Holomorphic modular forms and cocycles

We study generating series of torus integrals that contain all so-called modular graph forms relevant for massless one-loop closed-string amplitudes. By analysing the differential equation of the generating series we construct a solution for their low-energy expansion to all orders in the inverse string tension α. Our solution is expressed through initial data involving multiple zeta values and certain real-analytic functions of the modular parameter of the torus. These functions are built from real and imaginary parts of holomorphic iterated Eisenstein integrals and should be closely related to Brown's recent construction of real-analytic modular forms. We study the properties of our real-analytic objects in detail and give explicit examples to a fixed order in the α-expansion. In particular, our solution allows for a counting of linearly independent modular graph forms at a given weight, confirming previous partial results and giving predictions for higher, hitherto unexplored weights. It also sheds new light on the topic of uniform transcendentality of the α-expansion.

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Generating series of all modular graph forms from iterated Eisenstein integrals

Gerken

Kleinschmidt

Schlotterer

2020

J. High Energ. Phys.

169

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Holomorphic modular forms and cocycles

Abstract: This is a slightly expanded version of my lecture at the conference Modular forms are everywhere at Bonn, May 2017, taking into account remarks by Don Zagier and Shaul Zemel, and suggestions of the referees.

Cited by 1 publication

References 10 publications

Generating series of all modular graph forms from iterated Eisenstein integrals

Generating series of all modular graph forms from iterated Eisenstein integrals

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