Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms

TL;DR

This paper classifies all invertible anomalies and topological terms in various Standard Model and GUT frameworks using advanced mathematical tools, revealing new anomaly conditions and constraints on quantum gauge dynamics.

Contribution

It provides a comprehensive mathematical classification of anomalies and topological terms in BSM physics, introducing new anomaly matching conditions and topological constraints.

Findings

Computed bordism groups and topological invariants for various symmetry groups.

Discovered new anomaly matching conditions beyond the Witten anomaly.

Proposed constraints on quantum gauge dynamics and potential new topological sectors.

Abstract

We classify and characterize all invertible anomalies and all allowed topological terms related to various Standard Models (SM), Grand Unified Theories (GUT), and Beyond Standard Model (BSM) physics. By all anomalies, we mean the inclusion of (1) perturbative/local anomalies captured by perturbative Feynman diagram loop calculations, classified by $\mathbb{Z}$ free classes, and (2) non-perturbative/global anomalies, classified by finite group $\mathbb{Z}_N$ torsion classes. Our work built from [arXiv:1812.11967] fuses the math tools of Adams spectral sequence, Thom-Madsen-Tillmann spectra, and Freed-Hopkins theorem. For example, we compute bordism groups $\Omega^{G}_d$ and their invertible topological field theory invariants, which characterize $d$d topological terms and $(d-1)$d anomalies, protected by the following symmetry group $G$: $Spin\times \frac{SU(3)\times SU(2)\times U(1)}{\mathbb{Z}_q}$ for SM with $q=1,2,3,6$; $\frac{Spin \times Spin(n)}{\mathbb{Z}_2^F}$ or $Spin \times Spin(n)$ for SO(10) or SO(18) GUT as $n=10, 18$; $Spin \times SU(n)$ for Georgi-Glashow SU(5) GUT as $n=5$; $\frac{Spin\times \frac{SU(4)\times(SU(2)\times SU(2))}{\mathbb{Z}_{q'}}}{\mathbb{Z}_2^F}$ for Pati-Salam GUT as $q'=1,2$; and others. For SM with an extra discrete symmetry, we obtain new anomaly matching conditions of $\mathbb{Z}_{16}$, $\mathbb{Z}_{4}$, and $\mathbb{Z}_{2}$ classes in 4d beyond the familiar Witten anomaly. Our approach offers an alternative view of all anomaly matching conditions built from the lower-energy (B)SM or GUT, in contrast to high-energy Quantum Gravity or String Theory Landscape v.s. Swampland program, as bottom-up/top-down complements. Symmetries and anomalies provide constraints of kinematics, we further suggest constraints of quantum gauge dynamics, and new predictions of possible extended defects/excitations plus hidden BSM non-perturbative topological sectors.