Periods and Superstring Amplitudes

TL;DR

This paper explores the mathematical structure of string scattering amplitudes, linking them to advanced concepts in algebraic geometry and number theory, and discusses their representation as periods on moduli spaces with connections to multiple zeta values.

Contribution

It reviews recent mathematical developments connecting string amplitudes to periods, hypergeometric functions, and motivic multiple zeta values, highlighting the relation between real and complex integrals.

Findings

String amplitudes are described by periods on moduli spaces of Riemann spheres.

A link between hypergeometric functions and motivic multiple zeta values is established.

Complex integrals can be expressed as single-valued projections of real iterated integrals.

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-sheet given by the topology of a disk or sphere, respectively. Formally, for scattering of N strings this leads to N-3-dimensional iterated real integrals along the compactified real axis or N-3-dimensional complex sphere integrals, respectively. As a consequence the physical observables are described by periods on M_{0,N} - 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 M_{0,N}(R) and the decomposition of motivic multiple zeta values. Furthermore, a relation expressing complex integrals on M_{0,N}(C) as single-valued projection of iterated real integrals on M_{0,N}(R) is exhibited.