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.

learn more… | top users | synonyms

4
votes
1answer
73 views

Finding explicit unimodular transformations for Chern-Simons K-matrices

An invertible, symmetric matrix with integer entries, $K$, that encodes the braiding and statistics of an Abelian topologically ordered state, is equivalent to another such matrix, $K'$, if there ...
0
votes
1answer
121 views

Partition functions for a (3+1)-d TQFT

It is well known that for a Chern-Simons theory defined on an arbitrary (2+1)-d oriented manifold, its partition function can be evaluated based on Witten's surgery method. My question is: is there a ...
1
vote
0answers
32 views

Doubts about Chern-Simons state as a solution of the Hamiltonian constraint in quantum gravity

I've been doing some work with both Baez's Knots, gauge fields and gravity (1) and Gambini, Pullin's Loops, knots, gauge Theories and quantum gravity (2), lately. I have basically two problems: I ...
0
votes
0answers
33 views

Simplify $K$-matrix

2+1D Abelian topologically ordered states are believed to be described by multicomponent $U(1)$ Chern-Simons theories, with Lagrangian \begin{equation} \mathcal{L}=\frac{K_{IJ}}{4\pi}\epsilon^{\mu\nu\...
0
votes
0answers
48 views

Explicit derivative of Chern-Simons current

I know that for a Chern-Simons 3-form $\omega=\operatorname{Tr}\left[F\wedge A-\frac{1}{3}A\wedge A\wedge A\right]$, with $F=A\wedge A +\operatorname{d}A$, I should get $\operatorname{d}\omega=\...
1
vote
0answers
42 views

How to write Fractional Quantum Hall States with Symmetric Polynomials?

Is there a link between Fractional Quantum Hall States and Symmetric Polynomials. In papers of Xiao Gang Wen [1], [2] work out a few examples: $ \Phi_{1/2} = \prod_{i < j} (z_i - z_j)^2$ is the ...
1
vote
1answer
72 views

chern number as an obstruction to choose a smooth gauge

In condensed matter physics, I heard that if chern number of a band $n$ is non zero, it is impossible to choose a gauge such that $\psi_{nk}$ is smooth in the whole brillouin zone. However, it is ...
1
vote
0answers
51 views

Coordinate variation of a Wilson loop

In Chern Simons Gauge Theory as a String Theory, Witten derives the general coordinate variation of a Wilson loop, i.e., equation 3.11. My question is, how does one derive this? I only managed to ...
1
vote
0answers
30 views

Charged rotating sphere in Maxwell-ChernSimons theory

Does anybody know what the electromagnetic field of a charged rotating sphere in five dimensional Maxwell-Chern-Simons theory is? This theory has action \begin{equation*} S = - \frac{1}{4} \int d^5 ...
1
vote
2answers
132 views

Chern-Simons theory

The Chern-Simons 3-form is given by $\omega_3={\rm Tr} \left[ A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right]$ where $A$ is a connection one-form in the adjoint representation of a non-Abelian gauge ...
1
vote
0answers
54 views

Chern Simons Theory over S^3 as integral - what is domain of integration?

I found these nice lecture notes Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories so I am hoping to understand some parts of the Chern Simons theory better. ...
5
votes
1answer
90 views

Variations of actions of (lie algebra valued) differential forms

I have always found it a bit difficult to understand the variation of an action written in differential form language. For example, take the action $$\int tr A\wedge A\wedge A$$ where $A=A_\mu dx^\...
1
vote
0answers
39 views

Thermal AdS3 in Chern Simons

I am currently working with (2+1) gravity in Chern-Simons formulation and I have a question about thermal AdS. The way I understand that one retrieves thermal AdS is by Wick rotation and ...
1
vote
0answers
72 views

Massive Dirac model Chern number 1/2

Why is massive Dirac model have a Chern number as half? I know this is something related to anomalies. And for fermions we have half, for bosons we have integer. But I failed to find any good ...
0
votes
0answers
37 views

Local Translation of frame field- Geometric Picture

In order to phrase my question I review my geometrical picture of the first order formulation of gravity: Given some d-dimensional manifold $\mathcal{M}$ one constructs the Frame bundle $FM$, a ...
2
votes
2answers
267 views

Topological susceptibility

In QCD we have strong CP violation (and hence a $\theta$-dependence of the theory) only if the topological susceptibility of the vacuum is nonzero: $<F\tilde{F},F\tilde{F}>_{q \rightarrow 0} =...
0
votes
0answers
57 views

Chern-Simons function

The Chern-Simons function on the space of connections, mod the gauge transformations, on a 3-manifold can be defined by an integral. I study mathematics as profession, so I want to know what is the ...
3
votes
0answers
121 views

Choice of framing in Gravitational Chern-Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ...
1
vote
0answers
82 views

Can we quantize Maxwell-Chern-Simon Theory through Gupta-Bleuler approach?

