Modular Forms and String Theory

Eric D'Hoker obtained his Ph.D. in Physics from Princeton, and is currently Distinguished Professor of Physics at UCLA and Fellow of the American Physical Society. He was previously a Simons Fellow, a Dyson Distinguished Visiting Professor at Princeton's Institute for Advanced Study, and has served as President of the Aspen Center for Physics. Justin Kaidi obtained his Ph.D. in Physics from UCLA. After two years as a Research Assistant Professor at the Simons Center for Geometry and Physics at Stony Brook University, he joined the University of Washington as an Assistant Professor. He is currently an Associate Professor of Physics at Kyushu University in Japan.

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.