  We generalize the eigenstate thermalization hypothesis to systems with global
symmetries. We present two versions, one with microscopic charge conservation
and one with exponentially suppressed violations. They agree for correlation
functions of simple operators, but differ in the variance of charged one-point
functions at finite temperature. We then apply these ideas to holography and to
gravitational low-energy effective theories with a global symmetry. We show
that Euclidean wormholes predict a non-zero variance for charged one-point
functions, which is incompatible with microscopic charge conservation. This
implies that global symmetries in quantum gravity must either be gauged or
explicitly broken by non-perturbative effects.
