AdS perturbations, isometries, selection rules and the Higgs oscillator

TL;DR

This paper explores how AdS isometries lead to selection rules that restrict energy transfer in small perturbations, connecting these rules to the Higgs oscillator's algebraic structure and simplifying previous formulations.

Contribution

It reveals a systematic, simpler approach linking AdS isometries to selection rules and introduces new representations of AdS mode functions related to the Higgs oscillator.

Findings

Selection rules restrict mode interactions in AdS perturbations.

AdS isometries are directly related to the algebraic structure of the Higgs oscillator.

New representations of AdS mode functions are developed, connecting to superintegrable systems.

Abstract

Dynamics of small-amplitude perturbations in the global anti-de Sitter (AdS) spacetime is restricted by selection rules that forbid effective energy transfer between certain sets of normal modes. The selection rules arise algebraically because some integrals of products of AdS mode functions vanish. Here, we reveal the relation of these selection rules to AdS isometries. The formulation we discover through this systematic approach is both simpler and stronger than what has been reported previously. In addition to the selection rule considerations, we develop a number of useful representations for the global AdS mode functions, with connections to algebraic structures of the Higgs oscillator, a superintegrable system describing a particle on a sphere in an inverse cosine-squared potential, where the AdS isometries play the role of a spectrum-generating algebra.