Having constructed our first examples of tachyon-free closed string models we now turn to the description of branes and open strings in type II superstring theory. We shall begin with a brief discussion of solitonic branes in type II supergravity. This part is rather descriptive. In particular, we do not intend to explain how these solutions are found. We shall then turn to the string theoretic analogue of solitonic branes. As we explained in the introductory chapter branes in string theory are objects on which open strings can end.We shall use this as a definition of D-branes in string theory and verify that the resulting objects share many features with the solitonic branes in supergravity. Finally, gauge theories on the world-volume of D-branes are discussed briefly.

Our quantization of open strings in this section will proceed through the lightcone gauge rather than the covariant formalism. This will give us a good opportunity to review the basic ideas of light-cone gauge quantization. But there is another reason for us to do so. In the last chapters we have spent some time to verify that the massless states of type II string theories form full multiplets of the Poincaré superalgebra. Extending this analysis to massive sectors would be a rather difficult task. Here we want to verify that all eigenspaces of the mass operator for open strings in the type II theory possess the same number of bosonic and fermionic components. To reach that goal, the light-cone gauge has clear advantages since it allows for a direct and simple construction of physical states.

Solitonic Branes in Type II Supergravity

At the end of the previous section we have seen N=2 supersymmetric supergravities in 10 dimensions. As we argued there, these theories may be considered as higher dimensional generalizations of EINSTEIN-MAXWELL theory. In fact, the action of type IIA/B supergravities contains a number of higher form potentials Cp+1. From the study of 4-dimensional gravity it is well known that there exist special solutions that describe charged black holes.

In the last chapter we have sketched some results from mathematics. We looked at complex manifolds with Hermitian metric and then imposed the KÄHLER condition (16.16) without any further motivation. After combining the KÄHLER condition with RICCI flatness we ended up with the class of CALABI-YAU manifolds. The results of mathematics ensure that the HODGE diamond of a CALABI-YAU manifold has a very special form. It contains only two non-trivial entries, h1,1 = h2,2 and h1,2 = h2,1. All other entries are either zero or one; see Figure 16.24. Let us finally also recall that the numbers hp,q count the number of harmonic (p, q) forms on the manifold.

We made very little attempt to connect the KÄHLER condition and its various consequences to our goal, namely to construct 4-dimensional string compactifications with N=2 supersymmetry in space-time. The aim of the present chapter is to fill this gap. In the first half we shall show that the massless spectrum of type II compactifications on CALABI-YAU manifolds is organized in multiplets of the 4-dimensional N=2 POINCARÉ superalgebra. The second part is devoted to a deeper understanding of the KÄHLER condition. Our strategy is to connect this condition to the GSO projection in curved backgrounds.

CALABI-YAU Spaces and 4D Supersymmetry

We now want to investigate how the 4-dimensional low-energy effective field theories depend on the CALABI-YAU space we compactify our type IIA/B string theory on. As we mentioned before, the resulting gravity theories come with an N=2 supersymmetry in four dimensions, i.e. they possess eight supercharges. Our main goal is to determine the type and number of massless N=2 supermultiplets that appear in a given compactification.

This needs a bit of preparation. To begin, let us recall the bosonic content of the relevant supermultiplets. As in all massless supermultiplets, half of the supercharges decouple. Hence, we are left with four supercharges that we combine into two creation and two annihilation operators. The two fermionic creation operators carry helicity. With this in mind we can easily determine the bosonic content of N=2 supermultiplets. We shall need

At the end of the last chapter we have seen that strongly coupled theories may have unexpected descriptions. These can look very different from the weakly coupled formulation of the model. In the case of type IIA/M-theory duality, for example, it was suggested that we describe strongly coupled type IIA superstring theory in terms of an 11-dimensional theory. The only candidates for a fundamental description of the latter were the M2- and M5-branes of 11-dimensional supergravity. Admittedly, the duality between the two theories remained rather speculative, mostly because it is not possible to give a proper construction of M-theory. In this section we shall see another example of a duality that involves two theories that are reasonably well defined, yet very different in nature. It relates (p + 1)- dimensional gauge theories with 10-dimensional type II string compactifications. As in the example of type IIA/M-theory, both theories live in a space-time of different dimension, and their fundamental degrees of freedom are quite distinct. Special attention will be paid to the simplest example of such a duality between 4-dimensional N=4 Super YANG-MILLS (SYM) theory and closed strings on AdS5 × S5. This will enable us to outline the formulation and the use of such dualities.

