Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
16 views

Effective theory of hierarchial fractional quantum hall state

In describing the effective field theory picture of the hierarchical fractional quantum Hall states in Tong's lecture notes, page 165 he gives the expression for filling fraction, quasi-particle ...
1
vote
0answers
53 views

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 ...
1
vote
0answers
58 views

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 ...
5
votes
2answers
50 views

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 ...
0
votes
0answers
30 views

Number of Physical States of a $U(1)$ Chern-SImons Theory on a Riemann Surface of Genus $g$

In A Duality Web in 2+1 Dimensions and Condensed Matter Physics, the authors claimed in Appendix B that for a $U(1)_{k}$ Chern-Simons theory defined on a Riemann surface $\Sigma$ of genus $g$, the ...
3
votes
0answers
42 views

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}{...
2
votes
1answer
70 views

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 ...
5
votes
1answer
68 views

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 ...
5
votes
0answers
95 views

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\...
1
vote
0answers
44 views

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 ...
5
votes
1answer
142 views

Intepreting Fermions as Differential Forms?

In this paper on path-integral quantization of Chern-Simons theory, on page 434 (equation 4.17), the authors used fermions to interpret wedge product and contractions of differential forms. Let $M$ ...
2
votes
0answers
119 views

Deriving Chern-Simons term from path integral representation of the first quantized non interacting many body Hamiltonian

This is an exercise from condensed matter filed theory book of altland and simons. Exercise Subject the first quantized many particle hamiltonian $H=\sum_{i=1}^{N}\frac{p^{i}{^{2}}}{2m}+V(x^i)$ to ...
2
votes
1answer
77 views

Hamiltonian Structure of Chern Simons Electrodynamics

I am reading the review paper "Aspects of Chern-Simons Theory" by Gerald Dunne https://arxiv.org/abs/hep-th/9902115 Starting from p. 17, Dunne works on the Hamiltonian structure of the CS ...
4
votes
0answers
95 views

Is $U(2)_{2, 1}$ Chern Simons Theory Completely Trivial?

I am using the method outlined in appendix C4 of a paper by Seiberg and Witten [1] to calculate the statistics of lines in $U(2)_{2, 1}$. However, this method shows that all lines are trivial. ...
5
votes
1answer
121 views

Second Chern class in 2D Haldane model from Atiyah-Singer Index Theorem?

I was reading through a physics-centered exposition of the Atiyah-Singer index theorem and I wondered what it would mean to talk about Haldane's model for the case of a manifold with a boundary. It is ...
8
votes
0answers
93 views

Lattice QFT of the Jones Polynomial

Start with a gauge theory with Chern-Simons action $$ S[A] = \frac{k}{4 \pi} \text{Tr} \intop_{M} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right) $$ and a Wilson loop observable in the ...
0
votes
1answer
69 views

About the Level of Non-Abelian Chern Simon theory

The Lagrangian of the non-Abelian Chern Simons theory is the following $$\mathcal{L}=\frac{k}{4\pi}\int \text{tr}(AdA+\frac{2}{3}AAA)$$ What is the definition of tr here? Namely, which representation ...
5
votes
0answers
123 views

Is it possible to couple an odd number of Dirac fermions, at finite density, to a massless gauge field in 2+1d?

In a beautiful paper by A. N. Redlich (PRL $\bf{52}$, 18 (1984)) on the parity anomaly, the author indicates that an odd number of Dirac fermions can never be coupled to a massless gauge field in 2+1d ...
2
votes
0answers
69 views

Understanding the fractional quantum Hall effect in Chern-Simons formalism described in Wen's book

So I study fractional quantum hall effect with Chern-Simons formalism by using Wen's book, this is an excellent book, but it assumes that you know field theory very well thus it has gaps between steps....
3
votes
0answers
75 views

Twisted Chern-Simons, and Twisted Wess-Zumino Term

I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten. Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...
3
votes
1answer
106 views

Chern-Simons action for non-abelian brane worldvolume and Tsetlyn's symmetric trace prescription

I am trying to reproduce the results of the (famous) Myer's paper "Dielectric Branes" https://arxiv.org/abs/hep-th/9910053. I am struggling a bit to obtain the numerical factors in the equations. When ...
12
votes
1answer
222 views

