Intertwining operator realization of the AdS/CFT correspondence

Abstract

We give a group-theoretic interpretation of the AdS/CFT correspondence as relation of representation equivalence between representations of the conformal group describing the bulk AdS fields φ their boundary fields φ 0 and the coupled to the latter boundary conformal operators O . We use two kinds of equivalences. The first kind is 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 φ 0 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 φ 0 and O , the limits being governed by the corresponding conjugated conformal weights d − Δ and Δ. Thus, from the viewpoint of the bulk-to-boundary correspondence the coupled fields φ 0 and O are generically on an equal footing. Our setting is more general since our bulk fields are described by representations of the Euclidean conformal group, i.e. the de Sitter group G = SO( d+1, 1), induced from representations τ of the maximal compact subgroup SO( d + 1) of G. From these large reducible representations we can single out representations which are equivalent to conformal boundary representations labeled by the conformal weight and by arbitrary representations μ of the Euclidean Lorentz group M = SO( d), such that μ is contained in the restriction of τ to M. Thus, our boundary ↔ bulk operators can be compared with those in the literature only when for a fixed At we consider a 'minimal' representation τ = τ( μ) containing μ. We also relate the boundary → bulk normalization constant to the Plancherel measure for G.