Towards closed strings as single-valued open strings at genus one

Abstract We relate the low-energy expansions of world-sheet integrals in genus-one amplitudes open- and closed-string states. The respective expansion coefficients are elliptic multiple zeta values (eMZVs) open-string case non-holomorphic modular forms dubbed ‘modular graph (MGFs)’ for closed strings. By inspecting differential equations degeneration limits suitable generating series integrals, we identify formal substitution rules mapping eMZVs open strings to MGFs Based on properties these rules, refer them as an single-valued map which generalizes genus-zero notion a acting MZVs seen tree-level relations between string.