String Theory Demystified

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Final Exam 283 68. When a single spatial dimension is compactifi ed on a circle of radius R, there are extra terms in the expression used to determine the mass of the state. What factors are these? Is the mass increased or decreased? 69. What is the effect of compactifi cation on the center of mass momentum? 70. Consider the compactifi cation of a single spatial dimension to a circle of radius R and consider the limiting behavior as R → 0. Why do the winding states go to the continuum limit? 71. How does T-duality relate winding and momentum states? 72. How does T-duality relate distance scales in different theories? 73. What string theories are related by T-duality? 74. A certain string theory has orientable strings. What does this mean? 75. What is the mass of a tachyon state in bosonic string theory? 76. How is Type I string theory different from all the other superstring theories? 77. What symmetry groups are associated with heterotic string theory? 78. How does the Kasner metric describe the behavior of space-time as the universe evoles? 79. What is the effect of the dilaton fi eld on the Kasner metric? 80. What is the self-dual point? 81. In the Horava-Witten model, how are the uncompactifi ed dimensions laid out? 82. What condition on the beta function of the dilaton fi eld must be satisfi ed for scale invariance? 83. What property of space-time is implied by conformal invariance as related to the dilaton fi eld? 84. How is the vanishing of the beta function related to general relativity? 85. Einstein’s relativity allows for the description of “extreme” black holes that carry electric charge. How is this extended in string theory? 86. Let K be the Kaluza-Klein excitation and n the winding number. How is a state ( Kn , ) in a theory with a compactifi ed dimension of radius R transformed under T-duality? 87. How are the number operators transformed under T-duality? 88. A state has mass m. How is the mass changed under T-duality? 89. How are the equations of motion related for the compactifi ed dimension under T-duality between a theory and its dual?
284 <strong>String</strong> <strong>Theory</strong> Demystifi ed 90. Consider an open bosonic string under a T-duality transformation. What happens to the momentum of the dual string along the compactifi ed dimension? 91. The string momenta along a certain spatial direction are found to be fractional. How are the fi elds described in the dual theory? µ 92. What type of boundary condition is ∂ X 0 = 0? σ σ= , π 93. In type II A superstring theory, what dimensions are allowed for Dp-branes? 94. If the value of the fi elds X µ are specifi ed on the boundary, what type of problem is being posed? In Probs. 95–96 let D = 10 and suppose that there is a pure electric background fi eld F = E. 0 i i 95. What is the form of the Born-Infeld action? 96. What is the maximum value of the electric fi eld? 97. How is the dilaton fi eld φ related to string coupling gs? The fi nal problems will be very challenging for many readers. For problems 98, 99, and 100, suppose that you have closed strings with periodic boundary 25 25 conditions for µ = 01 , ,..., 24but X ( στ , ) =− X ( σ+ , τ) s . Use the light-cone gauge (based on Polchinski 1.9). 98. How does the modal expansion change for X 25 as opposed to the usual modal expansion for closed bosonic strings? 99. Using Π µ + ∞ p µ 1 1 2 = ∂ X and ( n − ) = − ( − ) τ ∑ θ 2θ 1 fi nd the hamiltonian 24 8 s n= 1 s s i i i i from H = d ′ + X X ′ p ′ ∂ ∂ ⎛ 1 ⎞ + ∫ σ ⎝ ⎜ 2παΠΠ 4πα 0 σ σ 2πα ⎠ ⎟ . 100. What is the mass spectrum? Note that the level-matching condition is still satisfi ed.
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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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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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CHAPTER 13 D-Branes One of the most
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CHAPTER 13 D-Branes 223 The Space-T
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CHAPTER 13 D-Branes 225 Once again
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CHAPTER 13 D-Branes 227 and ( X i i
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CHAPTER 13 D-Branes 229 fi rst exci
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CHAPTER 13 D-Branes 231 a set of D-
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Page 254 and 255: CHAPTER 14 Black Holes Black holes,
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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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