### Superstrings

An introduction to life in six dimensions As always, there's more information at the 'official site'.

Plus, we may be making them in the laboratory soon!

*String theory is either a theory of everything - which automatically unites gravity with the other three forces in nature - or a theory of nothing, but finding the correct form of the theory is like searching for a needle in a stupendous haystack*Plus, we may be making them in the laboratory soon!

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A recipe for making strings in the lab

11 May 2005

Theoretical physicists in the Netherlands have proposed a way to make superstrings in the laboratory. If their idea can be put into practice, it would allow aspects of string theory to be explored in an experiment for the first time. The new approach relies on exploiting the properties of ultracold atomic gases (arxiv.org/abs/cond-mat/0505055).

String theorists attempt to explain all the fundamental particles as vibrations on tiny strings on length scales of about 10-33 metres. The theory naturally includes "supersymmetry" - a symmetry that connects particles with integer spin, known as bosons, to particles with half-integer spin, which are known as fermions. The particles that carry the fundamental forces of nature, such as the photon and the gluon, are bosons, while the quarks and leptons that make up matter are fermions. Although superstring theory is the leading candidate for a theory of everything, there is no experimental evidence to date for strings or supersymmetry.

Now Michiel Snoek, Masudul Haque, Stefan Vandoren and Henk Stoof of Utrecht University have proposed making a "non-relativistic Green-Schwarz superstring" by trapping an ultracold cloud of fermionic atoms along the core of a quantized vortex in a Bose-Einstein condensate (BEC). A BEC is a special state of matter in which all the particles are in the same quantum ground state. Bosonic atoms such as rubidium-87 can enter such as state because, unlike fermions, they do not obey the Pauli exclusion principle.

The bosonic part of the superstring would consist of a vortex line created by rapidly rotating a one-dimensional BEC in an optical lattice (see figure). Next, a gas of fermion atoms, such as potassium-40, would be trapped within this vortex, which is possible under certain conditions. Snoek and colleagues say that it should be possible to observe the supersymmetry between the fermions and bosons by carefully tuning the interactions between the two types of atom with a laser.

Quantized vortices were first seen in superfluid helium. They are formed inside a rotating superfluid when it begins to spin faster than a certain critical speed. In the mid-1990s it was suggested that these vortices could simulate the formation of cosmic strings in the early universe.

About the author

Belle Dume is science writer at PhysicsWeb

Superstrings

Feature: November 2003

String theory is either a theory of everything - which automatically unites gravity with the other three forces in nature - or a theory of nothing, but finding the correct form of the theory is like searching for a needle in a stupendous haystack

As I sit down to write this article I feel that I have taken on a task rather like trying to summarize the history of the world in 10 pages. It is just too large a subject, with too many lines of thought and too many threads to weave together. In the 34 years since it began, string theory has developed into an enormous body of knowledge that touches on every aspect of theoretical physics.

String space

String space

String theory is a theory of composite hadrons, an aspiring theory of elementary particles, a quantum theory of gravity, and a framework for understanding black holes. It is also a powerful technical tool for taming strongly interacting quantum field theories and, perhaps, a basis for formulating a fundamental theory of the universe. It even touches on problems in condensed-matter physics, and has also provided a whole new world of mathematical problems and tools.

All I can do with this gargantuan collection of material is to make my own guess about which aspects of string theory are most likely to form the core of a future physical theory, perhaps 100 years from now. It will come as no surprise to my friends that my choice revolves around those things that have most interested me in the last several years. No doubt many of them will disagree with my judgement. Let them write their own articles.

String theory is considered to be a branch of high-energy or elementary particle physics. However, a high-energy theorist from the 1950s, 1960s or 1970s would be surprised to read a recent string-theory paper and find not a single Feynman diagram, cross-section or particle decay rate. Nor would there be any mention of protons, neutrinos or Higgs bosons in the majority of current literature. What the reader would find are black-hole metrics, Einstein equations, Kaluza-Klein theories and plenty of fancy geometry and topology. The energy scales of interest are not MeV, GeV or even TeV, but energies at the Planck scale - the scale at which the classical concepts of space and time break down.

