Natural Inflation with Natural Trans-Planckian Axion Decay Constant from
  Anomalous $U(1)_X$
  We propose a natural inflation model driven by an imaginary or axionic
component of a K\"ahler modulus in string-inspired supergravity. The shift
symmetry of the axion is gauged under an anomalous $U(1)_X$ symmetry, which
leads to a modulus-dependent Fayet-Iliopoulos (FI) term. The matter fields are
stabilized by F-terms, and the real component of the modulus is stabilized by
the $U(1)_X$ D-term, while its axion remains light. Therefore, the masses of
real and imaginary components of the modulus are separated at different scales.
The scalar potential for natural inflation is realized by the superpotential
from the non-perturbative effects. The trans-Planckian axion decay constant,
which is needed to fit with BICEP2 observations, can be obtained naturally in
this model.
