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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)

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3 Answers

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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 an interesting exchange with Kevin Walker (which I hope I am not misrepresenting), it appears there are two different settings for the Crane-Yetter-Kauffman TFT. In the Atiyah-Segal-style axiomatics of extended TFT (the setting for the cobordism hypothesis), the results of Freed-Teleman indeed imply that the 4d CYK TFT only depends on the anomaly data of a modular tensor category, which is a single characteristic class. In particular it follows (combining their result with the cobordism hypothesis) that two modular tensor categories C, C' with the same anomaly invariant are Morita equivalent, in the sense that they define equivalent 3-categories, $$C-mod-mod \simeq C'-mod-mod.$$ On the other hand there is a different formalism for TFT, developed in Kevin's notes as well as his work with Scott Morrison. In this formalism the 4-dimensional CYK field theory DOES know the original modular tensor category (not just its Morita class) and hence can recover the corresponding Witten-Reshetikhin-Turaev theory. It would be very interesting to have a precise relationship between the two settings.

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When you say it's less info is that because you're talking about the 34 theory not the 234 theory? The oriented 234 theory is exactly the same info as the anomalous 23 theory, right? – Noah Snyder Jan 18 at 14:55
3 
 @Noah: no, I'm talking about the 01234 theory and it's still less info. the anomalous 23 (or 123) Reshetikhin-Turaevish theory is VALUED in the invertible field theory that Dan and Constatin construct, but that latter theory is completely determined by a single characteristic class, and just keeps track of the actual anomaly, not of the anomalous theory. – David Ben-Zvi Jan 18 at 16:52
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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.

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@Kevin: very interesting! definitely had missed that point. I'm confused though since the 4d TFT only depends on your modular tensor category C up to Morita equivalence - are you saying it's Morita rigid? Or put another way, I think you're saying we recover C=Z_{CYK}(disc) - but I only see the latter as an object in D=Z_{CYK}(S^1), and unless I give the extra structure of forgetful functor from D to categories how do I recover the underlying category of C? That forgetful functor (domain wall) seems to me equivalent to the data of the anomalous theory Z_{WRT} itself, and not part of Z_{CYK}? – David Ben-Zvi May 16 at 15:28
@David: I'm not sure I fully understand your question but I'll answer as best I can. I would say that $C$, or rather $Rep(C)$, is isomorphic to $Z_{CYK}(pt)$, not $Z_{CYK}(D^2)$. Perhaps part of the confusion is due to the fact that my TQFT framework is not the Atiyah-Segal framework. Compared to Atiyah-Segal, I have some extra structure at my disposal: "fields" or boundary conditions on manifolds. (The graph $\Gamma$ above is an example of such.) My Hilbert spaces are realized concretely as functions on boundary conditions. There is often... – Kevin Walker May 16 at 15:58
[continued] ... a preferred boundary condition corresponding to gluing together many copies of (iterated) identity morphisms of the input $n$-category, and evaluation at this preferred boundary condition gives a preferred map from Hilbert spaces to the ground ring. If W is a 4-manifold with boundary, then applying this preferred map to the element $Z_{CYK}(W)$ of the Hilbert space gives an element of the ground ring which is equal to $Z_{WRT}(\partial(W))$. – Kevin Walker May 16 at 16:04
I think we're simply working with different amounts of structure. In the Freed-Teleman setting (via cobordism hypothesis) Z_{CYK}(pt) is indeed C, but as an object of the Morita category, so is equivalent to any other modular tensor category with the same anomaly invariant. So in that setting you can't functorially recover C (hence WRT) itself from the CYK TFT, but only a single characteristic class. On the other hand if you give enough additional structure, you can rigidify from the Morita category to the category of braided tensor categories and then recover C. – David Ben-Zvi May 16 at 16:16
For a parallel example: given a (2-dualizable) associative algebra A, considered as an object of the Morita 2-category, we can construct a 2d TFT Z. But you can only recover the category A-mod functorially from Z, or A up to Morita equivalence, even though one could say A is Z(pt).. to see A itself you need extra structure: a specific boundary condition in the field theory Z, or equivalently a (compact) object of A-mod. In the Freed-Teleman language, that defines a theory relative to the 2d TFT Z, analogous to WRT being relative to CYK, and that relative theory is what knows A itself. – David Ben-Zvi May 16 at 16:20
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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.

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