BV-BFV approach to general relativity

Abstract

This thesis is devoted to the study of different formulations of General Relativity (GR) as a a fundamental theory of the gravitational interaction in the setting of Cattaneo, Mnev and Reshetikhin (CMR) on manifolds with boundary. The Batalin (Fradkin) Vilkovisky formalisms (BV and BFV) were joined by CMR to associate to a BV gauge theory on a space-time manifold $ \mathit{M}$ a correspondent BFV structure on its boundary $\partial \mathit{M}$, and a set of axioms for general gauge theories was proposed in this context, in order to have a neat quantisation scheme. The present work is aimed at testing the axioms on different, classically equivalent formulations of General Relativity, namely the Einstein Hilbert metric theory of gravity, the Palatini Holst tetrad formulation of GR and two BF-like theories that go under the name of Plebanski action8 and McDowell-Mansouri action9. We prove that only some of these formulations satisfy the CMR axioms, thus inducing a BV-BFV theory: the Einstein Hilbert theory, for all manifolds with boundary of dimension $ \mathit{d}$ + 1 $ \ne$ 2 with spacelike or timelke boundary components, and the BF-formulation of the McDowell-Mansoury action, under some natural regularity assumptions on the field $ \mathit{B}$. The classical canonical analysis for the Einstein Hilbert and the Palatini Holst actions is also discussed, and we show how the machinery is capable of recovering known results in a straightforward way, yielding in addition an explicit symplectic characterisation of the phase space of the theory. This is a first step in the programme of CMR quantisation of gauge theories on manifolds with boundary, applied to the fundamental, and still open case of General Relativity.