Abstract
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.
Highlights
Closed-string scattering amplitudes at perturbative one-loop order are formulated as integrals over the complex-structure parameter τ of the torus worldsheet
We study generating series of torus integrals that contain all so-called modular graph forms relevant for massless one-loop closed-string amplitudes
The families of modular invariants and more generally modular forms that can arise in this low-energy expansion have been studied from various perspectives [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],1 and they are known as modular graph forms (MGFs)
Summary
Closed-string scattering amplitudes at perturbative one-loop order are formulated as integrals over the complex-structure parameter τ of the torus worldsheet. Similar types of generating series have been constructed for one-loop open-string amplitudes, i.e. for a conjectural basis of integrals over punctures on the boundary of a cylinder or Mobius-strip worldsheet [38, 39] Their differential equations have been solved to yield explicitly known combinations of iterated integrals over holomorphic Eisenstein series Gk at all orders of the open-string α -expansions.. We shall here exploit that the first-order differential equations of closed-string generating series have the same structure as their open-string counterparts [30]: our main result is a solution of the closed-string differential equations that pinpoints a systematic parametrization of arbitrary MGFs in terms of iterated Eisenstein integrals and their complex conjugates.. Poincare-series representations of modular-invariant functions feature crucially in the Rankin-Selberg-Zagier method for integrals over τ [1, 62,63,64,65] and related work in the context of MGFs can be found in [4, 22, 25]
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