ABJ Correlators with Weakly Broken Higher Spin Symmetry

We consider four-point functions of operators in the stress tensor multiplet of the 3d $\mathcal{N}=6$ $U(N)_k\times U(N+M)_{-k}$ or $SO(2)_{2k}\times USp(2+2M)_{-k}$ ABJ theories in the limit where $M$ and $k$ are taken to infinity while $N$ and $\lambda\sim M/k$ are held fixed. In this limit, these theories have weakly broken higher spin symmetry and are holographically dual to $\mathcal{N}=6$ higher spin gravity on $AdS_4$, where $\lambda$ is dual to the bulk parity breaking parameter. We use the weakly broken higher spin Ward identities, superconformal Ward identities, and the Lorentzian inversion formula to fully determine the tree level stress tensor multiplet four-point function up to two free parameters. We then use supersymmetric localization to fix both parameters for the ABJ theories in terms of $\lambda$, so that our result for the tree level correlator interpolates between the free theory at $\lambda=0$ and a parity invariant interacting theory at $\lambda=1/2$. We compare the CFT data extracted from this correlator to a recent numerical bootstrap conjecture for the exact spectrum of $U(1)_{2M}\times U(1+M)_{-2M}$ ABJ theory (i.e. $\lambda=1/2$ and $N=1$), and find good agreement in the higher spin regime.