Symmetries of the Schr\"odinger Equation and Algebra/Superalgebra Duality

TL;DR

This paper explores the symmetries of the Schr"odinger equation, focusing on algebra/superalgebra duality, on-shell symmetry, and their implications for superconformal sigma-models, revealing new perspectives on spectrum-generating subalgebras.

Contribution

It introduces the concept of algebra/superalgebra duality and the notion of on-shell symmetry, providing novel insights into the symmetry structure of Schr"odinger equations and superconformal models.

Findings

Algebra/superalgebra duality offers different views of spectrum-generating subalgebras.

On-shell symmetry depends on representation and affects the association with $sl(2)$ generators.

Superconformal actions can be either supersymmetric or non-supersymmetric based on symmetry considerations.

Abstract

Some key features of the symmetries of the Schr\"odinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation-dependent notion of on-shell symmetry is introduced. The difference in associating the time-derivative symmetry operator with either a root or a Cartan generator of the $sl(2)$ subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric.