gr-qc/yymmddd AN ANALYSIS OF THE REPRESENTATIONS OF THE MAPPING CLASS GROUP OF A MULTI-GEON THREE-MANIFOLD

It is well known that the inequivalent unitary irreducible representations (UIR’s) of the mapping class group G of a 3-manifold give rise to “theta sectors” in theories of quantum gravity with fixed spatial topology. In this paper, we study several families of UIR’s of G and attempt to understand the physical implications of the resulting quantum sectors. The mapping class group of a three-manifold which is the connected sum of IR with a finite number of identical irreducible primes is a semi-direct product group. Following Mackey’s theory of induced representations, we provide an analysis of the structure of the general finite dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables one to draw several interesting qualitative conclusions about the geons’ behavior in different quantum sectors, without requiring an explicit knowledge of the UIR’s corresponding to the individual primes. An important general result is that the problem of finding the UIR’s of the so called particle subgroup (equivalently, the UIR’s of G in which the slide diffeomorphisms are represented trivially) is reduced entirely to the problem of finding the UIR’s of the internal diffeomorphism groups of the individual primes. Moreover, this reduction is entirely consistent with the geon picture, in which the UIR of the internal group of a prime determines the species of the corresponding quantum geon, and the remaining freedom in the overall UIR of G expresses the possibility of choosing an arbitrary statistics (bose, fermi or para) for the geons of each species. For UIR’s which represent the slides nontrivially, we do not provide a complete classification, but we find some new types of effects due to the slides, including partial or complete quantum breaking of internal symmetry and of particle indistinguishability. In connection with the latter, a new kind of statistics arises which is determined by representations of proper subgroups of the permutation group, rather than the group as a whole. Finally, we observe that for a generic 3-manifold there will be an infinity of inequivalent UIR’s and hence an infinity of “consistent” theories, when topology change is neglected.