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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Can we construct a Chern-Simons theory provided that the ground state is degenerate and gapped, like the Abelian fractional quantum Hall effect?

I am studying the Chern-Simons approach to fractional quantum Hall effect, which a special focus on the topological order in the context of Abelian fractional quantum Hall effect. To me, the logic to ...
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Knots in 3d pure gravity

Three-dimensional pure gravity on $\mathrm{AdS}_3$ can be described as a Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$,with action $$ S[A,\bar{A}] = \...
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Non-abelian Chern-Simons from fermion effective action

(1) Starting from the fermion effective action $$S_\text{eff}[A,m] = \log \det(i\gamma^\mu{\partial_\mu} + \gamma^\mu A_\mu + m)\tag{223}$$ once can do a loop expansion following https://arxiv.org/abs/...
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Non-Abelian Chern-Simons path integral on a torus

Is it possible to exactly evaluate the Chern-Simons path integral with a non-compact gauge group (say $SU(2)$) on a torus? I am asking this because 3d gravity is an $SL(2,\mathbb{R})$ Chern-Simons ...
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Chern-simons term to total derivative

I'm trying to prove $$Tr[G_{\mu\nu} \tilde{G}^{\mu\nu}]=2\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}Tr[A_{\nu}G_{\rho\sigma}-\frac{2}{3}iA_{\nu}A_{\rho}A_{\sigma}]$$ expanding the L.H.S. I don't know ...
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Why can't we define mathematical observables in asymptotic $dS$ or flat space for gravitational theories?

In higher spin currents, the boundary CFT is dual to an asymptotic $AdS$. I have heard that $dS$ is not quantizable. But I don't understand why we want it to be in the first place. Isn't Chern-Simons ...
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Is a Wilson line evaluated in zero cosmological constant equal to correlation functions in 2D CFT's?

The path Integral defined by the Wilson lines over some connection $A\subset Hol$ for correlation functions dominated by the vacuum block is $e^{2iL_0}$ evacuated at <0|$e^{2iL_0}$|0>. Does this ...
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What does it mean for correlation functions to be dominated by the vacuum block for a 2D CFT?

In a 2D CFT, correlation functions dominated by the vacuum block have no conical defects. You can calculate the OPE and determine the correlation function using the D-S equations and cancel out UV ...
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Evaluating the $A \land A \land A$ in the Chern-Simons action

I am trying to evaluate $A \land A \land A$, but I am a bit confused on how exactly to do it and produce the usual notation used in physics. I am trying to use the definition of the wedge product of ...
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Normalization of $U(1)$ gauge fields

In G. W. Moore, “Introduction to Chern-Simons theories.” 2019 TASI School. [Online]. Available: https://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf the $U(1)$ gauge field has a ...
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Chern-Simon level quantization and quantum Hall effect

It is well-known that integer and fractional quantum Hall effect can be effectively described by $U(1)$ abelian Chern-Simon theory. In both cases, quantization(fractionalization) of Hall resistance is ...
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Motivation for 3D Quantum Gravity

I was briefly going through the idea of 3d quantum gravity on nLab, where it is stated that: The case of dimension 3 is noteworthy, because in this case the quantum theory can be and has been fairly ...
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Regarding a possible duality between (2+1)D gravity and Chern-Simons Theory

Is there a duality between (2+1)D gravity and Chern-Simons Theory? Or they merely have related features? If so, of which type and why?
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Is the "Push-Down" Quantization of Chern-Simons Theory part of a more general approach to Quantization?

I've recently started reading Axelrod, Della Pietra and Witten's original paper about the quantization of Chern-Simons theory. I'd like to know if the "push-down" quantization strategy they ...
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Propagator, mass and electrostatic potential of an modified 2+1 dimensional Maxwell action

Consider the modified Maxwell action in 2 + 1 dimensions $$S=\int d^3x[-\frac{1}{4} F^{\mu\nu} F_{\mu\nu}+\frac{\theta}{2}\epsilon^{\alpha\mu\nu}A_\alpha F_{\mu\nu}] .$$ The action invariant under ...
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Chern-Simons Lagrangian and gauge-fixing

Main question: Consider (2+1)D Chern-Simons action $$S = \int dt d^2\mathbf r \frac{k}{4\pi} \epsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda.$$ Assuming the Coulomb gauge $\nabla\cdot \mathbf a ...
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Are central charges equal or similar to irreducible spinor representations?

First of, if any of the following below does not make sense, please feel free to leave a comment =) Central charges in Chern Simons in in the Virasoro conformal blocks play an important role for ...
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Chern-Simons term for a non-abelian gauge multiplet

In equation (20.9) of Freedmann and Van Proeyen's Supergravity, it is stated that for the following Chern-Simons term: $$S_{\mathrm{CS}} = C_{IJK}\int A^I\wedge F^J \wedge F^K$$ to be invariant under ...
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Topological defects in general and Chern-Simons in particular

I'm trying to gain intuition on some physical concepts that I cannot yet fully understand, and I think many of you can help me. Is it correct to think of of a topological defect as the addition ad hoc ...
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Fukui method for non-square lattice

How to generalize the Fukui method in his 2005 article "Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances" to the case where the lattice is ...
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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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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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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 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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