Phase space of gravity on space-like boundaries

The Palatini-Cartan (PC) action and the Macdowell-Mansouri (MM) action are classically equivalent to the Einstein-Hilbert (EH) action. Though these are classically equivalent their phase spaces carry di erent structure. In particular the phase space of PC gravity on a space-like boundary of a manifold with d≥4 does not reduce smoothly into Cauchy data as the symplectic form ωPC has a kernel. These are highly non-trivial examples of gravity theories that resist similarity with EH gravity at a stage that preceeds quantisation. We study the phase space of the MM theory in it's BF formulation on a boundary. We then use Cartan decomposition on the BF elds to compare the symmetries of the MM theory to the symmetries of the PC elds.