The Fate of Discrete 1-Form Symmetries in 6d

TL;DR

This paper investigates the conditions under which 1-form symmetries in 6d supersymmetric gauge theories are consistent, revealing how fractional instanton charges and string states influence the realization or breaking of these symmetries.

Contribution

It provides explicit criteria for the consistency of 1-form symmetries in 6d theories, linking fractional instanton charges, string states, and string compactification examples.

Findings

Fractional instanton charges can break 1-form symmetries.

String states can explicitly break or preserve 1-form symmetries.

F-theory models show gauging of 1-form symmetries via Mordell--Weil torsion.

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.