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New D(2, 1; α) mechanics with spin variables

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Abstract

We elaborate on a novel superconformal mechanics model possessing D(2, 1; α) symmetry and involving extra U(2) spin variables. It is the one-particle case of the \( \mathcal{N} = 4 \) superconformal matrix model recently proposed in ar**v:0812.4276 [hep-th], and it generalizes to arbitrary α≠0 the OSp(4|2) superconformal mechanics of ar**v:0905.4951 [hep-th]. As in the latter case, the U(2) spin variables are described by a Wess-Zumino action and define the first Hopf map S 3 → S 2 in the target space. Upon quantization, they represent a fuzzy sphere. We find the classical and quantum generators of the D(2, 1; α) superalgebra and their realization on the physical states. The super wavefunction encompasses various multiplets of the SU(2) R and SU(2) L subgroups of D(2, 1; α), with fixed isospins. The conformal potential is determined by the external magnetic field in the Wess-Zumino term, whose strength is quantized like in the OSp(4|2) case. As a byproduct, we reveal new invariant subspaces in the envelo** algebra of D(2, 1; α) for our quantum realization.

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References

  1. P. Claus et al., Black holes and superconformal mechanics, Phys. Rev. Lett. 81 (1998) 4553 [hep-th/9804177] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. S. Fedoruk, E. Ivanov and O. Lechtenfeld, Supersymmetric Calogero models by gauging, Phys. Rev. D 79 (2009) 105015 [ar**v:0812.4276] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  3. F. Delduc and E. Ivanov, Gauging N = 4 supersymmetric mechanics, Nucl. Phys. B 753 (2006) 211 [hep-th/0605211] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. G.W. Gibbons and P.K. Townsend, Black holes and Calogero models, Phys. Lett. B 454 (1999) 187 [hep-th/9812034] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  5. J. Michelson and A. Strominger, The geometry of (super)conformal quantum mechanics, Commun. Math. Phys. 213 (2000) 1 [hep-th/9907191] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. J. Michelson and A. Strominger, Superconformal multi-black hole quantum mechanics, JHEP 09 (1999) 005 [hep-th/9908044] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  7. A. Maloney, M. Spradlin and A. Strominger, Superconformal multi-black hole moduli spaces in four dimensions, JHEP 04 (2002) 003 [hep-th/9911001] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. R. Britto-Pacumio, J. Michelson, A. Strominger and A. Volovich, Lectures on superconformal quantum mechanics and multi- black hole moduli spaces, in Progress in string theory and M-theory, Cargese (1999), pg. 235, [hep-th/9911066] [SPIRES].

  9. G. Papadopoulos, Conformal and superconformal mechanics, Class. Quant. Grav. 17 (2000) 3715 [hep-th/0002007] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  10. V. de Alfaro, S. Fubini and G. Furlan, Conformal invariance in quantum mechanics, Nuovo Cim. A 34 (1976) 569 [SPIRES].

    Article  ADS  Google Scholar 

  11. D.Z. Freedman and P.F. Mende, An exactly solvable N particle system in supersymmetric quantum mechanics, Nucl. Phys. B 344 (1990) 317 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. N. Wyllard, (Super)conformal many-body quantum mechanics with extended supersymmetry, J. Math. Phys. 41 (2000) 2826 [hep-th/9910160] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. 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] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  14. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  15. A. Gala**sky, O. Lechtenfeld and K. Polovnikov, Calogero models and nonlocal conformal transformations, Phys. Lett. B 643 (2006) 221 [hep-th/0607215] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  16. A. Gala**sky, O. Lechtenfeld and K. Polovnikov, N=4 superconformal Calogero models, JHEP 11 (2007) 008 [ar**v:0708.1075] [SPIRES];

    Article  MathSciNet  ADS  Google Scholar 

  17. A. Gala**sky, O. Lechtenfeld and K. Polovnikov, N=4 mechanics, WDVV equations and roots, JHEP 03 (2009) 113 [ar**v:0802.4386] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  18. S. Fedoruk, E. Ivanov and O. Lechtenfeld, OSp(4|2) superconformal mechanics, JHEP 08 (2009) 081 [ar**v:0905.4951] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  19. S. Krivonos and O. Lechtenfeld, SU(2) reduction in N = 4 supersymmetric mechanics, Phys. Rev. D 80 (2009) 045019 [ar**v:0906.2469] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  20. E.A. Ivanov, S.O. Krivonos and V.M. Leviant, Geometric superfield approach to superconformal mechanics, J. Phys. A 22 (1989) 4201 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  21. E. Ivanov and O. Lechtenfeld, N = 4 supersymmetric mechanics in harmonic superspace, JHEP 09 (2003) 073 [hep-th/0307111] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  22. F. Delduc and E. Ivanov, Gauging N = 4 supersymmetric mechanics. II: (1,4,3) models from the (4,4,0) ones, Nucl. Phys. B 770 (2007) 179 [hep-th/0611247] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. E. Ivanov, S. Krivonos and O. Lechtenfeld, New variant of N = 4 superconformal mechanics, JHEP 03 (2003) 014 [hep-th/0212303] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. E. Ivanov, S. Krivonos and O. Lechtenfeld, N = 4, D = 1 supermultiplets from nonlinear realizations of D(2,1:alpha), Class. Quant. Grav. 21 (2004) 1031 [hep-th/0310299] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. L.D. Faddeev and R. Jackiw, Hamiltonian reduction of unconstrained and constrained systems, Phys. Rev. Lett. 60 (1988) 1692 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. G.V. Dunne, R. Jackiw and C.A. Trugenberger, Topological (Chern-Simons) quantum mechanics, Phys. Rev. D 41 (1990) 661 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  27. R. Floreanini, R. Percacci and E. Sezgin, σ-models with purely Wess-Zumino-Witten actions, Nucl. Phys. B 322 (1989) 255 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  28. P.S. Howe and P.K. Townsend, Chern-Simons quantum mechanics, Class. Quant. Grav. 7 (1990) 1655 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. A.P. Polychronakos, Integrable systems from gauged matrix models, Phys. Lett. B 266 (1991) 29 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  30. E. Ivanov, L. Mezincescu and P.K. Townsend, Fuzzy CP(n|m) as a quantum superspace, contribution to Symmetries in Gravity and Field Theory, conference for Jose-Adolfo de Azcarraga’s 60th birthday, Salamanca Spain (2003), [hep-th/0311159] [SPIRES].

