20
votes

### Fully extended TQFT and lattice models

It may take a bit of extraction, but positive answers to both of your questions follow from my results joint with Gaiotto in Condensations in higher categories (arXiv:1905.09566). In that paper we ...

- 49.9k

10
votes

Accepted

### How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

One way to construct the duality functor ${\cal C} \to {\cal C^{\rm op}}$ is through the notion of a pairing of $\infty$-categories (see HA, Definition 5.2.1.5). In particular, in this case we're ...

- 8,876

10
votes

Accepted

### Is there a PL, or topological, bordism hypothesis?

This is addressed in Remark 2.4.30 of Jacob's paper. The PL case has a very nice description but the topological case does not. In particular, there's no difference between framed bordisms in the PL ...

- 26.7k

10
votes

### Lagrangian of Reshetikhin-Turaev TFT's

I don't think there's a way to extract a Lagrangian from the Reshetikhin-Turaev construction. There's certainly not
a unique way to do so.
Physicists believe that most QFTs are "non-Lagrangian,&...

- 6,586

8
votes

### Lagrangian of Reshetikhin-Turaev TFT's

It's an open conjecture by Moore and Seiberg (originally in the context of conformal field theory) that every MTC can be obtained from Chern-Simons theory of simple Lie groups with known constructions....

- 456

8
votes

Accepted

### Is Turaev-Viro-Barrett-Westbury stronger than homotopy?

I think the following example seems to work: consider a special class of TVBW, the Dijkgraaf-Witten invariant whose input are a finite $G$ and a 3-cocycle from $H^3(G, C^*)$ (together they define a ...

- 456

7
votes

Accepted

### Does the notion of a "coherent state" exist in TQFTs? (ETQFTs?)

Yes they do. In the geometric quantization approach to the 3d Chern-Simons TQFT, the vector space assigned to a closed surface is a space of holomorphic sections of a certain line bundle. In this ...

- 1,771

7
votes

### (3,2,1)-TQFTs and Verlinde algebras

A somehow detailed answer could be as follows (thanks to Alessandro Valentino, who is a coauthor of this answer (but me alone is to blame for mistakes and inaccuracies in it)). Kevin Walker's notes ...

- 6,299

7
votes

Accepted

### Are there 4d state sum models, extended TQFTs or chain mail invariant that detect smooth structures?

This MO answer by Arun Debray gives an example in the unoriented case where two specific homeomorphic manifolds can be distinguished by a specific TFT of this kind.
In general all these constructions ...

- 26.1k

6
votes

Accepted

### What are the topological phases of quantum Hall systems?

Fermionic modular categories and the 16-fold way classifies the topological phases of the fractional quantum Hall effect. The Laughlin states (Abelian anyons at filling factor $1/Q$, $Q$ odd) are ...

- 153k

5
votes

Accepted

### Fully dualizable objects in classical field theories

All of the objects in these iterated span categories are fully dualizable. See this paper by Rune Haugseng.

- 110k

4
votes

### Extended TFT with coefficients in spans in any $\infty$-topos

Here an argument using the assumption that $\mathbf{H}$ has an $\infty$-site $\mathcal{S}$ of definition all whose objects are étale contractible.
The proof of prop. 3.2.8 arXiv:0905.0465 ...

- 18.9k

3
votes

### What are the topological phases of quantum Hall systems?

We have a paper that contains lists of simple fermionic topological orders in 2+1D: https://arxiv.org/abs/1507.04673 . For fermionic topological without symmetry, there is no filling fraction. So our ...

- 4,462

1
vote

Accepted

### How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?

Based on the discussion in the comments with Ian Agol, here's a draft answer. I would welcome corrections/confirmation from anyone who knows more.
Let $M$ be an orientable manifold with possibly ...

- 2,213

1
vote

### How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

I don't consider this an actual answer to the question, as it uses a hypothesis that is, to my knowledge, not yet proven. I hope that there is a simpler answer to this question that is more elementary ...

1
vote

### Are fully extended TQFTs generalized cohomology theories?

If you mean 'generalised cohomology'as in 'all but dimension axiom', then no. Those animals are additive functors, while tqft's are multiplicative.

- 251

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