W algebra and eigenvalues of the low-order string constraint operators for the discrete KP hierarchy

Abstract

Firstly, for any p ∈ N , the string constraint operators without the redundant variables are calculated, which are imposed by the string equation on the τ -function of the p -reduced discrete Kadomtsev–Petviashvili (dKP) hierarchy. Furthermore, the commutators of the string constraint operators without the redundant variables are calculated. It is found that although the string constraint operators without the redundant variables are different from the original string constraint operators with the redundant variables, they also form a Virasoro algebra and a W algebra after a transformation. Moreover, using the algebraic structure, eigenvalues of the low-order string equation constraint operators are calculated. Based on the obtained eigenvalues, it is also proved that the τ -function of the p -reduced dKP hierarchy constrained by the string equation is a vacuum vector for a Witt algebra. 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.