The Planck energy is equal to h-bar5/G, where h-bar is Planck's constant divided by 2π, c is the speed of light and G is the gravitational constant, and it corresponds to masses that are some 19 orders of magnitude larger than the proton mass. This is the energy of the universe when it was just 10-43s old, and it will probably be forever out of range of any particle accelerator. To understand physics at the Planck scale we need a quantum theory of gravity.

In the days when my career was beginning, a typical colloquium on high-energy physics would often begin by stating that there are four forces in nature - electromagnetic, weak, strong and gravitational - followed by a statement that the gravitational force is much too weak to be of any importance in particle physics so we will ignore it from now on. That has all changed.

Today the other three forces are described by the gauge theories of quantum chromodynamics (QCD) and quantum electrodynamics (QED), which together make up the Standard Model of particle physics. These quantum field theories describe the fundamental forces between particles as being due to the exchange of field quanta: the photon for the electromagnetic force, the W and Z bosons for the weak force, and the gluon for the strong force. In the string-theory community, however, the electromagnetic, strong and weak forces are generally considered to be manifestations of certain "compactifications" of space from 10 or 11 dimensions to the four familiar dimensions of space-time. But before I report on the status of string theory, I want to tell you how it came about that so many otherwise sensible high-energy theorists became interested in quantum gravity.

Why quantum gravity?

Elementary particles have far too many properties - such as spin, charge, colour, parity and hypercharge - to be truly elementary. Particles obviously have some kind of internal machinery at some scale. Protons and mesons reveal their "parts" at the modestly small distance of about 10-15 m, but quarks, leptons and photons hide their structure much more effectively. Indeed, no experiment has ever seen direct evidence of size or structure for any of these particles.

The first indication that the true scale of elementary particles might be somewhere in the neighbourhood of the Planck scale came in the 1970s. Howard Georgi and Sheldon Glashow, then at Harvard University, showed that the very successful, but somewhat contrived, Standard Model could be elegantly unified into a single theory by enlarging its symmetry group. The new construction was astonishingly compact and most particle theorists assumed that there must be some truth to it. But its predictions for the coupling constants - the constants that describe the strengths of the strong, weak and electromagnetic interactions - were wrong.

Georgi, along with Helen Quinn and Steven Weinberg, also at Harvard, soon solved this problem when they realized that the coupling constants are not really constants at all - they vary with energy. If the known couplings are extrapolated they all intersect the predictions of the unified theory at roughly the same scale. Moreover, this scale is close to the Planck scale. The implication of this was clear: the scale of the internal machinery of elementary particles is the Planck scale. And since the gravitational constant, G, appears in the definition of the Planck energy, to many of us this inevitably meant that gravitation must play an essential role in determining the properties of particles.

The earliest attempts to reconcile gravity and quantum mechanics - notably by Richard Feynman, Paul Dirac and Bryce DeWitt, who is now at the University of Texas at Austin - were based on trying to fit Einstein's general theory of relativity into a quantum field theory like the hugely successful QED. The goal was to find a set of rules for calculating scattering amplitudes in which the photons of QED are replaced by the quanta of the gravitational field: gravitons. But gravitational forces become increasingly strong as the energy of the participating quanta increases, and the theory proved to be wildly out of control. Attempting to treat the graviton as a point particle simply gave rise to far too many degrees of freedom at short distances.

In a sense the failure of this "quantum gravity" theory was a good sign. The theory itself gave no insight into the internal machinery of elementary particles, and it offered no explanation for the other forces of nature. At best it was more of the same: an effective (but not very) description of gravitation with no deeper insight into the origin of particle properties. At worst, it was mathematical nonsense.

