Some remarks on the free fields realization of the bosonic string on $AdS_{3}$

We discuss the classical solutions of the equations of motion and thepossible boundary condition for a bosonic string with Kalb-Ramond background inAdS3. It turns out that there exists three different physical sectors and thatit is also possible to describe the motion on an extremal black holebackground. The existence of three sectors clearly shows how one of the spectraproposed is incomplete. We consider also the classical 'canonical'transformation which maps the string fields to the classical Wakimoto ones. Itturns however out that the Wakimoto fields are not free because of the boundaryconditions and in order to have the usual mode expansion with reasonablebehaviour under complex conjugation it is necessary to consider thecomplexification of AdS3 and then add some constraints. Furthermore they coveronly half AdS3 and we need different patches to cover the whole space and tomake the above mentioned transformation really canonical.