${\mathbb{Z}}_2^3$-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.

References in corpus (7)

• Superconformal mechanics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie Symmetries of the Lévy-Leblond Equations
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the classical theory
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the quantization
• Supersymmetric Mechanics in Superspace
• Super-de Sitter and alternative super-Poincaré symmetries
• Superconformal Quantum Mechanics from M2-branes

Cited by in corpus (5)

• Irreducible representations of ${\mathbb{Z}}_2^2$-graded $\mathcal{N}=2$ supersymmetry algebra and ${\mathbb{Z}}_2^2$-graded supermechanics
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• Graded colour Lie superalgebras for solving Lévy-Leblond equations
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• $\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction