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

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.

24 pages; major changes and typos corrected; sections 2 to 5 have been rewritten; Proposition 3 has been added

References in corpus (7)

• Nonsingular solutions of Hitchin's equations for noncompact gauge groups
• On vanishing theorems for Higgs bundles
• Geometry of Solutions of Hitchin Equations on R^2
• On Gieseker stability for Higgs sheaves
• Integrable (2k)-Dimensional Hitchin Equations
• On an integrable deformation of Kapustin-Witten systems
• On a functional of Kobayashi for Higgs bundles