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?
 A: 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.
A: 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.
