Self-dual and anti-self-dual structure of the de Sitter gauge theory and Ashtekar's variables.

Abstract

A Yang-Mills-like theory of gravity, which is a de Sitter gauge theory, is presented starting from a Clifford algebra $C(3, 1)$ generated by the Dirac matrices. The self-dual and anti-self-dual structure of the quadratic terms in the curvature with respect to internal as well as external indices is investigated and then a Yang-Mills-like Lagrangian for the de Sitter gauge theory is obtained which involves no torsion term and returns to Ashtekar's Lagrangian naturally modulo topological and cosmological terms.