We propose a new procedure, TFT$_\pm, to allocate non-unitary topological quantum field theory (TQFT) pairs. [\mathcal{T}_{\rm rank \;0}]$, (2+1)D interaction $\mathcal{N}=4$ superconformal field theory (SCFT) $\mathcal{T}_{\rm rank \;0}$ rank 0, i.e. No and Higgs branches. Topological theory arises from certain degeneracy limits of SCFT. Modular data for non-unitary TQFTs are extracted from degenerate limit supersymmetric partition functions. As an important dictionary, $F = \max_\alpha \left(- \log |S^{(+)}_{0\alpha}| \right) = \max_\alpha \left(- \log |S^ {(-)}_{0\alpha}|\right)$, where $F$ is the circular three-sphere free energy of $\mathcal{T}_{\rm rank \;0 }. $ and $S^{(\pm)}_{0\alpha}$ are the first columns of the modular S matrix of TFT$_\pm$. Derive the lower bound of $F$ from the dictionary. $F \geq -\log \left(\sqrt{\frac{5-\sqrt{5}}{10}} \right) \simeq 0.642965$, which is true for any rank-0 SCFT. This bound is saturated by the minimal $\mathcal{N}=4$ SCFT proposed by Gang-Yamazaki. Gang-Yamazaki’s related topology theory is both Lee-Yang TQFT. Explicitly compute the (Rank 0 SCFT)/(Nonunitary TQFT) correspondence for an infinite number of examples.
