Geometrical structure of Weyl invariants for spin three gauge field in general gravitational background in $d=4$

TL;DR

This paper constructs and analyzes Weyl invariant actions for a linearized spin three gauge field in four-dimensional gravitational backgrounds, exploring their invariance properties and potential linear combinations for gauge invariance.

Contribution

It introduces all Weyl invariant actions for spin three fields in 4D backgrounds and investigates their gauge invariance and possible linear combinations.

Findings

Constructed Weyl invariant actions including the square of the generalized Weyl tensor.

Identified two non-gauge-invariant Weyl actions linear in the background Weyl tensor.

Discussed conditions for a linear combination to be gauge invariant in Ricci flat backgrounds.

Abstract

We construct all possible Weyl invariant actions in $d=4$ for linearized spin three field in a general gravitational background. The first action is obtained as the square of the generalized Weyl tensor for a spin three gauge field in nonlinear gravitational background. It is, however, not invariant under spin three gauge transformations. We then construct two other nontrivial Weyl but not gauge invariant actions which are linear in the Weyl tensor of the background geometry. We then discuss existence and uniqueness of a possible linear combination of these three actions which is gauge invariant. We do this at the linear order in the background curvature for Ricci flat backgrounds.