Questions tagged [topological-field-theory]

Use this tag for topological field theory (Tft) and topological string theory (tst) questions.

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4 votes
1 answer
86 views

Linking of a sphere with a Wilson line

In the papers such as Ref.[I] and Ref.[II], they have introduced the operator, $$ U_\alpha (M_{d-2}) = e^{\frac{i\alpha}{g^2}\int_{M_{d-2}}*F} . $$ They said that the Wilson loop: $$W_n(\gamma)=e^{in\...
4 votes
1 answer
67 views

Topological and non-topological defects?

The meaning of topological defect is only known intuitively to me. One explanation is it is some discontinuity in a system that cannot be removed. But I would like to know the precise mathematical ...
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3 votes
1 answer
60 views

$Z_n$ gauge theory from $U(1)$

In Appendix A of the paper, "Generalized Global Symmetries" by Gaiotto et al., they have considered an action, which for the purpose of the question, can be taken to be $$S=\frac{n}{2\pi}\...
2 votes
0 answers
39 views

What does it mean a theory is the gauging of a current?

What does it mean when people say that the Chern-Simons theory $$\mathcal{L}\sim\operatorname{Tr} \left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$$ is the "gauging of the the topological ...
1 vote
0 answers
108 views

What are $\mathcal{F}_g$ in string theory?

I was reading an article and came up on $\mathcal{F}_g$. Namely, it was in the following equation, $$\psi_{top} = \exp(\sum_g \mathcal{F}_g)$$ where I believe the $g$ denotes the genus of the topology ...
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1 vote
0 answers
68 views

Chern-Simons Realization of Dijkgraaf-Witten Theory

There is a realization of $Z_N$ Chern-Simons theory (Dijkgraaf-Witten theory) using an instance of $U(1) \times U(1)$ Chern-Simons theory. As explained on page 38 of https://arxiv.org/abs/2007.05915 , ...
  • 1,151
0 votes
0 answers
96 views

Resources for Topological Quantum Field Theory in Condensed Matter?

I'm interested in some resources (ideally books) for self-study of topological quantum field theory in condensed matter. I'd prefer resources that are more mathematically rigorous, so I don't mind ...
0 votes
0 answers
8 views

If two decays have the same topological effects are they the same?

What i mean is, if two decay processes have the same decay products (and initial state) and very similar topology can we treat them as the same? Or would you be able to detect which decay mode had ...
2 votes
2 answers
124 views

If quantum gravity is a TQFT, why isn't the Wheeler-De Witt equation satisfied automatically?

It is often said that QG is a topological QFT: given a bordism between $D$-manifolds $\Sigma_1$ and $\Sigma_2$, QG assigns a unitary between the Hilbert spaces associated with $\Sigma_1$ and $\Sigma_2$...
2 votes
1 answer
102 views

Can we write the effective field theory for the toric code model?

If not, then why not? If yes, then what is the effective field theory?
0 votes
1 answer
41 views

Filling factors and implementation for non-Abelian models

Currently reading through Pachos' Introduction to Topological Quantum Computation, and perusing other related articles and papers online. Have seen in many places that the 5/2 filling factor for ...
0 votes
0 answers
55 views

Calculate the energy gap using Green's function

Can I calculate the energy gap of the given Hamiltonian by Green's function? Is there any basic code in MATLAB to do that?
3 votes
0 answers
144 views

What is a topological quantum field theory?

I've seen reasonably tangible explanations of lots of "topological things" in physics like topological spin liquids, the surface code and the Kitaev chain. Although calculations in quantum ...
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2 votes
0 answers
48 views

(In)finite lattice in quantum statistical mechanics: validity of phase classifications and TQFT [closed]

I would like to understand the motivation for studying quantum statistical mechanics, such as spin models, on an infinite lattice, or in other word, in the operator algebraic framework. I learned that ...
0 votes
0 answers
37 views

What's string operators in a Chern-Simons theory?

In a $\mathbb{Z}_2$ lattice gauge theory or a toric-code model, we have e particles, m particles and their composition, and we have string operators which can create two anyons at the ends of a string,...
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0 votes
0 answers
33 views

Why $T$-transformation is a symmetry in $d=4$ pure Maxwell theory?

The Maxwell $\theta$-term $\mathcal{S}_{\theta}=\int d^4x\frac{i\theta}{32\pi^2}\epsilon_{\mu\nu\rho\lambda}F^{\mu\nu}F^{\rho\lambda}$ has the identification for theories have $\theta$ and $\theta + 2\...
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1 vote
2 answers
50 views

Why doesn't the second Chern form $C_2$ vanish in 4D Euclidean space?