  31. L. Mezincescu, Super Chern-Simons quantum mechanics, in proceedings of the International Workshop “Supersymmetries and Quantum Symmetries” (SQS’03), Dubna, Russia, 24–29 July 2003, [hep-th/0405031] [SPIRES].

  32. L. Frappat, P. Sorba and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161 [SPIRES].

  33. I.A. Bandos, E. Ivanov, J. Lukierski and D. Sorokin, On the superconformal flatness of AdS superspaces, JHEP 06 (2002) 040 [hep-th/0205104] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  34. T. Hakobyan, S. Krivonos, O. Lechtenfeld and A. Nersessian, Hidden symmetries of integrable conformal mechanical systems, Phys. Lett. A 374 (2010) 801 [ar**v:0908.3290] [SPIRES].

    ADS  Google Scholar 

  35. E. Ivanov and J. Niederle, Bi-harmonic superspace for N = 4 mechanics, Phys. Rev. D 80 (2009) 065027 [ar**v:0905.3770] [SPIRES].

    ADS  Google Scholar 

  36. J. Van Der Jeugt, Irreducible representations of the exceptional Lie superalgebras D(2, 1α), J. Math. Phys. 26 (1985) 913 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals Math. 48 (1947) 568 [SPIRES].

    Article  MathSciNet  Google Scholar 

  38. A.M. Perelomov, Algebraical approach to the solution of one-dimensional model of N interacting particles, Teor. Mat. Fiz. 6 (1971) 364 (in Russian).

    MathSciNet  Google Scholar 

  39. J.A. de Azcarraga, J.M. Izquierdo, J.C. Perez Bueno and P.K. Townsend, Superconformal mechanics and nonlinear realizations, Phys. Rev. D 59 (1999) 084015 [hep-th/9810230] [SPIRES].

    ADS  Google Scholar 

  40. V.P. Akulov, I.A. Bandos and D.P. Sorokin, Particle in harmonic N = 2 superspace, Sov. J. Nucl. Phys. 47 (1988) 724 [Yad. Fiz. 47 (1988) 1136] [SPIRES].

    MathSciNet  Google Scholar 

  41. V.P. Akulov, D.P. Sorokin and I.A. Bandos, Particle mechanics in harmonic superspace, Mod. Phys. Lett. A 3 (1988) 1633 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  42. E.A. Ivanov, S.O. Krivonos, A.I. Pashnev, Partial supersymmetry breaking in N=4 supersymmetric quantum mechanics, Class. Quant. Grav. 8 (1990) 19.

    Article  MathSciNet  ADS  Google Scholar 

  43. J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis and H.S. Reall, All supersymmetric solutions of minimal supergravity in five dimensions, Class. Quant. Grav. 20 (2003) 4587 [hep-th/0209114] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  44. J.P. Gauntlett, R.C. Myers and P.K. Townsend, Black holes of D = 5 supergravity, Class. Quant. Grav. 16 (1999) 1 [hep-th/9810204] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. A.M. Perelomov, Integrable systems of classical mechanics and Lie algebras, Birkhauser Verlag, Boston U.S.A. (1990), pg. 307 [Nauka Fizmatlit, Russia 1990, pg. 240 (in Russian)].

    Google Scholar 

  46. J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, D-branes and spinning black holes, Phys. Lett. B 391 (1997) 93 [hep-th/9602065] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  47. P.K. Townsend, Killing spinors, supersymmetries and rotating intersecting branes, in proceedings of the 22 nd Johns Hopkins Workshop on Novelties of String Theory, Goteborg, Sweden, 20–22 Aug 1998 [hep-th/9901102] [SPIRES].

  48. P.K. Townsend, Surprises with angular momentum, Annales Henri Poincaré 4 (2003) S183 [hep-th/0211008] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  49. G.W. Gibbons and P. Rychenkova, Cones, tri-Sasakian structures and superconformal invariance, Phys. Lett. B 443 (1998) 138 [hep-th/9809158] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  50. R. Casalbuoni, On the quantization of systems with anticommutating variables, Nuovo Cim. A 33 (1976) 115 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  51. R. Casalbuoni, The classical mechanics for Bose-Fermi systems, Nuovo Cim. A 33 (1976) 389 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Bogoliubov Laboratory of Theoretical Physics, JINR, 141980, Dubna, Moscow region, Russia

    S. Fedoruk & E. Ivanov

  2. Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, D-30167, Hannover, Germany

    O. Lechtenfeld

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  1. S. Fedoruk
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  2. E. Ivanov
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  3. O. Lechtenfeld
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Correspondence to O. Lechtenfeld.

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Ar**v ePrint: 0912.3508

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Fedoruk, S., Ivanov, E. & Lechtenfeld, O. New D(2, 1; α) mechanics with spin variables. J. High Energ. Phys. 2010, 129 (2010). https://doi.org/10.1007/JHEP04(2010)129

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