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

TL;DR

The paper constructs the complete type IIA superspace whose bosonic body is $AdS_4\times CP^3$ and which includes all 32 fermionic directions, revealing that this background is not a coset. By performing a Hopf-fibration–driven lift from the $D=11$ coset $OSp(8|4)/SO(7)\times SO(1,3)$ and a subsequent KK reduction with a tailored Lorentz rotation, the authors obtain explicit expressions for the 10D supervielbeins and RR/NS–NS superfields as functions of all 32 fermions, and they derive the complete Green–Schwarz-type actions for the type IIA string and D-branes in this background. The work shows that the $OSp(6|4)/U(3)\times SO(1,3)$ coset subsector yields a classically integrable sigma-model, but integrability of the full AdS$_4\times CP^3$ GS string is not guaranteed and remains an important open problem, with a twisted subspace ${\cal M}_{4,8}$ offering a potential testing ground. Overall, the explicit 32-fermion geometry provides a robust framework to study classical and quantum dynamics of strings and branes in this holographic setting and to probe the integrability structure of the ABJM correspondence.

Abstract

We lift the bosonic AdS_4 x 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) 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 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 x CP^3 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 CP^3 superspace is not a supercoset requires the use of more general methods to determine whether the superstring in the complete AdS_4 x CP^3 superbackground is classically integrable.