Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models

Using 3D-3D correspondence, we construct 3D dual bulk field theories for general Virasoro minimal models $M(P,Q)$. These theories correspond to Seifert fiber spaces $S^2((P,P-R),(Q,S),(3,1))$ with two integers $(R,S)$ satisfying $PS-QR=1$. In the unitary case, where $\mathrm{\mid }P-Q\mathrm{\mid }=1$, the bulk theory has a mass gap and flows to a unitary topological field theory (TQFT) in the IR, which is expected to support the chiral Virasoro minimal model at the boundary under an appropriate boundary condition. For the non-unitary case, where $\mathrm{\mid }P-Q\mathrm{\mid }>1$, the bulk theory flows to a 3D $\mathcal{N}=4$ rank-0 superconformal field theory, whose topologically twisted theory supports the chiral minimal model at the boundary. We also provide a concrete field theory description of the 3D bulk theory using $T[SU(2)]$ theories. Our proposals are supported by various consistency checks using 3D-3D relations and direct computations of various partition functions.