Vacuum Geometry and the Search for New Physics

TL;DR

The paper addresses how the vacuum geometry of $\mathcal{N}=1$ SUSY gauge theories, and in particular subsectors of the MSSM, can reveal new physics through special geometric structure. It develops an algorithmic framework based on Gröbner-basis methods to compute the vacuum moduli space $\mathcal{M}$ from gauge-invariant operators and F-flatness, with implementation in $\texttt{Macaulay2}$ applied to MSSM-like sectors. A key result is that the electroweak sector yields a five-dimensional moduli space $\mathcal{M}$ (an affine cone over a four-dimensional base $\mathcal{B}$) where $\mathcal{B}$ is the Veronese surface in $\mathbb{P}^5$, and that certain deformations of the superpotential can drastically alter this special geometry, including reductions to $\mathbb{C}$ or to lower-dimensional cones. The authors propose that such special vacuum geometry serves as a phenomenological selection principle and may carry signatures of underlying string physics, motivating a broader program to compute the full MSSM moduli space and explore UV completions.

Abstract

We propose a new guiding principle for phenomenology: special geometry in the vacuum space. New algorithmic methods which efficiently compute geometric properties of the vacuum space of N=1 supersymmetric gauge theories are described. We illustrate the technique on subsectors of the MSSM. The fragility of geometric structure that we find in the moduli space motivates phenomenologically realistic deformations of the superpotential, while arguing against others. Special geometry in the vacuum may therefore signal the presence of string physics underlying the low-energy effective theory.