3d $\mathcal{N}=$ 4 rank-0 SCFT from punctured lens space

Abstract: {\it Gang-Kim-Stubbs} theory $\mathcal{T}_n$ --- a pioneering 3d bulk description of $M(2n+3,2)$ Virasoro minimal model as $\mathcal{N}=4$ rank-0 superconformal field theory upon topological A-twist --- is derived from the compactification of a pair of parallel M5-branes on {\it lens space} $L(2n+3,2)$ with a single vertex removed. From this perspective, we propose a family of 3d $\mathcal{N}=2$ abelian gauge theories arising from the punctured $L(2n+3,1)$ lens space whose infrared phases realize the unitary member in the Galois orbit of $M(2n+3,2)$ modular tensor category. We also conjecture self-mirror rank-0 fixed points from amphichirality condition of the lens space.

• The paper derives the Gang–Kim–Stubbs theory from the minimal triangulation of the punctured lens space L_{n,1}, yielding a U(1)^n Chern–Simons theory with K_{ij}=2 min(i,j) and monopole superpotentials.
• Punctured lens spaces L_{n,n+1} produce an infinite family of U(1)^{2n} theories that are proposed to flow to unitary TQFTs in the Galois orbit of the Virasoro minimal model M(2n+3,2).
• The paper conjectures that T[L_{n,k}] is a self-mirror rank-0 SCFT precisely when 2k²=n+1 modulo 2n+3, with index checks supporting the criterion while reducible flat connections and triangulation minimality remain open issues.

The paper constructs three-dimensional $\mathcal{N}=2$ abelian Chern-Simons matter theories via the Dimofte-Gaiotto-Gukov (DGG) construction applied to punctured lens spaces $L_{n,k} = L(2n+3,2k)\setminus\{v\}$, where a single vertex is removed. Its central result is a precise derivation of the Gang-Kim-Stubbs (GKS) theory $\mathcal{T}_n$ — previously proposed as a 3d bulk description of the $M(2n+3,2)$ Virasoro minimal model — from the minimal triangulation of $L_{n,1}$, together with an infinite family of gauge theories from $L_{n,n+1}$ flowing to unitary TQFTs in the Galois orbit of that minimal model, and a conjecture identifying self-mirror rank-0 SCFTs through the amphichirality of the underlying lens space (2608.13300).

Background: rank-0 SCFTs and the GKS theory

Three-dimensional rank-0 SCFTs possess neither Higgs nor Coulomb branch, yet exhibit emergent $\mathcal{N}=4$ or $\mathcal{N}=5$ supersymmetry without a manifest Lagrangian. Their topological A- and B-twists capture non-unitary 3d TQFT structures, while their boundary VOAs are obtained via supersymmetric localization. The GKS theory $\mathcal{T}_n$ is a 3d $\mathcal{N}=2$ $L_{n,k} = L(2n+3,2k)\setminus\{v\}$0 gauge theory with $L_{n,k} = L(2n+3,2k)\setminus\{v\}$1 chirals of charge $L_{n,k} = L(2n+3,2k)\setminus\{v\}$2 under the $L_{n,k} = L(2n+3,2k)\setminus\{v\}$3-th gauge factor, CS level matrix $L_{n,k} = L(2n+3,2k)\setminus\{v\}$4, and a superpotential of half-BPS monopole operators with flux vectors $L_{n,k} = L(2n+3,2k)\setminus\{v\}$5. The superpotential breaks all topological symmetries except one combination $L_{n,k} = L(2n+3,2k)\setminus\{v\}$6, which becomes the axial symmetry in the IR. The A-twisted sector supports $L_{n,k} = L(2n+3,2k)\setminus\{v\}$7 and the B-twisted sector the affine $L_{n,k} = L(2n+3,2k)\setminus\{v\}$8 VOA.

The minimal model $L_{n,k} = L(2n+3,2k)\setminus\{v\}$9 also arises as the chiral algebra of the 4d $\mathcal{T}_n$0 Argyres-Douglas SCFT, and $\mathcal{T}_n$1 is expected to be its $\mathcal{T}_n$2 twisted circle reduction. In class-$\mathcal{T}_n$3 language this reduction corresponds to fibering the punctured-sphere Gaiotto curve over a circle, producing precisely the punctured lens space $\mathcal{T}_n$4 with wrapping number $\mathcal{T}_n$5. This motivates deriving $\mathcal{T}_n$6 directly from the 3d/3d correspondence.

