AdS${}_3$/AdS${}_2$ degression of massless particles

We study a 3d/2d dimensional degression which is a Kaluza-Klein type mechanism in AdS${}_3$ space foliated into AdS${}_2$ hypersurfaces. It is shown that an AdS${}_3$ massless particle of spin $s=1,2,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\infty }$ degresses into a couple of AdS${}_2$ particles of equal energies $E=s$. Note that the Kaluza-Klein spectra in higher dimensions are always infinite. To formulate the AdS${}_3$/AdS${}_2$ degression we consider branching rules for AdS${}_3$ isometry algebra o$(2,2)$ representations decomposed with respect to AdS${}_2$ isometry algebra o$(1,2)$. We find that a given o$(2,2)$ higher-spin representation lying on the unitary bound (i.e. massless) decomposes into two equal o$(1,2)$ modules. In the field-theoretical terms, this phenomenon is demonstrated for spin-2 and spin-3 free massless fields. The truncation to a finite spectrum can be seen by using particular mode expansions, (partial) diagonalizations, and identities specific to two dimensions.