# Questions tagged [tqft]

Topological quantum field theory.

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### Equivariant L-infinity structure associated to a DGBV algebra

Let $V$ be a differential graded Batalin-Vilkovisky (DGBV) algebra over say $\mathbb{C}$. This means that $V$ is a commutative differential graded algebra (CDGA), with differential $\partial$ of ...

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### Chern-Simons and framing dependence$.$

I posted this question to physics.SE last week (cf. here), but it got not attention. I hope it is not too trivial to post it here.
According to ref.1, the correlation functions of a Chern-Simons ...

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448 views

### How is Chern-Simons theory related to Floer homology?

Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional
$$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$
...

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167 views

### Projecting GxG onto subspace with tied irreducible representations

Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.
I could alternatively ...

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### How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?

In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$").
Physically this modular fusion category describes the ...

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### Generalization of Drinfeld double to comodule algebras

Let $ \mathcal C $ be a monoidal category. Then $ \mathcal C $ is both a left and right module category over itself. Moreover, the Drinfeld centre of $ \mathcal C $ can be defined as the category of ...

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### What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?

Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$
superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $...

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284 views

### Physical consequences of cobordism hypothesis?

Let $C$ be a symmetric monoidal $n$-category. An extended framed $C$-valued TQFT is a symmetric monoidal functor from the framed bordism category $\mathrm{Cob}^{fr}_n(n)$ to $C$.
The cobordism ...

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### What is the “classical limit” of Khovanov homology?

Let me first explain what I mean by the "classical limit".
For quantum group invariants of links and webs (such as colored Jones polynomials), the "classical limit" means the limit $k\rightarrow +\...

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### Framing dependence of HOMFLY polynomial

I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...

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### Twisted Chern-Simons, and Twisted Wess-Zumino Term

I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten.
Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...

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158 views

### What is the mathematical structure of 2d TQFT from the 2d foam category (instead of 2d cobordism category)?

It is well-known that the category of 2d TQFTs is equivalent to the category of commutative Frobenius algebras.
What about functors from the 2d foam category (instead of 2d cobordism category) to ...

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**1**answer

114 views

### 2-morphisms for Bord(n)

I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, and have a (I guess) pretty basic question for my understanding of the (...

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155 views

### Define the 3d Chern-Simons TQFT on a discrete simplicial complex

Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction ...

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372 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 ...

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### 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....

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276 views

### Compactification of 6d (2, 0) SCFT on 4-manifolds

This question is about the 6d (2, 0) superconformal field theory (also called 'theory X' by some people). This SCFT, which can be considered as a relative quantum field theory (see here for a ...

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215 views

### Is the instanton homology for webs and foams a categorified Chern-Simons?

In their paper, Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my ...

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219 views

### The Fock Space vs the Hilbert space in the context of Quantum Field Theory

Mathematically the definitions are as follows : if $H_n$ is a $n-$dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ ...

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158 views

### commutative “weakly” Frobenius algebras and 2d TQFT

Fix a field $k$. A classic result written up carefully by Abrams in the article "Two-Dimensional Topological Quantum Field Theories and Frobenius Algebras"
says that there is a bijective ...

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### Why are Levin-Wen/Turaev-Viro models said to be non-chiral?

I'd like to bring together the following two notions of "non-chiral":
On the abstract algebraic side, a modular fusion category describing the anyon content of some physical system is said to be non-...

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201 views

### Knot Factorization Homology inputs

Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf
If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...

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168 views

### Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal?

By Morita equivalent I mean that there is an invertible bi-module between the two fusion categories. [Feel free to replace the Drinfeld centers being "equal" by an appropriate categorial notion of "...

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### Topological field theories and their path integrals

Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten ...

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198 views

### How to define the direct sum of TQFTs $(\infty,1)$-categorically?

Let $\mathit{Bord}_d$ be the symmetric monoidal category of $(d-1)$-manifolds and bordisms between them.
Let $\mathcal{C}$ be the symmetric monoidal category of $k$-modules. Then, for two symmetric ...

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73 views

### Open Gromov-Witten invariants via lozalization with $\mathbb{C}^{*}$ (not $S^1$) action

Amplitudes of open A-model on a Calabi-Yau 3-fold $X$ with branes are given by the open Gromov-Witten invariants of $X$. It is known how to compute them if there is a toric action on a manifold, which ...

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### Can non-chiral 3D TQFTs be extended to non-orientable manifolds whereas chiral ones cannot?

As far as I know, when talking about TQFT, one usually means TQFTs on oriented manifolds with boundary (cobordisms)
It appears to me that the Turaev-Viro-Barrett-Westbury state-sum construction can ...

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181 views

### Is Yetter's invariant multiplicative under connected sum?

Classical formulation
Consider the (untwisted) Dijkgraaf-Witten invariant, defined for an oriented, connected, closed manifold $M$ and a finite group $G$:
$$DW_G(M) := \lvert \operatorname{Hom}(\...

