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

TL;DR

This paper investigates how massless particles in AdS$_3$ space decompose into particles in AdS$_2$, revealing a specific spectral structure and representation branching rules relevant for higher-spin theories.

Contribution

It introduces a detailed analysis of the AdS$_3$/AdS$_2$ degression mechanism, including the decomposition of higher-spin representations and field-theoretical demonstrations for spin-2 and spin-3 fields.

Findings

Massless particles in AdS$_3$ decompose into pairs of particles in AdS$_2$ with equal energies.

Higher-spin representations on the unitary bound split into two equal modules upon degression.

The spectral truncation to finite modes is achieved through mode expansions and algebraic identities.

Abstract

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 $s=1,2,...,\infty$ 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.