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

We briefly describe the construction of St\"{a}\-kel-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 Kill\-ing-Yano tensors of the homogeneous Sasaki-Einstein space $T^{1,1}$. We discuss the integrability of geodesics and construct explicitly the action-angle variables. Two pairs of frequencies of the geodesic motions are resonant giving way to chaotic behavior when the system is perturbed.


Introduction
The renewed interest in Sasaki geometries has arisen in connection with some recent developments in mathematics and theoretical physics [1]. Sasakian geometry represents the natural odd-dimensional counterpart of the Kähler geometry. Sasaki structure in (2n−1) dimensions is sandwiched between the Kähler metric cone in n complex dimensions and the transverse Kähler structure of complex dimension (n − 1). In particular the Kähler cone is Ricci flat, i.e. Calabi-Yau manifold, if and only if the corresponding Sasaki manifold is Einstein.
The interest in physics for Sasaki-Einstein (SE) geometry is connected with its relevance in AdS/CFT duality. New nontrivial infinite family of toric SE manifolds were constructed [2] and many new insights were obtained for AdS/ CFT correspondence. These SE spaces are denoted by Y p,q and characterized by the two coprime positive integers p and q with q < p. A vastly greater number of SE spaces was constructed in [3] and denoted by L p,q,r where p, q and r are coprime positive integers with 0 < p ≤ q , 0 < r < p+q and with p and q each coprime to r and to s = p + q − r. These metrics have U (1) × U (1) × U (1) isometry. In the special case p + q = 2r the isometry of these metrics is enlarged to SU (2) × U (1) × U (1) which is the isometry of the spaces Y p,q = L p−q,p+q,p . Another special limit is p = q = r = 1 and the metric becomes the homogeneous T 1,1 space with SU (2) × SU (2) × U (1) isometry.
The symmetries of SE spaces play an important role in connection with the study of integrability properties of geodesic motions and separation of variables of the Hamilton-Jacobi or quantum Klein-Gordon, Dirac equations.
Higher order symmetries associated with Stäckel-Killing (SK) and Killing-Yano (KY) tensors generate conserved quantities polynomial in momenta describing the so called dynamical or hidden symmetries. In general it is a difficult task to solve straightly the generalized Killing equations satisfied by KY and SK tensors. Using the geometrical structure of a SE manifold, its connection with the complex structure of the Calabi-Yau metric cone it is possible to produce explicitly the complete set of Killing tensors and consequently the integrals of the geodesic motions.
One of the interesting aspect of AdS/CFT correspondence is integrability which allows obtaining many new classical solutions of the theory (see the review [4]). By focusing on the integrability of the geodesics on SE spaces we gain a better understanding of the geometries produced by D-branes on non-flat bases.
The standard way to decide if a system is integrable is to find integrals of motion. A dynamical system is called integrable if the number of functionally independent integrals of motion is equal to the number of degrees of freedom.
In the context of AdS/CFT duality type IIB strings on AdS 5 × S 5 with Ramond-Ramond fluxes is related to N = 4 SU (N ) gauge theory. An interesting generalization of this duality between gauge theory and strings is to consider backgrounds of the form AdS 5 × X 5 where X 5 is in a general class of five-dimensional Einstein spaces admitting U (1) fibration. While the type IIB string theory on AdS 5 × S 5 is classically integrable [5] there are many nonintegrable AdS/CFT dualities in which the string world-sheet theory exhibits a chaotic behavior. This is the case when the internal space is a SE manifold like T 1,1 or Y p,q [6].
The purpose of this paper is to analyze the integrability of geodesics of the homogeneous regular SE 5-dimensional space T 1,1 . In the light of the AdS/CFT correspondence, AdS × T 1,1 represents the first example of a supersymmetric holographic theory based on a compact manifold which is not locally S 5 [7].
Using the multitude of Killing vector (Kv) fields and SK tensors it is possible to construct the conserved quantities for geodesic motions on T 1,1 . However, the number of functionally independent constants of motion is only 5, implying the complete integrability of geodesics, but not superintegrability.
In order to understand the peculiarities of the geodesic motions in T 1,1 space we shall perform the action-angle formulation of the phase space. The descrip-tion of the integrability of geodesics in T 1,1 in these variables gives us a comprehensive geometric description of the dynamics.
The existence of the action-angle variables is very important both for the theory of near-integrable systems (KAM theory) and for the quantization of integrable systems (Bohr-Sommerfeld rule). The action-angle variables define an n-dimensional surface which is a topological torus (Kolmogorov-Arnold-Moser (KAM) tori) [8]. The KAM theorem states that when an integrable Hamiltonian is perturbed by a small nonintegrable piece most tori survive but suffer small deformations. On the other hand the resonant tori which have rational ratios of frequencies get destroyed and motion on them becomes chaotic. The use of action-angle variables provides a powerful technique to quickly obtain the frequencies of the periodic motions without finding a complete solution to the motion of the system. The paper is organized as follows. In the next two Sections we recall some definitions and known results concerning the Killing tensors and SE spaces. For the completeness and clarity of the exposition, in Section 4 we present the constants of motion on T 1,1 space proving the integrability of geodesics. In Section 5 we construct explicitly the action-angle variables and the frequencies of the motions. Finally in Section 6 we present some concluding remarks and discuss the issues regarding the presence of resonant frequencies. For the convenience of the reader, some details concerning the evaluation of some integrals from Section 5 are deferred to the Appendix.

