Hitchin system

An algebraically completely integrable Hamiltonian system defined on the cotangent bundle to the moduli space of stable vector bundles (of fixed rank and degree; cf. also Vector bundle) over a given Riemann surface $X$ of genus $g\geq2$. Hitchin's definition of the system [a9] greatly enhanced the theory of spectral curves [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. [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 of this curve.

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

Moreover, by moving the curve $X$ in moduli, Hitchin [a10] achieved geometric quantization by constructing a projective connection over the spaces of bundles, whose associated heat operator generalizes the 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 [a7], [a6]. A connection of Hitchin's Hamiltonians with KP-flows (cf. also KP-equation) is given in [a4] and [a11].

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