Generalised $Ω$-backgrounds in twisted $N=4$ super Yang-Mills

We introduce rigid supergravity backgrounds for 4d $\mathcal{N}=4$ super Yang-Mills which deform its topological twists and generalise the $Ω$-deformation of $\mathcal{N}=2$ gauge theories. We derive the twisted supersymmetry variations of the $\mathcal{N}=4$ Weyl multiplet, and the $\mathcal{N}=4$ matter multiplet coupled to a rigid bosonic background, for each of the three inequivalent topological twists. Other, more exotic backgrounds are also discussed.