Advertisement

Annales Henri Poincaré

, Volume 19, Issue 5, pp 1587–1610 | Cite as

Generalised Spin Structures in General Relativity

  • Bas Janssens
Open Access
Article
  • 186 Downloads

Abstract

Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalised spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math Phys 72:103–118, 1980).

Notes

Acknowledgements

This work was supported by the NWO Grant 613.001.214 ‘Generalised Lie algebra sheaves’. I would like to thank the anonymous referee for several comments that helped improve the structure of the paper.

References

  1. 1.
    Alexanian, G., Balachandran, A.P., Immirzi, G., Ydri, B.: Fuzzy \(\mathbb{C}{{\rm P}}^2\). J. Geom. Phys. 42(1–2), 28–53 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avis, S.J., Isham, C.J.: Generalized spin structures on four dimensional space–times. Commun. Math. Phys. 72, 103–118 (1980)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Berg, M., DeWitt-Morette, C., Gwo, S., Kramer, E.: The pin groups in physics: C, P and T. Rev. Math. Phys. 13, 953–1034 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Back, A., Freund, P.G.O., Forger, M.: New gravitational instantons and universal spin structures. Phys. Lett. B 77, 181–184 (1978)ADSCrossRefGoogle Scholar
  5. 5.
    Baez, J., Huerta, J.: The algebra of grand unified theories. Bull. Am. Math. Soc. 47(3), 483–552 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balachandran, A.P., Immirzi, G., Lee, J., Prešnajder, P.: Dirac operators on coset spaces. J. Math. Phys. 44(10), 4713–4735 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chakraborty, B., Parthasarathy, P.: On instanton induced spontaneous compactification in \(M^4\times { C}{{\rm P}}^2\) and chiral fermions. Class. Quantum Gravity 7(7), 1217–1224 (1990)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Da̧browski, L., Percacci, R.: Spinors and diffeomorphisms. Commun. Math. Phys. 106(4), 691–704 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eck, D.J.: Gauge-natural bundles and generalized gauge theories. Mem. Am. Math. Soc. 33(247), vi+48 (1981)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Eichhorn, J., Heber, G.: The configuration space of gauge theory on open manifolds of bounded geometry. In: Budzyński, R., Janeczko, S., Kondracki, W., Künzle A.F. (eds.) Symplectic Singularities and Geometry of Gauge Fields (Warsaw, 1995), vol. 39 of Banach Center Publications, pp. 269–286. Polish Academy of Sciences, Warsaw (1997)Google Scholar
  11. 11.
    Eichhorn, J.: Spaces of Riemannian metrics on open manifolds. Results Math. 27(3–4), 256–283 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Epstein, D.B.A., Thurston, W.P.: Transformation groups and natural bundles. Proc. Lond. Math. Soc. 38(3), 219–236 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Forger, M., Römer, H.: Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem. Ann. Phys. 309, 306–389 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gotay, M.J., Marsden, J.E.: Stress–energy–momentum tensors and the Belinfante–Rosenfeld formula. Contemp. Math. 132, 367–392 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hermann, R.: Spinors, Clifford and Cayley Algebras. Interdisciplinary Mathematics, vol. VII. Department of Mathematics, Rutgers University, New Brunswick (1974)zbMATHGoogle Scholar
  16. 16.
    Hawking, S.W., Pope, C.N.: Generalized spin structures in quantum gravity. Phys. Lett. B 73, 42–44 (1978)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Huet, I.: A projective Dirac operator on \(\mathbb{C}P^2\) within fuzzy geometry. J. High Energy Phys. 1102, 106 (2011)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Janssens, B.: Transformation and uncertainty. Some thoughts on quantum probability theory, quantum statistics, and natural bundles. Ph.D. thesis, Utrecht University (2010), arxiv:1011.3035
  19. 19.
    Janssens, B.: Infinitesimally natural principal bundles, 2016. J. Geom. Mech. 8(2), 199–220 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Key, J.S., Cornish, N.J., Spergel, D.N., Starkman, G.D.: Extending the WMAP bound on the size of the universe. Phys. Rev. D 75, 084034 (2007)ADSCrossRefGoogle Scholar
  21. 21.
    Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  22. 22.
    Lecomte, P.B.A.: Sur la suite exacte canonique associée à un fibré principal. Bull. S. M. F. 113, 259–271 (1985)zbMATHGoogle Scholar
  23. 23.
    Lawson, H .B., Michelsohn, M.-L.: Spin Geometry, 2nd edn. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  24. 24.
    Luminet, J., Weeks, J.R., Riazuelo, A., Lehoucq, R., Uzan, J.: Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background. Nature 425, 593–595 (2003)ADSCrossRefGoogle Scholar
  25. 25.
    Matteucci, P.: Einstein–Dirac theory on gauge-natural bundles. Rep. Math. Phys. 52(1), 115–139 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Morrison, S.: Classifying spinor structures. Master’s thesis, University of New South Wales (2001)Google Scholar
  27. 27.
    Müller, O., Nowaczyk, N.: A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory. Lett. Math. Phys. 107(5), 933–961 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nijenhuis, A.: Theory of the geometric object. Doctoral thesis, Universiteit van Amsterdam (1952)Google Scholar
  29. 29.
    Nijenhuis, A.: Geometric aspects of formal differential operations on tensors fields. In: Proceedings of the International Congress of Mathematicians, 1958, pp. 463–469. Cambridge University Press, New York (1960)Google Scholar
  30. 30.
    Nijenhuis, A.: Natural bundles and their general properties. Geometric objects revisited. In: Differential geometry (in honor of Kentaro Yano), pp. 317–334. Kinokuniya, Tokyo (1972)Google Scholar
  31. 31.
    Pope, C.N.: Eigenfunctions and \({{\rm Spin}}^{c}\) structures in \({ C}P^{2}\). Phys. Lett. B 97(3–4), 417–422 (1980)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Palais, R.S., Terng, C.L.: Natural bundles have finite order. Topology 16, 271–277 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Palese, M., Winterroth, E.: Covariant gauge-natural conservation laws. Rep. Math. Phys. 54(3), 349–364 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Roukema, B.F., Bulinski, Z., Szaniewska, A., Gaudin, N.E.: Optimal phase of the generalised Poincaré dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps. Astron. Astrophys. 486, 55–72 (2008)ADSCrossRefGoogle Scholar
  35. 35.
    Salvioli, S.E.: On the theory of geometric objects. J. Diff. Geom. 7, 257–278 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Schouten, J.A., Haantjes, J.: On the theory of the geometric object. Proc. Lond. Math. Soc. S2–42(1), 356 (1936)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Watamura, S.: Spontaneous compactification of \(d=10\) Maxwell–Einstein theory leads to \(\text{ SU }(3)\times \text{ SU }(2)\times \text{ U }(1)\) gauge symmetry. Phys. Lett. B. 129(3, 4), 188–192 (1983)ADSCrossRefGoogle Scholar
  38. 38.
    Whiston, G.S.: Lorentzian characteristic classes. Gen. Relativ. Gravit. 6(5), 463–475 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Witten, E.: Search for a realistic Kaluza–Klein theory. Nucl. Phys. B 186(3), 412–428 (1981)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  41. 41.
    Wundheiler, A.: Objekte, Invarianten und Klassifikation der Geometrie. Abh. Sem. Vektor Tenzoranal. Moskau 4, 366–375 (1937)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands

Personalised recommendations