Complete integrability of geodesic motion in Sasaki-Einstein toric spaces

Complete integrability of geodesic motion in Sasaki-Einstein toric spaces Mihai Visinescu Department of Theoretical Physics National Institute for Physics and Nuclear Engineering Horia Hulubei Bucharest, Romania p ADICS.2015 Belgrade, September 7 12, 2015

References M. Visinescu, Mod. Phys. Lett. A 27, 1250217 (2012) M. Visinescu, G. E. Vîlcu, SIGMA 8, 058 (2012) M. Visinescu, J. Geom. Symm. Phys. 25, 93 104 (2013) A. M. Ionescu, V. Slesar, M. Visinescu, G. E. Vîlcu, Rev. Math. Phys. 25, 1330011 (2013) V. Slesar, M. Visinescu, G. E. Vîlcu, Phys. Scr. 89, 125205 (2014) V. Slesar, M. Visinescu, G. E. Vîlcu, EPL Europhys. Lett. 110, 31001 (2015) E. M. Babalic, M. Visinescu, Mod. Phys. Lett. A, in press V. Slesar, M. Visinescu, G. E. Vîlcu, Annals of Physics 361, 548 562 (2015)

Outline 1. Killing forms 2. Sasakian geometry 3. Hidden symmetries on Sasaki-Einstein spaces 4. Symplectic and complex approaches to Calabi-Yau cone 5. Delzant construction 6. Example: Complete integrability on T 1,1 space 7. Example: Complete integrability on Y p,q spaces 8. Outlook

Killing forms (1) Let (M, g) be an n-dimensional Riemannian manifold with the metric g and let be its Levi-Civita connection. The geodesic quadratic Hamiltonian is: H = 1 2 gµν P µ P ν, where P µ are canonical momenta conjugate to the coordinates x µ, P µ = g µν ẋ ν with overdot denoting proper time derivative. Let us recall that in classical mechanics a Hamiltonian system with Hamiltonian H and integrals of motion K j is called completely integrable (or Liouville integrable) if it allows n integrals of motion H, K 1,..., K n 1 which are well-defined functions on the phase space, in involution {H, K j } = 0, {K j, K k } = 0, j, k = 1,, n 1, and functionally independent. A system is superintegrable if it is completely integrable and allows further functionally independent integrals of motion.

Killing forms (2) Killing vector field K µ preserves the metric g if it satisfies Killing s equation (µ K ν) = 0. Here and elsewhere a round bracket denotes a symmetrization over the indices within. In the presence of a Killing vector, the system of a free particle admits a conserved quantity K = K µ ẋ µ, which commutes with the Hamiltonian in the sense of Poisson brackets: {K, H} = 0.

Killing forms (3) A Stäckel-Killing tensor is a covariant symmetric tensor field satisfying the generalized Killing equation The quantities (λ K µ1,...,µ r ) = 0. K SK = K µ1...µ r ẋ µ1 ẋ µr, are constants of geodesic motion. Killing forms or Killing-Yano tensors are differential r-forms satisfying the equation (µ f ν1 )ν 2...ν r = 0. Moreover, a Killing form ω is said to be a special Killing form if it satisfies for some constant c the additional equation for any vector field X on M. X (dω) = cx ω,

Killing forms (4) An important connection between these two generalizations of the Killing vectors. Given two Killing-Yano tensors f (1) ν 1 ν 2...ν r and f (2) ν 1 ν 2...ν r there is a Stäckel-Killing tensor of rank 2 K µν (1,2) = f (1) µλ 2...λ r f (2) λ 2...λ r ν + f (2) µλ 2...λ r f (1) λ 2...λ r ν. This fact offers a method to generate higher order integrals of motion by identifying the complete set of Killing-Yano tensors.

Sasakian geometry (1) A (2n 1)-dimensional manifold Y is a contact manifold if there exists a 1-form η (called a contact 1-form) on Y such that η (dη) n 1 0. A simple and direct definition of the Sasakian structures is the following: A compact Riemannian manifold (Y, g) is Sasakian if and only if its metric cone (X = C(Y ) = R + Y, ḡ = dr 2 + r 2 g) is Kähler. Here r (0, ) may be considered as a coordinate on the positive real line R +. The Sasakian manifold (Y, g) is naturally isometrically embedded into the metric cone via the inclusion Y = {r = 1} = {1} Y C(Y ).

