In previous chapters we became increasingly aware of the intimate relation of D-branes to both spacetime geometry and to gauge theory, via the collective description of their low energy dynamics. In fact, we have already seen that we can reinterpret many aspects of the spacetime geometry in which the brane moves by reference to the vevs of scalars in the world-volume gauge theory. In this chapter we explore this in much more detail, by using D-branes to probe a number of string theory backgrounds, and find that they allow us to get a new handle on quite detailed properties of the geometry. In addition, we will find that D-branes can take on the properties of a variety of familiar objects, such as monopoles and instantons, depending upon the situation.

D-branes as probes of ALE spaces

One of the beautiful results which we uncovered soon after constructing the type II strings was that we can ‘blow-up’ the 16 fixed points of the T4/ℤ2 ‘orbifold compactification’ to recover string propagation on the smooth hyper-Kähler manifold K3. (We had a lot of fun with this in section 7.6.) Strictly speaking, we only recovered the algebraic data of the K3 manifold this way, and it seemed plausible that the full metric geometry of the space is recovered, but how can we see this directly?

We can recover the metric data by using a brane as a short distance ‘probe’ of the geometry.

As we saw in chapter 12, there is an extremely tantalising picture of the fate of string theory at strong coupling, obtained using certain ‘duality’ transformations. In fact, D-branes were rather useful, as they allowed for an explicit constructive method for finding evidence of the products of duality, for example exhibiting stable states which must exist – with special properties – on both sides of the duality.

One major task is to try to understand how to write better formulations of the physics of strong coupling. There are two main goals to be achieved by this. The first is simply to find better ways of finding new and interesting backgrounds (vacua) for string theory, with techniques which allow for better handing of strongly coupled regions of the solution. The second is to attempt to find the ‘correct’ manner in which to describe the complete M-theory from which all string theories are supposed to arise as weakly coupled limits.

Both ‘Matrix theory’ and ‘F-theory’ are ideas in these directions, putting together the strongly coupled brane and string data in ways which allow for new geometric ways of describing and connecting string vacua, and giving insights into the next generation of formulations of the physics. In this chapter we shall uncover aspects of both, while learning much more about the properties of various branes.

The type IIB string and F-theory

One of the remarkable dualities which we observed in chapter 12 was the ‘self-duality’ of the type IIB superstring theory.

It is hoped that we have learned rather a lot about string theory in this book, and that the role of D-branes and other extended objects has been fascinating, entertaining, and instructive. It was with great pain that we had to sacrifice a tremendous amount of material in order to keep this book close to a sensible length, while retaining enough to succeed in telling a coherent story.

It is tempting to sit and reflect upon what great lessons we have learned, although it is not clear that this is a useful exercise at this stage, so we will be brief in our remarks. The main and most unambiguous lesson is that extended objects are vitally important to our understanding of string theory, and possibly (probably) whatever the final form of the fundamental quantum theory of space and time turns out to be. While extended objects are universally accepted as important, it is still (at a stretch) a matter of taste whether someone wants to go further and accept that it is unambiguously true that ‘string theory is not a theory of strings’. The author believes it to be so, but will not insist that the reader take a position, since it seems that nobody can yet say what string or M-theory actually are theories of.

Whatever the final theory turns out to be, and whether or not once it is found it turns out be directly relevant to nature at all, it is clear that we have many new tools to work with which should keep us busy for some time to come in various areas.

Einstein's theory of the classical relativistic dynamics of gravity is remarkable, both in its simple elegance and in its profound statement about the nature of spacetime. Before we rush into the diverse matters which concern and motivate the search which leads to string theory and beyond, such as the nature of the quantum theory, the unification with other forces, etc., let us remind ourselves of some of the salient features of the classical theory. This will usefully foreshadow many of the concepts which we will encounter later.

The classical dynamics of geometry

Spacetime is of course a landscape of ‘events’, the points which make it up, and as such it is a classical (but of course relativistic) concept. Intuition from quantum mechanics points to a modification of this picture, and there are many concrete mechanisms in string theory which support this expectation and show that spacetime is at best a derived object or effective description. We shall see some of these mechanisms in the sequel. However, since string theory (as currently understood), seems to be devoid of a complete definition that does not require us to refer to spacetime, the language and concepts we will employ will have much in common with those used by professional practitioners of General Relativity, and of classical and quantum Field Theory. In fact, it will become clear to the newcomer that success in the physics of string theory is greatly aided by having technical facility in both of those fields.

We have already stated that since the D-brane is a dynamical object, and couples to gravity, it should have a mass per unit volume. This tension will govern the strength of its response to outside influences which try to make it change its shape, absorb energy, etc. We have already computed a recursion relation (5.11) for the tension, which follows from the underlying T-duality which we used to discovere D-branes in the first place.

