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

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Chapter 3Superstrings“One <strong>of</strong> the problems has to do with the speed <strong>of</strong> lightand the difficulties involved in trying to exceed it. Youcan’t. Nothing travels faster than the speed <strong>of</strong> light withthe possible exception <strong>of</strong> bad news, which obeys it ownspecial laws. The Hingefreel people <strong>of</strong> Arkinto<strong>of</strong>le Minordid try to build spaceships that were powered by badnews but they didn’t work particularly well and were soextremely unwelcome whenever they arrived anywherethat there wasn’t really any point in being there.”Douglas AdamsHitchhiker’s Guide to the GalaxyThe bosonic string is a versatile object, but as things usually go with prototypes, it isnot exactly there yet. It mainly has two flaws. First, its spectrum always contains atachyon, something we do not exactly appreciate (Adams seems to rightly suggest it isthe quantum <strong>of</strong> bad news). Second, it is purely bosonic, meaning it can not possiblydescribe fermions. Being that fermions in large part make up the world as we know it,this is <strong>of</strong> course a serious issue.So the question now becomes how we can possibly alter bosonic string theory suchthat it also incorporates fermions. The answer turns out to be to introduce a new symmetrycalled supersymmetry, that relates, at least on some level, bosons and fermions.Using this symmetry, one can expand the bosonic string action with an extra fermionicterm, thus creating what is called the superstring. As in the case <strong>of</strong> the bosonic string,the equations <strong>of</strong> motion <strong>of</strong> this superstring will lead to several possible boundary conditions,resulting in different types <strong>of</strong> superstrings. Possibilities to combine boundaryconditions for open strings will however increase, and have considerable consequencesfor the spectrum <strong>of</strong> said strings, resulting in different types, or flavours, <strong>of</strong> superstringtheories.38
CHAPTER 3. SUPERSTRINGS 39First <strong>of</strong> all, it should be pointed out that two main methods exist to introducesupersymmetry to the bosonic string theory. The first method is called the “Ramond-Neveu-Schwarz” (RNS) method, and starts from a position <strong>of</strong> making the world sheetsupersymmetric. The second method is called the “Green-Schwarz” (GS) method, andstarts from a position <strong>of</strong> making space-time supersymmetric. Both methods have theiradvantages and disadvantages, however the second one has the drawback <strong>of</strong> being vastlymore technical and hard to quantize. Therefore, we will disregard it completely in thisdocument, and will concentrate instead on the RNS formalism. Both formalisms areequivalent for a 10-dimensional space-time, which turns out to be the critical dimensionfor superstring theory. Parties interested in the GS formalism are referred to e.g.Chapter 5 in both [2] and [9].We will not be concerned with specific details and applications, but rather focus onpresenting the method that allows one to write a SUSY string action, what this entailsfor the spectrum <strong>of</strong> said strings, and how this results in different superstring theories.3.1 Superstring actionBut let us start at the beginning. If we want to include fermions in our theory, the firstthing we need is something that behaves like a fermion. To this end, we introduce fieldsψ α µ with µ ∈ {0, . . .,D − 1} and α ∈ {0, 1}, which are physical objects that behave liketwo-component spinors on the world sheet (hence their spinor index α), and as vectorsin D-dimensional space-time (hence their Lorentz index µ). Note that we introduceobjects that behave as fermions on the world sheet. We do not yet have objects thatbehave as fermions in spacetime, but we will come to that soon enough.If we want to add these new fields to our existing bosonic string action, options arelimited as to what we can do. Since we are assuming a supersymmetric relation betweenthe bosonic fields X µ and ψ µ , give by the simultaneous transformations{δX µ = ¯ǫψ µ ,δψ µ = ρ α ∂ α X µ (3.1)ǫ,in which ǫ represents an infinitesimal parameter <strong>of</strong> this supersymmetry transformation,we should <strong>of</strong> course add as many fermionic fields as we had bosonic fields, meaning onefor each space-time dimension. The form <strong>of</strong> the action that imposes itself is that <strong>of</strong> astandard Dirac action for massless fermions, hence the superstring action readsS = − 1 ∫2πα ′ d 2 σ ( ∂ α X µ ∂ α X µ + ¯ψ µ ρ α )∂ α ψ µ . (3.2)This action is symmetric under Eq. 3.1, and in it, ρ α represents the two-dimensionalDirac matrices, obeying the Clifford algebra 1 defined by{ρ α , ρ β} = 2η αβ½2×2. (3.3)1 See e.g. Chapter 3 in [22].
Page 1: Master ThesisDBI <
Page 5 and 6: ContentsAcknowledgements 1Samenvatt
Page 7: 7.1.4 Perturbed equations o
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 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
Page 60 and 61: CHAPTER 3. SUPERSTRINGS 52associate
Page 62 and 63: CHAPTER 4. CONFORMAL INVARIANCE 54t
Page 64 and 65: CHAPTER 4. CONFORMAL INVARIANCE 56w
Page 66 and 67: CHAPTER 4. CONFORMAL INVARIANCE 58i
Page 68 and 69: Chapter 5Branes“Many speak <stron
Page 70 and 71: CHAPTER 5. BRANES 62but if the stri
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
Page 83 and 84: Part IIIResearch75
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
Page 91 and 92: CHAPTER 6. SETTING 83through the in
Page 93 and 94: Chapter 7A First Example“Le temps
Page 95 and 96: CHAPTER 7. A FIRST EXAMPLE 877.1.3
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CHAPTER 7. A FIRST EXAMPLE 897.1.4
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CHAPTER 7. A FIRST EXAMPLE 91with
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CHAPTER 7. A FIRST EXAMPLE 937.2.3
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CHAPTER 7. A FIRST EXAMPLE 95This t
Page 105 and 106:
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
Page 116 and 117:
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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