Sasakian geometry (2) For any choice of the contact 1-form η there exists a unique vector field K η such that η(k η ) = 1, K η dη = 0. Such a vector field K is called the characteristic vector field or the Reeb vector field on Y, has unit length and, in particular, is nowhere zero. The integral curves of K are geodesics and the corresponding foliation F K is called the Reeb foliation. The Reeb vector field K lifts to a Killing vector field on C(Y ) ( K J r ), r where J is the complex structure on the cone manifold.

Sasakian geometry (3) Let Y be a Sasaki-Einstein manifold of dimension dim R Y = 2n 1 and its Kähler cone X = C(Y ) is of dimension dim R X = 2n, (dim C X = n). Sasaki-Einstein geometry is naturally sandwiched between two Kähler-Einstein geometries as follows: Let (Y, g) be a Sasaki manifold of dimension 2n 1. Then the following are equivalent 1. (Y, g) is Sasaki-Einstein with Ric g = 2(n 1)g; 2. The Kähler cone (C(Y ), ḡ) is Ricci-flat Ricḡ = 0; 3. The transverse Kähler structure to the Reeb foliation F K is Kähler-Einstein with Ric T = 2ng T.

Hidden symmetries on Sasaki-Einstein spaces (1) The Killing forms of the toric Sasaki-Einstein manifold Y are described by the Killing forms Ψ k = η (dη) k, k = 0, 1,, n 1. Besides these Killing forms, there are n 1 closed conformal Killing forms (also called -Killing forms) Φ k = (dη) k, k = 1,, n 1.

Hidden symmetries on Sasaki-Einstein spaces (2) Moreover in the case of the Calabi-Yau cone, the holonomy is SU(n) and there are two additional Killing forms of degree n. We express the volume form of the metric cone in terms of the Kähler form ω = 1 2 d(r 2 η) dv = 1 n! ωn. Here ω n is the wedge product of ω with itself n times. The volume of a Kähler manifold can be also written as dv = in 2 n ( 1)n(n 1)/2 Ω Ω, where Ω is the complex volume holomorphic (n, 0) form of C(Y ). The additional (real) Killing forms are given by the real respectively the imaginary part of the complex volume form. [Semmelmann, 2003]

Hidden symmetries on Sasaki-Einstein spaces (3) In order to extract the corresponding additional Killing forms of the Einstein-Sasaki spaces we make use of the fact that for any p-form ψ on the space Y we can define an associated p + 1-form ψ C on the cone C(Y ): ψ C := r p dr ψ + r p+1 p + 1 dψ. ψ C is parallel if and only if ψ is a special Killing form with constant c = (p + 1).

Symplectic and complex approaches to Calabi-Yau cone (1) Let (y, φ) be the symplectic coordinates on the Calabi-Yau manifold X and the symplectic (Kähler) form is ω = dy i dφ i. If (X, ω) is toric, the standard n-torus T n = R n /2πZ n acts effectively on X τ : T n Diff (X, ω), preserving the Kähler form. / φ i generate the T n action, φ i being the angular coordinates along the orbit of the torus action φ i φ i + 2π. T n -invariant Kähler metric on X is ds 2 = G ij dy i dy j + G ij dφ i dφ j, where G ij is the Hessian of the symplectic potential G(y) in the y coordinates and G ij = (G ij ) 1. G ij = 2 G y i y j, 1 i, j n,

Symplectic and complex approaches to Calabi-Yau cone (2) The almost complex structure is ( 0 G ij J = G ij 0 ). Associated to (X, ω, τ) there is a moment map µ : X R n µ(y, φ) = y, i.e. the projection on the action coordinates: y i = 1 2 r 2 η,. φ i

Symplectic and complex approaches to Calabi-Yau cone (3) Let us write the Reeb vector in the form: K = b i. φ i In the symplectic coordinates (y, φ) we have r r = 2y i, y i and the components of the Reeb vector are b i = 2G ij y j.

