A pure connection formulation with real fields for Gravity

Abstract

We study an SO(1, 3) pure connection formulation in four dimensions for real-valued fields, inspired by the Capovilla, Dell and Jacobson complex self-dual approach. By considering the CMPR BF action, also, taking into account a more general class of the Cartan-Killing form for the Lie algebra so(1, 3) and by refining the structure of the Lagrange multipliers, we integrate out the metric variables in order to obtain the pure connection action. Once we have obtained this action, we impose certain restrictions on the Lagrange multipliers, in such a way that the equations of motion led us to a family of torsionless conformally flat Einstein manifolds, parametrized by two numbers. Finally, we show that, by a suitable choice of parameters, that self-dual spaces (Anti-) De Sitter can be obtained.

Comments on the symmetry breaking condition in MacDowell-Mansouri action

“…A higher dimensional internal symmetry group based on MM approach for pure connection formulations of gravity for real fields, inspired in [9] may be considered in order to obtain a family of torsionless conformally flat Einstein manifolds, this work is in progress and will be reported elsewhere.…”

“…On the other hand, the usual way that elementary interactions are described it's by considering the connection associated with an internal symmetry group where the spacetime is non-dynamical, but let us remember that in GR gravity is a dynamical entity. Various attempts have been made to construct Yang-Mills type gauge theories of gravity, fortunately there have been advances towards this direction, giving some very interesting formulations that are known in the literature as pure connection actions for gravity [5][6][7][8][9], where the fundamental field is the connection gauge field for a corresponding symmetry group G. So the metric is not longer the main field for describe gravity, instead it becomes a derived object so GR is a consequence of the proposed gauge theory.…”