Topological quantum field theory.

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### Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?

In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory ...

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**2**answers

185 views

### How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?

Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original ...

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**0**answers

197 views

### Reference Request for TQFTs

I had originally asked this question on Math StackExchange but have not obtained any answers, so I decided to post this here. (I have flagged the MSE post to be moved to MathOverflow, for the ...

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92 views

### translation invariance of the Laughlin wave function

This is a translation into math of the following question, posted on PhysicsOverflow.
Let $H:=L^2(\mathbb C)$.
For every $N$, let $\psi_N\in\Lambda^N H\cong (L^2(\mathbb C^N))^{S_N}$ be the function
...

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**2**answers

443 views

### High dimensional topological field theory

In the article Topological Field Theories in 2 dimension, Constantin Teleman has the following commentary " By constrast, in higher dimension, there seem to be no interesting theories: all examples ...

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255 views

### What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...

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543 views

### Classifying TQFTs with 1d vector spaces

To what extent have people classified $n$-dimensional TQFTs that assign a 1-dimensional vector space to every compact oriented $(n-1)$-manifold?
I have some vague reasons to suspect that the ...

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**1**answer

557 views

### Learning roadmap to TQFT from a mathematics perspective

I had asked a question on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posting it below.
...

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**1**answer

222 views

### How do we handle the symmetry condition in nCob and TQFTs?

A $(n+1)$-topological quantum field theory $\mathcal{T}$ is a rigid symmetric monoidal functor from the category $(n+1)$-Cob of $n$-manifolds and $(n+1)$-cobordisms to FdVect.
My question is about the ...

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vote

**1**answer

253 views

### Comparison of Different Types of QFT

As far as I can tell, there are a number of major types of quantum field theory. For example,
Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT).
Topological ...

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**1**answer

150 views

### Geometric Intuition of $P^+$ in Modular Tensor Categories

I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...

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193 views

### Does the notion of a “coherent state” exist in TQFTs? (ETQFTs?)

In the quantum harmonic oscillator, there exists a family of states called coherent states which form an overcomplete set of states. They are regarded as "the states most resembling classical states", ...

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443 views

### Interpreting the CS/WZW correspondence

It is understood that there is a correspondence between the 3d Chern-Simons topological quantum field theory (TQFT) and the 2d Wess-Zumino-Witten conformal quantum field theory (CQFT). A good summary ...

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**1**answer

130 views

### How to obtain $Z(\Sigma_f)=\text{Trace}\ \Sigma(f)$ in TQFT?

I am studying TQFT and have a question on one standard property.
A remark in Wikipedia (see the link above) says:
If for a closed manifold $M$ we view $Z(M)$ as a numerical invariant, then for a ...

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**1**answer

155 views

### Mapping class group vs automorphism group in cobordism category

Let $3Cob$ be the category whose objects are closed surfaces and whose morphisms are diffeomorphism classes of cobordisms.
By sending a diffeomorphism $\phi$ of a surface $X$ to its associated ...

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**0**answers

105 views

### Where is there a treatment of double field theory other than in local coordinates?

The n-lab seems to lack a treatment of double field theory. Where is there a treatment other than in local coordinates? Or at least one which identifies the coordinates as local coordinates for a ...

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**1**answer

238 views

### Is there a reasonable definition of TQFTs for n-cobordisms with connected inputs/outputs?

A question that's been on my mind for a while is whether any precise statement to the effect of "Heegaard Floer homology is a TQFT," for some reasonable definition of TQFT, can be made.
Of course, a ...

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245 views

### Provide a citation for the “spine lemma”?

I'm looking for a citable reference for the following (perhaps folkloric?) result on topological field theories.
(There are obviously generalizations to other dimensions; I'm happy with just the ...

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votes

**1**answer

268 views

### Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...

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207 views

### How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...

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**1**answer

349 views

### Anomalies in the definition of Turaev's TQFT

In his book Quantum invariants of knots and 3-manifolds page 124, Turaev defined a TQFT $\tau$ axiomatically.
For a cobordism $(M, \partial_{-}M, \partial_{+}M)$, a TQFT assignes a $k$-homomorhism ...

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**2**answers

343 views

### How unique are extensions of TQFTs to lower dimension?

Say I have an "ordinary" TQFT $F$ of dimension $n$, assigning groups or vector spaces to closed $(n-1)$-manifolds and linear maps to cobordisms. Consider the different ways $F$ can be obtained from a ...

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**4**answers

473 views

### Morita equivalence for *-algebras

This is a reference request. I'm looking for a definition of Morita equivalence of *-algebras, as described below. If anyone thinks that this is not the right way to define Morita equivalence of ...

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**0**answers

289 views

### 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 ...

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485 views

### 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 ...

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**1**answer

411 views

### 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, ...

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**1**answer

257 views

### 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 ...

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996 views

### 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 ...

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541 views

### 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) + ...

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**1**answer

294 views

### 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 ...

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**0**answers

141 views

### 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 ...

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votes

**1**answer

377 views

### 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 ...

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**1**answer

342 views

### 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 ...

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votes

**1**answer

272 views

### 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 ...

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**0**answers

273 views

### 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 ...

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**0**answers

158 views

### 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 ...

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votes

**1**answer

859 views

### 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, ...

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**0**answers

285 views

### 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 ...

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votes

**1**answer

275 views

### 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 ...

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**3**answers

1k views

### 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 ...

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252 views

### 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 ...

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511 views

### 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 ...

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348 views

### 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 ...

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581 views

### 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 ...

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**1**answer

635 views

### 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
...

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**1**answer

389 views

### How to interpret sections over the $SU(2)$ character variety as sections over the $SL(2,\mathbb{C})$ 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 $SU(2)$ and some ...

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292 views

### 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 ...

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1k views

### 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, ...

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**2**answers

2k views

### 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" ...

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**1**answer

444 views

### 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?