Elliptic modular graph forms I: Identities and generating series

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.

69 pages, v2: typos corrected, matches published version

References in corpus (7)

• All-order differential equations for one-loop closed-string integrals and modular graph forms
• Generating series of all modular graph forms from iterated Eisenstein integrals
• Tetrahedral modular graph functions
• Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality
• Poisson equations for elliptic modular graph functions
• Relations between elliptic modular graphs
• Two dialects for KZB equations: generating one-loop open-string integrals

Cited by in corpus (11)

• Coaction and double-copy properties of configuration-space integrals at genus zero
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Modular graph forms from equivariant iterated Eisenstein integrals
• A double copy from twisted (co)homology at genus one
• To the cusp and back: Resurgent analysis for modular graph functions
• Higher-genus Fay-like identities from meromorphic generating functions
• Basis decompositions of genus-one string integrals
• Advanced tools for basis decompositions of genus-one string integrals
• Poisson equation for genus two string invariants: a conjecture
• Integrating simple genus two string invariants over moduli space
• A construction of single-valued elliptic polylogarithms