In mathematics, a Higgs bundle is a pair
(E,\varphi)
\varphi
\varphi\wedge\varphi=0
\varphi
\varphi\wedge\varphi=0
A Higgs bundle can be thought of as a "simplified version" of a flat holomorphic connection on a holomorphic vector bundle, where the derivative is scaled to zero. The nonabelian Hodge correspondence says that, under suitable stability conditions, the category of flat holomorphic connections on a smooth projective complex algebraic variety, the category of representations of the fundamental group of the variety, and the category of Higgs bundles over this variety are actually equivalent. Therefore, one can deduce results about gauge theory with flat connections by working with the simpler Higgs bundles.
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.
Of particular interest in the theory of Higgs bundles is the notion of a stable Higgs bundle. To do so,
\varphi
In Hitchin's original discussion, a rank-1 subbundle labelled L is
\varphi
\varphi(L)\subsetL ⊗ K
K
(E,\varphi)
\varphi
L
E
\operatorname{deg}