Integrability of geodesics and action-angle variables in Sasaki-Einstein space $T^{1,1}$

TL;DR

This paper explores the integrability of geodesic motion on the Sasaki-Einstein space T^{1,1}, constructing explicit action-angle variables and analyzing resonance phenomena that could lead to chaos.

Contribution

It provides an explicit construction of Killing and Killing-Yano tensors on T^{1,1} and demonstrates the integrability of geodesics with explicit action-angle variables.

Findings

Geodesic integrals expressed via Killing tensors.

Explicit action-angle variables constructed.

Resonant frequencies suggest potential for chaos.

Abstract

We briefly describe the construction of St\"{a}\-kel-Killing and Killing-Yano tensors on toric Sasaki-Einstein manifolds without working out intricate generalized Killing equations. The integrals of geodesic motions are expressed in terms of Killing vectors and Kill\-ing-Yano tensors of the homogeneous Sasaki-Einstein space $T^{1,1}$. We discuss the integrability of geodesics and construct explicitly the action-angle variables. Two pairs of frequencies of the geodesic motions are resonant giving way to chaotic behavior when the system is perturbed.