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February 13, 2009

Lurie on TQFT and the Cobordism Hypothesis

Posted by John Baez

David Ben-Zvi notes that we can now see these talks by Jacob Lurie:

Lecture notes should be forthcoming. For a brief summary of the cobordism hypothesis and its generalizations, visit the nLab. Experts should improve this page!

The website recommends playing the ‘flash movie’, but that didn’t work for me. How about you?

Posted at February 13, 2009 8:09 PM UTC

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Re: Lurie on TQFT and the Cobordism Hypothesis

The flash movie didn’t work for me either.

The Quicktime version is 100MB but after downloading it my Quicktime didn’t recognise the format.

The MP4 version works but it’s 225MB (55 minute lecture)

Posted by: Colin Backhurst on February 15, 2009 11:50 AM | Permalink | Reply to this

Re: Lurie on TQFT and the Cobordism Hypothesis

Flash, no luck. Quicktime (on Windows) worked fine, so didn’t try MP4. Impressive lecturer.

Posted by: Charlie C on February 15, 2009 9:17 PM | Permalink | Reply to this

Re: Lurie on TQFT and the Cobordism Hypothesis

Ok the QuickTime version works for me too now (I was on version 7.5.5 and it needed version 7.6)

Posted by: Colin Backhurst on February 16, 2009 12:32 AM | Permalink | Reply to this

Re: Lurie on TQFT and the Cobordism Hypothesis

See also the paper – On the Classification of Topological Field Theories.

And two more on the ArXiv from Lurie today:

(Infinity,2)-Categories and the Goodwillie Calculus I and Derived Algebraic Geometry V: Structured Spaces.

Posted by: David Corfield on May 5, 2009 10:04 AM | Permalink | Reply to this

structured generalized spaces

And two more on the ArXiv from Lurie today:

[…] Derived Algebraic Geometry V: Structured Spaces.

I started an entry on this notion of structured space, highlighting the simple underlying idea and sketching the basic examples

[[structured generalized space]].

This is very close to our discussions in the context of comparative smootheology and [[generalized smooth space]].

David Spivak’s construction of “smooth derived geometry”, which we discussed here is a special case that Jacob Lurie mentions very briefly at the end.

What I keep finding a bit odd is the fact that we see remarks emphasized that an ordinary smooth manifold with a fancy structure (∞,1)(\infty,1)-sheaf of smooth ∞\infty-functions on it may model generalized things like orbifolds and higher generalizations, while the in some sense more direct fact that already (∞,1)(\infty,1)-sheaves on DiffDiff model these things (being smooth ∞\infty-stacks generalizing [[differentiable stacks]]) is not mentioned.

Posted by: Urs Schreiber on May 6, 2009 2:48 PM | Permalink | Reply to this

Re: structured generalized spaces

How could anyone keep up with the pace? Now we have Topological Quantum Field Theories from Compact Lie Groups by Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, and Constantin Teleman.

Posted by: David Corfield on May 7, 2009 10:26 AM | Permalink | Reply to this

quantization in QFT by abstract nonsense

How could anyone keep up with the pace? Now we have Topological Quantum Field Theories from Compact Lie Groups by Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, and Constantin Teleman.

From a quick look at the article, methinks the construction sketched in their section 3 “Finite path integrals” is pretty much the quantization prescription described in

Nonabelian cocycles and their σ\sigma-model QFTs ([pdf ], [Lab], [blog]).

Their category CC is my FF. The morphisms in their category Fam(C)Fam(C) of correspondences with functors to CC/FF are, as far as I understand their sketch, the [[bibrane]] appearing in def. 7.5 and composition is the “fusion” of bibranes. (They give the component description.)

Their “classical QFT” functor is the top-front part of the functor described by the big diagram on page 41, where for their section 3 the general X^\hat X is taken to be BG\mathbf{B}G.

