SU(2$|$1) supersymmetric mechanics on curved spaces

TL;DR

This paper develops SU(2|1) supersymmetric mechanics on curved manifolds, deriving conditions for supercharges and Hamiltonians, and constructing models on Kähler and isotropic spaces, unifying previous one-dimensional models.

Contribution

It introduces a general framework for SU(2|1) supersymmetric mechanics on curved spaces, including new solutions to curved WDVV equations and a unified approach to existing models.

Findings

Derived extended curved WDVV equations for supersymmetric systems.

Constructed all interactions for real Kähler manifolds.

Provided admissible prepotentials for isotropic spaces.

Abstract

We present SU$(2|1)$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds within the Hamiltonian approach. The structure functions including prepotentials entering the supercharges and the Hamiltonian obey extended curved WDVV equations specified by the manifold's metric and curvature tensor. We consider the most general $u(2)$-valued prepotential, which contains both types (with and without spin variables), previously considered only separately. For the case of real K\"{a}hler manifolds we construct all possible interactions. For isotropic ($so(n)$-invariant) spaces we provide admissible prepotentials for any solution to the curved WDVV equations. All known one-dimensional SU$(2|1)$ supersymmetric models are reproduced.