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Klein and conformal superspaces, split algebras and spinor orbits

    https://doi.org/10.1142/S0129055X17500118Cited by:2 (Source: Crossref)

    We discuss 𝒩=1 Klein and Klein-conformal superspaces in D=(2,2) space-time dimensions, realizing them in terms of their functor of points over the split composition algebra s. We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras. This leads to a natural interpretation of the sections of the spinor bundle in the critical split dimensions D=4,6 and 10 as s2, s2 and 𝕆s2, respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-conformal superspace structure.

    AMSC: 17A35, 17C60, 32C11, 15A66

    References

    • 1. Y. A. Gol’fand and E. P. Likhtman, Extension of the algebra of Poincaré group and violation of p invariance, JETP Lett. 13 (1971) 323–326. ISI, ADSGoogle Scholar
    • 2. D. V. Volkov and V. P. Akulov, Is the neutrino a Goldstone particle? Phys. Lett. B 46 (1973) 109–110. Crossref, ISI, ADSGoogle Scholar
    • 3. J. Wess and B. Zumino, Supergauge transformations in four dimensions, Nucl. Phys. B 70 (1974) 39–50. Crossref, ISI, ADSGoogle Scholar
    • 4. A. Salam and J. Strathdee, Super-gauge transformations, Nucl. Phys. B 76 (1974) 477–482. Crossref, ISI, ADSGoogle Scholar
    • 5. A. Salam and J. Strathdee, Unitary representations of supergauge symmetries, Nucl. Phys. B 80 (1974) 499–505. Crossref, ISI, ADSGoogle Scholar
    • 6. S. Ferrara and B. Zumino, Supergauge invariant Yang–Mills theories, Nucl. Phys. B 79 (1974) 413–421. Crossref, ISI, ADSGoogle Scholar
    • 7. D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Progress toward a theory of supergravity, Phys. Rev. D 13 (1976) 3214–3218. Crossref, ISI, ADSGoogle Scholar
    • 8. S. Deser and B. Zumino, Consistent supergravity, Phys. Lett. B 62 (1976) 335–337. Crossref, ISI, ADSGoogle Scholar
    • 9. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, 2 Vols., Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1987). Google Scholar
    • 10. J. Polchinski, String Theory, 2 Vols. (Cambridge University Press, 1998). CrossrefGoogle Scholar
    • 11. V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes, Vol. 1 (Amer. Math. Soc., 2004). CrossrefGoogle Scholar
    • 12. L. Balduzzi, C. Carmeli and R. Fioresi, The local functor of points of supermanifolds, Expos. Math. 28 (2010) 201–217. Crossref, ISIGoogle Scholar
    • 13. L. Balduzzi, C. Carmeli and R. Fioresi, A comparison of the functors of points of supermanifolds, J. Algebra Appl. 12 (2013) 1407–1415. Link, ISIGoogle Scholar
    • 14. A. S. Shvarts, On the definition of superspace, Teoret. Mat. Fiz. 60(1) (1984) 37–42. ISIGoogle Scholar
    • 15. A. Voronov, Maps of supermanifolds, Teoret. Mat. Fiz. 60(1) (1984) 43–48. Google Scholar
    • 16. F. A. Berezin, Introduction to Superanalysis (D. Reidel Publishing Company, Holland, 1987). CrossrefGoogle Scholar
    • 17. Y. I. Manin, Topics in Non Commutative Geometry (Princeton University Press, 1991). CrossrefGoogle Scholar
    • 18. Y. I. Manin, Gauge Field Theory and Complex Geometry, translated by N. Koblitz and J. R. King (Springer-Verlag, 1988). Google Scholar
