Cite as: doi:10.26081/K67S5V

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A modular graph function maps a Feynman graph for a conformal
scalar field in the plane to a modular function. Modular graph functions
generalize non-holomorphic Eisenstein series for genus one, and Kawazumi-Zhang
and Faltings invariants for higher genus. They arise naturally in string theory,
where their integral over the moduli space of compact Riemann surfaces
determines the strength of the effective interactions in the low energy
expansion of string theory. They satisfy remarkable algebraic and differential
identities.
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