Equivariant Cohomology

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.

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22 Responses to Equivariant Cohomology

  1. fp says:

    I guess the ‘cosmology’ in the 4th paragraph is a typo and should be ‘cohomology’.

  2. Peter Woit says:

    Oops, many people might like an application of this to cosmology, but that definitely was a typo…

  3. doctorgero says:

    …and I’m getting embarassed that I’m not hearing any authoritative rumors about this (other than about Tao and Perlman…

    Apparently, it is ‘Perelman’. :-)

  4. Authoritative Bigshot says:

    Bhargava.

  5. nonblogger says:

    I’ve noticed there are four slots for fields medalists’ talks, so perhaps four medals. If indeed Perelman and Tao get one that’s two left. Recent young EMS prize winners make up for a good shortlist
    http://www.math.kth.se/4ecm/prizes.ecm.html
    Looking at his recent publication record, I’d bet on Okounkov.
    There are also the lists of recent Clay Fellows http://www.claymath.org/fas/research_fellows/
    and award winners
    http://www.claymath.org/research_award/
    I’ve looked at recent AMS prize winners but most seem too old.

  6. bobo says:

    what about Zhu and Cao? the 2 dudes who completed the proof of the poincare conjecture.

  7. comentator says:

    My bet is T.Tao,A.Borodin,G.Perelman (though the age limit could be a problem in Perelman case). This could be surprising that in such a short time between publication of a paper and vetting of it and the award of a prize is given in the case of Zhu and Cao.

  8. woit says:

    doctorgero,

    Thanks, typo fixed.

    About possible Fields Medalists:

    Zhu and Cao: No way, and, knowing Cao, I’m pretty sure he’s over 40 anyway.

    Okunkov: Seems unlikely. In that field I’d think his Princeton colleague Rahul Pandharipande would be a more likely candidate.

    Bhargava: Hmm, that actually sounds plausible. Maybe “Authoritative Bigshot” really is one…

  9. Harry says:

    Wikipedia claims Perelman is just 40 this year :

    http://en.wikipedia.org/wiki/Grigori_Perelman

    (Not sure to what extent it’s reliable, though..)

  10. Zelah says:

    Who is A Borodin?

  11. Brett says:

    A great Georgian-Russian composer and chemist?

    Or more likely, this guy: http://www.cs.toronto.edu/~bor/

  12. comentator says:

    Alexei Borodin teaches in Caltech. Representation theory.done groundbreaking work in group theory (big groups) aspects related to representation theory.

  13. Peter Woit says:

    Brett and Zelah,

    The reference must be to Alexei Borodin, now at Caltech, see

    http://www.claymath.org/fas/research_fellows/Borodin/

    The list of Clay research fellows does probably contains many of the plausible candidates for the Fields.

  14. Matt says:

    There are also the prize winners from the 3rd ECM
    http://www.emis.de/ECM3/prizes.html
    Seidel, for his work on mirror symmetry and symplectic topology, and Gaitsgory, for his work on geometric Langlands, seem viable candidates. Since low dimensional topology/geometry always seems a popular subject, how about Zoltan Szabo?
    Both Borodin and Bhargava are eligible to claim the prize four years hence. This could affect their bids – in 2002, only 2 medals were distributed even though Tao was also a favorite then.
    Does anyone know who is on the medal committee this go-round?

  15. Anon says:

    “This kind of mathematics, growing out of the equivariant index theorem … 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.”

    Could you elaborate on that? How is equivariant index theory used there?

  16. Peter Woit says:

    Hi anon,

    This is a complicated story. Index theory itself hasn’t been much used, but there has been a lot of use of equivariant cohomology and localization formulae. For some surveys of this, see Kefeng Liu’s web-site

    http://www.math.ucla.edu/~liu/

    where he has various papers and surveys, including one by Yau, of the subject.

    A lot of this goes back to work by Givental, you can find his papers at his web-site

    http://math.berkeley.edu/~giventh/

  17. D R Lunsford says:

    Peter,

    Isn’t there a “for dummies” exposition of this stuff? How about putting it in historical context?

    -drl

  18. Peter Woit says:

    Danny,

    Sorry, I don’t know of any really good simpler expositions of this material, especially for physicists. There is some literature in the context of TQFT and supersymmetric quantum mechanics for physicists, look at papers from the early nineties by Blau and Thompson. Unfortunately most of the physics literature doesn’t really deal with the natural mathematical context for all this. Much of this is due to Atiyah, if you really want to understand this and its history, his collected works are a good place to look. But this is really not easy for a physicist to read and absorb.

  19. Johan Richter says:

    Is it decided already who will win the Fields medals? Or do they wait to the last minute to make that decision?

  20. Peter Woit says:

    Johan,

    I’m sure it’s already decided. I’ve heard the winners are notified some months in advance, so are those mathematicians chosen to deliver the addresses describing the work of the Fields medalists.

  21. Deane says:

    I am fairly certain that the Fields Medalists have already been chosen, but the names are a closely guarded secret.

    If you look at the citation counts on Mathscinet, then, of the names posted here, Tao, Okounkov, and Pandharipande are way ahead of everybody else. Perelman is definitely on the short list, too. But we’re all just guessing.

  22. Deane says:

    “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.”

    Amen!

    Terence Tao has lots of expository notes on his web site; the few that I’ve looked at are short and elegantly written.