Hidden symmetries of near-horizon extremal Kerr-AdS-NUT geometries

We study hidden symmetries, the symmetries associated with the Killing tensors, of the near horizon geometry of odd-dimensional Kerr-AdS-NUT black hole in two limits: generic extremal and extremal vanishing horizon (EVH) limits. Starting from Kerr-AdS-NUT black hole in ellipsoidal coordinates which admits integrable geodesic equations, we obtain the near horizon extremal/EVH geometries and their principal and Killing tensors by taking the near horizon limit. We explicitly demonstrate that geodesic equations are separable and integrable on theses near horizon geometries. We also compute the constants of motion and read the Killing tensors of these near horizon geometries from the constants of motion. As we expected, they are the same as the Killing tensors given by taking the near horizon limit.


Introduction
The exact symmetries in general relativity framework are usually known as isometries that are given by the Killing vectors. However, symmetries of a given metric can be generated by Killing tenors, as well. In this case, they are called hidden symmetries as they are not manifested in the isometries. In some special cases, hidden symmetries reduce to the isometries when Killing tensors can be trivially written as the product of Killing vectors.
The symmetries of metric are reflected in the motion of probe particles on that background metric such that each of the Killing vector or tensor gives a constant of motion. If the number of (independent) constants of motion is equal to the degrees of freedom of probe particle, its equations are integrable. If it has more independent constants of motion, the system is super-integrable.
The Killing tensors of four dimensional Kerr black hole and its generalization to ddimensional Kerr-AdS-NUT have been studied in [1,2]. These symmetric second rank tensors where the metric functions are 2 (1 − g 2 y 2 ν ) , Considering y µ = (x α , ir) with α = 1, . . . , n − 1, one can see that r is radial direction and x α and φ i with i = 1, . . . , n are related to the angular variables.
Note that M n is just equal to the mass parameter M, while the remaining M α are NUT parameters, denoted by L α . a i denotes the rotation parameters.
Writing the metric in terms of r coordinate, one can easily see that the horizon location is given by Entropy and temperature of this horizon are (2.4) in which A n is the volume of a unit n-sphere and G N is d-dimension Newton's constant.
We note that ∂ ∂t and ∂ ∂φ i are the Killing directions and the horizon is generated by this Killing vector, where the horizon angular velocity along each of φ i directions, Ω i , is given by Additionally, this geometry has n-number of second rank Killing tensors. Killing tensors are symmetric and a rank r Killing tensor K µ 1 ···µr satisfies, The Killing tensors of Kerr-AdS-NUT geometry in the coordinate system that metric is written, are [2], The orthonormal vierbins e µ ,ê µ (µ = 1, . . . , n) andê 0 are and their dual vectors e µ ,ê µ ,ê 0 are In this coordinates, the Killing tensors (4.1) are also simplified to Moreover, this geometry has more rich structure that admits Killing-Yano tensors. We remind the reader that a Killing-Yano tensor of rank q, Y µ 1 ···µq , is an anti-symmetric tensor and solves (2.14) It is easy to show that the contraction of two Killing-Yano tensor in this way, gives a (symmetric) Killing tensor. The Hodge dual of a (d − 2)-rank Killing-Yano tensor is closed conformal Killing-Yano tensor of second rank, called principal tensor, that satisfies here, ξ is a primary Killing vector and is defined by Since the principal tensor, h, is closed, a potential b is locally associated with it, The importance of principal tensor is that its orthogonal (non-degenerate) eigenvectors gives a coordinate basis in which geodesic equations are separable.
The principal tensor of Kerr-AdS-NUT black hole is and its local potential b is 3 Near horizon extremal geometry The extremal limit of this black hole is given by vanishing the temperature in (2.4). In this case, the horizon becomes degenerate and Note that r h is the solution to the equation (2.3). The near horizon transformations are also as follows, Applying this transformations and the constraint (3.1) to the metric (2.1) in the λ → 0 limit, we find the near horizon extremal geometry, The tilde over a function implies that it is evaluated at r = r h .

