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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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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Why does the Holst term not affect gravitational dynamics?

The general first-order Palatini action in four dimensions is given by $$S[e,\omega]=\frac{1}{2\kappa}\int_{\mathcal{M}} F_{IJ}[\omega]\wedge\left(\star+\frac{1}{\gamma}\right)\left(e^I\wedge e^J\...
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Meaning of mass-deformation in string theory and quantum field theories

I was reading some papers in the ABJM theory. I keep reading the term mass deformation but am not sure what it really means. I think the papers assume the reader is familiar with the term. Example ...
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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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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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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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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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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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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 ...