Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold ${\mathbb{p}}^1$

In this paper, we establish the convergence for Gromov-Witten invariant of elliptic orbifold ${\mathbb{P}}^1$ with type $(3,3,3),(4,4,2)$ and $(6,3,2)$. We also prove the mirror theorems of Gromov-Witten theory for those orbifolds and FJRW theory of elliptic singularities. Using T.Milanov and Y. Ruan’s work, we prove the Landau-Ginzburg/Calabi-Yau correspondence of all genera for the above three types of elliptic orbifold ${\mathbb{P}}^1$.