Generalised Spin Structures in General Relativity

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).

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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Avis, S.J., Isham, C.J.: Generalized spin structures on four dimensional space–times. Commun. Math. Phys. 72, 103–118 (1980)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  MATH  Google 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)

    ADS  Article  Google Scholar 

  5. 5.

    Baez, J., Huerta, J.: The algebra of grand unified theories. Bull. Am. Math. Soc. 47(3), 483–552 (2010)

    MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  Article  MATH  Google Scholar 

  8. 8.

    Da̧browski, L., Percacci, R.: Spinors and diffeomorphisms. Commun. Math. Phys. 106(4), 691–704 (1986)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Eck, D.J.: Gauge-natural bundles and generalized gauge theories. Mem. Am. Math. Soc. 33(247), vi+48 (1981)

    MathSciNet  MATH  Google 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)

  11. 11.

    Eichhorn, J.: Spaces of Riemannian metrics on open manifolds. Results Math. 27(3–4), 256–283 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Epstein, D.B.A., Thurston, W.P.: Transformation groups and natural bundles. Proc. Lond. Math. Soc. 38(3), 219–236 (1979)

    MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Gotay, M.J., Marsden, J.E.: Stress–energy–momentum tensors and the Belinfante–Rosenfeld formula. Contemp. Math. 132, 367–392 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Hermann, R.: Spinors, Clifford and Cayley Algebras. Interdisciplinary Mathematics, vol. VII. Department of Mathematics, Rutgers University, New Brunswick (1974)

    MATH  Google Scholar 

  16. 16.

    Hawking, S.W., Pope, C.N.: Generalized spin structures in quantum gravity. Phys. Lett. B 73, 42–44 (1978)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Huet, I.: A projective Dirac operator on \(\mathbb{C}P^2\) within fuzzy geometry. J. High Energy Phys. 1102, 106 (2011)

    ADS  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    ADS  Article  Google Scholar 

  21. 21.

    Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)

    Book  MATH  Google 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)

    MATH  Google Scholar 

  23. 23.

    Lawson, H .B., Michelsohn, M.-L.: Spin Geometry, 2nd edn. Princeton University Press, Princeton (1994)

    MATH  Google 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)

    ADS  Article  Google Scholar 

  25. 25.

    Matteucci, P.: Einstein–Dirac theory on gauge-natural bundles. Rep. Math. Phys. 52(1), 115–139 (2003)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Morrison, S.: Classifying spinor structures. Master’s thesis, University of New South Wales (2001)

  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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Nijenhuis, A.: Theory of the geometric object. Doctoral thesis, Universiteit van Amsterdam (1952)

  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)

  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)

  31. 31.

    Pope, C.N.: Eigenfunctions and \({{\rm Spin}}^{c}\) structures in \({ C}P^{2}\). Phys. Lett. B 97(3–4), 417–422 (1980)

    ADS  MathSciNet  Article  Google Scholar 

  32. 32.

    Palais, R.S., Terng, C.L.: Natural bundles have finite order. Topology 16, 271–277 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Palese, M., Winterroth, E.: Covariant gauge-natural conservation laws. Rep. Math. Phys. 54(3), 349–364 (2004)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  Article  Google Scholar 

  35. 35.

    Salvioli, S.E.: On the theory of geometric objects. J. Diff. Geom. 7, 257–278 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Schouten, J.A., Haantjes, J.: On the theory of the geometric object. Proc. Lond. Math. Soc. S2–42(1), 356 (1936)

    MathSciNet  MATH  Google 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)

    ADS  Article  Google Scholar 

  38. 38.

    Whiston, G.S.: Lorentzian characteristic classes. Gen. Relativ. Gravit. 6(5), 463–475 (1975)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Witten, E.: Search for a realistic Kaluza–Klein theory. Nucl. Phys. B 186(3), 412–428 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  40. 40.

    Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  41. 41.

    Wundheiler, A.: Objekte, Invarianten und Klassifikation der Geometrie. Abh. Sem. Vektor Tenzoranal. Moskau 4, 366–375 (1937)

    MATH  Google Scholar 

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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.

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Correspondence to Bas Janssens.

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Communicated by James A. Isenberg.

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Janssens, B. Generalised Spin Structures in General Relativity. Ann. Henri Poincaré 19, 1587–1610 (2018). https://doi.org/10.1007/s00023-018-0667-5

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