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Anomaly and Cobordism Constraints Beyond the Standard Model: Topological Force

Juven Wang

TL;DR

The paper uses Freed–Hopkins cobordism to classify invertible quantum anomalies and applies this framework to Standard Model and GUT constraints, including extensions with discrete symmetries. It shows that familiar perturbative anomalies cancel for SM and SU(5) GUT content, while a residual $\mathbb{Z}_{16}$ global anomaly associated with a ${\mathbb{Z}_{4,X}}$ discrete symmetry can remain, potentially requiring new hidden sectors. To saturate this missing anomaly, the author proposes scenarios involving a 5d invertible TQFT or a 4d noninvertible TQFT (and symmetry extensions), suggesting a novel Topological Force that mediates interactions with hidden topological sectors and can address neutrino masses and dark matter. The Ultra Unification program combines SM/GUT with topological sectors and higher-dimensional bulk dynamics, drawing on 4d–5d theories of quantum gravity and TQFTs to yield a richer, anomaly-consistent landscape with experimentally testable implications for neutrinos, dark matter, and potentially new long-range topological interactions.

Abstract

Standard lore uses local anomalies to check the kinematic consistency of gauge theories coupled to chiral fermions, e.g. Standard Models (SM). Based on a systematic cobordism classification, we examine constraints from invertible quantum anomalies (including all perturbative local and nonperturbative global anomalies) for gauge theories. We also clarify the different uses of these anomalies: including (1) anomaly cancellations of dynamical gauge fields, (2) 't Hooft anomaly matching conditions of background fields of global symmetries, and others. We apply several 4d $\mathbb{Z}_{n}$ anomaly constraints of $n=16,4,2$ classes, beyond the familiar Feynman-graph perturbative $\mathbb{Z}$ class local anomalies. As an application, for (SU(3)$\times$SU(2)$\times$U(1))/$\mathbb{Z}_q$ SM (with $q=1,2,3,6$) and SU(5) Grand Unification with 15n chiral Weyl fermions and with a discrete baryon minus lepton number $X=5({\bf B}- {\bf L})-4Y$ preserved, we discover a new hidden gapped sector previously unknown to the SM and Georgi-Glashow model. The gapped sector at low energy contains either (1) 4d non-invertible topological quantum field theory (TQFT, above the energy gap with heavy fractionalized anyon excitations from 1d particle worldline and 2d string worldsheet, inaccessible directly from Dirac or Majorana mass gap of the 16th Weyl fermions [i.e., right-handed neutrinos], but accessible via a topological quantum phase transition), or (2) 5d invertible TQFT in extra dimensions. Above a higher energy scale, the discrete $X$ becomes dynamically gauged, the entangled Universe in 4d and 5d is mediated by Topological Force. Our model potentially resolves puzzles, surmounting sterile neutrinos and dark matter, in fundamental physics.

Anomaly and Cobordism Constraints Beyond the Standard Model: Topological Force

TL;DR

The paper uses Freed–Hopkins cobordism to classify invertible quantum anomalies and applies this framework to Standard Model and GUT constraints, including extensions with discrete symmetries. It shows that familiar perturbative anomalies cancel for SM and SU(5) GUT content, while a residual global anomaly associated with a discrete symmetry can remain, potentially requiring new hidden sectors. To saturate this missing anomaly, the author proposes scenarios involving a 5d invertible TQFT or a 4d noninvertible TQFT (and symmetry extensions), suggesting a novel Topological Force that mediates interactions with hidden topological sectors and can address neutrino masses and dark matter. The Ultra Unification program combines SM/GUT with topological sectors and higher-dimensional bulk dynamics, drawing on 4d–5d theories of quantum gravity and TQFTs to yield a richer, anomaly-consistent landscape with experimentally testable implications for neutrinos, dark matter, and potentially new long-range topological interactions.