Symplectic and complex approaches to Calabi-Yau cone (4) The moment map exhibits the Kähler cone as a Lagrangian fibration over a strictly convex rational polyhedral cone C R n by forgetting the angular coordinates φ i C { y R n l a (y) 0, a = 1,..., d }, with the linear function l a (y) = (y, v a ), where v a are the inward pointing normal vectors to the d facets of the polyhedral cone. The set of vectors {v a } is called a toric data. v a = va i, va i Z, φ i

Symplectic and complex approaches to Calabi-Yau cone (5) The standard complex coordinates are w i on C\{0}. Log complex coordinates are z i = log w i = x i + iφ i and in these complex coordinates the metric is ds 2 = F ij dx i dx j + F ij dφ i dφ j, where F ij is the Hessian of the Kähler potential. Note also that in the complex coordinates z i the complex structures and the Kähler form are: J = ( 0 I I 0 ), ω = ( 0 Fij F ij 0 ).

Symplectic and complex approaches to Calabi-Yau cone (6) The symplectic potential G and Kähler potential F are related by the Legendre transform ( ) G F(x) = y i G (y = F/ x). y i Therefore F and G are Legendre dual to each other F(x) + G(y) = j F x j G y j at x i = G y i or y i = F x i. It follows from that F ij = G ij (y = F/ x).

Delzant construction (1) The image of X under the moment map µ is a certain kind of convex rational polytope in R n called Delzant polytope. A convex polytope P in R n is Delzant [Delzant, 1988] if (a) there are n edges meeting at each vertex p; (b) the edges meeting at the vertex p are rational, i.e. each edge is of the form 1 + tu i, 0 t where u i Z n ; (c) the u i,..., u n in (b) can be chosen to be a basis of Z n. Delzant construction associates to every Delzant polytope P R n a closed connected symplectic manifold (M, ω) together with the Hamiltonian T n action and the moment map µ.

Delzant construction (2) Using the Delzant construction the general symplectic potential has the following form in terms of the toric data: where G = G can + G b + h, G can = 1 l a (y) log l a (y), 2 a G b = 1 l b (y) log l b (y) 1 2 2 l (y) log l (y), a with l b (y) = (b, y), l (y) = a (v a, y) and h is a homogeneous degree one function of variables y i λ i, t being some constants. h = λ i y i + t,

Delzant construction (3) For a Calabi-Yau manifold X, by an appropriate SL(n, Z) transformation, it is possible to bring the normal vectors of the polyhedral cone in the form v a = (1, w a ). The (n, 0) holomorphic form of the Ricci-flat metric on the Calabi-Yau cone is Ω = e iα (det F ij ) 1/2 dz 1 dz n, with α a phase space which is fixed by requiring that Ω is a closed form. The complex coordinates can be chosen such that Ω = e x 1+iφ 1 dz 1 dz n = dw 1 dw n /(w 2... w n ).

Complete integrability on T 1,1 space (1) One of the most familiar example of homogeneous toric Sasaki-Einstein five-dimensional manifold is the space T 1,1 = S 2 S 3 endowed with the following metric ds 2 (T 1,1 ) = 1 6 (dθ2 1 + sin2 θ 1 dφ 2 1 + dθ2 2 + sin2 θ 2 dφ 2 2 ) + 1 9 (dψ + cos θ 1dφ 1 + cos θ 2 dφ 2 ) 2. The global defined contact 1-form is η = 1 3 (dψ + cos θ 1dφ 1 + cos θ 2 dφ 2 ). The Reeb vector field K has the form K = 3 ψ, and is easy to see that η(k) = 1.

Complete integrability on T 1,1 space (2) It is convenient to work with angle coordinates Φ 1 := ψ, Φ 2 := 1 2 φ 1 + 1 2 φ 2 + 1 2 ψ, Φ 3 = 1 2 φ 1 1 2 φ 2 + 1 2 ψ. writing the basis for an effectively acting T 3 in the form: e i := Φ i, Considering the momentum map, we get the action coordinates y 1 = 1 6 r 2 (cos θ 1 + 1), y 2 = 1 6 r 2 (cos θ 2 cos θ 1 ), y 3 = 1 6 r 2 (cos θ 1 + cos θ 2 ).

Complete integrability on T 1,1 space (3) The Reeb vector field written in the above basis has the components K = (3, 3/2, 3/2), consistent with the determination from toric data using Z -minimization or a-maximization. We consider the inward pointing normal vectors to the cone v 1 = [1, 1, 1], v 2 = [1, 0, 1], v 3 = [1, 0, 0], v 4 = [1, 1, 0].

