Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 071, 14 pages      arXiv:2103.10638

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

Shunya Doi and Naruhiko Aizawa
Department of Physical Science, Osaka Prefecture University, Nakamozu Campus, Sakai, Osaka 599-8531, Japan

Received March 24, 2021, in final form July 14, 2021; Published online July 20, 2021

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.

Key words: graded Lie superalgebras; superconformal mechanics.

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