Bootstrapping 3D Fermions

TL;DR

This paper advances the conformal bootstrap approach for 3D fermionic theories by developing new computational tools, deriving bounds on operator dimensions and central charge, and relating findings to known models like Gross-Neveu.

Contribution

It introduces an embedding formalism for 3D spinors and computes conformal blocks for fermion 4-point functions, enabling new bounds on operator dimensions and central charge.

Findings

Bounds on operator dimensions in fermionic OPEs

Bounds on the central charge $C_T$

Features matching Gross-Neveu model dimensions

Abstract

We study the conformal bootstrap for a 4-point function of fermions $\langle\psi\psi\psi\psi\rangle$ in 3D. We first introduce an embedding formalism for 3D spinors and compute the conformal blocks appearing in fermion 4-point functions. Using these results, we find general bounds on the dimensions of operators appearing in the $\psi \times \psi$ OPE, and also on the central charge $C_T$. We observe features in our bounds that coincide with scaling dimensions in the Gross-Neveu models at large $N$. We also speculate that other features could coincide with a fermionic CFT containing no relevant scalar operators.