Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Nonlinear symmetries of perfectly invisible PT-regularized conformal and superconformal mechanics systems

Abstract

We investigate how the Lax-Novikov integral in the perfectly invisible PT-regularized zero-gap quantum conformal and superconformal mechanics systems affects on their (super)-conformal symmetries. We show that the expansion of the conformal symmetry with this integral results in a nonlinearly extended generalized Shrödinger algebra. The PT-regularized superconformal mechanics systems in the phase of the unbroken exotic nonlinear \( \mathcal{N} \) = 4 super-Poincaré symmetry are described by nonlinearly super-extended Schrödinger algebra with the osp(2|2) sub-superalgebra. In the partially broken phase, the scaling dimension of all odd integrals is indefinite, and the osp(2|2) is not contained as a sub-superalgebra.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].

  2. [2]

    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].

  3. [3]

    V.P. Akulov and A.I. Pashnev, Quantum superconformal model in (1, 2) space, Theor. Math. Phys. 56 (1983) 862 [INSPIRE].

  4. [4]

    S. Fubini and E. Rabinovici, Superconformal quantum mechanics, Nucl. Phys. B 245 (1984) 17 [INSPIRE].

  5. [5]

    E.A. Ivanov, S.O. Krivonos and V.M. Leviant, Geometry of Conformal Mechanics, J. Phys. A 22 (1989) 345 [INSPIRE].

  6. [6]

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

  7. [7]

    D.Z. Freedman and P.F. Mende, An Exactly Solvable N Particle System in Supersymmetric Quantum Mechanics, Nucl. Phys. B 344 (1990) 317 [INSPIRE].

  8. [8]

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

  9. [9]

    S. Bellucci, A. Galajinsky, E. Ivanov and S. Krivonos, AdS 2 /CFT 1 , canonical transformations and superconformal mechanics, Phys. Lett. B 555 (2003) 99 [hep-th/0212204] [INSPIRE].

  10. [10]

    S. Bellucci, A. Galajinsky and S. Krivonos, New many-body superconformal models as reductions of simple composite systems, Phys. Rev. D 68 (2003) 064010 [hep-th/0304087] [INSPIRE].

  11. [11]

    S. Bellucci, A.V. Galajinsky and E. Latini, New insight into WDVV equation, Phys. Rev. D 71 (2005) 044023 [hep-th/0411232] [INSPIRE].

  12. [12]

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

  13. [13]

    A. Galajinsky, O. Lechtenfeld and K. Polovnikov, N = 4 superconformal Calogero models, JHEP 11 (2007) 008 [arXiv:0708.1075] [INSPIRE].

  14. [14]

    S. Krivonos and O. Lechtenfeld, Many-particle mechanics with D(2, 1 : α) superconformal symmetry, JHEP 02 (2011) 042 [arXiv:1012.4639] [INSPIRE].

  15. [15]

    J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].

  16. [16]

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].

  17. [17]

    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

  18. [18]

    P. Claus, M. Derix, R. Kallosh, J. Kumar, P.K. Townsend and A. Van Proeyen, Black holes and superconformal mechanics, Phys. Rev. Lett. 81 (1998) 4553 [hep-th/9804177] [INSPIRE].

  19. [19]

    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] [INSPIRE].

  20. [20]

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

  21. [21]

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

  22. [22]

    G.F. de Teramond, H.G. Dosch and S.J. Brodsky, Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding, Phys. Rev. D 91 (2015) 045040 [arXiv:1411.5243] [INSPIRE].

  23. [23]

    S.J. Brodsky, G.F. de Téramond, H.G. Dosch and C. Lorcé, Universal Effective Hadron Dynamics from Superconformal Algebra, Phys. Lett. B 759 (2016) 171 [arXiv:1604.06746] [INSPIRE].

  24. [24]

    E. D’Hoker and L. Vinet, Dynamical Supersymmetry of the Magnetic Monopole and the 1/r 2 Potential, Commun. Math. Phys. 97 (1985) 391 [INSPIRE].

