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 $U(N)_k$ $$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-1} $$ satisfy the Kac-Moody commutation relations of their mode expansion $$ [J^a_n, J^b_m] = \sum_c f^{a,b}_c J^c_{n+m} + k \cdot n \delta_{n+m,0} \mathrm{tr}( L^a L^b ) $$.

These currents generate the algebra of operators of the WZW theory.

[EDIT: THIS IS NOT QUITE TRUE. See the below explanation.]

The algebra of operators of WZW theory (say for $U(N)_k$) is generated of two kinds of operators.

- The currents $J^a(z)$.
- The 'primary fields' $\Phi^\rho$. Mathematically, $\Phi^\rho(0)$ is the intertwiner $M_{\mathrm{Id}} \to M_{\rho}$. The vacuum module $M_{\mathrm{Id}}$ has highest-weight vector $|\mathrm{Id}\rangle$. $M_{\rho}$ is generated by a highest weight vector $|\rho\rangle$.
The currents $J$ preserve the module $M_\rho \to M_\rho$. In general, $\Phi^\rho(z)$ is also an intertwining operator from $M_{\mathrm{Id}} \to M_{\rho}$. It is related to $\Phi^\rho(z)$ by conjugation with the evolution operator $e^{z \cdot H}$, where the Hamiltonian $H$ is a function of the currents $J$.

In particular, the primary fields $\Phi^\rho$ satisfy the OPEs $$J^a(y)\Phi^\rho(z) = - \frac{\rho(L^a)\Phi^\rho(z)}{y-z} + \text{reg.}$$ where $\rho$ is an `integrable' representation of the algebra and $\Phi^\rho(z)$ live in the representation space of the algebra.

So in principle, it should be possible to write all primary fields purely in terms of WZW currents. Or it should be possible to write correlations in terms of current correlations. Has this been done before?

EDIT: In lieu of Nikita's response, I think a better question would be *can nonzero correlations of vertex operators be written in terms of the current algebra*? Although by 'NOTE 2' below, maybe even this isn't necessary.

In the case of products of vertex operators (say for a single boson, which corresponds to a $U(1)_1$ WZW model), vertex correlations with some neutrality condition $\sum_i {\alpha_i} = 0$, the correlator $\langle e^{i \alpha_1 \phi(x_1)} \cdots e^{i \alpha_n (x_n)} \rangle$ can be written in terms of the currents $\epsilon_{\mu\nu} \partial^\nu \phi$ since the argument of the exponential can be written as

$$\sum_i \alpha_i \phi(x_i) = - \alpha_1 \int_{x_1}^{x_2} dx^\mu \partial_\mu \phi(x) - (\alpha_1 + \alpha_2) \int_{x_2}^{x_3} dx^\mu \partial_\mu \phi(x) - \cdots - (\alpha_1 + \alpha_2 + \cdots + \alpha_{n-1}) \int_{x_{n-1}}^{x_n} dx^\mu \partial_\mu \phi(x).$$

As such, these products of vertex operators are written as a 'non-local' integral of currents away from the insertion points. However, note that these representations are not 'universal' in the sense that there may be many ways to write these integrals.

For $U(1)_k$, primary fields correspond to restricting $\alpha_i \in \frac{1}{\sqrt{k}} \mathbb{Z}$, so such a presciption does indeed exist. The question is now to generalize the above to $U(N)_k$.

NOTE: Regarding $U(1)_k$, a somewhat more aesthetic way to write the correlations are in terms of $k$ independent boson fields $\{\phi^{(1)} \cdots \phi^{(k)}\}$, $e^{i \frac{n}{\sqrt{k}} \phi(x)} \mapsto e^{i \frac{n}{k} \sum_{i=1}^{k} \phi^{(i)}(x)}$, which trades a nasty factor of $1/\sqrt{k}$ for a factor $1/k$. This is also nice because the current of $U(1)_k$ can be written as $J = \sum_{i=1}^{k} \epsilon_{\mu \nu} \partial^\nu \phi^{(i)}$, the diagonal algebra of $k$ boson currents. So, the notion of $k$ independent layers of the currents should somehow play a role in such constructions.

NOTE 2: Note that if we impose $\phi(z) \xrightarrow{z \to \infty} 0$, we'd have $\phi(z) = \int_{\infty}^{z} dx^{\mu} \partial_\mu \phi(x)$, so even the operators $e^{\alpha \phi(z)}$ by themselves have such a representation.

correlatorsof primary fields. I think my question translates as follows. Take primaries $\Phi_1(z_1), \cdots, \Phi_n(z_n)$ corresponding to punctures $z_i$ and modules $M_i$. How does one write the projection to the vacuum sector of the operator $\Phi_1(z_1) \cdots \Phi_n(z_n)$ in terms of the VOA? This is only nonzero if $M_1 \otimes \cdots \otimes M_n$ decomposes with an identity representation. $\endgroup$