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

TL;DR

This paper employs the gauged linear sigma model to explicitly compute and sum instanton expansions in topological sigma models with toric and Calabi-Yau target spaces, providing new insights into mirror symmetry and algebraic solutions.

Contribution

It introduces a method to sum instanton expansions in linear models for toric varieties and Calabi-Yau hypersurfaces, clarifying algebraic solutions and confirming mirror symmetry conjectures.

Findings

Explicit instanton sums for toric and Calabi-Yau models

Reproduction and clarification of Batyrev's algebraic solution

Proof of the monomial-divisor mirror map conjecture

Abstract

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.