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

Kavli Affiliate: Masahito Yamazaki

| First 5 Authors: Dongmin Gang, Sungjoon Kim, Kimyeong Lee, Myungbo Shim, Masahito Yamazaki

| Summary:

We propose a novel procedure of assigning a pair of non-unitary topological
quantum field theories (TQFTs), TFT$_pm [mathcal{T}_{rm rank ;0}]$, to a
(2+1)D interacting $mathcal{N}=4$ superconformal field theory (SCFT)
$mathcal{T}_{rm rank ;0}$ 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 $F = max_alpha left(- log |S^{(+)}_{0alpha}|
right) = max_alpha left(- log |S^{(-)}_{0alpha}|right)$, where $F$ is
the round three-sphere free energy of $mathcal{T}_{rm rank ;0 }$ and
$S^{(pm)}_{0alpha}$ 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.

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