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.
Highlights
One missing ingredient for the conformal bootstrap program [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] for correlation functions with operators with spin in d dimensions is the explicit knowledge of seed conformal blocks exchanging mixed-symmetry tensors
In this paper the projectors to traceless mixed-symmetry tensors that appear in the correlator of four stress tensors were derived in terms of Gegenbauer polynomials
Knowledge of the explicit form of the projectors led us to a single universal recursion relation in l1 for seed conformal blocks, given by (3.21)
Summary
One missing ingredient for the conformal bootstrap program [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] for correlation functions with operators with spin in d dimensions is the explicit knowledge of seed conformal blocks exchanging mixed-symmetry tensors. These recursion relations are shown to hold for the seed conformal blocks of all the correlators in table 1 and conjectured to hold for any seed conformal block of bosonic operators These relations are derived, making use of the integral representations of the conformal blocks in the shadow formalism, where the projector appears explicitly. In appendix A we derive a differential operator that generates projectors to traceless mixed-symmetry tensors for Young diagrams of two rows. This operator is a generalization of a well known operator for traceless symmetric tensors. Correlator φ1φ2φ3φ4 φ1J2μφ3J4ν φ1T2μν φ3T4ρσ J1μJ2ν J3ρJ4σ J1μ T2ν ρ J3σ T4λκ T1μν T2ρσT3λκT4τ ω
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