21
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 ...
17
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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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?
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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15
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 ...
- 6,820
14
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 ...
- 21.8k
13
votes
Accepted
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 ...
- 27.8k
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 ...
- 12.4k
13
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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13
votes
Accepted
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 ...
- 137k
13
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&...
- 23.2k
12
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 ...
- 27.8k
12
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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12
votes
Accepted
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 ...
- 1,691
12
votes
Accepted
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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11
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 ...
- 27.8k
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 ...
- 4,017
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 $\Sigma\times M_4$...
11
votes
Accepted
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 $M$ ...
- 5,550
11
votes
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Alternative approaches to topological QFTs
It sounds like what you would like is a rigorous version of Witten's original Feynman integral & Wilson loop approach. This is not a totally unreasonable thing to ask for, since QFTs have been ...
- 5,879
11
votes
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Undergraduate research in Topological Quantum Field Theory
I am not sure if this is an answer or a request for clarification.
There are a lot of different topics in math (and in physics) that go by the name 'topological quantum field theory'. Beyond the ...
- 5,879
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 ...
- 3,327
11
votes
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Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?
It turns our the most serious disparity is your first bullet point, not the second: the key difference is that the cobordism hypothesis is about fully extended TFT. A sub-issue is that in the fully ...
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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
What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols?
This is really an answer to the question in the comment of Noah's answer.
Let $F$ be a solution to the pentagon equations for some fusion category $\mathcal C$. Pivotal structures on $\mathcal C$ ...
- 1,089
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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10
votes
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Is Segal's notion of conformal field theory a quantum field theory in the sense of Wightman axioms?
My understanding is that Segal invented his formalism (which was then adapted by Atiyah) by thinking about the same thing Wightman was thinking about: formalising the theory of local operators. In ...
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10
votes
Accepted
An algebra with more than one Frobenius algebra structure
If $A$ is a Frobenius $K$-algebra and $\lambda\colon A\to K$ is a Frobenius form, then the Frobenius forms are the mappings of the form $a\mapsto \lambda(ua)$ with $u$ a unit of $A$.
One way to see ...
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9
votes
Accepted
Which manifolds are sensitive to the cocycle in the Dijkgraaf-Witten model?
The example $G = \mathbb Z/2$ and $M = \mathbb{RP}^3$ works.
The inclusion $\mathbb Z/2\to\{\pm 1\}\subset\mathrm U(1)$ induces an isomorphism $H^3(B\mathbb Z/2, \mathbb Z/2)\to H^3(B\mathbb Z/2, \...
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