Liveblogging: Jacob Lurie on 2-d TQFT

We seem to still get a lot of google searches for this post. Jacob has an expository article out now that does a much better job of addressing this material than my liveblogging. You should read that paper instead.

Jacob Lurie is in town giving two topology talks. The first one is on classifying 2-d extended TQFT (a topic near and dear to my heart), and the second is a more leisurely introduction to extended 2-dimensional TQFTs . As is often the case when Jacob is in town, the room is rather packed.

At the moment I’m liveblogging the second talk, for the first talk go past the flip.

Jacobs 2nd talk is starting now, and since Peter Teichner just described it as “the talk where you start from the beginning” I’m going to try to continue the liveblogging, and hopefully it’ll make the earlier talk make more sense.

In this talk, Jacob is describing his joint work with Mike Hopkins on extended TQFT inspired by Kevin Costello’s papers.

Jacob starts off recalling Atiyah’s celebrated definition that an n-dimensional TQFT is a tensor functor from nCob to complex vector spaces. The “functor” part here means that gluing cobordisms corresponds to composition of linear maps. The “tensor” part says that F(M \cup N) = F(M) \otimes F(N).

Then he recalls the well-known result that 2-dimensional TQFTs are classified by Frobenius algebras. To see this, you first consider the vector space assigned to a circle. Then a pair of pants gives a multiplication on this space, and a disc gives a trace. Using the relations between cobordisms you can see that these algebraic structures fit together to make a Frobenius algebra.

The moral of this story is that we should understand n-dimensional TQFT you want to understand it on some simple pieces, and then take your manifold and chop it up into those simple pieces. This is nice, but unfortunately you can’t chop things too finely. You aren’t allowed to chop it up in ways that have corners. This suggests another definition.

Definition: An extended TQFT (in dimension n) is a rule

  • closed n-manifold –> complex number
  • closed (n-1)-manifold –> vector space
  • bordism of (n-1)-manfiold –> map of vector spaces
  • closed (n-2)-manifold –> linear category
  • bordism (n-2)-manifold –> linear functor

The “…” is not intended to mean that it is easy to keep going, only that you’re meant to try. But since we’re only talking about low-dimensional topology and “here low means n<2” we don’t really need to understand the “…”.

This definition can be summarized as “An extended TQFT is a functor between n-categories.”

At this point there’s a bit of a digression in which Rob Kirby wants to know why we should think about this hard problem of what an n-category is when we don’t have any examples in dimensions above 3. Jacob says “I’m the wrong man to ask, I only understand what’s going on in dimension less than 2.”

After that digression he moves on to describe the Baez-Dolan Cobordism Hypothesis (paraphrased by Jacob): “Extended TQFTs are “easy to describe/construct.” Elaborating a bit further he says that you only need to describe the TQFT on very small building blocks, and then n-category theory will do all the work for you. Rather than making the conjecture more precise he’s going to give examples where the conjecture is known to hold.

(non-)example (n=2): We restrict our attention to a smaller category where we only allow certain bordisms allowed by string topology based on some manifold M. To a circle we assign the homology of the loop space on M. To a pair of pants we assign the Chas-Sullivan product on homology. (To a disc we don’t get anything, since that’s a bordism that isn’t allowed.)

But rather than just assigning homology, we’d rather assign the chain complex itself. Unfortunately given a bordism you only get a chain homotopy between the corresponding complexes. Nonetheless we can cook up out of this more operations on F(circle) associated to higher homology of Bord(M,N).

A better way to restate this is that Bord(M,N) = Map(F(M), F(N)) where the latter space of chain complexes is thought of as a topological space. So our TQFT here is actually a functor of (\infty, 2)-categories! That is the 2-morphism spaces aren’t just a set, they’re actually topological spaces, and the functor respects this topological structure.

Now we get down to the question at hand. Define the monoidal (\infty, 2)-category 2Bord defined by

  • The objects of 2Bord are oriented (compact) 0-manifolds
  • The morphisms of 2Bord are bordisms between 0-manifolds
  • The space of 2-morphisms from f to g is the classifying space of bordisms from f to g which are trivial on their boundary

We want to classify tensor functors from this (\infty, 2)-category to other (\infty, 2)-categories. By Baez-Dolan we should expect this question to have an easy answer: all we need to know is where a point goes!

A point corresponds to some object C. The point with the opposite orientation corresponds to a dual to C (using a line segment as the map), so we need to require that C be dualizable. Then we can figure out where a circle goes just by making the circle out of two segments. So the circle goes to the “dimension” of C, which is an element of End(C).

This is already enough to classify 1-dimensional extended TQFTs! Exciting. Now we need to figure out how to promote 1-dim extended TQFTs to 2-dimensional ones.

So where is a disc going to go? Well, it must land in 2Hom(1_1, dim C). Using the circle action on dim C (given by the circle action on the circle) we know that the disc lands in the circle fixed points of 2Hom(1_1, dim C).

The punchline is that this is all that you need to know. The only data is a dualizable object and a circle fixed point in 2Hom(1_1, dim C). You may need to check lots of relations, but you don’t need any more data than that.

This fact allows Jacob to give a quick description of string topology, and a proof that it is homotopy invariant.  Since I don’t understand string topology, I’ll stop here.

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Embedded TQFT?

So, a subject rather near and dear to the hearts of many of my fellow co-bloggers is that of 1+1-dimensional TQFT: that is, of monoidal functors from the category of 1-manifolds with morphisms given by smooth cobordisms to the category of vector spaces over your favorite field k.

There’s a rather remarkable theorem about such functors, which really deserves a post of its own for proper explanation, but I’ll spoil the surprise here.

