Mapping class groups for D = 2 + 1 quantum gravity and topological quantum field theories

Abstract

A series of disjunctional techniques combining Cerf's original work on diffeomorphism extensions and the Smale-Hatcher Conjecture are used to determine the mapping class groups or homeotopy groups of three-manifolds in which Chern-Simons-Witten and 3-dimensional quantum gravity theories live. Four cases of interest are being considered, two of which focus on three-manifolds with topology M = Z g X R (g ⩾ 2) . We show that when the 3-manifold M g is endowed with only one boundary component, its mapping class group is trivial; whereas the mapping class group is non-zero when M g has two boundary components. As for proper 3-manifolds such as handlebody H g ( g ⩾ 2), we prove that the mapping class group is trivial when H g possesses one boundary component. In contrast, when endowed with no boundary component at all, H g has a highly non-trivial mapping class group. The original impetus for the present work draws on recent developments in 3-dimensional physics, most notably on Chern-Simons-Witten theories and their quantum gravity incarnations. In these theories, mapping class groups have been at the forefront of determining suitable quantizations, solving operator ordering problems, finding a suitable set of diffeomorphism-invariant observables, and detecting and cancelling global gravitational anomalies. The problem of finding a suitable set of mapping class group induced observables is singled out. In particular, we provide a generalization of the Nelson-Regge construction of diffeomorphism-invariant observables for 2 + 1 quantum gravity by showing that the set of mapping class group induced observables corresponds to the centralizer of non-trivial mapping class groups. For proper 3-handlebodies with non-trivial homeotopy groups, it is argued that the lack of a global time direction is a significant obstacle in the physical description of topological quantum field theories, including their quantum gravity incarnation. Because of the triviality of some 3-homeotopy groups, the message derived from the present work is that there are a limited number of 3-manifolds (proper or otherwise) in which one can define consistent theories of quantum gravity and/or Chern-Simons-Witten theories.