For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov–Witten potential, which serves as the quantum-corrected Landau–Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov–Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.
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The second author thanks Kyoung-Seog Lee for valuable discussions on the computation of Jacobian ideals. The fourth author expresses his gratitude to Yefeng Shen and Jie Zhou for useful discussions on the mirror maps for elliptic curve quotients. The work of C.H. Cho was supported by the National Research Foundation of Korea (NRF) grant No. 2010-0019516 and by No. 2012R1A1A2003117 . The work of S.-H. Kim was supported by the National Research Foundation of Korea (NRF) grant No. 2013R1A1A1058646 .
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