Proof of a modular relation between 1-, 2-and 3-loop Feynman diagrams on a torus

The coefficients of the higher-derivative terms in the low energy expansion of genus-one graviton Type II superstring scattering amplitudes are determined by integrating sums of non-holomorphic modular functions over the complex structure modulus of a torus. In the case of the four-graviton amplitude, each of these modular functions is a multiple sum associated with a Feynman diagram for a free massless scalar field on the torus. The lines in each diagram join pairs of vertex insertion points and the number of lines defines its weight w, which corresponds to its order in the low energy expansion. Previous results concerning the low energy expansion of the genus-one four-graviton amplitude led to a number of conjectured relations between modular functions of a given w, but different numbers of loops <= w 1. In this paper we shall prove the simplest of these conjectured relations, namely the one that arises at weight w = 4 and expresses the three-loop modular function D-4 in terms of modular functions with one and two loops. As a byproduct, we prove three intriguing new holomorphic modular identities. (C) 2017 Elsevier Inc. All rights reserved.