Strings as hadrons

We all know that science is full of surprising twists, but the discovery of string theory was particularly serendipitous. The theory grew out of attempts in the 1960s to describe the interactions of hadrons - particles that contain quarks, such as the proton and neutron. This was a problem that had nothing to do with gravity. Gabriele Veneziano, now at CERN, and others had written down a simple mathematical expression for scattering amplitudes that had certain properties that were fashionable at that time. It was soon discovered by Yoichiro Nambu of the University of Chicago and myself, and in a slightly different form by Holger Bech Nielsen at the Niels Bohr Institute, that these amplitudes were the solution of a definite physical system that consists of extended 1D elastic strings.

For the two years that followed, string theory was the theory of hadrons. One of the spectacular discoveries made in this early period was that the mathematical infinities that occur in quantum field theory are completely absent in string theory. However, from the very beginning there were big problems in interpreting hadrons as strings. For example, the earliest version of the theory could only accommodate bosons, whereas many hadrons - including the proton and neutron - are fermions.

The distinction between bosons and fermions is one of the most important in physics. Bosons are particles that have integer spins, such as 0, h-bar and 2h-bar, whereas fermions have half-integer spins of h-bar/2, 3h-bar/2 and so on. All fundamental matter particles, such as quarks and leptons, are fermions, while the particles that carry fundamental forces - the photon, W and Z, and so on - are all bosons.

Fermionic versions of string theory were soon discovered and, moreover, they turned out to have a surprising symmetry called supersymmetry that is now totally pervasive in high-energy physics. In supersymmetric theories all bosons have a fermionic superpartner and vice versa. The early development of "superstring" theory was due to pioneering work by John Schwarz of Caltech, Andrei Neveu of the University of Montpellier II, Michael Green of Cambridge and Pierre Ramond of the University of Florida, and much of the subsequent technical development was carried out in a famous series of papers by Green and Schwarz in the 1980s.

Another apparently serious problem with the string theory of hadrons concerned dimensions. Although the original assumptions in string theory were simple enough, the mathematics proved internally inconsistent, at least if the number of dimensions of space-time was four. The source of this problem was quite deep, but, strangely, if space-time has 10 dimensions it contrives to cancel out. The reasons were not at all easy to understand, but the extraordinary mathematical consistency of superstring theory in 10 dimensions was compelling. However, so was the obvious fact that space-time has four dimensions, not 10.

Thus by about 1972 theorists were beginning to question the relevance of string theory for hadrons. In fact, there were other serious physical shortcomings in addition to the bizarre need for 10 dimensions. A mathematical string can vibrate in many patterns, which represent a different type of particle, and among these are certain patterns that represent massless particles. But most dangerous of all were massless particles with two units of spin angular momentum ("spin-two"). There are certainly spin-two hadrons, but none that have anything like zero mass. Despite all efforts, the massless spin-two particle could not be removed or made massive.

Eventually, mathematical string theory gave way to QCD as a theory of hadrons, which had its own explanation of the string-like behaviour of these particles without the bad side effects. For most high-energy theorists, string theory had lost its reason for existence. But a few bold souls saw opportunity in the debacle. A massless spin-two field might not be good for hadronic physics, but it is just what was needed for quantum gravity, albeit in 10D. This is because just as the photon is the quantum of the electromagnetic field, the graviton is the quantum of the gravitational field. But the gravitational field is a symmetric tensor rather than a vector, and this means the graviton is spin-two, rather than spin-one like the photon. This difference in spin is the principal reason why early attempts to quantize gravity based on QED did not work.

A theory of everything

The massless spin-two graviton led to a radical shift in perspective among theorists. The focus of mainstream high-energy physics at the time was on energy scales anywhere from the hadronic scale of a few GeV to the weak interaction scale of a few hundred GeV. But to explore the idea that string theory governs gravity, the energy scale of string excitations has to jump from the hadronic scale to the Planck scale. In other words, with barely a blink of the eye, string theorists would leapfrog 19 orders of magnitude, and therefore completely abandon the idea that progress in physics proceeds incrementally. Heady stuff, but also the source of much irritation in the rest of the physics community.

