Mirror symmetric Gamma conjecture for Fano and Calabi-Yau manifolds

TL;DR

This paper explores the mirror symmetric Gamma conjecture, showing how it relates the Gamma class of Fano and Calabi-Yau manifolds to the asymptotics of mirror periods, and extends the conjecture to various geometric contexts.

Contribution

It generalizes the mirror symmetric Gamma conjecture to broader classes of manifolds and demonstrates implications via Laplace transformation.

Findings

Gamma class determines asymptotics of mirror periods

Conjecture extends to total spaces of anti-nef line bundles

Implications for nef complete intersections

Abstract

The mirror symmetric Gamma conjecture roughly speaking says that the Gamma class of a manifold determines the asymptotics of (exponential) periods of the mirror. We recast the method in [Iri11] in a more general context and show that the mirror symmetric Gamma conjecture for a Fano manifold F implies, via Laplace transformation, that for the total space K_F of the canonical bundle or for anticanonical sections in F. More generally, we discuss the mirror symmetric Gamma conjecture for the total space of a sum of anti-nef line bundles over F or for nef complete intersections in F.