Solving M-theory with the Conformal Bootstrap

TL;DR

This paper employs the conformal bootstrap to analyze 3d $ ext{SCFTs}$ related to M2-branes, deriving precise operator data and proposing an OPE coefficient minimization principle for certain theories.

Contribution

It introduces a novel combination of bootstrap, localization, and large N expansion to study M2-brane SCFTs and conjectures an OPE minimization principle for specific models.

Findings

Calculated OPE coefficients exactly at small N

Performed all-order large N expansion

Reconstructed low-lying operator dimensions and OPE coefficients

Abstract

We use the conformal bootstrap to perform a precision study of 3d maximally supersymmetric ($\mathcal{N}=8$) SCFTs that describe the IR physics on $N$ coincident M2-branes placed either in flat space or at a $\C^4/\Z_2$ singularity. First, using the explicit Lagrangians of ABJ(M) \cite{Aharony:2008ug,Aharony:2008gk} and recent supersymmetric localization results, we calculate certain half and quarter-BPS OPE coefficients, both exactly at small $N$, and approximately in a large $N$ expansion that we perform to all orders in $1/N$. Comparing these values with the numerical bootstrap bounds leads us to conjecture that some of these theories obey an OPE coefficient minimization principle. We then use this conjecture as well as the extremal functional method to reconstruct the first few low-lying scaling dimensions and OPE coefficients for both protected and unprotected multiplets that appear in the OPE of two stress tensor multiplets for all values of $N$. We also calculate the half and quarter-BPS operator OPE coefficients in the $SU(2)_k \times SU(2)_{-k}$ BLG theory for all values of the Chern-Simons coupling $k$, and show that generically they do not obey the same OPE coefficient minimization principle.