The ordered exponential representation of GKM using the $W_{1+\infty}$ operator

TL;DR

This paper introduces an ordered exponential representation of the monomial generalized Kontsevich model (GKM) using $W_{1+ abla}$ operators, maintaining KP integrability and connecting to known tau-function representations.

Contribution

It provides a novel $W_{1+ abla}$ operator-based exponential form for monomial GKM, linking it to KP hierarchy and tau-function constraints.

Findings

Representation preserves KP integrability.

Reduces to cut-and-join forms for specific tau-functions.

Connects $W_{1+ abla}$ operators with known models.

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 $W_{1+\infty}$ operators that preserves the KP integrability. In fact, this representation is naturally the solution of a $W_{1+\infty}$ constraint that uniquely determines the tau-function. Furthermore, we show that, for the cases of Kontsevich-Witten and generalized BGW tau-functions, their $W_{1+\infty}$ representations can be reduced to their cut-and-join representations under the reduction of the even time independence and Virasoro constraints.