DBI Analysis of Open String Bound States on Non-compact D-branes

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3 Superstrings 383.1 Superstring action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 <strong>Bound</strong>ary conditions and expansions . . . . . . . . . . . . . . . . . . . . 413.3.1 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 <strong>Open</strong> strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Quantization and spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.1 <strong>Open</strong> strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.2 GSO projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.3 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Conformal Invariance 534.1 Conformal Field Theory Jr. . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 <strong>String</strong> Theory vs. QFT vs. General Relativity . . . . . . . . . . . . . . . 564.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Branes 605.1 Dp-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1.1 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1.2 D-branes in Type IIA and Type IIB theories . . . . . . . . . . . 685.1.3 T-duality in Type IIA and Type IIB theories . . . . . . . . . . . 705.1.4 Dirac-Born-Infeld Action . . . . . . . . . . . . . . . . . . . . . . 705.2 NS5-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73III Research 756 Setting 766.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.2 What’s in a title? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2.1 Compact vs. non-compact D-branes . . . . . . . . . . . . . . . . 776.2.2 <strong>Bound</strong> states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2.3 <strong>DBI</strong> approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3 Geometrical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.4 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 How to recognize bound states . . . . . . . . . . . . . . . . . . . . . . . 837 A First Example 857.1 D1-brane on the cigar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1.1 Metric and action . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1.2 Classical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.1.3 Adding perturbations . . . . . . . . . . . . . . . . . . . . . . . . 87ii
7.1.4 Perturbed equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . 897.1.5 Solutions <strong>of</strong> the PDE . . . . . . . . . . . . . . . . . . . . . . . . . 897.2 D1-brane on the trumpet . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2.1 Metric and action . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2.2 Classical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2.3 Adding perturbations . . . . . . . . . . . . . . . . . . . . . . . . 937.2.4 Perturbed equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . 947.2.5 Solutions <strong>of</strong> the PDE . . . . . . . . . . . . . . . . . . . . . . . . . 948 Computations 968.1 Metric and field strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.2 Classical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.3 Field strength variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.4 Regularized Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.5 Equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.6 Road to differential equation . . . . . . . . . . . . . . . . . . . . . . . . 1018.7 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.8 Hypergeometric equation? . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Conclusion & Final Thoughts 105Bibliography 107List <strong>of</strong> Abbreviations 109iii
Page 1: Master ThesisDBI <
Page 5: ContentsAcknowledgements 1Samenvatt
Page 10 and 11: SamenvattingDe zogehete snaartheori
Page 12 and 13: Chapter 1Introduction“Smokey, my
Page 14: CHAPTER 1. INTRODUCTION 6sight seem
Page 20 and 21: Chapter 2Bosonic String</st
Page 22 and 23: CHAPTER 2. BOSONIC STRINGS 14moment
Page 24 and 25: CHAPTER 2. BOSONIC STRINGS 162.3 Eq
Page 26 and 27: CHAPTER 2. BOSONIC STRINGS 18we see
Page 28 and 29: CHAPTER 2. BOSONIC STRINGS 20which
Page 30 and 31: CHAPTER 2. BOSONIC STRINGS 22which
Page 32 and 33: CHAPTER 2. BOSONIC STRINGS 24from w
Page 34 and 35: CHAPTER 2. BOSONIC STRINGS 26to use
Page 36 and 37: CHAPTER 2. BOSONIC STRINGS 28This a
Page 38 and 39: CHAPTER 2. BOSONIC STRINGS 30World
Page 40 and 41: CHAPTER 2. BOSONIC STRINGS 32This g
Page 42 and 43: CHAPTER 2. BOSONIC STRINGS 34<stron
Page 44 and 45: CHAPTER 2. BOSONIC STRINGS 36For th
Page 46 and 47: Chapter 3Superstrings“One <strong
Page 48 and 49: CHAPTER 3. SUPERSTRINGS 40In D dime
Page 50 and 51: CHAPTER 3. SUPERSTRINGS 423.3.1 Clo
Page 52 and 53: CHAPTER 3. SUPERSTRINGS 443.4 Quant
Page 54 and 55: CHAPTER 3. SUPERSTRINGS 46freedom t
Page 56 and 57:
CHAPTER 3. SUPERSTRINGS 48hence, fe
Page 58 and 59:
CHAPTER 3. SUPERSTRINGS 50miraculou
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CHAPTER 3. SUPERSTRINGS 52associate
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CHAPTER 4. CONFORMAL INVARIANCE 54t
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CHAPTER 4. CONFORMAL INVARIANCE 56w
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CHAPTER 4. CONFORMAL INVARIANCE 58i
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Chapter 5Branes“Many speak <stron
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CHAPTER 5. BRANES 62but if the stri
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CHAPTER 5. BRANES 64meaning that th
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CHAPTER 5. BRANES 66i.e. our origin
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CHAPTER 5. BRANES 68The presence <s
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CHAPTER 5. BRANES 70heuristic argum
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CHAPTER 5. BRANES 72where F 12 is t
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Part IIIResearch75
Page 85 and 86:
CHAPTER 6. SETTING 77it does. This
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CHAPTER 6. SETTING 79of</st
Page 89 and 90:
CHAPTER 6. SETTING 81It was mention
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CHAPTER 6. SETTING 83through the in
Page 93 and 94:
Chapter 7A First Example“Le temps
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CHAPTER 7. A FIRST EXAMPLE 877.1.3
Page 97 and 98:
Page 99 and 100:
CHAPTER 7. A FIRST EXAMPLE 91with
Page 101 and 102:
Page 103 and 104:
CHAPTER 7. A FIRST EXAMPLE 95This t
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CHAPTER 8. COMPUTATIONS 97and the i
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CHAPTER 8. COMPUTATIONS 998.4 Regul
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CHAPTER 8. COMPUTATIONS 1018.5 Equa
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CHAPTER 8. COMPUTATIONS 103which de
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Chapter 9Conclusion & Final Thought
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BIBLIOGRAPHY 108[16] Polchinski J.,
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⌈I’m so glad you cameI’m so g
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DBI Analysis of Open String Bound States on Non-compact D-branes

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