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A New Method for Finding Vacua in String Phenomenology

James Gray, Yang-Hui He, Anton Ilderton, Andre Lukas

TL;DR

The paper develops an algorithmic, algebraic-geometry–driven framework to locate stabilized vacua in four-dimensional ${ m\ N}=1$ effective theories from string compactifications, extending prior perturbative methods to include non-perturbative superpotential terms via dummy variables for exponentials. It systematically reduces the extremization problem to polynomial systems using saturation and primary decompositions, and resolves non-polynomial elements by projecting/transcendental-elimination aided by numerical root counting, ensuring no vacua are missed. The authors demonstrate the approach on heterotic and IIB examples, obtaining both supersymmetric and non-supersymmetric AdS vacua, as well as de-Sitter turning points, and they show how to derive parameter-space constraints that must hold for specific vacua types. The work offers a practical pathway to scan flux vacua spaces and motivates a potential black-box software tool for automated vacuum searches across string-theory landscapes.

Abstract

One of the central problems of string-phenomenology is to find stable vacua in the four dimensional effective theories which result from compactification. We present an algorithmic method to find all of the vacua of any given string-phenomenological system in a huge class. In particular, this paper reviews and then extends hep-th/0606122 to include various non-perturbative effects. These include gaugino condensation and instantonic contributions to the superpotential.

A New Method for Finding Vacua in String Phenomenology

TL;DR

The paper develops an algorithmic, algebraic-geometry–driven framework to locate stabilized vacua in four-dimensional effective theories from string compactifications, extending prior perturbative methods to include non-perturbative superpotential terms via dummy variables for exponentials. It systematically reduces the extremization problem to polynomial systems using saturation and primary decompositions, and resolves non-polynomial elements by projecting/transcendental-elimination aided by numerical root counting, ensuring no vacua are missed. The authors demonstrate the approach on heterotic and IIB examples, obtaining both supersymmetric and non-supersymmetric AdS vacua, as well as de-Sitter turning points, and they show how to derive parameter-space constraints that must hold for specific vacua types. The work offers a practical pathway to scan flux vacua spaces and motivates a potential black-box software tool for automated vacuum searches across string-theory landscapes.

Abstract

One of the central problems of string-phenomenology is to find stable vacua in the four dimensional effective theories which result from compactification. We present an algorithmic method to find all of the vacua of any given string-phenomenological system in a huge class. In particular, this paper reviews and then extends hep-th/0606122 to include various non-perturbative effects. These include gaugino condensation and instantonic contributions to the superpotential.

Paper Structure

This paper contains 12 sections, 36 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The General Method
  3. Finding Vacua for Perturbative Superpotentials
  4. Finding Vacua for Non-Perturbative Superpotentials
  5. Equations for the Transcendental Variables
  6. Solutions for the Transcendental Variables
  7. Examples of Finding Supersymmetric and Non-Supersymmetric Vacua
  8. A Heterotic Example
  9. An Example with de-Sitter Turning Points
  10. Some IIB Examples
  11. Constraints on Flux Parameters
  12. Conclusions

Figures (4)

  • Figure 1: A reducible variety with a plane 'obscuring' a point. In this case, projection onto the $n$-$x$ plane will miss the isolated point. However, primary decomposition will avoid this problem since the point and the plane will be separated as different prime ideals.
  • Figure 2: The non-supersymmetric saddle point at the field values given in (\ref{['keepme']}), plotted in the $(t,\tau)$ plane.
  • Figure 3: The de Sitter saddle point described in the text, plotted in the $s$, $t$ plane.
  • Figure 4: The IIB AdS saddle point at $\tau=0$, $t\simeq35.396$, $s\simeq 16,369$, $\sigma=0$ plotted in the $\tau,s$ plane