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Questions tagged [chern-simons-theory]

Chern-Simons theory is an example of a topological quantum field theory. Its describes the field dynamics through the so-called Chern-Simons-form, hence its name.

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What is the trace in the Chern-Simons action

I have been looking at the Chern-Simons Lagrangian in $(2+1)$-dimensional spacetime $M$ in terms of a gauge field $A$: $$ S[A] = \frac{k}{4 \pi}\int_M \text{Tr}(A \wedge \text{d}A+ \frac{2}{3}A \...
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Gauge and global symmetries in Chern-Simons/WZW correspondence

I am trying to understand how bulk gauge symmetry in 3d Chern-Simons theory becomes a global symmetry in the boundary 2d WZW theory. In particular, I am trying to understand the papers by Elitzur et ...
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Induced Chern Simons terms

In a three dimensional U(1) gauge theory with massive fermions, one can integrate out the fermion fields with the result that the effective action for the gauge field now has an additional Chern ...
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Chern-Simons equation of motion

How do I get the equation of motion of the Chern-Simons Lagrangian below? Is there the product rule at work? Do I have to sum over the indices?
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Gapping out edge modes by backscattering

I was reading this paper by Michael Levin about protected edge states without symmetry. In the introduction, he makes the argument that backscattering terms or other perturbations gap out left and ...
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Quantization of Chiral Boson

I am trying to understand the edge modes of fractional quantum Hall(FQH) effect from ChernSmions theory picture. Chern-Simons action with a boundary along $y$ produces the following action $ \...
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Statistics of a quasiparticle

I was trying to understand the Chern-Simons theory description of the fractional quantum Hall effect. I was trying to follow this article. After integrating out Lagrangian the mutual statistics of two ...
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Integer quantum Hall conductance and time-reversal symmetry

If we have a (2+1)-dimensional electronic gapped system with a unique ground state and it has a nonzero integer quantum Hall conductance, then the system (or its ground state) must break the time-...
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Path integral measure in Chern-Simons/WZW correspondence

The relationship between 3d Chern-Simons theory on the product of the disk and the real line ($D\times \mathbb{R}$) and the chiral WZW model on $S^1\times \mathbb{R}$ was shown in Elitzur et al Nucl....
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Aharony-Bernman-Jafferis-Maldacena (ABJM) and k=1 Chern Simons matter

I have read recently that the partition function / half-BPS wilson vev (w/ NG probe) of a Chern-Simons matter theory with N=6 U(N)k x U(N)-k super-conformal symmetry (ABJM) on S3 is proportional to ...
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Maxwell-Chern-Simons equation: Translating from differential form to component form

I am trying to solve the scalar-coupled Maxwell-CS equations (which is one of the equation of motions in $N=2$ supergravity coupled to 3 vector multiplets), which is written in this form in the ...
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Is a constant electric field CP violating?

Consider, for instance, a fundamental massless three-form field $C_{\alpha\beta\gamma}$ in the Coulomb phase: $$ \mathcal L = E_{\mu\alpha\beta\gamma}E^{\mu\alpha\beta\gamma} + C_{\alpha\beta\gamma}J^...
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Effective theory of hierarchial fractional quantum hall state

In describing the effective field theory picture of the hierarchical fractional quantum Hall states in Tong's lecture notes, page 165 he gives the expression for filling fraction, quasi-particle ...
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Lifting 3d Chern-Simons theory to 4d

For simplicity, let us only consider abelian Chern-Simons theory. The usual way of lifting 3d Chern-Simons theory to 4d is achieved through the Stokes' theorem. Say, if the original Chern-Simons ...
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About Witten's path integral formulation of Jones polynomial

In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the ...
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Braiding matrix from CFT first principles

Various CFT models are known to produce representations of braid groups. A famous example is the $SU(2)$ WZW model at level $k$, for which the braiding matrix for the case of two fundamental irreps ...
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Number of Physical States of a $U(1)$ Chern-SImons Theory on a Riemann Surface of Genus $g$

In A Duality Web in 2+1 Dimensions and Condensed Matter Physics, the authors claimed in Appendix B that for a $U(1)_{k}$ Chern-Simons theory defined on a Riemann surface $\Sigma$ of genus $g$, the ...
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Chern-Simons Gravity term in 3D and equations of motion

In the book "Quantum Gravity in 2+1 dimensions" by Steven Carlip he writes down a possible modification to the Einstein-Hilbert Action in 3d (eq. 1.16 to eq. 1.18) \begin{equation} I_{GCS}=-\frac{1}{...
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Chern-Simons and framing dependence$.$

According to ref.1, the correlation functions of a Chern-Simons theory are topological invariants, up to the so-called framing, that is, the trivialisation of $TM\oplus TM$. The origin of this framing ...
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How does extending a Chern-Simons theory to the bulk fix potential singularities?

