## October 25, 2006

### Lauda on Topological Field Theory and Tangle Homology

#### Posted by John Baez

Aaron Lauda an undergrad here at U.C. Riverside, but now he’s finished his Ph.D at Cambridge and is a postdoc at Columbia. I just got a copy of his thesis:

It’s a great tour of these subjects!

I met Aaron Lauda when he was undergraduate here at U. C. Riverside and he started attending the Quantum Gravity Seminar. It soon became clear he was an exceptional fellow. We wrote a paper on 2-groups while he was getting a Master’s in physics here, and then he went on to Cambridge University to do a math Ph.D. with Martin Hyland. Hyland is an expert on category-theoretic logic, including game semantics, which people are busily rediscovering here. But he has broad interests, so Lauda wound up doing his thesis on Frobenius algebras, categorified Frobenius algebras, and their applications to topological quantum field theory and things like Khovanov homology.

Now Lauda is a postdoc at Columbia, working with Mikhail Khovanov on the challenge of categorifying quantum groups! This is something I’ve wondered about ever since I read Crane and Frenkel’s paper on the subject. The representations of a quantum group form a braided monoidal category with duals. The “free” example of such a gadget is the category of tangles. So, we get invariants of tangles from quantum groups. So, Crane and Frenkel dreamt a beautiful dream: maybe a categorified quantum group will have a braided monoidal 2-category of representations, which might give an invariant of 2-tangles - that is, 2d surfaces in 4d space!

Khovanov is a student of Frenkel, and he’s taken some huge steps towards making this dream a reality: Khovanov homology is a way of categorifying the Jones polynomial, which comes from the simplest quantum group in the world, namely quantum $\mathrm{SL}(2)$. Lauda’s thesis, based in part on work with Hendryk Pfeiffer, digs further into the heart of the subject. But with Lauda, Khovanov and Hendryk Pfeiffer all working on this sort of thing, I’m optimistic we’ll find out even more soon.

Suppose this stuff works. Then physicists have an interesting question to ponder. Quantum group invariants arise naturally from Chern-Simons theory, a 3d topological quantum field theory. Do the categorified quantum group invariants come from a 4d topological quantum field theory? What’s the Lagrangian for this theory? Is it just BF theory?

Posted at October 25, 2006 9:18 PM UTC

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### Re: Lauda on Topological Field Theory and Tangle Homology

I’m not specifically sure how n-categories fit into the framework of QFT research. Can someone explain their specific relevance - especially for well known QFTs like Yang-Mills theories?

Thanks,
QF Theorist

Posted by: QF Theorist on October 27, 2006 7:03 AM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

It’s a really fancy way of expressing the locality and gluing properties of the path integral, I think. Perhaps overly fancy; I’m somewhat sympathetic to Kevin Walker’s attempt to derive all the fun categorical stuff. My guess is that it all can be phrased in terms of (co?)simplicial sets of some sort, too – the nerve of a whatever category is always something simplicial as I remember it, so they’re not unrelated.

Posted by: Aaron Bergman on October 27, 2006 7:29 AM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

It’s a really fancy way of expressing the locality and gluing properties of the path integral

And hence of replacing the path integral by something well defined: saying that a $d$-dimensional quantum field theory is a suitable functor from $d\mathrm{Cob}$ to $\mathrm{Vect}$ is specifying precisely those conditions that we would like a path integral to satisfy.

In his latest lecture M. Hopkins made that very explicit by giving a precise homotopy-theoretic way for thinking of the symbols

(1)$\exp(i S) d\mu$

that appear in the path integral.

So a QFT is a functor from cobordisms to vector spaces, and in some cases we can use the path integral as a (more or less heuristic) way to construct such functors.

Perhaps overly fancy;

As with any power tool, it is overly fancy as long as applied to simple enough situations.

For QFT, there is unfortuantely only a small region in theory space lying between trivial and intractable. Systems in that region include in particular topological theories in more or less arbitrary and conformal field theories in two dimensions.

For both these cases the functorial conception of field theory has lead to insights which you can hardly even state without using the language of categories. (My favorite example is this theorem.)

My guess is that it all can be phrased in terms of (co?)simplicial sets of some sort, too – the nerve of a whatever category is always somthing simplicial as I remember it, so they’re not unrelated.

Categories internal to $C$ are the same as simplical $C$-objects, functors correspond to simplicial maps. This is established by identifying sequences of composable morphisms of length $p$ with $p$-simplices.

Similar statements hold for various flavors of $n$-categories (and $\omega$-categories), where in many cases the categorical structure is defined in terms of simplicial notions.

For instance, Mike Hopkins thinks that topological field theories should be thought of as living in the world of weak omega-categories defined in terms of complicial sets.

By geometrically realizing simplicial sets one passes from categories to topological spaces. Many things that can be said in terms of $n$-categories can and are being said in terms of homotopy theory.

Posted by: urs on October 27, 2006 9:42 AM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

You won’t find anyone applying $n$-categories to Yang-Mills theory yet, except in 2d spacetime.