Principal and Killing tensors
The Killing vectors of this geometry includes the generators of rotation along ϕ i , 4) and the generators of the SL(2, R), as follows These ξ i 's satisfy sl(2, R) algebra, It is clear that ζ i 's commute with ξ i 's. A trivial 3 second rank Killing tensor of this geometry is the Casimir of sl(2, R) algebra (3.8) The non-trivial Killing tensors of this geometry has been studied earlier [17] and are given byK where k = 0, . . . , n − 1 and the functionsÃ . (3.10) It is worth mentioning that these Killing tensors are invariant under the rotation and SL(2, R) generated by (3.4) and (3.5), respectively As discussed in [17], a combination of these Killing tensors can be written in terms of Killing vectors so is reducible, In the next part, we will see that this property plays an important role in the integrability of geodesics on this geometry.
However, it is not clear from [17] that this near horizon geometry has the principal tensor or not. Apparently, the answer is negative since the principal potential b, seems divergent in the near horizon limit. However, we claim that b can still be well-defined in the near horizon limit if we use this freedom: b is defined up to a shift like with constant C µ 's, that does not affect the principal tensor h, as h = db. In the following, we show that the term which blows up in the near horizon limit is a constant times dt and can be absorbed using this freedom.
To apply the near horizon transformations (3.2) to b in (2.20), we should write it in terms of dt and dφ i using the coordinate transformation given in appendix A. After a shift like C dt, it results in and ε i and Ξ are defined in (2.2). In the near horizon limit (3.2), it takes this form Here, prime is derivative with respect to r coordinate. As a result of the calculations in appendix B, (b 0 + i b i Ω i ) r h is constant and can be absorbed by appropriate C. Therefore, it cancels the divergent term and results We note thatb 0 andb i are functions of x α throughÃ (k) n andÃ (k+1) , respectively and do not depend on ρ.

Integrability of geodesic equations
The simplest constant of motion for time-like geodesics is, Using the inverse metric of near horizon geometry given by the k = 0 of (3.9) and the projection of the Casimir element, I, onto the momentum space, we have . (3.21) The angular Hamiltonian, E, which is defined by can be rewritten in terms of angular variables using equation (3.20).
Similar to the analysis of [19,21], we see that that the Hamilton-Jacobi equations are also separable on near horizon extremal geometry of odd dimensional Kerr-AdS-NUT black hole. Using the identity (C.3), the equation (3.20) can be conveniently rewritten as n µ=1 1 n ν=1 Recalling the identity (C.1), we can rewrite the expression (3.23) in this form, here, ν k 's are some arbitrary and independent constants which can be considered as constants of motion. To find ν k 's, we should reverse the equation, , summing over µ and using the identities (C.2),(C.4), we have In addition to these (n − 1) constants of motion, m 2 0 is also a constant of motion. This can be considered as ν 0 by the shift, We note the reader that the range of k was [1, n − 1] initially and did not include k = 0. However, we extended it to include k = 0.
Recalling (3.24) for the definition of R α and E for (3.22), we have where k runs over [0, n − 1] now. Considering these constants, ν k , as the contraction of Killing tensors, K µν (k) , with momentum p ν , one can readily see that the resultant Killing tensors are the same as Killing tensors in (3.9) that we obtained by taking the near horizon limit. For instance, the Killing tensor related to ν 0 is the metric itself (since A (0) = A (0) µ = 1). In addition to these n constants of motion made of Killing tensors, there are n number of constants of motion associated with Killing vectors ζ i , of the form ζ µ i p µ and 2 other from the Cartan and Casimir elements of SL(2, R). As a result of (3.11), these 2n + 2 constants are Poisson commuting. However, all these constants of motion are not independent and there is a constraint between them as pointed in equation (3.12). Altogether, geodesic equations on the near horizon extremal geometry of d = 2n + 1 dimensional Kerr-AdS-NUT black hole has 2n + 1 independent, commuting constant of motion, therefore is integrable.

