There has been considerable interest recently in Sasaki–Einstein (SE) geometry [1]. In dimension five, an infinite family of explicit SE metrics Yp,q on S2×S3 has been constructed, where p and q are two coprime positive integers, with q<p [2].

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 (1)η∧(dη)n−1≠0. The Reeb vector field &xgr; dual to η satisfies (2) where is the operator dual to the wedge product.

A contact Riemannian manifold (Y2n−1,gY2n−1) is Sasakian if its metric cone C(Y2n−1)=Y2n−1×ℝ+ with the metric (3)ds2(C(Y2n−1))=dr2+r2ds2(Y2n−1) is Kähler [3]. Here r∈(0,∞) may be considered as a coordinate on the positive real line ℝ+. If the Sasakian manifold is Einstein, the metric cone is Ricci-flat and Kähler, i.e., Calabi–Yau.

The orbits of the Reeb vector field &xgr; may or may not close. If the orbits of the Reeb vector field &xgr; are all closed, then &xgr; integrates to an isometric U(1) action on (Y2n−1,gY2n−1). Since &xgr; is nowhere zero, this action is locally free. If the U(1) action is in fact free, the Sasakian structure is said to be regular. Otherwise it is said to be quasi-regular. If the orbits of &xgr; are not all closed, the Sasakian structure is said to be irregular and the closure of the 1-parameter subgroup of the isometry group of (Y2n−1,gY2n−1) is isomorphic to a torus Tn [1].

The homogeneous SE metric on S2×S3, known as T1,1, represents an example of regular Sasakian structure with SU(2)×SU(2)×U(1) isometry. The Yp,q spaces have isometry SU(2)×U(1)×U(1) and for 4p2−3q2 a square they are examples of quasi-regular SE manifolds. The geometries Yp,q with 4p2−3q2 not a square are irregular SE spaces.

In a recent paper, Ref. [4], the constants of motion for geodesic motions in the 5-dimensional spaces Yp,q have been explicitly constructed. This task was achieved using the complete set of Killing vectors and Killing–Yano tensors of these toric SE spaces. A multitude of constants of motion have been generated, but only five of them are functionally independent implying the complete integrability of geodesic flow on Yp,q spaces.

The complete integrability of geodesics permits us to construct explicitly the action-angle variables. The formulation of an integrable system in these variables represents a useful tool for developing perturbation theory. The action-angle variables define an n-dimensional surface that is a topological torus (Kolmogorov–Arnold–Moser (KAM) tori) [5].

Our motivation for studying the action-angle parametrization of the phase space for geodesic motions in SE spaces comes from recent studies of the nonintegrability and chaotic behavior of some classical configurations of strings in the context of AdS/CFT correspondence. It was shown that certain classical string configurations in AdS5×T1,1 [6] or AdS5×Yp,q [7] are chaotic. Either numerical simulations or an analytic approach through the Kovacic algorithm were used [8].

The purpose of this paper is to describe geodesic motions in the SE spaces Yp,q in the action-angle formulation. We find that the Hamiltonian (energy) involves only four action variables, which have corresponding frequencies different to zero. One of the fundamental frequencies is zero, foreshadowing chaotic behavior when the system is perturbed.

The paper is organized as follows. In the next section we give the necessary preliminaries regarding the metric and the constants of motion for geodesics on Yp,q spaces. In Sect. 3 we perform the separation of variables and give an action-angle parametrization of the phase space. The paper ends with conclusions in Sect. 4.

The AdS/CFT correspondence represents an important advancement in string theory. A large class of examples consists of type IIB string theory on the background AdS5×Y5, with Y5 a 5-dimensional SE space. In the framework of AdS/CFT correspondence, Yp,q spaces have played a central role as they provide an infinite class of dualities.

We write the metric of the 5-dimensional Yp,q spaces [2,9,10] as (4)dsYp,q2=1−c y6(dθ2+sin2θ dϕ2)+1w(y)q(y)dy2+q(y)9(dψ−cosθ dϕ)2 +w(y)[ d&agr;+ac−2y+c y26(a−y2)(dψ−cosθ dϕ) ]2, where (5)w(y)=2(a−y2)1−cy,  q(y)=a−3y2+2cy3a−y2. This metric is Einstein with RicgYp,q=4gYp,q for all values of the constants a,c. For c=0 the metric takes the local form of the standard homogeneous metric on T1,1 [11]. Otherwise the constant c can be rescaled by a diffeomorphism and in what follows we assume c=1.

