space of functions on $\mathbb C$ by means of the differential operators $$\mathcal D^+_j=x^2\partial_x-2jx, \qquad\qquad \mathcal D^0_j=x\partial_x-j,\qquad\qquad\mathcal D^-_j=\partial_x,\nonumber$$ together with their complex conjugates. $\mathcal P_j$ is then extended to a representation of the full current algebra by requiring $J_n^a \mathcal P_j=\bar J_n^a \mathcal P_j=0$ for $n>0$, and freely generating $\mathcal R_j$ by acting with $J_n^a$ and $\bar J_n^a$ for $n<0$.

Representations $P_j$ and $P_{-1-j}$, and thus $\mathcal P_j$ and $\mathcal P_{-1-j}$, are isomorphic, and so are their affine extensions $\mathcal R_j$ and $\mathcal R_{-1-j}$. The form of the unitary intertwining operator will be given explicitly below. It is useful to extend ${\cal V}_{\mbox{\scriptsize H}_3^+}$ by setting $\mathcal C=-1/2+i\mathbb R$ in Eq. (\[decomposition\]) instead of $\mathcal C^+$, and then quotienting the model by the aforementioned equivalence.

Every state $|j,x\rangle \in \mathcal V_{\mbox{\scriptsize H}_3^+}$ is associated with a vertex $\Phi_j(x|z)$, $x,z \in \mathbb C$, by virtue of the state-operator correspondence, [*i.e.*]{} $$\label{stateoperator}
|j,x\rangle=\lim_{z\rightarrow0}\Phi_j(x|z)|0\rangle,\qquad\qquad\langle j,x|=\lim_{z\rightarrow\infty}|z|^{4\Delta}\langle0|\Phi_{-1-j}(x|z).$$ The vertex operators satisfy the following OPE with the currents, $$J^a(z)\Phi_j(x|w)\sim-\frac{D^a_j\Phi_j(x|w)}{z-w},\nonumber$$ and similarly for the antiholomorphic currents. The operator $\Phi_j(x|z)$ is not only an affine primary but also a primary for the Sugawara-Virasoro algebra, its conformal weight being $$\Delta_0=-b^2j(1+j),\qquad b^2=\frac{1}{k-2}.\nonumber$$ Semiclassically, $\Phi_j(x|z)$ can be identified with the wave 