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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, written down by Nigel Hitchin in 1987.[1] Hitchin's equations are locally equivalent to the harmonic map equation for a surface into the symmetric space dual to the structure group.[2] They also appear as a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to two dimensions, and solutions to Hitchin's equations give examples of Higgs bundles and of holomorphic connections. The existence of solutions to Hitchin's equations on a compact Riemann surface follows from the stability of the corresponding Higgs bundle or the corresponding holomorphic connection, and this is the simplest form of the Nonabelian Hodge correspondence.

The moduli space of solutions to Hitchin's equations was constructed by Hitchin in the rank two case on a compact Riemann surface and was one of the first examples of a hyperkähler manifold constructed. The nonabelian Hodge correspondence shows it is isomorphic to the Higgs bundle moduli space, and to the moduli space of holomorphic connections. Using the metric structure on the Higgs bundle moduli space afforded by its description in terms of Hitchin's equations, Hitchin constructed the Hitchin system, a completely integrable system whose twisted generalization over a finite field 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.[3][4]

Definition

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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.

Derivation

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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.

References

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  1. ^ a b c Hitchin, Nigel J. (1987). "The self-duality equations on a Riemann surface". Proceedings of the London Mathematical Society. 55 (1): 59–126. doi:10.1112/plms/s3-55.1.59. MR 0887284.
  2. ^ Donaldson, Simon (2004). "Mathematical uses of gauge theory" (PDF). Encyclopaedia of Mathematical Physics.
  3. ^ Hitchin, Nigel (1987), "Stable bundles and integrable systems", Duke Mathematical Journal, 54 (1): 91–114, doi:10.1215/S0012-7094-87-05408-1
  4. ^ Ngô, Bao Châu (2006), "Fibration de Hitchin et structure endoscopique de la formule des traces" (PDF), International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 1213–1225, MR 2275642
  5. ^ Wentworth, R. and Wilkin, G. eds., 2018. The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles (Vol. 36). World Scientific.
  6. ^ Gothen, Peter B.; García-Prada, Oscar; Bradlow, Steven B. (2007), "What is... a Higgs bundle?" (PDF), Notices of the American Mathematical Society, 54 (8): 980–981, MR 2343296