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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Energy-Momentum Tensor for lagrangian with Chern Simons term

I am trying to find the energy-momentum tensor for the effective field theory that is used to describe the fractional quantum hall effect(Ref: [Zhang 1992][1]). The effective Lagrangian is \begin{...
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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, ...
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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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Factor of half in the braiding statistics of Chern-Simons theory

My question is about the factor of $1/2$ discrepancy in the braiding phase of quasiparticles, from what one would naively expect. Why should it be that there is a certain flux at the location of each ...
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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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Spontaneous symmetry breaking induced by pure Chern-Simons field?

Maybe this is more of a reference request, but I'd also like an explanation of the example (if there is one). An answer with just a reference would suffice though. Simply put, are there any known $(2+...
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Chern-Simons, holography, and bibliography

I am looking for review papers / online talks on Chern-Simons theory with particular focus on the gravitational dual description within the AdS/CFT framework.
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Center symmetry on wavefunction

Assume we know the wavefunctions of a SU(N) Chern-Simons (or YM) on a 3-mfld $M$, perhaps using holomorphic quantization. How do the center symmetry transformations act on the wavefunctions? Is it ...
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Topologically massive $p$-form gauge fields in arbitrary dimensions

$\newcommand{\d}{\mathrm{d}}$In $d=2p+1$ dimensions one can have topologically massive $p$-form abelian gauge fields $A\in\Omega^p(X_{2p+1})$ by considering a Maxwell–Chern–Simons action: $$S[A] = \...
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Batalin-Vilkovisky (BV) form of the Chern-Simons Action

As seen in Section 4 of Chapter 5 of Costello, K. "Renormalization and Effective Field Theory", or in section 5.2 $L_\infty$-Algebras of Classical Field Theories and the Batalin-Vilkovisky ...
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Verlinde formula requires compact Riemann surface

Assume a Chern-Simons theory on a 3-mfld of the form $\Sigma_g \times S^1$, where $\Sigma_g$ is a Riemann surface of genus $g$. In his paper, Witten shows that one can use the Verlinde formula to get ...
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Chern-Simons theory for FQHE

Recently I have read something about describing FQHE within chern simons field theories. According to Atland's text book,Condensed matter field theory, one can map interacting fermions to composite ...
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Chern-Simons (CS) theory

I have a question about Constructuion of Chern-Simon Action. In its paper "Non-commutative geometry and string field theory", Witten construct the Action of the String Field Theory inspiring ...
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$F$-symbols for compact Lie groups

Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices $$ B\begin{bmatrix}j_1&j_2\\i&k \end{bmatrix}\colon \bigoplus_p V_{j_1p}^i\otimes V_{j_2k}^p\to V_{...
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Wilson loop expectation value in $RP^3$ using Dehn surgery

I am currently reading Guadagnini's The link invariants of Chern-Simons field theory, the part where he computes some examples of expectation values for different spaces. For $S^2 \times S^1$, he ...
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Expectation value of Wilson loop in Chern-Simons theory

I have read Witten's paper, and I am interested on computing the expectation value of a Wilson loop with a representation $R$ on Chern-Simons theory in $d=3$. I am especially interested in cases for $...
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Winding number is an integer

In computing the variation of the action in Chern-Simons, and in other contexts, we get the following expression that is named the winding number, where $U$ comes from a gauge transformation: $$ W[U] =...
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Chern-Simons Path integral restricting to small gauge transformations

How does one compute the Chern-Simons path integral in 2+1 dimensions considering only small gauge transformations? Is this even a well-defined theory?
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Chern-Simons theory with only small gauge transformations

Usually when one derives the variation of the Chern-Simons action in 2+1 dimensions, one has a term that is proportional to the winding number. Then one argues that the coupling constant must be an ...
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Why is this Chern-Simons theory gauge invariant?

I am reading the lecture notes in https://arxiv.org/abs/hep-th/9902115 and in it, it says that the Lagrangian $$\mathcal{L}_{\mathrm{CS}}=\frac{\kappa}{2} \epsilon^{\mu \nu \rho} A_{\mu} \partial_{\nu}...
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How does anomaly inflow work in terms of the eta invariant?

I'm trying to understand the non-perturbative picture of anomaly inflow, mainly following these two articles by Witten and Yonekura: [1] - https://arxiv.org/pdf/1909.08775.pdf , [2] - https://arxiv....
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Integrating over non-trivial fiber bundles - Chern-Simons Theory

I have been reading Tong's notes on QHE and Gauge Theories, specifically the part about quantizing the Abelian U(1) Chern-Simons level at finite temperature in the presence of a monopole (These ...
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Index on a compact manifold

How can the integral of a topological term (like the Nieh-Yan term) on all of a compact manifold be nonzero whereas it's a total derivative and the manifold has no boundary? I assume the manifold can ...
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How do $\theta$-terms not violate gauge invariance?

