Signatures of global symmetry violation in relative entropies and replica wormholes

Abstract: It is widely believed that exact global symmetries do not exist in theories that admit quantum black holes. Here we propose a way to quantify the degree of global symmetry violation in the Hawking radiation of a black hole by using certain relative entropies. While the violations of global symmetry that we consider are non-perturbative effects, they nevertheless give $\mathcal{O}(1)$ contributions to the relative entropy after the Page time. Furthermore, using "island" formulas, these relative entropies can be computed within semi-classical gravity, which we demonstrate with explicit examples. These formulas give a rather precise operational sense to the statement that a global charge thrown into an old black hole will be lost after a scrambling time. The relative entropies considered here may also be computed using a replica trick. At integer replica index, the global symmetry violating effects manifest themselves as charge flowing through the replica wormhole.