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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.

Wess–Zumino–Witten (WZW) models describe σ-models summarizing flavor-chiral anomalies in low-energy physics, with topological significance. Such terms trivialize the torsional curvature of the respective manifolds, leading to infrared fixed points of the RG. They extend to a type of two-dimensional conformal field theory with a symmetry based on the affine Lie algebra built from the corresponding Lie algebra, or any conformal field theory whose symmetry algebra is an affine Lie algebra.

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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 free Majorana fermions. The $SU(2)_1$ model has…
octonion
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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) $SO(2)_1$ conformal field theory (CFT), whereas…
Ruben Verresen
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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_{S^2}d^2z\ \operatorname{tr}(g^{-1}\partial_\mu g…
Gold
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$F$-symbols for compact Lie groups

Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices $$ B\begin{bmatrix}j_1&j_2\\i&k \end{bmatrix}\colon \bigoplus_p V_{j_1p}^i\otimes V_{j_2k}^p\to V_{j_2q}^i\otimes V_{j_1k}^q $$ which implement the duality…
Delmastro
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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 on $S^2$ is $$ B = \exp \left( \frac{4 \pi i}{k +…
Prof. Legolasov
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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 famously observed that the central brane-charge…
Urs Schreiber
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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 principle for WZW models. However, we can…
alpha
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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 $${\cal…
Gold
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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}'(\partial^{\mu}g^{-1}\partial_\mu g) +…
Kaixiang Su
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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 there a generalization of this for chiral…
AccidentalFourierTransform
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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 of colors. To fix this, one can add WZW term to…
Nikita
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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 what is, in practice, the integrablity condition…
Ariel
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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 $c =1$ Bosonic theory on a circle
ann marie cœur
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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 that A non-abelian bosonisation introduced by…
Joshuah Heath
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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 WZW currents of $U(N)_k$ $$ J(z) = \sum_{n \in…
Joe
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