Hitchin's equations and integrability of BPS Z(N) strings in Yang-Mills theories

TL;DR

This paper demonstrates that BPS Z(N) string equations in Yang-Mills theories are equivalent to Hitchin's equations and integrable systems, providing a unified framework for solutions across various gauge groups and vacua.

Contribution

It establishes the equivalence between BPS Z(N) string equations and Hitchin's equations, and constructs general solutions for arbitrary simple gauge groups with non-trivial centers.

Findings

BPS Z(N) string equations are equivalent to Hitchin's equations.

Different vacua lead to various integrable field equations.

Special vacuum yields affine Toda field theory.

Abstract

We show that Z(N) string's BPS equations are equivalent to the Hitchin's equations (or self-duality equation) and also to the zero curvature condition. We construct a general form for BPS Z(N) string solutions for arbitrary simple gauge groups with non-trivial center. Depending on the vacuum solutions considered, the Z(N) string's BPS equations reduce to different two dimensional integrable field equations. For a particular vacuum we obtain the equation of affine Toda field theory.