Questions tagged [topological-quantum-field-theory]

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.

Filter by
Sorted by
Tagged with
0
votes
1answer
42 views

A matrix ring over $\mathbb{C}$ is also a Frobenius algebra over $\mathbb{R}$

I am new to the Frobenius Algebra course. One of my textbook exercises ask to prove: A matrix ring over $\mathbb{C}$ is also a Frobenius algebra over $\mathbb{R}$ My attempt: I know that I can give a ...
1
vote
1answer
59 views

What are the dualizable objects in the category of Hilbert spaces?

Let $\mathbf{Hilb}$ be the category of Hilbert spaces and continuous linear maps. Turn it into a symmetric monoidal category using the tensor product of Hilbert spaces. What are the dualizable objects?...
6
votes
0answers
201 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. ...
9
votes
0answers
152 views

Motivation for quantum cohomology rings

I can't seem to find a good source for the motivation for defining the big quantum cohomology ring with its quantum product. Collecting the Gromov-Witten invariants in a generating function seems like ...
1
vote
1answer
48 views

Do those Hermitian and unitary matrices form a basis for the underlying complex vector space?

I conjecture that in the complex vector space $\mathbb{C}^{2^N \times 2^N}$, where $N$ is a positive integer, there is a basis $\mathcal{B}$ whose elements are Hermitian and unitary. However, I don't ...
1
vote
1answer
122 views

Is the category of PL manifolds equivalent to the category of topological manifolds in dimension 2,3?

Balsam and Kirillov write on page 10 in their paper Turaev Viro invariants as an extended TQFT: "Note that in dimensions 2 and 3, the category of PL (piecewise linear) manifolds is equivalent to ...
1
vote
0answers
75 views

Pivotal category: Why that name?

Question Why are pivotal categories called pivotal? What I can think of: In a (right) rigid monoidal category $\mathbf C$ one draws a morphism $\varphi \in Hom_{\mathbf C}(X, Y)$ in string diagrams ...
3
votes
1answer
80 views

Functorial isomorphisms between bracket spaces

1. Context This question came up while reading Passegger's Notes on Turaev-Viro-Barrett-Westbury invariants and TQFT. Let $\mathbf C$ be a spherical fusion category. Passegger defines a functor that ...
2
votes
1answer
78 views

String diagrams in a spherical fusion category

1. Context This question is about the proof of Lemma 1.9 on page 7 of Passegger's Notes on Turaev-Viro-Barrett-Westbury invariants and TQFT. As the statement and proof of the lemma contain string ...
3
votes
1answer
85 views

Spherical fusion categories: A certain functor

1. Context Let $C$ be a spherical fusion category over an algebraically closed field $k$ of characteristic zero. Denote by $Vec$ the category of finite-dimensional vector spaces. Currently, I am ...
1
vote
0answers
105 views

A path to reaching the frontiers of Quantum Field Theories.

I am very excited about all the ideas surrounding quantum field theories (with or without the use of infinity categories). When I look around I see the following interesting topics: Extended TQFTs ...
5
votes
0answers
68 views

Weird axiom in definition of TFT in Bakalov-Kirillov? What, then, is a modular functor?

In "Lectures on tensor categories and modular functors" by Bakalov, Kirillov, the definition of a $(d+1)$-dimensional TFT $\tau$ is given in section 4.2. Let $k$ be a field. The very last ...
1
vote
0answers
47 views

On manifolds with a polytope decomposition

How are manifolds with a polytope decomposition defined? I stumbled upon them when reading about the category of cobordisms with polytope decomposition. This category is defined as having as objects &...
3
votes
1answer
128 views

What is the zipper?

1. Defining an "open-closed TFT" Consider the following category of open-closed cobordisms $Cob_2^{o/c}$: Objects are compact oriented smooth one-dimensional manifolds possibly with ...
0
votes
1answer
35 views

For an algebra $A$, show that the bimodule $A \otimes_{A \otimes A^{\text{op}}} A \cong \frac{A}{[A,A]}$.

If A is an associative k-algebra, and $A^{\text{op}}$ represents the opposite k-algebra (i.e. $a*_{A^{\text{op}}} b := b\cdot a)$. In the following we consider ${\cal A_1}=A$ as a right $A\otimes A^{...
5
votes
1answer
162 views

Why do we define TQFTs as functors to vector spaces instead of Hilbert spaces?

