Inflation with Fayet-Iliopoulos Terms

Two of the most attractive realizations of inflation in supergravity are based upon the presence of a constant Fayet-Iliopoulos (FI) term. In D-term hybrid inflation it is the FI term itself which sets the energy scale of inflation. Alternatively, the breaking of a U(1) symmetry induced by the FI term can dynamically generate the quadratic potential of chaotic inflation. The purpose of this note is to study the possible UV embedding of these schemes in terms of the `field-dependent FI term' related to a string modulus field which is stabilized by a non-perturbative superpotential. We find that in settings where the FI term drives inflation, gauge invariance prevents a decoupling of the modulus from the inflationary dynamics. The resulting inflation models generically contain additional dynamical degrees of freedom compared to D-term hybrid inflation. However, the dynamical realization of chaotic inflation can be obtained in complete analogy to the case of a constant FI term. We present a simple string-inspired toy model of this type.