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

TL;DR

This paper constructs 3D dual theories for 2D Virasoro minimal models using 3D-3D correspondence, revealing different IR behaviors for unitary and non-unitary cases and providing explicit field theory descriptions.

Contribution

It introduces a novel 3D bulk theory framework for Virasoro minimal models, including explicit constructions and consistency checks, extending the 3D-3D correspondence.

Findings

Unitary models flow to topological field theories in IR.

Non-unitary models flow to superconformal theories supporting minimal models.

Explicit $T[SU(2)]$ based descriptions of bulk theories.

Abstract

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 $|P-Q|=1$, 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 $\mathcal{N}=4$ 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.