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 .
Furuta's \10/8-th's" theorem gives a bound on the magnitude of the signature of a smooth spin 4-manifold in terms of the second Betti number. We show that in the presence of a Z=2 p action, his bound can be strengthened. As applications, we give new genus bounds on classes with divisibility and we give a classiication of involutions on rational cohomology… (More)
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 .
Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we also count the number of curves in the fixed linear system |C| passing through g − 2 generic points. These two… (More)
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 Z3 act on C 2 by non-trivial opposite characters. Let X = [C 2 /Z3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials F X and F Y are equal after a change of variables — verifying the Crepant Resolution Conjecture for the pair (X , Y). Our computations involve Hodge integrals… (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)
Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C 3 /G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R,… (More)