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Now since R = ( gs N )1/ 4 we can write N dof =
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In ve-dimensions, the Newton gravitational constant is G5 = Hence we nd that N dof = A G5
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This agrees with the holographic principle, and is the same as the result obtained for black holes with the exception of the factor of 1/4.
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In this section we describe connections between the supergravity theory of the bulk and the SYM of the boundary. We can convert between bulk variables and SYM variables as follows. Let ESYM be energy on the boundary and M be the energy in the bulk. They are related as ESYM = RM Temperature is related in the same way: TSYM = RT where T is the temperature in the bulk. Now consider a thermal Yang-Mills state with temperature TSYM. The entropy is S = N 2 (TSYM )3 A thermal state of temperature TSYM corresponds to an AdS Schwarzschild black hole at the center of the AdS ball.
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CHAPTER 15 The Holographic Principle
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Using TSYM = RT and R =
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( gs N )1/ 4 we obtain (TR)3 = Sgs2 R8
Now if we take S = A/4G then we nd
3 TSYM =
A R3
Now we regulate the SYM so that the maximum TSYM is 1/d. Then we nd the maximum area to be Amax = R3 3
Regulation of the super Yang-Mills theory on the boundary gives a holographic description with one bit per Planck area. An interesting result derived by Susskind and Witten is the IR-UV connection. This relates IR divergences in the bulk to UV divergences on the boundary. Consider a string in the bulk that ends on the boundary. The ends of the string correspond to a point charge in the Yang-Mills theory. Now, just thinking back to the self-energy of an electron, you will realize that a point charge in the Yang-Mills theory has a divergent in nite self-energy. This is an UV divergence. The divergence of the bulk string is proportional to 1/d, while d plays the role of a short distance regulator for UV divergence in SYM theory. The energy of the string is linearly divergent at the boundaries. Since this divergence is softer, we say that it is an IR divergence. The propagator for a particle of mass m in the bulk is given by =
X1 X 2
where we have relgulated the area using A R3 and 1. Super Yang-Mills theory is a conformal eld theory. Remember Chap. 5 We learned how to calculate operator product expansions there. For super Yang-Mills theory: Y ( X1 )Y ( X 2 ) = p X1 X 2
You can see that you can transform between these two expressions. What this means is that a propagator for a particle of mass m in the bulk can be transformed into a power law in the conformal eld theory on the boundary.
String Theory Demysti ed
Summary
In this chapter we provided a brief and heuristic introduction to two interesting ideas that have sprung from string theory: the holographic principle and the AdS/ CFT correspondence. These two ideas are related. The holographic principle tells us that for an enclosed volume, the informational content of the volume can be described by an equivalent theory that lives on the bounding surface area. This notion is codi ed in black hole mechanics where the entropy of the black hole is proportional to the area of the horizon, not the volume it encloses. The AdS/CFT correspondence describes a ve-dimensional universe where ve-dimensional supergravity in the bulk is equivalent to a super Yang-Mills conformal eld theory on the boundary.
Quiz
A solution of supergravity gives the metric for a D-brane as: ds 2 = F ( z )(dt 2 dx 2 ) F ( z ) 1 dz 2 ags N where F ( z ) = 1 + 4 z
1/ 2
ags N z4
1. Find an expression for F(z) in the limit
2. Using your answer to Prob. 1, nd a new expression for the metric. 3. The holographic principle can be best described by (a) The informational content of a region is encoded in its volume. (b) The informational content of a region can be described entirely by the surface area. (c) Fields living in the bulk are not equivalent to elds living on the bounding surface. 4. In AdS/CFT correspondence, the number of degrees of freedom available to the super Yang-Mills theory on the boundary is (a) Independent of the AdS geometry. (b) Related to the string coupling strength only. (c) Is related to the string coupling strength and the fundamental string length. (d) Is related to the fundamental string length only.
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