A Class of Nonholomorphic Modular Forms Ii: Equivariant Iterated Eisenstein integrals

Abstract

We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.

Generating series of all modular graph forms from iterated Eisenstein integrals

“…All the integration kernels in this differential equation are of the form (τ −τ ) j G k (τ ) with k ≥ 4 and 0 ≤ j ≤ k−2. Hence, our kernels line up with those of Brown's holomorphic and single-valued iterated Eisenstein integrals [46,48,49].…”

“…The results of this work provide a link to recent developments in the mathematics literature: Brown constructed a class of non-holomorphic modular forms from iterated Eisenstein integrals and their complex conjugates which share the algebraic and differential properties of MGFs [46,48,49]. We expect the combinations of iterated Eisenstein integrals in our parametrization of MGFs to occur in Brown's generating series of single-valued iterated Eisenstein integrals that drive his construction of modular forms: at the level of the respective generating series, single-valued iterated Eisenstein integrals and closedstring integrals both obey differential equations of Knizhnik-Zamolodchikov-Bernard-type in τ .…”

“…. , k−2} ties in with Brown's iterated Eisenstein integrals [46,48,49]. More specifically, (1.5) will be shown to admit an all-order solution for the original integrals (1.1)…”

“…construction [46,49,75] of non-holomorphic modular forms, the series Y τ η is engineered to simplify the holomorphic derivative (2.33) at the expense of the more lengthy expression (2.37) for the antiholomorphic one.…”

Modular graph functions and odd cuspidal functions. Fourier and Poincaré series

“…The underlying nature of these identities remains to be fully uncovered, but it is already clear that they generalize to modular graph forms some of the algebraic identities which exist between multiple zeta values (see for example [17][18][19][20][21][22]). Direct connections between modular graph functions and single-valued elliptic polylogarithms were obtained in [6,[23][24][25]. The role of multiple zeta values in string amplitudes has been investigated extensively in [26][27][28][29][30].…”