Supergravity Analysis of Hybrid Inflation Model from D3--D7 System

TL;DR

This work analyzes a string-theory realization of hybrid inflation using a D3–D7 system in Type IIB on a local ALE×C geometry, embedded into a Calabi–Yau to include dynamical gravity. The inflaton resides in a ${\cal N}=2$ vector multiplet and the potential is governed by momentum maps $P^x_\Lambda$, yielding a naturally flat direction even with gravity, thanks to the special-geometry constraints and the translational symmetry in the internal directions. Through a detailed construction with a $K3\times T^2$ toy model and the Calabi–Visentini basis, the paper demonstrates that short-distance stringy corrections do not spoil flatness, and identifies explicit moduli-stabilization conditions (via 3-form fluxes) that produce a nonzero FI structure while keeping the slow-roll parameter $\eta$ small, e.g. $\eta \simeq 2-\frac{g_3^2\mathrm{vol}(K3)}{4(g_1+g_2)^2}$ with $g_3^2\mathrm{vol}(K3)\approx 8(g_1+g_2)^2$. The analysis also shows that integrating out heavy moduli can be arranged to preserve the inflaton flatness under suitable flux choices, and discusses generalizations to generic Calabi–Yau manifolds where an ${\cal N}=1$ moduli-stabilization scenario can maintain a nonzero FI parameter and a flat inflationary potential, potentially aided by a discrete $R$-symmetry. Overall, the paper provides a concrete string-theoretic route to viable slow-roll inflation within ${\cal N}=2$-compatible supergravity and its ${\cal N}=1$ low-energy limit.

Abstract

The 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 D = 4 supergravity description. We adopt the D3--D7 system of Type IIB string theory compactified on K3 x T^2/Z_2 orientifold for definiteness. It turns out that the slow-roll parameter can be sufficiently small despite the non-minimal Kahler potential of the model. The conditions for this to happen are clarified in terms of string vacua. We also find that the geometry obtained by blowing up singularity, which is necessary for the positive vacuum energy, is stabilized by introducing certain 3-form fluxes.