I know the construction of a 3D topological quantum field theory (TQFT) from a modular tensor category.
I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I would like to know how to construct it.
I'd like to study it but I don't know how to search references. Is there any good word to search this 4D TQFT? Or could you suggest references?
I also want to know if there is another mathematically constructed 4D TQFT and how it is called.
Thank you in advance.
(This question was asked in math.stackexchange but no answer was given.here)
Best Answer
There is a recent construction of a fully extended 4d TQFT from a modular tensor category, due to Dan Freed and Constantin Teleman (using Lurie's proof of the cobordism hypothesis). It is described in Freed's lecture notes from the Segal 70th birthday conference here: https://people.maths.ox.ac.uk/tillmann/ASPECTS.html
The idea is that braided tensor categories are naturally objects of a "Morita" 4-category (morphisms are algebra objects in bimodule categories, 2-morphisms are bimodules categories for these, 3-morphisms are functors of those, and 4-morphisms are natural transformations --- the quick mnemonic is that braided counts for two, category counts for one, together we get three, and three-categories form a four-category ---- a baby version of this is that algebras form a two-category, while monoidal categories (algebras in categories) form a three category).
Freed and Teleman show that modular categories are "superduper finite" (aka fully dualizable) objects of this category, ie satisfy the conditions of the cobordism hypothesis to define a functor from the 4d-bordism category. In fact much more is true -- this field theory is an invertible field theory... basically it means it's completely characterized by a single characteristic class of four manifolds, the "anomaly" of the original modular tensor category.
So in fact you shouldn't think of this 4d field theory as more information --it's LESS information than the 3d field theory attached to the MTC, but rather it's the anomaly information needed to completely define the three-dimensional field theory (which they use to extend Chern-Simons theory to a point eg.)
Edit: As a result of some interesting exchanges with Kevin Walker and Dan Freed I believe things are a little more complicated than I had initially understood. The results of Freed-Teleman indeed imply that the 4d CYK TFT is an invertible field theory, i.e. that modular tensor categories are invertible objects in the Morita 4-category of braided tensor categories. This means that the entire field theory can be described by a map of spectra -- namely the sphere spectrum (classifying space of the framed cobordism category) mapping to the space of invertible objects in the Morita category. However, it's not clear exactly what this target space IS --- what's much easier to see I believe is the OTHER space attached to the Morita category, namely its classifying space (where we invert morphisms to make a groupoid, rather than restrict to invertible morphisms as well as invertible objects). The latter map is close to the classical notion of anomaly as far as I understand, but the map that truly classifies modular tensor categories up to Morita equivalence is the former, about which it appears not much is known.