The octonionically-induced ${\cal N}=7$ exceptional $G(3)$ superconformal quantum mechanics

TL;DR

This paper constructs a unique ${ m N}=7$ superconformal quantum mechanics with $G(3)$ symmetry using octonion-inspired methods, providing explicit models, their spectra, and embedding techniques within Clifford algebra frameworks.

Contribution

It introduces the first ${ m N}=7$ superconformal quantum mechanics with $G(3)$ symmetry, derived via octonionic methods, and details its deformed oscillator and spectrum.

Findings

Constructed the ${ m N}=7$ superconformal model with $G(3)$ symmetry.

Derived the covariant embedding of $g_2$ within Clifford algebra.

Computed the spectrum of the $G(3)$ deformed oscillator.

Abstract

Both the ${\cal N}=7$ superconformal quantum mechanics possessing the exceptional $G(3)$ Lie superalgebra as dynamical symmetry and its associated deformed oscillator with $G(3)$ as spectrum-generating superalgebra are presented. This superconformal quantum mechanics, uniquely defined up to similarity transformations, is obtained via the octonionically-induced "quasi-nonassociative" method employed to derive the exceptional ${\cal N}=8$ $F(4)$ model. To construct the $G(3)$ theories, the covariant embedding of the $7$-dimensional representation of the Lie algebra $g_2$ within the $8\times 8$ matrices spanning the $Cl(0,7)$ Clifford algebra is derived. The Hilbert space of the $G(3)$ deformed oscillator is given by a $16$-ple of square-integrable functions of a real space coordinate. The spectrum of the theory is computed.