  25. [25]

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

  26. [26]

    S. Cacciatori, D. Klemm and D. Zanon, W algebras, conformal mechanics and black holes, Class. Quant. Grav. 17 (2000) 1731 [hep-th/9910065] [INSPIRE].

  27. [27]

    R. Britto-Pacumio, J. Michelson, A. Strominger and A. Volovich, Lectures on Superconformal Quantum Mechanics and Multi-Black Hole Moduli Spaces, NATO Sci. Ser. C 556 (2000) 255 [hep-th/9911066] [INSPIRE].

  28. [28]

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

  29. [29]

    E.E. Donets, A. Pashnev, V.O. Rivelles, D.P. Sorokin and M. Tsulaia, N = 4 superconformal mechanics and the potential structure of AdS spaces, Phys. Lett. B 484 (2000) 337 [hep-th/0004019] [INSPIRE].

  30. [30]

    M.S. Plyushchay, Monopole Chern-Simons term: Charge monopole system as a particle with spin, Nucl. Phys. B 589 (2000) 413 [hep-th/0004032] [INSPIRE].

  31. [31]

    P.K. Ghosh, Conformal symmetry and the nonlinear Schrödinger equation, Phys. Rev. A 65 (2002) 012103 [cond-mat/0102488] [INSPIRE].

  32. [32]

    M. Günaydin, K. Koepsell and H. Nicolai, The Minimal unitary representation of E 8(8), Adv. Theor. Math. Phys. 5 (2002) 923 [hep-th/0109005] [INSPIRE].

  33. [33]

    B. Pioline and A. Waldron, Quantum cosmology and conformal invariance, Phys. Rev. Lett. 90 (2003) 031302 [hep-th/0209044] [INSPIRE].

  34. [34]

    H.E. Camblong and C.R. Ordonez, Anomaly in conformal quantum mechanics: From molecular physics to black holes, Phys. Rev. D 68 (2003) 125013 [hep-th/0303166] [INSPIRE].

  35. [35]

    C. Leiva and M.S. Plyushchay, Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence, Annals Phys. 307 (2003) 372 [hep-th/0301244] [INSPIRE].

  36. [36]

    B. Pioline and A. Waldron, Automorphic forms: A Physicist’s survey, in Proceedings, Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization, Les Houches, France, March 9–21, 2003, pp. 277–302, (2007) [DOI:https://doi.org/10.1007/978-3-540-30308-4_7] [hep-th/0312068] [INSPIRE].

  37. [37]

    D. Gaiotto, A. Strominger and X. Yin, Superconformal black hole quantum mechanics, JHEP 11 (2005) 017 [hep-th/0412322] [INSPIRE].

  38. [38]

    S. Meljanac and A. Samsarov, Universal properties of conformal quantum many-body systems, Phys. Lett. B 613 (2005) 221 [Erratum ibid. B 620 (2005) 200] [hep-th/0503174] [INSPIRE].

  39. [39]

    C. Duval, G.W. Gibbons and P. Horvathy, Celestial mechanics, conformal structures and gravitational waves, Phys. Rev. D 43 (1991) 3907 [hep-th/0512188] [INSPIRE].

  40. [40]

    A. Anabalon, J. Gomis, K. Kamimura and J. Zanelli, N = 4 superconformal mechanics as a non linear realization, JHEP 10 (2006) 068 [hep-th/0607124] [INSPIRE].

  41. [41]

    M. Günaydin, A. Neitzke, B. Pioline and A. Waldron, Quantum Attractor Flows, JHEP 09 (2007) 056 [arXiv:0707.0267] [INSPIRE].

  42. [42]

    T. Hakobyan and A. Nersessian, Lobachevsky geometry of (super)conformal mechanics, Phys. Lett. A 373 (2009) 1001 [arXiv:0803.1293] [INSPIRE].

  43. [43]

    J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].

  44. [44]

    F. Correa, V. Jakubsky and M.S. Plyushchay, Aharonov-Bohm effect on AdS 2 and nonlinear supersymmetry of reflectionless Poschl-Teller system, Annals Phys. 324 (2009) 1078 [arXiv:0809.2854] [INSPIRE].

