News on SU(2|1) Supersymmetric Mechanics

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 191)


We report on a recent progress in exploring the SU(2|1) supersymmetric quantum mechanics. Our focus is on the harmonic SU(2|1) superspace formalism which provides a superfield description of the multiplet \((\mathbf{4, 4, 0})\) and its “mirror” version. We present the \(\sigma \)-model and Wess–Zumino type actions for these multiplets, in both the superfield and the component approaches. An interesting new feature as compared to the flat \(\mathcal{N}=4, d=1\) case is the absence of the explicit SU(2|1) invariant Wess–Zumino term for the ordinary \((\mathbf{4, 4, 0})\) multiplet and yet the existence of such a term for the mirror multiplet. The superconformal subclass of the SU(2|1) invariant \((\mathbf{4, 4, 0})\) actions is also described. Its main distinguishing features are the “trigonometric” realization of the \(d=1\) conformal group SO(2, 1) and the oscillator-type potential terms in the component actions.



Evgeny Ivanov thanks the organizers of the 11-th International Workshop “Lie Theory and Its Applications in Physics” and, especially, Vladimir Dobrev for the kind hospitality in Varna. The authors acknowledge a partial support from the RFBR grant 15-02-06670 and a grant of the Heisenberg - Landau program.


  1. 1.
    Ivanov, E., Sidorov, S.: Deformed Supersymmetric Mechanics. Class. Quant. Grav. 31, 075013 (2014) arXiv:hep-th/1307.7690
  2. 2.
    Festuccia, G., Seiberg, N.: Rigid Supersymmetric Theories in Curved Superspace. J. High Energy Phys. 1106, 114 (2011) arXiv:hep-th1105.0689; Dumitrescu, T.T., Festuccia, G., Seiberg, N.: Exploring Curved Superspace. J. High Energy Phys. 1208, 141 (2012) arXiv:hep-th1205.1115
  3. 3.
    Ivanov, E., Sidorov, S.: Super Kähler oscillator from \(SU(2|1)\) superspace. J. Phys. A47, 292002 (2014) arXiv:hep-th/1312.6821
  4. 4.
    Ivanov, E., Sidorov, S., Toppan, F.: Superconformal mechanics in \(SU(2|1)\) superspace. Phys. Rev. D91, 085032 (2015) arXiv:hep-th/1501.05622
  5. 5.
    Ivanov, E., Sidorov, S.: \(SU(2|1)\) mechanics and harmonic superspace., Class. Quant. Grav., 2016 (in press)
  6. 6.
    Smilga, A.V.: Weak supersymmetry. Phys. Lett. B585, 173 (2004) arXiv:hep-th/0311023
  7. 7.
    Bellucci, S., Nersessian, A.: (Super)Oscillator on CP(N) and Constant Magnetic Field. Phys. Rev. D67, 065013 (2003) arXiv:hep-th/0211070 CrossRefGoogle Scholar
  8. 8.
    Bellucci, S., Nersessian, A.: Supersymmetric Kahler oscillator in a constant magnetic field. In: Proceedings of 5th International Workshop on Supersymmetries and Quantum Symmetries (SQS 03), eds. Evgeny Ivanov and Anatoly Pashnev, 379-384 (2004) arXiv:hep-th/0401232
  9. 9.
    Papadopoulos, G.: New potentials for conformal mechanics. Class. Quant. Grav. 30, 075018 (2013) arXiv:hep-th/1210.1719
  10. 10.
    Holanda, N.L., Toppan, F.: Four types of (super)conformal mechanics: D-module reps and invariant actions. J. Math. Phys. 55, 061703 (2014) arXiv:hep-th/1402.7298
  11. 11.
    Ivanov, E., Sidorov, S.: New Type of N \(=\) 4 Supersymmetric Mechanics. In: Springer Proc. Math. Stat. 111, ed. Vladimir Dobrev, 51-66 (2014)Google Scholar
  12. 12.
    Ivanov, E., Lechtenfeld, O.: \({\cal{N}}=4\) Supersymmetric Mechanics in Harmonic Superspace. J. High Energy Phys. 0309, 073 (2003) arXiv:hep-th/0307111
  13. 13.
    Galperin, A.S., Ivanov, E.A., Ogievetsky, V.I., Sokatchev, E.S.: Harmonic Superspace. Cambridge Univ. Press, 2001, 306 ppGoogle Scholar
  14. 14.
    Delduc, F., Ivanov, E.: \({\cal{N}}=4\) mechanics of general \(({\bf 4, 4, 0})\) multiplets. Nucl. Phys. B855, 815 (2012) arXiv:hep-th/1107.1429
  15. 15.
    Fedoruk, S.A., Ivanov, E.A., Smilga, A.V.: \({\cal{N}} = 4\) mechanics with diverse \(({\bf 4, 4, 0})\) multiplets: Explicit examples of HKT, CKT and OKT geometries. J. Math. Phys. 55, 052302 (2014) arXiv:hep-th/1309.7253
  16. 16.
    Frappat, L., Sciarrino, A., Sorba, P.: Dictionary on Lie algebras and superalgebras. Academic Press, 2000, arXiv:hep-th/9607161
  17. 17.
    Fedoruk, S., Ivanov, E., Lechtenfeld, O.: Superconformal Mechanics. J. Phys. A45, 173001 (2012) arXiv:hep-th/1112.1947
  18. 18.
    Fedoruk, S., Ivanov, E., Lechtenfeld, O.: Supersymmetric Calogero models by gauging. Phys. Rev. D79, 105015 (2009) arXiv:hep-th/0812.4276
  19. 19.
    Delduc, F., Ivanov, E.: Gauging \({\cal {N}}=4\) Supersymmetric Mechanics. Nucl. Phys. B753, 211 (2006) arXiv:hep-th/0605211
  20. 20.
    Assel, B., Cassani, D., Di Pietro, L., Komargodski, Z., Lorenzen, J., Martelli, D.: The Casimir Energy in Curved Space and its Supersymmetric Counterpart. J. High Energy Phys. 1507, 043 (2015) arXiv:hep-th/1503.05537
  21. 21.
    Ivanov, E., Sidorov, S.: Long multiplets in supersymmetric mechanics.
  22. 22.
    Asplund, C.T., Denef, F., Dzienkowski, E.: Massive quiver matrix models for massive charged particles in AdS.

Copyright information

© Springer Nature Singapore Pte Ltd. 2016

Authors and Affiliations

  1. 1.Bogoliubov Laboratory of Theoretical PhysicsJINRDubnaRussia

Personalised recommendations