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 similarly write the current algebra for any 2d CFT as a Kac-Moody algebra(explained below), for example in Equation 2.60, Section 2.6, Page 37 of [1]. Is it then possible to make any 2d CFT into a WZW model by designing a Verma module which is a representation of the Kac-Moody algebra for the given CFT (For example this is done for free bosons in Section 15.6 of the Yellow book) ?
[1] Introduction to Conformal Field Theory: With Applications to String Theory by Plauschinn and Ralph Blumenhagen
Argument why conserved current forms a current algebra -
Consider quasi-primary chiral fields, $\phi_i(z)$ and $\phi_j(z)$, with mode expansion given by $\phi_i(z) = \sum_{m} \phi_{(i)m}z^{-m-h_i}$ and similarly for $\phi_{j}$. Then using contour integral representation of the modes and OPE of two chiral quasi-primary fields(expressed in terms of other quasi-primary fields)
$$ \left[\phi_{(i) m}, \phi_{(j) n}\right]=\sum_{k} C_{i j}^{k} p_{i j k}(m, n) \phi_{(k) m+n}+d_{i j} \delta_{m,-n}\left(\begin{array}{c} m+h_{i}-1 \\ 2 h_{i}-1 \end{array}\right) $$ where \begin{aligned} &p_{i j k}(m, n)=\sum_{r, s \in Z_{0}^{+}} C_{r, s}^{i j k} \cdot\left(\begin{array}{c} -m+h_{i}-1 \\ r \end{array}\right) \cdot\left(\begin{array}{c} -n+h_{j}-1 \\ s \end{array}\right) \\ &r+s=h_{l}+h_{j}-h_{k}-1 \\ &C_{r, s}^{i j k}=(-1)^{r} \frac{\left(2 h_{k}-1\right) !}{\left(h_{i}+h_{j}+h_{k}-2\right) !} \prod_{t=0}^{s-1}\left(2 h_{i}-2-r-t\right) \prod_{u=0}^{r-1}\left(2 h_{j}-2-s-u\right) \end{aligned} Now, let us assume that our theory has N - chiral quasi-primary fields of conformal dimension 1(which [1] calls current). It can be shown using the above relation that $$\left[j_{(i) m}, j_{(j) n}\right]=\sum_{k} C_{i j}^{k} p_{111}(m, n) j_{(k) m+n}+d_{i j} m \delta_{m,-n}$$ which after rotating the fields and rescaling them becomes into $$ \left[j^{(i)}_{m}, j^{(j)}_{n}\right] = \sum_{l} f^{ijl} j^{l}_{m+n} + k m \delta^{ij} \delta_{m,-n} $$ where j with superscripts are the new fields. This is the affine Kac-Moody algebra structure.