In this chapter we shall see in detail just how to compute the value of the tension for the D-brane, and also for the orientifold plane. While the numbers that we will get will not (at face value) be as useful as the analogous quantities for the supersymmetric case, the structure of the computation is extremely important. The computation puts together many of the things that we have learned so far in a very elegant manner which lies at the heart of much of what will follow in more advanced chapters.

Along the way, we will see that D-branes can be constructed and studied in an alternative formalism known as the ‘boundary state’ formalism, which is essentially conformal field theory with certain sorts of boundaries included. For much of what we will do, it will be a clearly equivalent way of formulating things which we also say (or have already said) based on the spacetime picture of D-branes.

As we have seen, branes of various sorts are solutions of string theory which are localised to some extent, and have well-defined mass and charge per unit volume. Since these masses and charges are measured at infinity, meaning that the branes are sources of fields from the massless sector, we might expect that they must be actually be solutions of the low energy equations of motion: the gravity sector and other fields such as the various antisymmetric tensor fields, and possibly the dilaton. These field configurations can be thought of as representing interesting backgrounds in which the string can propagate. It has become increasingly important in many recent research areas to consider the details of such solutions, and we shall begin exploring this highly developed technology in the present chapter.

A look at black holes in four dimensions

Before we launch into a description of the solutions associated to branes, it is a good idea to start with something more familiar in order to gain some intuition about how the solutions work. We will start in four dimensions with a familiar system: Einstein's gravity coupled to Maxwell's electromagnetism. The more advanced reader may wish to skip directly to section 10.2 if the following is too elementary, but beware, since we shall be uncovering and emphasising probably less familiar features in order to prepare for analogous properties of branes in higher dimensions.

As we learned in section 10.2, there are effectively two descriptions of the low energy dynamics of branes. One description uses the collective dynamics of the effetive world-volume field theory. In the case of N coincident D-branes, this is captured in string theory by the open string sectors which give a U(N) gauge theory with sixteen supercharges. The other description treats the brane as a soliton-like source of the various low energy closed string fields in the superstring theory. As such it has a description in terms of a classical solution of the low energy field equations. In both cases, we must remember that there is a whole tower of stringy dynamics which sits on top of this low energy physics, and we must understand in which situations this tower can be made irrelevant, or at least kept under control by a sensible expansion. To control string loops, we must work in a regime where gs is small, so that we can trust the classical action that we wrote down for the supergravity. Similarly, working in the α′ → 0 limit ensures that we can safely ignore the possibility of the massive string states introducing corrections to our supergravity, and in the open string sector, that the truncation to gauge theory of the full Born–Infeld, etc., action, is sensible.

The careful reader has patiently suspended disbelief for a while now, allowing us to race through a somewhat rough presentation of some of the highlights of the construction of consistent relativistic strings. This enabled us, by essentially stringing lots of oscillators together, to go quite far in developing our intuition for how things work, and for key aspects of the language.

Without promising to suddenly become rigourous, it seems a good idea to revisit some of the things we went over quickly, in order to unpack some more details of the operation of the theory. This will allow us to develop more tools and language for later use, and to see a bit further into the structure of the theory.

Conformal invariance

We saw in section 2.2.8 that the use of the symmetries of the action to fix a gauge left over an infinite dimensional group of transformations which we could still perform and remain in that gauge. These are conformal transformations, and the world-sheet theory is in fact conformally invariant. It is worth digressing a little and discussing conformal invariance in arbitrary dimensions first, before specialising to the case of two dimensions. We will find a surprising reason to come back to conformal invariance in higher dimensions much later, so there is a point to this.

In chapter 5, we saw a number of interesting terms arise in the Dp-brane world-volume action which had interpretations as smaller branes. For example, a U(1) flux was a D(p - 2)-brane fully delocalised in the world-volume, while for the non-Abelian case, we saw a D(p - 4)-brane arise as an instanton in the world-volume gauge theory. Interestingly, while the latter breaks half of the supersymmetry again, as it ought to, the former is still half BPS, since it is T-dual to a tilted D(p+1)-brane.

It is worthwhile trying to understand this better back in the basic description using boundary conditions and open string sectors, and this is the first goal of this chapter. After that, we'll have a closer look at the nature of the BPS bound and the superalgebra, and study various key illustrative examples.

Dpand Dp′from boundary conditions

Let us consider two D-branes, Dp and Dp′, each parallel to the coordinate axes. (We can of course have D-branes at angles, but we will not consider this here.) An open string can have both ends on the same D-brane or one on each. The p-p and p′-p′ spectra are the same as before, but the p-p′ strings are new if p ≠ p′. Since we are taking the D-branes to be parallel to the coordinate axes, there are four possible sets of boundary conditions for each spatial coordinate Xi of the open string, namely NN (Neumann at both ends), DD, ND, and DN.