I know the second Chern form cannot vanish identically, but I can't see what is wrong with a simple reasoning that gets that it vanishes. Take a standard $SU(N)$ Yang-Mills theory in a 4-dimensional ...
1 vote
0 answers
51 views

How to derive the canonical momentum of a single spin in the magnetic field in classical mechanics?

The classical spin can be denoted as $\vec{S}=S\, \vec{n}$, where $\vec{n}=(\sin{\theta}\cos{\phi},\sin{\theta}\sin{\phi},\cos{\theta})$. The magnet field $\vec{h}=(0,0,h)$. The Hamiltonian is $H=-hS\...
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2 votes
1 answer
77 views

Tension of a String

does anybody know a general relation between the tension of a string and it's energy density? I am at the moment learning about topological cosmic strings and calculated the energy density, now I do ...
  • 111
1 vote
1 answer
98 views

Quantum Double Model and Chern-Simons with finite gauge group

Is there a relationship between Kitaev's quantum double model for a finite group $ G $ and a Chern Simons theory with finite gauge group $ G $. They are apparently both related to quantum groups and ...
1 vote
0 answers
81 views

How does LQG solve the problem of the quantization of gravity? [closed]

How does LQG solve the problem of the quantization of gravity? Not only is it a discretization of space with groups of holonomy and diffemorphisms as curved backgrounds?,is that I see that they use ...
2 votes
0 answers
94 views

Topological invariant for the Toric code

My understanding is that the Toric code is a model with topologically non-trivial ground state. The ground state is degenerate on a Torus and is robust to local perturbations. The model has anyonic ...
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3 votes
0 answers
75 views

Defect network in conformal field theory and topological field theory?

Recently, I am trying to read the paper Generalized Global Symmetries. In the Preliminaries part, authors formulated ordinary symmetries by network of defects (PP6-7). It seems to be related to ...
5 votes
0 answers
100 views

Is there a physical interpretation of Poincaré duality?

Is there a known interpretation of Poincaré duality in terms of a physical equivalence between (maybe topological) sectors of different (probably susy) quantum field theories?
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1 vote
1 answer
219 views

Chern-simons term to total derivative

I'm trying to prove $$Tr[G_{\mu\nu} \tilde{G}^{\mu\nu}]=2\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}Tr[A_{\nu}G_{\rho\sigma}-\frac{2}{3}iA_{\nu}A_{\rho}A_{\sigma}]$$ expanding the L.H.S. I don't know ...
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15 votes
4 answers
506 views

What is the topological space in “topological materials/phases of matter”?

I’m embarrassed to admit that after sitting in on several “topological physics” seminars, I still don’t understand the basic ideas of the area. In particular, when physicists talk about the “topology” ...
  • 3,122
1 vote
1 answer
104 views

Distinguishable ways of splitting/fusing anyons

I have a difficulty in understanding the possibility that two simple anyons can fuse into one simple anyon in distinguishable ways: \begin{eqnarray} a\times b= 2c. \end{eqnarray} Let us put it in the ...
  • 653
0 votes
1 answer
41 views

Singularity and charts-problems met in Dirac quantization condition

Context: 45'23'' in a lecture given by Professor Wu, https://www.koushare.com/video/videodetail/4619. Consider a vector field $\vec{A}(\vec{x})$, with $\nabla\times\vec{A}(\vec{x})=\vec{B}(\vec{x})=g\...
  • 592
2 votes
0 answers
106 views

How to get the generators of $\mathfrak{so}(3)$ in the paper by Fidkowski and Kitaev?

In the paper by Fidkowski and Kitaev, they aim to study the interaction of 8 parallel Majorana wires, and they work on $\mathfrak{so(8)}$ Lie Algebra. They first start with just 4 parallel Majorana ...
2 votes
1 answer
289 views

How we can get the "Fermion Parity" and "Ground states" for Majorana fermions in Bernevig's talk PiTP 2015?

I have two questions regarding the talk, Topological Superconductors, Majorana...and Interactions, by Bernevig in PiTP 2015. How he gets the "Fermion Parity" for the ground states in the ...
1 vote
1 answer
140 views

Evaluating the $A \land A \land A$ in the Chern-Simons action

I am trying to evaluate $A \land A \land A$, but I am a bit confused on how exactly to do it and produce the usual notation used in physics. I am trying to use the definition of the wedge product of ...
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1 vote
2 answers
137 views

Chern-Simon level quantization and quantum Hall effect

It is well-known that integer and fractional quantum Hall effect can be effectively described by $U(1)$ abelian Chern-Simon theory. In both cases, quantization(fractionalization) of Hall resistance is ...
  • 391
0 votes
0 answers
51 views

Topological Field Theory for Physicists [duplicate]

I was wondering if anyone knows good resources for Topological Field Theories aimed at physicists. In particular, I am looking for references which are full of examples, starting with simple toy ...
0 votes
1 answer
105 views

What is the difference between topological theta term and Wess-Zumino-Witten term?

