Exploring the Vacuum Geometry of N=1 Gauge Theories

TL;DR

The paper introduces a computational algebraic-geometry framework to explicitly construct the vacuum moduli spaces of N=1 gauge theories, applying it to MSSM subsectors. By representing vacua as F-term solutions modulo the complexified gauge group and describing the space via holomorphic gauge-invariant operators, the authors use Gröbner-basis methods to obtain explicit affine varieties. A key finding is that the electroweak sector with renormalizable terms yields a cone over the Veronese surface, a highly special geometry that becomes trivial under certain deformations, linking low-energy phenomenology to high-energy structure. The work suggests that vacuum geometry could serve as a predictive tool for UV completions, while also highlighting computational limits and the value of building a geometry-based catalog across sectors and generations.

Abstract

Using techniques of algorithmic algebraic geometry, we present a new and efficient method for explicitly computing the vacuum space of N=1 gauge theories. We emphasize the importance of finding special geometric properties of these spaces in connecting phenomenology to guiding principles descending from high-energy physics. We exemplify the method by addressing various subsectors of the MSSM. In particular the geometry of the vacuum space of electroweak theory is described in detail, with and without right-handed neutrinos. We discuss the impact of our method on the search for evidence of underlying physics at a higher energy. Finally we describe how our results can be used to rule out certain top-down constructions of electroweak physics.