Let $\mathrm{Cob}_n$ be the category with objects closed oriented $n-1$-manifolds and morphisms being cobordisms identified upto boundary preserving diffeomorphism $\mathrm{Vect}_\mathbb C$ be the ...
2
votes
0answers
14 views

Why does a certain integral on a 3-manifold depend only on its boundary data?

I am reading Dan Freed's lectures on Quantum Groups on Path Integrals. I am picking up the required math as I go along and I am finding certain calculations hard to follow. This is regarding the ...
2
votes
0answers
51 views

Where can I get an introduction to the Colored HOMFLY Polynomial?

Where can I get an free introduction to the Colored HOMFLY Polynomial? It would be great If I could get an exposition of the invariant with regards to rational links, but I'm not holding my breath. I ...
2
votes
1answer
68 views

A computation in a commutative Frobenius algebra

Given a commutative Frobenius algebra (in the category of vector spaces over $\mathbb C$) $A$ with multiplication $m\colon A\otimes A\to A$, comultiplication $c\colon A\to A\otimes A$, identity $1\in ...
0
votes
1answer
35 views

Analogue of Temperley-Lieb algebra of higher rank

In Piunikhin’s Turaev-Viro and Kauffman invariants for 3-manifolds coincide, the exact relation between Kauffman algebra and the representation of the quantum group $U_q(sl_2)$ is shown. This seems ...
1
vote
0answers
83 views

Witten's description of WZW conformal blocks

I am reading this paper by Witten - Geometric Langlands From Six Dimensions. In section 4.1, he gives a description of the vector space of conformal blocks of the current algebra associated to a ...
2
votes
1answer
74 views

TQFTs as rigid monoidal functors

In contrast to the usual Atiyah definition of a topological quantum field theory as a monoidal functor between the category of n-dimensional Cobordisms and the category of vector spaces, in some ...
2
votes
1answer
44 views

How to show associativity in 2Cob follows from Frobenius relation

I am working through the book "Frobenius Algebras and 2D TQFTs" and am stuck on an exercise: Show that the Frobenius relations and the (co)unit relations imply the (co)associativity relations. It's ...
4
votes
0answers
75 views

Construction of a Frobenius algebra object

I am working with a rigid, abelian braided category $\mathcal{C}$, which has a natural Hopf algebra object defined via the coend $C=\int^{X \in \mathcal{C}}X^{*}\otimes X$, where $X^{*}$ is the dual ...
2
votes
1answer
61 views

Sufficiency of relations in $\mathbf{2Cob}$ - about TQFTs

I am in the midst of proving that 2D TQFTs are in one-to-one correspondence to commutative Frobenius algebras. A TQFT is a symmetric monoidal functor $Z$ from the category of cobordisms $\mathbf{2Cob}...
1
vote
0answers
26 views

Lattice construction of 2D topological field theory: Frobenius algebras vs. associative algebras

I have a basic confusion about 2D topological field theories (TFTs). In the lattice construction of 2D TFTs introduced by Fukuma et al (https://arxiv.org/abs/hep-th/9212154) only associative algebras ...
1
vote
0answers
86 views

Reference request: 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 ...
0
votes
1answer
28 views

Examples of codim-2 objects in extended TQFT

I'm scratching my head trying to understand what an extended TQFT associates to $(n-2)$-hypersurfaces. Here's some intuition that I've developed. For an $(n-1)$-hypersurface chopped into $(n-2)$-...
0
votes
1answer
67 views

Temperley-Lieb Diagrams and Representations of U_q(sl_2)

A Temperley-Lieb diagram is a crossingless matching of $2n$ points. We think of this matching as living in a rectangle, with $n$ points on top and the other $n$ on the bottom. To $n$ points we can ...
6
votes
1answer
132 views

Example of "practical" applications of Donaldson Invariants

I'm studying Donaldson Invariants from chapter 9 of The wild world of 4-manifolds by Scorpan, and I'm looking for an example where they're used to distinguish two 4-manifolds which are homeomorphic ...
2
votes
0answers
45 views

Is the category of $2$-topological quantum field theories locally small?