    • 19. R. Fioresi and E. Latini, The symplectic origin of conformal and Minkowski superspaces, J. Math. Phys. 57 (2016) 022307, 12pp. Crossref, ISIGoogle Scholar
    • 20. C. Carmeli, L. Caston and R. Fioresi, Mathematical Foundation of Supersymmetry, with an appendix with I. Dimitrov, EMS Ser. Lect. Math. (European Math. Soc., Zurich, 2011). CrossrefGoogle Scholar
    • 21. R. Fioresi, M. A. Lledó and V. S. Varadarajan, The Minkowski and conformal superspaces, J. Math. Phys. 48 (2007) 113505. Crossref, ISI, ADSGoogle Scholar
    • 22. A. Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898) 309–316. Google Scholar
    • 23. J. C. Baez and J. Huerta, Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C∗-algebras, eds. R. DoranG. FriedmanJ. Rosenberg, Proc. Symp. Pure Math., Vol. 81 (Amer. Math. Soc., 2010), pp. 65–80. CrossrefGoogle Scholar
    • 24. M. Cederwall, Jordan algebra dynamics, Nucl. Phys. B 302 (1988) 81–103. Crossref, ISI, ADSGoogle Scholar
    • 25. M. Cederwall, Octonionic particles and the S(7) symmetry, J. Math. Phys. 33 (1992) 388–393. Crossref, ISI, ADSGoogle Scholar
    • 26. M. Cederwall, Introduction to division algebras, sphere algebras and twistors, arXiv:hep-th/9310115. Google Scholar
    • 27. J. M. Evans, Supersymmetric Yang–Mills theories and division algebras, Nucl. Phys. B 298 (1988) 92–108. Crossref, ISI, ADSGoogle Scholar
    • 28. J. C. Baez and J. Huerta, Division algebras and supersymmetry II, Adv. Theor. Math. Phys. 15 (2011) 1373–1410. Crossref, ISIGoogle Scholar
    • 29. J. Huerta, Division algebras and supersymmetry III, Adv. Theor. Math. Phys. 16 (2012) 1485–1589. Crossref, ISIGoogle Scholar
    • 30. J. Huerta, Division algebras and supersymmetry IV, arXiv:1409.4361 [hep-th]. Google Scholar
    • 31. Z. Bern, J. J. M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602. Crossref, ISI, ADSGoogle Scholar
    • 32. A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy, Super Yang–Mills, division algebras and triality, JHEP 1408 (2014) 080. Crossref, ISI, ADSGoogle Scholar
    • 33. A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy, Yang–Mills origin of gravitational symmetries, Phys. Rev. Lett. 113 (2014) 231606. Crossref, ISI, ADSGoogle Scholar
    • 34. A. Anastasiou, L. Borsten, M. J. Hughes and S. Nagy, Global symmetries of Yang–Mills squared in various dimensions, JHEP 1601 (2016) 148. Crossref, ISI, ADSGoogle Scholar
    • 35. H. Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964) 145–190. CrossrefGoogle Scholar
    • 36. J. Tits, Algébres Alternatives, Algébres de Jordan et Algébres de Lie Exceptionnelles, Indag. Math. 28 (1966) 223–237. CrossrefGoogle Scholar
    • 37. C. H. Barton and A. Sudbery, Magic squares and matrix models of Lie algebras, Adv. in Math. 180 (2003) 596–647. Crossref, ISIGoogle Scholar
    • 38. J. C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002) 145–205. Crossref, ISIGoogle Scholar
    • 39. V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Graduate Text in Mathematics (Springer-Verlag, 1984). CrossrefGoogle Scholar
    • 40. C. H. Barton and A. Sudbery, Magic squares of Lie algebras, arXiv:math/0001083 [math.RA]. Google Scholar
    • 41. T. Dray, J. Huerta and J. Kincaid, The magic square of Lie groups: The 2 × 2 case. Lett. Math. Phys. 104 (2014) 1445–1468. Crossref, ISI, ADSGoogle Scholar
