group symmetries, the spectral flow automorphisms generally give rise to non-equivalent representations when acting on a current module. An exception is given by the case of the spectral flow image of a representation induced by a lowest-weight discrete series with spin $j$, which is isomorphic to one built upon a highest-weight discrete series with a “reflected” spin $-k/2-j$ and an additional unit of flow. This module isomorphism, referred to as the series identification, restricts the discrete representations allowed in the spectrum to be either those induced by the lowest or by the highest-weight series, while constraining the spin to lie in the real interval $$-\frac{k-1}{2}<j<-\frac{1}{2}.\nonumber$$

It was conjectured in [@Maldacena:2000hw] that the full spectrum of the SL$(2,\mathbb R)$-WZNW consists of the standard spectrum as well as their spectral flow images, running over all possible values of $\omega$. Explicitly, the spectrum of the model $\mathcal V_{\mbox{\scriptsize SL}(2,\mathbb R)}$ decomposes as $$\begin{aligned}
&& \mathcal V_{\mbox{\scriptsize SL}(2,\mathbb R)} = \bigoplus_{\omega\in\mathbb Z}\left[\int_{-\frac{k-1}{2}}^{-\frac{1}{2}} dj \, \mathcal D^{\omega}_j + \int_{\mathcal C} dj \int_0^1 d\alpha \, \mathcal C^{\omega}_{j\alpha}\right],\nonumber\end{aligned}$$ where $\mathcal D^{\omega}_j$ and $\mathcal C^{\omega}_{j\alpha}$ are the spectral flow images of $\mathcal D^{+}_j$ and $\mathcal C_{j\alpha}$, respectively.

A suitable realization of $\mathcal V_{\mbox{\scriptsize SL}(2,\mathbb R)}$ is obtained by 