Fortuity and relevant deformation

We investigate the supercharge cohomology of an $\mathcal{N}=1$ relevant deformation of $\mathcal{N}=4$ super Yang-Mills. By introducing a field redefinition, we integrate out massive fields in a cohomological sense. Then, we construct the monotone cohomologies corresponding to the Kaluza-Klein particles of the dual supergravity solution. Some of the monotone cohomologies obey stringy exclusion principle analogous to that of $AdS_3$. Relatedly, they vanish on the diagonal field configurations, unlike $\mathcal{N}=4$ monotone cohomologies. We also construct infinitely many fortuitous cohomologies for gauge group SU(2). We find that unlike $\mathcal{N}=4$ fortuitous cohomologies, they can either be non-vanishing or vanishing on the diagonal fields. By undoing the field redefinition and taking a suitable UV limit, we show that non-vanishing ones reduce to monotone cohomologies of $\mathcal{N}=4$ SYM, while vanishing ones reduce to fortuitous cohomologies of $\mathcal{N}=4$ SYM. This implies that the fortuity can arise due to the relevant deformation, while monotonicity is not.