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 D4​ in terms of
modular functions with one and two loops. As a byproduct, we prove three
intriguing new holomorphic modular identities.Comment: 38 pages, 9 figures, in version 2: Appendix D added and corrections
made in section