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

Abstract

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 distinct quantum sectors ("θ sectors") in quantum theories of gravity based on a product space–time of the form ℝ×ℳ. In this paper, we study the UIR's of G in an effort to understand the physical implications of these quantum sectors. The mapping class group of a three-manifold which is the connected sum of ℝ3 with a finite number of irreducible primes is a semidirect 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 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 classification of 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 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 efforts due to the slides, including quantum breaking of internal symmetry and of particle indistinguishability. In connection with the latter, a novel kind of statistics arises which is determined by representations of proper subgroups of the permutation group, rather than of the group as a whole. Finally, we observe that for a generic three-manifold there will be an infinity of inequivalent UIR's and hence an infinity of "consistent" theories, when topology change is neglected.