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A New Double-Scaling Limit of N=4 Super Yang-Mills Theory and PP-Wave Strings

C. Kristjansen, J. Plefka, G. W. Semenoff, M. Staudacher

TL;DR

The paper investigates a double-scaling limit of N=4 SYM that holographically corresponds to a pp-wave background in string theory. Using exact matrix-model techniques and combinatorial analysis, it shows that non-planar diagrams survive this limit and computes all-genus contributions to two- and three-point functions of chiral primaries, as well as the momentum-carrying operators that map to string excitations. It demonstrates non-planar mixing among these operators, requiring a redefinition that preserves orthogonality up to subleading order, and reveals non-planar corrections to their one-loop anomalous dimensions, signaling string-loop effects in the dual pp-wave description. The results provide a concrete realization of double-scaling dynamics in a holographic setting and establish precise scaling functions governing genus contributions in correlators.

Abstract

The metric of a spacetime with a parallel plane (pp)-wave can be obtained in a certain limit of the space AdS^5xS^5. According to the AdS/CFT correspondence, the holographic dual of superstring theory on that background should be the analogous limit of N=4 supersymmetric Yang-Mills theory. In this paper we shall show that, contrary to widespread expectation, non-planar diagrams survive this limiting procedure in the gauge theory. Using matrix model techniques as well as combinatorial reasoning it is demonstrated that a subset of diagrams of arbitrary genus survives and that a non-trivial double scaling limit may be defined. We exactly compute two- and three-point functions of chiral primaries in this limit. We also carefully study certain operators conjectured to correspond to string excitations on the pp-wave background. We find non-planar linear mixing of these proposed operators, requiring their redefinition. Finally, we show that the redefined operators receive non-planar corrections to the planar one-loop anomalous dimension.

A New Double-Scaling Limit of N=4 Super Yang-Mills Theory and PP-Wave Strings

TL;DR

The paper investigates a double-scaling limit of N=4 SYM that holographically corresponds to a pp-wave background in string theory. Using exact matrix-model techniques and combinatorial analysis, it shows that non-planar diagrams survive this limit and computes all-genus contributions to two- and three-point functions of chiral primaries, as well as the momentum-carrying operators that map to string excitations. It demonstrates non-planar mixing among these operators, requiring a redefinition that preserves orthogonality up to subleading order, and reveals non-planar corrections to their one-loop anomalous dimensions, signaling string-loop effects in the dual pp-wave description. The results provide a concrete realization of double-scaling dynamics in a holographic setting and establish precise scaling functions governing genus contributions in correlators.

Abstract

The metric of a spacetime with a parallel plane (pp)-wave can be obtained in a certain limit of the space AdS^5xS^5. According to the AdS/CFT correspondence, the holographic dual of superstring theory on that background should be the analogous limit of N=4 supersymmetric Yang-Mills theory. In this paper we shall show that, contrary to widespread expectation, non-planar diagrams survive this limiting procedure in the gauge theory. Using matrix model techniques as well as combinatorial reasoning it is demonstrated that a subset of diagrams of arbitrary genus survives and that a non-trivial double scaling limit may be defined. We exactly compute two- and three-point functions of chiral primaries in this limit. We also carefully study certain operators conjectured to correspond to string excitations on the pp-wave background. We find non-planar linear mixing of these proposed operators, requiring their redefinition. Finally, we show that the redefined operators receive non-planar corrections to the planar one-loop anomalous dimension.

Paper Structure

This paper contains 17 sections, 114 equations, 5 figures.

Table of Contents

  1. Introduction and Conclusions
  2. Double scaling limit of chiral primary two-point functions
  3. Three-point functions
  4. Computing momentum correlators
  5. Computing anomalous dimensions
  6. Notation
  7. One-loop integrals
  8. Cancellation of leading radiative corrections to 2-point correlator of chiral primary operators
  9. The first radiative correction to the anomalous dimension of unprotected operators to leading order in $\frac{J^2}{N}$
  10. The first radiative correction to the anomalous dimension of unprotected operators to two orders in $\frac{J^2}{N}$
  11. Complex matrix model technology
  12. Evaluating $C_{4Z}$,$C_{2\phi Z}$,$C_{4\phi}$ and $C_{SE}$
  13. $C_{4Z}$
  14. $C_{2\phi Z}$
  15. $C_{4\phi}$
  16. ...and 2 more sections

Figures (5)

  • Figure 1:
  • Figure 2: The contractions
  • Figure 3: The relevant graphs of the self energy, gluon exchange and four-point interaction contributing to the leading radiative corrections of two-point scalar field trace operators.
  • Figure 4: Feynman diagram for the free field limit of the correlator $\langle 0|{\rm Tr}Z^J(x)\, {\rm Tr} \bar{Z}^J(0) |0\rangle$. There are $J$ scalar propagators connecting the points $x$ and $0$.
  • Figure 5: Feynman diagrams contributing to the order $g^2_{{\rm YM}}$ corrections to the correlator $\langle 0| {\rm Tr}Z^J(x)\, {\rm Tr} \bar{Z}^J(0) |0\rangle$. There are either $J$ scalar lines with one insertion of a scalar self-energy sub-diagram or two lines connected by either a vector line with two three-point vertices or one four-point scalar vertex.