Another reason for annoyance was somebody's idea to start referring to string theory as a "theory of everything". Even string theorists found this irritating, but there is actually a technical sense in which string theory can either be a theory of everything or a theory of nothing. One of the problems in describing hadrons with strings was that it proved impossible to allow for the hadrons to interact with other fields, such as electromagnetic fields, as they clearly do experimentally. This was a deadly flaw for a theory of hadrons, but not for a theory in which all matter, including photons, are strings. In other words, either all matter is strings, or string theory is wrong. This is one of the most exciting features of the theory.

But what about the problem of dimensions? Here again, a sow's ear was turned into a silk purse. The basic idea goes back to Theodor Kaluza in 1919, who tried to unify Einstein's gravitational theory with electrodynamics by introducing a compact space-like fifth dimension. Kaluza discovered the beautiful fact that the extra components of the gravitational field tensor in 5 dimensions behaved exactly like the electromagnetic field plus one additional scalar field. Somewhat later, in 1938, Oskar Klein and then Wolfgang Pauli generalized Kaluza's work so that the single compact dimension was replaced by a 2D space. If the 2D space is the surface of a sphere then a remarkable thing happens when Kaluza's procedure is followed. Instead of electrodynamics, Klein and Pauli discovered the first "non-Abelian" gauge theory, which was later rediscovered by Chen Ning Yang and Robert Mills. This is exactly the same class of theories that is so successful in describing the strong and electromagnetic interactions in the Standard Model.

One may ask whether particles move in the extra dimensions. For example, can a particle that appears to be standing still in our usual 3D space have velocity or momentum components in the compact dimensions? The answer is yes, and the corresponding components of momentum define new conserved quantities (figure 1). What is more, these quantities are quantized in discrete units. In short, they are "charges" similar to electric charge, isospin and all the other internal quantum numbers of elementary particles. The answer to the problem of dimensions in string theory is obvious: six of the 10 dimensions should be wrapped up into some very small compact space, and the corresponding quantized components of momenta become part of the internal machinery of elementary particles that determines their quantum numbers.

Life in six dimensions

Much of the development of string theory is therefore concerned with 6D spaces. These spaces, which can be thought of as generalized Kaluza-Klein compactification spaces, were originally studied by mathematicians and are known as Calabi-Yau spaces. They are tremendously complicated and are not completely understood. But in the process of studying how strings move on them, physicists have created an unexpected revolution in the study of Calabi-Yau spaces.

In particular, it was discovered that a compactification radius of size R is completely equivalent to a space with size 1/R from the point of view of string theory. This connection, which is known as T-duality, has a mathematically profound generalization called mirror symmetry, which states that there is an equivalence between small and large spaces (see box above). Mirror symmetry of Calabi-Yau spaces - which are not only of different sizes but have completely different topologies - was completely unsuspected before physicists began studying quantum strings moving on them.

I wish it was possible to draw a Calabi-Yau space but they are tremendously complicated. They are six-dimensional, which is three more than I can visualize, and they have very complicated topologies, including holes, tunnels and handles. Furthermore, there are thousands of them, each with a different topology. And even when their topology is fixed there are hundreds of parameters called moduli that determine the shape and size of the various dimensions. Indeed, it is the complexity of Calabi-Yau geometry that makes string theory so intimidating to an outsider. However, we can abstract a few useful things from the mathematics, one of them being the idea of moduli.

The simplest example of a modulus is just the compactification radius, R, when there is only a single compact dimension. In more complicated cases, the moduli determine the sizes and shapes of the various features of the geometry. The moduli are not constants but depend on the geometry of the space itself, in the same way that the radius of the universe changes with time in a manner that is controlled by dynamical equations of motion. Since the compact dimensions are too small to see, the moduli can simply be thought of as fields in space that determine the local conditions. Electric and magnetic fields are examples of such fields but the moduli are even simpler: they are scalar fields (i.e. they have only one component), rather than vector fields. String theory always has lots of scalar-field moduli and these can potentially play important roles in particle physics and cosmology.

