1. Relativistic and nonrelativistic holography

Abstract

The AdS/CFT correspondence was introduced by Maldacena 20 years ago [433]. Soon important contributions were made by Gubser-Klebanov-Polyakov [293] and by Witten [572]. We recall the two ingredients of the AdS/CFT correspondence [433, 293, 572]: 1. the holography principle, which is very old, and means the reconstruction of some objects in the bulk (which may be classical or quantum) from some objects on the boundary; 2. the reconstruction of quantum objects, like 2-point functions on the boundary, from appropriate actions on the bulk. Here we give a group-theoretic interpretation of the AdS/CFT correspondence as a relation of a representation equivalence between representations of the conformal group describing the bulk AdS fields ø, their boundary fields ø₀ and the boundary conformal operators O coupled to the latter. We use two kinds of equivalences. The first kind is the equivalence between the representations describing the bulk fields and the boundary fields and it is established here. The second kind is the equivalence between conjugated conformal representations related by Weyl reflection, e. g. the coupled fields ø₀ and O. Operators realizing the first kind of equivalence for special cases were actually given by Witten and others-here they are constructed in a more general setting from the requirement that they are intertwining operators. The intertwining operators realizing the second kind of equivalence are provided by the standard conformal two-point functions. Using both equivalences we find that the bulk field has in fact two boundary fields, namely, the coupled fields ø₀ and O, the limits being governed by the corresponding conjugated conformal weights. In this chapter we give a group-theoretic interpretation of relativistic holography as equivalence between representations in three cases: 1. the Euclidean conformal (or de Sitter) group; 2. the anti de Sitter group SO(3,2); 3. the Schrödinger group. In each case we give explicitly boundary-to-bulk operators and we show that these operators and the easier bulk-to-boundary operators are intertwining operators. Furthermore, we show that each bulk field has two boundary (shadow) fields with conjugated conformal weights. These fields are related by another intertwining operator, given by a two-point function on the boundary.