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...o-CY mirror theorem will be established in section four. 1.2. Acknowledgements. The genus-0 Gromov-Witten theory of the above elliptic orbifold P1 has been studied extensively by Satake and Takahashi =-=[25]-=-. Among other things, they established the genus-0 LG-to-CY mirror symmetry for the above examples. The main focus of this article is the higher genus cases, which has not yet been studied in the lite...

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...rbifold structure Z2. Equivalently it could be obtained as the global quotient of an elliptic curve by a Z2 action. An explicit treatment of the genus zero part of its orbifold GW theory was given in =-=[ST]-=-. We have to introduce several objects to present it here. Definition 2.1. The functions ϑi(z, τ) for τ ∈ H and z ∈ C represented by the following Fourier expansions: ϑ1(z, τ) = i ∞∑ n=−∞ (−1)ne(n−1/2...

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...enius manifolds between the one constructed from the Gromov–Witten theory for P1A and the one constructed from the pair (fA, ζA). For the cases that χA > 0 by [10, 12, 9] and the cases that χA = 0 by =-=[19]-=-, same statements as Theorem 4.1 are shown. Therefore, combining them with Theorem 4.1, it is shown that, for arbitary triplet of positive integers A, there exists the classical mirror symmetry betwee...

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... considered in this programme, and new results have recently been obtained [16, 20]. The elliptic singularities base spaces also turn out to be mirror to certain orbifold quantum cohomology manifolds =-=[29]-=-. The results obtained here show that the G-function extends holomorphically over the caustic for these more complicated examples. How the hidden symmetry I acts on such dispersive hierarchies is an i...

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...ective line which is a quotient of elliptic curve, that is the orbifold projective line P12,2,2,2 with four Z2-singular points, which will not be considered in this paper.) Recently, Satake-Takahashi =-=[ST]-=- computed the full genus-0 Gromov-Witten potential for P13,3,3 and P12,2,2,2 making use of algebraic argument (for e.g. WDVV-equations). Furthermore, Krawitz-Shen [KS] independently computed the poten...

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...from the invariant theory of the elliptic Weyl group of type D (1,1) 4 depending on the particular choice of a vector in a two dimensional vector space (this is a direct consequence of Theorem 2.7 in =-=[SatT]-=-). A Frobenius structure varies according to a choice of this vector which is identified with a cycle in the homology group of the elliptic curve (see also Example 2 in Section 3.3 in [Sai]). We intro...