Now in Boston, Weinberg describes how his earlier work on current algebra led to effective field theory. With the Vietnam War going on, JASON work focuses on the war effort. In 1967, Weinberg takes up a lectureship at MIT and published his most-cited paper, “A Model of Leptons,” which heralded electroweak theory. He attends the Solvay conference in Brussels in 1967, but misses being in the group photo. Back in Boston, Weinberg discusses making friends through his election to the American Academy of Arts and Sciences. He becomes involved in an independent study of the US’ anti-ballistic missile program, concluding that this would hasten the arms race between the US and the Soviet Union.

The goal of this book is to exhibit the profound and myriad interrelations between the mathematics of modular forms and the physics of string theory. Our presentation is intended to be informal but mathematically precise, logically complete, and reasonably self-contained. The exposition is kept as simple as possible so as to be accessible to adventurous undergraduates, motivated graduate students, and dedicated professionals interested in the interface between theoretical physics and pure mathematics. Assuming little more than a knowledge of complex function theory, we introduce elliptic functions and elliptic curves as a lead-in to modular forms and their various deep generalizations. Following an economical introduction to string theory, its perturbative expansion, toroidal compactification, and supergravity limit are used to illustrate the power of modular invariance in physics. Dualities and their realization via modular forms in Yang–Mills theories with extended supersymmetry are studied both via the Seiberg–Witten solution and via their superconformal phase. Appendices are included to review foundational topics, and 75 exercises with detailed solutions give the reader ample opportunity for practice.

Back in Berkeley, Weinberg reconsiders how we understand some of the theories of physics, that is, why they actually are true. He begins teaching a general relativity course starting from physical principles, rather than the usual geometric approach. These course notes later became the basis for his book Gravitation and Cosmology. Around this time, Louise was pregnant, so Weinberg avoided opportunities to travel to spend more time at home. He begins working on functional analysis, but discovers the Russian Faddeev has already done foundational work in this area. Weinberg then reexamines what he knows about quantum field theory, and jettisons the Heisenberg–Pauli canonical formalism, taking particles as his starting point. This led him to a clearer understaning of antimatter. He embarks on a series of papers about massless particles. in 1964, he is promoted to full professor. Louise applied to Harvard Law School, prompting a move to Cambridge, Mass.

In the 1980s, two groups of physicists in Europe and America began to lay plans for a high energy proton accelerator that could settle the question of electroweak symmetry breaking. In 1984, Weinberg is appointed to the SSC Board of Overseers, and this work would occupy his time for much of the next decade. Weinberg testifies before Congress in favor of the SSC project and starts to appreciate the role of pork-barrel politics in the siting decisions. The Texas site of Waxahachie, near Dallas, is approved in 1988. The project’s construction funding is approved but faces ongoing challenges from other competing areas of science. Changes in the specifications, required by the science goals, lead to increases in the costs, resulting in bad press. In 1993, the funding was cut, and the SSC was killed off. Weinberg writes a successful trade book, Dreams of a Final Theory.

In Chapter 2, the dependence of elliptic functions on the points in the torus was studied for a fixed lattice. In this chapter, it is the dependence on the lattice that will be investigated. The modular group SL(2,Z) is introduced as the group of automorphisms of the lattice, and its generators, elliptic points, and cusps are identified. The hyperbolic geometry of the Poincaré upper half plane is reviewed, and the fundamental domain for SL(2,Z) is constructed. Modular forms and cusp forms are defined and shown to form a polynomial ring. They are related to holomorphic Eisenstein series, the discriminant function, the Dedekind eta-function, and the j-function and are expressed in terms of Jacobi theta-functions. The Fourier and Poincaré series representations of Eisenstein series are analyzed as well.

