The definition may be phrased for a connection on a vector bundle or principal bundle, with the two perspectives being essentially interchangeable. Here the definition of principal bundles is presented, which is the form that appears in Hitchin's work.[1][5][6]
Let be a principal -bundle for a compact real Lie group over a compact Riemann surface. For simplicity we will consider the case of or , the special unitary group or special orthogonal group. Suppose is a connection on , and let be a section of the complex vector bundle , where is the complexification of the adjoint bundle of , with fibre given by the complexification of the Lie algebra of . That is, is a complex -valued -form on . Such a is called a Higgs field in analogy with the auxiliary Higgs field appearing in Yang–Mills theory.
For a pair , Hitchin's equations[1] assert that
where is the curvature form of , is the -part of the induced connection on the complexified adjoint bundle , and is the commutator of -valued one-forms in the sense of Lie algebra-valued differential forms.
Since is of type , Hitchin's equations assert that the -component . Since , this implies that is a Dolbeault operator on and gives this Lie algebra bundle the structure of a holomorphic vector bundle. Therefore, the condition means that is a holomorphic -valued -form on . A pair consisting of a holomorphic vector bundle with a holomorphic endomorphism-valued -form is called a Higgs bundle, and so every solution to Hitchin's equations produces an example of a Higgs bundle.
Hitchin's equations can be derived as a dimensional reduction of the Yang–Mills equations from four dimension to two dimensions. Consider a connection on a trivial principal -bundle over . Then there exists four functions such that
where are the standard coordinate differential forms on . The self-duality equations for the connection , a particular case of the Yang–Mills equations, can be written
where is the curvature two-form of . To dimensionally reduce to two dimensions, one imposes that the connection forms are independent of the coordinates on . Thus the components define a connection on the restricted bundle over , and if one relabels , then these are auxiliary -valued fields over .
If one now writes and where is the standard complex -form on , then the self-duality equations above become precisely Hitchin's equations. Since these equations are conformally invariant on , they make sense on a conformal compactification of the plane, a Riemann surface.