Closed strings as single-valued open strings: A genus-zero derivation

TL;DR

This paper provides a straightforward derivation showing that closed-string amplitudes can be expressed as single-valued versions of open-string amplitudes, based on integrals over genus-zero moduli spaces.

Contribution

It offers an elementary, independent derivation of the relation between closed and open string amplitudes via the single-valued map, rooted in genus-zero moduli-space integrals.

Findings

Closed-string integrals are single-valued versions of open-string integrals.

The relation is derived from genus-zero moduli-space integrals.

The derivation confirms the mathematical connection between open and closed string amplitudes.

Abstract

Based on general mathematical assumptions we give an independent, elementary derivation of a theorem by Francis Brown and Cl\'ement Dupont which states that tree-level amplitudes of closed and open strings are related through the single-valued map `sv'. This relation can be traced back to the underlying moduli-space integrals over punctured Riemann surfaces of genus zero. The sphere integrals $J$ in closed-string amplitudes and the disk integrals $Z$ in open-string amplitudes are shown to obey $J = {\rm sv} \, Z$.