Gamma conjecture and tropical geometry

TL;DR

This paper explores the Gamma-conjecture in mirror symmetry for Calabi-Yau manifolds, proposing a tropical geometry approach that links Riemann zeta-values to tropicalization errors in mirror period computations.

Contribution

It introduces a novel tropical geometric perspective on the Gamma-conjecture, connecting transcendental coefficients to tropicalization errors in mirror symmetry.

Findings

Riemann zeta-values appear as error terms in tropicalization.

A new approach to Gamma-conjecture using tropical geometry.

Insights into the integral structures in quantum cohomology.

Abstract

Hodge-theoretic mirror symmetry for a Calabi-Yau mirror pair says that the variation of Hodge structure arising from quantum cohomology of a Calabi-Yau manifold and that arising from deformation of complex structures on the dual Calabi-Yau manifold can be identified with each other, and it has been conjectured (Gamma-conjecture) that the Gamma-integral structure in quantum cohomology corresponds to a natural integral structure on the mirror side. Here the Gamma-integral structure is defined via the topological K-group and the Gamma-class, a characteristic class with transcendental coefficients containing the Riemann $\zeta$-values. In this article, we explain an approach to the Gamma-conjecture using tropical geometry and observe that the Riemann $\zeta$-values arise as error terms of tropicalization in the computation of mirror periods. This is based on joint work [AGIS] with Abouzaid, Ganatra and Sheridan.