Notes on super Killing tensors

Open Access
Regular Article - Theoretical Physics

Abstract

The notion of a Killing tensor is generalised to a superspace setting. Conserved quantities associated with these are defined for superparticles and Poisson brackets are used to define a supersymmetric version of the even Schouten-Nijenhuis bracket. Superconformal Killing tensors in flat superspaces are studied for spacetime dimensions 3,4,5,6 and 10. These tensors are also presented in analytic superspaces and super-twistor spaces for 3,4 and 6 dimensions. Algebraic structures associated with superconformal Killing tensors are also briefly discussed.

Keywords

Extended Supersymmetry Superspaces Higher Spin Symmetry 

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]
    M.F. Sohnius, The conformal group in superspace, in Proceedings, Quantum Theory and The Structure Of Time and Space, vol. 2 (in memoriam Werner Heisenberg), München Germany (1977), pg. 241 and München Max-Planck-inst. Phys., MPI-PAE-PTH 32-76, Germany (1976) [INSPIRE].
  2. [2]
    A. Ferber, Supertwistors and conformal supersymmetry, Nucl. Phys. B 132 (1978) 55 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    W. Lang, Construction of the minimal superspace translation tensor and the derivation of the supercurrent, Nucl. Phys. B 179 (1981) 106 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    L. Bonora, P. Pasti and M. Tonin, Cohomologies and anomalies in supersymmetric theories, Nucl. Phys. B 252 (1985) 458 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    K.-I. Shizuya, Supercurrents and superconformal symmetry, Phys. Rev. D 35 (1987) 1848 [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    P.S. Howe and G.G. Hartwell, A superspace survey, Class. Quant. Grav. 12 (1995) 1823 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    I.L. Buchbinder and S.M. Kuzenko, Ideas and methods of supersymmetry and supergravity: or a walk through superspace, IOP, Bristol U.K. (1995) [INSPIRE].
  8. [8]
    K. Coulembier, P. Somberg and V. Soucek, Joseph-like ideals and harmonic analysis for \( \mathfrak{o}\mathfrak{s}\mathfrak{p}\left(m\Big|2n\right) \), Int. Math. Res. Not. 2014 (2014) 4291 [arXiv:1210.3507].MathSciNetMATHGoogle Scholar
  9. [9]
    W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M. Scheunert, W. Nahm and V. Rittenberg, Classification of all simple graded Lie algebras whose Lie algebra is reductive. 1, J. Math. Phys. 17 (1976) 1626 [INSPIRE].
  11. [11]
    M. Scheunert, W. Nahm and V. Rittenberg, Classification of all simple graded Lie algebras whose Lie algebra is reductive. 2. Construction of the exceptional algebras, J. Math. Phys. 17 (1976) 1640 [INSPIRE].
  12. [12]
    V.G. Kac, A sketch of Lie superalgebra theory, Commun. Math. Phys. 53 (1977) 31 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977) 8 [INSPIRE].CrossRefMATHGoogle Scholar
  14. [14]
    I. Bars, B. Morel and H. Ruegg, Kac-Dynkin diagrams and supertableaux, J. Math. Phys. 24 (1983) 2253 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    D. Leites, Indecomposable representations of Lie superalgebras, math/0202184.
  16. [16]
    M.G. Eastwood, Higher symmetries of the Laplacian, Annals Math. 161 (2005) 1645 [hep-th/0206233] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    A. Salam and J.A. Strathdee, Supergauge transformations, Nucl. Phys. B 76 (1974) 477 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    S. Ferrara, J. Wess and B. Zumino, Supergauge multiplets and superfields, Phys. Lett. B 51 (1974) 239 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    V.I. Ogievetsky and E. Sokatchev, Superfield equations of motion, J. Phys. A 10 (1977) 2021 [INSPIRE].ADSGoogle Scholar
  20. [20]
    J. Wess and B. Zumino, Superspace formulation of supergravity, Phys. Lett. B 66 (1977) 361 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    V. Ogievetsky and E. Sokatchev, Structure of supergravity group, Phys. Lett. B 79 (1978) 222 [Czech. J. Phys. B 29 (1979) 68] [INSPIRE].
  22. [22]
    W. Siegel and S.J. Gates, Jr., Superfield supergravity, Nucl. Phys. B 147 (1979) 77 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    V.P. Akulov and D.V. Volkov, Goldstone fields with spin 1/2, Theor. Math. Phys. 18 (1974) 28 [Teor. Mat. Fiz. 18 (1974) 39] [INSPIRE].
