AdS Weight Shifting Operators

TL;DR

<3-5 sentence high-level summary> The paper develops AdS weight shifting operators that mirror and extend the CFT weight shifting machinery to the AdS bulk, enabling systematic shifts in both dimension and spin of AdS harmonic functions and propagators. By establishing precise bulk-boundary correspondences and crossing relations, the authors show how spinning Witten diagrams—tree-level and certain loops—can be reduced to scalar diagrams, with the conformal-block data recovered via a new bulk operator formalism. This framework unifies boundary and bulk perspectives, introduces a diagrammatic language for AdS objects, and applies to mixed-symmetry tensors, ultimately providing a practical toolkit for computing spinning AdS/CFT correlators. The results promise extensions to fermionic cases and higher-point (quartic) couplings, and hint at deep connections to infinite-dimensional $6j$ symbols in loop dynamics.

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.