No-boundary θ sectors in spatially flat quantum cosmology

Gravitational θ sectors are investigated in spatially locally homogeneous cosmological models with flat closed spatial surfaces in 2+1 and 3+1 spacetime dimensions. The metric ansatz is kept in its most general form compatible with Hamiltonian minisuperspace dynamics. Nontrivial θ sectors admitting a semiclassical no-boundary wave function are shown to exist only in 3+1 dimensions, and there only for two spatial topologies. In both cases the spatial surface is nonorientable and the nontrivial no-boundary θ sector unique. In 2+1 dimensions the nonexistence of nontrivial no-boundary θ sectors is shown to be of topological origin and thus to transcend both the semiclassical approximation and the minisuperspace ansatz. The relation to the necessary condition given by Hartle and Witt for the existence of no-boundary θ states is discussed.