Questions tagged [topological-quantum-field-theory]
Topological quantum field theory.
200
questions
2
votes
0answers
100 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 ...
7
votes
0answers
108 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 ...
5
votes
1answer
209 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 $...
4
votes
2answers
164 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 ...
1
vote
0answers
63 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 ...
10
votes
1answer
349 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 ...
13
votes
2answers
285 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 ...
5
votes
0answers
140 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 ...
7
votes
0answers
100 views
Are there attempts to numerically finding algebraic structures over finite-dimensional vector spaces?
By "algebraic structure" I mean a finite set of linear operators between tensor products of copies of one (or more) finite-dimensional (complex or real) vector spaces, fulfilling a set of ...
4
votes
0answers
133 views
Classification of special symmetric Frobenius algebras over real vector spaces
Is there a general classification of special symmetric Frobenius algebras over real vector spaces? I know that $n\times n$ matrix algebras, the quaternions, the complex numbers, the trivial algebra, ...
18
votes
1answer
355 views
How aggressive is the fibrant replacement of $\mathrm{Bord}_n$?
Lurie (On the Classification of Topological Field Theories), with some corrections by Calaque and Scheimbauer (A note on the $(\infty,n)$-category of cobordisms), famously constructed a symmetric ...
12
votes
1answer
226 views
Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?
Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...
7
votes
1answer
314 views
Grothendieck group of the category of boundary conditions of topological field theory
In this paper also the journal front page, eq. 2.14, it introduces
the Grothendieck group $K^0$ of the category of boundary conditions of topological field theory.
My question is that
what exactly ...
3
votes
0answers
83 views
$e^{2\pi ic_{-}/8}$ and $e^{2\pi ic_{-}/24}$ in unitary modular category (UMC)
Background
Unitary modular categories (UMC) do not capture the central charge $c_{-}$ of the topological quantum field theory (TQFT). However, there is a relation that fixes,
$c_{-}\bmod
8$:
\begin{...
11
votes
0answers
254 views
Is there a bestiary of “derived 2-vector spaces”?
The appendix "A bestiary of 2–vector spaces" of Bartlett, Douglas, Schommer-Pries, Vicary, "Modular categories as representations of the 3-dimensional bordism 2-category" analyzes ...
11
votes
1answer
138 views
Are algebras with invertible linear duals always Frobenius?
Let $A$ be a finite dimensional algebra over a ground field $k$. The linear dual $A^* = Hom_k(A,k)$ is naturally an $A$-$A$ bimodule. I am interested in those algebras such that $A^*$ is an invertible ...
7
votes
1answer
134 views
An explicit expression for the naturality of the Serre automorphism in the bicategory of algebras
By the cobordism hypothesis, there is an $O(2)$-action on the maximal subgroupoid $\hat{\mathcal{C}}$ of the subcategory of fully dualizable objects in a bicategory $\mathcal{C}$. The $SO(2)$-part of ...
7
votes
0answers
173 views
2d TQFTs with values in simplicial sets and Reedy categories
Let $Cob$ be the category such that
$Obj(Cob)$ is $\emptyset\sqcup\mathbb{N}$, with $n$ seen as the union of $(n+1)$ circles numbered from $0$ to $n$,
morphisms are (homeomorphism classes of ...
11
votes
0answers
307 views
What is the motivation for a Frobenius manifold?
A Frobenius manifold is a type of manifolds with extra structure.
The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the ``Saito'' examples arising from singularities (...
7
votes
1answer
633 views
Stacks for a string theory student
First, I'm a string theory student hoping to grasp some math involved in some physics developments.
I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the ...
6
votes
0answers
147 views
References for superhomology
This question concerns topological string theory.
It was known sice its outset, that the BRST-cohomology ("observables") of the weakly coupled topological string B-model on a Calabi-Yau ...
3
votes
0answers
74 views
Does every special $C^*$-Frobenius algebra have a unit?
I have a rather basic question about $C^*$-Frobenius algebras (also called Q-systems). Any pointers or references will be most helpful!
We are given a finite-dimensional complex Hilbert space $\mathbb{...
5
votes
1answer
236 views
How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?
Chen and Yang have a more general version of the volume conjecture that they state for all hyperbolic $3$-manifolds (Conjecture 1.1 of [2]) including those with boundary. To do this, they have to ...
7
votes
0answers
176 views
Physical and mathematical significance of the NS-2 brane
This question is about topological string theory and it was also posted in Physics Stack Exchange.
The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph ...
9
votes
3answers
590 views
Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?
There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
9
votes
0answers
228 views
What is the current state of research in Chern-Simons theory?
I'm a PhD student in mathematical physics looking for some orientation. As asked in the title, I would like to know the current state of research in Chern-Simons theory. More specifically, what are ...
1
vote
1answer
125 views
A computation in a commutative Frobenius algebra
This is already posted here https://math.stackexchange.com/questions/3695584/a-computation-in-a-commutative-frobenius-algebra but I didn't get any answers.
