Critical points and supersymmetric vacua, III: String/M models

TL;DR

This paper rigorously counts and analyzes the distribution of supersymmetric vacua in type IIb string theories compactified on Calabi-Yau 3-folds, providing proofs and growth estimates for vacua satisfying flux and tadpole constraints.

Contribution

It provides the first rigorous proofs of the Ashok-Douglas and Denef-Douglas counting formulas for supersymmetric vacua in string theory.

Findings

Confirmed the exponential growth of vacua with bounded curvature in moduli space.

Provided van der Corput style estimates for remainder terms in counting formulas.

Established the distribution of vacua satisfying the tadpole constraint.

Abstract

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold $X$ with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas and Denef-Douglas are given, together with van der Corput style remainder estimates. We also give evidence that the number of vacua satisfying the tadpole constraint in regions of bounded curvature in moduli space is of exponential growth in $b_3(X)$.