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

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT<jats:sub>±</jats:sub>[<jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{T} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math></jats:alternatives></jats:inline-formula><jats:sub>rank 0</jats:sub>], to a (2+1)D interacting <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 4 superconformal field theory (SCFT) <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{T} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math></jats:alternatives></jats:inline-formula><jats:sub>rank 0</jats:sub> 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 <jats:italic>F</jats:italic> = max<jats:sub><jats:italic>α</jats:italic></jats:sub> (<jats:italic>−</jats:italic> log|<jats:inline-formula><jats:alternatives><jats:tex-math>$$ {S}_{0\alpha}^{\left(+\right)} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>α</mml:mi> </mml:mrow> <mml:mfenced> <mml:mo>+</mml:mo> </mml:mfenced> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula>|) = max<jats:sub><jats:italic>α</jats:italic></jats:sub> (<jats:italic>−</jats:italic> log|<jats:inline-formula><jats:alternatives><jats:tex-math>$$ {S}_{0\alpha}^{\left(-\right)} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>α</mml:mi> </mml:mrow> <mml:mfenced> <mml:mo>−</mml:mo> </mml:mfenced> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula>|), where <jats:italic>F</jats:italic> is the round three-sphere free energy of <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{T} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math></jats:alternatives></jats:inline-formula><jats:sub>rank 0</jats:sub> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {S}_{0\alpha}^{\left(\pm \right)} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>α</mml:mi> </mml:mrow> <mml:mfenced> <mml:mo>±</mml:mo> </mml:mfenced> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> is the first column in the modular S-matrix of TFT<jats:sub>±</jats:sub>. From the dictionary, we derive the lower bound on <jats:italic>F</jats:italic>, <jats:italic>F</jats:italic> ≥ <jats:italic>−</jats:italic> log <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \left(\sqrt{\frac{5-\sqrt{5}}{10}}\right) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:msqrt> <mml:mfrac> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>−</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> </mml:mrow> <mml:mn>10</mml:mn> </mml:mfrac> </mml:msqrt> </mml:mfenced> </mml:math></jats:alternatives></jats:inline-formula> ≃ 0<jats:italic>.</jats:italic>642965, which holds for any rank 0 SCFT. The bound is saturated by the minimal <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> ...