The number theory of superstring scattering amplitudes

The Feynman diagram expansion of scattering amplitudes in perturbative superstring theory can be written (for closed strings) as a series of integrals over compactified moduli spaces of Riemann surfaces with marked points, indexed by the genus. Therefore in genus 0 it is reasonable to find, as it often happens in QFT computations, periods of M_{0,N}, which are known to be multiple zeta values. In this talk I want to report on recent advances in the genus 1 amplitude, which are related to the development of 2 different generalizations of classical multiple zeta values, namely elliptic multiple zeta values and conical sums.