Hitchin's equations

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In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, first written down by Nigel Hitchin in 1987.^[1] Hitchin's equations appear as a dimensional reduction of the Yang–Mills equations from four dimensions to two dimensions, and solutions to Hitchin's equations give examples of Higgs bundles. The existence of solutions to Hitchin's equations is equivalent to the stability of the corresponding Higgs bundle structure, and this is the simplest form of the Nonabelian Hodge correspondence for Higgs bundles.

The moduli space of solutions to Hitchin's equations, the Higgs bundle moduli space, was constructed by Hitchin and was one of the first examples of a hyperkähler manifold constructed. Using the metric structure on this moduli space afforded by its description in terms of Hitchin's equations, Hitchin constructed the Hitchin system, a completely integrable system which was used by Ngô Bảo Châu in his proof of the fundamental lemma in the Langlands program, for which he was afforded the 2010 Fields medal.^[2][3]

Definition[]

The definition may be phrased for a connection on a vector bundle or principal bundle, which 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][4][5]

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.

Derivation[]

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 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 ${\displaystyle F=\sum _{i<j}F_{ij}dx^{i}\wedge dx^{j}}$ 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 ${\displaystyle \Phi ={\frac {1}{2}}\phi dz}$ 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.

References[]