Questions tagged [wess-zumino-witten]

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7
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1answer
157 views

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_{...
0
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1answer
36 views

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
3
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1answer
36 views

Deformations of CFTs

Do you know of any textbooks/lecture notes that treat marginal and or relevant deformations of CFT's / WZW-models?
1
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0answers
42 views

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*}...
1
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0answers
38 views

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} \...
1
vote
1answer
54 views

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 $...
2
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0answers
44 views

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(\...
2
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0answers
53 views

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 ...
3
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0answers
71 views

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. ...
1
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0answers
33 views

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 ...
6
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0answers
65 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_{...
4
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0answers
75 views

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 ...
7
votes
1answer
541 views

$SU(2)$ 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 ...
2
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0answers
51 views

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 $AdS_3/CFT_2$? ...
2
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0answers
58 views

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 ...
4
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0answers
56 views

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 ...
1
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0answers
60 views

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}) + ...
1
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1answer
104 views

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}{...
3
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2answers
165 views

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 ...
3
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0answers
132 views

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 $c =1$ ...
6
votes
2answers
303 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 ...
2
votes
0answers
89 views

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 ...
3
votes
2answers
581 views

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 ...
2
votes
1answer
133 views

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 ...
7
votes
2answers
299 views

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) $...