Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms

TL;DR

This paper demonstrates that Gromov-Witten invariants of specific elliptic orbifold lines are quasi-modular forms, using Givental's formalism and Frobenius structures, contributing to the Landau-Ginzburg/Calabi-Yau correspondence.

Contribution

It proves the quasi-modularity of GW invariants for certain elliptic orbifold lines using Frobenius structures, advancing the understanding of the Landau-Ginzburg/Calabi-Yau correspondence.

Findings

GW invariants are quasi-modular forms for specified elliptic orbifold lines

Application of Givental's higher genus reconstruction formalism

Connection between Frobenius structures and Gromov-Witten theory

Abstract

In this paper we prove that the GW invariants of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are quasi-modular forms. Our method is based on Givental's higher genus reconstruction formalism applied to the settings of Saito's Frobenius structures for simple elliptic singularities. Our results are part of a larger project whose goal is to prove the Landau-Ginzburg/Calabi-Yau correspondence for simple elliptic singularities. The correspondence describes a relation between Gromov-Witten theory (of a certain hypersurface) and Fan-Jarvis-Ruan-Witten theory (of a certain Landau-Ginzburg potential). Roughly, the main statement is that the Saito's Frobenius manifold for simple elliptic singularities has some special points such that locally near these points the Frobenius structure governs one of the two theories. The local part of the correspondence is established in a companion article by M. Krawitz and Y. Shen, while here we describe the global picture.