The Fate of Discrete 1-Form Symmetries in 6d

TL;DR

The paper develops a concrete framework for testing discrete center 1-form symmetries in 6d ${ m N}=(1,0)$ theories on their tensor branches by analyzing the GSWS coupling to center backgrounds and the induced charges on BPS strings. It shows that fractional instanton backgrounds generate fractional string charges which clash with Dirac quantization, thereby obstructing the realization of certain 1-form symmetries; massive string excitations subsequently break these symmetries, signaling a deep link between nonperturbative states and global symmetry realization. Through explicit SCFT, SUGRA, and LST examples, the authors map when 1-form symmetries survive, are broken by string modes, or can be gauged, with Mordell–Weil torsion in F-theory providing a geometric route to gauging via a diagonal center. The results tie into swampland ideas: in gravity-coupled 6d theories, global 1-form symmetries are generally absent or gauged, and torsion constrains the allowed global gauge group structure, potentially predicting novel consistency conditions for string vacua. The work also clarifies how circle reductions to 5d capture the origin of these anomalies and strengthens the link between geometric data (torsion) and nonperturbative spectra across dimensions.

Abstract

Recently introduced generalized global symmetries have been useful in order to understand non-perturbative aspects of quantum field theories in four and lower dimensions. In this paper we focus on 1-form symmetries of weakly coupled 6d supersymmetric gauge theories coupled to dynamical tensor multiplets. We study the consistency of global 1-form symmetries corresponding to the center of the gauge groups, or subgroups thereof, by activating their background fields, which makes the instanton density fractional. In 6d, an instanton background for a given gauge theory sources BPS strings via tadpole cancellation. The non-trivial 1-form symmetry background configurations contribute to the charge of the BPS strings. However, Dirac quantization imposes restrictions on the consistent 1-form backgrounds, since they can in general lead to and induce fractional charges, thus making (part of) the putative higher-form symmetry inconsistent. This gives explicit criteria to determine whether the discrete 1-form symmetries are realized. We implement these criteria in concrete examples originating from string compactifications. We also corroborate this by finding that a non-trivial fractional contribution is related to states which explicitly break the global 1-form symmetry appearing as massive excitations of the 6d BPS strings. For 6d theories consistently coupled to gravity, this hints at a symmetry breaking tower of states. When the fractional contributions are absent, the F-theory realization of the theories points to the gauging of the 1-form symmetry via the presence of non-trivial Mordell--Weil torsion.