Equivalence between various versions of the self-dual action of the Ashtekar formalism

TL;DR

This paper examines the relationships and differences among various formulations of the self-dual action in the Ashtekar formalism, aiming to facilitate its extension to higher dimensions like eight.

Contribution

It provides a detailed analysis of the equivalences and distinctions between different versions of the self-dual action in the Ashtekar formalism, which is useful for higher-dimensional generalizations.

Findings

Identifies equivalences between different self-dual actions

Highlights differences relevant for formalism development

Lays groundwork for eight-dimensional Ashtekar formalism

Abstract

Different aspects of the self-dual (anti-self-dual) action of the Ashtekar canonical formalism are discussed. In particular, we study the equivalences and differences between the various versions of such an action. Our analysis may be useful for the development of an Ashtekar formalism in eight dimensions.