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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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 ...
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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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Chern-Simons forms: interpretation and generalizations

Studying again differential geometry, anomalies and topology, I wondered if there is ANY physical interpretations (in terms of QFT or even classical field theory) of the Chern-Simons forms, via vacuum,...
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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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Bulk-edge correspondence in 4D synthetic system

I have been following this paper, which discuss the bulk-edge correspondence in the Harper Model. The Harper model is a model realized on a 2D square lattice, with the following Hamiltonian (see ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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Free homotopy of loops

Given a topological space $X$, a loop based at $x_{0}$ is defined to be a continuous function $$\gamma \colon [0,1]\to X$$ such that the starting point $γ(0) $ and the end point $γ(1) $ are both equal ...
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Euler-Lagrange equation in a differential form notation

Treating the Lagrangian density as a $d$-form in $d$-dimensional spacetime, how can one write the Euler-Lagrangian equation basis independently in the form notation? If possible, can you also apply ...
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Boundary term for Chern-Simons action

As discussed in David Tong's lecture series on the edge modes in the quantum Hall effect (http://www.damtp.cam.ac.uk/user/tong/qhe.html) (page 203), varying the 2+1D Chern-Simons action yields: $$\...
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Why is $T^*S^3$ a conifold?

So, I was reading the famous Gopakumar Vafa paper, and they mention that $T^*S^3$ is a conifold. Why is this the case? I would naively expect $T^*S^3$ to be basically the same everywhere ($S^3$ is a ...
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Can one quantize Chern-Simons theory in the covariant phase space formalism?

The covariant phase space, which coincides with the space of solutions to the equations of motion, gives a notion of phase space which does not rely on a decomposition of spacetime of the form $M=\...
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Self-loops in Chern-Simons theory

Consider the dumbbell graph decorated with propagator $P$ one the edges and with integration variables $x$ and $y$ on the vertices. We associate to it the following integral: $$ I = \int_{x,y} P(x,x)P(...
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Why doesn't the $3d$ gravitino have a quantized "level"?

The action for the $3d$ gravitino is $$S_g=-\int d^3x\bar{\Psi}_{\mu}\gamma^{\mu\lambda\nu}\partial_{\lambda}\Psi_{\nu}$$ Where $\gamma^{\mu\lambda\nu}=-\epsilon^{\mu\lambda\nu}$. This has a striking ...
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Topological Quantum Field Theory with Symmetries and Knot Quandles

It is well known that Chern-Simons theory provides an intrinsically three dimensional way to compute knot invariants like the Jones Polynomial. 3D TQFTs also have an algebraic description in terms of ...
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Are Chern-Simons theories classified by bordism groups?

For a long time it was thought that anomalies for a group $G$ were classified by $H^n(BG)$, although it is now understood that they are in fact classified by $\Omega^n(BG)$. On the other hand, ...
3 votes
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Chern-Simons Level Quantization for non-compact groups?

Usually for Chern-Simons theory with compact gauge group, which we can take to be $SU(2)$ for simplicity, one has that level $k$ (or coupling) has to be quantized in order for the action to be gauge ...
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Chern number for nonintracing hamiltonian while bands crossing

Is it possible to define and calculate chern number for two bands while they're crossing each other?
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