Jordan Frame Supergravity and Inflation in NMSSM
  We present a complete explicit N=1, d=4 supergravity action in an arbitrary
Jordan frame with non-minimal scalar-curvature coupling of the form $\Phi(z,
\bar z)\, R$. The action is derived by suitably gauge-fixing the superconformal
action. The theory has a modified Kaehler geometry, and it exhibits a
significant dependence on the frame function $\Phi (z, \bar z)$ 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.
