Superstrings

Feature: November 2003

Black holes

Figure 2

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 = Ak_Bc³/4h-barG, where A is the surface area of the horizon and k_Bis 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.

Figure 3

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).

Figure 4

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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