  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.
