Generalized Hirota Equations in Models of 2D Quantum Gravity

TL;DR

This paper derives Hirota-type bilinear equations for partition functions in 2D quantum gravity models, extending KP hierarchy equations to random lattice models related to $sl(2)$ integrable systems.

Contribution

It introduces a new set of bilinear functional equations for partition functions, generalizing Hirota equations to models of 2D quantum gravity on random lattices.

Findings

Derived bilinear equations for $sl(2)$ models

Connected equations to KP integrable hierarchy

Applicable to statistical models on random lattices

Abstract

We derive a set of bilinear functional equations of Hirota type for the partition functions of the $sl(2)$ related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota equations for the KP integrable hierarchy, which are satisfied by the partition function of the ensemble of planar graphs.