Algorithmic Algebraic Geometry and Flux Vacua

TL;DR

This paper introduces a computational algebraic geometry approach to systematically analyze flux compactifications in supergravity, enabling efficient identification of vacua and their properties.

Contribution

It presents a novel, algorithmic method to determine flux constraints and find all isolated vacua in flux compactifications, improving upon traditional techniques.

Findings

Developed a procedure for flux constraint calculation.

Created a stepwise method to find all isolated vacua.

Applied methods to examples previously difficult with conventional approaches.

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.