Questions tagged [topological-quantum-field-theory]
Topological quantum field theory.
179
questions
5
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1answer
131 views
How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?
Chen and Yang have a more general version of the volume conjecture that they state for all hyperbolic $3$-manifolds (Conjecture 1.1 of [2]) including those with boundary. To do this, they have to ...
7
votes
0answers
139 views
Physical and mathematical significance of the NS-2 brane
This question is about topological string theory and it was also posted in Physics Stack Exchange.
The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph ...
1
vote
0answers
70 views
Understanding the transgression map in AKSZ-BV construction
The transgression map in AKSZ construction is defined as follows:
Let $M$ be a closed oriented $n$-manifold and $\mathcal M = T[1]{M}$.
$\Omega^{0}(\mathcal M) = C^{\infty}(\mathcal M) \cong \Omega^\...
8
votes
3answers
464 views
Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?
There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
9
votes
0answers
193 views
What is the current state of research in Chern-Simons theory?
I'm a PhD student in mathematical physics looking for some orientation. As asked in the title, I would like to know the current state of research in Chern-Simons theory. More specifically, what are ...
1
vote
1answer
96 views
A computation in a commutative Frobenius algebra
This is already posted here https://math.stackexchange.com/questions/3695584/a-computation-in-a-commutative-frobenius-algebra but I didn't get any answers.
Given a commutative Frobenius algebra (in ...
6
votes
2answers
258 views
1d TQFT minus connection =?
Correct me if I am wrong but I believe at least conceptually (maybe even rigorously) data of a 1-dimensional TQFT and of a vector bundle with connection are equivalent.
Going into more detail (and ...
3
votes
0answers
52 views
Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?
Let us consider abelian and non-abelian 3d quantum Chern-Simons theory path integrals:
abelian Chern-Simons theory on non-spin manifolds ---
$$
\int [DA]\exp(i \frac{k}{2\pi} \int_X (A \wedge dA ))
...
7
votes
2answers
232 views
Path integral derivation of extended TQFT
I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of ...
6
votes
0answers
112 views
Why does the inverse Alexander polynomial appear in the MMR conjecture?
In an attempt to better understand why the inverse Alexander polynomial appears in the MMR conjecture, I was reading the paper [1] of Bar-Natan and Garoufalidis giving their proof of the conjecture ...
8
votes
2answers
373 views
B-model and Hochschild cohomology
In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\...
8
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1answer
161 views
Generators and relations for the 2-dimensional unoriented cobordism category
It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of ...
8
votes
1answer
222 views
DW, state sum models, and fully extended TQFTs
I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-...
4
votes
1answer
137 views
Mapping class group of torus, why is $(ST)^3=S^2$?
In the context of topological quantum field theories, I am interested in the mapping class group of a torus. Here I can consider the torus as a square with identified edges and also decorated with ...
3
votes
0answers
108 views
Why is the RT invariant from $\mathcal Z(\mathcal C)$ the (norm) square of the one from $\mathcal C$?
The relationship between Turaev-Viro/state-sum invariants and Reshetikhin-Turaev/surgery invariants is roughly that
$$\tau_{TV, \mathcal C}(M) = |\tau_{RT, \mathcal C}(M)|^2.$$
Here $\mathcal C$ is a ...
5
votes
1answer
176 views
Classification of $\operatorname{Rep}D(H)$
Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
4
votes
0answers
165 views
References for topological quantum field theory
I have been studying TQFTs and I am mainly reading
Lurie's proof of the cobordism hypothesis: https://www.math.ias.edu/~lurie/papers/cobordism.pdf
Walker's notes: https://canyon23.net/math/tc.pdf
...
4
votes
0answers
88 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, ...
4
votes
1answer
151 views
Group representation with algebra structure
I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.
Let $G$ be a finite group. Its finite-...
8
votes
1answer
852 views
What do physicists mean by a topological quantum gravity theory
This is a jargon-like question.
The fact that this is posted here rather in a physics forum indicates two things
I know too little physics.
An explanation with more mathematics flavors will be ...
6
votes
2answers
422 views
1-dimensional pure gauge theory
I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie,
and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard
even in the first section ($n = 1$), which was "trivial but ...
2
votes
0answers
226 views
Tracking down an elusive book
A few weeks ago I had a very engaging talk with a faculty member, where he told me lots of interesting things about quantum algebras, know theory and Reshetikhin-Turaev invariants (this field is not ...
10
votes
2answers
209 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
163 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 ...
11
votes
1answer
356 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-...
16
votes
1answer
709 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
147 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
375 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 ...
6
votes
0answers
140 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
177 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
351 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
127 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
131 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 ...
18
votes
1answer
910 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)$$
...
5
votes
0answers
181 views
Projecting GxG onto subspace with tied irreducible representations
Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.
I could alternatively ...
5
votes
1answer
152 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 ...
10
votes
3answers
282 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 ...
17
votes
2answers
622 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
563 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
170 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
225 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
247 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
199 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 ...
5
votes
1answer
121 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
votes
0answers
223 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
437 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
190 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....
12
votes
1answer
374 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
252 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 ...
8
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0answers
485 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 $F_{-1}$ ...