All of this raises an interesting question: what determines the compactification moduli in the real world of experience? Is there some principle that selects a special value of the moduli of a particular Calabi-Yau space and therefore determines the parameters of the theory, such as the masses of particles, the coupling constants of the forces, and so on? The answer seems to be no: all values of the moduli apparently give rise to mathematically consistent theories. Whether or not this is a good thing, it is certainly surprising.

Ordinarily we might expect the vacuum or ground state of the world to be the state of lowest energy. Furthermore, in the absence of very special symmetries, the energy of a region of space will depend non-trivially on the values of the fields in that region. Finding the true vacuum is then merely an exercise in computing the energy for a given field configuration and minimizing it. This is, to be sure, a difficult task, but it is possible in principle. In string theory, however, we know from the beginning that the potential energy stored in a given configuration has no dependence on the moduli fields.

The reason that the field potential is exactly zero for every value of the moduli is that string theory is supersymmetric. Supersymmetry has both desirable and undesirable consequences. Its most obvious drawback is the requirement that for every fermion there is a boson with exactly the same mass, which is clearly not a property of our world.

A more subtle difficulty involves the aforementioned fact that the vacuum energy is independent of the moduli. As well as telling us that we cannot determine the moduli by minimizing the energy, supersymmetry also tells us that the quanta of the moduli fields are exactly massless. No such massless fields are known in nature and, furthermore, such fields are very dangerous. Indeed, massless moduli would probably lead to long-range forces that would compete with gravity and violate the equivalence principle - the cornerstone of general relativity - at an observable level.

On the plus side, the vanishing vacuum energy that is implied by supersymmetry ensures that the cosmological constant vanishes. If it were not for supersymmetry, the vacuum would have a huge zero-point energy density that would make the radius of curvature of space-time not much bigger than the Planck scale - a most undesirable situation. Supersymmetry also stabilizes the vacuum against various hypothetical instabilities, and it allows us to make exact mathematical conclusions. Indeed, T-duality and mirror symmetry are examples of those exact consequences.

Throughout the 1980s and early 1990s progress in string theory largely consisted of working out the detailed rules of perturbation theory for the five known versions of the theory, which would allow theorists to arrive at actual solutions (figure 2). These perturbative rules were generalizations of the Feynman diagrams of QED and QCD in which the "world lines" of point particles are replaced by "world sheets" that are traced out by moving strings. The study of world-sheet physics created a huge body of knowledge about 2D quantum field theory, but it did not offer much insight into the inner workings of quantum gravity. At best, string theory provided an especially consistent way to introduce a small distance scale and thereby regulate the divergences that had plagued the older attempts at quantizing gravity.

Personally I found the whole enterprise dry, overly technical and, above all, disappointing. I felt that a quantum theory of gravity should profoundly affect our views of space-time, quantum mechanics, the origin of the universe, and the mysteries of black holes. But string theory was largely silent about all these matters. Then in 1993 all this began to change, and the catalyst was the awakening interest in Stephen Hawking's earlier speculations about black holes.

The starting point for Hawking's speculations was the thermal behaviour of black holes, which built on earlier work by Jacob Bekenstein of the Hebrew University in Israel. Rather than the cold, dead objects that they were originally thought to be, black holes turned out to have a heat content and to glow like black bodies. Because they glow they lose energy and evaporate, and because they have a temperature and an energy, they also have an entropy. This entropy, S, is defined by the Bekenstein-Hawking equation: S = AkBc3/4h-barG, where A is the surface area of the horizon and kBis Boltzmann's constant.

After realizing that black holes must evaporate by the emission of black-body radiation, Hawking raised an extremely profound question: what happens to all the detailed information that falls into a black hole? Once it falls through the horizon it cannot subsequently reappear on the outside without violating causality. That is the meaning of a horizon. But the black hole will eventually evaporate, leaving only photons, gravitons and other elementary particles as products of the decay. Hawking concluded that the information must ultimately be lost to our world. But one of the fundamental principles of quantum mechanics is that information is never lost, because the information in the initial state of a quantum system is permanently imprinted in the quantum state.

