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.
"LG/CY correspondence for elliptic orbifold curves via modularity." J. Differential Geom. 109 (2) 291 - 336, June 2018. https://doi.org/10.4310/jdg/1527040874