Combinatorics of KP hierarchy structural constants

TL;DR

This paper investigates the combinatorial structure of KP hierarchy constants, revealing their properties through a matrix model and proposing new directions for KP deformations based on these coefficients.

Contribution

It introduces a combinatorial framework for KP hierarchy constants and constructs a matrix model to analyze their properties and potential deformations.

Findings

Identifies combinatorial properties of KP coefficients

Constructs an eigenvalue matrix model for KP correlators

Suggests new directions for KP hierarchy deformations

Abstract

Following Natanzon-Zabrodin, we explore the Kadomtsev-Petviashvili hierarchy as an infinite system of mutually consistent relations on the second derivatives of the free energy with some universal coefficients. From this point of view, various combinatorial properties of these coefficients naturally highlight certain non-trivial properties of the KP hierarchy. Furthermore, this approach allows us to suggest several interesting directions of the KP deformation via a deformation of these coefficients. We also construct an eigenvalue matrix model, whose correlators fully describe the universal KP coefficients, which allows us to further study their properties and generalizations.