Closed strings as single-valued open strings: A genus-zero derivation
Oliver Schlotterer, Oliver Schnetz
TL;DR
This work proves that genus-zero moduli-space integrals governing open- and closed-string tree amplitudes are related by the single-valued map: $J={\rm sv}\,Z$. By focusing on the genus-zero, Betti–deRham duality between disk boundary cycles and anti-holomorphic sphere factors, the authors develop an elementary, inductive derivation that avoids reliance on motivic machinery while clarifying how single-valued MZVs arise in closed-string amplitudes. They show that the even-$\zeta$ contributions cancel in sphere integrals and that the remaining MZV content matches the single-valued image of the open-string disk integrals, i.e. ${\rm sv}$ acts on the $\alpha'$-expansion in $s_{ij}$. The results solidify the open/closed string connection, illuminate selection rules for MZVs, and offer a framework potentially extensible to loop and higher-genus settings, where elliptic and modular structures become relevant.
Abstract
Based on general mathematical assumptions we give an independent, elementary derivation of a theorem by Francis Brown and Clément 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 $J = {\rm sv} \, Z$.