Abstract
We derive new Poincaré-series representations for infinite families of non-holomorphic modular invariant functions that include modular graph forms as they appear in the low-energy expansion of closed-string scattering amplitudes at genus one. The Poincaré series are constructed from iterated integrals over single holomorphic Eisenstein series and their complex conjugates, decorated by suitable combinations of zeta values. We evaluate the Poincaré sums over these iterated Eisenstein integrals of depth one and deduce new representations for all modular graph forms built from iterated Eisenstein integrals at depth two. In a companion paper, some of the Poincaré sums over depth-one integrals going beyond modular graph forms will be described in terms of iterated integrals over holomorphic cusp forms and their L-values.
Highlights
The low-energy expansion of string scattering amplitudes at genus one introduced infinite classes of non-holomorphic so-called modular graph forms (MGFs) [1–3]
The Poincaré series are constructed from iterated integrals over single holomorphic Eisenstein series and their complex conjugates, decorated by suitable combinations of zeta values
We evaluate the Poincaré sums over these iterated Eisenstein integrals of depth one and deduce new representations for all modular graph forms built from iterated Eisenstein integrals at depth two
Summary
The low-energy expansion of string scattering amplitudes at genus one introduced infinite classes of non-holomorphic so-called modular graph forms (MGFs) [1–3]. It was already known that several two-loop MGFs can be reduced to one-loop ones and odd zeta values [1, 5], which illustrates that the notion of depth and loop order are not always lined up In those cases of both depth and loop order two, Poincaré-series can be viewed as interpolating between double integrals over holomorphic Eisenstein series and double sums over lattice momenta: our choices of seed functions involve a single sum over SL(2, Z) transformations (akin to one lattice momentum) of a depth-one integral. We identify modular invariant combinations of iterated integrals of holomorphic modular forms and their complex conjugates (conjecturally examples of Brown’s equivariant iterated Eisenstein integrals [33, 35, 36]) that cannot be represented in terms of MGFs. Among the real MGFs of depth two, the most prominent instances are the two-loop lattice sums Ca,b,c(τ ) [1] built from integrals over a+b+c closed-string Green functions. Poincaré sums over imaginary seed functions already generate iterated integrals over ∆12(τ ) at transcendental weight 7, see sections 4.4 and 5.5 for further comments and part II for a detailed analysis
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