Abstract
A Killing-Yano tensor is an antisymmetric tensor obeying a first-order differential constraint similar to that obeyed by a Killing vector. In this article we consider generalisations of such objects, focusing on the conformal case. These generalised conformal Killing-Yano tensors are of mixed symmetry type and obey the constraint that the largest irreducible representation of \( \mathfrak{o} \)(n) contained in the tensor constructed from the first-derivative applied to such an object should vanish. Such tensors appear naturally in the context of spinning particles having N0 = 1 worldline supersymmetry and in the related problem of higher symmetries of Dirac operators. Generalisations corresponding to extended worldline supersymmetries and to spacetime supersymmetry are discussed.
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References
K. Yano, Some remarks on tensor fields and curvature, Ann. Math. 55 (1952) 328.
P. Sommers, On Killing tensors and constants of motion, J. Math. Phys. 14 (1973) 787.
S. Tachibana, On conformal Killing tensor in a Riemannian space, Tohoku Math. J. 21 (1969) 56.
P. Krtous, D. Kubiznak, D.N. Page and V.P. Frolov, Killing-Yano Tensors, Rank-2 Killing Tensors and Conserved Quantities in Higher Dimensions, JHEP 02 (2007) 004 [hep-th/0612029] [INSPIRE].
B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968) 1559 [INSPIRE].
R. Penrose, Naked singularities, Annals N. Y. Acad. Sci. 224 (1973) 125 [INSPIRE].
G. Papadopoulos, Killing-Yano equations and G-structures, Class. Quant. Grav. 25 (2008) 105016 [arXiv:0712.0542] [INSPIRE].
G. Papadopoulos, Killing-Yano Equations with Torsion, Worldline Actions and G-Structures, Class. Quant. Grav. 29 (2012) 115008 [arXiv:1111.6744] [INSPIRE].
Y. Chervonyi and O. Lunin, Killing(-Yano) Tensors in String Theory, JHEP 09 (2015) 182 [arXiv:1505.06154] [INSPIRE].
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].
M. Cariglia, Hidden Symmetries of the Dirac Equation in Curved Space-Time, Springer Proc. Phys. 157 (2014) 25 [arXiv:1209.6406] [INSPIRE].
O.P. Santillan, Hidden symmetries and supergravity solutions, J. Math. Phys. 53 (2012) 043509 [arXiv:1108.0149] [INSPIRE].
J.W. van Holten and R.H. Rietdijk, Symmetries and motions in manifolds, J. Geom. Phys. 11 (1993) 559 [hep-th/9205074] [INSPIRE].
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].
M.G. Eastwood, Higher symmetries of the Laplacian, Annals Math. 161 (2005) 1645 [hep-th/0206233] [INSPIRE].
J. Kress, Generalised Conformal Killing-Yano Tensors: Applications to Electrodynamics, Doctoral Thesis, University of Newcastle, Department of Mathematics (1997).
P.S. Howe and U. Lindström, Notes on Super Killing Tensors, JHEP 03 (2016) 078 [arXiv:1511.04575] [INSPIRE].
W. Nahm, Supersymmetries and their Representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].
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].
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].
V.G. Kac, A Sketch of Lie Superalgebra Theory, Commun. Math. Phys. 53 (1977) 31 [INSPIRE].
V.G. Kac, Lie Superalgebras, Adv. Math. 26 (1977) 8 [INSPIRE].
M.F. Sohnius, The Conformal Group in Superspace, in Feldafing 1976, proceedings of the Quantum Theory and The Structure Of Time and Space. Volume 2: In Memoriam Werner Heisenberg, Carl Hanser Verlag, Munich Germany (1977), pp. 241-252 [MPI-PAE/PTh 32/76] [INSPIRE].
F.A. Berezin and M.S. Marinov, Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics, Annals Phys. 104 (1977) 336 [INSPIRE].
R. Casalbuoni, Relativity and Supersymmetries, Phys. Lett. B 62 (1976) 49 [INSPIRE].
A. Barducci, R. Casalbuoni and L. Lusanna, Supersymmetries and the Pseudoclassical Relativistic electron, Nuovo Cim. A 35 (1976) 377 [INSPIRE].
L. Brink, S. Deser, B. Zumino, P. Di Vecchia and P.S. Howe, Local Supersymmetry for Spinning Particles, Phys. Lett. B 64 (1976) 435 [Erratum ibid. B 68 (1977) 488] [INSPIRE].
L. Brink, P. Di Vecchia and P.S. Howe, A Lagrangian Formulation of the Classical and Quantum Dynamics of Spinning Particles, Nucl. Phys. B 118 (1977) 76 [INSPIRE].
V.D. Gershun and V.I. Tkach, Classical And Quantum Dynamics Of Particles With Arbitrary Spin, JETP Lett. 29 (1979) 288 [Pisma Zh. Eksp. Teor. Fiz. 29 (1979) 320] [INSPIRE].
