A Toolkit for Perturbing Flux Compactifications

TL;DR

The paper develops a perturbative framework for type IIB flux compactifications around ISD backgrounds, revealing a triangular structure in the n-th order equations that enables an iterative Green's-function solution on warped Calabi–Yau cones. By constructing explicit scalar, flux, and metric Green's functions and detailing harmonic seeds on the Sasaki–Einstein base, the authors provide a systematic procedure to compute corrections to a warped throat induced by coupling to a compact bulk and to estimate the sizes of warp-mediated effects. A central result is a general radial-scaling formula: n-th order corrections decompose into products of n harmonic-mode seeds, ensuring that throat perturbation theory is controlled by the magnitudes of these modes. The method has broad applicability to local model-building, inflationary scenarios, and exploring non-supersymmetric AdS/CFT pairs, with clear criteria for consistency and truncation, and explicit guidance for cones with known harmonic spectra such as the conifold.

Abstract

We develop a perturbative expansion scheme for solving general boundary value problems in a broad class of type IIB flux compactifications. The background solution is any conformally Calabi-Yau compactification with imaginary self-dual (ISD) fluxes. Upon expanding in small deviations from the ISD solution, the equations of motion simplify dramatically: we find a simple basis in which the n-th order equations take a triangular form. This structure implies that the system can be solved iteratively whenever the individual, uncoupled equations can be solved. We go on to demonstrate the solution of the system for a general warped Calabi-Yau cone: we present an algorithm that yields an explicit Green's function solution for all the supergravity fields, to any desired order, in terms of the harmonic functions on the base of the cone. Our results provide a systematic procedure for obtaining the corrections to a warped throat geometry induced by attachment to a compact bulk. We also present a simple method for determining the sizes of physical effects mediated through warped geometries.