WHAT BECOMES OF THE VACUUM ANGLE IN GENERALLY COVARIANT GAUGE THEORIES AND THE VIELBEIN APPROACH TO GRAVITY

Abstract

We examine the fate of the gauge theory vacuum angles (like the QCD θ) when the topology of the spatial slice M is nontrivial and diffeomorphism invariance is included. The analogues of the θ angles are then defined by the unitary irreducible representations of the semidirect product [Formula: see text] and [Formula: see text] being the groups of asymptotically trivial diffeomorphisms and gauge transformations for M. Features from spatial and gauge group topologies thus get nontrivially mixed up. As a consequence, a spectrum of θ angles or their appropriate analogues can occur in a quantum theory. We also find the remarkable result that the vacuum angles are quantized for certain M like ℝ2#T2. The new variable approach to gravity or any approach using vielbeins resembles a gauge theory with diffeomorphism invariance and hence can be subjected to our analysis. In particular, in such approaches as well as a spectrum of gravity θ angles or their suitable analogues can appear in a quantum theory.