    • 42. T. Dray, C. A. Manogue and R. A. Wilson, A symplectic representation of E7, Comment. Math. Univ. Carolin. 55 (2014) 387–399. Google Scholar
    • 43. J. Kincaid and T. Dray, Division algebra representations of SO(4,2), Mod. Phys. Lett. A 29 (2014), 1450128. Link, ISI, ADSGoogle Scholar
    • 44. T. Dray and C. A. Manogue, Octonionic Cayley spinors and E6, Comment. Math. Univ. Carolin. 51 (2010) 193–207. Google Scholar
    • 45. T. N. Bailey, M. G. Eastwood and A. R. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994) 1191–1217. Crossref, ISIGoogle Scholar
    • 46. A. R. Gover, A. Shaukat and A. Waldron, Tractors, mass and Weyl invariance, Nucl. Phys. B 812 (2009) 424–455. Crossref, ISI, ADSGoogle Scholar
    • 47. S. Curry and A. R. Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity, arXiv:1412.7559 [math.DG]. Google Scholar
    • 48. A. Rod Gover, E. Latini and A. Waldron, Poincare–Einstein holography for forms via conformal geonetry in the bulk, Mem. Amer. Math. Soc. 235 (2015) no. 1106, vi+95 pp. ISIGoogle Scholar
    • 49. A. Čap and A. R. Gover, Tractor bundles for irreducible parabolic geometries, in Global Analysis and Harmonic Analysis, Sémin. Congr., Vol. 4 (Soc. Math. France, 2000), pp. 129–154. Google Scholar
    • 50. C. Fefferman and C. R. Graham, Conformal invariants, in The Mathematical Heritage of Cartan (Lyon, 1984), Numero Hors Serie (Asterisque, 1985), pp. 95–116. Google Scholar
    • 51. S. M. Kuzenko, Conformally compactified Minkowski superspaces revisited, JHEP 1210 (2012) 135. Crossref, ISI, ADSGoogle Scholar
    • 52. D. Klemm and M. Nozawa, Geometry of Killing spinors in neutral signature, Class. Quant. Grav. 32 (2015) 185012. Crossref, ISI, ADSGoogle Scholar
    • 53. C. M. Hull, Duality and the signature of space-time, JHEP 9811 (1998) 017. Crossref, ADSGoogle Scholar
    • 54. C. M. Hull, Timelike T-duality, de Sitter space, large N gauge theories and topological field theory, JHEP 9807 (1998) 021. Crossref, ADSGoogle Scholar
    • 55. S. Ferrara, Spinors, superalgebras and the signature of space-time, hep-th/0101123. Google Scholar
    • 56. R. L. Bryant, Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor, in Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Sémin. Congr., Vol. 4 (Soc. Math. France, Paris, 2000), pp. 53–94. Google Scholar
    • 57. M. Dunajski, Anti-self-dual four manifolds with a parallel real spinor, Proc. Roy. Soc. Lond. A 458 (2002) 1205–1222. Crossref, ISI, ADSGoogle Scholar
    • 58. M. Dunajski, Einstein–Maxwell-dilaton metrics from three-dimensional Einstein–Weyl structures, Class. Quant. Grav. 23 (2006) 2833–2840. Crossref, ISI, ADSGoogle Scholar
    • 59. M. Dunajski and S. West, Anti-self-dual conformal structures in neutral signature, math/0610280 [math.DG]. Google Scholar
    • 60. S. Hervik, Pseudo-Riemannian VSI spaces II, Class. Quant. Grav. 29 (2012) 095011. Crossref, ISIGoogle Scholar
    • 61. H. Ooguri and C. Vafa, Selfduality and 𝒩 = 2 string magic, Mod. Phys. Lett. A 5 (1990) 1389–1398. Link, ISI, ADSGoogle Scholar
    • 62. R. Penrose, Twistor algebra, J. Math. Phys. 8 (1967) 345–366. Crossref, ISI, ADSGoogle Scholar
