Background field formalism for chiral matter and gauge fields conformally coupled to supergravity

We expand the generic model involving chiral matter, super Yang-Mills gauge fields, and supergravity to second order in the gravity and gauge prepotentials in a manifestly covariant and conformal way. Such a class of models includes conventional chiral matter coupled to supergravity via a conformal compensator. This is a first step toward calculating one-loop effects in supergravity in a way that does not require a perturbative expansion in the inverse Planck scale or a recourse to component level calculations to handle the coupling of the K\"ahler potential to the gravity sector. We also consider a more restrictive model involving a linear superfield in the role of the conformal compensator and investigate the similarities it has to the dual chiral model.