Integrability of the geodesic flow on the resolved conifolds over Sasaki-Einstein space $T^{1,1}$

TL;DR

This paper demonstrates the complete integrability of geodesic flow on the resolved conifolds over the Sasaki-Einstein space T^{1,1} and its Calabi-Yau cone, with implications for understanding geometric structures in string theory.

Contribution

It explicitly constructs constants of motion and proves integrability for geodesics on the resolved conifold and its cone, contrasting different smoothing methods.

Findings

Geodesic flow on the resolved conifold is completely integrable.

Complete integrability is preserved under small resolution.

Deformation of the conifold breaks integrability.

Abstract

Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds over Sasaki-Einstein space $T^{1,1}$. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki-Einstein space $T^{1,1}$ and its Calabi-Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution the geodesic motion on the resolved conifold remains completely integrable. Instead, in the case of the deformation of the conifold the complete integrability is lost.