On the propagator of a scalar field in AdS x S and in its plane wave limit

TL;DR

This paper analyzes scalar propagators in AdS x S backgrounds, deriving explicit formulas for special cases, exploring their source structures, and relating them to plane wave limits, with implications for understanding field behavior in curved spacetimes.

Contribution

It provides explicit formulas for scalar propagators in AdS x S, analyzes their source structures, and connects these to plane wave limits, including a new summation theorem for special functions.

Findings

Explicit propagator formulas for conformally flat cases.

Relation of AdS x S propagators to plane wave limits.

A new theorem for summing Legendre and Gegenbauer functions.

Abstract

We discuss the scalar propagator on generic AdS x S backgrounds. For the conformally flat situations and masses corresponding to Weyl invariant actions, the propagator is powerlike in the sum of the chordal distances with respect to AdS(d+1) and S(d'+1). In these cases we analyze its source structure. In all other cases the propagator depends on both chordal distances separately. There an explicit formula is found for certain special mass values. For pure AdS we show how the well known propagators in the Weyl invariant case can be expressed as linear combinations of simple powers of the chordal distance. For AdS(5) x S(5) we relate our propagator to the expression in the plane wave limit and find a geometric interpretation of the variables occurring in the known explicit construction on the plane wave. As a byproduct of comparing different techniques, including the KK mode summation, a theorem for summing certain products of Legendre and Gegenbauer functions is derived.