Supersymmetric extensions of Schr\"odinger-invariance

TL;DR

This paper explores supersymmetric extensions of Schr"odinger invariance, constructing superalgebras and calculating two-point functions, thereby broadening the understanding of dynamic symmetries in quantum equations.

Contribution

It introduces new supersymmetric models extending Schr"odinger symmetry and systematically constructs related infinite-dimensional Lie superalgebras.

Findings

Super-Schr"odinger and (3|2)-supersymmetric models derived.

Largest finite-dimensional Lie superalgebras identified within a Poisson algebra framework.

Two-point functions of superfields computed for key subalgebras.

Abstract

The set of dynamic symmetries of the scalar free Schr\"odinger equation in d space dimensions gives a realization of the Schr\"odinger algebra that may be extended into a representation of the conformal algebra in d+2 dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant, but as an additional coordinate. An analogous construction also holds for the spin-1/2 L\'evy-Leblond equation. A N=2 supersymmetric extension of these equations leads, respectively, to a `super-Schr\"odinger' model and to the (3|2)-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras osp(2|2) *_s sh(2|2) and osp(2|4), respectively. The Schr\"odinger algebra and its supersymmetric counterparts are found to be the largest finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically constructed in a Poisson algebra setting, including the Schr\"odinger-Neveu-Schwarz algebra sns^(N) with N supercharges. Covariant two-point functions of quasiprimary superfields are calculated for several subalgebras of osp(2|4). If one includes both N=2 supercharges and time-inversions, then the sum of the scaling dimensions is restricted to a finite set of possible values.