On $2k$-Hitchin's equations and Higgs bundles: a survey

TL;DR

This survey explores the geometric aspects of $2k$-Hitchin's equations within Higgs bundles, reviewing foundational concepts and analyzing properties, reductions, and related functionals to deepen understanding of these equations.

Contribution

It provides a comprehensive review of $2k$-Hitchin's equations, their reduction to simpler forms, and introduces a related functional, advancing the geometric understanding of Higgs bundles.

Findings

$2k$-Hitchin's equations reduce to two equations for Higgs bundles

A new functional related to $2k$-Hitchin's equations is introduced and analyzed

Connections between complex geometry, Yang-Mills theory, and Higgs bundles are clarified

Abstract

We study the $2k$-Hitchin equations introduced by Ward \cite{Ward 2} from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and $2k$-Hitchin's equations, we review some elementary facts on complex geometry and Yang-Mills theory. Then we study some properties of holomorphic vector bundles and Higgs bundles and we review the Hermite-Yang-Mills equations together with two functionals related to such equations. Using some geometric tools we show that, as far as Higgs bundles is concern, $2k$-Hitchin's equations are reduced to a set of two equations. Finally, we introduce a functional closely related to $2k$-Hitchin's equations and we study some of its basic properties.