Conformal correlators of mixed-symmetry tensors

TL;DR

This paper extends the embedding formalism for conformal field theories to include mixed-symmetry tensors, providing new tools for analyzing correlation functions, tensor structures, and conformal blocks with potential applications in scattering amplitudes.

Contribution

It introduces a generalized polarization vector formalism for mixed-symmetry tensors and develops an algorithm for counting tensor structures in conformal correlators.

Findings

Derived the tensor structures allowed in n-point correlators.

Provided an algorithm for counting tensor structures.

Showed how to compute conformal blocks for mixed-symmetry tensors.

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.