Gamma conjecture via mirror symmetry

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 [29] of Vasily Golyshev and the present authors claims that the principal asymptotic class $A_F$ equals the Gamma class $\widehat{\Gamma}_F$ 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.