DW, state sum models, and fully extended TQFTs

I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-bundles are replaced by higher bundles over higher algebraic structures). Forgive about my naiveness, but I immaturely suspect maybe three of them are the same. I hope I will get an answer one day, and this post is my first step.

I don't have much access to the experts of this field, and therefore am not sure how much this has been developed. Perhaps the answers are in written papers. In any case, I think a complete answer is too much to hope. If you have any relevant paper in mind, please point them out with short comments. Thank you so much in advance!

1. Crane-Yetter and Dijkgraaf-Witten

Crane-Yetter theory is a well-known $$4$$-D state sum model. According to Manuel Bärenz's edit on nLab, it can be realized as a generalized DW theory, based on quantum groups instead of finite groups.

Q1.1 Can you give a formal reference for that statement?

Q1.2 Can (m)any other state sum models be interpreted as a generalized DW theory? We have to be flexible here: fields can be higher bundles.

Q1.3 In contrast, can generalized Dijkgraaf-Witten theories be realized as state sum models?

2. Dijkgraaf-Witten and Fully Extended TQFTs

By Domenico Fiorenza's edit on nLab (sec. 2), Dijkgraaf-Witten models are fully extended.

Q2.1 Is there a formal reference for this statement?

Q2.2 Can fully extended TQFTs be realized as generalized Dijkgraaf-Witten models?

3. Fully Extended TQFTs and State Sum Models

Q3.1 I have been trying to find evidence why CY is fully extended and what the point is associated to, but in vain. The best answer I have heard is that physicists believe state sum models should automatically be fully extended. If this is true, I really want to know how and why it should work.

Q3.2 On the other hand, by cobordism hypothesis (proved by Lurie 2009), any fully extended TQFT is determined by the assignment at the point. I have a feeling that if the target category is "finite" enough, then this might be interpreted as a state sum model. Would you share your understanding? (EDIT: an answer by Kevin Walker suggested that some standard techniques bring you a state sum model from fully extended ones. Another possibly related post can be found here).

• For Q1.1 I would recommend "An Introduction to Spin Foam Models of Quantum Gravity and BF Theory, by John C. Baez: . arxiv.org/abs/gr-qc/9905087. DW with trivial cocycle could be thought of as a BF-theory with a finite group, at least in the classical level. – Joao Faria Martins Apr 20 at 6:27
• Continuing the previous comment. One gets CY by doing the construction over SU(2), and then passing to quantum SU(2) at a root of unity in order to obtain a finite sum. – Joao Faria Martins Apr 20 at 6:33
• If you have a look e.g. at "Topological Higher Gauge Theory - from BF to BFCG theory F. Girelli, H. Pfeiffer, E. M. Popescu: " you can find discussion of why Yetter homotopy 2-type TQFT can be seen as a DW with a finite 2-group. arxiv.org/abs/0708.3051". The relation to Higher Gauge theory (hence considering 2-bundles) is also discussed there in an references. – Joao Faria Martins Apr 20 at 6:37
• I tried to write up a partial answer to these questions, but as far as I know very few of them have any concrete answers where one can point to proofs in the literature, or even statements. For example, a proof that Crane-Yetter theories are fully extended would require an understanding of the $\mathrm{SO}_4$-action on a certain 4-category, which is difficult and still open. Many questions about fully extended TFT (e.g. 2.2) depend on the choice of target, but are nonetheless open for any reasonable target. – Arun Debray Apr 20 at 17:42

Let me try to answer your questions at least in part. My apologies for references I've missed. For an overview of the ideas without references, you might enjoy Pavel Safranov's talks at the intro conference at MSRI.

1.What exactly do you mean by Dijkgraaf-Witten? If you asked me what Dijkgraaf-Witten was (and this is more a statement about myself than about mathematical definitions) I would have said that it was Crane-Yetter theory for a pointed braided tensor category Vec(A,q). Without this I can't answer your questions.

2.1 This is certainly "known" but I'm having a little trouble tracking down the exact reference. I think Freed-Hopkins-Lurie-Teleman for finite groups is doing what you want, based on earlier work of Freed.

2.2 I really don't understand what you're asking here. If Dijkgraaf-Witten is a special case of fully extended TFTs, then by the definition of generalization fully extended TFTs are generalized Dijkgraaf-Witten...

3.1 Crane-Yetter attached to a modular tensor category C is a fully local 4-dimensional TFT which assigns to the point C inside the Haugseng-Johnson-Freyd-Scheimbauer Morita 4-category of E2 algebras in the 2-category of LFP categories. This is "known" to experts, but not fully in print anywhere. There is an incomplete construction given in unfinished notes by Walker, this uses somewhat non-standard definitions for higher categories and TFTs so even when its completed one can argue about whether it translates into other definitions. There's also some unfinished work of Freed-Teleman in this direction. The 012-dimensional part of Crane-Yetter is constructed and computed in work of Ben-Zvi-Brochier-Jordan building on Ayala-Francis's factorization homology. I have work with Brochier-Jordan and also with us and Safranov showing that non-degenerate braided fusion categories are fully dualizable and actually invertible, and thus by the cobordism hypothesis give framed fully local TFTs which we think of as a framed version of Crane-Yetter, but we don't have serious calculations of what this theory yields in high dimensions. Furthermore, as Arun points out in comments, in order to turn this into the usual Crane-Yetter you need to understand how the ribbon structure on a modular tensor category gives an SO(4)-fixed point structure. You can see some results in this direction in my MSRI talk on joint work in progress with Douglas-Schommer-Pries, but we're still a ways off from giving a full answer.

3.2 A good reference for how to get a state-sum out of a fully extended theory is Orit Davidovich's PhD thesis. In principal there's no problem doing this, but in practice there's plenty of interesting questions along these lines to which we don't yet know the answer. For example, you should be able to use my work with Douglas-Schommer-Pries to get a framed Turaev-Viro state sum model attached to a (not necessarily spherical) fusion category. But we don't even have a guess for what exactly that should look like. Or, after you've analyzed the SO(3)-fixed points enough, you should be able to show that the TFTs coming from spherical fusion categories via the oriented version of the cobordism hypothesis and our work agrees with Turaev-Viro thereby showing that Turaev-Viro theories are fully extended. I expect within 5-10 years we will have all of this understood, but we don't yet, and there may be other approaches that are more direct.

Two other helpful references (with extensive bibliographies) which I didn't mention specifically are recent papers by Schommer-Pries about invertible TFTs and Reutter's paper on semisimple theories.

• My apologies to my very late response. I was overwhelmed by many references, thinking of to reply after some digestion, and that just five weeks. Though I have many more questions.. I would wait for longer for they to make sense and unify. – Student May 25 at 21:37
• One thing I should have made clearer: I meant by a DW theory to be a TQFT in which a field is a (higher) bundle, or you can say it's a sigma model with target space being $K(G,n)$.. – Student May 25 at 21:38