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The Geometry of Generations

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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.

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Authors and Affiliations

  1. Department of Mathematics, City University London, London, EC1V 0HB, UK

    Yang-Hui He & Cyril Matti

  2. School of Physics, NanKai University, Tianjin, 300071, People’s Republic of China

    Yang-Hui He

  3. Merton College, University of Oxford, Oxford, OX1 4JD, UK

    Yang-Hui He

  4. Centre for Theoretical Physics, NITheP, and School of Physics, University of the Witwatersrand, WITS, Johannesburg, 2050, South Africa

    Vishnu Jejjala

  5. Department of Physics, Northeastern University, Boston, MA, 02115, USA

    Brent D. Nelson

  6. ICTP, Strada Costiera 11, 34014, Trieste, Italy

    Brent D. Nelson

  7. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    Michael Stillman

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  1. Yang-Hui He
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  2. Vishnu Jejjala
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  3. Cyril Matti
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  4. Brent D. Nelson
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  5. Michael Stillman
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Correspondence to Yang-Hui He.

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Communicated by N. A. Nekrasov

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He, YH., Jejjala, V., Matti, C. et al. The Geometry of Generations. Commun. Math. Phys. 339, 149–190 (2015). https://doi.org/10.1007/s00220-015-2416-7

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