Citation: Freed, Daniel S. Lectures on field theory and topology. Vol. 133. American Mathematical Soc., 2019.
Web: https://web.ma.utexas.edu/users/dafr/CBMS133.pdf
Tags: Expository, TQFT, Invertible-phases, MTC-reconstruction
In these lecture notes, Dan Freed gives an overview of his work on invertible field theory, and in general is a good exposition on Freed's TQFT/QFT philosophy. It's supposed to be a good primer to read before you dive into the more technical work,
> Freed, Daniel S., and Michael J. Hopkins. "Reflection positivity and invertible topological phases." Geometry & Topology 25.3 (2021): 1165-1330.
A more targeted reference for condensed-matter focused people is
> Freed, Daniel S. "Short-range entanglement and invertible field theories." arXiv preprint arXiv:1406.7278 (2014).
This is a heavy set of lecture notes to read. Instead of talking about structures like orientability and framing in the normal ways, Freed introduces the notion of a "flabby Hn-structure". He describes these as follows: "The model retains minimal information—the long range physics—which on the one hand is robust enough to determine the phase and on the other hand is flabby enough to be amenable to topological techniques". There's a lot of content in here to comment on.
One of the nice sections talks about how to extend the cobordism-style definition of TQFT beyond TQFTs. Namely, how to make the formalism include some geometric structure. A simple example is a version of TQFT but each connected component of every bordism is assigned a volume form (top-dimensional degree differential form). It is a theorem of Moser that if two top degree forms have the same integral then there is a diffeomorphism of the ambient manifold that takes one form to the other. Thus, the data of the form is essentially the data of a number, with some compatibility structure with the rest of the TQFT data. For 2D TQFTs, this gives a lovely classification of QFTs with volume form in terms of commutative Frobenius algebras with a small amount of extra data (a one-paramater family of elements of the algebra). This was first observed by Segal, and proved in this paper:
> Runkel, Ingo, and Lóránt Szegedy. "Area-dependent quantum field theory with defects." arXiv preprint arXiv:1807.08196 (2018).
A more general reference about putting geometric structures on TQFTs is in Ayala's PhD thesis,
> Ayala, David. Geometric cobordism categories. Stanford University, 2009.
Another lovely section in this paper is the discussion of the cobordism hypothesis. Freed states and proves the cobordism hypothesis for 0 categories and 1 categories. They are very convincing, in the sense that the 0-categorical statement is trivial (but could have in principle failed!) and the 1-categorical statement is reasonable but requires some insight. It makes sense that the n-cobordism hypothesis is true but highly non-trivial. Definitely my favorite reference I've ever seen for the baby versions of the cobordism hypothesis. Freed also talks about this in another paper:
> Freed, Daniel. "The cobordism hypothesis." Bulletin of the American Mathematical Society 50.1 (2013): 57-92.
There's also a nice quote about the codomain of TQFT in here. "There is no canonical choice for the codomain symmetric monoidal n-category C of an extended n-dimensional field theory. There are physic inspired desiderata". He then goes on to say a little bit about what those desiderata are. Also, he says clearly that "We believe that every field theory of physical relevance should be fully extended". I find this controversial/disagreeable.
One of the main conclusions is that invertible TQFTs should be classified by homotopy classes of stable maps from the Madsen-Tillman spectrum to the Anderson dual of the sphere spectrum. Once you prove this result, of course, it would be nice to be able to use it to actually compute the space of invertible maps in different dimensions. A wonderful (detailed) discussion of these computations is found in the following paper:
> Beaudry, Agnès, and Jonathan A. Campbell. "A guide for computing stable homotopy groups." Topology and quantum theory in interaction 718 (2018): 89-136.
It's good to be aware of the fact that this work has many gaps, and leaves much to be desired from a physical perspective. Here are a few representative quotes from this paper to demonstrate the situation: "Also, while we enumerate groups which should house invariants of SRE phases, we do not argue either that the effective field theory is a complete invariant or that every possible effective field theory is realized by a microscopic system." and "In particular, there are a few ingredients in our proposal (choice of tangential structure, choice of target spectrum) which involve leaps of faith and can easily be adjusted if further microscopic implications are discovered".
A good supporting reference to read in tandem with this paper is
> Freed, Daniel. "The cobordism hypothesis." Bulletin of the American Mathematical Society 50.1 (2013): 57-92.