Tetrahedral modular graph functions

TL;DR

This paper investigates the mathematical structure of modular invariant integrands in string theory, focusing on tetrahedral three-loop diagrams, and derives their Laplacian action and spectral properties.

Contribution

It provides explicit formulas for the Laplacian acting on tetrahedral modular graph functions and analyzes their eigenvalues using group and representation theory.

Findings

Closed-form expressions for Laplacian action

Identification of eigenvalues and degeneracies

Insights into modular invariance in string amplitudes

Abstract

The low-energy expansion of one-loop amplitudes in type II string theory generates a series of world-sheet integrals whose integrands can be represented by world-sheet Feynman diagrams. These integrands are modular invariant and understanding the structure of the action of the modular Laplacian on them is important for determining their contribution to string scattering amplitudes. In this paper we study a particular infinite family of such integrands associated with three-loop scalar vacuum diagrams of tetrahedral topology and find closed forms for the action of the Laplacian. We analyse the possible eigenvalues and degeneracies of the Laplace operator by group- and representation-theoretic means.