In week255, hear what happened at the 2007 Abel Symposium in Oslo. Read explanations of Jacob Lurie and Ulrike Tillman’s talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen’s talks on string topology, Stephan Stolz’s talk on cohomology and quantum field theory, and Fabien Morel’s talk on A1-homotopy theory.
But first, take a tour of the Paris Observatory:
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Lurie’s DAG I—II has just appeared in revised form on the arXiv
and IV, brand new but promised, is here at last.
(I’m confused about the case n = 0, for reasons having to do with the “degree” I mentioned earlier.)
I think this is 0 unless the manifold is 0-dimensional, in which case it counts points.
That would certainly resolve the ‘paradox’ I was encountering, namely that we want an oriented -manifold to have canonically associated to it some element of its th deRham cohomology group!
Any idea why as we move higher up this tower we get periodic cohomology theories, but not in this degenerate case?
Hm, I think that the degree of the QFT of the super D(-1)-brane gives the degree in deRham cohomology one finds.
Same pattern as we move up the dimensional ladder: the way K-theory arises from susy quantum mechanics: we get for the ungraded case (or degree 0), when we swtich on grading and look at degree 1, and so on.
Unless I am misremembering something, that is.
Voevodsky’s “Russian solution” reminds me of an old story (parable?) from the time of the space race.
NASA knew that astronauts needed to take notes while up in space, but existing pen technology basically worked by letting gravity draw ink down from a reservoir to the nib to make a mark on the paper. Obviously, this wouldn’t work in orbit. So they spent ten years and twelve million dollars to develop a new pen that would write under any gravitational conditions – a “space pen”. The Russians, faced with a similar difficulty for their cosmonauts, used a pencil.
He said that every mathematician has an “n-category number”. Your n-category number is the largest n such that you can think about n-categories for a half hour without getting a splitting headache.
Lol!
…leading up to a conjecture for the 3-vector space that Chern-Simons theory assigns to a point.
Wow. Are notes for this talk available? (I couldn’t seem to access that section of the website). What is the conjecture for the 3-vector space that Chern-Simons theory assigns to a point? My thesis project revolves around this stuff.
John — thanks for the great report!!! (Any general impressions from Toen’s talk?)
Bruce — I’m not sure what Dan said in Oslo, and maybe John will answer this better, but from what I understand the first guess as to what to assign to a point in the FHT K-theoretic Chern-Simons is the category of -modules. To the circle they assign , the -equivariant K-theory spectrum of twisted by a level, and you’d like a category whose Hochschild homology looks like this. The first proposal then is to take the “group-algebra of , valued in K-theory” (and twisted by the level). This is an algebra by convolution (well really an or spectrum) and we can look at its category of modules (really module spectra) (which will be a stable category, or rather stable -category if you like).
(I think of this group algebra (as I think they do) as the K-theoretic shadow of some category of sheaves on , endowed with convolution making it monoidal, and its modules as the shadow of some 2-category of module categories, but I don’t think this is worked out yet.)
I was told there are some subtleties with this formulation of having to do with completions – from the of this category you get not the honest but maybe its Borel version, so you need to modify the category of modules in a way that depends on the level or something, but I may have misunderstood (I heard this hiking at altitude…). Maybe the notes from Oslo will be clearer. Hope this helps.
Alas, I accidentally missed Toen’s talk while having a long conversation with Stephan Stolz. 
Dan Freed only suggested a 2-category for the point in the special case of ‘Chern–Simons theory with a finite gauge group ’ — i.e. the Dijkgraaf–Witten model — and even then, only in the untwisted case.
In this case he said should be the 2-category of -modules. Here is the ‘categorified group algebra’, namely the category of -graded vector spaces, made monoidal in the obvious way.
I discussed and its higher-dimensional generalizations here, including how to twist the monoidal structure by a cocycle, and how to get a TQFT out of this stuff. I believe Freed’s idea is essentially equivalent, though it looks a bit different at first, since I don’t attach to a point; I just attach to an edge.
So, I think most of the remaining fun lies in generalizing to a full-fledged compact Lie group equipped with an element of .
Oh, how I wish I could have been there.
At least I can take comfort in the fact that while you all were talking about how to 3-quatize Chern-Simons, I was staying here with Jens Fjelstad actually doing it. ;-)
(Okay, I admit there still remains more to be done…)
Concerning the question “What exactly does extended CS theory assign to a -dimensional space?”, and regarding Bruce’s remark that he has a good idea about what the answer is (for Dijkgraaf-Witrten theory, ast least, I think):
I happen to mention this in the motivating introduction of this talk. The point I would like to make is that:
a) answering this question should not be a matter of guessing and making consistency checks.
b) rather, if CS theory is suppoosed to be a 3-functor, then
b i) there should better be a 3-functorial way to conceive the CS action, i.e. to rewrite the Chern-Simons functional as a 3-transport;
b ii) and there should better be a systematic way to pass from such a 3-functorial action to the corresponding 3-functorial QFT
b iii) such that this spits out the right result, not just the values on points, etc, but the entire quantum 3-functor.
Incidentally, that talk of mine is precisely about the issue of identifying the right classical 3-transport which underlies Chern-Simons.
Unfortunately Bruce’s result, which he alluded to, is still top secret, but we have a pretty good idea how it can be obtained by systematically quantizing CS-theory in the context of the “quantum charged -particle” which I keep talking about #.
Incidentally, it’s all about push-forward. The quantum propagation -functor should be a certain push-forward to a point of the classical parallel transport -functor.
This one can usually explicitly compute, at least in principle, for all dimensions below top dimension. For instance one can find the D-branes that an open string couples to this way #.
For the top level dimension it seems naively like one runs into the need to invent a measure, as you mention. However, I think there are indications that if we really strictly follow the n-categorical dao, everything will take care of itself. The measure arises automatically from the way colimits work. (I checked this for a toy example of the 1-dimensional case here.)
(I am way behind with writing this stuff up. But I’ll keep mentioning it, since from time time somebody reads this, chimes in and starts collaborating. Last time this happened with David Roberts. I think we got a pretty cool paper out of that collaboration.)
Re: This Week’s Finds in Mathematical Physics (Week 255)
Anyway, Lurie and Hopkins have worked out the … Unfortunately this work is not yet written up.
With Hopkins’ record, don’t hold your breath. On the other hand, Lurie did write a 619-page book as a grad student. This irrestible-force-meets-immovable object scenario could be a topic for wagering.