A detailed analysis of the SE metric Yp,q [2] showed that for 0≤θ≤&pgr; and 0≤ϕ≤2&pgr;, the first two terms of Eq. (4) give the metric on a round two-sphere. The 2-dimensional (y,ψ)-space defined by fixing θ and ϕ is fibred over this two-sphere. The range of y is fixed so that 1−y>0, a−y2>0, which implies w(y)>0. Also it is demanded that q(y)≥0 and that y lies between two zeros of q(y), i.e., y1≤y≤y2 with q(yi)=0. To be more specific, the roots yi of the cubic equation (6)a−3y2+2y3=0 are real, one negative (y1) and two positive, the smallest being y2. All of these conditions are satisfied if the range of a is (7)0<a<1. Taking ψ to be periodic with period 2&pgr;, the (y,ψ)-fiber at fixed θ and ϕ is topologically a two-sphere. Finally, the period of &agr; is chosen so as to describe a principal S1 bundle over B4=S2×S2. For any p and q coprime, the space Yp,q is topologically S2×S3 and one may take [2,11] (8)0≤&agr;≤2&pgr;ℓ, where (9)ℓ=q3q2−2p2+p(4p2−3q2)1/2.

To put the formulas in simpler forms, in what follows we also introduce (10)f(y)=a−2y+y26(a−y2), (11)p(y)=w(y)q(y)6=a−3y2+2y33(1−y).

The conjugate momenta to the coordinates (θ,ϕ,y,&agr;,ψ) are (12)Pθ=1−y6θ˙,Py=16p(y)y˙,P&agr;=w(y)(&agr;˙+f(y)(ψ˙−cosθϕ˙)),Pψ=w(y)f(y)&agr;˙+[ q(y)9+w(y)f2(y) ](ψ˙−cosθϕ˙),Pϕ=1−y6sin2θϕ˙−cosθPψ=1−y6sin2θϕ˙−cosθw(y)f(y)&agr;˙−cosθ[ q(y)9+w(y)f2(y) ]ψ˙ +cos2θ[ q(y)9+w(y)f2(y) ]ϕ˙, with overdot denoting proper time derivative.

The Hamiltonian describing the motion of a free particle is (13)H=12g&mgr;νP&mgr;Pν, which for the Yp,q metric (4) and using the momenta (12) has the form (14)H=12{ 6p(y)Py2+61−y(Pθ2+1sin2θ(Pϕ+cosθPψ)2)+1−y2(a−y2)P&agr;2+9(a−y2)a−3y2+2y3(Pψ−a−2y+y26(a−y2)P&agr;)2 }.

Starting with the complete set of Killing vectors and Killing–Yano tensors of the SE spaces Yp,q, it is possible to find quite a lot of integrals of motions [4,12,13]. However the number of functionally independent constants of motion is only five, implying the complete integrability of geodesic flow on Yp,q spaces. For example, we can choose as independent conserved quantities the energy (15)E=H, the momenta corresponding to the cyclic coordinates (ϕ,ψ,&agr;) (16)Pϕ=cϕ,Pψ=cψ,P&agr;=c&agr;, where (cϕ,cψ,c&agr;) are some constants, and the total SU(2) angular momentum (17)J→2=Pθ2+1sin2θ(Pϕ+cosθPψ)2+Pψ2.

The connection between completely integrable systems and toric geometry in the symplectic setting is described by the classical Liouville–Arnold theorem [5,14]. A dynamical system defined by a given Hamiltonian H on a 2n-dimensional symplectic manifold (M2n,&ohgr;) is called Liouville integrable if it admits n functionally independent first integrals in involution. In other words, there are n functions F=(f1=H,f2,…,fn) such that df1∧…∧fn≠0 almost everywhere and (18){fi,fj}=0, ∀i,j.

Let Fc=(f1=E,f2=c2,…,fn=cn) by a common invariant level set. If Fc is regular, compact, and connected, then it is diffeomorphic to the n-dimensional Lagrangian torus. For n degrees of freedom the motion is confined to an n-torus (19)Tn=S1×S1×⋯×S1︸  times. These are called invariant tori and never intersect taking into account the uniqueness of the solution to the dynamical system, expressed as a set of coupled ordinary differential equations.

In a neighborhood of Fc there are action-angle variables J, w mod 2&pgr;, such that the symplectic form becomes (20)&ohgr;=∑i=1ndJi∧dwi, and the Hamiltonian H depends only on actions J1,…,Jn. An action variable Ji specifies a particular n-torus Tn and is constant since the tori are invariant. The location on the torus is specified by n angle variables wi. Even though the system is integrable, the dynamics on the singular set (where the differentials of the integrals f1,…,fn are dependent) can be quite complicated [15].

