Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes

TL;DR

This paper proves that the quantum cohomology algebra of smooth Fano toric varieties with facet-symmetric polytopes is semisimple, based on the non-degeneracy of the superpotential's critical points.

Contribution

It establishes the non-degeneracy of superpotential critical points for a broad class of smooth Fano toric varieties with facet-symmetric polytopes, ensuring semisimplicity.

Findings

Quantum cohomology is semisimple for these varieties.

Critical points of the superpotential are non-degenerate.

The result applies to all smooth Fano toric varieties with facet-symmetric dual polytopes.

Abstract

The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, W, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of W are non-degenerate. In this paper we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.