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

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