Hawking's view was that conventional quantum mechanics must be violated during the formation and evaporation of the black hole. He rightly understood that if this is true, the rules of quantum mechanics must be drastically modified as the Planck scale is approached. The importance of this for particle physics, particularly for unified theories, should have been obvious. But initially Hawking's idea generated little interest among high-energy theorists, apart from myself and Gerard 't Hooft at the University of Utrecht. We were convinced that by modifying the rules of quantum mechanics in the way advocated by Hawking, all hell would break loose, such as causing empty space to quickly heat up to stupendous temperatures and energy densities. We were sure that Hawking was wrong. By the early 1990s, however, the issue was becoming critical, especially to string theorists. String theory by its very definition is based on the conventional rules of quantum mechanics and if Hawking was right, the entire foundation of the theory would be destroyed.

Over the last decade the apparent clash between standard quantum principles and black-hole evaporation has been resolved, favouring, I should add, the views of 't Hooft and myself. The formation and evaporation of a black hole is similar to many other process in nature in which a collision between particles gives rise to a very rich and chaotic spectrum of intermediate states. In the case of a black hole, the collisions are between the original protons, neutrons and electrons in a collapsing star. Roughly speaking a black hole is nothing but a very excited string with a total length that is proportional to the area of its horizon. During the collision or collapse process, all the energy of the initial state goes into forming a single long, tangled string, and the entropy of the configuration is the logarithm of the number of configurations of a random-walking quantum string.

The correspondence between string configurations and black-hole entropy was checked for all of the various kinds of charged and neutral black holes that occur in compactifications of string theory. In most of the cases the entropy of the string configuration could be estimated and it agreed with the Bekenstein-Hawking entropy to within a factor of order unity.

But string theorists wanted to do better. The Bekenstein-Hawking formula for the entropy of a black hole is very precise: the entropy is one quarter of the horizon area, measured in Planck units, for every kind of black hole, be it static, rotating, charged or even higher-dimensional. Surely the universal factor of a quarter should be computable in string theory? The key to a precise calculation was obvious. Certain black holes called extremal black holes - which are the ground states of charged black holes that carry electric and magnetic charges - are especially tractable in a supersymmetric theory. The only problem was that in 1993 no-one knew how to build an extremal black hole out of the right type out of strings. This had to wait a couple of years for the discovery of entities called D-branes.

Brane world

In 1995 Joe Polchinski of the University of California in Santa Barbara electrified the string-theory community with a major discovery that has subsequently impacted every field of physics. As we have seen, T-duality is the strange symmetry that interchanges the Kaluza-Klein momenta and winding numbers of a closed string (see figure 1). But what happens to an open string? Obviously the idea of a winding number does not make sense for such a string. What actually happens to open stings under T-duality is that the free ends become fixed on surfaces called D-branes.

D-branes come in various dimensions; 2D branes, for example, can also be called membranes (figure 3). They have an energy or mass per unit surface area and are localized physical objects in their own right. In a sense they seem to be no less fundamental than the strings themselves. To an outsider, D-branes may seem to be arbitrary additions to the theory. They are not. Their existence is absolutely essential to the mathematical consistency of the theory. In addition to allowing T-duality to act on an open string in Type I string theory, they are necessary for implementing the deep dualities that link the five different kinds of string theory together.

But from the point of view of black holes, the importance of D-branes is that you can build extremal black holes from them. In fact, just by placing a large number of D-branes at the same location you can build an extremal supersymmetric black hole. And because of the special properties of supersymmetric systems, the statistical entropy of that black hole can be precisely computed. The result, which was first derived by Andrew Strominger and Cumrun Vafa at Harvard in 1996, is that the entropy is equal to exactly one quarter of the horizon area in Planck units! This suggested that the microscopic degrees of freedom implied by the Bekenstein-Hawking entropy are the degrees of freedom describing strings, and was a major boost for the superstring community.

