AdS/CFT intuiton

Main post: http://psiepsilon.wordpress.com/2013/07/20/intuition-behind-adscft/ Many of us may have heard of the AdS/CFT correspondence. $$ \mbox{ } $$ A \(D\) - dimensional string theory in Anti-de-Sitter (AdS) space is exactly equivalent to \(D-1\) dimensional Conformal field theory (CFT), such as Quantum Yang-Mills theory, etc. $$ \mbox{ } $$ That just sounds a bit crazy right? How can a string theory be equivalent to a mere CFT, of all things? $$ \mbox{ } $$ But in reality, the confusion only arises from the way it is phrased. It should be phrased in terms of the Holographic principle. Then you ask, "What is this Holographic principle?". $$ \mbox{ } $$ The information stored inside a reigon is completely described by the information on its boundary. $$ \mbox{ } $$ Ugh...... The information inside a water bottle (which is the information in the water) is equivalent to the information on the bottle's surface itself, which is the information in plastic? Is this alchemy, or something? $$ \mbox{ } $$ But holography is a law of nature and there's nothing wrong about it. Let us start with some obvious examples. $$ \mbox{ } $$ 1. Stokes's theorem $$ \mbox{ } $$ Ok, consider a field originating from a certain point. To make things simple, let us say its a vector field, and it is actually the field of forces (field of the electromagnetic force, as opposed to an electromagnetic field, but the latter would work too). Now, let us say there is some 2-dimensional surface \(S\) , with a boundary curve \( C \) . The work done by the force field along this curve, is given by: $$\oint \vec f\cdot\mbox{d}\vec r$$ This really just follows from \( \mbox{d} W=\vec F\cdot\mbox{ d} \vec r \). You may already start to see where this is going! What is the flux of the curl of the force field through the surface? We know that it is, equal to: $$\iint_S\left(\nabla\times\vec f\right)\cdot\hat n \mbox{ d}S $$ (Admittedly, it is foolish to say the integral of the "curl of the force field", because it has a very limited physical meaning, unless you use stokes' theorem). $$ \mbox{ } $$ Now what does Stokes' theorem, more specifically, the Kevin-Stokes' theorem, say? $$\oint \vec f\cdot\mbox{d}\vec r = \iint_S\left(\nabla\times\vec f\right)\cdot\hat n \mbox{ d}S $$ In other words, the flux of the curl through the surface, is exactly equivalent to the work done on the boundary, which is, the curve! . $$ \mbox{ } $$ 2. Gauss's theorem Consider the sum of the divergences within a volume \( V \). Then, Gauss's theorem tells us that that is equivalent to : $$\iint_S\vec f \cdot \hat n \mbox{ d}S = \iiint_V \nabla\cdot\vec f\mbox{ d}V$$ I.e. a property of the reigon is equivalent to a property of the surface. $$ \mbox{ } $$ 3. Black holes $$ \mbox{ } $$ Consider two observers, observer A, and observer B, . Observer B is falling into a black hole, whereas observer A is outside. Then, for simplicity, say, the black hole, is Schwarzschild, so that the time dilation is then: $$\frac{\mbox{d}t}{\mbox{d} \tau} = \frac1{\sqrt{1-\frac{r_s}r}}$$ Which is an obvious result from the Schwarzschild metric. $$ \mbox{ } $$ Then, this means that Observer A is going to observe that Observer B's time scales get shrunk, so that Observer B will appear to move towards the black hole slower, and slower, and finally stop at the event horizon. However, for Observer B himself, everything will appear normal, from his reference frame. I.e. what is going on inside the black hole (as observer B observes it) seems to go on on the surface of the black hole (the event horizon, of the black hole, now you know why it's called an "event horizon".) for Observer A. This is also a resolution to the Hawking information Paradox. The information is encoded on the event horizon, which is why it doesn't disappear. $$ \mbox{ } $$ So, this just means that the information inside a reigon is completely encoded on to its boundary. So, this means, that,... ? $$ \mbox{ } $$ It is the Holographic principle. $$ \mbox{ } $$ Now, what about AdS/CFT? $$ \mbox{ } $$ \( D\) - dimensional Anti-de-Sitter space has a \(D-1\)-dimensional boundary, which is governed by a Conformal field theory, and the Anti-de-Sitter space itself, is governed by a string theory. $$ \mbox{ } $$ So, in other words, $$ \mbox{ } $$ String theory in Anti-de-Sitter space is exactly equivalent to a Conformal Field Theory on its boundary.