Three-dimensional TQFTs from Argyres--Douglas theories via the 3d/3d correspondence

Abstract: We study the twisted dimensional reduction of 4d $\mathcal{N}=2$ Argyres--Douglas theories via the 3d/3d correspondence, focussing on $(A_1, A_{2n})$ theories realised as the class-$\mathcal{S}$ theory $T[\mathcal{C}]$ for a curve $\mathcal{C}$ with one irregular puncture. We argue the resulting 3d $\mathcal{N}=4$ rank-0 superconformal field theory (SCFT) arises as the IR fixed point of a Dimofte--Gaiotto--Gukov (DGG) abelian Chern--Simons-matter (ACSM) theory $T[M_3<sup>{(k)}]$ for the lens space $M_3<sup>{(k)}=L(2n+3,2k)$, obtained by fibering $\mathcal{C}$ over the circle with twist $k \in \mathbb{Z}<em>{2n+3}<sup>\times$. The triangulation of $M_3<sup>{(k)}$, derived from the 4d BPS spectrum, determines the ACSM theory including its superpotential. $T[M_3<sup>{(k)}]$ flows to the expected SCFT when the monopole superpotential is near-maximal in a precise sense, while the maximal superpotential gives $T_A[M_3<sup>{(k)}]$, which flows directly to a non-unitary TQFT, the topological $A$-twist of the SCFT. Geometrically, the SCFT and TQFT points are related by shifting conical singularities between edges of the triangulation, and we propose a correspondence between singular edges and SCFT points predicting when the SCFT is a unitary TQFT. The TQFTs from $M_3<sup>{(k)}$ can support 2d vertex operator algebras (VOAs) on their holomorphic boundary, including the Schur-sector VOA of the 4d SCFT for $k=1$. For $n=k=1$, our construction reproduces the minimal 3d $\mathcal{N}=4$ SCFT of Gang and Yamazaki from $L(5,2)$, where $T_A[M_3<sup>{(k)}]$ is the Yang--Lee TQFT supporting $M(2,5)$ on its boundary. We check our proposals in detail, reconstructing the modular structure of the IR TQFTs from partition functions on Seifert manifolds, and matching phases to determine VOA central charges $c</em>{\text{2d}} \bmod 8$ from the ACSM data, including the gravitational Chern--Simons level.