At about the same time as D-branes were discovered, another very important development took place. As I mentioned, the coupling constant of string theory is not really a constant at all, and in many respects it is very similar to the compactification moduli. String theorists took a surprisingly long time to make the connection, but it turns out that 10D string theory is itself a Kaluza-Klein compactification of an 11D theory that became known as "M-theory".

M-theory appears to underlie all string theories (figure 2). The five different versions of string theory are just different ways of compactifying its 11 dimensions. But M-theory is not itself a string theory. It has membranes but no strings, and the strings only appear when the 11th dimension is compactified. Furthermore, the momentum in the compact 11th direction (the Kaluza-Klein momentum) is identified as the number of D0-branes - i.e. zero-dimensional branes, or points - in a particular type of string theory.

This connection between Kaluza-Klein momentum and D0-branes led to another breakthrough. In 1996 myself, Tom Banks and Steve Shenker (at Rutgers University), and Willy Fischler (at the University of Texas) realized that M-theory could be cast in a form no more complicated than the quantum mechanics of a system of non-relativistic particles, i.e. D0-branes. The resulting theory, which is called Matrix theory, is an exact and complete quantum theory that describes the microscopic degrees of freedom of M-theory. As such it is the first precise formulation of a quantum theory of gravity.

Duality

Matrix theory was just one example of how D-branes can be used to formulate a theory of quantum gravity. Soon after its discovery, Juan Maldacena, who is now at the Institute for Advanced Study (IAS) in Princeton, came up with a new direction to explore. Ed Witten of the IAS and others had previously shown that D-branes have their own dynamics. But it turned out that the fluctuations and motions of a D-brane can be quantized in the form of a gauge theory that is restricted to the D-brane itself. The theory that lives on a coincident collection of D3-branes, for example, is a supersymmetric non-Abelian gauge theory. In other words, it is a supersymmetric version of QCD - the theory describing quarks and gluons. In a sense, string theory is returning to its roots as a possible description of hadrons (See Physics World May 2003 pp35-38).

Maldacena realized that in an appropriate limit the theory of D3-branes should be a complete description of string theory - not just on the branes, but in the entire geometry in which the branes are embedded. A gauge theory would therefore also be a description of quantum gravity in a particular background space-time. This space-time is called anti-de Sitter space, which, roughly speaking, is a universe inside a cavity. The walls of the cavity behave like reflecting surfaces so that nothing escapes it (figure 4).

This "duality" between quantum field theory and gravity is an exact realization of what is called the holographic principle. This strange principle, formulated by 't Hooft and myself, grew from our debate with Hawking regarding the validity of quantum mechanics in the formation and evaporation of black holes.

According to the holographic principle, everything that ever falls into a black hole can be described by degrees of freedom that reside in a thin layer just above the horizon. In other words, the full 3D world inside the horizon can be described by the 2D degrees of freedom on its surface. Even more generally, it should be possible to describe the physics of any region of space in terms of holographic degrees of freedom that reside on the boundary of that region. This leads to a drastic reduction of the number of degrees of freedom in a field theory, and most theorists found it very hard to swallow until Maldacena's work came along. Maldacena's duality replaces a gravitational theory in anti-de Sitter space by a field theory that lives on its boundary in a very precise way. In other words, the 3 + 1-dimensional boundary field theory is a holographic description of the interior of 4 + 1-dimensional anti-de Sitter space.

The D-brane revolution has been very far reaching. Matrix theory and the Maldacena duality are both formulations of quantum gravity that conform to the standard rules of quantum mechanics, and should therefore lay to rest any further questions about black holes violating these rules.

Googles of possibilities

I would like to end by discussing the future of string theory, not as a mathematical subject but as a framework for particle physics and cosmology. The final evaluation of string theory will rest on its ability to explain the facts of nature, not on its own internal beauty and consistency. String theory is well into its fourth decade, but so far it has not produced a detailed model of elementary particles or a convincing explanation of any cosmological observation. Many of the models that are based on specific methods of compactifying either 10D string theory or 11D M-theory have a good deal in common with the real world. They have bosons and fermions, for example, and gauge theories that are similar to those in the Standard Model. Furthermore, unlike any other theory, they inevitably include gravity. But the devil is in the details, and so far the details have eluded string theorists.

