From Feynman integrals to Kloosterman sums

Certain Feynman integrals in two-dimensional quantum field theory are given by moments of the Bessel functions. In an attempt to get a grasp of their properties, Broadhurst and Roberts explored the analogy with Kloosterman sums over finite fields. I will explain their conjectures and what we can prove in joint work with Claude Sabbah and Jeng-Daw Yu.