Faddeev-Popov Determinant of Chern-Simons Theory

I am asking this question in order to figure out the expression of the Faddeev-Popov determinant given by Edward Witten is his paper "Quantum Field Theory and Jones Polynomial". Starting from the ...
0
votes
0answers
49 views

3d TQFT for Fibonacci (Yang -Lee) anyons

What is the 3d TQFT whose Wilson line produces Fibonacci (Yang -Lee) anyons? I heard that 3d $SO(3)_3$ Chern-Simons theory produces the correct physics for Fibonacci anyon ($e$). How to show it? If ...
1
vote
0answers
84 views

2D Chern Simons action by integrating out fermions

In Qi, Hughes, and Zhang's paper (https://arxiv.org/abs/0802.3537), they show how the Chern number appears as a coefficient of response function. Given the Hamiltonian (49) of a (2+1) or (4+1)D ...
6
votes
1answer
172 views

The Hilbert space of Chern-Simons on a torus, part one$.$

There is a key result in 2+1 dimensional Chern-Simons theory, which was first discussed in ref.1.: the Hilbert space of the theory, when quantised on $T^2\times\mathbb R$, is isomorphic to $$ \frac{\...
2
votes
0answers
75 views

Gauge anomaly from conformal dimension?

According to ref.1, the Chern-Simons theory $\mathrm{SU}(N)_k$ has a $\mathbb Z_N$ one-form symmetry with anomaly $$ \eta=\exp\left[-2\pi i \frac{k}{N}\right]\tag{4.12} $$ which, apparently, can be ...
3
votes
1answer
61 views

Does a SUSY Chern-Simons term prevent the dualising of the gauge potential to a scalar?

In 3D $\mathcal{N}=2$ supersymmetric field theory with abelian gauge fields, the gauge field $A_{\mu}$ is often dualised to a real scalar $\gamma$. Does a Chern-Simons term prevent this dual ...
2
votes
0answers
70 views

Does a Chern-Simons term break the $F \rightarrow \star F$ symmetry?

When is the electro-magnetic duality $F \rightarrow \star F$ a symmetry of a theory? I know it holds for free Yang-Mills, but would for instance a Chern-Simons term break it or a coupling to matter? ...
0
votes
0answers
43 views

Flux attachment to dynamical gauge field

It is a question about flux attachment. When I attach fluxes to dynamical gauge fields, something weird happened: an extra Hall conductivity term. We start from the action \begin{equation} \mathcal{L}...
2
votes
0answers
64 views

Formalization of the concept of a topological charge

I want to write precisely in mathematical terms what a topological charge is. This is what I have, but I am not sure of how correct it is. Let $M$ be spacetime. Quantization of $M$ in some QFT will ...
2
votes
2answers
115 views

Braiding matrix in Chern-Simons theories$.$

Consider a Chern-Simons system with gauge group $G$ and level $k$. Such a system can be used to model anyons, where the latter are identified with the integrable representations of $G$. One of the ...
3
votes
0answers
127 views

Which Chern-Simons TQFTs are spin?

Refs.1&2 prove several level/rank dualities among different 3d Chern-Simons theories. An important point is that some dualities involve, on one side, a theory that depends on the spin structure, ...
3
votes
0answers
150 views

Wess-Zumino-Witten vs. Yang-Mills-Chern-Simons and Kac-Moody$.$

There is a really nice (holographic) duality between 2d Wess-Zumino-Witten and 3d Yang-Mills-Chern-Simons models (cf. Ref.1). For example, for a given gauge group $G$, the spectrum of both theories is ...
8
votes
1answer
205 views

Equation of motion from $D=3$ Lorentz Chern-Simons action

In three dimensions, the well known Lorentz Chern-Simons action is $$ S_{\text{CS}}=\int\text{d}^3x\varepsilon^{\mu\nu\rho}\bigg(\omega_{\mu}{}^{ab}R_{\nu\rho ab}+\frac{2}{3}\omega_{\mu a}{}^{b}\...
1
vote
0answers
65 views

About the Lagrangian for Chern-Simons-Matter theories

I am particularly thinking of the theory described in section $6$ (starting page 31) of this paper, https://arxiv.org/abs/1104.0680. The exact Lagrangian has never been explicitly stated here but ...
2
votes
0answers
52 views

Why is the semiclassical approximation of the abelian Chern-Simons theory exact?

