Questions tagged [topological-quantum-field-theory]

Topological quantum field theory.

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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
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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 ...
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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
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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
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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 ...
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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
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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 ...
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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
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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 ...
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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
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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
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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
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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 ...
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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
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1answer
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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-...
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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 ...
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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
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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-...
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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 (...
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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
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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 ...
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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
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
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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/...
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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 ...
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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
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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 $$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ ...
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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
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 ...
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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 ...
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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
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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 ...
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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 +\...
3
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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 $sl(n)$ ...
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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 $P\rightarrow M$ ...
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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
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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 (...
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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
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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
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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
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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
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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 ...
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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 $F_{-1}$ ...