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

TL;DR

The paper builds a unified bulk-boundary framework for Virasoro minimal models M(P,Q) via the 3D-3D correspondence, associating Seifert fiber spaces with 3D bulk theories ${\cal T}_{(P,Q)}$ that flow to either unitary TQFTs (for $|P-Q|=1$) or 3D ${\cal N}=4$ rank-0 SCFTs (for $|P-Q|>1$). It provides explicit field-theoretic constructions using Dehn surgery and $T[SU(2)]$-based building blocks, and identifies the IR bulk theories with a decoupled TFT structure described by $D(\vec{k})$ factors, yielding a concrete bulk-boundary dictionary that reproduces the chiral minimal-model data at the boundary. The approach is tested through non-hyperbolic 3-manifolds and a suite of examples including Ising, Lee-Yang, and tricritical Ising models, with consistency checks from partition functions, indices, and modular data. This framework paves the way for systematic bulk realizations of other minimal models and for analyzing boundary conditions that realize the full chiral algebras in a controlled 3D setting, while suggesting rich dualities and mirror structures to explore further.

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.

Figures (4)

• Figure 1: A Dehn surgery representation of $S^2 (\vec{p},\vec{q}):=S^2 ((p_1, q_1),(p_2,q_2),(p_3, q_3))$.
• Figure 2: Generalized quiver diagrams for $T[\Sigma_{0,3}\times S^1]$, $T[SU(2)]$ and $T[S^2 (\vec{p},\vec{q})]$. The difference between $T_{\rm full}[S^2 (\vec{p},\vec{q})]$ and $T_{\rm irred}[S^2 (\vec{p},\vec{q})]$ arises only from different choices of the $T[\Sigma_{0,3}\times S^1]$ theory, either $T_{\rm full}[\Sigma_{0,3}\times S^1]$ or $T_{\rm irred}[\Sigma_{0,3}\times S^1]$.
• Figure 3: A quiver diagram for $D(\vec{k})$ theory.
• Figure 4: Three basic dualities (modulo topological sectors) guaranteeing the independence of $D(p,q)$ on the choices of $\vec{k}$. The 2nd and 3rd moves follow from the $SL(2,\mathbb{Z})$ structure of the duality wall theory. The 1st move comes from the IR duality between $(T[SU(2)]/(SU(2)_R)_{\pm 1} )\sim (\textrm{empty theory only with background CS level $\mp 1$ for }SU(2)_L)$. See the computation in \ref{['SCI for TSU2/SU(2)']} for a non-trivial check of the duality using the superconformal index. The same moves were used in constructing $T_{\rm full}[M]$ for 3-manifolds $M$ associated to plumbed graphs in Gukov:2017kmk.