Questions tagged [topological-quantum-field-theory]
Topological quantum field theory.
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Is there a version of the 2d cobordism hypothesis for surfaces with non-empty incoming and outgoing boundary?
Question: Is there a condition on an object $x$ of an $(\infty,2)$-category $\mathcal C$ which is equivalent to $x = Z(pt_+)$ for a unique TFT $Z$ from the $(\infty,2)$-category of framed bordisms ...
11
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A proof of the gluing axiom of a TQFT
I posted the following question on math stackexchange but I have not received any answer.
So I hope people here can help me.
In the book Lectures on tensor categories and modular functors by Bakalov ...
9
votes
1
answer
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Trace of a functor (or dimension of a category) in extended 2d TQFTs
In an extended 2d TQFT $Z$, a point (with orientation + or -) is assigned a category $Z(+)$ or $Z(-)$. This category should be as close to a vector space as possible: $\mathbb{C}$-linear, monoidal, ...
6
votes
1
answer
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Does the following "symmetric" 2nd cohomology group of a finite group with coefficients in $Z_2$ always vanish?
Let $G$ be a finite group. Usually, a 2-cocycle on $G$ with values in $\mathbb{Z}\_2 = \{+1, -1\}$ is a collection of signs $\epsilon_{g,h} \in \{+1, -1\}$, $g,h \in G$, satisfying the cocycle ...
14
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3
answers
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Motivation and unsolved problems of TQFT
I have been studying topological quantum field theory by mainly reading the Turaev's book.
I'd like to know if there are unsolved problems that motivate mathematicians to study TQFT, like Riemann's ...
10
votes
4
answers
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Understand Witten's "QFT and Jones Polynomials" - how does he get to the twisted Dirac operator L_{-}?
Hi,
this is my first post here, so I hope I am asking the question the right way.
I am trying to understand to following piece of algebra:
In his paper, Witten claims that $\int_M Tr(B \wedge DB) + \...
12
votes
2
answers
537
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S-matrix for the HOMFLY/Hecke category
This question concerns the HOMFLY-PT category, closely related to Hecke algebras. (See here for example.)
The minimal idempotents of this category are indexed by pairs $(\lambda_+, \lambda_-)$ of ...
6
votes
0
answers
197
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S-matrix for the BMW category
This question concerns the Birman-Murakami-Wenzl category, or equivalently the tangle category associated to the 2-variable Kauffman polynomial. (See here for example.)
The minimal idempotents of ...
6
votes
1
answer
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Examples of calculations of Turaev-Reshetikhin TQFT of cobordisms with boundaries have genera greater than 1
I am studying Turaev-Reshetikhin TQFT. I describe the definition of the invariant $\tau(M)$ of a cobordism $(M, \partial_{-}M, \partial_{+}M)$ in the previous question breifly. Framings in the ...
3
votes
1
answer
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Framings in the definition of Reshetikhin-Turaev TQFT
I posted the following question at Mathe Stack Exchange.link text But it has not yet answered. I am sorry if you check both sites but I also want people here to look at this problem.
I am studying ...
5
votes
1
answer
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Set of physical states of FQHE on closed Riemann surface = ?
Disclaimer. One might argue that my question is off topic as it is clearly a question about physics... But I'd like a mathematically phrased answer,
and I expect that only a mathematician can offer an ...
9
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0
answers
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Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?
I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner.
They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...
0
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0
answers
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Finding a ribbon graph for a mapping class group action
Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$.
This action $\epsilon$ is ...
11
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1
answer
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Reference request for TQFT, functoriality
I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.
It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;
The function $(M, \partial_{-}M, \...
1
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0
answers
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Isomorphism of cobordisms
Let $(M, \partial_{-}M, \partial_{+}M)$ be a decorated 3-cobordism, where $M$ be a (topological) 3-manifold and $\partial M=-\partial_{-}M \cup \partial_{+}M$.
(decorated in a sense of Turaev, Quantum ...
2
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1
answer
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A special ribbon graph presents a cylinder.
I am reading "Quantum Invariants of Knots and 3-Manifolds" by Turaev. I have a dificulty to understand the proof of Lemma 2.6 on page 172.
The lemma says that a special ribbon graph drawn on page 167 ...
15
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3
answers
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Reshetikhin-Turaev as a 3-2-1-theory
I keep reading that the Reshetikhin-Turaev construction actually yields a 3-2-1 tqft. I know the construction that associates to a suitably decorated surface a vector space built up from a hom-space ...
8
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0
answers
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Completion of n-fold Segal spaces
During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for 2-...
7
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2
answers
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Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group
One method for finding the Hilbert spaces corresponding to surfaces in Chern-Simons TQFT is by geometrically quantizing the phase space, which is just the character variety of the surface. I know that ...
11
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0
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When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?
I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...
14
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2
answers
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Homotopy Fixed Points of SO(2) on Fully Dualizable Algebras
Note: by fixed points, I always mean homotopy fixed points.
As explained in Jacob Lurie's paper on the cobordism hypothesis, we have an action of O(2) on the $\infty $-groupoid $X$ given by ...
15
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1
answer
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Are bundle gerbes bundles of algebras?
The category of line bundles (possibly with connection)
on a smooth manifold M can be defined in two different ways:
The first definition uses transition functions that satisfy a cocycle condition
(...
3
votes
1
answer
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How to interpret sections over the $\mathrm{SU}(2)$ character variety as sections over the $\mathrm{SL}(2,\mathbb{C})$ character variety?
The motivation for this question comes from the Volume Conjecture of Kashaev-Murakami-Murakami. The Jones family of invariants of knots and $3$-manifolds can all be defined using $\mathrm{SU}(2)$ and ...
