String correlators in AdS$_3$ from FZZ duality

Motivated by recent works in which the FZZ duality plays an important role, we revisit the computation of correlation functions in the sine-Liouville field theory. We present a direct computation of the three-point function, the simplest to the best of our knowledge, and give expressions for the $N$-point functions in terms of integrated Liouville theory correlators. This leads us to discuss the relation to the $H^+_3$ WZW-Liouville correspondence, especially in the case in which spectral flow is taken into account. We explain how these results can be used to study scattering amplitudes of winding string states in AdS$_3$.