A solvable string on a Lorentzian surface

TL;DR

This paper investigates a class of nonlinear sigma models with solvable Vessiot groups, enabling explicit solutions for harmonic maps from Minkowski space into certain Lorentzian metrics, including a blow-up example.

Contribution

It introduces a new class of Darboux integrable sigma models with solvable Vessiot groups and derives explicit solution formulas for harmonic maps into Lorentzian 2-metrics.

Findings

Existence of harmonic maps with finite-time blow-up.

Derivation of a hyperbolic Weierstrass representation formula.

Reduction of the Cauchy problem to quadrature due to solvable Vessiot group.

Abstract

It is shown that there are nonlinear sigma models which are Darboux integrable and possess a solvable Vessiot group in addition to those whose Vessiot groups are central extensions of semi-simple Lie groups. They govern harmonic maps between Minkowski space $\mathbb{R}^{1,1}$ and certain complete, non-constant curvature 2-metrics. The solvability of the Vessiot group permits a reduction of the general Cauchy problem to quadrature. We treat the specific case of harmonic maps from Minkowski space into a non-constant curvature Lorentzian 2-metric, $\boldsymbol{\lambda}$. Despite the completeness of $\boldsymbol{\lambda}$ we exhibit a Cauchy problem with real analytic initial data which blows up in finite time. We also derive a hyperbolic Weierstrass representation formula for all harmonic maps from $\mathbb{R}^{1,1}$ into $\boldsymbol{\lambda}$.