Comments on contact terms and conformal manifolds in the AdS/CFT correspondence

TL;DR

This paper investigates contact terms in correlation functions of exactly marginal operators within the AdS/CFT framework, demonstrating that holographic methods accurately reproduce the expected conformal manifold structures and operator dimensions.

Contribution

It provides a comprehensive holographic analysis of contact terms and conformal manifolds, employing holographic RG and perturbative solutions to match CFT predictions.

Findings

Holographic RG captures the structure of contact terms in correlation functions.

Computed three- and four-point functions match CFT expectations.

Expressed anomalous dimensions of double trace operators in geometric terms.

Abstract

We study the contact terms that appear in the correlation functions of exactly marginal operators using the AdS/CFT correspondence. It is known that CFT with an exactly marginal deformation requires the existence of the contact terms with their coefficients having a geometrical interpretation in the context of conformal manifolds. We show that the AdS/CFT correspondence captures properly the mathematical structure of the correlation functions that is expected from the CFT analysis. For this purpose, we employ holographic RG to formulate a most general setup in the bulk for describing an exactly marginal deformation. The resultant bulk equations of motion are nonlinear and solved perturbatively to obtain the on-shell action. We compute three- and four-point functions of the exactly marginal operators using the GKP-Witten prescription, and show that they match with the expected results precisely. The cut-off surface prescription in the bulk serves as a regularization scheme for conformal perturbation theory in the boundary CFT. As an application, we examine a double OPE limit of the four-point functions. The anomalous dimensions of double trace operators are written in terms of the geometrical data of a conformal manifold.