Abstract
Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker-Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker-Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series.
Highlights
A modular graph function (MGF) maps a decorated graph to an SL(2, Z) invariant function on the upper half complex plane H1
Elliptic modular graph functions and forms are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker-Eisenstein series
We shall prove the generalization of the holomorphic subgraph reduction procedure to the case of Elliptic modular graph functions and forms (eMGFs), using the integral formulation of eMGFs and the Fay identities between the coefficient functions of the Kronecker-Eisenstein series
Summary
A modular graph function (MGF) maps a decorated graph to an SL(2, Z) invariant function on the upper half complex plane H1. A Mathematica package is available for the systematic implementation of identities among MGFs in [11] and brought into wider context in the PhD thesis [15] Their reduction to iterated Eisenstein integrals via generating series [16, 17] exposes all their relations and a connection to elliptic MZVs in open-string computations [18] and may clarify the connection with Brown’s construction of non-holomorphic modular forms [19, 20]. Following the definition of eMGFs in these various formulations, we proceed to deriving algebraic and differential relations for the characters and for the eMGFs, in close parallel to the corresponding derivations in the case of MGFs. In particular, we shall prove the generalization of the holomorphic subgraph reduction procedure to the case of eMGFs, using the integral formulation of eMGFs and the Fay identities between the coefficient functions of the Kronecker-Eisenstein series. Additional technical details and comments have been relegated to appendices B–D
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