Hilbert Schemes, Separated Variables, and D-Branes

Details Hilbert Schemes, Separated Variables, and D-Branes Hilbert Schemes, Separated Variables, and D-Branes

1999-01-20

arxiv.org/abs/hep-th/9901089v3

Commun.Math.Phys. 222 (2001) 299-318

Physics

High Energy Physics

High Energy Physics - Theory

harvmac, 27 pp. big mode; v2. typos and references corrected

Scientific paper

10.1007/s002200100503

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.

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