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

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CHAPTER 6. SETTING 82But we can perform a T-duality on the circular ψ dimension <strong>of</strong> the trumpet, resultingin the cigar geometry (see Fig. 6.4), and the change <strong>of</strong> our D4-brane to a D5-brane.In factorized terms, our D3-“Minkowski”-brane remains unchainged, but the D1-branein the trumpet geometry now becomes a D2-brane covering the entire cigar geometry.This geometry is given by⎧⎨ds 2 = α ′ k [ dρ 2 + tanh 2 ρ dτ 2] ,⎩e 2Φ = e2Φ 0ch 2 ρ . (6.10)As far as the presence <strong>of</strong> the NS5-branes goes in these geometries, on the trumpet onecan consider them to be at infinity on the exploding plane at r = 0, and at the originon the cigar.6.4 RoadmapWe have now come to a point where we managed to isolate our test subject in a geometrywe know how to handle. So how should we proceed from here on?We cannot simply write down the <strong>DBI</strong> action for a D-brane in some geometry, aD2-brane in the cigar geometry in our specific case, and start varying around. Whatwe first need to do is to find the classical solution <strong>of</strong> the D-brane in this geometry.The classical solution is what one finds when one varies the unperturbed <strong>DBI</strong> D-braneaction with respect to the coordinates or fields one ultimately wants to vary, or moreprecisely can vary. This solution effectively extremizes the <strong>DBI</strong> action, thus constitutingthe equilibrium solution around which we will want to perturb. As a specific example,consider a D1-brane in the cigar and trumpet geometries. As will be shown in the nextchapter, the classical solutions for these configurations are given bytrumpet:cigar:sin(ψ − ψ 0 ) = Cchρ , (6.11)sin (τ − τ 0 ) = Cshρ . (6.12)Minimizing the energy <strong>of</strong> the D-brane in these configurations, these equations also tell ushow the brane will position itself in these geometries, i.e. they represent the embeddings<strong>of</strong> the D-brane in the geometry. In these embeddings, we see the appearance <strong>of</strong> twointegration constants, C and ψ 0 and τ 0 for the trumpet and cigar respectively. ψ 0 andτ 0 are angles with no physical interpretation; they simply define an <strong>of</strong>fset compared tothe origin in that direction, which amounts to rotating the brane around geometriesthat are symmetrical in that direction, hence not changing the physics. The constant Chowever has a very distinct interpretation: it can be considered as a measure <strong>of</strong> distance<strong>of</strong> the D1-brane to the NS5-ring, as is shown in Figure 6.5.Once we have found the classical solution, we can start thinking about varying theaction. For this, one simply introduces perturbations <strong>of</strong> coordinates and/or fields (e.g.the Maxwell field living on the D-brane). These perturbations will enter the action
CHAPTER 6. SETTING 83through the induced metric, Kalb-Ramond field and Maxwell field strength, these lasttwo <strong>of</strong> course only if present. Specifically, given the general pullback form,T αβ = ∂Xµ∂X α ∂X ν∂X β T µν, (6.13)with T µν some arbitrary tensor, partial derivatives <strong>of</strong> the perturbations will appear. Assuch, when one computes the determinant <strong>of</strong> the resulting matrix (recall the general√−gαβ + B αβ + 2πα ′ F αβ term in the <strong>DBI</strong> action), and expands the square root, oneis typically left with a number <strong>of</strong> zero, first and second order partial derivatives <strong>of</strong> theexpansions. By computing the equations <strong>of</strong> motion <strong>of</strong> this Lagrangian density, oneobtains a differential equation for the perturbations. By solving this equation, one cananalyse the spectrum <strong>of</strong> the perturbations.Summarized in short, to obtain the spectrum <strong>of</strong> the perturbations <strong>of</strong> a D-brane usingthe <strong>DBI</strong> action, and hence <strong>of</strong> the low-energy excitations <strong>of</strong> the D-brane, one shouldfirst write down the <strong>DBI</strong> action for the D-brane in the geometry under consideration,compute the classical solution, introduce perturbations, compute the equations <strong>of</strong> motion<strong>of</strong> these perturbations, and combine them into a differential equation. The solutions <strong>of</strong>this differential equation tells you the spectrum <strong>of</strong> the perturbations.6.5 How to recognize bound statesThe last question left to answer is how to known whether or not the spectrum containsbound states.The crucial point is that in order to solve the equations <strong>of</strong> motion for the perturbations,one always proposes a factorized solution containing a quantum mechanical e iEtterm to remove the always present ∂ 2 t term. This term is always present, because theperturbations are always considered to be time dependent, which by virtue <strong>of</strong> the pullbackmechanism amounts to such a second time derivative. This introduces the energy<strong>of</strong> the fluctuations in the differential equation.Using E 2 = m 2 + p 2 , we can relate this energy to the mass <strong>of</strong> the particles. If onemanages to solve the differential equations resulting from the equations <strong>of</strong> motion, onecan also derive which values <strong>of</strong> E 2 are allowed by the solution. A priori, if nothingbounds the particles, this energy is arbitrary since nothing bounds their impulse, i.e.p ∈ [0, +∞[, resulting in a continuous spectrum. However, for a bound state p = 0, andso the small mass (E 2 = m 2 ) <strong>of</strong> the particle will reveal itself through a discrete value <strong>of</strong>the energy laying below the continuum spectrum.
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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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Page 42 and 43: CHAPTER 2. BOSONIC STRINGS 34<stron
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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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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
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Page 85 and 86: CHAPTER 6. SETTING 77it does. This
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DBI Analysis of Open String Bound States on Non-compact D-branes

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