Complete integrability on T 1,1 space (4) In order to introduce the complex coordinates on conifold we need the symplectic potential G which is given by the sum G can + G b. Applying the Legendre transform x i = G we finally obtain the y i following complex coordinates z 1 = 3 log r + log sin θ 1 + log sin θ 2 + iψ, z 2 = 3 2 log r + log sin θ 1 2 + log cos θ 2 2 + i 2 (ψ + φ 1 + φ 2 ), z 3 = 3 2 log r + log sin θ 1 2 + log sin θ 2 2 + i 2 (ψ φ 1 φ 2 ). For the sake of simplicity in the above relations we have ignored some additive constants.

Complete integrability on T 1,1 space (5) The complex holomorphic volume form is Ω = exp(z 1 )dz 1 dz 2 dz 3 = r 2 dr Ψ + r 3 3 dψ. Decomposing Ψ = RΨ + iiψ, the special real Killing forms are RΨ = cos ψ dθ 1 dθ 2 + sin θ 2 sin ψ dθ 1 dφ 2 sin θ 1 sin ψ dθ 2 dφ 1 sin θ 1 sin θ 2 cos ψ dφ 1 dφ 2, IΨ = sin ψ dθ 1 dθ 2 sin θ 2 cos ψ dθ 1 dφ 2 + sin θ 1 cos ψ dθ 2 dφ 1 sin θ 1 sin θ 2 sin ψ dφ 1 dφ 2.

Complete integrability on T 1,1 space (6) Using the contact 1-form η we get the Killing forms: Ψ 1 = 1 9 (sin θ 1dψ dθ 1 dφ 1 + sin θ 2 dψ dθ 2 dφ 2 cos θ 1 sin θ 2 dθ 2 dφ 1 dφ 2 + cos θ 2 sin θ 1 dθ 1 dφ 1 dφ 2 ), Ψ 2 = 2 27 sin θ 1 sin θ 2 dψ dθ 1 dθ 2 dφ 1 dφ 2. As expected the 5-form Killing-Yano tensor Ψ 2 is proportional to the volume element of the space T 1,1.

Complete integrability on T 1,1 space (7) The conjugate momenta to the coordinates (θ 1, θ 2, φ 1, φ 2, ψ) are: P θ1 = 1 6 θ 1, P θ2 = 1 6 θ 2, P φ1 = 1 6 sin2 θ 1 φ 1 + 1 9 cos2 θ 1 φ 1 + 1 9 cos θ 1 ψ+ 1 9 cos θ 1 cos θ 2 φ 2, P φ2 = 1 6 sin2 θ 2 φ 2 + 1 9 cos2 θ 2 φ 2 + 1 9 cos θ 2 ψ + 1 9 cos θ 1 cos θ 2 φ 1, P ψ = 1 9 ψ + 1 9 cos θ 1 φ 1 + 1 9 cos θ 2 φ 2,

Complete integrability on T 1,1 space (8) The conserved Hamiltonian takes the form: [ H =3 Pθ 2 1 + Pθ 2 2 + 1 sin 2 (P φ1 cos θ 1 P ψ ) 2 θ 1 + 1 ] sin 2 (P φ2 cos θ 2 P ψ ) 2 + 9 θ 2 2 P2 ψ = 1 12 ( θ 2 1 + sin2 θ 1 φ 2 1 + θ 2 2 + sin2 θ 2 φ 2 2 ) + 1 18 ( ψ + cos θ 1 φ1 + cos θ 2 φ2 ) 2. Taking into account the isometries of T 1,1, the momenta P φ1, P φ2 and P ψ are conserved.