Shortly after moving to Berkeley, Weinberg slips a disc and is bedbound. He reads Chandrasekhar’s stellar physics book, which helped spark his interest in astrophysics. They decide to stay on the San Francisco side of the bay. At that time, Berkeley was the world’s leading center of experimental research on elementary particles and the newly commissioned Bevatron was the latest particle accelerator. Weinberg resolves to do some work that will be useful to Berkeley experimenters and sets about studying muon physics. In Spring 1960, he is offered and accepts a tenure-track position as an assistant professor. He is invited to join JASON, the group of defense consultants. He begins teaching and learns that he loves it. He decides to take a year abroad via an Alfred Sloan Fellowship and he and Louise buy a round-the-world ticket.

In the summer of 1991, Weinberg receives the National Medal of Science from President George H. W. Bush. He describes various visits and internationl trips. Through the early 1990s at the University of Texas at Austin, he taught a course on the quantum theory of fields, which was published as a two-volume treatise on “The Quantum Theory of Fields.” (A third volume on supersymmetry would follow in 2000.) Around this time, he begins publishing popular science in The New York Review of Books. He gives the dedicatory address at the opening of the university’s Hobby–Eberly Telescope.

Disheartened by the cancelation of the Superconducting Super Collider, Weinberg turns his attention to the cosmological constant. It must behave like a vacuum energy density, and can be adjusted to cancel the energy in fluctuating fields. Today the effective vacuum energy density, including the cosmological constant, has come to be called “dark energy.”

In this chapter, we discuss the modular properties of quantum field theories of scalar fields that take values in a d-dimensional torus with a flat metric and a constant anti-symmetric tensor. The problem is of great interest in quantum field theory and string theory in view of the fact that such toroidal compactifications admit solutions using free-field theory methods on the worldsheet, preserve Poincaré supersymmetries and may be used to relate different perturbative string theories via T-duality. Toroidal compactifications produce large duality groups, which we shall derive and which generalize the full modular group SL(2,Z). The quantum field theories of toroidal compactification on a worldsheet torus for a singular modulus is shown to be a rational conformal field theory.

Weinberg takes a summer job in the Atomic Beam Group at Princeton, calculating the trajectories of beam particles through the experimental equipment. He describes the culture of close relations of graduate students in physics with the younger faculty and its emphasis on research rather than course work. He details the various courses he took, along with the personalities of the Princeton professors at the time. Sam Treiman agrees to be his PhD advisor for a thesis on strong interactions in decay processes.

Weinberg returns to the “The Cosmological Constant Problem” and suggests an anthropic principle solution. Anthropic reasoning could make it possible for us to calculate the effective vacuum energy. Observations of dark energy in 1998 show that the expansion of the universe is accelerating. This observational result is not inconsistent with the notion of a possible multiverse – the issue has not been settled.

Weinberg collaborates with Ed Witten. He becomes the youngest member of the Saturday Club of Boston. Weinberg signs up to write The Discovery of Subatomic Particles. After their continued separation due to teaching, Weinberg grows to like Austin more and more, with its social scene that crossed from academia into the public sphere. He negotiates with the Universioty of Texas for a position in Austin as the Josey Regental Chair in Science beginning in 1982. He joins the Headliners Club in Austin. Weinberg helps found the Jerusalem Winter School in Theoretical Physics. He begins exploring physical theories in higher dimensions. He attends the Shelter Island Conference in 1983. He is elected to the Philosophical Society of Texas and joined the Town and Gown Club in Austin, but quits the latter over its male-only stance, to help form a rival, the Tuesday Club (of Austin). In mid-1980s, he becomes seriously interested in string theory.

Congruence subgroups form a countable infinite class of discrete non-Abelian subgroups of SL(2,Z) and play a particularly prominent role in deriving the arithmetic properties of modular forms. In this chapter, we study various aspects of congruence subgroups, including their elliptic points, cusps, and topological properties of the associated modular curve. Jacobi theta-functions, theta-constants, and the Dedekind eta-function are used as examples of modular forms under congruence subgroups that are not modular forms under the full modular group SL(2,Z).