Anti-self-dual gravity and supergravity from a pure connection formulation

TL;DR

This paper presents a pure connection formulation for anti-self-dual gravity and supergravity, deriving Einstein spaces and supergravity equations from a gauge-invariant action with constraints.

Contribution

It introduces a novel complex pure connection action that unifies anti-self-dual gravity and supergravity within a gauge-invariant framework.

Findings

Derives anti-self-dual Einstein spaces from the connection formulation.

Extends the framework to anti-self-dual supergravity with cosmological constant.

Shows the equations of motion follow from the Bianchi identity and integrability conditions.

Abstract

We introduce a complex pure connection action with constraints which is diffeomorphism and gauge invariant. Taking as an internal group $SU(2)$, we obtain, from the equations of motion, anti-self-dual Einstein spaces together with the zero torsion condition thanks to Bianchi identity. By applying the same procedure, we take as internal symmetry the super group $OSp(1|2)$ and by means of the Bianchi identity and integrability conditions, the equations of motion are those that come from anti-self-dual supergravity $N=1$ with cosmological constant sector.