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Mapping class groups for \(D=2+1\) quantum gravity and topological quantum field theories. (English) Zbl 0990.83519

Summary: 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 \(M=\Sigma_{g}\times\mathbb{R}\) \((g\geqslant 2)\). 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 \(H_{g}\) \((g\geqslant 2)\), 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.

MSC:

83C45 Quantization of the gravitational field
57M50 General geometric structures on low-dimensional manifolds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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