  24. [24]
    Y.I. Manin, Gauge field theory and complex geometry, Grundlehren der mathematischen Wissenschaften 289, Springer, Berlin Germany (1988) [INSPIRE].
  25. [25]
    P. Dolan and N.S. Swaminarayan, Solutions of the geodesic deviation equation obtained by using hidden symmetries, Proc. Roy. Irish Acad. A 84 (1984) 133.MathSciNetMATHGoogle Scholar
  26. [26]
    P.S. Howe, Weyl superspace, Phys. Lett. B 415 (1997) 149 [hep-th/9707184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    M. Cederwall, U. Gran, B.E.W. Nilsson and D. Tsimpis, Supersymmetric corrections to eleven-dimensional supergravity, JHEP 05 (2005) 052 [hep-th/0409107] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    S.M. Kuzenko, Supersymmetric spacetimes from curved superspace, PoS (CORFU2014) 140 [arXiv:1504.08114] [INSPIRE].
  29. [29]
    W. Siegel, Solution to constraints in Wess-Zumino supergravity formalism, Nucl. Phys. B 142 (1978) 301 [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    P.S. Howe and R.W. Tucker, Scale invariance in superspace, Phys. Lett. B 80 (1978) 138 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    L. Brink and J.H. Schwarz, Quantum superspace, Phys. Lett. B 100 (1981) 310 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    W. Siegel, Hidden local supersymmetry in the supersymmetric particle action, Phys. Lett. B 128 (1983) 397 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    D.P. Sorokin, V.I. Tkach, D.V. Volkov and A.A. Zheltukhin, From the superparticle Siegel symmetry to the spinning particle proper time supersymmetry, Phys. Lett. B 216 (1989) 302 [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    D.P. Sorokin, V.I. Tkach and D.V. Volkov, Superparticles, twistors and Siegel symmetry, Mod. Phys. Lett. A 4 (1989) 901 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    J.A. Schouten, Über Differentialkonkomitanten zweier kontravarianten Grössen (in German), Indag. Math. 2 (1940) 449.MathSciNetGoogle Scholar
  36. [36]
    J.A. Schouten, On the differential operators of the first order in tensor calculus, in Cremonese. Convegno Int. Geom. Diff., Italy (1953), pg. 1.
  37. [37]
    A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, Indag. Math. 17 (1953) 390.MathSciNetMATHGoogle Scholar
  38. [38]
    P. Dolan, A. Kladouchou and C. Card, On the significance of Killing tensors, Gen. Rel. Grav. 21 (1989) 427.ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    D.V. Soroka and V.A. Soroka, Generalizations of Schouten-Nijenhuis bracket, Proc. Inst. Math. NAS Ukraine 50 (2004) 1480 [hep-th/0401088] [INSPIRE].MathSciNetMATHGoogle Scholar
  40. [40]
    M. Dubois-Violette and P.W. Michor, A common generalization of the Fröhlicher-Nijenhuis bracket and the Schouten bracket for symmetric multivector fields, alg-geom/9401006 [INSPIRE].
  41. [41]
    G.W. Gibbons, R.H. Rietdijk and J.W. van Holten, SUSY in the sky, Nucl. Phys. B 404 (1993) 42 [hep-th/9303112] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    O.P. Santillan, Hidden symmetries and supergravity solutions, J. Math. Phys. 53 (2012) 043509 [arXiv:1108.0149] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    P.S. Howe and P.K. Townsend, The massless superparticle as Chern-Simons mechanics, Phys. Lett. B 259 (1991) 285 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    P.S. Howe, K.S. Stelle and P.K. Townsend, Superactions, Nucl. Phys. B 191 (1981) 445 [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S.M. Kuzenko, J.-H. Park, G. Tartaglino-Mazzucchelli and R. Unge, Off-shell superconformal nonlinear σ-models in three dimensions, JHEP 01 (2011) 146 [arXiv:1011.5727] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    S.M. Kuzenko, On compactified harmonic/projective superspace, 5D superconformal theories and all that, Nucl. Phys. B 745 (2006) 176 [hep-th/0601177] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    K.A. Intriligator and W. Skiba, Bonus symmetry and the operator product expansion of N =4 super Yang-Mills,Nucl. Phys. B 559 (1999) 165[hep-th/9905020] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    A.A. Rosly, Super Yang-Mills constraints as integrability conditions, in Group theoretical methods in physics, M.A. Markov ed., Nauka, Moscow Russia (1983), pg. 263.Google Scholar
