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Abstract

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