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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Does the total Zak phase always sum to zero?

In 2D, the sum of the Chern numbers over all bands is zero. However, this result relies on the ability to define a Berry curvature, which is only possible in $d \geq 2$ dimensions. In 1D it is ...
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BF action as difference of Chern-Simons terms

I believe this boils down to lack of familiarity on my part with wedge products of forms, but I've been looking at https://arxiv.org/abs/hep-th/9505027, and the idea that the BF action $$S_{BF} = \...
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Link complement states in Chern Simons theory

I have been trying to understand how to get an explicit quantum state from a given link in Chern-Simons. Lets say the compact gauge group being SU(2) (this seems to be the most widely studied). I ...
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Is it actually true that Chern-Simons theory is topological?

Chern-Simons theory has action $$\tag{1} S = \frac{k}{4\pi}\int_X tr(A\wedge dA + \frac{2}{3}A\wedge A\wedge A).$$ Here, $X$ is some compact 3-manifold, perhaps with boundary, and $A$ is a connection ...
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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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Decomposition of Two-Particle Statistic in Chern-Simons Theory

In Fradkin's book "Field Theories of Condensed Matter Physics," the statistic of two particles under the Chern-Simons theory is examined. While I understand how the writhing numbers $R(\...
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Meaningful topological invariants and quantities for the description of 3D topological insulators

I'm currently trying to understand the classification of 3D topological insulators (like Bi2Se3). Most reviews dealing with this topic start with the introduction of the Quantum Hall effect since this ...
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Effective Theory for Matter Fields Coupled to a Chern-Simons Field

Assuming that matter fields coupled with the background Chern-Simons field (or Maxwell-Chern-Simons field), I want to obtain the effective theory in terms of matter fields only, by integrating out a ...
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Topological Insulator [closed]

What effect on the Brillouin zone (torus) after applying the magnetic field? As in real space, pressure deforms the torus and up to a certain pressure, this remains invariant topologically. Similar to ...
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Vanishing Chern-Simons partition function

I was reading again the article "Generalized Global Symmetries" and I notice that in the beginning of page 22, they argue that after gauging the $\mathbb{Z}_k$ one-form symmetry, of Chern-...
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3d gravity and Chern-Simons: where does the $i$ come from?

According to @NiharKarve, in this stack-exchange post: Regarding a possible duality between (2+1)D gravity and Chern-Simons Theory, there is the relation: \begin{equation} S_\text{EH} \simeq \frac{\...
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Effective action for chiral - anti-chiral interaction in 4d Chern-Simons

So the question I have is regarding the derivation of eqn (2.9) of 'Gauge theory and Integrability III' by Costello and Yamazaki - https://arxiv.org/abs/1908.02289. Beginning with 4d Chern-Simons, we ...
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How to calculate the Chern number of the edge modes?

We know that for each bulk medium, it is possible to calculate the Chern number of each mode. But at the interface of two different materials, there exists some new edge modes. We need to consider the ...
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Action for boundary term in Chern-Simons theory (David Tong's note)

This question is about obtaining the boundary action from Chern-Simons theory. While reading David Tong's chapter 6 on quantum Hall effect, I cannot derive an equation between (6.9) and (6.10) of the ...
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Relation between pfaffians and Chern number

I have a 2D topological superconductor in symmetry class D, only particle-hole symmetry, nothing else. I can calculate the Chern number using $$ C = \frac{1}{2\pi}\int_{BZ}\mathrm{d}^2{k}\ \mathcal{F} ...
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Axion interaction with a Chern Simons term

In the Lagrangian of self-dual Yang-Mills theory, an interaction term is added where an axion like field is coupled with a Chern-Simons term to cancel an anomaly in the twistor space. The interaction ...
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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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Gauge fixing in derivation of fractional QHE action

I'll copy the text from a relevant question: This follows the discussion in Altland and Simons Condensed Matter Field Theory -- section 9.5 on deriving the Chern-Simons action for FQHE. Starting with ...
Kouta Dagnino's user avatar
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Prerequisites for Chern-Simons approach to the Fractional Quantum Hall Effect

I am interested in learning the Chern-Simons approach to the fractional quantum Hall effect right from the basics. I have learnt about Lie groups and Weyl quantisation and am currently learning ...
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Edge state protection in Chern insulator

I have a confusion about the nature of topologically protected boundary states in the Chern insulator. Since the Chern insulator does not require any symmetries to be present in the system, what is ...
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Gauge invariance of the Abelian Chern-Simons term

Given the Abelian form of the Chern - Simons action we have $$ \mathcal{S}_{CS}[A]\equiv\frac{k}{4\pi}\int d^3x\epsilon^{\mu\nu\rho}A_\mu\partial_\nu A_\rho $$ If we would like to check whether this ...
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Test charge in 2+1 dimensions

Given a Chern-Simons theory,as in this resource(page 4), in 2+1 dimensions we can define the electric and magnetic fields as $$ E_i=-\partial_iA_0-\partial_0A_i\;\;\;B=\epsilon^{ij}\partial_iA_j $$ ...
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Why is gravitational Chern-Simons action invariant under conformal transformation?

We know the action of topologically massive gravity in 3-dimentional spacetime is \begin{equation} \label{eq:EH} S=S_{\mathrm{EH}}+S_{\mathrm{CS}}=\frac{1}{\kappa^2}\int d^3x\sqrt{-g}R+\frac{1}{\mu}\...
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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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How to Wick rotate differential forms?

The bosonic sector of the Cremmer-Julia-Sherk (CJS) 11D supergravity action is $$ eR-\frac12 F\wedge *F-\frac16 A\wedge F\wedge F,$$ where $F=dA$ is a 4-form field strength. How would one perform a ...
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"Correct" gauge for Chern-Simons terms in 5d?

