Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties

We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth $V$, our results reproduce and clarify an algebraic solution of the $V$ model due to Batyrev. In addition, we find an algebraic relation determining the solution for $M$ in terms of that for $V$. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the $M$ model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.