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Journal Article

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

MPS-Authors
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Gerken,  Jan E.
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Kleinschmidt,  Axel
Quantum Gravity and Unified Theories, AEI Golm, MPI for Gravitational Physics, Max Planck Society;

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1911.03476.pdf
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Citation

Gerken, J. E., Kleinschmidt, A., & Schlotterer, O. (2020). All-order differential equations for one-loop closed-string integrals and modular graph forms. Journal of High Energy Physics, 2020 (1): 64. doi:10.1007/JHEP01(2020)064.


Cite as: https://hdl.handle.net/21.11116/0000-0005-4D24-D
Abstract
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.