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

TL;DR

This paper explores the relationship between open and closed string amplitudes at genus one, introducing an elliptic single-valued map that connects elliptic multiple zeta values to modular graph forms, advancing understanding of string dualities.

Contribution

It introduces an elliptic single-valued map that generalizes the genus-zero single-valued map, linking open-string elliptic multiple zeta values to closed-string modular graph forms at genus one.

Findings

Identification of formal substitution rules between open and closed string integrals.

Establishment of an elliptic single-valued map extending genus-zero concepts.

Insights into the differential equations and degeneration limits of genus-one integrals.

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.