Questions tagged [topological-quantum-field-theory]

Topological quantum field theory.

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An algebra with more than one Frobenius algebra structure

Can a (finite dimenaional) $\mathbb{K}$-algebra $A$ be equipped with more than one Frobenius structure $\lambda:A \to \mathbb{K}$? Of course we identify two structures $\lambda$ and $\lambda'$ if they ...
Béla Fürdőház 's user avatar
1 vote
1 answer
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Associated graded algebras and symmetric Frobenius algebras

Let $A$ be a filtered algebra and let $G$ be its associated graded algebra. As discussed in this question, if $G$ is Frobenius, then $A$ is also Frobenius. If $G$ is a symmetric Frobenius algebra, ...
Béla Fürdőház 's user avatar
6 votes
1 answer
354 views

Is Segal's notion of conformal field theory a quantum field theory in the sense of Wightman axioms?

For context, I heard that the Atiyah-Segal formalism of quantum field theory is equivalent to the traditional Wightman approach (at least within the scope of certain theories). However, I could not ...
Ezzeddine El Sai's user avatar
3 votes
0 answers
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Gluing result in a TQFT of Donaldson

I am reading about a (2+1)-dimensional TQFT defined by Donaldson in this paper, see also here. Below is a short summary of the construction (homology is over $\mathbb{Z}$). To closed, connected, ...
contingent's user avatar
15 votes
0 answers
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"Next steps" after TQFT?

(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.) Recently, I've been ...
Nicholas James's user avatar
8 votes
1 answer
211 views

Stably-framed cobordism $(\infty,n)$-category

In Lurie's treatment of the cobordism hypothesis, the domain is $\mathsf{Bord}^{fr}_n$, the symmetric monoidal $(\infty,n)$-category of $k$-bordisms with $n$-framing for $0\leq k\leq n$. If I ...
Leo's user avatar
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State-sum for 4d TQFT from fusion 2-categories and invariants of Morita equiavalence classes beyond Drinfeld center

If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-...
Andrea Antinucci's user avatar
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0 answers
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$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$. Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
Pulcinella's user avatar
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A name for "anti-symmetric" Frobenius algebras?

Let $V$ be an $N$-dim vector space and denote its exterior algebra by $\Lambda^{\ast}(V)$. The algebra $\Lambda$ has an obvious Frobenius algebra structure $B(-,-)$ given by wedgeing and then ...
Didier de Montblazon's user avatar
5 votes
1 answer
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Defining extended TQFTs *with point, line, surface, … operators*

$\newcommand\Cob{\mathrm{Cob}}\newcommand\Vect{\mathrm{Vect}}\DeclareMathOperator\Rep{Rep}$The ordinary definition of a TQFT is: Defnition: A $d$-dimensional TQFT is a symmetric monoidal functor $\Cob^...
Pulcinella's user avatar
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Are dagger-categories / categories with duality related to unoriented field theories?

Recall that a dagger-category is a category $C$ equipped with an identity-on-objects, involutive anti-equivalence $\dagger: C^{op} \to C$. I believe that the correct $\infty$-categorical ...
Tim Campion's user avatar
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Classifying of low-dimensional Frobenius algebras

Does there exist a classification of finite-dimensional Frobenius algebras in low dimensional cases. This question is motivated by the following analogous question for Hopf algebras.
Didier de Montblazon's user avatar
17 votes
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What is the status of the cobordism hypothesis?

Let $\mathscr{C}$ be a symmetric monoidal (weak) $n$-category. A framed extended TQFT of dimension $n$ with values in $\mathscr{C}$ is a symmetric monoidal functor from the framed bordism $n$-category ...
Miguel I. Solano's user avatar
6 votes
1 answer
121 views

Commutative Frobenius algebra with non-invertible window element, but not square zero

For any commutative Frobenius algebra $A$ there is an associated window element $\omega \in A$. If $\mu: A \otimes A \to A$ denotes the multiplication, $1 \in A$ the unit, $b: A \otimes A \to k$ the ...
Chris Schommer-Pries's user avatar
9 votes
1 answer
237 views

Physics application of Wilson surface observables

There is some work which generalises the usual Wilson loop in QFT to higher dimensions and constructs non-abelian Wilson surface functionals in the context of non-abelian gerbes. It seems to me that ...
Hollis Williams's user avatar
4 votes
2 answers
555 views

A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?