Stäckel-Killing and Killing-Yano tensors
SK and KY tensors stand as a natural extension of the Kv fields which are linked to the continuous isometries that leave the metric invariant.
On a Riemannian manifold (M, g) with local coordinates x µ and metric g µν the geodesics can be obtained as the trajectories of test-particles with propertime Hamiltonian In the presence of a Kv the system of a free particle with Hamiltonian (1) admits a conserved quantity linear in the momenta.
A SK tensor of rank r is a symmetric tensor defined on the manifold M where ∇ is the Levi-Civita connection. If M admits a SK tensor, the system of a free particle possesses a conserved quantity of higher order in the momenta A different generalization of the Kv's is represented by the KY tensors. A KY tensor is a differential p-form defined on M satisfying the equation Giving two KY tensors Ψ µ1,...,µr and Σ µ1,...,µr the partial contracted product generates a SK tensor of rank 2: This property offers a method to generate higher order integrals of motion (4) by identifying the complete set of KY tensors.
It is worth mentioning that in general is is a hard task to find the complete set of SK or KY tensors trying to solve directly eqs. (3), (6). In some cases it is possible to produce the complete set of KY tensors taking advantage of geometrical properties of the space. That is the case of toric SE spaces for which the explicit construction of KY tensors is possible [9].

Sasaki-Einstein spaces
Recall that a (2n − 1)-dimensional manifold M is a contact manifold if there exists a 1-form η, called a contact 1-form, on M such that everywhere on M [10]. For every choice of contact 1-form η there exists a unique vector field K η , called Reeb vector field, that satisfies η(K η ) = 1 and K η dη = 0 .
The Reeb vector field K η is a Kv field on M , has unit length and its integral curve is a geodesic. A contact Riemannian manifold is Sasakian if its metric cone is Kähler. Here r ∈ (0, ∞) may be considered as a coordinate on the positive real line R + . As part of the connection between Sasaki and Kähler geometries it is worth noting that in the case of a SE manifold, Ric g = 2(n − 1)g, the metric cone is Ricci flat Ricḡ = 0, i.e. a Calabi-Yau manifold.
The KY tensors on a SE manifold are described by the Killing forms These KY tensors do not exhaust the complete set of KY tensors on SE spaces. The Calabi-Yau metric cone has holonomy SU (n) and admits two additional parallel forms given by the real and imaginary parts of the complex volume form [9]. In order to complete the set of KY tensors on a SE manifold it is necessary to use the relation between the KY tensors on SE space and its metric cone. More precisely, for any p-form Ψ on the space M we can define an associated (p + 1)-form Ψ C on the cone C(M ) [9].
In spite of the existence of a multitude of constants of motion, the number of functionally independent is only 5 [13,15], exactly the dimension of the SE space T 1,1 . In the next Section we shall use the complete integrability of geodesics to solve the Hamilton-Jacobi equation by separation of variables and construct the action-angle variables.

