Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences

TL;DR

This paper proves that the quantum cohomology rings of minuscule and cominuscule homogeneous spaces are semisimple at q=1, revealing algebraic automorphisms and dualities with implications for Gromov-Witten invariants.

Contribution

It establishes semisimplicity of quantum cohomology rings at q=1 for these spaces and links algebra automorphisms to known dualities, with formulas for Gromov-Witten invariants.

Findings

Quantum cohomology rings are semisimple at q=1.

Complex conjugation acts as an algebra automorphism.

Derived Vafa-Intriligator type formulas for Gromov-Witten invariants.

Abstract

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q=1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in a previous paper. We deduce Vafa-Intriligator type formulas for the Gromov-Witten invariants.