  45. [45]

    P.D. Alvarez, J.L. Cortes, P.A. Horvathy and M.S. Plyushchay, Super-extended noncommutative Landau problem and conformal symmetry, JHEP 03 (2009) 034 [arXiv:0901.1021] [INSPIRE].

  46. [46]

    A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].

  47. [47]

    F. Correa, H. Falomir, V. Jakubsky and M.S. Plyushchay, Hidden superconformal symmetry of spinless Aharonov-Bohm system, J. Phys. A 43 (2010) 075202 [arXiv:0906.4055] [INSPIRE].

  48. [48]

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

  49. [49]

    C. Chamon, R. Jackiw, S.-Y. Pi and L. Santos, Conformal quantum mechanics as the CFT 1 dual to AdS 2, Phys. Lett. B 701 (2011) 503 [arXiv:1106.0726] [INSPIRE].

  50. [50]

    Z. Kuznetsova and F. Toppan, D-module Representations of N = 2, 4, 8 Superconformal Algebras and Their Superconformal Mechanics, J. Math. Phys. 53 (2012) 043513 [arXiv:1112.0995] [INSPIRE].

  51. [51]

    S. Fedoruk, E. Ivanov and O. Lechtenfeld, Superconformal Mechanics, J. Phys. A 45 (2012) 173001 [arXiv:1112.1947] [INSPIRE].

  52. [52]

    K. Andrzejewski, J. Gonera and P. Maślanka, Nonrelativistic conformal groups and their dynamical realizations, Phys. Rev. D 86 (2012) 065009 [arXiv:1204.5950] [INSPIRE].

  53. [53]

    J. Gonera, Conformal mechanics, Annals Phys. 335 (2013) 61 [arXiv:1211.4403] [INSPIRE].

  54. [54]

    J. Molina-Vilaplana and G. Sierra, An xp model on AdS 2 spacetime, Nucl. Phys. B 877 (2013) 107 [arXiv:1212.2436] [INSPIRE].

  55. [55]

    M.S. Plyushchay and A. Wipf, Particle in a self-dual dyon background: hidden free nature and exotic superconformal symmetry, Phys. Rev. D 89 (2014) 045017 [arXiv:1311.2195] [INSPIRE].

  56. [56]

    S.J. Brodsky, G.F. de Teramond, H.G. Dosch and J. Erlich, Light-Front Holographic QCD and Emerging Confinement, Phys. Rept. 584 (2015) 1 [arXiv:1407.8131] [INSPIRE].

  57. [57]

    K. Andrzejewski, J. Gonera, P. Kosiński and P. Maślanka, On dynamical realizations of l-conformal Galilei groups, Nucl. Phys. B 876 (2013) 309 [arXiv:1305.6805] [INSPIRE].

  58. [58]

    M. Masuku and J.P. Rodrigues, De Alfaro, Fubini and Furlan from multi Matrix Systems, JHEP 12 (2015) 175 [arXiv:1509.06719] [INSPIRE].

  59. [59]

    I. Masterov, Remark on higher-derivative mechanics with l-conformal Galilei symmetry, J. Math. Phys. 57 (2016) 092901 [arXiv:1607.02693] [INSPIRE].

  60. [60]

    K. Andrzejewski, Quantum conformal mechanics emerging from unitary representations of SL(2, ℝ), Annals Phys. 367 (2016) 227 [arXiv:1506.05596] [INSPIRE].

  61. [61]

    R. Bonezzi, O. Corradini, E. Latini and A. Waldron, Quantum Mechanics and Hidden Superconformal Symmetry, Phys. Rev. D 96 (2017) 126005 [arXiv:1709.10135] [INSPIRE].

  62. [62]

    J. Mateos Guilarte and M.S. Plyushchay, Perfectly invisible \( \mathcal{P}\mathcal{T} \) -symmetric zero-gap systems, conformal field theoretical kinks and exotic nonlinear supersymmetry, JHEP 12 (2017) 061 [arXiv:1710.00356] [INSPIRE].