    • 63. E. Witten, Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys. 252 (2004) 189–258. Crossref, ISI, ADSGoogle Scholar
    • 64. M. Rios, Extremal black holes as qudits, arXiv:1102.1193 [hep-th]. Google Scholar
    • 65. W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149–166. Crossref, ISI, ADSGoogle Scholar
    • 66. R. Fioresi and M. A. Lledó, The Minkowski and Conformal Superspaces: The Classical and Quantum Descriptions (World Scientific Publishing, 2015). LinkGoogle Scholar
    • 67. A. Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, J. Phys. A 17 (1984) 939–955. Crossref, ADSGoogle Scholar
    • 68. T. Kugo and P. K. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B 221 (1983) 357–380. Crossref, ISI, ADSGoogle Scholar
    • 69. D. Cervantes, R. Fioresi, M. A. Lledó and F. Nadal, Quadratic deformation of Minkowski space, Fortschr. Phys. 60 (2012) 970–976. Crossref, ISIGoogle Scholar
    • 70. D. Cervantes, R. Fioresi and M. A. Lledó, The quantum chiral Minkowski and conformal superspaces, Adv. Theor. Math. Phys. 15 (2011) 565–620. Crossref, ISIGoogle Scholar
    • 71. D. Cervantes, R. Fioresi and M. A. Lledó, On chiral quantum superspaces, in Supersymmetry in Mathematics and Physics, Lecture Notes in Math., Vol. 2027 (Springer, 2011), pp. 69–99. CrossrefGoogle Scholar
    • 72. R. Fioresi, Quantizations of flag manifolds and conformal space time, Rev. Math. Phys. 9 (1997) 453–465. Link, ISIGoogle Scholar
    • 73. R. Fioresi, Quantum deformation of the flag variety, Comm. Algebra 27 (1999) 5669–5685. Crossref, ISIGoogle Scholar
    • 74. I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras (Springer, 1983). Google Scholar
    • 75. K. Carmody, Circular and hyperbolic quaternions, octonions, sedionions, Appl. Math. Comput. 84(1) (1997) 27–47. Google Scholar
    • 76. M. Günaydin and O. Pavlyk, Spectrum generating conformal and quasiconformal U-duality groups, supergravity and spherical vectors, JHEP 1004 (2010) 070. Crossref, ISI, ADSGoogle Scholar
    • 77. T. Dray and C. A. Manogue, The Geometry of the Octonions (World Scientific, 2015). LinkGoogle Scholar
    • 78. T. Springer and F. D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups (Springer, 2013). Google Scholar
    • 79. J. M. Evans, Trialities and exceptional Lie algebras: Deconstructing the magic square, arXiv:0910.1828 [hep-th]. Google Scholar
    • 80. B. L. Cerchiai, S. Ferrara, A. Marrani and B. Zumino, Charge orbits of extremal black holes in five dimensional supergravity, Phys. Rev. D 82 (2010) 085010. Crossref, ISI, ADSGoogle Scholar
    • 81. S. L. Cacciatori, B. L. Cerchiai and A. Marrani, Magic coset decompositions, Adv. Theor. Math. Phys. 17 (2013) 1077–1128. Crossref, ISIGoogle Scholar
    • 82. L. Andrianopoli, R. D’Auria, S. Ferrara, A. Marrani and M. Trigiante, Two-centered magical charge orbits, JHEP 1104 (2011) 041. Crossref, ISI, ADSGoogle Scholar
    • 83. K. McCrimmon, A Taste of Jordan Algebras (Springer, 2004). Google Scholar
    • 84. R. Iordanescu, Jordan Structures in Analysis, Geometry and Physics (Editura Academiei Române, 2009). Google Scholar
    • 85. N. Jacobson, Structure and Representations of Jordan Algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX (American Mathematical Society, Providence, R.I., 1968). CrossrefGoogle Scholar
