The group approach to AdS space propagators: A fast algorithm
Thorsten Leonhardt, Ruben Manvelyan, Werner Ruhl
TL;DR
The paper addresses the computation of two-point functions in $d+1$-dimensional AdS space by refining a group-theoretic approach to bulk-to-bulk propagators for symmetric traceless tensors. It develops an algorithm that expresses these propagators as linear combinations of Legendre functions of the second kind, leveraging Gegenbauer polynomials to enforce tracelessness and performing angular integrations to obtain a compact radial structure. The propagator decomposes into $l+1$ terms, each tied to covariant bitensors $L_1,...,L_4$ and assembled into $Q_k(\zeta)$ with coefficients $R^{(l,k)}_{r_1,r_2,r_3,r_4}(\mu)$, with explicit results and a Maple-based tool provided up to $l=4$. The work yields a practical, implementable method for AdS propagators with correct boundary behavior, matching conformal two-point functions, and sets the stage for higher-point extensions.
Abstract
In this letter we show how the method of [4] for the calculation of two-point functions in d+1-dimensional AdS space can be simplified. This results in an algorithm for the evaluation of the two-point functions as linear combinations of Legendre functions of the second kind. This algorithm can be easily implemented on a computer. For the sake of illustration, we displayed the results for the case of symmetric traceless tensor fields with rank up to l=4.