# Klein four-group and Darboux duality in conformal mechanics

## TL;DR

This paper explores the Klein four-group symmetry in conformal mechanics, its breaking, and its relation to Darboux duality, leading to new insights into spectral transformations and algebraic structures of deformed models.

## Contribution

It introduces the use of Klein four-group symmetry and Darboux duality to generate and analyze rationally deformed AFF models and their algebraic properties.

## Key findings

- Klein four-group symmetry undergoes partial or complete breaking in eigenstates.
- Darboux duality generates spectrally shifted rationally deformed models.
- Finite-gap structures and nonlinear algebraic symmetries are identified in deformed systems.

## Abstract

We study the Klein four-group $(K_4)$ symmetry of the time-dependent Schr\"odinger equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling constant $g=\nu(\nu+1)\geq -1/4$. We show that it undergoes a complete or partial (at half-integer $\nu$) breaking on eigenstates of the system, and is the automorphism of the $\mathfrak{osp}(2,2)$ superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of $\mathcal{N}=2$ Poincar\'e supersymmetry. We exploit the $K_4$ symmetry and its relation with the conformal symmetry to construct the dual Darboux transformations which generate spectrally shifted pairs of the rationally deformed AFF models. Two distinct pairs of intertwining operators originated from Darboux duality allow us to construct complete sets of the spectrum generating ladder operators that identify specific finite-gap structure of a deformed system and generate three distinct related versions of nonlinearly deformed $\mathfrak{sl}(2,{\mathbb R})$ algebra as its symmetry. We show that at half-integer $\nu$, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.