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

1. Front Matter

   Pages i-xiii

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2. Eta Quotients and Rademacher Sums

   • Kevin Acres, David Broadhurst

   Pages 1-27

3. On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms

   • Luise Adams, Stefan Weinzierl

   Pages 29-49

4. Iterative Non-iterative Integrals in Quantum Field Theory

   • Johannes Blümlein

   Pages 51-77

5. Analytic Continuation of the Kite Family

   • Christian Bogner, Armin Schweitzer, Stefan Weinzierl

   Pages 79-91

6. A Four-Point Function for the Planar QCD Massive Corrections to Top-Antitop Production in the Gluon-Fusion Channel

   • Roberto Bonciani, Matteo Capozi, Paul Caucal

   Pages 93-106

7. From Modular Forms to Differential Equations for Feynman Integrals

   • Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi

   Pages 107-131

8. One-Loop String Scattering Amplitudes as Iterated Eisenstein Integrals

   • Johannes Broedel, Oliver Schlotterer

   Pages 133-159

9. Expansions at Cusps and Petersson Products in Pari/GP

   • Henri Cohen

   Pages 161-181

10. CM Evaluations of the Goswami-Sun Series

    • Madeline Locus Dawsey, Ken Ono

    Pages 183-193

11. Automatic Proof of Theta-Function Identities

    • Jie Frye, Frank Garvan

    Pages 195-258

12. The Generators of all Polynomial Relations Among Jacobi Theta Functions

    • Ralf Hemmecke, Cristian-Silviu Radu, Liangjie Ye

    Pages 259-268

13. Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms

    • Fredrik Johansson

    Pages 269-293

14. Multi-valued Feynman Graphs and Scattering Theory

    • Dirk Kreimer

    Pages 295-325

15. Interpolated Sequences and Critical L-Values of Modular Forms

    • Robert Osburn, Armin Straub

    Pages 327-349

16. Towards a Symbolic Summation Theory for Unspecified Sequences

    • Peter Paule, Carsten Schneider

    Pages 351-390

17. Differential Equations and Dispersion Relations for Feynman Amplitudes

    • Ettore Remiddi

    Pages 391-414

18. Feynman Integrals, Toric Geometry and Mirror Symmetry

    • Pierre Vanhove

    Pages 415-458

19. Modular and Holomorphic Graph Functions from Superstring Amplitudes

    • Federico Zerbini

    Pages 459-484

20. Some Algebraic and Arithmetic Properties of Feynman Diagrams

    • Yajun Zhou

    Pages 485-509

This book includes review articles in the field of elliptic integrals, elliptic functions and modular forms intending to foster the discussion between theoretical physicists working on higher loop calculations and mathematicians working in the field of modular forms and functions and analytic solutions of higher order differential and difference equations. 

• elliptic integrals and functions
• iterated modular functions
• iterated integrals
• massive high loop Feynman integral
• differential equations

“The book is explicitly concerned with ‘the mathematics of modular forms [as it is] of central importance for the analytic solution of Feynman diagrams,’ but modular formers should be equally interested in the material offered in this compendium.” (Michael Berg, MAA Reviews, May 19, 2019)

• Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

  Johannes Blümlein

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

  Carsten Schneider, Peter Paule