Vertex Operators of the KP hierarchy and Singular Algebraic Curves

TL;DR

This paper explores how vertex operators generate solutions to the KP hierarchy that correspond to singular algebraic curves, revealing their role in creating singularities and connecting soliton solutions with algebraic geometry.

Contribution

It demonstrates that vertex operator actions produce solutions linked to singular algebraic curves and elucidates their geometric and solitonic properties.

Findings

Vertex operators create singular points on algebraic curves.

Solutions correspond to torsion free sheaves on singular curves.

Solitons can be constructed on quasi-periodic backgrounds using these solutions.

Abstract

Quasi-periodic solutions of the KP hierarchy acted by vertex operators are studied. We show, with the aid of the Sato Grassmannian, that solutions thus constructed correspond to torsion free rank one sheaves on some singular algebraic curves whose normalizations are the non-singular curves corresponding to the seed quasi-periodic solutions. It means that the action of the vertex operator has an effect of creating singular points on an algebraic curve. We further check, by examples, that solutions obtained here can be considered as solitons on quasi-periodic backgrounds, where the soliton matrices are deterimed by parameters in the vertex operators.