Operators with large R charge in N=4 Yang-Mills theory

TL;DR

This paper confirms through perturbative calculations in N=4 Yang-Mills theory that the anomalous dimensions of certain operators match string theory predictions in the plane wave limit, supporting the gauge/string duality.

Contribution

It provides perturbative evidence that the anomalous dimensions of operators with large R charge in N=4 SYM agree with string theory results, validating the duality at weak coupling.

Findings

Anomalous dimensions are finite as R charge J approaches infinity.

Two-loop calculations match string theory predictions.

Summation of specific Feynman diagrams supports the duality at all orders.

Abstract

It has been recently proposed that string theory in the background of a plane wave corresponds to a certain subsector of the N=4 supersymmetric Yang-Mills theory. This correspondence follows as a limit of the AdS/CFT duality. As a particular case of the AdS/CFT correspondence, it is a priori a strong/weak coupling duality. However, the predictions for the anomalous dimensions which follow from this particular limit are analytic functions of the 't Hooft coupling constant $\lambda$ and have a well defined expansion in the weak coupling regime. This allows one to conjecture that the correspondence between the strings on the plane wave background and the Yang-Mills theory works at the level of perturbative expansions. In our paper we perform perturbative computations in the Yang-Mills theory that confirm this conjecture. We calculate the anomalous dimension of the operator corresponding to the elementary string excitation. We verify at the two loop level that the anomalous dimension has a finite limit when the R charge $J\to \infty$ keeping $\lambda/J^2$ finite. We conjecture that this is true at higher orders of perturbation theory. We show, by summing an infinite subset of Feynman diagrams, under the above assumption, that the anomalous dimensions arising from the Yang-Mills perturbation theory are in agreement with the anomalous dimensions following from the string worldsheet sigma-model.