Exactly solvable strings in Minkowski spacetime

TL;DR

This paper demonstrates that the equations governing cohomogeneity-one Nambu-Goto strings in Minkowski spacetime are integrable and exactly solvable, classifying all cases and providing explicit solutions.

Contribution

It classifies cohomogeneity-one strings in Minkowski spacetime and proves the integrability and exact solvability of their equations of motion across all classes.

Findings

All geodesic equations are integrable.

Explicit solutions are derived for all string classes.

Integrability is linked to Killing vectors and tensors.

Abstract

We study the integrability of the equations of motion for the Nambu-Goto strings with a cohomogeneity-one symmetry in Minkowski spacetime. A cohomogeneity-one string has a world surface which is tangent to a Killing vector field. By virtue of the Killing vector, the equations of motion can be reduced to the geodesic equation in the orbit space. Cohomogeneity-one strings are classified into seven classes (Types I to VII). We investigate the integrability of the geodesic equations for all the classes and find that the geodesic equations are integrable. For Types I to VI, the integrability comes from the existence of Killing vectors on the orbit space which are the projections of Killing vectors on Minkowski spacetime. For Type VII, the integrability is related to a projected Killing vector and a nontrivial Killing tensor on the orbit space. We also find that the geodesic equations of all types are exactly solvable, and show the solutions.