Questions tagged [topological-field-theory]

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

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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 , ...
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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 ...
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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 ...
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2 answers
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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 $\Sigma_2$...
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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?
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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 ...
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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?
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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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How is TQFT connected to 1+1D and 2+1D quantum gravity?

Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical ...
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What is the relationship between QFT for physicists and TQFT for mathematicians?

What is the relationship between QFT for physicists and TQFT for mathematicians and what is the physical motivation to study TQFTs?, What does it mean for a quantum field theory to be topological? ...
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(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 ...
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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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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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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 ...
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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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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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” ...
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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 ...
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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\...
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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 ...
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Why stacking two $p+ip$ superconductors or superconductors with Chern number 1 ($C=1$) is a quantum hall state?

My question is based on the lecture by Bernevig in PiTP 2015 on "Category Theory and the Kitaev 16 Fold Way"41:00. Why by stacking two superconductors with Chern number $C^{(1)}=1$ we have ...
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1 answer
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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 ...
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1 answer
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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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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 ...
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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 ...
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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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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?
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1 answer
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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 ...
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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 ...
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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 ...
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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 ...
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2 votes
1 answer
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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 ...
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1 vote
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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 ...
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3 votes
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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 ...
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3 votes
1 answer
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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 ...
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10 votes
2 answers
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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 $$ S = \frac{n}{2\pi} \int \phi\, \mathrm{d} a \tag{1}$$ ...
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3 votes
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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, ...
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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 ...
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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 ...
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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 ...
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1 vote
1 answer
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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 ...
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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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5 votes
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Criteria to Define a (Classical) Topological Field Lagrangian? + Conjecture

I have a question concerning topological field theories. I'd rather keep the discussion at the classical level, so as to concentrate on the feature of topological evolution, which is what interests me ...
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4 votes
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The argument for a mass gap for the $O(3)$ Heisenberg ferromagnet

One possible argument for asymptotic freedom in the 2D $O(3)$ ferromagnetic Heisenberg model is the existence of so-called instantons, discovered in the 1975 paper of Belavin and Polyakov. This is ...
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