Single Field Inflation in Supergravity with a $U(1)$ Gauge Symmetry
  A single field inflation based on a supergravity model with a shift symmetry
and $U(1)$ extension of the MSSM is analyzed. We show that one of the real
components of the two $U(1)$ charged scalar fields plays the role of inflaton
{with} an effective scalar potential similar to the "new chaotic inflation"
scenario. Both non-anomalous and anomalous (with Fayet-Iliopoulos term) $U(1)$
are studied. We show that the non-anomalous $U(1)$ scenario is consistent with
data of the cosmic microwave background and recent astrophysical measurements.
A possible kinetic mixing between $U(1)$ {and} $U(1)_{B-L}$ is considered in
order to allow for natural decay channels of the inflaton, leading to a
reheating epoch. Upper limits on the reheating temperature thus turn out to
favour an intermediate ($\sim {\cal O}(10^{13})$ GeV) scale $B-L$ symmetry
breaking.
