Pure Spinor Formulation of the Superstring and Its Applications

The pure spinor formalism for the superstring has the advantage over the more conventional Ramond-Neveu-Schwarz formalism of being manifestly spacetime supersymmetric, which simplifies the computation of multiparticle and multiloop amplitudes and allows the description of Ramond-Ramond backgrounds. In addition to the worldsheet variables of the Green-Schwarz-Siegel action, the pure spinor formalism includes bosonic ghost variables which are constrained spacetime spinors and are needed for covariant quantization using a nilpotent BRST operator. In this review, several applications of the formalism are described including the explicit computation in D = 10 superspace of the general disk amplitude with an arbitrary number of external massless states, genus one amplitudes with up to seven external states, genus-two amplitudes with up to five external states, and the low-energy limit of the genus-three amplitude with up to four external states. The pure spinor formalism has also been used to covariantly quantize the superstring in an AdS5 × S5 background and might be useful for proving the AdS-CFT correspondence in the limit of small AdS radius. This is an overview written for the “Handbook of Quantum Gravity,” eds. C. Bambi, L. Modesto and I. Shapiro.