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Plane wave limit of local conserved charges

Andrei Mikhailov

TL;DR

The paper investigates how the infinite family of local Pohlmeyer charges organizes into an action variable for the classical string in $AdS_5\times S^5$ by exploiting the plane-wave (BMN) limit. Using Bäcklund transformations and the generating function $\mathcal{E}(\gamma)$, it shows that in the plane-wave regime the Pohlmeyer charges reduce to the local integrals of motion of the free massive theory, and that the string's oscillator number is an explicit infinite linear combination of these charges. A key result is an integral representation linking the action variable to $\mathcal{E}(\gamma)$, and the identification of an infinite set of improved currents $\mathcal{G}_k$ that express the action variable as a sum of local charges. This construction provides a concrete bridge between classical integrability, plane-wave limits, and the computation of anomalous dimensions within the AdS/CFT correspondence.

Abstract

We study the plane wave limit of the Backlund transformations for the classical string in AdS space times a sphere and obtain an explicit expression for the local conserved charges. We show that the Pohlmeyer charges become in the plane wave limit the local integrals of motion of the free massive field. This fixes the coefficients in the expansion of the anomalous dimension as the sum of the Pohlmeyer charges.

Plane wave limit of local conserved charges

TL;DR

The paper investigates how the infinite family of local Pohlmeyer charges organizes into an action variable for the classical string in by exploiting the plane-wave (BMN) limit. Using Bäcklund transformations and the generating function , it shows that in the plane-wave regime the Pohlmeyer charges reduce to the local integrals of motion of the free massive theory, and that the string's oscillator number is an explicit infinite linear combination of these charges. A key result is an integral representation linking the action variable to , and the identification of an infinite set of improved currents that express the action variable as a sum of local charges. This construction provides a concrete bridge between classical integrability, plane-wave limits, and the computation of anomalous dimensions within the AdS/CFT correspondence.

Abstract

We study the plane wave limit of the Backlund transformations for the classical string in AdS space times a sphere and obtain an explicit expression for the local conserved charges. We show that the Pohlmeyer charges become in the plane wave limit the local integrals of motion of the free massive field. This fixes the coefficients in the expansion of the anomalous dimension as the sum of the Pohlmeyer charges.

Paper Structure

This paper contains 14 sections, 51 equations.

Table of Contents

  1. Introduction.
  2. $AdS_5\times S^5$ and the plane wave.
  3. A special choice of coordinates in $AdS_5\times S^5$.
  4. An approximate solution for the classical string.
  5. Relation between $\mu$ and angular momentum.
  6. Bäcklund transformations and Pohlmeyer charges.
  7. Bäcklund transformations.
  8. Pohlmeyer charges.
  9. A relation between ${\cal E}_+$ and ${\cal E}_-$.
  10. An action variable in the plane wave limit.
  11. Action variable and local conserved charges of the free massive theory.
  12. Integral formula.
  13. Expansion in powers of $n\over \mu$.
  14. An infinite linear combination of local conserved charges.