Conformal and Superconformal Mechanics Revisited

TL;DR

This paper establishes the classical and superconformal equivalence between different models of conformal mechanics, providing explicit transformations and constructing a novel off-shell superfield action for the N=2 superextension of the Bertotti-Robinson particle.

Contribution

It explicitly relates conformal mechanics models and constructs a new off-shell superfield action for N=2 superconformal mechanics.

Findings

Classical equivalence between De Alfaro-Fubini-Furlan and Bertotti-Robinson conformal mechanics.

Construction of an off-shell superfield action for N=2 superconformal mechanics.

On-shell equivalence to the AdS$_2$ superparticle action.

Abstract

We find, at the Lagrangian off-shell level, the explicit equivalence transformation which relates the conformal mechanics of De Alfaro, Fubini and Furlan to the conformal mechanics describing the radial motion of the charged massive particle in the Bertotti-Robinson AdS$_2\times S^2$ background. Thus we demonstrate the classical equivalence of these two systems which are usually regarded as essentially different ``old'' and ``new'' conformal mechanics models. We also construct a similar transformation for N=2, $SU(1,1|1)$ superconformal mechanics in N=2 superfield formulation. Performing this transformation in the action of the N=2 superconformal mechanics, we find an off-shell superfield action of N=2 superextension of Bertotti-Robinson particle. Such an action has not been given before. We show its on-shell equivalence to the AdS$_2$ superparticle action derived from the spontaneous partial breaking of $SU(1,1|1)$ superconformal symmetry treated as the N=2 AdS$_2$ supersymmetry.