A Chern-Simons Pandemic

TL;DR

The paper analyzes generalized global symmetries in quantum gravity, showing that 4d theories with 2-forms or axions can endow black holes with global hair, risking remnants. It demonstrates that gauging or breaking these symmetries—most generically via cubic Chern-Simons terms—resolves the problem, and proposes a Swampland criterion: any consistent theory with antisymmetric tensor gauge fields must include such CS terms to break generalized global symmetries. The authors support this conjecture through holographic, flux-compactification, and M-/string-theory constructions, including AdS/CFT examples and domain-wall physics, and extend the discussion to 3-form symmetries. They argue that a proliferation of CS terms (the Chern-Simons pandemic) is a generic requirement for quantum gravity consistency, with broad implications for Einstein gravity, supergravity, and familiar gauge theories, effectively excluding certain theories unless properly completed.

Abstract

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.

Figures (2)

• Figure 1: Triple Chern-Simons terms mean that we must include into the path integral configurations with a bubble in which $G_0\neq0$. Outside of the bubble, in the green area, $\phi\sim\phi+c$ leaves the action invariant, but in the blue area it does not. As a result, $\phi\rightarrow\phi+c$ is no longer a symmetry.
• Figure 2: Schematic depiction of the transition that takes place across the domain wall described in the main text. As we cross the domain wall, the $D6$ brane approaches the $O6$ plane and reconnects differently with its orientifold image. In the first picture, the $BF$ couplings of the $D6$ brane and its image cancel; in the third, they add up.