Complete integrability on T 1,1 space (9) Two total SU(2) angular momenta are also conserved: J 2 1 =P 2 θ 1 + 1 = 1 36 sin 2 (P φ1 cos θ 1 P ψ ) 2 + Pψ 2 θ [ 1 ] θ 1 2 + sin2 θ 1 φ 2 1 + 1 [ ψ 2 + cos 2 θ 1 φ 2 1 81 + cos2 θ 2 φ 2 2 +2 cos θ 1 φ 1 ψ + 2 cos θ 2 φ 2 ψ + 2 cos θ 1 cos θ 2 φ 1 φ 2 ] ; J 2 2 =Pθ 2 2 + 1 sin 2 (P φ2 cos θ 2 P ψ ) 2 + Pψ 2 θ 2 = 1 [ ] θ 2 2 36 + sin2 θ 2 φ 2 2 + 1 [ ψ 2 + cos 2 θ 1 φ2 81 1 + cos 2 θ 2 φ2 2 +2 cos θ 1 φ 1 ψ + 2 cos θ 2 φ 2 ψ + 2 cos θ 1 cos θ 2 φ 1 φ 2 ].

Complete integrability on T 1,1 space (10) The first Stäckel-Killing tensor (K 1) µν is constructed using the real part of the Killing form Ψ (K 1) µν = (RΨ) µλ (RΨ) λ ν 1 0 0 0 0 0 1 0 0 0 = 6 0 0 sin 2 θ 1 0 0 0 0 0 sin 2 θ 2 0. 0 0 0 0 0 The next Stäckel-Killing tensor is constructed from the imaginary part of Ψ (K 2) µν = (IΨ) µλ (IΨ) λ ν, and we find that this tensor has the same components as K 1.

Complete integrability on T 1,1 space (11) The mixed combination of RΨ and IΨ produces the Stäckel-Killing tensor (K 3) µν = (RΨ) µλ (IΨ) λ ν + (IΨ) µλ(rψ) λ ν, but it proves that all components of this tensor vanish. Finally we construct the Stäckel-Killing tensor from the Killing form Ψ 1 (K 4) µν = (Ψ 1 ) µλσ (Ψ 1 ) λσ ν 1 0 0 0 0 = 4 0 1 0 0 0 3 0 0 1 3 (3 + cos2 4 θ 1 ) 3 cos θ 4 1 cos θ 2 3 cos θ 1 0 0 4 3 cos θ 1 1 cos θ 2 3 (3 + cos2 4 θ 2 ) 3 cos θ. 2 4 0 0 3 cos θ 4 1 3 cos θ 4 2 3

Complete integrability on T 1,1 space (12) Finally we investigate the integrability of the geodesic motion on T 1,1 and for this purpose we construct the Jacobian: J = (H, P φ 1, P φ2, P ψ, J1 2, J2 2, K 1, K 4) (θ 1, θ 2, φ 1, φ 2, ψ, θ 1, θ 2, φ 1, φ 2, ψ). The rank of this Jacobian is 5 implying the complete integrability of the geodesic motion on T 1,1. Not all constants of motion are functionally independent. We can choose the subset (H, P φ1, P φ2, P ψ, J1 2 ) as functionally independent constants of motion and the constants of motion J2 2, K 1 and K 4 are combinations of the chosen subset of constants: 1 6 K 1 = 12H 2 3 (9P ψ) 2, 3 4 K 4 = 12H + 2 3 (9P ψ) 2, 6 J 2 2 = 2H + 3P2 ψ 6 J 2 1.

Complete integrability on Y p,q spaces (1) Infinite family Y (p, q) of Einstein-Sasaki metrics on S 2 S 3 provides supersymmetric backgrounds relevant to the AdS/CFT correspondence. The total space Y (p, q) of an S 1 -fibration over S 2 S 2 with relative prime winding numbers p and q is topologically S 2 S 3. Explicit local metric of the 5-dim. Y (p, q) manifold is given by the line element [Gauntlett, Martelli, Sparks, Waldram, 2004] where dses 2 = 1 c y (dθ 2 + sin 2 θ dφ 2 1 ) + 6 w(y)q(y) dy 2 + q(y) (dψ cos θ dφ)2 9 + w(y) [dα + ac 2y + c y 2 2 6(a y 2 [dψ cos θ dφ]], )

Complete integrability on Y p,q spaces (2) w(y) = 2(a y 2 ) 1 cy, q(y) = a 3y 2 + 2cy 3 a y 2 and a, c are constants. For c = 0 the metric takes the local form of the standard homogeneous metric on T 1,1. Otherwise the constant c can be rescaled by a diffeomorphism and in what follows we assume c = 1. For 0 < α < 1 we can take the range of the angular coordinates (θ, Φ, Ψ) to be 0 θ 2π, 0 Φ 2π, 0 Ψ 2π. Choosing 0 < a < 1 the roots y i of the cubic equation a 3y 2 + 2y 3 = 0, are real, one negative (y 1 ) and two positive (y 2, y 3 ). If the smallest of the positive roots is y 2, one can take the range of the coordinate y to be y 1 y y 2.

