Introduction

In the course of the twentieth century fundamental notions concerning the physical world were radically altered by a succession of unifying insights. In the early part of the century quantum theory provided the framework for understanding the structure of the atom, its nucleus, and the “elementary” particles of which it is made. This understanding, combined with the special theory of relativity, led, by the early 1970s, to a comprehensive account of three of the fundamental forces – the weak, strong, and electromagnetic forces – in the “Standard Model.” Meanwhile, Einstein’s 1915 general theory of relativity replaced Newton’s theory of gravity, leading to a unification of the force of gravity with spacetime geometry, and explanations of the evolution of the Universe from the time of the Big Bang and of weird astrophysical phenomena, such as black holes. Yet, with each advance ever-deeper questions were posed and remained unanswered. What determines the species of observed elementary particles, such as the electron, neutrinos, quarks, etc., and what principle leads to a unified description of all of them? How can quantum theory be consistent with general relativity? What is the meaning of space and time at extremely small distances, where quantum theory plays a dominant role? How did the Universe begin and how will it end?

An optimist should regard such unresolved puzzles as the seeds from which further insights will arise. This spirit of incompleteness is a distinguishing feature of this chapter. Here we will see that many, if not all, of these deep questions might be resolved by string theory. Although there is no single mathematical equation that summarizes string theory – indeed, it is not yet a complete theory – it has many intriguing interrelations with some of the most modern areas of mathematics. Its compelling qualities suggest that string theory has the potential to overcome key problems present in previous physical theories and to overcome them in a manner that is surprisingly simple and at the same time very novel.