Higher Form Symmetries TFT in 6d

TL;DR

This work constructs the symmetry TFT for the 2-form and 1-form higher-form symmetries of 6d $(1,0)$ SCFTs with a single tensor multiplet, revealing a 3-group mixing between the two symmetry types. The derivation proceeds on the tensor branch by coupling to background fields, is corroborated by holographic IIA reductions, and is cross-checked via a 5d Coulomb-branch perspective and circle compactification, including explicit results for rank-1 theories and 5d $SU(p)_q$ and non-Lagrangian sectors. A detailed account of the circle reduction shows how absolute theories vs relative theories arise and how the anomaly structure emerges in 5d KK theories, with a consistent picture across 6d and 5d descriptions. The fate of these higher-form symmetries upon coupling to gravity is analyzed, demonstrating that the 2-form (and certain 1-form) sectors are often broken or gauged by the gravitational sector, while in other constructions a gauged 3-group persists, tying together geometric, holographic, and field-theoretic perspectives on symmetry and anomaly in higher dimensions.

Abstract

Symmetries and anomalies of a $d$-dimensional quantum field theory are often encoded in a $(d+1)$-dimensional topological action, called symmetry topological field theory (TFT). We derive the symmetry TFT for the 2-form and 1-form symmetries of 6d $(1,0)$ field theories, focusing on theories with a single tensor multiplet (rank 1). We implement this by coupling the low-energy tensor branch action to the background fields for the higher-form symmetries and by looking at the symmetry transformation rules on dynamical and background fields. These transformation rules also imply a mixing of the higher-form symmetries in a 3-group structure. For some specific and related higher rank cases, we also derive the symmetry TFT from the holographic dual IIA supergravity solutions. The symmetry TFT action contains a coupling between the 2-form symmetry and the 1-form symmetry backgrounds, which leads to a mixed anomaly between the 1-form symmetries of the 5d KK-theory obtained by circle compactification. We confirm this by a pure 5d analysis provided by the 5d effective low-energy Coulomb branch Lagrangian coupled to background fields. We also derive the symmetry TFT for 5d $SU(p)$ supersymmetric gauge theories with Chern-Simons level $q$ and for 5d theories without non-abelian gauge theory description at low-energy. Finally, we discuss the fate of the 2-form and 1-form symmetry of rank 1 6d field theories when coupled to gravity.