    • 86. P. Jordan, J. von Neumann and E. P. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934) 29–64. CrossrefGoogle Scholar
    • 87. P. Budinich, From the geometry of pure spinors with their division algebras to Fermion’s physics, Found. Phys. 32 (2002) 1347–1398. Crossref, ISI, ADSGoogle Scholar
    • 88. P. Budinich, Internal symmetry from division algebras in pure spinor geometry, in Symmetry in Nonlinear Mathematical Physics, Proceedings of Institute of Mathematics of NAS of Ukraine, Vol. 50, Part 2 (Institute of Mathematics, Kyiv, 2004) pp. 654–665. Google Scholar
    • 89. P. Charlton, The geometry of pure spinors, with applications, PhD thesis, University of Newcastle, Department of Mathematics (1997). Google Scholar
    • 90. R. D’Auria, S. Ferrara, M. A. Lledó and V. S. Varadarajan, Spinor algebras, J. Geom. Phys. 40 (2001) 101–128. Crossref, ISI, ADSGoogle Scholar
    • 91. É. Cartan, Leçons sur la Theorie des Spineurs (Hermann, 1937). Google Scholar
    • 92. C. Chevalley, The Algebraic Theory of Spinors (Columbia University Press, 1954). CrossrefGoogle Scholar
    • 93. N. Berkovits, Super-Poincaré covariant quantization of the superstring, JHEP 0004 (2000) 018. Crossref, ADSGoogle Scholar
    • 94. N. Berkovits, ICTP lectures on covariant quantization of the superstring, arXiv:hep-th/0209059. Google Scholar
    • 95. J. Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970) 997–1028. Crossref, ISIGoogle Scholar
    • 96. V. G. Kac and E. B. Vinberg, Spinors of 13-dimensional space, Adv. Math. 30 (1978) 137–155. Crossref, ISIGoogle Scholar
    • 97. V. L. Popov, Classification of spinors of dimension fourteen, Trans. Moscow Math. Soc. 1 (1980) 181–232. Google Scholar
    • 98. X.-W. Zhu, The classification of spinors under GSpin14 over finite fields, Trans. Amer. Math. Soc. 333 (1992) 95–114. ISIGoogle Scholar
    • 99. L. V. Antonyan and A. G. Èlashvili, Classification of spinors in dimension sixteen, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 70 (1982) 4–23. Google Scholar
    • 100. S. Giler, P. Kosinski and J. Rembielinski, On SO(p,q) pure spinors, Acta Phys. Pol. B 18 (1987) 713–727. ISIGoogle Scholar
    • 101. P. Furlan and R. Raczka, Nonlinear spinor representations, J. Math. Phys. 26 (1985) 3021–3032. Crossref, ISI, ADSGoogle Scholar
    • 102. R. L. Bryant, Remarks on spinors in low dimension, Unpublished notes (1999). Google Scholar
    • 103. M. Günaydin, G. Sierra and P. K. Townsend, Exceptional supergravity theories and the magic square, Phys. Lett. B 133 (1983) 72–76; M. Günaydin, G. Sierra and P. K. Townsend, The geometry of 𝒩 = 2 Maxwell–Einstein supergravity and Jordan algebras, Nucl. Phys. B 242 (1984) 244–268. Google Scholar
    • 104. M. Günaydin, H. Samtleben and E. Sezgin, On the magical supergravities in six dimensions, Nucl. Phys. B 848 (2011) 62–89. Crossref, ISI, ADSGoogle Scholar
    • 105. C. Carmeli, R. Fioresi and S. D. Kwok, SUSY structures, representations and Peter–Weyl theorem for S1|1, J. Geom. Phys. 95 (2015) 144–158. Crossref, ISI, ADSGoogle Scholar
    • 106. C. Carmeli, R. Fioresi and S. D. Kwok, The Peter–Weyl theorem for SU(1|1), P-Adic Numbers, Ultrametric Anal. Appl. 7 (2015) 266–275. CrossrefGoogle Scholar