The Fate of Discrete 1-Form Symmetries in 6d

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 tensor multiplets. We study their interplay with large gauge transformations for dynamical tensor fields. In a non-trivial background for the global 1-form symmetry, this leads to an ambiguity of the effective field theory partition function. This anomaly is eliminated by the inclusion of BPS strings. However, the non-trivial 1-form background can induce fractional string charges which are not compatible with Dirac quantization, and hence the symmetry is absent. The anomalous term therefore serves as a tool to detect whether the discrete 1-form symmetries are realized, which we demonstrate in explicit examples originating from string compactifications. We also corroborate this by finding that a non-trivial ambiguity is related to states which explicitly break the global 1-form symmetry, which appear as generally massive excitations of the 6d BPS strings. For 6d theories consistently coupled to gravity, this ambiguity of the partition function hints at the presence of a symmetry breaking tower of states. When the ambiguity is absent, the F-theory realization of the theories points to the gauging of the 1-form symmetries via the presence of non-trivial Mordell--Weil torsion.