A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
Didier de Montblazon's user avatar
4 votes
0 answers
223 views

CFT as an axiomatic field theory

I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every ...
Andi Bauer's user avatar
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13 votes
1 answer
592 views

Practical consequences of the geometric cobordism hypothesis

As far as I understand, the cobordism hypothesis provides a construction of all (appropriately defined) fully-extended TQFTs. In particular, given a fully-dualizable object in a certain category, one ...
Confused Physicist's user avatar
3 votes
2 answers
182 views

Classification of crossed $G$-algebras

Added later: As Viktor Ostrik points out in a comment, what I'm looking for is a classification of so-called crossed $G$-algebras corresponding to homotopy TQFTs with homotopy target space $K(G, 1)$ ...
Andi Bauer's user avatar
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5 votes
1 answer
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Relation between TQFT representations and factorizable sheaves

I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks. More ...
Josh Lam's user avatar
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1 vote
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Different modular data with same T-matrix

Modular data (MD) is an invariant of a modular fusion category $\mathcal{C}$. It is a couple of symmetric matrices $S, T \in M_r(\mathbb{C})$ with: $r$ the rank of $\mathcal{C}$, $S$ invertible, $T$ ...
Sebastien Palcoux's user avatar
13 votes
2 answers
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Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh

In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these notes or this lecture) some important aspects of the Langlands correspondence are stated in the language of topological ...
Anton Hilado's user avatar
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7 votes
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Sensitivity of topological field theories

I am struggling to find references or studies that explore the overall sensitivity of topological field theories as an invariant of smooth manifolds. There is the paper by Davis that explores how ...
Jack Romo's user avatar
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6 votes
3 answers
320 views

Original reference for generators and relations of 2-dimensional TQFT

What is the original reference where it was first proven that the generators and relations of the 2-dimensional cobordism category are those of commutative Frobenius algebras? I've seen this article ...
Andi Bauer's user avatar
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3 votes
0 answers
120 views

Riemann-Hilbert-type correspondence for locally constant factorization algebras

This is related to a previous post, but a bit softer and should probably stand on its own. In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there ...
Markus Zetto's user avatar
6 votes
1 answer
318 views

$\mathbb{E}_M$ as colimit of little cubes operads

In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}_M$ can be identified with the colimit of a diagram of $\infty$...
Markus Zetto's user avatar
4 votes
0 answers
217 views

Summary of different types of TQFT?

For the purposes of this question, a TQFT comprises the following data: An "upper dimenison" $n \in \mathbb N$. A "lower dimension" $0 \leq l \leq n$. A choice of structure ...
Tim Campion's user avatar
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9 votes
1 answer
683 views

Two vague questions about TFT

Question 1. Take a smooth projective Calabi-Yau $X$. Then $D^b(X)$ is a fully-dualizable category and there's an associated 2d TFT. This the usual 2d B-model with target $X$. But $D^b(X)$ is actually ...
Ed Segal's user avatar
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2 votes
0 answers
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On the "integrality condition" of the bilinear form in the Chern-Simons action

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\mathcal{M}$ be a principal $G$-bundle over a smooth orientable manifold $\mathcal{M}$. Furthermore, let $\langle\cdot,\cdot\...
G. Blaickner's user avatar
9 votes
2 answers
1k views

Undergraduate research in Topological Quantum Field Theory

I'm really interested in Topological Quantum Field Theory (TQFT) and am currently planning to focus on it in my undergraduate thesis. My university, unfortunately, does not allow double majors in ...
Gab Romero's user avatar
3 votes
0 answers
74 views

Does a factorization of a modular fusion category imply some "factorization" of TFTs?

Mueger showed in this paper that if $C$ is a modular fusion category and $D$ is a modular fusion subcategory of $C$, then $C$ is equivalent to $D \boxtimes M_C(D)$ as ribbon categories, where $M_C(D)$ ...
Jo Mo's user avatar
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3 votes
1 answer
116 views

Closed-form expressions for the Kashaev invariant via surgery

For a knot $K$, let $J_N(K)$ denote the $N$th Kashaev invariant of $K$. This is the same as the $N$th colored Jones polynomial evaluated at an $N$th root of unity (or $2N$th depending on your ...
Calvin McPhail-Snyder's user avatar
7 votes
0 answers
141 views

What are the generators and relations of the conformal cobordism category?

