On conformal higher spins in curved background

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.

24 pages. v2,v3: minor corrections and remarks added

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