An Application Of Symplectic Virtual Localization

In this paper, we use the symplectic virtual localization formula to compute the Gromov-Witten invariants of ccrtain Calabi-Yau manifolds W_k. After localizing the Gromov-Witten invariants, we compute the equivariant Euler class of the virtual normal bundle of the fixed loci and the equivariant Euler class of obstruction bundle. Finally, we enumerate all the possible graphs correspond-ing to the fixed loci of the moduli space under certain S~1 action and compute the genus O Gromov-Witten invariants when d is less than 5.