# Weiss-Zumino-Witten models
Weiss-Zumino-Witten (WZW) models are completely solvable models of CFTs.
Given a Lie group $G$, the action for the corresponding WZW model is given by$\begin{aligned}S&=\frac{k}{4\pi} \int_{\mathrm{S}^2} \mathrm{~d}^2 z \operatorname{tr}\left(g^{-1} \partial_\mu g g^{-1} \partial^\mu g\right)\\&-\frac{k}{12 \pi} \int_B \mathrm{~d}^3 y \, \mathrm{i}\epsilon_{\alpha \beta \gamma} \operatorname{tr}\left(g^{-1} \partial^\alpha g g^{-1} \partial^\beta g g^{-1} \partial^\gamma g\right),\end{aligned}$where $g: S^2\to G$ is a map from $S^2$ to the group manifold $G$, and $B$ is a 3-ball with boundary $\partial B=S^2$. The second term is called the Weiss-Zumino term and is perturbatively independent of how the domain of the map $g$ is extended into $B$; non-perturbatively, the action can change by multiples of $2\pi \mathrm{i}$, but as long as $k$ is an integer, $e^{-S}$ is single-valued and therefore well-defined.
From the action, one finds conserved currents. There is a holomorphic current $J \equiv-k \partial g g^{-1}$ and an anti-holomorphic current $\bar{J} \equiv k g^{-1} \bar{\partial} g$. Focus now on the holographic current. Expanding it in modes,$J^a(z) \equiv \sum_{n \in \mathbb{Z}} J_n^a z^{-n-1},$leading to the following commutation relations:$\left[J_m^a, J_n^b\right]=k m \delta^{a b} \delta_{m+n, 0}+i f^{a b}{}_c J_{m+n}^c,$known as the affine [[0069 Kac-Moody algebra|kac-Moody algebra]].
Using [[0095 Sugawara construction|Sugawara construction]], one can obtain the stress tensor and show that the theory is actually conformal by deriving the [[0032 Virasoro algebra|Virasoro algebra]] satisfied by the stress tensor. One can also get the Virasoro algebra by letting $L_m=\frac{1}{2\left(k+h^{\vee}\right)}\sum_{n \in \mathbb{Z}}: J_n^a J_{m-n}^a:$and finding the commutation relations between $L_m