String Theory Demystified

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CHAPTER 2 Equations of Motion 37 homomorphic—the donut has a hole but a sphere does not. The bottom line is there is no way to transform the donut into the sphere. If a geometric shape is homeomorphic to a sphere, then the Euler characteristic is χ = V − E+ F = 2 (2.29) Many shapes have an Euler characteristic which vanishes. Some examples of this include a torus, a möbius strip, and a Klein bottle. Another example is a cylinder, which also has χ = 0 (see Fig. 2.2). Why is this interesting for us? If the worldsheet of a string has a vanishing Euler characteristic, then it is possible to write the auxiliary fi eld hαβ as a two-dimensional fl at space metric. That is, we take [using the choice of coordinates for the worldsheet as ( τ, σ ) ] Klein bottle h = αβ − ⎛ 1 0⎞ ⎝ ⎜ 0 1 ⎠ ⎟ (2.30) Tours Mobius strip Cylinder Figure 2.2 Some surfaces with a vanishing Euler characteristic. When the Euler characteristic vanishes, we can defi ne the auxiliary fi eld such that it has a representation of the fl at space Minkowski metric.
38 <strong>String</strong> <strong>Theory</strong> Demystifi ed Now notice that with this choice, h= det h =−1. We also have αβ αβ h ∂ X⋅∂ X =−∂ X⋅∂ X +∂ X⋅∂ X =− X+ X′ α β τ τ σ σ 2 2 In this case, we are able to write the Polyakov action in the remarkably simple form S P T 2 2 2 = ∫ d σ( X − X′ ) 2 (2.31) EXAMPLE 2.3 Find the equations of motion using the Polyakov action as written in Eq. (2.27) when the auxiliary fi eld takes the form of the fl at space metric. SOLUTION In this case we have So, we can write the lagrangian as Therefore, S P ∂ = ∂ ∂ L X ∂X T 2 αβ µ ν =− ∫ d σ −hh ∂ X ∂ X η α β 2 T 2 =− ∫ d σ( −∂X⋅∂X +∂ X⋅∂X) τ τ σ σ 2 T 2 µ ν µ ν =− ∫ d σ( −η ∂ X ∂ X + η ∂ X ∂ X ) µν τ τ µν σ σ 2 µ ν µ L =−η ∂ X ∂ X + η ∂ X ∂ X µν τ τ µν σ σ =− η X µ X ν + η µµ ν X′ X′ µ µ µν ∂L ∂ = ∂ X′ ∂ X′ µν µ µ µν µν ( − X µ Xν µ ν + X′ X′ )=− Xν η η η = −X µν µν ( − µ ν µ ν ν η X X + η X′ X µν ′ )= η X µν ′ = X′ µν ν µ µ
Page 2 and 3: Accounting Demystified Advanced Cal
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Page 8 and 9: Contents vii BRST in String Theory-
Page 10 and 11: Contents ix CHAPTER 14 Black Holes
Page 12 and 13: PREFACE String theory is the greate
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Page 16 and 17: CHAPTER 1 Introduction General rela
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Page 68 and 69: CHAPTER 3 Symmetries and Worldsheet
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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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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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