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

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Chapter 8Computations“Trilingue! hablo espanol, franceset le language des bêtesJ’accepte d’utiliser la manière forte quand elle s’y prête”La Cliqua (Rocca)Comme Une SarbacaneThis chapter will go through all necessary motions to compute the equations <strong>of</strong> motionfor the fluctuations <strong>of</strong> a D2-brane in a cigar geometry. Already giving away a majorspoiler, we will stumble on a problem for which no apparent solution seems to be athand.8.1 Metric and field strengthThe geometry <strong>of</strong> the cigar is given by{ds 2 = − 1 k dt2 + dρ 2 + tanh 2 ρ dτ 2 ,e −Φ= e −Φ 0chρ.(8.1)The metric thus isThe D2-brane action is given byS D2 = −T 2 e Φ 0⎡⎤−10 0kg µν = ⎢⎣0 1 0 ⎥⎦ . (8.2)0 0 tanh 2 ρ∫√dαdβchρ − det(g αβ + 2πF αβ ), (8.3)96
CHAPTER 8. COMPUTATIONS 97and the induced metric g αβ equalsg αβ =⎡⎢⎣⎤−k −1 0 00 (∂ α ρ) 2 + th 2 ρ (∂ α τ) 2 ∂ α ρ∂ β ρ + th 2 ρ∂ α τ∂ β τ ⎥⎦ . (8.4)0 ∂ α ρ∂ β ρ + th 2 ρ∂ α τ∂ β τ (∂ β ρ) 2 + th 2 ρ (∂ β τ) 2Using parameterization invariance to set{α= chρ,β = τ,(8.5)we find the induced metric to be⎡−1⎤0 0kg αβ =1⎢0⎣ α 2 0− 1 ⎥α 2 ⎦ , (8.6)− 10 0α 2and the induced field strengthF αβ =⎡⎢⎣0 0 00 0 F ατ0 −F ατ 0⎤⎥⎦ , (8.7)withF ατ =1√α 2 − 1 F ρτ. (8.8)8.2 Classical solutionTo find the embedding <strong>of</strong> the D2-brane, we use a Lagrange multiplier, as shown in [15],§4.2. We need to minimizeS D2 − √ λ (∫ ) ∫ √ ( ) 1 1F − N = A dαdτ αk k√2 α 2 + 4π2 Fατ2 − √ λ ∫Fdαdτ √ ατk α 2 − 1= √ A ∫ [ √1dαdτ + α 2 π 2 Fατ 2 − λ ]F√ ατ, (8.9)k A α 2 − 1with A = T 2 e −Φ 0. We find thatδ (S − λ . . .)δF ατ= A √k∫[4π 2 α 2 F ατdαdτ √ − λ ]δF. (8.10)1 + 4π 2 α 2 F ατ A
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Master ThesisDBI <
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ContentsAcknowledgements 1Samenvatt
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7.1.4 Perturbed equations o
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SamenvattingDe zogehete snaartheori
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Chapter 1Introduction“Smokey, my
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CHAPTER 1. INTRODUCTION 6sight seem
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Chapter 2Bosonic String</st
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CHAPTER 2. BOSONIC STRINGS 14moment
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CHAPTER 2. BOSONIC STRINGS 162.3 Eq
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CHAPTER 2. BOSONIC STRINGS 18we see
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CHAPTER 2. BOSONIC STRINGS 20which
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CHAPTER 2. BOSONIC STRINGS 22which
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CHAPTER 2. BOSONIC STRINGS 24from w
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CHAPTER 2. BOSONIC STRINGS 26to use
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CHAPTER 2. BOSONIC STRINGS 28This a
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CHAPTER 2. BOSONIC STRINGS 30World
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CHAPTER 2. BOSONIC STRINGS 32This g
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CHAPTER 2. BOSONIC STRINGS 34<stron
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CHAPTER 2. BOSONIC STRINGS 36For th
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Chapter 3Superstrings“One <strong
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CHAPTER 3. SUPERSTRINGS 40In D dime
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CHAPTER 3. SUPERSTRINGS 423.3.1 Clo
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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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Page 62 and 63: CHAPTER 4. CONFORMAL INVARIANCE 54t
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Page 72 and 73: CHAPTER 5. BRANES 64meaning that th
Page 74 and 75: CHAPTER 5. BRANES 66i.e. our origin
Page 76 and 77: CHAPTER 5. BRANES 68The presence <s
Page 78 and 79: CHAPTER 5. BRANES 70heuristic argum
Page 80 and 81: CHAPTER 5. BRANES 72where F 12 is t
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Page 85 and 86: CHAPTER 6. SETTING 77it does. This
Page 87 and 88: CHAPTER 6. SETTING 79of</st
Page 89 and 90: CHAPTER 6. SETTING 81It was mention
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DBI Analysis of Open String Bound States on Non-compact D-branes

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