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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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Varying the Chern-Simons action

Summary/TL;DR I want a detailed calculation of the derivation of classical equations of motion from the Chern-Simons action using differential forms, using variational derivatives. I mentioned "...
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Feynman rule from dynamical Chern-Simons

Consider the following action that \begin{equation} S = \int d^4x\sqrt{-g}\left(-\frac{1}{2}(\partial\phi)^2 + V(\phi) + \frac{2R}{\kappa^2} - \frac{\phi}{4f}{}^*RR\right) \end{equation} where \...
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Chern-Simons (K matrix) theory and ${\rm Spin}^{\mathbb C}$ connections

If I understand correctly (e.g. from this paper), an Abelian bosonic Chern-Simons theory defined on $T^2\times \mathbb R$ is specified by a $K$ matrix via e.g. $S \sim \int_M K_{IJ}A^I \wedge dA^J$. ...
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Abelian Chern-Simons large gauge transform

My question concerns the $U(1)$ Chern-Simons theory with the action $$S = \frac{k}{2\pi}\int A\wedge \mathrm{d}A.$$ In my lecture, it is stated that: A large gauge transformation involves taking $A\...
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Chern-Simons theory: Connection between Thermal and Quantum Partition Function

I have been reading the Quantum Hall Effect from Prof. David Tong's notes. In the section on Chern-Simons theory, he describes the connection between the Thermal Partition Function and the Quantum ...
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Computing variation of triple wedge term in Chern-Simons action

For simplicity, assume $G$ is simple, compact, connected, and simply-connected. The Chern-Simons action for a non-abelian structure group $G$ on a trivial bundle with closed base manifold is given by $...
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How is classical Chern-Simons theory topological?

Note: I am using "global" and "topological" somewhat interchangable. This seems to be the case in texts and papers, but please point out if this is inappropriate. Classical Chern-...
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Class of on-shell and gauge equivalent potentials in Chern-Simons theory

Let $(P, M, \pi, G)$ be a principal bundle with three dimensional manifold $M$ and compact, connected, simply-connected, and simple structure group $G$. We define a Lie algebra valued connection $1$ ...
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What is the meaning of the statistical gauge field in the fractional quantum Hall effect

I'm a grad student studying the fractional quantum Hall effect. To get started, I read chapter 9.5.1 of A. Altland and B. Simons' Condensed Matter Field Theory. They use the composite fermion (CF) ...
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How does Witten's path integral know about changing crossings?

At a crossing of a knot, if I change the crossing by swapping the two lines, the knot is changed, along with its Jones Polynomial. Witten's path integral $$ \int {D \mathcal{A}\ e^{i\mathcal{L}}\ W_R(...
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Faddeev-Popov integral for Chern-Simons theory

This question is with respect to this paper of Witten (PDF). According to this paper the following path integral holds for a $3D$ manifold $M$ $$\int \mathcal{D}B\mathcal{D}\phi\exp\left(\int_M\...
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Chern-Simons form of Euler class

Consider the Euler class for curvature $F_{AB} = d\omega_{AB}+\omega_A^{~~~C}\wedge\omega_{CB}$ where $\omega$ is the spin-connection given by $$\int_{\mathcal{M}} \epsilon^{ABCD}F_{AB} \wedge F_{CD} =...
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Understanding Chern-Simons on non-trivial manifold

I am studying abelian Chern-Simons theory on a non-trivial manifold. Could you let me know how accurate my understanding is? Here's what I figured out: The action of $U(1)$ leaves the action invariant ...
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De Rham current associated with knot in abelian CS theory on a generic manifold

I'm studying TQFT and I'm stucked on this part of the paper of my teacher: My teacher didn't explain a lot about it and I've never followed an advanced course on differential geometry or algebraic ...
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How to obtain the relation of eta invariant of the trivial gauge field and Chern-Simons invariant of the flat connection?

In Quantum Field Theory and the Jones Polynomial by Edward Witten(1989), how does $\eta(0)$ come from in this equation? $$\frac{1}{2}(\eta(A^{(\alpha)})-\eta(0))=\frac{c_2}{2\pi}I(A^{(\alpha)})$$ $c_2(...
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Basic Question on Differential Forms (Chern-Simons Level Quantization)

I came across the following post regarding the boundary term in Chern-Simons theory (specifically the level quantization of the theory). I am new to differential forms so the following questions may ...
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Gravitational Chern-Simons term in metric fields

The Chern-Simons gravitational term is $$ S= \int_{\mathcal{M}} d^3 x \epsilon^{\mu\nu\rho} \Gamma_{\mu\sigma}^\lambda \left( \partial_\nu \Gamma_{\rho\lambda}^\sigma +\frac{2}{3} \Gamma^\sigma_{\nu\...
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Normalization in the Abelian Chern-Simons action

In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
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Non-Abelian Chern-Simons Theory References

I am studying Chern-Simons theories and am fairly familiar with the usual Abelian $U(1)$ Chern-Simons theory. I am now looking to extend my knowledge to non-Abelian Chern-Simons and am having a hard ...
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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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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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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 ...
Satyendra Singh Nirvan's user avatar
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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-...
Lucas Queiroz's user avatar
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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{\...
Jeanbaptiste Roux's user avatar
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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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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 ...
Laplacian's user avatar
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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 ...
Pratik Chatterjee's user avatar
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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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2 answers
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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 ...
twisted manifold's user avatar
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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 $$ E_i=-\partial_iA_0-\partial_0A_i\;\;\;B=\epsilon^{ij}\partial_iA_j $$ ...
twisted manifold's user avatar
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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 ...
Alex Coglin's user avatar
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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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1 answer
187 views

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 ...
Ian Gershon Teixeira's user avatar
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1 answer
283 views

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 ...
Richard's user avatar
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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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