  The insertion of the spectral flow operator in a string scattering amplitude
on AdS$_3\times \mathcal{N}$ produces a change in the winding number of one of
the incoming (or outgoing) states, making it possible to compute amplitudes of
processes in which winding number in AdS$_3$ is not conserved. The insertion of
such operator, however, might seem artificial from the worldsheet theory
perspective, as it appears as an unintegrated vertex operator of conformal
dimension zero that does not represent any normalizable state. Here, we show
that the spectral flow operator naturally emerges in the Liouville field theory
description of the WZW correlation functions once it is combined with a series
of duality relations among conformal integrals. By considering multiple
insertions of spectral flow operators, we study the dependence on the moduli
for arbitrary number of them, and we show explicitly that the amplitude does
not depend on the specific locations of the accessory insertions in the
worldsheet, as required by consistency. This generalizes previous computations
in which particular cases were considered. This can also be thought of as an
alternative proof of the WZW-Liouville correspondence in the case of maximally
winding violating correlators.
