A pure connection formulation with real fields for Gravity

TL;DR

This paper develops a real-valued pure connection formulation of four-dimensional gravity, deriving a connection-based action that describes torsionless conformally flat Einstein manifolds, including (Anti-) De Sitter spaces, by refining previous complex self-dual approaches.

Contribution

It introduces a real-valued pure connection action for gravity, extending prior complex formulations, and characterizes a family of Einstein manifolds including (Anti-) De Sitter spaces.

Findings

Derived a pure connection action for real-valued gravity fields.

Identified conditions leading to torsionless conformally flat Einstein manifolds.

Showed how to obtain (Anti-) De Sitter spaces within this framework.

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 $\mathfrak{so(1,3)}$ 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.