# Geometry and integrability in $\mathcal{N}=8$ supersymmetric mechanics

## TL;DR

This paper develops $
abla=8$ supersymmetric mechanics on special K"ahler manifolds, revealing geometric structures and superintegrable deformations of oscillator and Coulomb systems with potential terms.

## Contribution

It constructs $
abla=8$ supersymmetric mechanics on special K"ahler manifolds and explores their geometric and integrable properties, extending previous $
abla=4$ models.

## Key findings

- Constructed $
abla=8$ supersymmetric mechanics with potential on special K"ahler manifolds.
- Identified superintegrable deformations of oscillator and Coulomb systems.
- Linked the models to curved WDVV equations and special K"ahler geometry.

## Abstract

We construct the $\mathcal{N}=8$ supersymmetric mechanics with potential term whose configuration space is the special K\"ahler manifold of rigid type and show that it can be viewed as the K\"ahler counterpart of $\mathcal{N}=4$ mechanics related to "curved WDVV equations". Then, we consider the special case of the supersymmetric mechanics with the non-zero potential term defined on the family of $U(1)$-invariant one-(complex)dimensional special K\"ahler metrics. The bosonic parts of these systems include superintegrable deformations of perturbed two-dimensional oscillator and Coulomb systems.