Modular Forms and String Theory

An introduction to modular forms with string theory and gauge theory applications for students and researchers in physics and mathematics.

Eric D'Hoker (Author), Justin Kaidi (Author)

9781009457538, Cambridge University Press

Hardback, published 12 December 2024

500 pages
25.2 x 17.7 x 3 cm, 1.016 kg

'This book provides a detailed account of modular form from a physics perspective, in the context of their application to string theory. It is written by two of the experts in the subject and gives a comprehensive mathematical physics account of modular forms. Certainly, this book is an essential reference for researchers working in this field.' Joseph Conlon, The Observatory

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.

1. Introduction
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
11. Singular moduli and complex multiplication
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 ϑ-functions and meromorphic forms
Appendix E Solutions to exercises.

Subject Areas: Statistical physics [PHS]