The ${\cal N} = 8$ Superconformal Bootstrap in Three Dimensions

TL;DR

This paper applies the conformal bootstrap approach to ${ m f N}=8$ superconformal field theories in three dimensions, deriving superconformal blocks, analyzing crossing symmetry constraints, and providing bounds on operator data that relate to known theories.

Contribution

It derives superconformal blocks for ${ m f N}=8$ theories, analyzes crossing equations, and numerically bounds OPE coefficients and operator dimensions, connecting to known models.

Findings

Bounds on OPE coefficients and operator dimensions as a function of central charge.

Near saturation of bounds by large N ABJM and free theories.

Parity symmetry in the stress-tensor multiplet OPE.

Abstract

We analyze the constraints imposed by unitarity and crossing symmetry on the four-point function of the stress-tensor multiplet of ${\cal N}=8$ superconformal field theories in three dimensions. We first derive the superconformal blocks by analyzing the superconformal Ward identity. Our results imply that the OPE of the primary operator of the stress-tensor multiplet with itself must have parity symmetry. We then analyze the relations between the crossing equations, and we find that these equations are mostly redundant. We implement the independent crossing constraints numerically and find bounds on OPE coefficients and operator dimensions as a function of the stress-tensor central charge. To make contact with known ${\cal N}=8$ superconformal field theories, we compute this central charge in a few particular cases using supersymmetric localization. For limiting values of the central charge, our numerical bounds are nearly saturated by the large $N$ limit of ABJM theory and also by the free $U(1)\times U(1)$ ABJM theory.