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.

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

It's related to knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry in mathematics.

See https://en.wikipedia.org/wiki/Topological_quantum_field_theory for more complete information.

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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 and the Cobordism Hypothesis Math Overflow,…
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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 category of cobordism in the above definition represent…
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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 Homological Algebra), self study of Geometry (Manifolds,…
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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 habitats for the ends of open strings and can be…
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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 Schroedinger equation: $$i\hbar\dfrac{\partial}{\partial…
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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 applications of TQFTs within both disciplines…
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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 a sensible thing to do (and is supposedly…
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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 perfect pairing $Z(\overline{M})\otimes Z(M)\to…
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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. But the central point is the same. One sentence…
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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 but not diffeomorphic. I know that these…
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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, his work was not considered rigorous enough to…
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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 axiom they state is called "Normalization" $\tau(S^d) = k$…
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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 category of complex vector spaces and linear…
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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 doesn't say which type of TQFT are dealt in the…
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(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 state what I learned as follows. Also I will only…
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