# The M-theory Archipelago

## TL;DR

This paper combines supersymmetric localization and numerical bootstrap to study 3d ${
m f N}=8$ SCFTs, especially the ABJM theory at level 1, deriving precise bounds on operator spectra and conjecturing the theory that saturates these bounds.

## Contribution

It introduces a mixed correlator bootstrap approach with localization input to analyze 3d ${
m f N}=8$ SCFTs, providing new bounds and insights into the operator spectrum.

## Key findings

- Derived precise islands in OPE coefficient space for ABJM at $k=1$.
- Bounded the scaling dimension of the lowest unprotected scalar operator.
- Conjectured the ABJ theory at $k=2$ saturates single correlator bounds.

## Abstract

We combine supersymmetric localization results and the numerical conformal bootstrap technique to study the 3d maximally supersymmetric (${\cal N} = 8$) CFT on $N$ coincident M2-branes (the $U(N)_k \times U(N)_{-k}$ ABJM theory at Chern-Simons level $k=1$). In particular, we perform a mixed correlator bootstrap study of the superconformal primaries of the stress tensor multiplet and of the next possible lowest-dimension half-BPS multiplet that is allowed by 3d ${\cal N} = 8$ superconformal symmetry. Of all known 3d ${\cal N} = 8$ SCFTs, the $k=1$ ABJM theory is the only one that contains both types of multiplets in its operator spectrum. By imposing the values of the short OPE coefficients that can be computed exactly using supersymmetric localization, we are able to derive precise islands in the space of semi-short OPE coefficients for an infinite number of such coefficients. We find that these islands decrease in size with increasing $N$. More generally, we also analyze 3d ${\cal N} = 8$ SCFT that contain both aforementioned multiplets in their operator spectra without inputing any additional information that is specific to ABJM theory. For such theories, we compute upper and lower bounds on the semi-short OPE coefficients as well as upper bounds on the scaling dimension of the lowest unprotected scalar operator. These latter bounds are more constraining than the analogous bounds previously derived from a single correlator bootstrap of the stress tensor multiplet. This leads us to conjecture that the $U(N)_2 \times U(N+1)_{-2}$ ABJ theory, and not the $k=1$ ABJM theory, saturates the single correlator bounds.