arXiv:2611.13126v1 Announce Type: new Abstract: We study mirror map derivatives as a means of relating the genus zero Gromov-Witten theories of an anticanonical Calabi-Yau hypersurface in a smooth, almost minimal Fano variety and the local Calabi-Yau formed from the total space of the Fano's canonical bundle with the euler-twisted theory of the ambient Fano. A precise relationship between reciprocal mirror map derivatives is obtained from the non-equivariant limit of quantum Serre duality for the euler-twisted theory of the ambient Fano as formulated by Iritani, Mann, & Mignon. This allows us to express the generating series of 3-point euler-twisted invariants as the derivative of the local mirror map with respect to the closed one. In low dimensions, this series is shown to be a modular form of Eisenstein type which has arisen before in this area. We prove integrality of the twisted invariants from this identification. The regularized small quantum $D$-module of the Fano is reducible, and we show it is equivalent to an inhomogeneous Picard-Fuchs equation governed by this generating series. The reciprocal mirror map derivative is then shown to be a generating function of local 3-point invariants, which recovers some known results for local Calabi-Yau threefolds and proves a conjecture of the second named author and A. Malmendier in previous work on local fourfolds. Explicit calculations for $\mathbb{P}^2,\mathbb{P}^3$ and $\mathbb{P}^4$ illustrate these relationships, which includes the vanishing of positive-degree ordinary genus-zero invariants for anticanonical elliptic curves and K3 surfaces, the recovery of the quintic threefold's instanton numbers, and finally some formulas for genus-zero invariants of local projective spaces.
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