String duality and M theory
To select the "best" theory, we have to introduce the idea of S-duality. Consider a strongly coupled physical system. There may be another set of variables in which the system is weakly coupled. The transformation from one set of variables to the other is called S-duality. In practice it is usually impossible to find the exact change of variables from the strongly coupled degrees of freedom to the weakly coupled ones. Unlike T-duality, S-duality is non-perturbative. This prevents us from proving it, but we can accumulate enough evidence to convince ourselves of its existence.
A major discovery is that all five consistent string theories are related by dualities such as T- and S- dualities. This puts the five string theories in a unified framework they are equivalent descriptions of the same physical system with different variables.
Figure 2 describes the parameter space of string theories the edges are points where the corresponding string is weakly coupled. In the interior the coupling is generically of order 1. One additional item in figure 2 is 11-dimensional M theory, which at low energies is described by 11-dimensional supergravity, a supersymmetric theory including gravity.
Unlike string theory, M theory does not have a coupling constant. It is reduced under various compactifications to the five string theories. As an example, consider M theory compactified on a circle of radius R. This is dual to Type IIA string theory with the string coupling proportional to R. The strong coupling limit of 10-dimensional Type IIA string theory corresponds to a large R and, amazingly, another dimension opens up, so that it is described by 11-dimensional theory.
Not much is known about M theory. With no coupling constant there is no systematic perturbative expansion. One attempt to provide a non-perturbative definition is Matrix Theory, based on supersymmetric quantum mechanics of infinite matrices.
Strings and quantum black holes
Classical black holes are completely black, swallowing whatever crosses their event horizon and emitting nothing. Hawking argued that quantum black holes emit radiation. This raises a long-standing puzzle. We can start with a pure quantum state to form a black hole and end with a mixed state of thermal radiation. This contravenes the basic rules of quantum mechanics. Black holes also carry entropy proportional to their event horizon. To understand these features of black holes requires a quantum theory of gravity, and string theory provides such a framework. The entropy of the black hole is a measure of how many quantum states it has. For certain black holes these can be identified as D-branes non-perturbative excitations of string theory on which open strings can end. (D-branes provide Dirichlet boundary conditions for the strings; figure 3.)
Open strings have gauge fields, so the D-branes define a gauge theory. There is a class of black holes made of D-branes, and these have a gauge theory description of their quantum properties. For these black holes it is possible to count the microscopic quantum states and derive the BekensteinHawking equation relating the black hole entropy to the area of the event horizon. It is also possible to compute the rate of Hawking radiation resulting from quantum scattering and give its microscopic explanation.
Strings and gauge theories
For an ordinary quantum field theory of a physical system in a volume V, the number of degrees of freedom is proportional to the volume.
A quantum theory of gravity is believed to possess a remarkable property called holography, which was introduced by Gerard 't Hooft and Leonard Susskind. Here the number of degrees of freedom of a quantum gravitational system in volume V is expected to be proportional to the area of the boundary of the volume. This means that the physics of a quantum gravitational system in d+1 dimensions is coded in a holographic way in d dimensions. As in an optical hologram, the information is coded in a complicated way.
Because string theory is our candidate for a theory of quantum gravity, it should exhibit such holography. This has been realized recently in Matrix Theory, and for strings on certain spaces with negative cosmological constant, known as Anti de Sitter spaces. "Holograms" are located on the boundaries of these spaces. In both cases the hologram that encodes the quantum gravity properties is a gauge theory. Remarkably, the secrets of the evaporation of quantum black holes are hidden in such a hologram the strong coupling regime of quantum field theory.
On one hand the theory of D-branes is a gauge theory. On the other, D-branes are massive objects that curve the space-time in which they are embedded. The consideration of the D-branes from these two viewpoints led to a conjecture by Juan Maldacena (Harvard) on the duality between gauge theories and string theory on Anti de Sitter spaces.
Of particular interest is the equivalence between gravity in five dimensions and gauge theory in four. The additional fifth dimension corresponds to the energy scale of the four-dimensional quantum field theory. Just as theories of three spatial dimensions and one of time became four-dimensional space-time theories, we may be led to add the energy scale and have five natural co-ordinates.
The duality conjecture has been extended by Ed Witten to non-supersymmetric gauge theories, such as pure quantum chromodynamics (QCD) the field theory of gluons. The Anti de Sitter spaces are now replaced by Anti de Sitter black holes. When the gravitational background is regular, the supergravity approximation of string theory can be used and exhibits the required properties of QCD, such as the colour confinement of quarks. To study QCD quantitatively requires singular gravitational backgrounds, and it seems that the key to long-standing questions, such as computation of the hadron spectrum, is hidden in the geometry of string theory.
Unification
In the supersymmetric extension of the Standard Model, the strong and electroweak gauge couplings seem to unify at an energy of 1016 GeV. The gravitational coupling does not unify with the others at this energy. Traditionally it is expected to unify at the Planck scale.
There is, however, another interesting possibility. The energy dependence of the gravitational coupling can be changed by using the extra dimensions. The unification energy of the gravitational coupling can be lowered if, as in the D-brane physics of figure 3, the gauge fields live in 3+1 space-time dimensions while the gravity also propagates in the other dimensions.
This can dramatically change the scales of string physics. The string scale cannot be less than 1 tera electron volt because string excitations have not been observed by accelerators. It is possible, however, that the string scale is at the tera electron volt scale, and that string excitations will be seen by the LHC. The compactification scale the size of the largest compact dimension can be even lower. Amazingly, it can be millimetre-sized.
Although this seems to be in contradiction with laboratory experience, in fact this is not the case. The gauge fields do not propagate in the compact dimensions, so they do not have KaluzaKlein excitations. There are KaluzaKlein excitations from the closed strings in figure 3, but they are weakly coupled and could not be observed. Larger dimensions are ruled out by experiments measuring gravitational effects at short distances.
The extra dimensions can be used to lower the unification energy so that a complete unification of all forces can occur already at tera electron volt energies.