A Chern-Simons Pandemic

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.