Projectors and seed conformal blocks for traceless mixed-symmetry tensors

TL;DR

This paper delivers the explicit construction of projectors to all $SO(d)$ irreps corresponding to traceless mixed-symmetry tensors that appear in seed conformal blocks of four stress-tensors in $d$ dimensions, expressing them in a closed Gegenbauer polynomial framework with $l_1$ as a central parameter. The authors develop a general algorithm to solve the tracelessness conditions, provide detailed results for two- and three-row Young diagrams (including $(l_1,1)$, $(l_1,1,1)$, $(l_1,2)$, $(l_1,2,1)$, $(l_1,2,2)$, and beyond), and compute normalization constants via shadow-formalism consistency. A universal recursion in $l_1$ for seed conformal blocks emerges from the Gegenbauer expansion, linking seed blocks to scalar blocks in higher dimensions and enabling potential closed-form expressions. The work also derives a differential operator generalizing Todorov-type constructions for two-row diagrams and demonstrates radial-coordinate recursions for practical expansion schemes, thereby significantly advancing the computation of mixed-symmetry seed blocks and their role in the conformal bootstrap. These results pave the way for systematic inclusion of higher-spin exchanges in bootstrap analyses and offer a robust toolkit for constructing and normalizing conformal blocks across dimensions.

Abstract

In this paper we derive the projectors to all irreducible SO(d) representations (traceless mixed-symmetry tensors) that appear in the partial wave decomposition of a conformal correlator of four stress-tensors in d dimensions. These projectors are given in a closed form for arbitrary length $l_1$ of the first row of the Young diagram. The appearance of Gegenbauer polynomials leads directly to recursion relations in $l_1$ for seed conformal blocks. Further results include a differential operator that generates the projectors to traceless mixed-symmetry tensors and the general normalization constant of the shadow operator.