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Elliptic modular graph forms I: Identities and generating series

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Kleinschmidt,  Axel
Quantum Gravity and Unified Theories, AEI Golm, MPI for Gravitational Physics, Max Planck Society;

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2012.09198.pdf
(Preprint), 701KB

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Citation

D'Hoker, E., Kleinschmidt, A., & Schlotterer, O. (in preparation). Elliptic modular graph forms I: Identities and generating series.


Cite as: https://hdl.handle.net/21.11116/0000-0007-CB80-3
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.