In the case of the geodesic motions on Yp,q, for the beginning, we fix a level surface F=(H,Pϕ,Pψ,P&agr;,J→2)=c of the mutually commuting constants of motion (15)–(17). The differentials of the chosen first integrals are real analytic [4]. Then it suffices to require their functional independence at least at one point [16] to apply the Liouville–Arnold theorem. Further we introduce the generating function for the canonical transformation from the coordinates (p,q), where p are the conjugate momenta (12) to the coordinates q=(θ,ϕ,y,&agr;,ψ), to the action-angle variables (J,w) as the indefinite integral (21)S(q,c)=∫F=cp⋅dq.

Since the Hamiltonian (14) has no explicit time dependence, we can write (22)S(q,c)=W(q,c)−Et, with the Hamilton characteristic function (23)W=∑i∫pi dqi.

In the case of geodesic motions in SE spaces Yp,q the variables in the Hamilton–Jacobi equation are separable and consequently we seek a solution of the Hamilton characteristic function of the form (24)W(y,θ,ϕ,ψ,&agr;)=Wy(y)+Wθ(θ)+Wϕ(ϕ)+Wψ(ψ)+W&agr;(&agr;).

The action variables J are defined as integrals over a complete period of the orbit in the (pi,qi) plane, (25)Ji=∮pi dqi=∮∂Wi(qi;c)∂qidqi (no summation). The Ji's form n independent functions of ci's and can be taken as a set of new constant momenta.

Conjugate angle variables wi are defined by the equations (26)wi=∂W∂Ji=∑j=1n∂Wj(qj;J1,⋯,Jn)∂Ji, having a linear evolution in time (27)wi=&ohgr;it+βi, with βi other constants of integration, and &ohgr;i frequencies associated with the periodic motion of qi.

The Hamilton characteristic functions associated with cyclic variables are (28)Wϕ=Pϕϕ=cϕϕ,Wψ=Pψψ=cψψ,W&agr;=P&agr;&agr;=c&agr;&agr;, where cϕ, cψ, c&agr; are the constants introduced in Eq. (16).

The corresponding action variables are (29)Jϕ=2&pgr;cϕ,Jψ=2&pgr;cψ,J&agr;=2&pgr;ℓc&agr;.

Taking into account Eqs. (14) and (23), the Hamilton–Jacobi equation becomes (30)E=12{ 6p(y)(∂Wy∂y)2+61−y[ (∂Wθ∂θ)2+1sin2θ(cϕ+cosθcψ)2 ] +1−y2(a−y2)c&agr;2+9(a−y2)a−3y2+2y3[ cψ−a−2y+y26(a−y2)c&agr; ]2 }.

This equation can be written as (31)(∂Wθ∂θ)2+1sin2θ(cϕ+cosθψ)2=1−y3E−p(y)(1−y)(∂Wy∂y)2−(1−y)212(a−y2)c&agr;2 −3(a−y2)(1−y)2(a−3y2+2y3)[ cψ−a−2y+y26(a−y2)c&agr; ]2.

We observe that the left-hand side of this equation depends only on θ and is independent of y. Therefore we may set (32)(∂Wθ∂θ)2+1sin2θ(cϕ+cosθcψ)2=cθ2, with cθ another constant. From the last equation we can evaluate the action variable (33)Jθ=∮dθ(cθ2−(cϕ+cψcosθ)2sin2θ)1/2.

The limits of integrations are defined by the roots θ− and θ+ of the expressions in the square root parentheses, and a complete cycle of θ involves going from θ− to θ+ and back to θ−.

This integral can be evaluated by elementary means or by using the complex integration method of residues, which turns out to be more efficient [14,17]. For the evaluation of the integral (33), we put cosθ=t, extend t to a complex variable z, and interpret the integral as a closed contour integral in the complex z-plane. Consider the integrand in Eq. (33): (34)(−(cθ2+cψ2)z2−2cϕcψz+cθ2−cϕ2)1/2z2−1=(−(cθ2+cψ2))1/2z2−1((z−t+)(z−t−))1/2, where the roots (35)t±=−cϕcψ±cθ(cθ2+cψ2−cϕ2)1/2cθ2+cψ2 are the turning points of the t-motion. They are real for (36)cθ2+cψ2−cϕ2≥0, and situated in the interval (−1,+1).

For z>t+ we specify the right-hand side of the square root from Eq. (34) as positive. We cut the complex z-plane from t− to t+ 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 (37)Jθ=2&pgr;[ (cθ2+cψ2)1/2−cϕ ].

