On conformal higher spins in curved background

TL;DR

This paper develops a covariant formulation for interacting conformal higher spin fields in curved backgrounds, demonstrating gauge invariance of the kinetic term in Bach-flat metrics and exploring background-dependent mixing effects.

Contribution

It generalizes previous results for spin 3 to all conformal higher spins, establishing gauge invariance of the kinetic term in Bach-flat backgrounds to first order.

Findings

Kinetic term is gauge-invariant in Bach-flat metrics to first order.

Generalization from spin 3 to all conformal higher spins.

Discussion on background-dependent mixing terms in the action.

Abstract

We address the question of how to represent an interacting action for the tower of conformal higher spin fields in a form covariant with respect to a background metric. We use a background metric to define a star product which plays a central role in the definition of the corresponding gauge transformations. By an analogy with the kinetic term in the 4-derivative Weyl gravity action expanded near an on-shell background one expects that the kinetic term in such an action should be gauge-invariant in a Bach-flat metric. We demonstrate this fact to first order in expansion in powers of the curvature of the background metric. This generalizes the result of arXiv:1404.7452 for spin 3 case to all conformal higher spins. We also comment on a possibility of extending this claim to terms quadratic in the curvature and discuss the appearance of background-dependent mixing terms in the quadratic part of the conformal higher spin action.