Hitchin system

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 ${\displaystyle G=\mathrm {GL} (n,\mathbb {C} )}$, the general linear group; then the Hamiltonians can be described as follows: the tangent space to the moduli space of G-bundles at the bundle F is

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

which by Serre duality is dual to

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

where ${\displaystyle K}$ is the canonical bundle, 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. They therefore define an integrable system in the symplectic or Arnol'd–Liouville sense.

The Hitchin fibration is the map from the moduli space of Hitchin pairs to characteristic polynomials, a higher genus analogue of the map Garnier used to define the spectral curves. Ngô (2006, 2010) used Hitchin fibrations over finite fields in his proof of the fundamental lemma.

• Yang–Mills equations
• Higgs bundle
• Nonabelian Hodge correspondence
• Character variety
• Hitchin's equations

• Chudnovsky, D.V. (1979), "Simplified Schlesinger systems", Lettere al Nuovo Cimento, 26 (14): 423–427, doi:10.1007/BF02817023, S2CID 122196561
• Garnier, René (1919), "Sur une classe de systemes différentiels abéliens déduits de la théorie des équations linéaires", Rend. Circ. Mat. Palermo, 43: 155–191, doi:10.1007/BF03014668, S2CID 120557738
• 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, S2CID 52064585