Hidden symmetries on toric Sasaki-Einstein spaces

Abstract

We describe the construction of Killing-Yano tensors on toric Sasaki-Einstein manifolds. We use the fact that the metric cones of these spaces are Calabi-Yau manifolds. The description of the Calabi-Yau manifolds in terms of toric data, using the Delzant approach to toric geometries, allows us to find explicitly the complex coordinates and write down the Killing-Yano tensors. As a concrete example we present the complete set of special Killing forms on the five-dimensional homogeneous Sasaki-Einstein manifold T 1,1 .

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

“…That is the case of the T 1,1 space with the Reeb vector field (14). More generally the U (1) action on M is only locally free and such structures are referred to as quasi-regular.…”

“…Finally there are two additional KY tensors related to the complex volume form of the metric cone C (T 1,1 ). All these constants of motion are explicitly given in [13,14].…”

Complete integrability of geodesics in toric Sasaki-Einstein space T 1,1 and action-angle variables

“…In spite of a multitude of conserved quantities, the number of functionally independent constants of motion is only five, exactly the dimension of the SE space T 1,1 . Therefore the geodesic motions in this space are completely integrable, but not superintegrable [9,10] .…”

Integrability of the geodesic flow on the resolved conifolds over Sasaki–Einstein space T1,1

“…Other constants of motion can be constructed using the Stäckel-Killing tensors of the T 1,1 space [6,12]. In spite of the presence of a multitude of conserved quantities, the number of functionally independent constants of motion is five.…”