P.S. Howe, S. Penati, M. Pernici and P.K. Townsend, Wave Equations for Arbitrary Spin From Quantization of the Extended Supersymmetric Spinning Particle, Phys. Lett. B 215 (1988) 555 [INSPIRE].
P.S. Howe, S. Penati, M. Pernici and P.K. Townsend, A Particle Mechanics Description of Antisymmetric Tensor Fields, Class. Quant. Grav. 6 (1989) 1125 [INSPIRE].
S.M. Kuzenko and Z.V. Yarevskaya, Conformal invariance, N extended supersymmetry and massless spinning particles in anti-de Sitter space, Mod. Phys. Lett. A 11 (1996) 1653 [hep-th/9512115] [INSPIRE].
P.S. Howe and U. Lindström, Super-Laplacians and their symmetries, JHEP 05 (2017) 119 [arXiv:1612.06787] [INSPIRE].
I. Bars, B. Morel and H. Ruegg, Kac-Dynkin Diagrams and Supertableaux, J. Math. Phys. 24 (1983) 2253 [INSPIRE].
D. Leites, Indecomposable representations of Lie superalgebras, math.RT/0202184v1.
P.S. Howe and U. Lindström, in preparation.
A.A. Rosly, Super Yang-Mills constraints as integrability conditions, in Group Theoretical Methods in Physics, M.A. Markov ed., Nauka, Moscow Russia (1983), p. 263.
A.A. Roslyi and A.S. Schwarz, Supersymmetry in a space with auxiliary dimensions, Commun. Math. Phys. 105 (1986) 645 [INSPIRE].
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].
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].
A. Karlhede, U. Lindström and M. Roček, Selfinteracting Tensor Multiplets in N = 2 Superspace, Phys. Lett. B 147 (1984) 297 [INSPIRE].
J. Grundberg and U. Lindström, Actions for Linear Multiplets in Six-dimensions, Class. Quant. Grav. 2 (1985) L33 [INSPIRE].
U. Lindström and M. Roček, Properties of hyperkähler manifolds and their twistor spaces, Commun. Math. Phys. 293 (2010) 257 [arXiv:0807.1366] [INSPIRE].
P.S. Howe and G.G. Hartwell, A Superspace survey, Class. Quant. Grav. 12 (1995) 1823 [INSPIRE].
J. Lukierski and A. Nowicki, General Superspaces From Supertwistors, Phys. Lett. B 211 (1988) 276 [INSPIRE].
G.G. Hartwell and P.S. Howe, (N, p, q) harmonic superspace, Int. J. Mod. Phys. A 10 (1995) 3901 [hep-th/9412147] [INSPIRE].
Y.I. Manin, Gauge Field Theory And Complex Geometry, Grundlehren der mathematischen Wissenschaften, volume 289, Springer, Berlin Germany (1988).
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].
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].
R. Penrose and W. Rindler, Spinors And Space-time. Volume 2: Spinor And Twistor Methods In Space-time Geometry, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1988) [ISBN: 9780521347860, 9780511868429] [INSPIRE].
R.J. Baston and M.G. Eastwood, The Penrose transform: Its interaction with representation theory, Oxford Mathematical Monographs, Clarendon Press, Oxford U.K. (1989).
R.S. Ward and R.O. Wells, Twistor geometry and field theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1991) [ISBN: 9780521422680, 9780521422680, 9780511869778] [INSPIRE].
P.J. Heslop and P.S. Howe, Aspects of N = 4 SYM, JHEP 01 (2004) 058 [hep-th/0307210] [INSPIRE].
W. Siegel, Fields, hep-th/9912205 [INSPIRE].
M. Hatsuda, Y.-t. Huang and W. Siegel, First-quantized N = 4 Yang-Mills, JHEP 04 (2009) 058 [arXiv:0812.4569] [INSPIRE].
C.Y. Ju and W. Siegel, Systematizing semishortening conditions, Phys. Rev. D 90 (2014) 125004 [arXiv:1302.2515] [INSPIRE].
C. Fronsdal, Massless Fields with Integer Spin, Phys. Rev. D 18 (1978) 3624 [INSPIRE].
E.S. Fradkin and M.A. Vasiliev, Superalgebra of Higher Spins and Auxiliary Fields, Int. J. Mod. Phys. A 3 (1988) 2983 [INSPIRE].
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 (1999), pp. 533-610 [hep-th/9910096] [INSPIRE].
A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].
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Howe, P.S., Lindström, U. Some remarks on (super)-conformal Killing-Yano tensors. J. High Energ. Phys. 2018, 49 (2018). https://doi.org/10.1007/JHEP11(2018)049
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DOI: https://doi.org/10.1007/JHEP11(2018)049