D-dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance Models
Gerhard Mack
TL;DR
The paper explores which structural aspects of D-dimensional conformal field theories with anomalous dimensions are independent of spacetime dimension by mapping CFT correlators to dual resonance model amplitudes via Mellin amplitudes. It shows that Mellin amplitudes encode exact duality, crossing, and factorization in a way that mirrors OPE data, with pole positions tied to operator twists and residues carrying D-dependent details. The work discusses dimensional reduction and induction, the geometric emergence of Anti de Sitter space, and concrete models (e.g., φ^3 in 6+ε and N=4 SYM) to illustrate rising Regge-like trajectories and the absence of shadow poles under certain conditions. It also outlines a path to derive Mellin amplitudes from string theory and AdS/CFT, linking worldsheet dynamics to CFT data and suggesting a broader stringy underpinning for CFT structure.
Abstract
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 analog of scattering amplitudes are called Mellin amplitudes; they depend on complex variables which substitute for the Mandelstam variables on which scattering amplitudes depend. The Mellin amplitudes satisfy exact duality - i.e. meromorphy 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 positions of the leading poles are given by the dimensions of fields in the OPE; their residues depend on D and determine satellites. Dimensional reduction and induction D goes to D-1 and D+1 are discussed. Dimensional reduction leads to the appearence of Anti de Sitter space.