On conformal correlators and blocks with spinors in general dimensions

TL;DR

This work develops a general framework for conformal correlators involving spinor and spinor-tensor fields in arbitrary dimensions using embedding-space methods. By introducing polarization spinors and deriving differential representations for all three-point structures, the authors express complex spinor/tensor correlators as derivatives of scalar correlators, enabling the construction of conformal blocks for four-point functions with spinor and scalar content via derivatives of scalar blocks: g^{a,b}_{Δ,ell}(u,v) = D^{12}_a D^{34}_b \bar{g}_{Δ,ell}(u,v). This approach extends the shadow-field formalism and lays groundwork for geodesic Witten diagrams with AdS spinor propagators, providing a unified route to spinor-based conformal blocks in any dimension and clarifying the structure of spinor and spinor-tensor three-point couplings. The results significantly advance the program of spinor-involving conformal blocks and their holographic realizations in AdS/CFT.

Abstract

We compute conformal correlation functions with spinor, tensor, and spinor-tensor primary fields in general dimensions with Euclidean and Lorentzian metrics. The spinors are taken to be Dirac spinors, which exist for any dimensions. For this, the embedding space formalism is employed and the polarisation spinors are introduced to simplify the computations. Three-point functions are rewritten in terms of differential operators acting on scalar-scalar-tensor correlation functions. This enables us to determine conformal blocks for four-point functions with scalar and spinor fields by acting the differential operators on scalar conformal blocks, which will be useful in finding their geodesic Witten diagrams.