  Monolayer FeSe exhibits the highest transition temperature among the iron
based superconductors and appears to be fully gapped, seemingly consistent with
$s$-wave superconductivity. Here, we develop a theory for the superconductivity
based on coupling to fluctuations of checkerboard magnetic order (which has the
same translation symmetry as the lattice). The electronic states are described
by a symmetry based ${\bf k}\cdot {\bf p}$-like theory and naturally account
for the states observed by angle resolved photoemission spectroscopy. We show
that a prediction of this theory is that the resultant superconducting state is
a fully gapped, nodeless, $d$-wave state. This state, which would usually have
nodes, stays nodeless because, as seen experimentally, the relevant spin-orbit
coupling term has an energy scale smaller than the superconducting gap.
