Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
1answer
15 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
28 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 ...
4
votes
0answers
57 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
36 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 ...
0
votes
1answer
43 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
72 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}...
1
vote
0answers
48 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'...
4
votes
1answer
361 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, ...
26
votes
0answers
361 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
130 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 ...
0
votes
1answer
80 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 ...
0
votes
1answer
61 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 ...
0
votes
1answer
80 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?
3
votes
1answer
89 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
107 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
202 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
163 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 ...
12
votes
0answers
809 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 ...
13
votes
1answer
682 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 ...
12
votes
1answer
408 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
197 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
245 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 ...
3
votes
0answers
123 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
3
votes
0answers
125 views

Wilson lines, boundary condisions, surface defects of TQFTs

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
3
votes
1answer
164 views

Various types of TQFTs

I am interested in topological quantum field theory (TQFT). It seems that there are many types of TQFTs. The first book I pick up is "Quantum invariants of knots and 3-manifolds" by Turaev. But it ...
7
votes
1answer
375 views

Evaluation and Coevaluation maps of a TQFT

In Lurie's On the Classification of Topological Field Theories, he states in Proposition 1.1.8 that for an oriented compact manifold $M$ and a TQFT $Z:\mathrm{Cob}(n)\to \mathrm{Vect}_k$, there is a ...
2
votes
1answer
167 views

Opposite Orientation of Boundary in Bordisms

In Lurie's "On the Classification of Topological Field Theories" (and certainly other places) he defines the category $\mathbf{Cob}(n)$ who objects are oriented $(n-1)$ manifolds. Given $M,N\in\mathrm{...
23
votes
2answers
979 views

Atiyah's definitions of Topological Quantum Field Theory

According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces. How does this definition relate with the physics of quantum mechanics? What does the ...
12
votes
0answers
321 views

What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...