  We derive general constraints on the existence of many-body localized (MBL)
phases in the presence of global symmetries, and show that MBL is not possible
with symmetry groups that protect multiplets (e.g. all non-Abelian symmetry
groups). Based on simple representation theoretic considerations, we derive
general Mermin-Wagner-type principles governing the possible alternative fates
of non-equilibrium dynamics in isolated, strongly disordered quantum systems.
Our results rule out the existence of MBL symmetry protected topological phases
with non-Abelian symmetry groups, as well as time-reversal symmetry protected
electronic topological insulators, and in fact all fermion topological
insulators and superconductors in the 10-fold way classification. Moreover,
extending our arguments to systems with intrinsic topological order, we rule
out MBL phases with non-Abelian anyons as well as certain classes of symmetry
enriched topological orders.
