Sinh-Gordon, Cosh-Gordon and Liouville Equations for Strings and Multi-Strings in Constant Curvature Spacetimes

TL;DR

This paper demonstrates that the fundamental quadratic form of classical string propagation in 2+1 dimensional constant curvature spacetimes satisfies the Sinh-Gordon, Cosh-Gordon, or Liouville equations, and constructs new multi-string solutions using elliptic functions.

Contribution

It shows that all three equations are necessary to describe generic string dynamics in these spacetimes and extends previous work limited to the Sinh-Gordon sector in de Sitter space.

Findings

All three equations are essential for complete string dynamics description.

Constructed new multi-string solutions using elliptic functions.

Results generalize to higher-dimensional constant curvature spacetimes.

Abstract

We find that the fundamental quadratic form of classical string propagation in $2+1$ dimensional constant curvature spacetimes solves the Sinh-Gordon equation, the Cosh-Gordon equation or the Liouville equation. We show that in both de Sitter and anti de Sitter spacetimes (as well as in the $2+1$ black hole anti de Sitter spacetime), {\it all} three equations must be included to cover the generic string dynamics. The generic properties of the string dynamics are directly extracted from the properties of these three equations and their associated potentials (irrespective of any solution). These results complete and generalize earlier discussions on this topic (until now, only the Sinh-Gordon sector in de Sitter spacetime was known). We also construct new classes of multi-string solutions, in terms of elliptic functions, to all three equations in both de Sitter and anti de Sitter spacetimes. Our results can be straightforwardly generalized to constant curvature spacetimes of arbitrary dimension, by replacing the Sinh-Gordon equation, the Cosh-Gordon equation and the Liouville equation by higher dimensional generalizations.