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

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.

38 pages, 9 figures, in version 2: Appendix D added and corrections made in section 4

References in corpus (2)

• On the modular structure of the genus-one Type II superstring low energy expansion
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops

Cited by in corpus (25)

• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
• Higher genus modular graph functions, string invariants, and their exact asymptotics
• Proving relations between modular graph functions
• Poisson equation for the Mercedes diagram in string theory at genus one
• Generating series of all modular graph forms from iterated Eisenstein integrals
• Simplifying the one loop five graviton amplitude in type IIB string theory
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality
• Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series
• Parity and the modular bootstrap
• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms
• Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes
• Modular graph forms from equivariant iterated Eisenstein integrals
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Elliptic modular graph forms I: Identities and generating series
• Poisson equations for elliptic modular graph functions
• Basis Decompositions and a Mathematica Package for Modular Graph Forms
• A simplifying feature of the heterotic one loop four graviton amplitude
• To the cusp and back: Resurgent analysis for modular graph functions
• Relations between elliptic modular graphs
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• Hierarchy of Modular Graph Identities
• From Modular Graph Forms to Iterated Integrals
• Integral of two-loop modular graph functions