On gauging finite subgroups

Abstract: We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\Gamma$-symmetric theory. Depending on how anomalous $\Gamma$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=\Gamma/A$ and a higher-form symmetry $\hat A$ with a mixed anomaly, where $\hat A$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\hat A$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\hat A)$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

• The paper classifies the symmetry configurations arising from gauging finite subgroups based on the anomaly status of the original group.
• It employs algebraic topology and group theory to distinguish between anomaly-free, tame, and wild cases in emerging QFT symmetries.
• The study offers insights relevant to condensed matter physics and highlights the role of higher-categorical structures in advanced quantum theories.

Overview of "On Gauging Finite Subgroups"

Yuji Tachikawa's paper focuses on the properties of quantum field theories (QFTs) derived from gauging finite subgroups within larger symmetry groups, specifically addressing finite normal Abelian subgroups. This investigation is pertinent to quantum field theories that possess symmetries which might include anomalies, and it explores how these symmetries manifest in the theories that emerge from gauging scenarios.

The study concerns itself with the symmetry structures of the gauged theories (denoted as T/A), which can lead to several possible configurations depending on the anomaly status of the initial symmetry group $\Pi$. Tachikawa identifies three primary configurations:

1. Anomaly-Free $\Pi$: In scenarios where $\Pi$ is free of anomalies, the symmetries of T/A form a direct product between $G = \Pi/A$ and a higher-form symmetry $\hat{A}^{(D-2)}$, with a mixed anomaly determined by the extension sequence $$0 \rightarrow A \rightarrow \Pi \rightarrow G \rightarrow 0$$. The outcome implies a relatively conventional interaction between these symmetry components.
2. Tame Anomalies in $\Pi$: If $\Pi$ exhibits what is categorized as a tame anomaly, the symmetry of T/A can no longer be expressed as a simple direct product. Instead, Tachikawa describes an extension of $G$ by $\hat{A}^{(D-2)}$ that results in G not forming a subgroup in the total symmetry, suggesting the appearance of (D-1)-groups or group-like higher categories.
3. Wild Anomalies in $\Pi$: For wild anomalies, T/A's symmetry structure can involve higher-categorical frameworks that do not conform to traditional group-like structures. This reflects a more intricate relationship between $G$ and $\hat{A}^{(D-2)}$.

The paper also bridges these results to broader considerations in theoretical physics, including relation to known concepts such as the $H^3(G;A)$ symmetry localization obstacle noted in condensed matter physics and earlier constructions by Kapustin, Thorngren, and others.

Mathematical and Physical Implications

The paper employs the lens of algebraic topology and group theory to address these complex interactions. By leveraging tools such as the Pontryagin dual and forms from cohomology theory, Tachikawa's analysis allows for a deep dive into the nature of anomalies and how they influence the resulting symmetries post-gauging.

Implications for Condensed Matter and Beyond: Gauged theories with mixed or non-group symmetries have implications in the study of topological phases in condensed matter physics, particularly in how these phases are understood in terms of symmetry localization anomalies. The results expand on how gauging a symmetry can lead to new insights into symmetry protection and related phenomena in physical systems.

Relevance to Higher-Categorical Structures: The unconventional symmetry types arising from wild anomalies reaffirm the need to consider higher-category theories in QFTs. This is profound for theoretical physics, as it guides exploration beyond traditional symmetries and groups, potentially elucidating new types of quantum phases or exotic gauge theories.

Speculative Outlook: As higher-categorical symmetries become more prominent in the description of physical systems, we might see implications in formulating theories related to quantum gravity and string theory, where such mathematical structures are naturally expected to play a role.

Conclusion

Yuji Tachikawa's examination expands the formal understanding of gauged finite subgroups in quantum field theories, providing a robust foundation to further explore symmetry anomalies and their implications across dimensions. This paper acts as a seed for deeper exploration into the relationship between anomalies, symmetry groups, and higher categorical structures, positioning itself as a significant contribution to theoretical physics by leveraging abstract mathematical constructs to address concrete questions in QFTs. Future research spurred by this work may deepen insights into the interplay of symmetries and anomalies in high-energy physics, condensed matter, and beyond.