Self-dual variables, positive-semidefinite action, and discrete transformations in four-dimensional quantum gravity

A positive-semidefinite Euclidean action for arbitrary four-topologies can be constructed by adding appropriate Yang-Mills and topological terms to the Samuel-Jacobson-Smolin action of gravity with (anti-)self-dual variables. Moreover, on shell, the (anti-)self-dual sector of the new theory corresponds precisely to all Einstein manifolds in four dimensions. The Lorentzian signature action and its analytic continuations are also considered. A self-contained discussion is given on the effects of discrete transformations C, P, and T on the Samuel-Jacobson-Smolin action, and other proposed actions which utilize self- or anti-self-dual variables as fundamental variables in the description of four-dimensional gravity.