The complete AdS 4 × CP 3 superspace for the type IIA superstring and D-branes

Abstract

We lift the bosonic AdS 4 × CP 3 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) × 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 CP 3, 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 × CP 3 superbackground. We show that the OSp(6|4)/U(3) × 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) × SO(1,3) subsector are classically integrable, the fact that the full AdS 4 × CP 3 superspace is not a supercoset requires the use of more general methods to determine whether the superstring in the complete AdS 4 × CP 3 superbackground is classically integrable. © 2009 SISSA.