Higgs bundle

Higgs bundles were first introduced by Hitchin in 1987, for the specific case where the holomorphic vector bundle E is over a compact Riemann surface. Further, Hitchin's paper mostly discusses the case where the vector bundle is rank 2 (that is, the fiber is a 2-dimensional vector space). The rank 2 vector bundle arises as the solution space to Hitchin's equations for a principal SU(2)-bundle. The theory on Riemann surfaces was generalized by Carlos Simpson to the case where the base manifold is compact and Kähler. Restricting to the dimension one case recovers Hitchin's theory. == Stability of a Higgs bundle ==

Of particular interest in the theory of Higgs bundles is the notion of a stable Higgs bundle. To do so, \varphi-invariant subbundles must first be defined. In Hitchin's original discussion, a rank-1 subbundle labelled L is \varphi-invariant if \varphi(L) \subset L \otimes K with K the canonical bundle over the Riemann surface M. Then a Higgs bundle (E, \varphi) is stable if, for each \varphi invariant subbundle L of E, \operatorname{deg} L with \operatorname{deg} being the usual notion of degree for a complex vector bundle over a Riemann surface. ==See also==