Using pure spinors, the superstring was recently quantized in a manifestly tendimensional super-Poincaré covariant manner and a covariant prescription was given for tree-level scattering amplitudes. In this paper, we prove that this prescription is cyclically symmetric and, for the scattering of an arbitrary number of massless bosons and up to four massless fermions, it agrees with the standard Ramond-Neveu-Schwarz prescription.

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The hybrid formalism is used to quantize the superstring compactified to twodimensional target-space in a manifestly spacetime supersymmetric manner. A quantizable sigma model action is then constructed for the Type II superstring in curved twodimensional supergravity backgrounds which can include Ramond-Ramond flux. Such curved backgrounds include Calabi-Yau four-fold compactifications with Ramond-Ramond flux, and new extremal black hole solutions in two-dimensional dilaton supergravity theory.These black hole solutions are a natural generalization of the CGHS model and might be possible to describe using a supergroup version of the SL(2,R)/U(1) WZW model. We also study some dynamical aspects of the new black holes, such as formation and evaporation.

We introduce a method to compute from first principles the anomalous dimension of short operators in N=4 super Yang-Mills theory at strong coupling, where they are described in terms of superstring vertex operators in an anti-de Sitter background. We focus on the Konishi multiplet, dual to the first massive level of the superstring, and compute the one-loop correction to its anomalous dimension at strong coupling, using the pure spinor formalism for the superstring.Comment: 4 pages; v2: corrections and improvements, conclusions unchange

It is proven that the pure spinor superstring in an AdS 5 × S 5 background remains conformally invariant at one loop level in the sigma model perturbation theory.