Exploring the Potential Energy Landscape Over a Large Parameter-Space

TL;DR

This paper compares symbolic and numerical methods for analyzing large polynomial systems, focusing on potential energy landscapes and applications in string theory compactifications.

Contribution

It demonstrates the effectiveness of Comprehensive Gröbner bases and cheater's homotopy in studying complex polynomial systems relevant to physics and geometry.

Findings

Efficient extraction of physical quantities like masses and couplings.

Successful application to flux compactification scenarios.

Comparison of symbolic and numerical approaches for large systems.

Abstract

Solving large polynomial systems with coefficient parameters are ubiquitous and constitute an important class of problems. We demonstrate the computational power of two methods--a symbolic one called the Comprehensive Gr\"obner basis and a numerical one called the cheater's homotopy-applied to studying both potential energy landscapes and a variety of questions arising from geometry and phenomenology. Particular attention is paid to an example in flux compactification where important physical quantities such as the gravitino and moduli masses and the string coupling can be efficiently extracted.