Anomalous (3+1)d Fermionic TQFTs Enable New Insights into Global Anomalies and Symmetry

The fundamental symmetries governing particle interactions can encounter subtle obstructions, known as anomalies, that threaten the consistency of physical theories, and researchers are actively seeking ways to resolve these issues. Zheyan Wan from the Beijing Institute of Mathematical Sciences and Applications, along with Juven Wang from the London Institute for Mathematical Sciences, and their colleagues investigate how these anomalies manifest in theories describing fundamental particles called Weyl fermions, potentially offering insights into the behaviour of materials like Weyl semimetals. Their work demonstrates a method to eliminate these anomalies by constructing novel topological quantum field theories, which describe interactions through extended phenomena rather than point-like particles, effectively replacing problematic fermions with a topologically ordered system. This achievement is particularly significant because it provides a potential explanation for anomalies observed within the Standard Model of particle physics, suggesting the existence of a hidden, “dark sector” that resolves inconsistencies without requiring the commonly invoked mechanism of spontaneous symmetry breaking.

Potentially cancelled anomalies arise in symmetry-preserving topological quantum field theories, which contain no local point-like particles, but only extended operators like lines and surfaces. This work focuses on the mixed gauge-gravitational nonperturbative global anomaly of Weyl fermions, relevant to both particle physics and condensed matter systems, charged under a discrete abelian internal symmetry in four-dimensional spacetime. These fermions form a spacetime-internal symmetry, and the research determines the minimal gauge group for this anomalous symmetry.

Topological Phases, Anomalies, and Mathematical Structures

This work establishes a deep connection between topological phases of matter, anomalies in physics, and mathematical concepts like cobordisms and K-theory. Topological phases are exotic states of matter exhibiting properties beyond traditional physics, characterized by topological order and robust behavior against local disturbances. Anomalies arise when a symmetry present in a classical theory is not preserved when transitioning to quantum mechanics, potentially causing dramatic consequences like particle instability. Cobordisms, a mathematical framework for studying manifolds, provide a powerful tool for classifying topological phases and understanding anomalies.

The research explores how discrete symmetries and their anomalies impact particle stability and the existence of dark matter, aiming to understand the structure of the Standard Model of particle physics and potential extensions. The study details anomalies within the Standard Model, specifically how discrete symmetries can be broken, and emphasizes the role of cobordisms in classifying these anomalies. It explains how cobordisms are used to organize and understand different types of topological order, focusing on fermionic symmetry-protected topological phases, which exhibit unique boundary properties and are connected to anomalies. The Dai-Freed theorem, a crucial result relating anomalies to topological phase classification, is applied to various physical systems.

The implications for proton stability are investigated, as anomalies could lead to proton decay, and extending the Standard Model with new particles and symmetries can resolve these anomalies. The research proposes that anomalies may also play a role in dark matter, with dark matter particles potentially stabilized by symmetries protected by topological order. Cosmological implications, including leptogenesis and topological defects contributing to dark matter, are also explored. The work introduces categorical symmetry, a refined description of symmetries using category theory, and delves into mathematical tools like eta invariants and K-theory to rigorously classify topological phenomena.

Recent developments and collaborations are highlighted, including research on boundary and symmetry-enriched topological order. Key concepts are simplified, defining topological order as invariance under continuous deformations, anomalies as symmetry breakdowns at the quantum level, and SPT phases as symmetry-protected topological phases with gapless boundary states. This research offers new insights into particle physics, potential for new materials, dark matter candidates, and cosmological models, highlighting the power of mathematical tools in solving physics problems.

Symmetry Anomalies Resolved by Topological Field Theories

Scientists have discovered a profound connection between fundamental symmetries in particle physics and topological quantum field theories, advancing understanding of anomalies in quantum field theories. The work centers on discrete symmetries, which can exhibit nonperturbative anomalies, subtle inconsistencies arising from quantum effects, and how these anomalies can be resolved using topological field theories lacking conventional particle interactions but possessing extended excitations. Researchers determined the minimal gauge group necessary for anomalous symmetry-preserving topological field theories, capable of canceling anomalies through a symmetry-extension construction. Experiments revealed that the Standard Model, considering 15 Weyl fermions per family and excluding a sterile neutrino, suffers from mixed gauge-gravitational global anomalies between baryon and lepton number symmetries and spacetime diffeomorphisms.

The team identified a corresponding minimal gauge fermionic topological quantum field theory that successfully cancels these anomalies, offering an alternative description of the Standard Model’s missing Weyl fermions. This TQFT can be interpreted as a gapped, topologically ordered dark sector, replacing the missing fermions without requiring conventional symmetry-breaking mechanisms. The study demonstrates that this alternative boundary state represents a possible deformation of a three-dimensional Weyl fermion or semimetal with discrete symmetries, offering a quantum phase transition pathway preserving the ‘t Hooft anomaly. Calculations show the anomaly index for the Standard Model is precisely −3 + nνR, where nνR represents the number of right-handed neutrinos. The research establishes a framework for classifying these anomalies using bordism groups and the TP group, providing a powerful tool for analyzing symmetry structures in quantum field theories and condensed matter systems. This breakthrough delivers a novel approach to resolving anomalies and opens new avenues for exploring the fundamental symmetries governing the universe.

Symmetries, Anomalies, and Emergent Topological Order

This research establishes a connection between anomalies in fundamental symmetries and the emergence of topological order, potentially offering insights into the nature of dark matter. Scientists investigated how certain symmetries, present in particle physics, can exhibit inconsistencies at a fundamental level, known as anomalies. They demonstrate that these anomalies can be resolved by introducing a topological field theory, which describes physics governed by extended objects rather than point-like particles. The team focused on anomalies arising from the interplay between internal symmetries and spacetime symmetries.

Applying this framework to the Standard Model, they identified anomalies related to baryon and lepton number symmetries and propose a novel solution involving a topologically ordered “dark sector”. This dark sector, described by the developed topological field theory, effectively replaces missing Weyl fermions within the Standard Model, avoiding the need for conventional symmetry-breaking mechanisms. The authors acknowledge that their results depend on specific assumptions about the underlying symmetries and the number of Weyl fermions considered. Future work could explore the implications for specific models of dark matter and investigate the potential for experimental signatures of the predicted topological order. They also note that the complexity of calculations limits the scope of analysis to relatively simple symmetry groups, and extending the approach to more intricate scenarios remains a challenge.