On quantum cohomology ring of elliptic $\mathbb{P}^1$ orbifolds

TL;DR

This paper computes the quantum cohomology ring of elliptic orbifold projective lines using orbi-curve counting and a classification theorem, providing an alternative approach to previous results.

Contribution

It introduces a new method based on orbifold coverings and Diophantine equations to compute quantum cohomology of elliptic orbifolds, reproducing known results with a different technique.

Findings

Quantum cohomology ring computed for elliptic orbifold $P^1$

New classification theorem relating holomorphic orbi-curves and orbifold coverings

Alternative derivation of Satake and Takahashi's results

Abstract

We compute quantum cohomology ring of elliptic $\mathbb{P}^1$ orbifolds via orbi-curve counting. The main technique is the classification theorem which relates holomorphic orbi-curves with certain orbifold coverings. The countings of orbi-curves are related to the integer solutions of Diophantine equations. This reproduces the computation of Satake and Takahashi in the case of $\mathbb{P}^1_{3,3,3}$ via different method.