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.
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