ABJ quadrality

Abstract

Abstract We study physical consequences of adding orientifolds to the ABJ triality, which is among 3d $$ \mathcal{N}=6 $$ N = 6 superconformal Chern-Simons theory known as ABJ theory, type IIA string in AdS 4 × ℂℙ3 and $$ \mathcal{N}=6 $$ 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 $$ \mathcal{N}=5 $$ N = 5 . We propose that adding the orientifolds to the $$ \mathcal{N}=6 $$ N = 6 Vasiliev theory leads to $$ \mathcal{N}=5 $$ N = 5 SUSY Vasiliev theory. It turns out that the $$ \mathcal{N}=5 $$ N = 5 case is more involved because there are two formulations of the $$ \mathcal{N}=5 $$ N = 5 Vasiliev theory with either O or USp internal symmetry. We show that the two $$ \mathcal{N}=5 $$ N = 5 Vasiliev theories can be understood as certain projections of the $$ \mathcal{N}=6 $$ 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 $$ \mathcal{N}=5 $$ 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 $$ \mathcal{N}=5 $$ N = 5 ABJ theory, string/M-theory and two $$ \mathcal{N}=5 $$ 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 $$ \mathcal{N}=5 $$ 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 $$ \mathcal{N}=6 $$ N = 6 Vasiliev theory and the parameters in the $$ \mathcal{N}=6 $$ N = 6 ABJ theory, which was conjectured in [1].