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

Understand

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.