.

(These are not meant to be two different ways of explaining the same thing!)

Thanks for clarifying, Allen. I meant u to be the Bott element in , i.e., confusion number 1.

“It’s what matters when you’re pushing forward

cohomology classes; the degree shifts by the real dimension.”

I tend to index cohomology by the degree divided by two. This way I get a map A^k\to H^k instead of H^{2k} – that 2 always annoys me. (A^* as defined in Fulton Intersection Theory).

On the other hand, I hadn’t even understood which K-theory you’re talking about. Thanks to everybody who clarified things.

I tend to index cohomology by the degree divided by two.Don’t work much with high genus curves, do you?

(I don’t either.)