Abstract. The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3-folds. We evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting. In the toric Calabi-Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent

We give a complete solution for the reduced Gromov-Witten theory of resolved surface singularities of type An, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the T-equivariant relative Gromov-Witten theory of the threefold An × P 1 which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the An surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type D,E. As a corollary, we present a new derivation of the stationary Gromov-Witten theory of P 1.

Abstract. We study the Gromov-Witten and Donaldson-Thomas correspondence conjectured in [MNOP1, MNOP2] for trivial elliptic fibrations. In particular, we verify the Gromov-Witten and Donaldson-Thomas correspondence for primary fields when the threefold is E × S where E is a smooth elliptic curve and S is a smooth surface with numerically trivial canonical class. 1.

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau threefolds given by a bundle over a two-sphere. This theory can be regarded as a q–deformation of Hurwitz theory, and it has a conjectural nonperturbative description in terms of q–deformed 2d Yang–Mills theory. We solve the planar model and find a phase transition at small radius in the universality class of 2d gravity. We give strong evidence that there is a double–scaled theory at the critical point whose all genus free energy is governed by the Painlevé I equation. We compare the critical behavior of the perturbative theory to the critical behavior of its nonperturbative description, which belongs to the universality class of 2d supergravity, and we comment on possible implications for nonperturbative 2d gravity. We also give evidence for a new open/closed duality relating these Calabi–Yau backgrounds to open strings with framing.

Open string amplitudes and large order behavior