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.