In the context of QCD (and more generally, any quantum gauge theory in even dimensions), the $\theta$-term is $$ \frac{\theta}{8\pi^2}\langle F_A\wedge F_A\rangle = \frac{\theta}{32\pi^2}\langle F_A^{\...
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Is it true that in abelian Chern-Simons theory diffeomorphisms differ from ordinary gauge transformations trivially?

In Henneaux's Lectures on the Antifield BRST Formalism for Gauge Theories, it is claimed in Exercise 1 that diffeomorphisms $\delta_\xi A_\mu=\xi^\rho\partial_\rho A_\mu+\partial_\mu\xi^\rho A_\rho$ ...
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free fermion- monopole operator in scalar $QED_3+$ Chern simons term equivalence proof?

In most papers discussing 3D Abelian bosonization duality, they say that monopole operator in scalar $QED_3+CS$ is dual to free fermions. How do they know it, because I have never seen an actual ...
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The value of Gravitational Chern Simons theory integration on some three manifolds

Consider the 3d gravitational Chern Simons theory $$S= \frac{k}{192 \pi} \int_{M_3} \mathrm{Tr}\left(\omega\; \mathrm{d} \omega + \frac{2}{3}\omega^3\right)$$ where $\omega$ is the spin-connection on ...
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Chern-Simons term in Coulomb or radiation gauge

In some of the literature (for example, below Eq. (A3) of this paper), the following is claimed to be the Chern-Simons term in the Coulomb gauge: \begin{equation} 2a_0(\partial_1a_2-\partial_2a_1) \...
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Topological invariants, what's that?

What's the difference between the Berry phase, the Euler number,the winding number and the Chern number? As far as I know they can all be computed by the same integral, but there seems to be some ...
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Normalization of the Chern-Simons action in the Dijkgraaf-Witten paper

I am trying to understand the seminal paper "Topological gauge theories and group cohomology" by Dijkgraaf and Witten. They consider an oriented three-manifold $M$, compact Lie group $G$ and a $G$-...
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Induced “ungauged” Chern-Simons terms from a massive Dirac fermion

It is a well-known fact that a massive Dirac fermion minimally coupled to a gauge field $A_\mu$ induces a Chern-Simons term when integrating out the fermion: \begin{align} i\bar{\psi}\gamma^\mu(\...
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Theory on domain walls

In Baryons in Quantum Chromodynamics, Zohar Komargodski have slide: I wanna understand: Why domein wall can have nontrivial worldvolume theory? When such solitonic objects have interior degrees of ...
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Kinds of Wilson Loops in a $U(1)$ Chern-Simons Theory

Pardon the potentially easy question, but I am currently reading through a paper by Seiberg and Witten [1], and while reading appendix C I'm not sure where some of these results are coming from. The ...
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Chern-Simons action as a topological invariant

It is stated that the Chern-Simons action is a topological invariant that is proportional to the Chern-Simons form. But the latter is just a conformal invariant. How do we reconcile these views? Both ...
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Wiedemann-Franz law generalized to quantum Hall effects in electronic systems

Wiedemann-Franz law states a relation in a conductor between the thermal and electric conductivities by their ratio as $\kappa/\sigma=LT$ where $T$ is the temperature and $L$ is the Lorenz number ...
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Factor of 2 issue in the non-gauge invariance of Chern-Simons theory with a boundary

It is well known that the Chern-Simons (CS) theory by itself is not gauge invariant in the presence of a spacetime boundary. Concretely, suppose the flat half space $\mathcal{M}$ with $x\in \mathbb{R},...
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Angular momentum in the Maxwell field theory/Chern-Simons theory?

I'm trying to calculate the angular momentum in the chern simons theory. But equivalently, I was trying a calculation of angular momentum in the Maxwell field theory, which will hopefully be ...
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Chern-Simons theory with a discrete gauge symmetry

Let us consider a Chern-Simons theory on a $3$-manifold $M$ (can be a spin manifold with a given spin structure if needed) with a discrete-symmetry gauge field e.g. $\mathbb{Z}_n$ symmetry. It can be ...
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What is meant by the vacuum structure of ABJM theory?

I was reading the paper Large $N$ behavior of mass deformed ABJM theory. It talks about the vacuum structure of the (mass deformed) ABJM thoery. What does vacuum structure mean in general or in ...
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How to know which topological invariant is in play?

I'm currently working on the Haldane model where I've worked through the math to find that when the condition $$ \frac{M}{t_2} = 3 \sqrt{3} sin (\phi) $$ is satisfied the gap closes, meaning there ...
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Chern-Simons theory on a plane/sphere with a single charge insertion

Consider the pure Chern-Simons theory on the plane $\mathbb{R}^2$ with a single charge insertion in some representation $\rho$ of the group $G$. What does the Hilbert space look like? Is it null or ...
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The relation between Chern-Simons Theory and Yang-Mills Theory

So from this page, I know that there is a relation between Chern-Simons Theory and Yang-Mills Theory, but I have difficulty proving the identities in the document. I was going to prove $$\partial_\...

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