# Higgs Bundles

# Hitchin's Equations

Let XX be a smooth compact Riemann surface, of genus g2g \geq 2. Higgs bundles originally came to be as solutions to Hitchin's equations, or "self-duality" equations on XX. These are self-dual, dimensionally-reduced Yang-Mills equations written on a smooth Hermitian bundle of rank r1r \geq 1 and degree 00 on XX. Denote this bundle by EE and the metric by hh, then the equations take the form

F(A)+ϕϕ=0,Aϕ=0. \begin{aligned} F(A) + \phi \wedge \phi^{\ast} = 0, \\ \overline{\partial}_{A} \phi = 0. \end{aligned}

Here, AA is a unitary connection on the bundle with respect to hh, FF is its curvature, and ϕ\phi is a smooth bundle map from EE to EωXE \otimes \w_{X}, called a Higgs field.

Definition

Let XX be a smooth compact Riemann surface, with genus g2g \geq 2. Let E\mcE be a holomorphic bundle on XX which has a holomorphic section ϕH0(X,End(E)ωX)\phi \in H^{0}(X, \End(\mcE) \otimes \w_{X}). We refer to the pair (E,ϕ)(\mcE,\phi) as a Higgs bundle, and we call the holomorphic section ϕ\phi a Higgs field.