Revisiting N=4 superconformal blocks

TL;DR

This work constructs explicit superconformal blocks for four-point functions of four half-BPS multiplets in four-dimensional N=4 SCFT by formulating and solving the two-particle Casimir eigenproblem of the full superconformal algebra $\mathfrak{psu}(2,2|4)$ on analytic superspace. The authors separate the blocks into contributions from long and short representations, obtaining a scalar block governed by a differential equation whose solution factorizes into a spacetime conformal block and an $SU(4)_R$ harmonic part, with Ward identities linking all components of the multiplet to the lowest component. They provide detailed formulas for both long and short/semi-short exchanges, including Jacobi-polynomial structures in the R-symmetry sector and hypergeometric expressions in spacetime, reproducing the Dolan–Osborn result for the lowest component. This construction enables full superconformal partial-wave decompositions of four-point functions and offers a concrete tool for numerical and analytical bootstrap studies, as well as perturbative data extraction in $\mathcal{N}=4$ SYM. The approach is poised for generalization to theories with less supersymmetry or different spacetime dimensions, potentially illuminating nonperturbative structure constants and operator spectra.

Abstract

We study four-point correlation functions of four generic half-BPS supermultiplets of N=4 SCFT in four dimensions. We use the two-particle Casimir of four-dimensional superconformal algebra to derive superconformal blocks which contribute to the partial wave expansion of such correlators. The derived blocks are defined on analytic superspace and allow us in principle to find any component of the four-point correlator. The lowest component of the result agrees with the superconformal blocks found by Dolan and Osborn.