  We study models of hybrid inflation in the framework of supergravity with
superconformal matter. F-term hybrid inflation is not viable since the inflaton
acquires a large tachyonic mass. On the contrary, D-term hybrid inflation can
successfully account for the amplitude of the primordial power spectrum. It is
a two-field inflation model which, depending on parameters, yields values of
the scalar spectral index down to n_s ~ 0.96. Generically, there is a tension
between a small spectral index and the cosmic string bound albeit, within
2-sigma uncertainty, the current observational bounds can be simultaneously
fulfilled.
