The string equation and the τ-function Witt constraints for the discrete Kadomtsev-Petviashvili hierarchy

Abstract

Using the tool of pseudo-difference operators, the string equation for the discrete Kadomtsev-Petviashvili (KP) hierarchy is introduced and a general expression for the string equation is formulated. Furthermore, the operators of the constraints that the string equation imposes on the τ-function of the p-reduced dKP hierarchy are presented. The algebra, which part of the constraints spans, and eigenvalues of part of the constraints corresponding to the τ-function are calculated. It is also shown that this algebra is exactly a Witt algebra not only for p = 2, but also for a general \documentclass[12pt]{minimal}\begin{document}$p\in \mathbb {N}$\end{document}p∈N, and that the τ-function of the p-reduced dKP hierarchy constrained by the string equation is a vacuum vector of a Witt algebra.