Lobachevsky geometry of (super)conformal mechanics

TL;DR

This paper reveals a geometric interpretation of conformal mechanics transformations as inversions in Lobachevsky space, extending the concept to superconformal mechanics using Lobachevsky superspace.

Contribution

It provides a simple geometric explanation for conformal mechanics transformations and extends this to superconformal mechanics with Lobachevsky superspace.

Findings

Conformal transformations correspond to inversions in Lobachevsky space.

Extension of geometric picture to N=2k superconformal mechanics.

Unified geometric framework for conformal and superconformal mechanics.

Abstract

We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane). We also extend this picture to the N=2k superconformal mechanics described in terms of Lobachevsky superspace.