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

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).