Differential equations for one-loop string integrals

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.

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