Massless geodesics in $AdS_5\times Y(p,q)$ as a superintegrable system

TL;DR

This paper identifies a new conserved quantity for massless geodesics in certain five-dimensional geometries, demonstrating superintegrability and linking geometric properties to dual gauge theory operators.

Contribution

It introduces a Carter-like constant for geodesic motion in $Y(p,q)$ geometries, establishing superintegrability in these Einstein-Sasaki spaces.

Findings

Existence of a new Carter-like constant for geodesics.

Geodesic equations are superintegrable in these geometries.

Results relate massless geodesics to dual BPS operators in gauge theory.

Abstract

A Carter like constant for the geodesic motion in the $Y(p,q)$ Einstein-Sasaki geometries is presented. This constant is functionally independent with respect to the five known constants for the geometry. Since the geometry is five dimensional and the number of independent constants of motion is at least six, the geodesic equations are superintegrable. We point out that this result applies to the configuration of massless geodesic in $AdS_5\times Y(p,q)$ studied by Benvenuti and Kruczenski, which are matched to long BPS operators in the dual N=1 supersymmetric gauge theory.