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A Qualitative Description of AdS/CFT Correspondence
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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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String Theory Demysti ed
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The Holographic Principle and M-Theory
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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.
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CHAPTER 15 The Holographic Principle
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But, gravity can propagate into the bulk. In the bulk, which is the interior volume of the AdS sphere, gravity is the only interaction. Inside the ball of the AdS geometry, the theory is supergravity. We won t get into supergravity in this book but you can look it up on the arXiv if interested in learning about it. The conformal theory that describes particles and their interactions is supersymmetric and is called super Yang-Mills theory or SYM for short. The gauge group for SYM is SU(N). So the AdS/CFT correspondence can be framed as follows: There is a super Yang-Mills theory with SU(N) on the surface of the ball. There is bulk supergravity in the interior of the ball. In string theory, the number of degrees of freedom for the SYM is constrained by three factors: The fundamental string length The string coupling The curvature of AdS space The number of degrees of freedom for SYM is N2 since the gauge group is SU(N) and it has a gauge coupling gYM. The constraint on N is quanti ed in the following relationship: R=
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( gs N )1/ 4
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The gauge-coupling is related to the string-coupling constant as: gYM 2 = gs Now we would like to introduce a cutoff in the bulk. We divide up the sphere into little cells such that the total number of cells in the sphere is 3 for some cell d. That is, We cut off the information storage capacity by replacing the continuum of space by cells of size d. There is a single degree of freedom in each cell. With the total number of degrees of freedom for the SYM theory proportional to N2, we nd that the total number of degrees of freedom with the cutoff is N dof = N2 N2 =A 3 R 3
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