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We study the equivariant Gromov-Witten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the Gromov-Witten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix. X(More)
For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of Sym n C 2 and Hilb n C 2 .
We compute the local Gromov-Witten invariants of the " closed vertex " , that is, a configuration of three P 1 's meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of P 3 and then to compute those invariants via the(More)
We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. 1. If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given by n|d 1 n 3 .
We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau threefolds. We define relative invariants for the local theory which give rise to a 1+1-dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we obtain a structure result for the local invariants. As an(More)
Let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system |C| for any g and n satisfying C · C = 2g + 2n − 2. When g = 0, this coincides with the enumerative problem studied by Yau and Zaslow who(More)