Abstract
We generalize Manton's construction of discrete monopoles in Minkowski space to their analog in CP(n). Topological charge, analogous to the first Chern number in the smooth bundle, is obtained for the corresponding discrete bundle and is shown to be Q=±1. We also discuss the discretization of the smooth sphere bundles over the real projective space RP(n) and the quaternionic projective space HP(n). Finally, we make a conjecture of the discretization of the smooth sphere bundles over the discrete projective spaces R 2k P(n) for all positive integers k and n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
MantonN. S., Comm. Math. Phys. 113, 241 (1987). See also Morrison, K., Phys. Lett. B233, 393 (1989).
For the case of the monopole, see the lectures by J. Preskill, Les Houches Summer School (1986); for the instanton see Belavin, A. A., Polyakov, A. M., Schwarz A. S., and Tyupkin, Yu. S., Phys. Lett. B59, 85 (1975); Jackiw, R. and Rebbi, C., Phys. Rev. D14, 517 (1976); Nowakowski, J. and Trautman, A., J. Math. Phys. 19, 1100 (1978); and finally, for the octhopfon, see Grossman, B., Kephart, T. W., and Stasheff, J. D., Comm. Math. Phys. 96, 431 (1984); ibid, (E) 100, 311 (1985); Tchrakian, D. H., Phys. Lett. B150, 350 (1985). For a recent review on these matters, see for example, Feshbach, M., Ann. Physics 192, 85 (1989).
For a recent review on fractional statistics, see MackenzieR. and WilczekF., Internat. J. Modern Phys. A 3(12), 2827 (1988).
GrossmanB. and KephartT. W., Lett. Math. Phys. 18, 157 (1989).
BarnesE. S. and WallG. E., J. Amer. Math. Soc. 1, 47 (1959).
SteenrodN., The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, New Jersey, 1951.
ChernS. S., Complex Manifolds without Potential Theory, (2nd edn), Springer-Verlag, New York, 1979.
CoxeterH. M. S., Regular Polytopes, Methuen, London 1948.
ConwayJ. H. and SloaneN. J. A., Sphere Packing, Lattices, and Groups, Springer-Verlag, New York, 1988, and references therein.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kephart, T.W., Yuan, T.C. Discrete monopoles and instantons over projective spaces. Lett Math Phys 21, 301–308 (1991). https://doi.org/10.1007/BF00398328
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00398328