Exploring 2-Group Global Symmetries (1802.04790v1)

Abstract: We analyze four-dimensional quantum field theories with continuous 2-group global symmetries. At the level of their charges such symmetries are identical to a product of continuous flavor or spacetime symmetries with a 1-form global symmetry $U(1)^{(1)}_B$, which arises from a conserved 2-form current $J_B^{(2)}$. Rather, 2-group symmetries are characterized by deformed current algebras, with quantized structure constants, which allow two flavor currents or stress tensors to fuse into $J_B^{(2)}$. This leads to unconventional Ward identities, which constrain the allowed patterns of spontaneous 2-group symmetry breaking and other aspects of the renormalization group flow. If $J_B^{(2)}$ is coupled to a 2-form background gauge field $B^{(2)}$, the 2-group current algebra modifies the behavior of $B^{(2)}$ under background gauge transformations. Its transformation rule takes the same form as in the Green-Schwarz mechanism, but only involves the background gauge or gravity fields that couple to the other 2-group currents. This makes it possible to partially cancel reducible 't Hooft anomalies using Green-Schwarz counterterms for the 2-group background gauge fields. The parts that cannot be cancelled are reinterpreted as mixed, global anomalies involving $U(1)_B^{(1)}$ and receive contributions from topological, as well as massless, degrees of freedom. Theories with 2-group symmetry are constructed by gauging an abelian flavor symmetry with suitable mixed 't Hooft anomalies, which leads to many simple and explicit examples. Some of them have dynamical string excitations that carry $U(1)_B^{(1)}$ charge, and 2-group symmetry determines certain 't Hooft anomalies on the world sheets of these strings. Finally, we point out that holographic theories with 2-group global symmetries have a bulk description in terms of dynamical gauge fields that participate in a conventional Green-Schwarz mechanism.

• The paper introduces a novel 2-group symmetry structure that fuses 0-form and 1-form symmetries into deformed current algebras with quantized constants.
• It reveals how unconventional Ward identities and anomaly cancellation mechanisms constrain symmetry breaking and renormalization group flows.
• Practical examples—such as multi-flavor QED and holographic duals—demonstrate the significant impact of 2-group symmetries in quantum field theories.

Analysis of the Paper on 2-Group Global Symmetries

The paper "Exploring 2-Group Global Symmetries" investigates the properties and implications of 2-group global symmetries in four-dimensional quantum field theories (QFTs). The concept of 2-group symmetries extends traditional global symmetries by incorporating higher-form symmetries, specifically focusing on the interaction between 0-form and 1-form symmetries. These interactions introduce novel aspects into the symmetry structures of QFTs, such as unconventional Ward identities and modified conservation laws.

Key Insights and Contributions

1. 2-Group Symmetry Structure: The paper outlines the fundamental structure of 2-group symmetries by elaborating on how these symmetries arise through the fusion of conventional flavor or spacetime symmetries with 1-form global symmetries. A defining feature of these symmetries is their "deformed" current algebras, which allow flavor currents or stress tensors to merge into a conserved 2-form current. This deformation is characterized by quantized structure constants.
2. Ward Identities and Anomaly Cancellation: The unconventional Ward identities linked with 2-group symmetries impose strong constraints on allowed patterns of spontaneous symmetry breaking and renormalization group flows. A particularly compelling aspect of the paper is its analysis of how 2-group symmetries modify the behavior of background gauge fields, similar to the Green-Schwarz mechanism, partially canceling 't Hooft anomalies using counterterms.
3. Examples and Applications: The authors provide explicit examples of QFTs where 2-group symmetries play a distinct role, such as multi-flavor massless QED. They demonstrate dynamical strings carrying 1-form symmetry charges and analyze 't Hooft anomalies within these frameworks, also examining holographic dual descriptions.
4. Implications for RG Flows: The paper reveals that the RG flows preserving exact 2-group symmetry have intricate behaviors, constrained by the symmetry's nature. Specifically, scenarios where entire 2-group symmetries are spontaneously broken reflect sophisticated physics not seen in simpler symmetry groups.
5. 2-Group and n-Group Extensions: Beyond 2-groups, the paper speculates on the potential for even more complex symmetry structures, like n-group symmetries, suggesting a hierarchy of symmetry types that could enrich theoretical physics even further.

Numerical and Theoretical Evaluation

Through the work's methodology, it presents rigorous mathematical formulation and quantization conditions for structure constants in 2-group symmetries. The paper provides a solid theoretical framework, clearing a path to better understand and manipulate anomalies and symmetry violations in various physical systems.

Future Prospects and Open Questions

This investigation opens new avenues in theoretical physics concerning the integration of higher-form symmetries in field theories. The exploration seems poised to influence future studies of quantum phases and anomalies, potentially affecting string theory and related areas.

Future research might explore:

• Further constructions of higher-dimensional theories exhibiting 2-group symmetry.
• Practical implications of 2-group anomalies in condensed matter and quantum materials.
• Development of computational methods to simulate 2-group symmetric systems.

In conclusion, the paper enriches the conceptual landscape of quantum field theories by incorporating and systematizing 2-group symmetries, presenting new vocabulary and approaches for handling gauge theories and their anomalies.