The Inversion Formula and 6j Symbol for 3d Fermions

TL;DR

This paper develops methods to compute 6j symbols for 3d fermionic conformal operators, relating them to scalar cases, and applies these to analyze mean field theory spectra and non-perturbative effects at finite spin.

Contribution

It introduces a novel approach using weight-shifting operators to relate fermionic 6j symbols to scalar ones and computes non-perturbative corrections to 3d fermionic CFT spectra.

Findings

Derived fermionic 6j symbols from scalar cases.

Computed MFT OPE coefficients for fermions.

Analyzed non-perturbative corrections at finite spin.

Abstract

In this work we study the $6j$ symbol of the $3d$ conformal group for fermionic operators. In particular, we study 4-point functions containing two fermions and two scalars and also those with four fermions. By using weight-shifting operators and harmonic analysis for the Euclidean conformal group, we relate these spinning $6j$ symbols to the simpler $6j$ symbol for four scalar operators. As one application we use these techniques to compute $3d$ mean field theory (MFT) OPE coefficients for fermionic operators. We then compute corrections to the MFT spectrum and couplings due to the inversion of a single operator, such as the stress tensor or a low-dimension scalar. These results are valid at finite spin and extend the perturbative large spin analysis to include non-perturbative effects in spin.