Non-unitary TQFTs from 3D $$ \mathcal{N} $$ = 4 rank 0 SCFTs
Dongmin Gang, Sung-Joon Kim, Kimyeong Lee, Myungbo Shim, Masahito Yamazaki
Abstract
A bstract We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT ± [ $$ \mathcal{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> rank 0 ], to a (2+1)D interacting $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 superconformal field theory (SCFT) $$ \mathcal{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> rank 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 α ( − log| $$ {S}_{0\alpha}^{\left(+\right)} $$ <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> |) = max α ( − log| $$ {S}_{0\alpha}^{\left(-\right)} $$ <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> |), where F is the round three-sphere free energy of $$ \mathcal{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> rank 0 and $$ {S}_{0\alpha}^{\left(\pm \right)} $$ <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> is the first column in the modular S-matrix of TFT ± . From the dictionary, we derive the lower bound on F , F ≥ − log $$ \left(\sqrt{\frac{5-\sqrt{5}}{10}}\right) $$ <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> ≃ 0 . 642965, which holds for any rank 0 SCFT. The bound is saturated by the minimal $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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.