  A natural question about Quantum Field Theory is whether there is a
deformation to a trivial gapped phase. If the underlying theory has an anomaly,
then symmetric deformations can never lead to a trivial phase. We discuss such
discrete anomalies in Abelian Higgs models in 1+1 and 2+1 dimensions. We
emphasize the role of charge conjugation symmetry in these anomalies; for
example, we obtain nontrivial constraints on the degrees of freedom that live
on a domain wall in the VBS phase of the Abelian Higgs model in 2+1 dimensions.
In addition, as a byproduct of our analysis, we show that in 1+1 dimensions the
Abelian Higgs model is dual to the Ising model. We also study variations of the
Abelian Higgs model in 1+1 and 2+1 dimensions where there is no dynamical
particle of unit charge. These models have a center symmetry and additional
discrete anomalies. In the absence of a dynamical unit charge particle, the
Ising transition in the 1+1 dimensional Abelian Higgs model is removed. These
models without a unit charge particle exhibit a remarkably persistent order: we
prove that the system cannot be disordered by either quantum or thermal
fluctuations. Equivalently, when these theories are studied on a circle, no
matter how small or large the circle is, the ground state is non-trivial.
