General Leznov-Savelev solutions for Pohlmeyer reduced AdS$_5$ minimal surfaces

TL;DR

This paper develops a Lax formulation for the Pohlmeyer reduced sigma model describing AdS$_5$ minimal surfaces, enabling formal solutions similar to affine Toda models through a Laznov-Savelev analysis.

Contribution

It introduces a conformal extension with a Lax connection valued in a ${ m Z}_4$-invariant subalgebra of $ ext{su}(4)$, providing a new method to obtain formal solutions for the model.

Findings

Lax connection is valued in a ${ m Z}_4$-invariant subalgebra of $ ext{su}(4)$.

Formal expressions for general solutions are derived using a Laznov-Savelev analysis.

The approach relies on a specific algebraic decomposition for exponentiated elements.

Abstract

We consider the Pohlmeyer reduced sigma model describing AdS$_5$ minimal surfaces. We show that, similar to the affine Toda models, there exists a conformal extension to this model which admits a Lax formulation. The Lax connection is shown to be valued in a ${\mathbb Z}_4$-invariant subalgebra of the affine Lie algebra $\widehat{su(4)}$. Using this, we perform a modified version of a Laznov-Savelev analysis, which allows us to write formal expressions for the general solutions for the Pohlmeyer reduced AdS$_5$ theory. This analysis relies on the a certain decomposition for the exponentiated algebra elements.