String Theory Demystified

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CHAPTER 7 RNS Superstrings 143 If we choose the R sector, then following the open string case On the other hand, if we choose the NS sector, then r = 0, ± 1, ± 2,… (7.34) 1 3 r =± , ± , (7.35) 2 2 We can choose either sector for left and right movers independently. If the sectors match for left and right movers, we obtain space-time bosons. If the sectors are different for left and right movers, then we obtain space-time fermions. That is, • Choosing the NS sector for left movers and the NS sector for right movers gives space-time bosons. • Choosing the R sector for left movers and the R sector for right movers gives space-time bosons. • Choosing the NS sector for left movers and the R sector for right movers gives space-time fermions. • Choosing the R sector for left movers and the NS sector for right movers gives space-time fermions. Super-Virasoro Generators When we quantize the theory we will need super-Virasoro operators. These are generalizations of what we have already worked out for bosonic string theory. We extend the idea in this case to include a fermionic operator. That is, B L → L + L ( ) ( F ) m m m It can be shown that the following defi nition will work: ∞ 1 1 L = r m b b m ∑α ⋅ α + − m−n n ∑( 2 ) (7.36) − r m+ r 2 4 n=−∞ r
144 <strong>String</strong> <strong>Theory</strong> Demystifi ed In addition, in superstring theory we have a second generator that arises from the supercurrent for the NS sector, while we take ∞ 2 π irσ G = d e J = ⋅b r ∫ σ − + ∑ α (7.37) m r+ m π π m=−∞ F = d m ∑α ⋅ (7.38) − n m+ n for the R sector. Here, m and n are integers while r =± 12 /, ± 32 /,…. Canonical Quantization n Now we are ready to quantize the theory, and canonical quantization is not so bad because fermions are simple to deal with. The condition on the modes for the bosonic string was the commutator: ⎡ ⎣α , α ⎤ mδ η m n⎦ m n, = + 0 µ ν µν (7.39) This relation is supplemented by a similar commutator for the α ′ s in the case of closed strings. For the supersymmetric theory, we need to supplement Eq. (7.39) with relations for the fermionic modes. You will recall from your studies of quantum fi eld theory that fermionic fi elds satisfy anticommutation relations. In our case the Majorana fi elds will satisfy the equal time anticommutation relation: µ ν µν { ψ ( σ, τ), ψ ( σ′ , τ)} = π η δ δ( σ − σ′ ) (7.40) A B AB In terms of the modes, we will have the following sets of anticommutation relations depending on the sector used: µ ν µν { b , b }= η δ µ ν µν { d , d }= η δ r s r+ s, 0 m n m+ n, 0 (7.41) The presence of the Minkowski metric in these equations mean that the theory will still be plagued by negative norm states that we will have to remove.
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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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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 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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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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