  We prove that in $(3+1)d$ quantum field theories with $\mathbb{Z}_N$
symmetry, certain anomalies forbid a symmetry-preserving vacuum state with a
gapped spectrum. In particular, this applies to discrete chiral symmetries
which are frequently present in gauge theories as we illustrate in examples.
Our results also constrain the long-distance behavior of certain condensed
matter systems such as Weyl-semimetals and may have applications to
crystallographic phases with symmetry protected topological order. These
results may be viewed as analogs of the Lieb-Schultz-Mattis theorem for
continuum field theories.
