Kähler–Sasaki geometry of toric symplectic cones in action-angle coordinates

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 Kähler cone. Kähler–Sasaki 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–Martelli–Sparks–Waldram in 2004, to an old family of extremal Kähler metrics constructed by Calabi in 1982.