7
votes
2
answers
640
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What is the state in the WRT TQFT associated to a handlebody?
Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the ...
18
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3
answers
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How to calculate the Witten-Reshetikhin-Turaev invariants from a triangulation?
I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, ...
23
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3
answers
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How many definitions are there of the Jones polynomial?
Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
3
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0
answers
267
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Group cohomology and TFTs
Ok, so I am still trying to make my way through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Actions", and I have a question (what's new). The authors start by showing ...
13
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1
answer
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Spin TQFT's in dimensions (1+1)
I don't seem to be able to find anything written about Spin TQFT's in dimension (1+1). Does anyone know any references? Or is there some reason it is uninteresting?
37
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5
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Usefulness of using TQFTs
What is a topological feature, that a (some) TQFT (e.g. in 3 or 4 dim) sees but homology/cohomology/homotopy groups don't? Or: what is an example where using classical theories is hard, but using a ...
1
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0
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topological B model
The topological A model was constructed by Witten in Comm. Math. Phys. Volume 118, Number 3 (1988), 411-449. I am looking for the original paper where topological B model was first introduced. I am ...
5
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2
answers
645
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What does it mean to extend a 2d (topological) conformal field theory to Deligne-Mumford space?
For 2D (topological) conformal field theory the corresponding moduli space is the space of Riemann surface with boundary, right? What does it mean by extending the theory to Deligne-Mumford space? How ...
13
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1
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Why Lagrangian cobordism?
There are a good number of quantum topology papers in which a TQFT-like set-up is constructed as a functor to the category of vector spaces from some category of cobordisms which satisfy some "...
2
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1
answer
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Are there non-supersymmetric and/or non-Calabi-Yau topological sigma models?
I am reading some aspects of Mirror Symmetry and in mirror symmetry the $N=2$ SCFT on a Calabi Yau Manifold can be divided into two sectors each of which is a topological sigma model, A-Model and B-...
14
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1
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Witten's topological twisting
I am reading the Witten's topological twisting for $N = 2$ Superconformal Field Theory(SCFT) http://arxiv.org/abs/hep-th/9112056
In this paper Witten constructed 2 TQFTs i.e. A-model and B-model from ...
36
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3
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What are D-branes, really?
In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
10
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1
answer
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The algebro-geometric counterpart of the Dijkgraaf-Witten model
Can the Dikgraaf-Witten model for a finite gauge group $G$ [Robbert Dijkgraaf and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393] be described in ...
4
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Does a Dehn twist in the mapping class group of an cobordism give a BV-operator in string topology?
In her article Higher string topology operations, Godin in particular construct for each surface with $n$ incoming and $m \geq 1$ outgoing boundary circles an operation $H_\ast(BMod(S);det^{\otimes d})...
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2
answers
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What's the current state of the classification of not-fully-extended TQFTs?
Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some ...
7
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1
answer
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Can string topology be a open-closed TCFT with the full set of branes?
String topology studies the algebraic structure of the homology of the free loop space $LM = Map(S^1,M)$ of a oriented closed manifold. One aspect of this structure is that the pair $(H_\ast(LM;\...
5
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1
answer
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What is the Gromov-Witten potential associated to String Topology?
Kevin Costello's article on the Gromov-Witten potential associated to a TCFT constructs for each TCFT, i.e. a functor from chains on Riemann surfaces with boundary ...
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When is a TQFT the dimensional reduction of a higher dimensional TQFT?
In Lurie's framework for TQFT's, a TQFT is a symmetric mondoial functor from $Cob_n(n)$ to some symmetric monoidal $n$-category $\mathcal{C}$. One can construct an $(n-1)$-dimensional TQFT from an $n$-...
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Resources for graphical languages / Penrose notation / Feynman diagrams / birdtracks?
There is an idea I've recently gotten interested in that doesn't seem to have a good agreed-upon name ("diagrammatic algebra?"). It centers around the use of two-dimensional diagrams of dots, ...
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1
answer
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Seiberg-Witten theory on 4-manifolds with boundary
What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist?
I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to ...
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1
answer
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What happened to the Vacuum Hypothesis in TQFT?
I remember that in the beginning, there was an axiom for $(n+1)$-dimensional
TQFT that said that the state space $V(\Sigma)$ assigned to an $n$-dimensional
oriented manifold is spanned by the ...
20
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1
answer
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Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups
Which $6j$-symbols for quantised enveloping algebras are known explicitly?
The quantum $6j$-symbols for $sl(2)$ are well-known. The references are
Masbaum and Vogel and
Frenkel and Khovanov.
What ...
20
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2
answers
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Cohomology rings and 2D TQFTs
There is a "folk theorem" (alternatively, a fun and easy exercise) which asserts that a 2D TQFT is the same as a commutative Frobenius algebra. Now, to every compact oriented manifold $X$ we can ...
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References for a certain generalization of Hochschild cohomology?
Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
39
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9
answers
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Why are fusion categories interesting?
In the same vein as Kate and Scott's questions, why are fusion categories interesting? I know that given a "suitably nice" fusion category (which probably means adding adjectives such as "unitary," "...
12
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2
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Turaev-Viro extended TQFT
Hi I am looking for any papers which extends the Turaev-Viro TQFT to a 3-2-1 theory (i.e. allows manifolds with corners) . I know this construction is known, but I cannot find a source. Please help.
...
12
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2
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What do decategorification and "compactification on a circle" have to do with each other?
Some physicists have told me that if you think about an extended n-dimensional TQFT $F$, then the decategorification is given by $F'(X)=F(X\times S^1)$, which I believe they call "compactification on ...