The starting point for string theory is the idea that the elementary constituents of the theory, which in quantum field theory are assumed to be point-like, are in fact one-dimensional objects, namely strings. As time evolves, a string sweeps out a Riemann surface whose topology governs the interactions that result from joining and splitting strings. The Feynman–Polyakov prescription for quantum mechanical string amplitudes amounts to summing over all topologies of the Riemann surface, for each topology integrating over the moduli of the Riemann surface, and for each value of the moduli solving a conformal field theory. Modular invariance plays a key role in the reduction of the integral over moduli to an integral over a single copy of moduli space and, in particular, is responsible for rendering string amplitudes well behaved at short distances. In this chapter, we present a highly condensed introduction to key ingredients of string theory and string amplitudes, relegating the important aspects of toroidal compactification and T-duality to Chapter 13 and a discussion of S-duality in Type IIB string theory to Chapter 14.

The trip first takes in Japan, then Hong Kong, and Singapore. In Singapore, he loses a case containing all of his notes, and learns to lecture without them. He visits some Indian institutes and then flies to Israel. After tourist stopovers in Istanbul and Athens, they arrive finally in London as guests of Abdus Salam. He begins work on a research project on bound states in strong forces, which leads nowhere. He gives invited talks throughout the UK. They spend some time in Italy before attending the Rochester Conference of 1962, in Geneva. Weinberg receives news that he has been promoted to associate professor. They return to New York via Lisbon, after 280 days away.

A Riemann surface is a connected complex manifold of two real dimensions or equivalently a connected complex manifold of one complex dimension, also referred to as a complex curve. In this appendix, we shall review the topology of Riemann surfaces, their homotopy groups, homology groups, uniformization, construction in terms of Fuchsian groups, as well as their emergence from two-dimensional orientable Riemannian manifolds. All these ingredients provide crucial mathematical background for two-dimensional conformal field theory on higher genus Riemann surfaces and its application to string theory.

In this chapter, we shall draw together a number of different strands of inquiry addressed in Chapters 5, 12, and 13. We shall study the interplay between superstring amplitudes, their low-energy effective interactions, Type IIB supergravity, and the S-duality symmetry of Type IIB superstring theory. We begin with a brief review of Type IIB supergravity which, in particular, provides the massless sector of Type IIB superstring theory. We then discuss how the SL(2,R) symmetry of Type IIB supergravity is reduced to the SL(2,Z) symmetry of Type IIB superstring theory via an anomaly mechanism. We conclude with a discussion of how the low-energy effective interactions induced by string theory on supergravity may be organized in terms of modular functions and modular graph forms under this SL(2,Z) symmetry, and match the predictions provided by perturbative calculations of Chapter 12.

In this chapter, we shall discuss modular forms for the congruence subgroups introduced in Chapter 6. We shall obtain the dimension formulas for the corresponding rings of modular forms and cusp forms, describe the fields of modular functions on the modular curves introduced in Chapter 6, and construct the associated Eisenstein series. Throughout the chapter, we shall make use of the correspondence between modular forms and differential forms, viewed as sections of holomorphic line bundles on the compact Riemann surface of the modular curve. We shall provide concrete examples of modular forms for the standard congruence subgroups and apply the results to the theorems of Lagrange and Jacobi on counting the number of representations of an integer as a sum of squares.

In this appendix, we collect some basic results in number theory, including the Chinese remainder theorem, its application to solving polynomial equations, the Legendre and Jacobi quadratic residue symbols, quadratic reciprocity, its application to solving quadratic equations modulo N, and a brief introduction to Dirichlet characters and Dirichlet L-functions.

We close with a brief introduction to Galois theory and illustrate the application of these mathematical ideas in physics through examples from conformal field theory.

In this chapter, we construct differential equations in the modular parameter and find solutions to these equations in simple cases. The solutions can generically be assembled into vector-valued modular forms, which have proven fruitful in recent works in mathematics and physics. We will establish that, in general, each component of a vector-valued modular form is a modular form for a congruence subgroup.

Closely related variants of modular forms, including quasi-modular forms, almost-holomorphic modular forms, Maass forms, non-holomorphic Eisenstein series, mock modular forms, and quantum modular forms, are introduced and their properties are analyzed.

The author describes his parents’ upbringing and move to New York around the time of the Great Depression. The young Weinberg is encouraged to read widely and later takes inspiration from Norse myths from the Poetic Edda.

In this penultimate chapter, we shall discuss dualities in Yang–Mills theories with extended supersymmetry in four-dimensional Minkowski space-time. We briefly review supersymmetry multiplets of states and fields and the construction of supersymmetric Lagrangian theories with N = 1, 2, and 4 Poincaré supersymmetries. We then discuss the SL(2,Z) Montonen–Olive duality properties of the maximally supersymmetric N = 4 theory and the low-energy effective Lagrangians for N = 2 theories via the Seiberg–Witten solution. We shall close this chapter with a discussion of dualities of N = 2 superconformal gauge theories, which possess interesting spaces of marginal gauge couplings. In some cases, these spaces of couplings can be identified with the moduli spaces for Riemann surfaces of various genera.

In this final appendix, we shall review the modular geometry of the Siegel half-space at higher rank, Riemann theta-functions of higher rank, the embedding of higher-genus Riemann surfaces into the Jacobian variety via the Abel map, and use these ingredients to construct the prime form, the Szego kernel, and other meromorphic functions and differential forms on higher-genus Riemann surfaces.

Weinberg takes the standard first-year courses for physics students, studying mechanics, heat, light, and electromagnetism. While these are not the fancy modern topics, they are the essential foundations for everything else he would learn in physics. He joins Telluride House, a fraternity. In his sophomore year, he lets his studies slide (except for physics and math). He pulls out of this slump with the determination not to waste any more time and forms the habit of being a compulsive worker. He lands a summer job at Bell Telephone Laboratories. In his senior year, he learns quantum mechanics. He decides to apply to graduate school to study for a PhD in physics, and to marry Louise Goldwasser.

Having finished her law degree, Louise takes up work at a Boston law firm; they are not returning to Berkeley after all, so Weinberg resigns his professorship there. At MIT, he continues teaching graduate courses on general relativity, with an emphasis on cosmology. He spends the spring of 1971 in Paris, making comparisons between the academic characters of Paris and Boston. Gerard ’t Hooft and Martinus Veltman renormalize Weinberg’s theory of leptons, showing an experimental route to proving the theory. Weinberg starts to consider the extension of the electroweak theory to strongly interacting theories. Electroweak theory starts to receive a lot more attention from theorists. His first book, Gravitation and Cosmology, is published in 1972. Weinberg is offered the Higgins Professorship at Harvard, and accepts.

Familiarity with chemistry from children’s toy kits leads Weinberg to investigate physics, the subject that underlies all of chemistry. He reads George Gamow’s Mr. Tompkins books, among others. He is admitted to the famous Bronx High School of Science, where he becomes friends with Shelly Glashow and Gary Feinberg, who would also become well-known physicists. He wins a New York state scholarship to Cornell.

Some immediate applications of the theory of elliptic functions and modular forms to problems of physical interest are presented, including the construction of the Green functions and functional determinants for the two-dimensional quantum field theories of the bc fields, the scalar field, and the spinor fields on the torus. In particular, it will be shown how the singular terms in the operator product expansion of holomorphic fields for the bc system essentially determine arbitrary correlation functions on the torus in terms of elliptic functions. Along the way, a brief but reasonably systematic introduction will be presented of two-dimensional conformal field theory methods.