In 3+1 QED we covariant quantize the Maxwell theory through Gupta-Bleuler method. But I have seen that MCS theory is explicitly covariant quantized using Nakanishi auxiliary field. Why cannot we take ...
0
votes
0answers
62 views

Equivalence between Chern-Simons action and first order formalism

I can not derive second line from Chern-Simons action \noindent $S_{cs}$=k$\int{Tr(A \wedge dA+\frac{2}{3}}A \wedge A\wedge A)$ \noindent =k$\int{\ e }^a\wedge $R[$\omega $] we have to use ...
-2
votes
1answer
173 views

2 + 1 dimensional gravity as an exactly soluble system [closed]

In Gomez's article (Higher spin part 2)http://arXiv:1307.3200, I face to gauge algebras. Gomez says we have some algebras for ISO(2,1)and SO(2,2),SO(3,1). and he tells us only ISO(d-1,1) and only ...
4
votes
1answer
299 views

effective field theory of the projective semion model

The "projective semion" model was considered in http://arxiv.org/abs/1403.6491 (page 2). It is a symmetry enriched topological (SET) phase. There is one non-trivial anyon, a semion $s$ which induces a ...
2
votes
2answers
280 views

Chern Simons action in 4 dimensions

I can not understand why we do not have a Chern-Cimons action for 4 or even forms? And why it not good theory for (3+1) dim?
1
vote
1answer
144 views

Why Levi-Civita term signal the breaking of parity and time reversal?

For example, referring to Zee's QFT book, in Chern-Simons matter theory, after writing a term $$\gamma {\varepsilon ^{\mu \nu \lambda }}{a_\mu }{\partial _\nu}{a_\lambda }$$ he said The ...
2
votes
0answers
314 views

Equations of motion in Maxwell-Chern-Simons theory [closed]

I've started with the Maxwell-Chern-Simons lagrangian (in 2+1 dimensions): $$L_{MCS}=-\frac{1}{4}F^{\mu \nu}F_{\mu\nu}+\frac{g}{2} \epsilon^{\mu \nu \rho}A_\mu\partial_\nu A_\rho$$ From this ...
3
votes
1answer
155 views

Calculation of the Abelian Induced Chern-Simons Term

In Gerald Dunne's paper "Aspects of Chern-Simons Theory" (http://arxiv.org/abs/hep-th/9902115) I'm a little confused as to how equation (225) on page 53 is obtained. Equation (225): \begin{equation}\...
6
votes
4answers
435 views

Is gravitational Chern-Simons action “topological” or not?

Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection: $$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a} $$ $$ S=...
3
votes
2answers
156 views

Relation between conformal and topological field theories

The Chern-Simons (CS) theory is a topological quantum field theory (TQFT). The question is, is a conformal field theory (CFT) a topological quantum theory? Or the reverse, topological quantum field ...
4
votes
0answers
187 views

Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
5
votes
1answer
192 views

No local degrees of freedom when connection is flat

I was studying Chern-Simons theory and variation of action gives us the flatness conditions $\mathrm{d} A + A \wedge A = 0$. I am wondering how to see that this implies there are no local degrees of ...
4
votes
0answers
239 views

Chern-Simons on a lattice and the framing anomaly

Can someone make or refer me to the argument for why $U(1)$ Chern-Simons theory in three dimensions cannot be defined by a lattice action? (Unlike Dijkgraaf-Witten theories, which are defined on the ...
15
votes
0answers
398 views

Positivity for the level of Chern-Simons theory

In many classical papers about Chern-Simons theory (see, e.g. [1]), it is claimed that the Chern-Simons theories with gauge group $G$ are classified by an element of $k\in H^4(BG,\mathbb Z)$, the so-...
3
votes
1answer
94 views

How Chern-Simons gauge field transform fermion to scalar?

