The complete AdS4 x CP3 superspace for the type IIA superstring and D- branes

We lift the bosonic AdS(4) x CP3 solution of type IIA supergravity preserving 24 supersymmetries to a D = 10 superspace which has 32 Grassmann-odd directions. The type IIA superspace is obtained from D = 11 via dimensional reduction of the coset superspace OSp(8| 4)/SO(7) x SO(1, 3) by realizing the latter as a Hopf fibration over the former. This construction generalizes to superspace the Hopf fibration of S-7 as a U(1) bundle over CP3, and is suitable for writing the explicit form of Green-Schwarz-type actions encoding the dynamics of the type IIA string and branes in the AdS(4) x CP3 superbackground. We show that the OSp(6|4)/U(3) x SO(1, 3) supercoset string action describes only a subsector of the complete Green-Schwarz superstring. Thus, even though the superstring equations of motion in the OSp(6|4)/U(3) x SO(1, 3) subsector are classically integrable, the fact that the full AdS(4) x CP3 superspace is not a supercoset requires the use of more general methods to determine whether the superstring in the complete AdS(4) x CP3 superbackground is classically integrable.