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.

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.

We described actions and equations of motion for general supergravities, out of which we derive extremal and black p-brane solutions. We find electric p-brane solutions and magnetic p-brane solutions and show the duality between them, then we generalize to black p-branes and Dp-branes in 10 dimensions, and then we discuss fundamental string and NS5-brane solutions in 10 dimensions. Tseytlin’s harmonic function rule for writing intersecting brane solutions is explained.

This chapter deals with the role practice theory has played and can play in developing routine dynamics and the community of scholars associated with Routine Dynamics. It provides a short introduction to practice theory. It presents an analysis of how scholars in the field of Routine Dynamics relied on practice theory to build the foundation of the field and how scholars have continued to engage practice theory as the field has grown. The chapter ends with suggestions for how practice theory could help Routine Dynamics address questions of wide social relevance.

Japanese society embraces a rich variety of cultural forms that reflect its tradition, stratification, and regional expanse. To examine the internal diversity of Japanese culture as thoroughly as possible, this chapter first looks at two manifestations of its duality and then analyzes mass culture, folk culture, and alternative culture as three major spheres of popular culture. After confirming its plurality, we turn at the end to the Japanese cultural presence in the transnational context.