Generalized Charges, Part I: Invertible Symmetries and Higher Representations (2304.02660v4)

Abstract: $q$-charges describe the possible actions of a generalized symmetry on $q$-dimensional operators. In Part I of this series of papers, we describe $q$-charges for invertible symmetries; while the discussion of $q$-charges for non-invertible symmetries is the topic of Part II. We argue that $q$-charges of a standard global symmetry, also known as a 0-form symmetry, correspond to the so-called $(q+1)$-representations of the 0-form symmetry group, which are natural higher-categorical generalizations of the standard notion of representations of a group. This generalizes already our understanding of possible charges under a 0-form symmetry! Just like local operators form representations of the 0-form symmetry group, higher-dimensional extended operators form higher-representations. This statement has a straightforward generalization to other invertible symmetries: $q$-charges of higher-form and higher-group symmetries are $(q+1)$-representations of the corresponding higher-groups. There is a natural extension to higher-charges of non-genuine operators (i.e. operators that are attached to higher-dimensional operators), which will be shown to be intertwiners of higher-representations. This brings into play the higher-categorical structure of higher-representations. We also discuss higher-charges of twisted sector operators (i.e. operators that appear at the boundary of topological operators of one dimension higher), including operators that appear at the boundary of condensation defects.

• The paper introduces a higher-categorical framework that extends traditional group representations to model generalized charges in QFT.
• It demonstrates how 0-form and 1-form symmetries yield higher-representations, capturing phenomena like symmetry fractionalization and 't Hooft anomalies.
• Their analysis provides actionable insights into mathematical structures, including braided monoidal functors, to reconcile physical and abstract symmetry concepts.

Overview of Generalized Charges, Part I: Invertible Symmetries and Higher Representations

The paper "Generalized Charges, Part I: Invertible Symmetries and Higher Representations" by Lakshya Bhardwaj and Sakura Schäfer-Nameki provides a comprehensive exploration of the generalized charges associated with invertible symmetries in quantum field theories (QFTs). Central to the paper is the notion that the traditional understanding of representation theory must be expanded through the lens of higher categories, particularly when dealing with generalized global symmetries. The authors delineate how higher-representations, sophisticated mathematical constructs, naturally encapsulate the action of symmetries on operators of various dimensions in QFTs.

Generalized Charges under 0-Form Symmetries

The paper introduces the concept of $q$-charges associated with $q$-dimensional operators influenced by a 0-form symmetry, $G^{(0)}$. Traditionally, these are represented by group representations; however, the authors extend this to assert that $q$-charges are $(q+1)$-representations, thereby involving the categorical structure of higher-representations. For instance, they elucidate how a line operator transforms under the 0-form symmetry group resulting in a 2-representation. These 2-representations capture not only the action but also anomalies like the 't Hooft anomaly of induced symmetries on the operator. Such a formulation aids in understanding phenomena such as symmetry fractionalization, where a bulk symmetry effectively splits into a more extensive symmetry on a defect.

Higher-Charges and Symmetry Fractionalization

Beyond the confines of group cohomology type, the authors introduce non-group cohomology type $q$-charges which manifest physically as symmetry fractionalization. This phenomenon describes scenarios where an invertible global symmetry applied to an operator results in a larger symmetry—possibly non-invertible—on its world volume. The paper explores this through examples, demonstrating symmetry fractionalization to both larger invertible groups and non-invertible structures like the Ising fusion category.

1-Form Symmetries and Related Charges

Transitioning from 0-form symmetries, the authors examine 1-form symmetries, where topological codimension-2 operators generate actions on line operators. For instance, they describe how 2-charges of such symmetries encode complex interactions between induced localized symmetries and the higher categorical structure. These narratives are captured through braided monoidal functors, providing an elegant mathematical framework that reconciles physical characteristics of symmetries in QFTs. The implementation of a 3-representation encapsulates interactions and potential anomalies posed by such symmetries.

Non-Genuine and Twisted Generalized Charges

Another key contribution of this work lies in the exploration of non-genuine charges, which are pictured within the structure of higher-representation theories. Such charges arise in the context of non-genuine operators that exist at the boundaries of higher-dimensional operators, necessitating an intricate description via layered higher-categories. Additionally, the paper outlines the intriguing behavior of twisted charges within twisted sectors of both symmetry generators and condensation defects, emphasizing the impact of 't Hooft anomalies in these cases.

Implications and Future Directions

This paper's methodology significantly informs our understanding of operator charges in QFTs, particularly through the introduction of higher-representations and an emphasis on symmetry's categorical scope. The insights offered on symmetry fractionalization have profound implications for bridging the gap between mathematical abstraction and physical phenomena such as anomalies and higher-categorical symmetries. Looking ahead, these findings suggest fruitful avenues for further exploration, especially when combined with non-invertible symmetries, as anticipated in Part II of this series. Future work could expand this foundational structure to broader classes of symmetries and spacetime dimensions, potentially uncovering deeper insights into the symbiotic relationship between symmetries and higher-category structures in QFTs.