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Slow evolution of nearly-degenerate extremal surfaces

Andrei Mikhailov

TL;DR

The paper analyzes nearly-degenerate extremal surfaces in AdS5×S5, showing that slow evolution under a small tension can be described by a Hamiltonian flow on the moduli space of parametrized null-surfaces. By treating the tension as a perturbation with ε^2 = λ/J^2, the authors derive an averaged, long-term evolution equation and connect it to RG flow in the dual gauge theory, providing explicit formulas for the evolution in terms of geodesic data on AdS5 and S5. The framework unifies the null-surface geometry, the Jacobi equation, and spinning-string dynamics, and it clarifies how anomalous dimensions of large-charge operators arise from the secular drift. The results offer a concrete bridge between fast-moving string dynamics and the continuous spin-chain limit, with explicit examples like the two-spin solution illustrating the approach.

Abstract

It was conjectured recently that the string worldsheet theory for the fast moving string in AdS times a sphere becomes effectively first order in the time derivative and describes the continuous limit of an integrable spin chain. In this paper we will try to make this statement more precise. We interpret the first order theory as describing the long term evolution of the tensionless string perturbed by a small tension. The long term evolution is a Hamiltonian flow on the moduli space of periodic trajectories. It should correspond to the renormgroup flow on the field theory side.

Slow evolution of nearly-degenerate extremal surfaces

TL;DR

The paper analyzes nearly-degenerate extremal surfaces in AdS5×S5, showing that slow evolution under a small tension can be described by a Hamiltonian flow on the moduli space of parametrized null-surfaces. By treating the tension as a perturbation with ε^2 = λ/J^2, the authors derive an averaged, long-term evolution equation and connect it to RG flow in the dual gauge theory, providing explicit formulas for the evolution in terms of geodesic data on AdS5 and S5. The framework unifies the null-surface geometry, the Jacobi equation, and spinning-string dynamics, and it clarifies how anomalous dimensions of large-charge operators arise from the secular drift. The results offer a concrete bridge between fast-moving string dynamics and the continuous spin-chain limit, with explicit examples like the two-spin solution illustrating the approach.

Abstract

It was conjectured recently that the string worldsheet theory for the fast moving string in AdS times a sphere becomes effectively first order in the time derivative and describes the continuous limit of an integrable spin chain. In this paper we will try to make this statement more precise. We interpret the first order theory as describing the long term evolution of the tensionless string perturbed by a small tension. The long term evolution is a Hamiltonian flow on the moduli space of periodic trajectories. It should correspond to the renormgroup flow on the field theory side.

Paper Structure

This paper contains 13 sections, 97 equations.

Table of Contents

  1. Introduction.
  2. Perturbation of the degenerate surfaces ruled by the orbits of the light-like Killing vector.
  3. Summary of this section.
  4. General facts about the nearly-degenerate surfaces.
  5. A special case of the inhomogeneous Jacobi equation.
  6. Example: the two-spin solution.
  7. The general case: $V$ is not a Killing vector field.
  8. Hamiltonian approach to the fast moving strings.
  9. Summary.
  10. Explicit evolution equations.
  11. Interpretation in the dual field theory.
  12. Comment on the special case when $\Sigma(0)$ is generated by the orbits of $V$.
  13. Direct derivation from the Jacobi equation.