Any such functor associates a vector space A to a single circle, and to the “pair of pants” cobordism, it assigns a map m:A\otimes A\to A, which one can check is a commutative multiplication.

Furthermore, the cap, thought of as a cobordism from the empty set to a circle gives a map i:k\to A, which gives a unit of this algebra. Thought of as a cobordism from the circle to the empty set, it gives us a map \mathrm{tr}:A\to k which we call the counit or Frobenius trace.

Theorem. A commutative algebra with counit (A,\mathrm{tr}) arises from a TQFT if and only if \mathrm{tr} kills no left ideal of A.

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Freedman on distinguishing manifolds with quantum topology

This week Mike Freedman was in Berkeley for the annual series of three Bowen lectures. The first two were about topological quantum computation and the fractional quantum Hall effect. Since I missed one of those because of closing arguments in the case I was on the jury for, I’ll instead only discuss his third talk which dove-tails nicely with my last post.

His motivating question was why do we look for topological quantum computation in 3 space-time dimensions (that is, by restricting to a system stuck in a plane) rather than the full 4 space-time dimensions? His explanation was the following: In 3-dimensions it seems that unitary TQFT can distinguish all manifolds, however in 4-dimensions they can not.
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Quantum Topology in Hanoi

I recently got back from an interesting trip to Vietnam, where I attended “Quantum Topology in Hanoi“. This conference was held at “VAST”, the Vietnamese Academy of Science and Technology, and organised by Thang Le and Stavros Garoufalidis of Georgia Tech.

Being in Vietnam was great fun, and most participants (including at least one of our readers) enjoyed the crazy bustle of life in Hanoi. Even the afternoon when the power went out, the backup generator failed, the airconditioning was off, and Dylan Thurston talked about the combinatorial model for knot Floer homology in 34°C wasn’t so bad. :-) After the conference, I went to Halong Bay with Dylan and Jana for the weekend, and then up to Sapa for two more days.

My talk was about the “lasagna operad” acting on Khovanov homology, and I decided to try Dror’s ‘single page handout’ approach, and even more adventurously, his freshly made ‘javascript handout browser’. You can see my slide, and Dror’s programming, here.

But what I’d like to talk about now is Nathan Geer’s talk, about “fake quantum dimensions”. I’ll start somewhere near the beginning, reminding you how to build tangle invariants out of braided tensor categories, then explain what goes wrong when quantum dimensions become zero, and finally what Nathan and co. propose to do about it. Continue reading

Back from Maui

If any of you were wondering where we had all gone this past week, there’s a simple answer: Maui! More than half of the contributors to this blog were far far away from the internet at “Subfactors in Maui.” This was a relentlessly laidback conference organized by Vaughan Jones (which means it was actually organized by our very own Scott M), entirely dominated by Berkeleyites; a little internet research shows that every mathematician there either received their Ph.D., did a postdoc, or is a faculty member at Berkeley.

I’m afraid we didn’t really keep up our previous standards of talk blogging, but I’ll plead lack of internet and a somewhat exotic collection of material. I can’t hope to do justice to the talks on subfactors (given by Vaughan Jones, Emily Peters, Dietmar Bisch, and Pinhas Grossman), though they were very interesting to me as someone who’s generally been faking it when it comes to subfactors (though I do REALLY want to know what arithmetically equivalent subfactors are) or Laurent Bartholdi’s talk on automatic groups and subfactors (though maybe I should, considering that I took a class on automatic groups with Laurent 4 years ago when he was at Berkeley). That leaves my talk, which was basically on material I’ve already covered, Scott M’s talk, which I’ll let him cover in his own sweet time, and the Station Q denizens Mike Freedman and Kevin Walker. Continue reading

Integral TQFT and mapping class group actions

Gregor Massbaum gave one of the shorter contributed talks at the conference in Faro, speaking in very general terms on his work on integral TQFT.

When I say TQFT here, I really mean the Witten-Reshetkhin-Turaev 3-d TQFT associated to a quantum group and a root of unity. Since I don’t want to get into the guts of what that is, let’s just accept for a moment that a 3-manifold invariant called WRT_\zeta exists. For each root of unity $\zeta$, it associates a complex number to each oriented compact 3-manifold.

Now, as I’m sure you’re all aware, a numerical invariant of 3 manifolds is a lot weaker than a 3-d TQFT. But it’s less weak than you might think.For example, we can get a TQFT out of such an invariant as follows:

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Help David learn Quantum Field Theory

For the few months, I’ve been trying to learn Quantum Field Theory, working mostly from Zee’s book Quantum Field Theory in a Nutshell and also from John Baez’s course notes. Note that my goal is to actually understand physics, in terms of actual physical objects interacting, not to just understand mathematical constructions. This is going to be the first of what I imagine will be a series of posts where I ask dumb questions and hope that AJ, John Baez or one of our other physcist readers will help me out. For my first question, I’m going to ask about something which has been bothering me since the very start of the book.

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Getting canopolises right.

Leaving aside the issue of thorny question of the correct plural form of canopolis, lets try and get the definition right!

The motivating idea here is that a planar algebra is a nice (the right?) formalism for tensor categories with duals, and that there’s a reasonable hope that the definition will extend nicely to something like an n-planar algebra (a noodle algebra, a foam algebra?), which we should think of, very roughly, as an n-category with lots of duals. A canopolis is a first step along the way – it adds an extra categorical dimension to a planar algebra, but without demanding any more duals.

The term canopolis was introduced by Bar-Natan, although there called a canopoly, then used by Ben and by me and Ari Nieh. It still doesn’t have a good definition written down anywhere. The idea is simple; a canopolis is meant to be a planar algebra of categories. I’m still not too sure how much devil there is in the details though, and would love some help. Continue reading