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

TL;DR

This paper proves a key relation connecting 1-, 2-, and 3-loop modular functions in genus-one superstring amplitudes, advancing understanding of their mathematical structure and revealing new modular identities.

Contribution

It establishes the first proof of a conjectured relation between multi-loop modular functions at weight 4, linking three-loop functions to lower-loop functions in string theory.

Findings

Proved the relation expressing the three-loop modular function D_4 in terms of lower-loop functions.

Derived three new holomorphic modular identities.

Enhanced understanding of the structure of genus-one superstring amplitudes.

Abstract

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 $\le w-1$. 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.