Questions tagged [wess-zumino-witten]
May use for both 4d phenomenological theories of flavor chiral anomalies, and 2d CFTs involving affine Lie Algebras. Wess–Zumino–Witten (WZW) models describe σ-models with flavor-chiral anomalies, of topological significance. Such terms trivialize the torsional curvature of the respective manifolds, leading to infrared fixed points of the RG.
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How to derive WZW model’s energy-momentum tensor? The result is of course the Sugawara construction
I want to know how to derive WZW’s energy-momentum tensor. We know WZW action is
$$
S_{WZW}[g]=\frac{k}{16\pi}\int d^2x Tr(\partial g^{-1}\partial g) - \frac{ik}{24\pi}
\int_B d^3y \epsilon_{abc} Tr(h^...
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Fermionic generator from coset current algebra
I am recently studying some chiral algebra from physics perspective. I found many chiral algebras have nice coset realization of current algebra. However, most of the literatures are very algebraic. ...
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Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model
According to P. Francesco et al. conformal field theory book the central charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which ...
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WZW primary fields / correlations in terms of current algebra?
Cross-posted from a Mathoverflow thread! Answer there for a bounty ;)
Given the
$\mathfrak{u}_N$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the ...
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What does the WZ term in a WZW action means for string theory on group manifolds?
Let $G$ be a semi-simple Lie group. By Cartan's criterion its Killing form $B(X,Y)$ on $\frak g$ is non-degenerate. We can use it to define an inner product on the whole group by left translation
$${\...
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Holographic derivation of correlations functions in $\text{AdS}_3$/$\text{CFT}_2$
Intro
I am interested in finding the correlations functions $\langle A^{a_1}_z(z_1) \cdots A^{a_k}_z(z_k) \rangle$ in the frame of $\text{AdS}_3$/$\text{CFT}_2$ correspondance for 3D gravity and two ...
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Link complement states in Chern Simons theory
I have been trying to understand how to get an explicit quantum state from a given link in Chern-Simons. Lets say the compact gauge group being SU(2) (this seems to be the most widely studied). I ...
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Relation between WZW model and gauge transformation
I came across this question while reading Chapter 15 of Conformal Field Theory by Di Francesco.
So the action of the Weiss-Zumino-Witten(WZW) model is as follows:
$$S = \frac{1}{4a^2}\int d^2x {\rm Tr}...
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Is there a character ring for quantum groups?
It is a well known fact that for any (reasonable) group $G$, the character ring and the representation ring are isomorphic,
$$
\chi_{R_1}(g)\chi_{R_2}(g)=\chi_{R_1\otimes R_2}(g),\qquad g\in G
$$
Is ...
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CFTs that are not modular invariant
Are there any 2d CFTs that do not have modular invariant partition functions? All the examples that I know of, like the free boson, WZW models, etc. have modular invariant partition functions.
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2d CFT and WZW model
I have been using Lorenz Eberhardt's 2019 ESI lecture notes on WZW model. Below Equation 3.5 on Page 8, it is written that the current algebra, which forms a Kac-Moody Algebra, is the main organizing ...
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Conformal dimension of conserved current and Current Algebra in CFT
In the 2019 ESI lecture notes on WZW model, right before equation (3.1) Lorenz Eberhardt claims that any conserved current of a CFT has conformal weight (1,0) or (0,1). Can someone please explain why ...
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String theory in ${\rm AdS}_3$ and the ${\rm SL}{(2,\mathbb{R})}$ WZW model on the worldsheet
The WZW model on the sphere $S^2$ with group $G$ and level $k$ is described by the action for a $G$-valued field $g : S^2\to G$ (see these notes by Lorenz Eberhardt):
$$S[g]=\dfrac{1}{4\lambda^2}\int_{...
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What is the difference between topological theta term and Wess-Zumino-Witten term?
It seems that they both proportional to some thing like $\vec{n}\cdot \partial_{\tau}\vec{n}\, \times\,\partial_{s}\vec{n}$.
References: Fradkin, Quantum field theory: an integrated approach
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Deformations of CFTs
Do you know of any textbooks/lecture notes that treat marginal and or relevant deformations of CFT's / WZW-models?
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Determinant of differential operator as exponential of Wess-Zumino-Witten action
I am currently reading this paper (Mass Gap and Confinement in (2+1)-Dimensional Yang-Mills Theory, Dimitra Karabali) and between equations (6) and (7) the following identity is used:
\begin{equation*}...
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Finding OPE from conformal Ward identity WZW model
I'm working through section 15.1.3. of Di Francesco's CFT textbook. I don't understand the steps going between (15.42) and (15.43). They say to substitute
$\delta_\omega J = \sum_{b,c} i f_{abc} \...
