Questions tagged [topological-quantum-field-theory]

Topological quantum field theory.

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1answer
91 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 ...
4
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142 views

Does Dijkgraaf-Witten theory have a time-reversal symmetry?

By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated ...
10
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1answer
265 views

Importance of the principal bundle in Chern-Simons theory

This is a very basic beginners question about Chern-Simons theory. The configurations that we sum over to get the partition function are given by a Lie-algebra valued 1-form $A$ on a topological 3-...
14
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1answer
442 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 (...
4
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0answers
131 views

Are Turaev-Viro invariants holonomic?

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...
10
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1answer
268 views

What are the applications of topological quantum field theory to continuous-time dynamical systems?

From wikipedia: In dynamics, all continuous time dynamical systems, with and without noise, are Witten-type TQFTs and the phenomenon of the spontaneous breakdown of the corresponding topological ...
5
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110 views

Can we define topological quantum field theories on Calabi-Yau manifolds?

Calabi Yau manifolds are Kähler manifolds with vanishing first Chern class. According to the conjecture of E. Calabi , for a Kähler manifold M , if $c_1 (M) = 0 $ , then M would admit a Ricci-flat ...
6
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1answer
163 views

Non-trivial factor splitting from vacuum in TQFTs. F-move not unity?

Consider a unitary modular TQFT, defined by the F and R moves. More specifically, a braided tensor category relevant for anyon models in 2D topologically ordered phases of matter. I am interested in ...
9
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1answer
295 views

Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?

My question: Does there exist a discrete gauge theory as TQFT detecting the figure-8 knot? By detecting, I mean that computing the path integral (partition function with insertions of the knot/...
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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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116 views

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 ...
17
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1answer
557 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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175 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 ...
5
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1answer
128 views

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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3answers
234 views

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 ...
15
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2answers
486 views

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 $...
6
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1answer
360 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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148 views

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 +\...
3
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1answer
210 views

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 $sl(n)$ ...
9
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209 views

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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176 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 ...
4
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1answer
118 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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195 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 ...
11
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1answer
392 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
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146 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....
11
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1answer
334 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 ...
9
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1answer
230 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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298 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 $F_{-1}$ ...
7
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1answer
163 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 ...
7
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139 views

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-...
5
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1answer
215 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 ...
6
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1answer
184 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 "...
4
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0answers
194 views

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 ...
8
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205 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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81 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 ...
4
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109 views

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 ...
8
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1answer
194 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}(\...
5
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2answers
230 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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0answers
133 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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160 views

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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0answers
85 views

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 \...
9
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1answer
274 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 ...
8
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1answer
165 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'...
6
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0answers
266 views

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 ...
10
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1answer
328 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 ...
10
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2answers
290 views

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 ...
6
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0answers
237 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 ...
12
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2answers
457 views

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 ...
4
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0answers
187 views

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}^{\...