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.
Highlights
Symmetry is one of the most enduring and fruitful tools in the analysis of Quantum Field Theory (QFT)
We describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory
In this paper we focus on the particular case of 0-form and 1-form global symmetries, and we address the question: What is the most general possible symmetry structure including a 0-form group G and a 1-form group A? We show that one general possibility is that G and A are combined into a higher-categorical structure known as a 2-group
Summary
Symmetry is one of the most enduring and fruitful tools in the analysis of Quantum Field Theory (QFT). The defect junction illustrated in figure 4 has a sharp meaning in this language: at codimension-3 intersections of 0-form symmetry defects, there is a flux for A described by the Postnikov class [β] This means that when the theory is coupled to G gauge fields, a non-trivial 1-form background for A is sourced. A gauge transformation of the 0-form background fields modifies the partition function by the insertion of a 1-form symmetry defect, which is a non-trivial operator in the theory. A discrete analog of this construction was recently described in [14]: starting from theories with only 0-form symmetries and appropriate mixed ’t Hooft anomalies, one can construct theories with 2-group global symmetry by gauging.
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