Geon Statistics and Uir's of the Mapping Class Group *

Quantum Gravity admits topological excitations of microscopic scale which can manifest themselves as particles — topological geons. Nontrivial spatial topology also brings into the theory free parameters analogous to the θ-angle of QCD. We show that these parameters can be interpreted in terms of geon properties. We also find that, for certain values of the parameters, the geons exhibit new patterns of particle identity together with new types of statistics. Geon indistinguishability in such a case is expressed by a proper subgroup of the permutation group and geon statistics by a (possibly projective) representation of the subgroup. This talk attempts to answer two questions concerning the effect of topology in generally covariant theories: “how many free parameters are there in quantum gravity?” and “do topological geons really act like particles?” By way of comparison, consider the standard model, which contains both continuous parameters (like the masses of the * To appear in Juan Carlos D’Olivo, Martin Klein, Hector Mendez (eds.), Proceedings of the VII Mexican School of Particles and Fields I Latin American Symposium on High Energy Physics Conference, held November, 1996, Mérida, México (American Institute of Physics).