Complete integrability on Y p,q spaces (3) Sasakian 1-form η is: η = 2ydα + 1 y (dψ cos θdφ). 3 Basis for an effectively acting T 3 action is e 1 = φ + ψ, e 2 = φ l 2 γ, e 3 = γ, where l = p q, α lγ with l = l(p, q) again a constant. Reeb vector has the components K = (3, 3, 32 (l + 13l ) )

Complete integrability on Y p,q spaces (4) Using the momentum map we get the action coordinates y 1 = r 2 (1 y)(1 cos θ), 6 y 2 = r 2 6 (1 y) cos θ + r 2 2 lly, y 3 = lr 2 y. Toric data for Y p,q v 1 = [1, 1, p], v 2 = [1, 0, 0], v 3 = [1, 1, 0], v 4 = [1, 2, p + q]. The symplectic potential in the case of Y p,q contains the function h, in contradistinction to the case of the homogeneous Sasaki-Einstein manifold T 1,1

Complete integrability on Y p,q spaces (5) However, it can be proved that one can derive a more simple expression 6 1 G = 2 v A, y log v A, y, A=1 by introducing two additional vectors v 5 and v 6 ( v 5 :=K v 1 v 3 = 1, 1, 1 2 p + 3 2 q 1 ), 2l v 6 := v 2 v 4 = ( 2, 2, p q).

Complete integrability on Y p,q spaces (6) Using the Legendre transform, the following complex coordinates are introduced on C(Y p,q ) (additive constants being ignored) ( ) z 1 = log z 2 = log r 3 sin θ ( p(y1 y3) z 3 = log r ( cos θ 2 r 3 cos 2 θ 2 y 3 3 2 y 2 + a 2 y 3 3 2 y 2 + a 2 p(1 y 3 ) 1 y 1 2(1 y (y y 3 ) 1 ) ) l + i (y y1 ) p + iψ, ) + i(φ ψ), ( l 2 φ l ) 2 ψ + γ.

Complete integrability on Y p,q spaces (7) The conjugate momenta to the coordinates (θ, φ, y, α, ψ) are: where P θ = 1 y θ, 6 P φ + cos θp ψ = 1 y sin 2 θ φ, 6 P y = 1 6p(y)ẏ, ( ( )) P α = w(y) α + f (y) ψ cos θ φ, [ ] q(y) ( P ψ = w(y)f (y) α + 9 + w(y)f 2 (y) ψ cos θ φ) f (y) = a 2y + y 2 6(a y 2 ) p(y) = w(y)q(y) 6, = a 3y 2 + 2y 3 3(1 y).

Complete integrability on Y p,q spaces (8) Using the momenta, the conserved Hamiltonian becomes: { H = 1 6p(y)Py 2 + 6 ( P 2 2 1 y θ + 1 sin 2 θ (P φ + cos θp ψ ) 2) + 1 y 2(a y 2 ) P2 α + 9(a y 2 ) (P a 3y 2 + 2y 3 ψ a 2y + y 2 ) } 2 6(a y 2 ) P α = 1 y 12 ( θ 2 + sin 2 θ φ 2 ) + ẏ 2 12p(y) + q(y) 18 ( ψ cos θ φ) 2 + w(y) 2 [ α + f (y)( ψ cos θ φ)] 2.

Complete integrability on Y p,q spaces (9) From the isometry SU(2) U(1) U(1) of the metric we have that the momenta P φ, P ψ and P α are conserved. P φ is the third component of the SU(2) angular momentum and P ψ, P α are associated to the U(1) factors. In addition, the total SU(2) angular momentum J 2 = P 2 θ + 1 sin 2 θ (P φ + cos θp ψ ) 2 + P 2 ψ is also conserved. The explicit form of the Killing-Yano tensor Ψ 1 is Ψ 1 = (1 y) 2 sin θ dθ dφ dψ 6dy dα dψ + 6 cos θ dφ dy dα 6(1 y)y sin θ dθ dφ dα.

