Jordan frame supergravity and inflation in the NMSSM

We present a complete explicit N=1, d=4 supergravity action in an arbitrary Jordan frame with nonminimal scalar-curvature coupling of the form (z, z) R. The action is derived by suitably gauge fixing the superconformal action. The theory has a modified K\"ahler geometry, and it exhibits a significant dependence on the frame function (z, z) and its derivatives over scalars, in the bosonic as well as in the fermionic part of the action. Under certain simple conditions, the scalar kinetic terms in the Jordan frame have a canonical form. We consider an embedding of the next-to-minimal supersymmetric standard model (NMSSM) gauge theory into supergravity, clarifying the Higgs inflation model recently proposed by Einhorn and Jones. We find that the conditions for canonical kinetic terms are satisfied for the NMSSM scalars in the Jordan frame, which leads to a simple action. However, we find that the gauge singlet field experiences a strong tachyonic instability during inflation in this model. Thus, a modification of the model is required to support the Higgs-type inflation.