Action-angle variables
We start with the Hamilton-Jacobi equation H(q 1 , · · · , q n ; ∂S ∂q 1 , · · · , ∂S ∂q n ; t) + ∂S ∂t = 0 (18) with the Hamilton's principal function S = S(q 1 , · · · , q n ; α 1 , · · · , α n ; t) depending on n constants of integration. Taking into account that the Hamiltonian (16) has no explicit time dependence, Hamilton's principal function can be written in the form [16] S(q, α, t) = W (q, α) − Et (20) where one of the constants of integration is equal to the constant value E of the Hamiltonian and W (q, α) is the Hamilton's characteristic function If the motion stays finite, each q i will go through repeated cycles and the canonical action variables are defined as integrals over complete period of the orbit in the (q i , p i ) plane J i 's form n independent functions of the constants α i and can be taken as a set of new constant momenta.
Conjugate angle variables w i are defined by the equations: having a linear evolution in time with β i other constants of integration and ω i are frequencies associated with the periodic motion of q i . In the case of geodesic motions in T 1,1 space the variables in the Hamilton-Jacobi equation are separable and we seek a solution with the characteristic function Since φ 1 , φ 2 , ν are cyclic variables we have where α φ1 , α φ2 , α ν are constants of integration. The action variables corresponding to the cyclic coordinates can be straightly obtained from (26) We observe that the dependences on θ 1 and respectively θ 2 has been separated into the expressions within the square brackets. The quantities in the square brackets must be constants which will be denoted by α 2 θ1 and α 2 θ2 . To find the action variables J θi , (i = 1, 2) we infer from (28): and the corresponding action variables are The limits of integrations are defined by the roots θ i− and θ i+ of the expressions in the square root signs and a complete cycle of θ i involves going from θ i− to θ i+ and back to θ i− . This integral can be evaluated by elementary means or using the complex integration method of residues. In the later case, putting cos θ i = t i , for the evaluation of the action variables J θi , we extend t i to a complex variable z and interpret the integral as a closed contour integral in the complex z-plane. The turning points of the t i -motion are which are real for and situated in the interval (−1, +1). We cut the complex z-plane from t i− to t i+ and the closed contour integral of the integrand is a loop enclosing the cut in a clockwise sense. 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: We notice that the constants J θi , J φi , J ν and E are connected by the relation: making manifest that the Hamiltonian depends only on the action variables. The number of independent constants of motion is five implying the complete integrability of geodesics on T 1,1 .
A particular advantage of the change to action-angle variables is that one can identify the fundamental frequencies of the system. In this regard let us observe that the Hamiltonian (34) depends on J θi and J φi in the combination J θi + J φi meaning that the frequencies of motion in θ i and φ i are identical: Finally, using the action variables from (23) and (25) we have the angle variables (see the Appendix): (38)

Conclusions
Motivated by the great interest of higher order symmetries and their applications in various field theories, in this paper we describe the construction of conserved quantities for geodesic motions on toric SE manifold T 1,1 . The integrability of geodesics permits us to give an action-angle formulation of the phase space variables. Such a formulation represents a useful tool for developing of perturbation theory. The KAM theorem describes how an integrable system reacts to small perturbations. The loss of integrability is characterized by the presence of resonant tori which have rational ratios of frequencies That is the case of the frequencies (35) which evinces that the frequencies corresponding to θ i and φ i coordinates are equal. This result is in accord with the numerical simulations [17] which show that certain classical string configurations in AdS × T 1,1 are chaotic. It is considered a string localized on the center of AdS 5 that wraps the directions φ 1 and φ 2 in T 1,1 . To numerically investigate the perturbation of the integrable Hamiltonian by a small non-integrable piece, Ref. [18] considered Poincaré sections. In [6] it is presented an analytical approach of the loss of the integrability using the techniques of differential Galois theory for normal variational equation. It is quite remarkable that while the point-like string (geodesic) equations are integrable in some backgrounds, the corresponding extended classical string motion is not integrable in general [6,19]. A similar situation encountered in [20] in the study of (non)-integrability of geodesics in D-brane background.
Since the Reeb vector field K η (9) is nowhere vanishing, its orbits define a foliation of the SE space M . If all the orbits close, K η generates a circle action on M . If moreover the action is free the SE manifold is said to be regular and is the total space of a U (1) principle bundle over a Kähler-Einstein space. 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. The geometries Y p,q with 4p 2 − 3q 2 a square are examples of such manifolds. If the orbits of the Reeb vector field do not close the Kv generates an action R on M , with the orbits densely filling the orbits of a torus and the SE space is said to be irregular. The manifolds Y p,q with 4p 2 − 3q 2 not a square represent examples of such geometries [12].
From these considerations it is obvious that there are significant differences between T 1,1 and Y p,q SE spaces. The construction of the action-angle variables for integrable geodesic motions in quasi-regular and irregular spaces is more intricate and deserve a special study [21].
Concluding, KY tensors provide a powerful tool for studying symmetries of black holes in higher dimensions and stringy geometries. It would be interesting to extend the action-angle formulation to quasi-regular and irregular 5-dimensional SE metrics and their higher dimensional generalizations relevant for the predictions of the AdS/CFT correspondence.
where ∆ i = 4a i c i − b 2 i < 0 taking from granted (32). For the partial derivative of W θi (40) with respect to action variable J φi we have where [22] I 2 (a, b, c; t) = dt t √ a + bt + ct 2 for a < 0. Finally for the partial derivative of W θi (40) with respect to action variable J ν we have