Citation: Davide Gaiotto, "Gapped phases of matter vs. topological field theories", talk at Perimeter Institute for Theoretical Physics
Web: https://pirsa.org/17070066
Tags: TQFT, Modular-tensor-categories, SPT/SETs, Fermionic-order, MTC-reconstruction
This is a lovely talk given by Davide Gaiotto at the Perimiter institute. He discusses to what extent topological field theories correspond to gapped phases of matter, reviewing some recent literature. Of course, this is a problem I think about a lot. What is the connection between lattice models and field theory, exactly? I think that Gaiotto's answer is correct. Namely, the connection is modular tensor categories. Via anyons in (2+1)D you can extract anyons, and those anyons are described by a modular category, and using that modular category you can use the Reshetikhin-Turaev theorem to construct a TQFT. For higher dimension, you would need to look at higher order (i.e. line-like, surface-like) excitations to recover the higher categorical structure.
Another nice part of this talk is the conclusion, where he gives a philosophical note about why we see some many generalized cohomology theories appearing in SPT classifications. His answer is as follows. Start with a G-SPT. Couple it to a G-flat connection. That is, add a connection on the links of the lattice, and make it trivial almost everywhere except for some domain walls. He then argues that the states on the domain walls will be invertible states of higher codimension. In this way, you get a network of invertible phases of different dimensions with some compatibility conditions. He then argues that this is the same thing as having a cocycycle valued in the spectrum of invertible phases. Replacing the spectrum of invertible phases with different spectra (such as the spectrum of fermionic phases) gives different generalized cohomology theories.