String Theory Demystified

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CHAPTER 8 Compactifi cation and T-Duality 159 T-duality is a symmetry which exists between different string theories. This symmetry relates small distances in one theory to large distances in another, seemingly different theory and shows that the two theories are in fact the same theory expressed from different viewpoints. This is an important recognition; before T-duality was discovered it was believed that there were fi ve different string theories, when in fact they were all different versions of the same theory that could be related to one another by transformations or dualities. One can transform between small and large distances when considering the compactifi ed dimension in one theory, and arrive at another dual theory. This is the essence of T-duality. We will see later that other dualities exist in string theory as well. T-duality relates type IIA and type IIB string theories, as well as the heterotic string theories. It applies to the type of compactifi cation that we have been studying in this chapter, namely the compactifi cation of a spatial dimension to a circle of radius R. The transformation that is used in T-duality is to transform the radius to a new large radius R ′ which is defi ned by the exchange ′ ↔ ′ α R (8.29) R The T-duality transformation also exchanges winding states characterized by a winding number n with high-momentum states in the other theory (Kaluza-Klein excitations). That is, n↔ K (8.30) The symmetry of T-duality, described by these exchanges, makes its appearance in the mass formula [Eq. (8.27)], which we reproduce here: 2 2 ⎛ ′ = ⎞ ⎛ + ⎞ ⎝ ′ ⎠ ⎝ ⎠ α 2 nR K m α R Now exchange R′ ↔ α ′ / R and n↔ K, then: We also have: K R + 2( N + N ) −4 R L n nR → ′ ( R′ ) = ′ α α ′ (8.31) ( ) nR K R ′ → α ′ ′ α α ′ T-Duality for Closed Strings K = R′ (8.32)
160 String Theory Demystifi ed So we see that the mass formula Eq. (8.27) is invariant under the exchange R′ ↔ α ′ / R and n↔ K. It assumes the form 2 2 ⎛ ′ ⎞ ⎛ ′ = ⎝ ⎜ ′ ⎠ ⎟ + ⎞ ⎝ ′ ⎠ α 2 nR K m α R + 2( N + N ) −4 R L That is, it keeps the same form but now with the new radius R ′ . That’s the math of the transformation. The physics is that if we started with a theory with a small compactifi ed dimension R, we have transformed to a dual theory with a large extra dimension R ′ . What this means for the string is that a string in a type IIA theory (with small compactifi ed dimension), which winds around the small compact dimension (with winding states) is dual to a string in type IIB theory (with the dimension transformed to a large dimension of radius R ′ ), which has momentum along that dimension. Each time the string in type IIA theory winds around the compact dimension, this corresponds to increasing the momentum in type IIB theory by one unit. 25 25 Now let’s examine how p and p , and by extension α and α , transform L R 0 0 under this symmetry. Recall Eq. (8.22) that states p 25 = L nR K ′ R + α Now exchange R′ ↔ α ′ / Rand n↔ K. We fi nd 25 K ⎛ ′ ⎞ n K nR p → L ′ ⎝ ⎜ R′ ⎠ ⎟ R R + ′ ( ′ ) = ′ + α ′ α α α ′ 25 So, p maintains the same form under the transformation—that is, it is invariant. L 25 25 25 We can indicate this by writing p = p . Now consider how p transforms under L L R the exchange R′ ↔ α ′ / R and n↔ K: 25 K nR pR = − R ′ 25 n K ⇒ pR → ( ′ R R′ ) − α ⎛ α ′ ⎞ ′ α ′ ⎝ ⎜ ⎟ = α ′ ⎠ ′ − nR K α R′ 25 25 That is, p =− p . This tells us that the exchange R R R′ ↔ α ′ / R and n↔K is equivalent to transformations applied to the zero-modes: 25 25 25 25 α →α and α →−α (8.33) 0 0 0 0
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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 3 fl at spac
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CHAPTER 1 Introduction 5 we say tha
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CHAPTER 1 Introduction 7 initial an
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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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Page 124 and 125: CHAPTER 5 Conformal Field Theory Pa
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Page 142 and 143: CHAPTER 7 RNS Superstrings The real
Page 144 and 145: CHAPTER 7 RNS Superstrings 129 EXAM
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Page 168 and 169: CHAPTER 8 Compactifi cation and T-D
Page 182 and 183: CHAPTER 9 Superstring Theory Contin
Page 202 and 203: CHAPTER 10 A Summary of Superstring
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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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CHAPTER 13 D-Branes 233 D-brane 1 a
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CHAPTER 13 D-Branes 235 Tachyons ca
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CHAPTER 13 D-Branes 237 We consider
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CHAPTER 14 Black Holes Black holes,
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CHAPTER 14 Black Holes 241 This equ
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CHAPTER 14 Black Holes 243 Here, G
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CHAPTER 14 Black Holes 245 When the
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CHAPTER 14 Black Holes 247 hole squ
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CHAPTER 14 Black Holes 249 Entropy
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CHAPTER 14 Black Holes 251 Recallin
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CHAPTER 14 Black Holes 253 Now, the
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CHAPTER 15 The Holographic Principl
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CHAPTER 16 String Theory and Cosmol
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Final Exam 1 1. Consider the lagran
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Final Exam 281 0 0 26. Find a simpl
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Final Exam 283 68. When a single sp
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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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