Pure-spinor superstrings in d=2,4,6

We continue the study of the d=2,4,6 pure-spinor superstring models introduced in [1]. By explicitly solving the pure-spinor constraint we show that these theories have vanishing central charge and work out the (covariant) current algebra for the Lorentz currents. We argue that these super-Poincare covariant models may be thought of as compactifications of the superstring on CY_{4,3,2}, and take some steps toward making this precise by constructing a map to the RNS superstring variables. We also discuss the relation to the so called hybrid superstrings, which describe the same type of compactifications.