Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on fast moving strings

Andrei Mikhailov

TL;DR

This work develops a perturbative framework for fast moving strings in AdS5×S5 by focusing on null-surfaces and their symmetries. It identifies a hidden U(1)L symmetry whose generator acts as an action variable and argues that it encodes the length of the dual gauge theory spin chain via an infinite linear combination of Pohlmeyer local charges, in harmony with integrability. The paper provides both a perturbative construction extending U(1)L to nearly degenerate extremal surfaces and a perturbative justification that U(1)L is in involution with the known local charges, thereby linking string dynamics to the spin chain description. It also outlines how the slow evolution of null-surfaces reduces to a Landau-Lifshitz type system, reinforcing the integrable structure bridging AdS string theory and N=4 SYM in the large R-charge regime.

Abstract

We review the recent work on the mechanics of fast moving strings in anti-de Sitter space times a sphere and discuss the role of conserved charges. An interesting relation between the local conserved charges of rigid solutions was found in the earlier work. We propose a generalization of this relation for arbitrary solutions, not necessarily rigid. We conjecture that an infinite combination of local conserved charges is an action variable generating periodic trajectories in the classical string phase space. It corresponds to the length of the operator on the field theory side.

Notes on fast moving strings

TL;DR

This work develops a perturbative framework for fast moving strings in AdS5×S5 by focusing on null-surfaces and their symmetries. It identifies a hidden U(1)L symmetry whose generator acts as an action variable and argues that it encodes the length of the dual gauge theory spin chain via an infinite linear combination of Pohlmeyer local charges, in harmony with integrability. The paper provides both a perturbative construction extending U(1)L to nearly degenerate extremal surfaces and a perturbative justification that U(1)L is in involution with the known local charges, thereby linking string dynamics to the spin chain description. It also outlines how the slow evolution of null-surfaces reduces to a Landau-Lifshitz type system, reinforcing the integrable structure bridging AdS string theory and N=4 SYM in the large R-charge regime.

Abstract

We review the recent work on the mechanics of fast moving strings in anti-de Sitter space times a sphere and discuss the role of conserved charges. An interesting relation between the local conserved charges of rigid solutions was found in the earlier work. We propose a generalization of this relation for arbitrary solutions, not necessarily rigid. We conjecture that an infinite combination of local conserved charges is an action variable generating periodic trajectories in the classical string phase space. It corresponds to the length of the operator on the field theory side.

Paper Structure

This paper contains 19 sections, 51 equations, 3 figures.

Table of Contents

  1. Introduction.
  2. Null-surfaces.
  3. The definition.
  4. The moduli space of null-surfaces.
  5. Parametrized null-surfaces.
  6. The symplectic structure.
  7. Nearly-degenerate extremal surfaces and the role of the engineering dimension.
  8. Definition of $U(1)_L$.
  9. Action of $U(1)_L$ on nearly-degenerate extremal surfaces.
  10. Particle on a sphere.
  11. String on a sphere.
  12. Length of the operator and local conserved charges.
  13. Local conserved charges.
  14. Local conserved charges are invariant under $U(1)_L$.
  15. A geometrical meaning of $U(1)_L$.
  16. ...and 4 more sections

Figures (3)

  • Figure 1: A null-geodesic in $AdS_5\times S^5$ is specified by the choice of an equator $\bf E$ in $S^5$, a time-like geodesic $\bf T$ in $AdS_5$ and a map $F:{\bf T}\to {\bf E}$ which maps the angular parameter $\psi$ on the equator to the time $t$ on the geodesic, up to a constant.
  • Figure 2: A picture of a null-surface in $AdS_5\times S^5$. A null-surface is a two-dimensional surface with the degenerate metric, ruled by the light rays. We have shown five light rays and a spacial contour with a parameter $\sigma$. One can visualize the null-surface as the surface swept by the spacial contour as it moves along the light rays.
  • Figure 3: The action of $U(1)_L$ on the null-surface. The symmetry acts only on the $S^5$-part of the null-string. Each point shifts by the same angle along the equator which is the projection to $S^5$ of the corresponding light ray.