$n$-categories are what you get when you try to understand operations of gluing together $n$-dimensional blobs of spacetime in an algebraic sort of way - in other words, to describe the relations they satisfy. So, when you try to compute a path integral over big blobs of spacetime in terms of smaller pieces and how these are glued together, you’re led to $n$-categories as a formalism for describing what’s going on.

So far this has mainly been done for topological quantum field theories, which are simpler, because we don’t need to worry about the geometry of the blobs - just their topology. 2d Yang-Mills is close to topological since we only need to worry about the topology and the area: this theory is invariant under all area-preserving transformations. 2d conformal field theories are also particularly suited to an $n$-categorical treament, because again, the conformally invariant geometrical data is fairly manageable.

If you want an easy point of access for this stuff, I’d suggest:

Posted by: John Baez on October 27, 2006 8:23 AM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

You won’t find anyone applying n-categories to Yang-Mills theory yet…

Crucial aspects of S-duality in 4-dimensional Yang-Mills theory, leading to the relation to geometric Langlands, appear naturally in a categorical language.

Kapranov has written about how lifting everything by one dimension in this context (from strings to membranes, that is) requires 2-vector bundles.

Another way to understand S-duality in 4-dimensional Yang-Mills theory is to realize YM as the compactification of 6-dimensional field theory. The connection 1-form of Yang-Mills comes from a connection 2-form in the 6-dimensional theory, to be thought of as the connection on a gerbe.

For the abelian case this is very well understood. Witten explains it in his contribution Conformal field theory in four and six dimensions to the proceedings of Segal’s 60th birthday conference (Lecture Note Series 308). On slide 71 of the corresponding talk Witten mentioned the obvious guess that for nonabelian Yang-Mills S-duality can hence be understood by compactifying a nonabelian gerbe from 6 to four dimensions. (I once commented on that here.)

The jury on that is still out, though. But people are trying. Jussi Kalkkinen argued that we should patch together Yang-Mills theory that differ by S-duality transformations to get a global picture, and that the structure thus appearing is that of a nonabelian gerbe.

So, while, as yet, nobody has ever tried to say sentences like “4-dimensional Yang-Mills theory is such and such a functor”, crucial and deep properties of Yang-Mills theory find their formulation and illumination in terms of categorical language.

Posted by: urs on October 27, 2006 10:24 AM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

You won’t find anyone applying n-categories to Yang-Mills theory yet, except in 2d spacetime.

Not in the sense of what Urs and JB are discussing, certainly.

However, some may find hep-th/0312112 interesting.

Posted by: Amitabha on October 27, 2006 12:37 PM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

You won’t find anyone applying n-categories to Yang-Mills theory yet, except in 2d spacetime.

Not in the sense of what Urs and JB are discussing, certainly.

As I said #, the two aspects of Yang-Mills that people have applied $n$-categories on are

1) the S-duality/Langlands connection

and

2) the relation of Yang-Mills to (nonabelian) gerbe theories.

However, some may find hep-th/0312112 interesting.

This concerns point 2).

Posted by: urs on October 27, 2006 1:40 PM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

Ok. I was thinking of point (2) in terms of your statement

So, while, as yet, nobody has ever tried to say sentences like “4-dimensional Yang-Mills theory is such and such a functor”

To say that, one would need to formulate YM gauge theory completely in terms of holonomies.

That was the reason I said `not in the sense of what Urs and JB are discussing’. Never mind … :-)

Posted by: Amitabha on October 27, 2006 4:51 PM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

So, while, as yet, nobody has ever tried to say sentences like “4-dimensional Yang-Mills theory is such and such a functor”…

To say that, one would need to formulate YM gauge theory completely in terms of holonomies.

Ordinary Yang-Mills can certainly be formulated in terms of holonomies. Its lattice formulation is just that.

Yang-Mills is a quantum theory whose configuration space is that of connections on some base space. These connections can be expressed in terms of functors, namely as functors from paths in base space to some associated vector bundle. So we could say that Yang-Mills theory is a quantum theory whose configuration space is a space of functors.

But that does not yet mean that we know how to formulate quantum Yang-Mills theory itself as a functor.

There are two different sorts of functors in the game here.

One (those being points in config space) are parallel transport functors.

The functor that would describe the path integral of YM would however encode propagation in QFT. Nobody (as far as I know) has tried to characterize these “YM propagation functors”.

In contrast to that, for instance, Segal has tried to characterize the propagation functors of quantum 2-dimensional conformal field theory. So in this sense the functorial description of 2D CFT is known, while that of $(d \gt 2)$-Yang-Mills theory is not.

Posted by: urs on October 27, 2006 5:07 PM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

John Baez:

“n-categories are what you get when you try to understand operations of gluing together n-dimensional blobs of spacetime in an algebraic sort of way - in other words, to describe the relations they satisfy.”

Is this exercise purely mathematical-physics at the moment? or are there (reasonable) connections to QFT that may have implications for (near) future calculational methods?