In A.Zee's "QFT in a nutshell"book Page 324, after he wrote down the general Lagrangian, he said "in previous chapter, we learned that by introducing a Chern-Simons gauge field we can transform $\psi$ ...
4
votes
2answers
286 views

SPTs and systems with Topological Order

I am an undergrad interested in Condensed Matter Theory. Particularly topological phases and systems exhibiting topological order. A potential research advisor doing a lot of work in Symmetry ...
4
votes
0answers
205 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-...
4
votes
0answers
455 views

Gauge Invariance of the Non-abelian Chern-Simons Term

I'm trying to prove that, under the gauge transformation $$A_{\mu} \rightarrow A_{\mu}^{\prime} = g^{-1} A_{\mu} g + g^{-1} \partial_{\mu} g,$$ the non-abelian Chern-Simons Lagrangian density: $$\...
5
votes
1answer
287 views

Anyons: Effect of braiding on fusion multiplicities

In the theory of non-abelian anyons, essential information is stored in the fusion multiplicities or Verlinde coefficients $N_{ab}^c$. Having the Pants Decomposition in mind, it is possible to use ...
8
votes
1answer
290 views

geometric quantization of the moduli space of abelian Chern-Simons theory

I wish to understand the statement in this paper more precisely: (1). Any 3d Topological quantum field theories(TQFT) associates an inner-product vector space $H_{\Sigma}$ to a Riemann surface $\...
2
votes
1answer
198 views

Completing the trace in the Non-abelian Chern-Simons Term

I've been having a little trouble proving that [Page 138 of "Introduction to Topological Quantum Computation" by Jiannis K. Pachos]: $S_{CS} = \dfrac{k}{4 \pi} \int_{M} d^{3}x \, \epsilon^{\mu \nu \...
5
votes
3answers
361 views

TQFT associates a category to a manifold

Any 3d TQFT (topological-quantum-field-theory) associates a number to a closed oriented 3-manifold, a vector space to a Riemann surface, a category to a circle, and a 2-category to a point. This is ...
5
votes
1answer
451 views

Chern-Simons Energy-Momentum Tensor

I'm assuming the following statement is true. I'm not finding any reference which shows that explicitly. Statement: Chern-Simons term is a topological one and does not contribute to the Energy-...
4
votes
1answer
191 views

Gravitational Chern-Simons theory for bosons and fermions

Q1: What is the difference of boson and fermions for their Gravitational Chern-Simons theory? I suppose in general if the metric is not flat, we have vierbein ${e_{\hat{b}}}^{\nu}$, with $$ g_{\mu\...
6
votes
1answer
143 views

Quantized coefficients of Chern-Simons action and F $\wedge$ F $\wedge \dots$

We know that for U(1) gauge field Chern-Simons action in 2+1 Dim(ension), we have an action $$ S=\alpha \int A \wedge dA $$ with $\alpha=k/(4\pi)$ for a proper level quantization. Here $k$ is the ...
5
votes
1answer
179 views

How to compute the propogator for Chern-Simons on a torus?

I'm looking to better understand Chern-Simons theory on a torus. We are given the action $$ S(\phi) = \int_E (\partial \phi)(\overline\partial \phi) + \frac{\lambda}{6}(\partial \phi)^3 $$ which ...
6
votes
1answer
600 views

Field strength vanishes iff $A_{\mu}$ is pure gauge

Is it true that the field strength $F_{\mu\nu}$ in a non-Abelian gauge theory with gauge group $G$ vanishes if, and only if, the gauge field $A_{\mu}$ is a pure gauge? I can show one implication. ...
6
votes
3answers
329 views

Propagator of Chern-Simons Abelian gauge theory

I need to compute the "topologically massive photon" propagator. I've started with : $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{\mu}{4}\epsilon^{\mu\nu\lambda}A_\mu\partial_\nu A_\...
6
votes
2answers
262 views

Mass generation by Chern-Simons theory

Why the mass generation via a Higgs mechanism is different from that of Chern-Simons theory? I haven't done any formal course in Quantum field theory,so how do I understand this just having some basic ...
2
votes
0answers
71 views
1
vote
1answer
210 views

How can I show non-Abelian CS term is a total derivative?

I want to show:$$ Tr\left (F\tilde{F} \right )=\partial_{\mu}K^{\mu }=\partial_{\mu}\left (\varepsilon _{\mu \nu \rho \sigma }Tr\left ( F_{\nu \varrho }A_{\sigma }-\frac{2}{3}A_{\nu }A_{\rho }A_{\...
1
vote
0answers
56 views

Degeneracy and the unitarity of a gauge theory with a non-compact gauge group

The topological ground state degeneracy(g.s.d.) provides useful information for a topological field theory(TQFT), such as this post shows some example. To count g.s.d., it seems to be equivalent to ...