Questions tagged [topological-quantum-field-theory]
Topological quantum field theory.
278 questions
4
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Difficulties in explicitly constructing the "pairing" bordism
Freed's notes give the following definition of oriented bordism.
Definition. Let $\Sigma_0$ and $\Sigma_1$ be the two oriented closed manifolds. An $ n $-dimensional bordism from $\Sigma_0$ to $\...
2
votes
1
answer
245
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Are in- and out-boundaries of oriented bordisms determined by the orientation?
I am trying to pin down the right category of $ 2 $-dimensional bordisms that lies beneath the equivalence between $ 2 $-dimensional topological quantum field theories and commutative Frobenius ...
2
votes
0
answers
254
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Dualizability in physics and the cobordism hypothesis
This is a physically motivated question, cross-posted from PhySE. Apologies in advance if there are inaccuracies in my formulation.
A fundamental observation is that objects representing physical ...
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4
votes
0
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52
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Nontrivial vertex basis transformation that keeps the value of F and R invariant in modular tensor categories
I'm thinking whether there exist non-trivial vertex basis transformations that can keep all the values of F and R symbols invariant. In particular, I am trying to solve the following equations.
$U(a,b;...
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4
votes
1
answer
108
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Basic properties of unitary Turaev-Viro skein spaces
The following kinds of things seem to be well-known, but I don't know of explicit references.
Consider Turaev-Viro TQFT based on a unitary spherical fusion category $\mathcal{C}$. To any surface $\...
- 173
0
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0
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356
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A conjecture for factorization of group-cohomology classes
Some arguments inspired by physics (in particular a very influential paper by Wang–Wen–Witten Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions on symmetry preserving gapped ...
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3
votes
1
answer
124
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Algebraic criteria for reflection‐positive $3+1D$ TQFT realizations of fusion categories in the "alterfold" framework
In Liu’s recent work, one constructs $3+1D$ unitary TQFTs via the alterfold functional–integral approach, imposing reflection positivity, homeomorphic invariance, and finiteness on a stratified ...
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4
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1
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Can we realize any triple linking number of 3-surfaces in 5-manifolds?
Given three homologically trivial 3-dimensional submanifolds $\Sigma_1,\Sigma_2, \Sigma_3 \subset X$ of a 5-dimensional compact (maybe oriented) manifold X (assume there are pairwise intersections but ...
- 820
1
vote
1
answer
587
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A correspondence between group extensions trivializing a cocycle, and a cocycle valued in the dual group [closed]
Some arguments inspired by physics suggest that the following theorem should be true:
Consider finite group $G$ and a group cohomology class $\omega \in H^n(G,U(1))$ ($n\geq 4$). Consider then a ...
- 820
4
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1
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266
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Extended relative AKSZ TQFT
In the paper:https://arxiv.org/pdf/1306.3235
Calaque defines the AKSZ TQFT with boundary condition as a functor,
$$
d\mathrm{Cob}^{bd}\to \mathrm{Symp}(X),
$$
from the $ d$-dimensional cobordism ...
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3
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1
answer
330
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Reference requests: Koszul duality
I find the book(page 277, 19.96) http://jacksontvandyke.com/notes/langlands_sp21/langlands.pdf by Jackson Van Dyke and David Ben-Zvi mentioned the following statement,
Suppose $X$ is (probably a ...
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2
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0
answers
255
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K-Theory of Character Varieties
Let $\Gamma_g$ be a surface group, and let $U(n)$ by the unitary group over $\mathbb C$. I am interested in studying the twisted, equivariant K-theory of the character variety, $C_{g,n}$, associated ...
- 449
8
votes
1
answer
388
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Explicit Data of Homotopy Fixed Points in Lurie's TFT
Lurie's classification theorem [1, Theorem 2.4.26] classifies fully extended
tfts for $G$-manifolds in terms of homotopy fixed points:
Let $C$ be a symmetric monoidal $(\infty, n)$-category with ...
