W-Symmetry of the Adelic Grassmannian

TL;DR

This paper constructs a geometric interpretation of the W_{1+infty} vertex algebra via the adelic Grassmannian, linking higher symmetries and transformations of the KP hierarchy to W_{1+infty}-geometry.

Contribution

It introduces a novel geometric framework for understanding W_{1+infty} symmetry through D-bundles and sheaves, unifying various aspects of integrable systems and algebraic geometry.

Findings

Provides a geometric construction of W_{1+infty} algebra

Connects higher symmetries to D-bundles on curves

Offers a new perspective on KP hierarchy transformations

Abstract

We give a geometric construction of the W_{1+infty} vertex algebra as the infinitesimal form of a factorization structure on an adelic Grassmannian. This gives a concise interpretation of the higher symmetries and Backlund-Darboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of "W_{1+infty}-geometry": the geometry of D-bundles on smooth curves, or equivalently torsion-free sheaves on cuspidal curves.