  49. [49]
    A.A. Roslyi and A.S. Schwarz, Supersymmetry in a space with auxiliary dimensions, Commun. Math. Phys. 105 (1986) 645 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained N = 2 matter, Yang-Mills and supergravity theories in harmonic superspace, Class. Quant. Grav. 1 (1984) 469 [Erratum ibid. 2 (1985) 127] [INSPIRE].
  51. [51]
    A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained off-shell N =3 supersymmetric Yang-Mills theory, Class. Quant. Grav. 2(1985) 155[INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    A. Karlhede, U. Lindström and M. Roček, Selfinteracting tensor multiplets in N = 2 superspace, Phys. Lett. B 147 (1984) 297 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    U. Lindström and M. Roček, Properties of hyper-Kähler manifolds and their twistor spaces, Commun. Math. Phys. 293 (2010) 257 [arXiv:0807.1366] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  54. [54]
    J. Lukierski and A. Nowicki, General superspaces from supertwistors, Phys. Lett. B 211 (1988) 276 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    G.G. Hartwell and P.S. Howe, (N, p, q) harmonic superspace, Int. J. Mod. Phys. A 10 (1995) 3901 [hep-th/9412147] [INSPIRE].
  56. [56]
    J.P. Harnad and S. Shnider, Isotropic geometry, twistors and supertwistors. 1. The generalized Klein correspondence and spinor flags, J. Math. Phys. 33 (1992) 3197 [INSPIRE].
  57. [57]
    J.P. Harnad and S. Shnider, Isotropic geometry and twistors in higher dimensions. 2. Odd dimensions, reality conditions and twistor superspaces, J. Math. Phys. 36 (1995) 1945 [INSPIRE].
  58. [58]
    R. Penrose and W. Rindler, Spinors and space-time. Vol. 2: spinor and twistor methods in space-time geometry, Cambridge University Press, Cambridge U.K. (1988) [INSPIRE].
  59. [59]
    R.J. Baston and M.G. Eastwood, The Penrose transform: its interaction with representation theory, Clarendon, Oxford U.K. (1989) [INSPIRE].
  60. [60]
    R.S. Ward and R.O. Wells, Twistor geometry and field theory, Cambridge University Press, Cambridge U.K. (1991).MATHGoogle Scholar
  61. [61]
    P.J. Heslop and P.S. Howe, Aspects of N = 4 SYM, JHEP 01 (2004) 058 [hep-th/0307210] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    B.M. Zupnik and D.V. Khetselius, Three-dimensional extended supersymmetry in the harmonic superspace (in Russian), Sov. J. Nucl. Phys. 47 (1988) 730 [Yad. Fiz. 47 (1988) 1147] [INSPIRE].
  63. [63]
    P.S. Howe and M.I. Leeming, Harmonic superspaces in low dimensions, Class. Quant. Grav. 11 (1994) 2843 [hep-th/9408062] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    J. Grundberg and U. Lindström, Actions for linear multiplets in six-dimensions, Class. Quant. Grav. 2 (1985) L33 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    P.S. Howe, K.S. Stelle and P.C. West, N = 1 D = 6 harmonic superspace, Class. Quant. Grav. 2 (1985) 815 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    P.J. Heslop, Aspects of superconformal field theories in six dimensions, JHEP 07 (2004) 056 [hep-th/0405245] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    A.K.H. Bengtsson, I. Bengtsson, M. Cederwall and N. Linden, Particles, superparticles and twistors, Phys. Rev. D 36 (1987) 1766 [INSPIRE].ADSMathSciNetGoogle Scholar
  68. [68]
    P.K. Townsend, Supertwistor formulation of the spinning particle, Phys. Lett. B 261 (1991) 65 [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. École Norm. Sup. 9 (1976) 1.MathSciNetMATHGoogle Scholar
  70. [70]
    A. Braverman and A. Joseph, The minimal realisation from deformation theory, J. Alg. 205 (1998) 13.MathSciNetCrossRefMATHGoogle Scholar
  71. [71]
    M. Eastwood, P. Somberg and V. Souček, Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras, J. Geom. Phys. 57 (2007) 2539.ADSMathSciNetCrossRefMATHGoogle Scholar
  72. [72]
    P.S. Howe and U. Lindström, in preparation.Google Scholar
  73. [73]
