On the propagator of a scalar field on AdS× S and on the BMN plane wave

We discuss the scalar propagator on generic AdSd+1 × S ′+1 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 AdSd+1 and S d′+1. 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 AdS5 × S we relate our propagator to the expression in the BMN 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. [email protected], [email protected], [email protected]