In this chapter we will quantise the classical superstring given in the previous chapter. As with the bosonic string we can achieve this in different ways. We can use the old covariant method of quantisation, we can solve the constraints in the light-cone formalism and then quantise the independent variables or we can use the BRST approach to quantise the theory. In this chapter we will follow the first approach, given for the R-NS strings in the papers in which they were discovered [6.1, 6.2], and in chapter 10 we will give the BRST approach. For the superstring we find a set of constraints which generate the superconformal group and so the physical state conditions will involve not only the Virasoro generators Ln, but also their superanalogues, the Grs or Fns.

One major difference from the bosonic string stems from the choice of boundary conditions for the spinors, which we labelled by the letters R and N−S in the previous chapter. It will turn out that for the open superstring the R sector contains space-time fermions while the NS sector contains space-time bosons. For the closed superstring, on the other hand, the (NS, NS) and (R, R) sectors describe space-time bosons, while the (NS, R) and (R, NS) sectors describe space-time fermions. For the open superstring we will show that one can find projectors in both the R and NS sectors which are consistent with the physical state conditions and which lead to a superstring theory that possesses space-time supersymmetry. One effect of this projector in the NS sector is to remove the tachyon. Consequently, as the R sector possesses no tachyons we find a theory which is tachyon free. We will also discuss a projector for the closed string that has similar properties.