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.
218
questions
0
votes
0answers
39 views
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{...
2
votes
0answers
51 views
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 ...
4
votes
0answers
57 views
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 ...
1
vote
0answers
44 views
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:
$$\...
4
votes
1answer
96 views
Why is $T^*S^3$ a conifold?
So, I was reading the famous Gopakumar Vafa paper, and they mention that $T^*S^3$ is a conifold. Why is this the case? I would naively expect $T^*S^3$ to be basically the same everywhere ($S^3$ is a ...
4
votes
0answers
58 views
Can one quantize Chern-Simons theory in the covariant phase space formalism?
The covariant phase space, which coincides with the space of solutions to the equations of motion, gives a notion of phase space which does not rely on a decomposition of spacetime of the form $M=\...
1
vote
0answers
44 views
Self-loops in Chern-Simons theory
Consider the dumbbell graph decorated with propagator $P$ one the edges and with integration variables $x$ and $y$ on the vertices.
We associate to it the following integral:
$$ I = \int_{x,y} P(x,x)P(...
2
votes
1answer
65 views
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 ...
2
votes
0answers
67 views
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 ...
6
votes
0answers
78 views
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, ...
asked Sep 5 '20 at 15:27
AccidentalFourierTransform
39.8k19 gold badges92 silver badges177 bronze badges
2
votes
0answers
37 views
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 ...
0
votes
0answers
44 views
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 ...
0
votes
1answer
56 views
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?
7
votes
0answers
80 views
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+...
2
votes
0answers
29 views
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.
0
votes
0answers
21 views
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 ...
2
votes
0answers
44 views
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:
$$S[A] = \...
2
votes
1answer
137 views
Batalin-Vilkovisky (BV) form of the Chern-Simons Action
As seen in Section 4 of Chapter 5 of Costello, K. "Renormalization and Effective Field Theory", or in section 5.2 $L_\infty$-Algebras of Classical Field Theories and the Batalin-Vilkovisky ...
2
votes
0answers
34 views
Verlinde formula requires compact Riemann surface
Assume a Chern-Simons theory on a 3-mfld of the form $\Sigma_g \times S^1$, where $\Sigma_g$ is a Riemann surface of genus $g$. In his paper, Witten shows that one can use the Verlinde formula to get ...
1
vote
0answers
48 views
Chern-Simons theory for FQHE
Recently I have read something about describing FQHE within chern simons field theories. According to Atland's text book,Condensed matter field theory, one can map interacting fermions to composite ...
4
votes
1answer
126 views
Chern-Simons (CS) theory
I have a question about Constructuion of Chern-Simon Action.
In its paper "Non-commutative geometry and string field theory", Witten construct the Action of the String Field Theory inspiring ...
6
votes
0answers
63 views
$F$-symbols for compact Lie groups
Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices
$$
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_{...
1
vote
0answers
41 views
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 ...
2
votes
0answers
66 views
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 $...
3
votes
1answer
136 views
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:
$$ W[U] =...
0
votes
0answers
21 views
Chern-Simons Path integral restricting to small gauge transformations
How does one compute the Chern-Simons path integral in 2+1 dimensions considering only small gauge transformations? Is this even a well-defined theory?
3
votes
0answers
51 views
Chern-Simons theory with only small gauge transformations
Usually when one derives the variation of the Chern-Simons action in 2+1 dimensions, one has a term that is proportional to the winding number. Then one argues that the coupling constant must be an ...
0
votes
1answer
166 views
Why is this Chern-Simons theory gauge invariant?
I am reading the lecture notes in https://arxiv.org/abs/hep-th/9902115 and in it, it says that the Lagrangian
$$\mathcal{L}_{\mathrm{CS}}=\frac{\kappa}{2} \epsilon^{\mu \nu \rho} A_{\mu} \partial_{\nu}...
4
votes
0answers
63 views
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....
5
votes
1answer
264 views
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 ...
0
votes
1answer
55 views
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 ...
2
votes
1answer
138 views
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
$$
\frac{\theta}{8\pi^2}\langle F_A\wedge F_A\rangle = \frac{\theta}{32\pi^2}\langle F_A^{\...
2
votes
1answer
47 views
Is it true that in abelian Chern-Simons theory diffeomorphisms differ from ordinary gauge transformations trivially?
In Henneaux's Lectures on the Antifield BRST Formalism for Gauge Theories, it is claimed in Exercise 1 that diffeomorphisms $\delta_\xi A_\mu=\xi^\rho\partial_\rho A_\mu+\partial_\mu\xi^\rho A_\rho$ ...
2
votes
0answers
45 views
free fermion- monopole operator in scalar $QED_3+$ Chern simons term equivalence proof?
In most papers discussing 3D Abelian bosonization duality, they say that monopole operator in scalar $QED_3+CS$ is dual to free fermions.
How do they know it, because I have never seen an actual ...
4
votes
0answers
78 views
The value of Gravitational Chern Simons theory integration on some three manifolds
Consider the 3d gravitational Chern Simons theory
$$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 ...
2
votes
1answer
57 views
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)
\...
1
vote
0answers
63 views
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 ...
4
votes
1answer
87 views
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$-...
2
votes
1answer
60 views
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(\...
6
votes
1answer
181 views
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 ...
6
votes
1answer
255 views
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 ...
1
vote
1answer
195 views
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 ...
0
votes
0answers
26 views
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 ...
5
votes
0answers
72 views
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},...
0
votes
1answer
129 views
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 ...
1
vote
0answers
41 views
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 ...
1
vote
0answers
40 views
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 ...
0
votes
1answer
89 views
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)
$$
is satisfied the gap closes, meaning there ...
4
votes
1answer
116 views
Chern-Simons theory on a plane/sphere with a single charge insertion
Consider the pure Chern-Simons theory on the plane $\mathbb{R}^2$ with a single charge insertion in some representation $\rho$ of the group $G$. What does the Hilbert space look like? Is it null or ...
1
vote
1answer
105 views
The relation between Chern-Simons Theory and Yang-Mills Theory
So from this page, I know that there is a relation between Chern-Simons Theory and Yang-Mills Theory, but I have difficulty proving the identities in the document.
I was going to prove $$\partial_\...
