Eigenvalues of the string constraints for the Kadomtsev-Petviashvili hierarchy

Abstract

For an arbitrary p ∈ ℕ, a new and computable method is proposed, which can, through a recursive algorithm, calculate the eigenvalues of the string constraint operators corresponding to the τ-function of the p-reduced Kadomtsev-Petviashvili (KP) hierarchy. It shows that not all of these eigenvalues equal 0. In this process, a connection between the W algebra and the p-reduced W algebra is introduced, which can be used to calculate the algebraic structure of the p-reduced W algebra. And it is proved that the p-reduced W algebra includes a Virasoro algebra as its subalgebra. In addition, based on the obtained eigenvalues, it is also showed that the τ-function of the p-reduced KP hierarchy constrained by the string equation is a vacuum vector for a Virasoro algebra for any p ∈ ℕ. When p = 2, it is coincident with the classical fact that the τ-function of the Korteweg-de Vries hierarchy constrained by the string equation is a vacuum vector for a Virasoro algebra.