  In this paper we formulate the pure spinor superstring theory on AdS_4 x
CP^3. By recasting the pure spinor action as a topological A-model on the
fermionic supercoset Osp(6|4)/SO(6)xSp(4) plus a BRST exact term, we prove the
exactness of the sigma-model. We then give a gauged linear sigma-model which
reduces to the superstring in the limit of large volume and we study its branch
geometry in different phases. Moreover, we discuss possible D-brane boundary
conditions and the principal chiral model for the fermionic supercoset.