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RG flow of 4d Nonlinear Sigma model with $SU(n)$ target space
Let's consider the 4d Nonlinear Sigma model with $SU(n)$ target space, without a topological term. The Lagrangian is
$$\frac{f^2}{16}\mathrm{Tr}(\partial_{\mu}U^{-1} \partial^{\mu}U)$$
where $U$ is a $...
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Symmetry generating commutator in Witten's treatment of WZW model
In 'Non-abelian Bosonization in Two Dimensions', Witten writes down the commutation relations between currents and fields of the WZW model in equation (33):
\begin{equation}
\bigg[\frac{1}{2\pi}\bigg(\...
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Why does Witten not include higher order quantum corrections when quantizing Poisson brackets in the WZW model?
In 'Non-abelian Bosonization in Two Dimensions', Witten argues that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
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WZW model - affine Kac-Moody current algebra from Quantum Group exchange algebra
In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model.
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Conformal dimension of conserved current of WZW model
The holomorphic part of the conserved current of a Wess-Zumino-Witten model is given by
$ J = - k \partial g g ^ { - 1 } $, where $ g $ is a map from $ S^ 2 $ to some Lie group.
It is claimed that ...
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$F$-symbols for compact Lie groups
Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices
$$
B\begin{bmatrix}j_1&j_2\\i&k
\end{bmatrix}\colon \bigoplus_p V_{j_1p}^i\otimes V_{j_2k}^p\to V_{...
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Spin of skyrmion
Baryons can be considered as solitions in Skyrme model(See also this post.):
Such Lagrangian haven't any information about number of colors. Bosonic or fermionic nature of baryons depends on number ...
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$SU(2)$ and $SO(3)$ WZW models
It seems that the $SU(2)_1$ and $SO(3)_1$ Wess-Zumino-Witten models are quite different despite the Lie algebras being identical. The $SO(3)_1$ model has central charge 3/2 and is equivalent to 3 ...
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Do large $N$ free fermion or WZW theories have a holographic dual in $AdS_3/CFT_2$?
I was wondering if for $N$ free Dirac fermions (or equivalently by bosonization, $N$ free bosons or an $SU(N)_1$ WZW theory plus an extra boson) have a holographic dual description via $AdS_3/CFT_2$?
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Is there an explicit mapping between N free bosonic fields and the $SU(N)_1$ WZW model + free boson?
Witten's nonabelian bosonization tells us that $N$ free Dirac fields can by written in terms of an $SU(N)_1$ WZW model and one free boson. But bosonization also tells us that we could just as well ...
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Integrability condition of perturbations of Wess-Zumino-Witten (WZW) models
When one tries to analyze the renormalization group of marginal perturbations of Wess-Zumino-Witten (WZW) model in 1+1d, only those "integrable perturbations" can be computed analytically. I wonder ...
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Renormalization of the Wess Zumino Witten term
I was learning about the Wess-Zumino-Witten model and I encountered the the following 2-dimensional Lagrangian
$$ \mathcal{L} = \frac{1}{4\lambda^2} \text{Tr}(\partial_\mu g ~\partial^{\mu} g^{-1}) + ...
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Factor of $1/2$ in the Sugawara construction
I'm trying to reproduce the Sugawara construction calculation using this reference (page 14).
The normal-ordering of two local operators is defined as
$$ N(XY)(w)=\frac{1}{2\pi i} \oint_w \frac{dx}{...
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Kac-Moody algebra from WZW model via Poisson brackets
In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
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Lattice model realization of $SU(2)$ WZW model at level $k$?
Is there any lattice model realization of the following model:
$c=1$ boson at the self-dual radius, or the $SU(2)$ WZW model at $k=1$.
This is a question inspired by: Orbifolds of the $c =1$ ...
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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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Physical meaning of the WZW action and Lagrangian
What is the (super)-WZW term physical meaning? I mean, what is the physical importance of the Wess-Zumino-Witten action/Lagrangian in superstrings/M-theory and or field theory (not stringy if ...
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When are we required to use the Wess-Zumino term?
I was recently reading about non-Abelian bosonization, and I had a question concerning the Wess-Zumino term. In particular, I have been reading this short introduction by Ivan Karmazin, which states ...
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Computing OPEs of primary fields
I'm trying to compute the OPEs of two specific primary fields in a WZW model. The issue is that I can't apply the state-field correspondence since these fields don't belong to the vacuum module (so I ...
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Is there any qualitative difference between the WZW $SO(2)_1$ and the WZW $SU(2)_1$ CFT?
Consider the anisotropic spin-$\frac{1}{2}$ Heisenberg chain $$H = \sum_{n=1}^N S^x_n S^x_{n+1}+S^y_n S^y_{n+1} + \Delta S^z_n S^z_{n+1}$$
which for $\Delta = 0$ realizes the Wess-Zumino-Witten (WZW) $...
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Extended SUSY from the kappa-symmetry WZW terms
In
José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443 (spire)
it was ...
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