Conformal correlators of mixed-symmetry tensors

TL;DR

The paper extends the embedding-space formalism to general mixed-symmetry tensors by introducing a composite polarization framework tied to Young diagram structure. It provides a systematic algorithm to count tensor structures in $n$-point CFT correlators via tensor-product coefficients, and develops explicit embedding-space encodings for two-, three-, and four-point functions, including concrete examples such as hooks and $p$-forms. A key advance is the derivation of the unitarity bound and conservation conditions in embedding space for mixed-symmetry tensors, together with a practical conformal-block construction for arbitrary exchanges via shadow integrals and monodromy projection. The authors reveal a matching between the number of conformal structures and massive scattering amplitudes in one higher dimension, enabling a Unified counting perspective that supports conformal bootstrap analyses and AdS/CFT connections, and they illustrate conformal blocks for hook and two-form exchanges with explicit formulas. The work lays groundwork for analyzing correlators involving stress tensors and higher-spin mixed-symmetry operators, with potential implications for universal bootstrap bounds and holographic dualities.

Abstract

We generalize the embedding formalism for conformal field theories to the case of general operators with mixed symmetry. The index-free notation encoding symmetric tensors as polynomials in an auxiliary polarization vector is extended to mixed-symmetry tensors by introducing a new commuting or anticommuting polarization vector for each row or column in the Young diagram that describes the index symmetries of the tensor. We determine the tensor structures that are allowed in n-point conformal correlation functions and give an algorithm for counting them in terms of tensor product coefficients. A simple derivation of the unitarity bound for arbitrary mixed-symmetry tensors is obtained by considering the conservation condition in embedding space. We show, with an example, how the new formalism can be used to compute conformal blocks of arbitrary external fields for the exchange of any conformal primary and its descendants. The matching between the number of tensor structures in conformal field theory correlators of operators in d dimensions and massive scattering amplitudes in d+1 dimensions is also seen to carry over to mixed-symmetry tensors.