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

Abstract

We briefly describe the construction of Stäkel-Killing and Killing-Yano tensors on toric Sasaki-Einstein manifolds without working out intricate generalized Killing equations. The integrals of geodesic motions are expressed in terms of Killing vectors and Killing-Yano tensors of the homogeneous Sasaki-Einstein space T 1,1 . We discuss the integrability of geodesics and construct explicitly the actionangle variables. Two pairs of frequencies of the geodesic motions are resonant giving way to chaotic behavior when the system is perturbed.

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

“…In spite of the presence of a multitude of conserved quantities, the number of functionally independent constants of motion is five. This implies the complete integrability of geodesic motions in T 1,1 which allows us to solve the Hamilton-Jacobi equation by separation of variables and construct the action-angle variables [7].…”

“…The limits of integrations are defined by the roots θ i− and θ i+ of the expressions in the square root parenthesis and a complete cycle of θ i involves going from θ i− to θ i+ and back to θ i− . An efficient technique for evaluating I θi is to extend θ i to a complex variable z i and interpret the integral as a closed contour integral in the z i plane [7,8]. At the end, we get…”

“…In a recent paper [6] it has been constructed the complete set of constants of motion for Sasaki-Einstein spaces T 1,1 and Y p,q proving the complete inte-grability of geodesic flows. Moreover it was used the action-angle formalism to study thoroughly the complete integrability of these systems [7,8].…”

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

“…The contour can be deformed to a large circular contour plus two contour integrals about the poles at z = ±1. After simple evaluation of the residues and the contribution of the large contour integral we finally get [11]:…”

“…This result is in accord with the numerical simulations [13,14] which show that certain classical string configurations in AdS × T 1,1 are chaotic. Concerning the angle variables (11) they are evaluated from Equation (14) and (17) [11]: where:…”