String Theory Demystified

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CHAPTER 13 D-Branes 229 fi rst excited state. We can act on the ground state with a i† a 1 or with a1 using a i† fi rst. The state is In this case i† + i a p , p ∞ p ∞ d 2 i† + i 1 ⎛ i † i a † ma p, p = naa ma a 1 ∑∑ + n n ∑ ∑ m ′ ⎝ ⎜ α n= 1 i= 2 m= 1 a= p+ 1 1 ⎛ = ′ a + i Now, a p , p = 0, so m 1 a m ⎞ − 1 ⎠ ⎟ a p , p 1 i† + i ∞ p ∞ d i † i i† + i a † a i† ∑∑na a a p p + n n 1 ∑ ∑ ma a a m m 1 α ⎝ ⎜ n= 1 i= 2 m= 1 a= p+ 1 † . Let’s consider + i i† + i , p , p − a p , p ∞ p ∞ d 1 ⎛ i † i i† + i a † i† a + i = ∑∑na a a p , p + ma a a p p −a n n 1 ∑ ∑ , m m α ′ ⎝ ⎜ n= 1 i= 2 m= 1 a= p+ 1 ∞ p 2 1 ⎛ i † i i† + i i† + i m = ∑∑ nanana1p , p − a1p , p α ′ ⎝ ⎜ n= 1 i= 2 1 1 ⎞ ⎠ ⎟ 1 p , p i† + i i i † i i† i† i Now we use the commutator ⎡⎣ a , a ⎤⎦ = δ to write aa = δ + a a. And so m n ∞ p 2 1 ⎛ i † i† i + i m = ∑∑ na δ + a a p , p a n n1 1 n α ′ ⎝ ⎜ 1 ⎛ = α ′ ⎝ ⎜ 1 = α ′ n= 1 i= 2 ∞ p ∑∑ n= 1 i= 2 δ mn ( ) − − i† + i i† + i na p , p a p , p n1n 1 i† + i i† + i ( a p , p − a p , p 1 1 )= 0 n 1 n1 1 i† 1 ⎞ ⎠ ⎟ + p , p i† + i Hence, the state a1p , p has mass m 2 = 0. These states are characterized by an index i which denotes coordinates on the brane. Since i= 2, ..., p, there are a total of ( p + 1) −2states. Recall that a photon in a (3 + 1) dimensional theory has two transverse states. So these states are photon states. i ⎞ ⎠ ⎟ n ⎞ ⎠ ⎟ ⎞ ⎠ ⎟
230 String Theory Demystifi ed The next possibility for a massless state is to act on the ground state with a a† 1 . a† + i You can show that the state a1p , p also has m 2 = 0. These states are called Nambu-Goldstone bosons. They represent scalar bosons which have arisen from a symmetry breaking of translation invariance in space-time. Excitations of the a† + i Nambu-Goldstone bosons a1p , p , correspond to displacements of the D-brane in space-time along the coordinate x a . The lesson of the string states we have found in the presence of a D-brane is that gauge fi elds live on the brane. It turns out that gravity is different. It is not restricted to the brane and can propagate in the bulk. D-Branes in Superstring Theory Thinking about superstring theory for a moment on a qualitative level, different types of branes live in different superstring theories. In type IIA theory, only branes with even spatial dimensions are possible. Since d = 9 in superstring theory, this means that type IIA superstring theory incorporates branes with the following spatial dimensions: p = 02468 , , , , We met the D0-brane when discussing the supersymmetric point particle in Chap. 9. Now consider type IIB string theory. The dimension of p must be odd, so the theory can contain branes with spatial dimensions: p =−113579 , , , , , The case of p =−1 might stand out as a little odd. This object is called an instanton. It is an object that is forever fi xed in time and space-time does not fl ow for an instanton (thus the name). When p = 9 we have a space-fi lling brane in superstring theory. Note that space-fi lling branes are possible in type IIB string theory but not in type IIA string theory. Multiple D-Branes Having a confi guration of multiple D-branes allows for something new—an open string can begin on one brane and end on a different brane. This leads to some interesting results and changes in the mass spectrum. In general we could consider
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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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Page 194 and 195: 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
Page 238 and 239: CHAPTER 13 D-Branes 223 The Space-T
Page 240 and 241: CHAPTER 13 D-Branes 225 Once again
Page 242 and 243: CHAPTER 13 D-Branes 227 and ( X i i
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Page 252 and 253: CHAPTER 13 D-Branes 237 We consider
Page 254 and 255: CHAPTER 14 Black Holes Black holes,
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Page 264 and 265: CHAPTER 14 Black Holes 249 Entropy
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Page 268 and 269: CHAPTER 14 Black Holes 253 Now, the
Page 270 and 271: CHAPTER 15 The Holographic Principl
Page 280 and 281: 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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