Quantum ring of singularity $X^{p}+XY^{q}$

TL;DR

This paper proves an isomorphism between the quantum ring of a specific quasi-homogeneous polynomial and the Milnor ring of its mirror dual, providing explicit constructions and discussing Frobenius algebra pairings.

Contribution

It establishes a concrete isomorphism between quantum and Milnor rings for a class of polynomials, with distinct methods based on the gcd condition.

Findings

Isomorphism between quantum and Milnor rings proved

Explicit construction provided for different gcd cases

Discussion on Frobenius algebra pairings included

Abstract

In this paper, we will prove that the quantum ring of the quasi-homogeneous polynomial $X^{p}+XY^{q}(p\ge 2,q>1)$ with some admissible symmetry group $G$ defined by Fan-Jarvis-Ruan-Witten theory is isomorphic to the Milnor ring of its mirror dual polynomial $X^{p}Y+Y^{q}$. We will construct an concrete isomorphism between them. The construction is a little bit different in case $(p-1,q)=1$ and case $(p-1,q)=d>1$. Some other problems including the correspondence between the pairings of both Frobenius algebras has also been discussed.