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

David J. Gross, Andrei Mikhailov, Radu Roiban

TL;DR

This work tests the BMN gauge/string duality by computing anomalous dimensions for operators with large R-charge in N=4 SYM. The authors establish the perturbative framework, demonstrate a finite J→∞ limit at two loops with the effective coupling λ/J^2, and conjecture an all-orders structure in which the anomalous dimension depends on hat{λ} = λ(e^{iφ}+e^{-iφ}-2). Through a partial resummation, they show the gauge-theory results agree with the string worldsheet predictions in the plane-wave background, providing evidence for perturbative gauge/string equivalence in this sector. The analysis highlights that the relevant small parameter is hat{λ} (or equivalently λ sinh^2(φ/2)) and opens avenues for exploring non-planar corrections and a possible derivation of interacting string dynamics from gauge theory.

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 $λ$ 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 $λ/J^2$ 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.

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

TL;DR

This work tests the BMN gauge/string duality by computing anomalous dimensions for operators with large R-charge in N=4 SYM. The authors establish the perturbative framework, demonstrate a finite J→∞ limit at two loops with the effective coupling λ/J^2, and conjecture an all-orders structure in which the anomalous dimension depends on hat{λ} = λ(e^{iφ}+e^{-iφ}-2). Through a partial resummation, they show the gauge-theory results agree with the string worldsheet predictions in the plane-wave background, providing evidence for perturbative gauge/string equivalence in this sector. The analysis highlights that the relevant small parameter is hat{λ} (or equivalently λ sinh^2(φ/2)) and opens avenues for exploring non-planar corrections and a possible derivation of interacting string dynamics from gauge theory.

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 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 keeping 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.

Paper Structure

This paper contains 21 sections, 67 equations, 22 figures.

Table of Contents

  1. Introduction.
  2. A new gauge fields $\leftrightarrow$ strings correspondence.
  3. Anomalous dimensions from quantum mechanics.
  4. Our paper.
  5. Anomalous dimension from Yang-Mills perturbation theory.
  6. General facts about the anomalous dimension.
  7. A prediction from the dual string theory.
  8. The anomalous dimension at one loop.
  9. The Feynman rules.
  10. Our notation for integrals.
  11. The anomalous dimension at one loop.
  12. One loop corrections to the propagator and the anomalous dimension.
  13. Two loops.
  14. Diagrams proportional to $e^{2i\varphi}$.
  15. Diagrams with $e^{i\varphi}$.
  16. ...and 6 more sections

Figures (22)

  • Figure 1: "Dynamical removal" of massless states.
  • Figure 2: Simplified diagramatic notation for operators.
  • Figure 3: Contribution to $c_{3,3}$.
  • Figure 4: Feynman rules relevant for our computations.
  • Figure 5: One loop diagram with the exchange of $Z$ and $\phi$.
  • ...and 17 more figures