Questions tagged [topological-quantum-field-theory]
Topological quantum field theory.
157
questions
4
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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
votes
0answers
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
votes
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
votes
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
votes
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
votes
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
votes
0answers
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
votes
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
votes
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/...
5
votes
0answers
99 views
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 ...
4
votes
0answers
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
votes
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
$$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$
...
5
votes
0answers
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
votes
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 ...
9
votes
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
votes
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
votes
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 ...
9
votes
0answers
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
votes
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
votes
0answers
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 $P\rightarrow M$ ...
7
votes
0answers
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
votes
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 (...
12
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0answers
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
votes
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
votes
0answers
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
votes
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
votes
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 ...
7
votes
0answers
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
votes
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
votes
0answers
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
votes
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
votes
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
votes
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
votes
0answers
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 ...
3
votes
0answers
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
votes
0answers
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
votes
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 $G$:
$$DW_G(M) := \lvert \operatorname{Hom}(\...
5
votes
2answers
230 views
What do “pivotal” and “spherical” mean for (unitary) fusion categories on the level of the $F$-symbols?
For me, a fusion category (over $\mathbb{C}$) is just a tensor $F$ (the associator, with $6$ simple-object labels and $4$ fusion space indices) and a tensor $d$ (the quantum dimensions, with one ...
3
votes
0answers
133 views
Example of a non-extendable TQFT?
All 3D TQFTs I know of are of Reshetikhin-Turaev type. These are fully extended and I wondered if there are known examples of TQFTs such that there is no once-extended TQFT extending it.
14
votes
0answers
353 views
Do TQFTs give a complete set of invariants of manifolds?
An $n$-dimensional TQFT is a representation of the category $n$Cob of $n$-dimensional cobordisms. TQFTs are important sources of invariants of manifolds, and such invariants are highly computable by ...
6
votes
0answers
160 views
Has the structure of the 2-dimensional pin$^{\pm}$ bordism categories been written down?
If $H\to\mathrm O$ is a tangential structure (e.g. orientation, spin), let $\mathsf{Bord}_2^H$ denote the category whose objects are 1-dimensional manifolds with $H$-structure and whose morphisms $M_1\...
5
votes
0answers
85 views
Does this $SU(2)$ Chern-Simons Superconformal Index Example have Modular Properties?
Without any regards to the physics or the geometry used to generate this result, let's examine the formula of Gukov (see p. 32):
$$ \mathcal{I}_{SU(2)}(q,t) = \frac{1}{2}\sum_{m \in \mathbb{Z}}\int \...
9
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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}^{\...