According to ref.1 (§A.3), the naive definition of Chern-Simons $$ S[A]=k\int_M \mathrm{CS}[A]\tag{A.17} $$ is ill-defined, because $A$ may have "Dirac string singularities". The solution is to extend ...
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A question about Witten's paper on QFT and Jones polynomial

So I have been reading the famous paper on quantum field theory and Jones polynomial and have the following questions: On P.31 (381), it was said that the eigenvalues of $B$ are $$λ_i = ±\exp(i\...
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How exactly does a spin TQFT depend on the spin structure?

Take a spin Chern-Simons TQFT, such as $U(N)$ or $SO(N)$ with odd level. Such system depends on the spin structure of the underlying manifold. But how exactly does the theory depend on the spin ...
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Intepreting Fermions as Differential Forms?

In this paper on path-integral quantization of Chern-Simons theory, on page 434 (equation 4.17), the authors used fermions to interpret wedge product and contractions of differential forms. Let $M$ ...
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Deriving Chern-Simons term from path integral representation of the first quantized non interacting many body Hamiltonian

This is an exercise from condensed matter filed theory book of altland and simons. Exercise Subject the first quantized many particle hamiltonian $H=\sum_{i=1}^{N}\frac{p^{i}{^{2}}}{2m}+V(x^i)$ to ...
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Hamiltonian Structure of Chern Simons Electrodynamics

I am reading the review paper "Aspects of Chern-Simons Theory" by Gerald Dunne https://arxiv.org/abs/hep-th/9902115 Starting from p. 17, Dunne works on the Hamiltonian structure of the CS ...
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Is $U(2)_{2, 1}$ Chern Simons Theory Completely Trivial?

I am using the method outlined in appendix C4 of a paper by Seiberg and Witten [1] to calculate the statistics of lines in $U(2)_{2, 1}$. However, this method shows that all lines are trivial. ...
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Second Chern class in 2D Haldane model from Atiyah-Singer Index Theorem?

I was reading through a physics-centered exposition of the Atiyah-Singer index theorem and I wondered what it would mean to talk about Haldane's model for the case of a manifold with a boundary. It is ...
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Lattice QFT of the Jones Polynomial

Start with a gauge theory with Chern-Simons action $$ S[A] = \frac{k}{4 \pi} \text{Tr} \intop_{M} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right) $$ and a Wilson loop observable in the ...
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About the Level of Non-Abelian Chern Simon theory

The Lagrangian of the non-Abelian Chern Simons theory is the following $$\mathcal{L}=\frac{k}{4\pi}\int \text{tr}(AdA+\frac{2}{3}AAA)$$ What is the definition of tr here? Namely, which representation ...
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Is it possible to couple an odd number of Dirac fermions, at finite density, to a massless gauge field in 2+1d?

In a beautiful paper by A. N. Redlich (PRL $\bf{52}$, 18 (1984)) on the parity anomaly, the author indicates that an odd number of Dirac fermions can never be coupled to a massless gauge field in 2+1d ...
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Understanding the fractional quantum Hall effect in Chern-Simons formalism described in Wen's book

So I study fractional quantum hall effect with Chern-Simons formalism by using Wen's book, this is an excellent book, but it assumes that you know field theory very well thus it has gaps between steps....
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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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Chern-Simons action for non-abelian brane worldvolume and Tsetlyn's symmetric trace prescription

I am trying to reproduce the results of the (famous) Myer's paper "Dielectric Branes" https://arxiv.org/abs/hep-th/9910053. I am struggling a bit to obtain the numerical factors in the equations. When ...
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Faddeev-Popov Determinant of Chern-Simons Theory

I am asking this question in order to figure out the expression of the Faddeev-Popov determinant given by Edward Witten is his paper "Quantum Field Theory and Jones Polynomial". Starting from the ...
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3d TQFT for Fibonacci (Yang -Lee) anyons

