  We prove the global well-posedness of the stochastic Abelian-Higgs equations
in two dimensions. The proof is based on a new covariant approach, which
consists of two parts: First, we introduce covariant stochastic objects. The
covariant stochastic objects and their multi-linear interactions are controlled
using covariant heat kernel estimates. Second, we control nonlinear remainders
using a covariant monotonicity formula, which is inspired by earlier work of
Hamilton.
