In the twenty-five years since the original publication of these two Volumes, there have been numerous developments in string theory. The curious twists and turns that marked its pre-1987 evolution have continued apace, and current research makes contact with a wide range of areas of mathematics and physics. In the following we will mention briefly some of these developments and then explain why we believe that these volumes are still useful.

Major insights into the non-perturbative structure of string theory followed from the discovery of non-perturbative duality symmetries of super-string theory. This led to the realization that the myriad of apparently distinct superstring theories that arise in ten or fewer dimensions actually are different perturbative approximations to the same underlying theory, which has come to be known as M-theory. Furthermore, M-theory has eleven-dimensional supergravity as another semiclassical limit. The understanding of these interconnections was aided by the simultaneous discovery of the properties of a family of dynamical objects called p-branes, which are extended objects that fill p spatial dimensions, as opposed to the 1 dimension of the string. p-branes can be viewed as solitons that are generalizations of the magnetic monopoles of conventional quantum field theory and the black holes of general relativity. Indeed, these discoveries have stimulated impressive advances in understanding the quantum and thermodynamic properties of large classes of black holes.

An important outcome of these considerations has been striking progress in understanding the nonperturbative structure of the quantum field theories that arise from string theory in various limits.

The particle spectrum of a string theory consists of a finite number of massless states and an infinite tower of massive excitations at a mass scale characterized by a fundamental parameter – the string tension or Regge slope. As has been explained in previous chapters, this parameter must be of order the Planck mass (1019 GeV) in order that the graviton interact with the usual Newtonian strength. If one wishes to give a phenomenological description of the consequences of string theory for lowenergy physics, it should not be necessary to describe explicitly what the massive states are doing. It is natural, instead, to formulate an effective action based entirely on fields that correspond to massless, or at least very light, degrees of freedom only. Such a description turns out to be useful not only for a phenomenological analysis, but even as a framework for addressing certain theoretical issues, such as the occurrence of anomalies.

The infinite set of point-particle fields that arise in string theory consists of a finite number of massless fields, which we collectively represent for the moment by ϕ0, and an infinite number of heavy fields collectively represented by ϕH In principle, it must be possible to describe string theory by a classical action S(ϕ0 ϕH (or, at the quantum level, a quantum effective action) governing these fields. At present, we do not have really satisfactory ways to formulate and understand the exact classical action S(ϕ0, ϕH).

In the first eleven chapters of this book we have attempted to introduce the reader to string theory as it is presently understood. Our focus now shifts to making contact with more familiar physics. In this chapter we develop some concepts in differential geometry that are useful in understanding general relativity and Yang-Mills theory even in four dimensions, but which are of particular utility in ten-dimensional physics. Our treatment in this chapter is comparatively elementary and aims mostly to develop the minimum material we require in chapters 13 and 14. In chapter 13, we will discuss supergravity theory in ten dimensions, which at least in perturbation theory is the low-energy limit of ten-dimensional superstring theory. In chapter 14, we will discuss some of the important ideas that arise in compactification from ten to four dimensions. The concluding chapters of this book, chapters 15 and 16, are devoted to more specialized mathematical background and more speculative ideas about compactification.

Spinors In General Relativity

A good place to start our discussion is to think about the coupling of spinors to a gravitational field. This problem is of great importance in string theories in which both fermions and gravity are present, and this alone would justify its consideration here. In addition, thinking about the coupling of spinors to a gravitational field forces us to examine issues whose analogs for Yang-Mills theory we will wish to consider later. The question of coupling spinors to general relativity was considered briefly in chapter 4, in connection with a discussion of two-dimensional supergravity, but here we will be more extensive.

Since superstring theories are necessarily ten-dimensional theories, any discussion of phenomenology must begin with a discussion of how apparent four-dimensional physics is related to underlying ten-dimensional physics. The present chapter is devoted to this question. We will carry out the discussion in the context of field theory, but with an emphasis on properties that depend only on qualitative assumptions, not numerical details, and so can remain valid in string theory. What we will try to accomplish in this chapter is not to develop detailed models of compactification but to set the stage and introduce some of the essential concepts.

Wave Operators in Ten Dimensions

Most of the preceding chapters have been devoted to string propagation in ten-dimensional flat Minkowski space M10, but henceforth we will consider ten-dimensional space-time to be some more general ten manifold M. We take M to be of the form M4 × K, where M4 is four-dimensional Minkowski space and K is a compact six manifold which is, unfortunately, as yet unknown. More precisely, we take the vacuum state to be a product M4 × K; it must have this form if we wish to maintain four-dimensional Poincaré invariance. Of course, physical fluctuations will not necessarily respect the product form of the vacuum configuration, but as in so many other areas of physics, understanding the ground state is the key to understanding the low-energy excitations.

Our discussions of string scattering amplitudes in the first volume of this book were limited to tree diagrams. These are the lowest-order approximations to string scattering amplitudes. In principle, quantum corrections to the tree level or classical results should be obtained by a perturbation expansion derived from string quantum field theory. Our present state of knowledge does not make this possible. Historically, loop diagrams were constructed by using unitarity to construct loop diagrams from tree diagrams. This unitarization of the tree diagrams led, in time, to the topological expansion, as sketched in chapter 1.

