The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes

We study the space of 3d ${\cal N} = 6$ SCFTs by combining numerical bootstrap techniques with exact results derived using supersymmetric localization. First we derive the superconformal block decomposition of the four-point function of the stress tensor multiplet superconformal primary. We then use supersymmetric localization results for the ${\cal N} = 6$ $U(N)_k\times U(N+M)_{-k}$ Chern-Simons-matter theories to determine two protected OPE coefficients for many values of $N,M,k$. These two exact inputs are combined with the numerical bootstrap to compute precise rigorous islands for a wide range of $N,k$ at $M=0$, so that we can non-perturbatively interpolate between SCFTs with M-theory duals at small $k$ and string theory duals at large $k$. We also present evidence that the localization results for the $U(1)_{2M}\times U(1+M)_{-2M}$ theory, which has a vector-like large-$M$ limit dual to higher spin theory, saturates the bootstrap bounds for certain protected CFT data. The extremal functional allows us to then conjecturally reconstruct low-lying CFT data for this theory.