The Calculation Problems Of Local Gromov-Witten Invariants

In recent years there have been some progresses in the calculations of Gromov-Witten invariants of local Calabi-Yau geometries by the localization method. M.Marin?o and C. Vafa have conjectured a formula for some Hodge integrals arising nat-urally in such calculations of Gromov-Witten invariants. This formula has been math-ematically proved by C.-C. Liu, K. Liu, J. Zhou and A. Okounkov, R. Pandharipande.It has been generalized to the two-partition case and the three-partition case by C.-C.Liu, K. Liu, J. Zhou. As a result, we now have an e?ective method to carry out thecomputations for some Gromov-Witten invariants.In this paper, we define the local Gromov-Witten invariants of two types of openCalabi-Yau three-folds by using localization techniques. One is the local Gromov-Witten invariants of O(k)⊕O(?k ? 2)→P1 (where k≥?1), as a generalizationof the local Gromov-Witten invariants of the resolved conifold, the other is the localGromov-Witten invariants of the canonical line bundles of toric surfaces KS→S ,as a generalization of earlier results when S is Fano. We compute them by localiza-tion,Marin?o-Vafa formula,two-partition Hodge integral formula and the chemistryof graphs. We also compute some Gopakumar-Vafa invariants.As applications of such calculations to geometric engineering, we identify thepotential function of the Gromov-Witten theory of O(k)⊕O(?k ? 2)→P1 with theequivariant Riemann-Roch indices of certain powers of the determinant of the tauto-logical sheaves on the Hilbert schemes. We also identify the potential function of theGromov-Witten theory of Hirzebruch surface Fm(m≥0) with the equivariant indicesof suitable bundles on the framed moduli spaces. One sees that the curve countingpartition function of local Calabi-Yau geometry of Hirzebruch surface Fm(m≥0) canbe identified with suitable partition functions of S U(2) instanton counting on C2.