Exploring 2-Group Global Symmetries

TL;DR

This paper investigates four-dimensional quantum field theories with 2-group global symmetries, revealing their unique algebraic structures, anomaly cancellation mechanisms, and implications for spontaneous symmetry breaking and holography.

Contribution

It introduces the concept of 2-group global symmetries in 4D QFTs, detailing their algebraic properties, anomaly cancellation via Green-Schwarz terms, and their realization through gauging abelian symmetries with mixed anomalies.

Findings

2-group symmetries involve fused current algebras with quantized structure constants.

Partial anomaly cancellation is achieved through Green-Schwarz counterterms.

Examples include theories with dynamical strings carrying specific charges.

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.