Random Polynomials and the Friendly Landscape

TL;DR

The paper develops a realistic, large-$N$ SUSY landscape framework in which metastable vacua and the cosmological constant can be studied with algebraic-geometric tools. By imposing radiative-stability constraints, enforcing an $\ ext{Z}_4$ R-symmetry and Fermat-form cubic superpotentials, the authors derive the vacuum-statistics structure and compute holomorphic moments of W across $2^N$ vacua. They present exact results for small $N$ and systematic perturbative expansions for large $N$ with small cross-terms, showing that the cosmological constant can exhibit scanning behavior when the variance of holomorphic observables dominates the mean. The methodology connects flux-ensemble physics, algebraic geometry, and large-$N$ techniques to yield a concrete toolkit for vacuum counting and anthropic considerations in string-inspired landscapes, with potential generalizations to real fields and toric/gauged geometries.

Abstract

In hep-th/0501082, a field theoretic ``toy model'' for the Landscape was proposed. We show that the considerations of that paper carry through to realistic effective Lagrangians, such as those that emerge out of string theory. Extracting the physics of the large number of metastable vacua that ensue requires somewhat more sophisticated algebro-geometric techniques, which we review.