Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of CP^1 orbifolds

TL;DR

This paper proves that the Hirota quadratic equations for the orbifold $C_{k,m}$ define an integrable hierarchy equivalent to the extended bigraded Toda hierarchy, confirming a conjecture linking the total descendent potential with a tau function.

Contribution

It establishes the equivalence between Hirota equations for orbifold potentials and the extended bigraded Toda hierarchy, confirming a key conjecture in the field.

Findings

Hirota quadratic equations define an integrable hierarchy

The hierarchy is equivalent to the extended bigraded Toda hierarchy

Confirms the conjecture relating orbifold potentials to tau functions

Abstract

We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this proves a conjecture of Milanov-Tseng that relates the total descendent potential of the orbifold $C_{k,m}$ with a tau function of the bigraded Toda hierarchy.