On finite symmetries and their gauging in two dimensions (1704.02330v2)

Abstract: It is well-known that if we gauge a $\mathbb{Z}_n$ symmetry in two dimensions, a dual $\mathbb{Z}_n$ symmetry appears, such that re-gauging this dual $\mathbb{Z}_n$ symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.

• The paper introduces a generalized framework using unitary fusion categories to reframe finite symmetries beyond traditional groups.
• It demonstrates how gauging a Zₙ symmetry produces dual phenomena, offering insights into non-Abelian structures and Wilson line behaviors.
• The approach refines anomaly analysis in two-dimensional topological quantum field theories, suggesting applications in condensed matter and quantum computing.

Finite Symmetries and Their Gauging in Two Dimensions

The paper under consideration discusses the intricate nature of symmetries in two-dimensional quantum field theories, aiming to broaden the conceptualization of symmetries beyond traditional group structures. Authored by Bhardwaj and Tachikawa, it bridges classical symmetry paradigms with modern quantum field theory (QFT) frameworks, exploring how symmetries are governed by unitary fusion categories rather than merely groups, especially in the context of non-Abelian activities. This article provides a comprehensive framework for understanding and manipulating these generalized symmetries, thereby extending their applicability and theoretical underpinnings.

• Main Thesis: The central theme of the paper is the generalization of symmetry concepts using unitary fusion categories. The authors argue that finite symmetries in two dimensions are better understood via the language of these categories, which encompass the traditional group-based symmetries and their representations, while also providing tools for more intricate underlying structures. This approach is inherently more flexible and powerful, accommodating the nuances and complexities that arise within modern QFT studies.
• Core Results:
  • The authors elaborate on the representation of a $\mathbb{Z}_n$ symmetry when gauged in two-dimensional theories, illustrating the emergence of a dual $\mathbb{Z}_n$ symmetry. This reflects not only a cyclic symmetry character but also insights into dual group phenomena.
  • The transition from group-based symmetries to category-defined symmetries provides a vital framework for addressing non-Abelian group scenarios, demonstrating utility in evaluating concepts like Wilson lines and generalized symmetries.
  • An in-depth exploration of how these ideas lead to a refined understanding of anomalies, providing a high-resolution lens for deciphering the anomaly structures within two-dimensional topological quantum field theories.
• Implications and Consequences: The utilization of unitary fusion categories for symmetry definitions is expected to enhance the landscape of both theoretical and practical applications within QFTs. On a theoretical level, this generalization invites a more nuanced treatment of quantum anomalies. Practically, these insights could facilitate the exploration of condensed matter systems exhibiting exotic phase transitions governed by non-trivial topological orders or anyonic behaviors.
• Future Directions and Developments: The paper nudges the research community towards unresolved questions and potential developments:
  • Extending Concepts to Higher Dimensions: The current paradigm is heavily centered on two-dimensional scenarios. Extrapolating these principles to higher-dimensional settings remains a formidable yet enticing challenge, potentially reshaping understanding across broader QFT contexts.
  • Constructive Frameworks and Applications: While the paper establishes a robust theoretical backbone, translating these generalized symmetry concepts into constructive frameworks for computational analysis or real-world systems (such as quantum computers leveraging anyonic statistics) could have substantial future impacts.

Overall, Bhardwaj and Tachikawa's work paves a sophisticated path forward in symmetry research within QFTs, advocating for a robust reconceptualization that is both mathematically rigorous and pragmatically valuable. The focus on unitary fusion categories redefines the dialogues around finite symmetries, setting the stage for novel explorations in complex quantum systems.