# On induced action for conformal higher spins in curved background

## TL;DR

This paper investigates the structure of the induced action for conformal higher spin fields in curved 4d backgrounds, focusing on spin 1 and 3 interactions, their gauge invariance, and contributions to the conformal anomaly.

## Contribution

It provides a detailed analysis of the mixed spin 1-3 terms in the induced action and their impact on gauge invariance and conformal anomalies in curved backgrounds.

## Key findings

- Identified curvature-dependent spin 1-3 mixing terms.
- Showed pure spin 3 kinetic term is not gauge-invariant beyond leading order.
- Calculated the contribution of mixing terms to the conformal anomaly c-coefficient.

## Abstract

We continue the investigation of the structure of the action for a tower of conformal higher spin fields in non-trivial 4d background metric recently discussed in arXiv:1609.09381. The action is defined as an induced one from path integral of a conformal scalar field in curved background coupled to higher spin fields. We analyse in detail the dependence of the quadratic part of the induced action on the spin 1 and spin 3 fields, determining the presence of a curvature-dependent mixed spin 1 - spin 3 term. One consequence is that the pure spin 3 kinetic term cannot be gauge-invariant on its own beyond the leading term in small curvature expansion. We also compute the non-zero contribution of the 1-3 mixing term to the conformal anomaly c-coefficient. One is thus to determine all such mixing terms before addressing the question of possible vanishing of the total c-coefficient in the conformal higher spin theory.