Quantum cohomology and closed-string mirror symmetry for toric varieties

TL;DR

This paper provides a new algebraic approach to compute the quantum cohomology of smooth, possibly non-compact toric varieties, establishing an isomorphism with a Jacobian ring via the Kodaira-Spencer map.

Contribution

It introduces a direct algebraic proof that the Kodaira-Spencer map yields an isomorphism for quantum cohomology of toric varieties, including non-compact cases, expanding previous compact-only results.

Findings

Quantum cohomology of smooth toric varieties is isomorphic to a Jacobian ring.

The approach works for non-compact and monotone cases.

Explicit presentations are obtained for monotone varieties.

Abstract

We give a short new computation of the quantum cohomology of an arbitrary smooth toric variety $X$, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The proof is based on the purely algebraic fact that a class of generalised Jacobian rings associated to $X$ are free as modules over the Novikov ring. In contrast to previous results of this kind, $X$ need not be compact. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.