As has been explained in chapters 1 and 7, the tree amplitudes for onmass-shell string states can be represented by functional integrals over Riemann surfaces that are topologically equivalent to a disk (for open strings) or a sphere (for closed strings). Higher-order corrections are identified with functional integrals over surfaces of higher genus. An important ingredient in the calculation of scattering amplitudes is the correlation function of vertex operators corresponding to the external particles emitted from the surface. The possible world-sheet topologies include surfaces with holes or “windows” cut out (for type I theories, where the surfaces have boundaries) or “handles” attached. For theories with oriented strings the surfaces must be orientable. Similarly, for theories containing only closed strings the surfaces must be closed.

The Early Days of Dual Models

In 1900, in the course of trying to fit to experimental data, Planck wrote down his celebrated formula for black body radiation. It does not usually happen in physics that an experimental curve is directly related to the fundamentals of a theory; normally they are related by a more or less intricate chain of calculations. But black body radiation was a lucky exception to this rule. In fitting to experimental curves, Planck wrote down a formula that directly led, as we all know, to the concept of the quantum.

In the 1960s, one of the mysteries in strong interaction physics was the enormous proliferation of strongly interacting particles or hadrons. Hadronic resonances seemed to exist with rather high spin, the mass squared of the lightest particle of spin J being roughly m2 = J/α′, where α′ ˜ l(GeV)−2 is a constant that became known as the Regge slope. Such behavior was tested up to about J = 11/2, and it seemed conceivable that it might continue indefinitely. One reason that the proliferation of strongly interacting particles was surprising was that the behavior of the weak and electromagnetic interactions was quite different; there are, comparatively speaking, just a few low mass particles known that do not have strong interactions.

The resonances were so numerous that it was not plausible that they were all fundamental.

In chapters 12 and 14 we developed some simple tools in differential geometry and used them to gain some insight concerning the compactification of hidden dimensions as well as some insight concerning phenomena on the string world sheet. We now turn our attention to some more specialized mathematical tools involving complex manifolds and algebraic geometry. Again, the motivation is twofold. The world sheet of a string is a complex manifold – a Riemann surface, in fact – and as string theory develops, the deeper study of world-sheet phenomena is likely to involve deeper aspects of algebraic geometry, which have already begun to enter in recent works on multiloop diagrams. Also, algebraic geometry has been a tool in recent attempts to formulate more realistic models of string compactification.

In this chapter, we develop some of the basic concepts of complex geometry, with examples selected for their role both in world-sheet phenomena and in the study of compactification. We will unfortunately not be able to describe in this book recent work on the application of algebraic geometry to multi-loop diagrams. This subject is probably not yet ripe for synthesis; and the requisite mathematical machinery is more extensive than we will be able to present even in this moderately lengthy chapter. By laying at least some of the elementary foundations we hope to facilitate the task of the reader who wishes to delve further elsewhere.

In this chapter we discuss one-loop amplitudes in superstring theory. The operator methods used for the calculations of one-loop amplitudes with on-shell external states in chapter 8 will be applied to the calculation of the corresponding diagrams in superstring theories. The resulting amplitudes are expressed as integrals over world sheets that are closed and orientable in the case of the type II or heterotic theories. Type I theories are based on unoriented open and closed strings, and as a result their world sheets need not be orientable and can have boundaries. As in the case of the bosonic theory, the operator approach automatically gives the correct measure factors by elementary algebraic manipulations. In approaches that explicitly compute the sum over geometries, considerable care is required to correctly define and evaluate infinite determinants that give the measure. Nonetheless, such approaches have a number of advantages and certainly appear preferable for the study of multiloop amplitudes.

In the case of one-loop amplitudes in superstring theories, the light-cone gauge calculations based on the space-time supersymmetric formalism of chapter 5 and covariant calculations based on the formalism with worldsheet supersymmetry of chapter 4 are quite different, just as in the case of the calculation of the tree diagrams. For many purposes the light-cone method with manifest space-time supersymmetry is more economical, because there are fewer diagrams to calculate for a given process. For example, in theories with oriented closed strings (type II and heterotic) there is only one one-loop diagram, which corresponds to a world sheet that is topologically a torus.

In string theory, just as in other theories, it is necessary to understand the free theory well before trying to describe interactions. Our first task in a systematic exposition of string theory is to understand thoroughly the propagation of a single free string in space-time at both the classical and quantum levels. We begin in this chapter with a study of bosonic strings. In the course of this discussion, we will approach the bosonic string from many different points of view, corresponding to many different formalisms that have been developed over the years. These include various approaches to covariant and to light-cone quantization. Each adds important ingredients to an overall understanding of string theory, so it really is useful to become familiar with all of them.

The Classical Bosonic String

It may be helpful, as in the introduction, to begin with a discussion of point particles. Thus, let us consider the motion of a point particle of mass m in a background gravitational field, i.e., in a curved Riemannian geometry described by a metric tensor gμν(x). The metric is assumed to have D − 1 positive eigenvalues and one negative eigenvalue corresponding to the Minkowski signature of D-dimensional space-time. We always use units in which ℏ = c = 1.

The action principle that describes the motion of a massive point particle is well-known, and already entered in the introduction.