Numerical Analyses on Moduli Space of Vacua

TL;DR

This paper introduces a novel computational approach combining numerical algebraic geometry and elimination theory to analyze the vacuum moduli space in supersymmetric field theories, enabling efficient extraction of geometric and physical properties.

Contribution

It develops a new, efficient, and parallelizable algorithm for studying the moduli space of vacua, integrating numerical and algebraic techniques for the first time.

Findings

Successfully applied to examples from gauge and string theory

Extracted dimension, branch structure, and Hilbert series

Demonstrated efficiency and parallelizability of the method

Abstract

We propose a new computational method to understand the vacuum moduli space of (supersymmetric) field theories. By combining numerical algebraic geometry (NAG) and elimination theory, we develop a powerful, efficient, and parallelizable algorithm to extract important information such as the dimension, branch structure, Hilbert series and subsequent operator counting, as well as variation according to coupling constants and mass parameters. We illustrate this method on a host of examples from gauge theory, string theory, and algebraic geometry.