Nonrenormalization of Flux Superpotentials in String Theory

TL;DR

This work addresses the perturbative non-renormalization of flux-induced superpotentials $W_{GVW}$ in Type IIB string vacua. It develops a symmetry-based derivation using holomorphy and surviving subgroups of $SL(2,R)$—notably $R$-invariance, PQ invariance, and $SL(2,Z)$—along with a spurion analysis of background fluxes to constrain the 4D effective action. The main result is that $W$ remains equal to the GVW form to all orders in perturbation theory, with no dependence on Kähler moduli in any $oldsymbol{ u}$-expansion and only restricted corrections to the gauge-kinetic function $f_{ab}$; this underpins robust modulus stabilization scenarios like KKLT. The analysis fills a gap in Type IIB discussions by providing a robust, symmetry-based justification of perturbative non-renormalization, while noting limitations such as the assumption of trivial dilaton variation and the potential impact of nonperturbative effects.

Abstract

Recent progress in understanding modulus stabilization in string theory relies on the existence of a non-renormalization theorem for the 4D compactifications of Type IIB supergravity which preserve N=1 supersymmetry. We provide a simple proof of this non-renormalization theorem for a broad class of Type IIB vacua using the known symmetries of these compactifications, thereby putting them on a similar footing as the better-known non-renormalization theorems of heterotic vacua without fluxes. The explicit dependence of the tree-level flux superpotential on the dilaton field makes the proof more subtle than in the absence of fluxes.