propagator unchanged, as expected.

Following [@Teschner:1997ft], states $|j,x\rangle \in \mathcal V_{\mbox{\scriptsize H}_3^+}$ and their duals will be referred to as “macroscopic states”. They are understood as distributions on dense subspaces of $\mathcal V_{\mbox{\scriptsize H}_3^+}$ and are delta-function normalizable, as it can be read from Eq. (\[2pt\]). Analytic continuations in $j$ of these states deserve the name of “microscopic states”, and their consideration is crucial for the study of the string in AdS$_3$. Moreover, in order to give account of long string configurations in AdS$_3$, we shall need to relax the strong constraint and consider vertex operators built on principal series with possibly different holomorphic and antiholomorphic spins. We will refer to them as microscopic states as well.

The SL$(2,\mathbb R)$-WZNW model and spectral flow {#sect3}
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The standard spectrum of the SL$(2,\mathbb R)$-WZNW model is obtained after imposing the Virasoro constraints on the affine extensions of $\mathcal C_{j\alpha}=C_{j\alpha}\otimes C_{j\alpha}$ and $\mathcal D^{\pm}_j=D^{\pm}_j\otimes D^{\pm}_j$, where $C_{j\alpha}$, $\alpha \in [0,1)$, $j\in\mathcal C$, denotes the principal continuous representations and $D^{\pm}_j$, $j<-1/2$, are the principal discrete series of $\mathfrak{sl}(2,\mathbb R)$ generated by the zero-modes $J^a_0$ and $\bar J^a_0$, respectively for the left and the right sectors. It is known from harmonic analysis that these 