Holography : 2 - D or not 2 - D ?

As was recently pointed out by Cadoni, a certain class of two-dimensional gravitational theories will exhibit (black hole) thermodynamic behavior that is reminiscent of a free field theory. In the current letter, a direct correspondence is established between these two-dimensional models and the strongly curved regime of (arbitrary-dimensional) anti-de Sitter gravity. On this basis, we go on to speculatively argue that two-dimensional gravity may ultimately be utilized for identifying and perhaps even understanding holographic dual-ities. There is a growing suspicion that the holographic principle may be a crucial element in linking together semi-classical gravity and the fundamental quantum theory. That is to say, the holographic storage of information can, perhaps, be viewed as a semi-classical manifestation of some deep, fundamental principle that has its origins in the (yet-to-be-understood) quantum nature of spacetime. (For a review on the holographic principle, see [1]. For a discussion on how it might connect with quantum gravity, see [2].) The essence of this holographic paradigm is that the entropy (or, equivalently, the accessible information) in a given region of spacetime should have a precise limit which can be formulated in terms of the " area " of a suitably defined surface. [For a d-dimensional space-time, this " area " would measure the volume of some (d − 2)-dimensional hypersurface.] In the " conventional " (i.e., flat-space) quantum world, such a bound has contradictory implications ; for instance, quantum field theory predicts that the entropy will vary extensively with the volume of the applicable region. Nonetheless, such expectations need not persist once gravitational interactions have been " turned on ". Indeed, the most strongly gravitating of objects — black holes — have a clear thermodynamic interpretation [3,4] which necessarily implies that S max ∝ A [5]; with S max being the maximal amount of entropy that can be stored in a region bounded by a surface of area A. Analogous bounds can be extrapolated to other scenarios (both strongly and weakly gravitating) by way of the so-called covariant entropy bound [6]. So far, no violations of this bound are known given reasonable conditions on what constitutes physically allowable matter [1]. Although not entropy bounds per se, dualities between (bulk) gravitational and (boundary) field theories provide another elegant realization of the holographic principle. The most notable of these being the duality that is known to exist between an anti-de Sitter spacetime 1