AN ANALYSIS OF THE REPRESENTATIONS OF THE MAPPING CLASS GROUP OF A MULTIGEON 3-MANIFOLD

It is known that the distinct unitary irreducible representations (UIR 's) of the mapping class group G of a three-manifold M give rise to di stinct quantum sectors (''theta sectors'') in quantum theories of grav ity based on a product space-time of the form R x M. In this paper, we study the UIR's of G in an effort to understand the physical implicat ions of these quantum sectors. The mapping class group of a three-mani fold which is the connected sum of R-3 with a finite number of irreduc ible primes is a semidirect product group. Following Mackey's theory o f induced representations, we provide an analysis of the structure of the general finite-dimensional UIR of such a group. Zn the picture of quantized primes as particles (topological geons), this general group- theoretic analysis enables one to draw several qualitative conclusions about the geons' behavior in different quantum sectors, without requi ring an explicit knowledge of the UIR's corresponding to the individua l primes. An important general result is that the classification of th e UIR's of the so-called particle subgroup (equivalently, the UIR's of G in which the slide diffeomorphisms are represented trivially) is re duced to the problem of finding the UIR's of the internal diffeomorphi sm groups of the individual primes. Moreover, this reduction is entire ly consistent with the geon picture, in which the UIR of the internal group of a prime determines the species of the corresponding quantum g eon, and the remaining freedom in the overall UIR of G expresses the p ossibility of choosing an arbitrary statistics (Bose, Fermi or para) f or the geons of each species. For UIR's which represent the slides non trivially, we do not provide a complete classification, but we find so me new types of effects due to the slides, including quantum breaking of internal symmetry and of particle indistinguishability. In connecti on with the latter, a novel kind of statistics arises which is determi ned by representations of proper subgroups of the permutation group, r ather than of the group as a whole. Finally, we observe that for a gen eric three-manifold there will be an infinity of inequivalent UIR's an d hence an infinity of ''consistent'' theories, when topology change i s neglected.