Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithmic Algebraic Geometry and Flux Vacua

James Gray, Yang-Hui He, André Lukas

TL;DR

This paper reframes the problem of finding vacua in flux-compactified 4D supergravities as an algorithmic algebraic-geometry task. By formulating the extrema of the scalar potential $V$ as a polynomial ideal problem, it enables primary (and saturation) decompositions and real-root analysis to enumerate all isolated vacua and extract their physical properties, including SUSY status, stability, and effective-theory validity. The authors develop practical tools for generating flux-constraint conditions and for systematically solving for vacua, demonstrated through toy models and concrete heterotic and M-theory examples. The approach leverages complex and real algebraic geometry, GTZ primary decomposition, and Sturm-query real root counting to yield results that are difficult or impossible with traditional methods, with clear path toward fully automated flux-vacua phenomenology. The work illustrates that perturbative flux vacua can be exhaustively analyzed and paves the way for broader incorporation of non-perturbative effects and automated scanning of string-phenomenology landscapes.

Abstract

We develop a new and efficient method to systematically analyse four dimensional effective supergravities which descend from flux compactifications. The issue of finding vacua of such systems, both supersymmetric and non-supersymmetric, is mapped into a problem in computational algebraic geometry. Using recent developments in computer algebra, the problem can then be rapidly dealt with in a completely algorithmic fashion. Two main results are (1) a procedure for calculating constraints which the flux parameters must satisfy in these models if any given type of vacuum is to exist; (2) a stepwise process for finding all of the isolated vacua of such systems and their physical properties. We illustrate our discussion with several concrete examples, some of which have eluded conventional methods so far.

Algorithmic Algebraic Geometry and Flux Vacua

TL;DR

This paper reframes the problem of finding vacua in flux-compactified 4D supergravities as an algorithmic algebraic-geometry task. By formulating the extrema of the scalar potential as a polynomial ideal problem, it enables primary (and saturation) decompositions and real-root analysis to enumerate all isolated vacua and extract their physical properties, including SUSY status, stability, and effective-theory validity. The authors develop practical tools for generating flux-constraint conditions and for systematically solving for vacua, demonstrated through toy models and concrete heterotic and M-theory examples. The approach leverages complex and real algebraic geometry, GTZ primary decomposition, and Sturm-query real root counting to yield results that are difficult or impossible with traditional methods, with clear path toward fully automated flux-vacua phenomenology. The work illustrates that perturbative flux vacua can be exhaustively analyzed and paves the way for broader incorporation of non-perturbative effects and automated scanning of string-phenomenology landscapes.

Abstract

We develop a new and efficient method to systematically analyse four dimensional effective supergravities which descend from flux compactifications. The issue of finding vacua of such systems, both supersymmetric and non-supersymmetric, is mapped into a problem in computational algebraic geometry. Using recent developments in computer algebra, the problem can then be rapidly dealt with in a completely algorithmic fashion. Two main results are (1) a procedure for calculating constraints which the flux parameters must satisfy in these models if any given type of vacuum is to exist; (2) a stepwise process for finding all of the isolated vacua of such systems and their physical properties. We illustrate our discussion with several concrete examples, some of which have eluded conventional methods so far.

Paper Structure

This paper contains 36 sections, 44 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Flux Vacua and Algebraic Geometry
  3. Classification of the Problem
  4. Mapping the Problem to Algebraic Geometry
  5. Nomenclature
  6. Techniques from Complex Algebraic Geometry
  7. Primary Decomposition
  8. Dimension and Flat-Directions
  9. Techniques from Real Algebraic Geometry
  10. Sign Conditions and Real Root Finding
  11. Stability of the vacua:
  12. Validity of the effective theory:
  13. Geometry of the vacua:
  14. Supersymmetry of the vacua:
  15. Saturations and Classification Revisited
  16. ...and 21 more sections

Figures (2)

  • Figure 1: The non-supersymmetric vacuum for the supergravity theory specified in (\ref{['inveqns']}) for our toy example. The fields are $T_1 = t_1+ i \tau_1$ and $T_2 = t_2 + i \tau_2$. We have plotted the potential in two slices through field space, viz., $t_1$-$t_2$ and $t_1$-$\tau_1$. A shift in the $V$ axis of $2.09279725 \times 10^{-3}$ has been performed so that the very shallow vacuum can be plotted effectively.
  • Figure 2: The non-supersymmetric extremum corresponding to the ideal $I_2$ in (\ref{['paulsolI']}), for the supergravity potential specified in (\ref{['paulKW']}). The fields are $T_i = -i t_i + \tau_i$ and $U = -i x+ y$; we have here plotted the slices in $(\tau_1, t_3)$ and $(t_2,y)$ coordinates. A shift in the $V$ axis of $4.07 \times 10^{-4}$ has been performed so that the very shallow vacuum can be plotted effectively.