9 - Modular graph functions and forms

Summary

Modular graph functions and modular graph forms map decorated graphs to complex-valued functions on the Poincaré upper half plane with definite transformation properties. Specifically, modular graph functions are SL(2,Z)-invariant functions, while modular graph forms may be identified with SL(2,Z)-invariant differential forms. Modular graph functions and forms generalize, and at the same time unify, holomorphic and non-holomorphic Eisenstein series, almost holomorphic modular forms, multiple zeta-functions, and iterated modular integrals. For example, non-holomorphic Eisenstein series may be associated with one-loop graphs and represent a special class of modular graph functions. The expansion of modular graph forms at the cusp includes Laurent polynomials whose coefficients are combinations of Riemann zeta-values and multiple zeta-values, while each modular graph form may be expanded in a basis of iterated modular integrals. Eisenstein series and modular graph functions and forms beyond Eisenstein series occur naturally and pervasively in the study of the low-energy expansion of superstring amplitudes. Here we shall present a purely mathematical approach with only minimal reference to physics.