Abstract

Standard lore uses local anomalies to check the kinematic consistency of gauge theories coupled to chiral fermions, e.g. Standard Models (SM). Based on a systematic cobordism classification, we examine constraints from invertible quantum anomalies (including all perturbative local and nonperturbative global anomalies) for gauge theories. We also clarify the different uses of these anomalies: including (1) anomaly cancellations of dynamical gauge fields, (2) 't Hooft anomaly matching conditions of background fields of global symmetries, and others. We apply several 4d anomaly constraints of classes, beyond the familiar Feynman-graph perturbative class local anomalies. As an application, for (SU(3)SU(2)U(1))/ SM (with ) and SU(5) Grand Unification with 15n chiral Weyl fermions and with a discrete baryon minus lepton number preserved, we discover a new hidden gapped sector previously unknown to the SM and Georgi-Glashow model. The gapped sector at low energy contains either (1) 4d non-invertible topological quantum field theory (TQFT, above the energy gap with heavy fractionalized anyon excitations from 1d particle worldline and 2d string worldsheet, inaccessible directly from Dirac or Majorana mass gap of the 16th Weyl fermions [i.e., right-handed neutrinos], but accessible via a topological quantum phase transition), or (2) 5d invertible TQFT in extra dimensions. Above a higher energy scale, the discrete becomes dynamically gauged, the entangled Universe in 4d and 5d is mediated by Topological Force. Our model potentially resolves puzzles, surmounting sterile neutrinos and dark matter, in fundamental physics.

Paper Structure

This paper contains 46 sections, 80 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Overview on Standard Models, Grand Unifications, and Anomalies
  3. SM and GUT: local Lie algebras to global Lie groups, and representation theory
  4. Classifications of anomalies and their different uses
  5. Dynamical Gauge Anomaly Cancellation
  6. Summary of anomalies from a cobordism theory and Feynman diagrams
  7. U(1)${}_Y^3$: 4d local anomaly from 5d $\text{CS}_1^{{\mathrm{U}}(1)}c_1({\mathrm{U}}(1))^2$ and 6d $c_1({\mathrm{U}}(1))^3$
  8. U(1)${}_Y$-SU(2)${}^2$: 4d local anomaly from 5d $\text{CS}_1^{{\rm U}(1)}c_2({\rm SU}(2))$ and 6d $c_1({\rm U}(1))c_2({\rm SU}(2))$
  9. U(1)${}_Y$-SU(3)${}_c^2$: 4d local anomaly from 5d $\text{CS}_1^{{\rm U}(1)}c_2({\rm SU}(3))$ and 6d $c_1({\rm U}(1))c_2({\rm SU}(3))$
  10. U(1)$_Y$-(gravity)$^2$: 4d local anomaly from 5d $\mu(\text{PD}(c_1({\rm U}(1))))$ and 6d $\frac{c_1({\rm U}(1))(\sigma-{\rm F} \cdot {\rm F} )}{8}$
  11. SU(3)$_c^3$: 4d local anomaly from 5d $\frac{1}{2}{\text{CS}_5^{{\rm SU}(3)}}$ and 6d $\frac{1}{2}{c_3({\rm SU}(3))}$
  12. Witten SU(2) anomaly: 4d $\mathbb{Z}_2$ global anomaly from 5d $c_2({\rm SU}(2))\tilde{\eta}$ and 6d $c_2({\rm SU}(2))\text{Arf}$
  13. Anomaly Matching for SM and GUT with Extra $({\mathbf{B}- \mathbf{L}})$ Symmetries
  14. SM and GUT with extra continuous symmetries and a cobordism theory
  15. ($\bf{B}-\bf{L}$)-U(1)$_Y^2$: 4d local anomaly
  16. ...and 31 more sections

Figures (5)

