  In this paper we study the consistency of generalized global symmetries in
theories of quantum gravity, in particular string theory. Such global
symmetries arise in theories with $(p+1)$-form gauge fields, and for spacetime
dimension $d\leq p+3$ there are obstructions to their breaking even by quantum
effects of charged objects. In 4d theories with a 2-form gauge field (or with
an axion scalar), these fields endow Schwarzschild black holes with quantum
hair, a global charge leading to usual trouble with remnants. We describe
precise mechanisms, and examples from string compactifications and holographic
pairs, in which these problems are evaded by either gauging or breaking the
global symmetry, via (suitable versions of) Stuckelberg or Kaloper-Sorbo
couplings. We argue that even in the absence of such couplings, the generic
solution in string theory is the breaking of the global symmetries by cubic
Chern-Simons terms involving different antisymmetric tensor fields. We
conjecture that any theory with (standard or higher-degree antisymmetric
tensor) gauge fields is in the Swampland unless its effective action includes
such Chern-Simons terms. This conjecture implies that many familiar theories,
like QED (even including the charged particles required by the Weak Gravity
Conjecture) or $\mathcal{N}=8$ supergravity in four dimensions, are
inconsistent in quantum gravity unless they are completed by these Chern-Simons
terms.