't Hooft's Analysis of the LargeNcLimit

The idea of a duality between gauge and string theories is not new. Even though the first concrete examples of gauge/string dualities were formulated only at the end of the 1990s, it had long been suspected that large Nc gauge theories may be re-written in terms of closed strings. We shall describe some of these arguments in this section.

We shall begin by reviewing an argument of G.'t Hooft [66] that shows that computations in gauge theory can be rearranged such that they look like an expansion in some model of closed strings. Suppose we are given a pure YANG-MILLS theory with gauge group U(Nc). Schematically, the action of this theory takes the form

Here we have omitted terms of fourth order since they are not relevant for what we are about to discuss. From the action we may read off the Feynman rules.

About half a century has passed since string theory was born, initially to explain the observed Regge trajectories of hadronic resonances. During these past decades, it has gone through several phases of dismissal and new appreciation, it has become a promising candidate for a quantum theory of gravity, and occasionally it has been popularized as a theory of everything. What most of these periods have in common are profound and fruitful interactions with rather different areas of theoretical physics, e.g., with quantum field theory, high-energy physics, gravity, mathematics, and even statistical and condensed matter physics. Indeed, there is a long list of string-inspired developments, a few of which we will have a chance to touch upon below. Among them are, e.g., new unified extensions of the standard model that are being tested in collider experiments, insights into black holes and their entropy, contributions to knot invariants or the theory of modular forms, and our understanding in particular of 2-dimensional critical systems, or the impressive recent progress in accessing both perturbative as well as non-perturbative effects in supersymmetric gauge theories. This interdisciplinary nature of string theory is its real strength, and it can serve as a good motivation to enter or explore the field.

While this textbook addresses primarily masters’ or early PhD students, it may also be useful for scholars from neighboring fields who are looking for a short exposition of basic string theory. Compared to most other textbooks in the field, the scope here is fairly modest. Rather than attempting to cover string theory from its roots to all the fascinating modern developments, the intention is merely to provide a set of concepts and tools that are common to a wide range of recent research directions. By now many books have been written on string theory, which include various advanced and topical directions, such as [27, 28, 55, 56, 6, 39, 74, 41, 1] to list just a few that can be turned to for further reading. In addition, let me also mention a few shorter introductions such as [65, 68, 2]. Finally, a beautiful book by Barton Zwiebach [80] covers much less material but addresses undergraduates.

The last few chapters of this book deal with the vast subject of dualities in string and field theory. By the nature of its topic, our presentation will be significantly less self-contained and more descriptive. On the other hand, we hope to provide some introduction into a fascinating field of modern string theory that continues to expand.

All field and string theories depend on a certain number of parameters or couplings such as the string length ls, the string coupling gs, moduli of a compactification manifold, etc. In most cases, we cannot compute the dependence on the couplings exactly. Instead, perturbative techniques give us access to the neighborhood of some special points of the full parameter space, namely to those for which a perturbative description is known. It is difficult to determine the features of a model outside such weakly coupled regimes. On the other hand, it is certainly natural to wonder about the behavior of a theory at strong coupling. Many things can happen as we crank up one or more of the coupling constants. In particular, new fundamental degrees of freedom may show up, as it is the case, e.g., in QCD, where strongly coupled physics is best described in terms of hadrons, while quarks and gluons are the fundamental objects at weak coupling. Our main goal in this chapter is to explore type II string theories outside their perturbative regime. Our analysis will lead us to intriguing new relations, or dualities, between various perturbative expansions of string theory.

T-Duality and Mirror Symmetry

Before we start varying the string coupling gs we shall briefly revisit one duality symmetry that we have seen before. To this end, let us go back to the case in which one coordinate of our string background is compactified to a circle of radius R. Such a direction gives rise to the following contributions to the closed string mass operator:

The first terms are associated with closed strings that move with some momentum and winding number n and w along the circular direction.