On three-point correlation functions in the gauge/gravity duality

TL;DR

The paper develops a framework linking 3-point couplings in a CFT to linear deformations by a marginal or irrelevant operator ${ t D}$, showing that divergences generated by the deformation encode the couplings $a_{{ t D}AB}$ via the deformed anomalous dimension matrix $oldsymbol{ m \Gamma}$. In ${ m N}=4$ SYM, integrability is used to compute couplings $a_{{ t L}AB}$ for the Lagrangian deformation in the SU(2) sector, with the key relation $2\pi^2 a_{{\tt L}AA}=-g^2\partial_{g^2}\sum_j \gamma_j(g^2)$. At strong coupling, AdS/CFT is employed by inserting a light supergravity field on the heavy string worldsheet, yielding $ig\langle{ t O}_A{ t O}_A{ t D}\big\rangle$ in terms of a bulk tadpole integral $I_{ t D}[X,s;y]$, and the results for several string configurations (point-like, circular, giant magnon, spinning) are shown to reproduce the RG expectations $2\pi^2 a_{{\tt L}AA} \approx -g^2\partial \Delta_A/\partial g^2$ in the appropriate limits. This dual weak/strong coupling analysis demonstrates a concrete route to determine three-point couplings from integrability and holography, with agreement to renormalization group arguments. The work also outlines open problems in extending to more general deformations and operator bases, potentially unveiling new integrability structures in 3-point functions.

Abstract

We study the effect of marginal and irrelevant deformations on the renormalization of operators near a CFT fixed point. New divergences in a given operator are determined by its OPE with the operator D that generates the deformation. This provides a scheme to compute the couplings a_DAB between the operator D and two arbitrary operators O_A and O_B. We exemplify for the case of N=4 SYM, considering the simplest case of the exact Lagrangian deformation. In this case the deformed anomalous dimension matrix is determined by the derivative of the anomalous dimension matrix with respect to the coupling. We use integrability techniques to compute the one-loop couplings a_LAB between the Lagrangian and two distinct large operators built with Magnons, in the SU(2) sector of the theory. Then we consider a_DAA at strong coupling, and show how to compute it using the gauge/gravity duality, when D is a chiral operator dual to any supergravity field and O_A is dual to a heavy string state. We exemplify for the Lagrangian and operators O_A dual to heavy string states, showing agreement with the prediction derived from the renormalization group arguments.

Figures (1)

• Figure 1: Witten diagram for a 3-point function that represents a heavy string state interacting with a light supergravity field. Note that the heavy string line actually spans a two-dimensional worldsheet, whose classical saddle point determines the behaviour of the partition function, as explained in Janik:2010gc. To leading order, the string worldsheet acts as a tadpole for the supergravity fluctuations.