# Questions tagged [extended-tqft]

The extended-tqft tag has no usage guidance.

42
questions

**3**

votes

**0**answers

82 views

### Shapes of cores of symmetric monoidal $(\infty,n)$-categories (with duals)

According to the cobordism hypothesis, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-categories with duals, then framed fully extended TQFTs with target $\mathcal{C}$ are an $\infty$-groupoid, ...

**10**

votes

**2**answers

206 views

### What are the topological phases of quantum Hall systems?

(Fractional) quantum Hall systems are $2+1$-dimensional models which are said to possess topological order. One (maybe even complete) set of invariants of topological phases in $2+1$ dimensions is ...

**16**

votes

**1**answer

688 views

### Fully extended TQFT and lattice models

I often read that fully extended TQFTs are supposed to classify topological phases of matter. So I would like to understand the formal nature of fully extended TQFTs on a more direct physical level (...

**15**

votes

**2**answers

351 views

### How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor
$$
d: \mathcal{C} \longrightarrow \mathcal{C}^{op}
$$
such that $d(x)$ is dual to $x$ for ...

**11**

votes

**1**answer

434 views

### Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...

**5**

votes

**0**answers

183 views

### Bosonic topological orders and unitary fully dualizable fully extended TQFT

I would like to ask if the following statement can be true:
bosonic topological orders in $n$-dimensional space-time 1-to-1 correspond to unitary fully dualizable fully extended TQFT in $n$-dimensions....

**6**

votes

**0**answers

253 views

### Freed-Hopkins-Lurie-Teleman topological boundary conditions v.s. Lagrangian subspace

This question concerns the comparison of topological boundary conditions of TQFTs on a manifold with some boundary.
For example, we can consider defining the TQFT on a $D^3$ ball with a topological ...

**5**

votes

**0**answers

213 views

### Analogue of Reshetikhin-Turaev construction for unoriented TQFTs

The Reshetikhin-Turaev construction takes a modular tensor category $\mathcal C$ and produces a 3-2-1 oriented TQFT $Z_{\mathcal C}$ such that $Z_{\mathcal C}(S^1) = \mathcal C$.
Is there an ...

**7**

votes

**1**answer

383 views

### Is Turaev-Viro-Barrett-Westbury stronger than homotopy?

I've heard that Reshetikhin-Turaev (RT) is stronger than homotopy, and it can distinguish certain homotopy-equivalent, but non-homeomorphic Lens spaces (I think $L(7,1)$ and $L(7,2)$). Now the Turaev-...

**7**

votes

**0**answers

178 views

### $S^3$ partition function from the 1-category of $S^1$ in Chern Simons theory

I am trying to learn about some categorical aspects of topological quantum field theories. For concreteness, I am considering Chern Simons theory with gauge group $G$ in three dimensions. As I ...

**4**

votes

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117 views

### Cobordism invariance and “reverse-extended TQFTs”

According to Atiyah, an $n$-dimensional TQFT is a functor from the bordism category of $(n-1)$-dimensional manifolds into $\mathrm{Hilb}$. That is, for every $(n-1)$-dimensional manifold it assigns a ...

**1**

vote

**0**answers

99 views

### Informal description of symmetric monoidal $(\infty,n)$-categories

I know the question of what is a symmetric monoidal category has shown up here.
I was wondering if there was a more informal way of describing a symmetric monoidal $(\infty, n)$-category as a "...

**9**

votes

**0**answers

264 views

### How do sutured TQFT fit into the larger TQFT picture?

In https://arxiv.org/abs/0807.2431, Honda--Kazez--Matic introduce a definition of (1+1 dimensional) sutured TQFT; see also e.g. Mathews http://arxiv.org/abs/1006.5433 and Fink https://arxiv.org/abs/...

**6**

votes

**0**answers

193 views

### Fully dualizability of dg-algebras

I am working in the context of fully extended TQFTs and, at the moment, I am trying to find fully dualizable objects in certain $(\infty, 2)-$categories. In particular I know that for the $(\infty, 2)-...

**7**

votes

**0**answers

262 views

### Is there a classification of 2d extended TQFTs with defects?

Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly ...

**7**

votes

**0**answers

423 views

### Is there a higher, “orientalish” version of geometric realisation?

Geometric realisation of simplicial sets can be roughly thought of like this:
In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...

