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Eta Quotients and Rademacher Sums

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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

Eta quotients on \(\varGamma _0(6)\) 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 \(\varGamma _0(6)\) 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.

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Acknowledgements

The second author thanks KMPB for hospitality and colleagues at conferences in Zeuthen, Bonn, St. Goar and Les Houches for advice and encouragement that emboldened our joint effort to tackle eta quotients beyond the remit of genus zero so far encountered in massive Feynman diagrams. We especially thank Johannes Blümlein for his question on the possibility of obtaining an explicit formula for Fourier coefficients of the Hauptmodul of \(\varGamma _0(6)\) and Freeman Dyson for urging us to try to emulate the notable work by Rademacher on partition numbers [29]. We thank Yajun Zhou and an anonymous referee for helpful suggestions that improved our presentation.

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Authors and Affiliations

  1. Monash University, Melbourne, VIC, 3800, Australia

    Kevin Acres

  2. Open University, Milton Keynes, MK7 6AA, UK

    David Broadhurst

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  1. Kevin Acres
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Correspondence to David Broadhurst .

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Editors and Affiliations

  1. Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

    Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

  3. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Peter Paule

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Acres, K., Broadhurst, D. (2019). Eta Quotients and Rademacher Sums . In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_1

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  • DOI: https://doi.org/10.1007/978-3-030-04480-0_1

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