### 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:

- Aaron Lauda, Open-closed topological quantum field theory and tangle homology, PhD thesis, Cambridge University, 2006.

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?

## 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

specificrelevance - especially for well known QFTs like Yang-Mills theories?Thanks,

QF Theorist