Questions tagged [topological-quantum-field-theory]
Topological quantum field theory.
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Why does the bordism category have duals?
The proof of the Cobordism Hypothesis outlined by Lurie seems to assume that the $(\infty, n)$-category $\mathbf{Bord}_n^{(X, \zeta)}$ has duals, i.e. duals for objects and adjoints for all $k$-...
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Example of pseudo $3$-manifold without any shape structure
I'm reading Andersen and Kashaev's A TQFT from quantum Teichmüller theory and the following condition in their definition of admissible oriented triangulated pseudo $3$-manifold confused me:
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Some fusion rings/categories I don't recognize
Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...
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Recommendation to understand mean field theorem
I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...
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Software for working with fusion categories
One way to describe fusion categories is via a fusion system: several lists of numbers that define the fusion ring, associator, braiding (if it exists), etc. Often, these sets of numbers are quite big,...
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Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?
Background
I am currently writing an MSc dissertation on TQFTs (and Khovanov homology, but that is unrelated to this question).
After having read most of Kock's book on the equivalence between 2D ...
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Reshetikhin-Turaev invariants from extended 3d TQFTs
Attached to any object $V\in \mathcal{C}$ of a ribbon category $\mathcal{C}$, Reshetikhin and Turaev have defined knot invariants
$$\tau_V(K)\ \in\ \text{End}_{\mathcal{C}}(1_{\mathcal{C}})$$
for ...
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Are there (non Lagrangian) algebras of Turaev-Viro TQFTs which cannot be completed to Lagrangian algebras?
Consider a 3d TQFT of the Turaev-Viro type, say TV$(\mathcal{C})$, where $\mathcal{C}$ is some fusion category. Equivalently, this is a TQFT admitting Lagrangian algebra objects $\mathcal{L}$ of the ...
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Explicit examples of 4-cocycles over finite 2-groups
By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent ...
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Modularity of the Drinfeld center of the category of G-graded vector spaces
Background: Let $G$ be a finite group, and $\mathrm{Vect}_G$ be the category of finite dimensional $G$-graded vector spaces over some algebraically closed field $k$ of char 0. It is well-known that $\...
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Knot invariants in WZW CFT via Holographic Principle
In the physics literature the Holographic Principle relates
theories in the bulk and the theories in the asymptotic boundary.
While the bulk theory is the 3D Chern-Simons theory, the
corresponding ...
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"Inclusion" between higher categories of framed bordisms?
Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds.
It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences.
If $n$ is large enough, ...
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What are some examples of 3-dualizable $(\infty,2)$ categories?
From the cobordism hypothesis, we know that (the space of) symmetric monoidal functors from the $(\infty,3)$ category of framed cobordisms into a symmetric monoidal $(\infty,3)$ category is the same ...
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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. ...