Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs

We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT${}_{\pm }[{\mathcal{T}}_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}0}]$, to a (2+1)D interacting $\mathcal{N}=4$ superconformal field theory (SCFT) ${\mathcal{T}}_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}0}$ of rank 0, i.e. having no Coulomb and Higgs branches. The topological theories arise from particular degenerate limits of the SCFT. Modular data of the non-unitary TQFTs are extracted from the supersymmetric partition functions in the degenerate limits. As a non-trivial dictionary, we propose that $F = \max_\alpha \left(- \log |S^{(+)}{0\alpha}| \right) = \max\alpha \left(- \log |S^{(-)}_{0\alpha}|\right)$,where$F$ is the round three-sphere free energy of ${\mathcal{T}}_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}0}$ and $S_{0\alpha }^{(\pm )}$ is the first column in the modular S-matrix of TFT${}_{\pm }$. From the dictionary, we derive the lower bound on $F$, $F \geq -\log \left(\sqrt{\frac{5-\sqrt{5}}{10}} \right) \simeq 0.642965$, which holds for any rank 0 SCFT. The bound is saturated by the minimal $\mathcal{N}=4$ SCFT proposed by Gang-Yamazaki, whose associated topological theories are both the Lee-Yang TQFT. We explicitly work out the (rank 0 SCFT)/(non-unitary TQFTs) correspondence for infinitely many examples.