[Paper Review] Anomaly and Cobordism Constraints Beyond the Standard Model: Topological Force

Juven Wang|arXiv (Cornell University)|Jun 30, 2020

Noncommutative and Quantum Gravity Theories148 references21 citations

TL;DR

This paper proposes that anomaly and cobordism constraints in 4d chiral gauge theories—beyond standard perturbative anomalies—reveal a hidden gapped sector in the Standard Model and SU(5) Grand Unified Theories. Using a cobordism classification of invertible quantum anomalies (including nonperturbative global anomalies), it identifies a new topological force mediated by a 5d invertible TQFT or 4d non-invertible TQFT, which stabilizes right-handed neutrinos via topological mass and potentially explains dark matter and neutrino masses without sterile neutrinos.

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)$ imes$SU(2)$ imes$U(1))/$\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.

Motivation & Objective

• To extend anomaly constraints beyond perturbative local anomalies to include global and nonperturbative anomalies via cobordism theory.
• To resolve long-standing issues in the Standard Model and Georgi-Glashow SU(5) GUT, such as the need for sterile neutrinos and dark matter, using topological field theory.
• To identify new dynamical sectors—specifically 4d non-invertible TQFTs or 5d invertible TQFTs—that emerge from anomaly cancellation in chiral gauge theories.
• To unify quantum gravity, topological order, and gauge theories by proposing a 'Topological Force' mediated by higher-dimensional TQFTs.
• To show that right-handed neutrinos are not truly sterile but couple to a new Z4,X gauge symmetry, enabling topological mass generation and new dark matter candidates.

Proposed method

• Applies cobordism classification of 4d and 5d invertible TQFTs to classify all possible anomalies (local and global) in chiral gauge theories.
• Uses characteristic classes (e.g., c1(U(1)), c2(SU(2)), c2(SU(3)), c3(SU(3))) and secondary invariants (e.g., η-invariant, Arf invariant) to compute anomalies in 5d and 6d Chern-Simons terms.
• Analyzes Z16, Z4, and Z2 anomalies from Witten's SU(2) anomaly and related global anomalies via the η-invariant and Pin+ bordism groups.
• Constructs symmetry extensions (e.g., Spin × Z4,X) and gauges discrete symmetries to realize anomaly-matching conditions in the infrared.
• Proposes a 5d invertible TQFT (iTQFT) and 4d non-invertible TQFT as low-energy effective theories, with defects hosting Majorana zero modes.
• Introduces a 'Topological Force' as a new long-range interaction mediated by topological order and extended objects (strings, vortices), linking 4d and 5d sectors.

Experimental results

Research questions

• RQ1 What are the full set of anomaly constraints—beyond perturbative local anomalies—that apply to chiral gauge theories like the SM and SU(5) GUT?
• RQ2 How can global and nonperturbative anomalies (e.g., Z16, Z4, Z2) in 4d chiral fermion systems lead to new, hidden gapped quantum phases?
• RQ3 Can the right-handed neutrino sector be stabilized via topological mass rather than Dirac/Majorana mass, and what are the implications for neutrino masses and dark matter?
• RQ4 What is the role of the discrete X = 5(B−L)−4Y symmetry in anomaly matching, and how does its gauging lead to a 5d iTQFT and topological force?
• RQ5 How can 4d and 5d TQFTs mediate a new 'Topological Force' that unifies quantum gravity, topological order, and gauge dynamics?

Key findings

• The paper identifies a new hidden gapped sector in the Standard Model and SU(5) GUT, composed of either a 4d non-invertible TQFT or a 5d invertible TQFT, arising from anomaly cancellation in the Z16, Z4, and Z2 classes.
• For SU(3)×SU(2)×U(1)Zq SM with q=1,2,3,6 and SU(5) GUT with 15n chiral Weyl fermions and preserved X=5(B−L)−4Y, the Z16 anomaly index ν=−3 mod 16 is canceled by a 4d TQFT with ν=2 or 4, or a 5d iTQFT with ν=−2.
• The right-handed neutrinos are not sterile to the Z4,X gauge field, as they carry odd Z4,X charge, enabling topological mass generation via vortex defects in the TQFT.
• The 4d TQFT hosts Majorana zero modes bound to vortices, which can be probed via braiding statistics and link invariants, offering a topological quantum computing pathway.
• A 5d invertible TQFT emerges above the GUT scale, with a ν=−2 anomaly index, and mediates a new 'Topological Force' that entangles 4d and 5d sectors via topological order.
• The model provides a candidate for dark matter: heavy, extended topological excitations (string-like) in the TQFT sector with masses near the GUT scale, potentially explaining the abundance of dark matter.