Segmented strings from a different angle

TL;DR

This paper introduces a novel discretization of segmented strings in AdS$_3$, linking their geometry to integrable Toda-type lattices and providing exact solutions and energy calculations.

Contribution

It generalizes segmented strings to AdS$_3$, showing their relation to integrable lattices and deriving exact area and energy formulas.

Findings

Exact discretization of string equations of motion in AdS$_3$

Expression of string area via cross-ratios and kink variables

Reduction to integrable Toda-type lattice equations

Abstract

Segmented strings in flat space are piecewise linear classical string solutions. Kinks between the segments move with the speed of light and their worldlines form a lattice on the worldsheet. This idea can be generalized to AdS$_3$ where the embedding is built from AdS$_2$ patches. The construction provides an exact discretization of the non-linear string equations of motion. This paper computes the area of segmented strings using cross-ratios constructed from the kink vectors. The cross-ratios can be expressed in terms of either left-handed or right-handed variables. The string equation of motion in AdS$_3$ is reduced to that of an integrable time-discretized relativistic Toda-type lattice. Positive solutions yield string embeddings that are unique up to global transformations. In the appendix, the Poincare target space energy is computed by integrating the worldsheet current along kink worldlines and a formula is derived for the integrated scalar curvature of the embedding.