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Generating AdS String Solutions

Antal Jevicki, Kewang Jin, Chrysostomos Kalousios, Anastasia Volovich

TL;DR

The paper develops a comprehensive, integrability-based framework for constructing classical string solutions in AdS spacetimes by employing the Pohlmeyer reduction to a generalized sinh-Gordon model and leveraging inverse scattering to build AdS_3 embeddings from sinh-Gordon data. It explicitly demonstrates a one-to-one correspondence between spikes of AdS_3 strings and sinh-Gordon solitons, showing that the long-string (GKP) limit corresponds to a two-soliton configuration and that solutions for solitons, antisolitons, and breathers can be generated via both the inverse scattering and the AdS dressing methods. The work provides detailed constructions of vacuum, one- and two-soliton, and breather solutions, including their Minkowskian and Euclidean embeddings, energy-spin relations, and boundary behavior. These results advance the toolbox for generating classical AdS string solutions and have potential implications for understanding strong-coupling aspects of AdS/CFT, such as gluon scattering amplitudes via minimal surfaces and their integrable structure.

Abstract

We use a Pohlmeyer type reduction to generate classical string solutions in AdS spacetime. In this framework we describe a correspondence between spikes in AdS_3 and soliton profiles of the sinh-Gordon equation. The null cusp string solution and its closed spinning string counterpart are related to the sinh-Gordon vacuum. We construct classical string solutions corresponding to sinh-Gordon solitons, antisolitons and breathers by the inverse scattering technique. The breather solutions can also be reproduced by the sigma model dressing method.

Generating AdS String Solutions

TL;DR

The paper develops a comprehensive, integrability-based framework for constructing classical string solutions in AdS spacetimes by employing the Pohlmeyer reduction to a generalized sinh-Gordon model and leveraging inverse scattering to build AdS_3 embeddings from sinh-Gordon data. It explicitly demonstrates a one-to-one correspondence between spikes of AdS_3 strings and sinh-Gordon solitons, showing that the long-string (GKP) limit corresponds to a two-soliton configuration and that solutions for solitons, antisolitons, and breathers can be generated via both the inverse scattering and the AdS dressing methods. The work provides detailed constructions of vacuum, one- and two-soliton, and breather solutions, including their Minkowskian and Euclidean embeddings, energy-spin relations, and boundary behavior. These results advance the toolbox for generating classical AdS string solutions and have potential implications for understanding strong-coupling aspects of AdS/CFT, such as gluon scattering amplitudes via minimal surfaces and their integrable structure.

Abstract

We use a Pohlmeyer type reduction to generate classical string solutions in AdS spacetime. In this framework we describe a correspondence between spikes in AdS_3 and soliton profiles of the sinh-Gordon equation. The null cusp string solution and its closed spinning string counterpart are related to the sinh-Gordon vacuum. We construct classical string solutions corresponding to sinh-Gordon solitons, antisolitons and breathers by the inverse scattering technique. The breather solutions can also be reproduced by the sigma model dressing method.

Paper Structure

This paper contains 12 sections, 75 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Pohlmeyer reduction for AdS strings
  3. Constructing string solutions from sinh-Gordon solutions
  4. Review of sinh-Gordon solutions
  5. String solutions
  6. Vacuum
  7. Long strings in $AdS_3$ as sinh-Gordon solitons
  8. One-soliton solutions
  9. Two-soliton solutions
  10. $AdS$ dressing method
  11. Breather solution
  12. Conventions

Figures (3)

  • Figure 1: The vacuum solution in (a) Minkowskian and (b) Euclidean worldsheet plotted in $AdS_3$ coordinates. (c) Top view of Minkowskian vacuum solution. The boundary of the worldsheet touches the boundary of $AdS$ space.
  • Figure 2: The one-soliton solution in (a) Minkowskian worldsheet plotted in $AdS_3$ coordinates. (b) Top view of the Minkowskian one-soliton solution. Please note the curvature of the string changes with the evolution of time.
  • Figure 3: The Minkowskian two-soliton solution with $v={1 \over \sqrt{5}}$ at different global time (a) $t=0$, (b) $t=\pi /4$.