The International Congress of Mathematicians will be taking place in Madrid relatively soon, in late August. One tradition at this conference is the announcement of the Fields Medals, and I’m getting embarassed that I’m not hearing any authoritative rumors about this (other than about Tao and Perelman); if you have any, please send them my way. One other tradition is to have speakers write up their talks in advance, with the proceedings available at the time of the conference, so already some write-ups of the talks to be given there have started appearing on the arXiv.
Last night, Michele Vergne’s contribution to the proceedings appeared, with the title Applications of Equivariant Cohomology. On her web-site she has a document she calls an exegesis of her scientific work, this gives some context for the equivariant cohomology paper. She also is co-author of a book called Heat Kernels and Dirac Operators, which has a lot more detail on some aspects of this subject. Finally, there has been a lot of nice recent work in this area by Paul-Emile Paradan.
Equivariant cohomology comes into play when one has a space with a group acting on it, and it mixes aspects of group (or Lie algebra) cohomology and the cohomology of topological spaces. There are various ways of defining it, the definition that Vergne works with is a bit more general than the one more commonly used. It involves both differential forms on the space, and generalized functions on the Lie algebra of the group.
The beauty of equivariant cohomology is that it often computes something more interesting than standard cohomology, and you can often do computations simply, since the results just depend on what is happening at the fixed points of the group action. There’s a similar story in K-theory: when you have a group action on a space, equivariant K-groups can be defined, with representatives given by equivariant vector bundles. Integration in K-theory corresponds to taking the index of the Dirac operator, and in the equivariant case this index is not just an integer, but a representation of the group. The index formula relates cohomology and K-theory, and one of Vergne’s main techniques is to work with the equivariant version of this formula.
In the case of a compact space with action of a compact group, there’s a localization formula that tells you how to integrate representatives of equivariant cohomology classes in terms of fixed point data. In many cases, this leads to a simple calculation, one famous example is the Weyl character formula, which can be gotten this way. New phenomena occur when the group action is free, and thus without fixed points. This was first investigated by Atiyah (see Lecture Notes in Math, volume 401), who found that he had to generalize the index theorem to deal with not just elliptic operators, but “transversally elliptic” ones. Such operators are not elliptic in the directions of orbits of the group action, but behavior of the index is governed by representation theory in those directions.
Vergne has been studying examples of this kind of situation, and it is here that generalized functions on the Lie algebra come into play. Integrating the kind of interesting equivariant cohomology classes that occur in the transversally elliptic index theory case over a space gives not functions but generalized functions on the Lie algebra. There’s a localization formula in this case due to Witten, who found it and applied it to 2d gauge theory in his wonderful 1992 paper Two Dimensional Gauge Theories Revisited.
This kind of mathematics, growing out of the equivariant index theorem, is strikingly deep and beautiful. It has found many applications in physics, from the ones in 2d gauge theory pioneered by Witten, to more recent calculations of Gromov-Witten invariants. It leads to a mathematically rigorous derivation of some of the implications of mirror symmetry in special cases, and a wide variety of other results related to topological strings. My suspicion is that it ultimately will be used to get new insight into the path integrals of gauge theory, not just in 2 dimensions but in 3 or 4.
Update: Vergne has another nice new paper on the arXiv. It’s some informal notes on the Langlands program which she describes as follows:
These notes are very informal notes on the Langlands program. I had some pleasure in daring to ask colleagues to explain to me the importance of some of the recent results on Langlands program, so I thought I will record (to the best of my understanding) these conversations, and then share them with other mathematicians. These notes are intended for non specialists. Myself, I am not a specialist on this particular theme. I tried to give motivations and a few simple examples.
It would be great if more good mathematicians wrote up informal notes like this about subjects they have learned something about, even if they are not experts. The notes are entitled All What I Wanted to Know About Langlands Program and Was Afraid to Ask.