# The component structure of conformal supergravity invariants in six dimensions

## TL;DR

This paper translates superspace conformal supergravity invariants into component form in six dimensions, providing explicit actions and exploring the structure of invariants in different supersymmetry settings.

## Contribution

It offers the first full component actions for 6D $N=(1,0)$ conformal supergravity invariants and analyzes the relation to $N=(2,0)$ invariants through truncation and uplift.

## Key findings

- Derived component actions for two conformal supergravity invariants.
- Constructed the supersymmetric $Fox F$ action in component form.
- Confirmed the uniqueness of the $N=(2,0)$ conformal supergravity invariant.

## Abstract

In the recent paper arXiv:1606.02921, the two invariant actions for 6D $N=(1,0)$ conformal supergravity were constructed in superspace, corresponding to the supersymmetrization of $C^3$ and $C\Box C$. In this paper, we provide the translation from superspace to the component formulation of superconformal tensor calculus, and we give the full component actions of these two invariants. As a second application, we build the component form for the supersymmetric $F\Box F$ action coupled to conformal supergravity. Exploiting the fact that the $N=(2,0)$ Weyl multiplet has a consistent truncation to $N=(1,0)$, we then verify that there is indeed only a single $N=(2,0)$ conformal supergravity invariant and reconstruct most of its bosonic terms by uplifting a certain linear combination of $N=(1,0)$ invariants.