The colimit operation on their page 14 computes the sections (as they remark, too), or more precisely the ordinary sections within the “generalized sections” appearing in def 5.56 and their formula (3.13) is the familiar formula for the pull-push operation in groupoidification, so that, I think, their “abstract nonsense realization of the path integral” coincides essentially with the prescription in my section 7.3, see last displayed formula before section 8.

Posted by: Urs Schreiber on May 7, 2009 2:16 PM | Permalink | Reply to this

Re: quantization in QFT by abstract nonsense

Which raises the question of whether your work is being sufficiently read in the right places. Or John and Jim’s for that matter – only HDA0 appears in the references, so nothing later than 1995.

Posted by: David Corfield on May 7, 2009 3:19 PM | Permalink | Reply to this

Re: quantization in QFT by abstract nonsense

David wrote:

Or John and Jim’s for that matter – only HDA0 appears in the references, so nothing later than 1995.

For what it’s worth, I’m delighted by the fact that Hopkins, Lurie and others have decided to take the hypotheses Jim and I propounded in HDA0 and turn them into precise theorems — and I’m very happy with how they’re crediting us. I don’t think any of our work after HDA0 needs to be cited, because it’s mainly about other stuff.

True, HDA4 proved a version of the tangle hypothesis for 2d tangles in 4 dimensions, but the techniques used there don’t easily generalize to higher-dimensional tangles — in that paper I compared the proof to moving a house across the country by taking it apart, mailing each piece to the new address, and reassembling it. If Lurie succeeds in proving the tangle hypothesis, it will be by completely different means.

I do however wish that someone working on Khovanov homology would use the definitions in HDA2 to prove that there’s a braided monoidal 2-category with duals underlying this theory. From a conversation with Scott Morrison in Glasgow a couple weeks ago, I got the feeling that this should be pretty straightforward (though not quick) using results in here:

Someone should give it a go!

Posted by: John Baez on May 7, 2009 3:51 PM | Permalink | Reply to this

Re: quantization in QFT by abstract nonsense

Can you extract from section 4.4 of On the Classification of Topological Field Theories how much of the Generalized Tangle Hypothesis has been solved?

Like the cobordism hypothesis, the tangle hypothesis can be generalized in many ways. For example, one can consider embedded submanifolds with tangential structure more complicated than that of a k-framing and embedded submanifolds with singularities. (p. 109)

No sign yet of fundamental (n + k)-categories of stratified spaces.

Posted by: David Corfield on May 7, 2009 4:20 PM | Permalink | Reply to this

Re: quantization in QFT by abstract nonsense

David Corfield wrote:

Can you extract from section 4.4 of On the Classification of Topological Field Theories how much of the Generaliz ed Tangle Hypothesis has been solved?

What do you mean by ‘solved’? If you mean ‘proved’, it’s worth noting that not even the Cobordism Hypothesis is proved in this paper. Indeed, the abstract says ‘this paper provides an informal sketch of a proof’. And if you look at the paper, you’ll see that the key notion that underlies the whole paper, namely that of an (∞,n)(\infty,n)-category, is not actually defined. (Apparently the definition can be found in Clark Barwick’s thesis, but this thesis has not been published and is not available online.) The beginning of Section 2 lists some other important definitions that need to be given before the Cobordism Hypothesis can become a precise conjecture, and the proof sketch can become a full-fledged proof.

This is not intended as a criticism: I think it’s great that Lurie is sketching the proof in a gentle expository paper before diving into the technical details. But since he’s clearly done more than he’s shown us here, it’s hard to know exactly how far he has gone beyond the Cobordism Hypothesis towards the Tangle Hypothesis and the Generalized Tangle Hypothesis.

No sign yet of fundamental (n + k)-categories of stratified spaces.

Actually I think Section 4.3 touches on this issue, at least in the ‘stable’ case. You might call Theorem 4.3.11 a ‘Generalized Cobordism Hypothesis’.

Posted by: John Baez on May 7, 2009 5:37 PM | Permalink | Reply to this

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