N = 3 Conformal Supergravity in Four Dimensions

TL;DR

This paper derives the invariant action for four-dimensional N=3 conformal supergravity using super form principles, explores two inequivalent embeddings, and verifies the supersymmetrization of key curvature terms.

Contribution

It provides the first explicit construction of the N=3 conformal supergravity action and embeds the N=3 Weyl multiplet within a new density formula, including a consistency check via N=4 truncation.

Findings

Derived the N=3 conformal supergravity action.

Identified two inequivalent embeddings leading to different invariant actions.

Confirmed the supersymmetrization of the Pontryagin density as a total derivative.

Abstract

In this paper, we derive the action for $N=3$ conformal supergravity in four space-time dimensions. We construct a density formula for $N=3$ conformal supergravity based on the super form action principle. Finally, we embed the $N=3$ Weyl multiplet in the density formula to obtain the invariant action for $N=3$ conformal supergravity. There are two inequivalent embeddings by changing a particular coefficient from real to imaginary. They lead to invariant actions, which will either be the supersymmetrization of the Weyl square term or the Pontryagin density in the eventuality of gauge fixing to Poincar\'{e} supergravity. As a consistency check of our formalism, we will show that the supersymmetrization of the Pontryagin density is a total derivative. We will demonstrate this for purely bosonic terms. We will also present the complete action for the supersymmetrization of the Weyl square term. We also discuss consistent truncation of $N=4$ Weyl multiplet to $N=3$ Weyl multiplet and use it for a robust check of our results using the earlier known results in $N=4$ conformal supergravity.