Abstract

Instanton numbers of Calabi—Yau threefolds are defined by Gromov—Witten theory. They ‘count’ curves of fixed degree on the manifold. The actual definition involves integration over the moduli space of curves, which gives a priori rational numbers. The mirror theorem allows one to express them in terms of solutions of a differential equation on the mirror manifold. However, the integrality of instanton numbers is not clear from this expression either. In 2003 Jan Stienstra outlined an approach to integrality using the p-adic Frobenius structure on the differential equation. In a recent series of papers with Frits Beukers we propose an explicit and rather elementary construction of the Frobenius structure, which allows us to prove integrality of instanton numbers in several key examples of mirror symmetry. In this talk I will speak about the beginnings of mirror symmetry and explain the ideas of our construction.