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

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CHAPTER 3. SUPERSTRINGS 50miraculously confirming our hopes that the NS and R sectors, after truncation, do indeedgive rise to an equal number <strong>of</strong> space-time bosons and fermions at each and every masslevel.Note that the match between both spectra does not prove space-time supersymmetry.But it is by any means a solid indication. Space-time supersymmetry can however beproven in the RNS formalism, but this is a highly non-trivial endeavour.3.4.3 Closed stringsAs we already saw earlier, for closed strings we can combine R and NS conditions infour distinct ways: (NS,NS), (NS,R), (R,NS) and (R,R). In order to obtain a space-timesupersymmetric spectrum though, we again need to perform the GSO projection. Forthe NS sector, this does not change much as far as our choices are concerned, howeverfor the R sector, this means we can choose between two chiralities, further increasingour number <strong>of</strong> possible choices.Essentially, this gives rise to two types <strong>of</strong> superstring theories: Type IIA and TypeIIB. Note that also in the superstring case, closed strings should satisfy the levelmatchingcondition, which still states that the left- and right-moving states shouldhave the same mass. From this we conclude that the full spectrum <strong>of</strong> both theories isobtained by tensoring left- and right-moving states <strong>of</strong> equal mass level.Type IIAThe Type IIA theory is the one that results when one chooses opposite chiralities for theleft- and right-moving R sectors. They should be chosen opposite, but which is whichdoes not matter. The massless states <strong>of</strong> this theory are summarized in Table 3.1. Thechiralities are denoted by R and ¯R. Only massless states are shown because first <strong>of</strong> allwe will only be interested in those, and second, the number <strong>of</strong> states at the first massivelevel is. . . massive.Looking at the massless states, our attention is grabbed by the (NS,NS) sector.Massless states in this sector are created by acting upon the vacuum state much in thesame way as for obtaining the massless bosonic closed string states (cfr. Eq. 2.138).This again suggests a decomposition in a tracless symmetric and antisymmetric part,and a part that is proportional to the unit matrix. In other words, we find back ourg µν , B µν and Φ fields!The (R,NS) and (NS,¯R) sectors give rise to two gravitinos with opposite chirality(one for each sector), and two dilatinos, also with opposite chirality. The fact that twogravitinos are present indicates that space-time is N = 2 supersymmetric. 8The (R,¯R) sector gives rise to 64 massless states (8×8). These can be decomposed ina vector gauge field A µ and an antisymmetric three-form gauge field A µνρ . Notice thatA µ contains 8 states, and that A µνρ contains ( 83)= 56 states, the sum <strong>of</strong> which indeed8 This means that there are two supersymmetric transformations, and hence the doublets (X, ψ) getexpanded to quadruplets (X 1, X 2, ψ 1, ψ 2).
CHAPTER 3. SUPERSTRINGS 51gives back our 64 massless states. These gauge fields are <strong>of</strong>ten called Ramond-Ramondfields for obvious reasons.Type IIBThe type IIB theory results from choosing identical chiralities for the R sector <strong>of</strong> bothleft- and right-movers. The massless states <strong>of</strong> this theory can also be found in 3.1. Weonly consider one chirality, namely R.The content <strong>of</strong> the (NS,NS) sector is exactly the same as for the Type IIA theory,and so also here we find back our g µν , B µν and Φ fields.The content <strong>of</strong> the (R,NS) and (NS,R) sectors is also the same, except for the factthat in the Type IIB theory the dilatinos and gravitinos have the same chirality.The (R,R) sector gives rise to quite different Ramond-Ramond gauge fields though,given that this time both chiralities are the same. It turns out that the massless TypeIIB (R,R) states can be regrouped in a zero-form scalar gauge field A, a two-formgauge(field A µν and a self-dual four-form gauge field A µνρσ . Note that also this time,8) (0 + 8)2 +1 82(4)= 1 + 28 + 35 = 64, as it <strong>of</strong> course should.Type IIAType IIB|0〉 RL × |0〉 ¯R |0〉 RL × |0〉 RR˜bi |0〉− 1 NSL × b j |0〉2− 1 NSR˜bi |0〉− 1 NSL × b j |0〉22− 1 NSR2˜bi |0〉− 1 NSL × |0〉 RR˜bi |0〉− 1 NSL × |0〉 RR22|0〉 ¯RL × b i |0〉− 1 NSR |0〉 RL × b i |0〉− 1 NSR22Table 3.1: Massless states for both Type II theories. The subscript L denotes left-movers,R stands for right-movers.Other superstring theoriesThe laws <strong>of</strong> logic dictate that if there are two Type II theories, there must be at leastone Type I theory. There indeed is such a theory, and it is a theory <strong>of</strong> unorientedstrings (remember: no B µν field, or any other antisymmetric state). For the sake <strong>of</strong>completeness we also mention the two heterotic string theories: E 8 × E 8 and SO(32).These are theories in which one <strong>of</strong> the left- and right-moving modes <strong>of</strong> the closed stringis supersymmetric, and the other one is bosonic. The names <strong>of</strong> these theories correspondto their symmetry groups.3.5 SummaryThis chapter has given a very brief introduction to the superstring, focusing mainlyon the construction <strong>of</strong> the open string spectrum and how to get rid <strong>of</strong> the problems
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Master ThesisDBI <
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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 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 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
Page 99 and 100: CHAPTER 7. A FIRST EXAMPLE 91with
Page 103 and 104: CHAPTER 7. A FIRST EXAMPLE 95This t
Page 105 and 106: CHAPTER 8. COMPUTATIONS 97and the i
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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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