A Survey of Elliptic Cohomology

There’s a beautiful survey paper about elliptic cohomology that Jacob Lurie, an AIM 5-year fellow in the math department at Harvard, has recently put on his home page. This paper has been discussed a bit already by David Corfield and by Urs Schrieber.

I don’t have time right now to try and write up something comprehensible about those parts of the elliptic cohomology story that I kind of understand, and in any case I want to spend more time reading Lurie’s paper. It brings into the elliptic cohomology story several of my favorite pieces of mathematics (Atiyah-Segal completion, Freed-Hopkins-Teleman), in a way that I don’t yet understand. But in any case there’s a lot of very beautiful and very new mathematics in this paper, mathematics that has tantalizing relations to quantum field theory.

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14 Responses to A Survey of Elliptic Cohomology

  1. Kea says:

    “…mathematics that has tantalizing relations to quantum field theory.”

    Curious comment: do you refer to section 5.2 or something else? Amazing set of ideas this guy has, I’ll agree. I’ve had the Adams book, for example, on my shelf for years and years, but I don’t imagine I shall ever be able to refer to it casually.

  2. Kea says:

    The only issue I have after browsing Lurie’s papers is that he seems to have solely a homotopy theorist’s take on higher categories, rather than a more open point of view which one would expect from someone discussing issues with such a broad scope. Admittedly, I am biased.

  3. woit says:

    Kea,

    You’re right that this is a homotopy theorist’s take on the subject, heavily influenced by the work of Mike Hopkins. But, for a homotopy theorist, Lurie is taking a quite broad point of view.

    The relations to QFT include not just the Freed-Hopkins-Teleman stuff that I’m fascinated by, but other things as well. The whole subject was heavily influenced by Witten’s writing down of a 2d QFT that gives the elliptic genus, and more recently Stolz and Teichner and others have been investigating the relation of 2d conformal field theory to elliptic cohomology. There’s more about this if you follow the link to Urs’s posting.

  4. Michael says:

    seems like there is an href missing from the HTML on Urs link. thanks.

  5. A.J. says:

    Hi Peter,

    You mentioned a while back (in a post about Freed-Hopkins-Teleman that you hadn’t quite been sold on stacks. I think that the Atiyah-Segal theorem might be one good reason to like them.

    In algebraic geometry, the classifying stack of G is the quotient stack pt/G. The nicest way to define this is to start with the action groupoid of G on a point; this is just the category with one object and one morphism for every element of G. If you take its nerve, you obtain the simplicial scheme with G^n in the n-th degree. Taking the geometric realization of this simplicial object gives you Milnor’s construction of BG. But this isn’t really the best thing to do; you’re throwing away information. What you should do instead is think of the functor of points of this simplicial scheme as a Category Fibered in Groupoids. This CFiG isn’t a stack however; it’s just a pre-stack, doesn’t satisfy effective descent. It assigns to a test scheme S, the groupoid Hom(S,pt), with morphisms given by the obvious G-action. To get a nice CFiG, we need to stackify, to study instead objects which are locally maps to the point, but might differ globally by the action of G. This “differing globally by the action of G” really means that we have a G-valued function on every non-empty intersection of local patches. This is exactly the data of a principal G-bundle.

    So what’s my point? Well, it’s that stacks are good magic. The natural definition of the quotient stack pt/G tells you precisely that maps into pt/G classify principal G-bundles. This magic happens because you made certain to keep track of how G acts on th e point. Which results in a nice thing: K(pt/G) is the Grothendieck group of the vector bundles on pt/G. But these are the same thing as G-equivariant vector bundles on the pt, i.e. G-modules. So the K-theory of the stacky pt/G is, in fact, canonically isomorphic to Rep(G). If I understand things right, the Atiyah-Segal theorem is basically an identification of the information you lose by passing to the geometric realization.

  6. The entry on elliptic cohomology that Peter is referring to is this one:

    http://golem.ph.utexas.edu/string/archives/000737.html

    Stay tuned for more on Stolz&Teichner in a couple of weeks when I report from this winter school:

    http://golem.ph.utexas.edu/string/archives/000712.html

  7. Hi AJ,

    I understand what you are saying, except for this last sentence

    So the K-theory of the stacky pt/G is, in fact, canonically isomorphic to Rep(G).

    I guess you mean that this K-theory is isomorphic to the Grothendieck group completion of the decategorification of Rep(G), no?

    BTW, if anyone is wondering what AJ is talking about: a chatty discussion of those quotient stacks is given here.

    A review of how to think of these quotients as groupoids is given here and here.

  8. A.J. says:

    Hi Urs,

    Yes, by Rep(G) I mean the representation ring, not the whole category. (I was using the same notation that Lurie used in his survey.)

    Nice expositions, by the way. Now if I could just find a chatty discussion of Quillen’s model categories…

  9. Now if I could just find a chatty discussion of Quillen’s model categories…

    If you find out anything at all, let me know. Thanks.

  10. D R Lunsford says:

    I wish I could get excited about this. Someone should explain this stuff in a way that is palatable to intuitive thinkers. This sort of exposition is almost unreadable to me.

    -drl

  11. Kea says:

    drl

    Unfortunately, whereas us poor physicists are just trying to understand, say, M-theory, this Lurie guy is on a fast rocket to trying to prove the Riemann Hypothesis (why else would he be working on higher toposes?) … it ain’t going to be easy for us sods to keep up.

  12. kielbasa says:

    “I wish I could get excited about this. Someone should explain this stuff in a way that is palatable to intuitive thinkers.”

    As opposed to “non-intuitive thinkers.”

  13. woit says:

    A.J.,

    Thanks a lot for your comment. That’s more or less exactly why I’ve gotten interested in stacks again recently (that K(pt/G)=R(G), rather than the completion). Chris Woodward had given me a more concrete explanation of this (that on the geometric realization you get K-theory classes that don’t quite come from representations, and thus need the completion), but your explanation is very helpful. Thanks again!!

  14. D R Lunsford says:

    kielbasa,

    Yes non-intuitive thinkers. It should be obvious who they are 🙂

    -drl

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