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

Abstract

We discuss the scalar propagator on generic AdS d+1 × S d ′ +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 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 × 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.

Bulk reconstruction of metrics with a compact space asymptotically

“…The series form of (B.18) may be understood as a Kaluza-Klein series expansion of the higher-dimensional bulk-to-bulk propagator. This expression reduces to a very compact form for conformally flat AdS n × S k , as is the case of (B.2), 27 if one chooses the scalar to be coupled to the metric in a Weyl invariant manner [32]. This is accomplished in (B.1) by choosing the mass of the scalar to be precisely…”

“…Although it is a natural object, G rarely appears in the literature (see [32] for an exception in global AdS n × S k ). Instead, propagators are commonly obtained after dimensionallyreducing spacetime and Kaluza-Klein expanding on the compact dimensions.…”

“…Without simplifying assumptions, the latter can be expressed as an infinite Kaluza-Klein series expansion as in (B.18). However for a conformally flat choice of radii in AdS n × S k and a specific mass term for the scalar corresponding to the Weyl invariant coupling to the scalar curvature, this infinite sum can be recast into the very simple form of (B.22) [32]. One example of this is the massless dilaton in AdS 5 × S 5 (with equal radii), since the scalar curvature vanishes.…”

“…The resulting propagator is simply a power-law in the total chordal distance along both AdS n and S k , viz. (see [32] for more details)…”

An infinite-dimensional family of black-hole microstate geometries

“…The scalar Green functions for AdS 3 × S 2 has been extensively discussed in [44] and here we simply follow their prescription. It also turns out that the particular AdS 3 × S 2 given by (3.17) has some extremely nice properties with respect to the procedures of [44].…”