View source for Hitchin system

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An algebraically completely integrable [[Hamiltonian system|Hamiltonian system]] defined on the cotangent bundle to the moduli space of stable vector bundles (of fixed rank and degree; cf. also [[Vector bundle|Vector bundle]]) over a given [[Riemann surface|Riemann surface]] $X$ of genus $g\geq2$. Hitchin's definition of the system [[#References|[a9]]] greatly enhanced the theory of spectral curves [[#References|[a8]]], which underlies the discovery of a multitude of algebraically completely integrable systems in the 1970s. Such systems are given by a Lax-pair equation: $L=[M,L]$ with $(n\times n)$-matrices $L$, $M$ depending on a parameter $\lambda$, the spectral curve is an $n$-fold covering of the parameter space and the system lives on a co-adjoint orbit in a loop algebra, by the Adler–Kostant–Symes method of symplectic reduction, cf. [[#References|[a1]]]. N.J. Hitchin defines the curve of eigenvalues on the total space of the canonical bundle of $X$, and linearizes the flows on the [[Jacobi variety|Jacobi variety]] of this curve.

The idea gave rise to a great amount of [[Algebraic geometry|algebraic geometry]]: moduli spaces of stable pairs [[#References|[a12]]]; meromorphic Hitchin systems [[#References|[a3]]] and [[#References|[a4]]]; Hitchin systems for principal $G$-bundles [[#References|[a5]]]; and quantized Hitchin systems with applications to the geometric Langlands program [[#References|[a2]]].

Moreover, by moving the curve $X$ in moduli, Hitchin [[#References|[a10]]] achieved geometric quantization by constructing a projective connection over the spaces of bundles, whose associated heat operator generalizes the [[Heat equation|heat equation]] that characterizes the Riemann theta-function for the case of rank-one bundles. The coefficients of the heat operator are given by the Hamiltonians of the Hitchin systems.

Explicit formulas for the Hitchin Hamiltonian and connection were produced for the genus-two case [[#References|[a7]]], [[#References|[a6]]]. A connection of Hitchin's Hamiltonians with KP-flows (cf. also [[KP-equation|KP-equation]]) is given in [[#References|[a4]]] and [[#References|[a11]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.R. Adams, J. Harnad, J. Hurtubise, "Integrable Hamiltonian systems on rational coadjoint orbits of loop algebras, Hamiltonian systems, transformation groups and spectral transform methods" , ''Proc. CRM Workshop, Montreal 1989'' (1990) pp. 19–32</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.A. Beilinson, V.G. Drinfel'd, "Quantization of Hitchin's fibration and Langlands program" A. Boutet de Monvel (ed.) et al. (ed.) , ''Algebraic and Geometric Methods in Math. Physics. Proc. 1st Ukrainian–French–Romanian Summer School, Kaciveli, Ukraine, Sept. 1-14 1993'' , ''Math. Phys. Stud.'' , '''19''' , Kluwer Acad. Publ. (1996) pp. 3–7 {{MR|1385674}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> F. Bottacin, "Symplectic geometry on moduli spaces of stable pairs" ''Ann. Sci. Ecole Norm. Sup. 4'' , '''28''' (1995) pp. 391–433 {{MR|1334607}} {{ZBL|0864.14004}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R. Donagi, E. Markman, "Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles" M. Francaviglia (ed.) et al. (ed.) , ''Integrable Systems and Quantum Groups. Lectures at the 1st session of the Centro Internaz. Mat. Estivo (CIME), Montecatini Terme, Italy, June 14-22 1993'' , ''Lecture Notes Math.'' , '''1620''' , Springer (1996) pp. 1–119 {{MR|1397273}} {{ZBL|0853.35100}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. Faltings, "Stable G-bundles and projective connections" ''J. Alg. Geometry'' , '''2''' (1993) pp. 507–568 {{MR|1211997}} {{ZBL|0790.14019}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> B. van Geemen, A.J. de Jong, "On Hitchin's connection" ''J. Amer. Math. Soc.'' , '''11''' (1998) pp. 189–228</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> B. van Geemen, E. Previato, "On the Hitchin system" ''Duke Math. J.'' , '''85''' : 3 (1996) pp. 659–683 {{MR|1422361}} {{ZBL|0879.14010}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> N.J. Hitchin, "The self-duality equations on a Riemann surface" ''Proc. London Math. Soc.'' , '''55''' (1987) pp. 59–126 {{MR|0887284}} {{ZBL|0634.53045}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> N.J. Hitchin, "Stable bundles and integrable systems" ''Duke Math. J.'' , '''54''' (1987) pp. 91–114 {{MR|0885778}} {{ZBL|0627.14024}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> N.J. Hitchin, "Flat connections and geometric quantization" ''Comm. Math. Phys.'' , '''131''' : 2 (1990) pp. 347–380 {{MR|1065677}} {{ZBL|0718.53021}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> Yingchen Li, M. Mulase, "Hitchin systems and KP equations" ''Internat. J. Math.'' , '''7''' : 2 (1996) pp. 227–244 {{MR|1382724}} {{ZBL|0863.58036}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> C.T. Simpson, "Moduli of representations of the fundamental group of a smooth projective variety I–II" ''Publ. Math. IHES'' , '''79/80''' (1994/5) pp. 47–129;5–79</TD></TR></table>