Hodge Integrals and Degenerate Contributions

Let X be a Calabi-Yau 3-fold containing a nonsingular genus g curve C ⊂ X. We study here the contribution of genus g + h covers of C to the genus g + h Gromov-Witten invariants of X. Degenerate contributions play an important role in the Gromov-Witten theory of Calabi-Yau 3-folds. In algebraic geometry, these contributions are related to Hodge integrals over the moduli space of curves Mg,n [FP]. In string theory, recent progress in the calculation of these contributions has been made by a link to Mtheory [GV1], [GV2] (see also [MM]). The mathematical results presented here agree exactly with the M-theoretic results of [GV2]. Consider the moduli space of maps Mg+h(X, d[C]). If g = 0 or 1, then for a general pair (C,X), this moduli space will have a component equal to Mg+h(C, d[C]). The contribution Cg(h, d) of C to the genus g+ h GromovWitten is thus well-defined for g = 0, 1 and all degrees d > 0. For curves of genus g ≥ 2, multiple covers of C are expected to deform in X away from C. However, in the degree 1 case, Mg+h(X, [C]) has a component equal to Mg+h(C, [C]) for all g and h. Hence, Cg(h, 1) is always well-defined. The contributions in case g = 0 have recently been calculated in algebraic geometry [FP] and string theory [GV1], [MM]: