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 ...
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20 votes
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What do physicists mean by a topological quantum gravity theory

Physicists here. The input for a physical theory is always some topological space and some structure (such as a metric) that depends on the specific context. The dynamics are invariant under the ...
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18 votes

Classifying TQFTs with 1d vector spaces

They are called "invertible field theories". See https://www.ma.utexas.edu/users/dafr/M392C-2012/index.html, lectures 17-24 for the relation to Madsen-Tillmann spectra.
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  • 266
17 votes
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How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?

Yes, the Walker-Wang model is related to the Crane-Yetter-Kauffman TQFT in the same way the Levin-Wen model is related to the Turaev-Viro TQFT. See, for example, the table on page 14 of the notes ...
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  • 11.9k
17 votes
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How aggressive is the fibrant replacement of $\mathrm{Bord}_n$?

The completeness condition is not really about making things invertible which weren't already. It is about where the information about invertible morphisms is stored. We can already see this with $(\...
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16 votes

Why is a Topological Field Theory equivalent to a Frobenius algebra?

Summary: the equivalence relies on the mathematical formalism of TQFTs, and sends a 2d TQFT $Z$ to the state space $Z(S^1)$, which is naturally a Frobenius algebra. One potential point of confusion ...
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  • 6,526
14 votes
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Why is a Topological Field Theory equivalent to a Frobenius algebra?

As the general relation between 2d TQFT and Frobenius algebras has already been given in another answer, let me describe the Frobenius algebras occurring in the A and B-models. 1) The A-model is ...
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  • 6,620
13 votes

Why is a Topological Field Theory equivalent to a Frobenius algebra?

Arun Debray's answer is good, with one addendum: 2d TQFT is the same thing as a commutative Frobenius algebra. But to truly understand what's going on, one also needs some pictures. A commutative ...
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  • 19.6k
13 votes

Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?

Short answer: Untwisted Dijkgraaf-Witten theories with non-abelian gauge groups (e.g. $S_3$) distinguish most knots from the unknot and from each other. Longer answer: Here's my understanding of ...
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  • 11.9k
12 votes
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How do we handle the symmetry condition in nCob and TQFTs?

As Oscar has explained in comments, with the most common definitions it's just not true that $M \sqcup N$ is exactly the same as $N \sqcup M$. But even if you were working with some version of the ...
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  • 26.6k
12 votes
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Mapping class group of torus, why is $(ST)^3=S^2$?

Flip the direction of rotation for $S$, or choose the other meridian for $T$. We can see this at the level of matrices. Define $$S_1 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad ...
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  • 1,661
12 votes
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Generators and relations for the 2-dimensional unoriented cobordism category

My initial answer was wrong, here's the correct version plus a reference: Turaev-Turner New generating morphisms: The Mobius strip $\emptyset \rightarrow S^1$ and the "orientation reversing" ...
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  • 26.6k
12 votes

Grothendieck group of the category of boundary conditions of topological field theory

To understand the possible spaces of boundary conditions for a TQFT, it is helpful to start in highest dimension. Suppose you have a $(d+1)$-dimensional nonanomalous TQFT $\mathcal Q$. (The anomalous ...
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12 votes

Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?

As Igor said, it's a bit late for that. If you just finished undergrad and you discovered a passion for QFT from a rigorous mathematical standpoint, what you should do, for example, is apply for the ...
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11 votes
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Twists, balances, and ribbons in pivotal braided tensor categories

Question 2: Given a pivotal braided category $\mathcal{C}$, there are 2 ways to endow $\mathcal{C}$ with twists under which $\mathcal{C}$ is a rigid balanced category. Conversely, given a rigid ...
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  • 5,087
11 votes
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Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal?

In the non-unitary setting ENO proved that if $Z(C)$ and $Z(D)$ are equivalent as braided tensor categories, then C and D are Morita equivalent. This is Theorem 3.1 of this paper. Note that they ...
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11 votes
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Physical consequences of cobordism hypothesis?

Yes. The physical motivation is that topological field theories, as examples of quantum field theories, should be fully local, meaning that one should be able to calculate any information about a (...
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  • 6,526
11 votes
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What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?

If you take the (2,0) theory and put it on a manifold which is $T^2 \times M_4$, it is known to reduce to $\mathcal{N} = 4$ super-Yang Mills theory on $M_4$. That theory exhibits S-duality, which has ...
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11 votes

What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?

In general, mathematical outputs of SUSY field theories often become more accessible after performing some twist, and the same is true of the 6d (2,0) SCFT. Considering the theory on $\Sigma\times M_4$...
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11 votes
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Importance of the principal bundle in Chern-Simons theory

In quantum Chern-Simons theory with gauge group $G$ (compact Lie), a field on a 3-manifold $M$ is a principal $G$-bundle with a connection $A$. The partition function/path integral associated to $M$ ...
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  • 5,133
11 votes

Two vague questions about TFT

I will only attempt to answer the first question. Given a symplectic manifold $M$, there is a TQFT in any odd dimension. In dimension 1 this is the topological quantum mechanics and in dimension 3 ...
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11 votes

Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh

The (fairly poetic and ill-formed) idea in this story is that the Kapustin-Witten story and the Langlands program are about the SAME four-dimensional TQFTs, but evaluated on different "manifolds&...
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10 votes
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Does the Turaev-Viro theory for the generalized $E_6$ subfactor for $\mathbb{Z}/7$ distinguish $L(7,1)$ and $L(7,2)$?

I think this paper of Wakui says that the answer is "No". The Turaev-Viro invariants associated to the generalized $E6$ subfactors for $\mathbb{Z}/7$ don't seem to distinguish $L(7,1)$ and $L(7,2)$.
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10 votes
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What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols?

Any unitary fusion category has a canonical spherical structure. See Example 2.12 of this paper and the references therein. It also follows from a more general result Prop 8.23 in this paper. So you ...
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  • 26.6k
10 votes
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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 ...
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  • 26.6k
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,&...
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  • 6,526
10 votes
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Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh

A curve $C$ over $\mathbb F_q$ has dimension $3$ in this perspective (which is why you get a vector space) and a local field has dimension $2$ (which is why you get a category. So one only has to go ...
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  • 117k
9 votes

What's the right way to think about "anomalies" in 3d TQFTs?

Here is my understanding from physics point of view. A quantum field theory is anomalous if it lacks of a UV completion. In other words there is no lattice theory in the same dimension, whose ...
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9 votes

High dimensional topological field theory

In high dimensions you always have available variants of Dijkgraaf-Witten theory (references at the bottom of the page). In dimension $n$ this is a TFT constructed from a finite group $G$ and a ...
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9 votes
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Segal's 1999 Stanford lecture notes on TQFT, where to find them?

For dead links, sometimes http://archive.org helps. In this case they might be these: http://web.archive.org/web/20000901075112/http://www.cgtp.duke.edu/ITP99/segal/
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