- 5,748
6
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0
answers
146
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Frobenius identity on finite-dimensional Hopf algebras
Let $(H,m_H,e_H,\Delta_H,\epsilon_H)$ be a finite-dimensional Hopf algebra over $\mathbb{C}$, where $m_H$ denotes multiplication, $e_H$ is the unit map, $\Delta_H$ is the comultiplication, and $\...
- 27.6k
3
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118
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Examples of (co)commutativity of Frobenius algebras via ambijunctions
This question is related to the paper "Frobenius algebras and ambidextrous adjunctions" by Aaron Lauda (https://arxiv.org/abs/math/0502550). Below $\Sigma\mathrm{Vect}$ is the one-object ...
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Convergence of an inverted Fourier series
Consider the following series for $x\notin \mathbb{Q}$:
$$f(x)=\sum_{n=1}^{\infty}\frac{1}{n^3\sin(n\pi x)}$$
When does it converge? By Khintchin theorem, I know that it converges almost surely but ...
- 1,351
1
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0
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106
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Integral formula of quantum dilogarithm
In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function:
\begin{equation}
\mathrm{D}_{\rm b}(x,n)=\prod_{...
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3
votes
1
answer
406
views
Do Frobenius subalgebras form a lattice?
A finite-dimensional, unital, associative algebra $A$ over a field $k$ is termed a Frobenius algebra if it is endowed with a nondegenerate bilinear form $\sigma : A \times A \to k$ satisfying the ...
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9
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0
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265
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Reduction of the $0$-handle data in Lurie's classification of TFT
A vital part of Jacob Lurie's classification of fully extended topological
field theories [1], very roughly, says that any representation of the
n-Cobordism category $Z: {\rm Cob}_{{n}} \to C$ depends ...
- 5,748
3
votes
2
answers
429
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Are $H^3(A,U(1))$ and $\operatorname{Ext}^1(A,A^\vee)$ isomorphic for $A$ finite Abelian?
Motivated by three-dimensional Dijkgraaf-Witten TQFTs for finite Abelian groups $A$, that are classified by $H^3(A,\mathbb{R}/\mathbb{Z})$, it seems natural that this group is (naturally) isomorphic ...
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2
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1
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Extending diffeomorphisms between surfaces
Suppose we have two smooth compact oriented surfaces $M_1$ and $M_2$ with boundary,both of them have genus $g$, and there are orientation preserving diffeomorphisms $\psi_1, \psi_2, \cdots, \psi_n$, ...
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0
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485
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Possible research topics for a beginner in Topological QFT?
I am highly interested in Topological Quantum Field Theory (TQFT) and am currently planning on doing a project on this topic this year. Some relevant background: Algebra (Groups, Rings, Fields, basics ...
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3
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0
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119
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Possible relation between causal-net condensation and algebraic K theory
Causal-net condensation is a natural construction which takes a symmetric monoidal category or permutative category $\mathcal{S}$ as input date and produces a functor $\mathcal{L}_\mathcal{S}: \mathbf{...
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1
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244
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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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8
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1
answer
454
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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 ...
- 323
1
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0
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197
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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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1
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491
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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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11
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1
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842
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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 ...
1
vote
0
answers
110
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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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4
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1
answer
340
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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 ...
- 820
3
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0
answers
151
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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 ...
- 3,105
6
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1
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487
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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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6
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0
answers
237
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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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0
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531
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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 ...
- 2,540
4
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2
answers
641
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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 ...
- 157
2
votes
1
answer
193
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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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8
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1
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809
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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 ...
3
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0
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152
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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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19
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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 ...
- 387
9
votes
1
answer
435
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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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0
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244
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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-...
- 820
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0
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283
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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 $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
- 6,181
0
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0
answers
95
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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 ...
7
votes
1
answer
750
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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^...
- 6,181
5
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1
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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 ...
- 66.7k
1
vote
0
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157
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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.
25
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0
answers
1k
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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 ...
- 351
6
votes
1
answer
268
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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 ...
- 28.7k
10
votes
1
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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 ...
- 5,479