On the solutions to the string equation

TL;DR

This paper characterizes solutions to the string equation involving differential operators, linking them to algebraic curves and exploring superanalogues, with implications for integrable hierarchies.

Contribution

It provides a simple description of solutions to the string equation and extends the analysis to superdifferential operators, connecting to various KP-hierarchies.

Findings

One-to-one correspondence between solutions and pairs of commuting differential operators.

Solutions can be described via moduli spaces of algebraic curves.

The superanalog of the string equation is invariant under multiple KP-hierarchies.

Abstract

The set of solutions to the string equation $[P,Q]=1$ where $P$ and $Q$ are differential operators is described.It is shown that there exists one-to-one correspondence between this set and the set of pairs of commuting differential operators.This fact permits us to describe the set of solutions to the string equation in terms of moduli spa- ces of algebraic curves,however the direct description is much simpler. Some results are obtained for the superanalog to the string equation where $P$ and $Q$ are considered as superdifferential operators. It is proved that this equation is invariant with respect to Manin-Radul, Mulase-Rabin and Kac-van de Leur KP-hierarchies.