Hilbert Schemes, Separated Variables, and D-Branes

TL;DR

This paper explores the geometric interpretation of Sklyanin's separation of variables, constructs Hilbert schemes related to integrable systems, and discusses their connections to D-branes and string duality.

Contribution

It provides a geometric framework for Sklyanin's separation of variables and constructs new Hilbert schemes linked to integrable systems and string theory.

Findings

Hilbert schemes of points are constructed for various surfaces.

Complex deformations of these schemes are shown to be integrable systems.

Connections between geometric constructions and D-branes are discussed.

Abstract

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.