Near horizon EVH geometry
In the previous section, it is assumed that the horizon area is non-zero. On the other hand, one can take a limit in which both horizon area and temperature vanish with the same rate. That is what is called EVH limit. As one can readily see from the form of entropy in the equation (2.4), horizon area vanishes once r h goes to zero. The EVH limit of Kerr-AdS-NUT black hole in odd dimensions has been studied in [15] with more details. It is given by such limit, where the parameterm is given bỹ (we used the notations of [15] for the λ 3 .) To obtain the near horizon limit of EVH black hole, we apply the EVH limit (4.1) and the following transformations to the metric 2.1, where, and assume that ǫ ≪ γ in the ǫ, γ → 0 limit. In this case, the near horizon of EVH black hole (NHEVH) reads as whereγ i andε i areγ We note that tilde on each quantity means that it is computed in the EVH and near horizon limit. Therefore, the metric functions arẽ One can check that this geometry is also a solution to the pure Einstein theory.

Principal and Killing tensors
The near horizon geometry of Kerr-AdS-NUT is given in the previous section. As is clear from (4.5), the metric includes an AdS 3 factor and is invariant under SO(2, 2) group. It can be viewed as two copies of SL(2, R). In the coordinates the generators of these two SL(2, R) are as follows The Casimir of each copy is One can simply check that both Casimirs are equal to Applying the EVH (4.1) and near horizon (4.3) limits to the second rank Killing tensors in (4.1) gives (4.13) One can simply check that the k = 0 case of them is just the inverse metric of NHEVH of Kerr-AdS-NUT in (4.5).
The principal tensor h, can be read from its potential b, defined in (2.20), by applying the near horizon and EVH limits, (4.3),(4.1), and taking the transformation (A.5) into account. This gives,b = n i=2b i dϕ i , (4.14) in whichb i 's are the b i 's in (3.15) that should be computed in the near horizon EVH limit.

Integrability of geodesic equations
Again, we start from the simplest constant of motion for geodesics, i.e.
Using the inverse metric of near horizon geometry given by the k = 0 of (4.13) and the projection of the Casimir element, I, onto the momentum space, we have where M ij α is defined by, (4.17) The angular Hamiltonian, E, which is defined by can be simplified using equation (4.16), Note that the structure of the reduced Hamiltonian in (4.19) is similar to the reduced Hamiltonian of NHEG of Kerr-AdS-NUT that we discussed in previous section.
Following the analysis of separability of Hamilton-Jacobi equations in [19,21] reveals that the Hamilton-Jacobi equations are also separable on near horizon EVH geometry of Kerr-AdS-NUT in odd dimensions. Using the identity (C.3), the angular Hamiltonian, E, given in (4.19), can be conveniently represented through Recalling the identity (C.1) we can rewrite the expression (4.20) in more useful form, here, ν k 's are some arbitrary and independent constants which can be considered as constants of motion. To find ν k 's, we should reverse the equation, , summation over α and using the identities (C.2),(C.4), we have , k = 1, · · · , n − 2 . This result can be rewritten by substituting E from (4.18) and noting thatÃ (n−1) is just α x 2 α , as , (4.25) In addition to these (n − 1) constants of motion, m 2 0 is also a constant of motion. This can be considered as ν 0 by the shift, We note the reader that the range of k was [1, n − 1] initially and did not include k = 0. However, we extended it to include k = 0. Recalling (4.21) for the definition of R α , we have where k runs over [0, n − 1], now. Considering these constants, ν k , as the contraction of Killing tensors, K µν (k) , with momentum p µ , one can readily see that the resultant Killing tensors are the same as Killing tensors in (4.13) that we obtained by taking the near horizon limit.
Similar to the extremal case, we have (2n + 1) independent constants of motion in this case. So the geodesic equations on the near horizon EVH Kerr-AdS-NUT geometry are also integrable and separable.

