Gamma conjecture via mirror symmetry

TL;DR

This paper explores the Gamma conjecture relating the principal asymptotic class of a Fano manifold to its Gamma class, demonstrating how mirror symmetry supports this conjecture for various classes of manifolds.

Contribution

It provides evidence for the Gamma conjecture via mirror symmetry, especially for toric varieties, complete intersections, and Grassmannians, and discusses its compatibility with hyperplane sections.

Findings

Gamma conjecture holds for toric varieties, complete intersections, and Grassmannians.

Mirror symmetry provides a framework to understand the Gamma class and oscillatory integrals.

The conjecture is compatible with hyperplane sections and relates to polynomial loop space insights.

Abstract

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's $\Gamma$-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.