Projectors and seed conformal blocks for traceless mixed-symmetry tensors

TL;DR

This paper derives explicit projectors for traceless mixed-symmetry tensors in conformal field theory, enabling recursive computation of seed conformal blocks and providing tools for analyzing four-point correlators of stress-tensors.

Contribution

It provides closed-form projectors for all irreducible SO(d) representations relevant to conformal blocks, including recursion relations and differential operators for these tensors.

Findings

Explicit projectors for traceless mixed-symmetry tensors derived

Recursion relations for seed conformal blocks established

Differential operators for generating projectors developed

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.