Higgs bundle

In mathematics, a Higgs bundle is a pair ${\displaystyle (E,\varphi )}$ consisting of a holomorphic vector bundle E and a Higgs field ${\displaystyle \varphi }$, a holomorphic 1-form taking values in the bundle of endomorphisms of E such that ${\displaystyle \varphi \wedge \varphi =0}$. Such pairs were introduced by Nigel Hitchin (1987),^[1] who named the field ${\displaystyle \varphi }$ after Peter Higgs because of an analogy with Higgs bosons. The term 'Higgs bundle', and the condition ${\displaystyle \varphi \wedge \varphi =0}$ (which is vacuous in Hitchin's original set-up on Riemann surfaces) was introduced later by Carlos Simpson.^[2]

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.