Extended supersymmetry with central charges in Dirac action with curved extra dimensions

Abstract

We discuss a new realization of N-extended quantum-mechanical supersymmetry (QM SUSY) with central charges hidden in the four-dimensional (4D) mass spectrum of higher dimensional Dirac action with curved extra dimensions. We show that this N-extended QM SUSY results from symmetries in extra dimensions, and the supermultiplets in this supersymmetry algebra correspond to the Bogomol'nyi-Prasad-Sommerfield states. Furthermore, we examine the model of the S 2 extra dimension with a magnetic monopole background and confirm that the N-extended QM SUSY explains the degeneracy of the 4D mass spectrum.So far, quantum-mechanical supersymmetry (QM SUSY) has attracted much attention and been applied to the various research areas, e.g. exactly solvable quantum mechanics [1][2][3][4][5], Berry phase [6][7][8], black holes and AdS/CFT [9-13], Sachdev-Ye-Kitaev model [14][15][16][17][18], extra dimensional models [19][20][21][22][23][24] and so on.It's extensions are also investigated. The N-extended supersymmetry is the extension which includes N independent supercharges in the supersymmetry algebra [25][26][27][28][29][30][31][32]. Each of supercharge corresponds to a square root of Hamiltonian, and they explain the degeneracy of the energy spectrum. In addition, the central extension which introduces central charges in the algebra is also studied [33][34][35][36]. 1 Central charges commute with all the operators in the algebra. As is well known, if there are central charges, the size of supermultiplets can be small compared with the regular representation [41,42]. Such multiplets are called short multiplets or Bogomol'nyi-Prasad-Sommerfield (BPS) states. 2 Since not so many models which realize arbitrary large N-extended QM SUSY with central charges are known, it is worth investigating a new realization of Nextended one.Here, we focus on the higher dimensional Dirac action with extra dimensions. In Refs. [23,24], it has been shown that the structure of the N = 2 QM SUSY is hidden in the four-dimensional (4D) mass spectrum of the higher dimensional Dirac action, and the Kaluza-Klein (KK) mode functions for the 4D right-handed and left-handed spinors form the supermultiplets. Furthermore, in the previous papers [45,46], we have revealed that this N = 2 QM SUSY can be extended to the N-extended QM SUSY from the reflection symmetries in the extra dimensions. Then we have found that the central charges appear as the result of the reflection symmetries, and the supermultiplets of this extended QM SUSY corresponds to the BPS states. These supercharges can explain the degeneracy of the 4D mass spectrum. However, previous works are only devoted to the case of the flat extra dimensions. Therefore, we should take into account the case of curved extra dimensions for a general discussion. Furthermore, we have only focused on the N-extended QM SUSY from the reflection symmetries. Then, we can expect the possibilities that further structures of N-extended QM SUSY from other symmetries are hidden in the 4D mass spectrum.In this paper, we dis...

${\mathbb Z}_2\times {\mathbb Z}_2$-graded mechanics: the classical theory

“…We recall that supersymmetry algebra with central extension naturally appears in higher dimensional Dirac actions with curved extra dimension, see [9], while commuting fermions also appear in the dual double field theory [10] (see also [11] and [12] for the relation to higher grading geometry). Therefore, one could expect that the Z 2 × Z 2 -graded quantum systems possess some physical relevance, at least in nonrelativistic or anyonic physics, where the spinstatistics connection does not necessarily holds.…”

$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$-graded mechanics: the classical theory

“…We recall that supersymmetry algebra with central extension naturally appears in higher dimensional Dirac actions with curved extra dimension, see [12], while commuting fermions also appear in the dual double field theory [13] (see also [14] and [15] for the relation to higher grading geometry). Therefore, one could expect that the Z 2 × Z 2 -graded quantum systems possess some physical relevance, at least in nonrelativistic or anyonic physics, where the spin-statistics connection does not necessarily hold.…”

The Z2×Z2-graded general linear Lie superalgebra

“…For some of the renewed physical interest, we refer to [9] and references therein. Very recent work consists of the importance of Z 2 × Z 2 graded Lie superalgebras in the analysis of the Lévy-Leblond equations [10,11], and various potential applications in the setting of supersymmetric and superconformal quantum mechanics [12][13][14][15][16].…”