$\mathbb{Z}_2^3$-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics

Abstract: Quantum mechanical systems whose symmetry is given by $\mathbb{Z}_2^3$-graded version of superconformal algebra are introduced. This is done by finding a realization of a $\mathbb{Z}_2^3$-graded Lie superalgebra in terms of a standard Lie superalgebra and the Clifford algebra. The realization allows us to map many models of superconformal quantum mechanics (SCQM) to their $\mathbb{Z}_2^3$-graded extensions. It is observed that for the simplest SCQM with $\mathfrak{osp}(1|2)$ symmetry there exist two inequivalent $\mathbb{Z}_2^3$-graded extensions. Applying the standard prescription of conformal quantum mechanics, spectrum of the SCQMs with the $\mathbb{Z}_2^3$-graded $\mathfrak{osp}(1|2)$ symmetry is analyzed. It is shown that many models of SCQM can be extended to $\mathbb{Z}_2^n$-graded setting.