The Geometry of Generations

TL;DR

This paper explores the algebraic geometric structure of particle generations within the MSSM, revealing complex geometric signatures that depend on the number of generations and offering insights into why three generations are unique.

Contribution

It introduces a novel algebraic geometric analysis of the MSSM's vacuum moduli space across multiple generations, highlighting geometric signatures related to the number of generations.

Findings

Moduli space exhibits Calabi-Yau, Grassmannian, and toric features.

Geometric signatures vary with the number of generations.

Three generations may have special geometric significance.

Abstract

We present an intriguing and precise interplay between algebraic geometry and the phenomenology of generations of particles. Using the electroweak sector of the MSSM as a testing ground, we compute the moduli space of vacua as an algebraic variety for multiple generations of Standard Model matter and Higgs doublets. The space is shown to have Calabi-Yau, Grassmannian, and toric signatures which sensitively depend on the number of generations of leptons, as well as inclusion of Majorana mass terms for right-handed neutrinos. We speculate as to why three generations is special.