I hear there’s a growing interest in n-category theory, but for your daily QF theorist how much would be necessary to absorb to make use of it calculationally?

Posted by: QF Theorist on October 27, 2006 2:15 PM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

If you’re a “daily quantum field theorist” whose goal is to compute stuff, I don’t think n-categories will help you much - yet.

Instead, I’d recommend learning about Kreimer’s work on renormalization and knot theory, and his work with Connes on renormalization and Hopf algebras. This kind of stuff has let Kreimer and Broadhurst do some 30-loop calculations in Yukawa theory! There’s also some fascinating work relating this to causal perturbation theory. I don’t understand this stuff, but it seems to be cool math with direct connections to the computational techniques of “daily quantum field theory”.

I sense a lot of interesting n-category theory lurking right beneath the surface of this work… because diagrams play such a vital role. But, this is probably something that people who already know and love n-categories should work on.

Posted by: John Baez on October 27, 2006 6:27 PM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

[…] his work with Connes on renormalization and Hopf algebras.

Maybe interestingly, the Hopf algebra used by Connes and Kreimer is nothing but the category algebra of the multicategory of Feynman diagrams, I think #.

Posted by: urs on October 29, 2006 2:46 PM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

Cool! Is there some slick way to see why the category algebra of a multicategory should be a Hopf algebra? I don’t see one.

Let me show you my line of thought, starting with some well-known stuff, so other people can join the fun.

Given a set $S$, there’s a vector space $\mathrm{hom}(S,\mathbb{C})$ consisting of functions $f : S \to \mathbb{C}$ This construction depends in a contravariant way on the set $S$, but covariantly on the ring $\mathbb{C}$. So, $\mathrm{hom}(S,\mathbb{C})$ is an algebra, with multiplication coming from multiplication in $\mathbb{C}$: $\mathbb{C} \times \mathbb{C} \to \mathbb{C}$ But if $S$ is a monoid, $\mathrm{hom}(S,\mathbb{C})$ is also a coalgebra, with comultiplication coming from multiplication in $S$: $S \times S \to S$ So, it’s a bialgebra - and if $S$ is a group, it’s a Hopf algebra. And so on.

(It’s fun to see what happens when $S$ is a ring, but I’ll avoid that digression.)

On the other hand, there’s also the vector space $\mathbb{C}[S]$ of formal linear combinations of elements of $S$, which depends covariantly on the set $S$.

This is a coalgebra, with comultiplication coming from the diagonal map $S \to S \times S$ And if $S$ is a monoid, $\mathbb{C}[S]$ becomes a bialgebra, with multiplication coming from the product $S \times S \to S$ This also works if $S$ is the set of morphisms of a category $C$ - then we get the category algebra of $C$. Multiplication of morphisms is not always defined, but when it’s not, we just say their product in $\mathbb{C}[S]$ is zero.

Now what if $C$ is a monoidal category? Does $\mathbb{C}[S]$ become a bialgebra? Hmm… it seems that instead, the tensor product gives a new map $S \times S \to S$ and thus another multiplication on $\mathbb{C}[S]$, not a comultiplication.

For a comultiplication, it seems I’d want our category to be comonoidal. If fact every category is comonoidal in a unique way. But now my line of thought is going in a strangely different direction than yours…

Posted by: John Baez on October 30, 2006 1:38 AM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

Maybe interestingly, the Hopf algebra used by Connes and Kreimer is nothing but the category algebra of the multicategory of Feynman diagrams, I think #.

Cool! Is there some slick way to see why the category algebra of a multicategory should be a Hopf algebra? I don’t see one.

I was being sloppy. As I review in the text behind the above link, the Hopf algebra that Connes-Kreimer use is the

- dual

- of the universal enveloping algebra

- of the Lie algebra of commutators

- in the Feynman diagram multicategory algebra.

Posted by: urs on October 30, 2006 8:53 AM | Permalink | Reply to this

### Re: Lauda on Topological Field Theory and Tangle Homology

André Joyal and Joachim Kock are moving into this area with
Feynman graphs, and nerve theorem for compact symmetric multicategories
:

We describe a category of Feynman graphs and show how it relates to compact symmetric multicategories (coloured modular operads) just as linear orders relate to categories and rooted trees relate to multicategories. More specifically we obtain the following nerve theorem: compact symmetric multicategories can be characterised as presheaves on the category of Feynman graphs subject to a Segal condition.

Posted by: David Corfield on August 20, 2009 9:04 PM | Permalink | Reply to this

### Annotations desired; Re: Lauda on Topological Field Theory and Tangle Homology

I skimmed through this the evening it appeared on arXiv, and went to bed fascinated and confused. Parts were clear to me; parts were in the deep half of the pool, where I can rarely swim without a flotation device. Will there be somewhere in the n-Category blog-cosmos and annotated version, with training wheels and water-wings? I have some Feynman graph experience, and some familiarity with some of Joyal’s vision.

Posted by: Jonathan Vos Post on August 20, 2009 9:46 PM | Permalink | Reply to this

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