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Quantum SU(2|1) supersymmetric Calogero-Moser spinning systems

  • Sergey Fedoruk
  • Evgeny Ivanov
  • Olaf Lechtenfeld
  • Stepan Sidorov
Open Access
Regular Article - Theoretical Physics

Abstract

SU(2|1) supersymmetric multi-particle quantum mechanics with additional semi-dynamical spin degrees of freedom is considered. In particular, we provide an \( \mathcal{N}=4 \) supersymmetrization of the quantum U(2) spin Calogero-Moser model, with an intrinsic mass parameter coming from the centrally-extended superalgebra \( \widehat{su}\left(2\Big|1\right) \). The full system admits an SU(2|1) covariant separation into the center-of-mass sector and the quotient. We derive explicit expressions for the classical and quantum SU(2|1) generators in both sectors as well as for the total system, and we determine the relevant energy spectra, degeneracies, and the sets of physical states.

Keywords

Extended Supersymmetry Field Theories in Lower Dimensions Matrix Models Superspaces 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys. 10 (1969) 2191 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    F. Calogero, Ground state of one-dimensional N body system, J. Math. Phys. 10 (1969) 2197 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    J. Moser, Three integrable Hamiltonian systems connnected with isospectral deformations, Adv. Math. 16 (1975) 197 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    M.A. Olshanetsky and A.M. Perelomov, Quantum Integrable Systems Related to Lie Algebras, Phys. Rept. 94 (1983) 313 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    L. Brink, T.H. Hansson and M.A. Vasiliev, Explicit solution to the N body Calogero problem, Phys. Lett. B 286 (1992) 109 [hep-th/9206049] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    L. Brink, T.H. Hansson, S. Konstein and M.A. Vasiliev, The Calogero model: Anyonic representation, fermionic extension and supersymmetry, Nucl. Phys. B 401 (1993) 591 [hep-th/9302023] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    G.W. Gibbons and P.K. Townsend, Black holes and Calogero models, Phys. Lett. B 454 (1999) 187 [hep-th/9812034] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    J. McGreevy, S. Murthy and H.L. Verlinde, Two-dimensional superstrings and the supersymmetric matrix model, JHEP 04 (2004) 015 [hep-th/0308105] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A. Dabholkar, Fermions and nonperturbative supersymmetry breaking in the one-dimensional superstring, Nucl. Phys. B 368 (1992) 283.ADSGoogle Scholar
  12. [12]
    A. Agarwal and A.P. Polychronakos, BPS operators in \( \mathcal{N}=4 \) SYM: Calogero models and 2D fermions, JHEP 08 (2006) 034 [hep-th/0602049] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S. Fedoruk and E. Ivanov, Gauged spinning models with deformed supersymmetry, JHEP 11 (2016) 103 [arXiv:1610.04202] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    A.V. Smilga, Weak supersymmetry, Phys. Lett. B 585 (2004) 173 [hep-th/0311023] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Bellucci and A. Nersessian, (Super)oscillator on CP N and constant magnetic field, Phys. Rev. D 67 (2003) 065013 [Erratum ibid. D 71 (2005) 089901] [hep-th/0211070] [INSPIRE].
  16. [16]
    S. Bellucci and A. Nersessian, Supersymmetric Kähler oscillator in a constant magnetic field, in Proceedings, 5th International Workshop on Supersymmetries and Quantum Symmetries (SQS’03), Dubna, Russia, July 24–29, 2003, pp. 379-384 (2004) [hep-th/0401232] [INSPIRE].
  17. [17]
    C. Römelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  18. [18]
    C. Romelsberger, Calculating the Superconformal Index and Seiberg Duality, arXiv:0707.3702 [INSPIRE].
  19. [19]
    E. Ivanov and S. Sidorov, Deformed Supersymmetric Mechanics, Class. Quant. Grav. 31 (2014) 075013 [arXiv:1307.7690] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    E. Ivanov and S. Sidorov, Super Kähler oscillator from SU(2|1) superspace, J. Phys. A 47 (2014) 292002 [arXiv:1312.6821] [INSPIRE].zbMATHGoogle Scholar
  21. [21]
    E. Ivanov and S. Sidorov, SU(2|1) mechanics and harmonic superspace, Class. Quant. Grav. 33 (2016) 055001 [arXiv:1507.00987] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    B. Assel, D. Cassani, L. Di Pietro, Z. Komargodski, J. Lorenzen and D. Martelli, The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    C.T. Asplund, F. Denef and E. Dzienkowski, Massive quiver matrix models for massive charged particles in AdS, JHEP 01 (2016) 055 [arXiv:1510.04398] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S. Fedoruk, E. Ivanov and O. Lechtenfeld, Supersymmetric Calogero models by gauging, Phys. Rev. D 79 (2009) 105015 [arXiv:0812.4276] [INSPIRE].ADSGoogle Scholar
