KP Equations, Strings, and the Schottky Problem

Abstract

The key point of the relationship between the Kadomtsev–Petviashvili (KP) theory and the characterization of jacobians of algebraic curves is the fact that the set A consisting of linear ordinary differential operators that commute with a given ordinary differential operator is itself a commutative algebra of transcendence degree 1 over the ground field. To prove the commutativity of A, fractional powers of differential operators are introduced, which are pseudo-differential operators. The chapter explains the algebraic structure of commuting ordinary differential operators and various exact solutions of the KP equation. It provides an overview of KP theory, which has many different roots in the long history of mathematics, solved a problem with another long history, and is now giving new dimensions in both mathematics and physics. The Schottky problem is a problem of finding a good characterization of jacobian varieties. As jacobians form an interesting special class of abelian varieties, historically a characterization always meant a characterization among abelian varieties. A natural approach to this problem is to perform a case study for low-genus jacobians. If the genus g is less than 4, then moduli of jacobians are opening dense in those of abelian varieties and there is no difficulty. The actual problem starts at g = 4. But already genus 5 is hard enough.