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

Abstract

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 $2(\epsilon -1)$nd order differential operator ${\Delta }_{\epsilon }$, which is non-local in odd dimensions $d=2(\epsilon +1)$. We generalize this construction to mixed correlators $⟨{\mathcal{O}}_{k_1}{\mathcal{O}}_{k_2}{\mathcal{O}}_{k_3}{\mathcal{O}}_{k_1+k_2+k_3-2\mathcal{E}}⟩$ 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 $\epsilon$, 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.

arXiv comment64 pages, 1 Mathematica notebook. v1: Comments are very welcome. v2: Typos fixed