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.

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

Let V be a nonsingular, complex, projective variety containing a nonsingular divisor W . The absolute Gromov-Witten theory of V is defined by integrating descendent classes over the moduli space of stable maps to V . The relative Gromov-Witten theory of the pair (V, W ) is defined by descendent integration over the space of stable relative maps to V with prescribed tangency data along W .We present here a systematic study of relative Gromov-Witten theory via universal relations. We find the relative theory does not provide new invariants: the relative theory is completely determined by the absolute theory. The relation between the relative and absolute theories is guided by a strong analogy to classical topology.Our results open new directions in the subject. For example, we present a complete mathematical determination of the Gromov-Witten theory (in all genera) of the Calabi-Yau quintic hypersurface in P 4 . Leray-HirschLet X be a nonsingular, complex, projective variety equipped with a line bundle L. Let Y be the projective bundle P(L ⊕ O X ), and let π be the projection map, π : Y → X.

We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in the literature.

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 GromovWitten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.