# Realization of Witten-type topological quantum field theory in condensed matter physics

It is well-known that some exotic phases in condensed matter physics are described by Schwarz-type TQFTs, such as Chern-Simons theory of quantum Hall states. My question is whether there are condensed matter systems that can realize Witten-type TQFTs?

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Wikipedia seems to only give one example of what a Witten-type TQFT is, and that is the WZW model which appears many places in condensed matter physics. I always thought that Witten type TQFT's are cohomological field theories, where there exist an operator $Q^2=0$ (so you have susy or BRST structure) and so on. I haven't seen any of these type theories in condensed matter physics –  Heidar Nov 6 '12 at 18:45
The type of TQFT's that Atiyah's axioms capture (where its a functor between certain appropriate categories) are the ones that usually appear in condensed matter context (at least in toy models). According to wikipedia, these include the Schwarz-type TQFT's but it is not clear that they describe the Witten-types fully. –  Heidar Nov 6 '12 at 18:47
@Heidar: That wikipedia article is wrong. The WZW model is not a topological field theory. It's conformal, but that's not the same thing. –  user1504 Nov 6 '12 at 23:13
@user1504 This is also the first time I have seen the WZW being called a topological field theory. The theory is gapless and has a lot of dynamical degrees of freedom, so its quite hard for me to see in what sense it could be topological. –  Heidar Nov 6 '12 at 23:16
It's simply not a topological field theory, and especially not one of Witten class. Whoever edited that wikipedia article didn't know what they were talking about. The WZW model isn't supersymmetric; it doesn't have a nilpotent supercharge. (It's intimately related to Chern-Simons theory, but that's a different story, and not relevant here.) –  user1504 Nov 6 '12 at 23:20

The TQFTs that Witten introduced are those obtained by a topological twist of a supersymmetric field theory. This includes notably the A-model and the B-model TQFTs.

Despite what seems to be suggested in the comments here and on Wikipedia, these are also "Schwarz type" (come from the Poisson sigma-model) and they do have a desciption in terms of functorial TQFT if only one allows what are called (infinity,1)-functors: they are "TCFTs" (i.e. non-compact 2d homotopy TQFTs).

Now, under homological Mirror symmetry these are related to other TCFTs known as Landau-Ginzburg models. And these do have applications in solid state physics.

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I very much appreciate this answer because it actually defines what the question is asking about. But these Landau-Ginzburg models appear how in solid state? As fixed points of disordered systems in two dimensions? –  BebopButUnsteady Jul 5 '13 at 16:32
Well, the Landau-Ginzburg theory is an old model for superconductivity and the behaviour of its potential term is what gives the Landau-Ginzburg model its name. But yeah, otherwise the relation is not super-close, I agree. –  Urs Schreiber Jul 5 '13 at 17:59
That's just a linguistic coincidence then... These models are by necessity supersymmetric and conformal (and even dimensional)? - the only place in solid state where I know of supersymmetric methods is in disordered systems/random matrices. If you don't know of a particular example, then that is fine, I just wanted to clarify. –  BebopButUnsteady Jul 5 '13 at 18:18

I don't think there are any Witten-type TQFTs which are directly relevant to condensed matter physics. Witten-type TQFTs are very strange beasts: they violate spin-statistics, they aren't unitary, etc. It'd be pretty tricky to find a physical system you could model with one in the usual way.

There are some indirect connections between Chern-Simons theory and Gromov-Witten theory, but that's all I can think of.

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