LG/CY Correspondence for Elliptic Orbifold Curves via Modularity

TL;DR

This paper establishes a modularity-based correspondence between Gromov-Witten and Fan-Jarvis-Ruan-Witten theories for elliptic orbifold curves, revealing they are different representations of the same quasi-modular forms connected by the Cayley transform.

Contribution

It proves the LG/CY correspondence for elliptic orbifold curves using modularity and relates two different representations of correlation functions via the Cayley transform.

Findings

Correlation functions are different representations of the same quasi-modular forms.

The Cayley transform relates these two representations.

The correspondence is established via modularity in enumerative geometry.

Abstract

We prove the Landau-Ginzburg/Calabi-Yau correspondence between the Gromov-Witten theory of each elliptic orbifold curve and its Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform.