On gauge fields - strings duality as an integrable system

TL;DR

This paper demonstrates that the equations of motion for a string theory in a negatively curved space are integrable, providing a geometric framework that connects gauge fields and string duality through a Lax pair formulation.

Contribution

It introduces a geometric framework showing the integrability of string equations of motion related to gauge/string duality, emphasizing the importance of boundary conditions.

Findings

Equations of motion are integrable via a Lax pair with spectral parameter.

The loop equation depends critically on the proper boundary conditions.

The framework links gauge fields and string duality through geometric methods.

Abstract

It was suggested in hep-th/0002106, that semiclassically, a partition function of a string theory in the 5 dimensional constant negative curvature space with a boundary condition at the absolute satisfy the loop equation with respect to varying the boundary condition, and thus the partition function of the string gives the expectation value of a Wilson loop in the 4 dimensional QCD. In the paper, we present the geometrical framework, which reveals that the equations of motion of such string theory are integrable, in the sense that they can be written via a Lax pair with a spectral parameter. We also show, that the issue of the loop equation rests solely on the properly posing the boundary condition.