Higher-form symmetries of 6d and 5d theories

TL;DR

This paper develops methods to determine higher-form symmetry groups in 5d and 6d superconformal theories, accounting for BPS excitations and geometric constructions, advancing understanding of their symmetry structures.

Contribution

It introduces new techniques for calculating higher-form symmetries in 5d and 6d theories, including an independent algorithm for 5d cases and geometric analysis for BPS states.

Findings

Reduced 1-form symmetry due to BPS excitations

Explicit geometric method for 6d 2-form symmetry

Algorithm for 5d 1-form symmetry in toric constructions

Abstract

We describe general methods for determining higher-form symmetry groups of known 5d and 6d superconformal field theories (SCFTs), and 6d little string theories (LSTs). The 6d theories can be described as supersymmetric gauge theories in 6d which include both ordinary non-abelian 1-form gauge fields and also abelian 2-form gauge fields. Similarly, the 5d theories can also be often described as supersymmetric non-abelian gauge theories in 5d. Naively, the 1-form symmetry of these 6d and 5d theories is captured by those elements of the center of ordinary gauge group which leave the matter content of the gauge theory invariant. However, an interesting subtlety is presented by the fact that some massive BPS excitations, which includes the BPS instantons, are charged under the center of the gauge group, thus resulting in a further reduction of the 1-form symmetry. We use the geometric construction of these theories in M/F-theory to determine the charges of these BPS excitations under the center. We also provide an independent algorithm for the determination of 1-form symmetry for 5d theories that admit a generalized toric construction. The 2-form symmetry group of 6d theories, on the other hand, is captured by those elements of the center of the abelian 2-form gauge group that leave all the massive BPS string excitations invariant, which is much more straightforward to compute as it is encoded in the Green-Schwarz coupling associated to the 6d theory.