A Perspective on Classical Strings from Complex Sine-Gordon Solitons

TL;DR

This work presents a comprehensive CsG-based construction of classical strings with large spins on $\mathbb{R}_t\times \mathrm{S}^3$ within $\mathrm{AdS}_5\times \mathrm{S}^5$, derived via Pohlmeyer reduction to CsG and Lamé equations. It provides explicit helical-string solutions for both one and two spins, categorized as Type (i) and Type (ii), and analyzes their closedness, conserved charges, and limiting behavior. The solutions interpolate between the GKP folded/circular strings and dyonic giant magnons (HM limit), with stationary and uniform-density limits recovered as special cases, revealing a unifying framework across BMN/Frolov-Tseytlin and HM regimes. The results point to a finite-gap interpretation with two-cut spectral data and suggest connections to gauge-theory solitons and potential extensions to three-spin configurations, highlighting the broader integrable-structure implications for AdS/CFT correspondence.

Abstract

We study a family of classical string solutions with large spins on R x S^3 subspace of AdS_5 x S^5 background, which are related to Complex sine-Gordon solitons via Pohlmeyer's reduction. The equations of motion for the classical strings are cast into Lame equations and Complex sine-Gordon equations. We solve them under periodic boundary conditions, and obtain analytic profiles for the closed strings. They interpolate two kinds of known rigid configurations with two spins: on one hand, they reduce to folded or circular spinning/rotating strings in the limit where a soliton velocity goes to zero, while on the other hand, the dyonic giant magnons are reproduced in the limit where the period of a kink-array goes to infinity.

Figures (6)

• Figure 1: Type (i) helical solution with a single spin. The diagram shows $k=0.68$ and $n=8$ case. Each turning points are located away from the equator, and each segment curves inwards.
• Figure 2: Type (i) helical solution with a single spin ; $\omega=0$ and $k=0.75$. This can be regarded as a folded string of FT03, in which case $n$ represents the number of folds.
• Figure 3: Type (i) helical solution with a single spin, in the limit $k\to 1$. The diagram shows $n=8$ case, and it can be understood as an array of $n=8$ giant magnons.
• Figure 4: Type (ii) helical solution with a single spin. The diagram shows $k=0.68$ and $m=8$ case. As compared to the type (i) case, each segment curves outwards about the northern pole.
• Figure 5: Type (ii) helical solution with a single spin, with $\omega=0$. This can be regarded as a circular string of FT03, in which case $m/2$ represents the winding number along a great circle.