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Anomalies in family unification models from bordism classification

Tsubasa Sugeno, Hiroki Wada

Abstract

We study anomalies in family unification models within the framework of the bordism classification of invertible field theories. These models are based on four-dimensional $\mathcal{N}=1$ supersymmetric nonlinear sigma models, in which the three generations of quarks and leptons arise as superpartners of the sigma model fields. We focus on models whose target spaces are constructed from the exceptional group $E_{7}$ and its subgroups. For the consistency of the theory, sigma model anomalies must be cancelled. We show the absence of global sigma model anomalies, which are encoded in the torsion part of the relevant bordism groups, by explicitly computing these groups using the Atiyah-Hirzebruch spectral sequence. In constructing family unification models, symmetries acting on the coset spaces are gauged, which may introduce additional anomalies. We identify the relevant bordism groups in this setting and demonstrate that no global anomalies arise when the isotropy subgroup of the coset space is gauged.

Anomalies in family unification models from bordism classification

Abstract

We study anomalies in family unification models within the framework of the bordism classification of invertible field theories. These models are based on four-dimensional supersymmetric nonlinear sigma models, in which the three generations of quarks and leptons arise as superpartners of the sigma model fields. We focus on models whose target spaces are constructed from the exceptional group and its subgroups. For the consistency of the theory, sigma model anomalies must be cancelled. We show the absence of global sigma model anomalies, which are encoded in the torsion part of the relevant bordism groups, by explicitly computing these groups using the Atiyah-Hirzebruch spectral sequence. In constructing family unification models, symmetries acting on the coset spaces are gauged, which may introduce additional anomalies. We identify the relevant bordism groups in this setting and demonstrate that no global anomalies arise when the isotropy subgroup of the coset space is gauged.

Paper Structure

This paper contains 49 sections, 165 equations, 1 figure, 18 tables.

Table of Contents

  1. Introduction and summary
  2. Anomalies in sigma model
  3. Review of sigma model anomalies
  4. Classification of anomalies in gauged sigma models
  5. Family unification model
  6. Conventions of Lie algebra and Lie group
  7. Lie algebra and its subalgebras
  8. Subgroups and in
  9. Construction of the family unification model
  10. model
  11. model
  12. Constraints on representations and charges
  13. Anomalies in family unification models
  14. Anomalies in model
  15. Sigma model anomalies in model
  16. ...and 34 more sections

Figures (1)

  • Figure 1: The Dynkin diagram of $\mathfrak{e}_{7}$. The numbering of the simple roots is the same as Francesco:1997.