Questions tagged [topological-field-theory]
Use this tag for topological field theory (Tft) and topological string theory (tst) questions.
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What are non-propagating fields?
I have read at different places that in 3 spacetime dimensions, there are NO propagating gravitational degrees of freedom. This seems to imply that we have only "non-propagating" degrees of ...
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Fundamental group of configuration space of gauge theories
If I consider the space of all the gauge fields $A_{\mu}$ (call this $A$) and a proper gauge group $\Omega_*$, I know that the fundamental group $\pi_1(A)=0$ and the for the gauge group, for example $...
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How do equations of motion in BF theory imply triviality of powers of observables?
Following the lectures of Nathan Seiberg at PiTP in 2015 https://www.youtube.com/watch?v=pqgNrVTQ4yM&t=666s, consider $U(1)$ BF theory in 2D
$$S(B,A)=\frac{n}{2\pi}\int_\Sigma B\text{d}A,$$
and ...
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Simplicity constraints from $SO(4)$ Plebanski action
The $SO(4)$ Plebanski action yields a first order formulation of Euclidean General Relativity as a constrained (topological) BF-theory. It depends on a $so(4)$ connection 1-form $\omega^{IJ} = \omega_{...
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Can we expect some nice "conservation laws" to be hold at each vertex of any Feynman diagram?
I'm reading two textbooks: A. Zee's QFT book and Bruce Bartlett's TQFT book.
In Zee's book, chapter 1 & 2 introduces Feynman diagram smoothly.
Although notations are slightly different, I'll ...
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Can any useful physical theories other than TQFTs be formulated on a smooth manifold without a metric structure?
The vast majority of physical theories are formulated on a spacetime that is mathematically represented by a pseudo-Riemannian manifold, i.e. a smooth manifold with a metric tensor structure. The ...
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Duality and corrections to second-order gravity without and with torsion terms
Recently, there appeared a paper by Giacomo Pollari, A Nieh-Yan-like topological invariant in General Relativity, where the action for gravity looks like:
$$S_g=S_{EHP}+S_{HO}+S_{PO}+S_{GB}+S_{NY}+S_{...
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What is the direct sum of 1d domain walls in Toric code model?
I have read this paper:"An invitation to topological orders and category theory" (https://arxiv.org/abs/2205.05565v2). In page 93 and page 109, they show the result of fusion of simple 1d ...
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Topological classification of (classical, Abelian) vortices on a lattice
Consider the XY model on the square lattice. A field configuration $\theta$ is specified by an element of the Abelian group $\mathbb{R}/2\pi \simeq U(1)$ at each vertex of the lattice. The gradient of ...
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Topological Insulators with different spin band
To obtain a topological band insulator, we usually start with two bands with either spin up or down. If these bands now get 'inverted', they will cross. When there is coupling of these two bands such ...
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Gauging the one form symmetries in the GKSW paper
Following is few of the many questions I have regarding the paper "Generalized Global Symmetry" by Gaitto et al. (https://arxiv.org/abs/1412.5148).
In Sec. 4 of the paper they considered the ...
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Topology active subgroup for the QCD vacuum
I've been reading about the nontrivial topological structure of the QCD vacuum and, when studying the different equivalence classes created by the pure gauge fields, all papers say that it is possible ...
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Anyon and state spaces
I start learning about anyons, but I'm confused by a few Hilbert spaces.
First of all, it is said that anyons are "excitations" with anyonic statistics. By that I would imagine they are ...
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Why topology indicates edge state?
I have learned some models with topology like Kitaev chain and SSH. All of them possess edge states if the bulk is topologically non-trivial. The reading materials did the calculation and verified ...
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Does Fibonacci anyon come from a representation category of Hopf algebra?
I have heard that the UMTC(unitary modular tensor category) of Fibonacci anyon comes from a quantum group, but the representation category of Hopf algebra is equipped with a forgetful functor to $\...
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Is this a suitable topological quantum field theory?
I thought the last days on how to construct topological quantum field theories (TQFTs) and now I have the following idea:
Suppose I have a 4-dimensional manifold $M$ with a 2-form bosonic field $B$ (...
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Closed String State
In this paper, it is said (pg. 77) that closed string states are singlets of the $GL(N,\mathbb{C})$ gauge group. How can I understand this statement?
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Dijkgraaf-Witten for a 2-Group
As a natural extension of my previous question Higher Dijkgraaf-Witten Theory on DW Theory for a 1-form symmetry, we can extend now to 2-groups.
How can we generalize the notion of gauging to a 2-...
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Hilbert Space in Categorical Gauged TQFT
I am trying to understand how gauge theory interacts with the categorical formulation of TQFT. I will formulate my doubts in two different questions.
I have understood gauging a TQFT in different ...
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Higher Dijkgraaf-Witten Theory
I am trying to understand higher-form symmetries in TQFT. In particular the higher-form version of Dijkgraaf-Witten Theory.
It is known that for a 0-form symmetry we can specify the principal G-bundle ...
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Role of fusion/splitting spaces in TQFT
In his book on topological quantum field theories Steven Simon writes that 2+1D TQFTs are objects that assign topologically invariant numbers to labeled links embedded in arbitrary 3-manifolds. They ...
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Energy of a flux tube
I was reading the paper Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of
Gauge Theories by Gukov and Kapustin. I don't understand many of the things in there. But, I think ...
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References for undergraduate Classical Mechanics [duplicate]
If you know of any russian text books that are available in English On Mechanics, the undergrad level please let me know. I have Kleppner and Kolenkow but I wish to read from a Russian Author
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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\...
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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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$Z_n$ 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
$$S=\frac{n}{2\pi}\...
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What does it mean a theory is the gauging of a current?
What does it mean when people say that the Chern-Simons theory $$\mathcal{L}\sim\operatorname{Tr} \left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$$
is the "gauging of the the topological ...
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What are $\mathcal{F}_g$ in string theory?
I was reading an article and came up on $\mathcal{F}_g$. Namely, it was in the following equation, $$\psi_{top} = \exp(\sum_g \mathcal{F}_g)$$ where I believe the $g$ denotes the genus of the topology ...
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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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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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(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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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 ...
