Periods and Superstring Amplitudes

Abstract

Scattering amplitudes which describe the interaction of physical states play an important role in determining physical observables. In string theory the physical states are given by vibrations of open and closed strings and their interactions are described (at the leading order in perturbation theory) by a world–heet given by the topology of a disk or sphere, respectively. Formally, for scattering of N strings this leads to –dimensional iterated real integrals along the compactified real axis or –dimensional complex sphere integrals, respectively. As a consequence the physical observables are described by periods on –the moduli space of Riemann spheres of N ordered marked points. The mathematical structure of these string amplitudes share many recent advances in arithmetic algebraic geometry and number theory like multiple zeta values, single–valued multiple zeta values, Drinfeld, Deligne associators, Hopf algebra and Lie algebra structures related to Grothendiecks Galois theory. We review these results, with emphasis on a beautiful link between generalized hypergeometric functions describing the real iterated integrals on and the decomposition of motivic multiple zeta values. Furthermore, a relation expressing complex integrals on as single–valued projection of iterated real integrals on is exhibited.