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) M ( P , Q ) . These theories correspond to Seifert fiber spaces S^2 ((P,P-R),(Q,S),(3,1)) S 2 ( ( P , P − R ) , ( Q , S ) , ( 3 , 1 ) ) with two integers (R,S) ( R , S ) satisfying PS-QR =1 P S − Q R = 1 . In the unitary case, where |P-Q|=1 | 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|;gt1 | P − Q | > 1 , the bulk theory flows to a 3D \mathcal{N}=4 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)] T [ S U ( 2 ) ] theories. Our proposals are supported by various consistency checks using 3D-3D relations and direct computations of various partition functions.