at the same point. Scattering amplitudes involving operators in the space-time representation and integrated over the string worldsheet acquire a similar interpretation as correlation functions on the dual two-dimensional CFT.

For unflowed primaries, the definition of the coordinate basis vertex operators comes from the Euclidean model as microscopic states, [*i.e.*]{} through analytic continuation. The corresponding correlators follow analogously from those of the H$_3^+$-WZNW model. Now, when dealing with spectral flowed primary fields the situation is more complicated since these operators generally lie in representations with energy unbounded from below. A solution for this issue was proposed in [@Maldacena:2001km]. An arbitrary lowest-energy state can be seen from a spectral flowed frame with $\omega > 0$ as the lowest-weight state of a certain discrete representation of the global algebra generated by the zero-modes with a spin $J$ being equal to $-m-k\omega/2$. Similarly, if the flow number $\omega$ is negative, the associated spectral flow automorphism maps the same state into the highest-weight state of a discrete representation with $J=m+k\omega/2$. The algebra generated by the $J_0^a$ is identified with the space-time isometry algebra acting on the background and the global $\mathfrak{sl}(2,\mathbb C)$ symmetry algebra of the CFT at the boundary. Therefore, vertex operators in 