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…