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

Zheyan Wan, Juven Wang

TL;DR

This work develops a complete cobordism-based classification of invertible anomalies and topological terms for SM, GUT, and BSM theories in four dimensions. By leveraging the Freed–Hopkins correspondence between iTQFTs and cobordism groups, and computing via Madsen–Tillmann–Thom spectra and the Adams spectral sequence, the authors obtain detailed Ω^G_d and TP_d(G) for a range of spacetime/internal symmetry groups, including various quotients of the SM gauge group and several GUTs. The results reveal new 4d anomaly structures tied to discrete symmetries (e.g., Z16, Z4, Z2) and illuminate how perturbative and global anomalies extend beyond familiar Witten-type cases, with concrete invariant generators and topological terms (Chern–Simons, η-invariants, Rokhlin invariants) across SM and GUT embeddings. This bottom-up cobordism perspective provides constraints on quantum dynamics, suggests possible extended defects or hidden BSM sectors, and complements high-energy Swampland/Quantum Gravity viewpoints by offering a lattice-regularizable, bottom-up framework for anomaly matching and topological order in particle physics models.

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 $Ω^{G}_d$ 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.

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

TL;DR

This work develops a complete cobordism-based classification of invertible anomalies and topological terms for SM, GUT, and BSM theories in four dimensions. By leveraging the Freed–Hopkins correspondence between iTQFTs and cobordism groups, and computing via Madsen–Tillmann–Thom spectra and the Adams spectral sequence, the authors obtain detailed Ω^G_d and TP_d(G) for a range of spacetime/internal symmetry groups, including various quotients of the SM gauge group and several GUTs. The results reveal new 4d anomaly structures tied to discrete symmetries (e.g., Z16, Z4, Z2) and illuminate how perturbative and global anomalies extend beyond familiar Witten-type cases, with concrete invariant generators and topological terms (Chern–Simons, η-invariants, Rokhlin invariants) across SM and GUT embeddings. This bottom-up cobordism perspective provides constraints on quantum dynamics, suggests possible extended defects or hidden BSM sectors, and complements high-energy Swampland/Quantum Gravity viewpoints by offering a lattice-regularizable, bottom-up framework for anomaly matching and topological order in particle physics models.

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 free classes, and (2) non-perturbative/global anomalies, classified by finite group 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 and their invertible topological field theory invariants, which characterize d topological terms and d anomalies, protected by the following symmetry group : for SM with ; or for SO(10) or SO(18) GUT as ; for Georgi-Glashow SU(5) GUT as ; for Pati-Salam GUT as ; and others. For SM with an extra discrete symmetry, we obtain new anomaly matching conditions of , , and 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.

Paper Structure

This paper contains 41 sections, 1 theorem, 109 equations, 50 figures, 30 tables.

Table of Contents

  1. Introduction
  2. Physics Guide
  3. Mathematics Preliminary
  4. Definition of bordism groups
  5. Spectra
  6. Adams spectral sequence
  7. Characteristic classes
  8. Lie algebra to Lie group and the representation theory
  9. Standard Models
  10. ${{\rm Spin}\times {{\rm SU}(3)\times {\rm SU}(2)\times {\rm U}(1)}}$ model
  11. ${{\rm Spin}\times \frac{{\rm SU}(3)\times {\rm SU}(2)\times {\rm U}(1)}{\mathbb{Z}_2}}$ model
  12. ${{\rm Spin}\times \frac{{\rm SU}(3)\times {\rm SU}(2)\times {\rm U}(1)}{\mathbb{Z}_3}}$ model
  13. ${{\rm Spin}\times \frac{{\rm SU}(3)\times {\rm SU}(2)\times {\rm U}(1)}{\mathbb{Z}_6}}$ model
  14. Comparison between Adams spectral sequence and Atiyah-Hirzebruch spectral sequence
  15. Standard Models with additional discrete symmetries
  16. ...and 26 more sections

Key Result

Lemma 1

Given a short exact sequence of $\mathcal{A}_2(1)$-modules then for any $t$, there is a long exact sequence

Figures (50)

  • Figure 1: Bordism between manifolds. Here $M$ and $M'$ are two closed $d$-manifolds, $N$ is a compact $d+1$-manifold whose boundary is the disjoint union of $M$ and $M'$, so $\partial N= M \sqcup M'$. If there are additional $G$-structures on these manifolds, then the $G$-structure on $N$ is required to be compatible with the $G$-structures on $M$ and $M'$. If there are additional maps from these manifolds to a fixed topological space, then the maps are also required to be compatible with each other. If these conditions are satisfied, then $M$ and $M'$ are called bordant and $N$ is called a bordism between $M$ and $M'$. This relation is an equivalence relation.
  • Figure 3: The $\mathcal{A}_2(1)$-module structure of $\operatorname{H}^*(\mathrm{B}({\rm SU}(3)\times {\rm SU}(2)\times {\rm U}(1)),\mathbb{Z}_2)$ below degree 6.
  • Figure 4: $\Omega_*^{{\rm Spin}\times {\rm SU}(3)\times {\rm SU}(2)\times {\rm U}(1)}$.
  • Figure 5: The $\mathcal{A}_2(1)$-module structure of $\operatorname{H}^*(\mathrm{B}({\rm SU}(3)\times {\rm U}(2)),\mathbb{Z}_2)$ below degree 6.
  • Figure 6: $\Omega_*^{{\rm Spin}\times {\rm SU}(3)\times {\rm U}(2)}$.
  • ...and 45 more figures

Theorems & Definitions (1)

  • Lemma 1: Lemma 11 of WanWang2018bns1812.11967