String Theory Demystified

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CHAPTER 9 Superstring Theory Continued 173 To write down the SUSY transformations, we simply compare with * and look for corresponding terms in the θ expansion. Doing this we fi nd µ µ δX= εψ µ α µ µ δψ =−iρ∂ α X ε + B ε µ α µ δB =−iερ ∂ ψ We can write the action in terms of superfi elds. Since the superfi elds are functions of the Grassman variables, we will have to utilize a new kind of integration, called Grassman integration. We take a brief detour to describe this now. α Grassman Integration Another important tool in the supersymmetry toolkit is the technique of Grassman integration. It turns out that the integration of anticommuting Grassman variables is quite a bit different (and actually a lot simpler, although less intuitive) than the ordinary integration of a function of real variables. Even so, we develop the notion of Grassman integration by our wish to preserve one important property of integration. If you integrate a function of a real variable over the entire number line, that integral is translation invariant. That is, let a be some real constant then it must be true that ∞ ∫ ∫ −∞ ∞ f( x) dx= f( x+ a) dx Now let φ( θ) be a function of the Grassman variable θ. We also want integration here to be translation invariant: ∫ ∫ −∞ dθφ( θ) = dθφ( θ + c) where c is a constant (a Grassman number in this case). To deduce the properties of Grassman integration while preserving translation invariance, we expand the function φ( θ) in Taylor: Then, φθ ( )= a+ bθ ∫ ∫ dθφ( θ) = dθ( a+ bθ)
174 String Theory Demystifi ed Now, let θ → θ+c and we obtain ∫ ∫ dθ[ a+ b( θ+ c)] = dθ( a+ bθ+ bc) ∫ ∫ = ( a+ bc) dθ+ b dθθ In order for this integral to be translation invariant, it cannot depend on c, and so we conclude that ∫ dθ = 0 when θ is a Grassman variable. By convention, the Grassman integral is normalized to one in the following way: So that altogether we have the rule ∫ ∫ dθθ = 1 dθ( a+ bθ) = b For double integration over two Grassman coordinates, there is only one rule to remember ∫ 2 d θθθ =−2i With these rules in hand, you are on your way to becoming a supersymmetry expert. A Manifestly Supersymmetric Action The action written in Chap. 7 [Eq. (7.2)] includes fermionic fi elds but supersymmetry is not manifest. This situation can be remedied by writing down an action in terms of superfi elds. The action we use is given by i S = d d DY DY ′ ∫ 2 2 µ σ θ 8πα µ
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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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Page 142 and 143: CHAPTER 7 RNS Superstrings The real
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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
Page 222 and 223: CHAPTER 12 Heterotic String Theory
Page 236 and 237: 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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