"Old" Conformal Bootstrap in the AdS/CFT Context

TL;DR

This paper revisits classical conformal bootstrap techniques within the modern AdS/CFT framework, deriving explicit correlators and spectra for scalar operators using Wightman-Witten diagrams and Schwinger-Dyson equations.

Contribution

It integrates historical conformal bootstrap methods with AdS/CFT, providing explicit correlators and spectra for primary operators in scalar field models.

Findings

Derived explicit four-point correlators for primary operators.

Calculated spectra of Bethe-Salpeter equations.

Connected old bootstrap techniques with modern AdS/CFT methods.

Abstract

The Schwinger-Dyson equations for Green functions and 3-vertexes in planar approximation used 50 years ago in the earlier theory of conformal bootstrap are written in frames of the AdS/CFT correspondence. The technique of "Wightman-Witten" diagrams is used in calculations of the bubble bulk diagrams that permitted to obtain the values of conformal dimensions of primary operators in two models of the interacting bulk scalar fields which conformal dimensions obey "extremal" relation. Simple expressions for the four-point tree "Wightman-Witten" correlators of these primary operators are obtained and spectra of corresponding Bethe-Salpeter equations are calculated.