Supergravity analysis of the hybrid inflation model from a D3-D7 system

Slow-roll inflation is a beautiful paradigm, yet the inflaton potential can hardly be sufficiently flat when unknown gravitational effects are taken into account. However, the hybrid inflation models constructed in $D=4,$ N=1 supergravity can be consistent with N=2 supersymmetry, and can be naturally embedded into string theory. This article discusses the gravitational effects carefully in the string model, using a $D=4$ supergravity description. We adopt the D3-D7 system of type IIB string theory compactified on a K3×T²/Z₂ orientifold for definiteness. It turns out that the slow-roll parameter can be sufficiently small despite the nonminimal K\"ahler potential of the model. The conditions for this to happen are given in terms of string vacua. We also find that the geometry obtained by blowing up the singularity, which is necessary for the positive vacuum energy, is stabilized by introducing certain three-form fluxes.