It is, of course, possible that string theory is the wrong theory, but I believe that would be a very premature judgement and probably incorrect. The problem does not seem to be a lack of richness, but rather the opposite. String theory contains too many possibilities. For most physicists, the ideal physical theory is one that is unique and perfect, in that it determines all that can be determined and that it could not logically be any other way. In other words, it is not only a theory of everything but it is the only theory of everything. To the orthodox string theorist, the goal is to discover the one true consistent version of the theory and then to demonstrate that the solution manifests the known laws of nature, such as the Standard Model of particle physics, with its empirical set of parameters.

But the more we learn about string theory the more non-unique it seems to be. There are probably millions of Calabi-Yau spaces on which to compactify string theory. Each space has hundreds of moduli and hundreds of subspaces on which branes can be wrapped, fluxes imposed upon and so on. A conservative estimate of the number of distinct vacua of the theory is in the googles - that is, more than 10100. The space of possibilities is called the Landscape, and it is huge. To mix metaphors, it is a stupendous haystack that contains googles of straws and only one needle. Worse still, the theory itself gives us no hint about how to pick among the possibilities (see "The string-theory landscape").

This enormous variety may, however, be exactly what cosmology is looking for. A common theme among cosmologists is that the observed universe may merely be a minuscule part of a vastly bigger universe that contains many local environments, or what Alan Guth at MIT calls "pocket universes". According to this view, so many pocket universes formed during the early inflationary epoch - each of which with its own vacuum structure - that the entire landscape of possibilities is represented. The reasons for this view are not just idle speculation but are rooted in the many accidental fine-tunings that are necessary for a universe that supports life. Thus it may be that the enormous number of possible vacuum solutions, which is the bane of particle physics, may be just what the doctor ordered for cosmology.

Further information

T-duality

In a single compact dimension there are two kinds of quantum numbers: momentum in the compact direction and the winding number. Both of these are quantized in integer multiples of a basic unit, and each has a certain energy associated with it. In the case of momentum, for example, the energy is just the kinetic energy of motion in the compact direction. The energy of a particle with n units of compact momentum is equal to n/R, where R is the circumference of the compact direction. Note that the energy grows as the size of the compact space gets smaller. On the other hand, the winding modes also have energy, which is the potential energy needed to stretch the string around the compact co-ordinate. If we call the winding number m, then the winding energy is equal to mR. In this case the energy decreases as the size of the compact direction decreases.

The surprising thing is that the spectrum of energies is unchanged if we change the compactification radius from R to 1/R, and at the same time interchange the Kaluza-Klein momentum and winding modes. In other words, just by looking at the spectrum of energies you could never tell the difference between a theory that is compactified on a space of size R or on one of size 1/R. As you try to make the compactification scale smaller than the natural string scale - i.e. the size of a vibrating string - the theory begins to behave as if the compactification radius was getting bigger. Physically, the smallest compactification value of R is the string scale. But from a mathematical viewpoint, two different spaces - one large, the other small - are completely equivalent. This equivalence is called T-duality.

About the author

Leonard Susskind is in the Department of Physics, Stanford University, 382 Via Pueblo Mall, CA 94305-4060, US, e-mail susskind@stanford.edu

Further reading

J Maldacena 1999 The large N limit of superconformal field theories and supergravity Int. J. Theor. Phys. 38 1113-1133

J Polchinski 1995 Dirichlet-branes and Ramond-Ramond charges Phys. Rev. Lett. 75 4724

J Polchinski 1998 String Theory (volume 2): Superstring Theory and Beyond (Cambridge University Press)

J H Schwarz et al. 1981 Superstring Theory (volume 1): Introduction (Cambridge University Press)

A Strominger and C Vafa 1996 Microscopic origin of the Bekenstein-Hawking entropy Phys. Lett. B 379 99

The official string theory website: superstringtheory.com/

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