2025/06/16 by Yang‐Hui He, Vishnu Jejjala, He, Yang-Hui +7
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Phenomenology (hep-ph) #High Energy Physics - Theory (hep-th) #Quantum Mechanics and Applications #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2506.13868
openalex publication_date 2025/06/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
A starting point in the study of the minimal supersymmetric Standard Model (MSSM) is the vacuum moduli space, which is a highly complicated algebraic variety: it is the image of an affine variety X ⊂ ℂ49 under a symplectic quotient map ϕ to ℂ973. Previous work computed the vacuum moduli space of the electroweak sector; geometrically this corresponds to studying a restriction of ϕ: ℂ13 \stackrelϕres\longrightarrow ℂ22. We analyze the geometry of the full vacuum moduli space for superpotentials W\rm minimal (without neutrinos) and W\rm MSSM (with neutrinos) in ℂ973. In both cases, we prove that X consists of three irreducible components X1, X2, and X3, and determine the images Mi of the Xi under ϕ. For W\rm minimal we show they have, respectively, dimensions 1, 15, and 29, and prove that each of the Mi is a rational variety, while for W\rm MSSM we show that M3 is the only component. Restricting the Mi to the electroweak sector, we recover known results. We describe the components of the vacuum moduli space geometrically in terms of incidence varieties to a product of Segre varieties.