Integrable (2k)-Dimensional Hitchin Equations

Abstract

This letter describes a completely-integrable system of Yang-MillsHiggs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k = 2 case are described.MSC: 81T13, 53C26.

On the Moyal deformation of Kapustin-Witten systems

“…It is already known that these set of equations are integrable, thus spoiling nothing when we perform such deformation; also, solutions to these equations together with its corresponding deformations were presented. The possibility of carrying out such deformation in Hitchin's systems on R 2 [29] and in dimensions greater than two [4] are being explored and will be reported in a future communication.…”

“…This form of KW equations have been considered recently by Gagliardo and Uhlenbeck [7], Dunajski and Hoegner [8] and Ward [4], in particular in the last two references the equations (3) are also called the non-abelian Seiberg-Witten equations. From now on, we will refer to (1) and (3) as the KW equations and the non-abelian Seiberg-Witten equations, respectively.…”

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

“…This article is a review article on Higgs bundles and a set of equations in mathematical physics, recently introduced by Ward [35] and which are usually known as the 2k-Hitchin equations. The purpose of the article is to begin an study of these equations using complex geometry.…”

“…In Section 1 we give an historical introduction to the Hitchin equations, Higgs bundles and the 2k-Hitchin equations; this section is not intended to be a rigorous or an exhaustive introduction, however we would like to give a general overview on these topics. More details on Higgs bundles and 2k-Hitchin equations can be found in the articles of Simpson [30,31] and Ward [35], respectively. A reader familiarized with supersymmetric Yang-Mills theories and string theory can be found an introduction to Higgs bundles in [37].…”

“…We also introduce a couple of functionals closely related to such equations; namely the full Yang-Mills-Higgs functional and the Kobayashi functional and we show that there exists a non trivial relation between these functionals. In Section 5 we review the 2k-Hitchin equations of Ward [35] from the point of view of a holomorphic vector bundle E −→ X, with X a compact Kähler manifold and we show that such equations can be seen as a set of four equations whose variables are pairs (h, Φ), with h an hermitian metric in E and Φ an holomorphic form of type (1, 0) of X with values in the bundle of endomorphisms of E. At this point, we will see that two of the equations can be "formally" solved if we consider the Chern connection and a Higgs field on E. Therefore, as far as Higgs bundles is concern, the 2k-Hitchin equations can be reduced to a set of only two equations defined in the space of hermitian metrics in the Higgs bundle. The remaining equations have some resemblance with Seiberg-Witten equations, and hence, following this resemblance we propose a natural functional H(h) associated to the 2k-Hitchin equations on a Higgs bundle; we call such a functional the non-abelian Seiberg-Witten functional.…”