The ordered exponential representation of GKM using the W1+∞ operator

Abstract

The generalized Kontsevich model (GKM) is a one-matrix model with arbitrary potential. Its partition function belongs to the KP hierarchy. When the potential is monomial, it is an r-reduced tau-function that governs the r-spin intersection numbers. In this paper, we present an ordered exponential representation of monomial GKM in terms of the W1+∞ operators that preserves the KP integrability. In fact, this representation is naturally the solution of a W1+∞ constraint that uniquely determines the tau-function. Furthermore, we show that, for the cases of Kontsevich-Witten and generalized BGW tau-functions, their W1+∞ representations can be reduced to their cut-and-join representations under the reduction of the even time independence and Virasoro constraints.