The conifold point

TL;DR

This paper proves that the quantum cohomology algebra of a toric Fano manifold contains a field as a direct summand, using properties of Laurent polynomials with positive coefficients and their Morse critical points.

Contribution

It establishes a new link between Laurent polynomial convexity and the algebraic structure of quantum cohomology for toric Fano manifolds.

Findings

Quantum cohomology of toric Fano contains a field as a summand

Positive Laurent polynomials have a unique positive Morse critical point

Evidence suggests similar results for all Fano manifolds

Abstract

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer to a question of Ostrover and Tyomkin: the quantum cohomology algebra of a toric Fano manifold contains a field as a direct summand. Moreover, it gives a good evidence that the same statement holds for any Fano manifold.