Hitchin's equations

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 ${\displaystyle P\to \Sigma }$ be a principal ${\displaystyle G}$-bundle for a compact real Lie group ${\displaystyle G}$ over a compact Riemann surface. For simplicity we will consider the case of ${\displaystyle G={\text{SU}}(2)}$ or ${\displaystyle G={\text{SO}}(3)}$, the special unitary group or special orthogonal group. Suppose ${\displaystyle A}$ is a connection on ${\displaystyle P}$, and let ${\displaystyle \Phi }$ be a section of the complex vector bundle ${\displaystyle {\text{ad}}P^{\mathbb {C} }\otimes T_{1,0}^{*}\Sigma }$, where ${\displaystyle {\text{ad}}P^{\mathbb {C} }}$ is the complexification of the adjoint bundle of ${\displaystyle P}$, with fibre given by the complexification ${\displaystyle {\mathfrak {g}}\otimes \mathbb {C} }$ of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$. That is, ${\displaystyle \Phi }$ is a complex ${\displaystyle {\text{ad}}P}$-valued ${\displaystyle (1,0)}$-form on ${\displaystyle \Sigma }$. Such a ${\displaystyle \Phi }$ is called a Higgs field in analogy with the auxiliary Higgs field appearing in Yang–Mills theory.

For a pair ${\displaystyle (A,\Phi )}$, Hitchin's equations[1] assert that

$${\displaystyle {\begin{cases}F_{A}+[\Phi ,\Phi ^{*}]=0\\{\bar {\partial }}_{A}\Phi =0.\end{cases}}}$$

where ${\displaystyle F_{A}\in \Omega ^{2}(\Sigma ,{\text{ad}}P)}$ is the curvature form of ${\displaystyle A}$, ${\displaystyle {\bar {\partial }}_{A}}$ is the ${\displaystyle (0,1)}$-part of the induced connection on the complexified adjoint bundle ${\displaystyle {\text{ad}}P\otimes \mathbb {C} }$, and ${\displaystyle [\Phi ,\Phi ^{*}]}$ is the commutator of ${\displaystyle {\text{ad}}P}$-valued one-forms in the sense of Lie algebra-valued differential forms.

Since ${\displaystyle [\Phi ,\Phi ^{*}]}$ is of type ${\displaystyle (1,1)}$, Hitchin's equations assert that the ${\displaystyle (0,2)}$-component ${\displaystyle F_{A}^{0,2}=0}$. Since ${\displaystyle {\bar {\partial }}_{A}^{2}=F_{A}^{0,2}}$, this implies that ${\displaystyle {\bar {\partial }}_{A}}$ is a Dolbeault operator on ${\displaystyle {\text{ad}}P^{\mathbb {C} }}$ and gives this Lie algebra bundle the structure of a holomorphic vector bundle. Therefore, the condition ${\displaystyle {\bar {\partial }}_{A}\Phi =0}$ means that ${\displaystyle \Phi }$ is a holomorphic ${\displaystyle {\text{ad}}P}$-valued ${\displaystyle (1,0)}$-form on ${\displaystyle \Sigma }$. A pair consisting of a holomorphic vector bundle ${\displaystyle E}$ with a holomorphic endomorphism-valued ${\displaystyle (1,0)}$-form ${\displaystyle \Phi }$ 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 ${\displaystyle A}$ on a trivial principal ${\displaystyle G}$-bundle over ${\displaystyle \mathbb {R} ^{4}}$. Then there exists four functions ${\displaystyle A_{1},A_{2},A_{3},A_{4}:\mathbb {R} ^{4}\to {\mathfrak {g}}}$ such that

$${\displaystyle A=A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}+A_{4}dx^{4}}$$

where ${\displaystyle dx^{i}}$ are the standard coordinate differential forms on ${\displaystyle \mathbb {R} ^{4}}$. The self-duality equations for the connection ${\displaystyle A}$, a particular case of the Yang–Mills equations, can be written

$${\displaystyle {\begin{cases}F_{12}=F_{34}\\F_{13}=F_{42}\\F_{14}=F_{23}\end{cases}}}$$

where ${\textstyle F=\sum _{i<j}F_{ij}dx^{i}\wedge dx^{j}}$ is the curvature two-form of ${\displaystyle A}$. To dimensionally reduce to two dimensions, one imposes that the connection forms ${\displaystyle A_{i}}$ are independent of the coordinates ${\displaystyle x^{3},x^{4}}$ on ${\displaystyle \mathbb {R} ^{4}}$. Thus the components ${\displaystyle A_{1}dx^{1}+A_{2}dx^{2}}$ define a connection on the restricted bundle over ${\displaystyle \mathbb {R} ^{2}}$, and if one relabels ${\displaystyle A_{3}=\phi _{1}}$, ${\displaystyle A_{4}=\phi _{2}}$ then these are auxiliary ${\displaystyle {\mathfrak {g}}}$-valued fields over ${\displaystyle \mathbb {R} ^{2}}$.

If one now writes ${\displaystyle \phi =\phi _{1}-i\phi _{2}}$ and ${\textstyle \Phi ={\frac {1}{2}}\phi dz}$ where ${\displaystyle dz=dx^{1}+idx^{2}}$ is the standard complex ${\displaystyle (1,0)}$-form on ${\displaystyle \mathbb {R} ^{2}=\mathbb {C} }$, then the self-duality equations above become precisely Hitchin's equations. Since these equations are conformally invariant on ${\displaystyle \mathbb {R} ^{2}}$, they make sense on a conformal compactification of the plane, a Riemann surface.