K\"ahler geometry for $su(1,N|M)$-superconformal mechanics

TL;DR

This paper develops a superconformal mechanics framework based on the $su(1,N|M)$ algebra using a phase superspace modeled on a super-analogue of the Lobachevsky plane, enabling new superintegrable systems with distinct supersymmetries.

Contribution

It introduces a novel geometric formulation of $su(1,N|M)$-superconformal mechanics on a superspace analogous to the Lobachevsky plane, and constructs new superintegrable systems with specific supersymmetries.

Findings

Constructed superconformal extensions of superintegrable systems.

Proposed superintegrable oscillator- and Coulomb-like systems.

Found oscillator-like systems admit deformed $ ext{N}=2M$ Poincaré supersymmetry.

Abstract

We suggest the $su(1,N|M)$-superconformal mechanics formulated in terms of phase superspace given by the non-compact analogue of complex projective superspace $\mathbb{CP}^{N|M}$. We parameterized this phase space by the specific coordinates allowing to interpret it as a higher-dimensional super-analogue of the Lobachevsky plane parameterized 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 proposed scheme allows 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 oscillator-like systems admit deformed $\mathcal{N}=2M$ Poincar\'e supersymmetry, in contrast with Coulomb-like ones.