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All-order differential equations for one-loop closed-string integrals and modular graph forms
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We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and second-order Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first- and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.
Forward citations
Cited by 2 Pith papers
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From Modular Graph Forms to Iterated Integrals
A tree-based algorithm converts modular graph forms into equivariant iterated Eisenstein integrals, is implemented for topologies up to four vertices, and is used to extract the alpha'^8 zeta3 zeta5 term of the four-g...
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Equivariant primitives of Eisenstein series for congruence subgroups
Equivariant primitives of Eisenstein series for principal congruence subgroups are shown to equal the corresponding non-holomorphic Eisenstein series, including new weight-two cases expressed via single-valued logarithms.
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