A note on the extended dToda hierarchy

TL;DR

This paper derives dispersionless Hirota equations for the extended dispersionless Toda hierarchy, linking them to topological recursion in the $CP^1$ model, and computes two-point functions involving Catalan numbers.

Contribution

It provides a novel derivation of dispersionless Hirota equations from topological recursion relations for the extended Toda hierarchy.

Findings

Dispersionless Hirota equations are derived from topological recursion.

Two-point functions are computed and expressed via Catalan numbers.

The work connects integrable hierarchies with topological field theory.

Abstract

We give a derivation of dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are nothing but a direct consequence of the genus-zero topological recursion relation for the topological $CP^1$ model. Using the dispersionless Hirota equations we compute the two point functions and express the result in terms of Catalan number.