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

Seungjoo Baek, Heesu Kang

TL;DR

This work provides a concrete bulk–boundary framework for 2D rational CFTs, constructing 3D bulk theories via the 3D–3D correspondence on Seifert manifolds M and identifying their IR phases with either gapped unitary boundary algebras or non-unitary rank-0 SCFT twists. For the unitary $ ext{SM}(P,Q)$ models, the bulk is proposed as $ ext{T}_{ ext{irred}}[M]$ with $M=S^2((P,P{-}R),(Q,S),(3,1))$ and $PS-QR=2$, with explicit checks matching irreducible PSL$(2,c)$ flat connections to NS-sector primaries; the bulk is also described in terms of glued $T[ ext{SU}(2)]$ blocks yielding a UV description. For the non-unitary case, the IR bulk is a 3D $ ext{N}=4$ rank-0 SCFT whose topological twist yields a non-unitary TQFT whose boundary data align with the corresponding non-unitary minimal models. The paper also extends the construction to $W_N(P,Q)$ minimal models by proposing $ ext{T}_{W_N(P,Q)}\nsimeq T_{N, ext{irred}}[M]$ with $M=S^2((P,P{-}R),(Q,S),(N{+}1,-2N{-}1))$ and $PS-QR=1$, though full checks are left for future work; partial tests are performed via embedded $SL(2,c)$ connections and comparisons to modular data. Collectively, these results illuminate a non-hyperbolic 3-manifold route to RCFT bulk duals beyond CS/WZW and lay groundwork for broader Seifert-manifold generalizations and parafermionic extensions.

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.

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

TL;DR

This work provides a concrete bulk–boundary framework for 2D rational CFTs, constructing 3D bulk theories via the 3D–3D correspondence on Seifert manifolds M and identifying their IR phases with either gapped unitary boundary algebras or non-unitary rank-0 SCFT twists. For the unitary models, the bulk is proposed as with and , with explicit checks matching irreducible PSL flat connections to NS-sector primaries; the bulk is also described in terms of glued blocks yielding a UV description. For the non-unitary case, the IR bulk is a 3D rank-0 SCFT whose topological twist yields a non-unitary TQFT whose boundary data align with the corresponding non-unitary minimal models. The paper also extends the construction to minimal models by proposing with and , though full checks are left for future work; partial tests are performed via embedded connections and comparisons to modular data. Collectively, these results illuminate a non-hyperbolic 3-manifold route to RCFT bulk duals beyond CS/WZW and lay groundwork for broader Seifert-manifold generalizations and parafermionic extensions.

Abstract

Building on the work of Gang, Kang, and Kim arXiv:2405.16377, we propose 3D bulk dual field theories for 2D supersymmetric minimal models and algebra minimal models . We associate to a Seifert fibered space with , and for a Seifert fibered space with , 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 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 building blocks. For , we provide non-trivial consistency checks -- matching between various bulk partition functions and boundary conformal data -- while for , we present preliminary checks and leave further consistency checks for future work.

Paper Structure

This paper contains 41 sections, 132 equations, 1 figure, 13 tables.

Table of Contents

  1. Introduction
  2. Bulk field theory for 2D $\mathcal{N}=1$ minimal model $SM(P, Q)$
  3. 2D $\mathcal{N}=1$ minimal model $SM(P, Q)$
  4. Bulk field theory as $T_{\text{irred}}[M]$ with $M=S^2((P,P-R),(Q,S),(3,1))$
  5. Conjecture
  6. Irreducible PSL$(2,\mathds{C})$ flat connections
  7. Example: $SM(2,8)$ from $S^{2}((2,1),(8,5),(3,1))$
  8. Example: $SM(3,5)$ from $S^{2}((3,4),(5,-1),(3,1))$
  9. Example: $SM(3,7)$ from $S^{2}((3,2),(7,3),(3,1))$
  10. Example: $SM(4,10)$ from $S^{2}((4,3),(10,3),(3,1))$
  11. Field theory description of $\mathcal{T}_{SM(P, Q)}$
  12. Example : $(P, Q)=(3,7)$
  13. Example : $(P, Q)=(4,10)$
  14. Comparison with the $\mathcal{T}_{SM(P, Q)}$ by Baek-Gang
  15. $SM(2,4r)$
  16. ...and 26 more sections

Figures (1)

  • Figure 1: Generalized quiver diagrams for $T_{N}[\Sigma_{0,3}\times S^{1}],T[SU(N)]$ and $T_{N}[S^{2}(\vec{p},\vec{q})]$. The difference between $T_{N,\text{full}}[S^{2}(\vec{p},\vec{q})]$ and $T_{N,\text{irred}}[S^{2}(\vec{p},\vec{q})]$ arises from different choices of the $T_{N}[\Sigma_{0,3}\times S^{1}]$ theory, either $T_{N,\text{full}}[\Sigma_{0,3}\times S^{1}]$ or $T_{N,\text{irred}}[\Sigma_{0,3}\times S^{1}]$.