means of vertex operators in the so-called $m$-basis, where the label $m$ is introduced in order to keep track of the eigenvalue of $J_0^3$ in the unflowed frame. This basis is the best suited for a Wick rotation from the $H_3^+$-WZNW model as well as to further include a spectral flow charge while computing correlation functions. We shall denote the affine primary fields realizing the unflowed spectrum by $\Phi_j(m|z)$ and their images under a spectral flow automorphism by $\Phi^{\omega}_j(m|z)$. The OPE of these fields with the currents and their conformal weights are given by $$J^{3}(z)\Phi^{\omega}_j(m|w) = \frac{m+k\omega/2}{z-w}\Phi^{\omega}_j(m|w),\nonumber$$ $$J^{\pm}(z)\Phi^{\omega}_j(m|w) = \frac{\mp j + m}{(z-w)^{1 \pm \omega}}\Phi^{\omega}_j(m\pm 1|w),\nonumber$$ $$\Delta=\Delta_0-\omega m - \frac{k}{4} \omega^2,\nonumber$$ and analogously for the antiholomorphic counterparts. Let us stress that spectral flowed primaries $\Phi^{\omega}_j(m|z)$ with non-trivial $\omega$ are not affine primaries. However, after the Sugawara construction in the spectral flowed frame, it can be proved that they are indeed conformal primary fields.

Unflowed vertex operators in the Lorentzian model are related to states in the H$_3^+$-WZNW model through the following Mellin-like transform, $$\label{mellin}
\Phi_j(m|z) = \int d^2x \, \left|x^{j+m}\right|^2 \Phi_{-1-j}(x|z),$$ where the integrated operator corresponds either to a macroscopic state if $\Phi_j(m|z)$ lies in a continuous series or to 