Kähler geometry for su(1,N|M) superconformal mechanics

Erik Khastyan,Sergey Krivonos,Armen Nersessian

doi:10.1103/physrevd.105.025007

Abstract

We suggest the $su(1,N|M)$ superconformal mechanics formulated in terms of phase superspace given by the noncompact analogue of complex projective superspace. We parametrized this phase space by the specific coordinates allowing us to interpret it as a higher-dimensional superanalogue of the Lobachevsky plane parametrized by lower half-plane (Klein model). Then we introduced the canonical coordinates corresponding to the known separation of the ``radial'' and ``angular'' parts of (super)conformal mechanics. Relating the ``angular'' coordinates with action-angle variables, we demonstrated that the proposed scheme allows us to construct the $su(1,N|M)$ supeconformal extensions of wide class of superintegrable systems. We also proposed the superintegrable oscillator- and Coulomb-like systems with a $su(1,N|M)$ dynamical superalgebra and found that oscillatorlike systems admit deformed $\mathcal{N}=2M$ Poincar\'e supersymmetry, in contrast with Coulomb-like ones.

Highlights

Kähler manifolds are the Hermitian manifolds, which possesses the symplectic structure obeying the specific compatibility condition with the Riemann structure [1]
We suggest the suð1; NjMÞ superconformal mechanics formulated in terms of phase superspace given by the noncompact analogue of complex projective superspace
Relating the “angular” coordinates with action-angle variables, we demonstrated that the proposed scheme allows us to construct the suð1; NjMÞ supeconformal extensions of wide class of superintegrable systems

Summary

INTRODUCTION

Kähler manifolds are the Hermitian manifolds, which possesses the symplectic structure obeying the specific compatibility condition with the Riemann (and/or complex) structure [1]. The similar formulation of some higher-dimensional systems was given in terms of suð; NÞ-symmetric Kähler phase space treated as the noncompact version of a complex projective space [5] In such an approach, all symmetries of the generic superintegrable conformal-mechanical systems acquire interpretation in terms of the powers of the suð; NÞ isometry generators. We consider the systems with suð; NjMÞ-symmetric ðNjMÞC-dimensional Kähler phase superspace (in what follows, we denote it by CfPNjM) and relate their symmetries with the isometry generators of the super-Kähler structure We construct this superspace, reducing the ðN þ 1jMÞC-dimensional complex pseudo-Euclidean superspace by the Uð1Þ-group action and identify the reduced phase superspace with the noncompact analogue of complex projective superspace constructed in [6].

NONCOMPACT COMPLEX PROJECTIVE SUPERSPACE

OSCILLATOR- AND COULOMB-LIKE SYSTEMS

Oscillatorlike systems

FUBINI-STUDY-LIKE KÄHLER STRUCTURE

CONCLUDING REMARKS