left-right symmetric combinations of unitary irreducible representations form a complete basis in $\mathcal L^2($AdS$_3)$.

This space of states, if thought of as the complete spectrum, gives rise to two problems. On the one hand, unitarity imposes a seemingly arbitrary upper bound on the mass of string states in AdS$_3$, so that the internal energy of the string could not be too high. On the other hand, it gives no account of long string configurations, known to be present in the theory from both the classical and semiclassical approaches.

Both puzzles were solved in [@Maldacena:2000hw], where the spectrum was proposed to be enhanced by the so-called spectral flow. Spectral flow automorphisms are parametrized by an integer number $\omega$, known as the spectral flow charge, classically related to the number of times a long string winds around the AdS$_3$ boundary. Given such number, the corresponding map is defined by $$\label{auto1}
    J^3_n \rightarrow J^3_n - \frac{k}{2}\omega \delta_{n,0},$$ $$J^{\pm}_n \rightarrow J^{\pm}_{n\pm \omega},$$ for $n\in\mathbb Z$, and similarly for the antiholomorphic currents. This current algebra isomorphism turns to be also a Virasoro-Sugawara automorphism with $$\label{auto3}
    L_n \rightarrow L_n+\omega J_n^3-\frac{k}{4}\omega^2\delta_{n,0},$$ for $n\in\mathbb Z$, and correspondingly for the $\bar L_n$.

Unlike rational models, which have underlying compact 