String Theory Demystified

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CHAPTER 15 The Holographic Principle 259 • 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 ∼ N 2 since the gauge group is SU(N) and it has a gauge coupling g YM . The constraint on N is quantifi ed in the following relationship: R= ( g N) s s / 14 The gauge-coupling is related to the string-coupling constant as: 2 g = g YM s 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 ∼ δ −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 N 2 , we fi nd that the total number of degrees of freedom with the cutoff is N dof N A N 2 = = 3 δ R 2 3
260 <strong>String</strong> <strong>Theory</strong> Demystifi ed Now since R= ( g N) s s / 14 we can write N dof 5 AR = g 8 2 s s In fi ve-dimensions, the Newton gravitational constant is Hence we fi nd that G N 5 g = 5 R dof = 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. More Correspondence 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 E SYM be energy on the boundary and M be the energy in the bulk. They are related as Temperature is related in the same way: 8 2 s s A G 5 E RM SYM = T RT SYM = where T is the temperature in the bulk. Now consider a thermal Yang-Mills state with temperature T SYM . The entropy is S = N TSYM 2 3 ( ) A thermal state of temperature T SYM corresponds to an AdS Schwarzschild black hole at the center of the AdS ball.
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Accounting Demystified Advanced Cal
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Copyright © 2009 by The McGraw-Hil
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Contents vii BRST in String Theory-
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Contents ix CHAPTER 14 Black Holes
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PREFACE String theory is the greate
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Preface xiii time as this one to he
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CHAPTER 1 Introduction General rela
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CHAPTER 1 Introduction 13 Figure 1.
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CHAPTER 1 Introduction 15 • Type
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CHAPTER 1 Introduction 17 Close up,
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CHAPTER 1 Introduction 19 5. The mi
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CHAPTER 2 The Classical String I: E
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CHAPTER 2 Equations of Motion 23 To
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CHAPTER 2 Equations of Motion 25 No
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CHAPTER 2 Equations of Motion 31 Th
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CHAPTER 2 Equations of Motion 33 is
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CHAPTER 2 Equations of Motion 37 ho
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CHAPTER 2 Equations of Motion 39 Th
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CHAPTER 2 Equations of Motion 41 wh
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CHAPTER 2 Equations of Motion 43 In
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CHAPTER 2 Equations of Motion 47 We
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CHAPTER 2 Equations of Motion 49 In
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CHAPTER 3 The Classical String II:
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CHAPTER 3 Symmetries and Worldsheet
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CHAPTER 4 String Quantization At th
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CHAPTER 4 String Quantization Here,
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CHAPTER 4 String Quantization That
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CHAPTER 4 String Quantization Using
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CHAPTER 4 String Quantization expec
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CHAPTER 4 String Quantization † A
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CHAPTER 4 String Quantization To ob
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CHAPTER 4 String Quantization lower
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CHAPTER 4 String Quantization this)
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CHAPTER 4 String Quantization Norma
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CHAPTER 5 Conformal Field Theory Pa
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CHAPTER 6 BRST Quantization So far
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CHAPTER 6 BRST Quantization 119 Cal
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CHAPTER 6 BRST Quantization 121 The
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CHAPTER 6 BRST Quantization 123 To
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CHAPTER 6 BRST Quantization 125 The
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CHAPTER 7 RNS Superstrings The real
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CHAPTER 7 RNS Superstrings 129 EXAM
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CHAPTER 7 RNS Superstrings 131 SOLU
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CHAPTER 7 RNS Superstrings 133 In t
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CHAPTER 7 RNS Superstrings 135 Now,
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CHAPTER 7 RNS Superstrings 141 OPEN
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CHAPTER 7 RNS Superstrings 151 3. A
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CHAPTER 8 Compactifi cation and T-D
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CHAPTER 9 Superstring Theory Contin
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CHAPTER 10 A Summary of Superstring
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CHAPTER 11 Type II String Theories
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CHAPTER 12 Heterotic String Theory
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Page 254 and 255: CHAPTER 14 Black Holes Black holes,
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Page 264 and 265: CHAPTER 14 Black Holes 249 Entropy
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Page 304 and 305: Quiz Solutions 289 Chapter 11 1. c
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Page 312 and 313: Books References Becker, K., M. Bec
Page 314 and 315: |NS〉 ⊕ |NS〉 Sector, 203 |NS
Page 316 and 317: INDEX 301 Cyclic model, 277 Cylinde
Page 318 and 319: INDEX 303 Mechanics, quantum. See Q
Page 320 and 321: INDEX 305 Schwarzschild metric, 242
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String Theory Demystified

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