4D TQFT from a modular tensor category 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)
 A: 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. 
A: Originally, the idea of a 4D TQFT was to be found in a Hopf Category as defined by Crane and Igor Frenkel. Crane and Yetter gave an example via certain cocycles over a finite group. Kauffman, Saito, and I explicitly constructed this, but never were able to compute with it. 
One should also look through Lurie's work to get explicit examples of braided monoidal 2-categories with duals. From David Ben-Zvi's description above, I would guess these are Morita 4 categories.  
A: The TQFT in question should probably be called the Crane-Yetter or Crane-Yetter-Kauffman TQFT.
Crane-Yetter-Kauffman didn't work it out as a fully extended theory, and didn't notice (so far as I can tell) the relation to Witten-Reshetikhin-Turaev theories, but they definitely were the first to write down the 4d part of the theory.
The CYK TQFT contains all of the information of the WRT TQFT.  (This disagrees with David Ben Zvi's answer, but I think the difference is due to our using different axiomatic frameworks for TQFTs, not disagreement about mathematical facts.)  More specifically,
$$
  Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma).
$$
Here $X$ is a manifold of dimension 1, 2 or 3 (not necessarily closed).  $X$ is equipped with extra structure ($p_1$ structure, signature structure, null-bordism structure, ...) which makes $\partial^{-1}(X)$ sufficiently unambiguous.  (For example, if $X$ is a closed 3-manifold, then the choice of $\partial^{-1}(X)$ only matters up to bordism.)  The $\Gamma$ on the left hand side is a collection of "Wilson loops" or more generally a Wilson (labeled) graph.  The $\Gamma$ on the right hand side is a boundary condition.  (Same graph, but different interpretation.)
For more details, see Chapter 9 of these notes.
One way of looking at this is as follows.  We expect, roughly, a correspondence
$$
  \mbox{$n$-category} \;\; \leftrightarrow \;\; \mbox{$(n{+}1)$-dimensional TQFT}.
$$
The input data for for a WRT TQFT is a modular tensor category, which is a particular type of 3-category.  But the WRT TQFT is a (2+1)-dimensional theory, not a (3+1)-dimensional theory, so something weird is going on here.  The natural thing to do with a modular tensor category is to build the (3+1)-dimensional CYK TQFT, which is fully extended (a "0-1-2-3-4" theory) and anomaly-free.  One then notices that the CYK theory is almost trivial for closed manifolds (more specifically, the dimensional reduction by $S^1$ is 2-Morita trivial), so we can derive from the CYK TQFT the (2+1)-dimensional WRT TQFT via the slogan
$$
  Z_{WRT}(X, \Gamma) = Z_{CYK}(\partial^{-1}(X); \Gamma).
$$
But note that since CYK is merely almost trivial and not completely trivial, the WRT TQFT acquires an anomaly (i.e. manifolds need to be equipped with extra structure).  Also, since it's hard to make sense of $\partial^{-1}(X)$ when $X$ is a point, the WRT theory is not fully extended; it's a 1-2-3 theory rather than a 0-1-2-3 theory.
I should also note that the input data for the CYK can be a premodular category ($S$-matrix perhaps degenerate).  When the input is premodular but not modular, then the CYK TQFT is not almost trivial and we cannot construct a (2+1)-dimensional TQFT as above.
