| Summary: | Abstract We study physical consequences of adding orientifolds to the ABJ triality, which is among 3d N=6 $$ \mathcal{N}=6 $$ superconformal Chern-Simons theory known as ABJ theory, type IIA string in AdS 4 × ℂℙ3 and N=6 $$ \mathcal{N}=6 $$ supersymmetric (SUSY) Vasiliev higher spin theory in AdS 4. After adding the orientifolds, it is known that the gauge group of the ABJ theory becomes O(N 1) × USp(2N 2) while the background of the string theory is replaced by AdS 4 × ℂℙ3/Z 2, and the supersymmetries in the both theories reduce to N=5 $$ \mathcal{N}=5 $$. We propose that adding the orientifolds to the N=6 $$ \mathcal{N}=6 $$ Vasiliev theory leads to N=5 $$ \mathcal{N}=5 $$ SUSY Vasiliev theory. It turns out that the N=5 $$ \mathcal{N}=5 $$ case is more involved because there are two formulations of the N=5 $$ \mathcal{N}=5 $$ Vasiliev theory with either O or USp internal symmetry. We show that the two N=5 $$ \mathcal{N}=5 $$ Vasiliev theories can be understood as certain projections of the N=6 $$ \mathcal{N}=6 $$ Vasiliev theory, which we identify with the orientifold projections in the Vasiliev theory. We conjecture that the O(N 1) × USp(2N 2) ABJ theory has the two vector model like limits: N 2 ≫ N 1 and N 1 ≫ N 2 which correspond to the semi-classical N=5 $$ \mathcal{N}=5 $$ Vasiliev theories with O(N 1) and USp(2N 2) internal symmetries respectively. These correspondences together with the standard AdS/CFT correspondence comprise the ABJ quadrality among the N=5 $$ \mathcal{N}=5 $$ ABJ theory, string/M-theory and two N=5 $$ \mathcal{N}=5 $$ Vasliev theories. We provide a precise holographic dictionary for the correspondences by comparing correlation functions of stress tensor and flavor currents. Our conjecture is supported by various evidence such as agreements of the spectra, one-loop free energies and SUSY enhancement on the both sides. We also predict the leading free energy of the N=5 $$ \mathcal{N}=5 $$ Vasiliev theory from the CFT side. As a byproduct, we give a derivation of the relation between the parity violating phase in the N=6 $$ \mathcal{N}=6 $$ Vasiliev theory and the parameters in the N=6 $$ \mathcal{N}=6 $$ ABJ theory, which was conjectured in [1].
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