String Theory Demystified

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CHAPTER 13 D-Branes 235 Tachyons can actually describe D-brane decay, so let’s say a little bit about that since it shows how they can fi t into the overall theory. Consider the action for a scalar fi eld. Suppose that: D µ S = ∫ d x( ∂ ϕ∂ ϕ + λϕ ) µ 2 Quadratic terms in the potential identify mass terms. In the above, we have: λ = m 2 Now, notice that the quadratic terms indicates a harmonic potential. We can use this to see why the presence of a tachyon indicates an instability of the vacuum. If m 2 > 0, then the potential V( ϕ ) opens upward, with the minimum located at ϕ = 0. On the other hand, if m 2 < 0 , the parabola opens downward. This means that the point ϕ = 0 is unstable. It’s like placing a ball at the top of a hill—a small perturbation will cause it to roll down the hill. These potentials are illustrated in Fig. 13.3. We can expand the potential energy V( ϕ ) about its critical points, which tell us where the maxima and minima are, to determine its behavior. To second order it’s going to assume the form V(f ) Tachyons and D-Brane Decay * * V( ϕ) = V( ϕ ) + λ( ϕ− ϕ ) + 2 m 2 > 0 m 2 > 0 Figure 13.3 A comparison of the potential for m 2 V(f ) f f > 0 and m 2 < 0 .
236 <strong>String</strong> <strong>Theory</strong> Demystifi ed where ϕ * * 2 is a critical point. The term λϕ ( − ϕ) is quadratic so is a mass term. In the case of a D-brane, the leading term is given by the tension T. If a tachyon lives on the D-brane, then V( ϕ) T α ϕ 1 = − + 2 ′ 2 So, if the potential strays away from ϕ = 0 this shows that the D-brane is losing energy. What happens is the D-brane decays away into closed string states. Generally speaking this is an artifact of bosonic string theory. In superstring theory there are stable D-brane states. However, in superstring theory you can have an anti-D-brane, which can be coincident with a D-brane. Like particles and antiparticles, they annihilate. This is because there are tachyon states stretched between them. We consider a simple example. The tachyon potential for a D1-brane coincident with an anti-D1-brane is V( ϕ) ( ) λ 2 = ϕ −ϕ0 2 * The fi rst step is to fi nd the critical points V ′ ( ϕ ) = 0. The fi rst derivative is Setting this equal to 0 we fi nd The second derivative of the potential is 2 2 ( 0 ) 2 2 V ′ ( ϕ) = 2λ ϕ −ϕ ϕ * ϕ = , ± ϕ 0 0 2 2 2 V ′′ ( ϕ) = 2λ( ϕ −ϕ )+ 4λϕ 0 Expanding the potential to second order about ϕ = a is 1 V = V( a) + V′ ( a)( ϕ− a) + V′′ ( a)( ϕ− a) + 2 2
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Accounting Demystified Advanced Cal
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Copyright © 2009 by The McGraw-Hil
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For more information about this tit
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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 5 we say tha
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CHAPTER 1 Introduction 9 a spin-2 b
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CHAPTER 1 Introduction 11 So if α
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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 29 ct
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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 35 Si
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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 117 It
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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 137 We c
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CHAPTER 7 RNS Superstrings 139 We o
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CHAPTER 7 RNS Superstrings 141 OPEN
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CHAPTER 7 RNS Superstrings 143 If w
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CHAPTER 7 RNS Superstrings 145 The
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CHAPTER 7 RNS Superstrings 147 From
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CHAPTER 7 RNS Superstrings 149 The
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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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Page 200 and 201: CHAPTER 9 Superstring Theory Contin
Page 202 and 203: CHAPTER 10 A Summary of Superstring
Page 210 and 211: CHAPTER 11 Type II String Theories
Page 222 and 223: CHAPTER 12 Heterotic String Theory
Page 236 and 237: CHAPTER 13 D-Branes One of the most
Page 238 and 239: CHAPTER 13 D-Branes 223 The Space-T
Page 240 and 241: CHAPTER 13 D-Branes 225 Once again
Page 242 and 243: CHAPTER 13 D-Branes 227 and ( X i i
Page 244 and 245: CHAPTER 13 D-Branes 229 fi rst exci
Page 246 and 247: CHAPTER 13 D-Branes 231 a set of D-
Page 248 and 249: CHAPTER 13 D-Branes 233 D-brane 1 a
Page 252 and 253: CHAPTER 13 D-Branes 237 We consider
Page 254 and 255: CHAPTER 14 Black Holes Black holes,
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Page 264 and 265: CHAPTER 14 Black Holes 249 Entropy
Page 266 and 267: CHAPTER 14 Black Holes 251 Recallin
Page 268 and 269: CHAPTER 14 Black Holes 253 Now, the
Page 270 and 271: CHAPTER 15 The Holographic Principl
Page 280 and 281: CHAPTER 16 String Theory and Cosmol
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Chapter 1 1. b 6. a 2. a 7. c 3. c
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Quiz Solutions 287 3. i( − ) 4.
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Quiz Solutions 289 Chapter 11 1. c
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1. P 2. E µ 2 ν 1 ( X′ ) X µ
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Final Exam Solutions 293 27. Supers
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Final Exam Solutions 295 73. Types
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Books References Becker, K., M. Bec
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|NS〉 ⊕ |NS〉 Sector, 203 |NS
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INDEX 301 Cyclic model, 277 Cylinde
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INDEX 303 Mechanics, quantum. See Q
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INDEX 305 Schwarzschild metric, 242
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String Theory Demystified

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