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?

  • 1
    $\begingroup$ 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 $\endgroup$ – Heidar Nov 6 '12 at 18:45
  • $\begingroup$ 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. $\endgroup$ – Heidar Nov 6 '12 at 18:47
  • $\begingroup$ @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. $\endgroup$ – user1504 Nov 6 '12 at 23:13
  • $\begingroup$ @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. $\endgroup$ – Heidar Nov 6 '12 at 23:16
  • 1
    $\begingroup$ 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.) $\endgroup$ – 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.

  • $\begingroup$ 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? $\endgroup$ – BebopButUnsteady Jul 5 '13 at 16:32
  • $\begingroup$ 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. $\endgroup$ – Urs Schreiber Jul 5 '13 at 17:59
  • 2
    $\begingroup$ 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. $\endgroup$ – BebopButUnsteady Jul 5 '13 at 18:18
  • $\begingroup$ Are Witten-type theories unitary? Is there a reference to explain that? $\endgroup$ – QGravity Jul 11 '18 at 21:38

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.

  • 1
    $\begingroup$ Why are Witten type TQFTs not unitary? $\endgroup$ – Marion Mar 12 '18 at 23:38
  • $\begingroup$ Why are Witten-type theories non-unitary? Can you please mention a reference? $\endgroup$ – QGravity Jul 11 '18 at 21:32
  • $\begingroup$ The question about unitary is actual. Could you provide some reference? $\endgroup$ – Nikita Feb 20 '20 at 21:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.