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

More documents

Recommendations

Info

CHAPTER 2. BOSONIC STRINGS 36For the first excited level though, we now have many more possibilities, since thesestates are given byα−1˜α i j −1 |0〉, (2.138)and both excitations are essentialy independent. This results in 24 2 = 576 masslessstates. However, the fact that these excitations are indepent has a huge consequence,namely it allows us to decompose the states in three distinct parts as follows:[α−1α i j −1 |0〉 = α[i −1˜αj] −1 |0〉 + α −1˜αj) (i−1 − 1D − 2 δij α−1˜α k −1k]|0〉+ 1D − 2 δij α k −1˜α k −1|0〉, (2.139)where [...] stand for anti-symmetrization and (...) stand for symmetrization. Thus we obtainan anti-symmetric part, a traceless symmetric part, and a part that is proportionalto the unit matrix. And now comes the moment you have all been waiting for: thegraviton has entered the building! Admittedly, we are skipping a few steps here, butwe can nevertheless already identify the traceless symmetric part <strong>of</strong> Eq. 2.139 with amassless spin two particle described by a traceless symmetric rank two tensor, usuallydenoted g µν . It turns out though that if one would investigate how this particle couplesto matter and to itself, it does indeed behave like a graviton (cfr. [14], Chapter 15).Furthermore, we also find ourselves in the lovely company <strong>of</strong> a massless antisymmetricrank two tensor, which is called the Kalb-Ramond field and denoted B µν , and a masslessscalar which responds to the name dilaton field and denoted Φ, both <strong>of</strong> which will playvery important roles later on.Note that in the case <strong>of</strong> the closed string, the generating function trick describedfor the open string case does not apply. The origin for this lies in the level matchingcondition, Eq. 2.102, combined with the commutators for the modes, Eq. 2.53. The commutationrelations show that an m-level operator weights more than an (n < m)-leveloperator, resulting in the fact that one cannot simply write down an elegant generatingfunction for this situation, but has to verify the level-matching on a level-per-level basis.2.7 Oriented vs. unoriented stringsWe note in passing that one could define a projection operation that projectsσ −→ π − σ, (2.140)i.e. it flips all points on the string around the centerpoint. The consequence <strong>of</strong> thisoperation for the mode expansions isopen strings: α µ m ←→ (−1) m α µ m, (2.141)closed strings: α µ m ←→ ˜α µ m. (2.142)<strong>String</strong>s unaffected by this operation are called unoriented, in contrast to their brethrenwho do feel the operation and are called oriented.
CHAPTER 2. BOSONIC STRINGS 37The first relation amounts to saying that the spectrum <strong>of</strong> the unoriented open stringcontains only states with even mode number. In particular, it contains no photon(represented by the massless states).An important consequence <strong>of</strong> the second relation is that as far as the spectrum <strong>of</strong>the unoriented closed bosonic string is concerned, only states symmetric in α and ˜αsurvive, implying that the antisymmetric B µν tensor does not. Giving away a spoiler bysaying that strings can be shown to acquire charge under the B µν field, this and previousparagraphs show that unoriented theories are vastly different from oriented ones.In this document, we will only consider oriented theories.2.8 SummaryIn this chapter, we started out by considering the simplest possible bosonic string action,namely the Nambu-Goto action. We noted that it was difficult to quantize, and sointroduced a world sheet metric so as to be able to switch to the Polyakov action. Wethen analysed the symmetries <strong>of</strong> the Polyakov action, which allowed us to choose agauge in which deriving the equations <strong>of</strong> motion <strong>of</strong> the bosonic fields was particularlyeasy. Demanding that the boundary term <strong>of</strong> the varied action vanish, we found thatit was possible to impose several boundary conditions on our string, which gave birthto different types <strong>of</strong> strings. By their nature, these strings, and more specifically thesolutions to their equations <strong>of</strong> motion, allowed to be Fourier decomposed. This led to acanonical quantization in which the classical modes where promoted to quantum raisingand lowering operators. However, there was an ordering ambuigity to be solved, andthis appeared to be simpler to do in the light-cone gauge. Switching to this gauge alsoallowed us to derive the number <strong>of</strong> space-time dimensions in order for the theory to beLorentz invariant, and finally, to easily construct the spectrum <strong>of</strong> the bosonic string(s).All this been said, what we have seen in this chapter is only a minuscule part <strong>of</strong> whatbosonic string theory is comprised <strong>of</strong>. One could write an entire book about the bosonicstring alone, an endeavour which has been succesfully accomplished most notably byPolchinski [18].
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 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
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 97 and 98:
Page 99 and 100:
CHAPTER 7. A FIRST EXAMPLE 91with
Page 101 and 102:
Page 103 and 104:
CHAPTER 7. A FIRST EXAMPLE 95This t
Page 105 and 106:
CHAPTER 8. COMPUTATIONS 97and the i
Page 107 and 108:
CHAPTER 8. COMPUTATIONS 998.4 Regul
Page 109 and 110:
CHAPTER 8. COMPUTATIONS 1018.5 Equa
Page 111 and 112:
CHAPTER 8. COMPUTATIONS 103which de
Page 113:
Chapter 9Conclusion & Final Thought
Page 116 and 117:
BIBLIOGRAPHY 108[16] Polchinski J.,
Page 119:
⌈I’m so glad you cameI’m so g
show all

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

Create successful ePaper yourself

Delete template?

Save as template?