I was told that in abelian Chern-Simons theory (say, with a general level matrix $K$), semiclassical approximation is exact because there is no trivalent vertex, which in non-abelian case makes the ...
0
votes
0answers
53 views

Conserved quantity in Graphene

The computation of the band structure of Graphene basically leads to the diagonalization of the following Hamiltonian: $$ H = -t \left( \begin{array}{cc} 0 & \epsilon(\vec{k}) \\ \epsilon^*(\vec{...
2
votes
0answers
57 views

Can topological degrees of freedoms interact?

Question Can topological degrees of freedom interact with: other topological dof's other local dof's Premise For concreteness, I can't understand why the following term will not be allowed in a ...
6
votes
0answers
292 views

Level-rank duality in WZW models and CS theories

Cross-posting from Physics Overflow: https://www.physicsoverflow.org/41281/level-rank-duality-in-wzw-models-and-cs-theories I know that the classical level-rank duality in the $\widehat{\mathfrak{sl}}...
4
votes
0answers
95 views

Precise justification for quantization of Chern-Simons level

Consider $U(1)$ Chern-Simons theory on some three-manifold M: $$S = \frac{k}{4\pi}\int_M A \wedge dA.$$ The standard argument for why we require $k\in \mathbb{Z}$ comes from demanding invariance under ...
1
vote
0answers
55 views

About 2+1 General Relativity [closed]

I'm currently studying on how to write GR as a CS theory, but i have a problem with one of the basic theorem of this subject. $$ e^\mu_{\ \ a} \epsilon^{\nu \rho \sigma} R_{\ \ \rho \sigma}^a = \det(...
1
vote
0answers
51 views

Are these 2D conformal field theories the same or related?

Consider the following 2D conformal field theories: 26D bosonic string theory compactified on the $\Lambda_{24}$ Leech lattice torus. 10D N=1 Superstring Theory (any of the 5 as they are all dual) ...
8
votes
1answer
173 views

Allowed anyons for Chern-Simons at level $k.$

Ref.1. proves that the allowed representations of Chern-Simons $\mathrm{SU}(2)_k$ are those with dimension $$ \dim(R)\le k+1\tag{7.53} $$ Question: Is the generalisation of $(7.53)$ to arbitrary $N$ ...
5
votes
1answer
160 views

Meaning of Yang-Baxter equation for classical $r$-matrix

I'm reading this [math/9802054] paper on the structure of the phase space of Chern-Simons TQFT. I'm stuck at the definition of the classical $r$-matrix, which goes as follows: This might sound dumb, ...
16
votes
1answer
518 views

The system is topological; so what?

Lately I've been studying topological quantum field theory, mainly from mathematically oriented sources and, while I understand why a mathematician would care about TQFT's, I'm not yet quite sure why ...
1
vote
0answers
58 views

Path-Integral of Charged Particle in Chern-Simons Gauge Fields

From the paper "Fermi-Bose Transmutations Induced by Gauge Fields" by Polyakov, http://inspirehep.net/record/22956 http://dx.doi.org/10.1142/S0217732388000398 the theory in 3D, $$\mathcal{L}=\sum_{...
1
vote
0answers
53 views

Quantum Spin Hall Effect : Dimensional reduction from Chern Simons 3 form

I have a problem matching the two definitions of the Z2 topological invariant for a 2D system, assuming Sz symmetry. I can either define it from 1D to 2D, as a Chern number per spin: $$ C_{\rm s} = \...
0
votes
0answers
44 views

What is spin/charge relation in condensed matter physics?

I have been studying the paper Gapped Boundary Phases of Topological Insulators via Weak Coupling. On page 13, it talks about the spin/charge relation. Let $a^{1}$, $a^{2}$, ... ,$a^{n}$ be $U(1)$-...
5
votes
2answers
156 views

Symplectic reduction to moduli space in Chern-Simons theory

In QFT and the Jones polynomial, Witten claims that it is possible to perform symplectic reduction from the distributional Poisson bracket on the unconstrained phase space to a symplectic structure on ...
6
votes
1answer
247 views

Path Integral of Chern-Simons Theory

Can the path-integral of Abelian Chern-Simons theory be valuated exactly? $$\int \mathcal{D}[A] \exp\left\{{\frac{i}{2\pi}\int A\wedge dA}\right\}$$ I found Witten's paper "Quantum Field Theory and ...