Derivation of $\mathcal{T}_n$7 from $\mathcal{T}_n$8

The author triangulates the $\mathcal{T}_n$9-gon — the Gaiotto curve of $M(2n+3,2)$0 — into $M(2n+3,2)$1 triangles and performs $M(2n+3,2)$2 edge flips. Interpreting each flip as introducing an ideal tetrahedron, and gluing the two polygons with an $M(2n+3,2)$3-unit rotation ($M(2n+3,2)$4), yields $M(2n+3,2)$5. The truncation tips of the tetrahedra leave an $M(2n+3,2)$6 boundary filled by a ball whose vertex $M(2n+3,2)$7 is removed; every lens space admits such a layered triangulation with one vertex. By the Matveev-Jaco-Rubinstein-Tillmann conjecture on minimal triangulations, the continued fraction $M(2n+3,2)$8 implies the minimal triangulation of $M(2n+3,2)$9 has exactly $L_{n,1}$0 tetrahedra, so the construction is minimal.

With polarization choice $L_{n,1}$1, the Neumann-Zagier matrix $L_{n,1}$2 satisfies $L_{n,1}$3, so the DGG theory admits a simple UV description: a $L_{n,1}$4 gauge theory with diagonally charged chirals, CS level matrix $L_{n,1}$5 equal to $L_{n,1}$6, and a superpotential of $L_{n,1}$7 monopole terms identical to those of $L_{n,1}$8. The DGG theory of the minimal triangulation of $L_{n,1}$9 therefore reproduces the GKS theory exactly, confirming its interpretation as the once-wrapped $L_{n,n+1}$0 twisted reduction of $L_{n,n+1}$1. A related derivation from the torus knot complement $L_{n,n+1}$2 exists in the literature up to a two-tetrahedron reduction; clarifying the relation between the two manifolds remains open.

Unitary TQFTs from $L_{n,n+1}$3

For $L_{n,n+1}$4, a sequence of $L_{n,n+1}$5 flips produces the minimal triangulation of $L_{n,n+1}$6 with $L_{n,n+1}$7 tetrahedra, again saturating the minimality bound. The resulting DGG theory is a $L_{n,n+1}$8 gauge theory with a sparse CS level matrix supported only when $L_{n,n+1}$9 and $\mathcal{N}=4$0, and a superpotential of $\mathcal{N}=4$1 monopole terms dressed by quadratic powers of chiral fields. This recovers the theories previously constructed for $\mathcal{N}=4$2 from higher-power monodromy traces and extends them to arbitrary $\mathcal{N}=4$3. The claim is that $\mathcal{N}=4$4 flows to a unitary TQFT whose modular data match the complex conjugate of $\mathcal{N}=4$5; since $\mathcal{N}=4$6 is the orientation reversal of $\mathcal{N}=4$7, its theory is the parity conjugate. Together with the rank-0 cases at other $\mathcal{N}=4$8, these theories form a Galois orbit of modular structures associated with $\mathcal{N}=4$9.

Self-mirror conjecture

A lens space is invariant under orientation reversal if and only if $\mathcal{N}=5$0; for $\mathcal{N}=5$1 this reads

$\mathcal{N}=5$2

Since orientation reversal acts as parity conjugation flipping the axial charge — the $\mathcal{N}=5$3 mirror action — the paper conjectures that $\mathcal{N}=5$4 is a self-mirror rank-0 SCFT if and only if this condition holds. Notably, the condition never holds for $\mathcal{N}=5$5 or $\mathcal{N}=5$6, so self-mirror theories are always genuine rank-0 SCFTs rather than unitary TQFTs. The first instances include $\mathcal{N}=5$7, recovering the Gang-Yamazaki theory, along with $\mathcal{N}=5$8. Superconformal indices computed for these cases satisfy $\mathcal{N}=5$9 up to order $\mathcal{T}_n$0, providing strong evidence for the conjecture. Explicit abelian CS-matter descriptions, F-maximization data for the mixing parameters, and the surviving axial symmetry charges are provided in the appendix.

Limitations and open questions

Two caveats are stated explicitly. First, the minimality of the triangulation rests on a conjecture rather than a theorem. Second, and more substantively, the DGG construction sees only irreducible flat $\mathcal{T}_n$1 connections and discards reducible ones; since closed lens spaces have only reducible flat connections, their DGG partition functions vanish due to spontaneously broken supersymmetry. For the punctured lens spaces, however, the number of supersymmetric vacua matches the number of non-trivial reducible flat connections, suggesting that the removed vertex plays a role in capturing reducible connections. Understanding this mechanism is left open, as is the precise geometric relation between $\mathcal{T}_n$2 and the torus knot complement used in earlier derivations. The self-mirror conjecture itself is verified only perturbatively in the index expansion.

Conclusion

The paper places the GKS family $\mathcal{T}_n$3 firmly within the 3d/3d correspondence by deriving it from the minimal triangulation of the punctured lens space $\mathcal{T}_n$4, realizes the unitary members of the Galois orbit of $\mathcal{T}_n$5 from $\mathcal{T}_n$6 for arbitrary $\mathcal{T}_n$7, and proposes a sharp amphichirality criterion for self-mirror rank-0 fixed points supported by index computations to high order. The residual questions concern the role of the removed vertex in accounting for reducible flat connections and a proof-level confirmation of both the minimality bound and the self-mirror criterion.