Note on Z2 symmetries of the Knizhnik-Zamolodchikov equation

Abstract

We continue the study of hidden Z2 symmetries of the four-point sl (2) k Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector ω=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS3 has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the Knizhnik-Zamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both Wess-Zumino-Novikov-Witten (WZNW) and Liouville conformal theories. © 2007 American Institute of Physics.