Note on constants of motion in conformal mechanics associated with near horizon extremal Myers–Perry black holes

Abstract

We investigate dynamics of probe particles moving in the near-horizon limit of (2N + 1)-dimensional extremal Myers-Perry black hole (in the cases of N = 3, 4, 5) with arbitrary rotation parameters. Very recently it has been shown [1] that in the most general case with nonequal nonvanishing rotational parameters the system admits separation of variables in N -dimensional ellipsoidal coordinates. We wrote down the explicit expressions of Liouville integrals of motion, given in [1] in ellipsoidal coordinates, in initial "Cartesian" coordinates in seven, nine and eleven dimensions, and found that these expressions hold in any dimension. Then, taking the limit where all of the rotational parameters are equal, we reveal that each of these N − 1 integrals of motion results in the Hamiltonian of the spherical mechanics of a (2N + 1)-dimensional MP black hole with equal nonvanishing rotational parameters.

Super 0-brane action on the coset space of D(2, 1; α) supergroup

“…is identical on the constraint surface (39). Finally, let us canonically redefine the bosonic coordinate z and its conjugate momentum p z…”

“…A plenty of such models have been constructed and investigated in [7]- [17] which all relied upon the d = 1, N = 4 superconformal group SU (1, 1|2). A related line of research concerns the study of (super)conformal particles propagating on near horizon black hole backgrounds [18]- [32] and the construction of novel (super)integrable systems associated with such geometries [33]- [39]. Worth mentioning also is the resent study of the AdS superparticles within the group-theoretic approach [40,41,42].…”

Integrability of geodesics in near-horizon extremal geometries: Case of Myers-Perry black holes in arbitrary dimensions

“…In this work, we continue our analysis of [1,23] and extend the analysis there to Near Horizon Extremal Myers-Perry [24] (NHEMP) black holes [21] in general odd and even dimensions. We discuss the separability of variables, constants of motion for "angular mechanics" associated with these systems and how they are related to the second rank Killing tensors of the background.…”

“…One can check that in odd dimensions in the special cases of F N −1 , F N −2 and F N −3 , the above reduce to the corresponding integrals of motion given in [23]. One can also check, that simply requiring the rotation parameters to be equal in these expressions, one does not recover all the integrals of the special case of a i = a, ∀i NHEMP.…”

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

“…It is worth mentioning that these Killing tensors are invariant under the rotation and sl(2, R) generated by Eqs. (24) and (25), respectively:…”