On spectral flow symmetry and Knizhnik-Zamolodchikov equation

It is well known that five-point function in Liouville field theory provides a representation of solutions of the SL(2,R)k Knizhnik-Zamolodchikov equation at the level of four-point function. Here, we make use of such representation to study some aspects of the spectral flow symmetry of slˆ(2)k affine algebra and its action on the observables of the WZNW theory. To illustrate the usefulness of this method we rederive the three-point function that violates the winding number in SL(2,R) in a very succinct way. In addition, we prove several identities holding between exact solutions of the Knizhnik-Zamolodchikov equation.