At first, looking at 2TQFT I see no reason to expect it to be locally small. But we know that 2TQFT is equivalent to the category of commutative Frobenius algebras cFA$_{\mathbb K}$, which is locally ...
2
votes
1answer
65 views

Field Theory Phase Factor vs Anomaly

In this paper on topological quantum field theories the authors discuss something called the anomaly in section 5. In Witten's paper on field theory and the Jone's polynomial he discusses something ...
4
votes
0answers
86 views

What is $H_{3}Spin(3)$, and how is this related with the twist of framing on a 3-manifold?

From the question, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action $$I(g)=\frac{1}{4\pi}\int_{M}\mathrm{Tr}(\omega\wedge d\omega+\frac{2}{3}...
2
votes
0answers
63 views

Mathematical Physics references in the spirit of Segal's lectures

There are a few lectures about topological quantum field theory of Graeme Segal on youtube. The first one is this. I am really interested in the mathematical descriptions of classical mechanics, ...
1
vote
0answers
55 views

Question on Turaev's paper about axioms for topological quantum field theory

I am currently reading Turaev's paper Axioms for topological quantum field theory. In couple of place, there is a paraphrase "... is natural with respect to $\mathfrak{U}$-homeomorphism" and I don'...
5
votes
1answer
1k views

What rigorous mathematical theorems has Edward Witten discovered?

I read that Ed Witten's 1990 Fields Medal was somewhat controversial among mathematicians, because even though no one questioned his deep conceptual understanding of important new mathematical ideas, ...
39
votes
0answers
708 views

How are topological invariants obtained from TQFTs used in practice?

Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory ...
4
votes
1answer
202 views

Examples of higher dimensional TQFTs

1-dimensional TQFT's assign to every 1-manifold (disjoint union of circles) a vector space and to every surface a linear map between the vector spaces that correspond to the boundary manifolds. So ...
1
vote
1answer
96 views

Enriched Categories In TQFT

I'm reading the On the Classification of Topological Field Theories and have a question about the use of enriching categories in the definition of a strict 2-category found on page 9. These are ...
1
vote
1answer
68 views

Topological Quantum Field Theory: Compact Implies Finite Boundaries

The following is from this paper. A bordism $B:\coprod_{i\in I} S^1 \to \coprod_{j\in J} S^1$ is just a “generalized pair of pants” - a pair of pants with one waist hole for each $i\in I$ and ...
1
vote
1answer
123 views

Topological Quantum Field Theory

For a topological quantum field theory, $Z:Cob(n)\to Vect(\mathbb{C})$ why is it that typically $Z(\emptyset)\cong \mathbb{C}$? Is that just the definition that makes everything work?
4
votes
1answer
134 views

Why is a Topological Field Theory equivalent to a Frobenius algebra?

How can a physicist understand a 2-dimensional topological field theory as a Frobenius algebra? Are there some explicit examples in order to understand this relation? The definition (e.g. on ...
3
votes
0answers
138 views

Is there a TQFT of links and their "cobordisms" (embedded surfaces in $S^3\times [0,1]$)?

Given two oriented links $K_1, K_2$ in $S^3$, it is an interesting problem to figure out whether there exists an oriented surface with boundary which is embedded (locally flatly) in $S\times [0,1]$ ...
2
votes
1answer
310 views

Extended Topological Quantum Field Theory (ETQFT) by Jacob Lurie

What is the functorial (categorical) definition of TQFT (Topological Quantum Field Theory), which Jacob Lurie "had extended", for his ETQFT ? Actually I just need to know what are basic tools, to ...
1
vote
0answers
172 views

Suggest a reading list to start TQFT

What would be books that would give the necessary prerequisities to study TQFT? I want to read something like Kock's Frobenius algebras and 2d TQFTs, I only know enough math that got me through a ...
17
votes
0answers
1k views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
14
votes
1answer
884 views

Topological Quantum Field theories

I was wondering about the following on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: Why is this so? Secondly, consider the ...
13
votes
1answer
492 views

Applications of TQFTs beyond physics

I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some ...
4
votes
0answers
228 views

(Topological quantum field theory) identifying objects of cobordism category

I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I ...
2
votes
0answers
283 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...