What is the 3d TQFT whose Wilson line produces Fibonacci (Yang -Lee) anyons? I heard that 3d $SO(3)_3$ Chern-Simons theory produces the correct physics for Fibonacci anyon ($e$). How to show it? If ...
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2D Chern Simons action by integrating out fermions

In Qi, Hughes, and Zhang's paper (https://arxiv.org/abs/0802.3537), they show how the Chern number appears as a coefficient of response function. Given the Hamiltonian (49) of a (2+1) or (4+1)D ...
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The Hilbert space of Chern-Simons on a torus, part one$.$

There is a key result in 2+1 dimensional Chern-Simons theory, which was first discussed in ref.1.: the Hilbert space of the theory, when quantised on $T^2\times\mathbb R$, is isomorphic to $$ \frac{\...
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Gauge anomaly from conformal dimension?

According to ref.1, the Chern-Simons theory $\mathrm{SU}(N)_k$ has a $\mathbb Z_N$ one-form symmetry with anomaly $$ \eta=\exp\left[-2\pi i \frac{k}{N}\right]\tag{4.12} $$ which, apparently, can be ...
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Does a SUSY Chern-Simons term prevent the dualising of the gauge potential to a scalar?

In 3D $\mathcal{N}=2$ supersymmetric field theory with abelian gauge fields, the gauge field $A_{\mu}$ is often dualised to a real scalar $\gamma$. Does a Chern-Simons term prevent this dual ...
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Does a Chern-Simons term break the $F \rightarrow \star F$ symmetry?

When is the electro-magnetic duality $F \rightarrow \star F$ a symmetry of a theory? I know it holds for free Yang-Mills, but would for instance a Chern-Simons term break it or a coupling to matter? ...
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Flux attachment to dynamical gauge field

It is a question about flux attachment. When I attach fluxes to dynamical gauge fields, something weird happened: an extra Hall conductivity term. We start from the action \begin{equation} \mathcal{L}...
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Formalization of the concept of a topological charge

I want to write precisely in mathematical terms what a topological charge is. This is what I have, but I am not sure of how correct it is. Let $M$ be spacetime. Quantization of $M$ in some QFT will ...
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Braiding matrix in Chern-Simons theories$.$

Consider a Chern-Simons system with gauge group $G$ and level $k$. Such a system can be used to model anyons, where the latter are identified with the integrable representations of $G$. One of the ...
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Which Chern-Simons TQFTs are spin?

Refs.1&2 prove several level/rank dualities among different 3d Chern-Simons theories. An important point is that some dualities involve, on one side, a theory that depends on the spin structure, ...
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Wess-Zumino-Witten vs. Yang-Mills-Chern-Simons and Kac-Moody$.$

There is a really nice (holographic) duality between 2d Wess-Zumino-Witten and 3d Yang-Mills-Chern-Simons models (cf. Ref.1). For example, for a given gauge group $G$, the spectrum of both theories is ...
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Equation of motion from $D=3$ Lorentz Chern-Simons action

In three dimensions, the well known Lorentz Chern-Simons action is $$ S_{\text{CS}}=\int\text{d}^3x\varepsilon^{\mu\nu\rho}\bigg(\omega_{\mu}{}^{ab}R_{\nu\rho ab}+\frac{2}{3}\omega_{\mu a}{}^{b}\...
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About the Lagrangian for Chern-Simons-Matter theories

I am particularly thinking of the theory described in section $6$ (starting page 31) of this paper, https://arxiv.org/abs/1104.0680. The exact Lagrangian has never been explicitly stated here but ...
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Why is the semiclassical approximation of the abelian Chern-Simons theory exact?

I was told that in abelian Chern-Simons theory (say, with a general level matrix $K$), semiclassical approximation is exact because there is no trivalent vertex, which in non-abelian case makes the ...
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Conserved quantity in Graphene

The computation of the band structure of Graphene basically leads to the diagonalization of the following Hamiltonian: $$ H = -t \left( \begin{array}{cc} 0 & \epsilon(\vec{k}) \\ \epsilon^*(\vec{...
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Can topological degrees of freedoms interact?

Question Can topological degrees of freedom interact with: other topological dof's other local dof's Premise For concreteness, I can't understand why the following term will not be allowed in a ...