Particle dynamics near extreme Kerr throat and supersymmetry

The extreme Kerr throat solution is believed to be non-supersymmetric. However, its isometry group SO(2,1) x U(1) matches precisely the bosonic subgroup of N=2 superconformal group in one dimension. In this paper we construct N=2 supersymmetric extension of a massive particle moving near the horizon of the extreme Kerr black hole. Bosonic conserved charges are related to Killing vectors in a conventional way. Geometric interpretation of supersymmetry charges remains a challenge.Comment: V2: 10 pages; discussion in sect. 4 and 5 extended, acknowledgements and references adde

Conformal mechanics inspired by extremal black holes in d=4

A canonical transformation which relates the model of a massive relativistic particle moving near the horizon of an extremal black hole in four dimensions and the conventional conformal mechanics is constructed in two different ways. The first approach makes use of the action-angle variables in the angular sector. The second scheme relies upon integrability of the system in the sense of Liouville.Comment: V2: presentation improved, new material and references added; the version to appear in JHE

Action-angle variables for the particle near extreme Kerr throat

We construct the action-angle variables for the spherical part of conformal mechanics describing the motion of a particle near extreme Kerr throat. We indicate the existence of the critical point $|p_\varphi|=mc R_{\rm Sch}$ (with $m$ being the mass of the particle, $c$ denoting the speed of light, $R_{\rm Sch}=2\gamma M /c^2$ being the Schwarzschild radius of a black hole with mass $M$, and $\gamma$ denoting the gravitational constant), where these variables are expressed in terms of elementary functions. Away from this point the action-angle variables are defined by elliptic integrals. The proposed formulation allows one to easily reconstruct the whole dynamics of the particle both in initial coordinates, as well as in the so-called conformal basis, where the Hamiltonian takes the form of conventional non-relativistic conformal mechanics. The related issues, such as semiclassical quantization and supersymmetrization are also discussed.Comment: 8 pages, PACS numbers: 04.70.Bw, 45.10.Na; we corrected a mistak

Generalizations of MICZ-Kepler system

We discuss the generalizations of the MICZ-Kepler system (the system describing the motion of the charged particle in the field of Dirac dyon), to the curved spaces, arbitrary potentials and to the multi-dyon background.Comment: 6 pages, talk given at Colloquium on Integrable models and Quantum symmetry, 14-16.07.2007, Prague. submitted in Rep. Math.Phy

Superconformal mechanics

We survey the salient features and problems of conformal and superconformal mechanics and portray some of its developments over the past decade. Both classical and quantum issues of single- and multiparticle systems are covered.Comment: 1+68 pages, invited review for Journal of Physics A; v2: revised text extended by 4 pages and 11 references, published versio

Massless geodesics in AdS5×Y(p,q)AdS_5\times Y(p,q) as a superintegrable system

A Carter like constant for the geodesic motion in the $Y(p,q)$ Einstein-Sasaki geometries is presented. This constant is functionally independent with respect to the five known constants for the geometry. Since the geometry is five dimensional and the number of independent constants of motion is at least six, the geodesic equations are superintegrable. We point out that this result applies to the configuration of massless geodesic in $AdS_5\times Y(p,q)$ studied by Benvenuti and Kruczenski, which are matched to long BPS operators in the dual N=1 supersymmetric gauge theory.Comment: 20 pages, no figures. Small misprint is corrected in the Killing-Yano tensor. No change in any result or conclusion

Saghatelian A. Spherical Mechanics for a Particle Near the Horizon of Extremal Black Hole

We describe canonical transformation, which links the Hamiltonian of a massive relativistic particle moving near the horizon of an extremal black hole to the conventional form of the conformal mechanics. Thus, like the nonrelativistic conformal mechanics, the investigation of the particle dynamics reduces to analyzing its ©spherical sectorª deˇned by the Casimir element of the conformal algebra. We present a detailed list of such systems originating from various types of black hole conˇgurations. PACS: 04.70.Bw Conformal mechanics associated with the near-horizon geometry of an extremal black hole is described by the triple which involves the Hamiltonian H, the generator of dilatations D, and the generator of special conformal transformations K. Under the Poisson brackets they form an so(2, 1) algebra Recently, we suggested a canonical transformation (p r , r, p a , ϕ a ) → (p R , R,p a ,φ a ), which links the Hamiltonian (1) to the conventional nonrelativistic form [1Ä3] (for related earlier studies, see