String Theory Demystified

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CHAPTER 8 Compactifi cation and T-Duality 157 Similarly, using L 0 = 1 together with Eq. (8.17) we obtain ′ = ′ α 2 α 25 25 m p p + 2N −2 2 2 L L L (8.21) Now using Eq. (8.8) together with Eqs. (8.9) and (8.10) we can write p 25 = L Which of course allows us to compute nR K ′ R + α 2 2 25 nR K nR K pL R R 2 ( ) = ′ + ⎛ ⎞ ⎝ ⎜ ⎠ ⎟ = ⎛ ⎞ ⎛ ⎞ ⎝ ⎜ ′ ⎠ ⎟ + α α ⎝ ⎜ ⎠ ⎟ 25 and similarly p = ( K/ R) − ( nR/ α ′ ) so that R 2 2 25 K nR nR K pR R R 2 ⎛ ⎞ ( ) = − ⎝ ⎜ ′ ⎠ ⎟ = ⎛ ⎞ ⎛ ⎞ ⎝ ⎜ ′ ⎠ ⎟ + α α ⎝ ⎜ ⎠ ⎟ This allows us to obtain the sum and difference formulas: 2 2 + 2 − 2 (8.22) nK α ′ (8.23) nK α ′ (8.24) 25 nR K p p L R R 2 25 2 2 2 ⎡⎛ ⎞ ⎛ ⎞ ⎤ ( ) + ( ) = 2 ⎢ ⎝ ⎜ ′ ⎠ ⎟ + ⎝ ⎜ ⎠ ⎟ ⎥ ⎣⎢ α ⎦⎥ (8.25) 25 nK p p L R 2 25 2 ( ) − ( ) = 4 α ′ (8.26) Using Eqs. (8.25) and (8.26) we can add Eqs. (8.20) and (8.21) to obtain 2 2 ′ = ⎛ ⎞ ⎝ ⎜ ′ ⎠ ⎟ + ⎛ 2 nR K α m ⎞ ⎝ ⎜ ⎠ ⎟ + 2( NR + NL) −4 (8.27) α R and subtracting Eqs. (8.21) from (8.20) gives N − N = nK (8.28) R L So notice we have extra terms in the formulas for mass [Eq. (8.27)] and the level matching condition [Eq. (8.28)] as compared to the formulas introduced for the
158 <strong>String</strong> <strong>Theory</strong> Demystifi ed bosonic string in Chap. 2. The extra terms are due to two components, the Kaluza- Klein excitations and the winding states of the string. The Kaluza-Klein excitations can be regarded as particles and so cannot be thought of as due to strings in a general sense. However the winding excitations can only come from strings, because only strings can wrap around a compactifi ed extra dimension. Now let’s look at the mass formula in Eq. (8.27) together with the relations for the momenta in Eqs. (8.22) and (8.25). Our task here is to consider limiting behavior. First we consider the case where R →∞. In this limit, the momentum goes to the continuum limit and the Kaluza-Klein excitations disappear. It is simple to show that pL = pR and hence the winding state 1 25 25 w= ( p − p R L R )→0as →∞ 2 The center of mass momentum Eq. (8.8) returns to the nonquantized, continuous momentum of the noncompactifi ed case. Now let’s think about the opposite limit where R → 0. You might also expect that this is like returning to the noncompactifi ed case. After all, taking the limit R → 0 is like making the extra dimension go away. In quantum fi eld theory we might expect the fi elds to completely decouple from that unseen extra dimension. However, things don’t quite work this way in string theory. As R → 0, we fi nd that the Kaluza-Klein modes become infi nitely massive and decouple from the theory. Since these can be regarded as particle states, maybe this isn’t so surprising. What’s left behind for the center of mass momentum are the winding states. First note that as R → 0 we obtain Hence p 25 25 (or to − pR p =−p R L → 0 but the winding term behaves in the following way: 1 25 25 25 w= ( p − p p R L R )→ as →0 L 2 if you like). Now the winding states, rather than the momentum states, form a continuum of states. This should not be so surprising, as R → 0 the circle gets smaller and smaller. So it gets easier and easier to wrap a string around it—that is, it costs less energy. When the circle is very small it doesn’t require a lot of energy to wrap the string around it. So you see as the radius gets very large or very small there is a trade-off between winding states and momentum. This trade-off leads us to a discussion of T-duality, the topic of the next section.
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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 122 and 123: 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
Page 210 and 211: 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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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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String Theory Demystified

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