Exploring 2-Group Global Symmetries

TL;DR

The paper develops a comprehensive framework for continuous 2-group global symmetries in 4D QFTs, revealing how a 1-form U(1)_B^(1) current can fuse with 0-form flavor or Poincaré currents through GS-like shifts of background fields. It introduces deformed current algebras and a quantized 2-group structure constant κ̂_A, deriving novel Ward identities and showing how these constrain RG flows, spontaneous breaking, and anomaly matching. By coupling to 2-group background fields and exploiting GS counterterms, the authors classify reducible 't Hooft anomalies, analyze their cancellation, and demonstrate how many explicit models—such as multi-flavor QED, fermionic Fermat-type theories, and Goldstone–Maxwell systems—realize 2-group symmetry. The work also connects to holographic descriptions with bulk Green–Schwarz mechanisms and explores higher-group generalizations, including nonabelian and Poincaré 2-groups, as well as the implications for strings, defects, and dimensional reduction. Overall, the results provide a versatile toolkit for understanding unconventional global symmetries and their interplay with anomalies, RG flows, and IR dynamics in 4D QFTs.

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.

Figures (2)

• Figure 1: Horizontal arrows represent the gauging of $U(1)_C^{(0)}$ in a parent theory $T_1$ with $U(1)_A^{(0)} \times U(1)_C^{(0)}$ flavor symmetry and a mixed $\kappa_{A^2 C}$ 't Hooft anomaly to produce a theory $T_2$ with $U(1)_A^{(0)} \times_{\widehat{\kappa}_A} U(1)_B^{(1)}$ 2-group symmetry. Vertical arrows represent the RG flows interpolating between $T_{1}^\text{UV} \rightarrow T_{1}^\text{IR}$ and $T_{2}^\text{UV} \rightarrow T_{2}^\text{IR}$. The diagram is commutative.
• Figure 2: Possible gaugings of a theory $T_2$ with $U(1)_A^{(0)} \times_{\widehat{\kappa}_A} U(1)_B^{(1)}$ 2-group symmetry and no 't Hooft anomalies. Theory $T_1$ is the parent theory with $U(1)_A^{(0)} \times U(1)_C^{(0)}$ flavor symmetry and a mixed $\kappa_{A^2 C} = - 2 \widehat{\kappa}_A$ 't Hooft anomaly; it can be obtained from $T_2$ by gauging $U(1)_B^{(1)}$. Theory $T_3$ results from $T_2$ by gauging $U(1)_A^{(0)} \times_{\widehat{\kappa}_A} U(1)_B^{(1)}$, or from $T_1$ by gauging $U(1)_C^{(0)}$.