**4**

votes

**0**answers

320 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 ...

**9**

votes

**1**answer

304 views

### Extended TFT with coefficients in spans in any $\infty$-topos

In the TFT classification article by Jacob Lurie (arXiv:0905.0465) the (∞,n)-category of correspondences (there: $Fam_n$) plays a key role, whose $k$-morphisms are $k$-fold spans of $\infty$-...

**15**

votes

**0**answers

700 views

### “extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...

**2**

votes

**0**answers

132 views

### target category of extended field theory

An A-S TFT is a functor from $\text{Bord}_{<n−1,n>}(\mathcal{F})$ to $\text{Vect}$ where $\mathcal{F}$ denotes a set of background fields, eg a spin structure. An extended theory is a functor ...

**11**

votes

**2**answers

856 views

### Relation between fully-extended TQFT and a “topless” TQFT

Consider 3-dimensional TQFTs for example. One version of them is the
3-2-1-0 fully extended TQFT. Do we have another version: 2-1-0 extended "TQFT"?
If yes, do we have an example of 2-1-0 extended ...

**8**

votes

**1**answer

283 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", ...

**4**

votes

**0**answers

225 views

### (∞,n)-category of triangulated cobordisms

What is an accepted definition of a (∞,n)-category of triangulated cobordisms?
Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...

**13**

votes

**2**answers

452 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 ...

**6**

votes

**0**answers

403 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 ...

**4**

votes

**0**answers

349 views

### Pull-back and push-forward of higher local systems

This is a follow up to the following two MO questions: q1,q2
What I'm interesting in understanding is the universal property (if any) of the morphism $Sum_n:Fam_n(\mathcal{C})\to \mathcal{C}$ ...

**11**

votes

**0**answers

403 views

### Higher holonomies for higher local systems

In Jacob Lurie's classification of tqfts, one finds a version of the cobordism hypothesis for $(X,\zeta)$-structure, where an $(X,\zeta)$ structure on a manifold $M$ is the datum of a continuous map $...

**12**

votes

**3**answers

2k 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 ...

**6**

votes

**1**answer

485 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

**1**answer

460 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 ...

**19**

votes

**2**answers

927 views

### Are fully extended TQFTs generalized cohomology theories?

Forgive the naiveness of this question. Whatever an $n$-vector space exactly is, one expects that the basic example of fully extended $n$-dimensional tqft is a symmetric monoidal functor $Cob_n\to n$-...

**8**

votes

**0**answers

592 views

### TQFTs with target category of higher type than the source

In the classical version of the Cobordism Hypothesis, such as, e.g., in Jacob Lurie's On the Classification of Topological Field Theories, one considers the $\infty$-category of symmetric monoidal ...

**13**

votes

**2**answers

825 views

### (3,2,1)-TQFTs and Verlinde algebras

Given a modular category $\mathcal{C}$ there are two natural ways to get a Frobenius algebra out of $\mathcal{C}$. One is to take the Verlinde algebra (or `fusion algebra') of $\mathcal{C}$. The other ...

**14**

votes

**2**answers

768 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 ...

**7**

votes

**0**answers

495 views

### Principal $G$-bundles as fully extended TQFTs, and $n$-representations

This is a follow up to this MO question: Fully dualizable objects in classical field theories
Assuming the notation there (which in turn come from Topological Quantum Field Theories from Compact Lie ...

**16**

votes

**3**answers

2k 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, ...

**16**

votes

**1**answer

859 views

### Fully dualizable objects in classical field theories

This is a follow up to this MO question: Free symmetric monoidal $(\infty,n)$-categories with duals
Freed-Hopkins-Lurie-Teleman define a classical field theory as a symmetric monoidal functor $I$ ...

**17**

votes

**1**answer

966 views

### Free symmetric monoidal $(\infty,n)$-categories with duals

The reading of (Hopkins-)Lurie's On the Classification of Topological Field Theories (arXiv:0905.0465) suggests that a stronger version of the cobordism hypothesis should hold; namely, that (under ...

**9**

votes

**1**answer

679 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 ...

**23**

votes

**1**answer

979 views

### 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 $n$-...

**12**

votes

**2**answers

1k views

### 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.
...

**11**

votes

**2**answers

538 views

### 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 ...