Multi-instantons and Multi-Axions in 4d QFT Enable Description of Non-Invertible Symmetries

The fundamental symmetries governing particle interactions represent a cornerstone of modern physics, and recent theoretical work explores symmetries that are more subtle than previously understood. Sungwoo Hong, Hyungyu Kim, and Sung Mook Lee, alongside Dongmin Seo, all from the Korea Advanced Institute of Science and Technology and CERN, investigate non-invertible symmetries in four-dimensional field theories, extending existing knowledge to encompass scenarios with multiple instantons and axions. This research establishes a framework that unifies diverse theoretical constructions and provides a concrete, Lagrangian-based description of these symmetries, offering insights into both their mathematical foundations and potential applications in particle phenomenology. By analysing how these symmetries act on fundamental objects like strings and lines, the team reveals a deeper understanding of the constraints governing particle interactions and opens new avenues for exploring physics beyond the Standard Model.

To generalise existing discussions to theories with multiple instantons and axions, and to make the subject more accessible to particle phenomenology, this work identifies the three-dimensional topological quantum field theories required to describe non-invertible symmetries in the presence of multi-instanton effects. Building on both the Adler-Bell-Jackiw anomaly construction and the half-space gauging approach, the researchers introduce a method termed partial gauging. They demonstrate that partial gauging of three-dimensional Chern-Simons theories naturally leads to anomaly inflow actions with general ABJ anomaly matrices, and generalized the half-space gauging construction to accommodate multiple gauge sectors, successfully computing correlation functions of boundary line operators and analyzing how non-invertible operators act on various species of ’t Hooft lines, interpreting the results in terms of the Witten effect.

Non-Invertible Symmetries and Topological Quantum Field Theories

Current research increasingly focuses on non-invertible symmetries, a relatively new area in quantum field theory where symmetries do not always behave in the traditional way. These symmetries involve unique fusion rules governing how charges combine, and can arise from topological defects like domain walls and cosmic strings. The mathematical framework of category theory provides a powerful language for describing these complex symmetries, closely linked to topological quantum field theories, which offer a mathematical framework for understanding systems with topological order, and the interplay between these theories and quantum anomalies is a key area of investigation. Researchers are exploring the connection between anomalies and anomaly inflow, where anomalies on a boundary can affect the bulk theory. Axions, hypothetical particles proposed as dark matter candidates and linked to the strong CP problem in particle physics, also feature prominently in this research. Studies investigate various axion models, their interactions with other particles, and the topological defects they can form, revealing a growing interest in non-invertible symmetries, suggesting they may be more common and important than previously thought.

Partial Gauging and Anomaly Inflow in Field Theories

Scientists are developing a new framework for understanding non-invertible symmetries in four-dimensional field theories, extending existing approaches to encompass theories with multiple instantons and axions. This work builds upon established methods like the Adler-Bell-Jackiw anomaly construction and half-space gauging, identifying the three-dimensional field theories necessary to describe non-invertible symmetries even when multi-instanton effects are present. A novel method, termed “partial gauging”, was introduced, demonstrating that partial gauging of three-dimensional Chern-Simons theories naturally leads to anomaly inflow actions with general ABJ anomaly matrices. For theories featuring multiple axions, researchers determined both non-invertible zero-form and one-form symmetries, as well as their actions on axion strings and ’t Hooft lines, effectively generalizing the established notion of the non-invertible Gauss law.

This framework unifies previously disparate constructions, providing a concrete, Lagrangian-based formulation of non-invertible symmetries, and extends naturally to theories with multiple instantons and axions, which are relevant to both theoretical and phenomenological investigations. The research demonstrates that the anomaly structure required to cancel four-dimensional ABJ anomalies can be realized using concrete three-dimensional topological quantum field theories, specifically Chern-Simons and BF theories, with partial gauging of a subgroup of the one-form symmetry. This approach provides a clear Lagrangian description of the topological quantum field theory, making the structure and properties of non-invertible symmetries more transparent, and may also find broader applications in beyond-the-Standard-Model scenarios, offering new avenues for exploring fundamental physics.

Partial Gauging And Anomaly Inflow Actions

This work presents a comprehensive framework for understanding non-invertible symmetries in four-dimensional quantum field theories, extending previous research to encompass theories with multiple instantons and axions. The team developed a method called partial gauging, which successfully describes these symmetries and naturally leads to anomaly inflow actions consistent with general anomaly structures. This approach unifies previously disparate constructions and provides a concrete, Lagrangian-based formulation for exploring these complex symmetries. They also determined the non-invertible symmetries present in theories with multiple axions, including their effects on axion strings and topological lines, effectively broadening the concept of the non-invertible Gauss law. While the analysis successfully extends to theories with multiple instantons and axions, the authors acknowledge that the full implications for beyond-the-Standard-Model scenarios require further investigation, suggesting future research directions include a more detailed exploration of these symmetries in realistic particle physics models and a deeper understanding of their connections to topological quantum field theories.