String Theory Demystified

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INDEX 305 Schwarzschild metric, 242 Schwarzschild radius, 250 Second quantization, 69 Sectors, 141. See also individual sectors by name Signatures of metrics, 23 Singularities, 241 SO(32) heterotic string theory action for, 208–209 overview, 192 quantization of, 209–214 Space-fi lling branes, 221 Space-time arena, 223–224 Space-time bosons, 143, 199–200 Space-time fermions, 143, 200 Space-time supersymmetry, 175–176, 180 Space-time translation, 56–57 Spatial dimensions in bosonic string theory, 188 Special conformal transformations, 99 Spin-1 gauge bosons, 8 Spin-1/2 fermions, 8 Spin-2 bosons, 8–9 Spinor index, 177 Splitting strings, 14 Standard model, 8 Standing waves on string, 45 Stress-energy tensor, 3, 52, 106, 136–139 String coordinates, 30 String quantization covariant closed string spectrum, 82–85 commutation relations for closed string, 74–75 commutation relations for open string, 75 open string spectrum, 75–82 overview, 70–73 light-cone, 85–87 overview, 69 String sigma model action defi ned, 36 light-cone coordinates, 42 symmetries of overview, 53–56 Poincaré transformations, 56–57 reparameterizations, 57–58 Weyl transformations, 58–59 String theory. See also individual types of string theory by name; Superstring theory basic analysis in, 10–11 BRST quantization in, 120–121 conformal fi eld theory in, 92–93 D-branes, 15–16 fundamental constants, 11–12 general relativity, 2–3 higher dimensions, 16–17 M-theory, 15 overview, 1–2, 12–14 quantizing gravitational fi eld, 8–10 quantum theory, 3–8 standard model, 8 types of, 14–15 unifi cation, 11–12 Strings in space-time, 28–32 Strong particle interactions, 8 Structure constants, 116 Sundrum, Raman, 273 Super Yang-Mills (SYM) theory, 257, 259–261 Supercharge, 169 Supercovariant derivative, 171 Supercurrents, 138 Superfi elds overview, 168–171 for worldsheet supersymmetry, 171–173 Super-partners, 190 Super-Poincaré group, 178 Superspace, 168–171 Superstring theory. See also RNS superstrings; Type II A string theory; Type II B string theory bosonic string theory, 188–189 canonical quantization, 184–185 D-branes in, 230 dualities, 192–193 Grassman integration, 173–174 Green-Schwarz action, 175–179 heterotic E 8 × E 8 , 192 heterotic SO(32), 192 light-cone gauge, 181–184 overview, 167–168, 187–188, 190–191 space-time supersymmetry, 180 superfi elds overview, 168–171 for worldsheet supersymmetry, 171–173 superspace, 168–171 supersymmetric action, 174–175 type I, 14, 191 Supersymmetry (SUSY) defi ned, 127 point particle, 176–179 space-time, 180 supersymmetric actions, 174–175 transformations, 132–133, 172–173 ways to describe, 210 Supersymmetry generator, 169 Super-Virasoro operators NS sector algebra, 145 overview, 143–144 R sector algebra, 146–147 Susskind, Leonard, 256 SUSY (supersymmetry) defi ned, 127 point particle, 176–179 space-time, 180 supersymmetric actions, 174–175 transformations, 132–133, 172–173 ways to describe, 210 SYM (Super Yang-Mills) theory, 257, 259–261 Symmetries. See also Worldsheet currents energy-momentum tensor, 52–53 hamiltonian, 67 overview, 51 of Polyakov action overview, 53–56
306 String Theory Demystifi ed Symmetries, of Polyakov action (Cont.): Poincaré transformations, 56–57 reparameterizations, 57–58 Weyl transformations, 58–59 Symmetry breaking, 17 T Tachyons D-brane decay and, 235–237 ground state of bosonic string theory, 81–82 GSO projection, 148 Taus, 17 Taylor expansion, 171–172 T-duality for closed strings, 159–161 D-branes, 164–165 open strings and, 162–164 overview, 192–193 Temperature of black holes, 247–248 ekpyrotic, 276 Tensors. See individual tensors by name Thermodynamics, 246 T’Hooft, Gerard, 256 Three-dimensional branes, 16 Topology, 17 Torus surface, 37f Total number operators, 77, 197 Trace, calculating for energy-momentum tensor, 52–53 Transformations BRST, 121–125 conformal, 90, 98–100 Möbius, 102 Poincaré conserved currents from, 63–66 overview, 56–57 scale, 91–92 special conformal, 99 supersymmetry, 132–133 Weyl, 58–59 Two-dimensional conformal group, 100–104 Two-component Majorana spinors, 130 Two-dimensional conformal fi eld theories, 90 Two-point function, 107 Type I string theory, 14, 191 Type II A string theory defi ned, 15, 191 massless spectrum of different sectors, 203–204 NS sectors, 195–196, 198–200 overview, 201–202 R sectors, 195–198 spin fi eld, 200–201 Type II B string theory defi ned, 15, 191 massless spectrum of different sectors, 203–204 NS sectors, 195–196, 198–200 overview, 203 R sectors, 195–198 spin fi eld, 200–201 U Uncertainty principle, 4, 9 Unifi cation, 11–12 Unoriented strings, 188–189 Uppercase letters in coordinates, 22 V Vacuum fi eld equations, 241 Vanishing Euler characteristic, 37 Vibration of strings, 14, 92–93 Virasoro algebra with central extension, 80, 104 Virasoro constraints, 78 Virasoro operators, 78, 80, 155–156 Virtual processes, 7 Visible branes, 273 Volume elements, 96 W Warp factor, 274 Wave equation solutions, 43–44 Wave function, 4–5 Weak particle interactions, 8 Weyl spinors, 149 Weyl transformations, 58–59 Wick expansion, 107–110 Wick rotations, 93 Winding contribution, 47 Winding mode, 154 Winding number, 47, 154 Winding states, 154 Witten algebra, 104 World line for point particle, 13f Worldsheet currents conserved currents from Poincaré invariance, 63–66 energy-momentum tensor, 52–53 hamiltonian, 67 overview, 51 supersymmetry transformations on, 132–133 symmetries of Polyakov action overview, 53–56 Poincaré transformations, 56–57 reparameterizations, 57–58 Weyl transformations, 58–59 transforming to fl at metric, 59–63 Worldsheet supersymmetry, 171–173 Worldsheet for closed strings, 13f, 29f, 106f World-volume, 222 Y Yang-Mills theory, 231, 257, 259–261 Z z plane, mapping world sheet of closed string to, 106f Zero-dimensional objects, 28–29 Zero modes of ghost states, 124 Zeroth law, 246 Zeroth mode, 233
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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 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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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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Page 316 and 317: INDEX 301 Cyclic model, 277 Cylinde
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String Theory Demystified

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