Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms (1910.14668v4)

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.

Anomalies and Topological Terms in Particle Physics via Cobordism Theory

This essay discusses an advanced paper in theoretical physics and mathematics conducted by Zheyan Wan and Juven Wang. It explores the cobordism groups associated with certain particle physics models, specifically Standard Models (SM), Grand Unified Theories (GUTs), and Beyond Standard Models (BSM), processing through anomalies and topological terms. The research merges quantum field theory aspects with topological insights to provide a structured analysis of anomalies relevant to high-energy physics.

Overview

The paper classifies all invertible anomalies and topological terms connected with various models like SMs, GUTs, and BSMs. Anomalies in a physical context refer to symmetry violations in quantum theories, including perturbative local anomalies arising from Feynman diagram loop calculations and non-perturbative global anomalies associated with torsion classes like $\mathbb{Z}_N$. Cobordism theory offers the mathematical structure to address these challenges.

The approach leverages the Freed-Hopkin's framework and generalizations to describe a one-to-one correspondence between invertible topological quantum field theories (iTQFTs) with symmetry and cobordism groups. For instance, the group $TP_d(G)$ is computed by examining bordism groups $\Omega_d^G$. These computations involve understanding topological spaces and their interactions through classifying spaces, spectra, and the Adams spectral sequence.

Key Results

Standard Models

1. Gauge Structure of SMs: For groups ${\times \frac{(3)\times (2)\times (1)}{_q}}$, various cobordism groups are computed for different $q$. Results show that $q=1$ focuses on signatures and Chern classes as key invariants, with potential anomalies related to Witten's $\mathbb{Z}_2$ classes. For $q=2$, these anomalies transform into local perturbation anomalies due to a subtler internal symmetry consideration.
2. Discrete Symmetries in SMs: Extending the SM to include discrete symmetries such as ${\times_{_2} _4}$ entails examining new anomaly conditions across $Z_4$, $Z_2$, and $Z_6$ symmetry implementations. These symmetries introduce subtle variations in anomaly presence concerning topological terms and reflection symmetries.

Grand Unified Theories

1. Pati-Salam and SO(n) GUTs: For high-dimensional GUT models, such as SO(10) or SO(18), cobordism computations reveal complex interactions among Stiefel-Whitney, Pontryagin, and Euler classes, indicating particular $\mathbb{Z}_2$ anomaly cancelations when symmetry groups are altered (e.g., reduction to U(2) from SU(2)).
2. SU(n) Unification: The SU(5) context showcases sophisticated algebraic topology where anomalies often bundled with Chern-Simons terms reflect the interaction between geometry and quantum physics. Chern classes pertinent to SU(5) were analyzed, highlighting incremented anomaly structures across dimensions.

Implications

The breadth of these findings suggests profound implications for theoretical physics, especially concerning anomaly matching, quantum gravity considerations, and string theory applications. Identifying and linking anomalies with topological terms could impact the construction of consistent quantum field theories in higher dimensions. Furthermore, the paper insinuates possible non-perturbative lattice gauge constructions in GUT models and their nuanced symmetry reconstructions under cobordism inference.

Conclusion

This rigorous research provides a mathematical foundation via cobordism theory to categorize and characterize complex anomalies and topological terms in particle physics. It unites elements of algebraic topology with quantum field theory, offering profound insights into anomaly implications across different models. The methodology serves as a potential gateway for further exploration of consistent high-energy physical theories, ensuring their compatibility within holistic quantum frameworks. This strengthens the union of theoretical physics and advanced mathematical topology, extending our understanding of the fabrications underpinning the universe.