Single-valued hyperlogarithms, correlation functions and closed string amplitudes

TL;DR

This paper explores the mathematical structure of closed string amplitudes at genus zero, revealing their connection to single-valued periods, hyperlogarithms, and multiple zeta values, with implications for string theory and number theory.

Contribution

It provides new proofs of global and local properties of string amplitude integrals, linking them to single-valued hyperlogarithms and establishing their number-theoretic nature.

Findings

Closed string integrals factorize into open string integrals (KLT formula).

Asymptotic expansion coefficients are in the ring of single-valued multiple zeta values.

Develops a theory of integration for single-valued hyperlogarithms.

Abstract

We give new proofs of a global and a local property of the integrals which compute closed string theory amplitudes at genus zero. Both kinds of properties are related to the newborn theory of single-valued periods, and our proofs provide an intuitive understanding of this relation. The global property, known in physics as the KLT formula, is a factorisation of the closed string integrals into products of pairs of open string integrals. We deduce it by identifying closed string integrals with special values of single-valued correlation functions in two dimensional conformal field theory, and by obtaining their conformal block decomposition. The local property is of number theoretical nature. We write the asymptotic expansion coefficients as multiple integrals over the complex plane of special functions known as single-valued hyperlogarithms. We develop a theory of integration of single-valued hyperlogarithms, and we use it to demonstrate that the asymptotic expansion coefficients belong to the ring of single-valued multiple zeta values.