A Note on Z_2 Symmetries of the KZ Equation

TL;DR

This paper explores hidden Z_2 symmetries in the four-point KZ equation, providing a simplified relation between spectral flowed and non-spectral flowed correlators, which aids in understanding string scattering in AdS_3.

Contribution

It introduces a formula linking spectral flow states to non-spectral flow four-point functions without solving complex five-point functions, utilizing functional relations and conformal theory connections.

Findings

Derived a simple expression for spectral flow correlators

Connected winding violating and conserving four-string scattering processes

Utilized functional relations and conformal theory links

Abstract

We continue the study of hidden Z_2 symmetries of the four-point sl(2)_k Knizhnik-Zamolodchikov equation iniciated in hep-th/0508019. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector w=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 AdS_3 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 non-violating 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, non diagonal functional relations between different solutions of the KZ 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 WZNW and Liouville conformal theories.