Toric data, Killing forms and complete integrability of geodesics in Sasaki-Einstein spaces $Y^{p,q}$

TL;DR

This paper demonstrates a method to identify special Killing forms on 5D Sasaki-Einstein spaces using symplectic geometry, confirming previous results and analyzing geodesic integrability.

Contribution

It introduces a general algorithm based on the Delzant construction for finding Killing forms on toric Sasaki-Einstein manifolds, validated against prior computations.

Findings

Complete list of Killing forms for $Y^{p,q}$ spaces obtained

Method confirms previous direct computation results

Discusses integrability of geodesic motion in these spaces

Abstract

In the present paper we show that the complete list of special Killing forms on the 5-dimensional Sasaki-Einstein spaces $Y^{p,q}$ can be extracted using the symplectic potential and the classical Delzant construction. The results achieved here agree with previous ones obtained by direct computation, proving the reliability of the method which stands in fact as a general algorithm for toric Sasaki-Einstein manifolds. Finally, we discuss the integrability of geodesic motion in $Y^{p,q}$ spaces.