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

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CHAPTER 8. COMPUTATIONS 104see what comes out anyway. Concentrating on the ∼ ∂ 2 ω term, we see we can put thisterm in the right form by applying the substitutionThe second order term hereby becomesω −→ −z ( 1 − C 2) + 1. (8.41)4 ( ω − C 2) (ω − 1) ( ∂ 2 ωf ) −→ −4 (1 − z)z. (8.42)The only thing left to do is multiply through by −4 −1 . Going through the same motionsfor the other terms, the fully transformed equation becomes (courtesy <strong>of</strong> Mathematica):0 ={ [2 − ( −5C 2 + 5 + kE 2) z + ( C 2 − 1 ) ( 6C 2 − 3 − 2kE 2) z 2 − ( C 2 − 1 ) 2kE 2 z 3+ L 2 (z − 1) ( 1 + ( C 2 − 1 ) z ) ] [2× 4z ( 1 + ( C 2 − 1 ) z ) ] } −12f−{ [− 2 + 6C 2 + 2z − 6C 2 z + 4C 4 z − C 2 ( 1 + ( C 2 − 1 ) z ) 3/2+ ( 1 + ( C 2 − 1 ) z ) ] [5/2× 2 ( C 2 − 1 )( 1 + ( C 2 − 1 ) z ) ] } −1∂ z f+ z (1 − z)∂ 2 zf. (8.43)This is decidely not a hypergeometric equation.
Chapter 9Conclusion & Final Thoughts“I know what I’ve done, but tell me what did I miss?”Queens Of The Stone AgeSick, Sick, SickWe derived the equations <strong>of</strong> motion for the fluctuations <strong>of</strong> a D2-brane in the cigargeometry, and managed to combine them into a second order partial differential equation,Eq. 8.37. We unfortunately found out that we run into problems when we try to rewritethis equation as a hypergeometric equation, preventing us from obtaining the spectrum<strong>of</strong> the D2-brane fluctuations, and concluding the research.The problems arise starting from Eq. 8.27, where the ρ derivative also acts on thefactor in front <strong>of</strong> the fluctuation. This happens a second time right after in Eq. 8.29,resulting in a number <strong>of</strong> α-dependent terms in the zero-order and first-order differentialterms that were absent from the D1-brane case. T-duality however strongly suggeststhat it should nevertheless be possible to rewrite Eq. 8.37 as a hypergeometric equation,although this seems a rather non-trivial task.105
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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
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CHAPTER 3. SUPERSTRINGS 46freedom t
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CHAPTER 3. SUPERSTRINGS 48hence, fe
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CHAPTER 3. SUPERSTRINGS 50miraculou
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CHAPTER 3. SUPERSTRINGS 52associate
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DBI Analysis of Open String Bound States on Non-compact D-branes

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