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.
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References
F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys. 10 (1969) 2191 [INSPIRE].
F. Calogero, Ground state of one-dimensional N body system, J. Math. Phys. 10 (1969) 2197 [INSPIRE].
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].
J. Moser, Three integrable Hamiltonian systems connnected with isospectral deformations, Adv. Math. 16 (1975) 197 [INSPIRE].
M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].
M.A. Olshanetsky and A.M. Perelomov, Quantum Integrable Systems Related to Lie Algebras, Phys. Rept. 94 (1983) 313 [INSPIRE].
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].
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].
G.W. Gibbons and P.K. Townsend, Black holes and Calogero models, Phys. Lett. B 454 (1999) 187 [hep-th/9812034] [INSPIRE].
J. McGreevy, S. Murthy and H.L. Verlinde, Two-dimensional superstrings and the supersymmetric matrix model, JHEP 04 (2004) 015 [hep-th/0308105] [INSPIRE].
A. Dabholkar, Fermions and nonperturbative supersymmetry breaking in the one-dimensional superstring, Nucl. Phys. B 368 (1992) 283.
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].
S. Fedoruk and E. Ivanov, Gauged spinning models with deformed supersymmetry, JHEP 11 (2016) 103 [arXiv:1610.04202] [INSPIRE].
A.V. Smilga, Weak supersymmetry, Phys. Lett. B 585 (2004) 173 [hep-th/0311023] [INSPIRE].
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].
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].
C. Römelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
C. Romelsberger, Calculating the Superconformal Index and Seiberg Duality, arXiv:0707.3702 [INSPIRE].
E. Ivanov and S. Sidorov, Deformed Supersymmetric Mechanics, Class. Quant. Grav. 31 (2014) 075013 [arXiv:1307.7690] [INSPIRE].
E. Ivanov and S. Sidorov, Super Kähler oscillator from SU(2|1) superspace, J. Phys. A 47 (2014) 292002 [arXiv:1312.6821] [INSPIRE].
E. Ivanov and S. Sidorov, SU(2|1) mechanics and harmonic superspace, Class. Quant. Grav. 33 (2016) 055001 [arXiv:1507.00987] [INSPIRE].
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].
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].
S. Fedoruk, E. Ivanov and O. Lechtenfeld, Supersymmetric Calogero models by gauging, Phys. Rev. D 79 (2009) 105015 [arXiv:0812.4276] [INSPIRE].
S. Fedoruk, E. Ivanov and O. Lechtenfeld, Superconformal Mechanics, J. Phys. A 45 (2012) 173001 [arXiv:1112.1947] [INSPIRE].
E. Ivanov and O. Lechtenfeld, N = 4 supersymmetric mechanics in harmonic superspace, JHEP 09 (2003) 073 [hep-th/0307111] [INSPIRE].
S. Fedoruk, E. Ivanov and S. Sidorov, Deformed supersymmetric quantum mechanics with spin variables, JHEP 01 (2018) 132 [arXiv:1710.02130] [INSPIRE].
J. Gibbons and T. Hermsen, A generalization of the Calogero-Mozer system, Physica D 11 (1984) 337.
S. Wojciechowski, An integrable marriage of the Euler equations with the Calogero-Mozer system, Phys. Lett. A 111 (1985) 101.
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].
A.P. Polychronakos, Generalized Calogero models through reductions by discrete symmetries, Nucl. Phys. B 543 (1999) 485 [hep-th/9810211] [INSPIRE].
A.P. Polychronakos, Calogero-Moser models with noncommutative spin interactions, Phys. Rev. Lett. 89 (2002) 126403 [hep-th/0112141] [INSPIRE].
A.P. Polychronakos, Physics and Mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].
A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992) 703 [hep-th/9202057] [INSPIRE].
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].
M. Feigin, O. Lechtenfeld and A.P. Polychronakos, The quantum angular Calogero-Moser model, JHEP 07 (2013) 162 [arXiv:1305.5841] [INSPIRE].
A.P. Polychronakos, Integrable systems from gauged matrix models, Phys. Lett. B 266 (1991) 29 [INSPIRE].
S. Hellerman and M. Van Raamsdonk, Quantum Hall physics equals noncommutative field theory, JHEP 10 (2001) 039 [hep-th/0103179] [INSPIRE].
A.P. Polychronakos, Quantum Hall states as matrix Chern-Simons theory, JHEP 04 (2001) 011 [hep-th/0103013] [INSPIRE].
A.M. Perelomov, Algebraical approach to the solution of one-dimensional model of n interacting particles, Teor. Mat. Fiz. 6 (1971) 364 [INSPIRE].
S. Bellucci, A.V. Galajinsky and E. Latini, New insight into WDVV equation, Phys. Rev. D 71 (2005) 044023 [hep-th/0411232] [INSPIRE].
A. Galajinsky, O. Lechtenfeld and K. Polovnikov, N = 4 superconformal Calogero models, JHEP 11 (2007) 008 [arXiv:0708.1075] [INSPIRE].
A. Galajinsky, O. Lechtenfeld and K. Polovnikov, N = 4 mechanics, WDVV equations and roots, JHEP 03 (2009) 113 [arXiv:0802.4386] [INSPIRE].
S. Krivonos and O. Lechtenfeld, Many-particle mechanics with D(2, 1 : α) superconformal symmetry, JHEP 02 (2011) 042 [arXiv:1012.4639] [INSPIRE].
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].
O. Lechtenfeld, K. Schwerdtfeger and J. Thürigen, N = 4 Multi-Particle Mechanics, WDVV Equation and Roots, SIGMA 7 (2011) 023 [arXiv:1011.2207] [INSPIRE].
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Fedoruk, S., Ivanov, E., Lechtenfeld, O. et al. Quantum SU(2|1) supersymmetric Calogero-Moser spinning systems. J. High Energ. Phys. 2018, 43 (2018). https://doi.org/10.1007/JHEP04(2018)043
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DOI: https://doi.org/10.1007/JHEP04(2018)043