Hirota Quadratic Equations for the Gromov--Witten Invariants of $\mathbb{P}_{n-2,2,2}^1$

TL;DR

This paper proves that the total descendant potential for Fano orbifold lines of type D is a tau-function of an extended Kac--Wakimoto hierarchy, extending known results from type A.

Contribution

It establishes that in the D-case, the total descendant potential is a tau-function of an extended Kac--Wakimoto hierarchy, expanding the class of orbifold lines with this property.

Findings

Total descendant potential for type D is a tau-function of an extended hierarchy.

Extension of Kac--Wakimoto hierarchies applies to D-type orbifold lines.

Generalization of known results from type A to type D.

Abstract

Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and $E$. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that in the A-case the Kac--Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac--Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac--Wakimoto hierarchy.