Given a commutative Frobenius algebra (in ...
6
votes
2answers
296 views
1d TQFT minus connection =?
Correct me if I am wrong but I believe at least conceptually (maybe even rigorously) data of a 1-dimensional TQFT and of a vector bundle with connection are equivalent.
Going into more detail (and ...
3
votes
1answer
310 views
Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?
Let us consider abelian and non-abelian 3d quantum Chern-Simons theory path integrals:
abelian Chern-Simons theory on non-spin manifolds ---
$$
\int [DA]\exp(i \frac{k}{2\pi} \int_X (A \wedge dA ))
...
9
votes
2answers
569 views
Path integral derivation of extended TQFT
I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of ...
6
votes
0answers
130 views
Why does the inverse Alexander polynomial appear in the MMR conjecture?
In an attempt to better understand why the inverse Alexander polynomial appears in the MMR conjecture, I was reading the paper [1] of Bar-Natan and Garoufalidis giving their proof of the conjecture ...
9
votes
2answers
495 views
B-model and Hochschild cohomology
In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\...
8
votes
1answer
208 views
Generators and relations for the 2-dimensional unoriented cobordism category
It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of ...
8
votes
1answer
347 views
DW, state sum models, and fully extended TQFTs
I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-...
4
votes
1answer
178 views
Mapping class group of torus, why is $(ST)^3=S^2$?
In the context of topological quantum field theories, I am interested in the mapping class group of a torus. Here I can consider the torus as a square with identified edges and also decorated with ...
3
votes
0answers
117 views
Why is the RT invariant from $\mathcal Z(\mathcal C)$ the (norm) square of the one from $\mathcal C$?
The relationship between Turaev-Viro/state-sum invariants and Reshetikhin-Turaev/surgery invariants is roughly that
$$\tau_{TV, \mathcal C}(M) = |\tau_{RT, \mathcal C}(M)|^2.$$
Here $\mathcal C$ is a ...
5
votes
1answer
180 views
Classification of $\operatorname{Rep}D(H)$
Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
4
votes
0answers
180 views
References for topological quantum field theory
I have been studying TQFTs and I am mainly reading
Lurie's proof of the cobordism hypothesis: https://www.math.ias.edu/~lurie/papers/cobordism.pdf
Walker's notes: https://canyon23.net/math/tc.pdf
...
4
votes
0answers
93 views
Shapes of cores of symmetric monoidal $(\infty,n)$-categories (with duals)
According to the cobordism hypothesis, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-categories with duals, then framed fully extended TQFTs with target $\mathcal{C}$ are an $\infty$-groupoid, ...
4
votes
1answer
154 views
Group representation with algebra structure
I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.
Let $G$ be a finite group. Its finite-...
8
votes
1answer
1k views
What do physicists mean by a topological quantum gravity theory
This is a jargon-like question.
The fact that this is posted here rather in a physics forum indicates two things
I know too little physics.
An explanation with more mathematics flavors will be ...
6
votes
2answers
551 views
1-dimensional pure gauge theory
I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie,
and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard
even in the first section ($n = 1$), which was "trivial but ...
2
votes
0answers
232 views
Tracking down an elusive book
A few weeks ago I had a very engaging talk with a faculty member, where he told me lots of interesting things about quantum algebras, know theory and Reshetikhin-Turaev invariants (this field is not ...
10
votes
2answers
277 views
What are the topological phases of quantum Hall systems?
(Fractional) quantum Hall systems are $2+1$-dimensional models which are said to possess topological order. One (maybe even complete) set of invariants of topological phases in $2+1$ dimensions is ...
4
votes
0answers
178 views
Does Dijkgraaf-Witten theory have a time-reversal symmetry?
By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated ...
11
votes
1answer
444 views
Importance of the principal bundle in Chern-Simons theory
This is a very basic beginners question about Chern-Simons theory. The configurations that we sum over to get the partition function are given by a Lie-algebra valued 1-form $A$ on a topological 3-...
18
votes
1answer
849 views
Fully extended TQFT and lattice models
I often read that fully extended TQFTs are supposed to classify topological phases of matter. So I would like to understand the formal nature of fully extended TQFTs on a more direct physical level (...
4
votes
0answers
162 views
Are Turaev-Viro invariants holonomic?
Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...
11
votes
1answer
465 views
What are the applications of topological quantum field theory to continuous-time dynamical systems?
From wikipedia:
In dynamics, all continuous time dynamical systems, with and without noise, are Witten-type TQFTs and the phenomenon of the spontaneous breakdown of the corresponding topological ...
6
votes
0answers
156 views
Can we define topological quantum field theories on Calabi-Yau manifolds?
Calabi Yau manifolds are Kähler manifolds with vanishing first Chern class. According to the conjecture of E. Calabi , for a Kähler manifold M , if
$c_1 (M) = 0 $ , then M would admit a Ricci-flat ...