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199 views

### What do “pivotal” and “spherical” mean for (unitary) fusion categories on the level of the $F$-symbols?

For me, a fusion category (over $\mathbb{C}$) is just a tensor $F$ (the associator, with $6$ simple-object labels and $4$ fusion space indices) and a tensor $d$ (the quantum dimensions, with one ...

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129 views

### Example of a non-extendable TQFT?

All 3D TQFTs I know of are of Reshetikhin-Turaev type. These are fully extended and I wondered if there are known examples of TQFTs such that there is no once-extended TQFT extending it.

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### Do TQFTs give a complete set of invariants of manifolds?

An $n$-dimensional TQFT is a representation of the category $n$Cob of $n$-dimensional cobordisms. TQFTs are important sources of invariants of manifolds, and such invariants are highly computable by ...

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### Has the structure of the 2-dimensional pin$^{\pm}$ bordism categories been written down?

If $H\to\mathrm O$ is a tangential structure (e.g. orientation, spin), let $\mathsf{Bord}_2^H$ denote the category whose objects are 1-dimensional manifolds with $H$-structure and whose morphisms $M_1\...

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### Does this $SU(2)$ Chern-Simons Superconformal Index Example have Modular Properties?

Without any regards to the physics or the geometry used to generate this result, let's examine the formula of Gukov (see p. 32):
$$ \mathcal{I}_{SU(2)}(q,t) = \frac{1}{2}\sum_{m \in \mathbb{Z}}\int \...

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255 views

### String cobracket from TFT

Let $M$ be a closed oriented manifold. Chas and Sullivan (https://arxiv.org/abs/math/0212358) introduced a Lie bialgebra structure on $H_\bullet^{S^1}(LM, M)$, $S^1$-equivariant homology of the loop ...

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161 views

### Can I recover a crossed module by its homomorphisms?

This is a follow up to this question.
Imagine there is a finitely presented crossed module $\mathcal{G} = (G,H, -\triangleright-\colon G \to \operatorname{Aut}(H), \delta\colon H \to G)$ which I don'...

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### Khovanov homology and Crane-Yetter TQFT

Crane-Yetter(-Kauffman) have constructed 4-dimensional TQFT in such a way that Reshetikhin-Turaev theory lives on the boundary $\partial M$ of a 4-manifold $M$. Therefore, Crane-Yetter TQFT can be ...

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302 views

### Which manifolds are sensitive to the cocycle in the Dijkgraaf-Witten model?

Often, TQFTs are defined in families, parametrised by some algebraic data. For example, the Turaev-Viro-Barrett-Westbury TQFTs are parametrised by spherical fusion categories, the Crane-Yetter TQFTs ...

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### Is there a 1-dimensional analogue of the correspondence between the Levin-Wen and Turaev-Viro models?

Given a spherical fusion category $\mathcal C$, the Levin-Wen model constructs a lattice field theory: to each oriented surface with a triangulation, it assigns a state space $\mathcal H$ and a ...

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### 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 ...

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### Is a unitary Hamiltonian TQFT the same as a unitary axiomatic TQFT?

Introduction
Axiomatic TQFTs
An axiomatic $n$-dimensional TQFT is a symmetric monoidal functor $\mathcal{Z}\colon \operatorname{Bord}_n \to \operatorname{Hilb}$ from $n$-dimensional oriented ...

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### How to find a geometric construction of a TQFT

Assume that the surface $\sum$ is equipped with the structure of a smooth algebraic curve over $\mathbb{C}$. We denote by $H^0(M_\sum,\mathcal{L}^{\otimes k})$ the space of sections of $\mathcal{L}^{\...

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### 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 ...

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295 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-...

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### $S^3$ 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 ...

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### Differential form TQFT for Walker-Wang model?

In terms of the TQFTs in continuous differential form gauge fields, what would the Walker-Wang lattice model describe? Obviously, there is a $BF$ theory part:
$$\frac{N}{2 \pi}\int B dA$$
if it ...

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**1**answer

383 views

### Phase transitions between Category Theories

question: What are the mathematical theories suitable to describe the "continuous Phase transitions between Category Theories"? The phase transitions mean that in terms of the quantum statistical ...

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**1**answer

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### 4-dimensional TQFT with/without requiring spin structure

My questions here are focused on $D$-dimensional topological quantum field theories (TQFTs) which are unitary and which have finite dimensional Hilbert space on a closed spatial manifold $M^{d-1}$. ...

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### Why is a Topological Field Theory equivalent to a Frobenius algebra?

How can a physicist understand a 2-dimensional topological field theory as a Frobenius algebra? Are there some explicit examples in order to understand this relation?
The definition (e.g. on ...

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203 views

### Higher category theory (invertible morphism, Topological categories, monoidal n-category) [closed]

I'm Looking at a paper using higher category theory. Would you please answer these questions ?
1) A weak n-category (or $\infty$-category) has objects and k-morphisms. What is an invertible morphism ?...

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### 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 ...