Z₂³-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics

Abstract

Quantum mechanical systems whose symmetry is given by Z 3 2 -graded version of superconformal algebra are introduced. This is done by finding a realization of a Z 3 2 -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 Z 3 2 -graded extensions. It is observed that for the simplest SCQM with osp(1|2) symmetry there exist two inequivalent Z 3 2 -graded extensions. Applying the standard prescription of conformal quantum mechanics, spectrum of the SCQMs with the Z 3 2 -graded osp(1|2) symmetry is analyzed. It is shown that many models of SCQM can be extended to Z n 2 -graded setting.

Integration on minimal Z22 -superspace and emergence of space

“…Our study of integration is motivated by recent extensive studies of Z 2 2 -supersymmetric theories [22][23][24][25][26][27][28][29][30][31][32][33][34]. The Z 2 2 -supersymmetry opens up a new area of research of great interest.…”

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

“…Among others, Z n 2 -graded Lie superalgebras and symmetries generated by them attract some interest recently (Z n 2 := Z 2 ו • •×Z 2 , n times). This is because such symmetries are found in the known quantum systems [5][6][7][8] and many simple systems having such symmetries are constructed [9][10][11][12][13][14][15][16][17][18][19]. It is also pointed out the existence of quantum mechanical observables which detect the particles obeying Z 2 2 -graded parastatistics [20,21].…”