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

TL;DR

This paper combines numerical bootstrap methods with exact localization results to analyze 3d ${ m f N}=6$ superconformal field theories, providing non-perturbative insights into their operator spectrum and dualities across different regimes.

Contribution

It derives superconformal blocks for 3d ${ m f N}=6$ SCFTs and uses localization data to produce rigorous bounds and interpolations between M-theory and string theory duals.

Findings

Bootstrap bounds are saturated by certain theories, indicating extremal spectra.

Localization results determine protected OPE coefficients for various parameters.

Evidence suggests the vector-like large-M limit relates to higher spin theories.

Abstract

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.