From modular forms to differential equations for Feynman integrals
Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi
TL;DR
This work develops an algebraic representation of modular forms tailored for elliptic Feynman integrals, showing that any weight-$k$ modular form on genus-zero congruence groups can be written locally as $f(\tau)=\mathrm{K}(\lambda(\tau))^{k}\,A(x(\tau))$ with $A$ algebraic. By constructing explicit bases for $M_{2k}$ on $\Gamma(2)$, $\Gamma_0(2)$, $\Gamma_0(4)$, $\Gamma_0(6)$, and $\Gamma_1(6)$, the authors express elliptic objects—such as elliptic multiple zeta values and kernels in differential equations for multiloop integrals—in terms of powers of the complete elliptic integral $\mathrm{K}$ multiplied by algebraic functions of Hauptmoduls. They apply this framework to rewrite elliptic multiple zeta values as iterated integrals over modular forms, derive a canonical differential-equation form for a class of hypergeometric functions, and show that sunrise and kite integrals can be described by iterated integrals of modular forms for $\Gamma_1(6)$. The approach clarifies the link between elliptic Feynman integrals and modular forms, providing algebraic, kernel-based tools that facilitate generalisations to multi-variable settings and connections to elliptic polylogarithms.
Abstract
In these proceedings we discuss a representation for modular forms that is more suitable for their application to the calculation of Feynman integrals in the context of iterated integrals and the differential equation method. In particular, we show that for every modular form we can find a representation in terms of powers of complete elliptic integrals of the first kind multiplied by algebraic functions. We illustrate this result on several examples. In particular, we show how to explicitly rewrite elliptic multiple zeta values as iterated integrals over powers of complete elliptic integrals and rational functions, and we discuss how to use our results in the context of the system of differential equations satisfied by the sunrise and kite integrals.