It seems that they both proportional to some thing like $\vec{n}\cdot \partial_{\tau}\vec{n}\, \times\,\partial_{s}\vec{n}$. References: Fradkin, Quantum field theory: an integrated approach
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1 vote
2 answers
206 views

Regarding a possible duality between (2+1)D gravity and Chern-Simons Theory

Is there a duality between (2+1)D gravity and Chern-Simons Theory? Or they merely have related features? If so, of which type and why?
0 votes
1 answer
39 views

Visualizing topological deformation and quantum mechanical interference

In section VI.1 of Zee's QFT, he says that for indistinguishable hard core particles in 2D, when comparing trajectories with different winding numbers: Since the classes cannot be deformed into each ...
  • 117
3 votes
0 answers
95 views

Excitations of the string-net Hamiltonian

A quite general 2D topological order can be constructed through the string-net theory. Here, if the input data is some braided fusion category $C$ (i.e. the $F$-symbols), the elementary excitations ...
  • 378
1 vote
1 answer
150 views

What is the role of the dilaton in Jackiw-Teitelboim 2D gravity?

I read that the Einstein Hilbert action is topological in 2 dimensions. (What does that mean?). To write down a non-trivial action one introduces the dilaton field in JT gravity. Does this field have ...
  • 285
5 votes
0 answers
104 views

Physics behind the Kobayashi-Hitchin correspondence

Let $X$ be a $d$-dimensional Kähler manifold with Kähler metric $\omega$. Now consider the following setups: Suppose $E \rightarrow X$ is hermitian vector bundle with hermitian connection $A$. In ...
2 votes
1 answer
62 views

Classification of topolgical phases when eigenstates belong to complex Grassmannian

I want to understand the paper which belongs to Ludwig (I put it below). I do not understand why exactly he got the new space $U(m+n)/U(m) \times U(n)$. My understanding from Grassmannian Manifold is ...
1 vote
0 answers
69 views

Physical origin of coisotropic branes

The paper "Remarks on A-branes, Mirror Symmetry, and the Fukaya category" develops the possibility of A-model open strings ending on a coisotropic submanifold equipped with a holomorphic ...
3 votes
0 answers
74 views

Mass deformations in D-brane systems

It is well known that the worldvolume theory of $N$ coincident D$p$-branes is given by the $U(N)$ Yang-Mills theory in $(p+1)$-dimensions. One important feature of this setup is the possibility of ...
3 votes
1 answer
147 views

Deriving the Topological Descent Equations

I am trying to show that in a cohomological TQFT, given a physical operator $\phi^{(0)}$, one can construct a chain of non-local physical operators. In doing so, I need to show that a certain set of ...
10 votes
2 answers
292 views

How to calculate a TQFT Gaussian path integral from Seiberg's "fun with free field theory"?

In his talk "Fun with Free Field Theory", Seiberg discusses a topological quantum field theory in $d+1$ dimensions with the action $$ S = \frac{n}{2\pi} \int \phi\, \mathrm{d} a \tag{1}$$ ...
3 votes
1 answer
93 views

A pedagogical semi-rigorous review of topological phases, topological order, and related subjects

I'm looking for a pedagogical review or book about topological phases, topological order, TQFTs, and related subjects. The ideal thing would be a mix of rigorous definitions and physical examples, ...
5 votes
1 answer
553 views

Why is Kitaev's toric code a $Z_2$ gauge theory?

I am reading Kitaev's 2003 paper. In the literature, it is often said that the model proposed in this paper is a $Z_2$ gauge theory. I don't quite see why it is the case. Where is the $Z_2$ gauge ...
  • 378
1 vote
1 answer
78 views

Topological defects in general and Chern-Simons in particular

I'm trying to gain intuition on some physical concepts that I cannot yet fully understand, and I think many of you can help me. Is it correct to think of of a topological defect as the addition ad hoc ...
  • 93
16 votes
3 answers
2k views

What is the physical importance of topological quantum field theory?

Apart from the fascinating mathematics of TQFTs, is there any reason that can convince a theoretical physicist to invest time and energy in it? What are/would be the implications of TQFTs? I mean is ...
1 vote
1 answer
171 views

Why do we call fracton by its name?

I am reading on fractons. In the literature, it is said that factons are fractionalized excitations. My understanding about fractons is that it is energetically costly to move fractons, and in this ...
  • 378
4 votes
0 answers
45 views

Partition functions of descendent SPTs of the Haldane chain

The Haldane chain can be viewed as a $1+1$ D SPT protected by an $SO(3)$ symmetry. If this SPT is put on a triangulated closed manifold $X$, its partition function can be written as $$ e^{i\pi\...

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