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 k · 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 has an energy scale smaller than the superconducting gap.The origin of superconductivity in iron based superconductors represents an important problem in condensed matter [1,2]. These materials have a relatively high superconducting transition temperature (T c ) and reveal unconventional states which are likely a consequence of electronic interactions. The most common explanation for superconductivity originates in a repulsive interaction between electron and hole pockets, leading to a superconducting gap that changes sign between these pockets [3,4]. In this context, superconductivity in single layer FeSe presents a conundrum [5,6]. Although it has the highest T c of the Fe-based superconductors, only electron pockets are present, so that the usual pairing interaction is not easily ascribable as the origin of superconductivity [5]. Furthermore, in spite of the evidence of electronic correlations in monolayer FeSe [6], the observed superconducting state is consistent with a fully gapped conventional s-wave pairing state [2,7].Understanding these apparent paradoxes is complicated by the complexity of existing theoretical models of ironbased superconductors. These models contain ten orbital and two spin degrees of freedom, which often obscures the underlying physics. Here, for monolayer FeSe, we introduce a simple symmetry-based effective k · p theory containing just two orbital degrees of freedom to describe the electronic excitations at the Fermi surface. We show that when these fermions are coupled to fluctuations associated with translation invariant checkerboard magnetic (CB-AFM) order, the resultant fully gapped, nodeless, d-wave superconducting state naturally produces the gap anisotropy seen in angle-resolved photoemission spectroscopy (ARPES) [2]. A key parameter in our k·p theory is a spin-orbit coupling (SOC) energy that is distinct from the usual on-site SOC. This SOC would usually require the nodeless d-wave state to develop nodes, but ARPES reveals this SOC is too small to allow for the nodes to develop. We then discuss how our theory generalizes to 3D K-dosed bulk FeSe and to (Li 0.8 Fe 0.2 )OHFeSe [9-11]. We do not include the role of interface phonons here but adhere to the viewpoint that these can enhance the T c found from other mechani...