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

CHAPTER 3. SUPERSTRINGS 48hence, fermions. Hence we see that if we keep all states <strong>of</strong> both sectors, we will obtain adifferent number <strong>of</strong> space-time fermions and bosons, while if space-time supersymmetrywere respected these numbers should be equal. Lucky for us, there is a way to obtainsuch a space-time supersymmetric spectrum, and we will soon enough see how. Butfirst, we finish contructing the unconstrainted R spectrum.Apart from the degeneracy in ground states, the spectrum is constructed in the sameway as always. Only this time, we can let the raising operators act on several groundstates. We thus find that the first excited level with mass α ′ M 2 = 1 can be reached byapplyingα i −1|a〉 ; α i −1|ā〉 ; b i −1|a〉 ; b i −1|ā〉. (3.45)We thus find that there are 8 × 16 × 2 = 256 states at the first mass level, 128 for eachchirality. The generating function reads+∞ ∏f R (x) = 16The first few terms in the expansion aren=1( 1 + xn1 − x n ) 8. (3.46)f R (x) = 16x 0 + 256x 1 + 2304x 2 + 15360x 3 + . . .. (3.47)Notice by comparing this to Eq. 3.42 that at corresponding mass levels, the Ramondsector contains twice as many states.3.4.2 GSO projectionAs we have just seen, the NS sector contains a tachyon, which is something we wouldvery much like to get rid <strong>of</strong>. Not only that, but moreover there is a different number <strong>of</strong>states in corresponding mass levels between the NS sector (which are space-time bosons)and R sector (which are space-time fermions). This is bad, because if supersymmetrywere to be preserved, both numbers would be equal.In order to solve this sneaky problem, we will use an ingenious little sorting device,called the GSO projection. 6 Looking back at Eqs. 3.42 and 3.47, one sees that it wouldturn out really great if one were to find a way to eliminate the half-integer mass levels<strong>of</strong> the NS sector, and half the states <strong>of</strong> the R sector. This second part does not seemtoo complicated; simply project out one <strong>of</strong> both chiralities. For the first requirement,we will need to find a way to separate half-integer from integer mass modes.To reach our objectives, we introduce two operators, one for the NS sector, and onefor the R sector, but both responding to the name <strong>of</strong> G-parity operator. For the NSsector, we define this operator as 7G = (−1) F+1 with F =∞∑b i −rb i r. (3.48)r= 1 26 Originally introduced by Gliozzi, Olive and Scherk.7 Some references include the “-1” term in the definition <strong>of</strong> F.