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June 18, 2008

Schommer-Pries on Classification of 2-Dimensional Extended TFT

Posted by Urs Schreiber

Today at the Hausdorff institute we heard a nice talk by Chris Schommer-Pries over from the Secret Blogging Seminar on a generators-and.relations presentation of an extended 2-dimensional cobordism category.

The bicategory that Chris considers is similar to, but different from, the week double category in

Jeffrey Morton
A Double Bicategory of Cobordisms With Corners
arXiv:math/0611930

which John discussed in week 242.

Chris defines a 2-category whose

- objects are disjoint unions of points

- morphisms are 1-dimensional cobordisms between these

- 2-morphisms go between cobordisms with the same boundaries and are (that’s my formulation of the pictures he drew) surfaces with a height function Σ[0,1]\Sigma \to [0,1] such that the preimage of 0 is the incoming 1-dimensional morphism, the preimage of 1 the outgoing one and each source-source point runs to a corresponding target-source point.

Chris proves a big theorem which says that this bicategory is equivalent to one obtained from a handful of generators modulo a bunch of relations.

The generators he presented include precisely those in the pictures on p. 27 of Baez, Dolan Higher-dimensional Algebra and Topological Quantum Field Theory, but also some more. I suppose that, hence, Chris is talking about the 2-category that places like this would be called 2Tang2Tang or the like. But I can’t tell for sure.

In any case, Chris invokes Morse theory, singularity theory and Cerf theory to prove that his generators and relations are sufficient. The general strategy here is quite simialr to the proof that Aaron Lauda and Hendryk Pfeiffer give for similar but different situation

Aaron D. Lauda, Hendryk Pfeiffer
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
arXiv:math/0510664 (blog)

In that article the sufficiency of the generators and relations is proven by showing that they allow to reduce every surface to one of a certain “normal form”. Chris Schommer-Pries emphasized that this only works because we have a good classification of 2-manifolds, whereas his proof immediately generalizes to higher dimensions.

In fact, he is working with Chris Douglas already on the d=3d=3 case. The singularity theory they use is under control up to d=7d=7 and hence they can already see how to generalize everything up to that dimension, the main problem being on the algebraic side, where one has to come to grips with “symmetric monoidal weak 7-categories”.

In fact, it seems that the number 7 appearing here which was attributed by Mike Hopkins to joint work by Arthur Bartels, Chris Douglas, and André Henriques is really the 7 in Chris Schommer-Pries’s work.

The nice thing about having a presentation in terms of generators and relations is that it allows to classify functors out of this 2-category Bord 2Bord^2 into some other bicategory.

This way Chris obtains the following nice corollary:

2-Functors from these extended 2-cobordisms to the bicategory of algebras, bimodules and bimodule homomorphisms Bord 2Bimod() Bord^2 \to Bimod(\mathbb{R}) are given, up to equivalence, by trace-Morita classes of separable symmetric Frobenius algebras.

These algebras are of course what the 2-functor assigns to a single point.

Notice that, while refined to a 2-category, Chris’s Bord 2Bord^2 is a “localized” version of “just” closed string worldsheets. He told me about a cool idea how to generalize his work to the open-closed case. That idea was very close to my heart, but I am not sure if I am allowed to talk about it here in public.

Posted at June 18, 2008 6:35 PM UTC

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4 Comments & 0 Trackbacks

Re: Schommer-Pries on Classification of 2-Dimensional Extended TFT

Hi Urs!

Thanks for this post.

Urs said:

In fact, it seems that the number 7 appearing here which was attributed by Mike Hopkins to joint work by Arthur Bartels, Chris Douglas, and André Henriques is really the 7 in Chris Schommer-Pries’s work.

I wouldn’t say that the number 7 is really coming from my work (joint with Chris Douglas for dimension >2). I know that Arthur, Chris and André have been working on the higher dimensional framed cases and that it is entirely possible that they solved this. I don’t really know the current status of their project.

But, I do know that their methods are very similar to mine (this is what got Chris and I to start collaborating in the first place). I wouldn’t be surprised if they ran into the same 7-dimensional barrier.

Posted by: Chris Schommer-Pries on June 18, 2008 8:22 PM | Permalink | Reply to this

Re: Schommer-Pries on Classification of 2-Dimensional Extended TFT

Hi Chris!

I could just ask you in person – but maybe it is actually better to do it here, since others will be interested, too:

myself and people I talked to have this open question:

what is a trace-Morita class of algebras?

:-)

Posted by: Urs Schreiber on June 19, 2008 6:11 PM | Permalink | Reply to this

Re: Schommer-Pries on Classification of 2-Dimensional Extended TFT

Ahh! right! Sorry about that. I was thinking about the more refined version where we talk about the bicategory of Extended TFTs. Let me explain…

What I want to say is that (symmetric monoidal) 2-functors into algebras, bimodules and intertwiners (up to equiv.) are in bijection with separable symmetric Frobenius algebras up to Morita equivalence. I think this conveys the right answer, as long as you don’t think too hard about it.

What I wrote was “trace-Morita” and what I was thinking ( I think I said this out load) was “trace-preserving Morita equivalence”. But that probably doesn’t help you. You’re going to ask what in the world is a trace preserving Morita-equivalence?

Before I get to that, let me ask you a warm-up question to get the juices flowing. Suppose I have a Frobenius algebra A (this is a structure on A, not a property) and another algebra B. Suppose I know B is Morita equivalent to A. Is B a Frobenius algebra? Can we give it a canonical Frobenius algebra structure? What if I give you the Morita equivalence? What if A is a symmetric Frobenius algebra? I suggest you stop and think about this for a minute before continuing on.

(time passes)

Ok. So it depends on what you mean by a Morita equivalence. If you mean the usual thing i.e. an A-B bimodule M, such that there exists a B-A bimodule such that there exist isom. etc. If that is what you mean then the answer is no. You cannot transfer the structure with just that much data. There will be no canonical choice of Frob. alg. structure on B.

However, if you have both the A-B bimodule and the B-A bimodule and the appropriate isomorphisms to the identity bimodules, then you can transfer the symmetric Frobenius algebra structure. When I say Morita equivalence I’m going to mean this whole package. Maybe someone has a name for this? coherent Morita equivalence?

So now I think what I wrote is probably starting to make more sense. If we have two symmetric Frobenius algebras, A and B, and a (coherent) Morita equivalence between them we can transfer the Frobenius algebra structure from A to B and compare it with the original structure on B.

Being a Frobenius algebra is several things. It is some properties and it is some structure. But the structure can all be put into the trace. One of the equivalent definitions of a symmertic Frobenius algebra is that it is an algebra with a trace such that a bunch of properties are satisfied.

So when I say “trace-preserving Morita equivalence” what I really mean is (coherent) Morita equivalence which preserves the symmetric Frobenius algebra structure, i.e. the trace.

Hope this helps.

PS: What you wrote looks like I was just talking about algebras over the reals, but really the ground ring can be any commutative ring. If I had been talking about the reals I wouldn’t have said “separable”, I would have said “semi-simple”!


Posted by: Chris Schommer-Pries on June 19, 2008 7:51 PM | Permalink | Reply to this

Re: Schommer-Pries on Classification of 2-Dimensional Extended TFT

Maybe someone has a name for this? coherent Morita equivalence?

Morita context maybe?

Thanks for the explanation, when I read “trace-Morita class” my mind immediately wandered off in a direction which turned out to be the wrong one.

Posted by: Jens on June 19, 2008 11:11 PM | Permalink | Reply to this

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