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).
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Alexanian, G., Balachandran, A.P., Immirzi, G., Ydri, B.: Fuzzy \(\mathbb{C}{{\rm P}}^2\). J. Geom. Phys. 42(1–2), 28–53 (2002)
Avis, S.J., Isham, C.J.: Generalized spin structures on four dimensional space–times. Commun. Math. Phys. 72, 103–118 (1980)
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)
Back, A., Freund, P.G.O., Forger, M.: New gravitational instantons and universal spin structures. Phys. Lett. B 77, 181–184 (1978)
Baez, J., Huerta, J.: The algebra of grand unified theories. Bull. Am. Math. Soc. 47(3), 483–552 (2010)
Balachandran, A.P., Immirzi, G., Lee, J., Prešnajder, P.: Dirac operators on coset spaces. J. Math. Phys. 44(10), 4713–4735 (2003)
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)
Da̧browski, L., Percacci, R.: Spinors and diffeomorphisms. Commun. Math. Phys. 106(4), 691–704 (1986)
Eck, D.J.: Gauge-natural bundles and generalized gauge theories. Mem. Am. Math. Soc. 33(247), vi+48 (1981)
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)
Eichhorn, J.: Spaces of Riemannian metrics on open manifolds. Results Math. 27(3–4), 256–283 (1995)
Epstein, D.B.A., Thurston, W.P.: Transformation groups and natural bundles. Proc. Lond. Math. Soc. 38(3), 219–236 (1979)
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)
Gotay, M.J., Marsden, J.E.: Stress–energy–momentum tensors and the Belinfante–Rosenfeld formula. Contemp. Math. 132, 367–392 (1992)
Hermann, R.: Spinors, Clifford and Cayley Algebras. Interdisciplinary Mathematics, vol. VII. Department of Mathematics, Rutgers University, New Brunswick (1974)
Hawking, S.W., Pope, C.N.: Generalized spin structures in quantum gravity. Phys. Lett. B 73, 42–44 (1978)
Huet, I.: A projective Dirac operator on \(\mathbb{C}P^2\) within fuzzy geometry. J. High Energy Phys. 1102, 106 (2011)
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
Janssens, B.: Infinitesimally natural principal bundles, 2016. J. Geom. Mech. 8(2), 199–220 (2016)
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)
Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)
Lecomte, P.B.A.: Sur la suite exacte canonique associée à un fibré principal. Bull. S. M. F. 113, 259–271 (1985)
Lawson, H .B., Michelsohn, M.-L.: Spin Geometry, 2nd edn. Princeton University Press, Princeton (1994)
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)
Matteucci, P.: Einstein–Dirac theory on gauge-natural bundles. Rep. Math. Phys. 52(1), 115–139 (2003)
Morrison, S.: Classifying spinor structures. Master’s thesis, University of New South Wales (2001)
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)
Nijenhuis, A.: Theory of the geometric object. Doctoral thesis, Universiteit van Amsterdam (1952)
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)
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)
Pope, C.N.: Eigenfunctions and \({{\rm Spin}}^{c}\) structures in \({ C}P^{2}\). Phys. Lett. B 97(3–4), 417–422 (1980)
Palais, R.S., Terng, C.L.: Natural bundles have finite order. Topology 16, 271–277 (1977)
Palese, M., Winterroth, E.: Covariant gauge-natural conservation laws. Rep. Math. Phys. 54(3), 349–364 (2004)
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)
Salvioli, S.E.: On the theory of geometric objects. J. Diff. Geom. 7, 257–278 (1972)
Schouten, J.A., Haantjes, J.: On the theory of the geometric object. Proc. Lond. Math. Soc. S2–42(1), 356 (1936)
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)
Whiston, G.S.: Lorentzian characteristic classes. Gen. Relativ. Gravit. 6(5), 463–475 (1975)
Witten, E.: Search for a realistic Kaluza–Klein theory. Nucl. Phys. B 186(3), 412–428 (1981)
Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill, New York (1967)
Wundheiler, A.: Objekte, Invarianten und Klassifikation der Geometrie. Abh. Sem. Vektor Tenzoranal. Moskau 4, 366–375 (1937)
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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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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DOI: https://doi.org/10.1007/s00023-018-0667-5