Kähler-sasaki Geometry of Toric Symplectic Cones in Action-angle Coordinates

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 Kähler cone. KählerSasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kähler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kähler-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-MartelliSparks-Waldram in 2004, to an old family of extremal Kähler metrics constructed by Calabi in 1982.