Spinning Conformal Blocks

TL;DR

This work develops a universal method to derive conformal blocks for the exchange of traceless symmetric tensors in four-point functions with spinning external operators. By leveraging the index-free embedding space formalism, the authors represent all such blocks as simple differential operators acting on the scalar conformal blocks, enabling complete coverage in three dimensions. The framework hinges on a differential basis for three-point functions (built from D_ij, H_ij, and related operators) that recasts spinning OPE structures as derivatives of scalar structures, so spinning partial waves reduce to derivatives of scalar blocks with shifted dimensions. The results advance the conformal bootstrap program by providing practical, embedding-space expressions for spinning blocks, facilitate analyses involving conserved currents and the stress tensor in CFTs, and offer a path toward holographic interpretations in AdS/CFT via Mellin representations. They also discuss special cases (parity-odd structures, d=3 peculiarities, and conservation constraints) and outline how antisymmetric or mixed-symmetry exchanges might be tackled with further development.

Abstract

For conformal field theories in arbitrary dimensions, we introduce a method to derive the conformal blocks corresponding to the exchange of a traceless symmetric tensor appearing in four point functions of operators with spin. Using the embedding space formalism, we show that one can express all such conformal blocks in terms of simple differential operators acting on the basic scalar conformal blocks. This method gives all conformal blocks for conformal field theories in three dimensions. We demonstrate how this formalism can be applied in a few simple examples.