Reduced superblocks at next-to-next-to-extremality for all maximally supersymmetric CFTs

We consider mixed four-point correlators of 1/2-BPS operators ${\mathcal{O}}_k$ in the maximally supersymmetric CFTs, i.e. the 3d $\mathcal{N}=8$, 4d $\mathcal{N}=4$, and 6d $\mathcal{N}=(2,0)$ theories. In arXiv:hep-th/0405180, Dolan, Gallot, and Sokatchev demonstrated that four-point correlators of identical ${\mathcal{O}}_k$ in these SCFTs can be expressed in terms of a number of unconstrained one- and two-variable reduced correlator" functions acted on by a $2(\varepsilon-1)$nd order differential operator $Δ_\varepsilon$, which is non-local in odd dimensions $d=2(\varepsilon+1)$. We generalize this construction to mixed correlators $\langle \mathcal{O}_{k_1}\mathcal{O}_{k_2}\mathcal{O}_{k_3}\mathcal{O}_{k_1+k_2+k_3-2\mathcal{E}}\rangle$ up to extremality $\mathcal{E}=2$. To construct superconformal blocks, we generalize the R-symmetry channel equations and use Jack polynomial expansions to recursively generate the full spectrum of conformal blocks in a superblock from a single channel. We observe that for each $\varepsilon$, this channel equation can be inverted to expand the reduced correlators in reduced superblocks" involving blocks with shifted external kinematics. These reduced blocks reproduce what is known in 4d, generalize the known $⟨{\mathcal{O}}_2{\mathcal{O}}_2{\mathcal{O}}_k{\mathcal{O}}_k⟩$ case in 6d, and offer a novel result in 3d.