We propose a novel procedure of assigning a pair of non-unitary topological
quantum field theories (TQFTs), TFTΒ±β[Trank0β], to a
(2+1)D interacting N=4 superconformal field theory (SCFT)
Trank0β 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=maxΞ±β(βlogβ£S0Ξ±(+)ββ£)=maxΞ±β(βlogβ£S0Ξ±(β)ββ£), where F is
the round three-sphere free energy of Trank0β and
S0Ξ±(Β±)β is the first column in the modular S-matrix of TFTΒ±β.
From the dictionary, we derive the lower bound on F, Fβ₯βlog(105β5βββ)β0.642965, which holds for
any rank 0 SCFT. The bound is saturated by the minimal 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.Comment: 60 pages, v2: minor corrections, references adde