Gary Horowitz, UC Santa Barbara
gary@horizon.physics.ucsb.edu
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).