For the action variable corresponding to the y-coordinate we have from Eq. (31), (38)∂Wy∂y={1−ya−3y2+2y3E−3a−3y2+2y3cθ2 −9(a−y2)(1−y)2(a−3y2+2y3)2cψ2+3(a−2y+y2)(1−y)2(a−3y2+2y3)2cψc&agr; −(1−y)(2a+a2−6ay−2y2+2ay2+6y3−3y4)8(a−3y2+2y3)2(a−y2)c&agr;2}1/2. It is harder to evaluate the action variable Jy in a closed analytic form taking into account the complicated expression (38). In fact, the closed form of Jy is not at all illuminating. More important is the fact that Jy depends only on four constants of motion: E, Jθ, J&agr;, Jψ. In consequence, the energy depends only on four action variables Jy, Jθ, J&agr;, Jψ representing a reduction in the number of action variables entering the expression for the energy of the system.

For the angular variable wϕ we have (39)wϕ=12&pgr;Jϕ+∂Wθ∂Jϕ.

Putting cosθ=t, the second term is (40)∂Wθ∂Jϕ=−12&pgr;∫dt(Jϕ+Jθ)t2+Jψt(1−t2)(−(Jϕ+Jθ)2t2−2JϕJψt+(Jθ2+2JθJϕ−Jψ2))1/2=12&pgr;∫dt1−t2dt2+et(a+bt+ct2)1/2, where (41)a=Jθ2+2JθJϕ−Jψ2,b=−2JθJψ,c=−(Jθ+Jϕ)2,d=Jθ+Jϕ,e=Jψ.

We require the following integrals [18]: (42)I1(a,b,c;t)=∫dt(a+bt+ct2)1/2=−1−carcsin(2ct+b−Δ) evaluated for c<0, Δ=4ac−b2<0, and (43)I2(a,b,c;t)+∫dtt(a+bt+ct2)1/2=1−aarctan(2a+bt2−a(a+bt+ct2)1/2) evaluated for a<0. That is the case of the constants a, b, c taking into account the constraint (36).

Using these integrals, for the angular variable wϕ we get (44)wϕ=12&pgr;Jϕ−d2&pgr;I1(a,b,c;cosθ) −d+e4&pgr;I2(a+b+c,b+2c,c;cosθ−1) −e−d4&pgr;I2(a−b+c,b−2c,c;cosθ+1).

The explicit evaluation of the angular variables wθ, wψ, w&agr;, wy is again intricate due to the absence of a simple closed form for the action variable Jy. However, it is remarkable that one of the fundamental frequencies (27) (45)&ohgr;i=∂H∂Ji, is zero, namely (46)&ohgr;ϕ=∂H∂Jϕ=0, since the action Jϕ does not enter the expression for the energy.

The topological nature of the flow of each invariant torus (19) depends on the properties of the frequencies &ohgr;i (27). There are essentially two cases [19]:

(1) The frequencies &ohgr;i are nonresonant: (47)ki&ohgr;i≠0 for all 0≠ki∈ℤn. Then, on this torus each orbit is dense and the flow is ergodic.

In the case of geodesics on Yp,q space, the frequencies are resonant (46) giving way to chaotic behavior when the system is perturbed. The analysis performed in Ref. [7] confirms the present results produced in the action-angle approach.

The action-angle formulation for Yp,q spaces gives us a better understanding of the dynamics of the geodesic motions in these spaces. In spite of the complexity of the evaluation of some variables, we are able to prove that the energy of the system depends on a reduced number of action variables signaling a degeneracy of the system.

This fact corroborates a similar result obtained in the case of geodesic motions in the homogeneous SE space T1,1 [17]. The metric on T1,1 may be written by utilizing the fact that it is a U(1) bundle over S2×S2. The evaluations of all action and angle variables was entirely done by putting them in closed analytic forms. In the case of the space T1,1, the isometry is SU(2)2×U(1) and there are two pairs of fundamental frequencies that are resonant. The degeneracy of these two pairs of frequencies may be removed by a canonical transformation to new action-angle variables. Finally, the Hamiltonian governing the motions on T1,1 can be written in terms of only three action variables for which the corresponding frequencies are different from zero.

In conclusion, the action-angle approach offers strong support for the assertion that certain classical string configurations in AdS5×Y5 with Y5 in a large class of Einstein spaces is nonintegrable [20,21]. It would be interesting to extend the action-angle formulations to other 5-dimensional SE spaces, as well as to their higher-dimensional generalizations, relevant for predictions of the AdS/CFT correspondence.