  We derive a necessary and sufficient criterion for when a two dimensional
gapped many-body system with Abelian anyons and a unitary $\mathbb{Z}_2$
symmetry has a protected gapless edge mode. Our criterion is phrased in terms
of edge theories --- or more specifically, chiral boson edge theories with
$\mathbb{Z}_2$ symmetry --- and it applies to any bosonic or fermionic system
whose boundary can be described by such an edge theory. At an operational
level, our criterion takes as input a chiral boson edge theory with
$\mathbb{Z}_2$ symmetry, and then produces as output a prediction as to whether
this edge theory can be gapped without breaking the symmetry. Like previous
work, much of our derivation involves constructing explicit perturbations that
gap chiral boson edge theories. Interestingly, however, we find that the
standard class of gapping perturbations --- namely cosine terms constructed
from null-vectors --- is not sufficient to gap some edge theories with
$\mathbb{Z}_2$ symmetry, and thus we are forced to go beyond the usual
null-vector analysis to establish our results.

Strongly Correlated Electrons