Complete integrability on Y p,q spaces (10) Let us call Ψ the Killing form on Y p,q related to the complex holomorphic (3, 0) form on C(Y p,q ). The real part of Ψ is: 1 y RΨ = ( cos ψ p(y) [ ] dθ dy + 6p(y) sin θ dφ dα + p(y) sin θ dφ dψ [ sin ψ sin θ dφ dy 6p(y)dθ dα p(y)dθ dψ + p(y) cos θ dθ dφ] ).

Complete integrability on Y p,q spaces (11) The imaginary part of Ψ is: 1 y IΨ = p(y) ( [ ] sin ψ dθ dy + 6p(y) sin θ dφ dα + p(y) sin θ dφ dψ [ + cos ψ sin θ dφ dy 6p(y)dθ dα p(y)dθ dψ + p(y) cos θ dθ dφ] ).

Complete integrability on Y p,q spaces (12) The first Stäckel-Killing tensor (K 1) µν is constructed using the real part of the Killing form Ψ: (K 1) µν = (RΨ) µλ (RΨ) λ ν The corresponding conserved quantity is K 1 = 6(1 y) θ θ + 3 + a 6y + 2y 3 + ( 3 + a + 6y 6y 2 + 2y 3 ) cos 2θ 1 y 24 (a + ( 3 + 2y)y 2 ) cos θ 1 y φ φ φ α 4 (a + ( 3 + 2y)y 2 ) cos θ 1 y 1 y + ( 3 + 2y)y 2 + 18 ẏẏ + 72a a + ( 3 + 2y)y 2 1 y a + ( 3 + 2y)y 2 + 24 α 1 y ψ a + ( 3 + 2y)y 2 + 2 1 y α α ψ ψ. φ ψ

Complete integrability on Y p,q spaces (13) The next Stäckel-Killing tensor will be constructed from the imaginary part of Ψ: (K 2) µν = (IΨ) µλ (IΨ) λ ν, and we find that this tensor has the same components as K 1. The mixed combination of RΨ and IΨ produces the Stäckel-Killing tensor (K 3) µν = (RΨ) µλ (IΨ) λ ν + (IΨ) µλ(rψ) λ ν, but it proves that all components of this tensor vanish.

Complete integrability on Y p,q spaces (14) Finally we construct the Stäckel-Killing tensor from the Killing form Ψ 1 : (K 4) µν = (Ψ 1 ) µλσ (Ψ 1 ) λσ ν. the corresponding conserved quantity is: [ K 4 = 18 6(1 y) θ θ 24 (a + ( 4 + 5y 2y 2 )y) cos θ φ α 1 y + 7+a 18y +12y 2 2y 3 +(1+a 6y +6y 2 2y 3 ) cos 2θ 1 y 4 (a (2 y)2 ( 1 + 2y)) cos θ φ 1 y ψ 1 y + (1 2y)y 2 + 18 ẏẏ + 72a a + ( 3 + 2y)y 2 1 y +24 a + ( 4 + 5y 2y 2 )y 1 y α α α ψ + 2 a (2 y)2 ( 1 + 2y) 1 y φ φ ] ψ ψ

Complete integrability on Y p,q spaces (15) Having in mind that K 1 = K 2 and K 3 vanishes, we shall verify if the set H, P φ, P ψ, P α, J 2, K 1, K 4 constitutes a functionally independent set of constants of motion for the geodesics of Y p,q constructing the Jacobian: J = (H, P φ, P ψ, P α, J 2, K 1, K 4) (θ, φ, y, α, ψ, θ, φ, ẏ, α, ψ) Evaluating the rank of this Jacobian we find: Rank J = 5, which means that the system is completely integrable. In spite of the presence of the Stäckel-Killing tensors K 1 and K 4 the system is not superintegrable, K 1 and K 4 being some combinations of the first integrals H, P φ, P ψ, P α, J 2..

Outlook Hidden symmetries on higher dimensional toric Sasaki-Einstein spaces Hidden symmetries of other spacetime structures Separability of Hamilton-Jacobi, Klein-Gordon, Dirac equations