Gary Horowitz, UC Santa Barbara
Last November, a young theorist named Juan Maldacena made a bold conjecture for a nonperturbative formulation of string theory (hep-th/9711200). He considered spacetimes which asymptotically approach Anti-de Sitter (AdS) space and stated that with this boundary condition, string theory is completely equivalent to an ordinary field theory in one less dimension which is conformally invariant i.e. a conformal field theory (CFT). At first sight, this seems crazy -- but it is not obviously wrong. When the string theory is weakly coupled and hence well understood, the field theory is strongly coupled, and vice versa. It is well known that perturbatively, string theory has many more degrees of freedom than an ordinary field theory. However, there have been earlier hints from studies of strings at high temperature that fundamentally, string theory should have many fewer degrees of freedom than it appears to. The conformal boundary at infinity of is a time-like cylinder . For many purposes, one can think of the field theory as living on this boundary. The most natural field theory to live on this boundary is a CFT since the metric is only defined up to conformal transformations. In a sense, this theory is `holographic' since the fundamental degrees of freedom live on the boundary, but describe all the physics taking place inside.
There are actually a series of conjectures which differ by the precise asymptotic boundary conditions one imposes on the spacetime. Perhaps the simplest case applies to ten dimensional spacetimes which asymptotically approach . String theory with this boundary condition is conjectured to be completely equivalent to four dimensional, , supersymmetric Yang-Mills theory. The radii of the and are equal (i.e., their scalar curvatures are equal in magnitude, but of course opposite in sign) and proportional to in Planck units. To describe spacetimes with small curvature asymptotically, one needs the radii to be large and hence N to be large.
Maldacena was led to his conjectures by exploring the consequences of the recent description of quantum states of extreme and near extreme black holes in string theory. The near horizon geometry of the extreme Reissner-Nordström solution is . In higher dimensions, the near horizon geometries of extreme black holes and extended black `p-branes' are also products of AdS and spheres. It was found a few years ago, that the quantum states of a near extreme black hole could be described in terms of a gauge theory. The gauge theory excitations interacted with the usual string states which described strings propagating further from the black hole. Maldacena argued that if one takes a certain limit which removes the asymptotically flat region around the black hole and focuses on the near horizon geometry, the gauge theory completely decouples from the usual string modes. So it should describe all the physics of strings propagating near the horizon.
Since we do not have another nonperturbative definition of string theory, one could simply take the gauge theory as the definition of the theory. However, to prove the conjecture, one must show that there is an expansion of the gauge theory which reproduces the perturbative expansion of string theory about . More than twenty years ago, 't Hooft showed that the expansion of a gauge theory indeed resembles a string theory. People are now trying to make this correspondence more precise. It is easy to see that the symmetries agree. The isometry group of is , so perturbative string theory on this background will have these symmetries. Four dimensional Yang-Mills theory is invariant under the conformal group . The supersymmetry implies that in addition to the gauge field, there are four fermions and six scalars, all taking values in the adjoint of . There is an symmetry which rotates the six scalars, so the bosonic symmetries agree. It turns out that the supersymmetries agree as well.
The low energy string excitations are described by a supergravity theory. It has been shown that the energy of linearized supergravity modes on agrees precisely with the energy of states in the gauge theory. This can be verified even though the gauge theory is strongly coupled, since the supergravity states correspond to states in the gauge theory which are protected against quantum corrections. Some perturbative interactions have also been checked and shown to agree. The gauge theory is believed to describe all finite energy excitations about , including black holes. It is clear from earlier work that the gauge theory has enough states to reproduce the entropy of black holes. It is a simple exercise to check that a large five dimensional Schwarzschild-AdS black hole has and , exactly like a four dimensional field theory.
There is currently a tremendous amount of activity in this area, and the subject is developing rapidly in many directions. For example, Witten suggested that one could break supersymmetry, and use this conjecture to study strongly coupled non-supersymmetric gauge theories ( hep-th/9802150, hep-th/9803131). In fact, a simple picture of confinement is emerging based on the geometry on certain asymptotically AdS spacetimes. Conversely, efforts are being made to use the gauge theory to study black hole evaporation. One immediate consequence seems to be that the evaporation will be unitary, since the underlying gauge theory is unitary and its time is equivalent to the asymptotic AdS time. An application to cosmology has also been suggested (G. Horowitz, D. Marolf hep-th/9805207). I have mentioned only a few of the hundreds of papers which have appeared. There was also a recent discussion in Physics Today (August 1998, p. 20).