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

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 $J=\mathrm{s}\mathrm{v}Z$.