Introduction

In this lecture we will try to describe electromagnetic and strong interactions of hadrons in the same framework which follows from general quantum field theory considerations without the introduction of quarks or other exotic objects.

We will assume that there exist point-like constituents in the sense of quantum field theory which are, however, strongly interacting. It is convenient to refer to these particles as partons.

One of the most striking results of the mid-1990s was the realisation that all of the superstring theories are in fact dual to one another at strong coupling. This also brought eleven dimensional supergravity into the picture and started the search for M-theory, the dynamical theory within which all of those theories would fit as various effective descriptions of perturbative limits. All of this is referred to as the ‘Second Superstring Revolution’. Every revolution is supposed to have a hero or heroes. We shall consider branes to be cast in that particular role, since they (and D-branes especially) supplied the truly damning evidence of the strong coupling fate of the various string theories.

We shall discuss aspects of this in the present section. We simply study the properties of D-branes in the various string theories, and then trust to that fact that as they are BPS states, many of these properties will survive at strong coupling.

Type IIB/type IIB duality

D1-brane collective coordinates

Let us first study the D1-brane. This will be appropriate to the study of type IIB and the type I string by Ω-projection. Its collective dynamics as a BPS soliton moving in flat ten dimensions is captured by the 1+1 dimensional world-volume theory, with 16 or 8 supercharges, depending upon the theory we are in. (See figure 12.1.(a).)

In view of the exciting developments in our understanding of those particular aspects of fundamental physics that string theory seems to capture, it seems appropriate to collect together some of the key tools and ideas which helped move things forward. The developments included a true revolution, since the physical perspective changed so radically that it undermined the long-standing status of strings as the basic fundamental objects, and instead the idea has arisen that a string theory description is simply a special (albeit rather novel and beautiful) corner of a larger theory called ‘M-theory’. This book is not an attempt at a history of the revolution, as we are (arguably) still in the midst of it, especially since we are in the awkward position of not knowing even one satisfactory intrinsic definition of M-theory, and have implicit knowledge of it only through interconnections of its various limits.

All revolutions are supposed to have a collection of characters who played a crucial role in it, ‘heroes’ if you will. Hence, one would be expected to proceed to list here the names of various individuals. While I was lucky to be in a position to observe a lot of the activity at first hand and collect many wonderful anecdotes about how some things came to be, I will decline to start listing names at this juncture.

In this chapter we shall study the spectrum of strings propagating in a spacetime that has a compact direction. The theory has all of the properties we might expect from the knowledge that at low energy we are placing gravity and field theory on a compact space. Indeed, as the compact direction becomes small, the parts of the spectrum resulting from momentum in that direction become heavy, and hence less important, but there is much more. The spectrum has additional sectors coming from the fact that closed strings can wind around the compact direction, contributing states whose mass is proportional to the radius. Thus, they become light as the circle shrinks. This will lead us to T-duality, relating a string propagating on a large circle to a string propagating on a small circle. This is just the first of the remarkable symmetries relating two string theories in different situations that we shall encounter here. It is a crucial consequence of the fact that strings are extended objects. Studying its consequences for open strings will lead us to D-branes, since T-duality will relate the Neumann boundary conditions we have already encountered to Dirichlet ones, corresponding to open strings ending on special hypersurfaces in spacetime.

Fields and strings on a circle

Let us remind ourselves of what happens in field theory, for the case of placing gravity on a spacetime with a compact direction.

In a number of the previous chapters, we probed various systems while remaining largely in the limit where D-branes are pointlike in their transverse directions. However, we learned in chapter 10 that D-branes have an intrinsic geometry of their own, which can be seen when we place a lot of them together to produce a large back-reaction on the spacetime geometry, or if we were to turn up the string coupling (for fixed string tension) such that Newton's constant is strong. Both sorts of situation can and will be forced upon us later, so it is worthwhile trying to understand what we can learn by probing the supergravity geometry with different types of branes (we have already probed extremal p-branes with Dp-branes in section 10.3). If we choose things such that there is some supersymmetry preserved, we can use it to help us learn many useful things.

Probingpwith D(p - 4)

Let us probe the geometry of the extremal p-branes with a D(p - 4)-brane. From our analysis of chapter 11, we know that this system is supersymmetric. Therefore, we expect that there should still be a trivial potential for the result of the probe computation, but there is not enough supersymmetry to force the metric to be flat. There are actually two sectors within which the probe brane can move transversely.

We saw in the previous chapter that the ‘holographic’, duality between AdS5 physics and the physics of the conformally invariant four dimensional Yang–Mills theory can be extended to the properties of solutions which are only asymptotically AdS5, in keeping with the basic dictionary of the correspondence. We studied the properties of Schwarzschild and Reissner–Nordstrom black holes in AdS, arising naturally as limits of non-extremal and spinning D3-branes, and found that their properties make considerable physical sense in the holographically dual field theory.

It is very clear that this duality between gravitational physics and that of gauge theory is potentially a powerful tool for studying gauge theory. The prototype example is, of course, a highly specialised sort of gauge theory, since it has sixteen supercharges, and is conformally invariant. Of great interest is the study of gauge theories which might be closer to the theories we use to model interactions in particle physics, such as QCD. Perhaps there are gravitational duals of such theories. More generally, of course, we would like to also find and study full string theory duals, if we want to study more than just very large N. At the time of writing, this is subject of considerable research effort.

In this chapter we shall have a brief look at extending the intuition we have developed about the AdS/CFT correspondence a bit further, and address the issue of studying less symmetric gauge theories by deforming the AdS/CFT example.