Hidden symmetries of integrable conformal mechanical systems

Abstract

We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N=4 supersymmetric "angular" Hamiltonian system one may construct a new system with full N=4 superconformal D(1,2;\alpha) symmetry.Comment: 9 pages revte

The tetrahexahedric angular Calogero model

“…A number of classical features of angular rational Calogero model (center-of-massreduced or not) have been recently analyzed by an Armenian group [22][23][24][25][26][27]. Intertwining relations for the quantum model were investigated by M. Feigin already in [28], but more recently the energy spectra and eigenstates have been constructed [29], and the algebra of angular Dunkl operators has been studied [30].…”

Conformal mechanics inspired by extremal black holes in d = 4

“…The advantage of the conformal basis is that it allows one to accumulate all specific information about the original model in a lower-dimensional integrable system governed by the effective coupling I. Moreover, with such a formulation at hand, one can reduce the problem of constructing an N = 4 superconformal extension of the original system to that for the angular part [12].…”

“…At this point, the Hamiltonian can be put in the conventional conformal mechanics form by introducing the new radial coordinates as in [12] (for related earlier studies see [14,15])…”

“…A relation between the original conformal mechanics and its angular sector is far from trivial. For example, constants of motion of the initial conformal mechanics and those of its angular part are related in a rather intricate way [11,12,13]. One can also reverse the argument and construct new integrable conformal mechanical systems starting from known ones.…”

“…Conformal basis also provides a useful means for supersymmetrization of conformal mechanics. Given N = 4 supersymmetric extension of the Hamiltonian system ( 1 2 ω αβ du α ∧ du β , I(u)), one can immediately construct its generalization which enjoys the full D(1, 2; α) symmetry [12].…”