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

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CHAPTER 3. SUPERSTRINGS 443.4 Quantization and spectrumTurning to the quantization <strong>of</strong> these strings, we will again start by promoting the expansionmodes to quantum operators using the canonical formalism, switching to (spacetime)light-cone gauge in order to construct the spectra.First thing to note is that the behaviour <strong>of</strong> the bosonic degrees <strong>of</strong> freedom has notchanged, and as a consequence, also the quantization <strong>of</strong> their modes remains the same,i.e.[α µ m, α ν n] = [˜α µ m, ˜α ν n] = mδ m+n,0 η µν ; [α µ m, ˜α ν n] = 0. (3.29)We could have instead chosen to use a instead <strong>of</strong> α, in which case the above relationsbecome[a µ m, a ν n] = [ã µ m, ã ν n] = δ m+n,0 η µν ; [a µ m, ã ν n] = 0. (3.30)The behaviour <strong>of</strong> the fermionic modes is very similar, the biggest difference being thatthey obey anticommutation relations instead <strong>of</strong> commutation relations, as a consequence<strong>of</strong> them being fermionic. As such, the quantization prescription becomes{. . .,...} −→ −i {. . .,...} , (3.31)where the left anticommutator represents the classical one, and the one on the right isthe quantum version. We also note that the fact that ψ µ are Majorana spinors getsinherited by the modes in that sense that they need to satisfy the condition(b µ r) † = b µ −r ; (d µ r) † = d µ −r , (3.32)and analogous for their tilded brethren. The result <strong>of</strong> all this are the following relations:}{ }{b µ r,b ν s} ={˜bµ r ,˜b ν s = η µν δ r+s ; b µ r,˜b ν s = 0, (3.33){{d µ m, d ν n} = ˜dµ m , ˜d}{νn = η µν δ m+n ; d µ m, ˜d}νn = 0. (3.34)Notice that the b µ and ˜b µ operators do not contain a zero mode.Before moving on to the spectrum, and although we will not go into the details, wemention that also in the superstring case one can expand the energy-momentum tensor,as well as the supercurrent coming from the global world sheet supersymmetry. Themodes <strong>of</strong> these expansions form the super-Virasoro algebra. Furthermore, also analogousto the bosonic string case, upon normal ordering <strong>of</strong> the modes one induces two orderingcontants a R and a NS , respectively for the Ramond and Neveu-Schwarz sectors. Byplaying a clever game involving the super-Virasoro algebra and physical states, one canderive the results that a R = 0, a NS = 1 2and D = 10 for superstrings. The same resultcould be obtained by first constructing the spectrum <strong>of</strong> the strings in the space-timelight-cone gauge, and then applying a zeta-regularization technique similar to the onewe saw in Chapter 2 for the bosonic string. We will consider these results as known,and immediatly use them in what follows.It is worth pointing out that Eq. 3.2 still describes one, and only one, string. Wedo not get two strings: one which behaves like a boson, and one which behaves like a
CHAPTER 3. SUPERSTRINGS 45fermion. We still have only one string, only this time around this string has bosonic andfermionic degrees <strong>of</strong> freedom on the world sheet. This means that general states <strong>of</strong> thissingle string will be obtained by acting with both bosonic and fermionic raising operatorson the vacuum. Hence, when we state that a R = 0 and a NS = 1 2, we in fact state thatthe normal ordering constant that results from the sum <strong>of</strong> the bosonic and fermionicdegrees <strong>of</strong> freedom equals 0 when considering R modes, and 1 2when considering NSmodes.On a sidenote, remark that the fact that a R = 0 can be intuitively understoodby realizing that both α and d are integer moded operators, but that since one obeyscommutation relations and the other anticommutation relations, their normal orderingcontants will carry opposite signs, and hence annihilate.Moving on to the construction <strong>of</strong> the spectra, we will now switch to space-time lightconegauge. This is possible because <strong>of</strong> residual gauge symmetries left after gauge fixingthe world sheet metric, similar to the bosonic string case. Moreover, a similar thing alsoapplies to the fermionic maps, as a result <strong>of</strong> which we can setX + (σ, τ) = x + + p + τ, (3.35)ψ + (σ, τ) = 0. (3.36)If one were to go one step further, one would find that one could still relate the X −and ψ − modes to the modes <strong>of</strong> the transverse oscillators, thereby effectively eliminatingtwo sets <strong>of</strong> modes, and keeping only the transverse (raising and lowering) modes. Thezero modes are still present for all directions, where applicable. As for the bosonic case,transverse modes will be denoted with a superscript i . Using the previously stated factthat D = 10 for superstring theories, we are thus left with eight transverse directions.3.4.1 <strong>Open</strong> stringsSpecializing to the case <strong>of</strong> open superstrings, we need to differentiate between the Ramondand Neveu-Schwarz sectors.Neveu-Schwarz sectorApplying the mass-shell condition to the NS sector results in the mass formulaα ′ M 2 =+∞∑n=1α i −nα i n ++∞∑r= 1 2rb i −rb i r − 1 2 , (3.37)where summation over i is understood, and where as anticipated earlier we alreadyincorporated the result a NS = 1 2. This immediatly tells us thatα ′ M 2 |0〉 NS = − 1 2 |0〉 NS, (3.38)where |0〉 NS describes the vacuum <strong>of</strong> the open string NS sector. We find our old foo,the tachyon, back with a vengeance. It appears that simply adding fermionic degrees <strong>of</strong>
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Page 14: CHAPTER 1. INTRODUCTION 6sight seem
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Page 22 and 23: CHAPTER 2. BOSONIC STRINGS 14moment
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Page 26 and 27: CHAPTER 2. BOSONIC STRINGS 18we see
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Page 34 and 35: CHAPTER 2. BOSONIC STRINGS 26to use
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Page 38 and 39: 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
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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
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Page 74 and 75: CHAPTER 5. BRANES 66i.e. our origin
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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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CHAPTER 7. A FIRST EXAMPLE 95This t
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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
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BIBLIOGRAPHY 108[16] Polchinski J.,
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DBI Analysis of Open String Bound States on Non-compact D-branes

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