Anomalies in family unification models from bordism classification

TL;DR

This paper investigates anomalies in family unification models using bordism classification, focusing on supersymmetric sigma models with target spaces related to the exceptional group E7, and confirms anomaly cancellation through explicit bordism group computations.

Contribution

It provides a bordism-based analysis of anomalies in family unification models, demonstrating the absence of global sigma model anomalies and additional anomalies when gauging symmetries.

Findings

No global sigma model anomalies are present, as shown by bordism group calculations.

Gauging symmetries on coset spaces does not introduce additional anomalies.

Explicit computation of bordism groups confirms anomaly cancellation in these models.

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.