According to a definition by Segal, a $2$-dimensional CFT is a symmetric monoidal functor from the category of oriented conformal cobordisms to the cateogry of projective complex vectorspaces. Coming ...
Andi Bauer's user avatar
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5 votes
1 answer
117 views

Non-cyclotomic modular fusion categories

In a recent talk (see [1]) Richard Ng asks whether the invariants of 3-manifolds derived from any modular fusion category are cyclotomic integers. In tqft, such an invariant is computed from the F-...
Sebastien Palcoux's user avatar
11 votes
2 answers
636 views

Reference on the Chern-Simons theory and WZW models for mathematicians

I would like to ask if there are any beginner friendly references for learning CS theory and WZW models. It seems that most mathematical texts on the subjects begin with convenient definitions that ...
WJL's user avatar
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5 votes
0 answers
243 views

Factorization homology and topological conformal field theories

My question concerns some of the results of Costello's "Topological conformal field theories and Calabi-Yau categories" and how they are related/ can be rederived via the description of (...
Markus Zetto's user avatar
4 votes
0 answers
128 views

Infinite dimensional topological quantum field theories?

A topological quantum field theory is a functor $Z:Cob(n) \to Hilb$ mapping from the $n$-dimensional cobordisms to $Hilb$, the category of Hilbert spaces with morphisms being bounded linear operators. ...
Juan Sebastian Lozano's user avatar
10 votes
1 answer
236 views

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

A state sum model is a smooth invariant defined on smooth triangulated, or PL manifolds, by summing a local partition function over labels attached to the elements of the triangulation. Typical ...
Manuel Bärenz's user avatar
25 votes
1 answer
2k views

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

Edition On July 15 2021, the description of the question has been considerably modified to meet the requirement of making this question more OP-independent and thus more useful for general readers. ...
Qi Tianluo's user avatar
7 votes
1 answer
321 views

Are there non-homeomorphic 3-manifolds with the same Turaev-Viro-Barrett-Westbury invariants?

The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but ...
Sebastien Palcoux's user avatar
2 votes
1 answer
197 views

Non-extendable 3D TQFTs

Non-extendable 2D TQFTs correspond to finite dimensional Frobenius algebras [1]. How about 3D TQFTs? While the answer is clear for the extended ones (e.g. (3,2,1) TQFTs almost correspond to modular ...
Student's user avatar
  • 4,648
8 votes
2 answers
407 views

Formula for the anomalies of spin Chern-Simons theories?

$\newcommand{\SH}{\mathit{SH}}\newcommand{\Z}{\mathbb Z}$Let $G$ be a compact Lie group and $\lambda\in H^4(BG;\Z)$. The data $(G, \lambda)$ determine a 3d topological field theory called Chern-Simons ...
Arun Debray's user avatar
  • 6,676
3 votes
0 answers
155 views

Lifting in String Theory and QFT

I'm posting this here instead of Physics Stack as my question is on the precise mathematical meaning of a word which is often used in the physics literature. In theoretical physics (especially string ...
Hollis Williams's user avatar
7 votes
0 answers
127 views

Endofunctors of the surface category

Let $\mathrm{Cob}_2$ be the symmetric monoidal $(\infty,1)$-category whose objects are closed oriented $1$-manifolds and whose morphisms are compact oriented surface bordisms. (Higher morphisms are ...
Jan Steinebrunner's user avatar
5 votes
1 answer
368 views

Uses for (Framed) E2 algebras twisted by braided monoidal structure

$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$ If $\C$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $...
Dmitry Vaintrob's user avatar
4 votes
2 answers
286 views

Are there examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras?

Are there any examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? Or can one show that none exist? I went through this list of all complex associative ...
Andi Bauer's user avatar
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1 vote
0 answers
77 views

Ordering in Cobordism Category

Let $Cob^{3}$ denote the cobordism category of $1$ dimensional manifolds i.e the objects are finite disjoint union of circles and morphisms are represented by surfaces. Is it possible to treat the ...
Monkey.D.Luffy's user avatar
11 votes
1 answer
481 views

Alternative approaches to topological QFTs

A while ago I read the paper 'Quantum Field Theory and the Jones Polynomial' by Edward Witten. This article uses a lot of concepts from physics like BRST symmetry and the Chern-Simons action which ...
Hollis Williams's user avatar
14 votes
2 answers
433 views

Lagrangian of Reshetikhin-Turaev TFT's

One of the results from the Reshetikhin-Turaev package is that given a modular tensor category $\mathscr{C}$ one can construct a TFT $Z$. In the case where $\mathscr{C}$ is the category of positive ...
Alonso Perez Lona's user avatar
5 votes
0 answers
170 views

References for computing $n$-point correlations in Chern-Simons theory

I am interested in learning how to compute $n$-point correlation functions in Chern-Simons theory, thought of as a TQFT (similar to Witten's work linking that theory to knot theory). I am mostly ...
Malkoun's user avatar
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