On spin 3 interacting with gravity

Understand

Recently Boulanger and Leclercq have constructed cubic four derivative $3-3-2$ vertex for interaction of spin 3 and spin 2 particles.

• This vertex is trivially invariant under the gauge transformations of spin 2 field, so it seemed that it could be expressed in terms of (linearized) Riemann tensor.
• And indeed in this paper we managed to reproduce this vertex in the form $R \partial \Phi \partial \Phi$, where $R$ -- linearized Riemann tensor and $\Phi$ -- completely symmetric third rank tensor.
• Then we consider deformation of this vertex to $(A)dS$ space and show that such deformation produce "standard" gravitational interaction for spin 3 particles (in linear approximation) in agreement with general construction of Fradkin and Vasiliev.