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Vacuum moduli space of the minimal supersymmetric standard model

Yang-Hui He1,2,*, Vishnu Jejjala3,†, Brent D. Nelson4,‡, Hal Schenck5,6,§, and Michael Stillman7,6,∥

  • *Contact author: yh@lims.ac.uk
  • †Contact author: v.jejjala@wits.ac.za
  • ‡Contact author: b.nelson@northeastern.edu
  • §Contact author: hks0015@auburn.edu
  • ∥Contact author: mike@math.cornell.edu
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Phys. Rev. D 114, 035013 – Published 10 August, 2026

DOI: https://doi.org/10.1103/vkmb-4jnn

Abstract

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 [Y.-H. He et al., The geometry of generations, Commun. Math. Phys. 339, 149 (2015)] computed the vacuum moduli space of the electroweak sector; geometrically this corresponds to studying a restriction of ϕ: C13→ϕresC22. We analyze the geometry of the full vacuum moduli space for superpotentials Wminimal (without neutrinos) and WMSSM (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 Wminimal we show they have, respectively, dimensions 1, 15, and 29, and prove that each of the Mi is a rational variety, while for WMSSM we show that M3 is the only component. Restricting the Mi to the electroweak sector, we recover the results in He et al. [The geometry of generations, Commun. Math. Phys. 339, 149 (2015)]. We describe the components of the vacuum moduli space geometrically in terms of incidence varieties to a product of Segrè varieties.

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Phys. Rev. D 114, 035013– Published 10 August, 2026
  • Received 20 April 2026
  • Accepted 8 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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