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

Abstract

Both the $\mathcal{N}=7$ 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 $\mathcal{N}=8$ $F(4)$ 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.

arXiv comment17 pages

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