Non-hyperbolic 3-manifolds and bulk field theories for supersymmetric/$W_N$ minimal models

TL;DR

This paper constructs 3D bulk dual theories for 2D supersymmetric and $W_N$ minimal models using Seifert fibered spaces and the 3D-3D correspondence, exploring their IR phases and boundary conformal theories.

Contribution

It introduces explicit bulk duals for supersymmetric and $W_N$ minimal models via Seifert fibered spaces and proposes UV gauge theories with consistency checks.

Findings

Bulk theories flow to gapped or superconformal phases in IR.

Boundary theories match with boundary conformal data.

Proposed UV gauge descriptions support the bulk-boundary correspondence.

Abstract

Building on the work of Gang, Kang, and Kim arXiv:2405.16377, we propose 3D bulk dual field theories for 2D $\mathcal{N}=1$ supersymmetric minimal models $SM(P, Q)$ and $W_{N}$ algebra minimal models $W_{N}(P, Q)$. We associate to $SM(P, Q)$ a Seifert fibered space $S^2((P,P-R),(Q,S),(3,1))$ with $PS-QR=2$, and for $W_{N}(P, Q)$ a Seifert fibered space $S^2((P,P{-}R),(Q,S),(N{+}1,-2N{-}1))$ with $PS-QR=1$, and realize the bulk theory via the 3D-3D correspondence. For the unitary series, the bulk theory flows in the IR to a gapped phase which, under suitable boundary conditions, supports the unitary chiral minimal model on the boundary. For the non-unitary series, the bulk theory flows to the 3D $\mathcal{N}=4$ superconformal field theory whose topological twist yields a non-unitary topological field theory supporting the non-unitary chiral minimal model on the boundary under appropriate boundary conditions. We also propose UV gauge theory descriptions of the bulk theories obtained by gluing $T[SU(n)]$ building blocks. For $SM(P, Q)$, we provide non-trivial consistency checks -- matching between various bulk partition functions and boundary conformal data -- while for $W_N(P, Q)$, we present preliminary checks and leave further consistency checks for future work.