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 S² ( (P, R-P), (Q, -S), (3, 1) ) with two integers (R, S) satisfying PS-QR =1. In the unitary case, where |P-Q|=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 |P-Q|>1, the bulk theory flows to a 3D 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 TSU (2) theories. Our proposals are supported by various consistency checks using 3D-3D relations and direct computations of various partition functions.