On 2k-Hitchin’s equations and Higgs bundles: A survey

Abstract

We study the [Formula: see text]-Hitchin’s equations introduced by Ward from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and [Formula: see text]-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 are concerned, [Formula: see text]-Hitchin’s equations are reduced to a set of two equations. Finally, we introduce a functional closely related to [Formula: see text]-Hitchin’s equations and we study some of its basic properties.