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

TL;DR

This paper develops a positive semi-definite Euclidean action for four-dimensional quantum gravity using self-dual variables, connecting it to Einstein manifolds and analyzing discrete symmetries and Lorentzian extensions.

Contribution

It introduces a new positive semi-definite action for 4D quantum gravity with self-dual variables, linking it to Einstein manifolds and exploring discrete transformations.

Findings

Constructed a positive semi-definite Euclidean action for arbitrary four-topologies.

On-shell, the theory's self-dual sector corresponds to all Einstein manifolds.

Analyzed effects of C, P, T transformations on the action and its Lorentzian extensions.

Abstract

A positive semi-definite 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.