Quantum Gravity

QUANTUM-GRAVITATIONAL ASPECTS 297What about quantization? From the X µ -part, one gets as before the commutationrelations (9.16)–(9.18) for the α µ n and the ˜αµ n . For the superstring, onehas in addition anticommutators for the fermionic modes. They are of the type[b µ m,b ν n] += η µν δ m,−n , (9.67)etc. One finds a SUSY extension of the Virasoro algebra, which again has acentral charge. The demand for the vanishing of the Weyl anomaly leads thistime to D = 10 dimensions for the superstring.Consider first the case of closed strings. 4 The NS–NS sector yields, as in thebosonic case, a tachyonic ground state. At M = 0, one finds again a graviton,a dilaton, and an antisymmetric tensor field (‘axion’ in four dimensions). TheR–R sector yields antisymmetric tensor gauge fields, while the NS–R sector givesspace–time fermions. The R–NS sector contains an exchange of left- and rightmovingfermions compared to the NS–R sector. Among these fermions are themassless gravitinos. In this sense, string theory contains space–time supergravity(SUGRA); see Section 2.3. An important notion (for both open and closedstrings) is the ‘GSO projection’ (named after Gliozzi, Scherk, and Olive). It removesthe tachyon and makes the spectrum supersymmetric. Moreover, it mustnecessarily be implemented in the quantum theory. In the R–R sector, the GSOprojection applied on ground states can yield states of the opposite or of thesame chirality. In the first case, one talks about type IIA superstring (which isnon-chiral), in the latter case, about type IIB superstring (which is chiral). TypesIIA and IIB are oriented closed superstrings with N = 2 SUSY. After the GSOprojection, there is no longer a tachyon in the NS–NS sector, but one still hasthe graviton, the dilaton, and the axion as massless states (for both types IIAand IIB). In the NS–R and the R–NS sectors, one is left with two gravitinos andtwo dilatinos (the SUSY partners of the dilaton), which have opposite chiralitiesfor type IIA and the same chirality for type IIB.In the case of open strings, one gets in the NS sector a tachyonic ground stateand a massless gauge boson. In the R-sector, all states are space–time spinors.Again, one gets rid of the tachyon by applying the GSO projection. This leads tothe type I superstring—the only consistent theory with open (and closed) strings(the strings here are non-oriented). It must have the gauge group SO(32) andhas N = 1 SUSY. In the closed-string sector of type I theory, one must projecttype IIB onto states which are invariant under worldsheet parity in order to getnon-oriented strings. There remain the graviton, the dilaton, a two-form field,one gravitino, and one dilatino. From the open-string sector, one gets masslessvector and spinor fields.In addition to types I, IIA, and IIB, there exists a consistent hybrid constructionfor closed strings combining the bosonic string with type II superstrings.This is referred to as ‘heterotic string’; the right-moving part is taken from typeII and the left-moving part is from the bosonic string. It possesses N = 1 SUSY.4 We neglect all massive states in our discussion.