Gromov-Witten invariants of Calabi-Yau manifolds with two Kähler parameters

Abstract

We study the Gromov-Witten theory of $K_{{\mathsf{P}}^1\times {\mathsf{P}}^1}$ and some Calabi-Yau hypersurface in toric variety. We give a direct geometric proof of the holomorphic anomaly euqation for $K_{{\mathsf{P}}^1\times {\mathsf{P}}^1}$ in the form predicted by B-model physics. We also calculate the closed formula of genus one quasimap invariants of Calabi-Yau hypersurface in ${\mathsf{P}}^{m-1}\times {\mathsf{P}}^{n-1}$ after restricting second Kähler parameter to zero. By wall-crossing theorem between Gromov-Witten and quasimap invariants, we can obtain the genus one Gromov-Witten invariants.

40 pages. arXiv admin note: text overlap with arXiv:1702.06096, arXiv:1803.01409