Laplace-eigenvalue equations for length three modular iterated integrals

TL;DR

This paper extends the theory of modular iterated integrals to length three, providing new Laplace-eigenvalue equations and exploring their potential in understanding modular graph functions in string theory.

Contribution

It introduces a method for constructing length three modular iterated integrals with unique Laplace-eigenvalue equations, expanding on previous length two results.

Findings

Length three modular iterated integrals with eigenvalue equations

Extension of Brown's length two case

Potential applications in string perturbation theory

Abstract

A space of modular iterated integrals sits inside the space of real analytic modular forms. We present a theorem for producing length three modular iterated integrals which are not simply combinations of real analytic Eisenstein series; each function has an associated Laplace-eigenvalue equation. This can be viewed as an extension of the length two case recently given by F. Brown, a review of which is included in this paper. We discuss how modular iterated integrals could help understand the modular graph functions which arise in string perturbation theory.