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.

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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 ...
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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?...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 &...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...