Strings and Quantum Gravity








RESEARCH > (A)dS/CFT Correspondence and Cosmology > The Penrose limit

The plane-wave limit of spacetime suggested by Penrose [49] a while ago and its supersymmetric generalization has recently attracted a lot of attention in conjunction with the AdS/CFT correspondence. This limit leads to a more direct understanding of the AdS/CFT correspondence because the background is simple enough for the sigma model (string theory) to be exactly solvable [50]. Other exactly solvable models studied earlier can be shown to be special cases of the Penrose limit [51].

On the other hand, the issue of holography is blurred in this limit, because the background is a plane wave or even flat Minkowski space on which no holographic principle exists. This is in contrast to the well-understood holography in the original AdS space whose limit we are taking. Since the AdS/CFT correspondence seems to have survived the limit, one is tempted to conjecture the existence of a holographic screen on which the CFT resides.

Dr. Siopsis considered the simplest case of AdSD+1 space whose Penrose limit is flat (D+1)-dimensional Minkowski space (MD+1)  [52]. This may be viewed as a special case of the Penrose limit of AdSD+1 x Sq in which we boost along a geodesic with vanishing spin in the q-dimensional sphere Sq. One may define string theory on this Minkowski space and study the CFT correspondence in a standard fashion. Since no holographic principle applies to Minkowski space, it is evident that an additional condition is needed for some kind of holography to emerge. Dr. Siopsis introduced such a condition by restricting the Hilbert space to the states that are scale-invariant. This is a constraint leading to the treatment of scale transformations as gauge transformations. In the resulting theory, one needs to fix the gauge. This can be done by restricting to a D-dimensional hypersurface of the Minkowski space MD+1. This hypersurface is arbitrary as long as it cuts all gauge orbits exactly once and all such choices are gauge-equivalent. Dr. Siopsis considered explicitly a hyperboloid (dSD space; note that supersymmetry is broken on dSD, but it gets restored in the flat space limit) and showed that the Poincaré algebra of Minkowski space turns into a SO(D,2) algebra once the Poisson brackets are replaced by Dirac brackets. In the flat-space limit of the hyperboloid, the SO(D,2) group turns into the conformal group of the resulting D-dimensional Minkowski space MD. This is then the ``holographic screen'' on which information on the corresponding CFT resides.

Thus, a conformal field theory was recovered on the D-dimensional Minkowski space which was obtained as the flat-space limit of a dSD hypersurface in the original (D+1)-dimensional Minkowski space. The restriction on dSD induced the replacement of Poisson brackets by Dirac brackets. Consequently, the (D+1)-dimensional Poincaré algebra turned into a SO(D,2) algebra which was identified with the conformal group on the D-dimensional Minkowski space. It should be emphasized that this ``holographic screen'' is merely a gauge choice and therefore arbitrary.

Dr. Siopsis extended the above results by considering the more general case of a spinning particle [53]. The Penrose limit of AdS x S along such geodesics is a Cahen-Wallace space (pp-wave). The ``holographic screen'' is similarly obtained as a gauge-fixing condition upon restricting the allowed operators defined on the Cahen-Wallace space.

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