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

TL;DR

This work develops a unified framework for reducing and organizing four-point functions of 1/2-BPS operators in maximally supersymmetric CFTs across 3d, 4d, and 6d. By extending the Dolan–Gallot–Sokatchev construction to mixed next-to-next-to-extremal correlators, it expresses the full superblocks in terms of a two-variable reduced correlator ${ m extcal{T}}^{oldsymbol{k}}$ and up to two single-variable functions $h^{oldsymbol{k}}_I$, with a differential operator $Δ_oldsymbol{ extvarepsilon}$ and Jack polynomials providing spectral data. The paper shows how to invert channel equations to obtain reduced blocks with shifted kinematics, and it provides explicit constructions for the six R-symmetry channels, assembling full superblocks in 6d, 4d, and 3d, as well as detailed reduced-block decompositions. It also discusses alternative formulations connecting to holography and chiral algebras, and outlines outlooks for higher extremality and half-maximal theories. Overall, the results offer a practical, dimension-unified route to analytic and numerical bootstrap studies in these highly constrained CFTs.

Abstract

We consider mixed four-point correlators of 1/2-BPS operators $\mathcal{O}_{k_i}$ in the maximally supersymmetric CFTs, i.e. the 3d $\mathcal{N}=8$, 4d $\mathcal{N}=4$, and 6d $\mathcal{N}=(2,0)$ theories. In \cite{Dolan:2004mu}, Dolan, Gallot, and Sokatchev demonstrated that four-point correlators of identical $\mathcal{O}_{k_i}$ in these SCFTs can be expressed in terms of unconstrained ``reduced correlators" $\mathcal{T}^{\{k_i\}}_{I,J}(U,V)$, $h^{\{k_i\}}_I(z)$ acted on by a $2(\varepsilon-1)^\text{nd}$ 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 this channel equation can be inverted to expand $\mathcal{T}^{\{k_i\}}_{I,J}$, $h^{\{k_i\}}_I$ in ``reduced" blocks with shifted kinematics $(\tildeΔ_{12},\tildeΔ_{34})=\left(Δ_{12},Δ_{34}-2(\varepsilon-1)\right)$. These reduced blocks reproduce what is known in 4d, generalize the known $\langle \mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{k}\mathcal{O}_{k}\rangle$ case in 6d, and offer a novel result in 3d.