Spectral curves and integrable systems

Abstract

We study the existence of Algebraically Completely Integrable Hamiltonian System (ACIHS) structures on the moduli spaces $M\\sb{L}(r,d)$ of stable L-twisted Higgs pairs over a smooth algebraic curve $\\Sigma.$ These spaces parametrize pairs consisting of a vector bundle of rank r and degree d and an endomorphism twisted by a line bundle L. An invariant polynomial map gives rise to a fibration of $M\\sb{L}(r,d)$ with a Jacobian of a spectral curve as a generic fibre. We show that a canonical family of ACIHS structures exists whenever the twisting line bundle L is bigger than the canonical line bundle. The Moduli space ${\\cal U}\\sb\\Sigma(r,d,D)$ of vector bundles with level structures over a divisor D in the linear system $\\IP H\\sp0(\\Sigma,L\\otimes K\\sp{-1})$ is key to understanding the geometry of our systems. The Poisson structure on $M\\sb{L}(r,d)$ is obtained via symplectic reduction from the canonical symplectic structure on $\\vert T\\sp*{\\cal U}\\sb\\Sigma(r,d,D)\\vert.$ Our construction specializes to the Hitchin system when L is the canonical line bundle, Beauville's system when the base curve is $P\\sp1.$ It is equivalent to a construction of Reyman and Semenov-Tian-Shansky when the base curve is elliptic and r,d are coprime. Many known ACIHS's, classical as well as recently discovered (e.g., the spaces of KP Elliptic solitons), embed as symplectic leaves of the systems with rational and elliptic base curves.