An indispensable resource for readers in physics and mathematics seeking a solid grasp of the mathematical tools shaping modern theoretical physics, this book comprises a practical introduction to the mathematical theory of modular forms and their application to the physics of string theory and supersymmetric Yang-Mills theory. Suitable for adventurous undergraduates, motivated graduate students, and researchers wishing to navigate the intersection of cutting-edge research in physics and mathematics, it guides readers from the theory of elliptic functions to the fascinating mathematical world of modular forms, congruence subgroups, Hecke theory, and more. Having established a solid basis, the book proceeds to numerous applications in physics, with only minimal prior knowledge assumed. Appendices review foundational topics, making the text accessible to a broad audience, along with exercises and detailed solutions that provide opportunities for practice. After working through the book, readers will be equipped to carry out research in the field.

Table of Contents

Part I Modular forms and their variants

2 Elliptic functions

3 Modular forms for SL(2, Z)

4 Variants of modular forms

5 Quantum fields on a torus

6 Congruence subgroups and modular curves

7 Modular forms for congruence subgroups

8 Modular derivatives and vector-valued modular forms

9 Modular graph functions and forms

Part II Extensions and applications

10 Hecke operators

12 String amplitudes

13 Toroidal compactification

14 S-duality of Type IIB superstrings

15 Dualities in N = 2 super Yang–Mills theories

16 Basic Galois theory

Part III Appendix

Appendix A Some arithmetic

Appendix B Riemann surfaces

Appendix C Line bundles on Riemann surfaces

Appendix D Riemann #-functions and meromorphic forms

Appendix E Solutions to exercises

Review

‘A mother and baby, or babies, will go through what may arguably be the toughest journey of their lives during labour and childbirth. The birth passage is short, but the list of potential problems that could occur in this journey is long. When a problem occurs, what one doesn’t often have is time. The ticking clock often determines the outcome. Having clear knowledge about what to do, how to do, and when to do can make all the difference. Sound knowledge about labour and delivery is at the core of effective care, and this book has a central contribution to make in getting you ready to face the delivery suite with confidence and purpose. The book provides clear information, guidance and instructions, all based on sound evidence and current consensus. Make time for this book; it will help you, your team, mothers and babies.’ Arri Coomarasamy, OBE, MBChB, MD, FRCOG, FMedSci, University of Oxford

About the Author

Sir Sabaratnam Arulkumaran is past President of the International Federation of Obstetrics & Gynaecology, the British Medical Association and the Royal College of Obstetricians and Gynaecologists. He is Professor Emeritus of Obstetrics & Gynaecology, St George's University of London; Visiting Professor at the Institute of Global Health, Imperial College, London; Foundation Professor of Obstetrics & Gynaecology at the University of Nicosia, Cyprus. Arul's contribution to women's health was recognised with three National Honours; UK – Knight Bachelor by the Queen in her Birthday Honours List in June 2009; Ranjana National award by the Sri Lankan President in 2019 & Active citizen award by the President of Cyprus in 2019.