2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories

TL;DR

This work demonstrates that six-dimensional quantum field theories harbor higher-form symmetries that fuse into nontrivial 2-group structures via mixed gauge–global anomalies. It shows 2-group symmetries naturally arise in IR theories with gauge dynamics, including many Little String Theories, but are absent in unitary SCFTs, where Green–Schwarz mechanisms cancel mixed anomalies on the tensor branch. The authors provide a concrete algorithm to compute ’t Hooft anomalies of most 6d SCFTs from tensor-branch spectra and prove a positivity bound for the a-type Weyl anomaly, linking anomaly data to UV completeness and IR dynamics. They also develop a detailed framework for background 2-group fields, compute structure constants for free fermions and supersymmetric 6d theories, and discuss implications for sigma models and string worldsheets, highlighting the role of 2-group symmetries in constraining 6d QFTs and LSTs with potential broader applications in higher-dimensional quantum field theory.

Abstract

We examine six-dimensional quantum field theories through the lens of higher-form global symmetries. Every Yang-Mills gauge theory in six dimensions, with field strength $f^{(2)}$, naturally gives rise to a continuous 1-form global symmetry associated with the 2-form instanton current $J^{(2)} \sim * \text{Tr} \left( f^{(2)} \wedge f^{(2)}\right)$. We show that suitable mixed anomalies involving the gauge field $f^{(2)}$ and ordinary 0-form global symmetries, such as flavor or Poincaré symmetries, lead to continuous 2-group global symmetries, which allow two flavor currents or two stress tensors to fuse into the 2-form current $J^{(2)}$. We discuss several features of 2-group symmetry in six dimensions, many of which parallel the four-dimensional case. The majority of six-dimensional supersymmetric conformal field theories (SCFTs) and little string theories have infrared phases with non-abelian gauge fields. We show that the mixed anomalies leading to 2-group symmetries can be present in little string theories, but that they are necessarily absent in SCFTs. This allows us to establish a previously conjectured algorithm for computing the 't Hooft anomalies of most SCFTs from the spectrum of weakly-coupled massless particles on the tensor branch of these theories. We then apply this understanding to prove that the $a$-type Weyl anomaly of all SCFTs with a tensor branch must be positive, $a > 0$.