AdS Weight Shifting Operators

TL;DR

This paper introduces new differential operators that shift weights in AdS and CFT representations, simplifying the computation of Witten diagrams and conformal partial waves, especially for fields with arbitrary spins.

Contribution

The paper constructs novel AdS weight shifting operators, establishes their crossing relations with CFT operators, and applies them to streamline Witten diagram calculations involving spinning fields.

Findings

Tree-level 4-point Witten diagrams with arbitrary spins reduced to scalar diagrams.

Method to derive conformal partial wave expansions using new operators.

Reduction of 1-loop diagrams with spinning fields to scalar diagrams with a single external spin.

Abstract

We construct a new class of differential operators that naturally act on AdS harmonic functions. These are weight shifting operators that change the spin and dimension of AdS representations. Together with CFT weight shifting operators, the new operators obey crossing equations that relate distinct representations of the conformal group. We apply our findings to the computation of Witten diagrams, focusing on the particular case of cubic interactions and on massive, symmetric and traceless fields. In particular we show that tree level 4-point Witten diagrams with arbitrary spins, both in the external fields and in the exchanged field, can be reduced to the action of weight shifting operators on similar 4-point Witten diagrams where all fields are scalars. We also show how to obtain the conformal partial wave expansion of these diagrams using the new set of operators. In the case of 1-loop diagrams with cubic couplings we show how to reduce them to similar 1-loop diagrams with scalar fields except for a single external spinning field (which must be a scalar in the case of a two-point diagram). As a bonus, we provide new CFT and AdS weight shifting operators for mixed-symmetry tensors.