Consider 5d Einstein-Maxwell-Chern-Simons gravity with action $$S=\frac{1}{16\pi G}\int d^5x\sqrt{-g}\left[R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{12\sqrt{3}}\frac{\epsilon^{\mu\nu\rho\sigma\lambda}...
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Transformation of Chern-Simons type action under large $SO(4)$ gauge transformation

For the Chern-Simons action $$S = \kappa \epsilon^{\mu\nu\delta}tr(A_\mu\partial_\nu A_\delta + \frac{2}{3}A_\mu A_\nu A_\delta)$$ under a large gauge transformation $$A_\mu \rightarrow g^{-1}A_\mu g +...
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Propagator of four-dimensional Chern-Simons theory

In https://arxiv.org/abs/1903.03601, on page 13, the propagator of 4d Chern-Simons theory is computed, in the gauge $D^iA_i=0$, where $D^i = (\partial_x,\partial_y,4\partial_z)$. The gauge-fixed ...
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$U(1)$ Chern-Simons from simple integer quantum Hall arguments, basics

I am trying to understand how the Chern-Simons term appear in $U(1)$ effective theory of integer quantum Hall effect (IQHE). I usually read in (90% of) lecture notes "$A dA $ is the Chern-Simons ...
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Derivative of the Chern-Simons form

I want to verify the relation of the Chern-Simons form $$ d \, \text{tr} (AdA+ \frac{2}{3} AAA) = \text{tr} FF$$ where $\omega \mu\equiv \omega \wedge \mu$ and $F=dA+AA$. Using the property $d\, \...
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Holography vs edge modes in Chern-Simons theories

There are two facts I have heard about Chern-Simons theories They are dual to a WZW theory on the boundary (e.g. Relation between CS/WZW and AdS/CFT). They require a chiral theory on the boundary due ...
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What the role of classical equation of motion in quantum field theory?

I've learnt quantum field theory for a semester but I still can't understand the role of classical equation of motion in QFT. I have looked up for several books. They all discuss classical field ...
Taveren Sa's user avatar
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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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Can we construct a Chern-Simons theory provided that the ground state is degenerate and gapped, like the Abelian fractional quantum Hall effect?

I am studying the Chern-Simons approach to fractional quantum Hall effect, which a special focus on the topological order in the context of Abelian fractional quantum Hall effect. To me, the logic to ...
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Knots in 3d pure gravity

Three-dimensional pure gravity on $\mathrm{AdS}_3$ can be described as a Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$,with action $$ S[A,\bar{A}] = \...
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Non-abelian Chern-Simons from fermion effective action

(1) Starting from the fermion effective action $$S_\text{eff}[A,m] = \log \det(i\gamma^\mu{\partial_\mu} + \gamma^\mu A_\mu + m)\tag{223}$$ once can do a loop expansion following https://arxiv.org/abs/...
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Non-Abelian Chern-Simons path integral on a torus

Is it possible to exactly evaluate the Chern-Simons path integral with a non-compact gauge group (say $SU(2)$) on a torus? I am asking this because 3d gravity is an $SL(2,\mathbb{R})$ Chern-Simons ...
Sounak Sinha's user avatar
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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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Why can't we define mathematical observables in asymptotic $dS$ or flat space for gravitational theories?

In higher spin currents, the boundary CFT is dual to an asymptotic $AdS$. I have heard that $dS$ is not quantizable. But I don't understand why we want it to be in the first place. Isn't Chern-Simons ...
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What does it mean for correlation functions to be dominated by the vacuum block for a 2D CFT?

In a 2D CFT, correlation functions dominated by the vacuum block have no conical defects. You can calculate the OPE and determine the correlation function using the D-S equations and cancel out UV ...
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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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Normalization of $U(1)$ gauge fields

In G. W. Moore, “Introduction to Chern-Simons theories.” 2019 TASI School. [Online]. Available: https://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf the $U(1)$ gauge field has a ...
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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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Motivation for 3D Quantum Gravity

I was briefly going through the idea of 3d quantum gravity on nLab, where it is stated that: The case of dimension 3 is noteworthy, because in this case the quantum theory can be and has been fairly ...
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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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Is the "Push-Down" Quantization of Chern-Simons Theory part of a more general approach to Quantization?

I've recently started reading Axelrod, Della Pietra and Witten's original paper about the quantization of Chern-Simons theory. I'd like to know if the "push-down" quantization strategy they ...
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Propagator, mass and electrostatic potential of an modified 2+1 dimensional Maxwell action

Consider the modified Maxwell action in 2 + 1 dimensions $$S=\int d^3x[-\frac{1}{4} F^{\mu\nu} F_{\mu\nu}+\frac{\theta}{2}\epsilon^{\alpha\mu\nu}A_\alpha F_{\mu\nu}] .$$ The action invariant under ...
Daniel Vainshtein's user avatar
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Chern-Simons Lagrangian and gauge-fixing

Main question: Consider (2+1)D Chern-Simons action $$S = \int dt d^2\mathbf r \frac{k}{4\pi} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda.$$ Assuming the Coulomb gauge $\nabla\cdot \mathbf a ...
eigenvalue's user avatar
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Are central charges equal or similar to irreducible spinor representations?

First of, if any of the following below does not make sense, please feel free to leave a comment =) Central charges in Chern Simons in in the Virasoro conformal blocks play an important role for ...
Burak Guner's user avatar
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Chern-Simons term for a non-abelian gauge multiplet

In equation (20.9) of Freedmann and Van Proeyen's Supergravity, it is stated that for the following Chern-Simons term: $$S_{\mathrm{CS}} = C_{IJK}\int A^I\wedge F^J \wedge F^K$$ to be invariant under ...
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