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 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.

43 pages, 3 figures, submitted to the proceedings of the conference "Primitive Forms and Related Subjects" at IPMU (Feb 2014), v2: exposition improved, discussion on odd cohomology added in the appendix, references updated, v3: final version, credit given to Sanda-Shamoto for the content in the appendix

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