Differential equations for one-loop string integrals
From Giulio Falcioni on July 8th, 2020
This talk is dedicated to mathematical structures in the low-energy expansion of one-loop string amplitudes. I will review basics of string perturbation theory and give examples of connections with number theory from the integration over moduli of punctured worldsheet. At one loop, the insertion of external states on the open- and closed-string worldsheets requires integration over punctures on a cylinder boundary and a torus, respectively. Suitable bases of such integrals will be shown to obey simple first-order differential equations in the modular parameter of the surface. These differential equations will be exploited to perform the integrals order by order in the inverse string tension, similar to modern strategies for dimensionally regulated Feynman integrals. Our method manifests the appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion. Moreover, infinite families of Laplace equations can be generated for the modular forms in closed-string low-energy expansions.