Matrix Theory, Hilbert Scheme and Integrable System

TL;DR

This paper reinterprets matrix theory using second quantized operators linked to the Hilbert scheme of points, connecting it to free boson Fock space, Calogero-Sutherland eigenfunctions, and Virasoro symmetry.

Contribution

It introduces a novel reinterpretation of matrix theory in terms of second quantized operators and relates it to the Hilbert scheme, free boson Fock space, and integrable systems.

Findings

Relation between poles and inner products in Fock space

Identification of deformation parameter with Calogero-Sutherland coupling

Discussion of Virasoro symmetry structure

Abstract

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relate the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These basis can be related to the eigenfunctions of Calogero-Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.