Holography in AdS Space

We will refer to AdS(5)⊗S(5) as the bulk space and the 4-dimensional boundary of AdS at y = 0 as the boundary. According to the Holographic Principle we should be able to describe everything in the bulk by a theory whose degrees of freedom can be identified with the boundary at y = 0. However the Holographic Principle requires more than that. It requires that the boundary theory has no more than 1 degree of freedom per Planck area. To see what this entails, let us compute the area of the boundary. Later we will regulate this divergence by moving in a little way from y = 0, but for the time being we can assume that the number of degrees of freedom per unit coordinate area is infinite. That suggests that the boundary theory might be a quantum field theory, and that is in fact the case. Another important fact involves the symmetry of AdS. It is obvious that the geometry is invariant under ordinary Poincare transformations of the 4-dimensional Minkowski coordinates t, xⁱ. In addition there is a “dilatation” symmetry

 (1.1)

On the other hand if we consider the representation of AdS we can see additional symmetry. For example the rotations of the sphere Ω are symmetries. The full symmetry group of AdS(5) is the group O(4|2).I n addition there is also the symmetry O(6) associated with rotations of the internal 5-sphere.

Since our goal is a holographic boundary description of the physics in the bulk spacetime it is very relevant to ask how the symmetries act on the boundary of AdS. Obviously, the 4-dimensional Poincare symmetry acts on the boundary straightforwardly. The dilatation symmetry also acts as a simple dilatation of the coordinates t, x. All of the transformations act on the boundary as conformal transformations which preserve light-like directions on the boundary. In fact the full AdS symmetry group, when acting on the boundary at y = 0 is precisely the conformal group of 4-dimensional Minkowski space.

The implication of this symmetry of the boundary is that the holographic boundary theory must be invariant under the conformal group. This together with the fact that the boundary has an infinite (coordinate) density of degrees of freedom suggests that the holographic theory is a Conformal Quantum Field Theory, and so it is.

As we mentioned, AdS(5) ⊗ S(5) is a solution of the 10-dimensional super gravity that describes low energy superstring theory. Indeed the space has more symmetry than just the conformal group and the O(6) symmetry of the internal 5-sphere.The additional symmetry is the so-called N = 4 supersymmetry. This symmetry must also be realized by the holographic theory. All of this leads us to the remarkable conclusion that quantum gravity in AdS(5) ⊗ S (5) should be exactly described by an appropriate super conformal Lorentz invariant quantum field theory associated with the AdS boundary.

In order to have a benchmark for the counting of degrees of freedom in AdS (5) ⊗ S(5) imagine constructing a cutoff field theory in the bulk. A conventional cutoff would involve a microscopic length scale such as the 10- dimensional Planck length l_p. One way to do this would be to introduce a spatial lattice in 9-dimensional space. It is not generally possible to make a regular lattice, but a random lattice with an average spacing l_p is possible. We can then define a simple theory such as a Hamiltonian lattice theory on the space. In order to count degrees of freedom we also need to regulate the area of the boundary of AdS which is infinite. To do so we introduce a surface L at r = 1 − δ. The total 9-dimensional spatial volume in the interior of L is easily computed using is seen to be critically divergent.

 (1.2)

The number of bulk lattice sites and therefore the number of degrees of freedom is

 (1.3)

In such a theory we also will find that the maximum entropy is of the same order of magnitude. However the Holographic Principle suggests that this entropy is overestimated.

The holographic bound discussed requires the maximum entropy and the number of degrees of freedom to be of order

 (1.4)

where A is the 8-dimensional area of the boundary L. This is also easily computed. We find

 (1.5)

In other words when R/l_p becomes large the holographic description requires a reduction in the number of independent degrees of freedom by a factor l_p/R. To say it slightly differently, the Holographic Principle implies a complete description of all physics in the bulk of a very large AdS space in terms of only l_p/R degrees of freedom per spatial Planck volume. Nonetheless the theory must be able to describe microscopic events in the bulk even when R becomes extremely large.