ℤ³₂-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics

Abstract

Quantum mechanical systems whose symmetry is given by the ℤ³₂-graded version of superconformal algebra are introduced. This is done by finding a realization of a ℤ³₂-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 ℤ³₂-graded extensions. It is observed that for the simplest SCQM with 𝔬𝔰𝔭(1|2) symmetry, there exist two inequivalent ℤ³₂-graded extensions. Applying the standard prescription of conformal quantum mechanics, the spectrum of the SCQMs with the ℤ³₂-graded 𝔬𝔰𝔭(1|2) symmetry is analyzed. It is shown that many models of SCQM can be extended to a ℤⁿ₂-graded setting.