Eta quotients and Rademacher sums

TL;DR

This paper explores eta quotients and Rademacher sums to evaluate sunrise integrals, revealing modular parametrizations, explicit Fourier coefficient formulas, and handling levels with various genera, advancing the understanding of modular forms in physics and mathematics.

Contribution

It introduces explicit Rademacher sum formulas for eta quotients across multiple levels and genera, and applies these to evaluate complex sunrise integrals.

Findings

Explicit Rademacher sum formulas for eta quotients at various levels.

Modular parametrizations of sunrise integrals at multiple loops.

Handling of eta quotients with different genera and explicit Fourier coefficient calculations.

Abstract

Eta quotients on $\Gamma_0(6)$ yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome $q$. Atkin-Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of an eta quotient that is a Hauptmodul for $\Gamma_0(6)$ and its generalization is found for all levels with genus 0, namely for $N = 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25$. There are elliptic obstructions at $N = 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49,$ with genus 1. We surmount these, finding explicit formulas for Fourier coefficients of eta quotients in thousands of cases. We show how to handle the levels $N=22, 23, 26, 28, 29, 31, 37, 50$, with genus 2, and the levels $N=30,33,34,35,39,40,41,43,45,48,64$, with genus 3. We also solve examples with genera $4,5,6,7,8,13$.