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We conjecture an equivalence between the Gromov-Witten theory of 3-folds and the holomorphic Chern-Simons theory of Donaldson and Thomas. For Calabi-Yau 3-folds, the equivalence is defined by the change of variables e iu = −q, where u is the genus parameter of Gromov-Witten theory and q is the Euler characteristic parameter of Donaldson-Thomas theory. The conjecture is proven for local Calabi-Yau toric surfaces.

For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.Comment: Typo fixe

Dedicated to the memory of Claude Itzykson Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 13 3. A rigidification of M g,n (P r , d) 14 4. The construction of M g,n (P r , d) 18 5. The construction of M g,n (X, β) 26 6. The boundary of M 0,n (X, β) 30 7. Gromov-Witten invariants 33 8. Quantum cohomology 36 9. Applications to enumerative geometry 40 10. Variations 47 References 51

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relative geometry. We derive a formula for the equivariant vertex measure in the degree 0 case and prove Conjecture 1 ′ of [14] in the toric case. A degree 0 relative formula is also proven. AcknowledgmentsWe thank J. Li for explaining his definition of relative Donaldson-Thomas theory to us. An outline of his ideas is presented in Section 3.2.1. We thank

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of P 1 \mathbb P^1 . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a complete and effective solution. The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of C 2 \mathbb C^2 , and the orbifold quantum cohomology of the symmetric product of C 2 \mathbb C^2 . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.