    K. Govil and M. Günaydin, Deformed twistors and higher spin conformal (super-)algebras in four dimensions, JHEP 03 (2015) 026 [arXiv:1312.2907] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    K. Govil and M. Günaydin, Deformed twistors and higher spin conformal (super-)algebras in six dimensions, JHEP 07 (2014) 004 [arXiv:1401.6930] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  75. [75]
    S. Fernando and M. Günaydin, Massless conformal fields, AdS d+1 /CF T d higher spin algebras and their deformations, Nucl. Phys. B 904 (2016) 494 [arXiv:1511.02167] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  76. [76]
    C. Fronsdal, Massless fields with integer spin, Phys. Rev. D 18 (1978) 3624 [INSPIRE].ADSGoogle Scholar
  77. [77]
    E.S. Fradkin and M.A. Vasiliev, Superalgebra of higher spins and auxiliary fields, Int. J. Mod. Phys. A 3 (1988) 2983 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  78. [78]
    M.A. Vasiliev, Higher spin gauge theories: star product and AdS space, in The many faces of the superworld, M.A. Shifman ed., World Scientific, Singapore (2000), pg. 533 [hep-th/9910096] [INSPIRE].
  79. [79]
    E. Sezgin and P. Sundell, Higher spin N = 8 supergravity, JHEP 11 (1998) 016 [hep-th/9805125] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  80. [80]
    E. Sezgin and P. Sundell, On curvature expansion of higher spin gauge theory, Class. Quant. Grav. 18 (2001) 3241 [hep-th/0012168] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  81. [81]
    E. Sezgin and P. Sundell, Supersymmetric higher spin theories, J. Phys. A 46 (2013) 214022 [arXiv:1208.6019] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  82. [82]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  83. [83]
    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].ADSMathSciNetCrossRefGoogle Scholar
  84. [84]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  85. [85]
    E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [INSPIRE].
  86. [86]
    I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O (N ) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  87. [87]
    A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].
  88. [88]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  89. [89]
    N. Boulanger, D. Ponomarev, E.D. Skvortsov and M. Taronna, On the uniqueness of higher-spin symmetries in AdS and CFT, Int. J. Mod. Phys. A 28 (2013) 1350162 [arXiv:1305.5180] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  90. [90]
    M. Bianchi, F.A. Dolan, P.J. Heslop and H. Osborn, N = 4 superconformal characters and partition functions, Nucl. Phys. B 767 (2007) 163 [hep-th/0609179] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  91. [91]
    M. Bianchi, P.J. Heslop and F. Riccioni, More on La Grande Bouffe, JHEP 08 (2005) 088 [hep-th/0504156] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  92. [92]
    S.J. Gates, Jr., S.M. Kuzenko and A.G. Sibiryakov, N = 2 supersymmetry of higher superspin massless theories, Phys. Lett. B 412 (1997) 59 [hep-th/9609141] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  93. [93]
    G.J. Weir, Conformal Killing tensors in reducible spaces, J. Math. Phys. 18 (1977) 1782.ADSCrossRefMATHGoogle Scholar
  94. [94]
    R. Rani, S.B. Edgar and A. Barnes, Killing tensors and conformal Killing tensors from conformal Killing vectors, Class. Quant. Grav. 20 (2003) 1929 [gr-qc/0301059] [INSPIRE].
  95. [95]
    G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986) 2693.ADSMathSciNetCrossRefMATHGoogle Scholar
  96. [96]
    M. Cariglia, G.W. Gibbons, J.W. van Holten, P.A. Horvathy, P. Kosinski and P.M. Zhang, Killing tensors and canonical geometry, Class. Quant. Grav. 31 (2014) 125001 [arXiv:1401.8195] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  97. [97]
    A. Lischewski, Charged conformal Killing spinors, J. Math. Phys. 56 (2015) 013510 [arXiv:1403.2311] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  98. [98]
    R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  99. [99]
    Y. Chervonyi and O. Lunin, Killing(-Yano) tensors in string theory, JHEP 09 (2015) 182 [arXiv:1505.06154] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College LondonLondonU.K.
  2. 2.Department of Physics and Astronomy, Theoretical PhysicsUppsala UniversityUppsalaSweden
  3. 3.Theoretical Physics, Imperial College LondonLondonU.K.

Personalised recommendations