Kahler-Sasaki geometry of toric symplectic cones in action-angle coordinates

TL;DR

This paper explores the Kahler-Sasaki geometry of toric symplectic cones using action-angle coordinates, connecting recent Sasaki-Einstein metrics to classical extremal Kahler metrics.

Contribution

It introduces a symplectic action-angle coordinates approach to toric Kahler-Sasaki geometry and relates new Sasaki-Einstein metrics to classical extremal Kahler metrics.

Findings

Develops a symplectic action-angle coordinates framework for toric Kahler-Sasaki geometry.

Establishes a connection between recent Sasaki-Einstein metrics and classical extremal Kahler metrics.

Provides a geometric interpretation linking different families of metrics.

Abstract

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kahler-Sasaki geometry. It also describes, as an application, how this approach can be used to relate a recent new family of Sasaki-Einstein metrics constructed by Gauntlett-Martelli-Sparks-Waldram in 2004, to an old family of extremal Kahler metrics constructed by Calabi in 1982.