Gromov-Witten Invariants of Calabi-Yau Manifolds With Two Kahler Parameters

Abstract

We study the Gromov–Witten theory of Kℙ1×ℙ1 and some Calabi–Yau hypersurfaces in toric varieties. We give a direct geometric proof of the holomorphic anomaly equation for Kℙ1×ℙ1 in the form predicted by B-model physics. We also calculate the closed formula of genus one quasimap invariants of Calabi–Yau hypersurfaces in ℙm−1×ℙn−1 after restricting the 2nd Kähler parameter to zero. By the wall-crossing theorem between Gromov–Witten and quasimap invariants, we thus obtain their genus one Gromov–Witten invariants.