Discussion and conclusion
In this work, we studied the principal and Killing tensors of near horizon extremal/EVH geometries of Kerr-AdS-NUT black hole in odd dimensions. Even dimensional case can be analyzed in a similar manner for the extremal case. Although the Killing tensors were given for the extremal case earlier [17,18], we improve the discussion of hidden symmetries by introducing the principal tensor for near horizon extremal/EVH geometries. The principal tensor is a closed form and locally accompanied by a potential. Then, this potential is defined up to an exact form. In the near horizon limit, we used this freedom to make the principal tensor finite. Existence of this tensor for a given metric makes geodesics/Klein-Gordon/Dirac equations separable on that background metric. We explicitly showed the separability of time-like geodesic equations on the mentioned near horizon geometries.
Finding the constants of motion associated with the geodesics, one can read the Killing tensors of the background metric. We observed that the obtained Killing tensors in this way are the same as the Killing tensors given by taking the near horizon limit.
It is well-known that the isometries enhance in the near horizon extremal limit and Killing vectors have sl(2, R) algebra. However, there is no extra structure among the given second rank Killing tensors and Killing vectors. Particularly, hidden symmetries (associated with the given second rank Killing tenors) do not enhance in the near horizon extremal/EVH limit. This statement should be revised for the equal angular momenta or null geodesics.
In spite of the fact that the Casimir of sl(2, R) gives an extra constant of motion for the geodesic equations on near horizon extremal geometry, this problem is still integrable (not super-integrable) since there is a relation between the constants associated with the Killing tensors, Casimir and Killing vectors. One may expect that for the EVH case where we have so(2, 2) as a subgroup of the isometries, we have more constants of motion. But this is not the case because two constants that the Casimirs of so(2, 2) provide, are equal. Therefore, geodesic problem on near horizon EVH geometry of Kerr-AdS-NUT black hole is also integrable.
These geometries have dual CFT descriptions from the AdS/CFT point of view. It would be interesting to find the meaning of the Killing tensors, hidden symmetries and integrability of geodesics in the CFT side.

Acknowledgments
The author is grateful to Hovhannes Demirchian, Armen Nersessian and especially M.M. Sheikh-Jabbari for discussions during our previous collaborations. I also thank the conference on "Gravity -New perspectives from strings and higher dimensions" where the project was initiated. I learned hidden symmetries from David Kubiznak there and thank him. This work is partially supported by ICTP program network scheme NT-04.

A A useful coordinate transformation
Using the transformationst on the Kerr-NUT-AdS metric (2.1) in odd dimensions (D = 2n + 1) and the definitions Then, it can be written as By using the relations, In this part, we restrict our attention to g = 0, L α = 0 and d = 5 case, i.e. to the near horizon extremal geometry of five dimensional Myers-Perry black hole [23]. This solution is described by two rotation parameter, a 1 , a 2 . After solving (2.3) for the horizon location in the extremal limit, (3.1), we find that The near horizon metric is and its functions arẽ As discussed in section4.1, this geometry has 2 second rank Killing tensors. One of them, K (0) , is the metric itself and another is The angular velocities are .
Then, b 0 and b i 's in (3.15) are equal to By applying the near horizon transformation, to the principal potential b in (3.14), (3.15), it is easy to see that which is constant. Therefore, choosing C = − a 2 b 2 2 (a+b) 2 will remove the divergent term of b in the near horizon limit. Then, we get . (B.10) The principal tensor, h = d b, reads dx ∧ (a 2 1 dϕ 1 − a 2 2 dϕ 2 ) . (B.11) The Hodge dual of h, gives a Killing-Yano tensor of this form The Killing tensor made of this Killing-Yano tensor, using (2.15), is proportional to K (1) given by (B.4).

B.2 Generic odd dimensions
As discussed in section 4.1, the principal potential, b, is divergent in the near horizon limit. However, it is defined up to a shift of the form with constant C µ 's. This shift does not affect h = d b. We use this freedom to make b finite in the near horizon limit. The divergent term arise from dt → dτ /λ. In the following, we show that its coefficient which is equal to (b 0 + i b i Ω i ) r h , is a constant. We start by substituting b 0 and b i 's from (3.15), . (B.14) The summation over k can be easily done by changing k = u − 1, where in the last line we used the definition y 2 0 ≡ 0. The contribution of the last expression of S 1 to B is a constant, B 0 , which is desirable. Therefore, by applying the change into the B, we have , (B.17) Then, using the identity (C.5), the expression in the bracket can be expanded in powers of (−d 2 M ). Using the identity (C.1) for the summation over M, simplifies B significantly and leads to which is obviously constant and can be absorbed by C 0 . Therefore, the near horizon expansion of b τ starts from ρ, as explained in (3.16) and gives b =b 0 ρ dτ + (−x 2 α )