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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TQFT vs CQFT vs QFT intro
What is a vague motivational intro to the relationship between topological quantum field theory, cohomological quantum field theory, and quantum field theory?
I am a beginner.
Here are the vague basic ...
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Clarification on a paper
I have a question about the following excerpt from p.93-94 in a paper of Donaldson:
Suppose $U_0, U_1$ are finite-dimensional vector spaces and $\Gamma$ is a linear subspace of $U_0 \oplus U_1$. Then,...
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Could the word "symmetry" represent different things in different contexts? (naive question)
I just wanted to bring up some discussion about an apparently essential concept for some fields in mathematics as so as for some in physics, as already mentioned in the title, I'm referring to the ...
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Natural candidates for energy function of knots in $S^3$?
Let $K\subseteq S^3$ be a knot.
In real-life, knots (like protein chains) $K$ moves around stochastically, and experimentally the lowest energy/highest entropy states are particularly simple from a ...
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cochains valued in a central extension in terms of the cochains valued in the subgroup and the quotient.
Consider a finite abelian group $G$ and a subgroup $H\subset G$, and denote by $A=G/H$ the quotient. Then $G$ is an extension of $A$ by $H$ determined by the short exact sequence
$$
1\rightarrow H\...
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How do you calculate the partition function on a manifold-with-corners in extended TQFT?
I'm a physicist trying to study Topological Quantum Field Theory (TQFT), so apologies if the following has some basic mistakes or misuse of terminology. When answering please bear in mind that I'm not ...
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State-sum construction of the Drinfeld center of a fusion 2-category
If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-...
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A map in group cohomology from $H^2(G,\widehat{G})$ to $H^3(G,U(1))$
Let $G$ be an abelian finite group, and denote by $\widehat{G}$ its Pontriajin dual, which is (non-canonically) isomorphic to $G$. I found the statement that there exists a map in group cohomology
\...
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Higher Dijkgraaf Witten Theory
I am trying to understand higher form symmetries in TQFT. In particular the higher form version of Dijkgraaf Witten Theory.
It is know that for a 0-form symmetry we can specify the principal G-bundle ...
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References about Quantum Theory
I’m interested in Quantum Theory. Can anyone recommend a good book (for mathematicians) that tackles varios topics (not necessarily formally).
I have in mind books that have the same idea of books ...
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Topological order at finite temperature
For the past few decades the study of so-called topological orders has remained an active area of research. By definition (following the motivation section of this recent invitation to the subject, ...
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Correspondence between G-covers and grupoid homomorphisms
I have a question concerning this article (https://math.berkeley.edu/~qchu/TQFT.pdf) about TQFTs and representation theory of finite groups.
In the beginning of the third section, it is stated that, ...
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Outcome of a concrete surgery operation in 3D
Consider the 3-sphere $S_3$ with an unlink loop $L$ whose tubular neighborhood is identified with the solid torus $B_2\times S_1$ with one twist, i.e., such that the image of $x\times S_1$ (where $x$ ...
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How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines?
Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$.
Is it possible to do so ...
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Braided coherence in braided monoidal categories
In MacLane's Categories for the Working Mathematician the author shows that the evaluation at 1 gives an equivalence of categories $\text{hom}_{BMC}(B,M)\simeq M_0$ where $B$ is the braid category, $M$...
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In what situations are Path Integrals well-defined?
In physics I have come across contexts where apparently path integrals are well-defined, and others where they are not. However I have no clear understanding of when and why they succeed or fail to be ...
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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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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^{...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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}...
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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 ...
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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 ...
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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)$-...
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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 ...
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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 ...
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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 ...
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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 ...
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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}...
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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, ...
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