Lifts of cycles in tropical hypersurfaces and the Gamma conjecture

TL;DR

This paper constructs lifts of tropical cycles in hypersurfaces within toric varieties, computes their intersection numbers, and explores their asymptotics in relation to the Gamma conjecture in mirror symmetry.

Contribution

It introduces a method to lift tropical cycles to topological cycles with torus fibrations and relates their intersection theory and period integrals to mirror symmetry.

Findings

Lifts of tropical cycles are explicitly constructed as topological cycles.

Intersection numbers of lifted cycles are computed via tropical intersection theory.

Asymptotic formulas for period integrals are derived, supporting the Gamma conjecture.

Abstract

For a complex hypersurface of dimension $d \geq 1$ in a toric variety, we construct lifts of tropical $(p, q)$-cycles with $p+q=d$ in the associated tropical hypersurface. The tropical cycles we consider are described by Minkowski weights, and their lifts are realized as topological cycles admitting a torus fibration structure over the tropical cycles. The intersection numbers of these lifted cycles are computed in terms of tropical intersection theory. We further derive the asymptotic formulas for the period integrals of the lifts in the tropical limit, which are closely related to the mirror symmetric Gamma conjecture. Throughout the paper, we assume that the tropicalization is dual to a unimodular triangulation of the Newton polytope.