Minimal surfaces in AdS space and Integrable systems

TL;DR

This paper explores the Pohlmeyer reduction of spacelike minimal surfaces in AdS$_5$, deriving the Lax pair in terms of the $A_3=D_3$ root system, and connects the area calculation to Hitchin systems and Painleve transcendents.

Contribution

It generalizes the affine Toda system for AdS$_5$ minimal surfaces using the $A_3=D_3$ root system and links the area to Hitchin moduli space, extending previous AdS$_4$ results.

Findings

Derived the Lax pair in terms of $A_3=D_3$ root system.

Connected the minimal surface area to Hitchin system moduli space.

Recovered solutions related to Painleve transcendents.

Abstract

We consider the Pohlmeyer reduction for spacelike minimal area worldsheets in AdS$_5$. The Lax pair for the reduced theory is found, and written entirely in terms of the $A_3=D_3$ root system, generalizing the $B_2$ affine Toda system which appears for the AdS$_4$ string. For the $B_2$ affine Toda system, we show that the area of the worlsheet is obtainable from the moduli space K\"ahler potential of a related Hitchin system. We also explore the Saveliev-Leznov construction for solutions of the $B_2$ affine Toda system, and recover the rotationally symmetric solution associated to Painleve transcendent.