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The framework of the holographic principle which comes out of string/M-theory is known as AdS/CFT (anti-de Sitter/conformal eld theory) correspondence. We can quantitatively describe the space-time using AdS space in ve dimensions. The ve-dimensional AdS model has a boundary with four dimensions that looks like at space with three spatial directions and one time dimension. The AdS/CFT correspondence involves a duality, something we re already familiar with from our studies of superstring theories. This duality is between two types of theories: Five-dimensional gravity Super Yang-Mills theory de ned on the boundary By super Yang-Mills theory we mean theory of particle interactions with supersymmetry. The holographic principle comes out of the correspondence between these two theories because Yang-Mills theory, which is happening on the boundary, is equivalent to the gravitational physics happening in the ve-dimensional AdS geometry. So the Yang-Mills theory can be colloquially thought of as a hologram on the boundary of the real ve-dimensional space where the ve-dimensional gravitational physics is taking place.

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Now let s make our description more quantitative. In the nal chapter of the book we discuss stringy cosmology. There we will encounter a model of space-time that has sprung out of string/M-theory that might in fact describe our actual universe. That same model has a nice application in the topic of this chapter as well. The model is a ve-dimensional AdS space. It can be described as follows. We start with a ve-dimensional AdS space. In a nutshell, this is a fourdimensional spatial ball and an in nite time axis. The radius of the ball is 0 r < 1. The radius of curvature is denoted by R, and we lump the remaining spatial dimensions together into a unit three-sphere denoted by . The metric which describes the AdS is written as ds 2 = R2 [(1 + r 2 )dt 2 4 dr 2 4r 2 d 2 ] (1 r 2 )2

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Note that there are different, equivalent ways to write this metric which you might encounter elsewhere. AdS space has negative curvature and acts like a cavity of size R with re ecting walls. Light or objects and re ect off the boundary and return to the center (see The Illusion of Gravity by Juan Maldacena in Scienti c American, November 2005, for a nice popular level description of AdS). For us, we are interested in superstring theory. The number of space-time dimensions in superstring theory is D = 10. So the complete space is AdS S 5 where S5 is a unit ve-sphere containing the remaining dimensions from string theory. If we denote the extra ve coordinates by y5 they are incorporated into our metric by 2 adding a term Rdy5 . We can imagine compactifying these dimensions to a very small size so that they can be effectively ignored. So the universe can be effectively treated as the ve-dimensional bulk which is the interior of the sphere and the boundary which is the surface. The surface has three spatial dimensions and time. In the M-theory picture, the world we know is in essence a shadow or hologram living on the boundary of a larger dimensional universe. The physics is divided as follows: The boundary conformal theory lives on the surface of the sphere at x = 1. These are the particles and interactions of the standard model, plus any supersymmetric extension of it. Gravity is everywhere.

CHAPTER 2: Blast-Off!

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