Superconformal D-Term Inflation
  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. 
W. Buchm\"uller, V. Domcke, K. Schmitz
hep-ph astro-ph.CO hep-th