On 2-Group Global Symmetries and Their Anomalies

TL;DR

This paper explores 2-group global symmetries in quantum field theories, detailing their structure, anomalies, and how they can be identified and coupled to backgrounds, with examples spanning gapped and gapless systems.

Contribution

It provides a systematic method to determine 2-group symmetries in QFTs, classifies their anomalies, and clarifies the relation to symmetry fractionalization.

Findings

A simple procedure to identify 2-group symmetries in QFTs

Classification of 't Hooft anomalies for 2-group symmetries

Connection between symmetry fractionalization and 2-group symmetries

Abstract

In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related 't Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the "obstruction to symmetry fractionalization" discussed in some condensed matter literature is really an instance of 2-group global symmetry.