Distributions of flux vacua
Frederik Denef, Michael R. Douglas
TL;DR
The paper develops a comprehensive statistical framework for the distribution and counting of flux vacua in type IIb string theory on Calabi–Yau manifolds. By formulating SUSY and nonsupersymmetric vacua as densities over moduli space and employing the Gukov–Vafa–Witten superpotential, it derives explicit density formulas, index densities, and attractor-point analogues, and applies them to concrete examples (T^6, conifold, mirror quintic). It reveals how vacua accumulate near special loci (large complex structure and conifold) and analyzes the impact of metastability, cosmological constants, and flux quantization on the landscape; in particular, it shows that near degenerations the density of vacua can be large and that metastable de Sitter vacua are highly constrained in these regimes. The results offer quantitative guidance for naturalness and phenomenology in the string landscape, and establish a tractable, geometry-dependent approach to vacuum statistics across a range of moduli spaces and flux sectors.
Abstract
We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points.