  • Figure 1: Examples of perturbative local anomalies of $\mathbb{Z}$ classes in 4d that can be captured by the free part of cobordism group $\Omega^{d=5}_{G} \equiv \mathrm{TP}_{d=5}(G)$ in (\ref{['eq:TPG']}), which is in fact descended from the free part of bordism group $\Omega_{d=6}^{G}$. ( 1) Dynamical gauge anomaly. ( 2) 't Hooft anomaly of background (Backgrd.) fields. ( 3) Adler-Bell-Jackiw (ABJ Adler1969gkABJBell1969tsABJ) type of anomalies. ( 4) Anomaly that involves two background fields of global symmetries and one dynamical gauge field.
  • Figure 2: General nonperturbative global anomalies not captured by Feynman graphs can still be characterized by generic curved manifolds with mixed gauge and gravitational background probes. The figure shows a 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 $M'$, namely $\partial N= M \sqcup M'$. 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 obeyed, then $M$ and $M'$ are called bordant equivalence and $N$ is called a bordism between $M$ and $M'$.
  • Figure 3: Examples of dynamical gauge anomaly cancellations in SM. In fact, the 5 perturbative local anomalies from perturbative one-loop triangle Feynman diagrams precisely match anomaly classification of $\mathbb{Z}^5$ obtained from the cobordism group calculations in Table \ref{['table:SU3SU2U1']} and Ref. WW2019fxh1910.14668.
  • Figure 4: Examples of anomaly constraint for SM (or GUT) with extra symmetries such as ${{ \mathbf{B}- \mathbf{L}}}$ or $X \equiv 5({ \mathbf{B}- \mathbf{L}})-4Y$. We only show perturbative local anomalies from perturbative one-loop triangle Feynman diagrams discussed in (\ref{['eq:Spinccobordism']}). We will explore nonperturbative global anomalies (not captured by Feynman diagrams) in later sections. Assume the gravity contributes as background field: $\bullet$ If ${{ \mathbf{B}- \mathbf{L}}}$ or $X$ is not gauged, (i), (ii), (iii), and (vi) are ABJ anomalies of Fig. \ref{['fig:triangle']} ( 3) and Remark \ref{['remark:ABJanomaly']}; (iv) and (v) are 't Hooft anomalies of Fig. \ref{['fig:triangle']} ( 2) and Remark \ref{['remark:tHooftanomaly']}. $\bullet$ If ${{ \mathbf{B}- \mathbf{L}}}$ or $X$ is gauged, (i)-(iv), (vi) are dynamical gauge anomalies of Fig. \ref{['fig:triangle']} ( 1) and Remark \ref{['remark:dynamicalanomaly']}; (v) is an anomaly of Fig. \ref{['fig:triangle']} ( 4) and Remark \ref{['remark:4anomaly']}. If these anomalies are not matched, we can still saturate the anomalies by proposing new sectors appending to the QFT; we will explore those new physics in Sec. \ref{['sec:HiddenTopologicalSectors']}.
  • Figure 5: (a) The Whitehead link formed by a disjoint union of two-component 1-loops, $\ell = \ell_1 \sqcup \ell_2$, is detectable by the Sato-Levine invariant. (See related QFTs explored in Putrov2016qdo1612.09298PWYGuoJW1812.11959.) (b) A schematic nontrivial link configuration in a 4d spacetime (inside the large black circle) that can carry an odd $\mathbb{Z}_{4,X}$ charge (also an odd $\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_2^F}$ charge) measured by a codimension 1 operator $\star {\cal J}_{{\mathbb{Z}_2}}$ (from the 5d bulk perspective) or $\star {\cal J}_{{\mathbb{Z}_4}}$ (from the 4d boundary perspective). The $\mathbb{Z}_{4,X}$ charge is trapped non-locally within the surface link. The surface link $L = \Sigma_{(1)}^2 \sqcup \Sigma_{(2)}^2$ contains two surfaces obtained from a spun version of Whitehead link. Let a $D^3$ ball contain the Whitehead link $\ell$. The $\Sigma_{(1)}^2$ rotates the $\ell_1$ along an ${S^1}'$ of $D^3 \times {S^1}'$ outside the $D^3$. The $\Sigma_{(2)}^2$ rotates the $\ell_2$ along the blue dashed arrow around the $\ell_1$ circle within the $D^3$. From a 4d boundary theory perspective, we have the ${\cal A}_{{\mathbb{Z}_4}}$ along a 1-line $I^1$ (drawn in the red dashed) Poincaré dual (PD) to the $\star {\cal J}_{{\mathbb{Z}_4}}$. From a 5d bulk theory perspective, we have the ${\cal A}_{{\mathbb{Z}_2}}$ along a 1-line $I^1$ PD to the $\star {\cal J}_{{\mathbb{Z}_2}}$.