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We investigate generating functions for the integrals over world-sheet toriappearing in closed-string one-loop amplitudes of bosonic, heterotic andtype-II theories. These closed-string integrals are shown to obey homogeneousand linear differential equations in the modular parameter of the torus. Wespell out the first-order Cauchy-Riemann and second-order Laplace equations forthe generating functions for any number of external states. The low-energyexpansion of such torus integrals introduces infinite families ofnon-holomorphic modular forms known as modular graph forms. Our resultsgenerate homogeneous first- and second-order differential equations forarbitrary such modular graph forms and can be viewed as a step towardsall-order low-energy expansions of closed-string integrals.

All-order differential equations for one-loop closed-string integrals and modular graph forms