The loop equation for the Burgers--KdV hierarchy

TL;DR

This paper derives a loop equation for the Burgers--KdV hierarchy's tau-function, linking it to intersection numbers on moduli spaces with boundaries, and provides new relations for open intersection numbers.

Contribution

It introduces a Dubrovin--Zhang type loop equation for the Burgers--KdV hierarchy's topological tau-function, advancing understanding of open intersection numbers.

Findings

Derived the loop equation for the hierarchy's tau-function

Established relations for open intersection numbers

Connected hierarchy theory with moduli space intersection theory

Abstract

The Burgers--KdV hierarchy was introduced towards understanding intersection numbers on the moduli space of Riemann surfaces with boundaries. The goal of this paper is to derive the Dubrovin--Zhang type loop equation for the topological tau-function of the Burgers--KdV hierarchy. As as application, we provide some relations for open intersection numbers.