Mapping-class groups of 3-manifolds in canonical quantum gravity

TL;DR

This paper explores the role of mapping-class groups of 3-manifolds as symmetry groups in canonical quantum gravity, highlighting their potential to encode topological information and influence superselection sectors.

Contribution

It analyzes the significance of mapping-class groups as gauge symmetries in quantum gravity and discusses their mathematical structure and physical implications.

Findings

Mapping-class groups act as gauge symmetries in quantum gravity.

Inequivalent representations lead to complex superselection structures.

Topological features influence physical states in quantum gravity.

Abstract

Mapping-class groups of 3-manifolds feature as symmetry groups in canonical quantum gravity. They are an obvious source through which topological information could be transmitted into the quantum theory. If treated as gauge symmetries, their inequivalent unitary irreducible representations should give rise to a complex superselection structure. This highlights certain aspects of spatial diffeomorphism invariance that to some degree seem physically meaningful and which persist in all approaches based on smooth 3-manifolds, like geometrodynamics and loop quantum gravity. We also attempt to give a flavor of the mathematical ideas involved.