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 m odels with target space a toric variety V or a Calabi-Yau hypersurface M subset of V. In the linear model the instanton moduli spaces are re latively 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 reprod uce and clarify an algebraic solution of the V model due to Batyrev. I n addition, we find an algebraic relation determining the solution for M in terms of that for V. Finally, we propose a modification of the l inear model which computes instanton expansions about any limiting poi nt in the moduli space, In the smooth case this leads to a (second) al gebraic solution of the M model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectur ed ''monomial-divisor mirror map'' of Aspinwall, Greene and Morrison.