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.

中文翻译：

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3d费米子的反演公式和6j符号

在这项工作中，我们研究了费米子算子的 $3d$ 共形群的 $6j$ 符号。特别是，我们研究了包含两个费米子和两个标量的 4 点函数以及包含四个费米子的函数。通过对欧几里得保形群使用权重移位算子和调和分析，我们将这些旋转的 $6j$ 符号与四个标量算子的更简单的 $6j$ 符号联系起来。作为一项应用，我们使用这些技术来计算费米算子的 $3d$ 平均场论 (MFT) OPE 系数。然后，我们计算由于单个算子（例如应力张量或低维标量）的反演而对 MFT 谱和耦合进行的校正。这些结果在有限自旋时有效，并将微扰大自旋分析扩展到包括自旋中的非微扰效应。