Hilbert Schemes, Separated Variables, and D-Branes

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on $T^{*}\Sigma$ for \Sigma = {\IC}, {\IC}^{*} or elliptic curve, and on ${\bf C}^{2}/{\Gamma}$ and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of $D$-branes and string duality.Comment: harvmac, 27 pp. big mode; v2. typos and references correcte

Duality in Integrable Systems and Gauge Theories

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.Comment: harvmac 45 pp.; v4. minor corrections, to appear in JHE