  63. [63]

    L. Inzunza and M.S. Plyushchay, Hidden superconformal symmetry: Where does it come from?, Phys. Rev. D 97 (2018) 045002 [arXiv:1711.00616] [INSPIRE].

  64. [64]

    H. Airault, H.P. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Commun. Pure Appl. Math. 30 (1977) 95.

  65. [65]

    M. Adler and J. Moser, On a Class of Polynomials Connected with the Korteweg-De Vries Equation, Commun. Math. Phys. 61 (1978) 1 [INSPIRE].

  66. [66]

    A. Gorsky and N. Nekrasov, Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory, Nucl. Phys. B 414 (1994) 213 [hep-th/9304047] [INSPIRE].

  67. [67]

    J.J. Duistermaat and F.A. Grünbaum, Differential equations in the spectral parameter, Commun. Math. Phys. 103 (1986) 177.

  68. [68]

    A.P. Veselov, Huygens’ principle and integrability, Prog. Math. 169 (1998) 259.

  69. [69]

    S.P. Novikov, S.V. Manakov, L.P. Pitaevskii and V.E. Zakharov, Theory of Solitons, Plenum, New York (1984).

  70. [70]

    I.M. Krichever, Baker-Akhiezer functions and integrable systems, in Integrability. The Seiberg-Witten and Whitham Equations, H.W. Braden and I.M. Krichever eds., Gordon and Breach Science Publishers, Amsterdam (2000).

  71. [71]

    E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its and V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994).

  72. [72]

    F. Correa and M.S. Plyushchay, Hidden supersymmetry in quantum bosonic systems, Annals Phys. 322 (2007) 2493 [hep-th/0605104] [INSPIRE].

  73. [73]

    F. Correa, L.-M. Nieto and M.S. Plyushchay, Hidden nonlinear supersymmetry of finite-gap Lame equation, Phys. Lett. B 644 (2007) 94 [hep-th/0608096] [INSPIRE].

  74. [74]

    J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. Lond. Math. Soc. s2-21 (1923) 420 [Proc. Roy. Soc. Lond. A 118 (1928) 557].

  75. [75]

    E.L. Ince, Ordinary differential equations, Dover (1956).

  76. [76]

    F. Correa, V. Jakubsky, L.-M. Nieto and M.S. Plyushchay, Self-isospectrality, special supersymmetry and their effect on the band structure, Phys. Rev. Lett. 101 (2008) 030403 [arXiv:0801.1671] [INSPIRE].

  77. [77]

    A. Arancibia, J. Mateos Guilarte and M.S. Plyushchay, Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems, Phys. Rev. D 87 (2013) 045009 [arXiv:1210.3666] [INSPIRE].

  78. [78]

    C. Leiva and M.S. Plyushchay, Superconformal mechanics and nonlinear supersymmetry, JHEP 10 (2003) 069 [hep-th/0304257] [INSPIRE].

  79. [79]

    F. Correa, M.A. del Olmo and M.S. Plyushchay, On Hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry, Phys. Lett. B 628 (2005) 157 [hep-th/0508223] [INSPIRE].

  80. [80]

    C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].

  81. [81]

    A. Mostafazadeh, Pseudo-Hermitian Representation of Quantum Mechanics, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 1191 [arXiv:0810.5643] [INSPIRE].

  82. [82]

    E.P. Wigner, Normal form of antiunitary operators, J. Math. Phys. 1 (1960) 409.

  83. [83]

    A. Arancibia, F. Correa, V. Jakubský, J. Mateos Guilarte and M.S. Plyushchay, Soliton defects in one-gap periodic system and exotic supersymmetry, Phys. Rev. D 90 (2014) 125041 [arXiv:1410.3565] [INSPIRE].

  84. [84]

    A. Arancibia and M.S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D 92 (2015) 105009 [arXiv:1507.07060] [INSPIRE].

  85. [85]

    C.M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].

