Hitchin system

In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the theory of Lie algebras and integrable system theory. It also plays an important role in geometric Langlands correspondence over the field of complex numbers; related to conformal field theory. A genus zero analogue of the Hitchin system arises as a certain limit of the Knizhnik–Zamolodchikov equations. Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system (or its meromorphic generalization or in a singular limit).

The Hitchin fibration is the map from the moduli space of Hitchin pairs to characteristic polynomials. Ngô (2006, 2010) used Hitchin fibrations over finite fields in his proof of the fundamental lemma.

Description

Using the language of algebraic geometry, the phase space of the system is a partial compactification of the cotangent bundle to the moduli space of stable G-bundles for some reductive group G, on some compact algebraic curve. This space is endowed with a canonical symplectic form. Suppose for simplicity that G=GL(n), the general linear group; then the hamiltonians can be described as follows: the tangent space to G-bundles at the bundle F is

${\displaystyle H^{1}(\operatorname {End} (F)),}$

which by Serre duality is dual to

${\displaystyle \Phi \in H^{0}(\operatorname {End} (F)\otimes K),}$

so a pair

${\displaystyle (F,\Phi )}$

called a Hitchin pair or Higgs bundle, defines a point in the cotangent bundle. Taking

${\displaystyle \operatorname {Tr} (\Phi ^{k}),\qquad k=1,\ldots ,\operatorname {rank} (G)}$

one obtains elements in

${\displaystyle H^{0}(K^{\otimes k}),}$

which is a vector space which does not depend on ${\displaystyle (F,\Phi )}$. So taking any basis in these vector spaces we obtain functions H_i, which are Hitchin's hamiltonians. The construction for general reductive group is similar and uses invariant polynomials on the Lie algebra of G.

For trivial reasons these functions are algebraically independent, and some calculations show that their number is exactly half of the dimension of the phase space. The nontrivial part is a proof of Poisson commutativity of these functions.

References

• Hitchin, Nigel (1987), "Stable bundles and integrable systems", Duke Mathematical Journal, 54 (1): 91–114, doi:10.1215/S0012-7094-87-05408-1
• 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
• Ngô, Bao Châu (2010), "Fibration de Hitchin et endoscopie", Inventiones Mathematicae, 164 (2): 399–453, arXiv:math/0406599, Bibcode:2006InMat.164..399N, doi:10.1007/s00222-005-0483-7, ISSN 0020-9910, MR 2218781