# 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 ...

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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, ...

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### 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 ...

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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, ...

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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 ...

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### 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 ...

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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-...

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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 $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...

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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 ...

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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^...

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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 ...

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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.

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $K(G, 1)$ ...

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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 ...

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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,
$T$ ...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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### $\mathbb{E}_M$ 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 $\infty$...

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### 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 ...

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### 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 ...

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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\...

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### 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 ...

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### 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 $M_C(D)$ ...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 ...

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### 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 (...

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### 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. ...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...