Abstract
We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT±[ mathcal{T} rank 0], to a (2+1)D interacting mathcal{N} = 4 superconformal field theory (SCFT) mathcal{T} 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 = maxα (− log| {S}_{0alpha}^{left(+right)} |) = maxα (− log| {S}_{0alpha}^{left(-right)} |), where F is the round three-sphere free energy of mathcal{T} rank 0 and {S}_{0alpha}^{left(pm right)} 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) ≃ 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.
Highlights
Supersymmetric quantum field theories with 8 supercharges (8 Qs) provide a fertile ground for many interesting research topics connecting various areas in theoretical and mathematical physics
Another peculiar fact about 3D N = 4 theories is that there exist non-trivial interacting superconformal field theory (SCFT) of rank 0, as studied by two of the authors of this present paper in [27]. This is in contrast with the case of D ≥ 4, where it is often implicitly assumed that there is no non-trivial interacting rank 0 SCFTs with 8 Qs, so that the classification program starts with rank 1. (Recently, 4D/5D rank 0 SCFTs were found through a geometrical engineering but it is yet unclear if they are interacting SCFTs [28].) Note that most of the classification schemes in previous studies do not work for rank 0 cases since the existence of Coulomb or Higgs branch operators is an crucial assumption in the analysis
As main result of the paper, we propose that for any rank 0 N = 4 SCFT Trank 0 we can associate a pair of non-unitary topological quantum field theories (TQFTs) denoted by TFT±[Trank 0], satisfying the following relation
Summary
Supersymmetric quantum field theories with 8 supercharges (8 Qs) provide a fertile ground for many interesting research topics connecting various areas in theoretical and mathematical physics. We initiate the classification of rank 0 3D N = 4 SCFTs by establishing the following correspondence: 3D N = 4 superconformal field theories of rank 0 ←→ A pair of 3D non-unitary topological quantum field theories (TQFTs). Trank 0 : a 3D N = 4 interacting SCFT with empty Coulomb and Higgs branches (2.1) −−−→ TFT±[Trank 0] : a pair of 3D non-unitary TQFTs. The basic dictionaries for the correspondence are summarized in table 1. We call a topological quantum field theory a non-spin (or bosonic) TQFT when its partition function is independent on the choice of the spin structure, and a spin (or fermionic). Trank 0 ⊗ Tspin top with an unitary fermionic topological field theory Tspin top is still a rank 0 SCFT not satisfying the above condition since the decoupled topological sector does not contribute to the superconformal index.
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