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

SIGMA 17 (2021), 071, 14 pages      arXiv:2103.10638      https://doi.org/10.3842/SIGMA.2021.071

$\mathbb{Z}_2^3$-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

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.

Key words: graded Lie superalgebras; superconformal mechanics.

pdf (418 kb)   tex (22 kb)  

References

  1. Aizawa N., Generalization of superalgebras to color superalgebras and their representations, Adv. Appl. Clifford Algebr. 28 (2018), 28, 14 pages, arXiv:1712.03008.
  2. Aizawa N., Amakawa K., Doi S., $\mathbb{Z}_2^n$-graded extensions of supersymmetric quantum mechanics via Clifford algebras, J. Math. Phys. 61 (2020), 052105, 13 pages, arXiv:1912.11195.
  3. Aizawa N., Amakawa K., Doi S., $\mathcal N$-extension of double-graded supersymmetric and superconformal quantum mechanics, J. Phys. A: Math. Theor. 53 (2020), 065205, 14 pages, arXiv:1905.06548.
  4. Aizawa N., Cunha I.E., Kuznetsova Z., Toppan F., On the spectrum-generating superalgebras of the deformed one-dimensional quantum oscillators, J. Math. Phys. 60 (2019), 042102, 18 pages, arXiv:1812.00873.
  5. Aizawa N., Kuznetsova Z., Tanaka H., Toppan F., $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie symmetries of the Lévy-Leblond equations, Prog. Theor. Exp. Phys. 2016 (2016), 123A01, 26 pages, arXiv:1609.08224.
  6. Aizawa N., Kuznetsova Z., Tanaka H., Toppan F., Generalized supersymmetry and Lévy-Leblond equation, in Physical and Mathematical Aspects of Symmetries, Editors J.-P. Gazeau, S. Faci, T. Micklitz, R. Scherer, F. Toppan, Springer, Cham, 2017, 79-84, arXiv:1609.08760.
  7. Aizawa N., Kuznetsova Z., Toppan F., The quasi-nonassociative exceptional $F(4)$ deformed quantum oscillator, J. Math. Phys. 59 (2018), 022101, 13 pages, arXiv:1711.02923.
  8. Aizawa N., Kuznetsova Z., Toppan F., ${\mathbb Z}_2\times {\mathbb Z}_2$-graded mechanics: the classical theory, Eur. Phys. J. C 80 (2020), 668, 14 pages, arXiv:2003.06470.
  9. Aizawa N., Kuznetsova Z., Toppan F., $\mathbb Z_2\times \mathbb Z _2$-graded mechanics: the quantization, Nuclear Phys. B 967 (2021), 115426, 30 pages, arXiv:2021.11542.
  10. Amakawa K., Aizawa N., A classification of lowest weight irreducible modules over $\mathbb{Z}_2^2$-graded extension of $\mathfrak{osp}(1|2)$, J. Math. Phys. 62 (2021), 043502, 18 pages, arXiv:2011.03714.
  11. Bellucci S., Krivonos S., Supersymmetric mechanics in superspace, in Supersymmetric Mechanics, Vol. 1, Lecture Notes in Phys., Vol. 698, Springer, Berlin, 2006, 49-96, arXiv:hep-th/0602199.
  12. Bruce A.J., On a $\mathbb{Z}_2^n$-graded version of supersymmetry, Symmetry 11 (2019), 116, 20 pages, arXiv:1812.02943.
  13. Bruce A.J., $\mathbb Z_2 \times \mathbb Z_2$-graded supersymmetry: 2-d sigma models, J. Phys. A: Math. Theor. 53 (2020), 455201, 25 pages, arXiv:2006.08169.
  14. Bruce A.J., Duplij S., Double-graded supersymmetric quantum mechanics, J. Math. Phys. 61 (2020), 063503, 13 pages, arXiv:1904.06975.
  15. Carrion H.L., Rojas M., Toppan F., Quaternionic and octonionic spinors. A classification, J. High Energy Phys. 2003 (2003), 040, 28 pages, arXiv:hep-th/0302113.
  16. de Alfaro V., Fubini S., Furlan G., Conformal invariance in quantum mechanics, Nuovo Cimento A 34 (1976), 569-612.
  17. Fedoruk S., Ivanov E., Lechtenfeld O., Superconformal mechanics, J. Phys. A: Math. Theor. 45 (2012), 173001, 59 pages, arXiv:1112.1947.
  18. Le Roy B., $\mathbb{Z}_n^3$-graded colored supersymmetry, Czechoslovak J. Phys. 47 (1997), 47-54, arXiv:hep-th/9608074.
  19. Okazaki T., Superconformal quantum mechanics from M2-branes, Ph.D. Thesis, California Institute of Technology, USA, Osaka University, Japan, arXiv:1503.03906.
  20. Okubo S., Real representations of finite Clifford algebras. I. Classification, J. Math. Phys. 32 (1991), 1657-1668.
  21. Papadopoulos G., Conformal and superconformal mechanics, Classical Quantum Gravity 17 (2000), 3715-3741, arXiv:hep-th/0002007.
  22. Poncin N., Towards integration on colored supermanifolds, in Geometry of Jets and Fields, Banach Center Publ., Vol. 110, Polish Acad. Sci. Inst. Math., Warsaw, 2016, 201-217.
  23. Ree R., Generalized Lie elements, Canadian J. Math. 12 (1960), 493-502.
  24. Rittenberg V., Wyler D., Generalized superalgebras, Nuclear Phys. B 139 (1978), 189-202.
  25. Rittenberg V., Wyler D., Sequences of $Z_{2}\oplus Z_{2}$ graded Lie algebras and superalgebras, J. Math. Phys. 19 (1978), 2193-2200.
  26. Scheunert M., Generalized Lie algebras, J. Math. Phys. 20 (1979), 712-720.
  27. Tolstoy V.N., Super-de Sitter and alternative super-Poincaré symmetries, in Lie Theory and its Applications in Physics, Springer Proc. Math. Stat., Vol. 111, Springer, Tokyo, 2014, 357-367, arXiv:1610.01566.
  28. Toppan F., $\mathbb Z_2 \times \mathbb Z_2$-graded parastatistics in multiparticle quantum Hamiltonians, J. Phys. A: Math. Theor. 54 (2021), 115203, 35 pages, arXiv:2008.11554.

Previous article  Next article  Contents of Volume 17 (2021)