  86. [86]

    P. Dorey, C. Dunning and R. Tateo, Spectral equivalences, Bethe Ansatz equations and reality properties in \( \mathcal{P}\mathcal{T} \) -symmetric quantum mechanics, J. Phys. A 34 (2001) 5679 [hep-th/0103051] [INSPIRE].

  87. [87]

    P. Dorey, C. Dunning and R. Tateo, Supersymmetry and the spontaneous breakdown of \( \mathcal{P}\mathcal{T} \) symmetry, J. Phys. A 34 (2001) L391 [hep-th/0104119] [INSPIRE].

  88. [88]

    M. Znojil, \( \mathcal{P}\mathcal{T} \) -symmetric harmonic oscillators, Phys. Lett. A 259 (1999) 220 [quant-ph/9905020] [INSPIRE].

  89. [89]

    F. Correa and A. Fring, Regularized degenerate multi-solitons, JHEP 09 (2016) 008 [arXiv:1605.06371] [INSPIRE].

  90. [90]

    A. Fring and M. Znojil, \( \mathcal{P}\mathcal{T} \) -symmetric deformations of Calogero models, J. Phys. A 41 (2008) 194010 [arXiv:0802.0624] [INSPIRE].

  91. [91]

    F. Correa and O. Lechtenfeld, \( \mathcal{P}\mathcal{T} \) deformation of angular Calogero models, JHEP 11 (2017) 122 [arXiv:1705.05425] [INSPIRE].

  92. [92]

    R. El-Ganainy, K.G. Makris, M. Khajavikhan, Z.H. Musslimani, S. Rotter and D.N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14 (2018) 11.

  93. [93]

    J.F. Cariñena and M.S. Plyushchay, ABC of ladder operators for rationally extended quantum harmonic oscillator systems, J. Phys. A 50 (2017) 275202 [arXiv:1701.08657] [INSPIRE].

  94. [94]

    J.F. Cariñena, L. Inzunza and M.S. Plyushchay, Rational deformations of conformal mechanics, Phys. Rev. D 98 (2018) 026017 [arXiv:1707.07357] [INSPIRE].

  95. [95]

    A. Arancibia and M.S. Plyushchay, Extended supersymmetry of the self-isospectral crystalline and soliton chains, Phys. Rev. D 85 (2012) 045018 [arXiv:1111.0600] [INSPIRE].

  96. [96]

    S.P. Novikov, The periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl. 8 (1974) 236.

  97. [97]

    L. Inzunza and M.S. Plyushchay, Hidden symmetries of rationally deformed superconformal mechanics, Phys. Rev. D 99 (2019) 025001 [arXiv:1809.08527] [INSPIRE].

  98. [98]

    F. Correa, O. Lechtenfeld and M. Plyushchay, Nonlinear supersymmetry in the quantum Calogero model, JHEP 04 (2014) 151 [arXiv:1312.5749] [INSPIRE].

  99. [99]

    S. Wojciechowski, Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983) 279.

  100. [100]

    V.B. Kuznetsov, Hidden symmetry of the quantum Calogero-Moser system, Phys. Lett. A 218 (1996) 212 [solv-int/9509001] [INSPIRE].

  101. [101]

    C. Gonera, A note on superintegrability of the quantum Calogero model, Phys. Lett. A 237 (1998) 365.

  102. [102]

    M.F. Rañada, Superintegrability of the Calogero-Moser system: Constants of motion, master symmetries, and time-dependent symmetries, J. Math. Phys. 40 (1999) 236.

Download references

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.

Author information

Correspondence to Mikhail S. Plyushchay.

Additional information

ArXiv ePrint: 1806.08740

Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guilarte, J.M., Plyushchay, M.S. Nonlinear symmetries of perfectly invisible PT-regularized conformal and superconformal mechanics systems. J. High Energ. Phys. 2019, 194 (2019). https://doi.org/10.1007/JHEP01(2019)194

Download citation

Keywords

  • Conformal and W Symmetry
  • Extended Supersymmetry
  • Integrable Hierarchies