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4
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1answer
89 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. ...
5
votes
0answers
80 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", ...
16
votes
3answers
253 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 ...
1
vote
1answer
95 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 ...
8
votes
1answer
128 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 ...
3
votes
0answers
96 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 ...
9
votes
1answer
200 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 ...
9
votes
0answers
222 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 ...
4
votes
0answers
126 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 ...
5
votes
0answers
170 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 ...
8
votes
1answer
303 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 ...
10
votes
1answer
219 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 ...
7
votes
2answers
295 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 ...
5
votes
0answers
255 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 ...
3
votes
0answers
183 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 ...
6
votes
1answer
335 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, ...
5
votes
1answer
233 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 ...
7
votes
3answers
853 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 ...
7
votes
4answers
518 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) + ...
10
votes
1answer
279 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 ...
4
votes
0answers
139 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 ...
6
votes
1answer
329 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 ...
3
votes
1answer
309 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 ...
5
votes
1answer
249 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 ...
7
votes
0answers
245 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 ...
0
votes
0answers
149 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 ...
8
votes
0answers
274 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, ...
0
votes
0answers
270 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 ...
2
votes
1answer
256 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 ...
9
votes
3answers
962 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 ...
6
votes
0answers
221 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 ...
6
votes
2answers
472 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 ...
12
votes
0answers
330 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 ...
12
votes
2answers
514 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 ...
9
votes
1answer
578 views

Are bundle gerbes bundles of algebras?

The category of line bundles (possibly with connection) on a smooth manifoldĀ M can be defined in two different ways: The first definition uses transition functions that satisfy a cocycle condition ...
2
votes
0answers
312 views

How to interpret sections over the SU(2) character variety as sections over the SL(2,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 ...
3
votes
0answers
262 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 ...
10
votes
3answers
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, ...
15
votes
2answers
1k 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" ...
12
votes
1answer
401 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?
27
votes
5answers
2k views

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 dont? Or: what is an example where using classical theories is hard, but using a ...
0
votes
0answers
221 views

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 ...
4
votes
1answer
443 views

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 ...
6
votes
1answer
777 views

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
votes
1answer
529 views

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 ...
7
votes
1answer
1k views

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 ...
24
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3answers
2k views

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 ...
7
votes
1answer
535 views

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 ...
3
votes
0answers
343 views

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 ...
20
votes
2answers
2k views

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 ...