Mapping-Class Groups of 3-Manifolds in Canonical Quantum Gravity

Abstract

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. Mathematics Subject Classification (2000). Primary 83C45; Secondary 57N10.

The Superspace of geometrodynamics

“…This behaviour can also be studied in more complicated examples [32,72]. For a more geometric understanding of the maps representing E and S, see [37].…”

Gravitation at the Josephson Junction

“…= /2, where is the scalar Ricci curvature of the three-dimensional hypersurface [40]. Combining these relations we end up with ∇ ⋅ → = 2 /4, which is equivalent to…”

Dynamical and Hamiltonian Formulation of General Relativity

“…The theta-structure then depends on the topology of Σ and can range from 'trivial' to 'very complicated'. See [73] for more details on the role and determination of these mapping-class groups and [74] for a more general discussion of the configuration space in GR, which, roughly speaking, is the quotient Riem(Σ)/Diff(Σ), often referred to as Wheeler's superspace [116] [52].…”