5 S ep 2 00 9 D - dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance

An exact correspondence is pointed out between conformal field theories in D dimensions and dual resonance models in D ′ dimensions, where D ′ may differ from D. Dual resonance models, pioneered by Veneziano, were forerunners of string theory. The analogs of scattering amplitudes in dual resonance models are called Mellin amplitudes; they depend on complex variables s ij which substitute for the Man-delstam variables on which scattering amplitudes depend. The Mellin amplitudes satisfy exact duality-i.e. meromorphy in s ij with simple poles in single variables, and crossing symmetry-and an appropriate form of factorization which is implied by operator product expansions (OPE). Duality is a D-independent property. The position of the leading poles in s 12 are given by the dimensions of fields in the OPE, but there are also satellites and the precise correspondence between fields in the OPE and the residues of these poles depends on D. Dimensional reduction and dimensional induction D → D ∓ 1 are discussed. Dimensional reduction leads to the appearence of Anti de Sitter space * Dedicated to Professor Ivan Todorov on the occasion of his 75 th anniversary