All-order differential equations for one-loop closed-string integrals and modular graph forms
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.
Cite
@article{arxiv.1911.03476,
title = {All-order differential equations for one-loop closed-string integrals and modular graph forms},
author = {Jan E. Gerken and Axel Kleinschmidt and Oliver Schlotterer},
journal= {arXiv preprint arXiv:1911.03476},
year = {2020}
}
Comments
54+24 pages, v2: typos corrected, version published in JHEP