  25. [25]
    S. Fedoruk, E. Ivanov and O. Lechtenfeld, Superconformal Mechanics, J. Phys. A 45 (2012) 173001 [arXiv:1112.1947] [INSPIRE].ADSzbMATHGoogle Scholar
  26. [26]
    E. Ivanov and O. Lechtenfeld, N = 4 supersymmetric mechanics in harmonic superspace, JHEP 09 (2003) 073 [hep-th/0307111] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    S. Fedoruk, E. Ivanov and S. Sidorov, Deformed supersymmetric quantum mechanics with spin variables, JHEP 01 (2018) 132 [arXiv:1710.02130] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    J. Gibbons and T. Hermsen, A generalization of the Calogero-Mozer system, Physica D 11 (1984) 337.ADSzbMATHGoogle Scholar
  29. [29]
    S. Wojciechowski, An integrable marriage of the Euler equations with the Calogero-Mozer system, Phys. Lett. A 111 (1985) 101.ADSCrossRefGoogle Scholar
  30. [30]
    A.P. Polychronakos, Generalized statistics in one-dimension, in Topological Aspects of Low-dimensional Systems: Proceedings, Les Houches Summer School of Theoretical Physics, Session 69, Les Houches, France, July 7–31 1998 (1999) [hep-th/9902157] [INSPIRE].
  31. [31]
    A.P. Polychronakos, Generalized Calogero models through reductions by discrete symmetries, Nucl. Phys. B 543 (1999) 485 [hep-th/9810211] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    A.P. Polychronakos, Calogero-Moser models with noncommutative spin interactions, Phys. Rev. Lett. 89 (2002) 126403 [hep-th/0112141] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    A.P. Polychronakos, Physics and Mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].ADSzbMATHGoogle Scholar
  34. [34]
    A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992) 703 [hep-th/9202057] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    J.A. Minahan and A.P. Polychronakos, Integrable systems for particles with internal degrees of freedom, Phys. Lett. B 302 (1993) 265 [hep-th/9206046] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    M. Feigin, O. Lechtenfeld and A.P. Polychronakos, The quantum angular Calogero-Moser model, JHEP 07 (2013) 162 [arXiv:1305.5841] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  37. [37]
    A.P. Polychronakos, Integrable systems from gauged matrix models, Phys. Lett. B 266 (1991) 29 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    S. Hellerman and M. Van Raamsdonk, Quantum Hall physics equals noncommutative field theory, JHEP 10 (2001) 039 [hep-th/0103179] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    A.P. Polychronakos, Quantum Hall states as matrix Chern-Simons theory, JHEP 04 (2001) 011 [hep-th/0103013] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    A.M. Perelomov, Algebraical approach to the solution of one-dimensional model of n interacting particles, Teor. Mat. Fiz. 6 (1971) 364 [INSPIRE].Google Scholar
  41. [41]
    S. Bellucci, A.V. Galajinsky and E. Latini, New insight into WDVV equation, Phys. Rev. D 71 (2005) 044023 [hep-th/0411232] [INSPIRE].ADSGoogle Scholar
  42. [42]
    A. Galajinsky, O. Lechtenfeld and K. Polovnikov, N = 4 superconformal Calogero models, JHEP 11 (2007) 008 [arXiv:0708.1075] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  43. [43]
    A. Galajinsky, O. Lechtenfeld and K. Polovnikov, N = 4 mechanics, WDVV equations and roots, JHEP 03 (2009) 113 [arXiv:0802.4386] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    S. Krivonos and O. Lechtenfeld, Many-particle mechanics with D(2, 1 : α) superconformal symmetry, JHEP 02 (2011) 042 [arXiv:1012.4639] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    S. Krivonos, O. Lechtenfeld and K. Polovnikov, N = 4 superconformal n-particle mechanics via superspace, Nucl. Phys. B 817 (2009) 265 [arXiv:0812.5062] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    O. Lechtenfeld, K. Schwerdtfeger and J. Thürigen, N = 4 Multi-Particle Mechanics, WDVV Equation and Roots, SIGMA 7 (2011) 023 [arXiv:1011.2207] [INSPIRE].zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Sergey Fedoruk
    • 1
  • Evgeny Ivanov
    • 1
  • Olaf Lechtenfeld
    • 2
  • Stepan Sidorov
    • 1
  1. 1.Bogoliubov Laboratory of Theoretical PhysicsJINRDubnaRussia
  2. 2.Institut für Theoretische Physik and Riemann Center for Geometry and PhysicsLeibniz Universität HannoverHannoverGermany

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