${\cal N}{=}\,4$ supersymmetric mechanics on curved spaces

TL;DR

This paper develops ${ m N}=4$ supersymmetric mechanics on curved Riemannian manifolds, extending flat space solutions, and reveals how curvature induces additional potential terms like the Higgs oscillator, leading to superintegrable models.

Contribution

It introduces a Hamiltonian framework for ${ m N}=4$ supersymmetric mechanics on curved spaces, generalizing flat space solutions and analyzing curvature effects on potentials.

Findings

Solutions of flat space equations can be lifted to curved manifolds.

Curvature induces Higgs-oscillator potentials on spheres and hyperboloids.

A superintegrable deformation of conformal mechanics emerges from this construction.

Abstract

We present ${\cal N}{=}\,4$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds constructed within the Hamiltonian approach. The structure functions entering the supercharges and the Hamiltonian obey modified covariant constancy equations as well as modified Witten-Dijkgraaf-Verlinde-Verlinde equations specified by the presence of the manifold's curvature tensor. Solutions of original Witten-Dijkgraaf-Verlinde-Verlinde equations and related prepotentials defining ${\cal N}{=}\,4$ superconformal mechanics in flat space can be lifted to $so(n)$-invariant Riemannian manifolds. For the Hamiltonian this lift generates an additional potential term which, on spheres and (two-sheeted) hyperboloids, becomes a Higgs-oscillator potential. In particular, the sum of $n$ copies of one-dimensional conformal mechanics results in a specific superintegrable deformation of the Higgs oscillator.