Quantum Cohomology Rings of Toric Manifolds

TL;DR

This paper computes the quantum cohomology rings of smooth projective toric manifolds, revealing how their structure depends on specific cohomology elements, and discusses properties expected to hold for Kähler manifolds.

Contribution

It provides explicit calculations of quantum cohomology rings for arbitrary toric manifolds and explores their properties related to Kähler manifolds.

Findings

Quantum cohomology rings depend on an element in H^2.

Explicit formulas for the multiplicative structure.

Properties expected to hold for Kähler manifolds.

Abstract

We compute the quantum cohomology ring $H^*_{\varphi}({\bf P}, {\bf C})$ of an arbitrary $d$-dimensional smooth projective toric manifold ${\bf P}_{\Sigma}$ associated with a fan $\Sigma$. The multiplicative structure of $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ depends on the choice of an element $avarphi$ in the ordinary cohomology group $H^2({\bf P}_{\Sigma}, {\bf C})$. There are many properties of the quantum cohomology rings $H^*_{\varphi}({\bf P}_{\Sigma}, {\bf C})$ which are supposed to be valid for quantum cohomology rings of K\"ahler manifolds